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Condenser Capacity and Hyperbolic Perimeter
CONDENSER CAPACITY AND HYPERBOLIC PERIMETER MOHAMED M. S. NASSER, OONA RAINIO, AND MATTI VUORINEN Abstract. We apply domain functionals to study the conformal capacity of condensers (G; E) where G is a simply connected domain in the complex plane and E is a compact subset of G. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of E. Novel computational algorithms based on implementations of the fast multipole method are combined with analytic techniques. Computational experiments are used throughout to, for instance, demonstrate sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed. 1. Introduction A condenser is a pair (G; E), where G ⊂ Rn is a domain and E is a compact non-empty subset of G. The conformal capacity of this condenser is defined as [14, 17, 20] Z (1.1) cap(G; E) = inf jrujndm; u2A G 1 where A is the class of C0 (G) functions u : G ! [0; 1) with u(x) ≥ 1 for all x 2 E and dm is the n-dimensional Lebesgue measure. The conformal capacity of a condenser is one of the key notions in potential theory of elliptic partial differential equations [20, 29] and it has numerous applications to geometric function theory, both in the plane and in higher dimensions, [10, 14, 15, 17]. The isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length, or, perimeter. Constrained extremal problems of this type, where the constraints involve geometric or physical quantities, have been studied in several thousands of papers, see e.g. -
Harmonic and Complex Analysis and Its Applications
Trends in Mathematics Alexander Vasil’ev Editor Harmonic and Complex Analysis and its Applications Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference. For further volumes: http://www.springer.com/series/4961 Harmonic and Complex Analysis and its Applications Alexander Vasil’ev Editor Editor Alexander Vasil’ev Department of Mathematics University of Bergen Bergen Norway ISBN 978-3-319-01805-8 ISBN 978-3-319-01806-5 (eBook) DOI 10.1007/978-3-319-01806-5 Springer Cham Heidelberg New York Dordrecht London Mathematics Subject Classification (2010): 13P15, 17B68, 17B80, 30C35, 30E05, 31A05, 31B05, 42C40, 46E15, 70H06, 76D27, 81R10 c Springer International Publishing Switzerland 2014 This work is subject to copyright. -
Roth's Theorem in the Primes
Annals of Mathematics, 161 (2005), 1609–1636 Roth’s theorem in the primes By Ben Green* Abstract We show that any set containing a positive proportion of the primes con- tains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We de- rive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes. 1. Introduction Arguably the second most famous result of Klaus Roth is his 1953 upper bound [21] on r3(N), defined 17 years previously by Erd˝os and Tur´an to be the cardinality of the largest set A [N] containing no nontrivial 3-term arithmetic ⊆ progression (3AP). Roth was the first person to show that r3(N) = o(N). In fact, he proved the following quantitative version of this statement. Proposition 1.1 (Roth). r (N) N/ log log N. 3 " There was no improvement on this bound for nearly 40 years, until Heath- Brown [15] and Szemer´edi [22] proved that r N(log N) c for some small 3 " − positive constant c. Recently Bourgain [6] provided the best bound currently known. Proposition 1.2 (Bourgain). r (N) N (log log N/ log N)1/2. 3 " *The author is supported by a Fellowship of Trinity College, and for some of the pe- riod during which this work was carried out enjoyed the hospitality of Microsoft Research, Redmond WA and the Alfr´ed R´enyi Institute of the Hungarian Academy of Sciences, Bu- dapest. -
All That Math Portraits of Mathematicians As Young Researchers
Downloaded from orbit.dtu.dk on: Oct 06, 2021 All that Math Portraits of mathematicians as young researchers Hansen, Vagn Lundsgaard Published in: EMS Newsletter Publication date: 2012 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Hansen, V. L. (2012). All that Math: Portraits of mathematicians as young researchers. EMS Newsletter, (85), 61-62. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. -
Klaus Friedrich Roth, 1925–2015 the Godfrey Argent Studio C
