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

the computations of the critical groups with an application to elliptic resonant problems at a higher eigenvalue in this paper we study the existence of a nontrivial solution of the elliptic problem which is a resonant type both at the origin and at infinity with the same higher eigenvalue and the nonlinear term is permitted to be unbounded our main ingredient is the computation of the critical groups

multiplicity results for asymptotically linear elliptic problems at resonance in this paper several new multiplicity results for asymptotically linear elliptic problem at resonance are obtained via morse theory and minimax methods some new observations on the critical groups of a local linking type critical point are used to deal with the resonance case at 0

geometrical implications of certain infinite dimensional decompositions we investigate the connections between the global structure of a banach space i e the existence of certain finite and infinite dimensional de compositions and the geometrical properties of the closed convex bounded sub sets of such a space i e the existence of extremal and other topologically distin guished points the global structures of various supposedly pathological examples of banach spaces constructed by r c james turn out to be more universal than expected for instance james tree type resp james matrix type decompositions characterize banach spaces with the point of continuity property resp the radon nikodym property moreover the convex point of continuity property is stable under the formation of james infinitely branching tree type sums of infinite dimensional factors we also give several counterex amples to various questions relating some topological and geometrical concepts in banach spaces

the bochner schoenberg eberlein property for commutative banach algebras especially fourier and fourier stieltjes algebras the classical bochner schoenberg eberlein theorem characterizes the continuous functions on the dual group of a locally compact abelian group g which arise as fourier stieltjes transforms of elements of the measure algebra m g of g this has led to the study of the algebra of bse functions on the spectrum of an arbitrary commutative banach algebra and of the concept of a bse algebra as introduced by takahasi and hatori since then bse algebras have been studied by several authors in this paper we investigate bse algebras in the general context on the one hand and on the other hand we specialize to fourier and fourier stieltjes algebras of locally compact groups

optimal cycles for low dissipation heat engines we consider the optimization of a finite time carnot engine characterized by small dissipations we bound the power with a simple inequality and show that the optimal strategy is to perform small cycles around a given working point which can be thus chosen optimally remarkably this optimal point is independent of the figure of merit combining power and efficiency that is being maximized furthermore for a general class of dissipative dynamics the maximal power output becomes proportional to the heat capacity of the working substance since the heat capacity can scale supraextensively with the number of constituents of the engine this enables us to design optimal many body carnot engines reaching maximum efficiency at finite power per constituent in the thermodynamic limit

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

an elliptic resonance problem with multiple solutions in this paper we study the existence of multiple nontrivial solutions for a semilinear elliptic equation at resonance by the minimax methods and morse theory

the second laws of quantum thermodynamics the second law of thermodynamics places constraints on state transformations it applies to systems composed of many particles however we are seeing that one can formulate laws of thermodynamics when only a small number of particles are interacting with a heat bath is there a second law of thermodynamics in this regime here we find that for processes which are approximately cyclic the second law for microscopic systems takes on a different form compared to the macroscopic scale imposing not just one constraint on state transformations but an entire family of constraints we find a family of free energies which generalize the traditional one and show that they can never increase the ordinary second law relates to one of these with the remainder imposing additional constraints on thermodynamic transitions we find three regimes which determine which family of second laws govern state transitions depending on how cyclic the process is in one regime one can cause an apparent violation of the usual second law through a process of embezzling work from a large system which remains arbitrarily close to its original state these second laws are relevant for small systems and also apply to individual macroscopic systems interacting via long range interactions by making precise the definition of thermal operations the laws of thermodynamics are unified in this framework with the first law defining the class of operations the zeroth law emerging as an equivalence relation between thermal states and the remaining laws being monotonicity of our generalized free energies

