Fuzzy generalized Bi-ideals of Γ-semigroups

In this paper, we introduce the concept of fuzzy generalized bi-ideal of a ??-semigroup, which is an extension of the concept of a fuzzy bi-ideal of a ??-semigroup and characterize regular ??-semigroups in terms of fuzzy generalized biideals.     

Characterizations of connected orthogonality graphs of projections of Rickart ∗-rings

In this paper, we study the orthogonality graphs (see Definition 1.2) of ortholattices. We provide a graph theoretic condition for an ortholattice to be orthomodular. We prove that, the orthogonality graphs of two orthomodular lattices are isomorphic if and only if the lattices are isomorphic. As an application, it is proved that the zero-divisor graph of a Rickart ?-ring is obtained by successively duplicating the vertices of the orthogonality graph of the lattice of projections in the ring. We characterize the finite Rickart ?-rings for which the orthogonality graph of projections is connected.     

Growth of generalized Temperley–Lieb algebras connected with simple graphs

We prove that the generalized Temperley–Lieb algebras associated with simple graphs Γ have linear growth if and only if the graph Γ coincides with one of the extended Dynkin graphs [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , or [(E)\tilde]₇ {\tilde E_7} . An algebra TL_{G, t} T{L_{\Gamma, \tau }} has exponential growth if and only if the graph Γ coincides with none of the graphs A_n {A_n} , D_n {D_n} , E_n {E_n} , [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , and [(E)\tilde]₇ {\tilde E_7} .     

A new class of generalized nonlinear functionals of white noise with application to random integral equations ∗

In this paper we introduce some new classes of generalized nonlinear functionals of White noise which have integral representation with kernels either belonging to any L_p (1 ≤ p ≤ ∞)space or any Sobolev space W ^{P, s} for any real s and(1 ≤ p ≤ ∞)Our results extend the class of generelized nonlinear functionals of white noise originally developed by Hida [6, 7, 8] and greatly broaden their scope for application. We conclude the paper with an example involving random integral equations arising from Heat equation excited by White noise on the boundary.     

A generalization of semi prime indeals in γ-rings

The purpose of this paper is to generalize the concept of semi prime ideals in Γ-rings. We use a general definition of a regularity F for Γ-rings to define and F- prime ideal. Relationships between F-semi prime ideals of a Γ-ring M and F-semi prime ideals of the operator rings R and L are discussed. D-regularity, f-regularity and λ-regularity for Γ-rings are introduced and studied against the background of the concept F-semi prime ideal. Finally, D-, λ- and f-regular Γ-rings are characterized.     

A note on Brown—McCoy radicals ofΓ-rings

In [1] we defined the Brown—McCoy radical,B(M), of a-ringM. In this note we show thatB is a special radical. The simplicial radical, defined by Kyuno [4] for-rings with left and right unities, is extended to arbitrary-rings. The simplicial radicalS is shown to be a generalization of the Brown—McCoy radical of a ring. In general,B(M) S(M). This work is supported financially by the Technikon Natal.     

∗-regularities of γ-rings

A general regularity for weak Nobusawa γ-rings is defined which takes into account the nature of the factor r-rings in this category. The class of

?-regularities is defined and studied. We show that ?-regularities represent only six different algebraic properties for elements of γ-rings. Right (left weak-regularity, Von Neumann-regularity, α-regularity and f-regularity are identified as special cases of ?-regularities. Various characterzations o the different types of regular γ-rings are given. Relationships amongst the operator rings are also, established for the above mentioned regularities     

The characterization of Γ-modules in terms of fuzzy soft Γ-submodules

In this paper, we focus on combining the theories of fuzzy soft sets with Γ-modules, and establishing a new framework for fuzzy soft Γ-submodules. The main contributions of the paper are 3-fold. First, we present the concepts of (R, S)-bi-Γ-submodules, quasi-Γ-submodules and regular Γ-modules. Meanwhile, some illustrative examples are given to show the rationality of the definitions introduced in this paper. Second, several new kinds of generalized fuzzy soft Γ-submodules are proposed, and related properties and mutual relationships are also investigated. Third, we discover some intrinsic connections between the generalized fuzzy soft Γ-submodules presented in this paper and crisp Γ-submodules, and describe the relationships between regular Γ-modules and the generalized fuzzy soft Γ-submodules presented in this paper.     

