decomposition of matrices into three involutions abstract in this paper we generalize some results of c s ballantine concerning products of three n n complex involutions we prove that each n n complex matrix a with determinant 1 and dimker a n 2 for all c is the product of three involutions on the other hand we show that if an n n complex matrix a is the product of three involutions then m 2n r 3 and m 3n 4 where m dimker a and r dimker a 3 for any 0 and 4 1 we also completely characterize products of three 5 5 complex involutions

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

numerical analysis of a nonlinear evolutionary system with applications in viscoplasticity we consider numerical approximations of a class of abstract nonlinear evolutionary systems arising in the study of quasi static frictional contact problems for elastic viscoplastic materials both semidiscrete and fully discrete schemes are analyzed strong convergence of both approximations is established under minimal solution regularity the results are applied to two particular frictional contact problems for viscoplastic bodies where the finite element method is employed to discretize the spatial domain under additional regularity assumptions on the exact solution some error estimates are derived

quantitative parametric connections between methods for generating noninferior solutions in multiobjective optimization abstract in multiobjective optimization the trade off information between different objective functions is probably the most important piece of information in a solution process to reach the most preferred solution generating methods are the methods in generating noninferior solutions different generating methods provide decision makers trade off information in different forms various generating methods are characterized in this paper and the quantitative parametric connections between these generating methods are established the result in this paper is then used to consolidate trade off information with different formats associated with different generating methods

fuzzy mathematical programming for multi objective linear fractional programming problem the present paper deals with a solution procedure for multi objective linear fractional programming problems an equivalent multi objective linear programming form of the problem has been formulated in the proposed methodology using fuzzy set theoretic approach a procedure has been explored the proposed solution procedure has also been used to solve numerical examples

on generalized quadratic matrices abstract in this paper a wide class of matrices is considered containing idempotent involutory nilpotent and several other types of matrices extending an approach considered by radjavi and rosenthal h radjavi p rosenthal on commutators of idempotents linear multilinear algebra 50 2 2002 121 124 we investigate the set q p of square matrices a c n n satisfying the equation a 2 a p for some complex numbers and and for some n n nonzero complex idempotent matrix p such the ap pa a special attention is paid to the moore penrose and group inverse of matrices belonging to q p

products of two unipotent matrices of index 2 abstract necessary and sufficient conditions are presented for a square matrix a over a general field f to be the product of two unipotent matrices of index 2 this generalizes a result established by wang and wu 1991 4 for the case where f is the complex field

sums and products of two quadratic matrices abstract necessary and sufficient conditions are given for a complex matrix t to be the sum of an idempotent and a square zero matrix these conditions are expressed in terms of 1 the block structure of t under similarity as well as 2 the elementary divisor structure of t this leads to a complete characterization of which complex matrices are a sum of two quadratic matrices also using a similar scheme we characterize when a matrix is the product of an involution and a unipotent matrix of index 2

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

on linear combinations of two idempotent matrices over an arbitrary field abstract given an arbitrary field k and non zero scalars and we give necessary and sufficient conditions for a matrix a m n k to be a linear combination of two idempotents with coefficients and this extends results previously obtained by hartwig and putcha in two ways the field k considered here is arbitrary possibly of characteristic 2 and the case is taken into account

an elastic viscoplastic quasistatic contact problem existence and uniqueness of a weak solution in this paper the contact between an elastic viscoplastic body and a deformable obstacle is studied the effect of the damage due to internal tension or compression and caused by the opening and growth of micro cracks and micro cavities is also considered the variational formulation leads to a coupled system of evolutionary equations an existence and uniqueness result is established by using approximate problems the pseudomonotone operators theory schauder fixed point theorem and a comparison result

an elastic viscoplastic quasistatic contact problem with damage abstract in this paper the contact between an elastic viscoplastic body and a deformable obstacle is studied the effect of the damage due to tension or compression is also considered the variational formulation leads to a coupled system of evolutionary equations for which an existence and uniqueness result is stated then a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an euler scheme to discretize the time derivatives error estimates are obtained and as a consequence the convergence of the algorithm is deduced finally some numerical simulations are presented to demonstrate the accuracy of the algorithm

