on existence of solutions to differential equations or inclusions remaining in a prescribed closed subset of a finite dimensional space abstract this paper is devoted to existence of trajectories to differential equations and inclusions remaining in a given closed or compact set topological properties of the boundary of this set enables us to conclude the existence of such trajectories in many cases

cored hypergraphs power hypergraphs and their laplacian h eigenvalues abstract in this paper we introduce the class of cored hypergraphs and power hypergraphs and investigate the properties of their laplacian h eigenvalues from an ordinary graph one may generate a k uniform hypergraph called the kth power hypergraph of that graph power hypergraphs are cored hypergraphs but not vice versa sunflowers loose paths and loose cycles are power hypergraphs while squids generalized loose s paths and loose s cycles for 2 s k 2 are cored hypergraphs but not power graphs in general we show that the largest laplacian h eigenvalue of an even uniform cored hypergraph is equal to its largest signless laplacian h eigenvalue especially we find out these largest h eigenvalues for even uniform squids moreover we show that the largest laplacian h eigenvalue of an odd uniform squid loose path and loose cycle is equal to the maximum degree i e 2 we also compute the laplacian h spectra of the class of sunflowers when k is odd the laplacian h spectra of the loose cycle of size 3 and the loose path of length 3 are characterized as well

equitable partition theorem of tensors and spectrum of generalized power hypergraphs abstract in this paper we present an equitable partition theorem of tensors which gives the relations between h eigenvalues of a tensor and its quotient equitable tensor and extends the equitable partitions of graphs to hypergraphs furthermore with the aid of it some properties and h eigenvalues of the generalized power hypergraphs are obtained which extends some known results including some results of yuan qi and shao 20

the proof of a conjecture on largest laplacian and signless laplacian h eigenvalues of uniform hypergraphs let a g l g and q g be the adjacency tensor laplacian tensor and signless laplacian tensor of uniform hypergraph g respectively denote by t the largest h eigenvalue of tensor t let h be a uniform hypergraph and h be obtained from h by inserting a new vertex with degree one in each edge we prove that q h q h denote by gk the kth power hypergraph of an ordinary graph g with maximum degree 2 we prove that q gk is a strictly decreasing sequence which implies conjecture 4 1 of hu qi and shao in 4 we also prove that q gk converges to when k goes to infinity the definition of kth power hypergraph gk has been generalized as gk s we also prove some eigenvalues properties about a gk s which generalize some known results some related results about l g are also mentioned

the largest laplacian and signless laplacian h eigenvalues of a uniform hypergraph a b s t r a c t in this paper we show that the largest laplacian h eigen value of a k uniform nontrivial hypergraph is strictly larger than the maximum degree when k is even a tight lower bound for this eigenvalue is given for a connected even uniform hypergraph this lower bound is achieved if and only if it is a hyperstar however when k is odd in certain cases the largest laplacian h eigenvalue is equal to the maximum degree which is a tight lower bound on the other hand tight upper and lower bounds for the largest signless laplacian h eigenvalue of a k uniform connected hypergraph are given for connected k uniform hypergraphs of fixed number of vertices respectively fixed maximum degree the upper respectively lower bound of their largest signless laplacian h eigenvalues is achieved exactly for the complete hypergraph respectively the hyperstar the largest laplacian h eigenvalue is always less than or equal to the

on some properties of the determinants of tensors abstract we use the hilbert s nullstellensatz hilbert s zero point theorem to give a direct proof of the formula for the determinants of the products of tensors by using this determinant formula and using tensor product to represent the transformations of the slices of tensors we prove some basic properties of the determinants of tensors which are the generalizations of the corresponding properties of the determinants for matrices we also study the determinants of tensors after two types of transposes we use the permutational similarity of tensors to discuss the relation between weakly reducible tensors and the triangular block tensors and give a canonical form of the weakly reducible tensors

morse equation of attractors for nonsmooth dynamical systems abstract this paper is concerned with a morse theory of attractors for finite dimensional nonsmooth dynamical systems described by differential inclusions with upper semi continuous righthand sides we first show that all open attractor neighborhoods of an attractor share the same homotopy type then based on this basic fact we introduce the concept of homology index for morse sets and establish morse inequalities and morse equation by using smooth morse lyapunov functions

