Erdős matching conjecture for almost perfect matchings

In 1965 Erd?s asked, what is the largest size of a family of k-element subsets of an n-element set that does not contain a matching of size $s+1$? In this note, we improve upon a recent result of Frankl and resolve this problem for $s>101k^3$ and $(s+1)k?n<(s+1)(k+\frac{1}{100k})$.     

A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random $m\times n$ rank-r matrix over the binary field ${\mathbb{F}}_2$, and let $\mathrm{wt}(\mathbf{M})$ be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as $m,n\to +\infty$ with r fixed and $m/n$ tending to a constant, we have that$\frac{\mathrm{wt}(\mathbf{M})-\frac{1-2^{-r}}{2}mn}{\sqrt{\frac{2^{-r}(1-2^{-r})}{4}(m+n)mn}}$ converges in distribution to a standard normal random variable.     

Semiclassical resolvent bounds for compactly supported radial potentials

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator $?h^2\mathrm{\Delta }+V(|x|)?E$ in dimension $n\ge 2$, where $h,E>0$, and $V:[0,\infty )\to \mathbb{R}$ is $L^{\infty }$ and compactly supported. The weighted resolvent norm grows no faster than $\exp ?(C{h}^{?1})$, while an exterior weighted norm grows $～h^{?1}$. We introduce a new method based on the Mellin transform to handle the two-dimensional case.     

On Serre dimension of monoid algebras and Segre extensions

Let R be a commutative noetherian ring of dimension d and M be a commutative, cancellative, torsion-free monoid of rank r. Then S-$dim(R[M])\le max\{1,dim(R[M])?1\}$. Further, we define a class of monoids ${\{{\mathfrak{M}}_n\}}_{n\ge 1}$ such that if $M\in {\mathfrak{M}}_n$ is seminormal, then S-$dim(R[M])\le dim(R[M])?n=d+r?n$, where $1\le n\le r$. As an application, we prove that for the Segre extension $S_{mn}(R)$ over R, S-$dim(S_{mn}(R))\le dim(S_{mn}(R))?\left[\frac{m+n?1}{min\{m,n\}}\right]=d+m+n?1?\left[\frac{m+n?1}{min\{m,n\}}\right]$.     

Asymptotic expansions of Gurland's ratio and sharp bounds for their remainders

In this paper, we establish a new asymptotic expansion of Gurland's ratio of gamma functions, that is, as $x\to \infty$,$\frac{\mathrm{\Gamma }(x+p)\mathrm{\Gamma }(x+q)}{\mathrm{\Gamma }{(x+(p+q)/2)}^2}=\exp ?[\sum _{k=1}^n\frac{B_{2k}\left(s\right)?B_{2k}(1/2)}{k(2k?1){(x+r_0)}^{2k?1}}+R_n(x;p,q)]$where $p,q\in \mathbb{R}$ with $w=|p?q|\ne 0$ and $s=(1?w)/2$, $r_0=(p+q?1)/2$, $B_{2n+1}\left(s\right)$ are the Bernoulli polynomials. Using a double inequality for hyperbolic functions, we prove that the function $x?{(?1)}^n{R}_n(x;p,q)$ is completely monotonic on $(?r_0,\infty )$ if $|p?q|<1$, which yields a sharp upper bound for $\left|R_n(x;p,q)\right|$. This shows that the approximation for Gurland's ratio by the truncation of the above asymptotic expansion has a very high accuracy. We also present sharp lower and upper bounds for Gurland's ratio in terms of the partial sum of hypergeometric series. Moreover, some known results are contained in our results when $q\to p$.