Let $R$ be a commutative ring. The purpose of this article is to introduce a new class of ideals of R called weakly irreducible ideals. This class could be a generalization of the families quasi-primary ideals and strongly irreducible ideals. The relationships between the notions primary, quasi-primary, weakly irreducible, strongly irreducible and irreducible ideals, in different rings, has bee...

A boolean function f(x1, ..., xn) is weakly symmetric if it is invariant under a transitive permutation group on its variables. A boolean function f(x1, ..., xn) is elusive if we have to check all x1,..., xn to determine the output of f(x1, ..., xn) in the worst-case. It is conjectured that every nontrivial monotone weakly symmetric boolean function is elusive, which has been open for a long ti...

• Prop. 8. If g imperfectly dominates f, then hβ(f,g) imperfectly dominates f. ... and yield a hierarchy of manipulability and efficiency. • Prop. 9./10. Given f SP, g manipulable and weakly less varying than f, g imperfectly dominates f, 0≤β<β‘≤1, then 1. hβ‘(f,g) is intensely and strongly more manipulable than hβ(f,g) 2. hβ‘(f,g) imperfectly dominates hβ(f,g). ... which preserve PSP, are comp...

Let M be a space of weakly*-measurable functions F : ? B(H) on measure (?,?,?), for which the function F*F is Gel'fand integrable and integral d? nuclear operator Hilbert H. We show that H*-module contains an orthonormal basis.

For a fixed graph F , a graph G is F -saturated if there is no copy of F in G, but for any edge e 6∈ G, there is a copy of F in G+ e. The minimum number of edges in an F -saturated graph of order n will be denoted by sat(n, F ). A graph G is weakly F -saturated if there is an ordering of the missing edges of G so that if they are added one at a time, each edge added creates a new copy of F . Th...

Let A and B be uniform algebras on first-countable, compact Hausdorff spaces X and Y , respectively. For f ∈ A, the peripheral spectrum of f , denoted by σπ(f) = {λ ∈ σ(f) : |λ| = ‖f‖}, is the set of spectral values of maximum modulus. A map T : A → B is weakly peripherally multiplicative if σπ(T (f)T (g)) ∩ σπ(fg) 6= ∅ for all f, g ∈ A. We show that if T is a surjective, weakly peripherally mu...

For a bounded linear operator on Hilbert space we define a sequence of the so-called weakly extremal vectors‎. ‎We study the properties of weakly extremal vectors and show that the orthogonality equation is valid for weakly extremal vectors‎. ‎Also we show that any quasinilpotent operator $T$ has an hypernoncyclic vector‎, ‎and so $T$ has a nontrivial hyperinvariant subspace‎.

For a Banach algebra $A$, $A''$ is $(-1)$-Weakly amenable if $A'$ is a Banach $A''$-bimodule and $H^1(A'',A')={0}$. In this paper, among other things,  we study the relationships between the $(-1)$-Weakly amenability of $A''$ and the weak amenability of $A''$ or $A$. Moreover, we show that the second dual of every $C^ast$-algebra is $(-1)$-Weakly amenable.

Let P be a polynomial over the ring of mod m integers. P weakly Abstract-1 represents Boolean function f : {0, 1}n → {0, 1} if there is a subset S ⊆ {0, 1, . . . , m − 1} such that f(x) = 0 if and only if P (x) ∈ S. The smallest degree of polynomials P weakly representing f is called the weak mod m degree of f . We give here an Ω(logn) lower bound for the weak degree of the generalized inner pr...

In this paper we introduce the notion of $varphi$-commutativity for a Banach algebra $A$, where $varphi$ is a continuous homomorphism on $A$ and study the concept of $varphi$-weak amenability for $varphi$-commutative Banach algebras. We give an example to show that the class of $varphi$-weakly amenable Banach algebras is larger than that of weakly amenable commutative Banach algebras. We charac...