Mathematics of Computation 1943—1993: a Half-Century of Computational Mathematics
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Elliptic Systems on Riemann Surfaces
Lecture Notes of TICMI, Vol.13, 2012 Giorgi Akhalaia, Grigory Giorgadze, Valerian Jikia, Neron Kaldani, Giorgi Makatsaria, Nino Manjavidze ELLIPTIC SYSTEMS ON RIEMANN SURFACES © Tbilisi University Press Tbilisi Abstract. Systematic investigation of Elliptic systems on Riemann surface and new results from the theory of generalized analytic functions and vectors are pre- sented. Complete analysis of the boundary value problem of linear conjugation and Riemann-Hilbert monodromy problem for the Carleman-Bers-Vekua irregular systems are given. 2000 Mathematics Subject Classi¯cation: 57R45 Key words and phrases: Generalized analytic function, Generalized analytic vector, irregular Carleman-Bers-Vekua equation, linear conjugation problem, holo- morphic structure, Beltrami equation Giorgi Akhalaia, Grigory Giorgadze*, Valerian Jikia, Neron Kaldani I.Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University 2 University st., Tbilisi 0186, Georgia Giorgi Makatsaria St. Andrews Georgian University 53a Chavchavadze Ave., Tbilisi 0162, Georgia Nino Manjavidze Georgian Technical University 77 M. Kostava st., Tbilisi 0175, Georgia * Corresponding Author. email: [email protected] Contents Preface 5 1 Introduction 7 2 Functional spaces induced from regular Carleman-Bers-Vekua equations 15 2.1 Some functional spaces . 15 2.2 The Vekua-Pompej type integral operators . 16 2.3 The Carleman-Bers-Vekua equation . 18 2.4 The generalized polynomials of the class Ap;2(A; B; C), p > 2 . 20 2.5 Some properties of the generalized power functions . 22 2.6 The problem of linear conjugation for generalized analytic functions . 26 3 Beltrami equation 28 4 The pseudoanalytic functions 36 4.1 Relation between Beltrami and holomorphic disc equations . 38 4.2 The periodicity of the space of generalized analytic functions . -
Package 'Igraphmatch'
Package ‘iGraphMatch’ January 27, 2021 Type Package Title Tools Graph Matching Version 1.0.1 Description Graph matching methods and analysis. The package works for both 'igraph' objects and matrix objects. You provide the adjacency matrices of two graphs and some other information you might know, choose the graph matching method, and it returns the graph matching results. 'iGraphMatch' also provides a bunch of useful functions you might need related to graph matching. Depends R (>= 3.3.1) Imports clue (>= 0.3-54), Matrix (>= 1.2-11), igraph (>= 1.1.2), irlba, methods, Rcpp Suggests dplyr (>= 0.5.0), testthat (>= 2.0.0), knitr, rmarkdown VignetteBuilder knitr License GPL (>= 2) LazyData TRUE RoxygenNote 7.1.1 Language en-US Encoding UTF-8 LinkingTo Rcpp NeedsCompilation yes Author Daniel Sussman [aut, cre], Zihuan Qiao [aut], Joshua Agterberg [ctb], Lujia Wang [ctb], Vince Lyzinski [ctb] Maintainer Daniel Sussman <[email protected]> Repository CRAN Date/Publication 2021-01-27 16:00:02 UTC 1 2 bari_start R topics documented: bari_start . .2 best_matches . .4 C.Elegans . .5 center_graph . .6 check_seeds . .7 do_lap . .8 Enron . .9 get_perm . .9 graph_match_convex . 10 graph_match_ExpandWhenStuck . 12 graph_match_IsoRank . 13 graph_match_Umeyama . 15 init_start . 16 innerproduct . 17 lapjv . 18 lapmod . 18 largest_common_cc . 19 matched_adjs . 20 match_plot_igraph . 20 match_report . 22 pad.............................................. 23 row_cor . 24 rperm . 25 sample_correlated_gnp_pair . 25 sample_correlated_ieg_pair . 26 sample_correlated_sbm_pair . 28 split_igraph . 29 splrMatrix-class . 30 splr_sparse_plus_constant . 31 splr_to_sparse . 31 Index 32 bari_start Start matrix initialization Description initialize the start matrix for graph matching iteration. bari_start 3 Usage bari_start(nns, ns = 0, soft_seeds = NULL) rds_sinkhorn_start(nns, ns = 0, soft_seeds = NULL, distribution = "runif") rds_perm_bari_start(nns, ns = 0, soft_seeds = NULL, g = 1, is_splr = TRUE) rds_from_sim_start(nns, ns = 0, soft_seeds = NULL, sim) Arguments nns An integer. -
On Reciprocal Systems and Controllability
