the connection matrix theory for morse decompositions the connection matrix theory for morse decompositions is introduced the connection matrices are matrices of maps between the homology indices of the sets in the morse decomposition the connection matrices cover in a natural way the homology index braid of the morse decomposition and provide information about the structure of the morse decomposition the existence of connection matrices of morse decompositions is established and examples illustrating applications of the connection matrix are provided

connected simple systems transition matrices and heteroclinic bifurcations given invariant sets a b and c and connecting orbits a b and b c we state very general conditions under which they bifurcate to produce an a c connecting orbit in particular our theorem is applicable in settings for which one has an admissible semiflow on an isolating neighborhood of the invariant sets and the connecting orbits and for which the conley index of the invariant sets is the same as that of a hyperbolic critical point our proof depends on the connected simple system associated with the conley index for isolated invariant sets

norms of inner derivations for multiplier algebras of c algebras and group c algebras abstract the derivation constant k a 1 2 has been previously studied for unital non commutative c algebras a this paper begins the study of k m a where m a is the multiplier algebra of a non unital c algebra a two results are obtained giving separate conditions on a which imply that k m a 1 these results are applied to a c g for a number of locally compact groups g including sl 2 r sl 2 c and several 2 step solvable groups in these cases k m a 1 on the other hand if g is a non abelian amenable sin group then k m a 1 2

a strictly contractive peaceman rachford splitting method with logarithmic quadratic proximal regularization for convex programming recently a strictly contractive peaceman rachford splitting method prsm was proposed for a separable convex minimization model whose variables are subject to some linear constraints and two additional generic constraints in general the strictly contractive prsm requires to solve two constrained minimization subproblems at each iteration in this paper we consider the case where the additional constraints on variables are positive orthants and apply the well developed logarithmic quadratic proximal lqp regularization to regularize the subproblems of the strictly contractive prsm a new algorithm in combination with the strictly contractive prsm and the lqp regularization is thus proposed the new algorithm only needs to solve two unconstrained subproblems at each iteration an inexact version allowing the unconstrained subproblems to be solved approximately subject to certain inexactness criterion is also studied we prove the global convergence and establish a worst case convergence rate measured

rigorous computation of the global dynamics of integrodifference equations with smooth nonlinearities topological tools such as conley index theory have inspired rigorous computational methods for studying dynamics these methods rely on the construction of an outer approximation a combinatorial representation of the system that incorporates small bounded error in this work we present an automated approach to constructing outer approximations for systems in a class of integrodifference operators with smooth nonlinearities chebyshev interpolants and galerkin projections form the basis for the construction while analysis and interval arithmetic are used to incorporate explicit error bounds this represents a significant advance to the approach given by day junge and mischaikow siam j appl dyn syst 3 2004 pp 117 160 extending the nonlinearities that may be studied from low degree polynomials to smooth functions and the studied portion of phase space from a simulated attracting region to the global maximal invariant set as a demonstration of the techniques a morse decomposition of the

inexact alternating direction methods for image recovery in the image processing community there have recently been many restoration and reconstruction problems that can be reformulated into linearly constrained convex programming models whose objective functions have separable structures these favorable reformulations have promoted impressive applications of the alternating direction method adm in the field of image processing at each iteration the computation of adm is dominated by solving two subproblems exactly however in many restoration and reconstruction applications it is either impossible or extremely expensive to obtain exact solutions of these adm subproblems this fact urges the development on inexact versions of adm which allow the generated adm subproblems to be solved approximately subject to certain inexactness criteria in this paper we develop some truly implementable inexact adms whose inexactness criteria controlling the accuracy of the adm subproblems are easily implementable the convergence of the new inexact adms will be proved numerical results on several image processing problems will be given to illustrate the effectiveness of the proposed inexact adms

