dimension of the orbit of marked subspaces given a nilpotent endomorphism we consider the manifold of invariant subspaces having a fixed segre characteristic in linear algebra appl 332 334 2001 569 the implicit form of a miniversal deformation of an invariant subspace with respect to the usual equivalence relation between subspaces is obtained here we obtain the explicit form of this deformation when the invariant subspace is marked and we use it to calculate the dimension of the orbit and in particular to characterize the stable marked subspaces those with open orbit moreover we study the rank of the endomorphisms in the quotient space by the subspaces in the miniversal deformation of the giving subspace

the perron generalized eigenspace and the spectral cone of a cone preserving map a unified treatment is offered to reprove known results on the following four highlights of the combinatorial spectral theory of nonnegative matrices or to extend or partly extend the results to the setting of a linear map preserving a polyhedral proper or proper cone the preferred basis theorem equivalent conditions for equality of the graph theoretic level characteristic and the spectral height characteristic the majorization relation between the two characteristics and the relation between the combinatorial properties of a nonnegative matrix and the positivity of the individual entries in its principal components this is achieved by employing the new concept of spectral cone of a cone preserving map and combining the cone theoretic methods developed in our previous papers on the geometric spectral theory of cone preserving maps with the algebraic analytic method introduced by hartwig neumann and rose and further exploited by neumann and schneider for nonnegative matrices

some nondifferentiable multiobjective programming under generalized d v type i univexity in this paper we introduce new classes of functions called d v type i univex by extending the definition of d v type i functions and consider a multiobjective optimization problem involving generalized d v type i univex functions a number of karush kuhn tucker type sufficient optimality conditions are obtained for a feasible solution to be a weak pareto efficient solution the mond weir type duality results are also presented the results obtained in this paper generalize and extend the previously known result in this area

matrix polynomials factorization via bisolvents abstract we reconsider the classification of all the factorizations of a matrix polynomial p as p q r with q a matrix polynomial and r t s a regular matrix pencil it is shown that the entire classification problem can be reduced to the simpler classification of factors r with commuting coefficients s t it is then shown that for these commuting factors s and t must satisfy a certain algebraic equation which we call the bisolvent equation this extends the generalized bezout theorem which associates monic factors i s with solutions s of a solvent equation in case p is regular the classification of commuting pairs s t of this type up to left equivalence is described in terms of enlarged standard pairs following a well known approach under a non derogatory generic condition on the roots of p the number of such pairs associated with degree minimal factorizations is finite and admits explicit description in terms of jordan chains

predecessor property full combinatorial column rank and the height characteristics of an m matrix abstract for a loopless acyclic transitive directed graph we give a new result about the relation between the t predecessor property of subsets of its levels and the property of having full combinatorial column rank for certain blocks in its adjacency matrix as a corollary we give new lower bounds for the i th height characteristic number of a singular m matrix an open question about determining the height characteristics of singular m matrices with a prescribed singular graph is also posed

characteristic invariant subspaces generated by a single vector abstract if f is an endomorphism of a finite dimensional vector space v over a field k then an invariant subspace x v is called hyperinvariant respectively characteristic if x is invariant under all endomorphisms respectively automorphisms that commute with f the characteristic hull of a subset w of v is defined to be the smallest characteristic subspace in v that contains w it is known that characteristic subspaces that are not hyperinvariant can only exist when k 2 in this paper we study subspaces x which are the characteristic hull of a single element in the case where k 2 we derive a necessary and sufficient condition such that x is hyperinvariant

stable subspaces of matrix pairs abstract let a b c n n c n m and let m be an a b invariant subspace in this paper the following results are presented i if dim m rank b n then m is stable ii if m im b 0 under some additional hypotheses necessary and sufficient conditions for the stability of m are given iii if m contains the controllability subspace of the pair a b sufficient conditions are given in terms of the brunovsky r numbers and the weyr characteristic of the eigenvalues of a quotient endomorphism associated to the noncontrollable part of the subspace m

