one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

convergence of numerical approximations to non linear continuity equations with rough force fields we prove quantitative regularity estimates for the solutions to non linear continuity equations and their discretized numerical approximations on cartesian grids when advected by a rough force field this allows us to not only recover the known optimal regularity for linear transport equations but also to obtain the convergence of a wide range of numerical schemes our proof is based on novel commutator estimates quantifying and extending to the non linear case the classical commutator approach of the theory of renormalized solutions

on the domain of hyperbolicity of the cumulant equations in this article we consider the influence of non equilibirum values of classical variables on the eigenvalues of the advection part of the cumulant equations real and finite eigenvalues are a neccessary condition for the cumulant equations to be hyperbolic which can be used to obtain estimates on admissible deviations from equilibrium for a model of particular order still to be valid we find that this condition puts no constraints on velocity and shear stress values but specific energy must be positive normal stress must be bounded by specific energy and heat flux not be too large

compactness for nonlinear continuity equations abstract we prove compactness and hence existence for solutions to a class of nonlinear transport equations the corresponding models combine the features of linear transport equations and scalar conservation laws we introduce a new method which gives quantitative compactness estimates compatible with both frameworks

weak entropy boundary conditions for isentropic gas dynamics via kinetic relaxation abstract we consider a kinetic bgk model relaxing to isentropic gas dynamics previously introduced by the authors but with dirichlet boundary condition on the incoming velocities we pass to the limit as the relaxation parameter tends to zero by compensated compactness inside the domain and obtain that the limit satisfies entropy inequalities on the boundary involving weak traces of entropy fluxes our method is very general and could be applied to any entropy satisfying bgk model as soon as we have strong compactness of the macroscopic variables inside the domain

the boltzmann equation for weakly inhomogeneous data we solve the initial value problem associated to the nonlinear boltzmann equation in the case in which the initial distribution has sufficiently small spatial gradients

fine properties of functions with bounded deformation the paper is concerned with the fine properties of functions in the space of functions with bounded deformation we analyse the set of lebesgue points and the set where these functions have one sided approximate limits moreover following the analogy with we decompose the symmetric distributional derivative into an absolutely continuous part a jump part and a cantor part the main result of the paper is a structure theorem for functions showing that these parts of the derivative can be recovered from the corresponding ones of the one dimensional sections moreover we prove that functions are approximately differentiable in almost every point of their domain

renormalized solutions to the vlasov equation with coefficients of bounded variation we prove that weak bounded solutions to the vlasov equation with bv coefficients have the renormalization property and we show that when the renormalization property holds for a general transport equation it also holds for only lipschitz nonlinearities

euler poisson systems as action minimizing paths in the wasserstein space this paper uses a variational approach to establish existence of solutions t vt for the 1 d euler poisson system by minimizing an action we assume that the initial and terminal points 0 t are prescribed in mathcal p 2 mathbb r the set of borel probability measures on the real line of finite second order moments we show existence of a unique minimizer of the action when the time interval 0 t satisfies t these solutions conserve the hamiltonian and they yield a path t t in mathcal p 2 mathbb r when t y t is a dirac mass the euler poisson system reduces to ddot y y 0 the kinetic version of the euler poisson that is the vlasov poisson system was studied in ambrosio and gangbo comm pure appl math to appear as a hamiltonian system

trading classical communication quantum communication and entanglement in quantum shannon theory in this paper we give tradeoffs between classical communication quantum communication and entanglement for processing information in the shannon theoretic setting we first prove a unit resource capacity theorem that applies to the scenario where only the above three noiseless resources are available for consumption or generation the optimal strategy mixes the three fundamental protocols of teleportation superdense coding and entanglement distribution we then provide an achievable rate region and a matching multiletter converse for the direct static capacity theorem this theorem applies to the scenario where a large number of copies of a noisy bipartite state are available in addition to consumption or generation of the above three noiseless resources our coding strategy involves a protocol that we name the classically assisted state redistribution protocol and the three fundamental protocols we finally provide an achievable rate region and a matching multiletter converse for the direct dynamic capacity theorem this theorem applies to the scenario where a large number of uses of a noisy quantum channel are available in addition to the consumption or generation of the three noiseless resources our coding strategy combines the classically enhanced father protocol with the three fundamental unit protocols

