rank equalities related to the generalized inverses at s 2 bt 1 s1 2 of two matrices a and b for given matrices a and b by applying the group inverse expressions of generalized inverses a t s 2 and b t 1 s 1 2 we present some rank equalities related to a t s 2 and b t 1 s 1 2 as applications we obtain a variety of rank equalities related to the moore penrose inverses the drazin inverses the group inverses and the weighted moore penrose inverses of a and b

representation of tripotents and representations via tripotents abstract let a be an algebra an element a a is called tripotent if a 3 a we study the questions if both a and b are tripotents then under what conditions are a b and ab tripotent under what conditions do a and b commute we extend the partial order from the hilbert space idempotents to the set of all tripotents and show that every normal tripotent is self adjoint for a m n c we describe the set of all finite sums of tripotents the convex hull of tripotents and the set of all tripotents averages we also give the new proof of rational trace matrix representations by choi and wu 2

robust stabilization of nonlinear multiple time delay large scale systems via decentralized fuzzy control to overcome the effect of modeling errors between nonlinear multiple time delay subsystems and takagi sugeno t s fuzzy models with multiple time delays a robustness design of fuzzy control is proposed in this work in terms of lyapunov s direct method a delay dependent stability criterion is hence derived to guarantee the asymptotic stability of nonlinear multiple time delay large scale systems based on this criterion and the decentralized control scheme a set of model based fuzzy controllers is then synthesized via the technique of parallel distributed compensation pdc to stabilize the nonlinear multiple time delay large scale system finally a numerical example with simulations is given to demonstrate the concepts discussed throughout this work

matrices a such that aa a a are nonsingular abstract in this paper we study the class of square matrices a such that aa a a is nonsingular where a stands for the moore penrose inverse of a among several characterizations we prove that for a matrix a of order n the difference aa a a is nonsingular if and only if r a r a c n 1 where r denotes the range space also we study matrices a such that r a r a

calculus for interval valued functions using generalized hukuhara derivative and applications this paper is devoted to studying differential calculus for interval valued functions by using the generalized hukuhara differentiability which is the most general concept of differentiability for interval valued functions conditions examples and counterexamples for limit continuity integrability and differentiability are given special emphasis is set to the class f t c g t where c is an interval and g is a real function of a real variable here the emphasis is placed on the fact that f and g do not necessarily share their properties underlying the extra care that must be taken into account when dealing with interval valued functions two applications of the obtained results are presented the first one determines a delta method for interval valued random elements in the second application a new procedure to obtain solutions to an interval differential equation is introduced our results are relevant to fuzzy set theory because the usual fuzzy arithmetic extension functions and mathematical analysis are done on a cuts which are intervals

on subspace distances determined by the frobenius norm abstract two measures of separation between two subspaces of a finite dimensional complex vector space are introduced and investigated the measures were inspired by known expressions for the cosines of the angle and minimal angle between subspaces and are based on the frobenius norm by using partitioned representations of a pair of orthogonal projectors several properties of the measures are identified and their links with other notions used to characterize a separation between subspaces are pointed out

on the matrix difference i a for an nxn complex matrix a and the nxn identity matrix i n the difference i n ais investigated by exploiting a partitioned representation several features of such a difference are identified in particular expressions for its moore penrose inverse in some specific situations are established and representations of the pertinent projectors are derived special attention is paid to the problem how certain properties of a and i n a are related the properties in question deal with known classes of matrices such as gp ep partial isometries bi ep normal projectors and nilpotent an important part of the paper is devoted to demonstrating how to obtain representations of orthogonal projectors onto various subspaces determined by a and or i n a several such representations are provided and a number of relevant conclusions originating from them are identified

multiobjective programming in optimization of interval objective functions a generalized approach abstract we generalize known concepts those introduced by ishihuchi and tanaka 1 and rommerlfanger et al 2 of the solution of the linear programming problem with interval coefficients in the objective function based on preference relations between intervals we unify all the discussed concepts as well as the corresponding solution methods into one general framework

