An Extension Principle for Fuzzy Logics

Let S be a set, P(S) the class of all subsets of S and F(S) the class of all fuzzy subsets of S. In this paper an “extension principle” for closure operators and, in particular, for deduction systems is proposed and examined. Namely we propose a way to extend any closure operator J defined in P(S) into a fuzzy closure operator J* defined in F(S). This enables us to give the notion of canonical extension of a deduction system and to give interesting examples of fuzzy logics. In particular, the canonical extension of the classical propositional calculus is defined and it is showed its connection with possibility and necessity measures. Also, the canonical extension of first order logic enables us to extend some basic notions of programming logic, namely to define the fuzzy Herbrand models of a fuzzy program. Finally, we show that the extension principle enables us to obtain fuzzy logics related to fuzzy subalgebra theory and graded consequence relation theory. Mathematics Subject Classification : 03B52.     

Axiomatic Aspects of N = 2 Vertex Superalgebras with Odd Formal Variables

We formulate the notion of “N = 2 vertex superalgebra with two odd formal variables” using a Jacobi identity with odd formal variables in which an N = 2 superconformal shift is incorporated into the usual Jacobi identity for a vertex superalgebra. It is shown that as a consequence of these axioms, the N = 2 vertex superalgebra is naturally a representation of the Lie superalgebra isomorphic to the three-dimensional algebra of superderivations with basis consisting of the usual conformal operator and the two N = 2 superconformal operators. In addition, this superconformal shift in the Jacobi identity dictates the form of the odd formal variable components of the vertex operators, and allows one to easily derive the useful formulas in the theory. The notion of N = 2 Neveu–Schwarz vertex operator superalgebra with two odd formal variables is introduced, and consequences of this notion are derived. In particular, we develop the duality properties which are necessary for a rigorous treatment of the correspondence with the underlying supergeometry. Various other formulations of the notion of N = 2 (Neveu–Schwarz) vertex (operator) superalgebra appearing in the mathematics and physics literature are discussed, and several mistakes in the literature are noted and corrected.     

On some notions of convergence for n‐tuples of operators

The aim of this paper is to show that we can extend the notion of convergence in the norm‐resolvent sense to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of S‐resolvent operator. With this notion, we can define bounded functions of unbounded operators using the S‐functional calculus for n‐tuples of noncommuting operators. The same notion can be extended to the case of the F‐resolvent operator, which is the basis of the F‐functional calculus, a monogenic functional calculus for n‐tuples of commuting operators. We also prove some properties of the F‐functional calculus, which are of independent interest. Copyright © 2013 John Wiley & Sons, Ltd.     

On <Emphasis Type="Italic">p</Emphasis>-convergent Operators on Banach Lattices

The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sánchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.     

Two maps on one surface

Two embeddings of a graph in a surface S are said to be “equivalent” if they are identical under an homeomorphism of S that is orientation‐preserving for orientable S. Two graphs cellularly embedded simultaneously in S are said to be “jointly embedded” if the only points of intersection involve an edge of one graph transversally crossing an edge of the other. The problem is to find equivalent embeddings of the two graphs that minimize the number of these edge‐crossings; this minimum we call the “joint crossing number” of the two graphs. In this paper, we calculate the exact value for the joint crossing number for two graphs simultaneously embedded in the projective plane. Furthermore, we give upper and lower bounds when the surface is the torus, which in many cases give an exact answer. In particular, we give a construction for re‐embedding (equivalently) the graphs in the torus so that the number of crossings is best possible up to a constant factor. Finally, we show that if one of the embeddings is replaced by its “mirror image,” then the joint crossing number can decrease, but not by more than 6.066%. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 198–216, 2001     

Extending Independent Sets to Bases and the Axiom of Choice

We show that the both assertions “in every vector space B over a finite element field every subspace V ? B has a complementary subspace S” and “for every family ?? of disjoint odd sized sets there exists a subfamily ?={F_j:j ?ω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ? every generating set includes a basis”.     

