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# Mathematics and Statistics

Let *k* ≥ 1. A *Sperner k-family* is a maximum-sized subset of a finite poset that contains no chain with *k* + 1 elements. In 1976 Greene and Kleitman defined a lattice-ordering on the set *S _{k}*(

*P*) of Sperner

*k*-families of a fifinite poset

*P*and posed the problem: “Characterize and interpret the join- and meet-irreducible elements of

*S*(

_{k}*P*),” adding, “This has apparently not been done even for the case

*k*= 1.”

In this article, the case *k* = 1 is done.

The aim of this paper is to prove some uncertainty inequalities for the continuous Hankel wavelet transform, and study the localization operator associated to this transformation.

In this note, we show that the result [1, Proposition 5.2] is inaccurate. We further give and prove the correct modification of such a result. Some applications are also given.

Let K = ℚ(α) be a number field generated by a complex root a of a monic irreducible polynomial ƒ (x) = x^{36} − m, with m ≠ ±1 a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≠ ±1 (mod 9) then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡±1 (mod 9), then the number field K is not monogenic.

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.