on angles and distances between subspaces abstract the angles and distances between two given subspaces of c n 1 are investigated on the basis of a joint decomposition of the corresponding orthogonal projectors several new results are established with the particular attention paid to the notions of inclinedness and minimal angle to demonstrate the usefulness of the approach utilized some results known to be valid in hilbert space are reestablished in c n 1 either in generalized form or with considerably shorter proofs than in the original sources

accelerated growth in outgoing links in evolving networks deterministic versus stochastic picture in several real world networks such as the internet world wide web etc the number of links grow in time in a nonlinear fashion we consider growing networks in which the number of outgoing links is a nonlinear function of time but new links between older nodes are forbidden the attachments are made using a preferential attachment scheme in the deterministic picture the number of outgoing links m t at any time t is taken as n t ensuremath theta where n t is the number of nodes present at that time the continuum theory predicts a power law decay of the degree distribution p k ensuremath propto k ensuremath 1 ensuremath 2 1 ensuremath ensuremath theta while the degree of the node introduced at time t i is given by k t i t t i ensuremath theta phantom rule 0 2em 0ex t t i 1 ensuremath theta 2 when the network is evolved till time t numerical results show a growth in the degree distribution for small k values at any nonzero ensuremath theta in the stochastic picture m t is a random variable as long as m t is independent of time the network shows a behavior similar to the barab asi albert ba model different results are obtained when m t is time dependent e g when m t follows a distribution p m ensuremath propto m ensuremath ensuremath lambda the behavior of p k changes significantly as ensuremath lambda is varied for ensuremath lambda g3 the network has a scale free distribution belonging to the ba class as predicted by the mean field theory for smaller values of ensuremath lambda it shows different behavior characteristic features of the clustering coefficients in both models have also been discussed

scaling properties of scale free evolving networks continuous approach the scaling behavior of scale free evolving networks arising in areas such as communications scientific citations collaborations etc is studied we derive universal scaling relations describing properties of such networks and indicate the limits of their validity we show that the main properties of scale free evolving networks may be described in the framework of a simple continuous approach the simplest models of networks growing according to a mechanism of preferential attachment of links to nodes are used we consider different forms of this preference and demonstrate that the range of preferential attachments producing scale free networks is wide we also obtain scaling relations for networks with nonlinear accelerating growth and describe the temporal evolution of the arising distributions size effects char22 the cutoffs of these distributions char22 introduce restrictions for the observation of power law dependences mainly we discuss the so called degree distribution i e the distribution of the number of connections of nodes a scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied we describe the effects of differences between nodes the aging of nodes changes the exponents of the distributions the appearance of a single node with high fitness changes the degree distribution of a network dramatically if its fitness exceeds some threshold value this node captures a finite part of all links of the network we show that permanent random damage to a growing scale free network char22 a permanent deletion of some links char22 radically changes the values of the scaling exponents results of other kinds of permanent damage are described

on the product of orthogonal projectors abstract in this paper we introduce a canonical form to represent the product of two orthogonal projectors this form is useful for deriving many known results which are scattered in the literature we summarize many of those results and present new ones

applications of cs decomposition in linear combinations of two orthogonal projectors abstract we study the spectrum and the rank of a linear combination of two orthogonal projectors we characterize when this linear combination is ep diagonalizable idempotent tripotent involutive nilpotent generalized projector and hypergeneralized projector also we derive the moore penrose inverse of a linear combination of two orthogonal projectors in a particular case the main tool used here is the cs decomposition

interval number and fuzzy number linear programmings abstract in this paper we focus on two kinds of linear programmings with fuzzy numbers they are called interval number and fuzzy number linear programmings respectively the problems of linear programmings with interval number coefficients are approached by taking maximum value range and minimum value range inequalities as constraint conditions reduced it into two classical linear programmings and obtained an optimal interval solution to it the problems of fuzzy linear programming with fuzzy number coefficients are approached in two ways i e fuzzy decisive set approach and interval number linear programming approach for several membership levels finally we gave the numerical solutions of the illustrative examples

on disjoint range matrices abstract for a square complex matrix f and for f being its conjugate transpose the class of matrices satisfying r f r f 0 where r denotes range column space of a matrix argument is investigated besides identifying a number of its properties several functions of f such as f f f f ff f f and f f are considered particular attention is paid to the moore penrose inverses of those functions and projectors attributed to them it is shown that some results scattered in the literature whose complexity practically prevents them from being used to deal with real problems can be replaced with much simpler expressions when the ranges of f and f are disjoint furthermore as a by product of the derived formulae one obtains a variety of relevant facts concerning for instance rank and range

