Related papers: On Oppenheim-type conjecture for systems of quadra…

Let $k$ be an algebraically closed field of characteristic $p>0$, $W$ the ring of Witt vectors over $k$ and ${R}$ the integral closure of $W$ in the algebraic closure ${\bar{K}}$ of $K:=Frac(W)$; let moreover $X$ be a smooth, connected and…

The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimension two, incorporating zeta functions with differential forms and targeting all monodromy eigenvalues, and also considering singular ambient…

We study the vanishing sets of slice regular polynomials in several quaternionic variables. We obtain a geometric description of the vanishing sets in two variables, which leads to a new version of the Strong Hilbert Nullstellensatz in the…

We prove the well-posedness and regularity of solutions in mixed-norm weighted Sobolev spaces for a class of second-order parabolic and elliptic systems in divergence form in the half-space $\mathbb{R}^d_+ = \{x_d > 0\}$ subject to the…

Let $\mathrm{R}$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by polynomials of degrees bounded by $d$ vanishes in…

Consider a closed connected hypersurface in $\mathbb{R}^n$ with constant signature (k,l) of the second quadratic form, and approaching a quadratic cone at infinity. This hypersurface divides $\mathbb{R}^n$ into two pieces. We prove that one…

An achievement set of a series is a set of all its subsums. We study the properties of achievement sets of conditionally convergent series in finite dimensional spaces. The purpose of the paper is to answer some of the open problems…

We determine set-theoretic defining equations for the variety of hypersurfaces of degree d in an N-dimensional complex vector space that have dual variety of dimension at most k. We apply these equations to the Mulmuley-Sohoni variety, the…

A representation theorem for non-semibounded Hermitian quadratic forms in terms of a (non-semibounded) self-adjoint operator is proven. The main assumptions are closability of the Hermitian quadratic form, the direct integral structure of…

We show that for the base two expansion \[ x=\sum_{i=1}^{\infty}2^{-(d_{1}(x)+d_{2}(x)+\dots+d_{i}(x))}\] with $x\in(0,1]$ and $d_{i}(x)\in\mathbb{N}$ the set $A=\{x|\lim_{i\to\infty}d_{i}(x)=\infty\}$ has Hausdorff dimension zero, this is…

We study the quantum fidelity (groundstate overlap) near quantum phase transitions of the Ising universality class in one dimensional (1D) systems of finite size L. Prominent examples occur in magnetic systems (e.g. spin-Peierls, the…

Let $\mathcal{H}_i$ be a finite dimensional complex Hilbert space of dimension $d_i$ associated with a finite level quantum system $A_i$ for $i = i, 1,2, ..., k$. A subspace $S \subset \mathcal{H} = \mathcal{H}_{A_{1} A_{2}... A_{k}} =…

The paper is concerned with Kropholler's conjecture on splitting a finitely generated group over a codimension-1 subgroup. For a subgroup H of a group G, we define the notion of "finite splitting height" which generalises the finite-height…

Let F be a non-archimedian local field of characteristic 0, and O the ring of integres in F. We give an explicit formula for the Siegel series of a half-integral matrix over O. This formula expresses the Siegel series of a half-integral…

We study the Cauchy problem for a system of cubic nonlinear Klein-Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the order…

In this work we examine the stability of some classes of integrals, and in particular with respect to homogenization. The prototypical case is the homogenization of quadratic energies with periodic coefficients perturbed by a term vanishing…

In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some…

Here are two of our main results: Theorem 1. Let X be a normal space with dim X=n and m\geq n+1. Then the space C*(X,R^m) of all bounded maps from X into R^m equipped with the uniform convergence topology contains a dense G_{\delta}-subset…

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More…

We study Liouville-type theorem for polyharmonic H\'enon-Lane-Emden system $(-\Delta)^mu=|x|^av^p,\; (-\Delta)^mv=|x|^bu^q$ when $m,p,q\geq 1, pq\ne 1$, and $a,b\geq 0$. It is a natural conjecture that the nonexistence of positive solutions…