contragredient equivalence a canonical form and some applications abstract let a and c be m by n complex matrices and let b and d be n by m complex matrices the pair a b is contragrediently equivalent to the pair c d if there are square nonsingular complex matrices x and y such that xay 1 c and ybx 1 d contragredient equivalence is a common generalization of four basic equivalence relations similarity consimilarity complex orthogonal equivalence and unitary equivalence we develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that a a t is contragrediently equivalent to c c t if and only if there are complex orthogonal matrices p and q such that c paq using this result we show that the following are equivalent for a given n by n complex matrix a 1 1 a qs for some complex orthogonal q and some complex symmetric s 2 2 a t a is similar to aa t 3 3 a a t is contragrediently equivalent to a t a 4 4 a q 1 a t q 2 for some complex orthogonal q 1 q 2 5 5 a pa t p for some complex orthogonal p we then consider a linear operator on n by n complex matrices that shares the following properties with transpose operators for every pair of n by n complex matrices a and b a preserves the spectrum of a b a a and c ab b a we show that a a is contragrediently similar to b b if and only if a x 1 bx 2 for some nonsingular x 1 x 2 that satisfy x 1 1 x 1 and x 1 2 x 2 we also consider a factorization of the form a xy where x 1 x and y y we use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of flanders concerning the relative sizes of the nilpotent jordan blocks of ab and ba we present a sufficient condition for the existence of square roots of ab and ba and close with a canonical form for complex orthogonal equivalence

the morse index repeller attractor pairs and the connection index for semiflows on noncompact spaces abstract this paper extends the morse index theory of c c conley to semiflows on a noncompact meric space x is assumed to satisfy a hypothesis related to conditional contraction we collect background material define quasi index pairs and the morse index of a compact isolated invariant set k and prove that the morse index is a connected simple system we study repeller attractor pairs in k define index triples and prove their existence and several properties leading to the concepts of the connection index the connection map and the splitting class finally we consider paths continuous families of pairs k and study continuations of the morse and the connection indices along such paths the present paper is a sequel to the author s previous work on the homotopy index for infinite dimensional semiflows trans amer math soc 269 1982 351 382

a general product of tensors with applications abstract we study a general product of two n dimensional tensors a and b with orders m 2 and k 1 this product satisfies the associative law and is a generalization of the usual matrix product using this product many concepts and known results of tensors can be simply expressed and or proved and a number of applications of it will be given using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables we define the similarity and congruence of tensors which are also the generalizations of the corresponding relations for matrices and prove that similar tensors have the same characteristic polynomials thus the same spectra we study two special kinds of similarity permutational similarity and diagonal similarity and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors we also define the direct product of tensors in matrix case it is also called the kronecker product and give its applications in the study of the spectra of two kinds of the products of hypergraphs we also give applications of this general product in the study of nonnegative tensors including a characterization of primitive tensors the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors

continuous selection of the solution map for one sided lipschitz differential inclusions abstract in the paper we deal with differential inclusions with one sided lipschitz osl continuous right hand sides and prove the existence of a continuous selection of the solution map which assigns to any point the set of solutions to the multivalued cauchy problem as an application we study the problem of the existence of viable trajectories in a prescribed closed subset of a banach space via the generalized wazewski retract method

conley type index applied to hamiltonian inclusions in the paper we develop the theory of a cohomological index of the conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional hilbert space an application to the existence of periodic orbits to asymptotically linear hamiltonian inclusions is presented

on the spectral radius of a class of non odd bipartite even uniform hypergraphs abstract in order to investigate the non odd bipartiteness of even uniform hypergraphs starting from a simple graph g we construct a generalized power of g denoted by g k s which is obtained from g by blowing up each vertex into a s set and each edge into a k 2 s set where s k 2 when s k 2 g k s is always odd bipartite we show that g k k 2 is non odd bipartite if and only if g is non bipartite and find that g k k 2 has the same adjacency respectively signless laplacian spectral radius as g so the results involving the adjacency or signless laplacian spectral radius of a simple graph g hold for g k k 2 in particular we characterize the unique graph with minimum adjacency or signless laplacian spectral radius among all non odd bipartite hypergraphs g k k 2 of fixed order and prove that 2 5 is the smallest limit point of the non odd bipartite hypergraphs g k k 2 in addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors

perron frobenius theorem for nonnegative multilinear forms and extensions we prove an analog of perron frobenius theorem for multilinear forms with nonnegative coefficients and more generally for polynomial maps with nonnegative coefficients we determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector

a simple minded approach to the index of periodic orbits abstract we define an index for periodic orbits of semiflows which satisfy a mild compactness condition on infinite dimensional anr spaces without requiring differentiability the index is a one dimensional homology class

the inverse rank and product of tensors abstract in this paper we give some basic properties for the left right inverse rank and product of tensors the existence of order 2 left right inverses of tensors is characterized we obtain some equalities and inequalities on the tensor rank we also show that the rank of a uniform hypergraph is independent of the ordering of its vertices and the laplacian tensor and the signless laplacian tensor have the same rank for odd bipartite even uniform hypergraphs

on the homotopy index for infinite dimensional semiflows absrac r in this paper we consider semiflows whose solution operator is eventually a conditional a contraction such semiflows include solutions of retarded and neutral functional differential equations of parabolic and certain other classes of partial differential equations we prove existence of nonsmooth isolating blocks and index pairs for such semiflows via the construction of special lyapunov functionals we show that index pairs enjoy all the properties needed to define the notion of a homotopy index thus generalizing earlier results of conley 2 finally using a result of maii 9 we prove that under additional smoothness assumptions on the semiflow the homotopy index is essentially a finite dimensional concept this gives a formal justification of the applicability of waiewski s principle to infinite dimensional problems several examples illustrate the theory

topological invariants and detection of periodic orbits abstract let f be a smooth flow on a manifold m and c m 0 be an isolated compact set of periodic orbits of f here we consider the following topological invariants of the pair f c the homology index i f c h 1 m the fuller index i f f c q and the p detection number d p f c z p the latter invariant is defined for a positive integer p which is relatively prime with the multiplicities of periodic orbits in c motivated by problems concerning numerical determination of periodic points we introduce the notion of p detectability we prove that i f c 0 implies that f c is 1 detectable but in general this is not the case if i f f c is nontrivial the condition d p f c 0 implies that f c is p detectable as a consequence we prove that if i f f c 0 then f c is p detectable provided p is a sufficiently large prime number we present some applications of these results

a cohomological index of fuller type for set valued dynamical systems abstract in the paper we develop the theory of a cohomological index of the fuller type detecting periodic orbits of a set valued dynamical system generated by a differential inclusion or a differential equation without the uniqueness of solutions the theory presented is applied to establish a general result on the existence of bifurcation of periodic orbits from an equilibrium point of a differential inclusion

leray functor and cohomological conley index for discrete dynamical systems we introduce the leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological conley index of an isolated invariant set of a homeomorphism on a locally compact metric space we prove the homotopy and additivity properties for this index and compute the index in some examples as one of applications we prove the existence of nonconstant bounded solutions of the euler approximation of a certain system of ordinary differential equations

a global bifurcation index for set valued perturbations of fredholm operators abstract in the paper the homotopy invariant detecting global bifurcations of solutions to multi parameter equations involving general set valued perturbations of linear fredholm operators of non negative index is introduced some applications to the existence problems for differential inclusions are provided

the influence of the marked reduced graph of a nonnegative matrix on the jordan form and on related properties a survey abstract for a nonnegative matrix p we discuss the relation of its marked reduced graph to that part of the jordan form that is associated with the perron frobenius root to the nonnegativity of the eigenvectors and generalized eigenvectors to the nonnegativity of solutions of linear equations and to the asymptotic growth of powers of the matrix results are often stated in terms of m matrices and standard results on irreducible matrices are assumed we give examples to illustrate the theorems surveyed

the contragredient equivalence application to solve some matrix systems in this paper we apply the contragredient equivalence to solve two matrix systems firstly we characterize and build all possible solutions of the matrix system p xy q yx giving a recursive formula for the number of contragrediently nonequivalent solutions and secondly we find the solution of the matrix system ax yc by xd

an index for periodic orbits of local semidynamical systems we define an index of fuller type counting the number of periodic orbits of a semiflow on an anr by a suitable approximation process

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation