The continuity of densities given by the weight functions $n^{\alpha }$, $\alpha \in [-1,\infty [$, with respect to the parameter $\alpha $ is investigated.
The aim of this paper is to introduce a central limit theorem and an invariance principle for weighted U-statistics based on stationary random fields. Hsing and Wu (2004) in their paper introduced some asymptotic results for weighted U-statistics based on stationary processes. We show that it is possible also to extend their results for weighted U-statistics based on stationary random fields.
For a finite group $G$ and a non-linear irreducible complex character $\chi $ of $G$ write $\upsilon (\chi )=\{g\in G\mid \chi (g)=0\}$. In this paper, we study the finite non-solvable groups $G$ such that $\upsilon (\chi )$ consists of at most two conjugacy classes for all but one of the non-linear irreducible characters $\chi $ of $G$. In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable $\varphi $-groups. As a corollary, we answer Research Problem $2$ in [Y. Berkovich and L. Kazarin: Finite groups in which the zeros of every non-linear irreducible character are conjugate modulo its kernel. Houston J. Math.\ 24 (1998), 619--630.] posed by Y. Berkovich and L. Kazarin.
In the paper, we prove two theorems on |A, δ|k summability, 1 ≤ k ≤ 2, of orthogonal series. Several known and new results are also deduced as corollaries of the main results.
Let D be an oriented graph of order n and size m. A γ-labeling of D is a one-to-one function f : V (D) → {0, 1, 2, . . . , m} that induces a labeling f ' : E(D) → {±1, ±2, . . . , ±m} of the arcs of D defined by f ' (e) = f(v) − f(u) for each arc e = (u, v) of D. The value of a γ-labeling f is val(f) = ∑ e∈E(G) f ' (e). A γ-labeling of D is balanced if the value of f is 0. An oriented graph D is balanced if D has a balanced labeling. A graph G is orientably balanced if G has a balanced orientation. It is shown that a connected graph G of order n ≥ 2 is orientably balanced unless G is a tree, n ≡ 2 (mod 4), and every vertex of G has odd degree.
We give a positive answer to two open problems stated by Boczek and Kaluszka in their paper \cite{BK}. The first one deals with an algebraic characterization of comonotonicity. We show that the class of binary operations solving this problem contains any strictly monotone right-continuous operation. More precisely, the comonotonicity of functions is equivalent not only to +-associatedness of functions (as proved by Boczek and Kaluszka), but also to their ⋆-associatedness with ⋆ being an arbitrary strictly monotone and right-continuous binary operation. The second open problem deals with an existence of a pair of binary operations for which the generalized upper and lower Sugeno integrals coincide. Using a fairly elementary observation we show that there are many such operations, for instance binary operations generated by infima and suprema preserving functions.