Related papers: An asymptotic formula for integer points on Markof…

When k > 1 and s is sufficiently large in terms of k, we derive an explicit multi-term asymptotic expansion for the number of representations of a large natural number as the sum of s positive integral k-th powers.

For a competition-diffusion blow-up system involving the fractional Laplacian of the form \begin{equation*}\label{syst1} -(-\Delta)^su=uv^2,\quad-(-\Delta)^sv=vu^2,\quad u,v>0 \ \mathrm{in} \ \mathbb{R}^N, \end{equation*} whith $s\in(0,1)$,…

We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form $\lambda_1x_1^2+\cdots +\lambda_nx_n^2\equiv \lambda_{n+1}\bmod{p^m}$, where $p$ is a fixed odd prime, $\lambda_1,...,\lambda_{n+1}$ are integer…

In this paper, we study the nonlinear Helmholtz equation with mixed dispersion \begin{equation*} \Delta^2 u-\beta k^2\, \Delta u+\alpha k^4 u=W(x)\, |u|^{p-2}u~\text{in}~\mathbb{R}^N, \end{equation*} where the weight function $W(x)$ is…

We consider the set $\mathcal M_n\left(\mathbb Z; H\right)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain and asymptotic formula on the number of matrices from $\mathcal M_n\left(\mathbb Z; H\right)$ with…

Let $\alpha$ and $\beta$ be irrational real numbers and $0<\F<1/30$. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy $\|q\alpha\|\cdot\|q\beta\|<\F$. If we choose $\F$ as a function of $Q$ we get…

For $H \ge N^{1-\frac{1}{2c}} \ln^2 N$, where $c$ is a fixed non-integer number satisfying $$ \|c\| \ge 3c\left(2^{[c]+1}-1\right)\frac{\ln \ln N}{\ln N}, \qquad c > \frac{4}{3}\left(1 + \frac{52\ln \ln N}{\ln N}\right), $$ we obtain an…

The paper presents a discussion on the asymptotic formula for the number of plane partitions of a large positive integer.

In this paper, we study asymptotic expansions of positive solutions of the conformal scalar curvature equation $$ - \Delta u = K(x) u^\frac{n + 2}{n - 2} ~~~~~~ \textmd{in} ~ B_1 \setminus \{ 0 \} $$ with an isolated singularity at the…

We consider a family of solutions to the Painlev\'e II equation $$ u''(x)=2u^3(x)+xu(x)-\alpha \qquad \textrm{with } \a \in \mathbb{R} \cut \{0\}, $$ which have infinitely many poles on $(-\infty, 0)$. Using Deift-Zhou nonlinear steepest…

For fixed integers $b\geq k$, a problem of relevant interest in computer science and combinatorics is that of determining the asymptotic growth, with $n$, of the largest set for which a $(b, k)$-hash family of $n$ functions exists.…

Let $r_{k}(n)$ denote the number of representations of the integer $n$ as a sum of $k$ squares. In this paper, we give an asymptotic for $r_{k}(n)$ when $n$ grows linearly with $k$. As a special case, we find that \[ r_{n}(n) \sim \frac{B…

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x^2+Ny^2 for a squarefree integer N.

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as $t \to \pm \infty$ $(x/t \sim {\cal O}(1))$ of the solution…

In this paper, we are concerned with normalized solutions of the Kirchhoff type equation \begin{equation*} -M\left(\int_{\R^N}|\nabla u|^2\mathrm{d} x\right)\Delta u = \lambda u +f(u) \ \ \mathrm{in} \ \ \mathbb{R}^N \end{equation*} with $u…

Using the matrix Riemann-Hilbert factorization approach for nonlinear evolution systems which take the form of Lax-pair isospectral deformations and whose corresponding Lax operators contain both discrete and continuous spectra, the…

We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% \begin{equation*} \left\{ \begin{array}{l} -M\left( \int_{\mathbb{R}^{3}}\left\vert \nabla u\right\vert…

We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (\Delta - \lambda^2) u = N[u], \end{equation*} where $\Delta = -\sum_j \partial^2_j$ is the…

We prove that $\alpha$-dissipative solutions to the Cauchy problem of the Hunter-Saxton equation, where $\alpha \in W^{1, \infty}(\mathbb{R}, [0, 1))$, can be computed numerically with order $\mathcal{O}(\Delta x^{{1}/{8}}+\Delta…

We revisit a formula for the number of plane partitions due to Almkvist. Using the circle method, we provide modifications to his formula along with estimates of the errors. We show that the improved formula continues to be an asymptotic…