Rough set theory is an excellent mathematical tool for the analysis of a vague description actions in decision problems. Now, this paper by considering notion equality algebra, lower and upper approximations are introduced some properties them given. Moreover, it proved that define interior operator closure operator, respectively. Also, using D-lower D-upper approximation, conditions nonempty s...

We point out that a simple test of the mechanism of CP violation can be performed by a measurement of the CP asymmetries in exclusive and inclusive radiative rare decays. We show that the rate asymmetries ∆Γ = Γ(B → fγ)− Γ(B+ → f̄γ) for certain final states f can be predicted in a theoretically clean way. Some implications for b → sl+l− decays are discussed.

The main theorem says that any module-finite (but not necessarily commutative) algebra Λ over a commutative Noetherian universally catenary ring R is catenary. Hence the ring Λ is catenary if R is Cohen-Macaulay. When R is local and Λ is a Cohen-Macaulay R-module, we have that Λ is a catenary ring, dim Λ = dimΛ/Q + htΛQ for any Q ∈ Spec Λ, and the equality n = htΛQ− htΛP holds true for any pair...

Let O1 be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order O in an indefinite quaternion division algebra over Q. Using the (classical) Selberg trace formula, we show that the eigenvalues of the automorphic Laplacian for O1 and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian defined on the Maaß newforms for the Hecke...

We address the problem of representing mathematical structures in a proof assistant which: 1) is based on a type theory with dependent types, telescopes and a computational version of Leibniz equality; 2) implements coercive subtyping, accepting multiple coherent paths between type families; 3) implements a restricted form of higher order unification and type reconstruction. We show how to expl...

We study a question with connections to linear algebra, real algebraic geometry, combinatorics, and complex analysis. Let p(x, y) be a polynomial of degree d with N positive coefficients and no negative coefficients, such that p = 1 when x+ y = 1. A sharp estimate d ≤ 2N − 3 is known. In this paper we study the p for which equality holds. We prove some new results about the form of these “sharp...

Using the notion of a Banach operator, we have obtained a decompositional property of a Hilbert space, and the equality of two invertible bounded linear multiplicative operators on a normed algebra with identity. 1. Introduction. This paper is a continuation of our earlier work [7] on Banach operators. We recall that if X is a normed space and α : X → X is a mapping, then following [4], α is sa...

We present an algorithm for large-scale equality constrained optimization. The method is based on a characterization of inexact sequential quadratic programming (SQP) steps that can ensure global convergence. Inexact SQP methods are needed for large-scale applications for which the iteration matrix cannot be explicitly formed or factored and the arising linear systems must be solved using itera...