characterizations and linear combinations of k generalized projectors abstract we extend generalized projectors introduced by gros and trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 to k generalized projectors and we characterize them obtaining results in the aforesaid paper as a consequence moreover we list all situations when a linear combination of commuting k generalized projectors is a k generalized projector the method for obtaining this result permits to give a revisited version of the main result by baksalary and baksalary on linear combinations of generalized projectors linear algebra appl 388 2004 17 24 in addition the case of orthogonal projectors is also analyzed

synchronizability in complex ad hoc dynamical networks with accelerated growth recent research works have been pursued in connection with network synchronizability under various constraints for different topological structures and evolving mechanisms however the fundamental question of how the synchronizability of the networks relates to accelerated growth and ad hoc property in the evolving processes remains underexplored here we study the ad hoc dynamical models with accelerated growth where the total number of edges increases faster than linearly with network size by adopting three attachment mechanisms random rewired and preferential attachment we investigate the second largest eigenvalues and the ratios of the extremal eigenvalues of coupled matrices in the accelerated models with different evolving parameters for the new models we demonstrate the robustness and fragility of synchronization against random and specific attacks by numerical simulations we find that accelerated growth represents a convenient tool for improving the synchronizability of an evolving network furthermore we show that not only network synchronization but also topological periodicity has robustness and fragility against random failures and specifical removal of the nodes respectively in particular when ad hoc property is suggested in the evolving networks we find that the deletion of nodes is easier to change network synchronizability and robustness compared to the addition of nodes

nonnegative idempotent matrices and the minus partial order abstract we describe the structure of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus order

history and generality of the cs decomposition abstract it is almost a quarter of a century since chandler davis and william kahan brought together the key ideas of what stewart later completed and defined to be the cs decomposition csd of a partitioned unitary matrix this paper outlines some germane points in the history of the csd pointing out the contributions of jordan of davis and kahan and of stewart and the relationship of the csd to the direct rotation of davis and kato the paper provides an easy to memorize constructive proof of the csd reviews one of its important uses and suggests a motivation for the csd which emphasizes how generally useful it is it shows the relation between the csd and generalized singular value decompositions and points out some useful nullity properties one form of the csd trivially reveals finally it shows how via the qr factorization the csd can be used to obtain interesting results for partitioned nonsingular matrices we suggest the csd be taught in its most general form with no restrictions on the two by two partition and initially with no mention of angles between subspaces

constructing projections on sums and intersections abstract in this note we list several formulas for computing orthogonal projections onto the linear sum and intersection of two subspaces of c n in terms of the projections on the individual subspaces

subsets and subspaces several rules known to be valid within the set theory are revisited and considered from the point of view of the question whether they remain satisfied when sets are replaced with subspaces from c n 1 it is shown that some of the set relationships hold true upon such a replacement however there is also an extensive class of relationships whose validity is no longer guaranteed in this class the commutativity of the orthogonal projectors onto the involved subspaces proves to play an important role some related results inspired by probability theory are established as well

functions of orthogonal projectors involving the moore penrose inverse several results scattered in the literature express an oblique projector having given onto and along spaces in terms of a pair of orthogonal projectors the results were established in various settings including finite and infinite dimensional vector spaces over either real or complex numbers but their common feature is that they are valid merely under the assumption of the nonsingularity of certain functions of the involved projectors in the present paper these results are unified and reestablished in a generalized form in a complex euclidean vector space with the generalization obtained by relaxing the nonsingularity assumption and use of the moore penrose inverse instead of the ordinary inverse additionally several new formulae of the type are provided

combinatorial algorithms for the minimum interval cost flow problem the aim of minimum the interval cost flow problem micfp is to find the least cost of the shipment of a commodity through a capacitated network in order to satisfy demands at certain nodes from available supplies at other nodes where there exists some vague in vector cost of problem interval cost is a common event in uncertainty environments where statistical data are applied moreover they almost play an essential role in fuzzy programming specially in the case of using their cuts in this paper a complete order on intervals is defined and efficient combinatorial algorithms for micfp are proposed digital simulation results show the performance of the proposed algorithms compared with real scenarios

