determinants on von neumann algebras mahler measures and ljapunov exponents for an ergodic automorphism of a probability space a p consider the semidirect product von neumann algebra r l z let u be the unitary operator corresponding to the action of then the finite sums of the form i aiu i with ai in l form a strongly dense subalgebra of r in this paper we calculate the fuglede kadison determinant detr for a certain class of elements the answer is given in terms of the ljapunov exponents i of a matrix a over the ring l which is attached to by a simple receipe the formula has the form

a geometrically based criterion to avoid infimum gaps in optimal control abstract in optimal control theory infimum gap means a non zero difference between the infimum values of a given minimum problem and an extended problem obtained by embedding the original family v of controls in a larger family w for some embeddings like standard convex relaxations or impulsive extensions the normality of an extended minimizer has been shown to be sufficient for the avoidance of infimum gaps a natural issue is then the search of a general hypothesis under which the criterium normality implies no gap holds true we prove that this criterium is actually valid as soon as v is abundant in w without any convexity assumption on the extended dynamics abundance which was introduced by j warga in a convex context and was later generalized by b kaskosz strengthens density the latter being not sufficient for the mentioned criterium to hold true

lyapunov stability for impulsive control affine systems abstract this article studies several notions of lyapunov stability for impulsive control affine systems in the setting of nonautonomous dynamical systems it presents some relations between the stability of an impulsive control affine system and the stability of its adjacent control system stability of compact sets and their components are specially investigated lyapunov functionals are employed to characterize each type of stability of closed sets

a new probabilistic relaxation scheme and its application to edge detection this paper presents a new scheme for probabilistic relaxation labeling that consists of an update function and a dictionary construction method the nonlinear update function is derived from markov random field theory and bayes formula the method combines evidence from neighboring label assignments and eliminates label ambiguity efficiently this result is important for a variety of image processing tasks such as image restoration edge enhancement edge detection pixel classification and image segmentation the authors successfully applied this method to edge detection the relaxation step of the proposed edge detection algorithm greatly reduces noise effects gets better edge localization such as line ends and corners and plays a crucial role in refining edge outputs the experiments show that our algorithm converges quickly and is robust in noisy environments

mixing automorphisms of compact groups and a theorem of schlickewei we prove that every mixing d by automorphisms of a compact connected abelian group is mixing of all orders

relatively spectral morphisms and applications to k theory abstract spectral morphisms between banach algebras are useful for comparing their k theory and their noncommutative dimensions as expressed by various notions of stable ranks in practice one often encounters situations where the spectral information is only known over a dense subalgebra we investigate such relatively spectral morphisms we prove a relative version of the density theorem regarding isomorphism in k theory we also solve swan s problem for the connected stable rank in fact for an entire hierarchy of higher connected stable ranks that we introduce

pontryagin s maximum principle for constrained impulsive control problems abstract necessary conditions in the form of pontryagin s maximum principle are derived for impulsive control problems with mixed constraints a new mathematical concept of impulsive control is introduced as a requirement for the consistency of the impulsive framework additionally this control concept enables the incorporation of the engineering needs to consider conventional control action while the impulse develops the regularity assumptions under which the maximum principle is proved are weaker than those in the known literature ekeland s variational principle and lebesgue s discontinuous time variable change are used in the proof the article also contains an example showing how such impulsive controls could be relevant in actual applications

stability of impulsive functional differential equations abstract in this paper the stability of impulsive functional differential equations in which the state variables on the impulses are related to the time delay is studied by using lyapunov functions and razumikhin techniques some criteria of stability asymptotic stability and practical stability for impulsive functional differential equations in which the state variables on the impulses are related to the time delay are provided some examples are also presented to illustrate the efficiency of the results obtained

ergodicity and mixing of non commuting epimorphisms we study mixing properties of epimorphisms of a compact connected finite dimensional abelian group x in particular we show that a set f with f dim x of epimorphisms of x is mixing if and only if every subset of f of cardinality dim x 1 is mixing we also construct examples of free non abelian groups of automorphisms of tori which are mixing but not mixing of order 3 and show that under some irreducibility assumptions ergodic groups of automorphisms contain mixing subgroups and free non abelian mixing subsemigroups