On reciprocal systems and controllability Timothy H. Hughes a aDepartment of Mathematics, University of Exeter, Penryn Campus, Penryn, Cornwall, TR10 9EZ, UK Abstract In this paper, we extend classical results on (i) signature symmetric realizations, and (ii) signature symmetric and passive realizations, to systems which need not be controllable. These results are motivated in part by the existence of important electrical networks, such as the famous Bott-Duffin networks, which possess signature symmetric and passive realizations that are uncontrollable. In this regard, we provide necessary and sufficient algebraic conditions for a behavior to be realized as the driving-point behavior of an electrical network comprising resistors, inductors, capacitors and transformers. Key words: Reciprocity; Passive system; Linear system; Controllability; Observability; Behaviors; Electrical networks. 1 Introduction Practical motivation for developing a theory of reci- procity that does not assume controllability arises from This paper is concerned with reciprocal systems (see, electrical networks. Notably, the driving-point behavior e.g., Casimir, 1963; Willems, 1972; Anderson and Vong- of an electrical network comprising resistors, inductors, panitlerd, 2006; Newcomb, 1966; van der Schaft, 2011). capacitors and transformers (an RLCT network) is nec- Reciprocity is an important form of symmetry in physi- essarily reciprocal, and also passive, 2 but it need not be cal systems which arises in acoustics (Rayleigh-Carson controllable (see C¸amlibel et al., 2003; Willems, 2004; reciprocity); elasticity (the Maxwell-Betti reciprocal Hughes, 2017d). Indeed, as noted by C¸amlibel et al. work theorem); electrostatics (Green's reciprocity); and (2003), it is not known what (uncontrollable) behav- electromagnetics (Lorentz reciprocity), where it follows iors can be realized as the driving-point behavior of an as a result of Maxwell's laws (Newcomb, 1966, p. -
Emergent Spacetimes
Emergent spacetimes Silke Christine Edith Weinfurtner Department of Mathematics, Statistics and Computer Science — Te Kura Tatau arXiv:0711.4416v1 [gr-qc] 28 Nov 2007 Thesis submitted for the degree of Doctor of Philosophy at the Victoria University of Wellington. In Memory of Johann Weinfurtner Abstract In this thesis we discuss the possibility that spacetime geometry may be an emergent phenomenon. This idea has been motivated by the Analogue Gravity programme. An “effective gravitational field” dominates the kinematics of small perturbations in an Analogue Model. In these models there is no obvious connection between the “gravitational” field tensor and the Einstein equations, as the emergent spacetime geometry arises as a consequence of linearising around some classical field. After a brief survey of the most relevant literature on this topic, we present our contributions to the field. First, we show that the spacetime geometry on the equatorial slice through a rotating Kerr black hole is formally equivalent to the geometry felt by phonons entrained in a rotating fluid vortex. The most general acoustic geometry is compatible with the fluid dynamic equations in a collapsing/ ex- panding perfect-fluid line vortex. We demonstrate that there is a suitable choice of coordinates on the equatorial slice through a Kerr black hole that puts it into this vortex form; though it is not possible to put the entire Kerr spacetime into perfect-fluid “acoustic” form. We then discuss an analogue spacetime based on the propagation of excitations in a 2-component Bose–Einstein condensate. This analogue spacetime has a very rich and complex structure, which permits us to provide a mass-generating mechanism for the quasi-particle excitations. -
Of Triangles, Gas, Price, and Men