Some inequalities involving quadratic forms of operator means

Let T and S be two self-adjoint positive invertible operators of a complex Hilbert space H. In this paper, we investigate some inequalities involving the quadratic forms of the weighted arithmetic and harmonic means of T and S. Application for operator entropies is also discussed.     

A Generalization of Hankel Operators

We introduce a class of operators, called λ-Hankel operators, as those that satisfy the operator equation S*X−XS=λX, where S is the unilateral forward shift and λ is a complex number. We investigate some of the properties of λ-Hankel operators and show that much of their behaviour is similar to that of the classical Hankel operators (0-Hankel operators). In particular, we show that positivity of λ-Hankel operators is equivalent to a generalized Hamburger moment problem. We show that certain linear spaces of noninvertible operators have the property that every compact subset of the complex plane containing zero is the spectrum of an operator in the space. This theorem generalizes a known result for Hankel operators and applies to λ-Hankel operators for certain λ. We also study some other operator equations involving S.     

REAL AND COMPLEX OPERATOR IDEALS

Abstract

The powerful concept of an operator ideal on the class of all Banach spaces makes sense in the real and in the complex case. In both settings we may, for example, consider compact, nuclear, or 2-summing operators, where the definitions are adapted to each other in a natural way. This paper deals with the question whether or not that fact is based on a general philosophy. Does there exists a one-to-one correspondence between “real properties” and “complex properties” defining an operator ideal? In other words, does there exist for every real operator ideal a uniquely determined corresponding complex ideal and vice versa?

Unfortunately, we are not abel to give a final answer. Nevertheless, some preliminary results are obtained. In particular, we construct for every real operator ideal a corresponding complex operator ideal and for every complex operator ideal a corresponding real one. However, we conjecture that there exists a complex operator ideal which can not be obtained from a real one by this construction.

The following approach is based on the observation that every complex Banach space can be viewed as a real Banach space with an isometry acting on it like the scalar multiplication by the imaginary unit i.     

Performance analysis of a call center with interactive voice response units

A Call center may be defined as a service unit where a group of agents handle a large volume of incoming telephone calls for the purpose of sales, service, or other specialized transactions. Typically a call center consists of telephone trunk lines, a switching machine known as the automatic call distributor (ACD) together with a voice response unit (VRU), and telephone sales agents. Customers usually dial a special number provided by the call center; if a trunk line is free, the customer seizes it, otherwise the call is lost. Once the trunk line is seized, the caller is instructed to choose among several options provided by the call center via VRU. After completing the instructions at the VRU, the call is routed to an available agent. If all agents are busy, the call is queued at the ACD until one is free. One of the challenging issues in the design of a call center is the determination of the number of trunk lines and agents required for a given call load and a given service level. Call center industries use the Erlang-C and the Erlang-B formulae in isolation to determine the number of agents and the number of trunk lines needed respectively. In this paper we propose and analyze a flow controlled network model to capture the role of the VRU as well as the agents. Initially, we assume Poisson arrivals, exponential processing time at the VRU and exponential talk time. This model provides a way to determine the number of trunk lines and agents required simultaneously. An alternative simplified model (that ignores the role of the VRU) will be to use anM|M|S|N queueing model (whereS is the number of agents andN is the number of trunk lines) to determine the optimalS andN subject to service level constraints. We will compare the effectiveness of this simplified model and other approximate methods with our model. We will also point out the drawbacks of using Erlang-C and Erlang-B formulae in isolation. Contributed paper for the First Madrid Conference on Queueing Theory, held in Complutense University, Madrid, Spain, July 2–5, 2002     

Integral trees of arbitrarily large diameters

In this paper, we construct trees having only integer eigenvalues with arbitrarily large diameters. In fact, we prove that for every finite set S of positive integers there exists a tree whose positive eigenvalues are exactly the elements of S. If the set S is different from the set {1} then the constructed tree will have diameter 2|S|.     

