Several logarithmic Caffarelli–Kohn–Nirenberg inequalities and applications

In this paper, based on the Caffarelli–Kohn–Nirenberg inequalities on the Euclidean space and the weighted Hölder inequality, we establish the logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities, and give applications for the weighted ultracontractivity of positive strong solutions to a kind of evolution equations. We also prove corresponding logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities on the Heisenberg group and related to generalized Baouendi–Grushin vector fields. Some applications are provided.     

Properties of Some Families of Meromorphic Multivalent Functions Associated with Generalized Hypergeometric Functions

We introduce and study two subclasses ?_([α_1])(A, B, λ) and ?_([α_1])~+ (A, B, λ) of meromorphic p-valent functions defined by certain linear operator involving the generalized hypergeometric function. The main object is to investigate the various important properties and characteristics of these subclasses of meromorphically multivalent functions. We extend the familiar concept of neighborhoods of analytic functions to these subclasses. We also derive many interesting results for the Hadamard products of functions belonging to the class ?_([α_1])~+(α, β, γ, λ).     

Self-complementary magic squares of doubly even orders

A magic square $M$ in which the entries consist of consecutive integers from $1,2,\dots ,n^2$ is said to be self-complementary of order$n$ if the resulting square obtained from $M$ by replacing each entry $i$ by $n^2+1?i$ is equivalent to $M$ (under rotation or reflection). We present a new construction for self-complementary magic squares of order $n$ for each $n\ge 4$, where $n$ is a multiple of $4$.     

Compatible Eulerian circuits in Eulerian (di)graphs with generalized transition systems

A transition in a graph is defined as a pair of adjacent edges. A transition system of an Eulerian graph refers to a set of partitions such that for each vertex of the graph, there corresponds to a partition of the set of edges incident to the vertex into transitions. A generalized transition system $F\left(G\right)$ over a graph $G$ defines a set of transitions over $G$. A compatible Eulerian circuit of an Eulerian graph $G$ with a generalized transition system $F\left(G\right)$ is defined as an Eulerian circuit in which no two consecutive edges form a transition defined by $F\left(G\right)$. In this paper, we further introduce the concept of weakly generalized transition system which is an extension of the generalized transition system and prove some Ore-type sufficient conditions for the existence of compatible Eulerian circuits in Eulerian graphs with (weakly) generalized transition systems and obtain corresponding results for Eulerian digraphs. Our conditions improve some previous results due to Jackson and Isaak, respectively.     

A new approach method to obtain the L1 generalized analytic Fourier–Feynman transform

In this paper, we first introduce the concept of the ?-product on function space. We then proceed to use this concept to obtain several integration formulas. In addition, we establish various relationships which exist. Also, we establish the relationships among the ?-product, the generalized convolution product and the first variation.     

Endpoint Estimates for Generalized Multilinear Fractional Integrals on the Non-homogeneous Metric Spaces

In this paper, some endpoint estimates for the generalized multilinear fractional integrals I_(α,m) on the non-homogeneous metric spaces are established.     

On one-relator products induced by generalized triangle groups II

This is the second of two papers in which we study a group which is the quotient of a free product of groups by the normal closure of a single word that is contained in a subgroup which has the form of a free product of two cyclic groups. We use known properties of generalized triangle groups, together with detailed analysis of pictures and of words in free monoids, to prove a number of results such as a Freiheitssatz and the existence of Mayer-Vietoris sequences for such groups under suitable hypotheses. The results generalize those in an earlier article of the second author and Shwartz.     

Vanishing and cocentralizing generalized derivations on Lie ideals

Let R be a prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid and L a not central Lie ideal of R. Suppose that F, G and H are generalized derivations of R, with F≠0, such that F(G(x)x?xH(x)) = 0, for any x∈L. In this paper we describe all possible forms of F, G and H.