lagrangian duality and related multiplier methods for variational inequality problems we consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities developing a simple lagrangian duality scheme which is combined with the recent logarithmic quadratic proximal lqp theory introduced by the authors we derive three algorithms for solving the variational inequality vi problem this provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms we prove primal dual and primal dual convergence under very mild assumptions eliminating all the usual assumptions used until now in the literature for related algorithms applications to complementarity problems are also discussed

a new inexact alternating directions method for monotone variational inequalities the alternating directions method adm is an effective method for solving a class of variational inequalities vi when the proximal and penalty parameters in sub vi problems are properly selected in this paper we propose a new adm method which needs to solve two strongly monotone sub vi problems in each iteration approximately and allows the parameters to vary from iteration to iteration the convergence of the proposed adm method is proved under quite mild assumptions and flexible parameter conditions

the derivation problem for group algebras if g is a locally compact group then for each derivation d from l 1 g into l 1 g there is a bounded measure m g with d a a a for a l 1 g derivation problem of b e johnson

using machine learning to predict catastrophes in dynamical systems nonlinear dynamical systems which include models of the earth s climate financial markets and complex ecosystems often undergo abrupt transitions that lead to radically different behavior the ability to predict such qualitative and potentially disruptive changes is an important problem with far reaching implications even with robust mathematical models predicting such critical transitions prior to their occurrence is extremely difficult in this work we propose a machine learning method to study the parameter space of a complex system where the dynamics is coarsely characterized using topological invariants we show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner we are able to predict with high accuracy the locations of critical transitions in parameter space

a majorized admm with indefinite proximal terms for linearly constrained convex composite optimization this paper presents a majorized alternating direction method of multipliers admm with indefinite proximal terms for solving linearly constrained 2 block convex composite optimization problems with each block in the objective being the sum of a nonsmooth convex function p x or q y and a smooth convex function f x or g y i e min x in mathcal x y in mathcal y p x f x q y g y mid a x b y c by choosing the indefinite proximal terms properly we establish the global convergence and the iteration complexity in the nonergodic sense of the proposed method for the step length tau in 0 1 sqrt 5 2 the computational benefit of using indefinite proximal terms within the admm framework instead of the current requirement of positive semidefinite ones is also demonstrated numerically this opens up a new way to improve the practical performance of the admm and related methods

conley index of poincare maps in isolating segments abstract we calculate the discrete time conley index of the poincare map of a time periodic ordinary differential equation in an isolated invariant set generated by a periodic isolating segment as an application we present results on the existence of bounded solutions of some planar equations

infinite series in cohomology attractors and conley index in this paper we study the cohomological conley index of arbitrary isolated invariant continua for continuous maps f u subseteq mathbb r d rightarrow mathbb r d by analyzing the topological structure of their unstable manifold we provide a simple dynamical interpretation for the first cohomological conley index describing it completely and relate it to the cohomological conley index in higher degrees a number of consequences are derived including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3 our approach exploits a certain attractor repeller decomposition of the unstable manifold reducing the study of the cohomological conley index to the relation between the cohomology of an attractor and its basin of attraction this is a classical problem that in the present case is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated to address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology if z is a metric space and k subseteq z is a compact global attractor for a continuous map g z rightarrow z we show how to interpret series of the form sum j ge 0 a j g j as endomorphisms of the cohomology group of the pair z k

chaos near a resonant inclination flip abstract horseshoes play a central role in dynamical systems and are observed in many chaotic systems however most points in a neighborhood of the horseshoe escape after finitely many iterations in this work we construct a new model by re injecting the points that escape the horseshoe we show that this model can be realized within an attractor of a flow arising from a three dimensional vector field after perturbation of an inclination flip homoclinic orbit with a resonance the dynamics of this model without considering the re injection often contains a cuspidal horseshoe with positive entropy and we show that for a computational example the dynamics with re injection can have more complexity than the cuspidal horseshoe alone

