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In the analysis of highly-oscillatory evolution problems, it is commonly assumed that a single frequency is present and that it is either constant or, at least, bounded from below by a strictly positive constant uniformly in time. Allowing…
We study the number of solutions of the general semigroup equation in one variable, $X^\al=X^\be$, as well as of the system of equations $X^2=X, Y^2=Y, XY=YX$ in $H\wr T_n$, the wreath product of an arbitrary finite group $H$ with the full…
The review presents the development of an approach of constructing approximate solutions to complicated physics problems, starting from asymptotic series, through optimized perturbation theory, to self-similar approximation theory. The…
This article deals with the asymptotic behavior of fourth order differential equation where the coefficients are perturbations of linear constant coefficient equation. We introduce a change of variable and deduce that the new variable…
The large time behavior of solutions to Cauchy problem for viscous Hamilton-Jacobi equation is classified. The large time asymptotics are given by very singular self-similar solutions on one hand and by self-similar viscosity solutions on…
We studied the asymptotic behavior of local solutions for strongly coupled critical elliptic systems near an isolated singularity. For the dimension less than or equal to five we prove that any singular solution is asymptotic to a…
We study a discrete variant of the Airy equation, formulated as an advance-delay equation, to reveal that discretization induces the higher-order Stokes phenomenon, which is not present in the continuous Airy function and is typically only…
For a stochastic difference equation $D_n=A_nD_{n-1}+B_n$ which stabilises upon time we study tail distribution asymptotics of $D_n$ under the assumption that the distribution of $\log(1+|A_1|+|B_1|)$ is heavy-tailed, that is, all its…
For typical first-order logical theories, satisfying assignments have a straightforward finite representation that can directly serve as a certificate that a given assignment satisfies the given formula. For non-linear real arithmetic…
We consider the number of partitions of $n$ whose Young diagrams fit inside an $m \times \ell$ rectangle; equivalently, we study the coefficients of the $q$-binomial coefficient $\binom{m+\ell}{m}_q$. We obtain sharp asymptotics throughout…
This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial conditions. The class of…
Optimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a…
In this paper the asymptotic distributions are exactly solved for linearly independent solutions considering problems of the second order and for the coefficients of asymptotic destribution the recurent formulas are obtained. Further, using…
We use numerical methods to study the behaviour of the Serre equations in the presence of steep gradients because there are no known analytical solutions for these problems. In keeping with the literature we study a class of initial…
In this paper, we mainly discuss asymptotic profiles of solutions to a class of abstract second-order evolution equations of the form $u''+Au+u'=0$ in real Hilbert spaces, where $A$ is a nonnegative selfadjoint operator. The main result is…
Existing theoretical stabilization results for linear, hyperbolic multi-dimensional problems are extended to the discretized multi-dimensional problems. In contrast to existing theoretical and numerical analysis in the spatially…
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 softened point mass model is commonly used in simulations of gaseous discs including self-gravity while the value of associated length \lambda remains, to some degree, controversial. This ``parameter'' is however fully constrained when,…
Constraint satisfaction problems have been studied in numerous fields with practical and theoretical interests. In recent years, major breakthroughs have been made in a study of counting constraint satisfaction problems (or #CSPs). In…
Inspired by the works of Dewar, Murty and Kot\v{e}\v{s}ovec, we establish some useful theorems for asymptotic formulas. As an application, we obtain asymptotic formulas for the numbers of skew plane partitions and cylindric partitions. We…