linear response theory for thermally driven quantum open systems this note is a continuation of our recent paper v jaksic y ogata and c a pillet the green kubo formula and onsager reciprocity relations in quantum statistical mechanics commun math phys in press where we have proven the green kubo formula and the onsager reciprocity relations for heat fluxes in thermally driven quantum open systems in this note we extend the derivation of the green kubo formula to heat and charge fluxes and discuss some other generalizations of the model and results of v jaksic y ogata and c a pillet the green kubo formula and onsager reciprocity relations in quantum statistical mechanics commun math phys in press

stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

dynamical systems associated to separated graphs graph algebras and paradoxical decompositions abstract we attach to each finite bipartite separated graph e c a partial dynamical system e c f where e c is a zero dimensional metrizable compact space f is a finitely generated free group and is a continuous partial action of f on e c the full crossed product c algebra o e c c e c f is shown to be a canonical quotient of the graph c algebra c e c of the separated graph e c similarly we prove that for any field k the algebraic crossed product l k ab e c c k e c alg f is a canonical quotient of the leavitt path algebra l k e c of e c the monoid v l k ab e c of isomorphism classes of finitely generated projective modules over l k ab e c is explicitly computed in terms of monoids associated to a canonical sequence of separated graphs using this we are able to construct an action of a finitely generated free group f on a zero dimensional metrizable compact space z such that the type semigroup s z f k is not almost unperforated where k denotes the algebra of clopen subsets of z finally we obtain a characterization of the separated graphs e c such that the canonical partial action of f on e c is topologically free

finite group extensions and the atiyah conjecture the atiyah conjecture for a discrete group g states that the l 2 betti numbers of a finite cw complex with fundamental group g are integers if g is torsion free and in general that they are rational numbers with denominators determined by the finite subgroups of g here we establish conditions under which the atiyah conjecture for a torsion free group g implies the atiyah conjecture for every finite extension of g the most important requirement is that h g z p is isomorphic to the cohomology of the p adic completion of g for every prime p an additional assumption is necessary e g that the quotients of the lower central series or of the derived series are torsion free we prove that these conditions are fulfilled for a certain class of groups which contains in particular artin s pure braid groups and more generally fundamental groups of fiber type arrangements free groups fundamental groups of orientable compact surfaces certain knot and link groups primitive positive one relator groups and products of these therefore every finite in fact every elementary amenable extension of these groups satisfies the atiyah conjecture as a consequence if such an extension h is torsion free then the group ring ch contains no non trivial zero divisors i e h fulfills the zero divisor conjecture in the course of the proof we prove that if these extensions are torsionfree then they have plenty of non trivial torsion free quotients which are virtually nilpotent all of this applies in particular to artin s full braid group therefore answering question b6 on www grouptheory org our methods also apply to the baum connes conjecture this is discussed in 47 where for example the baum connes conjecture is proved for the full braid groups msc 55n25 homology with local coecients 16s34 group rings laurent rings 57m25 knots and links

a simple method and its applications to nonlinear partial differential equations abstract in this paper a simple method is proposed for constructing more general exact solutions of nonlinear partial differential equations we choose the camassa and holm degasperis and procesi equation and the generalized b family equations to illustrate the validity and advantages of the method as a result many new and more general exact solutions are obtained some previous results are extended

double variational principle for mean dimension with potential abstract this paper contributes to the mean dimension theory of dynamical systems we introduce a new concept called mean dimension with potential and develop a variational principle for it this is a mean dimension analogue of the theory of topological pressure we consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function we prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property the basic idea of the proof is a dynamicalization of geometric measure theory

optimal control problem for viscous cahn hilliard equation abstract this paper is concerned with the viscous cahn hilliard equation which arises in the dynamics of viscous first order phase transitions in cooling binary solutions the optimal control under boundary condition is given and the existence of optimal solution to the equation is proved

global existence for a new periodic integrable equation we establish the local well posedness for a new periodic integrable equation we show that the equation has classical solutions that blowup in finite time as well as classical solutions which exist globally in time

from rate distortion theory to metric mean dimension variational principle the purpose of this paper is to point out a new connection between information theory and dynamical systems in the information theory side we consider rate distortion theory which studies lossy data compression of stochastic processes under distortion constraints in the dynamical systems side we consider mean dimension theory which studies how many parameters per iterate we need to describe a dynamical system the main results are new variational principles connecting rate distortion function to metric mean dimension

non commutative central limits non commutative central limit theorems are derived the ccr c algebra of fluctuations is analyzed in detail the stability of the central limit is studied by means of the notion of relative entropy

stability properties of steady water waves with vorticity we present two stability analyses for exact periodic traveling water waves with vorticity the first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth the second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small amplitude or long wave approximation 2006 wiley periodicals inc