homotopy invariants of repeller attractor pairs ii continuation of r a pairs on considere une famille continue a 1 parametre de flots dans le plan parametrise par 0 1 on etablit un cadre qui rend precise la notion de famille continue de paires de repulseur attracteurs on montre qu il y a des equivalences naturelles entre les suites d indices de 2 paires r a le long d un parcours de paires r a

indices of the iterates of r 3 homeomorphisms at lyapunov stable fixed points abstract given any positive sequence c n n n we construct orientation preserving homeomorphisms f r 3 r 3 such that fix f per f 0 0 is lyapunov stable and lim sup i f m 0 c m we will use our results to discuss and to point out some strong differences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms

on the o 1 t convergence rate of alternating direction method with logarithmic quadratic proximal regularization it was shown recently that the douglas rachford alternating direction method of multipliers can be combined with the logarithmic quadratic proximal regularization for solving a class of variational inequalities with separable structures this paper further shows a worst case o 1 t convergence rate for this algorithm where a general glowinski relaxation factor is used

conley index for random dynamical systems conley index theory is a very powerful tool in the study of dynamical systems in this paper we generalize conley index theory to discrete random dynamical systems our constructions are basically the random version of franks and richeson in j franks d richeson shift equivalence and the conley index trans amer math soc 352 2000 3305 3322 for maps and the relations of isolated invariant sets between time continuous random dynamical systems and corresponding time h maps are discussed two examples are presented to illustrate results in this paper

measuring noncommutativity in c algebras abstract it is well known that if a is a von neumann algebra then the norm of any inner derivation ad a is equal to twice the distance from the element a to the centre z a of the algebra more generally this property holds in a unital c algebra if and only if the ideal p q r is primal whenever p q and r are primitive ideals of a such that p z a q z a r z a in this paper we give a characterization in terms of ideal structure of those unital c algebras a for which the norm of any inner derivation ad a at least dominates the distance from a to the centre z a in doing so we show that if a does not have this property then it necessarily contains an element a with ad a 1 whose distance from z a is greater than or equal to 3 8 2 14 we also show how this number is related to the numbers 4 15 and 1 2 1 3 which have previously arisen in the study of norms of inner derivations the techniques used in this work include spectral theory the theory of primitive and primal ideals and constrained geometrical optimisation

isolating blocks and epsilon chains for maps abstract the relationship between the outputs and computer simulations of dynamical systems and their true orbit structures is explored epsilon chains are used as a mathematical representation of computer generated pseudo orbits the question when is a pseudo orbit shadowed by a true orbit is discussed isolated invariant sets are defined and are shown to be contained in special neighborhoods called isolating blocks the conley index is extended and applied to maps

an inexact lqp alternating direction method for solving a class of structured variational inequalities in this paper we present an inexact alternating direction method each iteration of the proposed method contains a prediction and a correction the predictor is obtained via solving the logarithmic quadratic proximal lqp system approximately under significantly relaxed accuracy criterion and the new iterate is computed directly by an explicit formula derived from the original lqp method under certain conditions the global convergence of the proposed method is proved

an inexact alternating direction method for solving a class of structured variational inequalities the alternating direction method is mainly adopted to solve large scale variational inequality problems with separable structure the method is effective because it solves the original high dimensional variational inequality problem by solving a series of much easier low dimensional subproblems in this paper we present an inexact alternating directions method compared with the quadratic proximal alternating direction methods the proposed method solves a series of related systems of nonlinear equations instead of a series of sub vis the inexact criteria are more relaxed than the ones used by he et al 7 the generated sequence is fejer monotone with respect to the solution set and the convergence is proved under suitable conditions