soficity amenability and dynamical entropy in a previous paper the authors developed an operator algebraic approach to lewis bowen s sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measure preserving transformations on a standard probability space we show here that these measure and topological entropy invariants both coincide with their classical counterparts when the acting group is amenable

isomorphism rigidity of irreducible algebraic ℤd actions an irreducible algebraic d action on a compact abelian group x is a d action by automorphisms of x such that every closed invariant subgroup y x is finite we prove the following result if d 2 then every measurable conjugacy between irreducible and mixing algebraic d actions on compact zero dimensional abelian groups is affine for irreducible expansive and mixing algebraic d actions on compact connected abelian groups the analogous statement follows essentially from a result by katok and spatzier on invariant measures of such actions cf 4 and 3 by combining these two theorems one obtains isomorphism rigidity of all irreducible expansive and mixing algebraic d actions with d 2

mahler measure and entropy for commuting automorphisms of compact groups we compute the joint entropy ofd commuting automorphisms of a compact metrizable group letr d u 1 1 d 1 1 be the ring of laurent polynomials ind commuting variables andm be anr d module then the dual groupx m ofm is compact and multiplication onm by each of thed variables corresponds to an action m of d by automorphisms ofx m every action of d by automorphisms of a compact abelian group arises this way iff r d our main formula shows that the topological entropy of alpha r d left langle f right rangle is given by h left alpha r d left langle f right rangle right log m left f right int limits 0 1 ldots int limits 0 1 log left f left e 2 pi it 1 ldots e 2 pi it d right right dt 1 ldots dt d where m f is the mahler measure off this reduces to the classical result for toral automorphisms via jensen s formula while the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7 3 4 2 occur if p is a non principal prime ideal we show h left alpha r d p right 0 using an analogue of the yuzvinskii thomas addition formula we computeh m for arbitraryr d modulesm and then the joint entropy for an action of d on a not necessarily abelian compact group using a result of boyd we characterize those m which have completely positive entropy in terms of the prime ideals associated tom and show this condition implies that m is mixing of all orders we also establish an analogue of berg s theorem proving that if m has finite entropy then haar measure is the unique measure of maximal entropy if and only if m has completely positive entropy finally we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy

integrality of l 2 betti numbers the atiyah conjecture predicts that the l 2 betti numbers of a finite cw complex with torsion free fundamental group are integers we establish the atiyah conjecture under the condition that it holds for g and that h lhd g is a normal subgroup for amalgamated free products g h h rtimes f here f is a free group and h rtimes f is an arbitrary semi direct product this includes free products g f and semi direct products g rtimes f we also show that the atiyah conjecture holds with an additional technical condition for direct and inverse limits of groups for which it is true as a corollary it holds for positive 1 relator groups with torsion free abelianization putting everything together we establish a new bigger class of groups for which the atiyah conjecture holds which contains all free groups and in particular is closed under taking subgroups direct sums free products extensions with torsion free elementary amenable quotient or with free quotient and under certain direct and inverse limits

necessary conditions for optimal impulsive control problems necessary conditions of optimality in the form of a maximum principle are derived for a class of optimal control problems certain of whose controls are represented by measures and whose state trajectories are functions of bounded variation state trajectories are interpreted as robust solutions of the dynamic equations a concept of solutions which takes account of the interaction between the measure control and the state variables during the jumps the maximum principle which is derived improves on earlier optimality conditions for problems of this nature by allowing nonsmooth data measurable time dependence and a possibly time varying constraint set for the conventional controls

comparison theorems for viscosity solutions of a system of quasivariational inequalities with application to optimal control with switching costs in this paper we prove a comparison principle between a viscosity sub and supersolution for a system of quasivariational inequalities and apply it to show that a continuous lower value vector function of an optimal switching cost control problem is characterized as the minimal nonnegative continuous viscosity supersolution of the sqvi

