7Th International ISAAC Congress Volume of Abstracts

7th International ISAAC Congress Volume of Abstracts

European Mathematical International London Mathematical Society Mathematical Union Society

7th International ISAAC Congress — Abstracts Edited by M. Ruzhansky and J. Wirth. Prepared and typeset using LATEX. Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2AZ Welcoming address

The ISAAC board, the Local Organising Committee and the Department of Mathematics at Imperial Col- lege London, are pleased to welcome you to the 7th International ISAAC Congress in London. The 7th International ISAAC congress continues the successful series of meetings previously held in the Delaware (USA) 1997; Fukuoka (Japan) 1999; Berlin (Germany) 2001, Toronto (Canada) 2003, Catania (Italy) 2005 and Ankara (Turkey) 2007. The success of such a series of congresses would not be possible without all the valuable contributions of all the participants. We acknowledge the ﬁnancial support for this congress given by the London Mathematical Society (LMS),

the International Mathematical Union (IMU), Commission on Development and Exchanges (CDE), and Developing Countries Strategy Group (DCSG), the Engineering and Physical Sciences Research Council (EPSRC),

the Oxford Centre in Collaborational and Applied Mathematics (OCCAM), the Oxford Centre for Nonlinear Partial Diﬀerential Equations (OxPDE), the Bath Institute for Complex Systems (BICS), the Imperial College London, Strategic Fund,

and the Department of Mathematics, Imperial College London.

ISAAC Board Man Wah Wong (Toronto, Canada), President of the ISAAC Heinrich Begehr (Berlin, Germany) Alain Berlinet (Montpellier, France) Bogdan Bojarski (Warsaw,Poland) Erwin Bruning (Durban, South Africa) Victor Burenkov (Padova, Italy) Okay Celebi (Istanbul, Turkey) Robert Gilbert (Newark, Delaware, USA) Anatoly Kilbas (Minsk, Belarus) Massimo Lanza de Cristoforis (Padova, Italy) Michael Reissig (Freiberg, Germany) Luigi Rodino (Torino, Italy) Michael Ruzhansky (London, UK) John Ryan (Fayetteville, Arkansas, USA) Saburou Saitoh (Aveiro, Portugal) Bert-Wolfgang Schulze (Potsdam, Germany) Joachim Toft (V¨axj¨o,Sweden) Yongzhi Xu (Louisville, Kentucky, USA) Masahiro Yamamoto (Tokyo, Japan) Shangyou Zhang (Newark, Delaware, USA)

Local Organising Committee Michael Ruzhansky (Chairman) Dan Crisan Brian Davies Jeroen Lamb Ari Laptev (President of the European Mathematical Society) Jens Wirth Boguslaw Zegarlinski with further support by Laura Cattaneo, Federica Dragoni, Nikki Elliott, James Inglis, Vasileios Kontis and Ioannis Papageorgiou as well as further student helpers.

Abstracts

Plenary talks 15 Sir John Ball : The Q-tensor theory of liquid crystals ...... 15 Louis Boutet de Monvel : Asymptotic equivariant index of Toeplitz operators and Atiyah-Weinstein conjecture 15 Brian Davies : Non-self-adjoint spectral theory ...... 16 Simon Donaldson : Asymptotic analysis and complex differential geometry ...... 16 Carlos Kenig : The global behavior of solutions to critical nonlinear dispersive and wave equations ...... 16 Vakhtang Kokilashvili : Nonlinear harmonic analysis methods in boundary value problems of analytic and harmonic functions, and PDE ...... 17 Nicolas Lerner : Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems ...... 17 Paul Malliavin : Non-ergodicity of Euler deterministic fluid dynamics via stochastic analysis ...... 18 Vladimir Maz’ya : Higher order elliptic problems in non-smooth domains ...... 18 Bert-Wolfgang Schulze : Operator algebras with symbolic hierarchies on stratified spaces ...... 18 Gunther Uhlmann : Visibility and Invisibility ...... 19 Masahiro Yamamoto : Practise of industrial mathematics related with the steel manufacturing process ... 20

Public lecture 21 Pierre-Louis Lions : Analysis, Models and Simulations ...... 21

Sessions 23

I.1. Complex variables and potential theory 23 Tahir Aliyev Azeroˇglo: Analytic functions in contour-solid problems ...... 23 Rauno Aulaskari : A non-α-normal function whose derivative has finite area integral of order less than 2/α 23 Cristina Ballantine : Global mapping properties of rational functions ...... 23 Bogdan Bojarski : Beltrami equations ...... 23 Matteo Dalal Riva : A functional analytic approach for a singularly perturbed non-linear traction problem in linearized elastostatics ...... 23 Peter Dovbush : Boundary behavior of Bloch functions and normal functions ...... 23 Anatoly Golberg : Spatial quasiconformal mappings and directional dilatations ...... 24 Dorin Ghisa : Global mapping properties of entire and meromorphic functions ...... 24 Daniyal Israfilov : Approximation in Morrey-Smirnov classes ...... 24 Dmitri Karp : Two-sided bounds for the logarithmic capacity of multiple intervals ...... 24 Olena Karupu : On boundary smoothness of conformal mapping ...... 24 Boris Kats : Structures of non-rectifiable curves and solvability of the jump problem ...... 25 Gabriela Kohr : The Loewner differential equations and univalent subordination chains in several complex variables ...... 25 Mirela Kohr : Boundary integral equations in the study of some porous media flow problems ...... 25 Massimo Lanza de Cristoforis : Singular perturbation problems in potential theory: a functional analytic approach ...... 25 Jamal Mamedkhanov : Classic theorems of approximation in a complex plane by rational functions .... 25 Sergiy Plaksa : Commutative algebras of monogenic functions and biharmonic potentials ...... 26 Osamu Suzuki : Fractal method for Clifford algebra and complex analysis ...... 26 Yunus Emre Yildirir : Approximation theorems in weighted Lorenz spaces ...... 26 El Hassan Youssfi : Hankel operator on generalized fock spaces ...... 26 Yuriy Zelinskiy : Continues mappings between domains of manifolds ...... 26

I.2. Differential equations: Complex and functional analytic methods, applications 26 Umit Aksoy : A hierarchy of polyharmonic kernel functions and the related integral operators ...... 27 Heinrich Begehr : Boundary value problems for complex partial differential equations ...... 27 Peter Berglez : On some classes of bicomplex pseudoanalytic functions ...... 27 Carmen Bolosteanu : Boundary value problems on Klein surfaces ...... 27 Ilya Boykov : Optimal methods for evaluation hypersingular integrals and solution of hypersingular integral equations ...... 27 Okay Celebi : Complex partial differential equations with mixed-type boundary conditions ...... 28 Natalia Chinchaladze : On a mathematical model of a cusped plate with big deflections ...... 28

5 Jin-Yuan Du : Mixed boundary value problem with a shift for some pair of metaanalytic function and analytic function ...... 28 Grigory Giorgadze : Generalized analytic functions on Riemann surfaces ...... 28 Sonnhard Graubner : Optimization of fixed point methods ...... 28 Azhar Hussain : Generating functions of the Laguerre-Bernoulli polynomials involving bilateral series and applications ...... 28 Alexander Kheyfits : Asymptotic behavior of subparabolic functions ...... 29 Giorgi Khimshiashvili : Elliptic Riemann-Hilbert problems for generalized Cauchy-Riemann systems .... 29 Nino Manjavidze : On some qualitative issues of the elliptic systems ...... 29 Alip Mohammed : Poisson equation with the Robin boundary condition ...... 29 Nusrat Rajabov : Investigation of one class of two-dimensional conjugating model and non model integral equation with fixed super-singular kernels in connection with hyperbolic equation ...... 29 Lutfya Rajabova : About one class of two-dimensional Volterra type integral equation with two interior sinqular lines ...... 30 Roman Saks : Explicit global solutions of 3D rotating Navier-Stokes equations ...... 30 Emma Samoylova : Methods of solutions of an singular integrodifferential equation ...... 30 Tynysbek Sharipovich Kal’menov : A boundary condition of the volume potential ...... 30 Durbudkhan Suragan : Eigenvalues and eigenfunctions of volume potential ...... 30 Zhaxylyk Tasmambetov : The ending solutions of Ince system with irregular features ...... 31 Ismail Taqi : Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces ...... 31 Yufeng Wang : On mixed boundary-value problems of polyanalytic functions ...... 31 Oleg N. Zhdanov : An algorithm of solving the Cauchy problem and mixed problem for the two-dimensional system of quasi-linear hyperbolic partial differential equations ...... 32 Shouguo Zhong : On solution of a kind of Riemann boundary value problem on the real axis with square roots 32 Zhongxiang Zhang : Some Riemann boundary value problems in Clifford analysis ...... 32

I.3. Complex-analytical methods for applied sciences 32 Vladimir Mityushev : R-linear problem and its applications to composites ...... 33 Michael Porter : Application of the spectral parameter power series method to conformal mapping problems 33 Sergei Rogosin : Recent results on analytic methods for 2D composite materials ...... 33

I.4. Zeros and Gamma lines – value distributions of real and complex functions 33 Grigor Barsegian : An universal value distribution: for arbitrary meromorphic function in a given domain 33 Petter Branden : A generalization of the Stieltjes-Van Vleck-Bocher theorem ...... 33 David Cardon : A criterion for the reality of zeros ...... 33 Marios Charalambides : New properties of a class of Jacobi and generalized Laguerre polynomials ...... 34 George Csordas : Meromorphic Laguerre operators and the zeros of entire functions ...... 34 Arturo Fernández: On the logarithmic order of meromorphic functions ...... 34 Steve Fisk : An introduction to upper (stable) polynomials in several variables ...... 34 Paul Gauthier : Perturbations of L-functions with or without non-trivial zeros off the critical line ..... 34 Rod Halburd : Tropical and number theoretic analogues of Nevanlinna theory ...... 34 Aimo Hinkkanen : Growth of analytic functions in unbounded open sets ...... 34 Kazuko Kato : Zeros de la fonction holomorphe et bornee dans un polyhedre analytique de C2 ...... 35 Victor Katsnelson : Steiner and Weyl polynomials ...... 35 Anand Prakash Singh : Spiraling Baker domains ...... 35 Anatoliy Prykarpatsky : The algebraic Liouville integrability and the related Picard-Fuchs type equations . 35 Armen Sergeev : Quantization of universal Teichmüller space: an interplay between complex analysis and quantum field theory ...... 35

II.1 Clifford and quaternion analysis 36 Hendrik de Bie : Clifford analysis for orthogonal, symplectic and finite reflection groups ...... 36 Cinzia Bisi : Möbiustransformations and Poincarédistance in the quaternionic setting ...... 36 Paula Cerejeiras : Wavelets invariant under reflection groups ...... 36 Fabrizio Colombo : Some consequences of the quaternionic functional calculus ...... 36 Kevin Coulembier : Orthogonality of Clifford-Hermite polynomials in superspace...... 36 Sirkka-Liisa Eriksson : Recent results on hyperbolic function theory ...... 37 Ming-Gang Fei : Symmetric properties of the Fourier transform in Clifford analysis setting ...... 37 Milton Ferreira : Factorization of Möbiusgyrogroups - the paravector case ...... 37 Peter Franek : Higher spin analogues of the Dirac operator in two variables and its resolution ...... 37 Ghislain R. Franssens : Cauchy kernels in ultrahyperbolic Clifford analysis – Huygens cases ...... 37 Graziano Gentili : Power series and analyticity over the quaternions ...... 37 Anastasia Kisil : Isomorphic action of SL(2, R) on hypercomplex numbers ...... 38 Rolf Soeren Krausshar : Construction of 3D mappings on to the unit ball with the hypercomplex Szego kernel 38 Lukas Krump : Explicit description of the resolution for 4 Dirac operators in dimension 6 ...... 38 Roman Lavicka : On polynomial solutions of Moisil-Theodoresco systems in Euclidean spaces ...... 38 Matvei Libine : Quaternionic analysis, representation theory and Physics ...... 38

6 Maria Elena Luna-Elizarrarás: Hyperholomorphic functions in the sense of Moisil-Thodoresco and their different hyperderivatives ...... 39 Mircea Martin : Dirac and semi-Dirac pairs of differential operators ...... 39 Heikki Orelma : A differential form approach to Dirac operators on surfaces ...... 39 Dixan PeñaPeña: CK-extension and Fischer decomposition for the inframonogenic functions ...... 39 Alessandro Perotti : A new approach to slice-regularity on real algebras ...... 39 Yuying Qiao : Clifford analysis with higher order kernel over unbounded domains ...... 39 Guangbin Ren : Complex Dunkl operators ...... 40 John Ryan : p-Dirac equations ...... 40 Irene Sabadini : Duality theorems for slice hyperholomorphic functions ...... 40 Tomas Salac : Explicit description of operators in the resolution for the Dirac operator ...... 40 Michael Shapiro : On the relation between the Fueter operator and the Cauchy-Riemann-type operators of Clifford analysis...... 40 Petr Somberg : Conformally invariant boundary valued problems for spinors and families of homomorphisms of generalized Verma modules...... 40 Frank Sommen : Clifford calculus in quantum variables ...... 41 Vladimir Soucek : On relative BGG sequences ...... 41 Caterina Stoppato : Regular Moebius transformations over the quaternions ...... 41 Adrian Vajiac : Singularities of functions of one and several bicomplex variables ...... 41 Fabio Vlacci : Multiplicities of zeroes and poles of regular functions ...... 41 Zuzana Vlasakova : Gauss-Codazzi-Ricci equations in Riemannian, conformal, and CR geometry ...... 42 Liesbet Van de Voorde : Compatibility conditions and higher spin Dirac operators ...... 42

II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras 42 Swanhild Bernstein : Wavelets on spheres ...... 42 Sebastian Bock : On special monogenic power and Laurent series expansions and applications ...... 42 Ruth Farwell : Spin gauge models ...... 43 Nelson Faustino : Further results in discrete Clifford analysis ...... 43 Thanasis Fokas : Integrability in multidimensions, complexification and quaternions ...... 43 Svetlin Georgiev : Note on the linear systems in quaternions ...... 43 Jacques Helmstetter : Minimal algorithms for Lipschitzian elements and Vahlen matrices ...... 43 Jeff Hogan : Clifford-Fourier transforms and hypercomplex signal processing ...... 43 Uwe Kähler: Discrete Clifford analysis by means of skew-Weyl relations ...... 44 Vladimir Kisil : Hypercomplex analysis in the upper half-plane ...... 44 Rolf Soeren Krausshar : Formulas for reproducing kernels of solutions to polynomial Dirac equations in the n annulus of the unit ball in R and applications to inhomogeneous Helmholtz equations ...... 44 Remi Leandre : The Ito transform for partial differential equations ...... 44 Dimitris Pinotsis : Quaternionic analysis and boundary value problems ...... 44 Vitalii Shpakivskii : Integral theorems in a commutative three-dimensional harmonic algebra ...... 44 Wolfgang Sprößig: Initial boundary value problems with quaternionic analysis ...... 45 Tolksdorf, Jürgen: Real bi-graded Clifford modules, the Majorana equation and the standard model action . 45 Nelson Vieira : The regularized Schrödinger semigroup acting on tensors with values in vector bundles ... 45

III.1. Toeplitz operators and their applications 45 Cristina Câmara: On the relations between the kernel of a Toeplitz operator and the solutions to some associated Riemann-Hilbert problems ...... 45 Luis Castro : Convolution type operators with symmetry in exterior wedge diffraction problems ...... 45 Miroslav Englis : Berezin transform on the harmonic Fock space ...... 46 Sergey Grudsky : Inside the eigenvalues of certain Hermitian Toeplitz band matrices ...... 46 d Turan Gürkanlı : Toeplitz operators of M(p, q, w)(R ) spaces ...... 46 Oleksandr Karelin : Presentation of the kernel of a special structure matrix characteristic operator by the kernels of two operators one of them is a scalar characteristic operator ...... 46 Edixon Rojas : Bounds for the kernel dimension of singular integral operators with Carleman shift ..... 46 Anabela Silva : Invertibility of matrix Wiener-Hopf plus Hankel operators with different Fourier symbols .. 46 Harald Upmeier : Flat Hilbert bundles and Toeplitz operators on symmetric spaces ...... 47 Nikolai Vasilevski : Commutative algebras of Toeplitz operators on the unit ball ...... 47 Kehe Zhu : Toeplitz operators on the Fock space ...... 47

III.2. Reproducing kernels and related topics 47 Belkacem Abdous : A general theory for kernel estimation of smooth functionals ...... 47 Som Datt Sharma : Weighted composition operators on some spaces of analytic functions ...... 47 Keiko Fujita : Integral formulas on the boundary of some ball ...... 48 John Rowland Higgins : Paley–Wiener spaces and their reproducing formulae...... 48 Darian Onchis : Irregular sampling in multiple-window spline-type spaces ...... 48 Kazuo Takemura : Free boundary value problem for (−1)M (d/dx)2M and the best constant of Sobolev inequality 48

7 III.3. Modern aspects of the theory of integral transforms 48 Liubov Britvina : Integral transforms related to generalized convolutions and their applications to solving integral equations ...... 48 Qiuhui Chen : Bedrosian identity for Blaschke products in n-parameter cases ...... 49 Dong Hyun Cho : Evaluation formulae for analogues of conditional analytic Feynman integrals over a function space ...... 49 Diana Dolićanin: An equation with symmetrized fractional derivatives ...... 49 Hiroshi Fujiwara : Numerical real inversion of the Laplace transform by reproducing kernel and multiple- precision arithmetic ...... 49 Anatoly Kilbas : Method of integral transforms in the theory of fractional differential equations ...... 49 Bong Jin Kim : Notes on the analytic Feynman integral over paths in abstract Wiener space ...... 50 Sanja Konjik : On the fractional calculus of variations ...... 50 Anna Koroleva : Integral transforms with extended generalized Mittag-Leffler function ...... 50 Ljubica Oparnica : Systems of differential equations containing distributed order fractional derivative .... 50 Juri M. Rappoport : Some aspects of modified Kontorovitch-Lebedev integral transforms ...... 50 Semyon Yakubovich : A new class of polynomials related to the Kontorovich-Lebedev transform ...... 50

III.4. Spaces of differentiable functions of several real variables and applications 50 Alexandre Almeida : Hardy spaces with generalized parameter ...... 51 Tsegaye Gedif Ayele : Iterated norms in Nikol’ski˘ı-Besovtype spaces with generalized smoothness ..... 51 p(.) n Ismail Aydın : Embeddings Properties of The Spaces Lw (R )...... 51 n Canay Aykol : On the boundedness of fractional B-maximal operators in the Lorentz spaces Lp,q,γ (R )... 51 Oleg Besov : Spaces of functions of fractional smoothness on an irregular domain ...... 51 Santiago Boza : Rearrangement transformations on general measure spaces ...... 52 Maria Carro : Last developments on Rubio de Francia’s extrapolation theory ...... 52 Gurgen Dallakyan : On transformation of coordinates invariant relative to Sobolev spaces with polyhedral anisotropy ...... 52 Ismail Ekincioglu : The boundedness of high order Riesz-Bessel transformations generated by the generalized shift operator in weighted Lpw spaces with general weights ...... 52 Vladimir Goldshtein : Composition Operators for Sobolev spaces of functions and differential forms ..... 53 Vagif Guliyev : Boundedness of the fractional maximal operator and fractional integral operators in general Morrey type spaces and some applications ...... 53 Mubariz Hajibayov : Weighted estimates of generalized potentials in variable exponent Lebesque spaces ... 53 Ritva Hurri-Syrjanen : Our talk is on vanishing exponential integrability for Besov functions...... 53 Gennady Kalyabin : New sharp estimates for function in Sobolev spaces on finite Interval ...... 53 Leili Kusainova : On real interpolation of weighted Sobolev spaces ...... 53 Elijah Liflyand : The Fourier transform of a radial function ...... 54 Yagub Mammadov : Necessary and sufficient conditions for the boundedness of Riesz potential in Morrey spaces associated with Dunkl operator ...... 54 Ana Moura Santos : Image normalization of Wiener-Hopf operators in diffraction problems ...... 54 Bohum´ırOpic : Weighted estimates for the averaging integral operator and reverse Hölder inequalities ... 54 Humberto Rafeiro : Characterization of the variable exponent Bessel potential spaces via the Poisson semigroup 55 Evgeniy Radkevich : On the Maxwell problem ...... 55 Natasha Samko : Weighted potential operators in Morrey spaces...... 55 Stefan Samko : Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces ...... 55 Kader Senouci : Equivalent semi-norms for Nikol’skii- Besov spaces on an interval ...... 55 Ayhan Serbetci : Stein-Weiss inequalities for the fractional integral operators in Carnot groups and applications 55 Javier Soria : Translation-invariant bilinear operators with positive kernels ...... 55 Sergey Tikhonov : Sharp inequalities for moduli of smoothness and K-functionals ...... 56 Boris V. Trushin : Sobolev embedding theorems for a class of anisotropic irregular domains ...... 56 Yusuf Zeren : Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces ...... 56

III.5. Analytic and harmonic function spaces 56 Miloud Assal : Multiplier theorem in the setting of Laguerre hypergroups and applications ...... 56 Boo Rim Choe : Progress on ﬁnite rank Toeplitz products ...... 56 Daniel Girela : Functions and operators in analytic Besov spaces ...... 56 Maria Jose Gonzales : Square functions ...... 56 Sanjiv Gupta : Convolutions of generic orbital measures in compact symmetric spaces ...... 57 H. Turgay Kaptano˘glu: Harmonic Besov spaces on the real unit ball: reproducing kernels and Bergman projections ...... 57 Young Joo Lee : Sums of Toeplitz products on the Dirichlet space ...... 57 Jasbir Singh Manhas : Weighted composition operators on weighted spaces of analytic functions ...... 57 Auxiliadora Marquez : Superposition operators between Qp spaces and Hardy spaces ...... 57 Malgorzata Michalska : Bounded Toeplitz and Hankel products on Bergman space ...... 57

8 Kyesook Nam : Optimal norm estimate of the harmonic Bergman projection ...... 57 Pekka Nieminen : Old and new on composition operators on VMOA and BMOA spaces ...... 58 Maria Nowak : On Libera and Cesaro operators ...... 58 Jordi Pau : Integration operators on weighted Bergman spaces ...... 58 Amol Sasane : Extension to an invertible matrix in Banach algebras of measures ...... 58 Benoit F. Sehba : Multiplication operators on weighted BMOA spaces ...... 58 Pawel Sobolewski : Inequalities for Hardy spaces on the unit ball ...... 58 M¨ubarizTapdıgo˘glu : On the Duhamel algebras ...... 58 Jari Taskinen : Toeplitz operators on Bergman spaces ...... 58 Luis Manuel Tovar : Hyperbolic weighted Bergman classes ...... 59 Dragan Vukotic : Multiplicative isometries and isometric zero-divisors ...... 59 Zhijian Wu : Area operators on analytic function spacess ...... 59 Hasi Wulan : Composition operators on BMOA ...... 59 Wen Xu : Lacunary series and QK spaces on the unit ball ...... 59 Congli Yang : Some results on ϕ-Bloch functions ...... 59 Kehe Zhu : Holomorphic mean Lipschitz spaces ...... 59 Nina Zorboska : Univalently induced closed range composition operators on the Bloch-type spaces ...... 59

III.6. Spectral theory 59 Mikhael Agranovich : Strongly elliptic second-order systems in Lipschitz domains: surface potentials, equations at the boundary, and corresponding transmission problems...... 60 Shavkat Alimov : On the spectral expansions associated with Laplace-Beltrami operator ...... 60 Victor Burenkov : Sharp spectral stability estimates for higher order elliptic operators ...... 60 Daniel Elton : Strong ﬁeld asymptotics for zero modes ...... 60 Leander Geisinger : A universal bound for the trace of the heat kernel ...... 60 Tigran Harutyunyan : The eigenvalues function of the family of Sturm-Liouville operators and its applications 61 Jan Janas : Generalized eigenvectors of some Jacobi matrices in the critical case ...... 61 Thomas Krainer : Trace expansions for elliptic cone operators ...... 61 Pier Domenico Lamberti : Stability estimates for eigenfunctions of elliptic operators on variable domains .. 61 Oleksii Mokhonko : Spectral theory of the normal operator with the spectra on an algebraic curve ...... 61 Jiri Neustupa : Spectral properties of operators arising from modelling of ﬂows around rotating bodies .... 62 Serge Richard : New formulae for the wave operators ...... 62 Benedetto Silvestri : Spectral bundles ...... 62 Alexander Strohmaier : Scattering theory for manifolds and the scattering length ...... 62 Yuriy Tomilov : Spectrum and wandering ...... 62 Tomio Umeda : Eigenfunctions at the threshold energies of magnetic Dirac operators ...... 62

IV.1. Pseudo-differential operators 62 Mikhael Agranovich : Strongly elliptic second-order systems in Lipschitz domains: Dirichlet and Neumann problems...... 63 Chikh Bouzar : Generalized ultradistributions and their microlocal analysis ...... 63 Ernesto Buzano : Some remarks on the Sjöstrand class ...... 63 Viorel Catana : The heat equation for the generalized Hermite and the generalized Landau operators .... 63 Leon Cohen : Generalization of the Weyl rule for arbitrary operators ...... 63 Elena Cordero : Sharp results for the STFT and localization operators ...... 63 Yasuo Chiba : Fuchsian mild microfunctions with fractional order and their applications to hyperbolic equations 63 Paulo Dattori da Silva : About Gevrey semi-global solvability of a class of complex planar vector fields with degeneracies ...... 64 Julio Delgado : Invertibility for a class of degenerate elliptic operators ...... 64 Kenro Furutani : Heat kernel of a sub-Laplacian and Grushin type operators ...... 64 Lorenzo Galleani : Time-frequency analysis of stochastic differential equations ...... 64 Gianluca Garello : Lp-microlocal regularity for pseudodifferential operators of quasi-homogeneous type ... 64 Claudia Garetto : Generalized Fourier integral operators methods for hyperbolic problems ...... 65 Juan Gil : Resolvents of regular singular elliptic operators on a quantum graph ...... 65 Todor Gramchev : Hyperbolic systems of pseudodifferential equations with irregular symbols in t admitting superlinear growth for |x| → ∞...... 65 Bernhard Gramsch : Analytic perturbations for special Fréchet operator algebras in the microlocal analysis . 65 Günther Hörmann: The Cauchy problem for a paraxial wave equation with non-smooth symbols ...... 65 EugénieHunsicker : Pseudodifferential operators on locally symmetric spaces ...... 65 Wataru Ichinose : On the continuity of the solutions with respect to the electromagnetic potentials to the Schrödingerand the Dirac equations ...... 65 Chisato Iwasaki : Calculus of pseudo-differential operators and a local index of Dirac operators ...... 66 Jon Johnsen : On the theory of type 1, 1-operators ...... 66 Yuryi Karlovych : Pseudo-differential operators with discontinuous symbols and their applications ...... 66 Thomas Krainer : On maximal regularity for parabolic equations on complete Riemannian manifolds .... 66 Roberto de Leo : On the cohomological equation in the plane for regular vector fields ...... 66

9 Yu Liu : Lp-boundedness and compactness of localization operators associated with Stockwell transform ... 66 Jean-AndréMarti : About transport equation with irregular coefficient and data ...... 67 Shahla Molahajloo : The Heat Kernel of τ-Twisted Laplacian ...... 67 Alessandro Morando : Regularity of characteristic initial-boundary value problems for symmetrizable systems 67 David Natroshvili : Application of pseudodifferential equations in stress singularity analysis for thermo- electro-magneto-elasticity problems: a new approach for calculation of stress singularity exponents ... 67 Alessandro Oliaro : Wigner type transforms and pseudodifferential operators ...... 67 Michael Oberguggenberger : Local regularity of solutions to PDEs by asymptotic methods ...... 68 Nusrat Rajabov : Modern results by theory of the three dimensional Volterra type linear integral equations with singularity ...... 68 Frederic Rochon : The adiabatic limit of the Chern character ...... 68 Bert-Wolfgang Schulze : Boundary value problems as edge problems ...... 68 Elmar Schrohe : Noncommutative residues and projections associated to boundary value problems ..... 68 JörgSeiler : On maximal regularity for mixed order systems ...... 69 Tatyana Shaposhnikova : Dirichlet problem for higher order elliptic systems with BMO assumptions on the coefficients and the boundary ...... 69 Hidetoshi Tahara : Gevrey regularities of solutions of nonlinear singular partial differential equations .... 69 Nenad Teofanov : Wave-front sets and SG type operators in Fourier-Lebesgue spaces ...... 69 Joachim Toft : Wave-front sets of Fourier Lebesgue types ...... 69 Ville Turunen : Pseudo-differential operators and symmetries ...... 70 Vladimir Vasilyev : Pseudo differential equations and boundary value problems in non-smooth domains .. 70 AndrásVasy : Diffraction at corners for the wave equation on differential forms ...... 70 Ingo Witt : Formation of singularities near Morse points ...... 70 Man Wah Wong : Phases of modified Stockwell transforms and instantaneous frequencies ...... 70 Hongmei Zhu : Generalized cosine transforms in image compression ...... 70

IV.2. Dispersive equations 71 Marcello D’Abbico : Lp–Lq estimates for hyperbolic systems ...... 71 Q-Heung Choi : Multiple solutions for non-linear parabolic systems ...... 71 Ferruccio Colombini : Local sovability beyond condition ψ ...... 71 Daniele Del Santo : Continuous dependence for backward parabolic operators with Log-Lipschitz coefficients 71 Marcello Ebert : On the loss of regularity for a class of weakly hyperbolic operators ...... 72 Daoyuan Fang : Zakharov system in infinite energy space ...... 72 Anahit Galstyan : Wave equation in Einstein-de Sitter spacetime ...... 72 Vladimir Georgiev : Stability of solitary waves for Hartree type equation ...... 72 Marina Ghisi : Hyperbolic-parabolic singular perturbations for Kirchhoff-equations ...... 72 Massimo Gobbino : Existence and uniqueness results for Kirchhoff equations in Gevrey-type spaces ..... 73 Torsten Herrmann : Precise loss of derivatives for evolution type models ...... 73 Fumihiko Hirosawa : Wave equations with time dependent coefficients ...... 73 Tacksun Jung : Critical point theory applied to a class of systems of super-quadratic wave equations .... 73 Lavi Karp : On the well-posdness of the vacuum Einstein equations ...... 73 Hideo Kubo : Generalized wave operator for a system of nonlinear wave equations ...... 74 Tokio Matsuyama : Strichartz estimates for hyperbolic equations in an exterior domain ...... 74 Kiyoshi Mochizuki : Uniform resolvent estimates and smoothing effects for magnetic Schrödingeroperators 74 Hideo Nakazawa : Decay and scattering for wave equations with dissipations in layered media ...... 74 Tatsuo Nishitani : On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 4 well- posedness ...... 74 Rainer Picard : On the structure of the material law in a linear model of poro-elasticity ...... 74 Marco Pivetta : Backward uniqueness for the system of thermo-elastic waves with non-lipschitz continuous coefficients ...... 74 Michael Reissig : The log-effect for 2 by 2 hyperbolic systems ...... 75 Jun-ichi Saito : The Boussinesq equations based on the hydrostatic approximation ...... 75 Ryuichi Suzuki : Blow-up of solutions of a quasilinear parablolic equation ...... 75 Hiroshi Uesaka : Blow-up and a blow-up boundary for a semilinear wave equation with some convolution nonlinearity ...... 75 Karen Yagdjian : Fundamental solutions for hyperbolic operators with variable coefficients ...... 75 Borislav Yordanov : Global existence in Sobolev spaces for a class of nonlinear Kirchhoff equations ..... 76

IV.3. Control and optimisation of nonlinear evolutionary systems 76 Lorena Bociu : Global well-posedness and long-time behavior of solutions to a wave equation ...... 76 Mahdi Boukrouche : Distributed optimal controls for second kind parabolic variational inequalities ..... 76 Muriel Boulakia : Controllability of a ﬂuid-structure interaction problem ...... 76 Marcello Cavalcanti : Uniform decay rate estimates for the wave equation on compact surfaces and locally distributed damping ...... 77 Moez Daoulatli : Rate of decay for non-autonomous damped wave systems ...... 77

10 Valeria Domingos Cavalcanti : On qualitative aspects for the damped Korteweg-de Vries and Airy type equations ...... 77 Matthias Eller : Optimal control of waves in anisotropic media via conservative boundary conditions .... 77 Genni Fragnelli : Stability for some nonlinear damped wave equations ...... 77 Anahit Galstyan : Global existence for the one-dimensional semilinear Tricomi-type equation ...... 77 Catherine Lebiedzik : Optimal control of a thermoelastic structural acoustic model ...... 78 Walter Littman : The Balayage method: Boundary control of a thermo-elastic plate ...... 78 Paola Loreti : Hopf-Lax type formulas and Hamilton-Jacobi equations ...... 78 Vyacheslav Maksimov : Investigation of boundary control problems by on-line inversion technique ...... 78 Patrick Martinez : Null controllability properties of some degenerate parabolic equations ...... 78 Maria Grazia Naso : Dissipation in contact problems: an overview and some recent results ...... 78 Luciano Pandolﬁ : Heat equations with memory: a Riesz basis approach ...... 79 Michael Renardy : A note on a class of observability problems for PDEs ...... 79 Roland Schnaubelt : Invariant manifolds for parabolic problems with dynamical boundary conditions .... 79 Ilya Shvartsman : On regularity properties of optimal control and Lagrange multipliers ...... 79 Daniela Sforza : Evolution equations with memory terms ...... 79 Daniel Toundykov : Stabilization of structure-acoustics interactions for a Reissner-Mindlin plate by localized nonlinear boundary feedbacks ...... 79 Julie Valein : Exponential stability of the wave equation with boundary time varying delay ...... 80 Masahiro Yamamoto : State estimation for some parabolic systems ...... 80 Jean-Paul Zolesio : Euler ﬂow and Morphing Shape Metric ...... 80

IV.4. Nonlinear partial differential equations 80 Piero D’Ancona : Evolution equations in nonflat waveguides ...... 80 Mersaid Aripov : Investigation of solutions of one not divergent type ...... 80 Davide Catania : Asymptotic behavior of subparabolic functions ...... 81 N Kuan-Ju Chen : On multiple solutions of concave and convex effects for nonlinear elliptic equation on R 81 Kazuyuki Doi : Nonlinear gauge invariant evolution of the plane wave ...... 81 Mohammad Dehghan : New approach to solve linear parabolic problems via semigroup approximation ... 81 Albert Erkip : Global existence and blow-up for the nonlocal nonlinear Cauchy problem ...... 81 Marius Ghergu : Qualitative properties for reaction-diffusion systems modelling chemical reactions ..... 81 Marco Antonio Taneco-Hernández: Scattering in the zero-mass Lamb system ...... 82 Soichiro Katayama : Global existence for systems of the nonlinear wave and Klein-Gordon equations in 3D 82 Hideo Kubo : Global existence for nonlinear wave equations exterior to an obstacle in 2D ...... 82 Petr Kucera : Remark on Navier-Stokes equations with mixed boundary conditions ...... 82 Ut van Le : Contraction-Galerkin method for a semi-linear wave equation with a boundary-like antiperiodic condition ...... 82 Sandra Lucente : p − q systems of nonlinear Schrodinger equations ...... 83 Satoshi Masaki : Semiclassical analysis for nonlinear Schrodinger equations ...... 83 Gianluca Mola : 3-D viscous Cahn-Hilliard equation with memory ...... 83 Itir Mogultay : A symmetric error estimate for Galerkin approximations of time dependant Navier-Stokes equations in two dimensions ...... 83 Masahito Ohta : Stability of standing waves for some systems of nonlinear Schrödingerequations with three-wave interactions ...... 83 Michael Reissig : Decay rates for wave models with structural damping ...... 83 Yoshihiro Shibata : Stability theorems in the theory of mathematical fluid mechanics ...... 84 LászlóSimon : On singular systems of parabolic functional equations ...... 84 Zdenek Skalak : Survey of recent results on asymptotic energy concentration in solutions of the Navier-Stokes equations ...... 84 Atanas Stefanov : Conditional stability theorems for Klein-Gordon type equations ...... 84 Sergio Spagnolo : A regularity result for a class of semilinear hyperbolic equations ...... 84 Kamal Soltanov : On nonlinear equations, fixed-point theorems and their applications ...... 84 Alessandro Teta : Dynamics of a quantum particle in a cloud chamber ...... 84 Yoshihiro Ueda : Half space problem for the damped wave equation with a non-convex convection term ... 84 Nicola Visciglia : On the time-decay of solutions to a family of defocusing NLS ...... 85 Karen Yagdjian : The semilinear Klein-Gordon equation in de Sitter spacetime ...... 85

IV.5. Asymptotic and multiscale analysis 85 Natalia Babych : On the essential spectrum and singularities of solutions for Laméproblem in cuspoidal domain ...... 85 Michel Bellieud : Torsion effects in elastic composites with high contrast ...... 85 Yves Capdeboscq : Enhanced resolution in structured media ...... 85 Juan Casado-Diaz : Homogenization of elliptic partial differential equations with unbounded coefficients in dimension two ...... 86 Mikhail Cherdantsev : Two-scale Γ-convergence and its applications to homogenisation of non-linear high- contrast problems ...... 86

11 Valentina Alekseevna Golubeva : Construction of the two-parametric generalizations of the Knizhnik- Zamolodchikov equations of Bn type ...... 86 Fabricio Macia : Long-time behavior for the Wigner equation and semiclassical limits in heterogeneous media 86 Peter Markowich : On nonlinear dispersive equations in periodic structures: Semiclassical limits and numerical schemes ...... 86 Karsten Matthies : Derivation of Boltzmann-type equations from hard-sphere dynamics ...... 87 Bernd Schmidt : Minimizing atomic conﬁgurations of short range pair potentials in two dimensions: crystallization in the Wulﬀ shape ...... 87 Valery Smyshlyaev : Homogenization with partial degeneracies: analytic aspects and applications ...... 87

V.1. Inverse problems 87 Abdellatif El Badia : An inverse conductivity problem with a single measurement ...... 87 Fabrizio Colombo : Global in time existence and uniqueness results for some integrodifferential identification problems ...... 88 Mourad Choulli : Stability estimate for an inverse problem for the magnetic Schrödingerequation from the Dirichlet-to-Neumann map ...... 88 Mikko Kaasalainen : Optimal combination of data modes in inverse problems: maximum compatibility estimate 88 Christian Daveau : On an inverse problem for a linear heat conduction problem ...... 88 Matti Lassas : Inverse problems for wave equation and a modified time reversal method ...... 88 Koung Hee Leem : Picard condition based regularization techniques in inverse obstacle scattering ...... 88 William Lionheart : Limited data problems in tensor tomography ...... 89 Marco Marletta : The finite data non-selfadjoint inverse resonance problem ...... 89 Tsutomu Matsuura : Numerical solutions of nonlinear simultaneous equations ...... 89 George Pelekanos : A fixed-point algorithm for determining the regularization parameter in inverse scattering 89 Roland Potthast : A time domain probe method for inverse scattering problems ...... 89 Saburou Saitoh : Explicit and direct representations of the solutions of nonlinear simultaneous equations . 89 Vassilios Sevroglou : Direct and inverse mixed impedance problems in linear elasticity ...... 89 Igor Trooshin : On inverse scattering for nonsymmetric operators ...... 90

V.2. Stochastic analysis 90 David Applebaum : Cylindrical Levy processes in Banach space ...... 90 Vlad Bally : Integration by parts for locally smooth laws and applications to jump type diffusions ...... 90 Dorje Brody : Information and asset pricing ...... 90 Michael Caruana : A (rough) pathwise approach to fully non-linear stochastic partial differential equations . 91 Dan Crisan : Solving backward stochastic differential equations using cubature methods ...... 91 Ana Bela Cruzeiro : Some results on Lagrangian Navier-Stokes flows ...... 91 Alexander Davie : A uniqueness problem for SDEs and a related estimate for transition functions ..... 91 Mark H. A. Davis : Risk-sensitive portfolio optimization with jump-diffusion asset prices ...... 91 Istvan Gyongy : Accelerated numerical schemes for nonlinear filtering ...... 91 Martin Hairer : Periodic homogenisation with an interface ...... 91 Lane Hughston : Wiener chaos models for interest rates and foreign exchange ...... 92 Saul Jacka : Minimising the time to a decision ...... 92 Mark Kelbert : Markov process representations for polyharmonic functions ...... 92 Wilfried Kendall : Networks and Poisson line patterns ...... 92 Vassili Kolokoltsov : The Levy-Khinchine type operators with variable Lipschitz continuous coefficients and stochastic differential equations driven by nonlinear Levy noise ...... 92 Thomas Kurtz : Equivalence of stochastic equations and martingale problems ...... 92 Xue-Mei Li : Aida’s logarithmic Sobolev inequality with weights and Poincare inequalities...... 93 Terence Lyons : Evolution equations for communities ...... 93 Aleksandar Mijatovic : On the martingale property of certain local martingale ...... 93 Khairia El-Said El-Nadi : On some stochastic dynamical systems and cancer ...... 93 Anastasia Papavasiliou : Statistical inference for rough differential equations ...... 93 Martijn Pistorius : First passage for stochastic volatility models ...... 94 Boris Rozovsky : Unbiased random perturbations of Navier-Stokes equation ...... 94 Marta Sanz-Solé: A Poisson equation with fractional noise ...... 94 Radu Tunaru : Constructing discrete exact approximations algorithms for financial calculus from weak convergence results ...... 94 Michael Tretyakov : Numerical methods for parabolic SPDEs based on the averaging-over-characteristics formula ...... 94 Elena Usoltseva : Consistent estimator in AFTM ...... 94

V.3. Coercivity and functional inequalities 95 Franck Barthe : Remarks on non-interacting conservative spin systems ...... 95 Sergey Bobkov : On weak forms of Poincare-type inequalities ...... 95 Messaoud Boulbrachene : L∞-Error estimate for variational inequalities with vanishing zero order term .. 95 Federica Dragoni : Convexity along vector fields and application to equations of Monge-Ampère type .... 95 Ivan Gentil : Φ-entropy inequalities for diffusion semigroups ...... 95

12 Alexander Grigoryan : On positive solutions of semi-linear elliptic inequalities on manifolds ...... 95 Martin Hairer : Hypoellipticity in inﬁnite dimensions ...... 96 Waldemar Hebisch : Logaritmic Sobolev inequality on nilpotent groups ...... 96 Nolwen Huet : Isoperimetry for spherically symmetric log-concave probability measures ...... 96 James Inglis : Operators on the Heisenberg group with discrete spectra ...... 96 Mikhail Neklyudov : Liggett inequality and interacting particle systems ...... 96 Felix Otto : A new criterion for a covariance estimate ...... 96 Ioannis Papageorgiou : The Log-Sobolev inequality for non quadratic interactions ...... 96 Cyril Roberto : Isoperimetry for product probability measures ...... 96

V.4. Dynamical systems 97 Marco Abate : Poincaré-Bendixsontheorems in holomorphic dynamics ...... 97 JoséFerreira Alves : On the liftability of absolutely continuous ergodic expanding measures...... 97 Flavio Abdenur : New results on stability and genericity ...... 97 Pierre Berger : Abundance of one dimensional non uniformly hyperbolic attractors for surface dynamics .. 97 Svetlana Aleksandrovna Budochkina : First integrals in mechanics of infinite-dimensional systems ..... 97 Keith Burns : Partial hyperbolicity and ergodicity ...... 97 Jean-RenéChazottes : On tilings, multidimensional subshifts of finite type and quasicrystals ...... 98 Yi-Chiuan Chen : On topological entropy of billiard tables with small inner scatterers ...... 98 Bau-Sen Du : On the nature of chaos ...... 98 Michael Field : Mixing for flows and skew extensions ...... 98 Jorge Freitas : Rates of mixing, large deviations and recurrence times ...... 98 Giovanni Forni : Limiting distributions for horocycle flows ...... 98 Valery Gaiko : Limit cycle problems and applications ...... 98 Thomas Jordan : Hausdorff dimension of Projections of McMullen-Bedford carpets ...... 99 Jan Cees van der Meer : Fourfold 1:1 resonance, relative equilibria and moment polytopes ...... 99 Matthew Nicol : A dynamical Borel-Cantelli lemma for a class of non-uniformly hyperbolic systems .... 99 Asad Niknam : Approximately inner C∗-dynamical systems ...... 99 Giovanni Panti : Dynamical systems arising in algebraic logic ...... 99 Chen-chang Peng : Existence of transversal homoclinic orbits for Arneodo-Coullet-Tresser map ...... 99 Martin Rasmussen : Bifurcations of random diffeomorphisms with bounded noise ...... 100 Felix Sadyrbaev : Bifurcations of period annuli and solutions of nonlinear boundary value problems ..... 100 JörgSchmeling : Large intersection properties of some invariant sets in number-theoretic dynamical systems 100 Mike Todd : Thermodynamic formalism for unimodal maps ...... 100 Qiudong Wang : Dynamics of periodically perturbed homoclinic solutions ...... 100

V.5. Functional differential and difference equations 100 Jarom´ırBaˇstinec: Oscillation and non-oscillation of solutions of linear second order discrete delayed equations ...... 100 Leonid Berezansky : New stability conditions for linear differential equations with several delays ...... 101 Aleksandr Boichuk : Boundary-value problems for differential systems with a single delay ...... 101 Josef Dibl´ık: Representation of solutions of linear differential and discrete systems and their controllability 101 Alexander Domoshnitsky : Maximum principles and nonoscillation intervals in the theory of functional differential equations ...... 101 Marcia Federson : Averaging for impulsive functional differential equations: a new approach ...... 101 Yakov Goltser : Some bifurcation problems in the theory quasilinear integro differential equations ...... 102 IstvánGyöri: Stability in Volterra type population model equations with delays ...... 102 Ferenc Hartung : On parameter dependence in functional differential equations with state-dependent delays 102 Zeynep Kayar : Lyapunov type inequalities for nonlinear impulsive differential systems ...... 102 Conall Kelly : Evaluating the stochastic theta method ...... 102 Gabor Kiss : Delay-distribution effect on stability ...... 102 Martina Langerová: Solutions of linear impulsive differential systems bounded on the entire real axis .... 102 Malgorzata Migda : Oscillatory and asymptotic properties of solutions of higher-order difference equations of neutral type ...... 103 Abdullah Ozbekler¨ : Principal and non-principal solutions of impulsive differential equations with applications103 Mihali Pituk : Nonnegative iterations with asymptotically constant coefficients ...... 103 Irena Rachunkova : On singular models arising in hydrodynamics ...... 103 Andrejs Reinfelds : Decoupling and simplifying of noninvertible difference equations in the neighbourhood of invariant manifold ...... 103 David W. Reynolds : Precise asymptotic behaviour of solutions of Volterra equations with delay ...... 103 Alexandra Rodkina : On local stability of solutions of stochastic difference equations ...... 104 Miroslava R˚uˇziˇcková: Convergence of the solutions of a differential equation with two delayed terms .... 104 Vladimir Mikhailovich Savchin : Inverse problems of the calculus of variations for functional differential equations ...... 104 Ewa Schmeidel : Existence and nonexistence of asymptotically periodic solutions of Volterra linear difference equations ...... 104

13 Andrei Shindiapin : Gene regulatory networks and delay equations ...... 104 Benzion Shklyar : The moment problem approach for the zero controllability of ecolution equations ..... 105 Svatoslav Stanek : Properties of maximal solutions of autonomous functional-differential equations with state-dependent deviations ...... 105 Stevo Stevic : Boundedness character of some classes of difference equations ...... 105 Milan Tvrdy : Continuous dependence of solutions of generalized ordinary differential equations on a parameter105 Mehmet Unal¨ : Lyapunov type inequalities on time scales: A survey ...... 105 A˘gacıkZafer : Interval criteria for oscillation of delay dynamic equations with mixed nonlinearities ..... 105

V.6. Mathematical biology 105 Robert Gilbert : Cancellous bone with a random pore structure ...... 106 Irina Alekseevna Gainova : New computer technologies for the construction and numerical analysis of mathematical models for molecular genetic systems ...... 106 Sandra Ilic : Application of the multiscale FEM in modeling the cancellous bone ...... 106 Mark D. Ryser : Bone growth and destruction at the cellular level: a mathematical model ...... 106

VI. Others 106 Ruben Airapetyan : The relationship between Bezoutian matrix and Newton’s matrix of divided differences and separation of zeros of interpolation polynomials ...... 106 Hadeel Alkutubi : Bayesian shrinkage estimation of parameter exponential distribution ...... 107 Mohammed Bokhari : Interpolation beyond the interval of convergence: An extension of Erdos-Turan Theorem107 Zoubir Dahmani : The ADM method and the Tanh method for solving some non linear evolutions equations 107 Anvar Hasanov : Boundary-value problems for generalized axially-symmetric Helmholtz equation ...... 107 Maximilian Hasler : Asymptotic extension of topological modules and algebras ...... 107 S. Moghtada Hashemiparast : Approximation of fractional derivatives ...... 107 Hailiza Kamarulhaili : Discrepancy estimate for uniformly distributed sequence ...... 108 Erdal Karapinar : Bounded linear operators on l-power series spaces ...... 108 Erkinjon Karimov : On a three-dimensional elliptic equation with singular coefficients ...... 108 Nabiullah Khan : A unified presentation of a class of generalized Humbert polynomials ...... 108 Lixia Liu : Direct estimate for modified beta operators ...... 108 Eduard Marusic-Paloka : Mathematical model of an undergorund nuclear waste disposal site ...... 108 Abdeslam Mimouni : Compact and coprime packedness and semistar operations ...... 108 S. A. Mohiuddine : Characterization of some matrix classes involving (σ, λ)-convergence ...... 109 Mohammad Mursaleen : Sequence spaces of invariant mean and some matrix transformations ...... 109 Ali Mussa : New convection theory for thermal plasma and NHD convection in rapidly rotating spherical configurations ...... 109 Kourosh Nourouzi : Characterizations of Isometries on 2-modular spaces ...... 109 Shariefuddin Pirzada : On r-imbalances in tripartite r-digraphs ...... 110 Hashem Parvaneh Masiha : Invariance conditions and amenability of locally compact groups ...... 110 Zaure Rakisheva : Motion stabilisation of a solid body with fixed point ...... 110 Lyazzat Sarybekova : A Lizorkin type theorem for Fourier series multipliers in regular systems ...... 110 Pedro A. Santos : Inverse-closedness problems in the stability of sequences in Banach Algebras ...... 110 Ridha Selmi : Smoothing effects for periodic NSE in critical Sobolev space ...... 110 Mariana Sibiceanu : Large deviations and almost sure convergence ...... 111 Tanfer Tanriverdi : The k- Model in Turbulence ...... 111 Johnson Olaleru : The equivalence between modified Mann (with errors), Ishikawa (with errors), Noor (with errors) and modified multi-step iterations (with errors) for non-Lipschitzian strongly successively pseudo-contractive operators ...... 111 Serap Oztop : A characterization for multipliers of weighted Banach valued Lp(G)-spaces ...... 111 Karlyga Zhilisbaeva : Stationary motion of the dynamical symmetric satellite in the geomagnetic field ... 111

Index 113

14 Plenary talks

The Q-tensor theory of liquid crystals

Sir John Ball Mathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, U.K. [email protected]

The lecture will survey what is known about the mathematics of the de Gennes Q-tensor theory for describing nematic liquid crystals. This theory, despite its popularity with physicists, has been little studied by mathematicians and poses many interesting questions. In particular the lecture will describe the relation of the theory to other theories of liquid crystals, speciﬁcally those of Oseen-Frank and Onsager/Maier-Saupe. This is joint work with Apala Majumdar and Arghir Zarnescu. —————— Sir John Ball, FRS, is Sedleian Professor of Natural Philosophy at the University of Oxford and director of the Oxford Centre for Nonlinear PDE. He was president of the International Mathematical Union from 2003 to 2006.

———————————— Asymptotic equivariant index of Toeplitz operators and Attiyah- Weinstein conjecture

Louis Boutet de Monvel Universit´ePierre et Marie Curie, Institut de Math´ematiquesde Jussieu, 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE [email protected]

The equivariant index of transversally elliptic equivariant operators was introduced by M.F. Atiyah (1974); it is a virtual trace class representation of a compact group, or equivalently the character of this representation, which is a central distribution. This does not make sense for general Toeplitz operators because the Toeplitz space where they act is only defined up to a finite dimensional space. The asymptotic index is an avatar of this, which works for Toeplitz operators : essentially it is a virtual trace class representation mod finite representations; equivalently its character is a singularity (distribution mod C∞). It still is compatible with many natural operations, in particular the direct image by homogeneous symplectic maps. With E. Leichtnam, X. Tang and A. Weinstein, we have used this theory to give a new natural proof of the Atiyah-Weinstein conjecture (which was proved by C. Epstein): let X,X0 be two compact strictly pseudo- convex boundaries (of complex domains): they carry natural cooriented contact structures. If f : X → X0 0 −1 is a contact isomorphism, we define the holomorphic pushforward Tf : u 7→ S (u ◦ f ) where u is the boundary value of a holomorphic function, and S0 is the Szegöprojector, i.e. the orthogonal projector on 2 0 ¯ the subspace of boundary values of holomorphic functions in L (X ) (ker ∂b). It is well known that Tf is a Fredholm operator; the Weinstein conjecture proposed a topological formula for its index. A particular case of this, proposed earlier by Atiyah, is the following: let V,V 0 be two smooth compact manifolds, and f a homogeneous symplectic isomorphism T ∗V − {0} → T ∗V 0 − {0} (equivalently a contact isomorphism between the cotangent spheres); then there exists an elliptic Fourier integral operator attached to f, whose index is given essentially by the same formula (this is a special case of the former because, if V is real analytic, the algebra of pseudodifferential operators acting on distributions is isomorphic to the algebra of Toeplitz operators acting on holomorphic boundary values on the boundary of a small tubular neighborhood of V in its complexification). One difficulty in this problem is that, since we are modifying the boundary CR structures (there are two of them), we are typically in the framework of general Toeplitz operators where the index is not well defined. Our way out was to construct a related G-elliptic operator where the index is repeated infinitely many times, but still well related geomerically to the problem, so the asymptotic index theory can be used.

15 Atiyah, M.F. Elliptic operators and compact groups. Lecture Notes in Mathematics, Vol. 401. Springer- Verlag, Berlin-New York, 1974.

Boutet de Monvel, L. Asymptotic equivariant index of Toeplitz operators, RIMS Kokyuroku Bessatsu (2008).

Boutet de Monvel, L.; Leichtnam E.; Tang, X. ; Weinstein A. Asymptotic equivariant index of Toeplitz operators and relative index of CR structures arXiv:0808.1365v1; to appear in the Duistermaat 65 volume, Progress in Math, Birkh¨auser.

Weinstein, A.: Some questions about the index of quantized contact transformations RIMS Kokyuroku No. 1014, pages 1-14, 1997.

—————— Louis Boutet de Monvel was awarded with the 2007 Medaille Emile Picard of the French Academy of Sciences.

———————————— Non-self-adjoint spectral theory

Brian Davies Department of Mathematics, King’s College London, Strand, London WC2R 2LS, U.K. [email protected]

Over the last twenty years there has been remarkable progress in understanding the spectral behaviour of highly non-self-adjoint operators, particularly diﬀerential operators, partly as the result of numerical experiments. The lecture will describe some of the discoveries that have been made, and theorems proved, and will contrast them with the very diﬀerent spectral behaviour of self-adjoint operators. Connections with so-called pseudospectral theory, that is bounds on the norms of the resolvent operators, will be explained and illustrated. —————— Brian Davies, FRS, is Professor of Mathematics at King’s College London. In 1998 he was awarded the Senior Berwick Prize of the LMS. Brian Davies was president of the London Mathematical Society from 2007 to 2009.

———————————— Asymptotic analysis and complex diﬀerential geometry

Simon Donaldson Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, U.K. [email protected]

A long-standing problem in complex differential geometry is to find various preferred metrics on a complex manifold. These include Kähler-Einstein,constant scalar curvature and extremal metrics. Finding such metrics comes down to solving highly nonlinear partial differential equations. For some manifolds solutions do not exist, and this is known to be related to the algebro-geometric notion of “stability”. The talk will give an overview of this area, emphasising the role of asymptotic analysis, applied to holomorphic sections of high powers of a complex line bundle. This gives a bridge between the analytical problems and algebraic geometry which is important in the general existence theory. The ideas can also be applied to construct numerical approximations to the desired metrics. —————— Simon Donaldson, FRS, holds a Royal Society Research Professorship at Imperial College London. He received a Fields Medal in 1986, was awarded with the Crafoord Prize 1994, the King Faisal International Prize in 2006 and the Nemmers Prize in Mathematics in 2008. He will receive the 2009 Shaw Prize in Mathematical Sciences.

————————————

16 The global behavior of solutions to critical nonlinear dispersive and wave equations

Carlos Kenig Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, IL 60637-1514, USA [email protected]

In this lecture we will describe a method (which I call the concentration-compactness/rigidity theorem method) which Frank Merle and I have developed to study global well-posedness and scattering for critical non-linear dispersive and wave equations. Such problems are natural extensions of non-linear elliptic problems which were studied earlier, for instance in the context of the Yamabe problem and of harmonic maps. We will illustrate the method with some concrete examples and also mention other applications of these ideas. —————— Carlos Kenig is Louis Block Distinguished Service Professor of the University of Chicago. He was awarded the 2008 Bˆocher Memorial Prize for his contributions to harmonic analysis and non-linear dispersive partial diﬀerential equations.

———————————— Nonlinear harmonic analysis methods in boundary value problems of analytic and harmonic functions, and PDE

Vakhtang Kokilashvili A. Razmadze Mathematical Institute, 1, M. Aleksidze st., 0193 Tbilisi, Georgia [email protected]

The goal of our lecture is to present a survey of recent results in the nonlinear harmonic analysis operator theory and their applications in the boundary value problems for harmonic and analytic functions and related integral operators. We plan to discuss the above mentioned problems in the frame of Banach function spaces with nonstandard growth condition. For the sake of presentation, we have split the talk in the following topics: • One and two-weight norm estimates for the Cauchy singular integrals on Carleson curves in variable exponent Lebesgue spaces. • The Riemann-Hilbert problem for holomorphic functions from weighted classes of the Cauchy type integrals with densities in Lp(·)(Γ) in simply connected domains with piecewise-smooth boundaries Γ. Our aim is to give a complete solvability picture; to reveal the inﬂuence on the solvability character of the geometry of a boundary, of a weight function, and of the values of the space exponent at angular points; to give explicit formulas for solutions. • The Riemann-Hilbert-Poincar´eproblem in the class of holomorphic functions whose mth order derivatives are representable by the Cauchy type integrals with densities from the variable exponent Lebesgue spaces with weights. The solvability criteria are given for the problem. The study of the problem is heavily based on the extension of I.Vekua’s integral representation of holomorphic function whose derivative is representable by the Cauchy type integral in simply connected domain with non-smooth boundary. • Baundary value problem with shift (the Hasemann BVP) for holomorphic functions in the domain with arc-chord condition. The solvability criteria and explicit formulas for solutions are established. Some part of the talk is based on joint research with V.Paatashvili. —————— Vakhtang Kokilashvili is Head of the Mathematical Analysis Department of the Razmadze Mathematical Institute. He was awarded the Razmadze Prize of the Georgian Academy of Sciences.

———————————— Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems

Nicolas Lerner Institut de Math´ematiquesde Jussieu, Universit´eParis 6, 175 rue du Chevaleret, 75013 Paris, France [email protected]

17 We prove that in any C-infinity neighborhood of an analytic Cauchy datum, there exists a smooth function such that the corresponding initial value problem does not have any classical solution for a class of first-order non-linear systems. We use a method initiated by G. Métivierfor elliptic systems based on the representation of solutions and on the FBI transform; in our case the system can be hyperbolic at initial time, but the characteristic roots leave the real line at positive times.