semiartinian v rings and semiartinian von neumann regular rings abstract semiartinian right v rings which we call right sv rings form a special class of von neumann regular rings we characterize these rings by the fact that every factor ring imbeds as a subring in a direct product of right full linear rings containing the socle if r is a semiartinian ring with all primitive factor rings artinian then the condition of being an sv ring is right left symmetrical and is equivalent to being regular on the other hand if r is a right and left sv ring then all primitive factor rings of r are artinian for right sv rings whose proper ideals are prime we show that the condition of being unit regular is equivalent to being directly finite on the other hand we show that there exists a directly finite right sv ring which is not unit regular furthermore we provide two constructions for any given ordinal the first one gives a prime unit regular right sv ring of loewy length 1 which is not a left v ring and is hereditary if is a natural number the second one gives a directly infinite right sv ring not a left v ring whose loewy length is 2 if is a natural number and is 1 otherwise these constructions are general enough to produce a wide supply of sv rings starting from given ones

on asymptotically equivalent shallow water wave equations abstract the integrable third order korteweg de vries kdv equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves however at quadratic order this asymptotic expansion produces an entire family of shallow water wave equations that are asymptotically equivalent to each other under a group of nonlinear non local normal form transformations introduced by kodama in combination with the application of the helmholtz operator these kodama helmholtz kh transformations are used to present connections between shallow water waves the integrable fifth order kdv equation and a generalization of the camassa holm ch equation that contains an additional integrable case the dispersion relation of the full water wave problem and any equation in this family agree to fifth order the travelling wave solutions of the ch equation are shown to agree to fifth order with the exact solution

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

pontryagin s maximum principle of optimal control governed by some non well posed semilinear parabolic differential equations this paper concerns about necessary conditions for optimal control problems governed by some semilinear parabolic differential equations which may be non well posed the two point boundary time variable state constraint involves the control set may be non convex

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

about the exactness of the linear response theory for quantum lattice systems with finite range potentials and integrable space clustering we prove the linearity of the response theory when dealing with fluctuation observables

unitary equivalence of temperature dynamics for ideal and locally perturbed fermi gas we consider the local perturbation v varepsilon sum limits x y in mathbb z v v x y chi omega x chi omega y a x a y a y a x of the ideal fermi gas on the lattice v where is a finite subset of v and is its indicator the invertibility of moller morphisms for small is proven it follows that in the cyclic gns representation with respect to kms states the dynamics of ideal and locally perturbed fermi gas are unitary equivalent

optimal solution for the viscous nonlinear dispersive wave equation in this paper we study the optimal control problem for the viscous nonlinear dispersive wave equation we first prove the existence and uniqueness of a weak solution to this equation in a short interval by using the galerkin method furthermore the existence of an optimal solution to the viscous nonlinear dispersive wave equation is proved

projective modules over the gerasimov sakhaev counterexample abstract we investigate the structure of the so called gerasimov sakhaev counterexample which is a particular example of a universal localization and classify both finitely and infinitely generated projective modules over it

conservation laws of the camassa holm equation we use the bi hamiltonian structure of the camassa holm equation to show that its conservation laws h n m are homogeneous with respect to the scaling m lambdam moreover a direct argument is presented proving that h 1 h 2 are of local character finally simple representations of the conservation laws in terms of their variational derivatives are derived and used to obtain a constructive scheme for computation of the h n s

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

the strong atiyah conjecture for virtually cocompact special groups we provide new conditions for the strong atiyah conjecture to lift to finite group extensions in particular we show that fundamental groups of compact special cube complexes satisfy these conditions so the conjecture holds for finite extensions of these groups

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

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

the regular algebra of a quiver let k be a fixed field we attach to each column finite quiver e a von neumann regular k algebra q e in a functorial way the algebra q e is a universal localization of the usual path algebra p e associated with e the monoid of isomorphism classes of finitely generated projective right q e modules is explicitly computed

the bieri neumann strebel invariants via newton polytopes we study the newton polytopes of determinants of square matrices defined over rings of twisted laurent polynomials we prove that such newton polytopes are single polytopes rather than formal differences of two polytopes this result can be seen as analogous to the fact that determinants of matrices over commutative laurent polynomial rings are themselves polynomials rather than rational functions we also exhibit a relationship between the newton polytopes and invertibility of the matrices over novikov rings thus establishing a connection with the invariants of bieri neumann strebel bns via a theorem of sikorav we offer several applications we reprove thurston s theorem on the existence of a polytope controlling the bns invariants of a 3 manifold group we extend this result to free by cyclic groups and the more general descending hnn extensions of free groups we also show that the bns invariants of poincare duality groups of type mathtt f in dimension 3 and groups of deficiency one are determined by a polytope when the groups are assumed to be agrarian that is their integral group rings embed in skew fields the latter result partially confirms a conjecture of friedl we also deduce the vanishing of the newton polytopes associated to elements of the whitehead groups of many groups satisfying the atiyah conjecture we use this to show that the l 2 torsion polytope of friedl luck is invariant under homotopy we prove the vanishing of this polytope in the presence of amenability thus proving a conjecture of friedl luck tillmann

