In earlier work, we established the a.e. convergence of certain Eisenstein series on arithmetic quotients of loop groups. In this article we prove that these series converge everywhere, and uniformly on certain bounded sets

Let $f$ be a primitive positive integral binary quadratic form of discriminant $-D$, and let $r_f(n)$ be the number of representations of $n$ by $f$ up to automorphisms of $f$. In this article, we give estimates and asymptotics for the quantity $\sum _{n\le x}{r}_f(n){}^{\beta }$ for all $\beta \ge 0$ and uniformly in $D=o(x)$. As a consequence, we get more-precise estimates for the number of integers which can be written as the sum of two powerful numbers

We generalize Griffiths's theorem on the Hodge filtration of the primitive cohomology of a smooth projective hypersurface using the local Bernstein-Sato polynomials, the $V$-filtration of Kashiwara [22] and Malgrange [27] along the hypersurface, and the Brieskorn module of the global defining equation of the hypersurface (see [4])

By the inverse method we show that the Korteweg–de Vries equation (KdV) $\partial {}_{t}v(x,t)=-{\partial }_x^{3}v(x,t)+6v(x,t)\partial {}_{x}v(x,t)$ $(x\in \mathbb{T},t\in \mathbb{R})$ is globally (in time) wellposed in the Sobolev space of distributions $H^{\beta }(\mathbb{T},\mathbb{R})$ for any $\beta \ge -1$

A. W. Reid [R, Theorem 2.1] showed that if ${\Gamma }_1$ and ${\Gamma }_2$ are arithmetic lattices in $G={\mathrm{PGL}}_2(\mathbb{R})$ or in ${\mathrm{PGL}}_2(\mathbb{C})$ which give rise to isospectral manifolds, then ${\Gamma }_1$ and ${\Gamma }_2$ are commensurable (after conjugation). We show that for $d\ge 3$ and $\mathcal{S}={\mathrm{PGL}}_d(\mathbb{R})/{\mathrm{PO}}_d(\mathbb{R})$ or for $\mathcal{S}={\mathrm{PGL}}_d(\mathbb{C})/{\mathrm{PU}}_d(\mathbb{C})$, the situation is quite different; there are arbitrarily large finite families of isospectral noncommensurable compact manifolds covered by $\mathcal{S}$. The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants

We apply spectral analysis of quotients of the Bruhat-Tits buildings of type ${\tilde{A}}_{d-1}$ to construct isospectral nonisomorphic Cayley graphs of the finite simple groups ${\mathrm{PSL}}_d({\mathbb{F}}_q)$ for every $d\ge 5$ ($d\ne 6$) and prime power $q>2$

Let $Q_n$ denote a random symmetric ($n\times n)$-matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values $0$ and $1$ with probability $1/2$). We prove that $Q_n$ is nonsingular with probability $1-O(n^{-1/8+\delta })$ for any fixed $\delta >0$. The proof uses a quadratic version of Littlewood-Offord-type results concerning the concentration functions of random variables and can be extended for more general models of random matrices