e Bull. London Math. Soc. 50 (2018) 529–560 C 2017 Authors doi:10.1112/blms.12143 OBITUARY Klaus Friedrich Roth, 1925–2015 The Godfrey Argent Studio C Klaus Friedrich Roth, who died in Inverness on 10 November 2015 aged 90, made fundamental contributions to different areas of number theory, including diophantine approximation, the large sieve, irregularities of distribution and what is nowadays known as arithmetic combinatorics. He was the first British winner of the Fields Medal, awarded in 1958 for his solution in 1955 of the famous Siegel conjecture concerning approximation of algebraic numbers by rationals. He was elected a member of the London Mathematical Society on 17 May 1951, and received its De Morgan Medal in 1983. 1. Life and career Klaus Roth, son of Franz and Matilde (n´ee Liebrecht), was born on 29 October 1925, in the German city of Breslau, in Lower Silesia, Prussia, now Wroclaw in Poland. To escape from Nazism, he and his parents moved to England in 1933 and settled in London. He would recall that the flight from Berlin to London took eight hours and landed in Croydon. Franz, a solicitor by training, had suffered from gas poisoning during the First World War, and died a few years after their arrival in England. Roth studied at St Paul’s School between 1937 and 1943, during which time the school was relocated to Easthampstead Park, near Crowthorne in Berkshire, as part of the wartime evac- uation of London. There he excelled in mathematics and chess, and one master, Mr Dowswell, observed interestingly that he possessed complete intellectual honesty. -
Complex Numbers and Colors
Complex Numbers and Colors For the sixth year, “Complex Beauties” provides you with a look into the wonderful world of complex functions and the life and work of mathematicians who contributed to our understanding of this field. As always, we intend to reach a diverse audience: While most explanations require some mathemati- cal background on the part of the reader, we hope non-mathematicians will find our “phase portraits” exciting and will catch a glimpse of the richness and beauty of complex functions. We would particularly like to thank our guest authors: Jonathan Borwein and Armin Straub wrote on random walks and corresponding moment functions and Jorn¨ Steuding contributed two articles, one on polygamma functions and the second on almost periodic functions. The suggestion to present a Belyi function and the possibility for the numerical calculations came from Donald Marshall; the November title page would not have been possible without Hrothgar’s numerical solution of the Bla- sius equation. The construction of the phase portraits is based on the interpretation of complex numbers z as points in the Gaussian plane. The horizontal coordinate x of the point representing z is called the real part of z (Re z) and the vertical coordinate y of the point representing z is called the imaginary part of z (Im z); we write z = x + iy. Alternatively, the point representing z can also be given by its distance from the origin (jzj, the modulus of z) and an angle (arg z, the argument of z). The phase portrait of a complex function f (appearing in the picture on the left) arises when all points z of the domain of f are colored according to the argument (or “phase”) of the value w = f (z). -
Nevanlinna Theory of the Askey-Wilson Divided Difference
NEVANLINNA THEORY OF THE ASKEY-WILSON DIVIDED DIFFERENCE OPERATOR YIK-MAN CHIANG AND SHAO-JI FENG Abstract. This paper establishes a version of Nevanlinna theory based on Askey-Wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane C. A second main theorem that we have derived allows us to define an Askey-Wilson type Nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero/pole- scarce. That is, their zeros/poles are indeed deficient in the sense of difference Nevanlinna theory. A natural consequence is a version of Askey-Wilosn type Picard theorem. We also give an alternative and self-contained characterisation of the kernel functions of the Askey-Wilson operator. In addition we have established a version of unicity theorem in the sense of Askey-Wilson. This paper concludes with an application to difference equations generalising the Askey-Wilson second-order divided difference equation. Contents 1. Introduction 2 2. Askey-Wilson operator and Nevanlinna characteristic 6 3. Askey-Wilson type Nevanlinna theory – Part I: Preliminaries 8 4. Logarithmic difference estimates and proofs of Theorem 3.2 and 3.1 10 5. Askey-Wilson type counting functions and proof of Theorem 3.3 22 6. ProofoftheSecondMaintheorem3.5 25 7. Askey-Wilson type Second Main theorem – Part II: Truncations 27 8. Askey-Wilson-Type Nevanlinna Defect Relation 29 9. Askey-Wilson type Nevanlinna deficient values 31 arXiv:1502.02238v4 [math.CV] 3 Feb 2018 10. The Askey-Wilson kernel and theta functions 33 11. -