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

optimal control of the viscous weakly dispersive degasperis procesi equation abstract in this paper we study the optimal control problem for the viscous weakly dispersive degasperis procesi equation we deduce the existence and uniqueness of a weak solution to this equation in a short interval by using the galerkin method then according to optimal control theories and distributed parameter system control theories the optimal control of the viscous weakly dispersive degasperis procesi equation under boundary conditions is given and the existence of an optimal solution to the viscous weakly dispersive degasperis procesi equation is proved

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

group ring elements with large spectral density given delta 0 we construct a group g and a group ring element s in mathbb z g such that the spectral measure mu of s fulfils mu 0 varepsilon frac c log varepsilon 1 delta for small varepsilon in particular the novikov shubin invariant of any such s is 0 the constructed examples show that the best known upper bounds on mu 0 varepsilon are not far from being optimal

central limit theorem for locally interacting fermi gas we consider a locally interacting fermi gas in its natural non equilibrium steady state and prove the quantum central limit theorem qclt for a large class of observables a special case of our results concerns finitely many free fermi gas reservoirs coupled by local interactions the qclt for flux observables together with the green kubo formulas and the onsager reciprocity relations previously established jop4 complete the proof of the fluctuation dissipation theorem and the development of linear response theory for this class of models

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

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

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

on turing dynamical systems and the atiyah problem main theorems of the article concern the problem of atiyah on possible values of l 2 betti numbers it is shown that all non negative real numbers are l 2 betti numbers and that many for example all non negative algebraic real numbers are l 2 betti numbers of simply connected manifolds with respect to a free cocompact action also an explicit example is constructed which leads to a simply connected manifold with a transcendental l 2 betti number with respect to an action of the threefold direct product of the lamplighter group mathbb z 2 mathbb z wr mathbb z the main new idea is embedding turing machines into integral group rings the main tool developed generalizes known techniques of spectral computations for certain random walk operators to arbitrary operators in groupoid rings of discrete measured groupoids

dynamics of fluctuations for quantum lattice systems for short range interactions and forl1 space clustering states it is proved that there exists a bonafide time evolution on the set of normal fluctuations this dynamics is applied to derive the notion of equilibrium state of the algebra of fluctuations

free lamplighter groups and a question of atiyah we compute the von neumann dimensions of the kernels of adjacency operators on free lamplighter groups and show that they are irrational thus providing an elementary constructive answer to a question of atiyah

optimal control of two and three dimensional incompressible navier stokes flows the focus of this work is on the development of large scale numerical optimization methods for optimal control of steady incompressible navier stokes flows the control is affected by the suction or injection of fluid on portions of the boundary and the objective function represents the rate at which energy is dissipated in the fluid we develop reduced hessian sequential quadratic programming methods that avoid converging the flow equations at each iteration both quasi newton and newton variants are developed and compared to the approach of eliminating the flow equations and variables which is effectively the generalized reduced gradient method optimal control problems are solved for two dimensional flow around a cylinder and three dimensional flow around a sphere the examples demonstrate at least an order of magnitude reduction in time taken allowing the optimal solution of flow control problems in as little as half an hour on a desktop workstation

infinite propagation speed for the degasperis procesi equation abstract we prove that any nontrivial classical solution of the degasperis procesi equation will not have compact support if its initial data has this property

optimal control of the viscous degasperis procesi equation this paper studies the problem for optimal control of the viscous degasperis procesi equation the existence and uniqueness of weak solution to the equation are proved in a short interval the optimal control of the viscous degasperis procesi equation under boundary condition is given and the existence of optimal solution to the equation is proved

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

state constrained optimal control problems governed by coupled nonlinear wave equations with memory this work considers the optimal control problems governed by coupled nonlinear wave equations with memory where the state constraint is a type of integral first we investigate the existence of the optimal solution for the optimal control problem and then we deduce the maximum principle for the optimal solution with state constraint moreover we give a concrete example which is covered by the main result

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

the regular algebra of a poset let k be a fixed field we attach to each finite poset p a von neumann regular k algebra q k p in a functorial way we show that the monoid of isomorphism classes of finitely generated projective q k p modules is the abelian monoid generated by p with the only relations given by p p q q whenever q p in p this extends the class of monoids for which there is a positive solution to the realization problem for von neumann regular rings