———————————— Non-ergodicity of Euler deterministic ﬂuid dynamics via stochastic analysis

Paul Malliavin Universit´ePierre et Marie Curie, Institut de Math´ematiquesde Jussieu, 4 place Jussieu, F-75252 Paris CEDEX 05, FRANCE

Unitary representation associated to the motion of an incompressible fluid on the Tori. Fourier analysis of vector fields with vanishing divergence. Ergodi-city implies existence of an infinitesimal Haar measure. Randomization of Euler deterministic dynamics. Stochastic differential geometry on the group of volume preserving diffeomorphism of the Tori. Jump process describing the evolution of the repartition of the energy between modes. Non ergodicity of Euler equation via the transfert of energy towards micro scale. —————— Paul Malliavin is famous for his contributions to stochastic analysis and stochastic differential geometry. Among other distinctions he received in 1974 the Prix Gaston Julia of the French Academy of Science and is member of the Royal Swedish Academy of Sciences.

———————————— Higher order elliptic problems in non-smooth domains BICS Lecture, with an introduction by Valery Smyshlyaev

Vladimir Maz’ya University of Liverpool and Linkoeping University [email protected]

We discuss sharp regularity results for solutions of the polyharmonic equation in an arbitrary open set. The absence of information about geometry of the domain puts the question of regularity beyond the scope of applicability of the methods devised previously, which typically rely on specific geometric assumptions. Positive results have been available only when the domain is sufficiently smooth, Lipschitz or diffeomorphic to a polyhedron. The techniques developed in the present work allow to establish the boundedness of derivatives of solutions to the Dirichlet problem for the polyharmonic equation under no restrictions on the underlying domain and to show that the order of the derivatives is maximal. Then we introduce an appropriate notion of polyharmonic capacity which allows us to describe the precise correlation between the smoothness of solutions and the geometry of the domain. This is a joint work with S.Mayboroda, Perdue University. —————— His honours include the prize of the Leningrad Mathematical Society 1962, Doctor honoris causa of the University of Rostock 1990, Humbold Prize 1999, Corresponding Fellow of the Royal Society of Edinburgh 2001, Member of Royal Swedish Academy of Sciences 2002, Verdaguer Prize of the French Academy of Sciences 2003, The Celsius Gold Medal of the Royal Society of Sciences at Uppsala 2004. He is author of more than 20 books and more than 430 articles.

———————————— Operator algebras with symbolic hierarchies on stratiﬁed spaces

Bert-Wolfgang Schulze Institute of Mathematics, University Potsdam, Am Neuen Palais 10, Potsdam, D-14469 Germany [email protected]

18 We establish operator algebras on certain categories of stratified spaces (“corner manifolds”, or “manifolds with singularities”) that are designed to express parametrices of elliptic operators in terms of symbolic hierarchies. Our calculus contains special cases such as (pseudo-differential) boundary value problems with/without the transmission property at the boundary, and also mixed, transmission, and crack problems. The boundaries or interfaces may be smooth or have again singularities (conical points, edges, etc.). Other examples are equations on a smooth manifold, where the coefficients may have jumps or poles of a specific kind along some interfaces, smooth or singular in the above-mentioned sense, for instance, the Laplacian plus a singular interaction potential from a many-particle system. It is typical in such problems that a concrete situation (for instance, for the Laplacian in a corner domain) may generate operator-valued amplitude functions of a relatively high generality, consisting of operator functions on configurations of lower singularity order, now depending on various variables and covariables along the singular lower-dimensional strata. The calculus also contains analogues of Green functions, known from “standard” elliptic boundary value problems. In the singular case those refer again to all singular strata, operating on infinite cones. Moreover, when a stratum is of dimension zero the operator functions globally act on compact (in general singular) bases of such cones, with meromorphic dependence on a complex covariable, where non-bijectivity points (turning into poles under inversion) contribute to the asymptotics of solutions. Ellipticity in such a scenario is defined as invertibility of such operator-valued symbols. This depends on chosen weights in the respective distribution spaces. When a stratum is of dimension at least 1, this cannot be achieved in general, unless we pose extra edge conditions (analogues of boundary conditions), here of trace and potential type. The latter are possible when an analogue of the Atiyah-Bott condition for the existence of Shapiro-Lopatinskij boundary conditions is satisfied; otherwise another concept, namely, with global projection conditions may work (at least for smooth boundaries or edges, cf. the well-known work of Atiyah, Patodi, Singer, and papers of many other authors, especially, Seeley, Grubb, and also by the author, partly in joint work with J. Seiler, where corresponding operator algebras are established in a Toeplitz operator framework, unifying the structures of the Shapiro-Lopatinskij and the global projection set-up). The construction of parametrices relies on the inversion of the components of the principal symbolic hierarchy, combined with algebraic operations. Those symbols take values in spaces of operators referring to lower singularity orders. At this point, in order to express parametrices within our spaces, we need the calculus as an algebra. The analysis which is doing all this is rich in detail. Many authors contributed to the pseudo-differential methods in this framework, especially, Melrose, Mendoza, Gil, Seiler, Schrohe, Witt, and Krainer.There are several monographs of the author, a few jointly with coauthors (Rempel, Egorov, Kapanadze, Harutyunyan) containing the basics of the approach, including applications, and more references. In order to keep the calculus manageable it is important to reduce the stuctures to a few “axiomatic” principles and then to proceed in an iterative way, beginning with the pseudo-differential calculus on a smooth manifold, and then successively building up the algebras for conical, edge, corner, . . . , higher singularities. The focus of our talk is just a program of that kind. We present such an iterative process to obtain operator algebras containing the desirable (“typical”) differential operators (corner-degenerate in streched coordinates), together with the parametrices of elliptic elements, where the above-mentioned examples are covered. One of the principles to make the calculus iterative is to impose a relatively simple behaviour of the growth of norms of parameter-dependent operators when the parameters tend to infinity, then to make the parameter-dependence “edge-degenerate” at infinity of an infinite cone, and then to observe that this behaviour survives the step to the next floor of singular calculus, cf. a joint article with Abed. The general structure theory is full of new challenges and “unexpected” problems, for instance, from the point of view of index theory, or extensions to non-elliptic operators. Moreover, in concrete cases other substantial aspects remain essential, namely, to compute several data as explicitly as possible, e.g., the index of operators on infinite cones, or the number of extra edge conditions, the right weights that depend on the individual operator, the asymptotics of solutions, including iterated asymptotics, or the variable and branching behaviour connected with the above-mentioned poles when those depend on edge variables and change multipicities (cf. earlier work of Bennish, or the author, and a cycle of papers in progress jointly with Volpato). —————— Bert-Wolfgang Schulze is author of more than 240 publications and 20 books. He received the Euler Medal of the Berlin Academy of Sciences in 1984 and is doctor honoris causa of the Vekua Institute of Applied Mathematics in Tbilisi.

———————————— Visibility and Invisibility

Gunther Uhlmann Department of Mathematics, C-449 Padelford Hall, Seattle, Washington 98195-4350, USA

19 [email protected]

We will describe the method of complex geometrical optics and its applications to ﬁnd acoustic, quantum, and electromagnetic parameters of a body by making measurements at the boundary of the body. We will also survey recent results on how to make objects invisible to acoustic, quantum and electromagnetic waves. —————— Gunther Uhlmann is Walker Family Endowed Professor of Mathematics at the University of Washington. He is Fellow of the American Academy of Arts and Sciences, corresponding member of the Chilean Academy of Sciences, Fellow of the Institute of Physics and will be Clay Senior Scholar at MSRI in 2010.

———————————— Practise of industrial mathematics related with the steel manufacturing process OCCAM Lecture on Applied Mathematics, with an introduction by John Ockendon

Masahiro Yamamoto University of Tokyo, Department of Mathematical Sciences, 3-8-1 Komaba Meguro Tokyo 153, Japan [email protected]

We will discuss several problems given by the steel industry. Those problems have originated from real working sites, are related for example with heat conduction processes and have been solved by the speaker and his research groups. Those problems can be modelled mathematically, on such a a theoretical basis, we have solved them practically as well as mathematically to satisfy demands by industry for lowering costs and improving securities. For more fruitful contribution in the industrial mathematics from the side of mathematicians, we will discuss also possible schemes.

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20 Public lecture

Analysis, Models and Simulations OxPDE Public Lecture on Nonlinear PDE, with an introduction by Sir John Ball

Pierre-Louis Lions College de France, 3 rue d’Ulm, 75005 Paris, France [email protected]

In this talk, we shall first present several examples of numerical simulations of complex industrial systems. All these simulations rely upon some mathematical models involving Partial Differential Equations and we shall briefly explain the nature, the history and the role of such equations. Then, some examples showing the importance of the mathematical analysis (i.e. understanding) of those models will be presented. And we shall conclude indicating a few trends and perspectives. —————— Pierre-Louis Lions is the son of the famous mathematician Jacques-Louis Lions and has himself become a renowned mathematician, making numerous important contributions to the theory of non-linear partial differential equations. He was awarded a Fields Medal in 1994, in particular for his work with Ron DiPerna giving the first general proof that the Boltzmann equation of the kinetic theory of gases has solutions. Other awards Lions has received include the IBM Prize in 1987 and the Philip Morris Prize in 1991. Currently he holds the position of Chair of Partial Differential Equations and their Applications at the prestigious Collège de France in Paris.

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This is joint work with Shamil Makhmutov and Jouni Rättyä. ——— Sessions Global mapping properties of rational functions Cristina Ballantine Dept. of Mathematics, College of the Holy Cross, 1 I.1. Complex variables and potential theory College Street, Worcester, Massachusetts 01610 United States Organisers: [email protected] Tahir Aliyev, Massimo Lanza de Cristoforis, Sergiy Plaksa, Promarz Tamrazov The talk is based on a joint work with Dorin Ghisa. The main result is the following theorem. Every rational This session is devoted to a wide range of directions n function f of degree n defines a partition Cb = ∪k=1Ωk of complex analysis, potential theory, their applications of the Riemann sphere such that the interior of every and related topics. Ωk is mapped conformally by f on Cb \ Lk, where Lk is part of a cut L. The mapping extends conformally to the —Abstracts— boundary of every Ωk except forsome points b1, b2, ..., bj , j ≤ n, in the neighborhood of which f has one of the Analytic functions in contour-solid problems forms:

αk Tahir Aliyev Azerogloˇ (i) f(z) = f(bk) + (z − bk) ϕk(z), or Gebze Institute of Technology, Istanbul Caddesi, P.K. (ii) f(z) = (z − b )−αk ϕ (z), 141, Gebze, Kocaeli, 41400 Turkey k k [email protected] where αk is an integer, αk ≥ 2, and ϕk is an analytic function with ϕk(bk) 6= 0. We generalize and strengthen certain contour-solid the- Actually, (Cb, f) is a branched covering Riemann surface orems. The generalization consists in considering of Cb having the branch points b1, b2, ..., bj . In the neigh- finely meromorphic functions besides holomorphic, and borhood of z = ∞ we have: strengthening is connected with taking into account ze- α (iii) f(z) = z ϕ(z), where α ∈ Z and ϕ is analytic with roes and the multivalence of functions. lim ϕ(z) finite and non-zero. z→∞ ——— If f is a polynomial, then every Ωk is bounded by arcs A non-α-normal function whose derivative has finite approaching asymptotically rays of the form zk(t) = i(γ+2kπ/n) area integral of order less than 2/α te , t > 0, γ ∈ R. We will present examples of color mapping visualiza- Rauno Aulaskari tions. University of Joensuu Department of Physics and Math- ematics Joensuu, Joensuu 80101 Finland ——— [email protected] Beltrami equations Bogdan Bojarski Let D be the unit disk {z : |z| < 1} in the complex plane. IM PAN, Sniadeckich 8, Warsaw, 00-956 Poland A function f, meromorphic in D, is normal, denoted by 2 # # [email protected] f ∈ N, if supz∈D(1 − |z| )f (z) < ∞, where f (z) = 0 2 |f (z)|/(1+|f(z)| ). For α > 1, a meromorphic function In the talk will be discussed some new approaches to the 2 α # f is called α-normal if supz∈D(1 − |z| ) f (z) < ∞. H. Beltrami equations and operators in the complex plane Allen and C. Belna [J. Math. Soc. Japan, 24 (1972) and on Riemann surfaces in connections with the gen- 128–132] have proved that there is an analytic function eral theory of quasiconformal mappings and automor- f1, defined in D, such that phic functions. ZZ 0 ——— |f1(z)| dxdy < ∞ D A functional analytic approach for a singularly perturbed non-linear traction problem in linearized elas- but f1 6∈ N. S. Yamashita [Ann. Acad. Sci. Fenn. Ser. Math. 4 (1978/1979) 293–298] sharpened this result by tostatics showing that for another analytic function f2 which does Matteo Dalal Riva not belong to N it holds Universita’ degli Studi di Padova, Via Trieste, 63 ZZ Padova, Italy/Padova/Veneto 35121, Italy 0 p |f2(z)| dxdy < ∞ (*) [email protected] D We consider an application of an approach based on po- for all p, 0 < p < 2. Further, H. Wulan [Progress in anal- tential theory and functional analysis to analyze a non- ysis Vol. I,II, World Sci. Publ. 2003, 229–234] studied linear traction problem of linearized elasticity in a do- more the function f and showed that f 6∈ S Q# 2 2 0 1. Further, the question if fs belongs or not to S # ——— 0

23 I.1. Complex variables and potential theory

Boundary behavior of Bloch functions and normal func- Approximation in Morrey-Smirnov classes tions Daniyal Israfilov Peter Dovbush Department of Mathematics, Faculty of Art and Sci- Institute of Mathematics and Computer Science, 5 ences, Balikesir University,10145 Balikesir, Turkey Academy Street, Kishinev, MD-2028 Moldova [email protected] [email protected] In this talk we discuss the constructive characteriza- We give the version of the Lindelöfprinciple which valid tion problems in the Morrey-Smirnov classes, defined on n 2 in bounded domains in C with C -smooth boundary. the finite domain G with a sufficiently smooth Jordan We also prove that if a Bloch function is bounded on a boundary Γ. The Morrey spaces, introduced by Morrey K-special curve ending at a given boundary point it is in 1938, have been studied intensively by various authors bounded on any admissible domain with vertex at the and together with weighted Lebesgue spaces play an im- same point. portant role in theory of partial equations, and also in the fluid dynamics. They also provide a large class of ——— examples of mild solutions to the Navier-Stokes system. Spatial quasiconformal mappings and directional dilata- Nowadays there are sufficiently wide investigations relat- tions ing to the fundamental problems in these spaces, in view of the differential equations, potential theory, maximal Anatoly Golberg and singular operator theory and others. Holon Institut of Technology, 52 Golomb St., Holon To the best of the author’s knowledge in the literature 58102 Israel there are no results relating to the approximation prob- [email protected] lems in the Morrey and Morrey-Smirnov classes, defined on the interval of the real line and on the sets of the n Two dilatations in R connected with a given direction complex plane. are considered. They are spatial counterparts of the an- In the current talk we discuss the direct and inverse gular and radial dilatations in the plane. We establish theorems in these spaces and obtain the constructive new geometric estimates of module of ring domains un- characterization of the generalized Lipschitz classes of der quasiconformal mappings. These estimates are writ- functions defined in the Morrey-Smirnov classes, in par- ten in the terms of integrals depending on the directional ticular. dilatations. The sharpness of the bounds is illustrated by certain examples. We also present new criteria for ——— quasiconformal mapping to be Lipschitz or weekly Lip- Two-sided bounds for the logarithmic capacity of mul- schitz continuous in a prescribed point. tiple intervals ——— Dmitri Karp Global mapping properties of entire and meromorphic Institute of Applied Mathematics, 7 Radio Street, Vladi- functions vostok, 690041, Russia [email protected] Dorin Ghisa York University, 5795 Young Street #1205, Toronto, On- Potential theory on the complement of a subset of the tario M2M 4J3 Canada real axis attracts a lot of attention both in function the- [email protected] ory and applied sciences (in particular in signal analysis). In the talk we discuss one aspect of the theory - We consider that an analytic function f is the canonical the logarithmic capacity of closed subsets of the real line. projection of a branched Riemann surface (Cb \ E, f) on We give simple but precise upper and lower bounds for Cb. Here E = E(f) is the set of (isolated, or non isolated) the logarithmic capacity of multiple intervals and a lower essential singular points of f. Then Cb \ E consists of all bound valid also for closed sets comprising an infinite regular points of f, as well as of a discrete set of poles. number of intervals. We discuss the existing methods The main result shows that, for wide classes of integer to compute the exact value of capacity and demonstrate and meromorphic functions f, disjoint unions Cb \ E = graphically the results of numerical comparisons of our ∞ ∪n=1Ωn exist, such that the interior of every (fundamen- estimates with exact values of capacities. The main ma- tal region)Ωn is mapped conformally by f on Cb \ Ln, chinery behind our results are separating transforma- where Ln is a part of a cut L. tion and dissymmetrization developed by V.N. Dubinin The global mapping properties of f concern these map- and a version of the latter by K. Haliste as well as some pings. If f is an infinite Blaschke product or Blaschke classical symmetrization and projection result for loga- quotient whose set E of the cluster points of poles is rithmic capacity. The results presented in the talk im- a generalized Cantor subset of the unit circle, then the prove some previous achievements by A.Yu. Solynin and fundamental regions Ωn accumulate to every point of E K. Shiefermayr. and f : Ωn → Cb is surjective. In particular, the state- The work reported here was done jointly with ment of the Big Picard Theorem is obvious for the points V.N. Dubinin and was supported by Far Eastern Branch of E, which are non isolated essential singular points of of the Russian Academy of Sciences (grants 09-III-A- f. The same is true about a theorem of Hadamard which 01-008 and 09-II-CO-01-003), Russian Basic Research says that an entire function of fractional order assumes Fund (grant 08-01-00028-a) and the Presidential Grant every finite value infinitely many times. for Leading Scientific Schools (grant 2810.2008.1). ——— ———

24 I.1. Complex variables and potential theory

On boundary smoothness of conformal mapping Boundary integral equations in the study of some porous media flow problems Olena Karupu National Aviation Univesity, Kosmonavta Komarova Mirela Kohr ave. 1 Kiev, Kiev 03058 Ukraine Faculty of Mathematics and Computer Science, Babes- [email protected] Bolyai University, 1 M. Kogalniceanu Str., Cluj-Napoca, Cluj 400084 Romania Let two simply connected domains bounded by the [email protected] smooth Jordan curves be given. Boundaries of these domains are characterized by the angles between the In this talk we present a survey of recent results concern- tangents to the curves and the positive real axis which ing existence and uniqueness in Sobolev spaces for trans- are considered as the functions of the arc length on the mission problems associated with Stokes and Brinkman n curves. The estimates on the boundaries of the domains equations on Lipschitz domains in R , by using potential for the general moduli of smoothness of arbitrary or- theory and indirect boundary methods. Various appli- der for the homeomorphisms between the closures of the cations in the study of viscous incompressible flows in considered domains, conformal in open domains are es- porous media or past porous bodies are also discussed. tablished. ——— In partial case when moduli of smoothness of arbitrary order for the functions characterizing boundaries of the Singular perturbation problems in potential theory: a domains satisfy Holder condition, moduli of smoothness functional analytic approach of the same order for the derivatives of the functions re- Massimo Lanza de Cristoforis alizing conformal mapping satisfy Holder condition with Dipartimento di Matematica Pura ed Applicata, Via Tri- the same index. este 63, Padova, Italy 35127, Italy [email protected] ——— Structures of non-rectifiable curves and solvability of This talk is dedicated to the analysis of boundary value the jump problem problems on singularly perturbed domains by an approach which is alternative to those of asymptotic anal- Boris Kats ysis and of homogenization theory. Kazan State Architecture and Civil Engineering Univer- In particular, we will consider a certain linear or non- sity, Zelenaya Street 1, Kazan, Tatarstan 420043 Russia linear boundary value problem on a domain with one or [email protected] possibly infinitely many holes, whose size is determined by a positive parameter and we will consider a family Let Γ be a given closed non-rectifiable curve on the com- of solutions depending on as approaches 0. Then we plex plane C. We consider so-called jump problem, i.e. shall represent the dependence on of the family of so- the boundary value problem for determination of a holo- lutions, or of corresponding functionals of the solutions morphic in C \ Γ function F (z) satisfying equality such as the energy integral, in terms of possibly singular −1 + − at 0 but known functions of such as or log , and F (t) − F (t) = g(t), t ∈ Γ, in terms of possibly unknown real analytic operators. where g(t) is a given jump. As the author have shown ——— earlier, for non-rectifiable curve Γ this problem has a solution if the jump g satisfies the Holder condition with Classic theorems of approximation in a complex plane exponent exceeding half of upper metric dimension of Γ. by rational functions In the present report we prove new conditions of solvabil- Jamal Mamedkhanov ity of the jump problem and new boundary properties of Z.Khalilov, 23, Baku State University, Department of its solutions in terms of decomposition of C \ Γ into infi- theory of functions and functional analysis Baku, Baku nite sum of disjoint domains with rectifiable boundaries. AZ-1148 / 994 Azerbaijan In particular, the self-similarity of Γ weakens the condi- [email protected] tions of solvability and improves smoothness of solutions of the jump problem. Jackson’s classic theorem on closed boundary domains related to approximation on the boundary of domain ——— of functions determined on the boundary by means of The Loewner differential equations and univalent sub- polynomials assumes analyticity of the given function in ordination chains in several complex variables the considered domain. If the function is not analytic in the considered domain, the approximation of the given Gabriela Kohr function on the boundary of the given domain, generally Faculty of Mathematics and Computer Science, Babes- speaking, is not possible. Bolyai University, 1 M. Kogalniceanu Str., Cluj-Napoca, Natural aggregates of approximation of function deter- Cluj 400084 Romania mined only on the boundary of the domain are gener- [email protected] 1 alized polynomials Pn(z, z ),Pn(z, z) that coincide with rational functions of the form In this talk we present a survey with recent results con- n cerning the Loewner differential equations and univalent X k 1 1 Rn(z) = ak(z + ) = Pn(z, ), subordination chains on the unit ball in n. Various ap- zk z C k=1 plications and examples are also presented. n ∗ X k k ——— Rn(z) = bk(z + z ) = Pn(z, z). k=1

25 I.2. Diﬀerential equations: Complex and functional analytic methods, applications

In this case as in the case of polynomial approximation Dirac operator on an infinite dimensional Clifford alge- J.Walsh problem is urgent. Namely, what necessary and bra and give a theory of hyperfunctions on the fractal sufficient conditions should satisfy a closed curve in or- boundary. der Jackson’s theorem and under some additional con- ——— ditions the Bernstein’s theorem be true on it. Approximation theorems in weighted Lorentz spaces ——— Commutative algebras of monogenic functions and bi- Yunus Emre Yildirir Balikesir University, Necatibey faculty of Education, harmonic potentials Department of Mathematics Balikesir, Central 10100 Sergiy Plaksa Turkey Institute of Mathematics of the National Academy of [email protected] Sciences of Ukraine, Tereshchenkivska str., 3, Kiev-4, 01601, Ukraine In this work, we deal with the simultaneous and con- [email protected] verse approximation of functions possessing derivatives of positive orders by trigonometric polynomials in the An associative commutative two-dimensional algebra B weighted Lorentz spaces with weights satisfying so called with unit 1 over the field of complex numbers is bihar- Muckenhoupt’s Ap-condition. monic if in B there exists a biharmonic basis {e1, e2} satisfying the conditions ———

2 2 2 2 2 Hankel operator on generalized fock spaces (e1 + e2) = 0, e1 + e2 6= 0. (*) El Hassan Youssfi I. Mel’nichenko proved that there exists the unique bi- Centre de Mathématiqueset Informatique, Universitéde harmonic algebra, and he constructed all biharmonic Provence, LAT UMR CNRS 6632, 39 Rue F.Joliot-Curie bases. Marseille, 13453 France Consider a biharmonic plane µ := {ζ := x e + y e }, 1 2 [email protected] where x, y are real. Inasmuch as divisors of zero don’t belong to the biharmonic plane, analytic functions in µ We consider Hankel operators H ¯ with antiholomorphic are defined in the same way as in the complex plane f symbol f¯on the generalized Fock space A2(µ )of square and have similar properties: Cauchy’s integral theorem, m integrable functions with respect to µm, the measure Cauchy’s integral formula, Taylor’s expansion, Morera’s m with weight e−|z| , m > 0 with respect to the Lebesgue theorem. n measure in C . We prove that Hf¯ is bounded if and only Differentiable functions m if f is a polynomial of degree at most 2 . We show that Hf¯ is compact if and only if f is a polynomial of degree Φ(ζ) = U1(x, y)e1 + U2(x, y)ie1+ m strictly smaller that 2 . We also establish that Hf¯ is in U3(x, y)e2 + U4(x, y)ie2 (**) the Schatten class Sp if and only if p > 2n and f is a (p−2n) defined in µ form an commutative algebra and their com- polynomial of degree strictly smaller than m 2p . ponents satisfy the biharmonic equation ——— „ ∂4 ∂4 ∂4 « ∆2U := + 2 + U(x, y) = 0 (***) Continuous mappings between domains of manifolds ∂x4 ∂x2∂y2 ∂y4 Yuriy Zelinskiy 2 (4) 2 2 2 owing to equality ∆ Φ = Φ (ζ)(e1 + e2) and equality Institute of Mathematics Ukr.Ac.Sci., Tereshchenkivska (*). str.3 Kyiv, Kyiv 01601 Ukraine We proved that every solution of equation (***) in a sim- [email protected] ply connected domain is the component U1 of analytic function (**) found explicitly. We proved also that ev- Let f be continues mapping between domains of man- ery analytic function in the plane µ is expressed via two ifolds, with disjoint images of the boundary and of the analytic functions of complex variable. We established domain interior, and certain degree k, then either the an isomorphism between algebras of analytic functions mapping is interior in the sense of Stoilow or there is a given in various biharmonic planes. point possessing at least |k| + 2 preimages. If addition- This is joint work with S. Grishchuk. ally in the last case the map f be zerodimensional in the domain interior, then the set of points possessing at ——— least |k| + 2 preimages contains open ball. Fractal method for Clifford algebra and complex analy- ——— sis Osamu Suzuki Department of Computer Sciences and System Analy- I.2. Differential equations: Complex and sis,College of Humanities and Sciences, Nihon Univer- functional analytic methods, applications sity, Sakurajousui 3-25-40, Tokyo, Setagaya-ku 156-0045 Japan Organisers: [email protected] Heinrich Begehr, Dao-Qing Dai, Jinyuan Du

Wavelet expansion is introduced on a fractal boundary Complex analytic and functional analytic methods are and its diﬀerential and integral calculus are given. Us- used extensively to treat complex ordinary and partial ing this analysis, we construct renormalization theory of diﬀerential equations. The main subject of the session

26 I.2. Differential equations: Complex and functional analytic methods, applications will be higher order partial differential equations. In- bicomplex Vekua equations. We give representations tegral representations, boundary value problems, singu- for these functions using suitable differential operators lar integral equations, properties of integral transforms, acting on T-holomorphic functions as well as on other polyharmonic Green, Robin, Neumann functions are re- bicomplex pseudoanalytic functions. For example the lated. Particular subjects will be special equations as functions investigated here are of interest in connection the Vekua equation, Poisson equation, Bitsadze equa- with the complexified stationary Schrödingerequation. tion, inhomogeneous biharmonic equation. Hyperana- ——— lytic function theory as a tool for treating elliptic systems in plane domains, systems in several complex vari- Boundary value problems on Klein surfaces ables, metaanalytic function theory, Riemann-Hilbert problem and applications e.g. for orthogonal polynomi- Carmen Bolosteanu als might be also discussed. Ordinary complex differen- University “Spiru Haret” Bucharest, Faculty of Account- tial equations and applications in mathematical physics ing and Finance, 223 Traian Street, Campulung Muscel, are an other subject of the session. Arges 115100 Romania [email protected]

—Abstracts— In this paper we introduce some basic notions and solve Riemann-Hilbert problem on the M¨obiusstrip and on A hierarchy of polyharmonic kernel functions and the the real projective plane endowed with a dianalytic related integral operators structure. We use the symmetry in the sense of Klein, Umit Aksoy showing how symmetric conditions on the boundary Atilim University, Department of Mathematics, Kizil- translate into symmetric solutions, which generate so- casar Mahallesi, Incek, Golbasi, Ankara 06836 Turkey lutions to similar problems for Klein surfaces. [email protected] ———

Iterations of the harmonic Green, Neumann and Robin Optimal methods for evaluation hypersingular integrals functions leads to hybrid polyharmonic Green functions. and solution of hypersingular integral equations A hierarchy of these polyharmonic kernel functions will Ilya Boykov be discussed. Taking these functions as a family of kernel Krasnay Str. 40, Penza, 440026 Russia functions, a class of integral operators are established. [email protected] Some properties concerning the related boundary value problems are investigated. Discussed the optimal with respect to accuracy algo- ——— rithms for evaluation of hypersingular integrals Boundary value problems for complex partial differen- 1 Z f(t)dt tial equations f˜(s) = , (t − s)p Heinrich Begehr −1 1 1 Freie UniversitätBerlin, Mathematisches Institut, Arn- Z Z ˜ f1(t1, t2)dt1dt2 imallee 3, Berlin 14195, Germany f1(s1, s2) = p p , (t1 − s1) 1 (t2 − s2) 2 [email protected] −1 −1 1 1 Z Z Some particular complex partial differential equations ˜ f2(t1, t2)dt1dt2 f2(s1, s2) = 2 2 p , are investigates as e.g. the Cauchy-Riemann equation ((t1 − s1) + (t2 − s2) ) −1 −1 the Beltrami equation, the polyanalytic and the poly- 1 1 harmonic equation. Integral representations of Cauchy- Z Z f (t , t )dt dt f˜ (s , s ) = 3 1 2 1 2 , Pompeiu type are presented. They are adjusted to 3 1 2 2 2 p+λ ((t1 − s1) + (t2 − s2) ) certain boundary value problems of Schwarz, Dirich- −1 −1 let, Neumann, Robin type. In particular some polyhar- where 0 < λ < 1, p, p , p = 2, 3,.... monic Green functions are introduced which occur when 1 2 Investigated a smooth of functions f˜ (s), f˜(s , s ), i = merging different polyharmonic Green, Neumann, Robin 1 i 1 2 1, 2, 3. Introduced a classes of functions Ψ˜ , Ψ˜ , i = functions of lower order. This process leads to new repre- i 1, 2, 3,. Functions f˜(s), f˜(s , s ), i = 1, 2, 3. be- sentation formulas and to certain related boundary value i 1 2 long to the classes Ψ˜ , Ψ˜ , i = 1, 2, 3, when functions problems. Their solutions and if appropriate also solv- i f(s), f (s , s ), i = 1, 2, 3, belong to the functional ability conditions are given. i 1 2 classes Ψ, Ψi, i = 1, 2, 3. Evaluated Babenko and Kol- ˜ ˜ ——— mogorov widths of functional classes Ψ, Ψi, i = 1, 2, 3, and constructed a local splines for approximation func- On some classes of bicomplex pseudoanalytic functions tions from functional classes Ψ˜ , Ψ˜ i, i = 1, 2, 3, which Peter Berglez are optimal algorithms for approximation conjugate ˜ ˜ Department of Mathematics, Graz University of Tech- functions f(s), fi(s1, s2), i = 1, 2, 3, from function sets ˜ ˜ nology, Steyrergasse 30, Graz 8010, Austria Ψ, Ψi, i = 1, 2, 3. [email protected] Given optimal with respect to accuracy algorithms for solution of many-dimensional weakly singular, singular and hypersingular integral equations We consider certain classes of bicomplex pseudoanalytic functions using the commutative ring of bicomplex num- Z Z h(t, τ)x(τ) x(t) + λ ... dτ = f(t), bers =∼ Cl (1, 0) =∼ Cl (0, 1). They obey specific v T C C D (r(τ − t))

27 I.2. Diﬀerential equations: Complex and functional analytic methods, applications

1 1 Z Z q(θ)h(t, τ)x(τ) solvability for the mixed boundary value problem. The a(t)x(t) + ... dτ = f(t), format of the expression of solution and the condition (r(τ − t))l −1 −1 of solvability here is rather dissimilar with that in ref-

1 1 erence (Jinyuan Du and Ying Wang. Mixed boundary Z Z h(t, τ)x(τ) value problem for some pair of metaanalytic function a(t)x(t) + ... dτ = f(t). (r(τ − t))(l+λ) and analytic function. Mathematical Methods in the −1 −1 Applied Sciences, 31(15): 1761 - 1779, 2008), but the Here D = [−1, 1]l, 0 ≤ v < l, 0 ≤ λ ≤ 1, θ = ((τ − equivalence for them is concretely proved in the end of 2 2 1/2 this article. t)/(r(τ −t)), r(τ −t) = ((τ1 −t1) +···+(τl −tl) ) , t = (t1, . . . , tl), τ = (τ1, . . . , τl). ——— Given applications singular and hypersingular integral equations in aerodynamics and electrodynamics. Generalized analytic functions on Riemann surfaces Grigory Giorgadze ——— Tbilisi State University, Faculty of Exact and Natural Complex partial differential equations with mixed-type Sciences, Chavchavadze ave. 3, Tbilisi, GE 0128 Geor- boundary conditions gia [email protected] Okay Celebi Yeditepe University, Department of Mathematics Kay- The Riemann-Hilbert monodromy problem (21st Hilbert isdagi Caddesi, Kadikoy Istanbul, 34755 Turkey problem) is to construct the Fuchsian system by the [email protected] marked points and given non-degenerate matrices for which the marked points will be poles of system and the We consider the linear elliptic complex partial differen- monodromy matrices coincide with given matrices. The tial equations of higher order with mixed-type bound- Riemann-Hilbert boundary value problem (the problem ary conditions involving combinations of Dirichlet, Neu- of linear conjugation) was posed by Riemann in the same mann and Robin conditions. The solvability of problems work as one of the methods of the solution of the mon- are discussed via the theory of singular integral equa- odromy problem. tions. Notwithstanding the fact that the problem of linear conjugation is studied for the generalized analytic functions ——— (vectors) and the progress achieved in research of singu- On a mathematical model of a cusped plate with big lar elliptic systems, the connection between Riemann- deflections Hilbert problem with singular elliptic systems and problem of linear conjugation for the generalized analytic Natalia Chinchaladze functions was not noted until now. We consider the I.Vekua Institute of Applied Mathematics of Tbilisi analogous problems and apply the methods of algebraic State University, 2 University St., Tbilisi 0186 Georgia topology. The analog of Fuchsian system in our ap- [email protected] proach will be Carleman-Bers-Vekua system with sindu- lar points. Research object will be the space of multiple- The talk deals with big deflections by the cylindrical valued solutions, the analog of the monodromy theo- bending of a cusped plate with the variable flexural rem, Chen iterated integral,Riemann-Roch theorem and rigidity vanishing at the cusped edge. The setting of Riemann-Hurwitz formula. boundary conditions at the plate edges depends on the geometry of sharpening of the cusped edges. All the ad- ——— missible classical bending boundary-value problems are Optimization of fixed point methods formulated. Existence and uniqueness theorems for the solutions of these boundary-value problems are proved. Sonnhard Graubner Kant-Gymnasium Leipzig, Scharnhorststrasse 15, ——— Leipzig, 04275 Germany [email protected] Mixed boundary value problem with a shift for some pair of metaanalytic function and analytic function The talk deals with operator equations with Lipschitz Jin-Yuan Du continuous right-hand sides. In the case the Lipschitz School of Mathematics and Statistics, Wuhan Univer- condition is only local one, fixed-point methods can be sity, Wuhan, 430072 China applied only in subdomains (polydisc) of the underlying [email protected] function space.The polydisc is optmal if it leads restriction of the norm of the corresponding operator which In this article, we reconsider the same mixed bound- is as small as possible. In the talk we prove a criteron ary value problem on the unit circumference for some leading to a uniquely determined optimal polydisc. pair of a metaanalytic function and an analytic func- ——— tion given in reference (Jinyuan Du and Ying Wang. Generating functions of the Laguerre-Bernoulli polyno- Mixed boundary value problem for some pair of meta- mials involving bilateral series and applications analytic function and analytic function. Mathematical Methods in the Applied Sciences, 31(15): 1761 - 1779, Azhar Hussain 2008). By adopting appropriate function transforma- Department of Physics, Veer Kunwar Singh University, tion, we directly turn the problem into two independent Ara, Bihar, 802301 India boundary value problems for analytic functions. Then [email protected] we obtain the expression of solution and the condition of

28 I.2. Diﬀerential equations: Complex and functional analytic methods, applications

The object of the paper derives a new class of gener- Building, 4700 Keele Street Toronto, Ontario M3J 1P3 ating functions for two variable Laguerre and Laguerre- Canada Bernoulli polynomials involving bilateral series by ap- [email protected] propriately specializing a number of known or new partly unilateral and partly bilateral generating functions are The inhomogeneous Robin/third boundary condition shown to follow as applications of the main results. with general coefficient for the Poisson equation on the unit disc is studied in terms of holomorphic functions us- ——— ing Fourier analysis. It is shown that against the usual Asymptotic behavior of subparabolic functions expectations this problem cannot have a unique solution unless the coefficient of the first order term in the Alexander Kheyfits boundary condition is a constant. For the case of general Bronx Community College, Department of Mathemat- coefficient, it is actually a problem with essential singu- ics and CS, 2155 University Avenue, Bronx, NY 10453 larity in the domain, but still well-posed under proper United States assumptions and the unique solution is given explicitly. [email protected] ——— For subsolutions of the heat equation, we prove Investigation of one class of two-dimensional conju- analogs of some classical results of complex analysis, gating model and non model integral equation with for example, Phragmén-Lindelöfprinciple and Valiron- fixed super-singular kernels in connection with hyper- Titchmarsh theorem. bolic equation ——— Nusrat Rajabov Elliptic Riemann-Hilbert problems for generalized Tajik National University Rudaki Av. 17 Dushanbe, Cauchy-Riemann systems Dushanbe 734025 Tajikistan [email protected] Giorgi Khimshiashvili I.Chavchavadze ave. 32, Ilia Chavchavadze Sate Univer- Let D denote the rectangle D = {a < x < a0, b0 < y < b}. sity, Tbilisi 0179 Georgia In D we consider the conjugating integral equation cor- [email protected] responding to two-dimentional model integral equation with fixed super-singular kernels, that is the integral Generalized Cauchy-Riemann systems introduced by equation E.Stein and G.Weiss are considered. For such systems, a Z a α,β λ natural formulation of Riemann-Hilbert problem will be T (V ) ≡V (x, y) + V (t, y)dt λ,µ (x − a)α suggested which gives a direct generalization of the clas- x µ Z y sical Rieman-Hilbert problem for holomorphic functions. − V (x, s)ds (b − y)β Using the concept of Baum-Douglas obstruction for first b0 order systems, a topological characterization of general- δ Z a Z y + dt V (t, s)ds ized Cauchy-Riemann systems systems possessing ellip- (x − a)α(b − y)β x b0 tic Riemann-Hilbert problems will be given. Moreover, = g(x, y), (*) it will be explained how to verify the resulting condition in terms of the associated Clifford algebra representa- where α = constant > 1, β = constant > 1, λ, µ, δ = tion, which enables us to obtain the complete list of such constant, and g(x, y) is a given function, V (x, y) is the systems. For such systems, it will be also shown that desired function. nonlinear Riemann-Hilbert problems with target man- To problem investigation one dimensional Volterra type ifolds satisfying certain conditions of transversality are integral equation in the case, when kernels have bound- described by Fredholm operators in appropriate func- ary and interior fixed singularity and super-singularity, tional spaces. investigated N. Rajabov [Volterra type integral equation with fixed boundary and interior singular and super- ——— singular kernels and its applications, Dushanbe, 2007, On some qualitative issues of the elliptic systems 222p]. Two-dimentional and three-dimentional Volterra type integral equation investigated N. Rajabov, L. Ra- Nino Manjavidze jabova [Dokl. Math., V. 71, No. 1, 2005, pp 111–114; V. Georgian Technical University, Department of Mathe- 409, No. 6, 2006, pp. 749–753; Math. Notes, Miskols, matics, Kostava str.77, Tbilisi 0175 Georgia V. 4, No. 1 (2003), pp. 65–76]. [email protected] In this lecture for different values of parametrs in the integral equation (*) existance theorems are proved for The theory of elliptic systems on the plane is the clas- inhomogeneous equation. In case, when δ 6= −λµ, has sical object of investigation. We introduce and analyze proved that for existence solution inhomogeneous equa- the generalized Cauchy-Lebesgue classes for some partic- tion (*) representable in the form ular cases of such systems. The analogs of the maximum α β modulus theorem are found. V (x, y) − exp[−λωa (x) + µωb (y)]× This is joint work with G.Akhalaia, G.Makatsaria. ∞ X α n −α × (exp(−ωa (x))) (x − a) Vn(y), ——— n=1 α α−1 −1 Poisson equation with the Robin boundary condition ωa (x) = [(α − 1)(x − α) ] , Alip Mohammed it is necessary and sufficientey infinity number solvabil- Department of Mathematics and Statistics, N520 Ross ity conditions to right part equation (*).

29 I.2. Diﬀerential equations: Complex and functional analytic methods, applications

——— Methods of solutions of an singular integrodifferential equation About one class of two-dimensional Volterra type integral equation with two interior sinqular lines Emma Samoylova People’s Friendship University of Russia, Mikluho- Lutfya Rajabova Maklaya street 6, Moscow 117198 Russia Tajik Technical University Academician Rajabovs Av. [email protected] 10 Dushanbe, Dushanbe 734025 Tajikistan [email protected] It is considered an integrodifferential equation with the singular integral in the sence of Cauchy-Lebesgue prin- Let D denote the rectangle D = D1 ∪D2 ∪D3 ∪D4, D1 = cipale value. Necessary and sufficiant conditions for the {a1 < x < a, b1 < y < b}, D2 = {a < x < a2, b1 < existance and unicvenesse of it solution in corresponding x < b}, D3 = {(x, y): a1 < x < a, b < y < b2}, functions spaces are obtained. D4 = {a < x < a2, b < y < b2},Γ1 = ——— {a1 < x < a2, y = b},Γ2 = {x = a, b1 < y < b2}. In D we consider the two dimensional linear Volterra A boundary condition of the volume potential type Integral Equation with two interior fixed singular Kernels: Tynysbek Sharipovich Kal’menov Institute of Mathematics, Computer Science, and Me- Z x A(t)u(t, y) Z b B(s)u(x, s) chanics, Ministry of Education and Science of Kaza- u(x, y) + α dt − β ds a (t − a) y (b − s) khstan, ul.Shevchenko 28., Almaty 050010 Kazakhstan Z x dt Z b C(t, s)u(t, s) [email protected] + α β ds = f(x, y), (*) (t − a) (b − s) n a y Let Ω ⊂ R be a finite domain with smooth boundary where A(x), B(y), C(x, y), f(x, y) are given functions S.The volume potential is considered in Ω,which and a ∈ Γ1, b ∈ Γ2. In this paper the solution to (*) is Z constructed in the case α = 1, β = 1. In this cases it u(x) = Kf(x) = εn(x − y)f(y)dy Ω is proved that, for certain values A(a), B(b), the homo- 1 geneous Integral Equation (*) has an infinite number of where ε2(x − y) = − 2π log |x − y|, εn(x − y) = 1 1 , n ≥ 3 be a principal fundamental solution linearly independent solutions, and for other certain val- (n−2)σn |x−y| ues A(a), B(b) homogeneous Integral Equation (*) has of the Laplace equation namely not solution, exepting zero. n 2 X ∂ (εn(x − y)) The non-homogeneous Integral Equation (*) for certain −4xεn(x − y) = − ∂x2 values A(a), B(b) has always a solution and its general i=1 i n solutions contain arbitrary functions of one variable. For n 2π 2 n in R ,σn = n is the area of the unit sphere in R ,and other numbers A(a), B(b) the non-homogeneous Integral Γ( 2 ) Equation (*) has a unique solution. Γ is the gamma-function. The volume potential can be applied not only to solve problems in the theory of grav- ——— itation but, in general, to solve a wide range of problems Explicit global solutions of 3D rotating Navier-Stokes in mathematical physics, in particular in electrostatics equations and magnetism. So finding its boundary condition has great theoretic and practical interests.The main result Roman Saks of this paper is the following theorem. Mathematical Institute of RAS, 112, Chernyshevski Theorem. The volume potential with Street, Ufa, 450077, Russia [email protected] −4xu(x) = f(x)

We study the Cauchy problem for 3D Navier-Stockes satisﬁes the boundary condition equations in a frame uniform rotation around the verti- Z 1 ∂εn(x − y) cal axis with a periodical conditions by xj coordinats. u(x) = u(y)dS(y) 2 ∂ny Investigation is based on Fourier development known S Z ∂u(y) and unknown vector function by eigen functions of the − εn(x − y) dS(y), curl and Stockes operators, using Galerkin equations. S ∂ny

This system has very simple explicit form and its linear x ∈ S, for each function f ∈ L2(Ω). Inversely, if a func- 2 part is diagonal. tion u ∈ W2 (Ω) satisfies the Poisson equation and the The familie of the explicit periodical solutions of lin- boundary condition then the function u is defined the vol- ear Stokes-Sobolev equations were found. This solutions ume potential. Where R ∂εn(x−y) u(y)dS(y) - in terms S ∂ny are also the global solutions of the nonlinear Navier- of principal value of Cauchy. Stockes equations. New families of these equations are constracted. ——— For given function we have made program of calculation Eigenvalues and eigenfunctions of volume potential its Fourier coefficients. We have calculated the coefficients of some Galerkin systems. There is also programm Durbudkhan Suragan of numerical solution of its Cauchy problem. Institute of Mathematics, Computer Science, and Me- Our research have been done with the support of grant chanics, Ministry of Education and Science of Kaza- of RFFR 09-01-00349. khstan, ul.Shevchenko 28., Almaty 050010 Kazakhstan [email protected] ———

30 I.2. Diﬀerential equations: Complex and functional analytic methods, applications

n Let Ω ≡ {|x| < δ, x ∈ R , n = 2, 3} with smooth bound- hypothesizes it has four linear-independent partial so- ary S. We have to find eigenvalues and eigenfunctions lutions. Ince type systems with regular and irregular of the volume potential features were studied in work and the possibility of con- Z structing of normal and normal-regular solutions near u(x) = λ εn(x − y)u(y)dy these features were presented. Ω The aim of this work is to establish using the concepts of rank, antirank m and range (v + 1) of coefficients, an where ε (x − y) = − 1 log |x − y|, ε (x − y) = 2 2π n essential and sufficient existence condition of the ending 1 1 , n ≥ 3 Note, we proved that the vol- (n−2)σn |x−y| solution with unknown parameters is rational function ume potential correspondingly satisfying the following or polynomial of two-variables. boundary condition Also, there is an order of proved theorems establishing Z Z 1 ∂εn(x − y) ∂u(y) necessary existence conditions of the ending solutions. u(x) = u(y)dS(y)− εn(x−y) dS(y), This is joint work with A.Zh. Tasmambetova (Aktobe). 2 S ∂ny S ∂ny x ∈ S. ——— Theorem. a) Let n = 2.Then eigenvalues and eigen- Fractional integrals and hypersingular integrals in vari- functions of the volume potential are represented corre- able order Holder spaces on homogeneous spaces (k) [µ ]2 j (k) r ikϕ Ismail Taqi spondingly λkj = δ2 and ukj = ckj Jk(µj δ )e , k = 0, 1, ..., j = 1, 2, ..., where c∗ are constants, (r, ϕ) Arab Open University - Kuwait Branch AlSharhabeel are polar coordinates, Jk is the Bessel function of the Street Khaitan, 92400 Kuwait (k) [email protected] first kind , µj are roots of the following equation

µ(k) Mittag-Leffler [Mittag-Leffler, G. (1903). Sur la nouvelle J (µ(k)) + j (J (µ(k)) − J (µk)) = 0. k j 2k k−1 j k+1 j function Eα(X), C.R. Acad. Sci. Paris, (Ser. II). 137, 554-558] introduced a function defined by an infinite se- b) Let n = 3 . Then eigenvalues and eigenfunctions ries ∞ of the volume potential are represented correspondingly X zk 1 (l+ ) 2 Eα(z) = , α > 0 [µ 2 ] c l+ 1 Γ(αk + 1) j √ljm 2 r m k=0 λkj = 2 and uljm = J 1 (µ )Yl (θ, ϕ), δ r l+ 2 j δ l = 0, 1, ..., j = 1, 2, ..., m = 0, ±1, ..., ±l where (r, θ, ϕ) and investigated some of its properties. This is an entire m are spherical coordinates, Yk is the spherical harmonic function of order 1/α. l+ 1 function, µ 2 are roots of the following equation Another function having similar properties to those of j Mittag-Leffler functions is given by l+ 1 2 ∞ l+ 1 µ l+ 1 l+ 1 k 2 j 2 2 X z Jl+ 1 (µj ) + (Jl− 1 (µj ) − Jl+ 3 (µj )) = 0. Eα,β (z) = , α > 0, β > 0. 2 l + 1 2 2 Γ(αk + β) k=0 ——— For β = 1, Eα,1 = Eα. The ending solutions of Ince system with irregular fea- Such functions arise naturally in the solution of frac- tures tional integral equations [Saxena, R., Mathai, A. and Haubold, H. (2002). On fractional kinetic equations, Zhaxylyk Tasmambetov Astrophysics and Space Science, 282, 281-287] and es- Aktobe State University after K. Zhubanov 263, Bratiev pecially in the study of the fractional kinetic equation, Zhubanov’s street, Aktobe city, 030000 Kazakhstan random walks, etc. [email protected] We study Mittag-Leffler type functions and derive some of their properties including integrals and recurrence re- The Ince system with irregular features: lations. We also study fractional equations of the form

( (0) (1) (4) (2) (5) (3) −1 p Zxx + p q Zxy + p Zx + q Zy + p Z = 0, N(t) − N0 = −c 0Dt N(t),

(0) (4) (1) (5) (2) (3) −ν q Zyy + p q Zxy + p Zx + q Zy + q Z = 0, and its generalization, where 0Dt is the Riemann- Liouville operator of fractional integration. where coeﬃcients p(i) = p(i)(x) and q(i) = q(i)(y) (i = 0, 5) are polynomials of ——— On mixed boundary-value problems of polyanalytic δi ζi (i) X j (i) X j functions p (x) = pij x , q (y) = qij x j=πi j=ξi Yufeng Wang School of Mathematics and Statistics, Wuhan Univer- type (π , δ , ξ , ζ (i = 0, 5) - certain numbers), is studied. i i i i sity, Wuhan 430072 China Let the system be collocated and let the integrability wh [email protected] condition be executable

p(0)q(0) − p(1)q(1)p(4)q(4) 6= 0. Recently, boundary value problems of higher-order complex partial differential equations have been widely in- Ince established that singular curves of this system are vestigated. For example, various kinds of boundary defined by the coefficients in the case of second-order value problems of two-order complex partial differen- private derivatives and in the case of certain additional tial equations, including the Poisson equation and the

31 I.3. Complex-analytical methods for applied sciences

Bitsadze equation, have been systematically discussed, On solution of a kind of Riemann boundary value prob- and the explicit expression of solution and the condition lem on the real axis with square roots of solvability have already been obtained. In addition, some boundary value problems of polyanalytic equation, Shouguo Zhong polyharmonic equation and metaanalytic function have School of Mathematics and Statistics, Wuhan Univer- also been discussed. sity, Wuhan 430072 China In this paper, under the appropriate decomposition [email protected] of polyanalytic functions, some mixed boundary-value problems of polyanalytic functions have been discussed, Solution of the Riemann boundary value problem on the and the explicit expression of solution and the condition real axis X with square roots of solvability have been obtained. pΨ+(x) = G(x)pΨ−(x) + g(x), x ∈ X ——— for analytic function is considered, which was solved un- An algorithm of solving the Cauchy problem and mixed der certain assumptions on the branch points appeared. problem for the two-dimensional system of quasi-linear hyperbolic partial diﬀerential equations ———

Oleg N. Zhdanov Some Riemann boundary value problems in Cliﬀord Siberian State Aerospace University “M.F. Reshetnyov”, analysis Krasnoyarsk, Russia [email protected] Zhongxiang Zhang School of Mathematics and Statistics, Wuhan Univer- sity, Wuhan 430072 China Let’s consider the system of homogeneous quasilinear [email protected] hyperbolic partial diﬀerential equations

1 2 j 1 2 j In this paper, we mainly study the Rm (m > 0) Rie- aij (u , u )∂xu + bij (u , u )∂yu = 0, i, j = 1, 2, (*) mann boundary value problems for functions with values in a Clifford algebra C(V3,3). We firstly prove a general- where aij , bij - smooth functions in area D. ized Liouville theorem for harmonic functions and bihar- There are 3 classical boundary problems for system (*): monic functions by combining the growth behaviour esti- the Cauchy problem, the Riemann problem and the an- mates with the series expansions for k-regular functions. mixed problem. Earlier Cauchy and Riemann problems We obtain the result under only one growth condition at were solved for some particular cases using conservation infinity by using the integral representation formulas for laws. And now we have algorithm for the solution of harmonic functions and biharmonic functions. By us- the Cauchy problem of system (*) in general. Attempts ing the Plemelj formula and the integral representation to solve the mixed prolem weren’t successful for a long formulas, a more generalized Liouville theorem for har- time. Our approach consists in applying to this system monic functions and biharmonic functions is presented. not only one conservation law, as was done in many pa- Combining the Plemelj formula, the integral representa- pers, but a family of such laws with functions depending tion formulas with the above generalized Liouville theo- on parameters. rem, we prove that the Rm (m > 0) Riemann boundary Let’s accurately formulate the problem. Let the func- value problems for regular functions, harmonic functions tion u be specified on the non-characteristic curve MN and biharmonic functions are solvable. The explicit so- in the plane C, and functions u, v be specified on a char- lutions are given. acteristic curve crossing MN. It is important that every characteristic crosses the curve MN only in one point ——— and is not tangent to it in any point. Our aim is to find the intersections of characteristics and the values of functions u and v in these points. I.3. Complex-analytical methods for applied We reduce the mixed problem to the Cauchy problem. sciences We choose a point on the curve MN and a point on the characteristic, and we have a system of algebraic Organisers: equations - corollary fact of conservation law. Using re- Viktor Mityushev, Sergei Rogosin sultant, we obtained one equation for the value of hte function v in the initial point. We find this value and The main attention will be paid to analytic-type results repeat the procedure with another points. It allows us in complex analysis, especially those which have appli- to find the intersections of characteristics and function cations in Mathematical Physics, Mechanics, Chemistry, values in these points with preassigned exactness us- Biology, Medicine, Economics etc. Among the meth- ing a well-known method described in [Kiryakov P. P., ods under consideration are: boundary value problems Senashov S. I., Yakhno A. N. Application of symmetries for holomorphic and harmonic functions and their gen- and conservation laws to differential equations solving. eralizations, singular integral equations, potential anal- Novosibirsk, 2001., p. 170]. As application we obtained ysis, conformal mappings, functional equations, entire the solution of systems, describing state of plane stress and meromorphic functions, elliptic and doubly peri- of Mises‘s plastic surroundings– a problem that is inter- odic functions etc. Applications in Fluid Mechanics, esting for mechanics for more than 100 years. Composite Materials, Porous Media, Hydro- Aero- and ——— Thermo-Dynamics, Elasticity, Elasto-Plasticity, will be the most considered at the session.