on the near differentiability property of banach spaces abstract let be a scalar measure of bounded variation on a compact metrizable abelian group g suppose that has the property that for any measure whose fourier stieltjes transform vanishes at the measure has radon nikod m derivative with respect to the haar measure on g then l pigno and s saeki showed that itself has radon nikod m derivative such property is not shared by vector measures in general we say that a banach space x has the near differentiability property if every x valued measure of bounded variation shares the above property we prove that banach spaces with the radon nikod m property have the near differentiability property while banach spaces with the near differentiability property enjoy the near radon nikod m property we also show that the banach spaces l 1 0 1 and l 1 h 0 1 have the near differentiability property lastly we show that banach spaces with the near differentiability property have type ii radon nikod m property whenever is a riesz subset of type 0 of g

applications of the theory of semi embeddings to banach space theory let x and y be banach spaces and t x y an injective bounded linear operator t is called a semi embedding if t maps the closed unit ball of x to a closed subset of y this concept was introduced by lotz peck and porta proc edinburgh math soc 22 1979 233 240 it is proved that if x semi embeds in y and x is separable then x has the radon nikodym property provided y does it is shown that if l1 semi embeds in y then y fails the schur property and contains a subspace isomorphic to l1 as a consequence of the proof it is shown that if x is a subspace of l1 either l1 embeds in x or l1 embeds in l1x the simpler result that l1 does not semi embed in c0 is treated separately this result is used to deduce the classic result of menchoff that there exists a singular probability measure on the circle with fourier coefficients vanishing at infinity some generalizations of the notion of semi embedding are given and several complements and open questions are discussed

fluctuation theorem for partially masked nonequilibrium dynamics we establish a generalization of the fluctuation theorem for partially masked nonequilibrium dynamics we introduce a partial entropy production with a subset of all possible transitions and show that the partial entropy production satisfies the integral fluctuation theorem our result reveals the fundamental properties of a broad class of autonomous as well as nonautonomous nanomachines in particular our result gives a unified fluctuation theorem for both autonomous and nonautonomous maxwell s demons where mutual information plays a crucial role furthermore we derive a fluctuation dissipation theorem that relates nonequilibrium stationary current to two kinds of equilibrium fluctuations

bound on thermoelectric power in a magnetic field within linear response for thermoelectric power generation in a multiterminal geometry strong numerical evidence for a universal bound as a function of the magnetic field induced asymmetry of the nondiagonal onsager coefficients is presented this bound implies inter alia that the power vanishes at least linearly when the maximal efficiency is approached in particular this result rules out that carnot efficiency can be reached at finite power which an analysis based on the second law only would in principle allow

the three space problem for l 1 we construct a subspace x of l1 such that x is an 1 sum of spaces isomorphic to 11 but such that l1 x does not contain a copy of l1 we also construct two banach spaces e1 e2 that do not contain a copy of li but such that e1 x e2 contains a copy of l1 moreover the projections this content downloaded from 207 46 13 150 on thu 28 jul 2016 05 31 04 utc all use subject to http about jstor org terms the three space problem for l 29 of li on each factor are one to one and the images of the unit ball of l1 are closed these examples settle questions of j lindenstrauss p pelczynski j bourgain and h p rosenthal equipe d analyse tour 46 u a au c n r s n0 754 universitt paris vi 4 place jussieu 75230 paris cedex 05 department of mathematics ohio state university 231 w 18th avenue columbus ohio 43210 this content downloaded from 207 46 13 150 on thu 28 jul 2016 05 31 04 utc all use subject to http about jstor org terms

nontrivial periodic solutions for strong resonance hamiltonian systems via a local linking theorem in this paper we first establish an existence result of critical points for a class of functionals defined on hilbert spaces by using a local linking idea then as an application of the existence result we obtain the existence of periodic solutions of strong resonance hamiltonian systems which are asymptotically linear both at infinity and at origin

the iterated aluthge transforms of a matrix converge abstract given an r r complex matrix t if t u t is the polar decomposition of t then the aluthge transform is defined by t t 1 2 u t 1 2 let n t denote the n times iterated aluthge transform of t i e 0 t t and n t n 1 t n n we prove that the sequence n t n n converges for every r r matrix t this result was conjectured by jung ko and pearcy in 2003 we also analyze the regularity of the limit function

