A comparative study of the numerical scales and the prioritization methods in AHP

In the analytic hierarchy process (AHP), a decision maker first gives linguistic pairwise comparisons, then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them, and finally derives a priority vector from the numerical pairwise comparisons. In particular, the validity of this decision-making tool relies on the choice of numerical scale and the design of prioritization method. By introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices (LPCMs), and by defining the deviation measures of LPCMs, we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods. Using these performance measure algorithms, we compare the most common numerical scales (the Saaty scale, the geometrical scale, the Ma–Zheng scale and the Salo–Hämäläinen scale) and the prioritization methods (the eigenvalue method and the logarithmic least squares method). In addition, we also discuss the parameter of the geometrical scale, develop a new prioritization method, and construct an optimization model to select the appropriate numerical scales for the AHP decision makers. The findings in this paper can help the AHP decision makers select suitable numerical scales and prioritization methods.     

Strong reciprocity and strong consistency in pairwise comparison matrix with fuzzy elements

The decision making problem considered in this paper is to rank n alternatives from the best to the worst, using the information given by the decision maker in the form of an \(n\times n\) pairwise comparison matrix. Here, we deal with pairwise comparison matrices with fuzzy elements. Fuzzy elements of the pairwise comparison matrix are applied whenever the decision maker is not sure about the value of his/her evaluation of the relative importance of elements in question. We investigate pairwise comparison matrices with elements from abelian linearly ordered group (alo-group) over a real interval. The concept of reciprocity and consistency of pairwise comparison matrices with fuzzy elements have been already studied in the literature. Here, we define stronger concepts, namely the strong reciprocity and strong consistency of pairwise comparison matrices with fuzzy intervals as the matrix elements (PCF matrices), derive the necessary and sufficient conditions for strong reciprocity and strong consistency and investigate their properties as well as some consequences to the problem of ranking the alternatives.     

Membership maximization prioritization methods for fuzzy analytic hierarchy process

Fuzzy analytic hierarchy process (FAHP) has increasingly been applied in many areas. Extent analysis method is the popular tool for prioritization in FAHP, although significant technical errors are identified in this study. With addressing the errors, this research proposes membership maximization prioritization methods (MMPMs) using different membership functions as the novel solutions. As a lack of research about effectiveness measurement on the crisp/fuzzy prioritization methods, this study proposes membership fitness index to evaluate the effectiveness of the prioritization methods. Comparisons with the other popular fuzzy/crisp prioritization methods including modified fuzzy preference programming, Direct least squares, and Eigen value are conducted and analyses indicate that MMPMs lead to much more reliable result in view of membership fitness index. A numerical example demonstrates the usability of MMPMs for FAHP, and thus MMPMs can effectively be applied to various decision analysis applications.     

矩阵空间上保弱伴随矩阵的线性映射

为了刻画矩阵空间上保弱伴随矩阵的线性映射f,引入了保弱伴随矩阵的概念,以矩阵的弱伴随矩阵为不变量,得到了当n≥3时数域F上从线性矩阵空间Mn×n（F）到Mm×m（F）的保弱伴随矩阵的线性映射f的形式．     

Inequalities involving upper bounds for certain matrix operators

In this paper, we considered the problem of finding the upper bound Hausdorff matrix operator from sequence spaces l_p(v) (ord(v, p)) intol _p (w) (ord(w, p)). Also we considered the upper bound problem for matrix operators fromd(v, 1) intod(w, 1), and matrix operators frome(w, ∞) intoe(v, ∞), and deduce upper bound for Cesaro, Copson and Hilbert matrix operators, which are recently considered in [5] and [6] and similar to that in [10].     

Computational matrix representation modules for linear operators with explicit constructions for a class of lie operators

A linear mapping from a finite-dimensional linear space to another has a matrix representation. Certain multilinear functions are also matrix-representable. Using these representations, symbolic computations can be done numerically and hence more efficiently. This paper presents an organized procedure for constructing matrix representations for a class of linear operators on finite-dimensional spaces. First we present serial number functions for locating basis monomials in the linear space of homogeneous polynomials of fixed degree, ordered under structured lexicographies. Next basic lemmas describing the modular structure of matrix representations for operators constructed canonically from elementary operators are presented. Using these results, explicit matrix representations are then given for the Lie derivative and Lie-Poisson bracket operators defined on spaces of homogeneous polynomials. In particular, they are comprised of blocks obtained as Kronecker sums of modular components, each corresponding to specific Jordan blocks. At an implementation level, recursive programming is applied to construct these modular components explicitly. The results are also applied to computing power series approximations for the center manifold of a dynamical system. In this setting, the linear operator of interest is parameterized by two matrices, a generalization of the Lie-Poission bracket.     

Quaternionic operators with finite matrix trace

In this paper we consider bounded liner operators in quaternionic Hilbert space, having finite and invariant matrix trace. We prove that any such operator is selfadjoint. Besides, we prove that dual space of the real normed space of all such operators is isomorphic to the Banach space of all selfadjoint operators.This research was supported by Science Fund of Serbia, through the Mathematical Faculty of Belgrade.     

