successive refinement with decoder cooperation and its channel coding duals we study cooperation in multiterminal source coding models involving successive refinement specifically we study the case of a single encoder and two decoders where the encoder provides a common description to both the decoders and a private description to only one of the decoders the decoders cooperate via cribbing i e the decoder with access only to the common description is allowed to observe in addition a deterministic function of the reconstruction symbols produced by the other we characterize the fundamental performance limits in the respective settings of noncausal strictly causal and causal cribbing we use a coding scheme referred to as forward encoding and block markov decoding which builds on one recently used by cuff and zhao for coordination via implicit communication finally we use the insight gained to introduce and solve some dual channel coding scenarios involving multiple access channels with cribbing

three positive solutions for a generalized laplacian boundary value problem with a parameter concerned is a generalized laplacian boundary value problem with a positive parameter first we apply the leggett williams fixed point theorem to establish sufficient conditions on the existence of at least three positive solutions for the parameter belonging to an explicit interval then under a little bit stronger assumptions we show that there are at least three positive symmetric solutions for the parameter in an open interval the obtained results are new even for laplacian boundary value problems and they are illustrated with an example

on quantum coding for ensembles of mixed states we consider the problem of optimal asymptotically faithful compression for ensembles of mixed quantum states although the optimal rate is unknown we prove upper and lower bounds and describe a series of illustrative examples of compression of mixed states we also discuss a classical analogue of the problem

duality between channel capacity and rate distortion with two sided state information we show that the duality between channel capacity and data compression is retained when state information is available to the sender to the receiver to both or to neither we present a unified theory for eight special cases of channel capacity and rate distortion with state information which also extends existing results to arbitrary pairs of independent and identically distributed i i d correlated state information s sub 1 s sub 2 available at the sender and at the receiver respectively in particular the resulting general formula for channel capacity c max sub p u x s sub 1 i u s sub 2 y i u s sub 1 assumes the same form as the generalized wyner ziv 1976 rate distortion function r d min sub p u x s sub 1 p x spl i cap u s sub 2 i u s sub 1 x 1 u s sub 2

trading quantum for classical resources in quantum data compression we study the visible compression of a source script e phii pi of pure quantum signal states or more formally the minimal resources per signal required to represent arbitrarily long strings of signals with arbitrarily high fidelity when the compressor is given the identity of the input state sequence as classical information according to the quantum source coding theorem the optimal quantum rate is the von neumann entropy s script e qubits per signal we develop a refinement of this theorem in order to analyze the situation in which the states are coded into classical and quantum bits that are quantified separately this leads to a trade off curve q r where q r qubits per signal is the optimal quantum rate for a given classical rate of r bits per signal our main result is an explicit characterization of this trade off function by a simple formula in terms of only single signal perfect fidelity encodings of the source we give a thorough discussion of many further mathematical properties of our formula including an analysis of its behavior for group covariant sources and a generalization to sources with continuously parametrized states we also show that our result leads to a number of corollaries characterizing the trade off between information gain and state disturbance for quantum sources in addition we indicate how our techniques also provide a solution to the so called remote state preparation problem finally we develop a probability free version of our main result which may be interpreted as an answer to the question how many classical bits does a qubit cost this theorem provides a type of dual to holevo s theorem insofar as the latter characterizes the cost of coding classical bits into qubits

markovianizing cost of tripartite quantum states we introduce and analyze a task that we call markovianization in which a tripartite quantum state is transformed to a quantum markov chain by a randomizing operation on one of the three subsystems we consider cases where the initial state is the tensor product of n copies of a tripartite state rho abc and is transformed to a quantum markov chain conditioned by b n with a small error using a random unitary operation on a n in an asymptotic limit of infinite copies and vanishingly small error we analyze the markovianizing cost that is the minimum cost of randomness per copy required for markovianization for tripartite pure states we derive a single letter formula for the markovianizing costs counterintuitively the markovianizing cost is not a continuous function of states and can be arbitrarily large even if the state is close to a quantum markov chain our results have an application in analyzing the cost of resources for simulating a bipartite unitary gate by local operations and classical communication

positive solutions for three point boundary value problems with dependence on the first order derivative abstract a new fixed point theorem in a cone is applied to obtain the existence of at least one positive solution for the second order three point boundary value problem x f t x x 0 0 x 0 0 x 1 x where f is a nonnegative continuous function 0 0 1

existence theorems for a second order three point boundary value problem abstract letf 0 1 r 2 r be continuous ande c 0 1 let i 0 1 ai r all of theai s having the same sign i 1 2 m 2 0 x t f t x t x t e t t 0 1 e x 0 0 x 1 i 1 m 2 a i x i bc m we discuss existence theorems for the problem e bcm under some sign condition offabout the origin our analysis is based on a nonlinear alternative of leray schauder

isoperimetric and weingarten surfaces in the schwarzschild manifold we show that any star shaped convex hypersurface with constant weingarten curvature in the desitter schwarzschild manifold is a sphere of symmetry moreover we study an isoperimetric problem for bounded domains in the doubled schwarzschild manifold we prove the existence of an isoperimetric surface for any value of the enclosed volume and we completely describe the isoperimetric surfaces for very large enclosed volume this complements work in h bray s thesis where isoperimetric surfaces homologous to the horizon are studied

