A self-dual harmonic 2{form on a 4{dimensional Riemannian manifold is symplectic where it does not vanish. Furthermore, away from the form’s zero set, the metric and the 2{form give a compatible almost complex structure and thus pseudo-holomorphic subvarieties. Such a subvariety is said to have nite energy when the integral over the variety of the given self-dual 2{form is nite. This article pr...

let  be a banach algebra and  a derivation. in this paper, it is proved, under certain conditions, that , where  is the jacobson radical of . moreover, we prove that if  is unital and  is a continuous derivation, then , where  denotes the set of all primitive ideals such that  is commutative,  denotes the set of all maximal (modular) ideals such that  is commutative, and  is the set of all non-...

If there exists a nonempty, proper subset S of the set of all (n+ 1)(n+ 2)/2 inertias such that S ⊆ i(A) is sufficient for any n×n zero-nonzero pattern A to be inertially arbitrary, then S is called a critical set of inertias for zero-nonzero patterns of order n. If no proper subset of S is a critical set, then S is called a minimal critical set of inertias. In [Kim, Olesky and Driessche, Criti...

We introduce X is without zero as a synonym of X has non empty elements. We introduce X has zero as an antonym of X has non empty elements. Let us mention that R has zero and N has zero. Let us observe that there exists a set which is non empty and without zero and there exists a set which is non empty and has zero. Let us observe that there exists a subset of R which is non empty and without z...

The classical matrix theory is deficient to express the vagueness of the real life. The fuzzy set theory has been successfully applied to bridge this gap. Much work has already been done on a two-person zero sum matrix game with fuzzy goals. In continuation, this paper is dedicated to define and study a multi-criteria two-person zero sum game with intuitionistic fuzzy goals. It is shown that so...

Let L be a lattice with the least element 0. An element x ∈ L is a zero divisor if x∧ y = 0 for some y ∈ L∗ = L \ {0}. The set of all zero divisors is denoted by Z(L). We associate a simple graph Γ(L) to L with vertex set Z(L)∗ = Z(L) \ {0}, the set of non-zero zero divisors of L and distinct x, y ∈ Z(L)∗ are adjacent if and only if x ∧ y = 0. In this paper, we obtain certain properties and dia...

We examine when a space X has a zero set universal parametrised by a metrisable space of minimal weight and show that this depends on the σ-weight of X when X is perfectly normal. We also show that if Y parametrises a zero set universal for X then hL(X) ≤ hd(Y ) for all n ∈ N. We construct zero set universals that have nice properties (such as separability or ccc) in the case where the space ha...

The strong measure zero sets of reals have been widely studied in the context of set theory of the real line. The notion of strong measure zero is straightforwardly effectivized. A set of reals is said to be of effective strong measure zero if for any computable sequence {εn}n∈N of positive rationals, a sequence of intervals In of diameter εn covers the set. We observe that a set is of effectiv...

We study the almost Mathieu operator: 1) + cos(2n +)u(n), on l 2 (Z), and show that for all ; , and (Lebesgue) a.e. , the Lebesgue measure of its spectrum is precisely j4 ? 2jjj. In particular, for jj = 2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational 's (and jj = 2) we show that the Hausdorr dimension of the spectrum is smaller than or equal to 1=2.