two level linear programming decentralized planning has long been recognized as an important decision making problem many approaches based on the concepts of large scale system decomposition have generally lacked the ability to model the type of truly independent subsystems which often exist in practice multilevel programming models partition control over decision variables among ordered levels within a hierarchical planning structure a planner at one level of the hierarchy may have his objective function and set of feasible decisions determined in part by other levels however his control instruments may allow him to influence the policies at other levels and thereby improve his own objective function this paper examines the special case of the two level linear programming problem geometric characterizations and algorithms are presented with some examples the goal is to demonstrate the tractability of such problems and motivate a wider interest in their study

bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices abstract let a monic polynomial pn x xn a1xn 1 an aj plane1d 49e j 1 2 n be given bounds for the eigenvalues and the computation of the singular values of some special companion matrices give bounds for the zeros of the polynomial pn which improve classical and recently found ones also some new bounds for the zeros are given

hierarchical optimization a satisfactory solution abstract hierarchical optimization or multi level programming techniques are extensions of stackelberg games for solving decentralized planning problems with multiple decision makers in a hierarchical organization they become more important for contemporary decentralized organizations where each unit or department seeks its own interests traditional approaches include vertex enumeration algorithms and approaches based on kuhn tucker conditions or penalty functions these are not only technically inefficient but also lead to a paradox that the follower s decision power dominates the leader s in this study concepts of memberships of optimalities and degrees of decision powers are proposed to solve the above problems efficiently in the proposed hierarchy decision process the leader first sets memberships of optimalities of his her possible objective values and decisions as well as his her decision power and then asks the follower for his her optima calculated in isolation under given constraints the follower s decision with the corresponding levels of optimalities and decision powers are submitted to and modified by the leader with considerations of overall benefit for the organization and distribution of decision power until a best preferred solution is reached

popoviciu s inequality for functions of several variables abstract t popoviciu 1965 13 has proved an interesting characterization of the convex functions of one real variable based on an inequality relating the values at any three points x 1 x 2 x 3 with the values at their means of different orders x 1 x 2 2 x 2 x 3 2 x 3 x 1 2 and x 1 x 2 x 3 3 the aim of our paper is to develop a higher dimensional analogue of the usual convexity based on his characterization

a companion matrix approach to the study of zeros and critical points of a polynomial in this paper we introduce a new type of companion matrices namely d companion matrices by using these d companion matrices we are able to apply matrix theory directly to study the geometrical relation between the zeros and critical points of a polynomial in fact this new approach will allow us to prove quite a number of new as well as known results on this topic for example we prove some results on the majorization of the critical points of a polynomial by its zeros in particular we give a different proof of a recent result of gerhard schmeisser on this topic the same method allows us to prove a higher order schoenberg type conjecture proposed by m g de bruin and a sharma

relationship between the zeros of two polynomials abstract in this paper we shall follow a companion matrix approach to study the relationship between zeros of a wide range of pairs of complex polynomials for example a polynomial and its polar derivative or sz nagy s generalized derivative we shall introduce some new companion matrices and obtain a generalization of the weinstein aronszajn formula which will then be used to prove some inequalities similar to sendov conjecture and schoenberg conjecture and to study the distribution of equilibrium points of logarithmic potentials for finitely many discrete charges our method can also be used to produce in an easy and systematic way a lot of identities relating the sums of powers of zeros of a polynomial to that of the other polynomial

interactive fuzzy programming for multi level linear programming problems with fuzzy parameters this paper presents interactive fuzzy programming for multi level linear programming problems with fuzzy parameters in fuzzy programming for multi level linear programming problems recently developed by lai et al since the fuzzy goals are determined for both an objective function and decision variables at the upper level undesirable solutions are produced when these fuzzy goals are inconsistent in order to overcome such problems after eliminating the fuzzy goals for decision variables interactive fuzzy programming for multi level linear programming problems with fuzzy parameters is presented in our interactive method after determining the fuzzy goals of the decision makers at all levels a satisfactory solution is derived efficiently by updating the satisfactory degrees of decision makers with considerations of overall satisfactory balance among all levels illustrative numerical examples for two level and three level linear programming problems are provided to demonstrate the feasibility of the proposed method