algorithms for linear programming problems with interval objective function coefficients this paper presents three algorithms for solving linear programming problems in which some or all of the objective function coefficients are specified in terms of intervals which algorithm is applicable depends upon a the number of interval objective function coefficients b the number of nonzero objective function coefficients and c whether or not the feasible region is bounded the algorithms output all extreme points and unbounded edge directions that are multiparametrically optimal with respect to the ranges placed on the objective function coefficients the algorithms are most suitable to linear programs in which the objective function coefficients are deterministic but are likely to vary from time period to time period as for example in blending problems

on a generalized core inverse abstract the paper introduces the concept of a generalized core inverse of a matrix which extends the notion of the core inverse defined by baksalary and trenkler 1 while the original core inverse is restricted to matrices of index one the generalized core inverse exists for any square matrix several properties of the new concept are identified with the derivations based essentially on partitioned representations of matrices some of the features of the generalized core inverse coincide with those attributed to the core inverse but there are also such which characterize the core inverse only and not its generalization

a class of rough multiple objective programming and its application to solid transportation problem this paper establishes a rough multiple objective programming model for a solid transportation problem furthermore a general model for rough multiple objective programming problem is presented properties of the feasible and efficient solutions of rough multiple objective programming problems are investigated in addition compromise solutions are obtained by using the interactive fuzzy satisfying method to solve the rough multiple objective solid transportation problem the rough simulation based genetic algorithm is proposed in which the rough simulation is embedded into the genetic algorithm finally an application of the solid transportation problem at xiluodu hydropower station is provided as an illustration

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

scale free property of complex ad hoc networks with accelerated growth a complex ad hoc network model with accelerated growth is proposed in this paper in the evolving process of the model at each time step with the addition probability c1c1 a new vertex with m t m t edges is added into the model which new adding edges are connected with the old vertices according to the preferential attachment rule and with the deletion probability c2 c1 c2 c1 a random chosen vertex is deleted and its associated edges are disappeared simultaneously the accelerated growth means that at each time step the number of new adding edges is the increasing function of the time tt we consider mainly m t t m t t where 0 10 1 to analyze the degree distribution of the model using the continuous approach we prove theoretically and numerically that the model follows the scale free degree distribution and generates both the barabasi and albert ba model and the model introduced by sarshar and roychowdhury sr in addition we present the appropriate numerical range of c1c1 c2c2 and according to theoretical analyzes and numerical simulation the model takes on a certain universality to complex evolving network and has the guiding significance to the application of the peer to peer networks and complex ad hoc networks

characterizations of ep matrices and weighted ep matrices abstract a square complex matrix a is said to be ep if a and its conjugate transpose a have the same range in this paper we first collect a group of known characterizations of ep matrix and give some new characterizations of ep matrices then we define weighted ep matrix and present a wealth of characterizations for weighted ep matrix through various rank formulas for matrices and their generalized inverses

multiobjective transportation problem with interval cost source and destination parameters in this paper we focus on the solution procedure of the multiobjective transportation problem motp where the cost coefficients of the objective functions and the source and destination parameters have been expressed as interval values by the decision maker this problem has been transformed into a classical motp where to minimize the interval objective function the order relations that represent the decision maker s preference between interval profits have been defined by the right limit left limit centre and half width of an interval the constraints with interval source and destination parameters have been converted into deterministic ones finally the equivalent transformed problem has been solved by fuzzy programming technique numerical examples have been provided to illustrate the solution procedure for three possible cases of the original problem

rough multiple objective programming abstract in this paper we focused on characterizing and solving the multiple objective programming problems which have some imprecision of a vague nature in their formulation the rough set theory is only used in modeling the vague data in such problems and our contribution in data mining process is confined only in the post processing stage these new problems are called rough multiple objective programming rmop problems and classified into three classes according to the place of the roughness in the problem also new concepts and theorems are introduced on the lines of their crisp counterparts e g rough complete solution rough efficient set rough weak efficient set rough pareto front weighted sum problem etc to avoid the prolongation of this paper only the 1st class where the decision set is a rough set and all the objectives are crisp functions is investigated and discussed in details furthermore a flowchart for solving the 1st class rmop problems is presented