32 I.4. Zeros and Gamma lines – value distributions of real and complex functions

—Abstracts— The numbers of zeros of certain classes of meromorphic functions are studied, particularly, in the classi- R-linear problem and its applications to composites cal Nevanlinna and Ahlfors theories. Some analogous results were obtained also for the Gamma-lines of func- Vladimir Mityushev Podchorazych 2 Krakow, Malopolska 30-084 Poland tions (i.e., preimages of curves). This enlarges the value [email protected] distribution, describes not only the numbers but also the locations of a-points and, unexpectedly, leads to new distribution type phenomena for the zeros in real analysis We develop the method of functional equation to derive and real algebraic geometry. Thus we are now in a stage analytical approximate formulae for the effective con- of formation of some methods working in both real and ductivity tensor of the two–dimensional composites with complex analysis. The zeros (a-points, fixed-point) and elliptical inclusions. The sizes, the locations and the ori- Gamma-lines arising in complex analysis (particularly entations of the ellipses can be arbitrary. The analytical meromorphic functions and solutions of ODE, harmonic formulae contains all above geometrical parameters in and polynomial mappings), real analysis, real and com- symbolic form. plex algebraic geometry will be subject of this session. ——— Application of the spectral parameter power series —Abstracts— method to conformal mapping problems An universal value distribution: for arbitrary meromor- Michael Porter phic function in a given domain Department of Mathematics, CINVESTAV-IPN, Li- Grigor Barsegian bramiento Norponiente 2000, Fracc. Real de Juriquilla Institute of Mathematics of the National Academy of Queretaro, 76230 Mexico Sciences, 24-b Bagramian ave. Yerevan, 375019 Arme- [email protected] nia [email protected] Many problems in conformal mapping of plane domains are determined by the Schwarzian derivative of the map- Some purely geometric results analogous to the second ping, a third-order nonlinear differential operator, and fundamental theorems in the classical Nevanlinna and it is well known that this can be rephrased in terms of Ahlfors theories are revealed. These analogs are valid for a second-order linear differential equation y00 + φy = 0. arbitrary analytic (meromorphic) functions in given do- For many mapping problems the coefficient function φ in mains unlike the classical results that are valid only for this equation depends on one or more real or complex pa- some known sub classes of functions that have “equidis- rameters; a typical formulation might be y00 + qy = λry. tributions”. The obtained results are sharp as for func- The global aspect of a mapping problem often translates tions in the complex plane (the classical case) as well as into boundary conditions (possibly nonlinear) on a real for functions in a given domain. interval and a spectral problem is thus presented. We ——— apply the recently developed spectral parameter power series (SPPS) method for Sturm-Liouville problems to A generalization of the Stieltjes-Van Vleck-Bocher the- gain insight into conformal mapping problems. In par- orem ticular we will calculate the complete parameter space Petter Branden for conformal mappings from the disk to a symmetric Department of Mathematics Royal Institute of Technol- circular quadrilateral with right angles. ogy Stockholm, Stockholm 100 44 Sweden ——— [email protected]

Recent results on analytic methods for 2D composite A classical theorem of Stieltjes, Van Vleck and Bôcher materials describes the polynomial solutions f(z), v(z) to the sec- Sergei Rogosin ond order differential equation Department of Mathematics and Mechanics, Belarusian d d Y 00 X Y 0 State University, Nezavisimosti ave, 4 Minsk, BY-220030 (z − αj )f (z) + βj (z − αi)f (z) + v(z)f(z) = 0 Belarus j=1 j=1 i6=j [email protected] where α1 < ··· < αd are real and β1, . . . , βd are positive. B. Shapiro has recently developed a Heine-Stieltjes It is a survey talk on the recent analytic results for 2D theory for linear differential operators of higher order. composite materials. Special attention will be paid to He conjectured a vast generalization of the Stieltjes– application of the boundary value problems for analytic Van Vleck–Bôcher theorem. We prove this conjecture functions, of the functional equations method and of the and describe the intricate structure of the zeros of the integral equation method. solutions. ——— ——— A criterion for the reality of zeros I.4. Zeros and Gamma lines – value David Cardon distributions of real and complex functions Department of Mathematics, Brigham Young Univer- sity, Provo, Utah 84604 United States Organisers: [email protected] Grigor Barsegian, George Csordas

33 I.4. Zeros and Gamma lines – value distributions of real and complex functions

I will discuss a necessary and sufficient condition for cer- VA 24061-0123 United States tain real entire functions to have only real zeros. [email protected] ——— Polynomials with real coefficients and all real roots have New properties of a class of Jacobi and generalized La- many interesting and useful properties. This talk will guerre polynomials introduce an elegant generalization to polynomials with complex coefficients in seeveral variables. These new Marios Charalambides polynomials are called upper (or stable) polynomials and Department of Business Administration, Frederick Uni- are defined by their non-vanishing on the upper half versity, 7 Yianni Frederickou street, Nicosia, Pallourio- plane. This is recent work of J. Borcea, P. Bränden, tisa 1036 Cyprus S. Fisk, B. Shapiro, A. Sokal, and D. Wagner. [email protected] ——— New properties of a class of Jacobi and generalized La- guerre polynomials are presented. The results give new Perturbations of L-functions with or without non-trivial classes of stable polynomials and polynomials with real zeros off the critical line negative roots. Implications of these results on the areas Paul Gauthier of geometry of polynomials and numerical analysis are Departement de mathematiques et de statistique, Uni- also discussed. versitéde Montreal, CP-6128 Centreville Montreal, Que- ——— bec H3Y1Y8 Canada [email protected] Meromorphic Laguerre operators and the zeros of entire functions Joint with X. Xarles. George Csordas There exist small perturbations of L-functions, satis- Department of Mathematics University of Hawaii, Hon- fying the appropriate functional equation, for which olulu, Hawaii 96822 United States the analogue of the Riemann hypothesis fails radically. [email protected] Moreover, this phenomenon is generic. Moreover, there exist small perturbations, for which the analogue of the The purpose of this lecture is to announce new re- Riemann hypothesis holds. sults pertaining to the following open problem. Char- ——— acterize the meromorphic functions, F (x), such that P∞ k Tropical and number theoretic analogues of Nevanlinna k=0 F (k)akx /k! is a transcendental entire function with only real zeros (or that the zeros all lie in the theory half-plane

——— Let G be a non-empty open set in the complex plane C with at least two finite boundary points. Let f : G → An introduction to upper (stable) polynomials in several C be a continuous function that is analytic in G. Let µ(t) variables be a non-negative non-decreasing function defined for Steve Fisk t ≥ 0 such that µ(2t) ≤ 2µ(t) for all t ≥ 0. Suppose Department of Mathematics, Virginia Tech, Blacksburg, that |f(z1) − f(z2)| ≤ µ(|z1 − z2|) for a fixed z1 ∈ ∂G

34 I.4. Zeros and Gamma lines – value distributions of real and complex functions

iθ and for all z2 ∈ ∂G. Suppose that for each unbounded B ⊂ G := { A(r)e | φ(r) < θ < ψ(r), r > r1} component D of G, if any, there is a positive number q for some r1 > 0. Similarly we define a negatively ori- such that f(z) = O(|z|q) as z → ∞ in D. We prove that ented Baker domain. By Spiraling Baker domain, we then one of the following holds: mean either positively oriented Spiraling Baker domain or a negatively oriented spiraling Baker domain. (i) For all z ∈ G we have |f(z ) − f(z )| ≤ Cµ(|z − 2 1 2 1 In this paper we show the existence of Spiraling Baker z |) where C = 3456. 2 domain and obtain several properties of these. (ii) The set G contains a neighbourhood of infinity, ——— so that G has exactly one unbounded component, The algebraic Liouville integrability and the related and f has a pole at infinity. Picard-Fuchs type equations Such problems have a long history, with contributions Anatoliy Prykarpatsky made by Hardy and Littlewood, Walsh, Sewell, Tamra- The AGH-University of Science and Technology, zov, Hayman, Gehring, and the speaker, among others. Krakow, Poland, and Ivan Franko State Pedagogical Earlier such a result, with an absolute constant C, had University, Drohobych, Lviv region, Ukraine 30 Aleja only been known when G is simply or doubly connected. Mickiewicz, N120-C Krakow, 30059 Krakow Poland ——— [email protected]

Zeros de la fonction holomorphe et bornee dans un We consider a completely integrable Liouville-Arnold polyhedre analytique de C2 Hamiltonian system on a cotangent canonically symplectic manifold (T ∗( n), ω(2)), n ∈ , possessing exactly Kazuko Kato R Z+ n ∈ functionally independent and Poisson commut- 2-407 takehanakinomoto-cho yamashina-ku kyoto-si, Z+ ing algebraic polynomial invariants H : T ∗( n) → , kyoto-fu 607-8083 Japan j R R j = 1, n. Due to the Liouville-Arnold theorem this [email protected] Hamiltonian system can be completely integrated by quadratures in quasi-periodic functions on its integral On cherche la condition nécessairepour les zèrosde la submanifold when taken compact. It is equivalent to fonction holomorphe et bornéedans un polyhédre∆ an- 2 the statement that this compact integral submanifold alytique de C . n is diffeomorphic to a torus T , that makes it possible Et, pour les zèrosvérifiant la condition nécessaire,on to integrate the system by means of searching the corre- construit la solution bornéedu deuxièmeproblèmde sponding integral submanifold imbedding mapping. The Cousin dans ∆. following theorems are stated. ——— Theorem. Every completely algebraically integrable Hamiltonian system admitting an algebraic submanifold Steiner and Weyl polynomials n ∗ n Mh ⊂ T (R ) possesses a separable canonical trans- Victor Katsnelson formation which is described by differential algebraic Herzl Departmet of Mathematics, the Weizmann Insti- Picard-Fuchs type equations whose solution is a set of tute, Rehovot, 76100 Israel some algebraic curves [email protected] Theorem. Consider a completely integrable Hamilto- ∗ n nian system on the coadjoint manifold T (R ) whose in- We introduce certain polynomials, so-called H.Weyl and n ∗ n tegral submanifold Mh ⊂ T (R ) is described by Picard- H.Minkowski polynomials, which have a geometric ori- Fuchs type algebraic equations. The corresponding in- gin. The location of roots of these polynomials is stud- n ∗ n tegrability embedding mapping πh : Mh → T (R ) is ied. a solution of a compatibility condition subject to the ——— differential-algebraic relationships on the corresponding canonical transformations generating function. Spiraling Baker domains ——— Anand Prakash Singh Department of Mathematics, University of Jammu, Quantization of universal Teichmüllerspace: an inter- Jammu-180006, INDIA play between complex analysis and quantum field the- [email protected] ory Armen Sergeev Let f be a transcendental entire function. For n ∈ N, Steklov Mathematical Institute, Gubkina 8, Moscow, let f n denote the nth iterate of f. Fatou set F (f) of f is 119991 Russia defined to be the set of all points z in the complex plane [email protected] n C such that the family {f }n≥1 forms a normal family in some neighbourhood of z. Julia set is defined to be Universal Teichmüllerspace T is the quotient of the the complement of Fatou set. group QS(S1) of quasisymmetric homeomorphisms of S1 A periodic component U of F (f) of period m is called modulo Möbiustransformations. It contains the quo- mn 1 1 a Baker domain if f (z) → ∞ as n → ∞ for all tient S of the group Diff+(S ) of diffeomorphisms of S z ∈ U. Further we define a Baker domain B as a posi- modulo Möbiustransformations. Both groups act natu- 1/2 1 tively oriented spiraling Baker domain if there exist pos- rally on Sobolev space H := H0 (S , R). itive continuous functions A(r), φ(r), ψ(r), of r all tend- Quantization problem for T and S arises in string theory ing to ∞ as r → ∞ such that φ(r) in non decreasing, where these spaces are considered as phase manifolds. 0 < ψ(r) − φ(r) < 2π and To solve the problem for a given phase space means to

35 II.1 Cliﬀord and quaternion analysis

fix a Lie algebra of functions (observables) on it and con- Calabria 87036 Italy struct its irreducible representation in a Hilbert (quan- [email protected] tization) space. For S an algebra of observables is given by Lie algebra In the space H of quaternions, we investigate the natural, 1 1 Vect(S ) of Diff+(S ). For quantization space we take invariant geometry of the open, unit disc ∆H and of the 1/2 1 open half-space +. These two domains are diffeomor- the Fock space F (H), associated with H = H0 (S , R). H 1 phic via a Cayley-type transformation. We first study Infinitesimal version of Diff+(S )-action on H generates an irreducible representation of Vect(S1) in F (H), yield- the geometrical structure of the groups of Möbiustrans- + ing quantization of S. formations of ∆H and H and identify original ways of For T the situation is more subtle since QS(S1)-action representing them in terms of two (isomorphic) groups of on T is not smooth. So there is no classical Lie al- matrices with quaternionic entries. We then define the gebra, associated to QS(S1). However, we can define cross-ratio of four quaternions, prove that, when real, it a quantum Lie algebra of observables Derq(QS), gener- is invariant under the action of the Möbiustransforma- ated by quantum differentials, acting on F (H). These tions, and use it to define the analogous of the Poincaré q + differentials arise from integral operators d h on H with distances and differential metrics on ∆H and H . As a kernels, given essentially by finite-difference derivatives spin-off, we directly deduce that there exists no isometry 1 of h ∈ QS(S ). between the quaternionic Poincarédistance of ∆H and the Kobayashi distance inherited by ∆H as a domain of ——— 2 C , in accordance with the well known classification of the non compact, rank 1, symmetric spaces. II.1 Clifford and quaternion analysis ——— Wavelets invariant under reflection groups Organisers: Irene Sabadini, Frank Sommen Paula Cerejeiras Department of Mathematics, University of Aveiro, We call for contributions in the fields of theoretical Aveiro, P-3810-193 Portugal quaternionic and Clifford analysis and, more in general, [email protected] hypercomplex analysis intended as the study of the function theory related to the Dirac operator and systems of For signal reconstruction over a sphere, two main ap- partial differential operators taking values in a Clifford proaches are used: the group-theoretical one (see, algebra. All the topics varying from the study of mono- for instance, Antoine/Vandergheynst or M. Ferreira) genic functions, its generalisations to higher spin such as where the authors use representations over homogeneous the Rarita-Schwinger system, Clifford analysis on super- spaces and the one using approximate identities and sin- space, Clifford-Radon and Fourier transforms, discrete gular kernels (see Freeden, or Swelden). However, both Clifford analysis to functions with values in more gen- rely on the Lorentz group and, therefore, are not suit- eral non-commutative structures are welcome. able for signals with predefined symmetries which involve reflections. To overcome this problem, we consider differential-difference operators associated to specific fi- —Abstracts— nite reflection groups, the so-called Dunkl operators. In Clifford analysis for orthogonal, symplectic and finite this setting we construct spherical Dunkl wavelets based reflection groups on approximate identities and we give practical examples. Hendrik de Bie ——— Department of Mathematical Analysis, Ghent Univer- sity, Krijgslaan 281, 9000 Ghent (Belgium) Some consequences of the quaternionic functional cal- [email protected] culus Fabrizio Colombo In recent work we have developed a theory of Clifford Dipartimento di Matematica, Politecnico di Milano, via analysis in superspace. This can be seen as Clifford anal- Bonardi 9 Milano, Mi 20133 Italy ysis invariant under the product of the symplectic with [email protected] the orthogonal group. Other authors have recently also studied Clifford analysis with respect to finite reflection We show some of the most recent results on the quater- groups (using Dunkl operators). nionic functional calculus for left and right linear quater- In this talk we will give a general and unified framework nionic operators defined on quaternionic Banach spaces. that can be used for these different symmetries. This approach allows us to deal both with bounded and We will also discuss some typical problems that depend unbounded operators. In particular we use such a func- on the symmetry at hand. These will include the Fischer tional calculus to study the quaternionic evolution op- decomposition, the Fourier transform and the Hermite erator. polynomials. We also discuss related quantum systems. ——— ——— Orthogonality of Clifford-Hermite polynomials in super- Möbiustransformations and Poincarédistance in the space. quaternionic setting Kevin Coulembier Cinzia Bisi Department of Mathematical Analysis, Ghent Univer- Dipartimento Matematica, Universita’ della Calabria, sity, Krijgslaan 281 Ghent 9000 Belgium Cubo 30b, Ponte P.Bucci, Arcavacata di Rende Cosenza, [email protected]

36 II.1 Cliﬀord and quaternion analysis

In previous work by De Bie and Sommen, the Clifford- vector space over the scalar field F = R or C. We will Hermite polynomials were generalized to superspace. In present the factorizations of the paravector unit ball by this talk we will construct an inner product for which gyro-subgroups and subgroups, generalizing the case of n these polynomials are orthogonal, using the Berezin in- the unit ball on Euclidean space R . The main differ- tegral. This inner product can moreover be used for ences between both cases are the replacement of the Spin quantum mechanics in superspace, as it restores the her- group by the Spoin group and the establishment of a ge- miticity of the anharmonic oscillator. ometric product for the paravector case, analogous to As an application we will also derive a Mehler formula the geometric product in the vector case. with O(m) × Sp(2n) symmetry. The Mehler formula ——— gives an expansion of the kernel of the fractional Fourier transform in terms of the super Clifford-Hermite poly- Higher spin analogues of the Dirac operator in two vari- nomials. This was already known in one dimension ables and its resolution (Hermite polynomials) and formally for O(m) (Clifford- Peter Franek Hermite polynomials), but the O(m) × Sp(2n) poses Mathematical Institute, Charles University Praha, some extra difficulties. Sokolovska 83 Prague, 8 186 75 Czech Republic ——— [email protected] Recent results on hyperbolic function theory A resolution of the Dirac operator in two variables is well Sirkka-Liisa Eriksson known and well understood. It consists of three invariant Department of Mathematics, Tampere University of operators (on of those of second order) expressed using Technology , P.O.Box 553, Tampere 33101 Finland the Dirac operators in two individual variables. We shall [email protected] discuss higher spin analogues of such resolutions. They are again complexes of three invariant operators acting The aim of this talk is to consider the hyperbolic version on functions with values in more complicated represen- of the standard Clifford analysis. The need for such a tation spaces. modification arises when one wants to make sure that ——— the power function xm is included. The leading idea is that the power function is the conjugate gradient of a Cauchy kernels in ultrahyperbolic Clifford analysis – harmonic function, defined with respect to the hyper- Huygens cases bolic metric of the upper half space. We present re- Ghislain R. Franssens sults and problems concerning power series presentation Belgian Institute for Space Aeronomy, Ringlaan 3, B- of hypermonogenic functions This work is done jointly 1180 Brussels, Belgium with professor Heinz Leutwiler, University of Erlangen- [email protected] Nürnberg, Department of Mathematics, Erlangen, Ger- many, email: [email protected]. p,q ` p+q ´ Let R , R ,P , with P the canonical quadratic ——— form of signature (p, q). Clifford Analysis (CA) over Rp,q, called Ultrahyperbolic Clifford Analysis (UCA), is Symmetric properties of the Fourier transform in Clif- a non-trivial extension of the familiar (Euclidean) CA ford analysis setting over Rn. Ming-Gang Fei Essential for stating integral representation theorems in Departamento de Matemática, Universidade de Aveiro, UCA is the determination of a reproducing (or Cauchy) p,q Campus Universitario de Santiago Aveiro, Aveiro 3810- kernel Cx0 of R , ∀p, q ∈ Z+, for the Dirac operator ∂.

193 Portugal Any such kernel can be obtained as Cx0 = ∂gx0 , with gx0 [email protected] a fundamental distribution of the ultrahyperbolic equa- p+q tion p,qgx0 = δx0 , x0 ∈ R . The complexity of UCA In this talk we present Fueter’s Theorem for Dunkl- is due to the fact that Cx0 is a rather complicated dis- monogenic functions. We show that if f is a holomor- tribution, whose form profoundly depends on the parity phic function in one complex variable, then for any un- of p and q. d γκ+(d−1)/2 Iﬀ p and q are odd is g proportional to a delta distribu- derlying space R1 the induced function ∆h f(x) x0 p,q is Dunkl-monogenic whenever γκ + (d − 1)/2 is a non- tion δ(P (x−x0)), having as support the null space of R negative integer, where ∆h is Dunkl Laplacian. To relative to x0, and then gx0 is said to satisfy Huygens’ this end Vekua-type systems for axial Dunkl-monogenic principle. In this talk, explicit expressions for the distri- functions are studied. butions gx0 and Cx0 will be presented for the Huygens

cases. We will see how δ(P (x−x0)) arises as a pullback of ——— the one-dimensional delta distribution δ and the matter Factorization of Möbius gyrogroups - the paravector of “regularization”, required for some of these distribu- case tions, will be carefully addressed. Milton Ferreira ——— Campus Universitáriode Santiago, Departamento de Power series and analyticity over the quaternions Matemática,Universidade de Aveiro, Aveiro 3810-193 Portugal Graziano Gentili [email protected] Dipartimento di matematica ”U.Dini”, viale Morgagni 67/a, 50134 Firenze, Italy We consider a Möbiusgyrogroup on the unit ball of [email protected] the vector space F ⊕ V, where V is a finite dimensional

37 II.1 Cliﬀord and quaternion analysis

We study power series and analyticity in the quater- This is joint work with D. Constales and D. Grob. nionic setting. We first consider a function f defined P n ——— as the sum of a power series q an in its domain n∈N of convergence, which is a ball B(0,R) centered at 0. Explicit description of the resolution for 4 Dirac opera- At each p ∈ B(0,R), f admits expansions in terms of tors in dimension 6 appropriately defined regular power series centered at P ∗n Lukas Krump p, (q − p) bn. The expansion holds in a ball n∈N Mathematical Institute of the Charles University, Σ(p, R − |p|) defined with respect to a (non-Euclidean) Sokolovska 83, Praha 8, 186 75 Czech Republic distance σ. We thus say that f is σ-analytic in B(0,R). [email protected] Furthermore, we remark that Σ(p, R − |p|) is not always an Euclidean neighborhood of p; when it is, we say that There are several approaches to the construction of a res- f is quaternionic analytic at p. It turns out that f is olution of several Dirac operators in higher dimensions. quaternionic analytic in a neighborhood A of B(0,R)∩R, Among them, the Penrose transform method gives satis- with A strictly contained in B(0,R) unless R is infinite. fying results in both stable and unstable cases. Recently We then extend these results to the larger class of this method was used to determine the shape of such quaternionic slice regular functions, enriching their the- resolution in many cases and the next step is an explicit ory. Indeed, slice regularity proves equivalent to σ- description of operators involved. This will be shown for analyticity and slice regular functions are quaternionic the unstable case of four operators in dimension six. analytic only in a neighborhood of the real axis. ——— ——— On polynomial solutions of Moisil-Theodoresco systems Isomorphic action of SL(2, R) on hypercomplex num- in Euclidean spaces bers Roman Lavicka Anastasia Kisil Mathematical Institute, Charles University Sokolovska Triniti College Cambridge, University Cambridge, Cam- 83 Praha 8, Praha 186 75 Czech Republic bridgeshire CB2 1TQ, United Kingdom [email protected] [email protected] Let k be a positive integer and 0 ≤ s ≤ m. Denote by We investigate the SL(2, ) invariant geodesic curves R Pk the space of real-valued k-homogeneous polynomials with the associated invariant distance function in m s in R . Moreover, Λ stands for the space of s-vectors parabolic geometry. Parabolic geometry naturally oc- m s N s over R and P = Pk Λ . We are interested in the k R curs as action of SL(2, R) on dual numbers and is placed following space in between the elliptic and the hyperbolic geometries s s ∗ (which arise from the action of SL(2, R) on complex and Hk = {P ∈ Pk ; dP = 0, d P = 0}. double numbers). Initially we attempt to use standard Here d and d∗ is the de Rham differential and its ad- methods of finding geodesics but they lead to degener- joint, respectively. Moreover, assume that r, p and q are acy in this set-up. Instead, by studying closely the two non-negative integers such that p < q and r + 2q ≤ m. related hypercomplex numbers we discover a unified ap- Putting proach to a more exotic and less obvious dual number’s q (r,p,q) M r+2j case. With aid of common invariants we describe the Pk = Pk , possible distance functions that turn out to have some j=p unexpected, interesting properties. the space ——— (r,p,q) (r,p,q) ∗ MTk = {P ∈ Pk ;(d + d )P = 0} Construction of 3D mappings on to the unit ball with is formed by all k-homogeneous polynomial solutions of the hypercomplex Szego kernel the Moisil-Theodoresco system of type (r, p, q). We show Rolf Soeren Krausshar that Department of Mathematics, Katholieke Universiteit q q−1 Leuven, Celestijnenlaan, 200-B Leuven, Vlaams Bra- (r,p,q) M r+2j M r+2j+1 MTk ' Hk ⊕ Hk−1 . bant, 3001 Belgium j=p j=p

[email protected] s Later on, the spaces Hk are considered as SO(m)- In this talk we present a hypercomplex generalization modules. We are interested in irreducibility, the high- of the Szego kernel method that allows us to construct est weights and dimensions of such modules. In par- 3 ticular, we give a formula for the dimension of the 3D mappings from some elementary domains of R onto (r,p,q) space MT . Moreover, we decompose the kernel of the unit sphere. More precisely, we consider an ap- k propriately chosen line integral over the square of the the Hodge laplacian on polynomial forms into SO(m)- hypercomplex Szego kernel. The latter one is approxi- modules. mated numerically by the monogenic Fueter polynomials These results were obtained jointly with R. Delanghe for rectangular domains, an L-shaped domain, circular and V. Souˇcek. cylinders and the double cone. In all these cases the line ——— integration provides an amazingly good mapping onto Quaternionic analysis, representation theory and the unit sphere. We also compare the quality of results Physics ontained with this method with the results that were obtained previously by using alternatively the Bergman Matvei Libine kernel method. Department of Mathematics, Indiana University, Rawles

38 II.1 Cliﬀord and quaternion analysis

n ∗ Hall, 831 East 3rd St Bloomington, IN 47405 United exterior differentiation acting on forms on R , and d is States its formal adjoint. [email protected] Our goal is to prove that any Dirac and semi-Dirac pair (D, D†) has two Cauchy-Pompeiu and two Bochner- This is a joint work with Igor Frenkel. Martinelli-Koppelman type integral representation for- I will describe our new developments of quaternionic mulas. analysis using representation theory of various real forms ——— of the conformal group as a guiding principle. These developments will lead to a solution of Gelfand-Gindikin A differential form approach to Dirac operators on sur- problem. Along the way we discover striking new con- faces nections between quaternionic analysis and mathematical physics. In particular, the Maxwell equations are Heikki Orelma realized as the quaternionic counterpart of the Cauchy Institute of Mathematics, Tampere University of Tech- formula for the second order pole. We also find a nology, P.O. Box 553, FI-33101 Tampere, Finland representation-theoretic meaning of the polarization of [email protected] vacuum and one-loop Feynman integrals. This talk is partially based on the joint paper with Igor In this talk we consider Dirac operators on surfaces. Sur- m Frenkel, “Quaternionic analysis, representation theory faces are k-dimensional embedded submanifolds of R . and physics”, Advances in Mathematics 218 (2008) pp Let F be a Clifford algebra-valued differential form and m 1806-1877; also available at arXiv:0711.2699. ∂x be the Dirac operator on R . F is called monogenic if it is a solution of the equation ——— L∂ F = 0, Hyperholomorphic functions in the sense of Moisil- x Thodoresco and their different hyperderivatives where L∂x F is the Lie derivative of F with respect to ∂ . The aim of this talk is to show that if F and ∂ Maria Elena Luna-Elizarraras´ x x ESFM-IPN, U.P.A.L.M. Av. IPN s/n Col.Lindavista are restricted to the k-surface S we obtain a Dirac type Mexico City, D.F. 07338 Mexico equation L F | = 0 [email protected] ∂x|S S on S. As an application we shall consider winding num- Any Moisil-Théodoresco-hyperholomorphic function is bers. also Fueter-hyperholomorphic, but its hyperderivative is always zero, so one could consider then that these This is joint work with Frank Sommen (Gent). functions are a kind of “constants” for the Fueter oper- ——— ator. It turns out that the skew-field of quaternions as a real linear space is wide enough, so it is possible to CK-extension and Fischer decomposition for the infra- give another type of hyperderivatives “consistent” with monogenic functions the Moisil-Théodoresco operator. In this talk we present these notions of different hyperderivatives and the rela- Dixan Pena˜ Pena˜ tion between them. Department of Mathematics, University of Aveiro, Cam- The talk is based on a joint work with M. A. Mac´ıas pus Universitario de Santiago, Aveiro 3810-193, Portu- Cedeñoand M. Shapiro. The research was partially gal supported by CONACYT projects as well as by Insti- [email protected] tuto Politécnico Nacional in the framework of COFAA and SIP programs. Let ∂x denote the generalized Cauchy-Riemann opera- m+1 tor in R . In this communication, we will present a ——— refinement of the biharmonic functions and at the same Dirac and semi-Dirac pairs of differential operators time an extension of the monogenic functions by considering the solutions of the sandwich equation ∂xf∂x = 0. Mircea Martin In this setting a CK-extension and a Fischer decompo- Department of Mathematics, Baker University, 8th and sition are studied. Grove, Baldwin City, Kansas 66006 United States [email protected] ——— A new approach to slice-regularity on real algebras n The Euclidean Dirac operator Deuc,n on R , n ≥ 2, is a differential operator with coefficients in the Clifford al- Alessandro Perotti n 2 Dept. Mathematics, Univ. of Trento, Via Sommarive gebra of R that has the defining property Deuc,n = −∆, n 14 Povo, Trento I-38100 Italy where ∆ = ∆euc,n is the Laplace operator on R . As generalizations of this class of operators we investi- [email protected] † n gate pairs (D, D ) of differential operators on R with coefficients in a Banach algebra A, such that either We rivisit the concept of primary functions introduced † † † † DD = µL∆ and D D = µR∆, or DD + D D = µ∆, by Rinehart in the ’60’s and apply it to the theory of where µL, µR, or µ are some elements of A. Such pairs slice regular functions introduced recently by Gentili, (D, D†) are called Dirac or semi-Dirac pairs of dif- Struppa and other authors. ferential operators. The typical examples of a Dirac or (Joint work with Riccardo Ghiloni, Trento, Italy) semi-Dirac pair on n are given by D = D† = d + d∗, R ——— or D = d and D† = −d∗, where d is the operator of

39 II.1 Cliﬀord and quaternion analysis

Clifford analysis with higher order kernel over un- Purpose of this talk is to provide a characterization of bounded domains the dual of the Rn-module of slice monogenic functions on a class of compact sets in the Euclidean space n+1. Yuying Qiao R We are able to establish a duality theorem which, since Yuhua east Road 113, College of Mathematics and Infor- holomorphic functions are a very special case of slice mation Science, Hebei Normal University, Shijiazhuang, monogenic functions, is the generalization of the classi- Hebei Province 050016 China cal Köthe’stheorem. The duality results are also dis- [email protected] cussed in the quaternionic setting. In this paper we talk Clifford analysis with higher or- ——— der kernel over unbounded domains. First we derive Explicit description of operators in the resolution for an higher order Cauchy-Pompeiu formula for the func- the Dirac operator tions with rth order continuous differentiability over an unbounded domain whose complementary set con- Tomas Salac tains nonempty open set. Then we obtain higher or- Faculty of Mathemtarics and Physics, Sokolovská83, der Cauchy integral formula for k-regular functions and Prague 8, 18675 Czech Republic prove Cauchy inequality. Based on the higher order [email protected] Cauchy integral formula, we define higher order Cauchy- A study of Dirac operator D in several variables is a type integrals and the Plemelj formula. traditional part of Clifford analysis. A lot of effort was ——— spent to find an analogue of the Dolbeaut complex, i.e. a resolution starting with the operator D. The resolu- Complex Dunkl operators tion is composed (in the stable range) from operators Guangbin Ren of the first and the second order. Using representation Departamento de Matemática- Universidade de Aveiro, theory, it is possible to write down an explicit form of Campus de Santiago, Aveiro 3810-193 Portugal the first order operators in the resolution. It is, however, [email protected] much more difficult to compute an explicit form of second order operators. In the lecture, we shall use Casimir Complex Dunkl operators for certain Coxeter groups are operators (recently introduced in study of parabolic ge- introduced. These complex Dunkl operators have the ometries) as a new tool helping to get these explicit for- commutative property, which makes it possible to es- mulae for the second order part of the resolution. tablish the corresponding complex Dunkl analysis. ——— ——— On the relation between the Fueter operator and the p-Dirac equations Cauchy-Riemann-type operators of Clifford analysis. John Ryan Michael Shapiro Department of Mathematics, University of Arkansas, ESFM-IPN, U.P.A.L.M. Av. IPN s/n Col.Lindavista Fayetteville, Arkansas 72703, United States Mexico City, D.F. 07338 Mexico [email protected] [email protected] The Moisil-Théodoresco operator has an explicitly given Associated to Laplacians there are first order operators relation with the classic Dirac operator of Clifford anal- called Dirac operators. For instance the Dirac operator ysis for Cl . It turns out that the Fueter operator does associated to the Laplacian in the complex plane is the 0,3 not have, as one would expect, a similar relation with Cauchy-Riemann operator. In euclidean space there is the corresponding classic Cauchy-Riemann operator but the euclidean Dirac operator Similar such operators ex- a modification of the latter is necessary. The aim of the ist for Laplace-Beltrami operators on Riemannian man- talk is to explain all this in detail thus establishing a di- ifolds. rect relation between, on one hand, what is usually called Besides the usual Laplace equation in euclidean space quaternionic analysis, and, on the other hand, Clifford there are the non-linear p-Laplace equations. These analysis. equations are covariant under Möbiustransformations This is joint work with J. Bory-Reyes. M. Shapiro was and are invariant when p = n. Here we shall intro- partially supported by CONACYT projects as well as duce non-linear p-Dirac equations. We shall demon- by Instituto PolitécnicoNacional in the framework of strate their link to the p-Laplacian in euclidean space COFAA and SIP programs. and demonstrate their covariance under Möbiustrans- formations. Other basic properties of these equations ——— will be investigated. We shall extend the p-Dirac and Conformally invariant boundary valued problems for p-Laplace equations to spin manifolds. spinors and families of homomorphisms of generalized This is joint work with Craig A. Nolder (Florida State Verma modules. University). Petr Somberg ——— Mathematical Institute of Charles University, Duality theorems for slice hyperholomorphic functions Sokolovska 83, Prague, Karlin 180 00 Czech Repub- lic Irene Sabadini [email protected] Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, Milano, Mi 20133 Italy On a conformal manifold M with boundary ∂M there is a [email protected] construction associating conformally invariant non-local

40 II.1 Clifford and quaternion analysis operators to the boundary valued problems for confor- ——— mally invariant operators on M with symbols given by On relative BGG sequences power of Laplace operator. These operators belong to one parameter families of conformally invariant oper- Vladimir Soucek ators, generalizing conformal Dirichlet-to-Robin opera- Sokolovska 83 Mathematical Institute, Charles Univer- tor. We will discuss generalization towards conformally sity Praha, Czech Republic 186 75 Praha Czech Republic invariant boundary valued problems for the spinor rep- [email protected] resentation. The Penrose transform is a perfect tool for a study of ——— generalised Dolbeault resolutions in the theory of sev- Clifford calculus in quantum variables eral Clifford variables. An important notion used in the definition of the Penrose transform is the relative BGG Frank Sommen resolution. Its construction is indicated in the book by Department of Mathematics, University of Ghent, Gal- Baston and Eastwood on the Penrose transform. They, glaan 2. B-9000 Gent, Belgium however, deserve a better attention; their construction [email protected] can be made more detailed using tools used for construction of the classical BGG sequences. Starting from the axioms of the algebra R(S) of abstract ——— vector variables over a set S (radial algebra): Regular Moebius transformations over the quaternions z(xy + yx) = (xy + yx)z, x, y, z ∈ S, together with the basic q-commutation relations for co- Caterina Stoppato ordinates: Dipartimento di Matematica “U. Dini”, Universitàdi i j j i x x = qij x x Firenze, Viale Morgagni 67/A, I-50134 Firenze, Italy [email protected] we arrive at the defining relations for the q-Clifford algebra: Let denote the real algebra of quaternions. We present e e + q e e = −2g , H i j ji j i ij quaternionic transformations that are included in the whereby gij is the q-metric which also consists of non- class of regular quaternionic functions introduced by G. commuting parameters. The partial derivatives ∂xj sat- Gentili and D.C. Struppa in recent years. Regularity isfy the same q-relations yields to properties that recall the complex case, although the diversity of the quaternionic setting intro- ∂ ∂ = q ∂ ∂ xi xj ij xj xi duces new phenomena. Specifically, the group Aut(H) of biregular functions → coincides with the group together with the q-Weyl relations: H H of regular affine transformations (namely, q 7→ qa + b j j with a, b ∈ and a 6= 0). Moreover, inspired by the ∂xi x = qjix ∂xi + δij . H classical quaternionic linear fractional transformations, This leads to the introduction of a reciprocal Clifford we define the class of regular fractional transformations. j basis e satisfying: This class strictly includes the set of regular injective i i functions from b = ∪ {∞} to itself. Finally, we study e e + q e e = −2δ , H H j ji j ij regular Moebius transformations, which map the unit which is linked to the original Clifford basis by relations ball B = {q ∈ H : |q| < 1} onto itself. All regular of the form (Einstein summation convention): bijections from B to itself prove to be regular Moebius transformations. e = g ek. j jk ——— The vector derivative (Dirac operator) is then given by Singularities of functions of one and several bicomplex j ∂x = ∂xj e and the basic rules of Clifford calculus may variables be derived. On the level of radial algebra these rules are the same as for standard Clifford analysis, which indi- Adrian Vajiac cates that the q-deformation aspect is only visible when Chapman University, Dept of Math/CS, One University calculations are expressed in coordinates. This raises Drive, Orange, CA 92866 United States the problem to define a kind of q-deformation on the [email protected] level of abstract vector variables. This can be done by In this talk we introduce the notion of regularity for defining the Dirac operator ∂ in a suitable way as an x functions of one, as well as several bicomplex variables. endomorphism on R(S). This may be done by assuming Moreover, using computational algebra techniques, we the operator relation ∂ x = −qx∂ + m + 2qE, whereby x x prove that regular functions of one bicomplex variable m is the dimension of space and E ∈ End(R(S)) is the have the property that their compact singularities can q-Euler operator given by the operator relations be removed. Ex − qxE = x, Ez = zE, x, z ∈ S. ——— However, the identities for the q-quantum lattice seem Multiplicities of zeroes and poles of regular functions to lead (in the first approximation) to a relation of the Fabio Vlacci form Department of Mathematics Ulisse Dini, viale Morgagni ∂ x + qx∂ = m + q(q + 1)E x x 67/a FIRENZE, FI 50134 Italy whereby Ex − q2xE = x. [email protected]

41 II.2 Analytical, geometrical and numerical methods in Cliﬀord- and Cayley-Dickson-algebras

The aim of this talk is to give a survey on some recent The construction of wavelets relies on translations and results which have been obtained for the description of dilations which are perfectly given in R. On the sphere the zero sets (and poles) of regular functions. In partic- translations can be considered as rotations but it diffi- ular we will focus our attention to define (and evaluate) cult to say what are dilations. For the 2-dimensional a multiplicity for zeroes and poles of regular functions. sphere there exist two different approaches. The first concept defines wavelets by means of kernels of spher- ——— ical integrals. The other approach is a purely group Gauss-Codazzi-Ricci equations in Riemannian, confor- theoretical approach and defines dilations as dilations in mal, and CR geometry the tangent plane. Surprisingly both concepts coincides for zonal functions. We will define wavelets on the 3- Zuzana Vlasakova dimensional sphere by means of kernels of integrals and Sokolovska 83, Faculty of Mathematics and Physics, demonstrate that wavelets constructed according to the Praha 8, 18675, Czech Republic group-theoretical approach for zonal functions meet our [email protected] definition. Typical examples arise quite easily from the Abel- We will remind the Gauss-Codazzi-Ricci equations in Poisson and Gauß-Weierstraß kernel. Riemannian geometry, and the work of David Calder- We will extend these kernels and wavelets into the bank with Francis Burstal and Diemer on similar equa- Clifford-algebra setting. We specifically define spheri- tions for conformal geometry. Then we introduce the CR cal wavelets of order m. geometry and explain that we can do the same thing also for this geometry (it is a complex analogue of conformal Theorem. The elements of {Ψρ, ρ > 0} are wavelets of geometry). order m (m ≥ 0) if the following admissibility conditions are satisfied: ——— Z ∞ 2 2 Ψe ρ(k)α(ρ) dxρ = (k + 1) , k = m + 1, m + 2, ... Compatibility conditions and higher spin Dirac opera- 0 tors Ψe ρ(k) = 0, k = 0, ..., m; ∀ρ ∈ (0, ∞) Liesbet Van de Voorde Z π ˛Z ∞ ˛ ˛ (2) ˛ 2 ˛ Ψρ (θ)α(ρ) dxρ˛ sin (θ) dxθ ≤ T, ∀R ∈ (0, ∞), Department of Mathematical Analysis, Clifford Re- ˛ ˛ search Group, Galglaan 2, 9000 Gent, Belgium 0 R [email protected] (T > 0, independent of R). (2) Here, Ψρ stands for Ψρ ∗ Ψρ.Ψ1 (ρ = 1) is the mother In this talk, we investigate polynomial solutions for gen- wavelet. eralized Rarita-Schwinger operators. We will explain that there are two types of solutions, and we will explic- ——— itly construct one of them using results on compatibility On special monogenic power and Laurent series expan- conditions for systems in several Dirac operators. This sions and applications is joint work with David Eelbode and Fred Brackx. Sebastian Bock ——— Bauhaus-University Weimar, Institute for Mathemat- ics/Physics, Coudraystraße 13B, Weimar, 99421 Ger- many II.2 Analytical, geometrical and numerical [email protected] methods in Clifford- and Cayley-Dickson-algebras The contribution focuses on some recently developed (orthogonal) monogenic power and Laurent-series ex- Organisers: pansions which are complete in the space of square inte- Klaus Gurlebeck,¨ Vladimir Kisil, grable quaternion-valued functions and have as similar Wolfgang Sproßig¨ properties as the respective complex series expansions based on the well known z-powers. Starting with the The mathematical use of above mentioned algebras Fourier series expansion we will show some structural reaches from hypercomplex analysis and differential ge- properties of the series expansion with respect to their ometry up to corresponding numerical methods. There- hypercomplex derivative and primitive. These special fore we call especially for contributions with applications characteristics of the used orthonormal basis enable fur- in gauge theories, mathematical physics, image process- ther the construction of a new Taylor type series expan- ing, robotics, cosmology, engineering sciences etc. sion which can be explicitly related to the corresponding Fourier series analogously as in the complex one- —Abstracts— dimensional case. We end up by showing some orthogonality results for the exterior domain and present the Wavelets on spheres corresponding Laurent series expansion for the domain of the spherical shell. Swanhild Bernstein These series expansions find applications in the descrip- Freiberg University of Mining and Technology, Institute tion of the hypercomplex derivative as well as the mono- of Applied Analysis, Prüferstr. 9, D-09596 Freiberg, genic primitive of a monogenic function which are rep- Germany resented as Fourier series, Taylor type and Laurent se- [email protected] ries. In this connection some further applications are presented.

42 II.2 Analytical, geometrical and numerical methods in Cliﬀord- and Cayley-Dickson-algebras

——— Recent progress in this direction will be reviewed. In particular integrable generalizations of KdV and NLS in Spin gauge models 4+2 will be presented and the question of their reduc- Ruth Farwell tion to 3+1 will be discussed. The role of quaternions Buckinghamshire New University, Queen Alexandra for generalizing these results to higher dimensions will Road, High Wycombe, Bucks HP11 2JZ United King- be investigated. dom ——— [email protected] Note on the linear systems in quaternions In 1999 we defined a form of spin gauge theory of particle Svetlin Georgiev interactions in which both standard ’left-hand’ and new Sofia University, Faculty of Mathematics and Informat- ’right-hand’ interaction terms occur. In the proceed- ics, Department of Differential Equations, Blvd James ings of the 2005 Toulouse conference we reported the Boucher 126, Sofia 1000 Bulgaria predictions of the value of the Weinberg angle and the [email protected] mass of the Top quark based on a particular ’two-sided’ model, and we introduced the concepts of the ’quark’ In this talk we will discuss the linear system and ’centroid’ representations. We also discussed new r n X X s s gravitational effects and the replacement of the graviton plmxmqlm = As, s = 1, 2, . . . , n, (*) by the ’frame field quantum’. l=1 m=1 Recently, we have studied a variety of other two-sided s s where n, r ≥ 1 are given constants, plm, qlm, As, models, and we present the predictions of another model, l = 1, . . . , r, m = 1, . . . , n, s = 1, . . . , n, are given real in which a different choice of spinor idempotent allows quaternions, xm, m = 1, . . . , n, are unkown real quater- us to introduce a new particle interaction term. nions. This is joint work with Roy Chisholm (Kent). Here a propose an algorithm for finding a solution to ——— the system (*). Also, we give necessary and sufficient condition for the solvability of the system (*) and some Further results in discrete Clifford analysis examples. Nelson Faustino ——— Departamento de Matemática,Campus Universitáriode Minimal algorithms for Lipschitzian elements and Santiago Aveiro, Aveiro 3810-193 Portugal Vahlen matrices [email protected] Jacques Helmstetter In this talk we will present the fundamentals of a higher 15 rue de l Oisans, St-Martin d’Heres, Isere 38400 France dimensional discrete function theory by combining the [email protected] Clifford algebra setting with the umbral calculus approach. If S is a closed algebraic manifold in a vector space V , Starting with the umbral version of Fischer decomposi- and if d is the codimention of S in V , an algorithm that tion, we decompose the space of umbral homogeneous allows us to test whether an element of V belongs to S polynomials in terms of umbral monogenic polynomials. by means of only d numerical verifications, is called a This allows us to build up in a combinatorial way the minimal algorithm. If Cl(M, q) is the Clifford algebra theory of discrete spherical monogenics as a refinement derived from a quadratic module (M, q), the Lipschitz of the theory of spherical harmonics. monoid Lip(M, q) is (in most cases but not in all cases) Furthermore, the interplay between discrete Clifford the monoid generated in Cl(M, q) by M. From the in- analysis and the physical model of the discrete harmonic variance property of Lipschitz monoids, a minimal algo- oscillator will be explored along this talk by means of the rithm can be deduced for the even and odd components canonical generators of Wigner Quantum Systems. of Lip(M, q). A minimal algorithm can also be deduced for the two components of the monoid of Vahlen matri- ——— ces. Integrability in multidimensions, complexification and ——— quaternions Clifford-Fourier transforms and hypercomplex signal Thanasis Fokas processing Department of Applied Mathematics and Theoretical Jeff Hogan Physics, Centre for Mathematical Sciences, Wilberforce School of Mathematical and Physical Sciences, Univer- Road Cambridge, Cambridgeshire CB3 0WA United sity of Newcastle V-128, University Drive Callaghan, Kingdom NSW 2308 Australia [email protected] [email protected]

One of the most important open problems in the area In this talk we attempt to synthesize recent progress of integrable nonlinear evolution equations has been the made in the mathematical and electrical engineering construction of integrable equations in 3+1, i.e. in three communities on topics in Clifford analysis and the pro- spatial and one temporal dimensions. The celebrated cessing of colour images, in particular the construction KdV and NLS equations are integrable evolution equa- and application of Clifford-Fourier transforms designed tions in 1+1; the KP and DS equations are physically to treat multivector-valued signals. Emphasis will be significant generalizations of the KdV and NLS in 2+1. placed on the two-dimensional setting where the appro- Do there exist analogous equations in 3+1? priate underlying Clifford algebra is the familiar set of

43 II.2 Analytical, geometrical and numerical methods in Clifford- and Cayley-Dickson-algebras quaternions. We’ll describe some results and problems where µ ∈]0, 1[. We further give explicit formulas for in the construction of discrete wavelet bases for colour the Szegökernel for solutions to polynomial Dirac equa- images, and the difficulties encountered in constructing tions of polynomial degree m < 3 in the annulus. As Clifford-Fourier kernels in dimensions 3 and higher. concrete application we give an explicit representation formula for the solutions of generalized Helmholtz and ——— Klein-Gordon type equation inside the annulus and with Discrete Clifford analysis by means of skew-Weyl rela- prescribed data at the boundary of the annulus. The tions solutions are represented in terms of integral operators that involve the explicit formulas of the Bergman kernel Uwe Kahler¨ that we computed. Department of Mathematics, University of Aveiro, Aveiro, P-3810-193 Portugal ——— [email protected] The Ito transform for partial differential equations

Recently one can observe an increased interest in higher Remi Leandre dimensional discrete function theories. This is not only Institut de Mathematiques. Universite de Bourgogne Bd driven by the numerical application of continuous meth- Alain. Savary Dijon, Cote d’Or 21078. France ods but also due to problems from combinatorics and [email protected] quantum physics. While there is now a well-established approach in the continuous case, by means of the so- We give an interpretation of the celebrated Ito formula called radial algebra (F. Sommen), unfortunately, a di- of stochastic analysis in various contexts where there is rect translation to the discrete case is problematic. In no convenient measure on a convenient path space. this talk we present an alternative approach based on a We begin by the case of a diffusion (the classical one), recent idea of F. Sommen of replacing the Weyl relations we study after the case of the heat-equation associated by skew-Weyl relations. We will construct the basic in- to an operator of order four on a torus, we continue by gredients for discrete Clifford analysis in this context studying the case of the Schroedinger equation associ- and illustrate its applicability. ated to a big order operator on a torus, we consider after the case on the wave equation on a torus and we ——— finish by studying the case of a Levy type operator asso- Hypercomplex analysis in the upper half-plane ciated to a big fractional power of the Laplacian on the linear space. Vladimir Kisil ——— School of Mathematics, Woodhouse Lane, University of Leeds, LS2 9JT, United Kingdom Quaternionic analysis and boundary value problems [email protected] Dimitris Pinotsis Department of Mathematics, University of Reading, Complex analysis seems to be the only non-trivial ana- RG6 6AX, UK lytic function theory in the two dimensional case. How- [email protected] ever one can employ the group SL(2,R) and its representation theory in order to build elements of analytic First, we will review some results appearing in the the- functions with complex, dual and double numbers. This ory of quaternions. Then, we will apply these results to is a part of “Erlangen Programme at Large” approach solve boundary value problems for linear elliptic equa- in analysis. tions in four dimensions. Further extensions of these ——— results will also be discussed. Formulas for reproducing kernels of solutions to poly- ——— nomial Dirac equations in the annulus of the unit ball n Integral theorems in a commutative three-dimensional in R and applications to inhomogeneous Helmholtz harmonic algebra equations Vitalii Shpakivskii Rolf Soeren Krausshar Institute of Mathematics of National Academy of Sci- Department of Mathematics, Katholieke Universiteit ences of Ukraine, Tereshchenkivska str., 3, Kiev-4, Leuven, Celestijnenlaan, 200-B Leuven, Vlaams Bra- 01601, Ukraine bant, 3001 Belgium [email protected] [email protected] An associative commutative three-dimensional algebra Pn ∂ Let D := ei be the Euclidean Dirac operator in i=1 ∂xi A3 with unit 1 is harmonic if in A3 there exists a har- n m R and let P (X) = amX + ... + a1X1 + a0 be a poly- monic basis {e1, e2, e3} satisfying the conditions nomial with arbitrary complex coefficients. Differential 2 2 2 2 equations of the form P (D)f = 0 are called polynomial e1 + e2 + e3 = 0, ej 6= 0 for j = 1, 2, 3. (*) Dirac equations with complex coefficients. There are three harmonic algebras exactly over the field In this talk we consider Hilbert spaces of Clifford algebra of complex numbers only, and all harmonic bases are valued functions that satisfy such a polynomial Dirac constructed by I. Mel’nichenko. equation in annuli of the unit ball in n. We deter- R We consider a harmonic algebra A3 containing the rad- mine a fully explicit formula for the associated Bergman ical with basis {ρ1, ρ2} and multiplication table: kernel for solutions of complex polynomial Dirac equa- 2 2 tions of any degree m in the annulus of radii µ and 1 ρ1 = ρ2, ρ2 = 0, ρ1ρ2 = 0.

44 III.1. Toeplitz operators and their applications

We proved that every locally bounded function differen- To do this end, we start by express the arising tenso- tiable in the sense of Gateaux (such a function is mono- rial spaces in terms of complexified Clifford algebras and genic) we construct a fiber bundle identification of our spaces with appropriated vector spaces of tensors and differen- Φ(ζ) = U1(x, y, z)e1 + U2(x, y, z)e2 + U3(x, y, z)e3 tial forms. We then establish the semi-groups for the family of regularized Schrödingeroperators and prove (here ζ = xe1 + ye2 + ze3 and x, y, z are real) has n- th Gateau derivative for any n. So, the components their dissipative property. We end with an application to the non-stationary Schröringerequation. U1,U2,U3 satisfy the three-dimensional Laplace equation ——— „ ∂2 ∂2 ∂2 « ∆3U := + + U(x, y, z) = 0 ∂x2 ∂y2 ∂z2 III.1. Toeplitz operators and their 00 2 2 2 applications owing to equality ∆3Φ = Φ (ζ)(e1 + e2 + e3) and equality (*). Organisers: For monogenic functions Φ(ζ) taking values in A3, we proved Cauchy’s theorems for surface integral and curvi- Sergei Grudsky, Nikolai Vasilevski linear integral. We proved also an analog of Cauchy’s The idea of the session is to bring together the ex- formula that yields Taylor’s expansion of monogenic perts actively working on Toeplitz operators acting on function. Morera’s theorem is also established. Thus, Bergman, Fock or Hardy spaces, as well as in various as in the complex plane, one can give different equiva- related areas where Toeplitz operators play an essential lent definitions of monogenic functions taking values in role, such as asymptotic linear algebra, quantisation, ap- the algebra A3. proximation, singular integral and convolution type op- This is joint work with S. Plaksa. erators, financial mathematics, etc. ——— We expect that the results presented, together with fruitful discussions, will serve as a snapshot of the cur- Initial boundary value problems with quaternionic anal- rent stage of the area, as well as for better understanding ysis of the priority directions and themes of future develop- Wolfgang Sproßig¨ ments. TU Bergakademie Freiberg, Institute of Applied Analy- sis, Prüferstr.9, Freiberg 09599 Germany —Abstracts— [email protected] On the relations between the kernel of a Toeplitz op- A quaternionic operator calculus is used to find represen- erator and the solutions to some associated Riemann- tations of the solution of several initial boundary value Hilbert problems problems in mathematical physics. Cristina Camaraˆ ——— Departamento de Matemática, Instituto Superior Técnico,Av. Rovisco Pais ,Lisboa, 1049-001 Portugal Real bi-graded Clifford modules, the Majorana equation [email protected] and the standard model action It is possible, in many cases, to determine some solution ¨ Tolksdorf, Jurgen to a Riemann-Hilbert problem associated to T , of the Max-Planck-Institute for Mathematics in the Sciences, G form Inselstraße 22, 04105 Leipzig, Germany ± n Gh+ = h−, h± ∈ (H ) . (*) [email protected] ∞ Such a solution can provide important information on n×n The fundamental grading involution that underlies the the properties of TG. Namely, for G ∈ (L∞(R)) , with det G = 1 (or admitting a bounded canonical fac- Dirac equation is provided by parity. In contrast, the ± Majorana equation is based on charge conjugation. To- torization), if h± = (h1±, h2±) are corona pairs in C , gether, these two grading involutions form what is called i.e., inf (|h1±(ξ)| + |h2±(ξ)|) > 0, (**) a Majorana module. On these modules there exist a nat- ± ξ∈C ural class of Dirac operators encoding the action func- it can be shown that TG is invertible. tional of the Standard Model of particle physics. In this talk, the question of what information can be ——— obtained, as regards the kernel of TG, from a solution to (*), is considered. Several classes of symbols are studied The regularized Schrödingersemigroup acting on ten- which, if n = 2, correspond to a situation where (**) is sors with values in vector bundles not, or may not, be satisfied. Nelson Vieira ——— Departamento de Matemática-Universidade de Aveiro, Campus Universitáriode Santiago, P-3810-193 Aveiro, Convolution type operators with symmetry in exterior Portugal wedge diffraction problems [email protected] Luis Castro Campus Universitario, Department of Mathematics, In this talk we apply known techniques from semigroup University of Aveiro, Aveiro 3810-193 Portugal theory and Clifford analysis to the homogeneous prob- [email protected] lem with initial condition of the Schrödingerequation.

