Mathematics and Statistics

Alfréd Rényi, the founding director of the Mathematical Institute of the Hungarian Academy of Sciences was the first mathematician who proved a density theorem for the zeros of Dirichlet’s 𝐿-functions with variable moduli. This was based on a refinement of the large sieve of Linnik, developed by Rényi himself. He used this to show a weaker form of the binary Goldbach conjecture. His density theorem was the first forerunner of the famous Bombieri–Vinogradov theorem. We give a simple alternative proof of a weaker form of the Bombieri–Vinogradov theorem, based only on classical facts about 𝐿-functions (including Siegel’s theorem) and a simple but ingenious idea of Halász, but without using any form of the large sieve.

In the present paper, we study the asymptotic properties of an exponential-type operator which was recently constructed. It is connected with 𝑝(𝑥) = 𝑥^4/3. The main result is a pointwise complete asymptotic expansion valid for locally smooth functions. All coefficients are derived and explicitly given.

Following previous observations on 𝑞-Appell and 𝑞-Lauricella functions, the purpose of this article is to find canonical 𝑞-difference equations for the four intermediate 𝑞-Lauricella functions $\left(k\right){\Phi }_{\text{AC}}^{\left(n\right)},\left(k\right){\Phi }_{\text{AD}}^{\left(n\right)},\left(k\right){\Phi }_{\text{BD}}^{\left(n\right)}\text{ and }\left(k\right){\Phi }_{\text{CD}}^{\left(n\right)}$ . The convergence regions for the above functions have already been considered in previous papers/studies. To save space, these 𝑞-difference equations are written in vector form. Furthermore, many more solutions of these 𝑞-difference equations for the two first functions are proved and the proofs are almost identical to another 𝑞-Lauricella function article. The reason is that the order of the four functions above is by order of symmetry; like in physics, the molecules (our parameters) strive to obtain maximum symmetry. Furthermore, a 𝑞-Laplace integral expressions for the first function $\left(k\right){\Phi }_{\text{AC}}^{\left(n\right)}$ in the form 𝑞-confluent functions is used to find more solutions.

We prove that, when 𝑛 goes to infinity, Kostant’s problem has negative answer for almost all simple highest weight modules in the principal block of the BGG category O for the Lie algebra sl_𝑛(ℂ).

Recent results have provided important functional generalizations, extensions and improvements of the Hardy and Levinson integral inequalities. However, they require some assumptions on the main functions, such as monotonicity or convexity assumptions, which remain somewhat restrictive. In this article, we propose two new ideas of functional generalizations, one based on a series expansion approach and the other on an integral approach. Both achieve the goal of offering adaptable generalizations and extensions of the Hardy and Levinson integral inequalities. They are formulated in two different general theorems, which are proved in detail. Several examples of new integral inequalities are derived.

Let {𝐿_𝑛}_≥0 be the sequence of Lucas numbers. In this paper, we determine all Lucas numbers that are palindromic concatenations of two distinct repdigits.

The Erdős Matching Conjecture states that the maximum size 𝑓 (𝑛, 𝑘, 𝑠) of a family $F\subseteq \left(\begin{array}{l}\left[n\right]\\ k\end{array}\right)$ that does not contain 𝑠 pairwise disjoint sets is max. $\left\{\left|A_{k,s}\right|,\left|B_{n,k,s}\right|\right\}$ , where $A_{k,s}=\left(\left[\begin{array}{c}sk-1\\ k\end{array}\right]\right)$ and $B_{n,k,s}=\left\{B\in \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right):B\cap \left[s-1\right]\ne \varnothing \right\}$ . The case 𝑠 = 2 is simply the Erdős-Ko-Rado theorem on intersecting families and is well understood. The case 𝑛 = 𝑠𝑘 was settled by Kleitman and the uniqueness of the extremal construction was obtained by Frankl. Most results in this area show that if 𝑘, 𝑠 are fixed and 𝑛 is large enough, then the conjecture holds true. Exceptions are due to Frankl who proved the conjecture and considered variants for 𝑛 ∈ [𝑠𝑘, 𝑠𝑘 + 𝑐𝑠,𝑘 ] if 𝑠 is large enough compared to 𝑘. A recent manuscript by Guo and Lu considers non-trivial families with matching number at most 𝑠 in a similar range of parameters.

In this short note, we are concerned with the case 𝑠 ≥ 3 fixed, 𝑘 tending to infinity and 𝑛 ∈ {𝑠𝑘, 𝑠𝑘 + 1}. For 𝑛 = 𝑠𝑘, we show the stability of the unique extremal construction of size $\left(\begin{array}{c}sk-1\\ k\end{array}\right)=\frac{s-1}{s}\left(\begin{array}{c}sk\\ k\end{array}\right)$ with respect to minimal degree. As a consequence we derive $\underset{k\to \infty }{\lim }\frac{f\left(sk+1,k,s\right)}{\left(\begin{array}{c}sk+1\\ k\end{array}\right)}<\frac{s-1}{s}-{\epsilon }_s$ for some positive constant 𝜀_𝑠 which depends only on 𝑠.

We study the property of Kelley and the property of Kelley weakly on Hausdorff continua. We extend results known for metric continua to the class of Hausdorff continua. We also present new results about these properties.

The aim of this paper is to study the interrelationship between various forms of (F, G)-shadowing property and represent it through the diagram. We show that asymptotic shadowing is equivalent to (ℕ₀, F _𝑐𝑓 )-shadowing property and that (ℕ₀, F _𝑐𝑓 )-shadowing implies (F _𝑐𝑓 , F _𝑐𝑓 )-shadowing. Necessary examples are discussed to support the diagram. We also give characterization for maps to have the (F, G)-shadowing property through the shift map on the inverse limit space. Further, we relate the (F, G)-shadowing property to the positively F _𝑠-expansive map. Also, we obtain the necessary and sufficient condition for the identity map to have (ℕ₀, F _𝑡)-shadowing property.

In this article, we present new results on specific cases of a general Young integral inequality established by Páles in 1990. Our initial focus is on a bivariate function, defined as the product of two univariate and separable functions. Based on this, some new results are established, including particular Young integral-type inequalities and some upper bounds on the corresponding absolute errors. The precise role of the functions involved in this context is investigated. Several applications are presented, including one in the field of probability theory. We also introduce and study reverse variants of our inequalities. Another important contribution is to link the setting of the general Young integral inequality established by Páles to a probabilistic framework called copula theory. We show that this theory provides a wide range of functions, often dependent on adjustable parameters, that can be effectively applied to this inequality. Some illustrative graphics are provided. Overall, this article broadens the scope of bivariate inequalities and can serve related purposes in analysis, probability and statistics, among others.