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

realization of all dold s congruences with stability the main goal of this paper is to prove that for each n 2 every sequence of integers satisfying dold s congruences is realized as the sequence of fixed point indices of the iterates of an orientation preserving r n homeomorphism at an isolated stable fixed point

zeta functions periodic trajectories and the conley index abstract we use the conley index to produce a simple sufficient condition stated in terms of the betti numbers of the cohomology conley index for the existence of periodic orbits for flows which admit a cross section

a descent lqp alternating direction method for solving variational inequality problems with separable structure based on the logarithmic quadratic proximal method a descent alternating direction method is introduced for structured variational inequalities in the paper the proposal method generates the new iteration by searching the optimal step size along the integrated descent direction from two descent directions global convergence of the proposal method is proved under certain assumptions to illustrate the proposal method and demonstrate its efficiency some applications and their numerical results are also provided

index filtrations and the homology index braid for partially ordered morse decompositions abstract on a morse decomposition of an invariant set in a flow there are partial orderings defined by the flow these are called admissable orderings of the morse decomposition the index filtrations for a total ordering of a morse decomposition are generalized in this paper with the definition and proof of existence of index filtrations for admissable partial orderings of a morse decomposition it is shown that associated to an index filtration there is a collection of chain complexes and chain maps called the chain complex braid of the index filtration the homology index braid of the corresponding admissable ordering of the morse decomposition is obtained by passing to homology in the chain complex braid keywords morse decomposition conley index index filtration

on preconditioned and relaxed avmm methods for quadratic programming problems with equality constraints abstract to iteratively compute a solution of the equality constraint quadratic programming problem by successively introducing relaxation parameters and skillfully adopting a preconditioning matrix we establish a preconditioned and relaxed alternating variable minimization with multiplier pravmm method which is a further generalization of the preconditioned alternating variable minimization with multiplier pavmm method proposed by bai and tao 2016 bit numer math 56 2016 399 422 based on rigorous matrix analysis we demonstrate the asymptotic convergence property of the pravmm method numerical results show that the pravmm method is feasible and effective for solving the equality constraint quadratic programming problems

norms of inner derivations for multiplier algebras of c algebras and group c algebras ii abstract the derivation constant k a 1 2 has been extensively studied for unital non commutative c algebras in this paper we investigate properties of k m a where m a is the multiplier algebra of a non unital c algebra a a number of general results are obtained which are then applied to the group c algebras a c g n where g n is the motion group r n so n utilizing the rich topological structure of the unitary dual g n it is shown that for n 3 k m c g n 1 2 n 2

detection of renewal system factors via the conley index let n be an isolating neighborhood for a map f if we can decompose n into the disjoint union of compact sets n 1 and n 2 then we can relate the dynamics on the maximal invariant set inv n to the shift on two symbols by noting which component of n each iterate of a point x inv n lies in we examine a method based on work by mischaikow szymczak et al for using the discrete conley index to detect explicit subshifts of the shift associated to n in essence we measure the difference between the conley index of inv n and the sum of the indices of inv n 1 and inv n 2

a variable penalty alternating directions method for convex optimization we study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints the method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both lagrangian and proximal terms the minimizers determine the new proximal terms and a simple update of the lagrangian terms follows we prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers another novelty is that we allow penalty matrices and these may vary per iteration this can be beneficial in applications since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties as an application we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems

detection and computation of high codimension bifurcations in diffuse predator prey systems abstract in this paper we consider reaction diffusion model with modified leslie gower and holling type ii functional response and by varying simultaneously three model parameters our model exhibits different types of complex dynamics this allowed us to delimit several bifurcation surfaces with higher codimension corresponding to turing turing transcritical turing bogdanov taken turing hopf andronov turing saddle node moreover by varying at least three bifurcation parameters we show that small variations in the ratio of the diffusion coefficients can significantly alter bifurcation structure finally the paper ends with the emergence of spatio temporal patterns via numerical simulations these simultaneous parameter variations leaded to spatio temporal local and global bifurcations which are always catastrophic our results give new insights about how simultaneous changes in environmental and life history parameters drive different distribution dynamics of predator prey populations this only underscores the importance of including global bifurcations in the analysis of food chain models such results can be used to improve ecological decision making on species conservation