efficiency versus speed in quantum heat engines rigorous constraint from lieb robinson bound a long standing open problem whether a heat engine with finite power achieves the carnot efficiency is investgated we rigorously prove a general trade off inequality on thermodynamic efficiency and time interval of a cyclic process with quantum heat engines in a first step employing the lieb robinson bound we establish an inequality on the change in a local observable caused by an operation far from support of the local observable this inequality provides a rigorous characterization of the following intuitive picture that most of the energy emitted from the engine to the cold bath remains near the engine when the cyclic process is finished using this description we prove an upper bound on efficiency with the aid of quantum information geometry our result generally excludes the possibility of a process with finite speed at the carnot efficiency in quantum heat engines in particular the obtained constraint covers engines evolving with non markovian dynamics which almost all previous studies on this topic fail to address

carnot cycle at finite power attainability of maximal efficiency we want to understand whether and to what extent the maximal carnot efficiency for heat engines can be reached at a finite power to this end we generalize the carnot cycle so that it is not restricted to slow processes we show that for realistic i e not purposefully designed engine bath interactions the work optimal engine performing the generalized cycle close to the maximal efficiency has a long cycle time and hence vanishing power this aspect is shown to relate to the theory of computational complexity a physical manifestation of the same effect is levinthal s paradox in the protein folding problem the resolution of this paradox for realistic proteins allows to construct engines that can extract at a finite power 40 of the maximally possible work reaching 90 of the maximal efficiency for purposefully designed engine bath interactions the carnot efficiency is achievable at a large power

quasinormality of powers of commuting pairs of bounded operators abstract we study jointly quasinormal and spherically quasinormal pairs of commuting operators on hilbert space as well as their powers we first prove that up to a constant multiple the only jointly quasinormal 2 variable weighted shift is the helton howe shift second we show that a left invertible subnormal operator t whose square t 2 is quasinormal must be quasinormal third we generalize a characterization of quasinormality for subnormal operators in terms of their normal extensions to the case of commuting subnormal n tuples fourth we show that if a 2 variable weighted shift w and its powers w 2 1 and w 1 2 are all spherically quasinormal then w may not necessarily be jointly quasinormal moreover it is possible for both w 2 1 and w 1 2 to be spherically quasinormal without w being spherically quasinormal finally we prove that for 2 variable weighted shifts the common fixed points of the toral and spherical aluthge transforms are jointly quasinormal

random embedding of ell_p n into ell_r n for any 0 n we give an explicit definition of a random operator s ell p n to mathbb r n such that for every 0 0 is a real number depending only on p and r one of the main tools that we develop is a new type of multidimensional esseen inequality for studying small ball probabilities

brown measure and iterates of the aluthge transform for some operators arising from measurable actions we consider the aluthge transform t t 1 2 u t 1 2 of a hilbert space operator t where t u t is the polar decomposition of t we prove that the map t t is continuous with respect to the norm topology and with respect to the sot topology on bounded sets we consider the special case in a tracial von neumann algebra when u implements an automorphism of the von neumann algebra generated by the positive part t of t and we prove that the iterated aluthge transform converges to a normal operator whose brown measure agrees with that of t and we compute this brown measure this proof relies on a theorem that is an analogue of von neumann s mean ergodic theorem but for sums weighted by binomial coefficients

radon nikodym property for the projective tensor product of kothe function spaces abstract let e be an order continuous kothe function space or an order continuous banach lattice and x be a dual banach space then e x the projective tensor product of e and x has the radon nikodym property if and only if both e and x do

remarks on multiple nontrivial solutions for quasi linear resonant problems in this paper morse theory and local linking are used to study the existence of multiple nontrivial solutions for a class of dirichlet boundary value problems with double resonance at infinity and at 0

existence and multiplicity results for classes of elliptic resonant problems in this paper we prove some new existence results of nontrivial solutions for classes of elliptic resonant problems we also establish several multiplicity results the methods used here are based on combining the minimax methods and the morse theory especially some new observations on the critical groups of a local linking type degenerate critical point

multiple periodic solutions for asymptotically linear ordinary differential equations with double resonance abstract in this paper using index theory and morse theory we investigate multiple periodic solutions of asymptotically linear ordinary differential equation with double resonance at infinity and with resonance at origin and obtain new results

