相关论文: A truncated inner product formula in the geometry …

Invariant ensemble, which are characterised by the joint distribution of eigenvalues $P(\lambda_1,\ldots,\lambda_N)$, play a central role in random matrix theory. We consider the truncated linear statistics $L_K = \sum_{n=1}^K f(\lambda_n)$…

Invariant ensembles of random matrices are characterized by the distribution of their eigenvalues $\{\lambda_1,\cdots,\lambda_N\}$. We study the distribution of truncated linear statistics of the form $\tilde{L}=\sum_{i=1}^p f(\lambda_i)$…

We count primitive lattices of rank $d$ inside $\mathbb{Z}^{n}$ as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subsapce that a lattice spans, namely its…

We apply symmetric function theory to study random processes formed by singular values of products of truncations of Haar distributed symplectic and orthogonal matrices. These product matrix processes are degenerations of Macdonald…

We study the problem of estimating the parameters of a Boolean product distribution in $d$ dimensions, when the samples are truncated by a set $S \subset \{0, 1\}^d$ accessible through a membership oracle. This is the first time that the…

In section 1 we give an improved lower bound on Hermite's constant $\delta_{2g}$ for symplectic lattices in even dimensions ($g=2n$) by applying a mean-value argument from the geometry of numbers to a subset of symmetric lattices. Here we…

Let $f \in S_{k}(\Gamma_{0}(N))$ be a normalized Hecke eigenform. We study the Fourier coefficients $\lambda_{f \otimes \cdots \otimes_{\ell} f}(n)$ of the $\ell$-fold product $L$-function for odd $\ell \ge 3$. Our focus is the distribution…

There is a natural left and right invariant Haar measure associated with the matrix groups GL${}_N(\mathbb R)$ and SL${}_N(\mathbb R)$ due to Siegel. For the associated volume to be finite it is necessary to truncate the groups by imposing…

Truncated Neumann series $S_k(A)=I+A+\cdots+A^{k-1}$ are used in approximate matrix inversion and polynomial preconditioning. In dense settings, matrix-matrix products dominate the cost of evaluating $S_k$. Naive evaluation needs $k-1$…

Let $f$ a smooth function with compact support defined on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$, we prove a formula for the Mellin transform of $f$, then we can define the micro-local lift $d\epsilon_{it}$ to…

We investigate the approximation of quadratic Dirichlet $L$-functions over function fields by truncations of their Euler products. We first establish representations for such $L$-functions as products over prime polynomials times products…

We develop the theory and properties of primitive unimodular $S$-arithmetic lattices in $\mathbb{Q}_S^d$ by giving integral formulas in the spirit of Siegel's primitive mean value formula and Rogers' and Schmidt's second moment formulas.…

Let $f$ and $g$ be two holomorphic or Hecke-Maass primitive cusp forms for $SL(2,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character of modulus $p$, an odd prime. A subconvex bound for the central values of the Rankin-Selberg…

Given a half-integral weight holomorphic Kohnen newform $f$ on $\Gamma_0(4)$, we prove an asymptotic formula for large primes $p$ with power saving error term for \begin{equation*} \sideset{}{^*} \sum_{\chi \hspace{-0.15cm} \pmod{p}} |…

The results of a series of theoretical studies are reported, examining the convergence rate for different approximate representations of $\alpha$-stable distributions. Although they play a key role in modelling random processes with jumps…

Let $K = \mathbb{Q}(i)$. We study the Petersson inner product of a Hermitian Eisenstein series of Siegel type on the unitary group $U_{5}(K)$, diagonally-restricted on $U_2(K)\times U_2(K)\times U_1(K)$, against two Hermitian cuspidal…

We study matrix integrals of the form $$\int_{\mathrm{USp(2n)}}\prod_{j=1}^k\mathrm{tr}(U^j)^{a_j}\mathrm d U,$$ where $a_1,\ldots,a_r$ are natural numbers and integration is with respect to the Haar probability measure. We obtain a compact…

We study the value distribution of the Sudler product $\prod_{n=1}^N |2 \sin (\pi n \alpha )|$ and the Diophantine product $\prod_{n=1}^N (2e\| n \alpha \|)$ for various irrational $\alpha$, as $N$ ranges in a long interval of integers. At…

The main objective of this thesis is a classification project for integral lattices. Using Kneser's neighbour method we have developed the computer program tn to classify complete genera of integral lattices. Main results are detailed…

An asymptotic formula with a square root error term is obtained for the number of elements with given trace and norm in a finite semisimple algebra over a finite field. This extends previous results from finite etale algebras (commutative…