global existence of weak solutions for compresssible navier stokes equations thermodynamically unstable pressure and anisotropic viscous stress tensor we prove global existence of appropriate weak solutions for the compressible navier stokes equations for more general stress tensor than those covered by p l lions and e feireisl s theory more precisely we focus on more general pressure laws which are not thermodynamically stable we are also able to handle some anisotropy in the viscous stress tensor to give answers to these two longstanding problems we revisit the classical compactness theory on the density by obtaining precise quantitative regularity estimates this requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation these two cases open the theory to important physical applications for instance to describe solar events virial pressure law geophysical flows eddy viscosity or biological situations anisotropy

a resource framework for quantum shannon theory quantum shannon theory is loosely defined as a collection of coding theorems such as classical and quantum source compression noisy channel coding theorems entanglement distillation etc which characterize asymptotic properties of quantum and classical channels and states in this paper we advocate a unified approach to an important class of problems in quantum shannon theory consisting of those that are bipartite unidirectional and memoryless

hamiltonian odes in the wasserstein space of probability measures in this paper we consider a hamiltonian h on p 2 r 2d the set of probability measures with finite quadratic moments on the phase space r 2d r d r d which is a metric space when endowed with the wasserstein distance w 2 we study the initial value problem d t dt j d v t t 0 where j d is the canonical symplectic matrix 0 is prescribed and v t is a tangent vector to p 2 r 2d at t belonging to h t the subdifferential of h at t two methods for constructing solutions of the evolutive system are provided the first one concerns only the case where 0 is absolutely continuous it ensures that fit remains absolutely continuous and v t h t is the element of minimal norm in h t the second method handles any initial measure 0 if we further assume that h is convex proper and lower semicontinuous on p 2 r 2d we prove that the hamiltonian is preserved along any solution of our evolutive system h t h 0

continuity of quantum conditional information we prove continuity of quantum conditional information s 12 2 with respect to the uniform convergence of states and obtain a bound which is independent of the dimension of the second party this can e g be used to prove the continuity of squashed entanglement

classical communication over a quantum interference channel calculating the capacity of interference channels is a notorious open problem in classical information theory such channels have two senders and two receivers and each sender would like to communicate with a partner receiver the capacity of such channels is known exactly in the settings of very strong and strong interference while the han kobayashi coding strategy gives the best known achievable rate region in the general case here we introduce and study the quantum interference channel a natural generalization of the interference channel to the setting of quantum information theory we restrict ourselves for the most part to channels with two classical inputs and two quantum outputs in order to simplify the presentation of our results though generalizations of our results to channels with quantum inputs are straightforward we are able to determine the exact classical capacity of this channel in the settings of very strong and strong interference by exploiting winter s successive decoding strategy and a novel two sender quantum simultaneous decoder respectively we provide a proof that a han kobayashi strategy is achievable with holevo information rates up to a conjecture regarding the existence of a three sender quantum simultaneous decoder this conjecture holds for a special class of quantum multiple access channels with average output states that commute and we discuss some other variations of the conjecture that hold finally we detail a connection between the quantum interference channel and prior work on the capacity of bipartite unitary gates

capacity theorems for quantum multiple access channels classical quantum and quantum quantum capacity regions in this paper we consider quantum channels with two senders and one receiver for an arbitrary such channel we give multiletter characterizations of two different two dimensional capacity regions the first region comprises the rates at which it is possible for one sender to send classical information while the other sends quantum information the second region consists of the rates at which each sender can send quantum information for each region we give an example of a channel for which the corresponding region has a single letter description one of our examples relies on a new result proved here perhaps of independent interest stating that the coherent information over any degradable channel is concave in the input density operator we conclude with connections to other work and a discussion on generalizations where each user simultaneously sends classical and quantum information

critical non sobolev regularity for continuity equations with rough velocity fields abstract we present a divergence free vector field in the sobolev space h 1 such that the flow associated to the field does not belong to any sobolev space the vector field is deterministic but constructed as the realization of a random field combining simple elements this construction illustrates the optimality of recent quantitative regularity estimates as it gives a straightforward example of a well posed flow which has nevertheless only very weak regularity