growth in an impulsive integral inequality abstract in this paper we study a finite sum integral inequality with a sequence of impulses making a sequential monotonization we give recursively defined functions to estimate solutions of the inequality piecewise in order to use the estimate to study boundedness and asymptotics of solutions for differential equations we regard the estimating function as a solution of a nonautonomous difference equation simplify the equation with composition of operators and reduce our discussion to qualitative analysis of the difference equation so that we give results on monotonicity boundedness and weighted boundedness for solutions of the inequality as applications we study boundedness and asymptotics of solutions for two impulsive differential equations

independence in topological and c dynamics we develop a systematic approach to the study of independence in topological dynamics with an emphasis on combinatorial methods one of our principal aims is to combinatorialize the local analysis of topological entropy and related mixing properties we also reframe our theory of dynamical independence in terms of tensor products and thereby expand its scope to c dynamics

almost block independence and bernoullicity of ℤ d actions by automorphisms of compact abelian groups we prove that a d action by automorphisms of a compact abelian group is bernoulli if and only if it has completely positive entropy the key ingredients of the proof are the extension of certain notions of asymptotic block independence from actions to d action and their equivalence with bernoullicity and a surprisingly close link between one of these asymptotic block independence properties for d actions by automorphisms of compact abelian groups and the product formula for valuations on global fields

when is the maximum principle for state constrained problems nondegenerate abstract it is well known that a form of the pontryagin maximum principle applies to optimal control problems involving a unilateral state constraint we discuss how the need arises to solve problems of this nature when the left endpoint is fixed and when the initial state lies in the boundary of the state constraint set region in such cases previous versions of the maximum principle convey no information since a trivial choice of multipliers may be made namely one in which the state constraint multiplier is a unit measure concentrated at the left endtime and all the other multipliers are zero we prove strengthened forms of the maximum principle applicable to such situations constraint qualifications are formulated under which multipliers exist besides the trivial ones

entropy and the variational principle for actions of sofic groups recently lewis bowen introduced a notion of entropy for measure preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy by applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants and we establish the variational principle in this context in the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group c algebra

fuglede kadison determinants and entropy for actions of discrete amenable groups consider a discrete group t and an element in the integral group ring zt then t acts from the left on the discrete additive group zt ztf by automorphisms of groups dualizing we obtain a left t action on the compact pontrjagin dual group xf zt ztf by continuous automorphisms of groups by definition xf is a closed subshift of s1 1 for measurable or topological actions of amenable groups a good entropy theory is available ow m thus from now on let t be a finitely generated discrete amenable group we are interested in determining the topological entropy hf of the t action on xf it agrees with the measure theoretic entropy of the t action on xf equipped with its haar probability measure in their paper fk from 1952 fuglede and kadison introduced a determinant for units in finite factor von neumann algebras more recently l ck has general ized their theory to nonunits and to group von neumann algebras which are not necessarily factors l2 he needed this to define l2 torsion for general covering spaces the von neumann algebra a t of the amenable group t is a certain completion of cr and hence we may consider the fuglede kadison l ck determinant detjvr of we refer to section 6 for the definition of a log strong f0lner sequence it is known that finitely generated virtually nilpotent groups have log strong f0lner sequences our main result is

approximating l2 invariants and the atiyah conjecture let g be a torsion free discrete group and let denote the field of algebraic numbers in we prove that g fulfills the atiyah conjecture if g lies in a certain class of groups d which contains in particular all groups that are residually torsion free elementary amenable or are residually free this result implies that there are no nontrivial zero divisors in g the statement relies on new approximation results for l2 betti numbers over g which are the core of the work done in this paper another set of results in the paper is concerned with certain number theoretic properties of eigenvalues for the combinatorial laplacian on l2 cochains on any normal covering space of a finite cw complex we establish the absence of eigenvalues that are transcendental numbers whenever the covering transformation group is either amenable or in the linnell class we also establish the absence of eigenvalues that are liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class 2003 wiley periodicals inc

