Representations of the celebrated Heisenberg commutation relations in quantum mechanics (and their exponentiated versions) form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we will try to bridge the two communit...

The Riemann-Wirtinger integral is a function defined by a definite integral on a complex torus whose integrand is a power product of the exponential function and theta functions. It was found as a special solution of the system of differential equations which governs the monodromy-preserving deformation of Fuchsian differential equations on the complex torus [11], [12]. The main purpose of this...

We continue a recently developed systematic approach to the Bousinesq (Bsq) hierarchy and its algebro-geometric solutions. Our formalism includes a recursive construction of Lax pairs and establishes associated Burchnall-Chaundy curves, Baker-Akhiezer functions and Dubrovin-type equations for analogs of Dirichlet and Neumann divisors. The principal aim of this paper is a detailed theta function...

Helton and Vinnikov showed that every rigidly convex curve in the real plane bounds a spectrahedron. This leads to the computational problem of explicitly producing a symmetric (positive definite) linear determinantal representation for a given curve. We study three approaches to this problem: an algebraic approach via solving polynomial equations, a geometric approach via contact curves, and a...

We demonstrate deterministic extensive chaos in the dynamics of large sparse networks of theta neurons in the balanced state. The analysis is based on numerically exact calculations of the full spectrum of Lyapunov exponents, the entropy production rate, and the attractor dimension. Extensive chaos is found in inhibitory networks and becomes more intense when an excitatory population is include...

One of the more fruitful observations in recent years was the discovery [19] that the (normalized) character χμ of the integrable highest weight g-module L(μ) of a nontwisted Kac-Moody algebra g = X (1) r is a modular form. Thanks to the structure of the affine Weyl group, these χμ can be written as ratios of theta functions, so on them exists a natural action of SL2(Z). In fact, this action de...

We present some recent progresses on Heun functions, gathering results from classical analysis up to elliptic functions. We describe Picard’s generalization of Floquet’s theory for differential equations with doubly periodic coefficients and give the detailed forms of the level one Heun functions in terms of Jacobi theta functions. The finite-gap solutions give an interesting alternative integr...

We give an analytical description of the locus of the two-gap elliptic potentials associated with the corresponding flow of the Calogero– Moser system. We start with the description of Treibich–Verdier two– gap elliptic potentials. The explicit formulae for the covers, wave functions and Lamé polynomials are derived, together with a new Lax representation for the particle dynamics on the locus....

The aim of the present study was to explore auditory cortical oscillation properties in developmental dyslexia. We recorded cortical activity in 17 dyslexic participants and 15 matched controls using simultaneous EEG and fMRI during passive viewing of an audiovisual movie. We compared the distribution of brain oscillations in the delta, theta and gamma ranges over left and right auditory cortic...