entropy and the combinatorial dimension we solve talagrand s entropy problem the l2 covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension this extends dudley s theorem on classes of 0 1 valued functions for which the shattering dimension is the vapnik chervonenkis dimension in convex geometry the solution means that the entropy of a convex body k is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of k this has a number of consequences including the optimal elton s theorem and estimates on the uniform central limit theorem in the real valued case

attainability of carnot efficiency with autonomous engines the maximum efficiency of autonomous engines with a finite chemical potential difference is investigated we show that without a particular type of singularity autonomous engines cannot attain the carnot efficiency this singularity is realized in two ways single particle transports and the thermodynamic limit we demonstrate that both of these ways actually lead to the carnot efficiency in concrete setups our results clearly illustrate that the singularity plays a crucial role in the maximum efficiency of autonomous engines

combinatorics of random processes and sections of convex bodies we find a sharp combinatorial bound for the metric entropy of sets in r n and general classes of functions this solves two basic combinatorial conjectures on the empirical processes 1 a class of functions satisfies the uniform central limit theorem if the square root of its combinatorial dimension is integrable 2 the uniform entropy is equivalent to the combinatorial dimension under minimal regularity our method also constructs a nicely bounded coordinate section of a symmetric convex body in r n in the operator theory this essentially proves for all normed spaces the restricted invertibility principle of bourgain and tzafriri

applications of morse theory to some nonlinear elliptic equations with resonance at zero abstract in this paper we study the existence and multiplicity of solutions for some nonlinear elliptic boundary value problems with resonance at zero by applying morse theory we do not impose additional global condition on the nonlinearities except for a subcritical growth condition

nontrivial critical points for asymptotically quadratic function abstract in this paper we study the existence of nontrivial critical points as applications we study the existence of nontrivial solutions of the dirichelet problem for nonlinear elliptic pde and periodic boundary value problem for nonlinear ope

gδ embeddings in hilbert space ii in this paper which is a continuation of 10 we exhibit some topological conditions on a banach space which ensure that it contains isometric copies of infinite dimensional conjugate spaces this result is used to identify a large class of banach spaces that are hereditarily separable duals a method of defining a jamestree sum of a countable number of banach spaces is given it is used to construct various counterexamples for instance there exists for each integer n a banach space that can be mapped into hilbert space via the composition of n but not n 1 g embeddings we also continue the investigation of the global structure of some geometrically defined banach spaces for example it is shown that a separable banach space x with the radon nikodym property r n p has a subspace y with a boundedly complete finite dimensional decomposition f d d such that xy has an f d d and the r n p

universal trade off relation between power and efficiency for heat engines for a general thermodynamic system described as a markov process we prove a general lower bound for dissipation in terms of the square of the heat current thus establishing that nonvanishing current inevitably implies dissipation this leads to a universal trade off relation between efficiency and power with which we rigorously prove that a heat engine with nonvanishing power never attains the carnot efficiency our theory applies to systems arbitrarily far from equilibrium and does not assume any specific symmetry of the model

radon nikodým properties associated with subsets of countable discrete abelian groups with any subset of a countable discrete abelian we associate with it three banach space properties these properties are radon nikodym type properties the relationship between these properties is investigated the results are applied to give new characterizations of riesz subsets and rosenthal subsets of countable discrete abelian groups

type infratype and the elton pajor theorem consider a sequence x i i n of norm one vectors in a banach space for a subsetj of 1 n consider the equivalence constant j between x i i j and the l1 basis and consider k min j cardj k we give a near optimal relationship between the rate of decay of k and the averagee of left sum nolimits i leqq n varepsilon i x i right over all choices of signs in particular we show that one can choosek such that for some universal constantk k e 2 kn and xi k geqq left frac kn e 2 k right 1 2 left log left frac kn e 2 k right right k this is optimal within the logarithmic term we also prove that forp 2 the notions of type and infratype coincide