45 III.1. Toeplitz operators and their applications

2d d We will use convolution type operators with symme- Lorentz space L(p, q, wdµ)(R ). M(p, q, w)(R ) is a Ba- try in a Bessel potential spaces framework to anal- nach space with the norm kfkM(p,q,w) = kVgfkpq,w. yse classes of problems of wave diffraction by a In this paper we discussed the boundedness of d plane angular screen occupying an infinite 270 degrees Toeplitz operator on M(p, q, w)(R ) under some as- wedge sector. The problems are subjected to differ- sumptions. We also proved that the Toeplitz oper- d d ent possible combinations of boundary conditions on ator T pg(F ) of M(2, p, w1)(R ) into M(2, p, w1)(R ) the faces of the wedge. Namely, under consideration is S2 with the Hilbert-Schmidt norm bounded by there will be boundary conditions of Dirichlet-Dirichlet, kT pg(F )kS2 CkF k(1,t) under some condition. Neumann-Neumann, Neumann-Dirichlet, impedance- This is joint work with Ay¸seSandik¸ci. Dirichlet, and impedance-Neumann types. Existence ——— and uniqueness results are proved for all these cases in the weak formulation. In addition, the solutions are pro- Presentation of the kernel of a special structure matrix vided within the spaces in consideration, and higher reg- characteristic operator by the kernels of two operators ularity of solutions are also obtained in a scale of Bessel one of them is a scalar characteristic operator potential spaces. The talk is based on a joint work with Oleksandr Karelin D. Kapanadze. Advanced Research Center on Industrial Engineering, ——— Autonomous University of the Hidalgo State, Pachuca, Hidalgo 42184 Mexico Berezin transform on the harmonic Fock space [email protected] Miroslav Englis We denote the Cauchy singular integral operator along Mathematics Institute AS CR Zitna 25, Prague 1, the upper part of the unit semicircle by Prague 11567 Czech Republic T+ 1 Z ϕ(τ) [email protected] (S ϕ)(x) = dτ T+ πi τ − x The standard Berezin-Toeplitz quantization is based on T+ the asymptotic expansion of the Berezin transform as and the identity operator on T+ by (IT+ ϕ)(t) = ϕ(t). the weight parameter tends to infinity. We discuss an By operator equalities, results about the integral opera- extension of this result to the case of the harmonic Segal- tors with endpoint singularities are extended to matrix n Bargmann-Fock space on C . characteristic operators 2 ——— DR+ = uIT+ + vST+ ,DT+ ∈ [L2(T+)] Inside the eigenvalues of certain Hermitian Toeplitz with the coefficients u, v of a special structure. The fol- band matrices lowing decomposition ˜ \ Sergey Grudsky ker DT+ = ker H F ker C, Department of Mathematics, CINVESTAV, Av. Insti- is found. Here operator C is a scalar characteristic op- tuto Politecnico Nacional 2508, Col. San Pedro Zaca- erator, C ∈ [L ( )], operator F is invertible operator, tenco, 07360 Mexico 2 T+ F ∈ [L ( ),L2( )]. Operators H˜ and C are con- [email protected] 2 T+ 2 T+ structed by an arbitrary nontrivial element of ker DT+ or by an arbitrary nontrivial element of the kernel of the While extreme eigenvalues of large Hermitian Toeplitz associated operator. matrices have been studied in detail for a long time, This is joint work with Anna Tarasenko. much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of ——— an n-by-n banded Hermitian Toeplitz matrix as n goes Bounds for the kernel dimension of singular integral op- to infinity and provides asymptotic formulas that are erators with Carleman shift uniform in j for 1 ≤ j ≤ n. The real-valued generating function of the matrices is as- Edixon Rojas sumed to increase strictly from its minimum to its max- Campus Universitario, Department of Mathematics, imum and then to decrease strictly back from the max- University of Aveiro, Aveiro 3810-193 Portugal imum to the minimum, having nonzero second deriva- [email protected] tives at the minimum and the maximum. Upper bounds for the kernel dimension of singular in- ——— tegral operators with preserving-orientation Carleman

d shift are obtained. This is implemented by using some Toeplitz operators of M(p, q, w)(R ) spaces estimations which are derived with the help of certain Turan Gurkanlı¨ explicit operator relations. In particular, the interplay Ondokuz Mayıs University, Faculty of Arts and Sciences, between classes of operators with and without Carle- Department of Mathematics, Kurupelit Samsun, 55139 man shifts has a preponderant importance to achieve Turkey the mentioned bounds. [email protected] ———

d d Invertibility of matrix Wiener-Hopf plus Hankel opera- Let g be a function in S(R )/0, where S(R ) is Schwartz d tors with diﬀerent Fourier symbols space, and 1 ≤ p, q ≤ ∞. The space M(p, q, w)(R ) denotes the subspace of all tempered distributions f such Anabela Silva that the Gabor transform Vgf of f is in the weighted Departamento de Matem´atica- Universidade de Aveiro,

46 III.2. Reproducing kernels and related topics

Campus de Santiago, Aveiro 3810-193 Portugal States [email protected] [email protected]

Based on different kinds of auxiliary operators and corre- We will discuss Toeplitz operators on the Fock space sponding operator relations, we will present conditions induced by positive measures. Problems considered in- which characterize the invertibility of matrix Wiener- clude boundedness, compactness, and membership in Hopf plus Hankel operators having different Fourier the Schatten classes. symbols in the class of almost periodic elements. ——— ——— Flat Hilbert bundles and Toeplitz operators on symmetric spaces III.2. Reproducing kernels and related topics Harald Upmeier Organisers: Department of Mathematics, University of Marburg, Alain Berlinet, Saburu Saitoh Hans-Meerwein-Strasse, Lahnberge Marburg, Hessen 35032 Germany Since the first works laying its foundations as a subfield [email protected] of Complex Analysis, the theory of reproducing kernels has proved to be a powerful tool in many fields of Pure In generalization of the classical Fock spaces we con- and Applied Mathematics. The aim of this session is struct a family of Hilbert spaces, viewed as a Hilbert to gather researchers interested in theoretical as well as bundle over a bounded symmetric domain (Cartan do- applied modern problems involving this theory. main) B, which is equivariant under a suitable, non- holomorphic, action of the holomorphic automorphism —Abstracts— group G of B (a semisimple Lie group). Geometrically, these Hilbert spaces live on the so-called Matsuki dual A general theory for kernel estimation of smooth func- associated with the G-orbits in the boundary of B. tionals We show that the Hilbert bundle carries a natural connection over B which is projectively flat, similar as the Belkacem Abdous well-known case for the metaplectic representation on Universite Laval Medecine Sociale et Preventive, Pavil- Fock space. The associated parallel transport (Bogoluy- lon de l’Est, Quebec, Qc G1K 7P4 Canada bov transformations) is also determined. In the talk we [email protected] emphasize relations to classical Fock spaces over real, complex and quaternion matrix spaces, although the ba- In this talk, we present a general framework for estimat- sic construction depends mainly on the Jordan algebraic ing smooth functionals of the probability distribution description of bounded symmetric domains. functions, such as the density, the hazard rate function, the mean residual time, the Lorenz curve, the spectral ——— density, the tail index, the quantile function and many Commutative algebras of Toeplitz operators on the unit others. This framework is based on maximizing a local ball asymptotic pseudo-likelihood associated to the empirical distribution function. An explicit solution of this Nikolai Vasilevski problem is obtained by means of reproducing kernels Department of Mathematics, CINVESTAV, Av. Insti- approach. Some asymptotic properties of the obtained tuto Politecnico Nacional 2508, Col. San Pedro Zaca- estimators are presented as well. tenco, 07360 Mexico [email protected] ———

∗ Weighted composition operators on some spaces of an- All known commutative C -algebras generated by alytic functions Topelitz operators on the unit disk are classiﬁed as follows. Given a maximal commutative subgroup of bi- Som Datt Sharma holomorphisms of the unit ball, the C∗-algebra gener- Department of Mathematics, University of Jammu, ated by Toeplitz operators, whose symbols are invariant Jammu-180006, India 66 Ashok Nagar, Canal Road, under the action of this subgroup, is commutative on Jammu, Jammu & Kashmir 180016 India each weighted Bergman space. somdatt [email protected] Surprisently there exist many other Banach algebras generated by Toeplitz operators which are commuta- Let D be the open unit disk in the complex plane C and tive on each weighted Bergman space. These last al- H(D) be the space of holomorphic functions on D. In gebras are non conjugated via biholomorphisms of the this article, we give a short and selective account of re- ∗ unit ball, non of them is a C -algebra, and for n = 1 all sults known about weighted compostion operator Wψ,ϕ of them collapse to commutative C∗-algebra generated deﬁned by by Toeplitz operators on the unit disk. Wψ,ϕf(z) = ψ(z)f(ϕ(z)), f ∈ H(D), ——— where ϕ is a holomorphic map of that takes into Toeplitz operators on the Fock space D D itself and ψ is any holomorphic map of D. Discriptions Kehe Zhu of weighted composition operators acting from Hardy Department of Mathematics and Statistics, 1400 Wash- spaces, weighted Bergman spaces, α-Bloch spaces and ington Ave, SUNY Albany, New York 12222 United A−α-spaces into other spaces of holomorphic functions

47 III.3. Modern aspects of the theory of integral transforms have been obtained by a number of authors during re- 4) Vienna, A-1090 Austria cent years. We provide a unified way of treating these [email protected] operators. ——— Irregular sampling in spline-type spaces has become a vivid research area, with many contributions in the re- Integral formulas on the boundary of some ball cent literature. We will describe efficient implementa- Keiko Fujita tions of operators related to spline-type spaces with fi- d Faculty of Culture and Education, Saga university, Saga nite sets of generators on R , covering both the case of 840-8502 regular and irregular sampling. In contrast to earlier [email protected] papers, which either treat the continuous setting using abstract methods (i.e. continuous Fourier transforms) or We have been studied integral representations for holo- deal with the discrete case when it comes to numerical morphic functions and complex harmonic functions on implementations, we are discussing the problem of con- structively realizing the abstract concepts with methods some balls, which we call the ”Np-balls”. One of Np- balls is the Lie ball. For holomorphic functions on that can be implemented on a computer, achieving a the Lie ball we know the Cauchy-Hua integral formula, small error of reconstruction in a certain given norm. whose integral is taken over the Shilov boundary of the In such a situation the trade-off between realizing in- Lie ball. A generalization of the Cauchy-Hua integral dividual iterative steps with high precision but at high formula was considered for holomorpic functions on sub- computational costs, versus the option of doing a larger number of iterations has to be analyzed. spaces of the Lie ball by M.Morimoto. Since Np-ball can be represented by a union of these subspace, the bound- Joint work with Prof. Hans Feichtinger. ary of the Np-ball can be represented by a union of the ——— boundaries of the subspaces. Considering the fact, we consider an integral representation for holomorphic func- Free boundary value problem for (−1)M (d/dx)2M and tions on the Np-ball by an iterated integral. the best constant of Sobolev inequality In this talk, we will review some integral formulas on Kazuo Takemura holomorphic functions on the Np-ball and treat some topics. Shinei 2-11-1, Narashino, Chiba 275-8576 Japan [email protected] ——— Paley–Wiener spaces and their reproducing formulae. Green function of free boundary value problem for (−1)M (d/dx)2M is found using Whipple’s formula. Its John Rowland Higgins Green function is constructed through so-called symmet- I.H.P., 4 rue du Bary, 11250 Montclar, France. ric orthogonalization method under a suitable solvability [email protected] conditions. As an application, we found the best constant of Sobolev inequality for M = 1, 2, 3, 4, 5 by inves- Classical Paley–Wiener space, denoted by PW, consists tigating an aspect of Green function as a reproducing of functions that are inverse Fourier transforms of those kernel. For M ≥ 6, this is still open. 2 members of L (R) that are null outside [−π, π]. It is well known that PW possesses two reproducing formu- ——— lae; a reproducing equation and a ‘discrete’ analogue, or sampling series, and that these make a remarkable ‘concrete – discrete’ comparison. It is shown that such III.3. Modern aspects of the theory of analogies persist in the setting of more general Paley– integral transforms Wiener spaces. ‘Operator’ versions of the reproducing equation and of the sampling series will be given that Organisers: are also comparable, but now in a slightly different way. Anatoly Kilbas, Saburu Saitoh The setting emerges from two sources, the approach to sampling theory via the reproducing kernel theory due to S. Saitoh, and the approach via harmonic analysis of I. Kluvánek,M.M. Dodson et al. The capacity for —Abstracts— amalgamation of these two sources has gone unnoticed hitherto. Integral transforms related to generalized convolutions The special case of multiplier operators with respect and their applications to solving integral equations to the Fourier transform acting on Paley–Wiener space Liubov Britvina will be considered. The Hilbert transform, and in two- Department of Theoretical and Mathematical Physics, dimensions the Riesz transforms, provide examples with Novgorod State University, ul.St.Petersburgskaya 41, possibilities of extension to higher dimensions and to Veliky Novgorod, Novgorod region 173003 Russia further classes of operators. [email protected] ——— The present research is devoted to some integral trans- Irregular sampling in multiple-window spline-type forms of convolution type. The definition of polyconvo- spaces lution, or generalized convolution, was first introduced Darian Onchis by V.A. Kakichev in 1967. Let A1, A2 and A3 be oper- Faculty of Mathematics, University of Vienna, Nord- ators. The generalized convolution of function f(t) and bergstrae 15 (UniversitätsZentrum Althanstrae, UZA k(t), under A1, A2, A3, with weighted function α(x),

48 III.3. Modern aspects of the theory of integral transforms

“ α ” We study equation is the function h(t) denoted by fA1 ∗ kA2 (t) for A3 which the following factorization property is valid: 2 Z 1 d ± α 2 u (t)+ ET u (t) φ(α)dα + F (t, u (t)) = 0, 0 < t < T »“ α ” – dt 0 (A3h)(x) = A3 fA1 ∗ kA2 (x) A3 ± α where, ET u (t) is the symmetrized Caputo fractional = α(x)(A1f)(x)(A2k)(x). derivative of u, φ(α), α ∈ (0, 1), is a positive integrable function or a positive compactly supported distribution Here we consider the generalized convolution for integral with the support in (0, 1) and F is a continuous func- transforms with the Bessel functions in the kernels. Us- tion in [0,T ] × and locally Lipschitz continuous with ing the differential properties of these convolutions we R respect to the second variable. construct some integral transforms and find their exis- This is joint work with T. Atanackovic, S. Konjik and tence conditions and inverse formulas. Natural applica- S. Pilipovic. tions to the corresponding class of convolution integral equations are demonstrated. ——— ——— Numerical real inversion of the Laplace transform by reproducing kernel and multiple-precision arithmetic Bedrosian identity for Blaschke products in n-parameter cases Hiroshi Fujiwara Kyoto University, Yoshida-Honmachi Sakyo-ku Kyoto, Qiuhui Chen Kyoto 606-8501 Japan Departamento de Matemática- Universidade de Aveiro, [email protected] Campus de Santiago, Aveiro 3810-193 Portugal [email protected] We consider the real inversion of the Laplace transform. It appears in engineering or physics, and it is ill-posed We establish a necessary and sufficient condition for in the sense of Hadamard. We introduce some repro- the amplitude function such that a Bedrosian identity ducing kernel Hilbert spaces and propose an inversion holds in the case when the phase function is determined algorithm employing Tikhonov regularization. The reg- by the boundary value of a Blaschke product with n- ularized equation is well-posed, and its discretization parameters. is expected to have the stability and convergence with ——— a suitable norm. However, theoretical stability is not equivalent to the stability of computational processes. Evaluation formulae for analogues of conditional ana- We propose the use of multiple-precision arithmetic to lytic Feynman integrals over a function space reduce the influence of rounding errors for reliable nu- Dong Hyun Cho merical computations. Multiple-precision arithmetic is Department of Mathematics, Kyonggi University, useful for regularization as approximation. Young-Tong-Gu Suwon, Kyonggido 443-760 South Ko- ——— rea [email protected] Method of integral transforms in the theory of fractional differential equations In this talk, we introduce two simple formulae for the Anatoly Kilbas conditional expectations over an analogue C[0, t] of the Faculty of Mathematics and Mechanics, Belarusian Wiener space, the space of continuous real-valued paths State University, Independence Avenue, 4 Minsk, 220030 on the interval [0, t]. Using these formulae, we estab- Belarus lish various formulae for analogues of the conditional [email protected] analytic Wiener and Feynman integral of the functionals in a Banach algebra which corresponds to the Ba- Our report deals with the method of integral trans- nach algebra on the classical Wiener space introduced forms in investigtation of differential equations with or- by Cameron and Storvick. Finally, we evaluate the ana- dinary and partial fractional derivatives. First we give logues of the conditional analytic Wiener and Feynman an overview of results in this field. Then we present ap- integral for the functional plication of one-dimensional Laplace, Mellin and Fourier Z t ff integral transforms to solution of ordinary diferential exp θ(s, x(s)) dη(s) equations with Riemann-Liouville and Caputo fracitonal 0 derivatives. Further we give application of Laplace which is defined on C[0, t] and is of interest in Feynman and Fourier integral transforms to obtain explicit solu- integration theories and quantum mechanics. tions of Cauchy-type and Cauchy problems for the two- and multi-dimensional diffusion-wave equations with the ——— Riemann-Liouville and Caputo partial fractional deriva- An equation with symmetrized fractional derivatives tives, respectively, and indicate conditions for the existence of classical solutions of these problems. Fi- Diana Dolicanin´ nally, we use Laplace and Fourier integral transforms Faculty of Technical Sciences, University of Priˇstina to deduce explicit solutions of fractional evolution equa- - Kosovska Mitrovica, Kneza Miloˇsa, 28000 Kosovska tions involving partial fractional derivatives of Riemann- Mitrovica, Serbia Liouville or Caputo with respect to time and partial Li- dolicanin [email protected] oville fractional derivative with respect to real axis, and indicate applications.

49 III.4. Spaces of diﬀerentiable functions of several real variables and applications

We note that explicit solutions of the considered where φ is ”weight” distribution with compact support fractional differential equations and Cauchy-type and and, Dγ denotes the Riemman-Liouvill fractional deriva- Cauchy problems for them are expressed in terms of spe- tive of order γ. cial functions of Mittag-Leffler, Wright and the so-called Differential equations of the form H-functions. Z 2 Z 2 γ γ ——— φ1(γ)D u dγ = φ2(γ)D v dγ (*) 0 0 Notes on the analytic Feynman integral over paths in abstract Wiener space are constitutive equations for viscoelastic body. We consider (*) coupled with nonlinear ordinary differ- Bong Jin Kim ential equation Department of Mathematics, Daejin University, Pocheon, Kyeonggi-Do 487-711 South Korea D2u(t) + v = f(t, u(t)) (**) [email protected] and show existence and uniqueness of the solution to In this talk, we study some results about analytic Feyn- the problem (*)–(**) with initial conditions u(0) = u0, man integral over paths in abstract Wiener space. 0 u (0) = v0 in classical and mild sence. ——— ——— On the fractional calculus of variations Some aspects of modified Kontorovitch-Lebedev inte- Sanja Konjik gral transforms Faculty of Agriculture, Department of Agricultural En- gineering, Trg D. Obradovica 8, Novi Sad, 21000 Serbia Juri M. Rappoport sanja [email protected] Vlasov street Building 27 apt.8 Moscow 117335 [email protected] The purpose of this talk is to study variational principles allowing Lagrangian density to contain derivatives of arbitrary real order. We derive a necessary condition for A proof of inversion formulas of the modified existence of a solution to a fractional variational problem Kontorovitch-Lebedev integral transforms is developed. and examine invariance under the action of transforma- The Parceval equations for modified Kontorovitch- tion groups. As the results we obtain the Euler-Lagrange Lebedev integral transforms are proved and sufficient equations, as well as infinitesimal criterion and Noether’s conditions for them are found. Some new representa- theorem, which in fact extend the well-known classical tions and properties of these transforms are justified. results. In addition, we also study the case when both The inequalities which give estimations for their ker- function and the order of fractional derivative are varied nels - the real and imaginary parts of the modified in the minimization procedure. Bessel functions of the second kind Re K1/2+iτ (x) and Im K1/2+iτ (x) for all values of the variables x and τ ——— are obtained. The applications of Kontorovitch-Lebedev transforms to the solution of some mixed boundary value Integral transforms with extended generalized Mittag- problems in the wedge domains are accomplished. The Leffler function solution of the appropriate dual and singular integral Anna Koroleva equations is considered. The numerical aspects of using Department of Mathematics and Mechanics, Nezavici- of these transforms are elaborated in detail. mosti ave 4, Minsk BY-220030 Belarus [email protected] ——— A new class of polynomials related to the Kontorovich- Asymptotic results for integral transforms with extended Lebedev transform generalized Mittag-Leffler function in the kernel are discussed. Inversion formulas for such transforms in Semyon Yakubovich weighted spaces of integrable functions are found. Department of Pure Mathematics, Faculty of Science, ——— University of Porto, Campo Alegre str. 687, Porto 4169- 007 Portugal Systems of differential equations containing distributed [email protected] order fractional derivative Ljubica Oparnica We consider a class of polynomials related to the kernel Serbian Academy of Science and Art, Kneza Mihaila 36, Kiτ (x) of the Kontorovich-Lebedev transformation. Al- Belgrade, 11000 Serbia gebraic and differential properties are investigated and [email protected] integral representations are derived. We draw a parallel and establish a relationship with the Bernoullis and Eu- Distributed order fractional derivatives has appeared as lers numbers and polynomials. Finally, as an application generalization of the finite sum of fractional derivatives we invert a discrete transformation with the introduced and has wide applications in technical sciences. polynomials as the kernel, basing it on a decomposition We define distributed order fractional derivative of dis- of Taylors series in terms of the Kontorovich-Lebedev 0 tribution u ∈ S R) supported in R+ by fomula operator. Z h φ(γ)Dγ u dγ, ϕi = hφ, hDγ u, ϕii, ϕ ∈ S0, ——— supp u

50 III.4. Spaces of diﬀerentiable functions of several real variables and applications

k k k k n III.4. Spaces of differentiable functions of such that ~ϕ = {ϕ1 , ϕ2 , ··· , ϕn} and G ⊂ R be an several real variables and applications open parallelepiped with sides parallel to the coordinate planes. Then Organisers: a. holds true the inclusion Viktor Burenkov, Stefan Samko ~ϕ ~ϕ ~ϕk Bθ (···Bθ (Lp(G)) ··· ) ,→ Bp,θ(G) | {z } This session intends to cover various aspects of the the- k ory of Real Variables Function Spaces (Lebesgue, Or- lich, Sobolev, Nikol’skii-Besov, Lizorkin-Triebel, Mor- b. under the additional assumption that there exists rey, Campanato, and other spaces with zero or non- bounded extension operator zero smoothness), such as imbedding properties, den- ~ϕk ` ´ ~ϕk ` n´ S : Bθ G → Bθ R , sity of nice functions, weight problems, trace problems, extension theorems, duality theory etc. Vari- holds true the equality of spaces ous generalizations of these spaces are welcome, such k B ~ϕ(···B ~ϕ(L (G)) ··· ) = B ~ϕ (G) as for example Orlicz-Sobolev spaces, in particular θ θ p p,θ | {z } generalized Lebesgue-Sobolev spaces of variable or- k der, Morrey-Sobolev spaces, Muiselak-Orlich spaces and with equivalence of norms. their Sobolev counterparts etc. Other topics: any in- lj equalities related to these spaces, properties of operators In these results: (1.) If we set ϕj (h) = h we get results of real analysis acting in such spaces and also various ap- which are in works of V.I. Burenkov. (2.) In case, when plications to partial differential equations and integral ϕ(h) satisfy additional condition equations. ϕ(h) ∃ > 0 : ↑ on (0,H] h —Abstracts— leads to increment of smoothness using iterated norms. This is joint work with Abraham N. Abebe. Hardy spaces with generalized parameter ——— Alexandre Almeida Embeddings Properties of The Spaces Lp(.) ( n) Department of Mathematics, University of Aveiro w R Aveiro, Aveiro 3810-193, Portugal Ismail Aydın [email protected] Sinop University, Faculty of Arts and Sciences, Depart- ment of Mathematics, 57000. Sinop-TURKIYE Hardy spaces with generalized parameter are introduced [email protected] following the maximal characterization approach. As We derive some of the basic properties of weighted vari- particular cases, they include the classical Hardy spaces p(.) n Hp and the Hardy-Lorentz spaces Hp,q. Real interpola- able exponent Lebesgue spaces Lw (R ) and investi- p2(.) n p1(.) n tion results with function parameter are obtained. Based gate continuous embeddings Lw2 (R ) ,→ Lw1 (R ) on them, the behavior of some classical operators is stud- with respect to variable exponents and weight functions ied in this generalized setting. under some conditions. This talk is based on joint work with A. Caetano. ——— ——— On the boundedness of fractional B-maximal operators in the Lorentz spaces L ( n) Iterated norms in Nikol’ski˘ı-Besovtype spaces with gen- p,q,γ R eralized smoothness Canay Aykol Ankara University, Faculty of Science, Department of Tsegaye Gedif Ayele Mathematics, Ankara, Tandogan 06100 Turkey Department of Mathematics, Addis Ababa University [email protected] P.O.Box 1176 Addis Ababa - Ethiopia. [email protected] In this study, sharp rearrangement inequalities for the fractional B-maximal function Mα,γ f are obtained in In works of V.I. Burenkov iterated norms of Nikol’ski˘ı- the Lorentz spaces Lp,q,γ and by using these inequal- l l Besov type in spaces Bθ(···Bθ(Lp(Ω)) ··· ), k times it- ities the boundedness conditions of the operator Mα,γ erated, were introduced. Using these norms, it was are found. Then, the conditions for the boundedness of proved that every classical solution of the partial dif- the B-maximal operator Mγ are obtained in Lp,q,γ . ferential equation with constant coefficients is infinitely ——— differentiable. In this paper we consider iterated norms ~ϕ ~ϕ Spaces of functions of fractional smoothness on an ir- of Nikol’ski˘ı-Besov type in spaces Bθ (···Bθ (Lp(Ω)) ··· ) with generalized smoothness ~ϕ belonging to some class regular domain of functions Φ(~σ, θ) and with the norm Oleg Besov Steklov Institute of Mathematics, Department of Func- kfk ~ϕ ~ϕ Bθ (···Bθ (Lp(Ω)) ··· ) tion Theory, 8 Gubkina Str, Moscow 119991 Russia | {z } k [email protected] and holds the following In 1938 S.L. Sobolev proved his well-known embedding theorem Theorem. Let 1 < p < ∞, 1 ≤ θ < ∞, ~σ = m (σ1, σ2, ··· , σn), ~ϕ = (ϕ1, ϕ2, ··· , ϕn) ∈ Φ(~σ, θ) Wp (G) ⊂ Lq(G), m ∈ N, 1 < p < q < ∞, (*)

51 III.4. Spaces of diﬀerentiable functions of several real variables and applications

n n m − + ≥ 0, (**) Mf where M is the Hardy-Littlewood maximal opera- p q tor and presetn several applications of it. In particular, n for domains G ⊂ R satisfying the cone condition. we shall give some applications to the setting of the two Relationship (**) (which determines the maximum pos- weights problem for Calderón-Zygmund operators. sible value of q in theorem (*)) is also a necessary con- This is a joint work with J. Soria and R. Torres. dition for the embedding. Sobolev’s result has been ex- ——— tended to more general domains. n Definition. Given σ ≥ 1, a domain G ⊂ R is said On transformation of coordinates invariant relative to to satisfy the σ-cone condition if, for some T > 0 and Sobolev spaces with polyhedral anisotropy 0 < κ0 ≤ 1, for any x ∈ G there exists a piecewise Gurgen Dallakyan smooth path Russian-Armenian State University, Yerevan, Armenia 0 [email protected] γ = γx : [0,T ] → G, γ(0) = x, |γ | ≤ 1 a.e., Let Rn be n -dimentional euclidean space of the points such that n with real coordinates, N0 - the set of multi-indices. n σ dist (γ(t), R \ G) ≥ κ0t for 0 < t ≤ T. Definition. Nonempty polyhedron ℵ ⊂ Rn with vertices in N n is said to be complete, if it has vertices at the ori- The author established in (2001) that embedding (*) on 0 gin and at all coordinate axes of N n . Complete polyhe- a domain with the flexible σ-cone condition holds if 0 dron ℵ is called completely regular (CR), if all the coor- σ(n − 1) + 1 n dinates of outward normals of the noncoordinate (n−1)- m − + ≥ 0. (***) p q dimentional faces of ℵ are positive. Let ℵ ⊂ Rn- (CR) polyhedron with vertices We construct two families of function spaces Ls(m)(G) p,θ e0 , e1 , e2, e3, ..., eM , where the vertices ej (j = s(m) 0 and Bp,θ (G) of fractional smoothness s > 0 on domains 1, ..., n) lie on the j-th coordinate axe, e = (0, ..., 0) ˛ j ˛ G satisfying the flexible σ-cone condition such that em- , l = max ˛ e ˛ . beddings 1≤j≤n n ℵ For any domain Ω ⊂ R denote by Wp (Ω) (1 < p < ∞) m s(m) m s(m) Wp (G) ⊂ Lq,θ (G),Wp (G) ⊂ Bq,θ (G), the Sobolev space with polyhedral anisotropy, i.e. the space of functions with finite norm s(m) s(m) L (G) ⊂ L (G),B (G) ⊂ L (G), X α p,θ q p,θ q k u k = k D uk . ℵ,Ω Lp(Ω) hold with the same loss of smoothness as in (***). α∈ℵ ——— Consider the m -dimentional manifold Γm ⊂ Ω. Definition. The piece σ ⊂ Γ we call ℵ -regular, if for Rearrangement transformations on general measure m some n -dimentional subdomain ω , σ ⊂ ω ⊂ Ω, there spaces exist transformation of coordinates invariant relative to ℵ 0 m Santiago Boza Wp (Ω), mapping σ onto σ ⊂ R . EPSEVG, Avda Victor Balaguer s/n. Vilanova i Geltrú. Remark. Note, that any bounded domain always has 08800 (SPAIN) pieces of boundary, which are not ℵ -regular. [email protected] Let Γ ∈ Cs be m -dimentional manifold, where s ≥ r, N ˛ j ˛ S For a general set transformation R between two mea- (r = max ˛e ˛), i.e., Γ = σk and each piece σk has 1≤j≤M k=1 sure spaces, we define the rearrangement of a measur- representation able function by means of the Layer’s cake formula. We x = ψ (x0) , i = 1, ..., m, study some functional properties of the Lorentz spaces ik i,k 0 n−m s defined in terms of R, giving a unified approach to the where x ∈ Gk ,Gk ∈ R , ψi,k ∈ C (Gk). classical rearrangement, Steiner’s symmetrization, the Theorem. Let ℵ ⊂ Rn be (CR) polyhedron, Ω ⊂ Rn multidimensional case, and the discrete setting of trees. satisfies the week condition of rectangle, Γ ∈ Cs -m di- ——— mentional manifold ( Γ ⊂ Ω). Let σ ⊂ Γ has the repre- 0 0 sentation xi = ψi(x ) ,i = 1, ..., m ,x = (xm+1, ..., xn) ∈ Last developments on Rubio de Francia’s extrapolation s G , ψi ∈ C (G). Then the piece σ is ℵ -regular, if theory 1) s ≥ r ; 2) l ≥ r ; Maria Carro ˛ i˛ Department of Applied Mathematics and Analysis, Uni- 3) for all i = 1, ..., m the condition ˛e ˛ = l holds. versity of Barcelona, Gran Via 585, Barcelona 08007 ——— Spain [email protected] The boundedness of high order Riesz-Bessel transformations generated by the generalized shift operator in Since in the early 80’s, J.L. Rubio de Francia devel- weighted Lpw spaces with general weights oped his celebrated extrapolation theorem, this theory Ismail Ekincioglu has been developed in order to cover many other situa- Kutahya Dumlupinar University, Kutahya, Turkey tions such as boundedness of operator on rearrangement [email protected] invariant spaces or multilinear operator. In this talk, we shall present a new estimate on the dis- In this study, the boundedness of the the high order tribution function of T f in terms of the distribution of Riesz-Bessel transformations generated by generalized

52 III.4. Spaces of differentiable functions of several real variables and applications shift operator in weighted Lpwv-spaces with general For generalized potential operators with the kernel a[%(x,y)] weights is proved. [%(x,y)]N on bounded measure metric space (X, µ, %) with ——— doubling measure µ satisfying the upper growth condition µB(x, r) ≤ CrN ,N ∈ (0, ∞), we prove weighted Composition Operators for Sobolev spaces of functions estimates in the case of radial type power weight w = and differential forms ν [%(x, x0)] . Under some natural assumptions on a(r) in Vladimir Goldshtein terms of almost monotonicity we prove that such poten- Department of Mathematics, Ben Gurion University of tial operators are bounded from the weighted variable p(·) the Negev, P.O.Box 653, Beer Sheva, 84105 Israel exponent Lebesgue space L (X, w, µ) into a certain 1 Φ [email protected] weighted Musielak-Orlicz space L (X, w p(x0) , µ) with the N-function Φ(x, r) defined by the exponent p(x) and Composition operators for Sobolev spaces with first the function a(r). derivatives will be discussed. For such spaces composition operators are induced by mappings with bounded ——— mean distortion. These classes of mappings represent Our talk is on vanishing exponential integrability for a generalization of quasiconformal mappings. Applica- Besov functions. tions to embedding theorems will be described. In the framework of so-called Lq,p-cohomology similar classes Ritva Hurri-Syrjanen of mappings play an important role for quasiisometrical Department of Mathematics and Statistics, University of an/or Lipschitz classification of complete noncompact Helsinki PL 68 (Gustaf Hallstrominkatu 2 b), Helsinki Riemannian manifolds with bounded geometry FI-00014 Finland [email protected] ——— Boundedness of the fractional maximal operator and Our talk is on vanishing exponential integrability for fractional integral operators in general Morrey type Besov functions. spaces and some applications ——— Vagif Guliyev F. Agayev str, 7 Rasim Mukhtarov str, 10 Baku, Baku New sharp estimates for function in Sobolev spaces on AZ 1069 Azerbaijan finite Interval [email protected] Gennady Kalyabin Peoples Friendship University of Russia, Miklukho- The theory of boundedness of fractional maximal oper- Maklaya Str 6, Moscow, 117198 Russia ator and fractional integral operators from one weighted [email protected] Lebesgue space to another one is by now well studied. These results have good applications in the theory The smallest constants in new kind of Kolmogorov type of partial differential equations. However, in the the- inequalities for intermediate derivatives are obtained, ory of partial differential equations, along with weighted namely: the quantities A (x) defined as Lebesgue spaces, general Morrey-type spaces also play r,k an important role, but until recently there were no sup{f (k)(x): kf (r)k ≤ 1; results, containing necessary and sufficient conditions L2(−1,1) f on the weight functions ensuring boundedness of the (s) aforementioned operators from one general Morrey-type f (±1) = 0, s ∈ {0, . . . , r − 1}}, space to another one (apart from the cases in which this are calculated for all natural r, k ∈ {0, . . . , r − 1} and follows directly from the appropriate results for weighted x ∈ (−1, 1). Lebesgue spaces). The case of power-type weights was In particular well studied C.B. Morrey 1938, D.R. Adams 1975, F. Chiarenza and M. Frasca 1987, but for general Morrey- (1 − x2)2r−1 A2 (x) = . type spaces only sufficient conditions were known (T. r,0 Γ2(r)22r−1(2r − 1) Mizuhara 1991, E. Nakai 1994, V.S. Guliyev 1994). In the talk a survey of results, containing necessary and As a Corollary it is established that for the first eigen- sufficient conditions for boundedness of fractional max- value of the boundary problem imal operator and fractional integral operators, will be (−D2)ry(x) = λy(x), y(s)(±1) = 0, s ∈ {0, . . . , r − 1}, given, and open problems will be discussed in detail. As applications, we establish the boundedness of some the asymptotic formula Schödinger type operators on general Morrey-type √ spaces related to certain nonnegative potentials belong- λ1,r ≈ 2π 2r(2r ing to the reverse Hölderclass. e)2r, r → ∞, ——— holds. Weighted estimates of generalized potentials in variable ——— exponent Lebesque spaces On real interpolation of weighted Sobolev spaces Mubariz Hajibayov Institute of Mathematics and Mechanics of NAS of Azer- Leili Kusainova baijan, F.Agayev 9, Baku, Azerbaijan AZ1141 Azerbai- L.N. Gumilev Eurasian National University Astana, Mu- jan naitpasov 5, Akmola 010008 Kazakhstan [email protected] [email protected]

53 III.4. Spaces of diﬀerentiable functions of several real variables and applications

n Let 1 < p < ∞, m ∈ N,Ω ∈ R an open set, and let υ maximal operator Mβ and Dunkl type fractional integral be a non-negative function locally integrable in Ω. We operator Iβ on the Dunkl-type Morrey spaces Lp,λ,α(R), m denote by Wp (υ) the weighted Sobolev space with the 1 ≤ p < ∞. finite norm ——— m |u; Wp (υ)| = |∇mu; Lp(Ω)| + |u; Lp(Ω; υ)|. Image normalization of Wiener-Hopf operators in diffraction problems In this talk we describe Peetre interpolation spaces `W m0 (υ ),W m1 (υ ´ for weights υ , which allow in- Ana Moura Santos p 0 p 1 θ,p i troducing local variable average characteristics. Here Dept. de Matematica, IST, Av. Rovisco Pais 1, Lisbon, 1049-001 Portugal 0 ≤ m1 < m0, 1 ≤ p < ∞, mip 6= n (i = 0, 1). Let d(x) be a positive bounded function in Ω such that [email protected] for some a > 1 and all x ∈ Ω Q (x) ⊂ Ω, where ad(x) In this work we discuss the normalization problem for Q (x) = {y ∈ n : |y − x | < d/2, i = 1, 2, ..., n}. Let d R i i Wiener-Hopf Operators (WHO), which arrives in certain Bs(υ) denotes Besov space with the finite norm (s > 0) p ill-posed boundary-transmission value problems on half- s s planes. We first consider a wave diffraction problem by a |u; Bp(υ)| = |u; bp| + |u; Lp(Ω; υ)|. junction of two infinite half-planes, and different combi- For certain class of weights υi (i = 0, 1) we have obtained nations of normal and oblique derivatives on the planes. the equality of type Then a generalization for higher order derivatives follows. For all studied diffraction problems, which are as- `W m0 (υ ),W m1 (υ )´ = Bs((υ∗)−sp), p 0 p 1 θ,p p sociated with not normally solvable WHO, we solve the ∗ ∗ normalization problem based on the image normaliza- where 0 < θ < 1, s = (1 − θ)m0 + θm1, υ = maxυi (x) i=0,1 tion technique previously developed for one half-plane. and ——— ∗ mip−n 0 < υi (x) = sup{d : d υi(Qd(x)) ≤ 1} ≤ d(x). Weighted estimates for the averaging integral operator d>0 and reverse Hölderinequalities Bohum´ır Opic ——— Institute of Mathematics, AS CR The Fourier transform of a radial function Zitná25ˇ 11567 Praha 1 Elijah Liflyand Czech Republic Department of Mathematics Bar-Ilan University Ramat- [email protected] Gan, Gush-Dan 52900 Israel [email protected] Let 1 < p < +∞ and let v be a weight on (0, +∞) satisfying v(x)xρ is equivalent to a non-decreasing func- This talk naturally consists of two parts. In the first tion on (0, +∞) for some ρ ≥ 0. Let A be the aver- one we survey the known results on representation of 1 R x aging operator given by (Af)(x) := x 0 f(t) dt, x ∈ the Fourier transform of a radial function as the one- (0, +∞), and let Lp(v) denote the weighted Lebesgue dimensional Fourier transform of a related function. One space of all measurable functions f on (0, +∞) for which of such results, due to Leray, gave an impact to obtaining “ ”1/p R +∞ |f(x)|pv(x) dx < +∞. a series of new such formulas. We discuss those already 0 obtained in a joint work with S. Samko as well as tenta- First, we prove that the following statements are equiv- tive formulas. Correspondingly, already obtained appli- alent: cations are given and certain conjectures are posed. (i) A is bounded on Lp(v); ——— (ii) A is bounded on Lp−ε(v) for some ε ∈ (0, p − 1); Necessary and sufficient conditions for the bounded- (iii) A is bounded on Lp(v1+ε) for some ε > 0; ness of Riesz potential in Morrey spaces associated with p ε Dunkl operator (iv) A is bounded on L (v(x)x ) for some ε > 0. Moreover, if A is bounded on Lp(v), then A is bounded Yagub Mammadov on Lq(v) for all q ∈ [p, +∞). Institute of Mathematics and Mechanics, Rasim Second, we show that the boundedness of the averaging Mukhtarov str. 10, Narimanov area Baku, AZ 1141 operator A on the space Lp(v) implies that, for all r > 0, Azerbaijan 1−p0 [email protected] the weight v satisfies the reverse Hölderinequality over the interval (0, r) with respect to the measure dt, The maximal function, fractional maximal function while the weight v satisfies the reverse Hölderinequality over the interval (r, +∞) with respect to the measure and fractional integrals associated with the Dunkl op- −p erator were studied extensively in Lebesgue spaces t dt. Third, assuming moreover that p ≤ q < +∞ and that w on R. We study the fractional maximal function (Dunkl-type fractional maximal function) and fractional is a weight on (0, +∞) such that integrals (Dunkl-type fractional integrals) associated [w(x)x]1/q ≈ [v(x)x]1/p for all x ∈ (0, +∞), with the Dunkl operator in the Dunkl-type Morrey we prove that the operator A is bounded on Lp(v) if and space Lp,λ,α(R) and Dunkl-type Besov-Morrey spaces p q s only if the operator A : L (v) → L (w) is bounded. Bpθ,λ,α(R). We obtain the necessary and sufficient conditions for the boundedness of Dunkl-type fractional ———

54 III.4. Spaces of diﬀerentiable functions of several real variables and applications

Characterization of the variable exponent Bessel poten- We admit variable complex valued orders α(x), where tial spaces via the Poisson semigroup <α(x) may vanish at a set of measure zero. To cover this case, we consider the action of potential operators Humberto Rafeiro to weighted generalized Hölderspaces with the weight Universidade do Algarve, Dep. Matemática,Campus de α(x). Gambelas Faro, Faro 8005-139 Portugal [email protected] ——— Equivalent semi-norms for Nikol’skii- Besov spaces on In this talk we give a characterization of the variable an interval exponent Bessel potential space in terms of the convergence of the Poisson semigroup. We show that Kader Senouci Grunwald-Letnikov construction with respect to the University City Zaaroura, 50 logts, Tiaret 14000 Algeria Poisson semigroup coincides with the Riesz fractional differentiation under some natural restrictions on the ex- [email protected] ponent p(x). Let 1 < p, θ 6 ∞, l > 0, k ∈ N, k > l, −∞ ≤ a 6 b 6 ——— l ∞. Recall thatf ∈ bp,θ(a, b), if f is measurable on (a, b) On the Maxwell problem and

1 Evgeniy Radkevich 0 b−a 1 θ k Mathematics Department, Moscow State University, Z “ ”θ dh B −l k C kfkbl (a,b) = h k∆hfkL (a,b−kh) Vorobievy Gori, Moscow 119992 Russia p,θ @ p h A [email protected] 0 is finite. TBA Theorem. 1 < p, θ ∞, l > 0,k ∈ , 0 < l < k,α ≥ ——— 6 N 1 0, α2 ≥ k. Then for an arbitrary interval (a, b) Weighted potential operators in Morrey spaces. 1 0 b−a 1 θ α1+α2 Natasha Samko Z θ B “ −l k ” dh C Department of Mathematics, University of Algarve, kfkbl (a,b) ∼ B h k∆hfkL (a+α h,b−α h) C , p,θ @ p 1 2 h A Campus de Gambelas, 8005-139 Faro, Portugal 0 [email protected] where the equivalence constants are independent of a and We study the weighted (p, λ)-(q, λ)-boundedness of b. Hardy and potential operators. We show that the weighted boundedness of potential operators is reduced ——— to the boundedness of weighted Hardy operators. In case of power weights or oscillating weights from the Bary- Stein-Weiss inequalities for the fractional integral op- Stechkin class we find conditions for weighted Hardy operators in Carnot groups and applications erators to be bounded in Morrey spaces. Ayhan Serbetci ——— Department of Mathematics, Ankara University, Tando- gan, Ankara 06100 Turkey Fractional integrals and hypersingular integrals in vari- [email protected] able order Holder spaces on homogeneous spaces Stefan Samko In this study we consider the fractional integral opera- Universidade do Algarve Campus de Gambelas Faro, Al- tor Iα on any Carnot group G (i.e., nilpotent stratified garve 8005-139 Portugal Lie group) in the weighted Lebesgue spaces Lp,ρ(x)β (G). [email protected] We establish Stein-Weiss inequalities for Iα, and obtain necessary and sufficient conditions on the parameters We consider non-standard Hölderspaces Hλ(·)(X) of for the boundedness of the fractional integral operator functions f on a metric measure space (X, d, µ), whose Iα from the spaces Lp,ρ(x)β (G) to Lq,ρ(x)−γ (G), and from Hölderexponent λ(x) is variable, depending on x ∈ X. the spaces L1,ρ(x)β (G) to the weak spaces WLq,ρ(x)−γ (G) We establish theorems on mapping properties of poten- by using the Stein-Weiss inequalities. In the limiting case p = Q , we prove that the tial operators of variable order α(x), from such a vari- α−β−γ able exponent Hölderspace with the exponent λ(x) to modified fractional integral operator Ieα is bounded another one with a ‘better’ exponent λ(x) + α(x), and from the space Lp,ρ(x)β (G) to the weighted BMO space similar mapping properties of hypersingular integrals of BMOρ(x)−γ (G), where Q is the homogeneous dimension variable order α(x) from such a space into the space with of G. the ‘worse’ exponent λ(x)−α(x) in the case α(x) < λ(x). As applications of the properties of the fundamental so- These theorems are derived from the Zygmund type es- lution of sub-Laplacian L on G, we prove two Sobolev- timates of the local continuity modulus of potential and Stein embedding theorems on weighted Lebesgue and hypersingular operators via such modulus of their densi- weighted Besov spaces in the Carnot group setting. As ties. These estimates allow us to treat not only the case an another application, we prove the boundedness of Iα λ(·) s s of the spaces H (X), but also the generalized Hölder from the weighted Besov spaces Bpθ,β (G) to Bqθ,−γ (G). spaces Hw(·,·)(X) of functions whose continuity modulus ——— is dominated by a given function w(x, h), x ∈ X, h > 0.

55 III.5. Analytic and harmonic function spaces

Translation-invariant bilinear operators with positive III.5. Analytic and harmonic function spaces kernels Organisers: Javier Soria Rauno Aulaskari, Turgay Kaptanoglu, Department of Applied Mathematics and Analysis, Uni- Jouni Ratty¨ a¨ versity of Barcelona, Gran Via 585, Barcelona 08007 Spain Anticipated topics are normal families, complex valued [email protected] function spaces and classes, function spaces and local We study the boundedness of bilinear convolutions oper- theory of complex differential equations, composition ators with positive kernels. We prove both necessary and operators between function spaces, boundary behaviour sufficient conditions and, by means of several counterex- etc.; Hardy, Bergman, Bloch, Besov, Lipschitz, Fock, Qp amples we show that near the endpoints the behavior of spaces of one and several holomorphic or harmonic variables, Toeplitz, Hankel, composition, Volterra, multipli- positive translation-invariant bilinear operators can be ∗ quite different than that of positive linear ones. cation operators, C or other algebras of such operators, Toeplitz algebras, reproducing kernel Hilbert spaces of ——— holomorphic or harmonic functions, and other similar Sharp inequalities for moduli of smoothness and K- topics. functionals —Abstracts— Sergey Tikhonov Centre de Recerca Matemàtica,Facultat de Ciències, Multiplier theorem in the setting of Laguerre hyper- UAB, Bellaterra, Barcelona 08193 Spain groups and applications [email protected] Miloud Assal We discuss the (p − p) and (p − q) sharp inequalities Department of Mathematics, Faculty of Sciences of Biz- (Jackson-type, Marchaud-type, Ulyanov-type, etc) for erte Zarzouna, Bizerte 7021, Tunisia moduli of smoothness/K-functionals. Corresponding [email protected] embedding theorems are studied. ——— In this work we study a multiplier theorem in the setting of Laguerre hypergroups and their applications to Sobolev embedding theorems for a class of anisotropic estimate the solution of Schrdinger equation in Hardy irregular domains spaces. Boris V. Trushin ——— MIAN (Departament of Function Theory), ul. Gubkina, d. 8, Moscow 119991 Russia Progress on finite rank Toeplitz products [email protected] Boo Rim Choe Sufficient conditions for the embedding of a Sobolev Department of Mathematics, Korea University, Anam- space in Lebesgue spaces and the space of continuous dong 5 ga 1, Seongbuk-gu, Seoul 136-713 South Korea functions on a domain depend on the integrability and [email protected] smoothness parameters of the spaces and on the geometric features of the domain. In our talk, Sobolev em- It has been conjectured that a product of Toeplitz op- bedding theorems will obtaine for a class of domains erators with function symbols, either on Hardy space or with irregular boundary. This new class includes the Bergman space, has finite rank, then one of the factor well-known classes of σ-John domains, domains with the must be the zero operator. flexible cone condition, and their anisotropic analogs. In this talk we survey recent results towards the conjec- The results can be extended to weighted spaces with ture as well as related results. power weights. ——— ——— Functions and operators in analytic Besov spaces Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces Daniel Girela Departamento de Análisis Matemático, Facultad de Yusuf Zeren Ciencias, Campus de Teatinos, Universidad de Málaga, Department of Mathematics of Harran University, Cam- Málaga29071 Spain pus of Osmanbey, SanliUrfa, Region 6300 Turkey [email protected] [email protected] In this talk we shall focus on structural and geometric We obtain necessary and sufficient conditions on the properties of the functions in analytic Besov, primarily parameters for the boundedness of the fractional max- on the univalent functions in such spaces, and in opera- imal operator Mα, and the Riesz potential operator n tors acting on them. Iα from the modified Morrey spaces Lep,λ(R ) to the n spaces Leq,λ(R ), 1 < p < q < ∞, and from the ——— n spaces Le1,λ(R ) to the weak modified Morrey spaces n Square functions W Leq,λ(R ), 1 < q < ∞. Maria Jose Gonzales ——— Department of Mathematics, Casem Rio San Pedro

56 III.5. Analytic and harmonic function spaces

Puerto Real, Cadiz 11560 Spain ——— [email protected] Weighted composition operators on weighted spaces of analytic functions We will study multiplicative versions of the usual martingale square function and of the Lusin area of a har- Jasbir Singh Manhas monic function. Sultan Qaboos University, Department of Mathematics & Statistics, College of Science, P.O. Box 36, Al-Khod ——— Muscat, Muscat 123 Oman Convolutions of generic orbital measures in compact [email protected] symmetric spaces Let V be an arbitrary system of weights on an open Sanjiv Gupta N connected subset G of C (N ≥ 1). Let HV0(G) and DOMAS, PO BOX-36 Al-Khodh-123 Sultan Qaboos HVb(G) be the weighted locally convex spaces of analytic University Muscat, Oman functions with topology defined by seminorms which are gupta s [email protected] weighted analogues of the supremum norm. Let Hv0(G) and Hvb(G) be the weighted Banach spaces of analytic We prove that in any compact symmetric space, G/K, functions defined by a single weight v. In this talk be- there is a dense set of a1, a2 ∈ G such that if µj = sides presenting the characterizations of weighted com- mK ∗ δa ∗ mk is the K-bi-invariant measure supported j position operators on HV0(G)( Hv0(G) )and HVb(G) on Kaj K, then µ1 ∗ µ2 is absolutely continuous with re- ( Hvb(G) ), we shall present some results pertaining to spect to Haar measure on G. Moreover, the product of topological structures ( e.g. component structure, Iso- double cosets, Ka1Ka2K, has non-empty interior in G. lated points, compact differences ) of weighted compo- ∞ ——— sition operators on the spaces H (D) and Hv0(D)( Hvb(D) ). Harmonic Besov spaces on the real unit ball: reproducing kernels and Bergman projections ——— H. Turgay Kaptanoglu˘ Superposition operators between Qp spaces and Hardy Department of Mathematics, Bilkent University, spaces Ankara 06800, Turkey Auxiliadora Marquez [email protected] Departamento de Analisis Matematico, Facultad de Ciencias, Campus de Teatinos, Malaga 29071 Spain p Weighted Bergman spaces bq are well-known spaces of [email protected] harmonic functions for which q > −1 and 1 ≤ p < ∞. p Besov spaces, also denoted bq , generalize them to all For any pair of numbers (s, p) with 0 ≤ s < ∞ and q ∈ R. Our Besov spaces consist of harmonic functions 0 < p ≤ ∞ we characterize the superposition operators n on the unit ball B of R so that their sufficiently high- which apply the conformally invariant Qs space into the order (t) derivatives are in Bergman spaces (q+pt > −1). Hardy space Hp and, also, those which apply Hp into We compute the reproducing kernels Rq(x, y) of the Qs. Besov spaces b2 with q ≤ −1. The kernels turn out to be q ——— weighted infinite sums of zonal harmonics, and also radial fractional derivatives of the Poisson kernel. The new Bounded Toeplitz and Hankel products on Bergman kernels give rise to generalized Bergman projections by space R 2 s way of Qsϕ(x) = ϕ(y) Rs(x, y) (1−|x| ) dν(y), where B p p Malgorzata Michalska s ∈ R. We prove that Qs : Lq → bq are bounded if and Instytut Matematyki UMCS, Pl. M. Curie Sklodowskiej only if q + 1 < p(s + 1). This requires new estimates 1, Lublin, woj. lubelskie, 20-031 Poland on the integral growth of Bergman kernels near ∂B. We [email protected] obtain various applications of the Qs. This is joint work with Se¸cil Gergün and A. Ersin We improve the sufficient condition for boundedness of Ureyen.¨ The work is supported by TUB¨ ITAK˙ under products of Toeplitz operators Tf Tg¯ on the Bergman Research Project Grant 108T329. space obtained by K. Stroethoff and D. Zheng in 1999. ——— Using our result we give a short proof of the sufficient and necessary condition for the boundedness of Tf T1/f¯ Sums of Toeplitz products on the Dirichlet space obtained also by Stroethoff and Zheng in 2002. We con- ∗ Young Joo Lee sider also the products of Hankel operators Hf Hg . Department of Mathematics, Chonnam National Uni- ——— versity, Gwangju, Yongbongdong 500-757, South Korea [email protected] Optimal norm estimate of the harmonic Bergman projection In this talk, we will consider a class of operators which Kyesook Nam contains finite sums of products of two Toeplitz opera- Department of Mathematics, Hanshin University, Osan- tors with harmonic symbols on the Dirichlet space of the si, Gyeonggi-do 447-791 South Korea unit disk. [email protected] We will give characterizations of when an operators in that class is zero or compact. Also, we solve the zero On the unit ball of the Euclidean n-spaces, we give an product problem for products of finitely many Toeplitz optimal norm estimate for one-parameter family of op- operators with harmonic symbols. erators associated with the weighted harmonic Bergman

57 III.5. Analytic and harmonic function spaces projections. Using this result, we obtain an optimal Houghton Street London, WC2A 2AE United Kingdom norm estimate for the weighted harmonic Bergman projections. [email protected] This is the joint work with Boo Rim Choe and Hyung- woon Koo. We will address the question of whether a left invertible matrix with entries in certain convolution Banach alge- ——— bras of measures supported in [0, +∞) can be completed Old and new on composition operators on VMOA and to an invertible matrix with entries from the same Ba- BMOA spaces nach algebra. The Banach algebras we consider arise naturally in control theory as classes of inverse Laplace Pekka Nieminen transforms of stable transfer functions, and the relevance Dept. of Mathematics and Statistics, Univ of Helsinki, of the problem of completion to an isomorphism in con- PO Box 68, Helsinki, 00014 Finland trol theory will also be explained. [email protected] ——— We review various compactness characterizations for Multiplication operators on weighted BMOA spaces analytic composition operators acting on the spaces Benoit F. Sehba VMOA and BMOA, and give some new formulations. Department of Mathematics, University of Glasgow, We also discuss the equivalence of weak compactness G12 8QW, Glasgow, UK and (norm) compactness for these operators. Joint work [email protected] with Jussi Laitila, Eero Saksman and Hans-Olav Tylli (Helsinki). We give some (test function) criteria for symbols of bounded multiplication operators for a special familly ——— of weighted BMOA spaces in the unit ball. On Libera and Cesaro operators ——— Maria Nowak Inequalities for Hardy spaces on the unit ball Instytut Matematyki UMCS, Pl. M. Curie Sklodowskiej Pawel Sobolewski 1, Lublin, woj. lubelskie, 20-031 Poland Instytut Matematyki UMCS, Pl. M. Curie Sklodowskiej [email protected] 1, Lublin, woj. lubelskie, 20-031 Poland [email protected] Let H(D) denote the class of functions holomorphic in the unit disk D. The Cesàrooperator C is defined on “ ” In 1988 (TAMS 103(3)) D. Luecking obtained the follow- P∞ 1 Pn ˆ n p H(D) by Cf(z) = n=0 n+1 k=0 f(k) z , where ing results for Hardy spaces H in the unit disk D ⊂ C. P∞ ˆ n The inequality f(z) = n=0 f(n)z . The Libera operator L, defined “ ˆ ” Z by Lf(z) = P∞ P∞ f(k) zn, can be considered p−s 0 s s−1 p n=0 k=n k+1 |h(z)| |h (z)| (1 − |z|) dA(z) ≤ CkhkHp as an extension of the conjugate operator C∗ defined on D holds for h ∈ Hp, p > 0 if and only if 2 ≤ s < p + 2. H(D) - the space of holomorphic functions defined in a neighborhood of . We obtain results on Libera opera- We obtain analogous results for the Hardy spaces on the D n tor acting on known spaces of holomorphic functions in unit ball of C , n ≥ 2. the unit disk. (Joint work with Miroslav Pavlovic) ——— ——— On the Duhamel algebras Integration operators on weighted Bergman spaces Mubariz¨ Tapdıgoglu˘ Isparta Vocational School, Suleyman Demirel Univer- Jordi Pau sity, Dogu Campus Isparta, Cunur 32260 Turkey Departament de MatemàticaAplicada i Anàlisi,Univer- [email protected] sitat de Barcelona, Gran Via de les Corts Catalanes 585, Barcelona, 08007 Spain We introduce the notion of Duhamel algebra. We prove [email protected] that under some natural conditions any Banach space of analytic functions in the unit disc D is the Duhamel For an analytic function g on the unit disc, we consider algebra and describe its all closed ideals. In particular, the operators we improve some results of Wigley. Z z ——— 0 Jgf(z) = f(ζ)g (ζ)dζ. 0 Toeplitz operators on Bergman spaces

We describe the boundedness and compactness of Jg on Jari Taskinen Bergman spaces with exponential weights, answering an P.O.Box 68, Department of Mathematics and Statistics open question posed by Aleman and Siskakis in 1997. University of Helsinki Helsinki, Helsinki FI-00014 Fin- land ——— [email protected] Extension to an invertible matrix in Banach algebras of We give suﬃcient conditions for boundedness and com- measures pactness of Toeplitz operators in the Bergman spaces on Amol Sasane the unit disc of the complex plane. We consider both the Mathematics Department, London School of Economics, cases 1 < p < ∞ and p = 1. The conditions concern a

58 III.6. Spectral theory kind of averages of the symbol on hypebolic rectangles. joensuu 80100 Finland The sufficient condition is also necessary in the case of [email protected] positive symbols, and it thus coincides with known results in this case. An approach to the Fredholm proper- In this paper, we give a necessary and sufficient condi- ties of Toeplitz operators is also given. tion for a kind of lacunary series on the unit ball to be ——— in Qp spaces for (m − 1)/m < p ≤ 1. The necessity is extended to more general QK spaces. This is a gener- Hyperbolic weighted Bergman classes alization of the result of Aulaskari, Xiao and Zhao for that on the unit disk. Luis Manuel Tovar Department of Mathematics, Esc. Sup. de Fsica y Mat. ——— I.P.N., Edificio 9, Unidad Prof. A.L.M. Zacatenco del I.P.N., Mexico City, 07738 Mexico Some results on ϕ-Bloch functions [email protected] Congli Yang Yliopistokatu 7 Metria Building (Y6), Joensuu 80101 A new class of like-hyperbolic Bergman class of analytic Finland functions in the unit complex disk is introduced, which [email protected] has several interesting properties and relationships with several classical weighted spaces, like Bloch, Dirichlet Let ϕ : [0, 1)→(0, ∞) be an increasing function, such and Qp. that ϕ(r)(1−r) → ∞, as r → 1−. An analytic function ——— f(z) in the unit disc is said to be ϕ-Bloch function if it’s derivative satisfies |f 0(z)| = O(ϕ(|z|)) as |z| → 1−. Multiplicative isometries and isometric zero-divisors This paper is devoted to the study of analytic ϕ-Bloch functions. we obtain some new characterizations for ϕ- Dragan Vukotic Bloch functions are established under certain regularity Departamento de Matematicas & ICMAT, Modulo C- conditions on ϕ. XV, Universidad Autonoma de Madrid, Madrid, 28049 Spain ——— [email protected] Holomorphic mean Lipschitz spaces For some Banach spaces of analytic functions in the unit Kehe Zhu disk (weighted Bergman spaces, Bloch space, Dirichlet- Department of Mathematics and Statistics, 1400 Wash- type spaces), we show that their isometric pointwise ington Ave, SUNY Albany, New York 12222 United multipliers are necessarily unimodular constants. As a States consequence, it follows that none of those spaces have [email protected] isometric zero-divisors. We also investigate the isometric coefficient multipliers. I will talk about the connections between holomorphic ——— mean Lipschitz spaces and several other classes of function spaces, including Bergman spaces, Besov spaces, Area operators on analytic function spacess and Bloch type spaces. The setting is the open unit ball in Cn. Zhijian Wu Department of Mathematics, The University of Al- ——— abama, Tuscaloosa, Alabama 35487 United States Univalently induced closed range composition operators [email protected] on the Bloch-type spaces We characterize non-negative measures µ on the unit Nina Zorboska disk D for which the area operator Aµ is bounded or Department of Mathematics, University of Manitoba, compact on Hardy and Bergman spaces. Winnipeg, Manitoba R3T 2N2 Canada [email protected] ——— Composition operators on BMOA We will show that if the closed range composition operator is univalently induced, then the inducing function Hasi Wulan has to be a disk automorphism, whenever the underly- Department of Mathematics, Shantou University Shan- ing space is a Bloch-type space Bα with alpha not equal tou, Guangdong 515063 China to one. [email protected] The proof uses a combination of methods and results from operator theory, complex analysis and the pseudo- We give a new and simple compactness criterion for com- hyperbolic geometry on the unit disk. position operators Cϕ on BMOA and the Bloch space in terms of the norms of ϕn in the respective spaces. ——— ———

Lacunary series and QK spaces on the unit ball III.6. Spectral theory Wen Xu Organisers: Yliopistokatu 7 Metria Building (Y6) 3rd ﬂoor Joensuu, Brian Davies, Ari Laptev, Yuri Safarov

59 III.6. Spectral theory

Anticipated topics are: Spectral theory of differential and the coefficients Aαβ are real-valued Lipschitz con- operators. Spectra of non-self-adjoint operators. Spec- tinuous functions satisfying Aαβ = Aβα and the uniform tral asymptotics. Scattering theory. General spectral ellipticity condition theory and related topics. X 2 Aαβ (x)ξαξβ ≥ θ|ξ| |α|=|β|=m —Abstracts— for all x ∈ Ω and for all ξα ∈ R, |α| = m, where θ > 0 Strongly elliptic second-order systems in Lipschitz do- is the ellipticity constant. We consider open sets Ω for mains: surface potentials, equations at the boundary, which the spectrum is discrete and can be represented and corresponding transmission problems. by means of a non-decreasing sequence of non-negative Mikhael Agranovich eigenvalues of finite multiplicity λ1[Ω] ≤ λ2[Ω] ≤ · · · ≤ λn[Ω] ≤ ... Here each eigenvalue is repeated as many times as its multiplicity and lim λn[Ω] = ∞ . [email protected] n→∞ The aim is sharp estimates for the variation |λn[Ω1] − We consider a strongly elliptic second-order system in a λn[Ω2]| of the eigenvalues corresponding to two open sets bounded Lipschitz domain Ω. For convenience, we as- Ω1,Ω2 with continuous boundaries, described by means n sume that Ω = Ω+ lies in the standard torus T = T of the same fixed atlas A. and consider the system in the domain Ω− = T \ Ω too. Three types of estimates will be under discussion: for Assuming that the Dirichlet and Neumann problems in each n ∈ N for some cn > 0 depending only on n, A, m, θ ± the variational setting in Ω are uniquely solvable in and the Lipschitz constant L of the coefficients Aαβ σ σ some spaces Hp or Bp , we describe properties of the surface potentials. We define these operators and derive |λn[Ω1] − λn[Ω2]| ≤ cndA(Ω1, Ω2), corresponding formulas following Costabel and McLean, where dA(Ω1, Ω2) is the so-called atlas distance of Ω1 to without using properties of the fundamental solution Ω2, (but do not assume that p = 2 and that the coefficients are smooth). Main results: boundedness of the surface |λn[Ω1] − λn[Ω2]| ≤ cnω(dHP (∂Ω1, ∂Ω2)), potentials, their invertibility at the boundary (in partic- where d (∂Ω , ∂Ω ) is the so-called lower Hausdoff- ular, of the single layer and hypersingular operators) in HP 1 2 Pompeiu deviation of the boundaries ∂Ω and ∂Ω and Besov spaces, and a description of their spectral proper- 1 2 ω is the common modulus of continuity of ∂Ω and ∂Ω , ties in these spaces (including the case p 6= 2). We also 1 2 and, under certain regularity assumptions on ∂Ω and describe applications to the corresponding transmission 1 ∂Ω , problems, general and spectral. 2 ——— |λn[Ω1] − λn[Ω2]| ≤ cnmeas (Ω1∆Ω2) , On the spectral expansions associated with Laplace- where Ω1∆Ω2 is the symmetric difference of Ω1 and Ω2. Beltrami operator Joint work with Dr P. D. Lamberti. Shavkat Alimov ——— Vuzgorodok National University of Uzbekistan, Strong field asymptotics for zero modes Tashkent, 100174 Uzbekistan shavkat [email protected] Daniel Elton Department of Mathematics and Statistics, Fylde Col- lege, Lancaster University, Lancaster LA1 4YF United The eigenfunction expansions associated with Laplace- Kingdom Beltrami operator on n-dimensional symmetrical man- [email protected] ifold Ω of rank 1 is considered. If the eigenfunction expansion of the piecewise smooth function, which de- Given a magnetic potential A one can consider the exis- pends on the geodesic distance from some point, con- tence of zero modes (or zero-energy L2 eigenfunctions) verges at this point, then considered function belongs of the Weyl-Dirac operator σ.(−i∇−tA) on 3; here t is to C(n−3)/2(Ω). This result is the generalization of the R a positive parameter, with the limit t → ∞ correspond- result, which was proved by M. Pinsky and W. O. Bray ing to the strong field (or, equivalently, semi-classical) for geodesic ball. regime. General O(t3) bounds on the number of zero ——— modes can be obtained. These bounds can be refined to O(t2) asymptotics for a special class of potentials A Sharp spectral stability estimates for higher order ellip- that are constructed from potentials on 2; a key step tic operators R involves localising the Aharonov-Casher theorem to ob- Victor Burenkov tain good estimates for the number of “approximate zero Via Trieste 63, Padova University, Padova, 35121, Italy modes” for two-dimensional Pauli operators. [email protected], [email protected] ———

We consider the eigenvalue problem for the operator A universal bound for the trace of the heat kernel

m X α “ β ” Leander Geisinger Hu = (−1) D A (x)D u , x ∈ Ω, αβ UniversitätStuttgart, FakultätMathematik und Physik, |α|=|β|=m IADM, Pfaffenwaldring 57, Stuttgart 70569, Germany subject to homogeneous Dirichlet or Neumann boundary [email protected] N conditions, where m ∈ N, Ω is a bounded open set in R

60 III.6. Spectral theory

We derive a unviersal upper bound for the trace of the I plan to report on recent joint work with Juan Gil and P −λkt heat kernel Z(t) = k e , where (λk)k∈N denote the Gerardo Mendoza on the expansion of the resolvent trace eigenvalues of the Dirichlet Laplace Operator in an open and the heat kernel for (nonselfadjoint) elliptic operators 2 set Ω ⊂ R with finite volume. The result improves an on manifolds with conical singularities. Our approach al- inequality of Kac and holds true without further assump- lows for the treatment of elliptic operators A of general tions on Ω. The proof is based on improved Berezin-Li- form without simplifying assumptions on the coefficients Yau inequalities with a remainder term. or the geometry near the singularities, and we achieve results for a wide range of closed extensions of A in the ——— appropriate metric L2-space. In particular, we obtain The eigenvalues function of the family of Sturm- results for selfadjoint and nonselfadjoint extensions of Liouville operators and its applications Hodge-Laplacians in the presence of warped conical singularities where conventional methods that are based on Tigran Harutyunyan separation of variables and special functions fail. Faculty of Math. and Mechanics of Yerevan State Uni- versity, Alek Manukyan 1, Yerevan 0049 Armenia ——— [email protected] Stability estimates for eigenfunctions of elliptic operators on variable domains In order to study the dependence of the eigenvalues of the Sturm-Liouville problem on parameters, defining the Pier Domenico Lamberti boundary conditions, we introduce the concept of the Dipartimento di Matematica Pura ed Applicata, Via Tri- eigenvalues function of the family of Sturm-Liouville op- este, 63 Padova, Padova 35121, Italy erators. [email protected] We find the necessary and sufficient conditions for some function (of two variables) to be the eigenvalues func- We prove stability estimates for the variation of resol- tion. vents and eigenfunctions of second order uniformly ellip- Actually, we solve the direct and inverse Sturm-Liouville tic operators subject to homogeneous boundary condi- problems. This solution particularly includes: tions upon variation of the domain. We consider classes of open sets Ω˜ parametrized by suitable bi-Lipschitz a) the new (more precise) asymptotic formulae for the homeomorphisms φ˜ defined on a fixed reference do- eigenvalues and normalized constants, main Ω. We obtain estimates expressed in terms of ˜ k∇φ − IkLp(Ω) for finite values of p. We apply these b) some new uniqueness theorems in the inverse prob- estimates in order to control the variation of the eigen- lems, functions via the measure of the symmetric difference Ω Ω.˜ We also discuss an application to the stability c) the constructive solution of the inverse problems M of the solutions to the Poisson problem. in known and some new statements. This is joint work with G. Barbatis and V.I. Burenkov. Also we introduce the concept of the eigenvalues func- ——— tion of the family of Dirac operators and solve similar problems for that case. Spectral theory of the normal operator with the spectra on an algebraic curve ——— Oleksii Mokhonko Generalized eigenvectors of some Jacobi matrices in the Kyiv National Taras Shevchenko University, 64 critical case Volodymyrska street, 01033 Kyiv, Ukraine Jan Janas [email protected] Sniadeckich 8 Warsaw, Warsaw 00-956 Poland [email protected] The Jacobi (three-diagonal) structure of self-adjoint multiplication operator is well-known. Berezansky Yu.M. and Dudkin M.E. proved that similar Jacobi The talk will be concerned with asymptotic behavior of structure is typical not only for self-adjoint operators but generalized eigenvectors of a class of Hermitian Jacobi also for arbitrary unitary and even for any bounded nor- matrices J in the critical case. The last means that the mal operators for which a cyclic vector exists. This leads fraction q /λ generated by the diagonal entries q of n n n to numerous applications of these objects just in the J and its subdiagonal elements λ has the limit ±2. In n same way as it is for the classical Jacobi matrices, e.g. other word, the limit transfer matrix as n → ∞ con- application to non-abelian difference-differential lattices tains a Jordan box (=double root in terms of Birkhoff- generated by Lax equation (Golinskii L.B., Mokhonko Adams theory). This is the situation where the asymp- O.A.). The following results will be presented. totic Levinson theorem does not work and one has to elaborate more special methods for asymptotic analysis. 1. Block Jacobi matrix of a bounded normal operator J acts in 1 ⊕ 2 ⊕ 3 ⊕ 4 ⊕· · · . If one knows that ——— C C C C the spectrum of J is a subset of a curve {z ∈ C : Trace expansions for elliptic cone operators p(z, z¯) = 0}, p ∈ C[x, y] then its structure can be 1 2 n n simplified: it acts over C ⊕C ⊕· · ·⊕C ⊕C ⊕· · · Thomas Krainer (dimension stabilization phenomenon). E.g. if a Penn State Altoona 3000 Ivyside Park Altoona, Penn- normal operator is in fact the unitary one then sylvania 16601 United States 1 2 2 it acts over C ⊕ C ⊕ C ⊕ · · · (CMV matrix [email protected] structure) and if it is self-adjoint then it acts over 1 1 `2 ' C ⊕ C ⊕ · · · .