kinetic relaxation to entropy based coupling conditions for isentropic flow on networks abstract we consider networks for isentropic gas and prove existence of weak solutions for a large class of coupling conditions first we construct approximate solutions by a vector valued bgk model with a kinetic coupling function introducing so called kinetic invariant domains and using the method of compensated compactness justifies the relaxation towards the isentropic gas equations we will prove that certain entropy flux inequalities for the kinetic coupling function remain true for the traces of the macroscopic solution these inequalities define the macroscopic coupling condition our techniques are also applicable to networks with arbitrary many junctions which may possibly contain circles we give several examples for coupling functions and prove corresponding entropy flux inequalities we prove also new existence results for solid wall boundary conditions and pipelines with discontinuous cross sectional area

a communication efficient nonlocal measurement with application to communication complexity and bipartite gate capacities two dual questions in quantum information theory are to determine the communication cost of simulating a bipartite unitary gate and to determine their communication capacities we present a bipartite unitary gate with two surprising properties 1 simulating it with the assistance of unlimited epr pairs requires far more communication than with a better choice of entangled state and 2 its communication capacity is far lower than its capacity to create entanglement this suggests that 1 unlimited epr pairs are not the most general model of entanglement assistance for two party communication tasks and 2 the entangling and communicating abilities of a unitary interaction can vary nearly independently the technical contribution behind these results is a communication efficient protocol for measuring whether an unknown shared state lies in a specified rank one subspace or its orthogonal complement

the capacity of the quantum channel with general signal states it is shown that the capacity of a classical quantum channel with arbitrary possibly mixed states equals the maximum of the entropy bound with respect to all a priori distributions this completes the recent result of hausladen jozsa schumacher westmoreland and wootters 1996 who proved the equality for the pure state channel

existence and uniqueness of maximal regular flows for non smooth vector fields in this paper we provide a complete analogy between the cauchy lipschitz and the diperna lions theories for ode s by developing a local version of the diperna lions theory more precisely we prove the existence and uniqueness of a maximal regular flow for the diperna lions theory using only local regularity and summability assumptions on the vector field in analogy with the classical theory which uses only local regularity assumptions we also study the behaviour of the ode trajectories before the maximal existence time unlike the cauchy lipschitz theory this behaviour crucially depends on the nature of the bounds imposed on the spatial divergence of the vector field in particular a global assumption on the divergence is needed to obtain a proper blow up of the trajectories

quantum rate distortion reverse shannon theorems and source channel separation we derive quantum counterparts of two key theorems of classical information theory namely the rate distortion theorem and the source channel separation theorem the rate distortion theorem gives the ultimate limits on lossy data compression and the source channel separation theorem implies that a two stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel in spite of their importance in the classical domain there has been surprisingly little work in these areas for quantum information theory in this paper we prove that the quantum rate distortion function is given in terms of the regularized entanglement of purification we also determine a single letter expression for the entanglement assisted quantum rate distortion function and we prove that it serves as a lower bound on the unassisted quantum rate distortion function this implies that the unassisted quantum rate distortion function is nonnegative and generally not equal to the coherent information between the source and distorted output in spite of barnum s conjecture that the coherent information would be relevant here moreover we prove several quantum source channel separation theorems the strongest of these are in the entanglement assisted setting in which we establish a necessary and sufficient condition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion

convergence of a high order semi lagrangian scheme with propagation of gradients for the one dimensional vlasov poisson system in this paper we give a proof of convergence of a new numerical method introduced in n besse and e sonnendrucker j comput phys 191 2003 pp 341 376 for the vlasov equation the numerical method is based on the semi lagrangian principle and the transport of the gradient of the statistical distribution function in order to get a high order and stable reconstruction these kinds of new schemes have been successfully implemented on unstructured meshes of four dimensional phase space cf n besse etude mathematique et numerique de l equation de vlasov sur des maillages non structures de l espace des phases these de l universite louis pasteur strasbourg france 2003 n besse j segre and e sonnendrucker transport theory statist phys 34 2005 pp 311 332 in order to make a rigorous proof of convergence of this method and simplify the convergence analysis we have considered the periodic one dimensional vlasov poisson system in phase space on a grid the distribution f t x v and the electric field are shown to converge to the exact solution values in h 1 norm the rate of convergence is of mathcal o delta t 2 frac delta x 4 alpha delta t frac delta v 4 alpha delta t alpha in n 2 alpha leq 1