inflations of self affine tilings are integral algebraic perron we prove that any expanding linear map phi mathbb r mathtt d rightarrow mathbb r mathtt d that is the inflation map in an inflation substitution process generating a self affine tiling of mathbb r mathtt d is integral algebraic and perron this means that phi is linearly conjugate to a restriction of an integer matrix to a subspace mathcal e satisfying a maximal growth condition that generalizes the characterization of perron numbers as numbers that are larger than the moduli of their algebraic conjugates the case of diagonalizable phi has been previously resolved by richard kenyon and boris solomyak and it is rooted in thurston s idea of lifting the tiling from the physical space mathbb r mathtt d to a higher dimensional mathematical space where the tiles their control points sit on a lattice the main novelty of our approach is in lifting the inflation substitution process to the mathematical space and constructing a certain vector valued cocycle defined over the translation induced mathbb r mathtt d action on the tiling space the subspace mathcal e is obtained then by ergodic averaging of the cocycle more broadly we assemble a powerful framework for studying self affine tiling spaces

invariant subspaces of nilpotent linear operators i let k be a field we consider triples v u t where v is a finite dimensional k space u a subspace of v and t v v a linear operator with t n 0 for some n and such that t u u thus t is a nilpotent operator on v and u is an invariant subspace with respect to t we will discuss the question whether it is possible to classify these triples these triples v u t are the objects of a category with the krullremak schmidt property thus it will be sufficient to deal with indecomposable triples obviously the classification problem depends on n and it will turn out that the decisive case is n 6 for n 6 we deal with what is called wild representation type so no complete classification can be expected for n 6 we will exhibit a complete description of all the indecomposable triples let k be a field and v a finite dimensional k vector space we consider a linear operator t v v which is nilpotent say satisfying t n 0 for some fixed natural number n and we are interested in the subspaces u of v which are t invariant this means that t u u to be more precise we are going to consider the corresponding triples v u t and ask for a classification of all the isomorphism classes in order to do so we consider these triples v u t as the objects of a category s n or sk n in case a reference to the base field

stability of controlled invariant subspaces abstract let a b c n n c n m and m be an a b invariant subspace let c a b be the controllability subspace of the pair a b if m c a b m im b under additional hypotheses necessary and sufficient conditions for the stability of the subspace m are given

a finite thue system with decidable word problem and without equivalent finite canonical system abstract we present a single axiom thue system with a decidable word problem for which there does not exist any finite equivalent canonical system however an equivalent finite canonical system for this thue system can be obtained if new symbols are introduced in the presentation this result settles an open question by jantzen 1982 who asked whether every thue system with a decidable word problem has an equivalent finite canonical system we also discuss relationships between thue systems and term rewriting systems

geometric characterization and classification of marked subspaces abstract given an endomorphism of a finite dimensional c vector space we obtain geometric conditions in order to characterize and to classify the marked invariant subspaces w e that is to say the invariant subspaces having some jordan basis which can be extended to a jordan basis of e both conditions are expressed in terms of the double family of subspaces ker f h im f d the starting point is a new criterion for extensibility of jordan bases

the lattice of characteristic subspaces of an endomorphism with jordan chevalley decomposition abstract given an endomorphism a over a finite dimensional vector space having jordan chevalley decomposition the lattices of invariant and hyperinvariant subspaces of a can be obtained from the nilpotent part of this decomposition we extend this result for lattices of characteristic subspaces we also obtain a generalization of shoda s theorem about the characterization of the existence of characteristic non hyperinvariant subspaces

a finiteness condition for rewriting systems abstract we present a purely combinatorial approach to the question of whether or not a finitely presented monoid has a finite canonical presentation our approach is based on the notion of finite derivation type which is a combinatorial condition satisfied by certain rewriting systems our main result states that if a monoid m has a finite canonical presentation then all finite presentations of m have finite derivation type by proving that a certain monoid s1 does not have finite derivation type we show in addition that the homological finiteness condition fp is not sufficient to guarantee that a finitely presented monoid with a decidable word problem has a finite canonical presentation