on the permissible arrangements of ritz values for normal matrices in the complex plane abstract this article discusses ritz and harmonic ritz values computed from a krylov subspace generated by a normal matrix we give necessary and sufficient conditions for a given tuple of complex numbers to represent ritz values the existence of a positive solution to a certain linear system with a cauchy matrix using this characterization we prove several localization results for the ritz values of normal matrices and discuss consequences for the convergence of the restarted arnoldi algorithm similar results are shown for the harmonic case as well

on the zeros of a polynomial and its derivatives if p z is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk then the gauss lucas theorem states that all zeros of p z lie in the same disk we study the following question what is the maximum distance from the arithmetic mean of all zeros of p z to a nearest zero of p z we obtain bounds for this distance depending on degree we also show that this distance is equal to 1 for polynomials of degree 3 and polynomials with real zeros

an efficient point algorithm for a linear two stage optimization problem this paper presents an algorithm using sensitivity analysis to solve a linear two stage optimization problem the underlying theory rests on a set of first order optimality conditions that parallel the kuhn tucker conditions associated with a one dimensional parametric linear program the solution to the original problem is uncovered by systematically varying the parameter over the unit interval and solving the corresponding linear program finite convergence is established under nondegenerate assumptions the paper also discusses other solution techniques including branch and bound and vertex enumeration and gives an example highlighting their computational and storage requirements by these measures the algorithm presented here has an overall advantage finally a comparison is drawn between bicriteria and bilevel programming and underscored by way of an example

interactive fuzzy programming for two level linear fractional programming problems this paper presents interactive fuzzy programming for two level linear fractional programming problems with the essentially cooperative decision makers in our interactive method after determining the fuzzy goals of the decision makers at both levels a satisfactory solution is efficiently derived by updating the satisfactory level of the decision maker at the upper level with considerations of overall satisfactory balance between both levels in an interactive process optimal solutions to the formulated programming problems are obtained by combined use of the bisection method and the phase one of linear programming and the variable transformation by charnes and cooper an illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method

multiple objective linear fractional programming this paper presents a simplex based solution procedure for the multiple objective linear fractional programming problem by 1 departing slightly from the traditional notion of efficiency and 2 augmenting the feasible region as in goal programming the solution procedure solves for all weakly efficient vertices of the augmented feasible region the article discusses the difficulties that must be addressed in multiple objective linear fractional programming and motivates the solution algorithm that is developed

interactive fuzzy programming for multilevel linear programming problems this paper presents interactive fuzzy programming for multilevel linear programming problems in fuzzy programming for multilevel linear programming problems recently developed by lai et al since the fuzzy goals are determined for both an objective function and decision variables at the upper level undesirable solutions are produced when these fuzzy goals are inconsistent in order to overcome such problems after eliminating the fuzzy goals for decision variables interactive fuzzy programming for multilevel linear programming problems is presented in our interactive method after determining the fuzzy goals of the decision makers at all levels a satisfactory solution is derived efficiently by updating the satisfactory degrees of decision makers at the upper level with considerations of overall satisfactory balance among all levels illustrative numerical examples for two level and for three level linear programming problems are provided to demonstrate the feasibility of the proposed method

asymptotic behavior for two dimensional quasi autonomous almost periodic evolution equations let a be a maximal monotone operator in r2and f t r r2 any s1 almostperiodic function if u is a solution of the evolution equation dudt au t f t which is bounded on r then u t r r2 is asymptotically almost periodic as t

ritz values of normal matrices and ceva s theorem abstract this paper investigates the behavior of ritz values of normal matrices we apply ceva s theorem a classical geometric result to understand the geometric relationship between pairs of ritz values for 3 3 normal non hermitian matrices and then analyze the implications for larger matrices we find that in the case of normal non hermitian matrices the geometric constraints on the placement of ritz values provide less freedom than the cauchy interlacing theorem in the hermitian case using our results we analyze the restarted arnoldi method with exact shifts applied to a 3 3 normal non hermitian matrix

weak log majorization mahler measure and polynomial inequalities abstract we explore the use of the weak log majorization order in the analytic theory of polynomials we examine the relationship between weak log majorization and mahler measure we also improve the weak log majorization form of the de bruijn springer mahler inequality