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

a volume comparison theorem for asymptotically hyperbolic manifolds we define a notion of renormalized volume of an asymptotically hyperbolic manifold moreover we prove a sharp volume comparison theorem for metrics with scalar curvature at least 6 finally we show that the inequality is strict unless the metric is isometric to one of the anti desitter schwarzschild metrics

empirical processes typical sequences and coordinated actions in standard borel spaces this paper proposes a new notion of typical sequences on a wide class of abstract alphabets so called standard borel spaces which is based on approximations of memoryless sources by empirical distributions uniformly over a class of measurable test functions in the finite alphabet case we can take all uniformly bounded functions and recover the usual notion of strong typicality or typicality under the total variation distance for a general alphabet however this function class turns out to be too large and must be restricted with this in mind we define typicality with respect to any glivenko cantelli function class i e a function class that admits a uniform law of large numbers and demonstrate its power by giving simple derivations of the fundamental limits on the achievable rates in several source coding scenarios in which the relevant operational criteria pertain to reproducing empirical averages of a general alphabet stationary memoryless source with respect to a suitable function class

coordination capacity we develop elements of a theory of cooperation and coordination in networks rather than considering a communication network as a means of distributing information or of reconstructing random processes at remote nodes we ask what dependence can be established among the nodes given the communication constraints specifically in a network with communication rates ri j between the nodes we ask what is the set of all achievable joint distributions p x1 xm of actions at the nodes of the network several networks are solved including arbitrarily large cascade networks distributed cooperation can be the solution to many problems such as distributed games distributed control and establishing mutual information bounds on the influence of one part of a physical system on another

the method of types information theory the method of types is one of the key technical tools in shannon theory and this tool is valuable also in other fields in this paper some key applications are presented in sufficient detail enabling an interested nonspecialist to gain a working knowledge of the method and a wide selection of further applications are surveyed these range from hypothesis testing and large deviations theory through error exponents for discrete memoryless channels and capacity of arbitrarily varying channels to multiuser problems while the method of types is suitable primarily for discrete memoryless models its extensions to certain models with memory are also discussed

duality between source coding and channel coding and its extension to the side information case we explore the information theoretic duality between source coding with side information at the decoder and channel coding with side information at the encoder we begin with a mathematical characterization of the functional duality between classical source and channel coding formulating the precise conditions under which the optimal encoder for one problem is functionally identical to the optimal decoder for the other problem we then extend this functional duality to the case of coding with side information by invoking this duality we are able to generalize the result of wyner and ziv 1976 relating to no rate loss for source coding with side information from gaussian to more arbitrary distributions we consider several examples corresponding to both discrete and continuous valued cases to illustrate our formulation for the gaussian cases of coding with side information we invoke geometric arguments to provide further insights into their duality our geometric treatment inspires the construction and dual use of practical coset codes for a large class of emerging applications for coding with side information such as distributed sensor networks watermarking and information hiding communication systems

characterization of large isoperimetric regions in asymptotically hyperbolic initial data let m g be a complete riemannian 3 manifold asymptotic to schwarzschild anti desitter and with scalar curvature r geq 6 building on work of a neves and g tian and of the first named author we show that the leaves of the canonical foliation of m g are the unique solutions of the isoperimetric problem for their area the assumption r geq 6 is necessary this is the first characterization result for large isoperimetric regions in the asymptotically hyperbolic setting that does not assume exact rotational symmetry at infinity

higher order m point boundary value problems on time scales in this paper we investigate the existence of positive solutions for nonlinear even order m point boundary value problems on time scales by means of fixed point theorems

the non regular ceo problem we consider the ceo problem for non regular source distributions such as uniform or truncated gaussian a group of agents observe independently corrupted versions of data and transmit coded versions over rate limited links to a ceo the ceo then estimates the underlying data based on the received coded observations agents are not allowed to convene before transmitting their observations this formulation is motivated by the practical problem of a firm s ceo estimating non regular beliefs about a sequence of events before acting on them agents observations are modeled as jointly distributed with the underlying data through a given conditional probability density function we study the asymptotic behavior of the minimum achievable mean squared error distortion at the ceo in the limit when the number of agents l and the sum rate r tend to infinity we establish a 1 r 2 convergence of the distortion an intermediate regime of performance between the exponential behavior in discrete ceo problems berger zhang and viswanathan 1996 and the 1 r behavior in gaussian ceo problems viswanathan and berger 1997 achievability is proved by a layered architecture with scalar quantization distributed entropy coding and midrange estimation the converse is proved using the bayesian chazan zakai ziv bound

the penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass in the asymptotically locally hyperbolic setting it is possible to have metrics with scalar curvature 6 and negative mass when the genus of the conformal boundary at infinity is positive using inverse mean curvature flow we prove a penrose inequality for these negative mass metrics the motivation comes from a previous result of p chru ciel and w simon which states that the penrose inequality we prove implies a static uniqueness theorem for negative mass kottler metrics