gδ embeddings in hilbert space abstract it is shown that a separable banach space x has the point of weak to norm continuity property resp the radon nikodym property if and only if there exists a compact g embedding resp an h embedding from x into l2 this solves several questions of j bourgain and h p rosenthal j funct anal 52 1983 it is also shown that every non relatively compact sequence in a banach space with property pc has a difference subsequence which is a boundedly complete basic sequence this solves a question of pelczynski and extends some results of w b johnson and h p rosenthal studia math 43 1972 77 92 various related questions asked in the above bourgain rosenthal reference and by g a edgar and r f wheeler pac j math 115 1984 and n ghoussoub and h p rosenthal math ann 264 1983 321 332 are also settled

speed limit for classical stochastic processes we consider the speed limit for classical stochastic markov processes with and without the local detailed balance condition we find that for both cases a trade off inequality exists between the speed of the state transformation and the entropy production the dynamical activity is related to a time scale and plays a crucial role in the inequality for the dynamics without the local detailed balance condition we use the hatano sasa entropy production instead of the standard entropy production our inequalities consist of the quantities that are commonly used in stochastic thermodynamics and explicitly show underlying physical mechanisms

lieb robinson bounds and the exponential clustering theorem we give a lieb robinson bound for the group velocity of a large class of discrete quantum systems which can be used to prove that a non vanishing spectral gap implies exponential clustering in the ground state of such systems

spectral gap and exponential decay of correlations we study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short range interactions on a wide class of lattices we prove that if two observables anticommute with each other at large distance then the nonvanishing spectral gap implies exponential decay of the corresponding correlation when two observables commute with each other at large distance the connected correlation function decays exponentially under the gap assumption if the observables behave as a vector under the u 1 rotation of a global symmetry of the system we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself rather than just the connected correlation function decays exponentially under the gap assumption on a lattice with a certain self similarity in fractal dimensions d 2 in particular if the system is translationally invariant in one of the spatial directions then this self similarity condition is automatically satisfied we also treat systems with long range power law decaying interactions

nontrivial solutions of operator equations and morse indices of critical points of min max type on montre que si les hypotheses du theoreme de point selle sont satisfaites si dim y 2 et si f 1 c contient seulement des points critiques non degeneres alors f 1 c contient au moins un point critique u 0 tel que 2 indice de morse u 0 dim y

existence and multiplicity results for some elliptic systems with discontinuous nonlinearities abstract in this paper we complete two tasks we first extend the critical point theorem obtained by li and willem s j li m willem applications of local linking to critical point theory j math anal appl 189 1995 6 32 to the nonsmooth case in which the energy functional is locally lipschitz and satisfies the weaker nonsmooth cerami condition then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results

fundamental relation between entropy production and heat current we investigate the fundamental relation between entropy production rate and the speed of energy exchange between a system and baths in classical markov processes we establish the fact that quick energy exchange inevitably induces large entropy production in a quantitative form more specifically we prove two inequalities on instantaneous quantities one is applicable to general markov processes induced by heat baths and the other is applicable only to systems with the local detailed balance condition but is stronger than the former one we demonstrate the physical meaning of our result by applying to some specific setups in particular we show that our inequality is tight in the linear response regime

on the closure of the complex symmetric operators compact operators and weighted shifts abstract we study the closure cso of the set cso of all complex symmetric operators on a separable infinite dimensional complex hilbert space among other things we prove that every compact operator in cso is complex symmetric using a construction of kakutani as motivation we also describe many properties of weighted shifts in cso cso in particular we show that weighted shifts which demonstrate a type of approximate self similarity belong to cso cso as a byproduct of our treatment of weighted shifts we explain several ways in which our result on compact operators is optimal

conjugate operators that preserve the nonconvergence of bounded martingales we show that the conjugate t of an operator t x y with x and y banach spaces satisfies the following dichotomy either t preserves the nonconvergence of bounded martingales in y or there exists a compact operator k x y such that the kernel n t k fails the radon nikod m property

on multiple solutions of a nonlinear dirichlet problem we prove that a semilinear elliptic boundary value problem has five solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues we also prove that if the region is a ball the semilinear elliptic problem has two solutions that change sign and are nonradial