practical aggregation operators for gradual trust and distrust trust and distrust are two increasingly important metrics in social networks reflecting users attitudes and relationships towards each other in this paper we study the indirect derivation of these metrics values for users that do not know each other but are connected through the network in particular we study bilattice based aggregation approaches and investigate how they can be improved by using ordered weighted averaging techniques or through the incorporation of knowledge defects experiments on a real world data set from couchsurfing org demonstrate that the best operators from a theoretical perspective are not always the most suitable ones in practice and that the sophisticated aggregation methods can outperform the more obvious bilattice based approaches

nonsingularity of the difference and the sum of two idempotent matrices abstract gro s and trenkler 1 pointed out that if a difference of idempotent matrices p and q is nonsingular then so is their sum and koliha et al 2 expressed explicitly a condition which combined with the nonsingularity of p q ensures the nonsingularity of p q in the present paper these results are strengthened by showing that the nonsingularity of p q is in fact equivalent to the nonsingularity of any combination ap bq cpq where a 0 b 0 a b c and the nonsingularity of p q is equivalent to the nonsingularity of any combination ap bq cpq where a 0 b 0 a b c

characterizing solutions of rough programming problems abstract in this paper we define a new kind of mathematical programming problems this kind in which the decision set is a rough set is called a rough programming problem a rough optimal solution and a rough saddle point will be characterized some illustrative examples are presented

goal programming problems with interval coefficients and target intervals abstract in conventional goal programming the coefficients of objective functions and constraints and target values are determined as crisp values however it is not frequent that the coefficients and the target values are known precisely in such cases the coefficients and target values should be represented by intervals reflecting the imprecision this paper treats goal programming problems in which coefficients and target values are given the intervals it is shown that four formulations of the problems can be considered the properties of the four formulated problems are investigated an example is given to demonstrate the differences between the four formulations

rank equalities and inequalities for kronecker products of matrices with applications some rank equalities and inequalities are established for kronecker products of matrices as applications upper and lower bounds are presented for the dimensions of the ranges of the two well known linear transformations t 1 x x axb and t 2 x ax xb

fuzzy logic methods in recommender systems here we consider methodologies for constructing recommender systems the approaches studied here differ from collaborative filtering they are based solely on the preferences of the single individual for whom we are providing the recommendation and make no use of the preferences of other collaborators we have called these reclusive methods another important feature distinguishing these reclusive methods from collaborative methods is that they require a representation of the objects considerable use is made of fuzzy set methods for the representation and subsequent construction of justifications and recommendation rules it is pointed out these reclusive methods rather than being competitive with collaborative methods are complementary

affine projections this paper presents a constructive method for projecting a vector onto an affine subspace of a vector space it also provides formulas for projecting onto the intersection and sums of such subspaces

on disjoint range operators in a hilbert space for a bounded linear operator m in a hilbert space h various relations among the ranges r m r m r m m and the null spaces n m n m are considered from the point of view of their relations to the known classes of operators such as ep co ep weak ep gp dr or sr particular attention is paid to the range projectors of the operators m m and some further characteristics of these projectors are derived as well

fuzzy linear programming problems with fuzzy numbers abstract linear programming problems with fuzzy parameters are formulated by fuzzy functions the ambiguity considered here is not randomness but fuzziness which is associated with the lack of a sharp transition from membership to nonmembership parameters on constraint and objective functions are given by fuzzy numbers in this paper our object is the formulation of a fuzzy linear programming problem to obtain a reasonable solution under consideration of the ambiguity of parameters this fuzzy linear programming problem with fuzzy numbers can be regarded as a model of decision problems where human estimation is influential