numerical analysis and simulations of a quasistatic frictional contact problem with damage in viscoelasticity we consider a quasistatic problem which models the bilateral contact between a viscoelastic body and a foundation taking into account the damage and the friction the damage which results from tension or compression is then involved in the constitutive law and it is modelled using a nonlinear parabolic inclusion the variational problem is formulated as a coupled system of evolutionary equations for which we state the existence of a unique solution then we introduce a fully discrete scheme using the finite element method to approximate the spatial variable and the euler scheme to discretize the time derivatives error estimates are derived and under suitable regularity hypotheses the convergence of the numerical scheme obtained finally a numerical algorithm and results are presented for some two dimensional examples

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

variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance friction and damage we consider a model for quasistatic frictional contact between a viscoelastic body and a foundation the material constitutive relation is assumed to be nonlinear the mechanical damage of the material caused by excessive stress or strain is described by the damage function the evolution of which is determined by a parabolic inclusion the contact is modeled with the normal compliance condition and the associated version of coulomb s law of dry friction we derive a variational formulation for the problem and prove the existence of its unique weak solution we then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions

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

powers of 1 cyclic conjugacy classes in the groups gln f and sln f abstract let a gl n f where f is a field we say that a is 1 cyclic if a is similar to a matrix of the form a diag a 1 a 2 a k where a i gl l i f is cyclic for 1 i k l 1 0 1 and l i 2 for 2 i k it is shown that if a gl n f is 1 cyclic where n 2 and f 4 then every nonscalar matrix m gl n f whose determinant equals det a 4 is the product of four matrices which are similar to a under matrices of sl n f the problem of expressing a scalar matrix as a product of similar 1 cyclic matrices is also discussed the above result is applied to problems of factorizing matrices in the group sl n f into products of unipotent matrices of index 2 and into products of matrices of fixed finite order

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

interactive fuzzy programming for multi level 0 1 programming problems through genetic algorithms abstract in this paper we propose interactive fuzzy programming for multi level 0 1 programming problems through genetic algorithms our method is supposed to apply to hierarchical decision problems in which decision making at each level is sequential from upper to lower level and decision makers are essentially cooperative after determining the fuzzy goals of the decision makers at all levels a satisfactory solution is derived efficiently by updating the satisfactory degrees of the decision makers at the upper level with considerations of overall satisfactory balance among all levels an illustrative numerical example for three level 0 1 programming problems is provided to demonstrate the feasibility of the proposed method

computational difficulties of bilevel linear programming we show using small examples that two algorithms previously published for the bilevel linear programming problem blp may fail to find the optimal solution and thus must be considered to be heuristics a proof is given that solving blp problems is np hard which makes it unlikely that there is a good exact algorithm

on linear combinations of two tripotent idempotent and involutive matrices abstract let a c 1 a 1 c 2 a 2 where c 1 c 2 are nonzero complex numbers and a 1 a 2 is a pair of two n n nonzero matrices we consider the problem of characterizing all situations where a linear combination of the form a c 1 a 1 c 2 a 2 is i a tripotent or an involutive matrix when a 1 and a 2 are commuting involutive or commuting tripotent matrices respectively ii an idempotent matrix when a 1 and a 2 are involutive matrices and iii an involutive matrix when a 1 and a 2 are involutive or idempotent matrices

factorization of matrices into quadratic ones iii abstract it is shown that every complex n n matrix t is the product of four quadratic matrices moreover if t is invertible then the number of required quadratic matrices can be reduced to three and three is sharp for n 4 complex matrices which are expressible as the product of an idempotent and an invertible quadratic matrix are also characterized

an algorithm for solving the general bilevel programming problem the conflict that naturally arises in a hierarchical system can often be modeled as a multistage optimization problem this paper considers the sequential uncooperative problem in which two decision makers wish to maximize their own objective functions over a feasible region defined by interactive strategy sets the resultant problem is known as the bilevel program such programs are inherently nonconvex and resistant to standard nonlinear programming solution techniques such as piecewise linearization and convex underestimating envelopes alternatively a grid search algorithm is offered which exhibits the desirable property of monotonicity the algorithm is based on two sets of necessary conditions previously developed and combined here to provide an operational check for stationarity and local optimality potential solutions are obtained from a parameterized master program whose feasible region approximates that of the original problem as the one dimensional parameter is varied over the unit interval