OF TRIANGLES, GAS, PRICE, AND MEN Cédric Villani Univ. de Lyon & Institut Henri Poincaré « Mathematics in a complex world » Milano, March 1, 2013 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemannian (= non-Euclidean) geometry At each location, the units of length and angles may change Shortest path (= geodesics) are curved!! Geodesics can tend to get closer (positive curvature, fat triangles) or to get further apart (negative curvature, skinny triangles) Hyperbolic surfaces Bernhard Riemann 1826-1866 List of topics named after Bernhard Riemann From Wikipedia, the free encyclopedia Riemann singularity theorem Cauchy–Riemann equations Riemann solver Compact Riemann surface Riemann sphere Free Riemann gas Riemann–Stieltjes integral Generalized Riemann hypothesis Riemann sum Generalized Riemann integral Riemann surface Grand Riemann hypothesis Riemann theta function Riemann bilinear relations Riemann–von Mangoldt formula Riemann–Cartan geometry Riemann Xi function Riemann conditions Riemann zeta function Riemann curvature tensor Zariski–Riemann space Riemann form Riemannian bundle metric Riemann function Riemannian circle Riemann–Hilbert correspondence Riemannian cobordism Riemann–Hilbert problem Riemannian connection Riemann–Hurwitz formula Riemannian cubic polynomials Riemann hypothesis Riemannian foliation Riemann hypothesis for finite fields Riemannian geometry Riemann integral Riemannian graph Bernhard -
Chapter 6 Riemann Solvers I
Chapter 6 Riemann solvers I The numerical hydrodynamics algorithms we have devised in Chapter 5 were based on the idea of operator splitting between the advection and pressure force terms. The advection was done, for all conserved quantities, using the gas velocity, while the pressure force and work terms were treated as source terms. From Chapter 2 we know, however, thatthecharacteristicsoftheEuler equations are not necessarily all equal to the gas velocity. We have seen that there exist an eigen- vector which indeed has the gas velocity as eigenvectors, λ0 = u,buttherearetwoeigenvectors which have eigenvalues λ = u C which belong to the forward and backward sound propa- ± ± s gation. Mathematically speaking one should do the advectioninthesethreeeigenvectors,using their eigenvalues as advection velocity. The methods in Chapter 5 do not do this. By extracting the pressure terms out of the advection part and adding them asasourceterm,theadvectionpart has been reduced essentially to Burger’s equation,andthepropagationofsoundwavesisentirely driven by the addition of the source terms. Such methods therefore do not formally propagate the sound waves using advection, even though mathematically they should be. All the effort we have done in Chapters 3 and 4 to create the best advection schemes possible will therefore have no effect on the propagation of sound waves. Once could say that for two out of three characteristics our ingeneous advection scheme is useless. Riemann solvers on the other hand keep the pressure terms within the to-be-advected sys- tem. There is no pressure source term in these equations. The mathematical character of the equations remains intact. Such solvers therefore propagateallthecharacteristicsonequalfoot- ing. -
Quasiconformal Mappings, from Ptolemy's Geography to the Work Of
Quasiconformal mappings, from Ptolemy’s geography to the work of Teichmüller Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Quasiconformal mappings, from Ptolemy’s geography to the work of Teich- müller. 2016. hal-01465998 HAL Id: hal-01465998 https://hal.archives-ouvertes.fr/hal-01465998 Preprint submitted on 13 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. QUASICONFORMAL MAPPINGS, FROM PTOLEMY'S GEOGRAPHY TO THE WORK OF TEICHMULLER¨ ATHANASE PAPADOPOULOS Les hommes passent, mais les œuvres restent. (Augustin Cauchy, [204] p. 274) Abstract. The origin of quasiconformal mappings, like that of confor- mal mappings, can be traced back to old cartography where the basic problem was the search for mappings from the sphere onto the plane with minimal deviation from conformality, subject to certain conditions which were made precise. In this paper, we survey the development of cartography, highlighting the main ideas that are related to quasicon- formality. Some of these ideas were completely ignored in the previous historical surveys on quasiconformal mappings. We then survey early quasiconformal theory in the works of Gr¨otzsch, Lavrentieff, Ahlfors and Teichm¨uller,which are the 20th-century founders of the theory. -