a bohr neugebauer property for abstract almost periodic evolution equations in banach spaces application to a size structured population model abstract this work continues the investigations done in the literature about the so called bohr neugebauer property for almost periodic differential equations in hilbert spaces we give a new sufficient condition to ensure that an almost periodic linear evolution equation in a banach space has the bohr neugebauer property more specifically we investigate when all the bounded solutions on r are almost periodic this new sufficient condition involves the essential growth bound of the c 0 semigroup generated by the autonomous part of the equation this enables us to investigate the bohr neugebauer property for partial differential equations with solutions living in purely banach spaces an application to a size structured population model will be studied using the theory of positive c 0 semigroups

goal programming with linear fractional criteria abstract in this paper we present an extension of goal programming to include linear fractional criteria the extension forms a natural link between goal programming gp and multiple objective linear fractional programming molfp

differentiators and the geometry of polynomials in 1959 davis introduced the concept of a differentiator of an operator on a finite dimensional hilbert space we prove that every such operator possesses a differentiator we also use the theory of differentiators to solve several problems in the geometry of polynomials for instance we answer in the affirmative a twenty year old unsolved conjecture of schoenberg a related conjecture of katsoprinakis and a fifty year old unsolved conjecture of de bruijn and springer

a fuzzy approach for bi level integer non linear programming problem abstract bi level programming a tool for modeling decentralized decisions consists of the objective of the leader at its first level and that is of the follower at the second level integer programming deals with the mathematical programming problems in which some or all the variables are to be integer this paper studies a bi level integer non linear programming problem with linear or non linear constraints and in which the non linear objective function at each level are to maximized the bi level integer non linear programming bli nlp problem can be thought as a static version of the stackelberg strategy which is used leader follower game in which a stackelberg strategy is used by the leader or the higher level decision maker hldm given the rational reaction of the follower or the lower level decision maker lldm this paper proposes a two planner integer model and a solution method for solving this problem this method uses the concept of tolerance membership function and the branch and bound technique to develop a fuzzy max min decision model for generating pareto optimal solution for this problem an illustrative numerical example is given to demonstrate the obtained results

circulants and critical points of polynomials abstract we prove that for any circulant matrix c of size n n with monic characteristic polynomial p z the spectrum of its n 1 n 1 submatrix c n 1 consisting of the first n 1 rows and columns of c consists of all critical points of p z using this fact we provide a simple proof for the schoenberg conjecture recently proved by r pereira and s malamud we also prove full generalization of a higher order schoenberg type conjecture proposed by m de bruin and a sharma and recently proved by w s cheung and t w ng in its original form i e for polynomials whose mass centre of roots equals zero in this particular case our inequality is stronger than it was conjectured by de bruin and sharma some schmeisser s like results on majorization of critical point of polynomials are also obtained

imbedding conditions for normal matrices abstract when can an n k n k normal matrix b be imbedded in an n n normal matrix a this question was studied for the first time 50 years ago by ky fan and gordon pall who gave the complete answer in the case k 1 since then a few authors obtained additional results in this note we show how an approach inspired by the hermitian case can throw some light on the problem

analogs of cauchy poincare and fan pall interlacing theorems for j hermitian and j normal matrices abstract the interlacing theorem of cauchy poincare states that the eigenvalues of a principal submatrix a 0 of a hermitian matrix a interlace the eigenvalues of a fan and pall obtained an analog of this theorem for normal matrices in this note we investigate analogs of cauchy poincare and fan pall interlacing theorems for j hermitian and j normal matrices the corresponding inverse spectral problems are also considered

interaction between hermitian and normal imbeddings abstract for several applications it is highly desirable to understand how the eigenvalues of an imbeddable matrix v av where v c n k is an isometry are distributed throughout the numerical range of a c n n there has been extensive study for a hermitian while a geometric description for the eigenvalues of imbeddings in non hermitian matrices remains a challenging problem toward this direction a subspace is introduced wherein all n n complex diagonal matrices for which a given isometry v c n k generates diagonal imbeddings are defined in particular conditions upon which a real diagonal matrix may be imbeddable in some normal are obtained including an application for higher rank numerical ranges finally a procedure determining whether two given sets of complex numbers may be realized as spectra of a pair of imbeddable normal matrices is established

