Related papers: Sets of inhomogeneous linear forms can be not isot…

We present several results on the number of irrational and linear independent values among $\zeta(s),\zeta(s+2),...,\zeta(s+2n)$, where $s>2$ is an odd integer and $n>0$ is an integer. The main tool in our proofs is a certain generalization…

This survey text deals with irrationality, and linear independence over the rationals, of values at positive odd integers of Riemann zeta function. The first section gives all known proofs (and connections between them) of Ap\'ery's Theorem…

Stanley sequences starting from the set $\{0, n\}$ where $n$ is a positive integer have long been conjectured to be divided into two types: the "regular" type where the growth rate is $\Theta(n^{\log_2(3)})$, and the "irregular" type where…

We show that there exist real numbers $\alpha_1,\alpha_2$ linearly independent over $\mathbb{Z}$ together with 1 such that for every non-zero integer vector $(m_1,m_2)$ with $m_1\ge 0$ and $m_2\ge 0$ one has $||m_1\alpha_1+m_2\alpha_2|| \ge…

We consider two-player random extensive form games where the payoffs at the leaves are independently drawn uniformly at random from a given feasible set C. We study the asymptotic distribution of the subgame perfect equilibrium outcome for…

In this note we present examples of localization functors (in the category of spaces) whose composition with certain cellularization functors is not idempotent, and vice versa.

We investigate the relation between Bell inequalities and nonlocal games by presenting a systematic method for their bilateral conversion. In particular, we show that while to any nonlocal game there naturally corresponds a unique Bell…

We study a combinatorial game derived from a problem in the German National Mathematics Competition. In this game, two players take turns removing numbers from a finite set of natural numbers, aiming to satisfy a certain divisibility…

We found, by Hurwitz's Zeta Function, a new functional equation for Riemann Zeta Function. Considering this equation for $s=2$ and $s=1$, we determine a relation between the values of Riemann zeta Function on positive integers. The Matrix…

We study asymmetric regular types. If $\frak p$ is regular and $A$-asymmetric then there exists a strict order such that Morley sequences in $\frak p$ over $A$ are strictly increasing (we allow Morley sequences to be indexed by elements of…

We present a systematic approach to the problem whether a topologically infinite-dimensional space can be made homogeneous in the Coifman--Weiss sense. The answer to the examined question is negative, as expected. Our leading representative…

We study the problem of representing integers as sums of prime numbers from a fixed Beatty sequence $B_{\alpha,\beta}$, where $\alpha>1$ is irrational and of finite type.

We discuss the phenomenon where an element in a number field is not integrally represented by a given positive definite quadratic form, but becomes integrally represented by this form over a totally real extension of odd degree. We prove…

We consider the problem of estimating the mean vector of a p-variate normal $(\theta,\Sigma)$ distribution under invariant quadratic loss, $(\delta-\theta)'\Sigma^{-1}(\delta-\theta)$, when the covariance is unknown. We propose a new class…

We prove a vertex-isoperimetric inequality for [n]^(r), the set of all r-element subsets of {1,2,...,n}, where x,y \in [n]^(r) are adjacent if |x \Delta y|=2. Namely, if \mathcal{A} \subset [n]^(r) with |\mathcal{A}|=\alpha {n \choose r},…

For an irrational real $\alpha$ and $\gamma\not \in \mathbb Z + \mathbb Z\alpha$ it is well known that $$ \liminf_{|n|\rightarrow \infty} |n| ||n\alpha -\gamma || \leq \frac{1}{4}. $$ If the partial quotients, $a_i,$ in the negative…

In this article, we consider the notion of almost irredundant sets: A subset $\mathcal{X}$ of a C*-algebra $\mathcal{A}$ is called almost irredundant if and only if for every $a\in \mathcal{X}$, the element $a$ does not belong to the…

We introduce a new and natural stationary set preserving forcing $\mathbb P^{c-c}({\lambda},{\mu})$ that (under $\mathsf{NS}_{\omega_1}$ precipitous + existence of $H_{\theta}^#$ for a sufficiently large regular ${\theta}$) increases the…

This paper studies asymmetric Ebert's Hat Problem with five players where the probability of the colors may be unequal. We obtain maximal winning probabilities and optimal winning strategies using the concept of adequate sets.

We study constrained versions of the Ingleton inequality in the entropic setting and quantify its stability under small violations of conditional independence. Although the classical Ingleton inequality fails for general entropy profiles,…