61 IV.1. Pseudo-diﬀerential operators

2. The Direct Spectral Problem (the generalized by geometric quantities. For vector valued wave equa- eigenvalue expansion theorem) and the Inverse tions our estimates depend on quite involved geometric Spectral Problem will be presented for this type quantities like lengths of homological systoles. of normal operators. ———

——— Spectrum and wandering Yuriy Tomilov Spectral properties of operators arising from modelling Chopina Str. 12/18 Department of Mathematics and of flows around rotating bodies Informatics, Nicholas Copernicus University, Torun and Jiri Neustupa Institute of Mathematics, PAN, Warsaw Torun, Torun Mathematical Institute of the Czech Academy of Sci- 87-100, Poland ences Zitna 25 Prague 1, Czech Republic 115 67 Czech [email protected] Republic [email protected] Let T be a bounded linear operator on a Hilbert space H. A vector x ∈ H is called weakly wandering for T if We give a description of the spectrum of a Stokes-type or there is an increasing sequence (nk) such that the vectors nk an Oseen-type operator which appears in mathematical T x are mutually orthogonal. By a well-known result models of flows of a viscous incompressible fluid around due to Krengel, every unitary operator on H without rotating bodies. The special attention is paid to the es- point spectrum has a dense subset of weakly wandering sential spectrum. The operator is considered in an Lq vectors. space. We will present several far-reaching extensions of the Krengel result. In particular, we will show that if T ——— is a power bounded operator on H with infinite peripheral spectrum and with empty peripheral point spectrum New formulae for the wave operators then the set of weakly wandering vectors for T is dense Serge Richard in H. Our spectral assumptions on T are in a sense best Department of Pure Mathematics and Mathematical possible. Statistics, Wilberforce Road, Cambridge CB3 0WB This is joint work with V. Müller(Prague). United Kingdom ——— [email protected] Eigenfunctions at the threshold energies of magnetic We review some new formulae recently obtained for the Dirac operators wave operators of various scattering systems. Different Tomio Umeda applications of these formulae will be presented. Department of Mathematical Science University of Hy- oto, Shosha 2167 Himeji, Hyogo 671-2201 Japan ——— [email protected] Spectral bundles This talk will be devoted to investigation of the eigen- Benedetto Silvestri functions at the threshold energies ±m of the magnetic ` ´ Dipartimento di Matematica Pura ed Applicata, Univer- Dirac operator H = α · − i∇x − A(x) + mβ, where sit di Padova, Via Trieste 63 Padova, 35121 Italy α = (α1, α2, α3) and β are Dirac matrices and m is [email protected] a positive constant. It will be considered three different cases of the vector potential A to decay at infinity. In all In this talk I will construct certain bundles hM, ρ, Xi the cases, it will be shown that zero modes of the Weyl- ` ´ and hB, η, Xi of Hausdorff locally convex spaces associ- Dirac operator σ · − i∇x − A(x) play crucial roles in ated to a given Banach bundle hE, π, Xi. Then I will the analysis of the eigenfunctions at the threshold of H. present conditions ensuring the existence of bounded Here σ = (σ1, σ2, σ3) denotes Pauli matrices. It turns Q Q selections U ∈ x∈X Mx and P ∈ x∈X Bx both out that many existing works on the Weyl-Dirac opera- continuous at a point x∞ ∈ X, such that U(x) is a tor can be utilized. Accordingly, various results on the C0−semigroup on Ex and P(x) is a spectral projector threshold eigenfunctions of the magnetic Dirac operator of the infinitesimal generator of the semigroup U(x), for H are obtained. every x ∈ X. This talk is based on joint work with Yoshimi Sait¯o, University of Alabama at Birmingham, U.S.A. ——— ——— Scattering theory for manifolds and the scattering length IV.1. Pseudo-differential operators Alexander Strohmaier Department of Mathematical Sciences, Loughborough Organisers: University, Loughborough, Leicestershire LE11 3TU, Luigi Rodino, Man Wah Wong United Kingdom [email protected] Topics related to pseudo-differential operators such as PDE, geometry, quantisation, wavelet transforms, lo- We define the so-called scattering length for Rieman- calisation operators on groups and symmetric domains, nian manifolds with cylindrical ends as the time delay mathematical physics, signal and image processing, that waves experience when scattered in the manifold. among others, are the embodiment of the special ses- We show that this scattering length can be estimated sion.

62 IV.1. Pseudo-diﬀerential operators

—Abstracts— Following Wong’s point of view (Wong M.W., The heat equation for the Hermite operator on the Heisenberg Strongly elliptic second-order systems in Lipschitz do- group, Hokkaido Math. Journal, vol. 34 (2005), 393- mains: Dirichlet and Neumann problems. 404), we give a formula for the one-parameter strongly −tLλ Mikhael Agranovich continuous semigroup e , t > 0, generated by the λ generalized Hermite operator L , for a fixed λ ∈ R\{0}, [email protected] in terms of the Weyl transforms. Then we use it to obtain an L2 estimate for the solution of the initial value λ This is a survey talk. We consider a strongly ellip- problem for the heat equation governed by L , in terms p tic second-order system in a bounded Lipschitz domain. of the L norm, 1 ≤ p ≤??, of the initial data. The coefficients have minimized smoothness. The aim of Similar results have also been derived for the general- ˜ the talk is to describe the investigation of the Dirichlet ized Landau operator A which was firstly introduced by and (under natural additional assumptions) Neumann M.A. De Gosson (M.A. De Gosson, Spectral Properties σ of a class of generalized Landau operators, Comm. Part. problems in the variational setting in the spaces Hp σ Diff. Equ., 33 (2008), 2096–2104), who has studied its and Bp . The main case: the principal symbol is Her- mitian. Then we can use the Savaréapproach to the spectral properties. analysis of the smoothness of solutions and combine it ——— with some tools of the interpolation theory, in particular, with Shneiberg’s results on the extrapolation of the in- Generalization of the Weyl rule for arbitrary operators vertibility of operators. The main results: conditions for Leon Cohen the unique solvability of the problems and some spectral City University-Hunter College, 695 Park. Ave, New results (including the case p 6= 2) for the corresponding York, 10471 United States operators. Applications to Neumann-to-Dirichlet oper- [email protected] ators. Some results are also true for general strongly elliptic systems. We compare this approach with the The Weyl rule generally deals with two operators whose deep approach based on the investigation of the surface commutator is a c-number. The generalization to arbi- potentials and corresponding equations at the boundary trary operators is of importance and offers interesting (Calderón,Jerisson, Kenig, Verchota and many other and challenging mathematical issues. We review the ba- mathematicians) in terms of the non-tangential conver- sic ideas, present new results and discuss the unsolved gence, maximal functions, Rellich-type identities, etc. problems. We also show how our generalization leads to the consideration of quasi-probability distributions for ——— arbitrary variables. In addition to the Weyl rule we con- Generalized ultradistributions and their microlocal anal- sider other rules of association between operators and ysis symbols. Chikh Bouzar ——— Department of Mathematics, Oran-Essenia University Sharp results for the STFT and localization operators B. P. 1925 EL MNAOUER Oran, Oran 31003 Algeria [email protected] Elena Cordero Departimento di Matematica, Universita di Torino, via We first introduce new algebras of generalized functions Carlo Alberto 10 Torino, TO 10123 Italy containing ultradistributions. We then develop a mi- [email protected] crolocal analysis suitable for these algebras. Finally, we p give an application through an extension of the well- We completely characterize the boundedness on L known Hörmander’stheorem on the wave front of the spaces and on Wiener amalgam spaces of the short- product of two distributions. time Fourier transform (STFT) and of a special class of pseudodifferential operators, called localization oper- ——— ators. Precisely, a well-known STFT boundedness re- p Some remarks on the Sjöstrandclass sult on L spaces is proved to be sharp. Then, sufficient conditions for the STFT to be bounded on the Wiener Ernesto Buzano amalgam spaces W (Lp,Lq) are given and their sharp- Dipartimento di Matematica, Universitàdi Torino, Via ness is shown. Localization operators are treated sim- Carlo Alberto 10, Torino 10123 Italy ilarly. Using different techniques from those employed [email protected] in the literature, we relax the known sufficient boundedness conditions for localization operators on Lp spaces ∞ ∞,1 We show that the bi-dual of the closure of C0 in M and prove the optimality of our results. More generally, is an extension of M ∞,1 as a subalgebra of the algebra we prove sufficient and necessary conditions for such op- of bounded operators on L2. erators to be bounded on Wiener amalgam spaces. ——— ——— The heat equation for the generalized Hermite and the Fuchsian mild microfunctions with fractional order and generalized Landau operators their applications to hyperbolic equations Viorel Catana Yasuo Chiba University Politehnica of Bucharest, Splaiul Indepen- 1404-1, Katakura-cho Hachioji, Tokyo 1920982 Japan dentei 313, Bucharest 060042 Romania [email protected] catana [email protected]

63 IV.1. Pseudo-diﬀerential operators

Kataoka introduced a concept of mildness in bound- Heat kernel of a sub-Laplacian and Grushin type oper- ary value problems. He defined mild microfunctions ators with boundary values. This theory has effective results Kenro Furutani in propagation of singularities of diffraction. Further- Department of Mathematics, Science University of more, Oaku introduced F-mild microfunctions and ap- Tokyo, 2641 Yamazaki Noda, Chiba 278-8510 Japan plied them to Fuchsian partial differential equations. [email protected] Based on these theories, we introduce Fuchsian mild microfunctions with fractional order. We show the proper- First, I will introduce a framework of a sub-Riemannian ties of such microfunctions and their applications to par- structure which is compatible with a submersion and tial differential equations of hyperbolic type. By using define Grushin type operators. My purpose is to con- a fractional coordinate transform and a quantized Leg- struct heat kernel for various Grushin type operators endre transform, degenerate hyperbolic equations are from known heat kernel in an explicit integral form. As transformed into equations with derivatives of fractional a typical example, I explain the original Grushin oper- order. We present a correspondence between solutions ator and its heat kernel constucted from the heat ker- for the hyperbolic equations and those for the trans- nel on three dimensional Heisenberg group. Then as a formed equations. generalization to dimension three, I define Grushin type 3 4 5 ——— operators on R , R and R from a sub-Laplacian on the 6−dimensional free nilpotent Lie group, and give their About Gevrey semi-global solvability of a class of com- heat kernels in terms of fiber integration. plex planar vector fields with degeneracies Also in the case that the submersion is a covering map from the Heisenberg group to Heisenberg manifolds, Paulo Dattori da Silva Faculdade de Filosofia, Ciênciase Letras de Ribeirão I will determine the spectral zeta function for a sub- Preto - Departamento de Fsica e Matemática,Avenida Laplacian on them in terms of Riemann zeta function. If possible, I will also show a heat kernel for a spherical dos Bandeirantes, 3900 - Monte Alegre Ribeirao Preto, 2 3 Sao Paulo 14040-901 Brazil Grushin operator on S and CP which come from a sub-Laplacian on S3 or S7, respectively. [email protected] ——— 1 1 Let Ω = (−, ) × S , where > 0 and S is the unit Time-frequency analysis of stochastic differential equa- circle. Let tions L = ∂/∂t + (a(x) + ib(x))∂/∂x, b 6≡ 0, (*) Lorenzo Galleani Politecnico di Torino, Corso Duca degli Abruzzi 24, be a complex vector field defined on Ω, where a and b Torino, TO 10129 Italy are real-valued s-Gevrey functions on (−, ), and s ≥ 1. [email protected] We will assume that Σ = {0} × S1 is the characteristic set of L and that L is tangent to Σ. In particular, L is el- Most of the stochastic processes used to model physical liptic on Ω \Σ and (a+ib)(0) = 0. Hence, we may write systems are nonstationary, and yet most of the theo- n m (a + ib)(x) = x a0(x) + ix b0(x) in Ω, with m, n ≥ 1, retical results on stochastic processes are related to the and a0, b0 smooth. stationary case. We consider a nonstationary random In this talk we shall present results about Gevrey solv- process defined as the solution of a stochastic differen- ability of L, given by (*), in a neighborhood of Σ, in the tial equation. We first transform the stochastic equa- following sense: there exists s0 > 1 such that for any f tion to the Wigner spectrum domain, where we obtain s belonging to a subspace of finite codimension of G (Ω) a deterministic differential equation. Then, by applying 0 there exists a solution, u ∈ Gs , to the equation Lu = f the Laplace transform, we obtain the exact solution of in a neighborhood of Σ. We will see that the interplay the deterministic equation. Finally, we rewrite the gen- between the order of vanishing of the functions a and b eral solution in a form which clarifies the structure of at x = 0 plays a role in the Gevrey solvability. Moreover, the nonstationary stochastic process, and which high- lost of regularity occurs. lights the connection to the classical results obtained by This is a joint work with Adalberto P. Bergamasco Fourier analysis. (ICMC/USP) and Marcelo R. Ebert (FFCLRP/USP). ——— ——— Lp-microlocal regularity for pseudodifferential operators Invertibility for a class of degenerate elliptic operators of quasi-homogeneous type Gianluca Garello Julio Delgado Universitàdi Torino, Department of Mathematics, Via Cra 82 Bis 49-03 Ciudad Real Cali, Valle 9999 Colombia Carlo Alberto 10 Torino, Torino I-107123 Italy [email protected] [email protected]

In this work we study fundamental solutions for a class Pseudodifferential operators whose symbols have de- of degenerate elliptic operators. The type of operator cay at infinitive of quasi-homogeneous are considered considered is obtained as a sum of operators of the form and their behavior on the wave front set of distribu- D2 + x2kD2 . The invertibility for an operator of type xi i xj tions in weighted Zygmund-Hölderspaces and weighted 2 2k 2 2 p Dx1 +x1 Dx2 on R is known, here we extend this result Sobolev spaces in L framework is studied. Then mi- to higher dimensions. crolocal properties for solutions of linear partial differential equations with coefficients in weighted Zygmund- ——— Hölderspaces are obtained.

64 IV.1. Pseudo-diﬀerential operators

——— The symmetric Hoermander class of type (1, 1) (inter- Generalized Fourier integral operators methods for hy- esting for paradifferential operators) is included in the perbolic problems theory of holomorphic Fredholm functions in connection with the Oka principle. This class is known to be not Claudia Garetto spectrally invariant. But commutator methods lead to Arbeitsbereich fürtechnische Mathematik, Universität the submultiplicativity of this symmetric Fréchet alge- Innsbruck Technikerstrasse 13 Innsbruck, Austria 6020 bra. Some relations to operator algebras on singular Austria and stratified spaces are given. Stochastic PDE lead to [email protected] holomorphic operator functions on infinite dimensional domains in DFN - spaces with basis such as the distri- The past decade has seen the emergence of a differential- bution space S0 of Schwartz. A series of open problems algebraic theory of generalized functions that answered is mentioned for Fréchet operator algebras connected to a wealth of questions on solutions to partial differential parameter dependent equations on singular rep. rami- equations involving non-smooth coefficients and strongly fied manifolds. singular data. In such cases, the theory of distributions does not provide a general framework in which solutions ——— exist due to inherent constraints in dealing with nonlinear operations. The Cauchy problem for a paraxial wave equation with An alternative framework is provided by the theory of non-smooth symbols Colombeau algebras of generalized functions. Gunther¨ Hormann¨ In this talk we solve hyperbolic equations, generated by Nordbergstraße 15 FakultätfürMathematik Wien, Wien highly singular coefficients and data, by means of gener- A-1090 Austria alized FIO techniques developed in the Colombeau con- [email protected] text. Finally, we provide a careful microlocal investigation of the solution by studying the microlocal mapping 2 properties of these operators. We discuss evolution systems in L for Schroedinger- type pseudodifferential equations with non-Lipschitz co- ——— efficients in the principal part. The underlying operator Resolvents of regular singular elliptic operators on a structure is motivated from models of paraxial approx- quantum graph imations of wave propagation in geophysics. Thus, the evolution direction is a spatial coordinate (depth) with Juan Gil additional pseudodifferential terms in time and low regu- Penn State Altoona, 3000 Ivyside Park, Altoona, Penn- larity in the lateral variables. We formulate and analyze sylvania 16601 United States the Cauchy problem in distribution spaces with mixed [email protected] regularity. We will discuss the pseudodifferential structure of the Solutions with low regularity in the operator symbol will resolvent of regular singular differential operators on a provide a basis for an inverse analysis which allows to graph. For second order operators, we give a simple, ex- infer the lack of lateral regularity in the medium from plicit, sufficient condition for the existence of a sector of measured data. minimal growth. In particular, we will discuss operators ——— with a singular potential of Coulomb type. Our analysis is based on the theory of elliptic cone operators. Pseudodifferential operators on locally symmetric ——— spaces Hyperbolic systems of pseudodifferential equations with Eugenie´ Hunsicker irregular symbols in t admitting superlinear growth for Department of Mathematical Sciences, Loughborough |x| → ∞. University, Loughborough, LE11 3TU United Kingdom [email protected] Todor Gramchev Dipartimento di Matematica e Informatica, Università di Cagliari, via Ospedale 72, 09124 Cagliari I will discuss recent work with D. Grieser of U. Olden- [email protected] burg on the first stages of the construction of a pseudodifferential operator calculus tailored to locally sym- We consider hyperbolic systems of pseudodifferential metric spaces and other noncompact spaces with similar equations with irregular symbols with respect to the structures. time variable t and admitting superlinear growth for |x| → ∞. We investigate the global well-posedness of the ——— Cauchy problem for such systems in the framework of On the continuity of the solutions with respect to the weighted spaces which generalize the Cordes type spaces electromagnetic potentials to the Schrödingerand the s ,s n H 1 2 (R ). Dirac equations ——— Wataru Ichinose Analytic perturbations for special Fréchetoperator al- Department of Mathematical Science, Shinshu Univer- gebras in the microlocal analysis sity Matsumoto, Nagano 390-8621 Japan Bernhard Gramsch [email protected] Institut fürMathematik, UniversitätMainz, Staudinger- weg 9, Mainz, 55099 Germany The initial problem to families of the Schrödingerequa- [email protected] tions and the Dirac equations with the electromagnetic

65 IV.1. Pseudo-differential operators potentials are studied, respectively. Assume that the bounded variation on dyadic shells. For some Banach solutions have the same initial data and that the elec- algebras of pseudo-differential operators acting on the p tromagnetic potentials converge. space L (R, w) and having symbols discontinuous with Then, it is proved that the solutions of Schrödingerequa- respect to spatial and dual variables, we construct a non- tions and the Dirac equations with the corresponding commutative Fredholm symbol calculi and give Fred- electromagnetic potentials also converge, respectively. holm criteria and index formulas for the operators in The proof follows from the uniqueness and the bound- these algebras. Applications to algebras of generalized edness of the solutions, and the functional method, ex. singular integral operators with shifts are considered. the abstract Ascoli-Arzelàtheorem, which will be seen ——— to be applied to nonlinear equations. On maximal regularity for parabolic equations on com- ——— plete Riemannian manifolds Calculus of pseudo-differential operators and a local index of Dirac operators Thomas Krainer Penn State Altoona 3000 Ivyside Park Altoona, Penn- Chisato Iwasaki sylvania 16601 United States Department of Mathematical Sciences, Shosha 2167 [email protected] Himeji, Hyogo 671-2201 Japan [email protected] In this talk I plan to demonstrate how recent advances in the theory of pseudodifferential operators lead to a I will show a method to obtain a local index of Dirac method to effectively establish optimal Lp–Lq a priori operators. This method depends on construction of the estimates for solutions to parabolic equations on certain fundamental solution to the Cauchy problem for heat complete Riemannian manifolds. equations by introducing a weight for symbols of pseudo- The approach is based on Weis’ functional analytic char- differential operators. acterization of maximal regularity in terms of the R- boundedness of the resolvent. In recent work, partly in ——— collaboration with Robert Denk (Univ. of Constance, On the theory of type 1, 1-operators Germany), we have shown that the approximation of resolvents of elliptic operators by parameter-dependent Jon Johnsen parametrices in suitable classes of pseudodifferential op- Mathematics Department, Aalborg University, Fredrik erators readily leads to the desired R-boundedness, thus Bajers Vej 7G, Dk-9220 Aalborg Øst, Denmark to maximal regularity. [email protected] In my talk I plan to survey our results and the basic underlying principles of the method. After an introduction with a brief review of celebrated contributions on type 1, 1-operators of G. Bourdaud ——— (1983,1988) and L. Hörmander(1988–89), their results On the cohomological equation in the plane for regular will be set in relation to the general definition of type vector fields 1, 1-operators, which was introduced at the ISAAC 2007 congress. Progress in the area will be described as time Roberto de Leo permits. INFN, Complesso Universitario di Monserrato Monser- rato (CA), Sardegna 09042 Italy ——— [email protected] Pseudo-differential operators with discontinuous symbols and their applications In this talk we present our recent results about the solvability of the equation Xf = g, where X is a vector Yuryi Karlovych field on the plane without zeros, in the cases when X Universidad Autónomadel Estado de Morelos, Facultad is generic and when it is Hamiltonian with respect to de Ciencias, Av. Universidad 1001, Cuernavaca, More- some symplectic form. This work slightly generalizes a los 62209 Mexico recent result of S.P. Novikov, which showed recently that [email protected] a generic vector field on a compact surface, seen as a 1- st order operator on the set of smooth functions, has an Applying a weighted analogue of the Litllewood-Paley infinite-dimensional cokernel. Our study is also related theorem and the boundedness of the maximal singular to aspects of pseudo-differential operators on the plane. integral operator S∗ related to the Carleson-Hunt theorem on almost everywhere convergence on all weighted ——— p Lebesgue spaces L (R, w), where 1 < p < ∞ and Lp-boundedness and compactness of localization oper- w ∈ Ap(R), we study the boundedness and compactness ators associated with Stockwell transform of pseudo-differential operators a(x, D) with non-regular ∞ symbols in the classes L (R,V (R)) and Λγ (R,Vd(R)) on Yu Liu p ∞ the spaces L (R, w). The Banach algebra L (R,V (R)) Department of Mathematics and Statistics, York Univer- consists of all bounded measurable V (R)-valued func- sity, 4700 Keele St., Toronto, Ontario M3J1P3 Canada tions on R where V (R) is the Banach algebra of all func- [email protected] tions on R of bounded total variation, and the Banach algebra Λγ (R,Vd(R)) consists of all Lipschitz Vd(R)- Localization operators associated with the Stockwell valued functions of exponent γ ∈ (0, 1) on R where transform, with respect to the filter symbol and the win- p Vd(R) is the Banach algebra of all functions on R of dows, are a class of operators defined on L (R). Under

66 IV.1. Pseudo-diﬀerential operators suitable conditions for the symbol and the windows, the Canada localization operators turn to be bounded and compact. [email protected]

——— For a family of τ-twisted Laplacians that includes the About transport equation with irregular coefficient and usual twisted Laplacian when τ = 1/2, we compute the data heat kernel for each τ-twisted Laplacian for [0, 1]. ——— Jean-Andre´ Marti Campus de Schoelcher, Laboratoire CEREGMIA, Uni- Regularity of characteristic initial-boundary value prob- versitédes Antilles et de la Guyane Schoelcher, Mar- lems for symmetrizable systems tinique B.P. 7209-97275 France Alessandro Morando [email protected] Department of Mathematics - University of Brescia, Via Valotti, 9, I-25133, Brescia, Italy We are interested in the study of the Cauchy problem for [email protected] transport equation in the formally simplified case where the coefficients α and β are discontinuous and even dis- We study the initial-boundary value problem for a lin- tributions. For the data u0, we suppose it is a distribu- ear Friedrichs symmetrizable system, with characteristic p q tion and even a more singular object like δx ⊗ δy we will boundary of constant rank. We assume the existence of give later a generalized meaning. Then the problem is the strong L2 solution satisfying a suitable energy esti- formally written as mate, but we do not assume any structural assumption sufficient for existence, such as the fact that the bound- 8 ∂ ∂ ∂ < u + α ⊗ 1xy u + β ⊗ 1xy u = 0, ary conditions are maximally dissipative or the Kreiss- (Pform) ∂t ∂x ∂y Lopatinski condition. We show that this is enough in : u | = u (= δp ⊗ δq.) {t=0} 0 x y order to get the regularity of solutions, in the natural We remark that the product and the restriction writ- framework of weighted anisotropic Sobolev spaces, pro- ten above are generally not defined in a distributional vided the data are sufficiently smooth. sense. Consequently we begin in associating to (Pform) ——— a generalized one (P ) well formulated in a convenient gen Application of pseudodifferential equations in stress sin- (C, E, P) algebra A ` 3´ and recall the definition and R gularity analysis for thermo-electro-magneto-elasticity main properties of this generalized multiparametric fac- problems: a new approach for calculation of stress sin- tor algebra. gularity exponents In our case, we construct such an algebra by means of independant regularizations involving three independant David Natroshvili parameters and obtain Georgian Technical University 77 M.Kostava st. Tbilisi, Tbilisi 0175 Georgia 8 ∂ ∂ ∂ < u + F u + G u = 0, [email protected] (Pgen) ∂t ∂x ∂y : u |{t=0}= H. We apply the potential method and the pseudodifferential equations technique to the mathematical model of where F and G (resp.H) are the classes in A ` 3´ R the thermo-electro-magneto-elasticity theory. We study (resp.A ` 2´) of the families regularizing the coefficients R mixed and crack type boundary value problems. Along (resp. the data). with the existence and uniqueness questions our main First we solve (P ) and examine the existence of a gen goal is a detailed theoretical investigation of singularities solution. To study more pecisely its singularities, we re- of the thermo-mechanical and electro-magnetic fields fer to a generalization of the asymptotic singular spec- near the crack edges and the curves where the boundary trum defined previously and adapted here to the three- conditions change their type. In particular, the most im- parametric case. The so-called ”(a, D0)-singular spec- portant question is description of the dependence of the trum” of u ∈ A( 3) propose a spectral analysis of the R stress singularity exponents on the material parameters. singularities: by means of an ”analyzing” function a we We reduce the three-dimensional mixed and crack can see where and why u is not locally (associated to type boundary value problems of the thermo-electro- a section of) D0.The localization of such singularities magneto-elasticity to the equivalent system of pseudo- of u is always the ”D0-singular support” of u, and the differential equations which live on proper parts of the asymptotic causis is described by a fiber Σ (u) (above X boundary of the elastic body under consideration. each X = (t, x, y) ∈ 3) which is the complement in 3 R R+ We show that with the help of the principal homoge- of a conic subset of 3 . R+ neous symbol matrices of the corresponding pseudodif- In our case, the D0-singularities of the data propagate ferential operators it is possible to determine explicitly along the ”regularized characteristic Γ of the problem the singularity exponents for physical fields. We give (P )” on which the fiber Σ (u) remains constant. gen X an efficient method for computation of these exponents. This joint work of V. Dvou, M. Hasler and J.-A. Marti Moreover, we establish that these exponents essentially of Universit Antilles-Guyane. depend on the material parameters, in general. ——— ——— The Heat Kernel of τ-Twisted Laplacian Wigner type transforms and pseudodifferential operators Shahla Molahajloo Department of Mathematics and statistics, York Uni- Alessandro Oliaro versity 4700 Keele street, Toronto, Ontario M3J1P3 Department of Mathematics, University of Torino, Via

67 IV.1. Pseudo-diﬀerential operators

Carlo Alberto, 10 Torino, TO I-10123 Italy where A, B, E, A1, B1, C1, D are constants, f(x, y, z)– [email protected] is a given function in Ω, Φ(x, y, z) is the desired function. Some cases equation (*) investigated N. Rajabov [Ac. We present some modifications of the Wigner transform of Sciences Dokl. V. 409, No 6, 2006, pp.749-753]. In (Wig), suggested by the connections of Wig with pseu- this lecture the general solution of the integral equa- dodifferential operators. We analyze some properties of tion (*) is constructed, using the connection equation these representations, in particular the positivity and (*) with one dimensional integral equation of the type the behaviour with respect to the cross terms. (*). In the case, when A1 = AB, B1 = AE, D = AC1, ——— then the problem is determination general solution equation (*) redused, to problems found general solution sin- Local regularity of solutions to PDEs by asymptotic gle one-dimensional integral equation and single two– methods dimensional integral equation of the type (*). In this ba- Michael Oberguggenberger sis in the case when C1 = EB and A < 0, B < 0, E < 0, Unit for Engineering Mathematics, University of Inns- find general solution equation (*) by three arbitrary bruck, A-6020 Innsbruck, Austria functions two variables. In the case when C1 6= EB, [email protected] find the solution equation (*) by means of one arbitrary function two variabe and infinity number arbitrary func- In the nonlinear theory of generalized functions, alge- tion one variabe. Select the cases, when equation (*) has bras of generalized functions are commonly constructed unique solution. by means of nets of smooth functions (uε)ε∈E depending on one or more parameters. Typically, these nets do ——— not converge as ε → 0, say, but exhibit a certain asymptotic behavior. This behavior not only determines the The adiabatic limit of the Chern character algebras to which such an object belongs to, but may Frederic Rochon describe local regularity properties. This type of regu- Department of Mathematics, 40 St. George Street, larity theory has become increasingly important in ap- Toronto, Ontario M5S 2E4 Canada plications to partial- and pseudodifferential operators. [email protected] This presentation is devoted to a general framework – the so-called asymptotic spectrum – for measuring the asymptotic behavior in algebras of generalized functions, Certain spaces of pseudo-differential operators can be using asymptotic scales and various topologies. It has used as classifying spaces for K-theory. In this context, been developed in joint work with A. Delcroix and J.-A. Bott periodicity can be realized by taking a certain adi- Marti [Asymptotic Analysis 59(2008), 169 – 199] and abatic limit. In this talk, we will indicate how natu- forms a nonlinear alternative to the wave front set ap- ral forms representing the universal Chern chararcter on proach. these spaces behave under such an adiabatic limit. This Various applications to propagation of singularities as a joint work with Richard Melrose. well as to regularity in Colombeau algebras and to jump discontinuities in hyperbolic systems will be given. ——— ——— Boundary value problems as edge problems

Modern results by theory of the three dimensional Bert-Wolfgang Schulze Volterra type linear integral equations with singularity Institute of Mathematics, University Potsdam, Am Nusrat Rajabov Neuen Palais 10, Potsdam, D-14469 Germany Tajik National University Rudaki Av. 17 Dushanbe, [email protected] Dushanbe 734025 Tajikistan [email protected] The calculus of pseudo-differential operators on a mani- Let Ω denote the parallelepiped fold with edges can be established in such a way that standard boundary value problems (BVPs) with the Ω = {(x, y, z): a < x < a0, b < y < b0, c < z < c0}, transmission property at the boundary appear as a spe- D1 = {(x, y): a < x < a0, b < y < b0, z = c}, cial case (up to some simple modifications). Also the D2 = {(x, z): a < x < a0, y = b, c < z < c0}, case without the transmission property can be formu- D3 = {(y, z): x = a, b < y < b0, c < z < c0}. In the domain Ω we consider the following integral equa- lated as a special case of the edge calculus (as is shown tion in a joint paper of the author with J. Seiler, 2009). The remarkable fact here is that the symbols of the respective Z x Φ(t, y, z) Z b Φ(x, s, z) Φ(x, y, z) + A dt + B ds (classical) pseudo-differential operators are not required a t − a y s − b to be of edge-degenerate form but are only smooth up to Z z Φ(x, y, τ) Z x dt Z y Φ(t, s, z) the boundary in the usual sense. In our talk we illustrate + E dτ + A1 ds c τ − c a t − a b s − b the specific properties of that theory for the case of sym- Z x dt Z z Φ(t, y, τ) bols with the anti-transmission property (recently sin- + B dτ 1 t − a τ − c glet out by the author to investigate specific asymptotics a c of solutions). Symbols with the transmission property Z y ds Z z Φ(x, s, τ) + C1 dτ together with those with the anti-transmission prope rty b s − b c τ − c span the full space of symbols that are smooth up to the Z x dt Z y ds Z z Φ(t, s, τ) + D dτ boundary. a t − a b s − b c τ − s ——— = f(x, y, z), (*)

68 IV.1. Pseudo-diﬀerential operators

Noncommutative residues and projections associated to 7, Chiyoda-ku Tokyo, Tokyo 102-8554 Japan boundary value problems [email protected]

Elmar Schrohe In this talk, I will consider the regularity of the solution Institut für Analysis, Leibniz Universität Hannover, of a nonlinear singular partial differential equation (E): Welfengarten 1, 30167 Hannover m j α [email protected] (t∂/∂t) u = F (t, x, {(t∂/∂t) (∂/∂x) u}j+|α|≤m,j 0, then clockwise 0 0 increased the interest for such spaces, we refer to the about the origin on the circle of radius r to r eiθ and 0 0 work of Concetti-Toft, Cordero-Nikola-Rodino, Okoud- back to infinity along the second ray. jou, Ruzhansky-Sugimoto-Tomita-Toft. ——— In particular, we refer to several papers of Pilipovic- Teofanov-Toft for the micro-local analysis of Fourier On maximal regularity for mixed order systems Lebesgue spaces. Jorg¨ Seiler In this lecture we study continuity properties on the School of Mathematics Loughborough University Lough- Fourier-Lebesgue spaces by observing the localized ver- 0 borough, Leicestershire LE113TU United Kingdom sion of the class S0,0. Furthermore, we prove an ex- [email protected] tension to operators whose symbols enjoy certain decay with respect to the x variable. These operators belong to the class of symbol global type operators, recently stud- I will discuss some results on maximal Lp-regularity for parabolic mixed order systems based on the so-called ied by Cappiello-Gramchev-Rodino, Coriasco-Rodino, Dasgupta-Wong.... At the end, a continuity result for H∞-calculus as well as on a calculus of Volterra pseudodifferential operators. This is a joint work with R. a class of elliptic operators is given. Denk and J. Saal. ——— ——— Wave-front sets of Fourier Lebesgue types Dirichlet problem for higher order elliptic systems with Joachim Toft BMO assumptions on the coefficients and the boundary Department of Mathematics and Systems Engineering, Vejdes plats 6,7, Vxj University, Vxj, Smland 351 95 Tatyana Shaposhnikova Sweden Department of Mathematics, Linkoeping Universitym [email protected] Linkoeping, Ostergotland SE-58183 Sweden [email protected] Roughly speaking, a wave-front set WF∗(f) of the distribution f with respect to “something”, gives infor- Given a bounded Lipschitz domain, we consider the mation where the distribution f has singularities with Dirichlet problem with boundary data in Besov spaces respect to this “something”, as well as what directions for divergence form strongly elliptic systems of arbitrary in these points of singularities, the singularities propa- order with bounded complex-valued coefficients. gates. The wave-front set with respect to smoothness The main result gives a sharp condition on the local (by Hörmander)for a distribution f is the set of pairs d d mean oscillation of the coefficients of the differential ((x, ξ) ∈ R × (R \ 0), where ξ is the directions were f operator and the unit normal to the boundary (auto- is non-smooth at x. matically satisfied if these functions belong to the space In this talk we introduce wave-front sets WF∗(f) = WF Lq (f) of the distribution f with respect to VMO) which guarantee that the solution operator asso- F (ω) ciated with this problem is an isomorphism. q (weighted) Fourier Lebesgue spaces F L(ω), where ω is an appropriate weight function. An advantage with such ——— wave-front sets is that we may examine micro-local prop- Gevrey regularities of solutions of nonlinear singular par- erties more close to differentiability up to a certain order, tial differential equations instead of complete smoothness only. Especially we show the usual property Hidetoshi Tahara [ Department of Mathematics, Sophia University, Kioicho WF∗(a(x, D)f) ⊆ WF∗(f) ⊆ WF∗(a(x, D)f) Char(a),

69 IV.1. Pseudo-differential operators when a(x, D) is an appropriate pseudo-differential oper- metric g. That is, we consider the d’Alembertian g 1 ator with smooth coefficients and Char(a) is the set of on differential forms, and u ∈ Hloc(X;ΛX) solving characteristic points for a. We remark that our Char(a) gu = 0 with relative or absolute boundary conditions, is smaller than what is usual. For example, that Char(a) ν ∧ u|S = 0, resp. ιν u|S = 0, at all boundary hypersur- can be chosen to be empty when a(x, D) is hypoelliptic. faces S, where ν = νS is the conormal of S. We show In particular, WF∗(a(x, D)f) = WF∗(f). that the appropriate wave front set WFb(u) of u is a Finally we remark that one may get the “usual” wave- union of maximally extended generalized broken bichar- front set (with respect to smoothness) by considering acteristics. sequences of wave-front sets of Fourier Lebesgue types. I will indicate the key ideas of the proof, such as mi- crolocalization with respect to the appropriate ps.d.o. ——— algebra, Ψb(X), and gaining b-regularity (i.e. conormal 1 Pseudo-differential operators and symmetries regularity) relative to Hloc(X;ΛX) via positive commutator estimates. Ville Turunen These results are analogous to those obtained by the Helsinki University of Technology, Institute of Mathe- author for the scalar wave equation and for the wave matics, P.O.Box 1100, FIN-02015 HUT, Finland equation on systems with Dirichlet or Neumann bound- [email protected] ary conditions. The main novelty is thus the presence of natural boundary conditions, which effectively make the This work is joint with M. Ruzhansky (Imperial College problem non-scalar, even ‘to leading order’, at corners London). We study pseudo-differential equations glob- of codimension ≥ 2. ally on compact Lie groups, without resorting to local charts. We obtain a full global symbol and global cal- ——— culus. This can be done by presenting functions on the Formation of singularities near Morse points group by Fourier series obtained from the representa- Ingo Witt tions of the group. A pseudo-differential operator can Mathematical Institute, University of Göttingen,Bun- be presented as a convolution operator valued mapping senstr. 3-5 Göttingen,37073 Germany on the group. [email protected] The complete treatise can be found in the following monograph: M. Ruzhansky, V. Turunen: Pseudo- Given a Morse function f and a Riemannian metric h on Differential Operators and Symmetries. Birkhäuser a C∞ manifold M, we study solutions u to the equation 2009. gu = 0, ——— where g is the wave operator associated with the Pseudo differential equations and boundary value prob- Lorentzian metric lems in non-smooth domains 2 g = kdfkh h − ζ df ⊗ df, Vladimir Vasilyev Bryansk State University, Bezhitskaya 14, Bryansk and ζ > 1 a constant. As turns out, at Morse points 241036, Russia initially regular solutions u start to form singularities [email protected] that are then propagated as usual. These singularities can be described in terms of certain classes of conormal One discusses a possibility for constructing theory of distributions. boundary value problems for pseudo differential equa- Such models arise in quantum field theory on curved tions under existence of so called wave factorization of space-times with changing topology. elliptic symbol in non-smoothness points at the bound- ——— ary. It leads to well-posed statements of boundary value problems in Sobolev-Slobodetskii spaces (both old and Phases of modified Stockwell transforms and instanta- new) in non-smooth domains. Some of such problems neous frequencies and their solvability have been described in author’s pa- Man Wah Wong pers earlier, some of them will be considered at first Department of Mathematics and Statistics, York Uni- time. One suggests also to consider discrete analogues versity, 4700 Keele Street Toronto, Ontario M3J 1P3 of such equations (and boundary value problems), for Canada which some preliminary results have been obtained by [email protected] the author. Modified Stockwell transforms are introduced to include ——— the classical Stockwell transforms and related wavelet Diffraction at corners for the wave equation on differ- transforms. We begin with highlighting the subtle dif- ential forms ferences between the Stockwell transforms and the Mor- let wavelet transforms. The focus of the talk, however, Andras´ Vasy is on the characteristics of the phases of modified Stock- Department of Mathematics, Stanford University, 450 well transforms in general and explicit formulas for in- Serra Mall, CA 94305-2125, USA stantaneous frequencies of signals in terms of the TT- [email protected] transforms, which are closely related to the Stockwell transforms. (This is joint work with Shahla Molaha- I will describe the propagation of smooth (C∞) and jloo.) Sobolev singularities for the wave equation on smooth ——— manifolds with corners X equipped with a Lorentzian

70 IV.2. Dispersive equations

Generalized cosine transforms in image compression This is a joint work with Sandra Lucente and Giovanni Taglialatela from University of Bari. Hongmei Zhu York University, Department of Mathematics and Statis- ——— tics, 4700 Keele ST, Toronto, ON M3J1P3 Canada [email protected] Multiple solutions for non-linear parabolic systems

The generalized cosine transforms are deﬁned by the so- Q-Heung Choi called C-functions, a new family of special functions that Dept. of Mathematics San 68 Miryong Dong Kunsan arise in connection with compact semi-simple Lie group National University , Kunsan 573-701 South Korea of rank 2. For groups A1 and A1xA1, the generalized [email protected] cosine transform coincide with the well-known 1D and 2D cosine transform. Here, we focus on the generalized We have a concern with the existence of solutions (ξ, η) cosine transform for group C2 and investigate its appli- for perturbations of the parabolic system with Dirichlet cations in image compression. boundary condition ——— ξt = −Lξ + µg(3ξ + η) − sφ1 − h1(x, t) in Ω × (0, 2π),

ηt = −Lη + νg(3ξ + η) − sφ1 − h2(x, t) in Ω × (0, 2π). IV.2. Dispersive equations We prove the uniqueness theorem when the nonlin- Organisers: earity does not cross eigenvalues. We also investigate Michael Reissig, Fumihiko Hirosawa multiple solutions (ξ(x, t), η(x, t)) for perturbations of the parabolic system with Dirichlet boundary condition The goal of the session is to discuss the state-of-the-art when the nonlinearity f 0 is bounded and f 0(−∞) < of qualitative properties of solutions of dispersive equa- 0 λ1, λn < (3µ + ν)f (+∞) < λn+1. tions. Among other things Strichartz decay estimates, This is joint work with Tacksun Jung. Strichartz estimates, and dispersive estimates are of interest. The question of the influence of low regularity ——— coefficients on the well-posedness of the Cauchy problem is another key topic. Local sovability beyond condition ψ Ferruccio Colombini —Abstracts— Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5 Pisa, PI 56127 Italy Lp–Lq estimates for hyperbolic systems [email protected] Marcello D’Abbico Universita di Bari, Dipartimento di Matematica, Via E It is well known that condition ψ (PSI) is necessary and Orabona 4, Bari, BA 70125 Italy sufficient in order to have local solvability for differen- [email protected] tial (pseudo-differential) operators of principal type with coefficients sufficiently regular. We establish Lp −Lq estimates for the solution of M ×M We study some cases when such conditions are not sat- systems with bounded time dependent coefficients: isfied. n X These are two joint papers with Ludovico Pernazza and D U = A (t) D U + B(t)U,U(0, x) = U (x) . t j xj 0 Fran¸coisTreves and with Paulo Cordaro and Ludovico j=1 Pernazza. In the equation setting, Reissig and others obtained such estimates by using WKB representation of the solutions. ——— We put f = (t + e3)`log(t + e3)´−γ , γ Continuous dependence for backward parabolic opera-  ˛ ff ∞ ˛ k m−k tors with Log-Lipschitz coefficients Tγ {m} = a ∈ C ˛ |Dt a(t)| ≤ Ck (fγ (t)) , ˛ Daniele Del Santo for some γ ∈ [0, 1]; let Aj ∈ Tγ {0}, B ∈ Tγ {−1}, and Dipartimento di Matematica e Informatica, Via Valerio ‚Z t ‚ 12/1, Trieste, 34127 Italy ‚ ‚ ` 3 ´γ ‚ B(r) dr‚ ≤ c1 log(t + e ) . [email protected] ‚ 0 ‚ P Theorem. Let A(t, ξ) = Aj (t)ξj . We assume that there exists a smooth, regular matrix N(t, ξ) such that We consider the following backward parabolic equation NAN −1 is diagonal and, if ζ = (D N + NB)N −1, then t X ˛Z t ˛ ∂tu + ∂xi (ai,j (t, x)∂xj u) ˛ ` ´ ˛ ˛ = ζjj (r, ξ) dr˛ ≤ c2 , j = 1,...,M. i,j ˛ 0 ˛ X + bj (t, x)∂xj u + c(t, x)u = 0 (*) We assume that det |A(t, ξ)| ≥ c3 > 0, for t ≥ t0. j Let U be the solution of the Cauchy Problem; then

n−1 1 1 n − ( − )+s0 on the strip [0,T ] × 3 (t, x). We suppose that q 2 p q R kU(t, ·)kL ≤ C(n, p) (1 + t) kU0kHNp,p , −1 −1 n for some s0 > 0, where 1 = p + q , 1 < p ≤ 2 and • for all (t, x) ∈ [0,T ] × R and for all i, j = 1 . . . n, Np ≥ n(1/p − 1/q). One can take s0 = 0 if γ = 0 and s0 = for any > 0 if γ ∈ (0, 1) and C = C(n, p, ε). ai,j (t, x) = aj,i(t, x);

71 IV.2. Dispersive equations

• there exists k > 0 such that, for all (t, x, ξ) ∈ Zakharov system in inﬁnite energy space [0,T ] × n × n, R R Daoyuan Fang 2 X −1 2 Zhejiang University, Hangzhou, China k|ξ| ≤ ai,j (t, x)ξiξj ≤ k |ξ| ; i,j [email protected]

∞ n We consider the Zakharov system in space dimension • for all i, j = 1, . . . , n, ai,j ∈ LL([0,T ],L (R )) ∩ 2 ∞ 2 n ∞ 2 n two. We will show a L -concentration result for the data L ([0,T ],Cb (R )) and bj , c ∈ L ([0,T ],Cb (R )), without finite energy, when blow-up of the solution hap- ∞ n (where a ∈ LL([0,T ],L (R )) means that the function pens, and a low regularity global well-posedness result. a is Log–Lipschitz–continuous with respect to time with The proof uses a refined I-method originally initiated by ∞ values in L , i.e. Colliander, Keel, Staffilani, Takaoka and Tao. A polynomial growth bound for the solution is also given. ka(t, ·) − ai,j (s, ·)k ∞ n sup L (R ) ≤ ∞). This talk is based on some joint works with Sijia Zhong |t − s|(1 + | log |t − s||) t,s∈[0,T ], 0<|t−s|≤1 and Hartmut Pecher. 0 2 n 0 1 n Let E := C ([0,T ],L (R )) ∩ C ([0,T [,H (R )) ∩ ——— C1([0,T [,L2( n)). R Wave equation in Einstein-de Sitter spacetime Our main reslut is the following. Anahit Galstyan Theorem. For all T 0 ∈ ]0,T [ and for all D > 0 there Department of Mathematics, University of Texas-Pan exist ρ0, M 0, N 0, δ0 > 0 such that if u ∈ E is a solution American, 1201 West University Drive, Edinburg, Texas of the equation (*) with supt∈[0,T ] ku(t, ·)kL2 ≤ D and 0 78539 United States ku(0, ·)k 2 ≤ ρ , then L [email protected] 0 δ0 0 −N | log ku(0,·)kL2 | sup ku(t, ·)kL2 ≤ M e . In this talk we introduce the fundamental solutions of t∈[0,T 0] the wave equation in the Einstein-de Sitter spacetime. (joint work with Martino Prizzi, Trieste University) The last one describes the simplest non-empty expanding model of the universe. The covariant d’Alembert’s ——— operator in the Einstein-de Sitter spacetime belongs to On the loss of regularity for a class of weakly hyperbolic the family of the non-Fuchsian partial differential opera- operators tors. In this talk we investigate initial value problem for this equation and give the explicit representation formu- Marcello Ebert las for the solutions. Universidade de São Paulo, Faculdade de Filosofia, The equation is strictly hyperbolic in the domain with Ciências e Letras, Dept. de Fisica e Matemática, positive time. On the initial hypersurface its coefficients Av. dos Bandeirantes, 3900 RibeirãoPreto, SãoPaulo have singularities that make difficulties in studying of 14040-901 Brazil the initial value problem. In particular, one cannot an- [email protected] ticipate the well-posedness in the Cauchy problem for the wave equation in the Einstein-de Sitter spacetime. In this work we consider the Cauchy problem The initial conditions must be modified to so-called n n weighted initial conditions in order to adjust them to 2 2 X 2 X 2 P u = ∂t u−λ (t) aij (t)∂xixj u +λ(t) ci(t)∂txi u the equation. i,j=1 i=1 We also show the Lp − Lq estimates for solutions. Thus, 0 we have prepared all necessary tools in order to study the = f(x, t, u, ∂tu, λ (t)∇xu), (*) solvability of semilinear wave equation in the Einstein-de Sitter spacetime. u(x, 0) = u0(x), ∂tu(x, 0) = u1(x) (**) This is a joint work with Tamotu Kinoshita (Univer- where P is weakly hyperbolic in a neighborhood of sity of Tsukuba, Japan) and Karen Yagdjian (UTPA, {t = 0}, that is, U.S.A.). ——— the roots of p(x, t, ξ, τ) in τ are real; (***) Stability of solitary waves for Hartree type equation here p = p(x, t, ξ, τ) is the principal symbol of P . Ex- amples show that, differently of the hyperbolic case, un- Vladimir Georgiev der (*), (**) and (***) the solution might not exist. In Department of Mathematics, University of Pisa, Largo addition to condition (***), various authors presented Bruno Pontecorvo 5 Pisa, PI 56127 Italy sufficient conditions, usually called Levi conditions, for [email protected] the Cauchy problem to be well posed in Sobolev spaces. Those type of conditions relate p with lower order terms We prove the stability of solitary manifold associated of P . In this work, we narrowed the bounds for the op- with the solitary solutions of Hatree type equation with timal Sobolevs loss of regularity under some sharp Levi external Coulomb type potential. conditions. ——— This work was done in collaboration with Rafael A. dos Hyperbolic-parabolic singular perturbations for Santos Kapp and Jos Ruidival dos Santos Filho, both Kirchhoff-equations from UFSCar(Brazil). Marina Ghisi ——— Department of Mathematics, University of Pisa, Largo

72 IV.2. Dispersive equations

Pontecorvo 5 Pisa, Pi 56127 Italy derive results for well-posedness with a (possible) loss [email protected] of regularity. On the other hand we discuss strategies how to show optimality of the results and sharpness of We consider the second order Cauchy problem the assumptions. Here Floquet theory and instability arguments form the core of our strategies. We distin- 00 0 1/2 2 0 u + g(t)u + m(|A u| )Au = 0, u(0) = u0, u (0) = u1 guish between optimality for the leading coefficients of the principal part and for coefficients of the remaining where > 0, g is a positive function, m is a nonprincipal part. negative C1 function, A is a self-adjoint non-negative operator with dense domain D(A) in a Hilbert space, ——— 1/2 and (u0, u1) ∈ D(A) × D(A ). Wave equations with time dependent coefficients We prove the global solvability of the Cauchy problem under different conditions on the functions m and g, in- Fumihiko Hirosawa cluding the case where m(0) = 0, and the case where Department of Mathematics, Yamaguchi University, g(t) tends to 0 as t tends to +infinity (weak dissipa- 753-8512, Japan tion). We also consider the behavior of solutions as t [email protected] tends to +infinity (decay estimates), and as tends to The total energy of the wave equation is conserved with 0. respect to time if the propagation speed is a constant, ——— but it is not true in general for time dependent propagation speeds. Indeed, it is considered in [F. Hirosawa, Existence and uniqueness results for Kirchhoff equa- Math. Ann. 339 (2007), 819-839] that the following tions in Gevrey-type spaces properties of the propagation speed are crucial for the Massimo Gobbino estimates of the total energy: oscillating speed, differ- m Dipartimento di Matematica Applicata, via Filippo ence from the mean, and the smoothness in C cate- Buonarroti 1c, Pisa, PI 56127 Italy gory. The main purpose of our talk is to derive a benefit [email protected] of a further smoothness of the propagation speed in the Gevrey class for the energy estimates. We consider the second order Cauchy problem ———

00 1/2 2 0 u + m(|A u| )Au = 0, u(0) = u0, u (0) = u1, Critical point theory applied to a class of systems of super-quadratic wave equations where m : [0, +∞) → [0, +∞) is a continuous function, Tacksun Jung and A is a self-adjoint nonnegative operator with dense Dept. of Mathematics San 68 Miryong Dong Kunsan domain on a Hilbert space. National University , Kunsan 573-701 South Korea In this conference we present three results. [email protected] • The first result is local existence for initial data in suitable spaces depending on the continuity mod- We show the existence of a nontrivial solution for a class ulus of the nonlinear term m. This spaces are a of the systems of the super-quadratic nonlinear wave natural generalization of Gevrey spaces to the ab- equations with Dirichlet boundary conditions and pe- stract setting. We also show that solutions with riodic conditions with super-quadratic nonlinear terms less regular data may exhibit an instantaneous at infinity which have continuous derivatives. We ap- derivative loss. proach the variational method and use the critical point theory which is the Linking Theorem for the strongly • The second result concerns uniqueness in the case indefinite corresponding functional. where the nonlinear term is not Lipschitz continu- This is joint work with Q-Heung Choi. ous. ——— • The last result concerns the global solvability. On the well-posdness of the vacuum Einstein equations Roughly speaking, we show that every initial datum in the spaces where local solutions exist is the Lavi Karp sum of two initial data for which the solution is P.O. Box 78 Karmiel, Galilee 21982 Israel actually global. [email protected]

——— The Cauchy problem of the vacuum Einstein’s equations determines a semi-metric g of a spacetime with van- Precise loss of derivatives for evolution type models αβ ishing Ricci curvature Rα,β and prescribe initial data. Torsten Herrmann under harmonic gauge condition, the equations Rα,β = 0 Faculty 1, TU Bergakademie Freiberg, Prüferstr. 9, are transferred into a system of quasi-linear wave equa- Freiberg, 09596 Germany tions which are called the reduced Einstein equations. [email protected] The initial data for Einstein’s equations are a proper Riemannian metric hab and a second fundamental form The goal of this talk is to present statements about Kab. However, these data must satisfy Einstein con- well-posedness for Cauchy problems for degenerate p- straint equations and therefore the pair (hab,Kab) can- evolution equations with time-dependent coefficients. not serve as initial data for the reduced Einstein equa- Degeneracy means that the p-evolution operators may tions. have characteristics of variable multiplicity. On the one Previous results in the case of asymptotically flat space- hand we are interested to apply phase space analysis to times provide a solution to the constraint equations in