on finite complete rewriting systems finite derivation type and automaticity for homogeneous monoids this paper investigates the class of finitely presented monoids defined by homogeneous length preserving relations from a computational perspective the properties of admitting a finite complete rewriting system having finite derivation type being automatic and being biautomatic are investigated for this class of monoids the first main result shows that for any consistent combination of these properties and their negations there is a homogeneous monoid with exactly this combination of properties we then introduce the new concept of abstract rees commensurability an analogue of the notion of abstract commensurability for groups in order to extend this result to show that the same statement holds even if one restricts attention to the class of n ary homogeneous monoids where every side of every relation has fixed length n we then introduce a new encoding technique that allows us to extend the result partially to the class of n ary multihomogenous monoids

brief paper a class of marked invariant subspaces with an application to algebraic riccati equations invariant subspaces of a matrix a are considered which are obtained by truncation of a jordan basis of a generalized eigenspace of a we characterize those subspaces which are independent of the choice of the jordan basis an application to hamilton matrices and algebraic riccati equations is given

characteristic subspaces and hyperinvariant frames abstract let f be an endomorphism of a finite dimensional vector space v over a field k an f invariant subspace is called hyperinvariant respectively characteristic if it is invariant under all endomorphisms respectively automorphisms that commute with f we assume k 2 since all characteristic subspaces are hyperinvariant if k 2 the hyperinvariant hull w h of a subspace w of v is defined to be the smallest hyperinvariant subspace of v that contains w the hyperinvariant kernel w h of w is the largest hyperinvariant subspace of v that is contained in w and the pair w h w h is the hyperinvariant frame of w in this paper we study hyperinvariant frames of characteristic non hyperinvariant subspaces w we show that all invariant subspaces in the interval w h w h are characteristic we use this result for the construction of characteristic non hyperinvariant subspaces

eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations abstract udy the perturbation theory of structured matrices under structured rank one perturbations and then focus on several classes of complex matrices generic jordan structures of perturbed matrices are identified it is shown that the perturbation behavior of the jordan structures in the case of singular j hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in 18 28 30 31 thus a generic structured perturbation would not be generic if considered as an unstructured perturbation in other settings of structured matrices the generic perturbation behavior of the jordan structures within the confines imposed by the structure follows the pattern of that of unstructured perturbations

some nondifferentiable multiobjective programming problems in this paper it is shown that a relaxation defining the class of generalized d v type i functions leads to a new class of multi objective problems which preserves the sufficient optimality and duality results in the scalar non differentiable case and avoids the major difficulty of verifying that the inequality holds for the same kernel function the results obtained in this paper generalize and extend the previously known results in this area

on a b t invariant subspaces having extendible brunovsky bases we consider a by invariant subspaces having a brunovsky basis which can be extended to a brunovsky basis of the whole space we obtain a geometrical characterization of this class of a by invariant subspaces and a complete family of numerical invariants to classify them 0 elsevier science inc 1997

lipschitz stability of controlled invariant subspaces abstract let a b c n n c n m and m be an a b invariant subspace in this paper the following results are presented i if m im b 0 necessary and sufficient conditions for the lipschitz stability of m are given ii if m contains the controllability subspace of the pair a b sufficient conditions for the lipschitz stability of the subspace m are given

homotopy bases and finite derivation type for subgroups of monoids abstract given a monoid defined by a presentation and a homotopy base for the derivation graph associated to the presentation and given an arbitrary subgroup of the monoid we give a homotopy base and presentation for the subgroup if the monoid has finite derivation type fdt and if under the action of the monoid on its subsets by right multiplication the strong orbit of the subgroup is finite then we obtain a finite homotopy base for the subgroup and hence the subgroup has fdt as an application we prove that a regular monoid with finitely many left and right ideals has fdt if and only if all of its maximal subgroups have fdt we use this to show that a finitely presented regular monoid with finitely many left and right ideals satisfies the homological finiteness condition fp 3 if all of its maximal subgroups satisfy the condition fp 3

the characteristic subspace lattice of a linear transformation abstract given a square matrix a m n f the lattices of the hyperinvariant hinv a and characteristic chinv a subspaces coincide whenever f g f 2 if the characteristic polynomial of a splits over f a can be considered nilpotent in this paper we investigate the properties of the lattice chinv j when f g f 2 for a nilpotent matrix j in particular we prove it to be self dual