optimal quantum source coding with quantum side information at the encoder and decoder consider many instances of an arbitrary quadripartite pure state of four quantum systems abcd alice holds the ac part of each state bob holds b while r represents all other parties correlated with abc alice is required to redistribute the c systems to bob while asymptotically preserving the overall purity we prove that this is possible using q qubits of communication and e ebits of shared entanglement between alice and bob provided that q ges 1 2i c d b and q e ges h c b proving the optimality of the luo devetak outer bound the optimal qubit rate provides the first known operational interpretation of quantum conditional mutual information we also show how our protocol leads to a fully operational proof of strong subaddivity and uncover a general organizing principle in analogy to thermodynamics that underlies the optimal rates

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

existence and uniqueness of martingale solutions for sdes with rough or degenerate coefficients abstract in this paper we extend recent results on the existence and uniqueness of solutions of odes with non smooth vector fields to the case of martingale solutions in the stroock varadhan sense of sdes with non smooth coefficients in the first part we develop a general theory which roughly speaking allows to deduce existence uniqueness and stability of martingale solutions for l d almost every initial condition x whenever existence and uniqueness is known at the pde level in the l setting and conversely if existence and uniqueness of martingale solutions is known for l d a e initial condition then existence and uniqueness for the pde holds in the second part of the paper we consider situations where on the one hand no pointwise uniqueness result for the martingale problem is known and on the other hand well posedness for the fokker planck equation can be proved thus the theory developed in the first part of the paper is applicable in particular we will study the fokker planck equation in two somehow extreme situations in the first one assuming uniform ellipticity of the diffusion coefficients and lipschitz regularity in time we are able to prove existence and uniqueness in the l 2 setting in the second one we consider an additive noise and assuming the drift b to have bv regularity and allowing the diffusion matrix a to be degenerate also identically 0 we prove existence and uniqueness in the l setting therefore in these two situations our theory yields existence uniqueness and stability results for martingale solutions

channel capacities via p summing norms abstract in this paper we show how the metric theory of tensor products developed by grothendieck perfectly fits in the study of channel capacities a central topic in shannon s information theory furthermore in the last years shannon s theory has been fully generalized to the quantum setting and revealed qualitatively new phenomena in comparison in this paper we consider the classical capacity of quantum channels with restricted assisted entanglement these capacities include the classical capacity and the unlimited entanglement assisted classical capacity of a quantum channel our approach to restricted capacities is based on tools from functional analysis and in particular the notion of p summing maps going back to grothendieck s work pisier s noncommutative vector valued l p spaces allow us to establish the new connection between functional analysis and information theory in the quantum setting

continuity equations and ode flows with non smooth velocity in this paper we review many aspects of the well posedness theory for the cauchy problem for the continuity and transport equations and for the ordinary differential equation ode in this framework we deal with velocity fields that are not smooth but enjoy suitable weak differentiability assumptions we first explore the connection between the partial differential equation pde and the ode in a very general non smooth setting then we address the renormalization property for the pde and prove that such a property holds for sobolev velocity fields and for bounded variation velocity fields finally we present an approach to the ode theory based on quantitative estimates

semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data in this paper we study the semiclassical limit of the schrodinger equation under mild regularity assumptions on the potential u which include born oppenheimer potential energy surfaces in molecular dynamics we establish asymptotic validity of classical dynamics globally in space and time for almost all initial data with respect to an appropriate reference measure on the space of initial data in order to achieve this goal we prove existence uniqueness and stability results for the flow in the space of measures induced by the continuity equation 2011 wiley periodicals inc

