Related papers: Strong normality and generalised Copeland--Erd\H{o…
In this paper, we establish some strong laws of large numbers (SLLN) for non-independent random variables under the framework of sublinear expectations. One of our main results is for blockwise $m$-dependent random variables, and another is…
The main purpose of this paper is to obtain strong laws of large numbers for arrays or weighted sums of random variables under a scenario of dependence. Namely, for triangular arrays $\{X_{n,k}, \, 1 \leqslant k \leqslant n, \, n \geqslant…
In this paper we study the genralized q-Euler numbers and polynomials. From our results, we derive some interesting congruences related tothe generalized q-Euler numbers.
We show that if p is an odd prime then $$\sum_{k=0}^{p-1}E_kE_{p-1-k}=1 (mod p)$$ and $$\sum_{k=0}^{p-3}E_kE_{p-3-k}=(-1)^{(p-1)/2}2E_{p-3} (mod p),$$ where E_0,E_1,E_2,... are Euler numbers. Moreover, we prove that for any positive integer…
We consider normalizing sequences that can give rise to nondegenerate limittheorems for Birkhoff sums under the iteration of a conservative map. Mostclassical limit theorems involve normalizing sequences that are polynomial,possibly with an…
In this paper we use computational methods to disprove a conjecture by Alaoglu and Erd\H{o}s regarding the superabundant numbers.
We prove two results of strong continuity with respect to the initial datum for bounded solutions to the Euler equations in vorticity form. The first result provides sequential continuity and holds for a general bounded solution. The second…
A.A. Suslin proved a normality theorem for an elementary linear group, which says that an elementary linear group of size bigger than or equal to 3 over a commutative ring with unity is normal in the general linear group of same size.…
We give infinite triangularization and strict triangularization results for algebras of operators on infinite dimensional vector spaces. We introduce a class of algebras we call Ore-solvable algebras: these are similar to iterated Ore…
The main purpose of this article is to provide new results on algebraic independence of values of Mahler functions and their generalizations. Simultaneously, we establish new measures of algebraic independence for these values. Among the…
We study the Euler class of smooth orientable infinite-type surface bundles with a section. For many such surfaces, we show that this cohomology class is nontrivial, and that the behavior of its powers depends on the genus and the type of…
A product of compact normal spaces is normal; the product of a countably infinite collection of non-trivial spaces is normal if and only if it is countably paracompact and each of its finite sub-products is normal; if all powers of a space…
Given a weight two modular form f with associated p-adic Galois representation V_f, for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of V_f by taking the Kummer images of Heegner points…
In this paper we prove that any strongly embedded subgroup of a K*-group G of finite Morley rank and odd type that does not interpret any bad field is solvable if its Pruefer 2-rank is at least 2. If the normal 2-rank of G is at least 3…
We resolve the topological version of the Erd\H{o}s Similarity conjecture introduced previously by Gallagher, Lai and Weber. We show that a set is topologically universal on ${\mathbb R}$ if and only if it is of strong measure zero. As a…
We obtain a generalization of the recent Kelley--Meka result on sets avoiding arithmetic progressions of length three. In our proof we develop the theory of the higher energies. Also, we discuss the case of longer arithmetic progressions,…
Solving a decades-old problem we show that Keisler's 1967 order on theories has the maximum number of classes. The theories we build are simple unstable with no nontrivial forking, and reflect growth rates of sequences which may be thought…
Letting $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$, Erd\H{o}s wondered if $\delta_1(n,m)$ is unimodular for fixed $n$. We prove this is false in general, as the sequence $(\delta_1(n,m))$ has…
For all positive powers of primes $p\geq 5$, we prove the existence of infinitely many linear congruences between the exponents of twisted Borcherds products arising from a suitable scalar-valued weight $1/2$ weakly holomorphic modular form…
Let $X$ be an absolutely continuous random variable from the integrated Pearson family and assume that $X$ has finite moments of any order. Using some properties of the associated orthonormal polynomial system, we provide a class of…