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

compact group automorphisms addition formulas and fuglede kadison determinants for a countable amenable group and an element f in the integral group ring z being invertible in the group von neumann algebra of we show that the entropy of the shift action of on the pontryagin dual of the quotient of z by its left ideal generated by f is the logarithm of the fuglede kadison determinant of f for the proof we establish an p version of rufus bowen s denition of topological entropy addition formulas for group extensions of countable amenable group actions and an approximation formula for the fuglede kadison determinant of f in terms of the determinants of perturbations of the compressions of f

entropy and mixing for amenable group actions for i a countable amenable group consider those actions of i as measurepreserving transformations of a standard probability space written asft g 2i acting on x f we sayft g 2i has completely positive entropy or simply cpe for short if for any flnite and nontrivial partition p of x the entropy h t p is not zero our goal is to demonstrate what is well known for actions of and even d that actions of completely positive entropy have very strong mixing properties let si be a list of flnite subsets of i we say the si spread if any particular 6 id belongs to at most flnitely many of the sets sis i1 i theorem 0 1 for ft g 2i an action of i of completely positive entropy and p any flnite partition for any sequence of flnite sets si i which spread we have

entropy geometry and disjointness for zero dimensional algebraic actions we show that many algebraic actions of higher rank abelian groups on zero dimensional compact abelian groups are mutually disjoint the proofs exploit differences in the entropy geometry arising from subdynamics and a form of abramov rokhlin formula for half space entropies

on the foundations of relaxation labeling processes a large class of problems can be formulated in terms of the assignment of labels to objects frequently processes are needed which reduce ambiguity and noise and select the best label among several possible choices relaxation labeling processes are just such a class of algorithms they are based on the parallel use of local constraints between labels this paper develops a theory to characterize the goal of relaxation labeling the theory is founded on a definition of con sistency in labelings extending the notion of constraint satisfaction in certain restricted circumstances an explicit functional exists that can be maximized to guide the search for consistent labelings this functional is used to derive a new relaxation labeling operator when the restrictions are not satisfied the theory relies on variational cal culus it is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality a procedure nearly identical to the relaxation operator derived under restricted circum stances serves in the more general setting further a local convergence result is established for this operator the standard relaxation labeling formulas are shown to approximate our new operator which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here observations about con vergence and generalizations to higher order compatibility relations are described

mean dimension mean rank and von neumann luck rank we introduce an invariant called mean rank for any module m of the integral group ring of a discrete amenable group as an analogue of the rank of an abelian group it is shown that the mean dimension of the induced action on the pontryagin dual of m the mean rank of m and the von neumann luck rank of m all coincide as applications we establish an addition formula for mean dimension of algebraic actions prove the analogue of the pontryagin schnirelmnn theorem for algebraic actions and show that for elementary amenable groups with an upper bound on the orders of finite subgroups algebraic actions with zero mean dimension are inverse limits of finite entropy actions

a measure conjugacy invariant for free group actions this paper introduces a new measure conjugacy invariant for actions of free groups using this invariant it is shown that two bernoulli shifts over a finitely generated free group are measurably conjugate if and only if their base measures have the same entropy this answers a question of ornstein and weiss

algebraic entropy for amenable semigroup actions abstract we introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups s on discrete abelian groups a by endomorphisms they extend the classical algebraic entropy for endomorphisms of abelian groups corresponding to the case s n we investigate the fundamental properties of the algebraic entropy and compute it in several examples paying special attention to the case when s is an amenable group for actions of cancellative right amenable monoids on torsion abelian groups we prove the so called addition theorem in the same setting we see that a bridge theorem connects the algebraic entropy with the topological entropy of the dual action by means of pontryagin duality so that we derive an addition theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups

impulsive control of nonlocal transport equations abstract the paper extends an impulsive control theoretical framework towards dynamic systems in the space of measures we consider a transport equation describing the time evolution of a conservative mass probability measure which represents an infinite ensemble of interacting particles the driving vector field contains nonlocal terms and it is affine in the control variable the control is assumed to be common for all the agents i e it is a function of time variable only the main feature of the addressed model is the admittance of shock impacts i e controls whose influence on each agent can be arbitrary close to dirac type distributions we construct an impulsive relaxation of this system and of the corresponding optimal control problem for the latter we establish a necessary optimality condition in the form of pontryagin s maximum principle