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

coding theorem and strong converse for quantum channels we present a new proof of holevo s 1973 1977 coding theorem for transmitting classical information through quantum channels and its strong converse the technique is largely inspired by wolfwitz s 1964 combinatorial approach using types of sequences as a byproduct of our approach which is independent of previous ones both in the coding theorem and the converse we can give a new proof of holevo s information bound

a discrete bgk approximation for strongly degenerate parabolic problems with boundary conditions abstract we consider a class of bgk systems with a finite number of velocities depending on a positive relaxation parameter that approximate strongly degenerate hyperbolic parabolic equations with initial boundary conditions we prove a priori estimates for the solutions of the systems showing that these functions converge towards the entropy solutions of strongly degenerate problems when the relaxation parameter goes to zero

on quantum fidelities and channel capacities for the discrete memoryless quantum channel we show the equivalence of two different notions of quantum channel capacity that which uses the entanglement fidelity as its criterion for success in transmission and that which uses the minimum fidelity of pure states in a subspace of the input hilbert space as its criterion as a corollary any source with entropy rate less than the capacity may be transmitted with high entanglement fidelity we also show that a restricted class of encodings is sufficient to transmit any quantum source which may be transmitted on a given channel this enables us to simplify a known upper bound for the channel capacity it also enables us to show that the availability of an auxiliary classical channel from encoder to decoder does not increase the quantum capacity

entanglement assisted capacity of a quantum channel and the reverse shannon theorem the entanglement assisted classical capacity of a noisy quantum channel c sub e is the amount of information per channel use that can be sent over the channel in the limit of many uses of the channel assuming that the sender and receiver have access to the resource of shared quantum entanglement which may be used up by the communication protocol we show that the capacity c sub e is given by an expression parallel to that for the capacity of a purely classical channel i e the maximum over channel inputs spl rho of the entropy of the channel input plus the entropy of the channel output minus their joint entropy the latter being defined as the entropy of an entangled purification of spl rho after half of it has passed through the channel we calculate entanglement assisted capacities for two interesting quantum channels the qubit amplitude damping channel and the bosonic channel with amplification attenuation and gaussian noise we discuss how many independent parameters are required to completely characterize the asymptotic behavior of a general quantum channel alone or in the presence of ancillary resources such as prior entanglement in the classical analog of entanglement assisted communication communication over a discrete memoryless channel dmc between parties who share prior random information we show that one parameter is sufficient i e that in the presence of prior shared random information all dmcs of equal capacity can simulate one another with unit asymptotic efficiency

entanglement assisted capacity of quantum multiple access channels we find a regularized formula for the entanglement assisted ea capacity region for quantum multiple access channels qmac we illustrate the capacity region calculation with the example of the collective phase flip channel which admits a single letter characterization on the way we provide a first principles proof of the ea coding theorem based on a packing argument we observe that the holevo schumacher westmoreland theorem may be obtained from a modification of our ea protocol we remark on the existence of a family hierarchy of protocols for multiparty scenarios with a single receiver in analogy to the two party case in this way we relate several previous results regarding qmacs

entanglement assisted communication of classical and quantum information in this paper we consider the problem of transmitting classical and quantum information reliably over an entanglement assisted quantum eaq channel our main result is a capacity theorem that gives a 3 d achievable rate region points in the region are rate triples consisting of the classical communication rate the quantum communication rate and the entanglement consumption rate of a particular coding scheme the crucial protocol in achieving the boundary points of the capacity region is a protocol that we name the classically enhanced father cef protocol the cef protocol is more general than other protocols in the family tree of quantum shannon theoretic protocols in the sense that several previously known quantum protocols are now child protocols of it the cef protocol also shows an improvement over a timesharing strategy for the case of a qubit dephasing channel this result justifies the need for simultaneous coding of classical and quantum information over an eaq channel our capacity theorem is of a multiletter nature requiring a limit over many uses of the channel but it reduces to a single letter characterization for at least three channels the completely depolarizing channel the quantum erasure channel and the qubit dephasing channel

solutions for linear conservation laws with velocity fields in l rm infty a space time integrated least squares stils method is derived for solving the linear conservation law with a velocity field in l rm infty an existence and uniqueness result is given for the solution of this equation a maximum principle is established and finally a comparison with a renormalized solution is presented

