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We report on the seminar on Abelian functions held via Zoom on December 4, 2023 and January 9, 2024. / Atsushi Nakayashiki. Introduction to Sato Grassmannian and theta functions. 1-21. / Takashi Ichikawa. Abelian integrals and solitons. 22-33.

We report on the seminar on Abelian functions held via Zoom on December 4, 2023 and January 9, 2024. / Atsushi Nakayashiki. Introduction to Sato Grassmannian and theta functions. 1-21. / Takashi Ichikawa. Abelian integrals and solitons. 22-33.

For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with a Dirichlet boundary condition imposed along the disk, Neugebauer and Meinel in the 1990s famously derived an explicit expression for the spacetime metric corresponding to the Bardeen-Wagoner uniformly rotating disk of dust. In this paper, we consider a similar boundary value problem for a rotating disk in which a Neumann boundary condition is imposed along the disk instead of a Dirichlet condition. Using the integrable structure of the Ernst equation, we are able to reduce the problem to a Riemann-Hilbert problem on a genus one Riemann surface. By solving this Riemann-Hilbert problem in terms of theta functions, we obtain an explicit expression for the Ernst potential. Finally, a Riemann surface degeneration argument leads to an expression for the associated spacetime metric., QC 20190830

We provide a complete spectral analysis of all self-adjoint operators acting on l(2)(Z) which are associated with two doubly infinite Jacobi matrices with entries given by q(-n+1) delta(m,n-1) + q(-n) delta(m,n+1) and delta(m,n-1) + alpha q(-n) delta(m,n) + delta(m,n+1), respectively, where q is an element of (0, 1) and alpha is an element of R. As an application, we derive orthogonality relations for the Ramanujan entire function and the third Jackson q-Bessel function.

En aquest treball estudiem la recent teoria de les formes modulars de mock i en veiem algunes aplicacions.

Two classical upper bounds on the Shannon capacity of graphs are the theta-function due to Lovász and the minrank parameter due to Haemers. We provide several explicit constructions of n-vertex graphs with a constant theta-function and minrank at least n^delta for a constant delta>0 (over various prime order fields). This implies a limitation on the theta-function-based algorithmic approach to approximating the minrank parameter of graphs. The proofs involve linear spaces of multivariate polynomials and the method of higher incidence matrices.

The two lacunary series $chi$+(z)=displaystyle sum_{ngeq 0}z^{2^{n}}, x-(z)=displaystyle sum_{ngeq 0}(-1)^{n}z^{2^{n}} and $xi$(z)=displaystyle sum_{ngeq 1}s_{n}z^{n}, b s_{2n}=s_{n}, s_{2n+1}=(-1)^{n} are all connected to the paperfolding. We study their behaviours at z=1 and some of their relations to modular functions. We also investigate the zeros of their partial sums.

Inter-universal Teichmüller theory may be described as a sort of arithmetic version of Teichmüller theory that concerns a certain type of canonical deformation associated to an elliptic curve over a number field and a prime number lgeq 5. We begin our survey of interuniversal Teichmüller theory with a review of the technical difficulties that arise in applying scheme-theoretic Hodge-Arakelov theory to diophantine geometry. It is precisely the goal of overcoming these technical difficulties that motivated the author to construct the nonscheme-theoretic deformations that form the content of inter-universal Teichmüller theory. Next, we discuss generalities concerning "Teichmüller-theoretic deformations" of various familiar geometric and arithmetic objects which at first glance appear one-dimensional, but in fact have two underlying dimensions. We then proceed to discuss in some detail the various components of the log-theta-lattice, which forms the central stage for the various constructions of inter-universal Teichmüller theory. Many of these constructions may be understood to a certain extent by considering the analogy of these constructions with such classical results as Jacobi' s identity for the theta function and the integral of the Gaussian distribution over the real line. We then discuss the "inter-universal" aspects of the theory, which lead naturally to the introduction of anabelian techniques. Finally, we summarize the main abstract theoretic and diophantine consequences of inter-universal Teichmüller theory, which include a verication of the ABC/Szpiro Conjecture.

Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands.

Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands.

We will derive linear differential relations satis ed by Wirtinger integrals by exploiting classical formulas of Jacobi's theta functions, forgetting that Wirtinger integrals are related to Gauss hypergeometric functions, although these linear differential relations are related to ones satis ed by Gauss hypergeometric functions.

The first half of this paper deals with the structure of the twisted homology group ssociated to the Wirtinger integral. A basis of the first homology group is given, and the vanishing f the other homology groups is proved (Theorem 1). The second half deals with the structure of the wisted cohomology groups associated to the Wirtinger integral. The isomorphism between the twisted ohomology groups and the cohomology groups associated to a subcomplex of the de Rham complex is stablished, and a basis of the first cohomology group of this subcomplex (therefore, of the first twisted ohomology group) is given (Theorem 2).