73 IV.2. Dispersive equations one type of Sobolev spaces, while initial data for the with Dirichlet boundary conditions evolution equations belong to a different type of Sobolev N spaces. The aim of our work is to resolve this incompati- w(x, 0, t) = w(x, π, t) = 0, (x, t) ∈ R × (0, ∞). bility and to show well-posedness of the reduced Einstein −1 For long-range type of dissipations, e.g., b0(1 + |x|) ≤ vacuum equations in one type of Sobolev spaces. N b(x, y) ≤ b1 in R × [0, π] for some b0, b1 > 0, the to- ——— tal energy decays as t goes to infinity. For short-range −1−δ type of dissipations, e.g., 0 ≤ b(x, y) ≤ b2(1 + |x|) Generalized wave operator for a system of nonlinear N in R × [0, π] for some b2 > 0 and δ > 0, scatter- wave equations ing solution exists. Although the proof for scattering Hideo Kubo is based on Kato’s smooth perturbation theory, the sin- Graduate School of Information Sciences, Tohoku Uni- gular points called thresholds in the spectrum cause to versity 6-3-09 Aramaki-Aza-Aoba, Aoba-ku Sendai , difficulty. To eliminate this, density argument using Miyagi 980-8579 Japan some approximate operators are employed. This is joint [email protected] work with Mitsuteru Kadowaki (Ehime University) and Kazuo Watanabe (Gskushuin University). In this talk we discuss the asymptotic behavior of so- ——— lutions to a system of nonlinear wave equations whose decaying rate is actually slower than that of the free so- On the Cauchy problem for non-effectively hyperbolic lutions. Desipte of that fact, we are able to construct operators, the Gevrey 4 well-posedness wave operators in a generalized sense. The proof is done Tatsuo Nishitani by finding a nice approximation and introducing a suit- Machikaneyama-cho 1-1 Toyonaka, Osaka 560-0043 able metric (that is not a norm in fact). Moreover, the Japan sacttering operators are defined in a generarized sence. [email protected] ——— The Cauchy problem for non-effectively hyperbolic oper- Strichartz estimates for hyperbolic equations in an ex- ators is discussed in the Gevrey classes. Our operators terior domain belong to the class of non-effectively hyperbolic oper- Tokio Matsuyama ators with symbols vanishing of order 2 on a smooth Tokai University 1117 Kitakaname Hiratsuka, Kanagawa submanifold of codimension 3 on which the canonical 259-1292 Japan symplectic 2-form has a constant rank. Assuming that [email protected] there is no null bicharacteristic issuing from a simple characteristic point and landing tangentialy on the dou- In this talk we will present Strichartz estimates for ble characteristic manifold, we prove that the Cauchy higher oder hyperbolic equations in an exterior domain problem is Gevrey s well-posed for any lower order term outside a star-shaped obstacle. whenever 1 ≤ s < 4. ——— ——— Uniform resolvent estimates and smoothing effects for On the structure of the material law in a linear model magnetic Schrödingeroperators of poro-elasticity Kiyoshi Mochizuki Rainer Picard Department of Mathematics, Chuo University, Kasuga, Institut für Analysis, FB Mathematik,TU Dresden, Bunnkyo 1-13-27, Tokyo 112-8551 Japan 01062 Dresden, Germany [email protected] [email protected]

Uniform resolvent estimates for magnetic Schrödinger A modification of the material law associated with operators in an exterior domain are obtained under the well-known Biot system first investigated by smallness conditions on the magnetic fields and scalar R.E. Showalter is re-considered in the light of a new potentials. The results are then used to obtain space- approach to a comprehensive class of evolutionary prob- time L2-estimates for the corresponding Schrödinger, lems.The particular material law is of the form

Klein-Gordon and wave equations. ∗ ∗ T = (C + trace λ trace ∂0) E − trace α p ——— connecting the stress tensor T with strain tensor E and Decay and scattering for wave equations with dissipa- ﬂuid pressure p via parameters λ, α and C as the tions in layered media isotropic elasticity tensor. Here ∂0 denotes the time Hideo Nakazawa derivative and trace the matrix trace operation with ∗ Chiba Institute of Technology, Shibazono 2-1-1 trace as its adjoint. This model is generalized to Narashino, Chiba 275-0023 Japan anisotropic media and well-posedness of the generalized [email protected] model is shown. ——— We consider wave equations with linear dissipations in some layered regions; Backward uniqueness for the system of thermo-elastic waves with non-lipschitz continuous coeﬃcients wtt(x, y, t) − ∆w(x, y, t) + b(x, y)wt(x, y, t) = 0, N Marco Pivetta (x, y, t) ∈ R × [0, π] × (0, ∞) Dipartimento di Matematica e Informatica, Via Valerio

74 IV.2. Dispersive equations

12/1, Trieste, Italy 34127 Italy Japan [email protected] [email protected]

Using the Carleman estimates developed by Koch and We consider nonnegative solutions of the Cauchy prob- Lasiecka [Functional analysis and evolution equations, lem for quasilinear parabolic equations

389-403, Birkhäuser,Basel, 2008] together with an ap- m proximation technique in the phase space, a uniqueness ut = ∆u + f(u), result for the backward Cauchy problem is proved for the where m > 1 and f(ξ) is a positive function in ξ > 0 sat- system of themoelastic waves having coefficients which R ∞ isfying f(0) = 0 and a blow-up condition 1 1/f(ξ) dξ < are in a class of log-Lipschitz-continuous functions. ∞. We study under what conditions on f(ξ) all nontriv- ——— ial solutions blow up. The log-effect for 2 by 2 hyperbolic systems ——— Michael Reissig Blow-up and a blow-up boundary for a semilinear wave Faculty 1, TU Bergakademie Freiberg, Prüferstr. 9, equation with some convolution nonlinearity Freiberg, 09596 Germany Hiroshi Uesaka [email protected] Department of Mathematics, College of Science and Technology, Nihon University, Tokyo 101-8308, Chiyo- In the talk we are interested to explain how to ex- daku Kanda Surugadai 1-8, Japan tend the Log-effect from wave equations with time- [email protected] dependent coefficients to 2 by 2 strictly hyperbolic systems ∂ U − A(t)∂ U = 0. From wave models we know t x We consider the Cauchy problem with a convolution that besides oscillations in the coefficients a possible in- nonlinearity, teraction of oscillations has a strong influence on H∞  2 q p 3 well- or ill-posedness. Moreover, the precise loss of (∂t − 4)u = u (V ∗ u ), in R × (0,T ), 3 (0.1) derivatives can be proved. In the case of systems the u(x, 0) = f(x), ∂tu(x, 0) = g(x) in R , situation is more complicate. Besides the effects of os- q p q R up(y,t) cillating entries of the matrix A = A(t) and interactions where u (V ∗ u )(x, t) = u (x, t)( 3 γ dy) with R |x−y| between the entries of A we have to take into consid- p, q > 1, 0 < γ < 3. eration the system character itself. We will prove by The blow-up boundary is defined by Γ = ∂{u < ∞} ∩ using tools from phase space analysis results about H∞ {t > 0}. well- or ill-posedness. The precise loss of regularity is of We can give several suitable conditions to initial data to interest. Moreover, we discuss the question if the loss show that of derivatives does really appear. These considerations 1. the Cauchy problem has a classical positive real- base on the interplay between the Ljapunov and energy valued local solution u, functional. Finally, we discuss the cone of dependence property for solutions to 2 by 2 systems. 2. u is monotone increasing in t for any fixed x and This is a joint talk with T.Kinoshita (Tsukuba). moreover satisfies ∂tu ≥ |∇u|,

——— 3. there exists a positive T (x) for any x such The Boussinesq equations based on the hydrostatic ap- that u keeps its regularity in (0,T (x)) and proximation limt%T (x) u(x, t) = ∞ . Jun-ichi Saito Then the blow-up boundary Γ exists and T (x) satisﬁes Minamisenju 8-17-1 Arakawa-ku, Tokyo 116-0003 Japan |T (x) − T (y)| ≤ |x − y|. j [email protected] ———

The Boussinesq equations is studied in the field of dy- Fundamental solutions for hyperbolic operators with namic meteorology. Atmospheric flow in meteorology variable coefficients are described by the Boussinesq equations. Due to the Karen Yagdjian fact that the aspect ratio Department of Mathematics, University of Texas-Pan characteristic depth American, 1201 W. University Drive, Edinburg, TX ε = characteristic width 78541-2999, USA [email protected] is very small in most geophysical domains, asymptotic models have been used. One of the models is the hydro- The goal of this talk is to give a survey of a new approach static approximation of the Boussinesq equations. in the constructing of fundamental solutions for the We consider the Boussinesq equations in the domains partial differential operators with variable coefficients with very small aspect ratio and prove the convergence and of some resent results obtaining by that approach. theorem for this model. This new approach appeals neither to the Fourier trans- ——— form, nor to the Microlocal Analysis, nor to the WKB- approximation. More precisely, the new integral trans- Blow-up of solutions of a quasilinear parablolic equation formation is suggested which transforms the family of Ryuichi Suzuki the fundamental solutions of the Cauchy problem for the School of Science and Engineering, Kokushikan Uni- operator with the constant coefficients to the fundamen- versity, 4-28-1 Setagaya, Setagaya-ku Tokyo, 154-8515 tal solutions for the operators with variable coefficients.

75 IV.3. Control and optimisation of nonlinear evolutionary systems

The kernel of that transformation contains Gauss’s hy- cavities, control of turbulence), geophysics (reconstruc- pergeometric function. tion of seismic data) and others. This approach was applied by the author and his Recent years have witnessed rapid development of new coauthors, T.Kinoshita (University of Tsukuba) and mathematical tools in both analysis and geometry that A.Galstyan (University of Texas-Pan American), to in- allow to obtain various PDE estimates of inverse type. vestigate in the unified way several equations such as These are enabling to establish properties such as con- the linear and semilinear Tricomi and Tricomi-type trollability, reconstruction of the data, stabilisation or equations, Gellerstedt equation, the wave equation in optimal feedback control. Einstein-de Sitter spacetime, the wave and the Klein- Gordon equations in the de Sitter and anti-de Sitter —Abstracts— spacetimes. The listed equations play important role in the gas dynamics, elementary particle physics, quantum Global well-posedness and long-time behavior of solu- field theory in the curved spaces, and cosmology. In par- tions to a wave equation ticular, for all above mentioned equations, we have ob- Lorena Bociu tained representation formulas for the initial-value prob- University of Nebraska-Lincoln, Department of Mathe- lem, the L − L -estimates, local and global solutions p q matics, 203 Avery Hall Lincoln, NE 68588 United States for the semilinear equations, blow up phenomena, self- [email protected] similar solutions and number of other results. ——— The model under consideration is the semilinear wave equation with supercritical nonlinear sources and damp- Global existence in Sobolev spaces for a class of non- ing terms and the aim is to discuss the wellposedness linear Kirchhoff equations of the system on finite energy space and the long-time Borislav Yordanov behavior of solutions. A distinct feature of the equa- Borislav Yordanov, 226 Swain Ct, Belle Mead, NJ 08502- tion is the presence of the double interaction of source 4239 United States and damping, both in the interior of the domain and [email protected] on the boundary. Moreover, the nonlinear boundary sources are driven by Neumann boundary conditions. The nonlinear Kirchhoff equation Since Lopatinski condition fails to hold for dimension greater or equal than 2, the analysis of the nonlineari- 2 utt − m(k∇ukL2 )∆u = 0 ties supported on the boundary, within the framework of weak solutions, is a rather subtle issue and involves is studied for initial data (u, ut)t=0 = (u0, u1) in the s n s−1 n strong interaction between the source and the damping. Sobolev spaces H (R ) × H (R ) with s ≥ 2 and for I will provide positive answers to the questions of local smooth perturbations m(ρ) of the Pokhozhaev function −2 existence and uniqueness of weak solutions and moreover m0(ρ) = (k1ρ + k0) with k0, k1 > 0. Global exis- give complete and sharp description of parameters cor- tence is shown when ku1kL2 is large and m is close to responding to global existence and blow-up of solutions m0 in a suitable metric. Moreover, the asymptotic be- in finite time. havior of solutions is found as t → ±∞. It turns out I will also discuss asymptotic energy-decay rates and that the norms k∇ukL2 grow like |t|, so the propagation blow-up of solutions originating in a potential well. speeds decrease like t−2 and the waves remain trapped in bounded regions. ——— This is joint work with Lubin Vulkov. Distributed optimal controls for second kind parabolic ——— variational inequalities Mahdi Boukrouche LaMUSE Saint-Etienne University, 23 Rue Dr Paul IV.3. Control and optimisation of nonlinear Michelon, Saint-Etienne, 42023, France evolutionary systems [email protected]

Organisers: Let ugi be the unique solution of a second kind parabolic Francesca Bucci, Irena Lasiecka variational inequality with second member gi (i = 1, 2). We establish, in the general case, the error estimate

The session is focused on new developments in the area between u3(µ) = µug1 + (1 − µ)ug2 and u4(µ) = of well-posedness, optimisation, and control of systems uµg1+(1−µ)g2 for µ ∈ [0, 1], and prove a monotony prop- described by evolutionary partial differential equations. erty between u3(µ) and u4(µ) using a regularization These include: non-linear wave and plate equations, method. Navier-Stokes and Euler equations, non-linear thermoe- For a given constant M > 0, and the cost functional we lasticity, viscoelasticity and electromagnetism. Of par- establish the existence of solutions for a family of control ticular interest to the session are interacting systems problems, over the the external force g for each parame- that involve PDE’s of different type describing the dy- ter h > 0. Using the monotony property between u3(µ) namics on two (or more) separate regions with a cou- and u4(µ), we establish the uniqueness of the solution pling on an interface between these regions. Particular for each control problem of the above family. We prove examples of such systemsare: structural acoustic inter- also the convergence of the optimalcontrols and states actions, fluid structure interactions, magnetostructure associated to this family of control problems governed interactions. These have a wide range of applications by a second kind parabolic variational inequalities. that include medicine (diagnostic imaging such as MRI, ——— ultrasound), engineering (noise reduction in an acoustic

76 IV.3. Control and optimisation of nonlinear evolutionary systems

Controllability of a fluid-structure interaction problem In addition, we investigate the existence of uniform decay rates for both, the Airy type equation Muriel Boulakia 175 rue du Chevaleret, 75013 Paris, France ut + uxxx + g(u) = 0, in [0,L] × (0, +∞), [email protected] posed in a bounded interval [0,L] and supplemented by a nonlinear damping g(u). By considering suitable as- We are interested by the controllability of a fluid- sumptions on g and on the initial data, general decay structure interaction problem. The fluid in governed by rates are proved in L2− level as well as exponential de- the incompressible Navier-Stokes equations and a rigid cay rates are established in H1−level. structure is immersed inside. The control acts on a fixed subset of the fluid domain. For small initial data, we ——— prove that this system is null controllable i.e. that the Optimal control of waves in anisotropic media via con- system can be driven at rest. This result is obtained servative boundary conditions with the help of a Carleman inequality proven for the Matthias Eller adjoint linearized system. Department of Mathematics, Georgetown University, ——— Washington, DC 20057 United States [email protected] Uniform decay rate estimates for the wave equation on compact surfaces and locally distributed damping An optimal boundary control problem for symmetric hyperbolic systems is considered. The quadratic cost func- Marcello Cavalcanti tional is of tracking type and the control acts through a Department of Mathematics - State University of conservative boundary condition. Some loss of regularity Maringa, Av. Colombo 5790, Maringa, PR 87020-900 is associated with these boundary conditions. This re- Brazil sults in certain choices for the underlying function spaces [email protected] in the cost functional. The loss of regularity occurs only near the boundary and it may be attributed to the oc- In this talk we present new contributions concerning uni- currence of surface waves. As examples we consider the form decay rates of the energy associated with the wave anisotropic Maxwell equations as well as the anisotropic equation on compact surfaces subject to a dissipation equations of elasticity. locally distributed. We present a method that gives us a sharp result in what concerns of reducing arbitrarily ——— the area where the dissipative effect lies. Stability for some nonlinear damped wave equations ——— Genni Fragnelli Dipartimento di Ingegneria dell’Informazione, Univer- Rate of decay for non-autonomous damped wave sys- sitàdegli Studi di Siena, via Roma 56, c.a.p. 53100 tems [email protected] Moez Daoulatli ISSATS, University of Sousse (& LAMSIN) CitéTaffala We prove stability results for a large class of abstract (Ibn Khaldoun), Sousse 4003 Tunisia nonlinear damped wave equations, whose prototype is [email protected] the usual wave equation 8 < utt + h(t)ut = ∆u + f(u) in (0, +∞) × Ω, We study the rate of decay of solutions of the wave sys- u(t, x) = 0 in (0, +∞) × ∂Ω, tems with time dependent nonlinear damping which is : u(0, x) = u0(x), ut(0, x) = u1(x) x ∈ Ω, localized on a subset of the domain. We prove that the where Ω is a bounded and smooth domain of N , N ≥ 1, asymptotic decay rates of the energy functional are ob- R u ∈ H1(Ω), u ∈ L2(Ω) and f : → . tained by solving nonlinear non-autonomous ODE. 0 0 1 R R At first, the damping is nonnegative, but it is allowed ——— to be zero either on negligible sets or even in a sequence of intervals. Then, also the case of a positive–negative On qualitative aspects for the damped Korteweg-de damping is treated. Vries and Airy type equations ——— Valeria Domingos Cavalcanti Department of Mathematics - State University of Global existence for the one-dimensional semilinear Maringa, Av. Colombo 5790, Maringa, PR 87020-900 Tricomi-type equation Brazil Anahit Galstyan [email protected] Department of Mathematics, University of Texas-Pan American, 1201 W. University Dr., Edinburg 78541, TX, This talk is concerned with the study of the damped U.S.A. Korteweg-de Vries equation posed in whole real line [email protected] ut + uxxx + uux + λ u = 0, in R × [0, +∞), λ > 0. In this talk the issue of global existence of the solutions of the Cauchy problem for one-dimensional semilinear 1 We establish two invariant subsets of H (R) where just weakly hyperbolic equations, appearing in the boundary one of the following statements holds: (i) solutions de- value problems of gas dynamics is investigated. We solve cay exponentially in H1− level or (ii) solutions do not the Cauchy problem trough integral equation and give decay to zero in H1− level as t goes to infinity. some sufficient conditions for the existence of the global

77 IV.3. Control and optimisation of nonlinear evolutionary systems weak solutions. The necessary condition for the exis- Russia tence of the similarity solutions for the one-dimensional [email protected] semilinear Tricomi-type equation will be presented as well. Our approach is based on the fundamental solu- A number of applied studies address such fundamental tion of the operator and the Lp − Lq estimates for the issues as linear Tricomi equation. (i) reconstruction of uncertain parameters of multi- ——— dimensional dynamical systems and Optimal control of a thermoelastic structural acoustic (ii) control of uncertain dynamical systems. model We discuss a technical approach intended to help solve Catherine Lebiedzik problems of this kind. The approach employs the on-line Department of Mathematics, Wayne State University, inversion theory adjoining theory of closed-loop control 656 W Kirby Detroit, MI 48202 United States and theory of ill-posed problems. On-line inversion al- [email protected] gorithms involve artificially designed dynamical models whose parameters track non-observable parameters of We consider point control of a structural acoustic model the system; it is important that the tracking quality is with thermoelastic effects. The key feature of this pa- insensitive to perturbations in the observation channels. per is that the two-dimensional plate modeling the active In combination with appropriate closed-loop regulators, wall of the acoustic chamber has clamped boundary con- on-line parameter tracking algorithms give raise to ro- ditions. For this case a new optimal regularity result has bust observer-controller patterns allowing one to guide recently become available. Using this new result for the the uncertain system close to the trajectories designed plate alone, we derive a sharp regularity result for the via an optimal feedback to a complete set of observed overall coupled system of wave and thermoelastic plate signals. The goal of this report is to demonstrate the equations. This allows for the study of optimal control essence and abilities of the on-line inversion technique; of the coupled system. for this purpose we consider three types of problems, namely, a problem of etalon motion tracking, a prob- ——— lem of game control, and a problem of dynamical input The Balayage method: Boundary control of a thermo- identification for a parabolic equation with the Neumann elastic plate and Dirichlet boundary condition. Walter Littman ——— University of Minnesota, School of Mathematics, 206 Null controllability properties of some degenerate Church Street, Southeast Minneapolis, Minnesota 55455 parabolic equations United States [email protected] Patrick Martinez Université Paul Sabatier Toulouse III, Institut de Mathématiques, 118 route de Narbonne, Toulouse, We discuss the null boundary controllablity of a linear 31062 France thermo-elastic plate. The method employs a smooth- [email protected] ing property of the system of PDEs which allows the boundary controls to be calculated directly by solving Motivated by several problems in fluid dynamics, biol- two Cauchy problems. ogy, or economics, we are interested in controllability ——— properties of parabolic equations degenerating at the boundary of the space domain. After considering the Hopf-Lax type formulas and Hamilton-Jacobi equations one dimensional case, this talk will mainly focus on the N-dimensional case: Paola Loreti u − div(A(x)∇u) = f(x, t)χ (x), x ∈ Ω, t > 0 Dipartimento di Metodi e Modelli Matematici per le t ω Scienze Applicate, via Antonio Scarpa n.16, 00161 where ω ⊂ Ω and the matrix A(x) is definite positive for Roma, Italia. all x ∈ Ω, and but has at least one eigenvalue equal to [email protected] 0 for all x ∈ ∂Ω. Mainly, we assume that - the least eigenvalue of the matrix A(x) behaves as Here we discuss Hopf-Lax type formulas related to the d(x, ∂Ω)α, where α ≥ 0, class of Hamilton-Jacobi equations - the degeneracy occurs in the normal direction: when x ∈ ∂Ω, the associated eigenvector is the unit outward ut(x, t) + αxDu(x, t) + H(Du(x, t)) = 0, vector. N When α ∈ [0, 2), we prove the null controllability via new in R × (0, +∞) with initial condition u(x, 0) = u0 in n Carleman estimates for the adjoint degenerate parabolic R , with α a positive, real number. The talk is based on some joint works with A. Avantag- equation. When α ∈ [2, +∞), we prove that the prob- giati. lem is not null controllable, using earlier results of Micu - Zuazua, Escauriaza-Seregin-Sverákˇ related to nondegen- ——— erate parabolic equations in unbounded domains. Investigation of boundary control problems by on-line These results were obtained in collaboration with P. inversion technique Cannarsa (Univ Tor Vergata, Roma 2), and J. Van- costenoble (Univ Toulouse 3). Vyacheslav Maksimov ——— S.Kovalevskaya 16 Ekaterinburg, Sverdlovsk 620219

78 IV.3. Control and optimisation of nonlinear evolutionary systems

Dissipation in contact problems: an overview and some and on its boundary combined with a nonlinear cou- recent results pled boundary condition. Such problems arise from free boundary value problems as the Stefan problem with Maria Grazia Naso surface tension after a suitable transformation. Besides Dipartimento di Matematica, Universitàdegli Studi di local well posedness and smoothing properties, we focus Brescia, Via Valotti, 9 Brescia, BS 25133 Italy on the qualitative behavior near an equilibrium. To that [email protected] purpose we construct locally invariant manifolds and establish their main properties. In this talk we investigate the longtime behaviour of a dynamic unilateral contact problem between two ther- ——— moelastic beams. Under suitable mechanical and ther- On regularity properties of optimal control and La- mal boundary conditions the evolution problem is shown grange multipliers to possess an energy decaying exponentially to zero, as Ilya Shvartsman time goes to infinity. Dept. of Mathematics and Computer Science, 777 W. ——— Harrisburg Pike Middletown, PA 17110 United States [email protected] Heat equations with memory: a Riesz basis approach In this talk we will go over classical and recent results Luciano Pandolfi on regularity properties (such as continuity, Holder and Politecnico di Torino, Dipartimento di Matematica, C.so Lipschitz continuity) of optimal controls and Lagrange Duca degli Abruzzi 24, Torino, 10129 Italy multipliers. [email protected] ——— In this talk we present recent results concerning a Riesz Evolution equations with memory terms basis approach to the heat equation with memory Daniela Sforza Z t Dipartimento di Metodi e Modelli Matematici per le θt(x, t) = N(t − s) [∆θ(x, s) − q(x)θ(x, s)] ds Scienze Applicate, via Antonio Scarpa 16, Roma, 00161 0 Italy (x ∈ [0, π]) and square integrable initial conditions. [email protected] We shall construct a special sequence {zn(t)} associated to this equations and we shall prove that it is a The purpose of the talk is to show some results concern- Riesz sequence on a suitable interval [0,T ], using Bari ing control problems for integro-differential equations of Theroem. These results are applied to the study of con- hyperbolic type. trol/observability problems. More precisely, we consider non-linear equations in Hilbert spaces with integral convolution terms and as- ——— sume the corresponding kernels to exhibit a polynomial A note on a class of observability problems for PDEs or exponential decay. We show that the solutions have the same decay behaviour as the kernel. Our main tool Michael Renardy is the multipliers method and we succeed in finding suit- Department of Mathematics, Virginia Tech, Blacksburg, able multipliers which work even in the presence of in- VA 24061-0123 United States tegral terms. [email protected] Besides, we provide a reachability result for a class of linear integro-differential problems. Our strategy is The question of observability arises naturally in the anal- founded on the so-called Reachability Hilbert Unique- ysis of control problems. If the solution of a PDE initial- ness Method, introduced by Lagnese - Lions, which boundary value problem is known to be zero in a part of amounts to proving Ingham type inequalities for the the domain, does this guarantee it is zero everywhere? Fourier series expansion of the solution of the adjoint The most popular techniques to establish such results problem. are based on local unique continuation results (Holm- To conclude, we observe that our abstract results may gren’s theorem) or Carleman estimates. The lecture will be used to treat some problems arising in the study of draw attention to a class of problems where the observed viscoelastic systems. region is bounded by characteristics, and local unique ——— continuation fails. Nevertheless, observability may hold. A problem of this nature arose in recent work by the Stabilization of structure-acoustics interactions for a author on control of viscoelastic flows. Reissner-Mindlin plate by localized nonlinear boundary feedbacks ——— Daniel Toundykov Invariant manifolds for parabolic problems with dynam- University of Nebraska-Lincoln Department of Mathe- ical boundary conditions matics, 203 Avery Hall Lincoln, NE 68588 United States [email protected] Roland Schnaubelt University of Karlsruhe, Department of Mathematics, This work addresses observability and energy decay for a Kaiserstrasse 89, Karlsruhe, 76128 Germany structural-acoustics model comprised of a wave equation [email protected] coupled with a Reissner-Mindlin plate. Both components of the dynamics are subject to localized boundary We study a class of nonlinear parabolic systems de- damping: the acoustic dissipative feedback is restricted scribed by coupled evolution equations on a domain to the flexible boundary and only a portion of the rigid

79 IV.4. Nonlinear partial differential equations wall; the plate is likewise damped on a segment of its For some types of parabolic systems, we consider in- boundary. equalities of Carleman’s type and prove conditional sta- The derivation of the “coupled” stabiliza- bility estimates for state determination problems such tion/observability inequalities requires weighted energy as a backward problem. multipliers related to the geometry of the domain, and special tangential trace estimates for the displacement ——— and the filament angles of the Reissner-Mindlin plate Euler flow and Morphing Shape Metric model. The behavior of the energy at infinity can be quantified by a solution to an explicitly constructed Jean-Paul Zolesio nonlinear ODE. The nonlinearities in the feedbacks may CNRS-INLN nd INRIA RTE Lucioles 1361 and 2004 include sub- and super-linear growth at infinity, in which Sophia Antipolis, France 06565 France case the decay scheme presents a trade-off between the [email protected] regularity of solutions and attainable uniform decay rates of the finite-energy. We extend the so-called ”Courant metric” into a new ”Tube metric” beetwen measurable sets and character- ——— ize the geodesic as variational solution to incompress- Exponential stability of the wave equation with bound- ible Euler flow. Such geodesic can modelise topological ary time varying delay changes. We make use of a new ”Sobolev perimeter”, and Sobolev Mean curvature for the moving boundary Julie Valein which turns to be Shape differentiable under smooth Université de Valenciennes et du Hainaut-Cambrésis transverse perturbation. Then working with L2 speed LAMAV - ISTV2 - Le Mont-Houy Valenciennes, Nord- vector fields (we don’t use any renormalization bene- Pas de Calais 59313 France fit) we succed in the existence of connecting tubes and [email protected] in variational solution to the usual incompressible Eu- ler flow under surface tension associated to the Sobolev We consider the wave equation with a time-varying de- perimeter. This technic applies to several situations lay term in the boundary condition in a bounded domain in [Shape-Morphing Metric by Variational Formulation n 2 Ω ⊂ R with a boundary Γ of class C . for Incompressible Euler Flow.J. of Control and Cy- We assume Γ = ΓD ∪ ΓN , with ΓD ∩ ΓN = ∅,ΓD 6= ∅, bernetics, vol 38 (2009), No. 4], [Control of Moving and we consider Domains...,. Int.Ser.Num.Math., vol. 155, 329-382, 8 u (x, t) − ∆u(x, t) = 0 in Ω × (0, +∞) Birkhauser Verlag Basel,2007]. > tt > u(x, t) = 0 on Γ × (0, +∞) > D ——— > ∂u (x, t) = −µ u (x, t) − µ u (x, t − τ(t)) <> ∂ν 1 t 2 t on ΓN × (0, +∞) > > u(x, 0) = u0(x) and ut(x, 0) = u1(x) in Ω > IV.4. Nonlinear partial differential equations > ut(x, t − τ(0)) = f0(x, t − τ(0)) :> in ΓN × (0, τ(0)), Organisers: (*) Vladimir Georgiev, Tohru Ozawa where τ(t) is the delay, µ1, µ2 > 0. We assume The Session intends to discuss various nonlinear partial differential equations in mathematical physics. 0 ≤ τ(t) ≤ τ, τ 0(t) ≤ d < 1, Among possible arguments the following ones shall be ∀ t > 0 and τ ∈ W 2,∞([0,T ]), ∀ T > 0. discussed: existence and qualitative properties of the solutions, existence of wave operators and scattering for Under √ these problems, stability of solitary waves and other spe- µ2 < 1 − dµ1, cial solutions. we prove the existence and uniqueness results of (*) by using the variable norm technique of Kato and we show —Abstracts— the exponential decay of an appropriate energy. Due to the time-dependence of the delay, we can not use an Evolution equations in nonflat waveguides observability estimate since the system is not invariant by translation in time. Hence we introduce a Lyapunov Piero D’Ancona Sapienza - Universitàdi Roma - Dipartimento di Matem- functional. atica P. Moro, 2 Roma, RM 00185 Italy We extend this result to a nonlinear version of the model. This is a joint work with Serge Nicaise and Cristina Pig- [email protected] notti. In a joint work with Reinhard Racke (Konstanz) we ——— prove smoothing and Strichartz estimates for evolution State estimation for some parabolic systems equations of Schroedinger or wave type on waveguides which are deformations, in a suitable sense, of flat waveg- k m Masahiro Yamamoto uides of the form O × R , O a bounded open set in R . University of Tokyo, Department of Mathematical Sci- For the proof, new weighted estimates for fractional pow- ences, 3-8-1 Komaba Meguro Tokyo 153, Japan ers of Schroedinger operators are required. [email protected] ———

80 IV.4. Nonlinear partial diﬀerential equations

Investigation of solutions of one not divergent type Miyagi 980-8579 Japan [email protected] Mersaid Aripov Mech.,Math, National University of Uzbekistan, Univer- We consider nonlinear gauge invariant evolution of the sitet 1, Tashkent, Tashkent 100174 Uzbekistan plane wave. In this talk, we deal with the power and log- [email protected] arithmic type nonlinearities. Although the plane wave does not decay at infinity, by an elementary and simple The properties of the weak solution of problem Cauchy argument we find an extremely smooth solution which and the first boundary value problem for one parabolic has an explicit expression. Additionally, we study the equation of not divergent type double nonlinearity and global behavior of the solution from its representation. with lower members are investigated. The researched equation is the best combination of forms of the equa- ——— tion of nonlinear diffusion, fast diffusion, the equation New approach to solve linear parabolic problems via to very fast diffusion and p-Laplace heat conductivity semigroup approximation equation. This equation describes various processes of nonlinear diffusion, heat conductivity, a filtration, mag- Mohammad Dehghan netic rheology, etc. Ferdowsi University of Mashhhad Azadi Square Mash- The method of investigation of the qualitative proper- had, Khorasan-e-Razavi 9177948974 Iran ties having physical sense weak solution on the basis [email protected] of a method of a nonlinear splitting and a method of We consider Linear Parabolic Problems (LPPs) whose the standard equations is offered. Two side estima- solutions can be expressed via semigroups. Computing tions of the solutions and free boundary, a condition the solutions of these LPPs depends on existing explicit of existence of global solutions (including case of crit- formulas for the corresponding semigroups. However, ical value of parameter and exponent) generalizing of in general explicit formulas are not available. The pro- known results of H. Fujite, A.A., Samarskii, S.P. Kur- posed approach defines a sequence of linear problems dyumov, A.P. Mikhajlov, V.A. Galaktionov, H.Vaskes, which are semidiscrete approximations of the consid- S. A. Posashkov are received. On the basis of the analy- ered LPP. The solutions of approximant linear problems sis of properties of solutions the numerical modeling and can be expressed via corresponding semigroups which visualization of solutions carried out. have explicit formulas. These solutions converge uni- ——— formly to the solution of LPP. So the corresponding semigroup of LPP can be approximated by semigroups Asymptotic behavior of subparabolic functions which have explicit formulas. The approximant linear Davide Catania problems are defined on the finite dimensional subspaces Universitàdi Brescia - Dip. Matematica Via Valotti, n. of the LPP solution space, via a hybrid finite-difference- 9 Brescia, BS 25133 Italy projection method. The accuracy of approximations, [email protected] order of convergency and their relations to the proposed hybrid method are discussed and some examples are pre- We consider an MHD-α model with regularized veloc- sented. ity for an incompressible fluid in two space dimension. ——— Such a model is introduced in analogy with the Navier– Stokes equation to study the turbulent behavior of flu- Global existence and blow-up for the nonlocal nonlinear ids in presence of a magnetic field, since this problem Cauchy problem is otherwise difficult to study, both analitically and nu- Albert Erkip merically. Sabanci University, Faculty of Engineering and Natural We prove local and global existence for the related Sciences, Orhanli, Tuzla Istanbul / 3495 Turkey Cauchy problem, where the velocity field is viscous, [email protected] while we have not any magnetic diffusivity. We study the Cauchy problem ——— u = (β ∗ (u + g (u))) x ∈ , t > 0 On multiple solutions of concave and convex effects for tt xx R N nonlinear elliptic equation on R u (x, 0) = φ (x) , ut (x, 0) = ψ (x) x ∈ R, Kuan-Ju Chen for a general class of nonlinear nonlocal wave equations Department of Applied Science, Naval Academy, arising in one-dimensional nonlocal elasticity. P.O.BOX 90175 Zuoying, Taiwan, R.O.C. The model involves a convolution integral operator with [email protected] a general kernel function β whose Fourier transform is nonnegative. We show that some well-known examples In this paper we consider the existence of multiple so- of nonlinear wave equations, such as Boussinesq-type N equations, follow from the present model for suitable lutions of the elliptic equation on R with concave and convex nonlinearities. choices of the kernel function. We establish global existence of solutions of the model ——— assuming enough smoothness on the initial data together Nonlinear gauge invariant evolution of the plane wave with some positivity conditions on the nonlinear term. Furthermore, conditions for finite time blow-up are pro- Kazuyuki Doi vided. Graduate School of Information Sciences, Tohoku Uni- ——— versity, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai,

81 IV.4. Nonlinear partial diﬀerential equations

Qualitative properties for reaction-diﬀusion systems The outgoing wave is a solution to the free wave equation modelling chemical reactions with some asymptotics states as initial data. We introduce corresponding nonlinear scattering operator, and Marius Ghergu obtain a necessary condition for the asymptotic states. School of Mathematical Sciences University College Also we prove the asymptotics completeness in the Lamb Dublin Belﬁeld, , Dublin 4 Dublin Ireland system. [email protected] This is joint work with A.E. Merzon and A.I. Komech.

In 1952 the British mathematician Alan M. Turing pub- ——— lished the foundation of reaction-diffusion theory for Global existence for systems of the nonlinear wave and morphogenesis, the development of form and shape in bi- Klein-Gordon equations in 3D ological systems. Since then, many Turing-type models Soichiro Katayama described by coupled reaction-diffusion equations have Department of Mathematics, Wakayama University, 930 been proposed for generating patterns in both organic Sakaedani Wakayama, Wakayama 640-8510 Japan and inorganic systems. [email protected] In this talk we present a qualitative study for reaction- diffusion systems of the type We consider the Cauchy problem for coupled systems of the nonlinear wave and Klein-Gordon equations in three space dimensions. We present a sufficient condition for u − d ∆u = a + bu + f(u)v in Ω × (0, ∞), t 1 global existence of small amplitude solutions to such sys- vt − d2∆v = c + du − f(u)v in Ω × (0, ∞), tems. u(x, 0) = u0(x), v(x, 0) = v0(x) on Ω, Our condition is much weaker than the strong null con- ∂u ∂u dition for this kind of coupled system, and our result (x, t) = (x, t) = 0 on ∂Ω × (0, ∞). ∂ν ∂ν is a natural extension of the global existence theorem for the nonlinear wave equations under the null con- Here Ω ⊂ RN (N ≥ 1) is a bounded domain, dition, as well as that for the Klein-Gordon equations a, b, c, d, d1, d2 ∈ R, u0, v0 ∈ C(Ω) are non-negative and with quadratic nonlinearities. Our result is applicable f ∈ C[0, ∞) ∩ C1(0, ∞) is a non-negative and nonde- to a certain kind of model equation in physics, such creasing such that f(0) = 0 and f > 0 in (0, ∞). as the Klein-Gordon-Dirac equations, the Klein-Gordon- The system encompasses two well known chemical mod- Zakharov equations, and the Dirac-Proca equations. els: the Brusselator and the Schnackenberg models ——— which are a rich source of varied spatio-temporal patterns. Global existence for nonlinear wave equations exterior We present several existence and stability results. A par- to an obstacle in 2D ticular attention is paid to the associated steady-state Hideo Kubo system where the crucial role played by the diffusion co- Graduate School of Information Sciences, Tohoku Uni- efficients d1, d2 and the behavior of the nonlinearity f is versity 6-3-09 Aramaki-Aza-Aoba, Aoba-ku Sendai , emphasized. Miyagi 980-8579 Japan The proofs rely on a-priori estimates combined with an- [email protected] alytical and topological methods. In this talk we discuss the global existence for the exte- ——— rior problem of nonlinear wave equations in two space di- Scattering in the zero-mass Lamb system mensions. The obstacle is assumed to be a star-shaped, so that the decay of the local energy is available. The Marco Antonio Taneco-Hernandez´ main difficulty compared with the three space dimen- Instituto de F´ısicay Matemáticas, Universidad Michoa- sional case is the weaker decay of solutions in 2D, as cana de San Nicolásde Hidalgo, Edificio C-3, Ciudad well as the lack of the sharp Huygens principle. How- Universitaria Av., Francisco J. Mujica s/n, Colonia Fe- ever, we are able to show the global existence for small licitas del Rio Morelia, Michoacán58040, Mexico initial data, provided the nonlinearity is of the cubic or- [email protected] der and fulfills the so-called null condition.

We consider nonlinear conservative Lamb system, which ——— is the wave equation coupled with a particle of zero mass: Remark on Navier-Stokes equations with mixed boundary conditions u¨(x, t) = u00(x, t), 0 0 Petr Kucera F (y(t)) + u (0+, t) − u (0−, t), y(t) = u(0, t), Czech Technical University, Fac. of Civil Engineering, ∂u 0 ∂u Dept. of Math., Thakurova 7, Prague 166 29 Czech Re- with x ∈ R \{0}, t ∈ R. Hereu ˙ := ∂t , u := ∂x d public and so on. The solutions u(x, t) take the values in R d [email protected] with d ≥ 1 and F := −∇V with V : R → R is a potential force field. For the first time we establish long We solve a system of the Navier-Stokes equations for in- time asymptotics in global energy norm for all finite en- compressible heat conducting fluid with mixed bound- ergy solutions. Namely, under some Ginzburg-Landau ary conditions (of the Dirichlet or non-Dirichlet type on type conditions to V , each solution from some functional different parts of the boundary). We suppose that the space decays to a sum of a stationary state, outgoing viscosity of the fluid depends on temperarure. wave and the rest which tends to zero in global energy norm as t → +∞. ———

82 IV.4. Nonlinear partial diﬀerential equations

Contraction-Galerkin method for a semi-linear wave A symmetric error estimate for Galerkin approximations equation with a boundary-like antiperiodic condition of time dependant Navier-Stokes equations in two dimensions Ut van Le Department of Mathematical Sciences, P.O. Box 3000, Itir Mogultay Oulu FI-90014 Finland Department of Mathematics, Yeditepe University, 26 [email protected] Agustos Yerlesimi Kayisdagi Caddesi Kayisdagi Istan- bul, 81120 Turkey We consider the unique solvability of initial-boundary [email protected] value problems of semi-linear wave equations with spacetime dependent coeﬃcients and special mixed non- A symmetric error estimate for Galerkin approximation homogeneous boundary values which make the so-called of solutions of the Navier-Stokes equations in two space boundary-like antiperiodic condition. The procedure in dimensions plus time is given. The ﬁnite dimensional this project is the combination of the Galerkin method function spaces are taken to be divergence free, and time and a contraction. is left continuous. The estimate is similar to known results for scalar parabolic equations. An application of ——— the result is given for mixed method formulations. A p − q systems of nonlinear Schrodinger equations short discussion of examples is included. Finally, there are some remarks about a partial expansion to three Sandra Lucente space dimensions. Dipartimento di Matematica, Via Orabona 4, Bari Note: This is a joint work with Prof. Todd F. Dupont 70124, Italy at the University of Chicago. [email protected] ———

In a joint work with L. Fanelli and E. Montefusco, we Stability of standing waves for some systems of nonlin- consider coupled nonlinear Schrödingerequations ear Schrödingerequations with three-wave interactions Masahito Ohta iut + ∆u ± N1(u, v) = 0, Department of Mathematics, Saitama University, 255 ivt + ∆v ± N2(u, v) = 0, Shimo-Ohkubo, Saitama, 338-8570 Japan with suitable semilinear terms N1(u, v) and N2(u, v) [email protected] having polynomial growth. We investigate on local and global existence critical exponents and describe the cor- We discuss orbital stability and instability of several responding solutions. types of standing waves for some three-component systems of nonlinear Schrödingerequations. ——— ——— Semiclassical analysis for nonlinear Schrodinger equations Decay rates for wave models with structural damping Satoshi Masaki Michael Reissig 6-3-09 Aza-aoba Aramki Aoba-ku Sendai, Miyagi 980- Faculty 1, TU Bergakademie Freiberg, Prüferstr. 9, 8579 Japan Freiberg, 09596 Germany [email protected] [email protected]

In this talk, we will present results on the behavior We consider the semiclassical limit of the nonlinear of higher order energies of solutions to the following Schrodinger equations. We approximate the solution by Cauchy problem for a wave model with structural damp- a function of phase-amplitude form, called WKB analy- ing: sis. We mainly treat the nonlocal nonlinearites. σ ——— utt − ∆u + b(t)(−∆) ut = 0, u(0, x) = u0(x), ut(0, x) = u1(x), 3-D viscous Cahn-Hilliard equation with memory σ ∈ (0, 1], b(t) = µ(1 + t)δ, µ > 0, δ ∈ [−1, 1]. Gianluca Mola Universitàdi Milano, Dipartimento di Matamatica, via We are interested in the influence of the structural dis- Saldini 50 Milano, MI 20133 Italy sipation (between external and visco-elastic damping) σ 2 2 [email protected] b(t)(−∆) ut on L − L estimates. Our main goal is to study under which conditions do we We deal with the memory relaxation of the viscous have a parabolic effect for the solutions, that is, the decay Cahn-Hilliard equation in 3-D, covering the well–known rates depend on the order of energy. hyperbolic version of the model. We study the longterm In the talk we will explain how hyperbolic or elliptic dynamic of the system in dependence of the scaling pa- WKB analysis comes in. The main tools are a correct rameter of the memory kernel ε and of the viscosity co- division of the extended phase space into zones, diago- efficient δ. In particular we construct a family of expo- nalization procedures, construction of fundamental so- nential attractors which is robust as both ε and δ go to lutions and a gluing procedure. Some open problems zero, provided that ε is linearly controlled by δ. complete the talk. This is joint work with Xiaojun Lu (Hangzhou). ——— ———

83 IV.4. Nonlinear partial diﬀerential equations

Stability theorems in the theory of mathematical fluid towards the stable manifold etc. Several outstanding mechanics open questions will be discussed as well. Yoshihiro Shibata ——— Department of Mathematics, Waseda University, A regularity result for a class of semilinear hyperbolic Ohkubo 3-4-1 Shinjuku-ku Tokyo, Tokyo 169-8555 equations Japan [email protected] Sergio Spagnolo Department of Mathematics, University of Pisa, Largo I would like to talk about some stability theorem of sta- Pontecorvo 5 Pisa, 56127 Italy tionary solutions of incompressible fluid flow with initial [email protected] disturbance. We first recall a former result of global wellposedness in ——— C-infinity (resp., in each Gevrey class) for a special kind On singular systems of parabolic functional equations of homogeneous, linear hyperbolic equations with analytic (resp., C-infinity) coefficients depending only on Laszl´ o´ Simon time. Then, we add to these equations an analytic semi- Pázmány P. sétány 1/C, L. EötvösUniversity, Institute linear term, and we prove that the resulting equations of Mathematics Budapest, Hungary H-1117 Hungary enjoy the following regularity property; each solution [email protected] which is real analytic at the initial time together with all its time derivatives, remains analytic as long as it is We shall consider initial-boundary value problems for bounded in C-infinity (resp., in some Gevrey class). a system consisting of a quasilinear parabolic functional equation and an ordinary differential equation with func- ——— tional terms. The parabolic equation may contain the On nonlinear equations, fixed-point theorems and their gradient with respect to the space variable of the un- applications known function in the ODE. It will be proved global existence of weak solutions, by using the theory of mono- Kamal Soltanov tone type operators and Schauder’s fixed point theo- Department of Mathematics, Faculty of Sciences, rem. Such problems are motivated by models describ- Hacettepe University, Beytepe Campus Ankara, ing reaction-mineralogy-porosity changes in porous me- Cankaya TR-06532 Turkey dia and polymer diffusion. [email protected]

——— In this work we investigated some class of the nonlinear Survey of recent results on asymptotic energy concen- operators and a nonlinear equations with such type op- tration in solutions of the Navier-Stokes equations erators in a Banach spaces. Here we obtained some new results on the solvability of the nonlinear equations, and Zdenek Skalak also a fixed-point theorems for continuous mappings. Thakurova 7, Czech Technical University, Prague, 16629 With use of the obtained here results we studied various Czech Republic boundary value problems (BVP) (and mixed problems) [email protected] for the different nonlinear differential equations. ——— We present some recent results on asymptotic energy concentration in solutions of the Navier-Stokes equa- Dynamics of a quantum particle in a cloud chamber tions. For example, if w is such a solution satisfying the strong energy inequality then there exists a ≥ 0 such Alessandro Teta that Dipartimento di matematica pura e applicata, Univer- sita’ di L’Aquila via Vetoio - loc. Coppito L’Aquila, lim ||Eλw(t)||/||w(t)|| = 1 t→∞ Abruzzo 67100 Italy [email protected] for every λ > a, where {Eλ; λ ≥ 0} denotes the resolution of identity of the Stokes operator. We consider the Schroedinger equation for a system com- ——— posed by a particle (the α-particle) interacting with two other particles (the atoms) subject to attractive poten- Conditional stability theorems for Klein-Gordon type 3 equations tials centered in a1, a2 ∈ R . At time zero the α-particle is described by a diverging spherical wave centered in the Atanas Stefanov origin and the atoms are in their ground state. The aim 1460, Jayhawk Blvd., Department of Mathematics, Uni- is to show that, under suitable assumptions on the phys- versity of Kansas, Lawrence, KS 66049, USA ical parameters of the system and up to second order in [email protected] perturbation theory, the probability that both atoms are ionized is negligible unless a2 lies on the line joining the We consider unstable ground state solutions of the Klein- origin with a1. Gordon equation with various power nonlinearities. The The work (in collaboration with G. Dell’Antonio and R. main result is a fairly precise construction of a stable Figari) is a fully time-dependent version of the original manifold in a close vicinity of the ground state. In par- analysis performed by Mott in 1929. ticular, we provide an asymptotic formula for the asymp- ——— totic phase, an estimate of the rate of the convergence

84 IV.5. Asymptotic and multiscale analysis

Half space problem for the damped wave equation with and nonlinear, partial or ordinary) with a small param- a non-convex convection term eter and/or multiple scales, and relevant applications. This includes singularly perturbed problems, problems Yoshihiro Ueda in thin domain or with singular boundaries, homoge- Graduate School of Sciences, Tohoku University 6-3 nization. The applications may include propagation and Aramaki-Aza-Aoba, Aoba-ku Sendai, Miyagi 980-8578 localization of waves, blow-up phenomena, metamateri- Japan als, etc. The relevant analytic issues are convergence [email protected] and relevant functional spaces, compactness and propagation of oscillations, asymptotic expansions with error We consider the initial-boundary value problem for bounds, etc. damped wave equations with a nonlinear convection term in the half space. In the case where the flux is —Abstracts— convex, it had already known that the solution tends to the corresponding stationary wave. In this talk, we show On the essential spectrum and singularities of solutions that even for a quite wide class of flux functions which for Laméproblem in cuspoidal domain are not necessarily convex, such the stationary wave is asymptotically stable. The proof is given by a technical Natalia Babych weighted energy method. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY United Kingdom ——— [email protected] On the time-decay of solutions to a family of defocusing NLS Within a Laméproblem of linear elasticity, we investigate singularities of solutions in the vicinity of an out- Nicola Visciglia ward cusp at the boundary. In case of a sharp cusp Dipartimento di Matematica, Universita di Pisa, Via F. 1 (the Hölderconstant is less or equal 2 ), we describe the Buonarroti 2, Pisa 56127 Italy essential spectrum that consists of a certain real ray ac- [email protected] cessing +∞. We analyse all possible local singularities of solutions and construct radiation conditions defining Let u(t, x) be any solution to the defocusing NLS with suitable spaces that guarantee a Fredholm type solv- α 4 pure power nonlinearity u|u| , where 0 < α < n−2 , and ability for the problem. We demonstrate that the sharp 1 n p with initial condition u(0, x) ∈ H (R ). Then the L outward cusp at the boundary is somewhat similar to norm of u(t, x) goes to zero as t → ∞ provided that infinity for unbounded domains. 2n This is joint work with Dr. I. Kamotski. 2 < p < n−2 . In particular we extend previous result due to Ginibre and Velo who have shown the property ——— above under the extra assumption 4 < α < 4 . n n−2 Torsion effects in elastic composites with high contrast ——— Michel Bellieud The semilinear Klein-Gordon equation in de Sitter Universitéde Perpignan, 52 avenue Paul Alduy, Perpig- spacetime nan, 66860 France [email protected] Karen Yagdjian Department of Mathematics, University of Texas-Pan In the context of linearized elasticity, we analyze as American, 1201 W. University Drive, Edinburg, TX ε → 0 a vibration problem for a two-phase medium 78541-2999, USA whereby an ε-periodic set of ”stiff” elastic fibers of elas- [email protected] tic moduli of the order 1 is embedded in a ”soft” elastic matrix of elastic moduli of the order ε2. We show that In this talk we present the blow-up phenomena for torsional vibrations take place at an infinitesimal scale. the solutions of the semilinear Klein-Gordon equation 2 p ——— gφ − m φ = −|φ| with the small mass m ≤ n/2 in de Sitter spacetime with the metric g. We prove that for Enhanced resolution in structured media every p > 1 large energy solutions blow up, while for the Yves Capdeboscq small energy solutions we give a borderline p = p(m, n) OxPDE Centre for Nonlinear Partial Differential Equa- for the global in time existence. The consideration is tions, University of Oxford, Mathematical Institute, Ox- based on the representation formulas for the solution ford, OX1 3LB United Kingdom of the Cauchy problem and on some generalizations of [email protected] Kato’s lemma. ——— In this talk, we show that it is possible to achieve a resolution enhancement in detecting a target inclusion if it is surrounded by an appropriate structured medium. This IV.5. Asymptotic and multiscale analysis work is motivated by the advances in physics concerning the so-called super resolution, or resolution beyond the Organisers: diffraction limit. We first revisit the notion of resolu- Ilia Kamotski, Valery Smyshlyaev tion and focal spot, and then show that in a structured medium, the resolution is conditioned by effective pa- BICS Mini-Symposium rameters. The minisymposium will focus on fundamental analyt- This is a joint work with Habib Ammari & Eric Bon- ical issues associated with differential equations (linear netier

85 IV.5. Asymptotic and multiscale analysis

——— The Knizhnik-Zamolodchikov equation associated with the root system B is investigated. This root system Homogenization of elliptic partial differential equations n has two orbits with respect Weyl group. By this reason with unbounded coefficients in dimension two KZ equation naturally contains two parameters. Sin- Juan Casado-Diaz gular locus of this equation consists from hyperplanes Dpto. de Ecuaciones Diferenciales y Analisis Numerico, xi − xj = 0, xi + xj = 0, xk = 0, i, j, k = 1, 2, . . . , n, n Facultad de Matematicas, C. Tarfia s/n Sevilla, Sevilla x = (x1, . . . , xn) ∈ C . The following inverse problem of 41012 Spain Riemann-Hilbert type is considered: given a representa- [email protected] tion of a fundamental group of complement to the singu- n lar locus in C to the orthogonal group of odd order. To This is work in collaboration with Marc Briane, where define the coefficients of the two-parametric differential we study the asymptotic behaviour of a given sequence of KZ equation as elements of tensor power of universal en- diffusion energies in L2(Ω) for a bounded open subset Ω velopping algebra for odd orthogonal Lie algebra. One- 2 of R . The corresponding diffusion matrices are assumed parametric case was investigated by A.Leibman. For to be coercive but any upper bound is considered. We coefficients were used Casimir elements of second order. prove that, contrary to the three dimension (or greater), In two-parametric case the coefficients are defined by the Γ-limit of any convergent subsequence of Fn is still using the families of Casimir elements of higher order a diffusion energy. We also provide an explicit represen- described by A. Molev. For construction of these ele- tation formula of the Γ-limit when its domains contains ments are used Capelli operators which permit to de- the regular functions with compact support in Ω. These scribe the centre of corresponding universal enveloping results are based on the uniform convergence satisfied by algebra. The invariants used in explicit form of coeffi- some minimizers of the equicoercive sequence Fn, which cients for the case o(5) are expressed by means of Pfaf- is specific to the dimension two. fian for matrix defined using the root system. ——— ——— Two-scale Γ-convergence and its applications to ho- Long-time behavior for the Wigner equation and semi- mogenisation of non-linear high-contrast problems classical limits in heterogeneous media Mikhail Cherdantsev Fabricio Macia School of Mathematics, Cardiff University, Senghennydd Universidad Politecnica de Madrid, DEBIN ETSI Road, Cardiff, CF24 4AG United Kingdom Navales, Avda. Arco de la Victoria, Madrid 28040 Spain [email protected] [email protected]

It is a resent results of Bouchitte, Felbacq, Zhikov and We study the semiclassical limit for a class of linear others that passing to the limit in high-contrast elliptic Schrödinger equations in an heterogeneous medium (for PDEs may lead to non-classical effects, which are due to instance, a Riemannian manifold) at time scales tending the two-scale nature of the limit problem. These have to infinity as the characteristic frequencies of the initial so far been studied in the linear setting, or under the as- data tend to zero. We are interested, in particular, in sumption of convexity of the stored energy function. It dealing with time scales larger than the Eherenfest time, seems of practical interest however to investigate the ef- for which the high frequency behavior is completely fect of high-contrast in the general non-linear case, such characterized by classical mechanics via Egorov’s the- as of finite elasticity. orem. Our analysis is performed by studying the high- With this aim in mind, we develop a new tool to frequency behavior of Wigner functions corresponding to study non-linear high-contrast problems, which may be solutions to the Schrodinger equation at very long times. thought of as a “hybrid” of the classical Γ-convergence We give a complete characterization of their structure (De Giorgi, Dal Maso, Braides) and two-scale conver- for systems arising as the quantization of a completely gence (Allaire, Briane, Zhikov). We demonstrate the integrable classical Hamiltonian flow. In particular, we need for such a tool by showing that in the high-contrast prove that in such systems the asymptotic behavior of case the minimising sequences may be non-compact in Wigner functions for times larger than Ehrenfest’s might Lp space and the corresponding minima may not con- no longer be determined by the classical flow. This is due verge to the minimum of the usual Γ-limit. We prove a to effects caused by resonances, that have to be studied compactness principle for high-contrast functionals with via a new object, the resonant Wigner distribution. respect to the two-scale Γ-convergence, which in particular implies convergence of their minima. We briefly ——— discuss possible applications of this new technique in On nonlinear dispersive equations in periodic structures: the mechanics of composites. (This is a joint work with Semiclassical limits and numerical schemes K.D. Cherednichenko.) Peter Markowich ——— DAMTP, University of Cambridge, Wilberforce Road, Construction of the two-parametric generalizations of Cambridge, CB3 0WA United Kingdom [email protected] the Knizhnik-Zamolodchikov equations of Bn type

Valentina Alekseevna Golubeva We discuss (nonlinear) dispersive equations, such as Steklov Mathematical Institute, Gubkina 8, Moscow the Schrdinger equation, the Gross-Pitaevskii equation 119991 Russia modeling Bose-Einstein condensation, the Maxwell- [email protected] Dirac system and semilinear wave equations. Semiclas- sical limits are analysed using WKB and Wigner tech-