on the change of the jordan form under the transition from the adjacency matrix of a vertex transitive digraph to its principal submatrix of co order one abstract let j n 1 n k be the set of matrices a such that is an eigenvalue of a and n 1 n k are the sizes of the jordan blocks associated with for a given index v of a denote by a v the principal submatrix of co order one obtained from a by deleting the v th row and column in the present paper all possible changes of the part of the jordan form corresponding to under the transition from a to a v are determined for matrices a j n 1 n k such that for the eigenvalue of both a and a there exists a jordan chain of the largest length n k whose eigenvector has nonzero v th entry in particular it is shown that for almost every matrix a j n 1 n k n 1 n k 1 are the sizes of jordan blocks for considered as an eigenvalue of a v moreover it is also proved that if a is the adjacency matrix of a vertex transitive digraph and k 2 then the change n 1 n k n 1 n k 2 2 n k 1 1 holds for the eigenvalue under the transition from a to a v in the case of k 1 is a simple eigenvalue of a and does not belong to the spectrum of a v

singular graph and extension of jordan chains of an m matrix abstract let s be a singular graph we study conditions on the vertices of s to extend jordan chains for the generalized nullspace of each minus m matrix a with singular graph s the conditions are the interlevel the strong interlevel and the strong predecessor properties

optimality and duality for multiple objective optimization under generalized type i univexity abstract in this paper we extend the classes of generalized type i vector valued functions introduced by aghezzaf and hachimi in j global optim 18 2000 91 101 to generalized univex type i vector valued functions and consider a multiple objective optimization problem involving generalized type i univex functions a number of kuhn tucker type sufficient optimality conditions are obtained for a feasible solution to be an efficient solution the mond weir and general mond weir type duality results are also presented

similarity of non everywhere defined linear maps abstract the equivalence between linear maps defined only on a subspace or by duality defined modulo a subspace is studied and applied to block similarity of rectangular matrices in that way we find the complete families of invariants obtained by brunovsky and zaballa and the results of gohberg and others for this particular case also we describe explicit methods for obtaining the reduced matrix and the corresponding bases

miniversal deformations of marked matrices abstract given the set of square matrices m m n m c that keep the subspace w c n x 0 c n m invariant we obtain the implicit form of a miniversal deformation of a matrix a m and we compute it explicitely when this matrix is marked this is if there is a permutation matrix p m n m c such that p 1 ap is a jordan matrix we derive some applications to tackle the classical carlson problem

differentiable structure of the set of controllable a b t invariant subspaces abstract given a b t hom c n c n m observable we prove that the set of a b t invariant subspaces having a fixed brunovsky kronecker structure is a connected manifold and we compute its dimension also we include some applications of these results

finite derivation type for rees matrix semigroups this paper introduces the topological finiteness condition finite derivation type fdt on the class of semigroups this notion is naturally extended from the monoid case with this new concept we are able to prove that if a rees matrix semigroup m s i j p has fdt then the semigroup s also has fdt given a monoid s and a finitely presented rees matrix semigroup m s i j p we prove that if the ideal of s generated by the entries of p has fdt then so does m s i j p in particular we show that for a finitely presented completely simple semigroup m the rees matrix semigroup m m s i j p has fdt if and only if the group s has fdt

a numerical algorithm for computing a basis for the root subspace at a nonderogatory eigenvalue of a multiparameter system abstract a numerical algorithm for the calculation of a basis for the root subspace at a nonderogatory eigenvalue of a finite dimensional multiparameter system is given it is based on the algorithm proposed by kosir 1 the perturbation theory for the algorithm is studied and a new stopping criterion is proposed at the end two numerical examples are given

finite complete rewriting systems and finite derivation type for small extensions of monoids abstract letsbe a monoid and lettbe a submonoid of finite index ins the main results in this article state thatscan be presented by a finite complete rewriting system iftcan andshas finite derivation type ifthas