EMS Newsletter September 2012 1 EMS Agenda EMS Executive Committee EMS Agenda
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. Matemática, Faculdade Facultad de Matematicas de Ciências, Edifício C6, Universidad Complutense Piso 2 Campo Grande Mathematical de Madrid 1749-006 Lisboa, Portugal e-mail: [email protected] Plaza de Ciencias 3, 28040 Madrid, Spain Eva-Maria Feichtner e-mail: [email protected] (2012–2015) Society Department of Mathematics Lucia Di Vizio (2012–2016) Université de Versailles- University of Bremen St Quentin 28359 Bremen, Germany e-mail: [email protected] Laboratoire de Mathématiques Newsletter No. 85, September 2012 45 avenue des États-Unis Eva Miranda (2010–2013) 78035 Versailles cedex, France Departament de Matemàtica e-mail: [email protected] Aplicada I EMS Agenda .......................................................................................................................................................... 2 EPSEB, Edifici P Editorial – S. Jackowski ........................................................................................................................... 3 Associate Editors Universitat Politècnica de Catalunya Opening Ceremony of the 6ECM – M. -
Roth's Orthogonal Function Method in Discrepancy Theory and Some
Roth’s Orthogonal Function Method in Discrepancy Theory and Some New Connections Dmitriy Bilyk Abstract In this survey we give a comprehensive, but gentle introduction to the cir- cle of questions surrounding the classical problems of discrepancy theory, unified by the same approach originated in the work of Klaus Roth [85] and based on mul- tiparameter Haar (or other orthogonal) function expansions. Traditionally, the most important estimates of the discrepancy function were obtained using variations of this method. However, despite a large amount of work in this direction, the most im- portant questions in the subject remain wide open, even at the level of conjectures. The area, as well as the method, has enjoyed an outburst of activity due to the recent breakthrough improvement of the higher-dimensional discrepancy bounds and the revealed important connections between this subject and harmonic analysis, proba- bility (small deviation of the Brownian motion), and approximation theory (metric entropy of spaces with mixed smoothness). Without assuming any prior knowledge of the subject, we present the history and different manifestations of the method, its applications to related problems in various fields, and a detailed and intuitive outline of the latest higher-dimensional discrepancy estimate. 1 Introduction The subject and the structure of the present chapter is slightly unconventional. In- stead of building the exposition around the results from one area, united by a com- mon topic, we concentrate on problems from different fields which all share a com- mon method. The starting point of our discussion is one of the earliest results in discrepancy theory, Roth’s 1954 L2 bound of the discrepancy function in dimensions d ≥ 2 [85], as well as the tool employed to obtain this bound, which later evolved into a pow- Dmitriy Bilyk Department of Mathematics, University of South Carolina, Columbia, SC 29208 USA e-mail: [email protected] 1 2 Dmitriy Bilyk erful orthogonal function method in discrepancy theory. -
Calculus Redux