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

a tracial quantum central limit theorem we prove a central limit theorem for non commutative random variables in a von neumann algebra with a tracial state any non commutative polynomial of averages of i i d samples converges to a classical limit the proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the baker campbell hausdorff expansion the result can be considered a generalization of johansson s theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity math co 9906120 math pr 9909104

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

the green kubo formula and the onsager reciprocity relations in quantum statistical mechanics we study linear response theory in the general framework of algebraic quantum statistical mechanics and prove the green kubo formula and the onsager reciprocity relations for heat fluxes generated by temperature differentials our derivation is axiomatic and the key assumptions concern ergodic properties of non equilibrium steady states

traveling wave solutions of the degasperis procesi equation we classify all weak traveling wave solutions of the degasperis procesi equation in addition to smooth and peaked solutions the equation is shown to admit more exotic traveling waves such as cuspons stumpons and composite waves

an efficient computational method of boundary optimal control problems for the burgers equation abstract the burgers equation is a simple one dimensional model of the navier stokes equation in the present paper we suggest an efficient method of solving optimal boundary control problems of the burgers equation which is practical as well as mathematically rigorous our eventual purpose is to extend this technique to the control problems of viscous fluid flows the present method is based on the karhunen loeve decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system employing these empirical eigenfunctions as basis functions of a galerkin procedure one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena and consequently reduce the burgers equation to a set of ordinary differential equations with a minimum degree of freedom the resulting low dimensional model of burgers equation is shown to simulate the original system almost exactly the present algorithm is well suited for the problems of control or optimization where one has to solve the governing equation repeatedly but one can also estimate the approximate solution space based on the range of control variables the present method of solving boundary control problems of burgers equation employing the lowdimensional model obtained by the karhunen loeve galerkin procedure is found to yield accurate results in a very efficient way when the minimization of the objective function is performed using a conjugate gradient method

coxeter groups are virtually special abstract in this paper we prove that every finitely generated coxeter group has a finite index subgroup that is the fundamental group of a special cube complex some consequences include every f g coxeter group is virtually a subgroup of a right angled coxeter group every word hyperbolic coxeter group has separable quasiconvex subgroups

central limit theorem for mixing quantum systems and the ccr algebra of fluctuations we analyse macroscopic fluctuations of an infinite quantum system and introduce the ccr c algebra of normal fluctuations a non commutative central limit theorem for mixing quantum systems is proved

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

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

sofic mean length abstract given a length function l on the r modules of a unital ring r for each sofic group we define a mean length for every locally l finite r module relative to a bigger r module we establish an addition formula for the mean length we give two applications the first one shows that for any unital left noetherian ring r r is stably direct finite the second one shows that for any z module m the mean topological dimension of the induced action on the pontryagin dual of m coincides with the von neumann luck rank of m

on the inverse spectral problem for the camassa holm equation abstract a key basis for seeking periodic solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y my the periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum the potential can then be reconstructed from the periodic spectrum a necessary and sufficient condition for exponential decrease of the widths 2n 2n 1for a sequence 0 m c1 r

groups with the same cohomology as their profinite completions abstract for any positive integer n a n is the class of all groups g such that for 0 i n h i g a h i g a for every finite discrete g module a we describe certain types of free products with amalgam and hnn extensions that are in some of the classes a n in addition we investigate the residually finite groups in the class a 2

global wellposedness of kdv in h 1 t r by the inverse method we show that the korteweg de vries equation kdv tv x t x3v x t 6v x t xv x t x t t r is globally in time wellposed in the sobolev space of distributions h t r for any 1

time domain decomposition of optimal control problems for the wave equation we consider the problem of boundary optimal control of a wave equation with boundary dissipation by the way of time domain decomposition of the corresponding optimality system we develop an iterative algorithm which shows that the decomposed optimality system corresponds to local in time optimal control problems which can be treated in parallel we show convergence of the algorithm finally we provide a time discretization which is reminiscent of an instantaneous control scheme we thereby also contribute to the problem of convergence of such schemes

a quantum central limit theorem for sums of independent identically distributed random variables we formulate and prove a general central limit theorem for sums of independent identically distributed noncommutative random variables

cohomological goodness and the profinite completion of bianchi groups the concept of cohomological goodness was introduced by j p serre in his book on galois cohomology 31 this property relates the cohomology groups of a group to those of its profinite completion we develop properties of goodness and establish goodness for certain important groups we prove for example that the bianchi groups i e the groups psl 2 o where o is the ring of integers in an imaginary quadratic number field are good as an application of our improved understanding of goodness we are able to show that certain natural central extensions of fuchsian groups are residually finite this result contrasts with examples of p deligne 5 who shows that the analogous central extensions of sp 4 z do not have this property