‎Let $V$ be an $n$-dimensional complex inner product space‎. ‎Suppose‎ ‎$H$ is a subgroup of the symmetric group of degree $m$‎, ‎and‎ ‎$chi‎ :‎Hrightarrow mathbb{C} $ is an irreducible character (not‎ ‎necessarily linear)‎. ‎Denote by $V_{chi}(H)$ the symmetry class‎ ‎of tensors associated with $H$ and $chi$‎. ‎Let $K(T)in‎ (V_{chi}(H))$ be the operator induced by $Tin‎ ‎text{End}(V)$‎. ‎Th...

‎let $v$ be an $n$-dimensional complex inner product space‎. ‎suppose‎ ‎$h$ is a subgroup of the symmetric group of degree $m$‎, ‎and‎ ‎$chi‎ :‎hrightarrow mathbb{c} $ is an irreducible character (not‎ ‎necessarily linear)‎. ‎denote by $v_{chi}(h)$ the symmetry class‎ ‎of tensors associated with $h$ and $chi$‎. ‎let $k(t)in‎ (v_{chi}(h))$ be the operator induced by $tin‎ ‎text{end}(v)$‎. ‎the...

Let G be a finite group and A be a normal subgroup of G. We denote by ncc(A) the number of G-conjugacy classes of A and A is called n-decomposable, if ncc(A) = n. Set KG = {ncc(A)|A⊳ G}. Let X be a non-empty subset of positive integers. A group G is called X-decomposable, if KG = X . Ashrafi and his co-authors [1,2,3,4,5] have characterized the X-decomposable nonperfect finite groups for X = {1...

Let K denote a field. A polynomial f(x) ∈ K[x] is said to be decomposable over K if f(x)= g(h(x)) for some polynomials g(x) and h(x)∈K[x] with 1< deg(h) < deg(f ). Otherwise f(x) is called indecomposable. If f(x)= g(xm) with m> 1, then f(x) is said to be trivially decomposable. In this paper, we show that xd+ax+b is indecomposable and that if e denotes the largest proper divisor of d, then xd+a...

Given a tournament T = (V, A), with each subset X of V is associated the subtournament T [X] = (X, A∩(X×X)) of T induced by X. A subset I of V is an interval of T provided that for any a, b ∈ I and x ∈ V \I, (a, x) ∈ A if and only if (b, x) ∈ A. For example, ∅, {x}, where x ∈ V , and V are intervals of T called trivial. A tournament is indecomposable if all its intervals are trivial; otherwise,...

‎Let $mathcal{A}$ be a commutative Banach algebra and $mathscr{X}$ be a left Banach $mathcal{A}$-module‎. ‎We study the set‎ ‎${rm Dec}_{mathcal{A}}(mathscr{X})$ of all elements in $mathcal{A}$ which induce a decomposable multiplication operator on $mathscr{X}$‎. ‎In the case $mathscr{X}=mathcal{A}$‎, ‎${rm Dec}_{mathcal{A}}(mathcal{A})$ is the well-known Apostol algebra of $mathcal{A}$‎. ‎We s...

Let $mathcal {N}_G$ denote the set of all proper‎ ‎normal subgroups of a group $G$ and $A$ be an element of $mathcal‎ ‎{N}_G$‎. ‎We use the notation $ncc(A)$ to denote the number of‎ ‎distinct $G$-conjugacy classes contained in $A$ and also $mathcal‎ ‎{K}_G$ for the set ${ncc(A) | Ain mathcal {N}_G}$‎. ‎Let $X$ be‎ ‎a non-empty set of positive integers‎. ‎A group $G$ is said to be‎ ‎$X$-d...

let $mathcal {n}_g$ denote the set of all proper‎ ‎normal subgroups of a group $g$ and $a$ be an element of $mathcal‎ ‎{n}_g$‎. ‎we use the notation $ncc(a)$ to denote the number of‎ ‎distinct $g$-conjugacy classes contained in $a$ and also $mathcal‎ ‎{k}_g$ for the set ${ncc(a) | ain mathcal {n}_g}$‎. ‎let $x$ be‎ ‎a non-empty set of positive integers‎. ‎a group $g$ is said to be‎ ‎$x$-d...

We use Shelah’s theory of possible cofinalities in order to solve some problems about ultrafilters. Theorem 1. Suppose that λ is a singular cardinal, λ′ < λ, and the ultrafilter D is κ-decomposable for all regular cardinals κ with λ′ < κ < λ. Then D is either λ-decomposable, or λ-decomposable. Corollary 2. If λ is a singular cardinal, then an ultrafilter is (λ, λ)-regular if and only if it is e...

Consider a (possibly infinite) exchangeable sequence X= {Xn : 1≤ n < N}, where N ∈ N ∪ {∞}, with values in a Borel space (A,A), and note Xn = (X1, . . . ,Xn). We say that X is Hoeffding decomposable if, for each n, every square integrable, centered and symmetric statistic based on Xn can be written as an orthogonal sum of n U statistics with degenerated and symmetric kernels of increasing order...