THE NEWSLETTER OF THE MATHEMATICAL ASSOCIATION OF AMERICA VOLUME 6 NUMBER 2 MARCH-APRIL 1986 Calculus Redux Paul Zorn hould calculus be taught differently? Can it? Common labus to match, little or no feedback on regular assignments, wisdom says "no"-which topics are taught, and when, and worst of all, a rich and powerful subject reduced to Sare dictated by the logic of the subject and by client mechanical drills. departments. The surprising answer from a four-day Sloan Client department's demands are sometimes blamed for Foundation-sponsored conference on calculus instruction, calculus's overcrowded and rigid syllabus. The conference's chaired by Ronald Douglas, SUNY at Stony Brook, is that first surprise was a general agreement that there is room for significant change is possible, desirable, and necessary. change. What is needed, for further mathematics as well as Meeting at Tulane University in New Orleans in January, a for client disciplines, is a deep and sure understanding of diverse and sometimes contentious group of twenty-five fac the central ideas and uses of calculus. Mac Van Valkenberg, ulty, university and foundation administrators, and scientists Dean of Engineering at the University of Illinois, James Ste from client departments, put aside their differences to call venson, a physicist from Georgia Tech, and Robert van der for a leaner, livelier, more contemporary course, more sharply Vaart, in biomathematics at North Carolina State, all stressed focused on calculus's central ideas and on its role as the that while their departments want to be consulted, they are language of science. less concerned that all the standard topics be covered than That calculus instruction was found to be ailing came as that students learn to use concepts to attack problems in a no surprise. -
NEWSLETTER No
NEWSLETTER No. 453 December 2015 JOINT ANNIVERSARY WEEKEND Report Pavel Exner, President of the European Mathematical Society and Terry Lyons, President of the London Mathematical Society he weekend 18 to 20 September 2015 saw lain Memorial Clock-tower (or ‘Old Joe’, T mathematicians from around the world the world’s tallest free-standing clock-tow- congregate at the University of Birming- er), participants divided between parallel ham, UK, for a conference in celebration of sessions on the themes of Algebra, Com- two birthdays: the 150th of the venerable binatorics, and Analysis, and reunited for London Mathematical Society (LMS), and plenary talks from some of mathematics’ the 25th of the relatively youthful European current leading lights. Mathematical Society. After warm greetings from Terry Lyons and Under the watch of the Joseph Chamber- Pavel Exner, the two societies’ respective (Continued on page 3) SOCIETY MEETINGS AND EVENTS • 10–11 December: Joint Meeting with the Edinburgh Mathematical Society, • 26 February: Mary Cartwright Lecture, London Edinburgh page 5 • 21 March: Society Meeting at the BMC, Bristol • 14 December: SW & South Wales Regional Meeting, • 21–25 March: LMS Invited Lectures, Loughborough Southampton page 29 page 22 • 15–16 December: LMS Prospects in Mathematics, • 8 July: Graduate Student Meeting, London Loughborough page 14 • 8 July: Society Meeting, London NEWSLETTER ONLINE: newsletter.lms.ac.uk @LondMathSoc LMS NEWSLETTER http://newsletter.lms.ac.uk Contents No. 453 December 2015 8 33 150th Anniversary Events -
Multiplicities in Hayman's Alternative
J. Aust. Math. Soc. 78 (2005), 37-57 MULTIPLICITIES IN HAYMAN'S ALTERNATIVE WALTER BERGWEILER and J. K. LANGLEY (Received 3 September 2002; revised 28 August 2003) Communicated by P. C. Fenton Abstract In 1959 Hayman proved an inequality from which it follows that if/ is transcendental and meromorphic in the plane then either/ takes every finite complex value infinitely often or each derivative /(k), k > 1, takes every finite non-zero value infinitely often. We investigate the extent to which these values may be ramified, and we establish a generalization of Hayman's inequality in which multiplicities are not taken into account. 2000 Mathematics subject classification: primary 3OD35. 1. Introduction Our starting point is the following result of Hayman [8, 9]. THEOREM 1.1. Let k e H and let a, b e C, b ^ 0. Let f be nonconstant and meromorphic in the plane. Then at least one of f — a and f(Ar) — b has at least one zero, and infinitely many iff is transcendental. Hayman's proof of Theorem 1.1 is based on Nevanlinna theory, the result deduced from the following inequality. Here the notation is that of [9], with S(r,f) denoting any term which is o(T(r,f)) as r -> oo, possibly outside a set of finite Lebesgue measure. THEOREM 1.2 ([8, 9]). Let ieN and let f be transcendental and meromorphic in the plane. Then, as r -» oo, (1) r(rJ) © 2005 Australian Mathematical Society 1446-7887/05 $A2.00 + 0.00 Downloaded from https://www.cambridge.org/core.