an interactive fuzzy satisficing method for multiobjective linear fractional programming problems abstract we present a new interactive fuzzy decision making method for solving multiobjective linear fractional programming problems by assuming that the decision maker dm has fuzzy goals for each of the objective functions through the interaction with the dm the fuzzy goals of the dm are quantified by eliciting the corresponding membership functions including nonlinear functions after determining the membership functions if the dm specifies reference membership values the minimax problem is solved by combined use of the bisection method and the linear programming method and the dm is supplied with the corresponding pareto optimal solution together with the trade off rates between the membership functions then by considering the current values of the membership functions as well as the trade off rates the dm responds by updating his her reference membership values in this way the compromise or satisficing solution for the dm can be derived efficiently from among a pareto optimal solution set by updating his her reference membership values on the basis of the proposed method a time sharing computer program is written and an illustrative numerical example is demonstrated along with the computer outputs

fuzzy approaches for multiple objective linear fractional optimization abstract the problem of finding a solution to a multiple objective linear fractional program arises in several real world situations in this paper we advocate that fuzzy sets theory provides a basis for solving this problem with sufficient consistency and rigorousness after representing imprecise aspirations of the decision maker by structured linguistic variables or converting the original problem via approximations or change of variables into a multiple objective linear program techniques of fuzzy linear programming may be used to reach a satisfactory solution it is shown that under reasonable restrictions this solution is efficient pareto optimal for the original problem numerical examples are also included for illustration

equilibrium points of logarithmic potentials induced by positive charge distributions i generalized de bruijn springer relations equilibrium points of logarithmic potentials induced by positive charge distributions i generalized de bruijn springer relations

integrators of matrices a notion of differentiation for matrices through differentiators where one is actually differentiating the characteristic polynomials can be found in the literature this concept is very useful in studying roots and critical points of complex polynomials here we introduce the inverse operation of integration for matrices through integrators surprisingly not all matrices are integrable in this sense it is seen that non derogatory matrices are best behaved with respect to this operation they are freely integrable

a converse to the jensen inequality its matrix extensions and inequalities for minors and eigenvalues we obtain a scalar inequality converse to the jensen inequality we also derive an operator converse to the jensen inequality as special cases we obtain inequalities similar to the kantorovich one as well as some operator generalizations of them using some exterior algebra we prove a generalization of the sylvester determinant theorem we also deduce some determinant analogs of the additive and multiplicative kantorovich inequalities

large a quasi jacobi form for j normal matrices and inverse eigenvalue problems abstract a quasi jacobi form for j unitarily diagonalizable j normal matrices is given extending a result for normal matrices due to malamud the inverse eigenvalue problem for j normal matrices satisfying certain prescribed spectral conditions is investigated it is shown that there exists unicity in the case of pseudo jacobi matrices

interactive compensatory fuzzy programming for decentralized multi level linear programming dmllp problems abstract this paper presents interactive compensatory fuzzy programming for decentralized multi level linear programming dmllp problems by adjusting the cooperative decision making process between the different levels and also between the decision makers of the same level our aim is to obtain a preferred compensatory compromise pareto optimal solution for dmllp for this the weights of objectives at each level are assigned by the next upper level decision maker dm by using analytic hierarchy process ahp or any other weighting methods the weight of any objective for whole system is equal to the product of the weights on the path tying it to the top decision maker dm 0 using these weights equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights werners compensatory fuzzy and operator is offered to solve dmllp problem the most important idea to be emphasized is that equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights thanks to this equivalence dmllp problem has been transformed to the multi objective linear programming molp problem at level 0 the equivalence is reflected to the compensatory model within the constraints and the equivalence also enables all dms to obtain proportional satisfactions with their weights as much as possible so in our compensatory model a reduction on equivalent satisfactory level of one dm can be compensated for by an increase in the equivalent satisfactory level of another dm furthermore being developed a finite interactive iterative procedure with maximum interaction step a set of compensatory solutions which are also pareto optimal is obtained depending on compensation parameter giving a theorem we will show that the solutions generated by werners compensatory fuzzy and operator do guarantee pareto optimality for our dmllp problem and comparing it with some other computational efficient compensatory fuzzy aggregation operators we will conclude that this operator is more appropriate for dmllp illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed interactive fuzzy compensatory method for dmllp