the operator factorization problems we survey various results concerning operator factorization problems more precisely we consider the following setting let h be a complex hilbert space and let g h be the algebra of all bounded linear operators on h for a given subset of 3 h we are interested in the characterization of operators in q h which are expressible as a product of finitely many operators in v and for each such operator the minimal number of factors in a factorization the classes of operators we consider include normal operators involutions partial isometries together with their various subclasses and other miscellaneous classes of operators most of the known results are for operators on finite dimensional spaces or finite matrices the paper concludes with some applications due to hochwald concerning the uniqueness of the adjoint operation on operators

a quasistatic contact problem for viscoelastic materials with friction and damage abstract we consider a mathematical model which describes the frictional contact between a deformable body and a foundation the process is quasistatic the material is assumed to be viscoelastic with long memory and the frictional contact is modelled with subdifferential boundary conditions the mechanical damage of the material is described by the damage function which is modelled by a nonlinear partial differential equation we derive the variational formulation of the problem which is a coupled system of a hemivariational inequality and a parabolic equation then we prove the existence of a unique weak solution to the model the proof is based on arguments of abstract stationary inclusion and a fixed point theorem

sums of commuting square zero transformations abstract we show that a linear transformation on a vector space is a sum of two commuting square zero transformations if and only if it is a nilpotent transformation with index of nilpotency at most 3 and the codimension of im t ker t in ker t is greater than or equal to the dimension of the space im t 2 we also characterize products of two commuting unipotent transformations with index 2

on properties of generalized quadratic operators abstract in this paper we investigate the set p of generalized quadratic operators a satisfying the equation a 2 a p for all complex numbers and and for an idempotent operator p such that ap pa a furthermore the close relationship between the operator a p and the idempotent operator p are established and expressions for the inverse the moore penrose inverse and the drazin inverse of a p are given some related results are also obtained

the length problem for a sum of idempotents abstract it was shown by r e hartwig and m s putcha and independently by p y wu that the complex square matrix t is a sum of finitely many idempotent matrices if and only if tr t is an integer and tr t rank t we determine the minimal number of required idempotents for such matrices in terms of their traces and sizes and we completely solve the problem for matrices t with size 2 3 4 or 5 also we give some sufficient necessary conditions for a complex matrix t to be the sum of three or more idempotents

duality methods for solving variational inequalities abstract methods of maximal monotone operators are used in order to study from a general point of view duality numerical algorithms for solving variational inequalities with classical algorithms such as uzawa s method for the standard and augmented lagrangian this paper presents some new algorithms which appear to have very good numerical performances

a frictionless contact problem for elastic viscoplastic materials with normal compliance numerical analysis and computational experiments in this paper we consider a frictionless contact problem between an elastic viscoplastic body and an obstacle the process is assumed to be quasistatic and the contact is modeled with normal compliance we present a variational formulation of the problem and prove the existence and uniqueness of the weak solution using strongly monotone operators arguments and banach s fixed point theorem we also study the numerical approach to the problem using spatially semi discrete and fully discrete finite elements schemes with implicit and explicit discretization in time we show the existence of the unique solution for each of the schemes and derive error estimates on the approximate solutions finally we present some numerical results involving examples in one two and three dimensions

joint optimization of product family configuration and scaling design by stackelberg game abstract product family design is generally characterized by two types of approaches module based and scale based while the former aims to enable product variety based on module configuration the latter is to variegate product design by scaling up or down certain design parameters the prevailing practice is to treat module configuration and scaling design as separate decisions or aggregate two design problems as a single level all in one optimization problem in practice optimization of scaling variables is always enacted within a specific modular platform and meanwhile an optimal module configuration depends on how design parameters are to be scaled the key challenge is how to deal with explicitly the coupling of these two design optimization problems this paper formulates a stackelberg game theoretic model for joint optimization of product family configuration and scaling design in which a bilevel decision structure reveals coupled decision making between module configuration and parameter scaling a bilevel mixed 0 1 non linear programming model is developed and solved comprising an upper level optimization problem and a lower level optimization problem the upper level seeks for an optimal configuration of modules and module attributes by maximizing the shared surplus of an entire product family the lower level entails parametric optimization of attribute values for optimal technical performance of each individual module a case study of electric motors demonstrates that the bilevel joint optimization model excels in leveraging optimal scaling in conjunction with optimal module configuration which is advantageous over the existing paradigm of product family scaling design that cannot change the product family configuration

on decomposing any matrix as a linear combination of three idempotents abstract in a recent article we gave a full characterization of matrices that can be decomposed as linear combinations of two idempotents with prescribed coefficients in this one we use those results to improve on a recent theorem of rabanovich we establish that every square matrix is a linear combination of three idempotents for an arbitrary coefficient field rather than just one of characteristic 0

a frictionless contact problem for elastic viscoplastic materials with normal compliance and damage abstract we study a quasistatic frictionless contact problem with normal compliance and damage for elastic viscoplastic bodies the mechanical damage of the material caused by excessive stress or strain is described by a damage function whose evolution is modelled by an inclusion of parabolic type we provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution to the model we then introduce and study a fully discrete scheme for the numerical solutions of the problem an optimal order error estimate is derived for the approximate solutions under suitable solution regularity numerical examples are presented to show the performance of the method

products of idempotent and square zero matrices abstract this paper presents necessary and sufficient conditions for a square matrix over an arbitrary field to be expressed as a product of idempotent and square zero matrices since the similarity structures of such factors are completely determined by their rank we also investigate the ranks that these factors may have the case where the factors are all idempotent or all square zero have been investigated by various authors some of whom are listed in the references of this paper

difference and similarity models of two idempotent operators in a recent work hartwig and putcha obtained a complete characterization of those finite matrices which can be expressed as the difference of two idempotents extending this result to operators on a possibly infinite dimensional hilbert space seems more difficult in this paper we initiate its study and obtain among other things 1 that not every nilpotent operator is the difference of two idempotents 2 that if t is the difference of two idempotents then the spectra of t and t differ at most by the two points k 1 and 3 a characterization of differences of two idempotents among normal operators in the second part of the paper we develop some similarity invariant models of two idempotents these are analogous to the known unitary equivalence invariant models for two orthogonal projections

product of invertible operators of quadratic type abstract suppose all invertible quadratic operators t are assumed to satisfy t 2 bt i 0 we show that a complex matrix t is the product of two invertible quadratic matrices if and only if t is similar to a matrix of the form d d 1 i n i m i 1 j i 2 i 2 i 2 i xy 2 i yx 1 where b b 2 4 2 0 1 and gb 2 are not eigenvalues of d n xy and yx are nilpotent and each j i is a nilpotent jordan block of even size also we show that an n n matrix t is the product of finitely many invertible quadratic matrices if and only if det t m where m is an integer for n odd and is an even integer for n even on the other hand for operators on an infinite dimensional hilbert space we characterize the products of two and four invertible quadratic operators among normal operators and show that every invertible operator is the product of six invertible quadratic operators

sums of two triangularizable quadratic matrices over an arbitrary field abstract let k be an arbitrary field and a b c d be elements of k such that the polynomials t 2 at b and t 2 ct d are split in k t given a square matrix m m n k we give necessary and sufficient conditions for the existence of two matrices a and b such that m a b a 2 aa bi n and b 2 cb di n prior to this paper such conditions were known in the case b d 0 a 0 and c 0 4 and in the case a b c d 0 1 here we complete the study which essentially amounts to determining when a matrix is the sum of an idempotent and a square zero matrix this generalizes results of wang 5 to an arbitrary field possibly of characteristic 2

sums of two square zero matrices over an arbitrary field abstract the problem to express an n n matrix a as the sum of two square zero matrices was first investigated by wang and wu 2 for matrices over the complex field this paper investigates the problem over an arbitrary field f it is shown that if char f 2 then a m n f is the sum of two square zero matrices if and only if a is similar to a matrix of the form n x x i 1 m c g i x 2 where n is nilpotent x is nonsingular and each c g i x 2 is a companion matrix associated with an even power poly nomial with nonzero constant term if f is of characteristic two the term x x falls away if f is of characteristic zero and algebraically closed the term i 1 m c g i x 2 falls away and the result of wang and wu is obtained