finite dimensional multiparameter spectral theory the nonderogatory case abstract a completeness theorem for nonderogatory eigenvalues of multiparameter systems is proved in a finite dimensional setting also a basis for the second root subspace of a simple eigenvalue is given weakly elliptic multiparameter systems are introduced it is shown that simple real eigenvalues of such systems are nonderogatory

necessary and sufficient conditions in constrained optimization additional conditions are attached to the kuhn tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization under appropriate constraint qualifications necessary and sufficient conditions are also given for optimality of the dual problem duality and converse duality are treated accordingly

description of characteristic non hyperinvariant subspaces over the field gf 2 given a square matrix a an a invariant subspace is called hyperinvariant respectively characteristic if and only if it is also invariant for all matrices t respectively nonsingular matrices t that commute with a shoda s theorem gives a necessary and sufficient condition for the existence of characteristic non hyperinvariant subspaces for a nilpotent matrix in gf 2 here we present an explicit construction for all subspaces of this type

on stability of invariant subspaces of commuting matrices we study the stability of joint invariant subspaces of a finite set of commuting matrices we generalize some of the results of gohberg lancaster and rodman for the single matrix case for sets of two or more commuting matrices we exhibit some phenomena different from the single matrix case we show that each root subspace is a stable invariant subspace that each invariant subspace of a root subspace of a nonderogatory eigenvalue is stable and that even in the derogatory case the eigenspace is stable if it is one dimensional we prove that a pair of commuting matrices has only finitely many stable invariant subspaces at the end we discuss the stability of invariant subspaces of an algebraic multiparameter eigenvalue problem

infine functions and nonsmooth multiobjective optimization problems in this paper we consider notion of infine functions and we establish necessary and sufficient optimality conditions for a feasible solution of a multiobjective optimization problem involving mixed constraints equality and inequality to be an efficient or properly efficient solution we also obtain duality theorems for wolf type and mond weir type duals under the generalized invexity assumptions

finite derivation type for semi direct products of monoids abstract squier introduced the notion finite derivation type which is a combinatorial condition satisfied by certain rewriting systems the main result in this paper states that the class of monoids of finite derivation type is closed under semi direct products

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

nondifferentiable multiobjective programming under generalized di invexity abstract in this paper we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints we introduce new concepts of d i invexity and generalized d i invexity in which each component of the objective and constraint functions is directionally differentiable in its own direction d i new fritz john type necessary and karush kuhn tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be weakly efficient efficient or properly efficient moreover we prove weak strong converse and strict duality results for a mond weir type dual under various types of generalized d i invexity assumptions

stratification and bundle structure of the set of conditioned invariant subspaces in the general case abstract we extend some known results on the smooth stratification of the set of conditioned invariant subspaces to a general pair c a c p n n without any assumption on the observability more precisely we prove that the set of c a conditioned invariant subspaces having a fixed brunovsky kronecker structure is a submanifold of the corresponding grassmann manifold with a fiber bundle structure relating the observable and nonobservable part and we then compute its dimension we also prove that the set of all c a conditioned invariant subspaces having a fixed dimension is connected provided that the nonobservable part of c a has at most one eigenvalue this condition is in general necessary

feedback invariants of restrictions a polynomial approach the possible feedback invariants of a linear time invariant dynamical system represented by a pair of matrices a b are studied when the only information about the system is that it has an a b invariant subspace s whose restriction is an autonomous system and the quotient is controllable this problem is shown to be related to the problem of designing a control matrix b 1 for a series connected system to have prescribed feedback invariants

mixed type duality for nondifferentiable multiobjective fractional programming under generalized d ρ φ θ type 1 univex function in this paper we introduced a new generalized class of d r h q type 1 univex function with an example we introduced the mixed duality problem mfxd of the nondifferentiable multiobjective fractional programming problem mfp where the objective and constraint functions involved are directionally differentiable also we established the duality results under generalized class of d r h q type 1 univex function

extensions of jordan bases for invariant subspaces of a matrix a characterization is obtained for the matrices a with the property that every some jordan basis of every a invariant subspace can be extended to a jordan basis of a these results are based on a criterion for a jordan basis of an invariant subspace to be extendable to a jordan basis of the whole space the criterion involves two concepts the constancy property and the depth property