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

lattice properties of the order for complex matrices abstract it is shown that the set c m n of complex m n matrices forms a lower semilattice under the partial ordering a b defined by a a a b aa ba where a denotes the conjugate transpose of a as a special case of a result for division rings it is further shown that over any field f for m n 2 and any proper involution of f2 2 the corresponding intersections a b all exist

on a subspace metric based on matrix rank abstract the metric between subspaces m n c n 1 defined by m n rk p m p n where rk denotes rank of a matrix argument and p m and p n are the orthogonal projectors onto the subspaces m and n respectively is investigated such a metric takes integer values only and is not induced by any vector norm by exploiting partitioned representations of the projectors several features of the metric m n are identified it turns out that the metric enjoys several properties possessed also by other measures used to characterize subspaces such as distance also called gap frobenius distance direct distance angle or minimal angle

column space equalities for orthogonal projectors in their paper y tian g p h styan rank equalities for idempotent and involutory matrices linear algebra appl 335 2001 101 117 tian and styan established several rank equalities involving a pair of idempotent matrices p and q subsequently these results are reinvestigated from the point of view of the following question provided that idempotent p q are hermitian which relationships given in the aforementioned paper remain valid when ranks are replaced with column spaces simultaneously some related results are established which shed additional light on the links between subspaces attributed to various functions of a pair of orthogonal projectors

a profit maximizing solid transportation model under a rough interval approach this study attempts to establish an innovative solid transportation problem that intends to maximize profit under the rough interval approximation methodology two transportation problems were constructed in this regard with interval coefficients corresponding to the upper approximations and the lower approximations of the rough intervals under study furthermore from the contingent solid transportation problems four different classical solid transportation problems were derived which were subsequently solved on the lingo iteration platform the concepts of completely satisfactory solution and rather satisfactory solution surely optimal range possibly optimal range and rough optimal range have been discussed with a perspective to its relevance to real world practical problems the rough chance constrained programming and the expected value operator for rough interval have been applied to solve the problem under study the distinct advantages of the proposed method over those existing have been outlined numerical examples have also been provided to illustrate the solution procedure and the methodologies adopted

fuzzy programming approach to multicriteria decision making transportation problem abstract the linear multiobjective transportation problem is a special type of vector minimum problem in which constraints are all equality type and the objectives are conflicting in nature all existing methods generate a set of nondominated solutions or construct a single compromise solution this paper presents an application of fuzzy linear programming to the linear multiobjective transportation problem it gives efficient solutions as well as an optimal compromise solution for this method an efficient fortran program has been developed based on the fuzzy linear programming algorithm which is an extended version of the simplex algorithm this method is compared with two existing interactive algorithms using two numerical examples

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

revisitation of the product of two orthogonal projectors abstract several results involving a product of two orthogonal projectors i e hermitian idempotent matrices are established by exploring a representation of the product as a partitioned matrix these results concern for instance rank trace range null space generalized inverses and spectral properties of the product and its various functions particular attention is paid to the conditions equivalent to the requirement that the product of two orthogonal projectors is an orthogonal projector itself and these characterizations refer to such known classes of matrices as hermitian involutory normal star dagger unitary as well as partial isometries and semi orthogonal projectors moreover some results dealing with the notions of parallel sum and spectral norm are obtained the variety of problems considered shows that the approach utilized in the paper provides a powerful tool of wide applicability

rank equalities for idempotent and involutary matrices abstract we establish several rank equalities for idempotent and involutary matrices in particular we obtain new formulas for the rank of the difference the sum the product and the commutator of idempotent or involutary matrices extensions to scalar potent matrices are also included our matrices are complex and are not necessarily hermitian

downlink training sequence design for fdd multiuser massive mimo systems we consider the problem of downlink training sequence design for frequency division duplex multiuser massive multiple input multiple output systems in the general case where users have distinct spatial correlations the training sequences leverage spatial correlations and are designed to minimize the channel estimation weighted sum mean square error mse under the assumption that users employ minimum mse estimators noting that the weighted sum mse function is invariant to unitary rotations of its argument a solution is obtained using a steepest descent method on the grassmannian manifold we extend the proposed design to scenarios with temporal correlations combined with kalman filters at the users and design sequences exploiting the multiuser spatio temporal channel structure finally we consider scenarios where only a limited number of bits b are available at the base station bs to inform the users of each chosen sequence e g sequences are chosen from a set of 2 b vectors known to the bs and users and develop a subspace version of matching pursuit techniques to choose the desired sequences simulation results using realistic channel models show that the proposed solutions improve user fairness with a proper choice of weights lead to accurate channel estimates with training durations that can be much smaller than the number of bs antennas and show substantial gains over randomly chosen sequences for even small values of b