an approach to find redundant objective function s and redundant constraint s in multi objective nonlinear stochastic fractional programming problems structural redundancies in mathematical programming models are nothing uncommon and nonlinear programming problems are no exception over the past few decades numerous papers have been written on redundancy redundancy in constraints and variables are usually studied in a class of mathematical programming problems however main emphasis has so far been given only to linear programming problems in this paper an algorithm that identifies redundant objective function s and redundant constraint s simultaneously in multi objective nonlinear stochastic fractional programming problems is provided a solution procedure is also illustrated with numerical examples the proposed algorithm reduces the number of nonlinear fractional objective functions and constraints in cases where redundancy exists

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

rank equalities for idempotent matrices with applications we show by elementary block matrix operations a variety of rank equalities related to sums of two idempotent matrices and give various consequences and applications

products of involutions in the stable general linear group abstract in the stable general linear group over an arbitrary field we prove that every element with determinant 1 is the product of three involutions and of no less in general we also obtain several results of the same flavor with applications to decompositions of automorphisms of an infinite dimensional vector space that are scalar multiples of finite rank perturbations of the identity

on two level optimization 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 have often existed in practice similarly tractable game theoretic approaches have not allowed for a continuous solution space in which one player s decision restricts the feasible choices of another this correspondence offers a formulation to help model this type of decision making hierarchy characterizations and algorithmic approaches for a class of two level problems are discussed

interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters in this paper we present interactive fuzzy programming for two level linear fractional programming problems with fuzzy parameters using the level sets of fuzzy parameters the corresponding nonfuzzy two level linear fractional programming problem is introduced in our interactive method after determining fuzzy goals of decision makers at both levels a satisfactory solution is derived efficiently by updating a minimal satisfactory level of the decision maker at the upper level with considerations of overall satisfactory balance between both levels the satisfactory solution well balanced between both levels is easily computed by combined use of the bisection method the phase one of the simplex method and the variable transformation method by charnes and cooper an illustrative numerical example for two level linear fractional programming problems with fuzzy parameters is provided to demonstrate the feasibility of the proposed method

damage gradient of damage and principle of virtual power a theory of continuum damage mechanics is developed within the framework of the principle of virtual power because the damage in a solid results from microscopic movements we decided to include the power of these microscopic movements in the power of the internal forces the microscopic velocities are related to the damage rate the power of the internal forces we choose depends on the damage velocity and its gradient to take into account the interactions models issued from this theory are presented they are coherent from the mechanical and mathematical points of view the numerical computations show no mesh sensitivity they describe with good agreement the main experimental properties concrete is chosen as an example to illustrate the theory a model using two damage quantities is also presented it yields the correct description of the unilateral phenomenon observed in concrete finally an extension of the existing models is proposed to describe the fatigue damage behaviour

matrices over finite fields as sums of periodic and nilpotent elements abstract we prove that every n n matrix m over a field of odd cardinality q has a decomposition of the form m e n such that e q e and n 3 0

sums of idempotent matrices abstract we show that any complex square matrix t is a sum of finitely many idempotent matrices if and only if trt is an integer and trt rank t moreover in this case the idempotents may be chosen such that each has rank one and has range contained in that of t we also consider the problem of the minimum number of idempotents needed to sum to t and obtain some partial results

a bm lambda cut and goal programming based algorithm for fuzzy linear multiple objective bilevel optimization bilevel programming techniques are developed to handle decentralized problems with two level decision makers which are leaders and followers who may have more than one objective to achieve this paper proposes a cut and goal programming based algorithm to solve fuzzy linear multiple objective bilevel flmob decision problems first based on the definition of a distance measure between two fuzzy vectors using cut a fuzzy linear bilevel goal flbg model is formatted and related theorems are proved then using a cut for fuzzy coefficients and a goal programming strategy for multiple objectives a cut and goal programming based algorithm to solve flmob decision problems is presented a case study for a newsboy problem is adopted to illustrate the application and executing procedure of this algorithm finally experiments are carried out to discuss and analyze the performance of this algorithm

