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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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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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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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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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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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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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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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In this paper, we establish some Landau–Kolmogorov type inequalities for differential operators generated by polynomials in the following form
P(D)fpK1(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 substitutingxi/x . Moreover, the explicit form of K1(ε,p) and K2(ε,m)

are given.

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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)u(f*>λ)C𝔼Φ2Cfυγ(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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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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