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Let $p$ and $q$ be anisotropic quadratic forms over a field $F$ of characteristic $\neq 2$, let $s$ be the unique non-negative integer such that $2^s < \mathrm{dim}(p) \leq 2^{s+1}$, and let $k$ denote the dimension of the anisotropic part…

A Schur ring (S-ring) over a group $G$ is called separable if every of its similaritities is induced by isomorphism. We establish a criterion for an S-ring to be separable in the case when the group $G$ is cyclic. Using this criterion, we…

We show that for every finite set of prime numbers S, there are at most finitely many singular moduli that are S-units. The key new ingredient is that for every prime number p, singular moduli are p-adically disperse. We prove analogous…

A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel…

Let $\D$ be a set of $n$ pairwise disjoint unit balls in $\R^d$ and $P$ the set of their center points. A hyperplane $\Hy$ is an \emph{$m$-separator} for $\D$ if each closed halfspace bounded by $\Hy$ contains at least $m$ points from $P$.…

We show that if two division $p$-algebras of prime degree share an inseparable field extension of the center then they also share a cyclic separable one. We show that the converse is in general not true. We also point out that sharing all…

Let $\Delta$ be a Delzant polytope in ${\mathbb R}^n$ and ${\bf b}\in{\mathbb Z}^n$. Let $E$ denote the symplectic fibration over $S^2$ determined by the pair $(\Delta, {\bf b})$. We prove the equivalence between the fact that $(\Delta,…

We consider point sets in the affine plane $\mathbb{F}_q^2$ where each Euclidean distance of two points is an element of $\mathbb{F}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean…

The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture in one dimension, which states that a bounded measurable subset of $\mathbb{R}$…

Wavelet analysis has been extended to the $p$-adic line $\mathbb{Q}_p$. The $p$-adic wavelets are complex valued functions with compact support. As in the case of real wavelets, the construction of the basis functions is recursive,…

We use a method involving elementary submodels and a partial converse of Foran lemma to prove separable reduction theorems concerning Suslin sigma-P-porous sets where "P" can be from a rather wide class of porosity-like relations in…

A point set $M$ in Euclidean plane is called an integral point set in semi-general position if all the distances between the elements of $M$ are integers, and $M$ does not contain collinear triples. We improve the lower bound for diameter…

Let $\mathcal{H}$ be a space of analytic functions on the unit ball $\mathbb B_d$ in $\mathbb C^d$ with multiplier algebra $\mathrm{Mult}(\mathcal{H})$. A function $f\in \mathcal{H}$ is called cyclic if the set $[f]$, the closure of…

Let $P \subset \mathbb{R}^{d}$ be a closed convex cone. Assume that $P$ is pointed, i.e. the intersection $P \cap -P=\{0\}$ and $P$ is spanning, i.e. $P-P=\mathbb{R}^{d}$. Denote the interior of $P$ by $\Omega$. Let $E$ be a product system…

Let H be a separable, infinite dimensional Hilbert space and let S be a countable subset of H. Then most positive operators on H have the property that every nonzero vector in the span of S is cyclic, in the sense that the set of operators…

This paper gives a criterion for a moduli point to be a point of non-transversal intersection of the hyperelliptic locus and the supersingular locus in the Siegel moduli stack $\mathfrak{A}_3 \times \mathbb{F}_p$. It is shown that for…

A cylinder $C^1_u$ is the set of infinite words with fixed prefix $u$. A double-cylinder $C^2_{[1,u]}$ is "the same" for bi-infinite words. We show that for every word $u$ and any automorphism $\varphi$ of the free group $F$ the image…

A locally compact stable plane of positive topological dimension will be called semiaffine if for every line $L$ and every point $p$ not in $L$ there is at most one line passing through $p$ and disjoint from $L$. We show that then the plane…

The p-rough numbers are those numbers all of whose prime factors are greater than p. These are exactly those numbers left after Eratosthenes sieve has been advanced from 2 through the prime p. Here we show that for fixed p there is a line…

Let $G$ be a finite group with a Sylow $p$-subgroup $P$. We prove that the principal $p$-blocks of $G$ and $N_G(P)$ are perfectly isometric under the assumption $G$ has a cyclic $p$-hyperfocal subgroup.