products of unipotent matrices of index 2 abstract we show that an n n complex matrix t is the product of two unipotent matrices of index 2 if and only if t is similar to a matrix of the form d d 1 i n i mi 1 ji where 0 and 1 are not eigenvalues of d n is nilpotent and each ji is a nilpotent jordan block of even size on the other hand t is the product of finitely many unipotent matrices of index 2 if and only if det t 1 in this case the minimal number of required unipotents is 1 if n 1 3 if n 2 and 4 if n 3

the product of two quadratic matrices abstract let p x x 1 k x where 2 2 and let v be a finite dimensional vector space over the field k a linear mapping m v v is called quadratic if p m 0 we characterize products of two quadratic linear mappings

a solution method for the linear static stackelberg problem using penalty functions a new solution technique is presented for the linear constrained static stackelberg problem for a given value of x the leader s decision vector the follower is at its rational reaction set when the duality gap of the second level problem becomes zero the outer problem is solved by appending to the leader s objective a function that minimizes the duality gap of the follower s problem this structure leads to the decomposition of the composite problem into a series of linear programs leading to an efficient algorithm it is proved that optimality is reached for an exact penalty function and the method is illustrated with some examples

on a question of m newman on the number of commutators abstract m newman asked whether there is an absolute constant c such that every matrix in slnr is the product of at most c commutators where r ranges over euclidean commutative rings and n 3 we give here a negative answer however if for the ring r every matrix in slmr is the product of a bounded number of commutators for some fixed m 3 then for all sufficiently large n every matrix in slnr is the product of six commutators

existence and uniqueness of solutions for a dynamic one dimensional damage model we consider a one dimensional dynamic model that describes the evolution of damage caused by tension in a viscoelastic material the process is modeled by a coupled set of two differential inclusions for the elastic displacement and damage fields we establish the existence of local weak solutions the existence result is derived from the a priori estimates obtained for a sequence of regularized truncated and time retarded approximations we also establish the existence of the unique weak solution of a simplified version of the model

fuzzy goal programming algorithm for solving decentralized bi level multi objective programming problems this paper presents fuzzy goal programming fgp algorithm for solving decentralized bi level multi objective programming dbl mop problems with a single decision maker at the upper level and multiple decision makers at the lower level a fgp algorithm for dbl multi objective linear programming dbl molp problems is proposed this algorithm is extended to solve bi level multi objective linear fractional programming dbl molfp problems in the proposed algorithm the membership functions for the defined fuzzy goals of all objective functions at the two levels as well as the membership functions for vector of fuzzy goals of the decision variables controlled by uldm are developed in the model formulation of the problem then fgp approach is used to achieve highest degree of each of the membership goals by minimizing their deviational variables and thereby obtaining the most satisfactory solution for all decision makers illustrative numerical examples are given to demonstrate the proposed algorithm

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

damageable contact between an elastic body and a rigid foundation in this work the contact problem between an elastic body and a rigid obstacle is studied including the development of material damage which results from internal compression or tension the variational problem is formulated as a first kind variational inequality for the displacements coupled with a parabolic partial differential equation for the damage field the existence of a unique local weak solution is stated then a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an euler scheme to discretize the time derivatives error estimates are derived on the approximate solutions from which the linear convergence of the algorithm is deduced under suitable regularity conditions finally three two dimensional numerical simulations are performed to demonstrate the accuracy and the behaviour of the scheme

on the spectra of some combinations of two generalized quadratic matrices let a and b be two generalized quadratic matrices with respect to idempotent matrices p and q respectively such that a p a p 0 a p p a a b q b q 0 b q q b b p q q p ab ba and a b p q p q a b with c let a b be diagonalizable the relations between the spectrum of the matrix a b and the spectra of some matrices produced from a and b are considered moreover some results on the spectrum of the matrix a b are obtained when a b is not diagonalizable finally some results and examples illustrating the applications of the results in the work are given