Related papers: The dynamics of the Ehrhard-M\"uller system with i…
We first give an exposition on holomorphic isometries from the Poincar\'e disk to polydisks and from the Poincar\'e disk to the product of the Poincar\'e disk with a complex unit ball. As an application, we provide an example of proper…
In this article, we derive estimates of Teichm\"uller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the…
We consider the motion of a particle described by an action that is a functional of the Frenet-Serret [FS] curvatures associated with the embedding of its worldline in Minkowski space. We develop a theory of deformations tailored to the FS…
Based on the d'Alembert-Lagrange-Poincar\'{e} variational principle, we formulate general equations of motion for mechanical systems subject to nonlinear nonholonomic constraints, that do not involve Lagrangian undetermined multipliers. We…
In this paper, we study the (1+3) dimensional massive Maxwell-Dirac system in the context of global existence and asymptotic behavior of solutions under the Lorenz gauge condition, as well as the modified and linear scattering phenomena for…
In the author's paper ''Poincar\'{e} series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity'' a relation is proved between the Poincar\'{e} series of the coordinate algebra of a two-dimensional quasihomogeneous…
In this paper we introduce a new dynamical system which we call Angular billiard. It acts on the exterior points of a convex curve in Euclidean plane. In a neighborhood of the boundary curve this system turns out to be dual to the Birkhoff…
This paper deals with fundamental properties of Poincar\'e half-maps defined on a straight line for planar linear systems. Concretely, we focus on the analyticity of the Poincar\'e half-maps, their series expansions (Taylor and…
The Poincar\'e-Alexander Theorem states that holomorphic mappings defined on an open subset of the unit ball of $C^n$ may, under certain conditions, be extended to a biholomorphism of the unit ball. In a complex manifold, every strongly…
The current series of two papers focus on a 3-dimensional Lotka-Volterra competition model of differential equations with seasonal succession, which exhibits that populations experience an external periodically forced environment. We are…
The f-invariant is a higher version of the e-invariant that takes values in the divided congruences between modular forms; it can be formulated as an elliptic genus of manifolds with corners of codimension two. In this thesis, we develop a…
We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semi-discrete variant). First, we linearize the equation around its arbitrary solution $u$. Then we construct a differential (respectively, difference)…
Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the…
"V - E + F = 2", the famous Euler's polyhedral formula, has a natural generalization to convex polytopes in every finite dimension, also known as the Euler-Poincar\'e Formula. We provide another short inductive proof of the general formula.…
Consider a general $3$-dimensional Lotka-Volterra system with a rational first integral of degree two of the form $H=x^i y^j z^k$. The restriction of this Lotka-Volterra system to each surface $H(x,y,z)=h$ varying $h\in \mathbb{R}$ provide…
We consider the breathing circle billiard, in which a point particle moves freely inside a disk. The radius varies periodically in time, with elastic reflections at the moving boundary. In this system the angular momentum is preserved, and…
This thesis establishes a geometric approach to the de Rham realization of the polylogarithm. As a central result we construct the logarithm sheaves of rational abelian schemes in terms of the birigidified Poincar\'e bundle with universal…
For several objects of interest in geometric complexity theory, namely for the determinant, the permanent, the product of variables, the power sum, the unit tensor, and the matrix multiplication tensor, we introduce and study a fundamental…
We study a Riemannian manifold equipped with a density which satisfies the Bakry--\'Emery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first…
We derive new Poincar\'e-series representations for infinite families of non-holomorphic modular invariant functions that include modular graph forms as they appear in the low-energy expansion of closed-string scattering amplitudes at genus…