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For nonautonomous linear difference equations in Banach spaces we show that a very general type of dichotomic behavior persists under small enough additive linear perturbations. By using a new approach, we obtain two general robustness…
A complete solution to the multiplier version of the inverse problem of the calculus of variations is given for a class of hyperbolic systems of second-order partial differential equations in two independent variables. The necessary and…
The restoration of an additive function defined on P parallelepipeds via its derivative with respect to P parallelepipeds is studied. The obtained theorem is applied to the questions of uniqueness of multiple series with regard to Haar and…
Recent studies of transport phenomena with complex potentials are explained by generic square root singularities of spectrum and eigenfunctions of non-Hermitian Hamiltonians. Using a two channel problem we demonstrate that such…
Infinite-dimensional differential algebraic equations (short DAEs) with input and output are studied. The concepts of operator nodes and system nodes are extended to systems which additionally may include algebraic constraints.…
We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…
Properties of solutions of generic hyperbolic systems with multiple characteristics with diagonalizable principal part are investigated. Solutions are represented as a Picard series with terms in the form of iterated Fourier integral…
The defect of valued field extensions is a major obstacle in open problems in resolution of singularities and in the model theory of valued fields, whenever positive characteristic is involved. We continue the detailed study of defect…
The method of separation of variables is significant, it has been applied to physics, engineering , chemistry and other fields. It allows to reduce the diffculity of problems by separating the variables from partial differential equation…
We study invariance and monotonicity properties of Kunita-type stochastic differential equations in $\RR^d$ with delay. Our first result provides sufficient conditions for the invariance of closed subsets of $\RR^d$. Then we present a…
We prove that separable extensions of noetherian rings and finite \'etale morphisms of noetherian schemes give rise to separable extensions of singularity categories.
This is an extended (factor 2.5) version of arXiv:math/0601371 and arXiv:0808.3486. We present new results in the theory of the classical $\theta$-functions of Jacobi: series expansions and defining ordinary differential equations (\odes).…
This paper is part of a series of papers in which the asymptotic theory and appropriate symbolic computer code are developed to compute the asymptotic expansion of the solution of an n-th order ordinary differential equation. The paper…
We study the topology of polynomial functions by deforming them generically. We explain how the non-conservation of the total ``quantity'' of singularity in the neighbourhood of infinity is related to the variation of topology in certain…
The conditions under which, the continuity equation can be substituted by an ordinary non differential equation, will be discussed. Since continuity equation is a fundamental equation, this result will be applicable in a vast area of…
We derive the discrete version of the classical Helmholtz condition. Precisely, we state a theorem characterizing second order finite differences equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide…
An algebraic definition of Gardner's deformations for completely integrable bi-Hamiltonian evolutionary systems is formulated. The proposed approach extends the class of deformable equations and yields new integrable evolutionary and…
We give an introduction to the valuation theoretical phenomenon of "defect", also known as "ramification deficiency". We describe the role it plays in deep open problems in positive characteristic: local uniformization (the local form of…
We study singularity confinement phenomena in examples of delay-differential Painlev\'e equations, which involve shifts and derivatives with respect to a single independent variable. We propose a geometric interpretation of our results in…
A simple and efficient variational method is introduced to accelerate the convergence of the eigenenergy computations for a Hamiltonian H with singular potentials. Closed-form analytic expressions in N dimensions are obtained for the matrix…