bounds and majorization relations for the critical points of polynomials abstract we apply several matrix inequalities to the derivative companion matrices of complex polynomials to establish new bounds and majorization relations for the critical points of these polynomials in terms of their zeros

the gauss lucas theorem and jensen polynomials a characterization is given of the sequences fyj o vvith the property that for any complex polynomial z 1akzk and convex region kcontaining the origin and the zeros of the zeros of 2 yi 0 for all n k

on the realizability of the critical points of a realizable list abstract the nonnegative inverse eigenvalue problem niep is to characterize the spectra of entrywise nonnegative matrices a finite multiset of complex numbers is called realizable if it is the spectrum of an entrywise nonnegative matrix monov conjectured that the kth moments of the list of critical points of a realizable list are nonnegative johnson further conjectured that the list of critical points must be realizable in this work johnson s conjecture and consequently monov s conjecture is established for a variety of important cases including ciarlet spectra sule manova spectra spectra realizable via companion matrices and spectra realizable via similarity by a complex hadamard matrix additionally we prove a result on differentiators and trace vectors and use it to provide an alternative proof of a result due to malamud and a generalization of a result due to kushel and tyaglov on circulant matrices implications for further research are discussed

interactive fuzzy programming for two level linear and linear fractional production and assignment problems a case study abstract in this paper we deal with actual problems on production and work force assignment in a housing material manufacturer and a subcontract firm we formulate two kinds of two level programming problems one is a profit maximization problem of both the housing material manufacturer and the subcontract firm and the other is a profitability maximization problem of them applying the interactive fuzzy programming for two level linear and linear fractional programming problems we derive satisfactory solutions to the problems after comparing the two problems we discuss the results of the applications and examine actual planning of the production and the work force assignment of the two firms to be implemented

inverse spectral problem for normal matrices and the gauss lucas theorem we establish an analog of the cauchy poincare interlacing theorem for normal matrices in terms of majorization and we provide a solution to the corresponding inverse spectral problem using this solution we generalize and extend the gauss lucas theorem and prove the old conjecture of de bruijn springer on the location of the roots of a complex polynomial and its derivative and an analog of rolle s theorem conjectured by schoenberg

semigroups and linear partial differential equations with delay abstract we prove the equivalence of the well posedness of a partial differential equation with delay and an associated abstract cauchy problem this is used to derive sufficient conditions for well posedness exponential stability and norm continuity of the solutions applications to a reaction diffusion equation with delay are given

maximal polynomials and the ilieff sendov conjecture in this paper we consider those complex polynomials which have all their roots in the unit disk one fixed root and all the roots of their first derivatives as far as possible from a fixed point we conjecture that any such polynomial has all the roots of its derivative on a circle centered at the fixed point and as many of its own roots as possible on the unit circle we prove a part of this conjecture and use it to define an algorithm for constructing some of these polynomials with this algorithm we investigate the 1962 conjecture of sendov and the 1969 conjecture of goodman rahman and ratti and independently schmeisser obtaining counterexamples of degrees 6 8 10 and 12 for the latter

generalized jacobi operators in krein spaces a special class of generalized jacobi operators which are self adjoint in krein spaces is presented a description of the resolvent set of such operators in terms of solutions of the corresponding recurrence relations is given in particular special attention is paid to the periodic generalized jacobi operators finally the spectral properties of generalized jacobi operators are applied to prove convergence results for pade approximants

weighting method for bi level linear fractional programming problems in this paper we consider the solution of a bi level linear fractional programming problem bllfpp by weighting method a non dominated solution set is obtained by this method in this article decision makers dms provide their preference bounds to the decision variables that is the upper and lower bounds to the decision variables they control we convert the hierarchical system into scalar optimization problem sop by finding proper weights using the analytic hierarchy process ahp so that objective functions of both levels can be combined into one objective function here the relative weights represent the relative importance of the objective functions