The Number of Equivalence Classes of Symmetric Sign Patterns
The number of equivalence classes of symmetric sign patterns Peter J. Cameron a;1 Charles R. Johnson b aSchool of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, U.K. bDepartment of Mathematics, College of William and Mary, Williamsburg, VA, USA Abstract This paper shows that the number of sign patterns of totally non-zero symmetric n-by-n matrices, up to conjugation by monomial matrices and negation, is equal to the number of unlabelled graphs on n vertices. Key words: symmetric matrix, sign pattern, enumeration, duality 1 Introduction By a (totally non-zero) sign pattern, we mean an n-by-n array S = (sij) of + and signs. With S, we associate the set off all real n-by-n matrices − S such that A = (aij) if and only if aij > 0 whenever sij = + and aij < 0 2 S whenever sij = . Many questions about which properties P are required (all elements of enjoy− P ) or allowed (some element of has P ) by a particular sign patternSS have been studied [3,4]. S One may also consider properties of pairs of sign patterns S1 and S2. For example, one such property of recent interest is commutativity of symmetric sign patterns [1]. We say that S1 and S2 allow commutativity (\commute", for short) if there exist symmetric matrices A1 1 and A2 2 such that 2 S 2 S A1A2 = A2A1. For a given n, a complete answer to the above question might be succinctly described by an undirected graph G, whose vertices are the sign patterns, with an edge between two distinct sign patterns if and only if they 1 Corresponding author; email: [email protected]. -
Riemann, Schottky, and Schwarz on Conformal Representation
Appendix 1 Riemann, Schottky, and Schwarz on Conformal Representation In his Thesis [1851], Riemann sought to prove that if a function u satisfies cer- tain conditions on the boundary of a surface T, then it is the real part of a unique complex function which can be defined on the whole of T. He was then able to show that any two simply connected surfaces (other than the complex plane itself Riemann was considering bounded surfaces) can be mapped conformally onto one another, and that the map is unique once the images of one boundary point and one interior point are specified; he claimed analogous results for any two surfaces of the same connectivity [w To prove that any two simply connected surfaces are conformally equivalent, he observed that it is enough to take for one surface the unit disc K = {z : Izl _< l} and he gave [w an account of how this result could be proved by means of what he later [1857c, 103] called Dirichlet's principle. He considered [w (i) the class of functions, ~., defined on a surface T and vanishing on the bound- ary of T, which are continuous, except at some isolated points of T, for which the integral t= rx + ry dt is finite; and (ii) functions ct + ~. = o9, say, satisfying dt=f2 < oo Ox Oy + for fixed but arbitrary continuous functions a and ft. 224 Appendix 1. Riemann, Schottky, and Schwarz on Conformal Representation He claimed that f2 and L vary continuously with varying ,~. but cannot be zero, and so f2 takes a minimum value for some o9. -
Solving an Inverse Eigenvalue Problem Subject to Triple Constraints
SOLVING AN INVERSE EIGENVALUE PROBLEM WITH TRIPLE CONSTRAINTS ON EIGENVALUES, SINGULAR VALUES, AND DIAGONAL ELEMENTS SHENG-JHIH WU∗ AND MOODY T. CHU† Abstract. An inverse eigenvalue problem usually entails two constraints, one conditioned upon the spectrum and the other on the structure. This paper investigates the problem where triple constraints of eigenvalues, singular values, and diagonal entries are imposed simultaneously. An approach combining an eclectic mix of skills from differential geometry, optimization theory, and analytic gradient flow is employed to prove the solvability of such a problem. The result generalizes the classical Mirsky, Sing-Thompson, and Weyl-Horn theorems concerning the respective majorization relationships between any two of the arrays of main diagonal entries, eigenvalues, and singular values. The existence theory fills a gap in the classical matrix theory. The problem might find applications in wireless communication and quantum information science. The technique employed can be implemented as a first-step numerical method for constructing the matrix. With slight modification, the approach might be used to explore similar types of inverse problems where the prescribed entries are at general locations. Key words. inverse eigenvalue problem, majorization relationships, projected gradient, projected Hessian, analytic gradient dynamics, AMS subject classifications. 