properties of the combinations of commutative idempotents this paper deals with idempotent operators i e p2 p and n potent operators i e pn p we characterize the invertibility the group invertibility and idempotency of the linear combinations of idempotent operators or n potent operators under certain commutativity property imposed on them

using rank formulas to characterize equalities for moore penrose inverses of matrix products a group of rank formulas related to the moore penrose inverses ab and b a are established various interesting consequences of these rank formulas are obtained the extensions of these results to the weighted moore penrose inverses ab m n and b p na m p are considered

eigenvalues of functions of orthogonal projectors by representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices several characterizations concerning eigenvalues of various functions of the pair are obtained these results substantially extend the ones already available in the literature additionally some related results dealing with the functions of a pair of orthogonal projectors are provided with the emphasis laying on the problem of invertibility

on the projectors ff and f f abstract a particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors namely ff and f f determined by a complex square matrix f and its moore penrose inverse f various functions of the projectors are considered from the point of view of their nonsingularity idempotency nilpotency or their relation to the known classes of matrices such as ep bi ep gp dr or sr this part of the paper was inspired by benitez and rakocevic j benitez v rakocevic matrices a such that aa a a are nonsingular appl math comput 217 2010 3493 3503 further characteristics of ff and f f with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues are derived as well besides establishing selected exemplary results dealing with ff and f f the paper develops a general approach whose applicability extends far beyond the characteristics provided therein

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

expansion formulas for the inertias of hermitian matrix polynomials and matrix pencils of orthogonal projectors abstract this paper gives a group of expansion formulas for the inertias of hermitian matrix polynomials a a 2 i a 2 and a a 3 through some congruence transformations for block matrices where a is a hermitian matrix then the paper derives various expansion formulas for the ranks and inertias of some matrix pencils generated from two or three orthogonal projectors and hermitian unitary matrices as applications the paper establishes necessary and sufficient conditions for many matrix equalities to hold as well as many inequalities in the lowner partial ordering to hold

on a pair of vector spaces the paper provides an extensive investigation of a pair of vectors spaces with a particular attention paid to perpendicular subspaces several new properties of the pair are identified some of which enabled to derive known facts about subspaces by means of simpler proofs the approach utilized exploits partitioned representations of orthogonal projectors and is based on the fact that there is one to one correspondence between an orthogonal projector and the subspace onto which it projects

distribution planning decisions using interactive fuzzy multi objective linear programming this work develops an interactive fuzzy multi objective linear programming i fmolp method for solving the fuzzy multi objective transportation problems with piecewise linear membership function the proposed i fmolp method aims to simultaneously minimize the total distribution costs and the total delivery time with reference to fuzzy available supply and total budget at each source and fuzzy forecast demand and maximum warehouse space at each destination additionally the proposed method provides a systematic framework that facilitates the fuzzy decision making process enabling a decision maker dm to interactively modify the fuzzy data and related parameters until a set of satisfactory solutions is obtained an industrial case is presented to demonstrate the feasibility of applying the proposed method to real transportation problems consequently the proposed method yields an efficient solution and overall degree of dm satisfaction with the determined objective values especially several significant management implications regarding the practical application of the proposed i fmolp method are presented

the index of a pair of projections we discuss pairs of self adjoint projections with p q j 2n 1 the trace ideal and prove that for m n tr p q 2m 1 tr p q 2n 1 dim ker q ran p dim ker p ran q is an integer we also prove that there exists a unitary v interchanging p and q if and only if this integer is 0

