On a density problem of Erdös by Safwan Akbik

“…For a positive integer n, let P(n) denotes the largest prime divisor of n and define the set: 𝒮(x)=𝒮={n≤x:n   does not divide   P(n)!}. Paul Erdös has proposed that |S|=o(x) as x→∞, where |S| is the number of n∈S. …”
Get full text

Analytic Erdös-Turán conjectures and Erdös-Fuchs theorem by L. Haddad, C. Helou, J. Pihko

“…The sequences of such coefficients have a lot of similarity with sequences of natural numbers considered in additive number theory. It is this analogy that we pursue, thus establishing many properties and giving equivalent statements to the well-known Erdös-Turán conjectures in terms of supported series and extending to them a version of Erdös-Fuchs theorem.…”
Get full text

Erdős–Rényi Poissonized by Curien, Nicolas

“…We introduce a variant of the Erdős–Rényi random graph where the number of vertices is random and follows a Poisson law. …”
Get full text

An Erdős-Révész Type Law for the Length of the Longest Match of Two Coin-Tossing Sequences by Karl Grill

“…., a sequence of independent variables, taking values 0 and 1 with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>. The famous Erdős-Rényi law of large numbers implies that the longest run of ones in the first <i>n</i> observations has a length <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><mi>n</mi></msub></semantics></math></inline-formula> that behaves like <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>log</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula>, as <i>n</i> tends to infinity (throughout this article, log denotes logarithm with base 2). …”
Get full text

Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences by Shunli Hao

“…We study the convergence rates in the law of large numbers for arrays of Banach valued martingale differences. …”
Get full text

New proof and generalization of some results on translated sums over k-almost primes by Laib, Ilias

“…A sequence $\mathcal{A}$ of strictly positive integers is said to be primitive if none of its terms divides the others, Erdős conjectured that the sum $f(\mathcal{A},0)\le f(\mathbb{N}_{1},0),$ where $\mathbb{N}_{1}$ is the sequence of prime numbers and $f(\mathcal{A},h)=\sum _{a\,\in \,\mathcal{A}}$ $\frac{1}{a(\log a+h) }$. …”
Get full text

The Intertemporal Evolution Model of Enterprise R&D Cooperative Network by Rongguo Qiu, Yutong Wang, Tingqiang Chen

“…Results show that (1) when the R&D network evolution reaches a stable state, the joint distribution of stock knowledge and the number of cooperative enterprises do not affect the initial R&D network topology. …”
Get full text

Finding product sets in some classes of amenable groups by Dimitrios Charamaras, Andreas Mountakis

“…In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. …”
Get full text