65F18, 90C52, 15A29, 15A45, 1. Introduction. Thefocusof this paperis on the existenceof a solution to a new type of inverse eigenvalue problem (IEP) where triple constraints of eigenvalues, singular values, and diagonal entries must be satisfied simultaneously. Before we present our results and explore some possible applications of this particular type of IEP,it might be fitting to givea brief recounton the generalscope of IEPs and why they are interesting, important, and challenging. -
Signature Matrices: the Eigenvalue Problem
University of Texas at El Paso DigitalCommons@UTEP Open Access Theses & Dissertations 2010-01-01 Signature Matrices: The iE genvalue Problem Valeria Aguirre Holguin University of Texas at El Paso, [email protected] Follow this and additional works at: https://digitalcommons.utep.edu/open_etd Part of the Mathematics Commons Recommended Citation Aguirre Holguin, Valeria, "Signature Matrices: The iE genvalue Problem" (2010). Open Access Theses & Dissertations. 2623. https://digitalcommons.utep.edu/open_etd/2623 This is brought to you for free and open access by DigitalCommons@UTEP. It has been accepted for inclusion in Open Access Theses & Dissertations by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. SIGNATURE MATRICES: THE EIGENVALUE PROBLEM VALERIA AGUIRRE HOLGUIN Department of Mathematical Sciences APPROVED: Piotr Wojciechowski, Ph.D. Emil Schwab, Ph.D. Vladik Kreinovich, Ph.D. Patricia D. Witherspoon, Ph.D. Dean of the Graduate School c Copyright by Valeria Aguirre Holgu´ın 2010 SIGNATURE MATRICES: THE EIGENVALUE PROBLEM by VALERIA AGUIRRE HOLGUIN THESIS Presented to the Faculty of the Graduate School of The University of Texas at El Paso in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Department of Mathematical Sciences THE UNIVERSITY OF TEXAS AT EL PASO May 2010 Abstract Dealing with matrices can give us a hard time, especially when their dimension is too big, but we also well know how valuable information a matrix may carry, and that is why we study them. When a matrix has a significant number of zeroes we realize how much easier all calculations are. -
Functional Diagnosability and Detectability of Nonlinear Models Based on Analytical Redundancy Relations Nathalie Verdière, Carine Jauberthie, Louise Travé-Massuyès
Functional diagnosability and detectability of nonlinear models based on analytical redundancy relations Nathalie Verdière, Carine Jauberthie, Louise Travé-Massuyès To cite this version: Nathalie Verdière, Carine Jauberthie, Louise Travé-Massuyès. Functional diagnosability and de- tectability of nonlinear models based on analytical redundancy relations. Journal of Process Control, Elsevier, 2015, 35, pp.1-10. 10.1016/j.jprocont.2015.08.001. hal-01198408 HAL Id: hal-01198408 https://hal.archives-ouvertes.fr/hal-01198408 Submitted on 15 Sep 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Functional diagnosability and detectability of nonlinear models based on analytical redundancy relations Nathalie Verdière1 , Carine Jauberthie2;3 , Louise Travé-Massuyès2;4 1 Normandie Univ, France; ULH, LMAH, F-76600 Le Havre; FR CNRS 3335, ISCN, 25 rue Philippe Lebon 76600 Le Havre, France 2 CNRS, LAAS, 7 avenue du Colonel Roche, F-31400 Toulouse, France 3 Université de Toulouse, UPS, LAAS, F-31400 Toulouse, France 4 Université de Toulouse, LAAS, F-31400 Toulouse, France (Corresponding author: [email protected]). Abstract This paper introduces an original definition of diagnosability for nonlinear dynamical models called functional di- agnosability. Fault diagnosability characterizes the faults that can be discriminated using the available sensors in a system.