86 V.1. Inverse problems niques, in particular for periodic structures, and connec- in the (tensorial) coefficients. The employed tools are tions to classical homogenisation problems for Hamilton- those of ”non-classical” (high contrast type) homogeni- Jacobi equations and hyperbolic conservation laws are sation. This leads to interesting effect physically, for ex- established. We present a new numerical technique for ample allowing ”directional localisation”, with no wave such PDE problems, based on Bloch decomposition, and propagation in certain directions, and mathematically show applications in semiconductor modelling, Bose- allows treating form a unified perspective ”classical”, Einstein condensation and Anderson localisation for ran- high-contrast homogenizations and intermediate cases. dom wave equations. We discuss some related analytic issues, including the need to develop appropriate versions of two-scale conver- ——— gence and of the theory of compensated compactness. Derivation of Boltzmann-type equations from hard- ——— sphere dynamics Karsten Matthies Department of Mathematical Sciences, University of V.1. Inverse problems Bath, Bath, BA2 7AY United Kingdom [email protected] Organisers: Yaroslav Kurylev, Masahiro Yamamoto The derivation of the continuum models from deterministic atomistic descriptions is a longstanding and fun- Inverse problems is a multidisciplinary subject having its damental challenge. In particular the emergence of irre- firm origin in application of mathematics to such prob- versible macroscopic evolution from reversible determin- lems as search for oil, gas and other mineral resources, istic microscopic evolution is still not fully understood. medical imaging, process monitoring in micro-biological, We study a classic system: N balls that interact with chemical and other industries, non-destructive testing of each other via a hard-core potential and show rigorously materials, to mention just few. Its mathematical un- that in the case of kinetic annihilation (particles annihi- derpinning stretches from discrete mathematics, to ge- late each other upon collision) the asymptotic behavior ometry, to computational methods with, however, the as N tends to infinity is correctly described by the Boltz- principal background being in analysis. In particular, mann equation without gain-term for non-concentrated the use of analytic methods makes it possible to address initial distributions. The mean-field description fails, such issues of IP as their strongly non-linear nature and when there are concentrations in the space or the veloc- severe ill-posdnesss. ity coordinates. In recent years, these relations have made it possible to This is joint work with Florian Theil. solve a number of long-standing inverse problems, including those with data on a part of the boundary, with ——— significantly reduced requirements on regularity and the Minimizing atomic configurations of short range pair number of measurements, etc. These were based on the potentials in two dimensions: crystallization in the advancing and employing such topics in analysis as Car- Wulff shape leman estimates for PDE’s, harmonic and quasiconformal analysis, global and geometric analysis, microlocal Bernd Schmidt calculus and stochastic/probabilistic methods. In this Zentrum Mathematik, TU Muenchen, Boltzmannstr. 3, section we intend to represent those progress by inviting Garching b. Muenchen, 85747 Germany the leading people in the area to give relevant talks. [email protected] —Abstracts— We investigate ground state configurations of atomic systems in two dimensions as the number of atoms tends An inverse conductivity problem with a single measure- to infinity for suitable pair interaction models. Suit- ment ably rescaled, these configurations are shown to crystal- lize on a triangular lattice and to converge to a macro- Abdellatif El Badia scopic Wulff shape which is obtained from an anisotropic LMAC, University of Compiegne, Compiegne, Oise surface energy induced by the microscopic atomic lat- 60200 France tice. Moreover, sharp estimates on the microscopic fluc- [email protected] tuations about the limiting Wullf shape are obtained. (Joined work with Y. Au Yeung and G. Friesecke.) We revisit in this paper the inverse boundary value problem of Calderon for a coated domain, where the ——— conductivity is constant in each subdomain. This ge- Homogenization with partial degeneracies: analytic as- ometric distribution of conductivity corresponds to the pects and applications well accepted model of heads in ElectroEncephaloGra- phy (EEG). For instance, the inmost interior domain Valery Smyshlyaev is occupied by the brain, and it is surrounded by the Department of Mathematical Sciences, University of skull and the scalp. The so-called spherical model, where Bath, Claverton Down Bath, BA2 7AY United King- these regions are concentric spherical layers, is also fre- dom quently used. We show for this distribution of conduc- [email protected] tivity that the inverse problem is completely solved with only one suitably chosen Cauchy data, instead of the We consider homogenization problems for a generic class whole Dirichlet-to-Neumann operator. The criterion of of (scalar or vector) operators with ”partial” degeneracy choice for these Cauchy data is completely set up in

87 V.1. Inverse problems the spherical model, using spherical harmonics. Also, e.g., regularization functions included). We illustrate a stability result is established. As for the numerical the method by showing that one can reconstruct a body method to compute the conductivity, we propose a least with sparse data of the boundary curves (profiles) and square procedure with a Kohn-Vogelius functional, and volumes (brightnesses) of its generalized projections. a boundary integral method for the direct problem. ——— This is joint work with T. Ha-Duong. On an inverse problem for a linear heat conduction ——— problem Global in time existence and uniqueness results for some integrodifferential identification problems Christian Daveau CNRS (UMR 8088) and Department of Mathematics, Fabrizio Colombo University of Cergy-Pontoise, 2 avenue Adolphe Chau- Dipartimento di Matematica, Politecnico di Milano, via vin, 95302 Cergy-Pontoise Cedex, France. Bonardi 9 Milano, Mi 20133 Italy [email protected] [email protected] In this talk, a boundary integral method is used to We show some results on the identification of memory solve an inverse linear heat conduction problem in two- kernels in some nonlinear equations such as the heat dimensional bounded domain. An inverse problem of equation with memory, the strongly damped wave equa- measuring the heat flux from partial (on a part of the tion with memory, the beam equation with memory and boundary) dynamic boundary measurements is consid- a peculiar model in the theory of combustion. ered. An additional restriction on the state variable is given This talk presents joint work with A. Khelifi. to determine both the state variable and the memory ——— kernels. We prove global in time uniqueness results and for suit- Inverse problems for wave equation and a modified time able nonlinearities we prove existence and uniqueness reversal method results for the solution of the identification problems associated to the models mentioned above. Matti Lassas Department of Mathematics and Statistics, P.O. Box ——— 68 (Gustaf Hallstromin katu 2b), Helsinki, University of Helsinki 00014 Finland Stability estimate for an inverse problem for the magnetic Schrödingerequation from the Dirichlet-to- [email protected] Neumann map A novel method to solve inverse problems for the wave Mourad Choulli equation is introduced. Suppose that we can send waves Department of Mathematics, Metz University, Ile du from the boundary into an unknown body with spatially Saulcy Metz, Lorraine 57000 France varying wave speed c(x). Using a combination of the [email protected] boundary control method and an iterative time reversal scheme, we show how to focus waves near a point x0 In this talk we consider the problem of stability esti- inside the medium and simultaneously recover c(x0) if mate of the inverse problem of determining the mag- the wave speed is isotropic. In the anisotropic case we netic field entering the magnetic Schrödingerequation can reconstruct the wave speed up to a change of coordi- n in a bounded smooth domain of R with input Dirich- nates. These results are obtained in collaboration with let data, from measured Neumann boundary observa- Kenrick Bingham, Yaroslav Kurylev, and Samuli Silta- tions. This information is enclosed in the dynamical nen. Also, we will disucss how the energy of a wave can Dirichlet-to-Neumann map associated to the solutions be focused near a single point in an unknown medium. of the magnetic Schrödingerequation. We prove in di- These results are done in collaboration with Matias Dahl mension n ≥ 2 that the knowledge of the Dirichlet- and Anna Kirpichnikova. to-Neumann map for the magnetic Schrödingerequa- ——— tion measured on the boundary determines uniquely the magnetic field and we prove a Hölder-type stability in Picard condition based regularization techniques in in- determining the magnetic field induced by the magnetic verse obstacle scattering potential. Koung Hee Leem ——— Dept. of Mathematics & Statistics, Southern Illinois University, Edwardsville, IL 62026 United States Optimal combination of data modes in inverse problems: maximum compatibility estimate [email protected]

Mikko Kaasalainen The problem of determining the shape of an obstacle Department of Mathematics and Statistics, PO Box 68, from far-field measurements is considered. It is well Helsinki, FI-00014 Finland known that linear sampling methods have been widely [email protected] used for shape reconstructions obtained via the singular system of an ill conditioned discretized far-field opera- We present an optimal strategy for weighting various tor. For our reconstructions we assume that the far field data modes in inverse problems. The solution, maxi- data are noisy and we present two novel regularization mum compatibility estimate, corresponds to the maxi- methods that are based on the Picard Condition and do mum likelihood estimate of the single-mode case (with, not require a priori knowledge of the noise level. Both

88 V.1. Inverse problems approaches yield results comparable to the ones obtained The factorization method is a fast inversion technique via the L-curve method and the discrepancy principle. for visualizing the profile of a scatterer from measurements of the far-field pattern. The mathematical basis ——— of this method is given by the far-field equation, which Limited data problems in tensor tomography is a Fredholm integral equation of the first kind in which the data function is a known analytic function and the William Lionheart integral kernel is the measured (and therefore noisy) far School of Mathematic, University of Manchester, Oxford field pattern. We present a Tikhonov parameter choice Rd, Manchester, M13 9PL United Kingdom approach based on a fast fixed-point method developed [email protected] by Bazan. The method determines a Tikhonov parameter associated with a point near the corner of the L-curve n photoelastic tomography one seeks to recover a trace in log-log scale and it works well even for cases where free symmetric second rank tensor from its truncated the L-curve exhibits more than one convex corner. The transverse ray transform. We present constructive performance of the method is evaluated by comparing uniqueness results in the case where realistic subsets of our reconstructions with those obtained via the L-curve data are known and numerical reconstruction methods. method. This is joint work with V Sharafutdinov and D Szotten. ——— ——— The finite data non-selfadjoint inverse resonance prob- A time domain probe method for inverse scattering lem problems Marco Marletta Roland Potthast Cardiff School of Mathematics Senghennydd Road Department of Mathematics, University of Reading, Cardiff, Wales CF24 4AG United Kingdom Whiteknights, PO Box 220, Berkshire, RG6 6AX, UK [email protected] [email protected] We consider Schrödingeroperators on [0, ∞) with compactly supported, possibly complex-valued potentials in 1 The goal of the talk is to discuss the development of L [0, ∞). It is known (at least in the case of a real- probe methods for inverse scattering problems in the valued potential) that the location of eigenvalues and time-domain. We will study wave scattering by three- resonances determines the potential uniquely. From the dimensional rough surface problems. Both the math- physical point of view one expects that large resonances ematics of these problems as well as the algorithmical are increasingly insignicant for the reconstruction of the solution of direct and inverse problems and the numer- potential from the data. We prove the validity of this ical analysis of algorithms provide a sincere challenges statement, i.e., we show conditional stability for nite since the methods developed for bounded objects can- data. As a by-product we also obtain a uniqueness result not be directly translated into the setting of unbounded for the inverse resonance problem for complex-valued po- scatterers. We survey recent results on the direct and tentials. inverse problems by Burkard, Chandler-Wilde, Heine- This is joint work with S. Naboko, S. Shterenberg and meyer, Lindner and the speaker. With the multi-section R, Weikard. method we present a numerical scheme for which con- ——— vergence both for direct and inverse scattering (using a multi-section Kirsch-Kress approach) can be shown. Numerical solutions of nonlinear simultaneous equa- The time-domain probe method is then formulated and tions discussed. Convergence for the reconstruction of sur- Tsutomu Matsuura faces can be shown and numerical examples are pre- Graduate School of Engineering, Gunma University 1-5- sented. 1 Tenjintyo Kiryu, Gunma 376-8515 Japan ——— [email protected] Explicit and direct representations of the solutions of In this paper we shall give practical and numerical rep- nonlinear simultaneous equations resentations of inverse mappings of 2-dimensional mappings (of the solutions of 2-nonlinear simultaneous equa- Saburou Saitoh tions) and show their numerical experiments by using Department of Mathematics, University of Aveiro, 3810- computers. We derive a concrete formula from a very 193 Aveiro, Portugal general idea for the representation of the inverse func- [email protected] tion ——— We shall present our recent results with Dr. Masato Yamada on practical, numerical and explicit represen- A fixed-point algorithm for determining the regulariza- tations of inverse mappings of n-dimensional mappings tion parameter in inverse scattering (of the solutions of n-nonlinear simultaneous equations) George Pelekanos and show their numerical experiments by using comput- Dept. of Mathematics & Statistics, Southern Illinois ers. We derive those concrete formulas from very general University, Edwardsville, IL 62026 United States ideas for the representation of the inverse functions. [email protected] ———

89 V.2. Stochastic analysis

Direct and inverse mixed impedance problems in linear —Abstracts— elasticity Cylindrical Levy processes in Banach space Vassilios Sevroglou University of Piraeus, Department of Statistics and In- David Applebaum surance Science, 80 Karaoli & Dimitriou Str., Piraeus, Dept of Probability and Statistics, Hicks Building Athens 18534, Greece Hounsfield Road, University of Sheffield Sheffield, York- [email protected] shire S3 7RH United Kingdom [email protected] Direct and inverse scattering problems with mixed boundary conditions in linear elasticity are considered. We formulate the direct scattering problem for a par- Cylindrical probability measures are finitely additive tially coated obstacle as well as the mathematical setting measures on Banach spaces that have sigma-additive for the inverse one. Uniqueness theorems are presented projections to Euclidean spaces of all dimensions. They and an inversion algorithm for the determination of the are naturally associated to notions of weak random vari- scattering obstacle is established. In particular, a lin- able and hence weak processes which may be good can- ear integral equation due to the linear sampling method didates to be the driving noise in stochastic evolution which arises from an application of the reciprocity gap equations. In this talk I’ll focus on cylindrical Levy pro- functional and the fundamental solution, connected with cesses. These have (weak) Levy-Ito decompositions and the appoximate solution of the inverse problem, is inves- an associated Levy-Khintchine formula. If the process tigated. Finally, a discussion about the validity of our is weakly square integrable, its covariance operator can method for mixed boundary value problems in elastic be used to construct a reproducing kernel Hilbert space scattering theory is presented. in which the process has a decomposition as an infinite series built from a sequence of uncorrelated bona fide ——— one-dimensional Levy processes. This series is used to On inverse scattering for nonsymmetric operators define cylindrical stochastic integrals from which cylin- Igor Trooshin drical Ornstein-Uhlenbeck processes can be constructed. Institute of Problems of Precise Mechanics and Control, (Based on joint work with Markus Reidel, Manchester) Russian Academy of Sciences, Rabochaya 24, Saratov, ——— 410028 Russia [email protected] Integration by parts for locally smooth laws and applications to jump type diffusions We consider a nonsymmetric operator AP in {L2(0, ∞)}2. defined by differential expression Vlad Bally Universite Marne-le-Vallée, Boulevard Descartes Cité 0 (AP u)(x) = Bu (x) + P (x)u(x), 0 < x < ∞ Descartes - Champs-sur-Marne Marne-la-Vallée,77454 where France [email protected] „0 1« „p (x) p (x)« B = ,P (x) = 11 12 , 1 0 p21(x) p22(x) with the domain Our main result concerns the regularity of the law of „ « solutions of jump type stochastic equations with discon- u1(x) 1 2 D = {u(x) = ∈ {H (R+)} ; u1(0) = hu2(0)}. tinuous coefficients. Since the coefficients are discontin- u2(x) uous the solution is not in the domain of the Malliavin An inverse problem of reconstruction of complex-valued differential operators and so the usual Malliavin calculus coefficients pij (x) from the scattering data of operator on the Poisson space does not work. AP is investigated.. The main tool in our approach is an integration by parts formula for finte dimensional random variables which ——— have a locally smooth law. This is an abstract version of the Malliavin calculus for simple functionals. Then V.2. Stochastic analysis we approximate a general functional (the solution of the equation in our case) by a sequence of simple functionals Organisers: and we use the integration by parts formula for them. Dan Crisan, Terence Lyons But the weights which appear in the integration by parts formula blow up. Nevertheless, using a balance argu- Stochastic analysis aims to provide mathematical tools ment we are able to estimate the Fourier transform of to describe and model high-dimensional random systems the functional and to conclude that the law has a smooth that arise in the study of stochastic differential equations density with respect to the Lebesgue measure. and stochastic partial differential equations, infinite dimensional stochastic geometry, random media and inter- ——— acting particle systems, super-processes, stochastic filtering, mathematical finance, etc. It has emerged as Information and asset pricing a core area of late 20th century mathematics and is Dorje Brody currently undergoing quite rapid scientific development. Department of Mathematics, Imperial College London, The section will provide a forum for researchers working 180 Queen’s Gate, London SW7 2AZ, UK on the different aspects of stochastic analysis to present [email protected] their findings, and to interact with people working in the wider area of analysis.

90 V.2. Stochastic analysis

A framework for asset price dynamics is introduced in ——— which the concept of noisy information about future cash A uniqueness problem for SDEs and a related estimate flows is used to derive the corresponding price processes. for transition functions In this framework an asset is defined by its cash-flow structure. With each cash flow we associate a mar- Alexander Davie ket information process, the values of which we assume School of Mathematics, University of Edinburgh, King’s are accessible to market participants. Each information Buildings, Mayfield Road, Edinburgh, EH9 3JZ United process consists of a sum of two terms; one contains Kingdom true information (the signal) about the value of the as- [email protected] sociated market factor, and the other represents noise. The market filtration is assumed to be that generated Existence and uniqueness theorems for (vector) stochas- by the aggregate of the independent information pro- tic differential equations dx = a(t, x)dt + b(t, x)dW are cesses. The price of an asset is given by the expectation usually formulated at the level of stochastic processes. of the discounted cash flows in the risk neutral measure, This talk will consider instead a uniqueness question for conditional on the information provided by the market an individual driving Brownian path W , when the equa- filtration. The work is done in corroboration with L.P. tion is interpreted using rough path theory, and in this Hughston, A. Macrina, as well as others including M. context a uniqueness theorem can be proved (for a.e. W ) Davis and R. Friedman. if b has suitable regularity and a is bounded Borel. The proof depends on an estimate for the transition function ——— of an associated diffusion process. A (rough) pathwise approach to fully non-linear ——— stochastic partial differential equations Risk-sensitive portfolio optimization with jump- Michael Caruana diffusion asset prices Department of Pure Mathematics & Mathematical Statistics, University of Cambridge, Wilberforce Road Mark H. A. Davis Cambridge, Cambridgeshire CB3 0WB United Kingdom Department of Mathematics, Imperial College, London [email protected] SW7 2AZ [email protected] In a series of papers, P. L. Lions and P. Souganidis proposed a pathwise theory for fully non-linear stochastic This paper considers a portfolio optimization problem partial differential equations. We present a (partial) ex- in which asset prices are represented by SDEs driven by tension towards certain spatial dependence in the noise Brownian motion and a Poisson random measure, with term. The main novelty is the use of rough path the- drifts that are functions of an auxiliary diffusion ‘fac- ory in this context. This joint work with P. Friz and H. tor’ process. The criterion, following earlier work by Oberhauser. Bielecki, Pliska, Nagai and others, is risk-sensitive optimization (equivalent to maximizing the expected growth ——— rate subject to a constraint on variance.) By using a Solving backward stochastic differential equations using change of measure technique introduced by Kuroda and cubature methods Nagai we show that the problem reduces to solving a certain stochastic control problem in the factor process, Dan Crisan which has no jumps. The main result of the paper is that Department of Mathematics, Imperial College London, the Hamilton-Jacobi-Bellman equation for this problem 180 Queen’s Gate, London SW7 2AZ, United Kingdom has a classical (C1,2) solution. The proof uses Bellman’s [email protected] ‘policy improvement’ method together with results on linear parabolic PDEs due to Layzhenskaya et al. This In the last decade, a new class of numerical methods is joint work with SébastienLleo. for approximating weak solutions of SDEs have been introduced by Kusuoka, Lyons, Ninomiya and Victoir. ——— These methods are based on the work of Kusuoka and Accelerated numerical schemes for nonlinear filtering Stroock who established refined gradient upper bounds for the associated semigroup using Malliavin Calculus Istvan Gyongy techiniques. In this talk, I will present an application of School of Mathematics, Edinburgh University, Mayfield these methods to the numerical solution of Road, Edinburgh, EH9 3JZ United Kingdom Backward SDEs and some applications to option pricing. [email protected] The talk is based on joint work with K. Manolarakis. Accelerated numerical schemes for deterministic and ——— stochastic PDEs are discussed. In particular, accelerated finite difference schemes for stochastic PDEs are Some results on Lagrangian Navier-Stokes flows presented and sufficient conditions are given under which Ana Bela Cruzeiro the convergence of finite difference approximations can Dep. Mathematics, IST and GFMUL, Av. Prof. Gama be accelerated to any given order of convergence by Pinto Lisboa, 2 1649-003 Portugal Richardson’s method. [email protected] The results are applied to numerical solutions of nonlinear filtering problems. The talk is based on joint work We consider stochastic Lagrangian trajectories associ- with N.V. Krylov. ated to the Navier-Stokes equation and some of its prop- ——— erties, namely stability.

91 V.2. Stochastic analysis

Periodic homogenisation with an interface We present the higher-order Feynman-Kac formula for solution of equation (∆ + V )mu = 0 in a domain Martin Hairer Courant Institute, 251 Mercer Street, New York, NY D. Probabilistic representation implies some a priori 10012 United States bounds on the growth of solutions when the domain D extends to d. The estimations of moments of random [email protected] R time to reach a high level for Bessel processes are used to It is well-known that, under a diffusive space-time rescal- establish a weak dependence of polyharmonic functions ing, a typical diffusion process with periodic coefficients of some boundary values. converges in law to a Brownian motion with a certain ——— effective diffusion tensor. The twist on this old problem considered in this talk is that we allow the presence Networks and Poisson line patterns of an interface that partitions the space into two half- Wilfried Kendall spaces. We then consider a diffusion whose coefficients Department of Statistics, University of Warwick, Coven- are periodic in each half-space (with possibly different try, West Midlands CV4 7AL United Kingdom periodic structures), with a smooth transition of order [email protected] one around the interface. In this talk, we will show that while the long-time large- scale limit of such a process converges to the expected How best to join up n nodes in a planar network? The homogenised Brownian motion on either side of the in- complete planar graph provides short connections at the terface, additional local time terms can appear on the in- expense of large network length. The Steiner tree min- terface. We will provide a complete description of these imizes total network length at the expense of generat- additional terms and outline the main ideas appearing ing potentially long connections. It turns out that a in the proofs. construction based on Poisson line processes does a re- markably good job of solving the frustrated optimization ——— problem based on the competing criteria of reducing to- Wiener chaos models for interest rates and foreign ex- tal network length versus providing connections which change are short in an average sense. An associated random graph presents intriguing questions of stochastic anal- Lane Hughston ysis motivated by this application: what is the typical Department of Mathematics, Imperial College London, behaviour of a geodesic? and what can one say about London, SW7 2BZ United Kingdom traffic flow in the graph? Answers involve consideration [email protected] of exponential functionals of Brownian motion and other We consider the general problem of modeling the Lévyprocesses, and a curious limiting object based on arbitrage-free dynamics of the nominal interest-rate a highly improper Poisson line process. term structure in the case when the discount-bond sys- (Joint work with David Aldous.) tem is driven by Brownian motion. We show that under ——— rather general assumptions the pricing kernel for such a system is given by the conditional variance of a random The Levy-Khinchine type operators with variable Lips- variable that admits a Wiener chaos expansion. The chitz continuous coefficients and stochastic differential resulting interest rate models can be classified in a hi- equations driven by nonlinear Levy noise erarchical fashion according to the degree of the chaos Vassili Kolokoltsov expansion. A number of specific models are constructed Department Statistics, University of Warwick, Coventry, according to this scheme, with extensions to foreign ex- West Midlands CV4 7AL United Kingdom change and other asset classes. [email protected] ——— Minimising the time to a decision Theory of stochastic integrals and SDEs driven by distribution dependent nonlinear Lévynoise is developed Saul Jacka yielding an effective construction of linear and nonlin- Department of Statistics, Zeeman Building University ear Markov semigroups and the corresponding processes of Warwick, Gibbet Hill Road Coventry, West Midlands with a given pseudo-differential (pre)generator. It is CV4 7AL United Kingdom shown that a conditionally positive integro-differential [email protected] operator (of the Lévy-Khintchine type) with variable coefficients (diffusion, drift and Lévymeasure) depend- We consider a stochastic control problem motivated by ing Lipschitz continuously on its parameters (position a conjecture of Peres. The problem is to choose (at each and/or its distribution) generates a linear or nonlinear time point) which of three absorbing BMs to run, in Markov semigroup, where the measures are metricized such a way as to minimise the time until at least two by the Wasserstein-Kantorovich metrics. This is a non- have been absorbed at either zero or one. trivial but natural extension to general Markov processes ——— of a long known fact for ordinary diffusions. Markov process representations for polyharmonic func- ——— tions Equivalence of stochastic equations and martingale Mark Kelbert problems Department of Mathematics, Swansea University, Sin- gleton Park Swansea, Wales SA2 8PP United Kingdom Thomas Kurtz [email protected] Department of Mathematics/UW-Madison, 480 Lincoln

92 V.2. Stochastic analysis

Drive, Madison, WI 53706 United States The theory of rough paths allows one to build rigorous [email protected] mathematical models for the evolution of these communities, even in the contiuum context. The fact that the solution of a martingale problem for a This is joint work with Thomas Cass. diffusion process gives a weak solution of the correspond- ——— ing Ito equation is well-known since the original work of Stroock and Varadhan. The result is typically proved On the martingale property of certain local martingale by constructing the driving Brownian motion from the Aleksandar Mijatovic solution of the martingale problem and perhaps an aux- Department of Mathematics, Imperial College London, iliary Brownian motion. This constructive approach is 180 Queen’s Gate London, England SW7 United King- much more challenging for more general Markov pro- dom cesses where one would be required to construct a Pois- [email protected] son random measure from the sample paths of the solution of the martingale problem. A soft approach to this The stochastic exponential equivalence will be given which begins with a joint mar- Zt = exp{Mt − M0 − (1/2)hM,Mit} tingale problem for the solution of the desired stochastic equation and the driving processes and applies a Markov of a continuous local martingale M is itself a continuous mapping theorem to show that any solution of the origi- local martingale. We present a necessary and sufficient nal martingale problem corresponds to a solution of the condition for the process Z to be a true martingale and to be a uniformly integrable martingale in the case where joint martingale problem. R t Mt = 0 b(Yu) dWu and Y is a one-dimensional diffusion ——— driven by a Brownian motion W . These conditions are deterministic and expressed only in terms of the func- Aida’s logarithmic Sobolev inequality with weights and tion b and the drift and diffusion coefficients of Y . As Poincare inequalities. an application of the result we describe a deterministic Xue-Mei Li necessary and sufficient condition for the existence of Department of Mathematics, University of Warwick, financial bubbles in the diffusion based models. Coventry, CV4 7AL United Kingdom ——— [email protected] On some stochastic dynamical systems and cancer Although a Logarithmic Sobolev inequality holds for the Khairia El-Said El-Nadi Brownian bridge measure on the Wiener space and for Department of Mathematics, Faculty of Science, Alexan- the Brownian motion measure on the path space over a dria University, Alexandria Egypt compact manifolds, it may not hold on a general loop khairia el [email protected] space. As noted big L. Gross Poincare inequalities do not hold on the Lie groups. A. Eberle gave an exam- Different models of tumor growth are considered. Some ple of a compact simply connected Riemannian mani- mathematical methods are developed to analyze the dy- fold on which the Poincare inequality does not hold for namics of mutations enabling cells in cancer patients to the Brownian bridge measure. For the Brownian bridge metastize. measure a positive result was obtained by Aida for the The mathematical models consist of some stochastic dy- Hyperbolic space H where he obtained a weak form Log- namical systems describing tumor cells and immune ef- arithmic Sobolev inequality with a weight function. We fectors. It is also considered a method to contrast the show that Aidas type weak logarithmic Sobolev inequal- ideal outcomes of some treatments. The results of the ity leads to a weak logarithmic sobolev inequality using considered model predict continuous under which some the non-homogeneous H1 norm together with an L∞ suitable treatment can be successful in returning an ag- norm. We also show that there is a precise passage from gressive tumor to its passive, non-immune evading state. weak Logarith-mic Sobolev inequality to weak Poincare The principle goal of this paper is to find ways to treat- inequality. As a corollary we obtain a Poincare inequal- ment the cancer tumor before they can reach an ad- ity for the Brownian bridge measure on loop spaces over vanced stage development. the hyperbolic space where the Bismut tangent space is ——— dened using the Levi-Civita connection. This is joint Statistical inference for rough differential equations work with Chenand Wu. Anastasia Papavasiliou ——— University of Warwick, Coventry, CV4 7AJ, UK. Evolution equations for communities [email protected] Terence Lyons Our goal is to estimate unknown parameters in the poly- Mathematical Institute, 24-29 St Giles, Oxford, Oxford- nomial vector field of a differential equation driven by shire OX1 3LB United Kingdom rough paths. We assume that we know the expected sig- [email protected] nature of the drivers and we observe several independent copies of the signature of the response. By approximat- There are many situations where one would like to model ing the theoretical expected signature of the response by the evolution of a large community, where each member that of the Picard iterations and its empirical signature has their own preferences. The members of the commu- by Monte-Carlo, we end up with a polynomial system nity evolves in ways which depend on the behaviour of whose solution is the “expected signature matching es- the ensemble of other community members as well as timator”. We prove its consistency and asymptotic nor- their own preferences. mality.

93 V.2. Stochastic analysis

At the second part of the talk, we will discuss in more We consider a stochastic elliptic SPDE on a bounded do- detail the computational challenges this approach poses. main driven by a fractional Brownian field with Hurst r 1 k More precisely, the degree of the polynomials grow as q , parameter H = (H1,...,Hk) ∈ [ 2 , 1[ . where q is the degree of the polynomial vector field and Firstly we give a meaning to the stochastic convolu- r the Picard iteration. This makes the computation of tion derived from the Green kernel. Using monotonicity the polynomial computationally very demanding. We methods, we prove existence and uniqueness of solution, will suggest ways for improving the efficiency by utiliz- along with regularity of the sample paths. Finally, for a ing a different definition for the product. Finally, we will given lattice scheme, we prove convergence to the solu- discuss applications to multiscale modelling. tion of the SPDE. ——— ——— Constructing discrete exact approximations algorithms First passage for stochastic volatility models for financial calculus from weak convergence results Martijn Pistorius Radu Tunaru South Kensington Campus, Imperial College, Depart- Cass Business School, Faculty of Finance, 106 Bunhill ment of Mathematics, London SW7 2AZ, UK Row, London EC1Y 8TZ United Kingdom [email protected] [email protected]

Barrier options are financial contracts that are acti- In financial calculus the calculation of moments in gen- vated or de-activated when the underlying price pro- eral and the expectation in particular is extremely im- cess crosses a specific level; they are among the most portant. This can be difficult in a multi-dimensional widely traded of exotic contracts. The pay-offs of bar- set-up and Monte Carlo methods are not always satis- rier options are path dependent and their valuation re- factory. In this paper we show how to construct exact quires the specification of the first-hitting-time distribu- discrete approximations schemes that can be used for a tion. In this talk, we present a new approach to obtain wide range of financial mathematics problems. The al- first passage probabilities for stochastic volatility mod- gorithms we present are derived from well-known weak els (i.e. diffusions whose coefficients are functions of convergence results. Theoretical results are adapted to a one-dimensional diffusion). We illustrate the results work for unbounded payoffs. In addition we show how by calculating the values and Greeks of barrier options, to circumvent the problem caused by the singularity of and compare the outcomes with Monte Carlo simula- the covariance matrix that appears with the CLT for tion results. The talk is based on joint work with Marc the multinomial distribution. The approximation grids Jeannin. developed here are shown to be dense in the set of real numbers, for the one-dimensional case. The results are ——— proved for the geometric Brownian set-up but it ca be easily adapted to other frameworks. Unbiased random perturbations of Navier-Stokes equation ———

Boris Rozovsky Numerical methods for parabolic SPDEs based on the Division of Applied Mathematics, 182 George Street, averaging-over-characteristics formula Providence, Rhode Island 02912 United States Michael Tretyakov [email protected] Department of Mathematics, University of Leicester, Le- icester LE1 7RH United Kingdom A random perturbation of a deterministic Navier-Stokes [email protected] equation is considered in the form of an Stochastic PDE with Wick product in the nonlinear term. The equation The method of characteristics (the averaging over the is solved in the space of generalized stochastic processes characteristic formula) and the weak-sense numerical in- using the Cameron-Martin version of the Wiener chaos tegration of ordinary stochastic differential equations are expansion. The generalized solution is obtained as an in- used to propose numerical methods for stochastic par- verse of solutions to corresponding quantized equations. tial differential equations (SPDEs). Their orders of con- An interesting feature of this type of perturbation is that vergence in the mean-square sense and in the sense of it preserves the mean dynamics: the expectation of the almost sure convergence are obtained. The developed solution of the perturbed equation solves the underlying approach is supported by numerical experiments. The deterministic Navier-Stokes equation. From the stand talk is based on a joint work with G.N. Milstein (Eka- point of a statistician it means that the perturbed model terinburg, Russia). is unbiased. ——— The talk is based on a joint work with R. Mikulevicius. Consistent estimator in AFTM ——— Elena Usoltseva A Poisson equation with fractional noise Vasilkovskaya str. 94, room 823 Kiev, 03022 Ukraine elena [email protected] Marta Sanz-Sole´ Facultat de Matemàtiques, Universitat de Barcelona, We consider a sample of n nonnegative i.i.d. random Gran Via de les Corts Catalanes 585, Barcelona 08028, variables Ti, i = 1, n from Accelerated Failure Time Spain Model. This model (AFTM) is described in general case [email protected] in the following way T log Ti = β0 + βX + ψεi, ψ > 0,

94 V.3. Coercivity and functional inequalities

where εi are i.i.d. with zero mean. A measurement er- and Statistics, Al Khod, Muscat 123 Oman ror is present in our model therefore instead of Xi we [email protected] observe the surrogate data Wi = Xi +Ui, where Ui form a centered i.i.d. sequence with ﬁnite second moments. In this paper, a new method for the ﬁnite element ap- Some lifetimes Ti may be censored, in that only a lower proximation of variational inequalities (VI) with vanish- bound for the lifetime is recorded. The distribution of Ti ing zero order term is introduced and analyzed. Error depends on unknown parameter ν, which is estimated. estimate in the maximum norm is derived and a basic In general AFTM, the adjusted Quasi-Likelihood equa- iterative scheme for the computation of the discrete so- tion leads to estimating equation: lution is also provided.

n ——— „ « T X 1 “ β0+β Xi ” Ti − e X = 0. Convexity along vector fields and application to equa- Xi i=1 tions of Monge-Ampèretype Under the present of measurement error, the adjusted Federica Dragoni unbiased Corrected Score estimating function for messy Department of Mathematics, Imperial College London, lifetime was constructed. In this report, based on the 180 Queen’s Gate, London SW7 2AZ, UK theory of estimating equation, it is proved that this esti- [email protected] mating function yields a consistent estimator under non- singular correlation matrix of regressor and some condi- We introduce and study a new notion of convexity and tions on censoring distribution. semiconvexity along vector fields. We give in particular characterizations in terms of inequalities (in the viscosity ——— sense) for the matrix of second derivatives with respect to the fields. Our notion of convexity is equivalent to the horiontal convexity in Carnot groups but holds for a V.3. Coercivity and functional inequalities far more general class of vector fields. As an application, we get comparison principles for subelliptic equations of Organisers: Monge-Ampèretype, extending a recent result of Bardi Dominique Bakry, Boguslav Zegarlinski and Mannucci for Carnot groups to more general vector fields. The key point is that convex functions along The session intend to present an overview of a recent vector fields satisfy a-priori gradient bounds similar to progress in coercive and functional isoperimetric inqual- those satisfied by euclidean convex funtions. ities and their applications to study long time behaviour This is joint work with Martino Bardi. of (sub-)elliptic problems, variety of probabilistic problems, analysis on groups as well as other related areas. ——— Φ-entropy inequalities for diffusion semigroups —Abstracts— Ivan Gentil CEREMADE, UniversitéParis-Dauphine, Pl. du Mal Remarks on non-interacting conservative spin systems De L. De Tassigny. 75016 PARIS, FRANCE. Franck Barthe [email protected] Institut de Mathématiquesde Toulouse, UniversitéPaul Sabatier, Toulouse, cedex 9 31062 France We obtain and study new Φ-entropy inequalities for dif- [email protected] fusion semigroups, with Poincaréor logarithmic Sobolev inequalities as particular cases. From this study we derive the asymptotic behaviour of a large class of linear We compute precise estimates for the spectral gap and Fokker-Plank type equations under simple conditions, log-Sobolev constants in the special particular case of widely extending previous results. The Γ criterion of gamma distributions. This turns out to be related to 2 D. Bakry and M. Emery [Diffusions hypercontractives. the Kannan-Lovász-Simonovits conjecture for simplices Séminaire de probabilités,XIX, 1983/84, Lecture Notes and Lp balls (joint work with Pawel Wolff). in Math., 1123, 177–206, (1985)] appears as a main tool ——— in the analysis, in local or integral forms. The presentation is based on [Bolley, F.; Gentil, I. Phi- On weak forms of Poincare-type inequalities entropy inequalities for diffusion semigroups Preprint, Sergey Bobkov (2008)]. 127 Vincent Hall, 206 Church St.S.E. Minneapolis, Min- ——— nesota 55455 United States [email protected] On positive solutions of semi-linear elliptic inequalities on manifolds We will discuss weak forms of Poincare-type inequali- Alexander Grigoryan ties for probability measures on Euclidean and abstract Department of Mathematics, University of Bielefeld, metric spaces. Bielefeld, D-33501 Germany [email protected] ——— We consider elliptic inequalities of the type ∆u+uσ ≤ 0 L∞-Error estimate for variational inequalities with van- on geodesically complete Riemannian manifolds and pro- ishing zero order term vide sharp sufficient conditions in terms of capacities and Messaoud Boulbrachene volumes for the non-existence of positive solutions. Joint Sultan Qaboos University, Department of Mathematics work with V.A.Kondratiev.

95 V.3. Coercivity and functional inequalities

——— spectral representation of the Heisenberg Laplacian. We then extend the result to other operators related to the Hypoellipticity in infinite dimensions heat kernel using functional inequalities. Martin Hairer ——— Courant Institute, 251 Mercer Street, New York, NY 10012 United States Liggett inequality and interacting particle systems [email protected] Mikhail Neklyudov Department of Mathematics, University of York, York One of Hörmander’s legacies is to provide a con- YO10 5DD, UK structive and essentially sharp criterion for checking [email protected] whether a second-order differential operator is hypoelliptic: Hörmander’sbracket condition. This result has In this talk we discuss one class of interacting parti- found numerous applications, one of them being the cle systems which correspond to degenerate generator. study of regularising properties for Markov semigroups In this case Poincaréinequality doesn’t hold even in a arising from diffusion processes. weak sense. Instead it is possible to show that Liggett It is natural to try to extend these results to an infinite- inequality holds in certain sense. dimensional setting, for example in the context of the study of Markov semigroups arising from stochastic par- ——— tial differential equations (SPDEs). We will present a A new criterion for a covariance estimate theory that allows to obtain such results in the setting of semilinear parabolic SPDEs with multilinear nonlin- Felix Otto earities. Applications of these results to linear response Institute for Applied Mathematics, University of Bonn, theory and ergodic theory will also be discussed. Endenicher Allee 60 Bonn, 53115 Germany [email protected] ——— Logaritmic Sobolev inequality on nilpotent groups We present a simple criterion for a covariance estimate for a spin system with continuous spin space which is Waldemar Hebisch formulated in terms of the single-site spectral gap and Mathematical Institute, University of Wroc law, the interaction between the sites. It is optimal in case Pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland of Gaussians. A typical application is the exponential [email protected] decay of correlations in case of weak interactions. This is joint work with Georg Menz. We discuss recent results about logaritmic Sobolev (LS) inequality and related coercive inequalities on nilpo- ——— tent groups. Our main result is LS inequality for mea- The Log-Sobolev inequality for non quadratic interac- −1 2 sures of form Z exp(−βd )dλ, where d is Carnot- tions Caratheodory distance on H-type group. We will also mention related results for heat kernel measure and more Ioannis Papageorgiou general groups. Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK ——— [email protected] Isoperimetry for spherically symmetric log-concave probability measures We are interested on the Logarithmic Sobolev q inequality for unbounded spin systems on the Lattice with in- Nolwen Huet teractions that are non quadradic. We present criteri- Institut de Mathématiquesde Toulouse, UniversitéPaul- ons that allow to extend the inequality from the one Sabatier (Toulouse III), 31062 Toulouse Cedex 9, France dimesional measure to the infinite dimensional Gibbs [email protected] measure, for variables on the real line as well as on the Heisenberg group. We prove an isoperimetric inequality for probabil- n ——— ity measures µ on R with density proportional to n exp(−φ(λ|x|)), where |x| is the Euclidean norm on R Isoperimetry for product probability measures and φ is a non-decreasing convex function. It applies in particular when φ(x) = xα with α ≥ 1. Under mild Cyril Roberto Universit de Marne la Valle, LAMA 5bv Descartes assumptions on φ, the inequality is dimension-free if λ Champs sur Marne, Marne la Valle 77454 France is chosen such that the covariance of µ is the identity. [email protected] ——— In this talk we shall give a short overview on the isoperi- Operators on the Heisenberg group with discrete spec- metric problem for product probability measures. tra An isoperimetric inequality is a lower bound on the James Inglis boundary measure of sets in terms of their measure. Department of Mathematics, Imperial College London, Finding the optimal sets (of given measure and of min- 180 Queen’s Gate, London SW7 2AZ, UK imal boundary measure) is very difficult, and the only [email protected] hope is to estimate the isoperimetric function. This is well understood on the line (Bobkov) and for the prod- We show that a certain class of hypoelliptic operators uct of standard Gaussian measures (Sudakov-Tsirel’son, on the Heisenberg group have discrete spectra, using a Borell). We shall start by recalling those known results.

96 V.4. Dynamical systems

Then, we shall explain how functional inequalities can New results on stability and genericity be used to get dimension free isoperimetric inequalities Flavio Abdenur for measures between exponential and Gaussian. Also, Rua Marques de Sao Vicente 225, Departamento de using the transport of mass technique we shall dervive Matematica, PUC-Rio Rio de Janeiro, Rio de Janeiro isoperimetric inequalities (depending on the dimension) CEP 22453-900 Brazil for measures with tails larger than exponential. [email protected] ——— I will briefly explain some very recent results on stability and genericity of different types of discrete dynamical V.4. Dynamical systems systems. In short: Organisers: 1. there exist non-hyperbolic diffeomorphisms which Jeroen Lamb, Stefan Luzzatto are ’weakly’ structurally stable in a very natural sense (joint w/ L. J. Diaz and E. Pujals) Dynamical Systems aims to provide the mathematical 2. generic partially hyperbolic transitive diffeomor- tools to describe and model deterministic systems that phisms are robustly transitive/robustly mixing arise in the study of ordinary Differential Equations and (joint w/ S. Crovisier) in the iteration of maps on smooth manifolds. A variety of ideas and techniques from Analysis and other areas 3. generic continuous maps have highly weird ergodic come together to provide existence and classification re- properties (joint w/ M. Andersson) sults regarding the dynamical properties of systems from I will discuss some or all of these three topics, as time al- geometrical, topological, probabilistic points of view. lows and interest arises. I might also discuss some other This section will provide a forum for high level re- related results, as whim intervenes. searchers working mainly in Bifurcation Theory and Er- godic Theory to present their recent research and to dis- ——— cuss open problems and technical issues. Abundance of one dimensional non uniformly hyperbolic attractors for surface dynamics —Abstracts— Pierre Berger Poincaré-Bendixsontheorems in holomorphic dynamics Mathematisches Forschungsinstitut Oberwolfach, Schwarzwaldstr. 9-11 (Lorenzenhof), Oberwolfach- Walke, 77709 Germany Marco Abate [email protected] Dipartimento di Matematica, Universitàdi Pisa, Largo Pontecorvo 5, Pisa 56127 Italy We present a (new) proof of the existence of a non uni- [email protected] formly hyperbolic attractor for a positive set of parameters a in the family of endomorphisms: I shall present a recent Poincaré-Bendixsontheorem de- (x, y) 7→ (x2 + a + y, 0) + B(x, y), scribing recurrence properties for geodesics of a mero- 2 morphic connections on the complex projective line. where B is any fixed C small function. For B = 0, this Then I shall explain how to use this theorem to study is the Jackoson theorem. For B = b · (0, x), we get the the dynamics of homogeneous vector fields in C2, and Benedicts-Carleson theorem for the Hnon map. why this approach provides tools for studying the local The proof is done thanks to analytical and probabilis- dynamics of holomorphic maps tangent to the identity. tic tools of (B-C) in the geometric and combinatorial (Joint work with F. Tovena.) formalism of Yoccoz puzzles generalized in a very algebraic way (pseudo-semi-group). These theorems are ——— notably generalized to the C2-case and to the endomorphisms. The theorem is an answer to question of Pesin- On the liftability of absolutely continuous ergodic ex- Yurchenko reaction-diffusion EDPs in applied mathe- panding measures. matics. Jose´ Ferreira Alves ——— Departamento de Matemtica Pura, Rua do Campo Ale- gre 687, 4169-007 Porto, Portugal First integrals in mechanics of infinite-dimensional sys- [email protected] tems Svetlana Aleksandrovna Budochkina We consider maps on a compact manifold of arbitrary Miklukho-Maklaya Str. 6, Moscow, 117198 Russia dimension possibly admitting critical points, discontinu- [email protected] ities or singularities. Under some mild nondegeneracy assumptions we show that the map admits an induced Equation of motion represented in the operator form is Gibbs-Markov map with integrable inducing times if and considered. Formulas for finding some first integrals of only if it has an ergodic invariant probability measure the equation of motion are given. which is absolutely continuous with respect to the Rie- ——— mannian volume and has all Lyapunov exponents positive. Partial hyperbolicity and ergodicity ——— Keith Burns Mathematics Department, Northwestern University,

97 V.4. Dynamical systems

n n Evanston, IL 60201 United States y in V such that lim supn→∞ d(f (x), f (y)) ≥ δ and n n [email protected] lim infn→∞ d(f (x), f (y)) = 0. ——— I will survey recent results which extend Hopf’s method for proving ergodicity to a large class of partially hyper- Mixing for flows and skew extensions bolic diffeomorphisms. Michael Field ——— Department of Mathematics, University of Houston, 4800 Calhoun Houston, Texas TX 77204-3008 United On tilings, multidimensional subshifts of finite type and States quasicrystals [email protected] Jean-Rene´ Chazottes CPHT, Ecole Polytechnique Palaiseau, Cedex 91128 Problems and results on rates of mixing and exponential France estimates. [email protected] ——— Rates of mixing, large deviations and recurrence times Subshifts of finite type (SFT) in dimension one (Z- actions) are well known objects used in the symbolic Jorge Freitas dynamics of hyperbolic dynamical systems. In dimen- Dep Matematica Pura, Fac Ciencias, Univ Porto, Rua d sion greater than one (Z -action, d ≥ 2), they are very do Campo Alegre, 687 Porto, 4169-007 Portugal rich and complicated objects for which few results are [email protected] known. We will see how SFTs arise naturally as sup- ports of ground states in lattice models in Statistical It is very well known that one can derive rates for the de- Physics and how they are connected to toy-models of cay of correlations of stationary stochastic processes aris- quasicrystals. We will also see how tiling dynamical sys- ing from dynamical systems admitting a Young tower. tems can be used to derive results on multidimensional These rates depend on the volume decay of the tail set SFTs. of the inducing times. In this work we exploit the connection between decay ——— of correlations of certain classes of observables and large On topological entropy of billiard tables with small inner deviations estimates of stochastic processes generated by scatterers the system. We also show the relation between the large deviations of the potential corresponding to the loga- Yi-Chiuan Chen rithm of the derivative and the volume decay of the tail Institute of Mathematics, Academia Sinica, 128 set of hyperbolic times (the set of points that resist to Academia Road, Section 2 Nankang, Taipei 11529 Tai- present hyperbolic behavior in short time range). wan Based on these considerations we obtain a converse of [email protected] L. S. Young’s result, namely, if we have a system with a certain rate of decay of correlations then the system An approach to studying the topological entropy of a admits a Young tower with the same type of rate for class of billiard systems will be presented. In this class, the volume of the tail of inducing times. Moreover, we any billiard table consists of strictly convex domain in can show how to obtain an estimate for the large devia- the plane and strictly convex inner scatterers. Using tions of a whole class of observable functions, when we the concept of anti-integrable limit, we show that a bil- only have an estimate for the large deviations for the liard system in this class generically admits a set of logarithm of the derivative. non-degenerate anti-integrable orbits which corresponds bijectively to a topological Markov chain of arbitrarily ——— large topological entropy. Consequently, we prove the Limiting distributions for horocycle flows topological entropy of the first return map to the scat- Giovanni Forni terers can be made arbitrarily large provided the inner Department of Mathematics, University of Maryland, scatterers are sufficiently small. College Park, MD 20742 United States ——— [email protected]

On the nature of chaos We give results on the existence and non-existence of Bau-Sen Du limiting distributions for horocycle ﬂows on the unit tan- Institute of Mathematics, Academia Sinica 128, Sec. 2, gent bundle of hyperbolic (negative constant curvature) Ian-Jiou-Yuan Rd., NanKang, Taipei 11529 Taiwan surfaces. This is joint work with A. Bufetov. [email protected] ———

Based on a very special property of the shift map Limit cycle problems and applications (Theorem 1), we believe that chaos should involve not Valery Gaiko only nearby points can diverge apart but also far- Belarusian State University of Informatics and Radio- away points can get close to each other. Therefore, electronics, L. Beda Str. 6-4 Minsk, 220040 Belarus we propose to call a continuous map f from an in- [email protected] ﬁnite compact metric space (X, d) to itself chaotic if there exists a positive number δ such that for any We establish the global qualitative analysis of planar point x and any nonempty open set V (not necessar- polynomial dynamical systems and suggest a new geo- ily an open neighborhood of x) in X there is a point metric approach to solving Hilbert’s Sixteenth Problem

98 V.4. Dynamical systems on the maximum number and relative position of their A dynamical Borel-Cantelli lemma for a class of non- limit cycles in two special cases of such systems. First, uniformly hyperbolic systems using geometric properties of four field rotation param- Matthew Nicol eters of a new canonical system, we present the proof Department of Mathematics University of Houston of our earlier conjecture that the maximum number of Houston, Texas 77204-3008 United States limit cycles in a quadratic system is equal to four and [email protected] their only possible distribution is (3:1). Then, by means of the same geometric approach, we solve the Problem We establish a dynamical Borel-Cantelli lemma for for Liénard’spolynomial system (in this special case, it shrinking balls for certain classes of non-uniformly hy- is considered as Smale’s Thirteenth Problem). Besides, perbolic dynamical systems. As an application we es- generalizing the obtained results, we present the solution tablish results on almost sure behavior of extremes for of Hilbert’s Sixteenth Problem on the maximum num- these classes of dynamical systems. ber of limit cycles surrounding a singular point for an This work is joint with Chinmaya Gupta and William arbitrary polynomial system and, applying the Wintner– Ott (both University of Surrey). Perko termination principle for multiple limit cycles, we develop an alternative approach to solving the Prob- ——— lem. By means of this approach we complete also the Approximately inner C∗-dynamical systems global qualitative analysis of a generalized Liénardcubic system, a neural network cubic system, a Liénard-type Asad Niknam piecewise linear system and a quartic dynamical system Departement of Mathematics, Ferdowsi University of which models the population dynamics in ecological sys- Mashad, Vakilahbad Boulvar Mashad, Khorahsan 1159- tems. 91775, Iran [email protected] ——— Hausdorff dimension of Projections of McMullen- In quantum statistical mechanics one often describe a ∗ Bedford carpets physical system in terms of a C -algebra A. The dynamics or time evoluton of the systems is given in terms Thomas Jordan of one parameter group of ∗-automophisims on A. We Department of Mathematics, The University of Bristol, study such C∗-dynamical systems. We prove that un- University Walk Clifton, Bristol BS8 1TW United King- der some restriction the dynamic is approximately in- dom ner. Moreover we construct a dynamical system which is [email protected] not approximatelly inner and therefore without ground state. Joint work with Andrew Ferguson and Pablo Shmerkin. ——— 2 Marstand’s Projection Theorem states that If E ⊂ R has Hausdorff dimension less than 1 then orthogonal pro- Dynamical systems arising in algebraic logic jections in almost all directions preserve this dimension. Giovanni Panti For general sets very little is known about exactly which Department of Mathematics, via delle Scienze, 208 directions preserve the dimension. We show that if E is Udine, UD 33100 Italy a type of self-affine set investigated by Bedford and Mc- [email protected] Mullen then orthogonal projections in all directions in (0, π/2) preserve the dimension. This is an extension Algebraic logic studies the algebras associated to cer- of a result on products of Cantors sets by Peres and tain logical systems. Standard examples are boolean Shmerkin. algebras, MV-algebras, Heyting algebras, associated to ——— classical logic, many-valued logic, intuitionistic logic, respectively. Typically, these algebras have dual spectral Fourfold 1:1 resonance, relative equilibria and moment spaces, and can be represented as algebras of functions polytopes on the spectrum: automorphisms of the algebras correspond then to dynamical systems on the dual. Jan Cees van der Meer We survey here the structure of the relevant dynamical Dept. of Mathematics and Computer Science, Eind- systems, the results that have been obtained and their hoven University of Technology, Den Dolech 2, Eind- significance, the open problems and directions for fur- hoven, N.B. 5612 AZ Netherlands ther research. [email protected] ——— A uniparametric 4-DOF family of perturbed Hamilto- Existence of transversal homoclinic orbits for Arneodo- nian oscillators in 1:1:1:1 resonance, with two additional Coullet-Tresser map rotational symmetries, is studied. These systems generalize several models of perturbed Keplerian systems. Af- Chen-chang Peng ter normalization the truncated normal form is reduced Department of Applied Mathematics, National Chiayi in stages to a one-degree-of-freedom system. In this re- University No.300 Syuefu Rd. Chiayi City, Taiwan duction process moment polytopes turn up describing 60004 Taiwan part of the relative equilibria for such systems. [email protected] Joint work with S. Ferrer, G. Diaz, J. Egea, J.A. Vera. In this talk, first we study difference equations ——— xk+n = F (xk+n−1, ··· , xk, b1xk−1, ··· , bmxk−m)

99 V.5. Functional differential and difference equations as C1-perturbation of the equation: be generalized to non-linear and also non-integer expansions of a real number. This talk is based on joined work xk+n = f(xk+n−1, ··· , xk) ≡ F (xk+n−1, ··· , xk, 0, ··· , 0). with T. Persson and D. Färm. We prove that if f has a snapback repeller then F has ——— a transversal homoclinic orbit for all |b | < for some i Thermodynamic formalism for unimodal maps > 0. Second, we study a class of two-dimensional maps (or called Mira map) and prove that there exist snapback Mike Todd repellers for the map near its anti-integrable limits. Fi- Departamento de MatemáticaPura, Rua do Campo Ale- nally, combining the above two results, we establish the gre, 687 Porto, 4169-007 Portugal existence of transversal homoclinic orbits in family of [email protected] Arneodo-Coullet-Tresser map near singularities. ——— Notions from thermodynamic formalism such as pressure, equilibrium states and large deviations can give a Bifurcations of random diffeomorphisms with bounded rich qualitative description of a dynamical system. Re- noise cently there has been a lot of activity in the development of thermodynamic formalism applied to non-uniformly Martin Rasmussen hyperbolic dynamical systems. These systems have been Department of Mathematics, Imperial College, London shown to exhibit a wide variety of phenomena, most in- SW7 2AZ, United Kingdom terestingly critical phenomena such as phase transitions. [email protected] In this talk I will give a fairly complete description of the possible behaviour of the class of unimodal interval We discuss iterates of random diffeomorphisms with maps, including the relation between phase transitions identically distributed and bounded noise. In this con- and the existence of a natural measure for the system. text, minimal forward invariant sets play an important role, since they support stationary measures, and when ——— the noise is interpreted as external control, minimal for- Dynamics of periodically perturbed homoclinic solu- ward invariant sets coincide with invariant control sets. tions Discontinuous bifurcations of minimal forward invariant sets are analysed, and a numerical method to approx- Qiudong Wang imate these sets is presented. The results are applied Department of Mathematics, University of Arizona, to study a bifurcation of the randomly perturbed Henon Tucson, Arizona 85721 United States map. This talk is based on joint work with Jeroen Lamb [email protected] (Imperial College) and Christian Rodrigues (University of Aberdeen). We study the dynamics of homoclinic tangles in periodi- ——— cally perturbed second order equations. Let µ be the size of the perturbation and Λµ be the homoclinic tangles. Bifurcations of period annuli and solutions of nonlinear We prove that (i) for infinitely many µ,Λµ contain noth- boundary value problems ing else but a horseshoe of infinitely many branches; (ii) for infinitely many µ,Λ contain nothing else but one Felix Sadyrbaev µ sink and one horseshoe of infinitely many branches; and Institute of Mathematics and Computer Science, Rainis (iii) there are positive measure set of µ so that Λ admits boul. 29 Riga, Latvia LV-1459 Latvia µ strange attractors with Sinai-Ruelle-Bowen measure. [email protected] ——— Differential equations of the type x00 + λf(x) = 0 are considered, where f(x) are polynomials. First bifurcations of period annuli (continua of periodic solutions) V.5. Functional differential and difference are studied under the change of coefficients of f(x). Sec- equations ondly, bifurcations of solutions to the Dirichlet problem x(a) = 0, x(b) = 0 are investigated under the change of Organisers: λ. Leonid Berezansky, Josef Dibl´ık, Agacık˘ Zafer ——— Scope of the session: Qualitative theory of functional Large intersection properties of some invariant sets in differential and difference equations: stability, bound- number-theoretic dynamical systems edness, oscillation, asymptotic behaviour, positive solutions, dynamic equations on time scales, applications to ¨ Jorg Schmeling population dynamics. Center of Mathematical Sciences, LTH, Box 118, Slveg- atan 18 Lund, 22100 Sweden [email protected] —Abstracts— Oscillation and non-oscillation of solutions of linear sec- In this talk we consider sets of real numbers that have a ond order discrete delayed equations given approximation property by rationals with denom- inators gn. We prove that these sets have large inter- Jarom´ır Baˇstinec section properties and are winning in a modified (α, β) Department of Mathematics, The Faculty of Electri- game or belong to Falconers s-class. This result will cal Engineering and Communication, Brno University

100 V.5. Functional differential and difference equations of Technology, Technická8, 616 00 Brno, Czech Repub- project APVV-0700-07 of Slovak Research and Develop- lic ment Agency.This is joint work with J. Dibl´ık,D. Khu- [email protected] sainov, M. R˚uˇziˇcková. ——— The phenomenon of the existence of a positive solution of difference equations is often encountered when analysing Representation of solutions of linear differential and dis- mathematical models describing various processes. This crete systems and their controllability is a motivation for an intensive study of the conditions Josef Dibl´ık for the existence of positive solutions of difference equa- Brno University of Technology, Brno, Czech Republic, tions. Such analysis is related to an investigation of the Kiev State University, Kiev, Ukraine case of all solutions being oscillating. In the talk, con- [email protected] ditions for the existence of a positive solution are given for a class of linear delayed discrete equations We study discrete controlled systems ∆x(n) = −p(n)x(n − 1) ∆x(k) = Bx(k − m) + bu(k), where n ∈ Z∞ := {a, a + 1,... }, a ∈ N is fixed, a where m ≥ 1 is a fixed integer, k ∈ Z∞,Zq := {s, s + ∆x(n) = x(n + 1) − x(n), p : Z∞ → (0, ∞). For the 0 s a 1, . . . , q}, B is a constant n × n matrix, x:Z∞ → Rn same class of equations, also conditions are given for all −m is unknown solution, b ∈ Rn is given nonzero vector and the solutions being oscillating. The results obtained in- u:Z∞ → R is input scalar function. Moreover, we con- dicate sharp sufficient conditions for the existence of a 0 sider the system of delayed linear differential equations positive solution or for the case of all solutions being os- of second order cillating. The investigation was supported by the grant 201/07/0145 of the Czech Grant Agency (Prague) and y00(t) + Ω2y(t − τ) = bu(t) by the Councils of Czech Government MSM 0021630529 and by MSM 00216 30503. and an initial problem y(t) = ϕ(t), y0(t) = ϕ0(t), This is joint work with Josef Diblik. t ∈ [−τ, 0] where τ > 0 and ϕ:[−τ, 0] → Rn is twice differentiable. Special matrix functions are defined: the de- ——— layed matrix sine and the delayed matrix cosine. These New stability conditions for linear differential equations matrix functions are applied to obtain explicit formulas with several delays for the solution of the initial problem and a controllability criterion. The investigation was supported by the Leonid Berezansky grant 201/08/0469 of the Czech Grant Agency (Prague), Department of Mathematics, Ben-Gurion University of by the Councils of Czech Government MSM 0021630519 the Negev, P.O. Box 653, Beer Sheva, Negev 84105 Is- and MSM 00216 30503 and by the project M/34-2008 of rael Ukrainian Ministry of Education. [email protected] This is joint work with Denys Khusainov, Blanka Morávková. New explicit conditions of asymptotic and exponential stability are obtained for the general scalar nonau- ——— tonomous linear delay differential equation with measur- Maximum principles and nonoscillation intervals in the able delays and coefficients. These results are compared theory of functional differential equations to known stability tests. Alexander Domoshnitsky ——— Ariel University Center, Department of Mathematics and Computer Science, Ariel, 44837 Israel Boundary-value problems for differential systems with [email protected] a single delay Aleksandr Boichuk Many classical topics in the theory of functional differ- Faculty of Science, Zilinaˇ University, Zilina,ˇ 01 026 Slo- ential equations, such as nonoscillation, differential in- vakia equalities and stability, were historically studied without [email protected] any connection between them. As a result, assertions associated with maximum principles for such equations Conditions are derived of the existence of solutions of lin- in contrast with the cases of ordinary and even partial ear Fredholm’s boundary-value problems for systems of differential equations do not add so much in problems ordinary differential equations with constant coefficients of existence and uniqueness of solutions to boundary and a single delay. Utilizing a delayed matrix exponen- value problems and stability for functional differential tial and a method of pseudo-inverse by Moore-Penrose equations. One of the goals of this talk is to present a matrices led to an explicit and analytical form of a cri- concept of the maximum principles for functional differ- terion for the existence of solutions in a relevant space ential equations. New results on existence and unique- and, moreover, to the construction of a family of linearly ness of solutions of boundary value problems are pro- independent solutions of such problems in a general case posed. Assertions about positivity og Green’s functions with the number of boundary conditions (defined by a are formulated. Tests of the exponential stability are linear vector functional) not coinciding with the number obtained on the basis of nonoscillation and positivity of of unknowns of a differential system with a single delay. the Cauchy function. This work was supported by the grant 1/0771/08 of the ——— Grant Agency of Slovak Republic (VEGA) and by the