On the Action of a Linear Operator over Sequences in a Banach Space

In this paper we study the action of a bounded linear operator over different kinds of sequences of a Banach space. Our work is mainly devoted to minimal and M - basic sequences. Plans and García Castellón have characterized the boundedness of a linear operator T by requiring the minimality of any sequence whose image is a minimal sequence (e.g. [P, 1969], [GC, 1990]). We extend these results to other types of sequences like M-basic, basic, strong M-basic, etc., We are also interested on conditions that ensure the minimality of the image of a given minimal sequence. Thus in Corollary 3.7 we characterize semi - Fredholm operators as those which transform every p-minimal sequence into q-minimal. In the last section we deal with M - basis whose image is M - basis or norming M - basis or basis or in general the “best” possible sequence.     

Limiting Absorption Principle for Stratified Operators with Thresholds: A General Method

Out problem is about propagation of waves in stratified strips. The operators are quite general, a typical example being a coupled elasto-acoustic operator H defined in ?² × I where I is a bounded interval of ? with coefficients depending only on z ∈ I. The “conjugate operator method” will be applied to an operator obtained by a spectral decomposition of the partial Fourier transform ? of H. Around each value of the spectrum (except the eigenvalues) including the thresholds, a conjugate operator is constructed which permits to get the ”good properties“ of regularity for H. A limiting absorption principle is then obtained for a large class of operators at every point of the spectrum (except eigenvalues).     

The Essential Spectrum and Banach Reducibility of Operator Weighted Shifts

For an operator weighted shift S, the essential spectrum σ _e (S) are described. Moreover, Banach reducibility of S is investigated and a condition for S* to be a Cowen-Douglas operator is characterized. Supported by MCME, Doctoral Foundation of the Ministry of Education and Science Foundation of Liaoning University     

On the functions of one selfadjoint integral operator

We construct a selfadjoint Akhiezer integral operator S on L ₂(?1, 1) with the spectrum {?1; 0; 1} and the property that every function φ(S) that is not a multiple of S fails to be an Akhiezer integral operator.     

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

Extensions,dilations, and spectral problems of singular Hamiltonian systems

In this paper, we construct a space of boundary values for minimal symmetric 1D Hamiltonian operator with defect index (1,1) (in limit‐point case at a(b) and limit‐circle case at b(a)) acting in the Hilbert space In terms of boundary conditions at a and b, all maximal dissipative, accumulative, and self‐adjoint extensions of the symmetric operator are given. Two classes of dissipative operators are studied. They are called “dissipative at a” and “dissipative at b.” For 2 cases, a self‐adjoint dilation of dissipative operator and its incoming and outgoing spectral representations are constructed. These constructions allow us to establish the scattering matrix of dilation and a functional model of the dissipative operator. Further, we define the characteristic function of the dissipative operators in terms of the Weyl‐Titchmarsh function of the corresponding self‐adjoint operator. Finally, we prove theorems on completeness of the system of root vectors of the dissipative operators.     

On extensions of some Flugede–Putnam type theorems involving (p,k)‐quasihyponormal,spectral, and dominant operators

A Hilbert space operator S is called (p, k)‐quasihyponormal if S *^k ((S *S)^p – (SS *)^p )S^k ≥ 0 for an integer k ≥ 1 and 0 < p ≤ 1. In the present note, we consider (p, k)‐quasihyponormal operator S ∈ B (H) such that SX = XT for some X ∈ B (K,H) and prove the Fuglede–Putnam type theorems when the adjoint of T ∈ B (K) is either (p, k)‐quasihyponormal or dominant or a spectral operator (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algebraic properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space

In this paper we investigate some algebra properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space in the Sobolev space. We completely characterize commuting dual Toeplitz operators with harmonic symbols, and show that a dual Toeplitz operator commutes with a nonconstant analytic dual Toeplitz operator if and only if its symbol is analytic. We also obtain the sufficient and necessary conditions on the harmonic symbols for SφSφψ= Sφψ.     

Commutative rings whose proper ideals are direct sums of uniform modules

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ? of the form ? = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.