A note on compact operators on matrix domains

We establish some identities or estimates for the operator norms and Hausdorff measures of noncompactness of linear operators given by infinite matrices that map the matrix domains of triangles in arbitrary BK spaces with AK, or in the spaces of all convergent or bounded sequences, into the spaces of all null, convergent or bounded sequences, or of all absolutely convergent series. Furthermore, we apply these results to the characterizations of compact operators on the matrix domains of triangles in the classical sequence spaces, and on the sequence spaces studied in [I. Djolovi?, Compact operators on the spaces and , J. Math. Anal. Appl. 318 (2) (2006) 658-666; I. Djolovi?, On the space of bounded Euler difference sequences and some classes of compact operators, Appl. Math. Comput. 182 (2) (2006) 1803-1811].     

Difference operators,Green's matrix,sampling theory and applications in signal processing

This paper introduces sampling representations for discrete signals arising from self adjoint difference operators with mixed boundary conditions. The theory of linear operators on finite-dimensional inner product spaces is employed to study the second-order difference operators. We give necessary and sufficient conditions that make the operators self adjoint. The equivalence between the difference operator and a Hermitian Green's matrix is established. Sampling theorems are derived for discrete transforms associated with the difference operator. The results are exhibited via illustrative examples, involving sampling representations for the discrete Hartley transform. Families of discrete fractional Fourier-type transforms are introduced with an application to image encryption.     

Transformations and inertia of solutions to linear matrix equations

Linear equations and operators in a space of matrices are investigated. The transformations of matrix equations which allow one to find the conditions of solvability and the inertial properties of Hermite solutions are determined. New families of matrices (collectives) are used in the theory of inertia and positive invertibility of linear operators and, in particular, in the problems of localization of matrix spectra and matrix beams.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 60–68, January, 1993.     

Generalized inversion of finite rank Hankel and Toeplitz operators with rational matrix symbols

The goal of the paper is a generalized inversion of finite rank Hankel operators and Hankel or Toeplitz operators with block matrices having finitely many rows. To attain it a left coprime fractional factorization of a strictly proper rational matrix function and the Bezout equation are used. Generalized inverses of these operators and generating functions for the inverses are explicitly constructed in terms of the fractional factorization.     

Some remarks on index of matrix singular integral operators in non-smooth domains

On manifolds with non-smooth boundaries the Mikhlin-Calderon-Zygmund operators with a matrix symbol are considered. The Noetherian property conditions (wave factorization) and reduction to Atiyah-Singer theorem are described. These results were obtained through joint work with I.V. Scherbenko. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relative oscillation theory for Dirac operators

We develop relative oscillation theory for one-dimensional Dirac operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established. As an application we extend a result by K.M. Schmidt on the finiteness/infiniteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators.     

Fredholmness and invertibility of Toeplitz operators with matrix almost periodic symbols

We consider Toeplitz operators with symbols that are almost periodic matrix functions of several variables. It is shown that under certain conditions on the group generated by the Fourier support of the symbol, a Toeplitz operator is Fredholm if and only if it is invertible.

     

Fundamental solutions and Liouville type theorems for nonlinear integral operators

In this article we study basic properties for a class of nonlinear integral operators related to their fundamental solutions. Our goal is to establish Liouville type theorems: non-existence theorems for positive entire solutions for Iu?0 and for Iu+up?0, p>1.We prove the existence of fundamental solutions and use them, via comparison principle, to prove the theorems for entire solutions. The non-local nature of the operators poses various difficulties in the use of comparison techniques, since usual values of the functions at the boundary of the domain are replaced here by values in the complement of the domain. In particular, we are not able to prove the Hadamard Three Spheres Theorem, but we still obtain some of its consequences that are sufficient for the arguments.     

Second-order linear differential equations in a Banach space and splitting operators

We consider a second-order linear differential equation whose coefficients are bounded operators acting in a complex Banach space. For this equation with a bounded right-hand side, we study the question on the existence of solutions which are bounded on the whole real axis. An asymptotic behavior of solutions is also explored. The research is conducted under condition that the corresponding “algebraic” operator equation has separated roots or provided that an operator placed in front of the first derivative in the equation has a small norm. In the latter case we apply the method of similar operators, i.e., the operator splitting theorem. To obtain the main results we make use of theorems on the similarity transformation of a second order operator matrix to a block-diagonal matrix.     

Spectral analysis of operator polynomials and higher-order difference operators

The study of the spectral properties of operator polynomials is reduced to the study of the spectral properties of the operator specified by the operator matrix. The results obtained are applied to higher-order difference operators. Conditions for their invertibility and for them to be Fredholm, as well as the asymptotic representation for bounded solutions of homogeneous difference equations are obtained.     

Causal operators and topological dynamics

Summary The translation of abstract causal operators along any function in their domain analogous to the Miller-Sell [21] translation of Volterra integral operators along solutions is established. A skew-product semi-flow with phase- space a convergence space is constructed via the shifting semi- flow and the translations of operators and dynamic properties arising from the nature of the semi- flow are investigated. The restriction of the semi- flow on a specific set is used to define the limiting equations along solutions of causal operator equations of the form x=Tx. Applications are given on implicit Volterra integral equations with an additional delay argument and new results on the asymptotic behavior of the solutions are given.This research was supported by the Deutscher Akademischer Austaunschdienst.     

Fractional integral operators in the complex matrix variate case

A formal definition of fractional integrals in the complex matrix variate case is given here. This definition will encompass all the various fractional integral operators introduced by various authors in the real scalar and matrix cases. The new definition is introduced in terms of M-convolutions of products and ratios of matrices in the complex domain. Their connections to statistical distribution theory, Mellin convolutions, M-transforms and Mellin transform are pointed out. Some basic properties are given and a pathway extension of the new definition is also given. The pathway extension will provide a switching mechanism to move among three different families of functions.