tight uniform continuity bounds for quantum entropies conditional entropy relative entropy distance and energy constraints we present a bouquet of continuity bounds for quantum entropies falling broadly into two classes first a tight analysis of the alicki fannes continuity bounds for the conditional von neumann entropy reaching almost the best possible form that depends only on the system dimension and the trace distance of the states almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators as applications we give new proofs with tighter bounds of the asymptotic continuity of the relative entropy of entanglement er and its regularization e r infty as well as of the entanglement of formation ef using a novel quantum coupling of density operators which may be of independent interest we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation aka entanglement cost e c e f infty second we derive analogous continuity bounds for the von neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint most importantly systems of multiple quantum harmonic oscillators while without an energy bound the entropy is discontinuous it is well known to be continuous on states of bounded energy however a quantitative statement to that effect seems not to have been known here under some regularity assumptions on the hamiltonian we find that quite intuitively the gibbs entropy at the given energy roughly takes the role of the hilbert space dimension in the finite dimensional fannes inequality

catalytic quantum error correction we develop the theory of entanglement assisted quantum error correcting eaqec codes a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to preshared entanglement conventional stabilizer codes are equivalent to self orthogonal symplectic codes in contrast eaqec codes do not require self orthogonality which greatly simplifies their construction we show how any classical binary or quaternary block code can be made into an eaqec code we provide a table of best known eaqec codes with code length up to 10 with the self orthogonality constraint removed we see that the distance of an eaqec code can be better than any standard quantum error correcting code with the same fixed net yield in a quantum computation setting eaqec codes give rise to catalytic quantum codes which assume a subset of the qubits are noiseless we also give an alternative construction of eaqec codes by making classical entanglement assisted codes coherent

entanglement cost of quantum channels the entanglement cost of a quantum channel is the minimal rate at which entanglement between sender and receiver is needed in order to simulate many copies of a quantum channel in the presence of free classical communication in this paper we show how to express this quantity as a regularized optimization of the entanglement formation over states that can be generated between sender and receiver our formula is the channel analog of a well known formula for the entanglement cost of quantum states in terms of the entanglement of formation and shares a similar relation to the recently shattered hope for additivity the entanglement cost of a quantum channel can be seen as the analog of the quantum reverse shannon theorem in the case where free classical communication is allowed the techniques used in the proof of our result are then also inspired by a recent proof of the quantum reverse shannon theorem and feature the one shot formalism for quantum information theory the postselection technique for quantum channels as well as sion s minimax theorem we discuss two applications of our result first we are able to link the security in the noisy storage model to a problem of sending quantum rather than classical information through the adversary s storage device this not only improves the range of parameters where security can be shown but also allows us to prove security for storage devices for which no results were known before second our result has consequences for the study of the strong converse quantum capacity here we show that any coding scheme that sends quantum information through a quantum channel at a rate larger than the entanglement cost of the channel has an exponentially small fidelity

global nonlinear stability for steady ideal fluid flow in bounded planar domains we prove stability of steady flows of an ideal fluid in a bounded simply connected planar region that are strict maximisers or minimisers of kinetic energy on an isovortical surface the proof uses conservation of energy and transport of vorticity for solutions of the vorticity equation with initial data in lp for p 4 3 a related stability theorem using conservation of angular momentum in a circular domain is also proved

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice iii from ab initio models to wkb for schrodinger poisson this work is concerned with the semiclassical approximation of the schrodinger poisson equation modeling ballistic transport in a 1d periodic potential by means of wkb techniques it is derived by considering the mean field limit of a n body quantum problem then k multivalued solutions are adapted to the treatment of this weakly nonlinear system obtained after homogenization without taking into account for pauli s exclusion principle numerical experiments display the behaviour of self consistent wave packets and screening effects

a hamiltonian preserving scheme for the liouville equation of geometrical optics with partial transmissions and reflections we construct a class of hamiltonian preserving numerical schemes for the liouville equation of geometrical optics with partial transmissions and reflections this equation arises in the high frequency limit of the linear wave equation with a discontinuous index of refraction in our previous work hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds j comput phys 214 2006 pp 672 697 we introduced the hamiltonian preserving schemes for the same equation when only complete transmissions or reflections occur at the interfaces these schemes are extended in this paper to the general case of partial transmissions and reflections the key idea is to build into the numerical flux the behavior of waves at the interface namely partial transmissions and reflections that satisfy snell s law of refraction with the correct transmission and reflection coefficients this scheme allows a hyperbolic stability condition under which positivity and stabilities in both l 1 and l infty norms are established numerical experiments are carried out to study the numerical accuracy