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Some problems in the theory and applications of stochastic processes can be reduced to solving integral equations. While explicit solutions for these equations are often elusive, valuable insights can be gained through their asymptotic…

We have an idea on the influence of a nonlinear term (tending to 0) on the prescribed scalar curvature equation to have an uniform estimate.

We obtain an asymptotic H\"older estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the…

If the step distribution in a renewal process has finite mean and regularly varying tail with index -{\alpha}, 1<{\alpha}<2, the first two terms in the asymptotic expansion of the renewal function have been known for many years. Here we…

This paper aims to derive explicit and computable error bounds for the asymptotic expansion of the Jacobi polynomials as their degree approaches infinity, using an integral method. The analysis focuses on the outer or oscillatory region of…

This paper studies a version of the counting problem in dynamical systems that is of interest, especially in conformal dynamical systems where the functions of the systems are angle preserving. Recently, M. Pollicott and M. Urba\'{n}ski…

We survey the classical results of the Dirichlet Approximation Theorem.

We consider abstract evolution equations with on-off time delay feedback. Without the time delay term, the model is described by an exponentially stable semigroup. We show that, under appropriate conditions involving the delay term, the…

Let $p$ be a prime and $n$ a positive integer such that $\sqrt{\frac p2} + 1 \leq n \leq \sqrt{p}$. For any arithmetic progression $A$ of length $n$ in $\mathbb{F}_p$, we establish an asymptotic formula for the number of directions…

Let $d(n)$ be the Dirichlet divisor function and $\Delta(x)$ denote the error term of the sum $\sum_{n\leqslant x}d(n)$ for a large real variable $x$. In this paper we focus on the sum $\sum_{p\leqslant x}\Delta^2(p)$, where $p$ runs over…

In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in the arithmetic progression.

We prove a new equidistribution estimate for the divisor function in arithmetic progression to moduli that have two small factors. We give two applications. First, we show an asymptotic formula for the divisor function over arithmetic…

In previous work the second author derived an asymptotic formula for the sum of the distances between centers of consecutive Ford spheres. In this paper we extend these results by proving asymptotic formulas for higher moments of the…

This note contains some asymptotic formulas for the sums of various residue classes of Euler's phi-function.

We establish asymptotic gain along with input-to-state practical stability results for disturbed semilinear systems w.r.t. the global attractor of the respective undisturbed system. We apply our results to a large class of nonlinear…

In this note, we will consider two classical volume problems related to elliptic integrals. The first problem has a neat formula by means of elliptic integrals. We remade it with details. In the second problem, we found a messy formula. On…

Is studied asymptotic expansion for solution of singularly perturbed equation for Markov random evolution in Rd. The views of regular and singular parts of solution are found.

This paper concerns the asymptotic behaviour of solutions of a linear convolution Volterra summation equation with an unbounded forcing term. In particular, we suppose the kernel is summable and ascribe growth bounds to the exogenous…

In this paper, insight is given in the techniques used to compute asymptotic expansions. In a broad fashion the technique is described. Most of the results apply to the paper "An expansion for the maximum likelihood estimator and its…

Is studied asymptotic expansion for solution of singularly perturbed equation for functional of Markovian evolution in Rd. The view of regular and singular parts of solution is found.