Volume 12, Issue 3

Using a generalized Riccati transformation, some new oscillation criteria of linear second order matrix differential system with damping are built by the method of integral average. These results are based on the information on a sequence of subintervals of 𝑡 0 , ).

Using the CartanLaptev invariant analytic method, an invariant affine normal (𝐸) is constructed, which is intrinsically connected with the distribution of hyperplane elements in the (𝑛 1)-dimensional affine space. For the normal (𝐸) we define a second kind normal and an invariant (𝑛 1)-dimensional plane lying in the plane of the element and not passing through the center and corresponding to this normal in the BompianiPantazi projectivity. An invariant point of intersection of the two-dimensional plane passing through the normals (𝐿) and (𝐸) with the second kind normal is found. The construction is carried out without assuming that the nonholonomy tensor is different from zero. Hence both nonholonomic and holonomic distributions are framed by the constructed normal.

In this paper, we show that every conjugacy class of the imprimitive complex reflection group 𝐺(𝑚, 1, 𝑛) can be represented by an admissible diagram. For this, we introduce a length function for elements of 𝐺(𝑚, 1, 𝑛) and study its properties. This then allows us to establish the admissible diagram for each conjugacy class of 𝐺(𝑚, 1, 𝑛). The corresponding results for Weyl groups and their conjugacy classes are well known.

In this study, we obtain a Voronovskaja type asymptotic estimate for 𝑞-BBH operators. Second purpose of this paper is to obtain the monotonicity properties of 𝑞-BBH operators.

Weakly prime ideals in a commutative ring with non-zero identity have been introduced and studied in Anderson and Smith, Houston J. Math. 29: 831840, 2003. Here we study the weakly primary ideals of a commutative ring. We define a proper ideal 𝑃 of 𝑅 to be weakly primary if 0 𝑝𝑞 𝑃 implies 𝑝 𝑃 or 𝑞 Rad(𝑃), so every weakly prime ideal is weakly primary. Various properties of weakly primary ideals are considered. For example, we show that a weakly primary ideal 𝑃 that is not primary satisfies Rad(𝑃) Rad(0). Also, we show that an intersection of a family of weakly primary ideals that are not primary is weakly primary.

We show that a norm version of Hardy's inequality holds in a variable exponent Sobolev space provided the maximal operator is bounded. Our proof uses recent local versions of the inequality for a fixed exponent. We give an example to show that our assumptions on the exponent are essentially sharp. In the one-dimensional case, we derive a necessary and a sufficient condition for Hardy's inequality to hold.

We describe a method for proving the existence of a differential field over which two given differential subfields are linearly disjoint. In the affirmative case, they are linear disjoint over their intersection, which can be computed using characteristic sets. Our method is based on the correspondence between field extensions and ideals.

We prove that there exist two absolutely negligible subsets A and B of the real line R , whose algebraic sum A B is an absolutely nonmeasurable subset of R . We also obtain some generalization of this result and formulate a relative open problem for uncountable commutative groups.

The linear conjugation problem for the class of exponentially doubly quasi-periodic functions is considered, when the jump line is doublyperiodic and consist of open arcs. Effective solutions are obtained by means of a Cauchy type integral with Weierstrass kernel.

It is shown that the uniqueness property of probability invariant measures is preserved under the operation of Cartesian products. A similar question is investigated for inductive limits of nonzero -finite invariant measures.

We prove a weakened version of a conjecture concerning the relation between the Grunsky constant and the Teichmller norm of extensions of univalent functions.

We study some set-theoretic properties of Schmidt's -ideal on , emphasizing its analogies and dissimilarities with both the classical - ideals on of Lebesgue measure zero sets and of Baire first category sets. We highlight the strict analogy between Schmidt's ideal on and Mycielski's ideal on 2 .

Suppose that is a nonconstant analytic self-mapping of the unit disk 𝐷. Necessary or sufficient conditions for the composition operator 𝐶 : 𝑄 𝑞 𝑄 𝑝 to be bounded or compact are given in terms of . The function-theoretic characterization of the compactness of the composition operator 𝐶 : 𝑄 𝑞 𝑄 𝑝,0 is also given.

In this paper, we introduce the concept of 𝑚𝑠𝑘-mappings and establish relationships between spaces with -compact-finite 𝑐𝑓𝑝-networks (𝑐𝑠-networks, 𝑐𝑠*-networks) and metric spaces by means of some covering mappings and 𝑚𝑠𝑘-mappings.

Optimal sufficient conditions are established for the existence and uniqueness of a nonnegative solution of the initial value problem 𝑢(𝑡) (𝑢) (𝑡) 𝑞(𝑡), 𝑢(𝑎) 𝑐 1 , 𝑢 (𝑎) 𝑐 2 , where is nonpositive 𝑎-Volterra operator.

Let 𝑓 denote a 2 periodic function in 𝐿0, 2, and , 𝑛 𝑍, be its Fourier coefficients. For a function 𝑓 of the generalized Wiener class BV (𝑝(𝑛) ) we have proved that

Based on a continuation theorem of Mawhin, periodic solutions are found for difference equations of the form 𝑘 𝑦𝑛 𝑓(𝑛, 𝑦 𝑛 , 𝑦 𝑛1 , . . . , 𝑦 𝑛𝑙 ), 𝑛 𝑍.

By using Ahlfors' theory of a covering surface, we establish the existence of a new singular direction for a meromorphic function 𝑓, namely a 𝑇 direction for 𝑓, for which the Nevanlinna characteristic function 𝑇(𝑟, 𝑓) is used as a comparison function. Then we prove that every 𝑇 direction is a Borel direction for meromorphic function with finite and positive regular growth order.

A procedure of the extension of orthogonal multiresolution analysis is introduced by assuming that 1 (𝑥) ia an orthogonal uniscaling function and constructing a new orthogonal multiscaling function (𝑥) 1 (𝑥), 2 (𝑥) 𝑇 . Moreover, an explicit formula of the orthogonal multiwavelets associated with (𝑥) is obtained. Finally, a construction example is given.

This note contains the following results for a ring A : (1) A is a quasi-Frobenius ring iff A is a left and right YJ-injective, left Noetherian ring whose prime factor rings are right YJ-injective iff every non-zero one-sided ideal of A is the annihilator of a finite subset of elements of A ; (2) if A is a right YJ-injective ring such that any finitely generated right ideal is either a maximal right annihilator or a projective right annihilator, then A is either quasi-Frobenius or a right p.p. ring such that every non-zero left ideal of A contains a non-zero idempotent; (3) a commutative YJ-injective Goldie ring is quasi-Frobenius; (4) if the Jacobson radical of A is reduced, every simple left A -module is either YJ-injective or flat and every maximal left ideal of A is either injective or a two-sided ideal of A , then A is either strongly regular or left self-injective regular with non-zero socle.