constructing invariant subspaces as kernels of commuting matrices abstract given an n n matrix a over c and an invariant subspace n a straightforward formula constructs an n n matrix n that commutes with a and has n ker n for q a matrix putting a into jordan canonical form j q 1 a q we get n q 1 m where m ker m is an invariant subspace for j with m commuting with j in the formula m p z t 1 p t the matrices z and t are m m and p is an n m row selection matrix if n is a marked subspace m n and z is an n n block diagonal matrix and if n is not a marked subspace then m n and z is an m m near diagonal block matrix strikingly each block of z is a monomial of a finite dimensional backward shift each possible form of z is easily arranged in a lattice structure isomorphic to and thereby displaying the complete invariant subspace lattice l a for a

nondifferentiable multiobjective programming under generalized d univexity abstract in this paper we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints we introduce four new classes of generalized convex functions by combining the concepts of weak strictly pseudoinvex strong pseudoinvex weak quasi invex weak pseudoinvex and strong quasi invex functions in aghezzaf and hachimi numer funct anal optim 22 2001 775 d invex functions in antczak europ j oper res 137 2002 28 and univex functions in bector et al univex functions and univex nonlinear programming proc admin sci assoc canada 1992 p 115 by utilizing the new concepts we derive a karush kuhn tucker sufficient optimality condition and establish mond weir type and general mond weir type duality results for the nondifferentiable multiobjective programming problem

computing the cardinality of the lattice of characteristic subspaces abstract we obtain the cardinality of the lattice of characteristic subspaces of a nilpotent jordan matrix when the underlying field is g f 2 the only field where the lattices of characteristic and hyperinvariant subspaces can be different if the characteristic polynomial of the matrix splits in the field the general case can be reduced to the nilpotent jordan case results are complex and highly combinatorial and include the design of an algorithm

matricial realizations of the solutions of the carlson problem abstract the klein theorem asserts the existence of at least one solution of the carlson problem when a related littlewood richardson sequence exists then we present an explicit construction of some matricial solutions

hermite indices and the action of the feedback group the controllability indices are a complete system of invariants for the feedback equivalence relation of controllable matrix pairs the hermite indices are invariant for the similarity of matrix pairs but they are not invariant by changing bases on the input space and performing state feedback the aim of this work is to partially characterize the hermite indices of a controllable matrix pair when transformations of the feedback group are performed on it

versal deformations of invariant subspaces we describe a miniversal deformation of invariant subspaces with regard to a fixed endomorphism by means of a technique which can be applied to obtain explicitly miniversal deformations in a general orbit space in addition we present an application to the problem of classifying invariant subspaces

simultaneous versal deformations of endomorphisms and invariant subspaces abstract we study the set m of pairs f v defined by an endomorphism f of f n and a d dimensional f invariant subspace v it is shown that this set is a smooth manifold that defines a vector bundle on the grassmann manifold we apply this study to derive conditions for the lipschitz stability of invariant subspaces and determine versal deformations of the elements of m with respect to a natural equivalence relation introduced on it

multiobjective programming under d invexity abstract in this paper a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable by considering the concept of pareto optimal solution and substituting d invexity for convexity the fritz john type and karush kuhn tucker type necessary optimality conditions and duality in the sense of mond weir and wolfe for nondifferentiable multiobjective programming are given

peak characteristic and nonnegative signature abstract we introduce the concepts of peak characteristic of an m matrix a and of peak signature and nonnegative signature of subsets of the generalized nullspace e a of a and we discuss the relations between them we also study completions of sets to height bases and to level bases our results are then applied to the study of the nonnegativity of height bases for e a in particular we show that a sequence can serve as the nonnegative signature of some height basis for e a if and only if it is less than or equal to the peak characteristic of a

optimality criteria in nonlinear programming involving nonconvex functions abstract in this paper a new class of functions called convex quasiconvex and pseudoconvex are defined and a number of sufficient optimality criteria are obtained for nonlinear programming problems involving these functions