101 V.5. Functional diﬀerential and diﬀerence equations

Averaging for impulsive functional diﬀerential equa- Lyapunov type inequalities for nonlinear impulsive dif- tions: a new approach ferential systems Marcia Federson Zeynep Kayar Av. Trabalhador Sao-carlense 400, CP 668, Sao Carlos, Middle East Technical University, Department of Math- SP 13560-970 Brazil ematics, Ankara, Cankaya 06531, Turkey [email protected] [email protected]

We obtain Lyapunov-type inequalities for systems of We consider a large class of functional differential equa- nonlinear impulsive differential equations. In particular, tions subject to impulse effects and state an averaging these sytems contain the Emden-Fowler-type systems result by means of the techniques of the theory of gen- and half linear systems in the special cases. In addi- eralized ordinary differential equations introduced by J. tion, as an application we make use of these inequalities Kurzweil. to derive some boundedness and disconjugacy criteria ——— and sufficient conditions for the asymptotic behaviour of solutions. Some bifurcation problems in the theory quasilinear integro differential equations ——— Evaluating the stochastic theta method Yakov Goltser Department of Computer Sciences and Mathematics, Conall Kelly Ariel University Center of Samaria, Ariel, 44837 Isser Department of Mathematics, University of the West In- Natanzon, 27/7, Pisgat Zeev, 97877 Jerusalem, Israel dies, Mona Kingston, Sn.Andrew 7, Jamaica [email protected] [email protected]

Our goal is to study parametrical perturbed nonlin- When a numerical method is applied to a differential ear quasiperiodic systems of differential and integro- equation, the result is a difference equation. Ideally the differential equations.Study bifurcation problems sim- dynamics of the difference equation should reflect those ilarly Hopf bifurcation, Bogdanov-Takkens bifurcation of the original as closely as possible, but in general this and bifurcation of invariant torus,based on the normal can be difficult to check. It is therefore useful to perform form theory and the truncated method for countable sys- a linear stability analysis: applying the method of inter- tems of ordinary differential equations. est to a linear test equation possessed of an equilibrium solution with known stability properties, and determin- ——— ing the asymptotic properties of the resultant difference equation for comparison. Stability in Volterra type population model equations We examine the issues that arise for this kind of analy- with delays sis in the context of stochastic differential equations, and Istvan´ Gyori¨ review the relevant literature. These issues have yet to Egyetem u. 10 Department of Mathematics, University be adequately addressed. We propose a new approach of Pannonia Veszprem, Veszprem County H-8200 Hun- and demonstrate its usage for the class of θ-Maruyama gary methods with constant step-size. [email protected] ———

In this talk some delay dependent and delay indepen- Delay-distribution effect on stability dent stability conditions will be given for differential Gabor Kiss equations arising in population dynamics. The proofs Department of Engineering Mathematics, University of are based on the construction of a Lyapunov functional Bristol Queen’s Building Bristol, South West England and some monotone techniques for nonautonomous sys- BS8 1TR United Kingdom tems. At the end of the talk we shall formulate some [email protected] open problems and conjectures. We consider the effect of delay distribution on retarted ——— functional differential equations with one delay. More On parameter dependence in functional differential specifically, we study the effect of delay distribution on equations with state-dependent delays the stability of solutions of first- and second-order equations by comparing the stability regions of the respective Ferenc Hartung equation with a single delay with that of the equation University of Pannonia Egyetem str 10 Veszprem, H- with distributed delays. 8200 Hungary [email protected] ——— Solutions of linear impulsive differential systems In this talk we study smooth dependence on parameters bounded on the entire real axis of solutions of several classes of functional differential Martina Langerova´ equations with state-dependent delays. As an applica- Dept. of Mathematics, Faculty of Science, University of tion of our results, we discuss the parameter estimation Zilina,ˇ Univerzitná1, 010 26 Zilina,ˇ Slovakia problems for FDEs with state-dependent delays using a [email protected] quasilinearization method. We consider the problem of existence and structure of ——— solutions bounded on the entire real axis of the linear

102 V.5. Functional differential and difference equations differential system with impulsive action at fixed points We shall discuss some results on the asymptotic be- of time haviour of the nonnegative solutions of systems of linear difference equations with asymptotically constant coef- x˙ = A(t)x + f(t), t 6= τ , i ficients. The main result describes the relationship be- ˛ ˛ n tween the nonnegative solutions of the perturbed system ∆x˛ = ai, t, τi ∈ R, i ∈ Z, ai ∈ R . t=τi and the positive eigenvalues and the corresponding non- Under the assumption that the corresponding homoge- negative eigenvectors of the limiting system. The proofs neous system is exponentially dichotomous on the semi- are based on Pringsheim’s Theorem and the Extended axes R+ and R− and by using the results of the well- Liouville Theorem from complex analysis. known Palmer lemma and the theory of pseudoinverse ——— matrices we establish necessary and sufficient conditions for the indicated problem. Co-authors: Oleksandr On singular models arising in hydrodynamics ˇ Boichuk, Jaroslava Skor´ıková. Irena Rachunkova This research was supported by the Grants 1/0771/08, Palacky University, Fakulty of Science, Dept. of Math- 1/0090/09 of the Grant Agency of Slovak Republic ematics, Tomkova 40, Olomouc, 77900 Czech Republic (VEGA) and APVV 0700-07. [email protected] ——— We investigate models arising in hydrodynamics. These Oscillatory and asymptotic properties of solutions of models have the form of the singular second order dif- higher-order difference equations of neutral type ferential equation Malgorzata Migda (p(t)u0(t))0 = p(t)f(u(t)) Institute of Mathematics, Poznan University of Technol- on the half-line. Here f is locally Lipsichtz on R and ogy, ul. Piotrowo 3A, Poznan, 60-965 Poland changes its sign and p is [email protected] continuous on [0, ∞) and p(0) = 0. A discrete formulation of this equation is investigated as well. We are inter- We consider higher-order linear difference equations ested in strictly increasing solutions and homoclinic solu- with delayed and advanced terms tions and provide conditions for p and f which guarantee m ∆ (xn − pxn−τ ) = qnxn−σ + hnxn+η the existence of such solutions. In particular cases a homoclinic solution determines an increasing mass density where p is a nonnegative number, τ, σ, η are positive in- in centrally symmetric gas bubbles which are surrounded tegers and (qn), (hn) are sequences of nonnegative real by an external liquid. numbers. ——— We give sufficient conditions under which all nonoscillatory solutions of the delayed part of the equation are Decoupling and simplifying of noninvertible difference unbounded and under which all nonoscillatory solutions equations in the neighbourhood of invariant manifold of the advanced part tend to zero as n → ∞. Andrejs Reinfelds We establish also sufficient conditions for the oscillation University of Latvia, Institute of Mathematics and Com- of all solutions of the full equation. puter Science; Rai¸nabulv¯aris 29, LV-1459, R¯ıga,Latvia ——— [email protected] Principal and non-principal solutions of impulsive differential equations with applications In Banach space X×E the system of difference equations Abdullah Ozbekler¨ x(t + 1) = g(x(t)) + G(x(t), p(t)), (*) Atılım University, Department of Mathematics, p(t + 1) = A(x(t))p(t) + Φ(x(t), p(t)) Kızılca¸sarKöyü, Incek˙ Gölba¸sı,Ankara 06836 Turkey [email protected] is considered. Sufficient conditions under which there is an local Lipschitzian invariant manifold u: X → E are In this work we first prove a theorem on the existence obtained. Using this result we find sufficient conditions of principal and nonprincipal solutions for second order of partial decoupling and simplifying of the system of differential equations having fixed moments of impulse noninvertible difference equations (*). actions. Next, by means of nonprincipal solution we give ——— new oscillation criteria for related impulsive differential Precise asymptotic behaviour of solutions of Volterra equations. Examples are provided with numerical simu- equations with delay lations to illustrate the importance of the study. David W. Reynolds ——— School of Mathematical Sciences, Dublin City Univer- Nonnegative iterations with asymptotically constant sity, Dublin 9, Ireland. coefficients [email protected]

Mihali Pituk This talk considers the rates at which solutions of Egyetem u. 10 Department of Mathematics, University Volterra equations with delay converge to asymptotic of Pannonia Veszprem, Veszprem County H-8200 Hun- equilibria. It is found that these convergence rates de- gary pend delicately on prescribed data. The results are es- [email protected] tablished using admissibility techniques. This work is motivated by logistic equations with inﬁnite delay.

103 V.5. Functional diﬀerential and diﬀerence equations

——— ——— On local stability of solutions of stochastic diﬀerence Existence and nonexistence of asymptotically periodic equations solutions of Volterra linear diﬀerence equations Alexandra Rodkina Ewa Schmeidel Department of Mathematics, University of the West In- Instytute of Mathematics, ul. Piotrowo 3A, Poznan, dies, Mona Kingston, Sn.Andrew 7, Jamaica Wielkopolska 60-965 Poland [email protected] [email protected]

We present results on the local stability of solutions of In this talk we investigate Volterra difference equation a stochastic difference equation with polynomial coeffi- of the form cients. Two cases are considered: when stochastic per- n X turbation are a state-independent and asymptotically x(n + 1) = a(n) + b(n)x(n) + K(n, i)x(i) fading and when stochastic perturbation are a state- i=0 dependent. where n ∈ N = {0, 1, 2,... }, a, b, x: N → R and ——— K : N × N → R, the special case of this equation is Volterra difference equation of convolution type Convergence of the solutions of a differential equation n with two delayed terms X x(n + 1) = Ax(n) + K(n − i)x(i). Miroslava R˚uˇzickovˇ a´ i=0 ˇ Faculty of science, University of Zilina, Slovak Republic This equation may be considered as a discrete analogue [email protected] of famous Volterra integrodifferential equation Z t In this contribution we deal with asymptotic behavior of 0 x (t) = Ax(t) + b(t − s)x(s)ds. solutions to a linear homogeneous differential equation 0 containing two discrete delays Such equation has been widely used as a mathematical y˙(t) = β(t)[y(t − δ) − y(t − τ)] (*) model in population dynamics. Both discrete equations represents a system in which the future state x(n + 1) + for t → ∞. We assume δ, τ ∈ R := (0, +∞), τ > δ, β : does not depend only on the present state x(n) but also + I−1 → R is a continuous function, I−1 := [t0 − τ, ∞), on all past states x(n − 1), x(n − 2), . . . , x(0). These sys- t0 ∈ R. Denote I := [t0, ∞) and the symbol “ ˙ ” de- tem are sometimes called hereditary. Given the initial notes (at least) the right-hand derivative. Similarly, if condition x(0) = x0, one can easy generate the solu- necessary, the value of a function at a point of I−1 is tion x(n, x0). Sufficient conditions for the existence of understood (at least) as value of the corresponding limit asymptotically periodic solutions of Volterra difference from the right. equation are presented. In addition we present sufficient The main results concern the asymptotic convergence of conditions for non-existence of an asymptotically peri- all solutions of Eq. (*). Especially we deal with so called odic solution satisfying some auxiliary conditions. The critical case with respect to the function β. When the results are illustrated by examples. function β is the constant function than this critical case is represented with the value β := (τ − σ)−1. The proof ——— of results is, except other, based on comparison of solu- Gene regulatory networks and delay equations tions of Eq. (*) with solutions of an auxiliary inequality which formally copies Eq. (*). Andrei Shindiapin This research was supported the Grant No 1/0090/09 Eduardo Mondlane University, Maputo, Mozambique of the Grant Agency of Slovak Republic (VEGA), by [email protected] the project APVV-0700-07 of Slovak Research and De- Gene regulatory networks consist of differential equa- velopment Agency and by the Slovak-Ukrainian project tions with smooth but steep nonlinearities (”sigmoids”). SK-UA-0028-07 (Ukrainian-Slovak project M/34 MOH As the number of genes may be rather large, any theoret- Ukraine 27.03.2008). ical or computer-based analysis of such networks can be This is joint work with Josef Dibl´ık. complicated. That is why a simplified approach based ——— on replacing sigmoids with step functions is widely used. However, this leads to some mathematical challenges, as Inverse problems of the calculus of variations for func- for instance analysis of stationary points belonging to tional differential equations the discontinuity set of the system (thresholds) cannot Vladimir Mikhailovich Savchin be done directly. Additional problems occur if one tries Peoples Friendship University of Russia, Mikluxo- to incorporate time delays into the network. The delay Maklaya street 6, Moscow, 117198 Russia effects naturally arise from the time required to com- [email protected] plete transcription, translation and diffusion to the place of action of a protein. We offer an algorithm of localiz- The problem of existense of solutions of inverse prob- ing stationary points in the presence of delays as well as lems of the calculus of variations for partial differencial stability analysis around such points. This algorithm is difference operators is investigated. Necessary and suf- combined with a method to study delay systems by re- ficient conditions of potentiality for such operators are placing them with an equivalent system of ordinary dif- obtained. Methods of construction of variational multi- ferential equations, commonly known as the linear chain plies are suggested. trick. However, a direct application of this ”trick” is not

104 V.6. Mathematical biology possible in our case, so that we suggest a modification Republic of it based on the general framework of representing de- [email protected] lay equations as ordinary differential equations using the integral transforms”. This is joint work with Arcady Ponosov. This contribution deals with systems of linear generalized linear differential equations of the form ———

The moment problem approach for the zero controlla- Z t bility of ecolution equations x(t) = xe+ d[A(s)] x(s)+g(t)−g(a), t ∈ [a, b], (*) a Benzion Shklyar 52 Golomb St., P.O.B. 305, Dept. of Appl. Math, Holon where −∞ < a < b < ∞,A is an n × n-complex ma- Institute of Technology, Holon, 58102 Israel trix valued function, g is an n-complex vector valued shk [email protected] function, A has a bounded variation on [a, b] and g is regulated on [a, b]. The integrals are understood in the The exact controllability to the origin for linear evolu- Kurzweil-Stieltjes sense. tion control equation is considered.The problem is inves- Our aim is to present some new results on continuous tigated by its transformation to infinite linear moment dependence of solutions to linear generalized differential problem. equations (*) on parameters and initial data. Controllability conditions for linear evolution control equations have been obtained. The obtained results are ——— applied to the zero controllability for partial differential and functional differential equations. Lyapunov type inequalities on time scales: A survey ——— Mehmet Unal¨ Properties of maximal solutions of autonomous Bahcesehir University, Çıra˘gan Caddesi, Osmanpa¸sa functional-differential equations with state-dependent Mektebi Sokak No. 4–6, Be¸sikta¸s,Istanbul 34353 Turkey deviations [email protected] Svatoslav Stanek Palacky University, Fakulty of Science, Dept. of Math- We survey Lyapunov type inequalities for linear and ematics, Tomkova 40, Olomouc, 77900 Czech Republic nonlinear dynamic equations on time scales. The in- [email protected] equalities contain the well-known classical Lyapunov in-

00 equalities as special cases. We also give some applica- Equations of the type x +x(t−kx)) = 0 are considered. tions to illustrate the importance of such inequalities. Here k is a positive parameter. It is described (i) the set 0 of all periodic solutions x satisfying x < 1/k and (ii) the ——— set of all maximal solutions x (that is, solutions which 0 have no extension) satisfying x ≥ 1/k. Interval criteria for oscillation of delay dynamic equa- ——— tions with mixed nonlinearities

Boundedness character of some classes of difference Agacık˘ Zafer equations Department of Mathematics Middle East Technical Uni- versity Cankaya, Ankara 06531 Turkey Stevo Stevic Mathematical Institute of the Serbian Academy of Sci- [email protected] ences, Knez Mihailova 36/III, Beograd, 11000 Serbia [email protected] We obtain interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlineari- Some results on the boundedness character of the pos- ties on an arbitrary time scale . All results are new itive solutions of the following two classes of difference T even for = and = . Analogous results for re- equations T R T Z lated advance type equations are also given, as well as xp extended delay and advance equations. The theory can x = A + n , n ∈ ; n+1 q r N0 be applied to second order delay dynamic equations re- xn−1xn−2 gardless of the choice of delta (∆) or nabla (∇) deriva-  p ff xn tives. xn+1 = max A, q r , n ∈ N0, xn−1xn−2 ——— where the parameters A, p, q and r are positive numbers, are presented. ——— Continuous dependence of solutions of generalized or- V.6. Mathematical biology dinary differential equations on a parameter Organisers: Milan Tvrdy Robert Gilbert Institute of Mathematics, Academy of Sciences of the Czech Republic, Zitna 25, Praha 1, CZ 115 67 Czech

105 VI. Others

—Abstracts— Due to the presence of the fluid and solid phase, the modeling of cancellous bone represents a complex, exten- Cancellous bone with a random pore structure sive task where the dynamic investigation and viscosity effects must be taken into consideration. The already es- Robert Gilbert tablished approach for the investigation of this material Department of Mathematics, University of Delaware, type is Biots method, originally developed for simulat- 317 Ewing Hall, Newark, DE 19716 United States ing saturated porous materials. In this contribution we [email protected] present the homogenization multiscale FEM as an alternative to Biots method. The motivation for this choice is We continue the study of acoustic wave propagation for decreasing the extent of the necessary laboratory inves- an elastic medium that is randomly fissured. Moreover, tigations. According to the multiscale FEM, the bone is the fissures are assumed to be statistically homogeneous. understood as the homogenized medium whose effective Although the underlying stochastic process does not nec- material parameters are obtained by the analysis of an essarily have to be ergodic, we assume for simplicity of appropriate representative volume element (RVE). This exposition that it is. This allows us to obtain an exis also the main topic of the presentation: a compari- plicit and computationaly easier auxillary problem in a son of the effective values obtained by studying different Representative Elementary Volume. In a later work we types of RVEs where the particular attention is paid to intend to study the more general case. the numerical values for Youngs modulus and attenu- This is joint work with Ana Vasilic. ation coefficient. The distinction between the models ——— pertains to the geometry of the solid frame of the RVE, the type of the applied elements as well as the type of New computer technologies for the construction and the coupling conditions on the interface of the phases. numerical analysis of mathematical models for molecular genetic systems ——— Irina Alekseevna Gainova Bone growth and destruction at the cellular level: a Sobolev Institute of Mathematics, Siberian Branch of mathematical model the Russian Academy of Sciences Acad. Koptyug av- Mark D. Ryser enue, 4 Novosibirsk, Novosibirsk region 630090 Russia McGill University, W. Burnside Hall, Room 1005, 805 [email protected] Sherbrooke Street, Montral, Quebec H3A 2K6 Canada [email protected] We have created an integrative computer system, which includes three program modules (Institute of Cytol- The process of bone destruction and subsequent growth ogy and Genetics, SB RAS): GeneNet, MGSgenera- is continually occurring in healthy bone tissue. This tor, MGSmodeller, and the software package STEP+ process is referred to as ’remodeling’ and plays a key (Sobolev Institute of Mathematics, SB RAS). The sys- role in many pathologies such as osteoporosis and os- tem is used to construct and numerically analyze mod- teoarthritis. We describe remodeling at the cellular level els describing dynamics of the molecular genetic sys- and discuss the cells and biochemical pathways involved. tems (MGS) functioning in pro- and eukaryotes. Us- We then develop a mathematical model for remodeling, ing module GeneNet we can reconstruct structure func- consisting of a system of coupled nonlinear PDEs. We tional organization of gene networks. We use MGSgen- discuss how physiological parameters may be obtained erator as an intermediate module in generation of math- through scaling of the equations and we comment on ematical models based on gene networks reconstructed their mathematical properties. Numerical experiments in GeneNet. Moreover, in the module MGSgenerator validating the model will be presented. This is joint we represent obtained mathematical models in the input work with Nilima Nigam (SFU) and Svetlana Komarova format of STEP+. Module MGSmodeller contains tools (McGill). for the gene network models to be developed and numerically analyzed. Package STEP+ is intended for the ——— numerical analysis of mathematical models represented by autonomous systems ODEs. We have tested our integrated system on the MGS model for intracellular auxin VI. Others metabolism in a plant cell. This work has been partially supported by the Siberian Organisers: Branch of the Russian Academy of Sciences (Interdisci- local organising committee plinary integration project Post-genomic bioinformatics: computer analysis and modeling of the molecular genetic systems, No. 119). —Abstracts— ——— The relationship between Bezoutian matrix and New- Application of the multiscale FEM in modeling the can- ton’s matrix of divided differences and separation of cellous bone zeros of interpolation polynomials Sandra Ilic Ruben Airapetyan Institute of Mechanics, Ruhr-University of Bochum, Kettering University, 1700 W Third Ave. Flint, Michi- Bochum, 44780 Germany gan 48504, United States [email protected] [email protected]

106 VI. Others

Let x1, . . . , xn be real numbers, Pn(x) = an(x − In our work, we use the ADM method for solving some n x1) ··· (x−xn). Denote by D g the matrix of generalized nonlinear evolution equations with time and space frac- divided differences of function g in Newton’s interpola- tional derivative. Then we use the Extended Tanh tion formula with nodes x1, . . . , xn and by Gn(x) the method to formally derive traveling wave solutions for Newton’s interpolation polynomial of function g. De- some evolution equations. The obtained solutions in- note by B = B(Pn(x),Gn(x)) the Bezoutian matrix of clude, also, kink soltuions. Pn and Gn. The relationship between the corresponding principal minors of the matrices Dng and B counted ——— from the left lower corner is establish. Then, it follows Boundary-value problems for generalized axially- that if these principal minors of the matrix of divided symmetric Helmholtz equation differences are positive or have alternating signs then the roots of the interpolation polynomial are real and Anvar Hasanov separated by the nodes of interpolation. 34 Durmon yoli, Tashkent branch of the Russian State University of oil and gas named after Gubkin, Tashkent, ——— Tashkent 100125, Uzbekistan Bayesian shrinkage estimation of parameter exponen- [email protected] tial distribution In this talk several main boundary-value problems such Hadeel Alkutubi the Dirichlet, Neumann problem and other problems will B-23-1 , The Heritage, JLN SB, Dagang Mines Resort be considered. The unique solvability of afore-mentioned City, Seri Kembangan, Serdang, 43300 Malaysia problems will be proved. [email protected] ——— In this paper, we would like to test the best estima- Asymptotic extension of topological modules and alge- tor (smallest MSE and MPE) of shrinkage estimator of bras parameter exponential distribution . To do this , we derived this estimators depend on Bayesian method with Maximilian Hasler Jeffreys prior information and square error loss func- Laboratoire AOC, Universit Antilles-Guyane, B.P. 7209, tion . To compared between estimators we used MSE campus de Schoelcher, Schoelcher, Martinique 97275 and MPE with respect of simulation study. We found France the shrinkage estimator between Bayes estimators under [email protected] different loss function is the best estimator. ——— Given a topological R-module or algebra E and an asymptotic scale M ⊂ RΛ, we exhibit a natural M- Interpolation beyond the interval of convergence: An extended topology on the sequence space EΛ, and define extension of Erdos-Turan Theorem the M-extension of E as the Hausdorff space associated with the subspace of nets for which multiplication is con- Mohammed Bokhari tinuous with respect to this topology. Department of Mathematics & Statistics, King Fahd Commonly used spaces of generalized functions are ob- University of Petroleum & Minerals, Dhahran, Saudi tained as special cases, but this new approach applies in Arabia many different situations. It also allows the iteration of [email protected] the construction, which is not possible with previously existing theories. An elegant result due to Erdos and Turan states that We use only the topology, i.e. neighbourhoods of zero, the sequence of Lagrange interpolants to a given contin- but not its explicit definition in terms of seminorms, in- uous function f at the zeros of orthogonal polynomials ductive or projective limits etc., which is particularly over a closed interval converges to f in the mean square convenient in non-metrizable spaces. sense. We introduce certain sequences of polynomials Many ideas commonly used in the context of generalized which preserve both interpolation as well as convergence functions (functoriality, association, sheaf structure, al- properties of Erdos-Teran Theorem. In addition, they gebraic analysis, . . . ) can be applied to a large extent. interpolate f at a finite number of pre-assigned points Reasoning on a category-theoretic level allows to estab- lying outside the underlying open interval. We shall in- lish several results so far only known for particular cases, troduce a method to construct the suggested polynomi- for the whole class of such spaces. als and also investigate their properties. Computational aspects will also be discussed. ——— ——— Approximation of fractional derivatives The ADM method and the Tanh method for solving S. Moghtada Hashemiparast some non linear evolutions equations Mathematics and Statistics, Jolfa Ave, Seyed Khandan, Tehran 193953358 Iran Zoubir Dahmani [email protected] Department of Mathematics, Faculty of Sciences, Uni- versity of Mostaganem Les HLM, 21 street les HLM Series represantations are presented to approximate the mostaganem, mostaganem 27000 Algeria fractional derivatives which have extensive application [email protected] in ordinary,partial difrential equations and specilly the stable probability distributions.The convergence of the

107 VI. Others series are considered and are applied to solving the equa- will be discussed. Here α, β, zeta are constants, more- tions,ﬁnally toillustrate the accuracy of the apprpxima- over 0 < 2α, 2β, ζ < 1. tions examples are solved. ——— ——— A uniﬁed presentation of a class of generalized Hum- Discrepancy estimate for uniformly distributed se- bert polynomials quence Nabiullah Khan Hailiza Kamarulhaili Department of Applied Mathematics, Z.H. College of School of Mathematical Sciences Universiti Sains Engineering and Technology, Aligarh Muslim University, Malaysia Minden, Penang 11800 Malaysia Aligarh 202002 India [email protected] nabi [email protected]

A general metrical result of discrepancy estimate related The principal object of this paper is to present a natural to uniform distribution of a sequence is proved . This further step toward the unified presentation of a class of work extends result of R.C. Baker where the sequence Humbert’s polynomials which generalizes the wellknown can be assumed to be real. The lighter version of this class of Gegenbauer, Humbert, Legendre, Tchebycheff, theorem will also be discussed in this talk. Pincherle, Horadam, Dave, Kinnsy, Sinha, Shreshtha, Horadam-Pethe, Djordjevie, Gould, Milovanović and ——— Djordjević,Pathan and Khan polynomials and many not Bounded linear operators on l-power series spaces so wellknown polynomials. We shall give some basic relations involving the generalized Humbert polynomials Erdal Karapinar and then take up several generating functions, hyperge- ATILIM University, Department of Mathematics, Kizil- ometric representations and expansions in series of some casar Koyu, INCEK ANKARA, 06836 Turkey relatively more familier polynomials of Legendre, Gegen- [email protected] bauer, Rice, Hermite, Jacobi, Laguerre Fasenmyer Sis- ter M. Celine, Bateman, Rainville and Khandekar. We Let A be the class of Banach space ` of scalar sequences also show that our results provide useful extensions of with a norm k · k` such that known results of Dilcher, Horadam, Sinha, Shreshtha, Milovanović-Djordjević,Pathan and Khan. (i) a = (ai) ∈ l∞, x = (ξi) ∈ ` ⇒ ax = (aiξi) ∈

`, kaxk` ≤ kakl∞ kxk`, ———

(ii) keik` = 1, ∀i ∈ N where ei = (δij )j∈N. Direct estimate for modified beta operators For a given ` ∈ A and a Köthematrix A, we de- Lixia Liu ` fine `-Köthespace K (ai,n) as a Fréchet space of all Yuhua east Road 113, College of Mathematics and Infor- scalar sequences x = (ξi) such that (ξiai,n) ∈ ` for mation Science, Hebei Normal University, Shijiazhuang, each n, endowed with the topology of Fr echet space, Hebei Province 050016 China determined by the canonical system of norms {kxkn = [email protected] k (ξiai,n) k`, n ∈ N}. 2 We write (E,F ) ∈ B, if every continuous linear map Use the Ditzian modulous of smoothness ωϕλ (f, t), (0 ≤ from E to F is bounded. In 1983, D.Vogt has character- λ ≤ 1), to study the pointwise direct results for modified ized those Fréchet spaces E for which (E,Kl∞ (A)) ∈ B Beta operators, which extend the approximation result holds. for Beta operators. This gives also a characterization of (E,Kc0 (A)) ∈ B. ——— We extend this results and prove that Mathematical model of an undergorund nuclear waste Theorem. For Fréchet space E and ` ∈ A, disposal site (E,Kl1 (A)) ∈ B ⇒ (E,K`(A)) ∈ B ⇒ (E,Kl∞ (A)) ∈ B. Eduard Marusic-Paloka Department of Mathematics, University of Zagreb, Bi- Theorem. For Fréchet space F and ` ∈ A, jenicka 30, Zagreb, 10000 Croatia (K`(A),F ) ∈ B ⇒ (Kl1 (A),F ) ∈ B. [email protected] ——— The goal of our research is to find an accurate model for On a three-dimensional elliptic equation with singular numerical simulations of the nuclear waste disposal site. coefficients The purpose of such model is to perform safety analysis of the site and find out its possible impact on the Erkinjon Karimov biosphere. Due to the large dimension of the site and Durmon yuli street 29, Akademgorodok Tashkent, very long lifetime of radioelements, realistic experiments Tashkent 100125 Uzbekistan are not possible. Thus, predictions based on numerical [email protected] simulations are all we have. Starting from the microscopic model given by the In this talk some questions such as finding fundamental reaction-diffusion-convection equation, using the asymp- solutions, investigations of main boundary-value prob- totic analysis and homogenization, we derive a macro- lems for an equation scopic model and discuss ity accuracy. 2α 2β 2ζ u + u + u + u + u + u ——— xx yy zz x x y y z z

108 VI. Others

Compact and coprime packedness and semistar opera- if and only if (i) φ(x) ≥ 0 when the sequence x = (xk) tions has xk ≥ 0 for all k, (ii) φ(e) = 1, where e = (1, 1, 1, ··· ), and (iii) φ(x) = φ((x )) for all x ∈ ` . Throughout Abdeslam Mimouni σ(k) ∞ this paper we consider the mapping σ which has no fi- Department of Mathematics and Statistics King Fahd nite orbits, that is, σp(k) 6= k for all integer k ≥ 0 and University of Petoleum and Minerals Dhahran, Estern p ≥ 1, where σp(k) denotes the pth iterate of σ at k. 31261 Saudi Arabia Note that, a σ-mean extends the limit functional on the [email protected] space c in the sense that φ(x) = lim x for all x ∈ c. ∞ In this paper we define a new sequence space Vσ (λ) In this talk we will present new developments on the which is related to the concept of σ-mean and the se- study of compact and coprime packedness of an inte- quence λ = (λn) described as above and characterize gral domain with respect to a star operation of finite ∞ ∞ the matrix classes (`∞,V (λ)) and (`1,V (λ)). character. Let R be an integral domain with quotient σ σ Let λ = (λn) be a non-decreasing sequence of positive field K and let ∗ be a star operation of finite type on numbers tending to ∞ such that λn+1 ≤ λn + 1, λ1 = 0. R.A ∗-ideal I is said to be ∗-compaclty (respectively Then we define the following sequence space and show ∗-coprimely) packed if whenever I ⊆ S P , where α∈Ω α that it is a BK-space: {Pα}α∈Ω is a family of ∗-prime ideals of R, I is actually ∞ contained in Pα (resp. (I + Pα)∗ ( R) for some α ∈ Ω; Vσ (λ) := {x ∈ `∞ : sup |τmn(x)| ≤ ∞}, and R is said to be ∗-compactly (resp. ∗-coprimely) m,n packed if every ∗-ideal of R is ∗-compactly (resp. ∗- where X coprimely) packed. In the particular case where ∗ = d τmn(x) = (1/(λm)) xσj (n). is the trivial operation, we obtain the so-called com- j∈`m pactly and coprimely packed domains. Our objectives is to study some ring-theoretic aspects of these notions ——— in different classes of integral domains, paying particu- New convection theory for thermal plasma and NHD lar attention to the the t-operation as the largest and convection in rapidly rotating spherical configurations well-known operation. Ali Mussa ——— King Abdulaziz City for Science and Technology Build- Characterization of some matrix classes involving (σ, λ)- ing # 2 King Abdullah Bin Abdulaziz Street Riyadh, convergence Riyadh 6086/11442 Saudi Arabia [email protected] S. A. Mohiuddine Department of Mathematics Aligarh Muslim University We extend Jones-Soward-Mussa (JSM) theory (2000): Aligarh, Uttar Pradesh 202002 India “analytic and computational solution for E → 0 and [email protected] P r/E → ∞”. We also make use of Zhang (2001) ansatz for: “E 1 arbitrary but fixed and 0 ≤ P r < ∞” Let σ be a one-to-one mapping from the set N of natural the so-called enhanced Nearly Geostrophic Inertial Wave numbers into itself. A continuous linear functional ϕ on (NGIW) approach. Such extension represented as a con- the space `∞ of bounded single sequences is said to be struction of a new MHD plasma convection and magne- an invariant mean or σ-mean if and only if (i) ϕ(x) ≥ 0 toconvection force theory. The flow field confinement in if x ≥ 0 (i.e. xk ≥ 0 for all k); (ii) ϕ(e) = 1, where the study assumed to be in spherical geometry config- e = (1, 1, 1, ··· ); (iii) ϕ(x) = ϕ((xσ(k))) for all x ∈ `∞. uration and our investigation is made under the basis Let λ = (λn) be a non-decreasing sequence of positive of magnetic balance and scaling theory. Furthermore, numbers tending to ∞ such that strong inertial turbulence can be achieved in presence of high Reynolds number so strong forces govern the flow λn+1 ≤ λn + 1, λ1 = 0. fields have to be sufficiently understood. Indeed, strong In this paper, first we define (σ, λ)-convergence and show rotation and strong magnetic field for the flow field in- λ side the spherical rotating geometry; take into consid- that V is a Banach space with kxk = sup |tmn(x)|, σ m,n eration the effect of the anticipated vigorous convection where V λ is the set of all (σ, λ)-convergent sequences σ and magnetoconvection in the flow field confinement. x = (xk). We also define and characterize (σ, λ)- conservative, (σ, λ)-regular and (σ, λ)-coercive matrices. ——— λ Further, we characterize the class (`1,Vσ ), where `1 is Characterizations of Isometries on 2-modular spaces the space of all absolutely convergent series. Kourosh Nourouzi ——— Department of Mathematics, K.N. Toosi University of Sequence spaces of invariant mean and some matrix Technology Tehran, Tehran 16315-1618 Iran transformations [email protected] Mohammad Mursaleen Let X be a real vector space of dimension greater than Department of Mathematics Aligarh Muslim University one. A real valued function ρ(·, ·) on X2 satisfying the Aligarh, UP 202002 India following properties is called a 2-modular on X, for all [email protected] x, y, z ∈ X: 1. ρ(x, y) = 0 if and only if x, y are linearly depen- Let σ be a one-to-one mapping from the set N of natural dent, numbers into itself. A continuous linear functional φ on the space `∞ is said to be an invariant mean or a σ-mean 2. ρ(x, y) = ρ(y, x),

109 VI. Others

3. ρ(−x, y) = ρ(x, y), of one of the major classical problems of theoretical mechanics, dynamics of a solid body with one fixed point 4. ρ(x, αy + βz) ≤ ρ(x, y) + ρ(x, z), for any in a gravity field. Motion of a solid body with one fixed nonnegative real numbers α, β with α + β = 1. point is described by the well-known system of Euler In this talk, we discuss on the characterization of isome- and Poisson equations. It is known the general solution tries defined on 2-modular spaces. exists if one considers two first terms of force function expansion into a series. By original change of variables ——— the system is reduced to the normal form with the first On r-imbalances in tripartite r-digraphs integral of norm type. The solution of this system is considered as the non-perturbed motion and it is inves- Shariefuddin Pirzada tigated on stability. The procedure is offered for ob- King Fahd University of Petroleum and Minerals, taining of asymptotically steady motion in general case. Dhahran, 31261 Saudi Arabia The controlling force nature was defined. This method [email protected] was applied to three cases with special restrictions on the bodys inertia moments, so-called generalized clas- A tripartite r-digraph(r ≥ 1) is an orientation of a tri- sical cases of Euler, Lagrange and Kovalevskaya. The partite multigraph that is without loops and contains numerical solution for the problem in Euler case was atmost r edges between any pair of vertices from dis- constructed. tinct parts. For any vertex x in a tripartite r-digraph + − ——— D(U, V, W ), let dx and dx denote the outdegree and + − A Lizorkin type theorem for Fourier series multipliers in indegree respectively of x. Define aui = dui − dui , + − + − regular systems bvj = dvj −dvj and cwk = dwk −dwk as the r-imbalances of the vertices ui in U, vj in V and wk in W respectively. Lyazzat Sarybekova In this paper, we characterize r-imbalances in tripartite Munaitpasov 7, Astana, 010010 Kazakhstan r-digraphs and obtain some results. [email protected]

——— A new Fourier series multiplier theorem of Lizorkin type Invariance conditions and amenability of locally com- is proved for the case 1 < q < p < ∞. The result is given pact groups for a general regular system and, in particular, for the trigonometrical system it implies an analogy of the orig- Hashem Parvaneh Masiha inal Lizorkin theorem. Department of Mathematics, Faculty of Science, K. N. Toosi University of Technology. No. 41, Kavian St., ——— Seyyed Khandan Bridge (N.), Shariati Ave., Tehran, Inverse-closedness problems in the stability of se- Tehran 16315-1613 Iran quences in Banach Algebras [email protected] Pedro A. Santos Departamento de Matemática, Instituto Superior Adler and Hamilton showed that a semigroup S is left Técnico,Av Rovisco Pais, Lisboa, 1049-001 Portugal amenable if and only if it satisfies the following invari- [email protected] ance condition. For any subsets A1,A2, ··· ,Ak of S and any s1, s2, ··· , sk ∈ S, there exists a nonempty finite We are concerned with the applicability of the finite sec- −1 subset E of S such that n(si Ai ∩ E) = n(Ai ∩ E), for tions method to operators belonging to the closed subal- −1 p i = 1, 2, ··· , k, where s A = {t ∈ S : st ∈ A} and n(A) gebra of L(L (R)), 1 < p < ∞, generated by operators is the number of elements in A. In this talk, we shall of multiplication by piecewise continuous functions in prove an analogous result for locally compact groups. R˙ and operators of convolution by piecewise continuous More precisely, we show that amenability of a locally Fourier multipliers. compact group G is equivalent to: For any λ-measurable The usual technique is to introduce a larger algebra of subsets A1,A2, ··· ,Ak of G, any g1, g2, ··· , gk ∈ G sequences, which contains the special sequences we are and any ε > 0, there exists a compact subset K of G interested and the usual operator algebra generated by −1 such that |λ(gi Ai ∩ K) − λ(Ai ∩ K)| < ελ(K), for the operators of multiplication and convolution. There i = 1, 2, ··· , k, where λ(A) denotes the left Haar mea- is a direct relationship between the applicability of the sure of A. In this paper, we suppose that G be a locally finite section method for a given operator and invertibil- compact group and λ a fixed left Haar measure on G. We ity of the corresponding sequence in this algebra. let X = {K ⊂ G : K is compact and λ(K) > 0}. For But, contrarily to the C∗ case and Banach analogue for ¯ 1 R f ∈ L∞(G), we define f(K) = λ(K) K fdλ, K ∈ X, Toeplitz operators, in our case several inverse-closedness then f¯ : X → R is well defined. problems must be solved. ——— ——— Smoothing effects for periodic NSE in critical Sobolev Motion stabilisation of a solid body with fixed point space Zaure Rakisheva Ridha Selmi Al-Faraby Kazak National University, Almaty, Kazak- Department of Mathematics, Faculty of Sciences of stan Gabes, 6072, TUNISIA zaure [email protected] [email protected]

The problem of a solid body dynamics in the central We prove smoothing eﬀects for 3D incompressible Navier Newton ﬁeld of forces is considered. It is generalization Stokes Equation for initial data belonging to critical

110 VI. Others

1 3 Sobolev Space H 2 (T ). on the recent results of Z.Y.Huang [Equivalence theo- Asymptotic behavior of the global solution when the rems of the convergence between Ishikawa and Mann it- time goes to +∞ is studied. erations with errors for generalized strongly successively pseudocontractive mappings without Lipschitzian as- ——— sumptions, J.Math.Anal.Appl. 329(2007),935-947], Z.Y. Large deviations and almost sure convergence Huang, F.W. Bu, M.A. Noor [On the equivalence of the convergence criteria between modified Mann-Ishikawa Mariana Sibiceanu and multistep iteration with errors for strongly pseudo- Gh.Mihoc-C.Iacob Institute of Mathematical Statistics contractive operators, Appl. Math. Compt. 181(2006), and Applied Mathematics, Romanian Academy, Calea 641-647], B.E.Rhoades, S.M.Soltuz [The equivalence be- 13 Septembrie Nr.13, Sector 5 Bucharest, 050711 Ro- tween the convergences of Ishikawa and Mann itera- mania tion for an asymptotically non-expansive in the inter- [email protected] mediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289(2004), 266-278] In our setup, the Large Deviation Principle for a se- and B.E.Rhoades, S.M.Soltuz [The equivalence between quence P (n) of probabilities on a separable Banach Mann-Ishikawa iterations and multi-step iteration, Non- space E, with a convex good rate function I is assumed, linear Anal.58(2004),219-228] among others. also the existence of the finite limits g(w) of the associ- ——— ated logarithmic moment generating function. We establish precise upper and lower bounds of the val- A characterization for multipliers of weighted Banach ues that P (n) assigns almost sure in the weak and strong valued Lp(G)-spaces topology of E, respectively, determined by the amounts Serap Oztop of the canonical dual product on E0 × N, N being the Istanbul University, Faculty of Sciences, Istanbul, nullifying set of the rate function. Vezneciler 34134 Turkey Also, we reveal the significance of the derivative of the [email protected] function g(tw) of real t for the almost sure convergence, in the situation when g is Gateaux differentiable on Let G be a locally compact group, 1 < p < ∞. The (E0, t(E0,E)). aim of this paper is to characterize the multipliers of ——— the weighted Banach valued intersection Lp(G) spaces as the space of multipliers of a certain Banach algebra. The k- Model in Turbulence ——— Tanfer Tanriverdi Harran University, Faculty of Arts and Sciences, Depart- Stationary motion of the dynamical symmetric satellite ment of Mathematics, Sanlurfa 63300, Turkey in the geomagnetic field [email protected] Karlyga Zhilisbaeva Al-Faraby Kazak National University, Almaty, Kazak- We prove analytically the existence of self-similar solu- stan tions for the k- model arising in the evolution of turbu- [email protected] lent bursts by employing the topological shooting technique where α > β with the some other conditions. Stationary solutions of the system of the satellite’s mo- The first author was supported by the Scientific and tion equations are of special interest for the various prob- Technological Research Council of Turkey (TUBITAK).¨ lems of space researches, and first of all for the satellite’s He is also thankful to the Oxford Center for Nonlinear magnetic stabilization. PDE, and to the Mathematical Institute of the Univer- In the paper stationary motions of the equatorial magne- sity of Oxford, for the hospitality they offered him during tized dynamically symmetric satellite round the centre his visit. of mass on a circular orbit are considered. Strong mag- This is joint work with Bryce McLeod (Oxford). nets are placed on the satellite’s board. Perturbations are taken into account, caused by insignificant deviation ——— of a satellite’s axis of dynamic symmetry and by mag- The equivalence between modified Mann (with errors), netization of its cover. Ishikawa (with errors), Noor (with errors) and modified The equations of the satellite’s perturbed motion in Eu- multi-step iterations (with errors) for non-Lipschitzian ler’s canonical variables are obtained. Conditions of sta- strongly successively pseudo-contractive operators tionary motion existence are defined, necessary and sufficient conditions of their stability are found with taking Johnson Olaleru into account of small perturbations. Mathematics Department, University of Lagos, Univer- sity of Lagos Road, Yaba, Lagos, Nigeria ——— [email protected]

In this paper, the equivalence of the convergence between modiﬁed Mann(with errors), Ishikawa(with errors), Noor(with errors) and modiﬁed multistep iteration(with errors) is proved for generalized strongly successively pseudocontractive mapping without Lips- chitzian assumption. Our results generalise and improve

111

Index

Abate, M., 97 Budochkina, S.A., 97 Dovbush, P., 24 D’Abbico, M., 71 Burenkov, V., 3, 51, 60 Dragoni, F., 3, 95 Abdenur, F., 97 Burns, K., 98 Du Bau-Sen, 98 Abdous, B., 47 Buzano, E., 63 Du Jinyuan, 26, 28 Agranovich, M., 60, 63 Airapetyan, R., 106 Câmara,C., 45 Ebert, M., 72 Aksoy, U., 27 Capdeboscq, Y., 85 Ekincioglu, I., 52 Alimov, S., 60 Cardon, D., 33 El-Nadi, K., 93 Aliyev, T., 23 Carro, M., 52 Eller, M., 77 Alkutubi, H., 107 Caruana, M., 91 Elliott, N., 3 Almeida, A., 51 Casado-Diaz, J., 86 Elton, D., 60 Alves, J.F., 97 Castro, L., 45 Englis, M., 46 D’Ancona, P., 80 Catana, V., 63 Eriksson, S.-L., 37 Applebaum, D., 90 Catania, D., 81 Erkip, A., 81 Aripov, M., 81 Cattaneo, L., 3 Assal, M., 56 Cavalcanti, M., 77 Fang Daoyuan, 72 Aulaskari, R., 23, 56 Celebi, O., 3, 28 Farwell, R., 43 Aydin, I., 51 Cerejeiras, P., 36 Faustino, N., 43 Aykol, C., 51 Charalambides, M., 34 Federson, M., 102 Chazottes, J.-R., 98 Fei, M.-G., 37 Chen Kuan-Ju, 81 Fernández,A., 34 Babych, N., 85 Chen Qiuhui, 49 Ferreira, M., 37 El Badia, A., 87 Chen Yi-Chiuan, 98 Field, M., 98 Bakry, D., 95 Cherdantsev, M., 86 Fisk, S., 34 Ball, J., 15 Chiba, Y., 63 Fokas, T., 43 Ballantine, C., 23 Chinchaladze, N., 28 Forni, G., 98 Bally, V., 90 Cho, D.H., 49 Fragnelli, G., 77 Barsegian, G., 33 Choe, B.R., 56 Franek, P., 37 Barthe, F., 95 Choi, Q-H., 71 Franssens, G.R., 37 Baˇstinec,J., 101 Choulli, M., 88 Freitas, J., 98 Begehr, H., 3, 26, 27 Cohen, L., 63 Fujita, K., 48 Bellieud, M., 85 Colombini, F., 71 Fujiwara, H., 49 Berezansky, L., 100, 101 Colombo, F., 36, 88 Furutani, K., 64 Berger, P., 97 Cordero, E., 63 Berglez, P., 27 Coulembier, K, 37 Gaiko, V., 98 Berlinet, A., 3, 47 Crisan, D., 3, 90, 91 Gainova, I.A., 106 Bernstein, S., 42 Cruzeira, A.B., 91 Galleani, L., 64 Besov, O., 51 Csordas, G., 33, 34 Galstyan, A., 72, 77 de Bie, H., 36 Garello, G., 64 Bisi, C., 36 Dahmani, Z., 107 Garetto, C., 65 Bobkov, S., 95 Dai Daoquin, 26 Gauthier, P., 34 Bociu, L., 76 Dalla Riva, M., 23 Gedif Ayele, T., 51 Bock, S., 42 Dallakyan, G., 52 Geisinger, L., 60 Boichuk, A., 101 Daoulatli, M., 77 Gentil, I., 95 Bojarski, B., 3, 23 Datt Sharma, S., 47 Gentili, G., 37 Bokhari, M., 107 Dattori da Silva, P., 64 Georgiev, S., 43 Bolosteanu, C., 27 Daveau, C., 88 Georgiev, V., 72, 80 Boukrouche, M., 76 Davie, A., 91 Ghergu, M., 82 Boulakia, M., 77 Davies, B., 3, 16, 59 Ghisa, D., 24 Boulbrachene, M., 95 Davis, M., 91 Ghisi, M., 73 Boutet de Monvel, L., 15 Dehgan, M., 81 Gil, J., 65 Bouzar, C., 63 Del Santo, D., 71 Gilbert, R., 3, 105, 106 Boykov, I., 27 Delgado, J., 64 Giorgadze, G., 28 Boza, S., 52 Diblik, J., 100, 101 Girela, D., 56 Branden, P., 33 Doi, K., 81 Gobbino, M., 73 Britvina, L., 48 Dolićanin,D., 49 Golberg, A., 24 Brody, D., 90 Domingos Cavalcanti, V., 77 Goldshtein, V., 53 Bruning, E., 3 Domoshnitzky, A., 101 Goltser, Y., 102 Bucci, F., 76 Donaldson, S., 16 Golubeva, V.A., 86

113 Index

Gonzalez, M.J., 57 Kelly, C., 102 Marquez, A., 57 Gramchev, T., 65 Kendall, W., 92 Marti, J.-A., 67 Gramsch, B., 65 Kenig, C., 17 Martin, M., 39 Graubner, S., 28 Khan, N., 108 Martinez, P., 78 Grigoryan, A., 95 Kheyfits, A., 29 Marusic-Paloka, E., 108 Grudsky, S., 45, 46 Khimshiashvili, G., 29 Masaki, S., 83 Guliyev, V., 53 Kilbas, A., 3, 48, 49 Matsuura, T., 89 Gupta, S., 57 Kim, B.J., 50 Matsuyama, T., 74 Gürlebeck, K., 42 Kisil, A., 38 Matthies, K., 87 Gürkanlı, A.T., 46 Kisil, V., 42, 44 Maz’ya, V., 18 Gyongy, I., 91 Kiss, G., 102 van der Meer, J. C., 99 Györi,I., 102 Kohr, G., 25 Michalska, M., 57 Kohr, M., 25 Migda, M., 103 Hairer, M., 92, 96 Kokilashvili, V., 17 Mijatovic, A., 93 Hajibayov, M., 53 Kolokoltsov, V., 92 Mimoiuni, A., 109 Halburd, R., 34 Konjik, S., 50 Mityushev, V., 32, 33 Hartung, F., 102 Kontis, V., 3 Mochizuki, K., 74 Harutyunyan, T., 61 Koroleva, A., 50 Mogultay, I., 83 Hasanov, A., 107 Krainer, T., 61, 66 Mohammed, A., 29 Hashemiparast, S.M., 107 Krausshar, R.S., 38, 44 Mohiuddine, S.A., 109 Hasler, M., 107 Krump, L., 38 Mokhonko, A., 61 Hebisch, W., 96 Kubo, H., 74, 82 Mola, G., 83 Helmstetter, J., 43 Kucera, P., 82 Molahajloo, S., 67 Herrmann, T., 73 Kurtz, T., 93 Morando, A., 67 Higgins, J.R., 48 Kurylev, Y., 87 Moura Santos, A., 54 Hinkkanen, A., 34 Kusainova, L., 54 Mursaleen, M., 109 Hirosawa, F., 71, 73 Mussa, A., 109 Hogan, J., 43 Lamb, J., 3, 97 Hörmann,G., 65 Lamberti, P., 61 Nakazawa, H., 74 Huet, N., 96 Langerová,M., 102 Nam, K., 57 Hughston, L., 92 Lanza de Cristoforis, M., 3, 23, Naso, M.G., 79 Hunsicker, E., 65 25 Natroshvili, D., 67 Neklyudov, M., 96 Hurri-Syrvanen, H., 53 Laptev, A., 3, 59 Neustupa, J., 62 Hussain, A., 28 Lasiecka, I., 76 Nicol, M., 99 Lassas, M., 88 Ichinose, W., 65 Nieminen, P., 58 Lavicka, R., 38 Ilic, S., 106 Niknam, A., 99 Le, U., 83 Inglis, J., 3, 96 Nishitani, T., 74 Leandre, R., 44 Israfilov, D., 24 Nourouzi, K., 109 Lebiedzik, C., 78 Iwasaki, C., 66 Nowak, M., 58 Lee, Y.L., 57 Jacka, S., 92 Leem, K.H., 88 Oberguggenberger, A., 68 Janas, J., 61 de Leo, R., 66 Ockendon, J., 20 Johnson, J., 66 Lerner, N., 17 Ohta, M., 83 Jordan, T., 99 Li Xue-Mei, 93 Olaleru, J., 111 Jung, T., 73 Libine, M., 39 Oliaro, A., 68 Liflyand, E., 54 Onchis, D., 48 Kaasalainen, M., 88 Lionheart, W., 89 Oparnica, L., 50 Kähler,U., 44 Lions, P.-L., 21 Opic, B., 54 Kalmenov, T.S., 30 Littman, W., 78 Orelma, H., 39 Kalyabin, G., 53 Liu Lixia, 108 Otto, F., 96 Kamarulhaili, H., 108 Liu Yu, 66 Ozawa, T., 80 Kamotski, I., 85 Loreti, P., 78 Ozbekler,¨ A., 103 Kaptanoglu, T., 56, 57 Lucente, S., 83 Oztop, S., 111 Karapinar, E., 108 Luna-Elizarrarás,M.E., 39 Karelin, O., 46 Luzzatto, S., 97 Pandolfi, L., 79 Karimov, E., 108 Lyons, T., 90, 93 Panti, G., 99 Karlovych, Y., 66 Papageorgiou, I., 3, 96 Karp, D., 24 Macia, F., 86 Papavasiliou, A., 93 Karp, L., 73 Maksimov, V., 78 Parvaneh Masoha, H., 110 Karupu, O., 25 Malliavin, P, 18 Pau, J., 58 Katayama, S., 82 Mamedkhanov, J., 25 Pelekanos, G., 89 Kato, K., 35 Mammadov, Y., 54 PeñaPeña,D., 39 Kats, B., 25 Manhas, J.S., 57 Peng, C., 99 Katsnelson, V., 35 Manjavidze, N., 29 Perotti, A., 39 Kayar, Z., 102 Markowich, P., 86 Picard, R., 74 Kelbert, M., 92 Marletta, M., 89 Pinotsis, D., 44

114 Index

Pirzada, S., 110 Schrohe, E., 69 Tretyakov, M., 94 Pistorius, M., 94 Schulze, B.-W., 3, 18, 68 Trooshin, I., 90 Pituk, M., 103 Sehba, B.F., 58 Trushin, B.V., 56 Pivetta, M., 75 Seiler, J., 69 Tunaru, R., 94 Plaksa, S., 23, 26 Selmi, R., 110 Turunen, V., 70 Porter, M., 33 Senouci, K., 55 Tvrdy, M., 105 Potthast, R., 89 Serbetci, A., 55 Prakash Sing, A., 35 Sergeev, A., 35 Ueda, Y., 85 Prykarpatsky, A., 35 Sevroglou, V., 90 Uesaka, H., 75 Sforza, D., 79 Uhlmann, G., 20 Quiao Yuying, 40 Shapiro, M., 40 Umeda, T., 62 Shaposhnikova, T., 69 Unal,¨ M., 105 Rättyä,J., 56 Shibata, Y., 84 Upmeier, H., 47 Rachunkova, I., 103 Shindiapin, A., 104 Usoltseva, E., 94 Radkevich, E., 55 Shklyar, B., 105 Rafeiro, H., 55 Shpakivskii, V., 44 Vajiac, A., 41 Rajabov, N., 29, 68 Shvartsman, I., 79 Valein, J., 80 Rajabova, L., 30 Sibiceanu, M., 111 Vasilevski, N., 45, 47 Rakisheva, Z., 110 Silva, A., 47 Vasilyev, V., 70 Rappoport, J., 50 Silvestri, B., 62 Vasy, A., 70 Rasmussen, M., 100 Simon, L., 84 Vieira, N., 45 Reinfelds, A., 103 Skalak, Z., 84 Visciglia, N., 85 Reissig, M., 3, 71, 75, 83 Smyshlyaev, V., 85, 87 Vlacci, F., 42 Ren Guangbin, 40 Sobolewski, P., 58 Vlasakova, Z., 42 Renardy, M., 79 Soltanov, K., 84 van de Voorde, L., 42 Reynolds, D., 103 Somberg, P., 40 Vukotic, D., 59 Richard, S., 62 Sommen, F., 36, 41 Roberto, C., 96 Wang Qiudong, 100 Soria, J., 56 Rochon, F., 68 Wang Yufeng, 31 Soucek, V., 41 Rodino, L., 3, 62 Wirth, J., 3 Spagnolo, S., 84 Rodkina, A., 104 Witt, I., 70 Sprößig,W., 42, 45 Rogosin, S., 32, 33 Wong, M.W., 3, 62, 70 Stanek, S., 105 Rojas, E., 46 Wu Zhijian, 59 Stefanov, A., 84 Rozovsky, B., 94 Wulan Hasi, 59 Stevic, S., 105 R˚uˇziˇcková, M., 104 Ruzhansky, M., 3 Stoppato, C., 41 Xu Wen, 59 Ryan, J., 40 Strohmaier, A., 62 Xu Yongzhi, 3 Ryan, M., 3 Suragan, D., 30 Ryser, M.D., 106 Suzuki, O., 26 Yagdjian, K., 75, 85 Suzuki, R., 75 Yakubovich, S., 50 Sabadini, I., 36, 40 Yamamoto, M., 3, 20, 80, 87 Sadyrbaev, F., 100 Tahara, H., 69 Yang Congli, 59 Safarov, Y., 59 Takemura, K., 48 Yildirir, Y.E., 26 Saito, J., 75 Tamrazov, P., 23 Yordanov, B., 76 Saitoh, S., 3, 47, 48, 89 Taneco-Hernández,M.A., 82 Youssfi, E.H., 26 Saks, R., 30 Tanriverdi, T., 111 Salac, T., 40 Tapdıgo˘glu,M., 58 Zafer, A., 100, 105 Samko, N., 55 Taqi, I., 31 Zegarlinski, B., 3, 95 Samko, S., 51, 55 Taskinen, J., 58 Zelinskiy, Y., 26 Samoylova, E., 30 Tasmambetov, Zh., 31 Zeren, Y., 56 Santos, P.A., 110 Teofanov, N., 69 Zhang Shangyou, 3 Sanz-Solé,M., 94 Teta, A., 84 Zhang Zhongxiang, 32 Sarybekova, L., 110 Tikhonov, S., 56 Zhdanov, O.N., 32 Sasane, A., 58 Todd, M., 100 Zhilisbaeva. K., 111 Savchin, V.M., 104 Toft, J., 3, 69 Zhong Shouguo, 32 Schmeidel, E., 104 Tolksdorf, J., 45 Zhu Hongmei, 71 Schmeling, J., 100 Tomilov, Y., 62 Zhu Kehe, 47, 59 Schmidt, B., 87 Toundykov, D., 79 Zolesio, J.-P., 80 Schnaubelt, R., 79 Tovar, L.M., 59 Zorboska, N., 59

115