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# The Erdős–Rényi Law and Strong Limit Theorems of Probability

Studia Scientiarum Mathematicarum Hungarica
Author: Andrei N. Frolov

Fifty years ago P. Erdős and A. Rényi published their famous paper on the new law of large numbers. In this survey, we describe numerous results and achievements which are related with this paper or motivated by it during these years.

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# Finite Groups with Some Subgroups of Sylow Subgroups s∗-Semipermutable

Studia Scientiarum Mathematicarum Hungarica
Authors: Qingjun Kong and Xiuyun Guo

We introduce a new subgroup embedding property in a finite group called s -semipermutability. Suppose that G is a finite group and H is a subgroup of G. H is said to be s -semipermutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K is s-semipermutable in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P | and study the structure of G under the assumption that every subgroup H of P with |H | = |D| is s -semipermutable in G. Some recent results are generalized and unified.

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# Ideals of Residuated Lattices

Studia Scientiarum Mathematicarum Hungarica
Authors: Liviu-Constantin Holdon and Arsham Borumand Saeid

In this article, we study ideals in residuated lattice and present a characterization theorem for them. We investigate some related results between the obstinate ideals and other types of ideals of a residuated lattice, likeness Boolean, primary, prime, implicative, maximal and ʘ-prime ideals. Characterization theorems and extension property for obstinate ideal are stated and proved. For the class of ʘ-residuated lattices, by using the ʘ-prime ideals we propose a characterization, and prove that an ideal is an ʘ-prime ideal iff its quotient algebra is an ʘ-residuated lattice. Finally, by using ideals, the class of Noetherian (Artinian) residuated lattices is introduced and Cohen’s theorem is proved.

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# Interpolation by Differences In H∞

Studia Scientiarum Mathematicarum Hungarica
Authors: Francesc Tugores and Laia Tugores

We pose an interpolation problem for the space of bounded analytic functions in the disk. The interpolation is performed by a function and its di˛erence of values in points whose subscripts are related by an increasing application. We impose that the data values satisfy certain conditions related to the pseudohyperbolic distance, and characterize interpolating sequences in terms of uniformly separated subsequences.

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# Nearly s-Semipermutable Subgroups

Studia Scientiarum Mathematicarum Hungarica
Author: Changwen Li

In this paper, we investigate the infiuence of nearly s-semipermutable subgroups on the structure of finite groups. Several recent results from the literature are improved and generalized.

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# On Some Continued Fractions and Series

Studia Scientiarum Mathematicarum Hungarica
Authors: Khalil Ayadi, Chiheb Ben Bechir, and Iheb Elouaer

We exhibit some explicit continued fraction expansions and their representation series in different fields. Some of these continued fractions have a type of symmetry, known as folding symmetry. We will extracted those whose are specialized.

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# Polynomial Entropy of the Logistic Map

Studia Scientiarum Mathematicarum Hungarica
Author: Milan Perić

We study the polynomial entropy of the logistic map depending on a parameter, and we calculate it for almost all values of the parameter. We show that polynomial entropy distinguishes systems with a low complexity (i.e. for which the topological entropy vanishes).

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# Some Landau–Kolmogorov Type Inequalities for Differential Operators Generated by Polynomials

Studia Scientiarum Mathematicarum Hungarica
Authors: Vu Nhat Huy, Nguyen Ngoc Huy, and Chu Van Tiep
In this paper, we establish some Landau–Kolmogorov type inequalities for differential operators generated by polynomials in the following form
$P(D)fp≤K1(ε,P)fq+K2(ε,m)Dm(P(D)f)p$

for all $ε>0$ , where 0 < gp ≤ ∞, and the differential operator P (D) is obtained from the polynomial P (x) by substituting$x→−i∂/∂x$ . Moreover, the explicit form of $K1(ε,p)$ and $K2(ε,m)$

are given.

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# A Unified Version of Weighted Weak Type Inequality for Martingale Maximal Operators

Studia Scientiarum Mathematicarum Hungarica
Authors: Yanbo Ren and Congbian Ma
Let ɣ and Φ1 be nondecreasing and nonnegative functions defined on [0, ∞), and Φ2 is an N -function, u, v and w are weights. A unified version of weighted weak type inequality of the form
$Φ1(λ)ℙu(f*>λ)≤C𝔼Φ2Cf∞υγ(λ)w$

for martingale maximal operators f is considered, some necessary and su@cient conditions for it to hold are shown. In addition, we give a complete characterization of three-weight weak type maximal inequality of martingales. Our results generalize some known results on weighted theory of martingale maximal operators.

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# Weak Solutions for Obstacle Problems with Weak Monotonicity

Studia Scientiarum Mathematicarum Hungarica
Authors: Farah Balaadich and Elhoussine Azroul

This paper is concerned with the existence of weak solutions for obstacle problems. By means of the Young measure theory and a theorem of Kinderlehrer and Stampacchia, we obtain the needed result.

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