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In this paper, we prove that the ∗-Ricci tensor of a real hypersurface in complex projective plane ℂ*P*
^{2} or complex hyperbolic plane ℂ*H*
^{2} is cyclic parallel if and only if the hypersurface is of type (A). We find some three-dimensional real hypersurfaces having non-vanishing and non-parallel ∗-Ricci tensors which are cyclic parallel.

We prove that

for all integers n ≥ 1 and ɵ ≤ 8 ≤ π. This result refines inequalities due to Jackson (1911) and Turán (1938).

Let *D* be a weighted oriented graph, whose underlying graph is *G*, and let *I (D)* be its edge ideal. If *G* has no 3-, 5-, or 7-cycles, or *G* is Kőnig, we characterize when *I (D)* is unmixed. If *G* has no 3- or 5-cycles, or *G* is Kőnig, we characterize when *I (D)* is Cohen–Macaulay. We prove that *I (D)* is unmixed if and only if *I (D)* is Cohen–Macaulay when *G* has girth greater than 7 or *G* is Kőnig and has no 4-cycles.

In cryptography one needs pseudorandom sequences whose short subsequences are also pseudorandom. To handle this problem, Dartyge, Gyarmati and Sárközy introduced weighted measures of pseudorandomness of binary sequences. In this paper we continue the research in this direction. We introduce weighted pseudorandom measure for multidimensional binary lattices and estimate weighted pseudorandom measure for truly random binary lattices. We also give lower bounds for weighted measures of even order and present an example by using the quadratic character of finite fields.

The symbol S(*X*) denotes the hyperspace of finite unions of convergent sequences in a Hausdor˛ space *X*. This hyper-space is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in S(*X*). Then we consider some cardinal invariants on S(*X*), and compare the character, the pseudocharacter, the sn-character, the so-character, the network weight and cs-network weight of *S*(*X*) with the corresponding cardinal function of *X*. Moreover, we consider rank *k*-diagonal on *S*(*X*), and give a space *X* with a rank *2*-diagonal such that *S*(*X*) does not *G _{δ}
*-diagonal. Further, we study the relations of some generalized metric properties of

*X*and its hyperspace

*S*(

*X*). Finally, we pose some questions about the hyperspace

*S*(

*X*).

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.

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.

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.

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.

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.