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

interactive solutions for the linear multiobjective transportation problem abstract mathematical programming problems that exhibit the mathematical structure of a transportation problem often arise in settings with multiple conflicting objectives existing procedures for analyzing these problems fall into two general categories these methods either generate all nondominated solutions or they construct a single compromise solution this paper presents two interactive algorithms which take advantage of the special form of the multiple objective transportation problem two examples are included to illustrate these algorithms and to demonstrate their viability

on oblique projectors abstract oblique projectors are useful in many contexts especially in the instrumental variable estimation in regression models in econometrics where the disturbance term tends to be correlated with predictor variables this paper addresses two important questions regarding oblique projectors 1 products of two oblique projectors and 2 decompositions of oblique projectors in 1 we investigate what determines the onto and the along spaces of an oblique projector defined as a product of two oblique projectors under a variety of conditions in 2 we examine various decompositions of oblique projectors when both predictor variables and instrumental variables consist of two distinct sets of variables these decompositions are analogous to those of orthogonal projectors

the difference and sum of projectors abstract the aim of this paper is to present new results on the invertibility of the sum of projectors and new relations between the nonsingularity of the difference and the sum of projectors we also give simple proofs without referring to rank theory of some results on the nonsingularity of the difference of projectors obtained by gros and trenkler siam j matrix anal appl 21 1999 390

scale free and stable structures in complex ad hoc networks unlike the well studied models of growing networks where the dominant dynamics consist of insertions of new nodes and connections and rewiring of existing links we study ad hoc networks where one also has to contend with rapid and random deletions of existing nodes and hence the associated links we first show that dynamics based only on the well known preferential attachments of new nodes do not lead to a sufficiently heavy tailed degree distribution in ad hoc networks in particular the magnitude of the power law exponent increases rapidly from 3 with the deletion rate becoming ensuremath infty in the limit of equal insertion and deletion rates we then introduce a local and universal compensatory rewiring dynamic and show that even in the limit of equal insertion and deletion rates true scale free structures emerge where the degree distributions obey a power law with a tunable exponent which can be made arbitrarily close to 2 the dynamics reported in this paper can be used to craft protocols for designing highly dynamic peer to peer networks and also to account for the power law exponents observed in existing popular services

degree correlations in growing networks with deletion of nodes in this paper we study the degree distribution and the two node degree correlations in growing networks generated via a general linear preferential attachment of new nodes together with a uniformly random deletion of nodes by using a continuum approach we show that under some suitable combinations of parameters deletion rate and node attractiveness the degree distribution not only loses its scale free character but can even be supported on a small range of degrees moreover we obtain new results on two vertex degree correlations showing that for degree distributions with finite variance such correlations can change under a nonselective removal of nodes

idempotency of linear combinations of two idempotent matrices abstract a complete solution is established to the problem of characterizing all situations where a linear combination of two different idempotent matrices p 1 and p 2 is also an idempotent matrix including naturally three such situations known in the literature viz if the combination is either the sum p 1 p 2 or one of the differences p 1 p 2 p 2 p 1 and appropriate additional conditions are fulfilled the solution asserts that in the particular case where p 1 and p 2 are complex matrices such that p 1 p 2 is hermitian these three situations exhaust the list of all possibilities and that this list extends when the above assumption on p 1 and p 2 is violated a statistical interpretation of the idempotency problem considered in this note is also pointed out

twosetsofnewcharacterizationsfornormaland ep matrices two sets of new characterizations for normal matrices and ep matrices are presented which are derived through ranks and generalized inverses of matrices

rough intervals enhancing intervals for qualitative modeling of technical systems the success of model based industrial applications generally depends on how exactly models reproduce the behavior of the real system that they represent however the complexity of industrial systems makes the construction of accurate models difficult an alternative is the qualitative description of process states for example by means of the discretization of continuous variable spaces in intervals in order to reach the required precision in the modeling of complex dynamic systems interval based representations usually produce qualitative models which are sometimes too large for practical use the approach introduced in this paper incorporates vague and uncertain information based on principles of the rough set theory as a way of enhancing the information contents in interval based qualitative models the resulting models are more compact and precise than ordinary qualitative models

a concept of the optimal solution of the transportation problem with fuzzy cost coefficients abstract in the paper a definition of the optimal solution of the transportation problem with fuzzy cost coefficients as well as an algorithm determining this solution are proposed

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved

multiobjective programming in optimization of the interval objective function abstract in this paper the mathematical programming problem whose objective function has interval coefficients is investigated to maximize the interval objective function the order relations which represent the decision maker s preference between interval profits are defined by the right limit the left limit the center and the width of an interval the maximization problem with the interval objective function is converted into a multiobjective problem using the order relations the minimization problem is also converted into a multiobjective problem using the order relations which represent the decision maker s preference between interval costs