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Let *G* be a finite group and *H* a subgroup of *G*. *H* is said to be *S*-quasinormal in *G* if *HP* = *PH* for all Sylow subgroups *P* of *G*. Let *H*
_{sG} be the subgroup of *H* generated by all those subgroups of *H* which are *S*-quasinormal in *G* and *H*
^{sG} the intersection of all *S*-quasinormal subgroups of *G* containing *H*. The symbol |*G*|_{p} denotes the order of a Sylow *p*-subgroup of *G*. We prove the followingTheorem A. *Let G be a finite group and p a prime dividing* |*G*|. *Then G is p-supersoluble if and only if for every cyclic subgroup H of*
*Ḡ* (*G*) *of prime order or order* 4 (*if p* = 2*)*, *Ḡ*
*has a normal subgroup T such that*
*H*
^{sḠ}
*and*
*H*∩*T*=*H*
_{sḠ}∩*T*.Theorem B. *A soluble finite group G is p-supersoluble if and only if for every* 2-*maximal subgroup E of G such that O*
_{p′} (*G*) ≦ *E and* |*G: E*| *is not a power of p, G has an S-quasinormal subgroup T with cyclic Sylow p-subgroups such that E*
^{sG} = *ET and* |*E* ∩ *T*|_{p} = |*E*
_{sG} ∩ *T*|_{p}.Theorem C. *A finite group G is p-soluble if for every* 2-*maximal subgroup E of G such that O*
_{p′} (*G*) ≦ *E and* |*G: E*| *is not a power of p, G has an S-quasinormal subgroup T such that E*
^{sG} = *ET and* |*E* ∩ *T*
_{p} = |*E*
_{sG} ∩ *T*|_{p}.

We introduce the concept of nil-McCoy rings to study the structure of the set of nilpotent elements in McCoy rings. This notion extends the concepts of McCoy rings and nil-Armendariz rings. It is proved that every semicommutative ring is nil-McCoy. We shall give an example to show that nil-McCoy rings need not be semicommutative. Moreover, we show that nil-McCoy rings need not be right linearly McCoy. More examples of nil-McCoy rings are given by various extensions. On the other hand, the properties of *α*-McCoy rings by considering the polynomials in the skew polynomial ring *R*[*x*; *α*] in place of the ring *R*[*x*] are also investigated. For a monomorphism *α* of a ring *R*, it is shown that if *R* is weak *α*-rigid and *α*-reversible then *R* is *α*-McCoy.

Suppose that *A* is either the Banach algebra *L*
^{1}(*G*) of a locally compact group *G*, or measure algebra *M*(*G*), or other algebras (usually larger than *L*
^{1}(*G*) and *M*(*G*)) such as the second dual, *L*
^{1}(*G*)**, of *L*
^{1}(*G*) with an Arens product, or *LUC*(*G*)* with an Arenstype product. The left translation invariant closed convex subsets of *A* are studied. Finally, we obtain necessary and sufficient conditions for *LUC*(*G*)* to have 1-dimensional left ideals.

Let *K* be a finite field and let *X** be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Gröbner bases, to compute the length and the dimension of *C*
_{X*} (*d*), the parameterized affine code of degree *d* on the set *X**. If *Y* is the projective closure of *X**, it is shown that *C*
_{X*} (*d*) has the same basic parameters that *C*
_{Y} (*d*), the parameterized projective code on the set *Y*. If *X** is an affine torus, we compute the basic parameters of *C*
_{X}* (*d*). We show how to compute the vanishing ideals of *X** and *Y*.

The smallest monoid containing a 2-testable semigroup is defined to be a 2-*testable monoid*. The well-known Brandt monoid *B*
_{2}
^{1}
of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid *B*
_{2}
^{1}
is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

## Реэюме

*s*,

*p*,

*q*,

*r*(эдесъ

*s*∈ (0, ∞)

^{n}, 1 ≤

*p*,

*r*,

*q*, ≤ ∞, 1 ≤

*n*≤

*k*,

*m*= (

*m*

_{1}, …,

*m*

_{n}) ∈ ℕ

^{n}:

*m*

_{1}+ … +

*m*

_{n}=

*k*).

## Abstract

The main objective of this paper is a study of some new multidimensional Hilbert type inequalities with a general homogeneous kernel. We derive a pair of equivalent inequalities, and also establish the conditions under which the constant factors included in the obtained inequalities are the best possible. Some applications in particular settings are also considered.

## Abstract

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallée Poussin sums on
the sets *C*
_{β}
^{q}
*H*
_{ω} of Poisson integrals of functions from the class *H*
_{ω} generated by convex upwards moduli of continuity *ω*(*t*) which satisfy the condition *ω*(*t*)/*t* → ∞ as *t* → 0. As an implication, a solution of the Kolmogorov-Nikol’skii problem for de la Vallée Poussin sums on the sets of Poisson
integrals of functions belonging to Lipschitz classes *H*
^{α}, 0 < *α* < 1, is obtained.

A system of *m* nonzero vectors in ℤ^{n} is called an *m*-icube if they are pairwise orthogonal and have the same length. The paper describes *m*-icubes in ℤ^{4} for 2 ≦ *m* ≦ 4 using Hurwitz integral quaternions, counts the number of them with given edge length, and proves that unlimited extension is possible in ℤ^{4}.

Motivated by a remark and a question of Nicholas Katz, we characterize the tangent space of the space of Fuchsian equations with given generic exponents inside the corresponding moduli space of logarithmic connections: we construct a weight 1 Hodge structure on the tangent space of the moduli of logarithmic connections such that deformations of Fuchsian equations correspond to the (1, 0)-part.