picturing the lattice of invariant subspaces of a nilpotent complex matrix abstract let q be a nilpotent transformation acting on a finite dimensional complex vector space a method is given by which a diagrammatic representation of lat q the lattice of invariant subspaces of q can be obtained basically the method consists of adding to the hasse diagram of hyperlat q the finite lattice of hyperinvariant subspaces of q in a specified way some applications are given

the eigenstructure of an arbitrary polynomial matrix computational aspects we give a new numerical method to compute the eigenstructure i e the zero structure the polar structure and the left and right null space structure of a polynomial matrix p these structural elements are of fundamental importance in several systems and control problems involving polynomial matrices the approach is more general than previous numerical methods because it can be applied to an arbitrary m n polynomial matrix p with normal rank r smaller than m and or n the algorithm is then shown to compute the structure of the left and right null spaces of p as well the speed and accuracy of this new approach are also discussed

characteristic and hyperinvariant subspaces over the field gf 2 let f be an endomorphism of a vector space v over a field k an f invariant subspace x v is called hyperinvariant respectively characteristic if x is invariant under all endomorphisms respectively automorphisms that commute with f if k 2 then all characteristic subspaces are hyperinvariant if k 2 then there are endomorphisms f with invariant subspaces that are characteristic but not hyperinvariant in this paper we give a new proof of a theorem of shoda which provides a necessary and sufficient condition for the existence of characteristic non hyperinvariant subspaces

d ρ η θ invexity in multiobjective optimization abstract in this paper a generalization of convexity is considered in the case of multiobjective optimization problems where the functions involved are non differentiable under d invexity assumptions on the functions involved weak strong and converse duality results are proved to relate weak pareto efficient solutions of the multiobjective programming problems pvp dvp and mwd we have also established the karush kuhn tucker sufficient optimality condition

pre invex functions in multiple objective optimization and called such functions convexlike if s is a convex set and if f is a convex function then clearly f is convexlike elster and nehse obtained a saddlepoint optimality condition for convexlike mathematical programs hayashi and komiya 9 also considered convexlike functions and developed a gordan type theorem of the alternative involving convexlike functions and in addition considered lagrangian duality for convexlike programs hanson 7 considered differentiable functions f s r for which there exists an n dimensional vector function q q u such that for all x u e s

structural stability of c a marked subspaces abstract given an observable pair of matrices c a we consider the manifold of c a invariant subspaces having a fixed brunovsky kronecker structure using arnold techniques we obtain the explicit form of a miniversal deformation of a marked c a invariant subspace with respect to the usual equivalence relation as an application we obtain the dimension of the orbit and we characterize the structurally stable subspaces those with open orbit

perturbed marked reduced forms of invariant subspaces abstract the classification of invariant subspaces is an open problem related to other important ones like the carlson problem here we obtain a reduced form of these invariant subspaces as a new tool to tackle these problems in particular it allows us to prove quite easily partial results already known the key point is assigning to each invariant subspace a marked one its marked type in order to partition the set of invariant subspaces in a finite number of subsets the marked classes each one containing only one marked subspace next we parametrize minimally each marked class by means of the so called pm reduced families so that representatives of an invariant subspace its pm reduced forms appear in just one of these families

d invexity and optimality conditions abstract in this paper a generalization of convexity called d invexity is introduced substituting d invex for convex we get some optimality conditions for nondifferentiable multiobjective programming the application is demonstrated by an example

block similarity invariants of restrictions to a b invariant subspaces we generalize some properties of invariant subspaces of endomorphisms to a b invariant subspaces in particular we show that in appropriate bases the pair a b can be written as

optimality conditions and duality for nondifferentiable multiobjective programming problems involving d r type i functions in this paper new classes of nondifferentiable functions constituting multiobjective programming problems are introduced namely the classes of d r type i objective and constraint functions and moreover the various classes of generalized d r type i objective and constraint functions are defined for directionally differentiable multiobjective programming problems sufficient optimality conditions and various mond weir duality results are proved for nondifferentiable multiobjective programming problems involving functions of such type finally it is showed that the introduced d r type i notion with r 0 is not a sufficient condition for wolfe weak duality to hold these results are illustrated in the paper by suitable examples