A novel approach towards fuzzy Γ-ideals in ordered Γ-semigroups

In many applied disciplines like computer science, coding theory and formal languages, the use of fuzzified algebraic structures especially ordered semigroups play a remarkable role. In this paper, we introduce a new concept of fuzzy Γ-ideal of an ordered Γ-semigroup G called an (∈, ∈ ?q _k )-fuzzy Γ-ideal of G. Fuzzy Γ-ideal of type (∈, ∈ ∨q _k ) are the generalization of ordinary fuzzy Γ-ideals of an ordered Γ-semigroup G. A new characterization of ordered Γ-semigroups in terms of an (∈, ∈ ∨q _k )-fuzzy Γ-ideal is given. We show that a fuzzy subset λ of an ordered Γ-semigroup G is an (∈, ∈ ∨q _k )-fuzzy Γ-ideal of G if and only if U (λ; t) is a Γ-ideal of G for all \(t \in \left( {0,\frac{{1 - k}} {2}} \right]\) . We also investigate some important characterization theorems in terms of this notion. Finally, regular ordered Γ-semigroups are characterized by the properties of their (∈, ∈ ∨q _k )-fuzzy Γ-ideals.     

Homological determination of Γ-modules

It is proved that if M is a profinitely generated -module which is free as a module over the ring of p-adic integers, then M is determined up to free direct factors by its homology. This result generalizes the theorem on homological determinacy of p-adic representations of a cyclic group [Ref. Zh. Mat., 3A, 318 (1971)].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeieniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 104–126, 1976.     

A generalized Dantzig—Wolfe decomposition principle for a class of nonconvex programming problems

Since Dantzig—Wolfe's pioneering contribution, the decomposition approach using a pricing mechanism has been developed for a wide class of mathematical programs. For convex programs a linear space of Lagrangean multipliers is enough to define price functions. For general mathematical programs the price functions could be defined by using a subclass of nondecreasing functions. However the space of nondecreasing functions is no longer finite dimensional. In this paper we consider a specific nonconvex optimization problem min {f(x):h _j (x)g(x),j=1, ,m, xX}, wheref(·),h _j (·) andg(·) are finite convex functions andX is a closed convex set. We generalize optimal price functions for this problem in such a way that the parameters of generalized price functions are defined in a finite dimensional space. Combining convex duality and a nonconvex duality we can develop a decomposition method to find a globally optimal solution.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.     

Limiting angles of Γ-martingales

Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m≥ 3 and that, outside a fixed compact set, the sectional curvatures are bounded above by −c ₁/{r ² ln r} and below by −c ₂ r ², where c ₁ and c ₂ are two positive constants and r is the geodesic distance from a fixed point. We show that, when κ≥ 1 satisfies certain conditions, the angular part of a κ-quasi-conformal Γ-martingale on M tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire sphere at infinity. This improves both a result of Le for Brownian motion and also results concerning the non-existence of κ-quasi-conformal harmonic maps from certain types of Riemannian manifolds into M. Received: 19 September 1997     

Blowup of solutions to generalized Keller–Segel model

The existence and nonexistence of global in time solutions is studied for a class of equations generalizing the chemotaxis model of Keller and Segel. These equations involve Lévy diffusion operators and general potential type nonlinear terms.     

Product of fuzzyH υ-ideals inH υ-rings

In this paper we define product between fuzzyH _υ-ideals of givenH _υ-rings. We consider the fundamental relation γ* defined on anH _υ-ring and give some properties of the fundamental relations and fundamental rings with respect to the product of fuzzyH _υ-ideals.     

On the Γ-convergence of matrix fields related to the adjugate Jacobian

Adjugate Jacobians of mappings $\text{f}{}_{\mathrm{j}}\text{:Ω?}\text{R}{}^2\text{→}\text{R}{}^2$ can be represented in terms of Jacobian matrices: $\text{adj}\text{Df}{}_{\mathrm{j}}\text{=}\text{A}{}_{\mathrm{j}}\text{(x)Df}{}^{\mathrm{t}}{}_{\mathrm{j}}$, for $\text{j=1,2,…}$, by mean of symmetric matrix fields $\text{A}{}_{\mathrm{j}}\text{(x)}$ with $\text{det}\text{A}{}_{\mathrm{j}}\text{(x)=1}$ a.e. Under suitable conditions, we prove that $\text{Df}{}_{\mathrm{j}}\text{?Df}$ weakly in $\text{L}{}^1{}_{\mathrm{<mtext>loc</mtext>}}\text{(Ω;}\text{R}{}^2\text{)}$ if and only if $\text{A}{}_{\mathrm{j}}\text{(x)}$Γ-converges to a matrix $\text{A}\text{(x)}$ with $\text{det}\text{A}\text{(x)=1}$ satisfying $\text{adj}\text{Df=}\text{A}\text{(x)Df}{}^{\mathrm{t}}$. To cite this article: C. Sbordone, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A theorem connected with results of Steinhaus and Smítal

We will prove that if A and B are subsets of the real line, each having positive outer Lebesgue measure, then A + B, the set of all numbers a + b with a ϵ A and b ϵ B, is “full,” in the sense of outer Lebesgue measure, in some interval K. This result is related to theorems of Steinhaus and Smítal.