a nondegenerate maximum principle for the impulse control problem with state constraints in this article a free time impulsive control problem with state constraints and equality and inequality constraints on the trajectory endpoints is considered a weakened maximum principle is obtained for problems with data measurable in the time variable being the time transversality conditions deduced with the help of some extra convexity assumption on the state constraints in the case of smooth problems a nondegenerate maximum principle is derived by using a penalty function method

a maximum principle for optimal processes with discontinuous trajectories a maximum principle is proved which governs solutions to dynamic optimization problems in which the controls driving the system may be impulsive and give rise to discontinuous trajectories the approach which involves approximating the problem by a conventional one and using ekeland s theorem is new it permits us to weaken very considerably the hypotheses under which maximum principles for such problems have previously been proved

an efficient edge detection algorithm using relaxation labeling technique abstract edge detection plays an important role in computer vision tasks and has received considerable attention in image processing literature to detect edges correctly and precisely contextual information is needed how to use contextual information is a key issue in this paper we introduce an edge detection method that will use edge contextual information of the whole image efficiency this new method tries to employ contextual information within a certain distance from the focus pixel at a time this distance keeps increasing recursively until the edge feature of a pixel is uniquely defined in this manner we can minimize the need for contextual information experimental results are presented to characterize the performance of our new method in terms of better connectedness of edges and less distortion and in terms of computational efficiency a detailed comparison of our method with the context free zero crossing edge operator that uses optimal exponential filter is discussed in this paper

some remarks on the entropy for algebraic actions of amenable groups in this short note we study the entropy for algebraic actions of certain amenable groups the possible values for this entropy are studied various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant we provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to these results extend previous work by lind and ward on p adic entropy

edge labeling using dictionary based relaxation an improved application of probabilistic relaxation to edge labeling is presented the improvement derives from the use of a representation of the edge process that is internally consistent and which utilizes a more complex description of edge structure the application uses a dictionary to represent permitted labelings of the entire context conveying neighborhood of each pixel details are given of the dictionary approach and the related representation of the edge process a comparison with other edge postprocessing strategies is provided

sofic mean dimension abstract we introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces these generalize the gromov lindenstrauss weiss mean dimensions for actions of countable amenable groups and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy

homoclinic points of algebraic ℤ actions an algebraic zd action is an action of zd by continuous automorphisms of a compact abelian group the dynamics of a single group automorphism have been investigated in great detail over the past several decades more recently the study of algebraic zd actions for d 2 has revealed a striking interplay between these actions and commutative algebra in 2 we summarize those parts of this interaction needed here the purpose of this paper is to study the homoclinic points of algebraic zd actions let al be an algebraic zd action on the compact abelian group x and let ox denote the additive identity of x a point x e x is homoclinic for ce if c nx ox as llnll oo the set ai x of all homoclinic points for oc is clearly a subgroup of x which we call the homoclinic group of oz in 3 we discuss some elementary properties of the homoclinic group including counltability of za x whenever oz is expansive our two main results are contained in 4 these are that if oz is an expansive algebraic ed action then 1 a x is nontrivial if and only if oz has strictly positive entropy and 2 a x is nontrivial and dense in x if and only if oz has completely positive entropy the second result is proved by first establishing in lemma 4 5 the density of a x for certain principal expansive actions by use of fourier analysis for these actions a x is generated by a single fundamental homoclinic point which can be computed explicitly this lemma is then combined with some commutative algebra to prove 2 recent work of kaminker and putnam 3 11 has suggested a general duality in the k theory of c algebras for a principal expansive action al on x we show that a x is isomorphic to the dual group of x providing a class of examples to which their duality theory applies ruelle has investigated expansive topological zd actions which satisfy an orbit tracing property called specification see 12 and 13 showing that there is a thermodynamic formalism for such actions in 5 we show that expansive algebraic zd actions with completely positive entropy always satisfy very strong specification properties thereby providing an extensive class of examples to which the thermodynamic formalism applies

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

harmonic models and spanning forests of residually finite groups abstract we prove a number of identities relating the sofic entropy of a certain class of non expansive algebraic dynamical systems the sofic entropy of the wired spanning forest and the tree entropy of cayley graphs of residually finite groups we also show that homoclinic points and periodic points in harmonic models are dense under general conditions

entropy determinants and l 2 torsion we show that for any amenable group and any z module m of type fl with vanishing euler characteristic the entropy of the natural action on the pontryagin dual of m is equal to the l 2 torsion of m as a particular case the entropy of the principal algebraic action associated with the module z z f is equal to the logarithm of the fuglede kadison determinant of f whenever f is a non zero divisor in z this conrms a conjecture of deninger as a key step in the proof we provide a general szeg o type approximation theorem for the fuglede kadison determinant on the group von neumann algebra of an amenable group as a consequence of the equality between l 2 torsion and entropy we show that the l 2 torsion of a non trivial amenable group with nite classifying space vanishes this was conjectured by luck finally we establish a milnor turaev formula for the l 2 torsion of a nite acyclic chain complex

homoclinic groups ie groups and expansive algebraic actions we give algebraic characterizations for expansiveness of algebraic actions of countable groups the notion of p expansiveness is introduced for algebraic actions and we show that for countable amenable groups a finitely presented algebraic action is 1 expansive exactly when it has finite entropy we also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms and show that the ie group determines the pinsker factor for such actions for an expansive algebraic action of a polycyclic by finite group on x it is shown that the entropy of the action is equal to the entropy of the induced action on the pontryagin dual of the homoclinic group the homoclinic group is a dense subgroup of the ie group the homoclinic group is nontrivial exactly when the action has positive entropy and the homoclinic group is dense in x exactly when the action has completely positive entropy

necessary conditions of optimality for vector valued impulsive control problems abstract a vector valued impulsive control problem is considered whose dynamics defined by a differential inclusion are such that the vector fields associated with the singular term do not satisfy the so called frobenius condition a concept of robust solution based on a new reparametrization procedure is adopted in order to derive necessary conditions of optimality these conditions are obtained by taking a limit of those for an appropriate sequence of auxiliary standard optimal control problems approximating the original one an example to illustrate the nature of the new optimality conditions is provided

a sampling method and approximation results for impulsive systems this paper studies an impulsive dynamical system that is in the form of a measure driven differential inclusion the employed solution concept depends upon a graph completion of the measure the first main result shows that a subsequence of discrete time trajectories graph converges to a solution and under lipschitz hypotheses that every solution can be obtained in this manner the second main result pursues a similar development with classical systems that approximate the given one

erratum to integrality of l2 betti numbers a gap has been found in the proof of proposition 3 of 3 consequently this result and the consequences drawn in the paper have to be considered as conjectures in particular it is not proved that the atiyah conjecture is stable under extensions with free quotient groups and under free products with free groups

weighted group algebras on groups of polynomial growth let g be a compactly generated group of polynomial growth and a weight function on g for a large class of weights we characterize symmetry of the weighted group algebra l 1 g in particular if the weight is sub exponential then the algebra l 1 g is symmetric for these weights we develop a functional calculus on a total part of l 1 g and use it to prove the wiener property

hamilton jacobi bellman equation and feedback synthesis for impulsive control there is an increasing number of applications whose trajectories are better modeled by discontinuous or impulsive trajectories thus we explore optimality conditions for impulsive control system expressed in terms of an hamilton jacobi bellman equation we use a measure driven differential inclusion to model the impulsive behavior since it provides a formal framework in which the control space is complete additionally we use the impulsive euler solution and invariance results to derive feedback optimal control synthesis

a remark on t valued cohomology groups of algebraic group actions we prove that for a weakly mixing algebraic action g x the nth cohomology group hn g x t after quotienting out the natural subgroup hn g t contains hn g x as a natural subgroup for n 1 if we further assume the diagonal actions 2 4 are t cocycle superrigid and h2 g x is torsion free as an abelian group then the above also holds true for n 2 applying it for principal algebraic actions when n 1 we show that h2 g zg is torsion free as an abelian group when g has property t as a direct corollary of sorin popa s cocycle superrigidity theorem we also use it when n 2 to answer negatively a question of sorin popa on the 2nd cohomology group of bernoulli shift actions of property t groups