large isoperimetric surfaces in initial data sets we study the isoperimetric structure of asymptotically flat riemannian 3 manifolds m g that are mathcal c 0 asymptotic to schwarzschild of mass m gt 0 refining an argument due to h bray we obtain an effective volume comparison theorem in schwarzschild we use it to show that isoperimetric regions exist in m g for all sufficiently large volumes and that they are close to centered coordinate spheres this implies that the volume preserving stable constant mean curvature spheres constructed by g huisken and s t yau as well as r ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume this confirms a conjecture of h bray our results are consistent with the uniqueness results for volume preserving stable constant mean curvature surfaces in initial data sets obtained by g huisken and s t yau and strengthened by j qing and g tian the additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work

the discrete memoryless multiple access channel with partially cooperating encoders corresp we introduce the communication situation in which the encoders of a multiple access channel are partially cooperating these encoders are connected by communication links with finite capacities which permit both encoders to communicate with each other first we give a general definition of such a communication process conference then by proving a converse and giving an achievability proof we establish the capacity region of the multiple access channel with partially cooperating encoders it turns out that the optimal conference is very simple

random coding bound and codes produced by permutations for the multiple access channel a random coding bound for the multiple access channel mac is given and it is shown to be obtainable universally for all mac s with common input and output alphabets for channels for which the random coding bound may be achieved by codes consisting of codewords of a single type for each sender this bound is also achievable by codes produced by permutations

distributed channel synthesis two familiar notions of correlation are rediscovered as the extreme operating points for distributed synthesis of a discrete memoryless channel in which a stochastic channel output is generated based on a compressed description of the channel input wyner s common information is the minimum description rate needed however when common randomness independent of the input is available the necessary description rate reduces to shannon s mutual information this paper characterizes the optimal tradeoff between the amount of common randomness used and the required rate of description we also include a number of related derivations including the effect of limited local randomness rate requirements for secrecy applications to game theory and new insights into common information duality our proof makes use of a soft covering lemma known in the literature for its role in quantifying the resolvability of a channel the direct proof achievability constructs a feasible joint distribution over all parts of the system using a soft covering from which the behavior of the encoder and decoder is inferred with no explicit reference to joint typicality or binning of auxiliary interest this paper also generalizes and strengthens this soft covering tool

the existence of countably many positive solutions for some nonlinear nth order m point boundary value problems in this paper we consider the existence of countably many positive solutions for nth order m point boundary value problems consisting of the equation u n t a t f u t 0 email protected 0 1 with one of the following boundary value conditions u 0 email protected i 1m 2k iu x i u 0 u n 2 0 0 u 1 0 and u 0 0 u 0 u n 2 0 0 u 1 email protected i 1m 2k iu x i where n 2 k i 0 i 1 2 m 2 0 1 and has countably many singularities in 0 12 the associated green s function for the nth order m point boundary value problem is first given and we show that there exist countably many positive solutions using holder s inequality and krasnoselskii s fixed point theorem for operators on a cone

rate region of the quadratic gaussian two encoder source coding problem we determine the rate region of the quadratic gaussian two encoder source coding problem this rate region is achieved by a simple architecture that separates the analog and digital aspects of the compression furthermore this architecture requires higher rates to send a gaussian source than it does to send any other source with the same covariance our techniques can also be used to determine the sum rate of some generalizations of this classical problem our approach involves coupling the problem to a quadratic gaussian ldquoceo problem rdquo

multiple positive solutions for singular nth order nonlocal boundary value problems in banach spaces in this paper we consider a class of singular nth order nonlocal boundary value problems in banach spaces the existence of multiple positive solutions for the problem is obtained by using the fixed point index theory of strict set contraction operators to demonstrate the applications of our results two examples are also given in the paper

unique isoperimetric foliations of asymptotically flat manifolds in all dimensions the question of isoperimetry what is the largest amount of volume that can be enclosed by a given amount of area can be traced back to antiquity 1 the first mathematically rigorous results are as recent as the nineteenth century the question of isoperimetry and its close relative the analysis of minimal surfaces are two of the model problems of the geometric calculus of variations the list of geometries where an explicit answer to the question of isoperimetry is available is short we provide an overview of available results in appendix h in this paper and 22 we extend this list by a class of riemannian manifolds m g for which we describe all large isoperimetric regions completely we refer the reader to sect 2 for the precise definitions of all terms in the statement of our first main theorem

two way source coding with a helper consider the two way rate distortion problem in which a helper sends a common limited rate message to both users based on side information at its disposal we characterize the region of achievable rates and distortions when the markov relation helper user 1 user 2 holds the main insight of the result is that in order to achieve the optimal rate the helper may use a binning scheme as in wyner ziv where the side information at the decoder is the further user namely user 2 we derive these regions explicitly for the gaussian sources with square error distortion analyze a tradeoff between the rate from the helper and the rate from the source and examine a special case where the helper has the freedom to send different messages at different rates to the encoder and the decoder the converse proofs use a technique for verifying markov relations via undirected graphs

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

insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds we construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3 manifold which does not have the convergence properties needed in order to prove a penrose type inequality this contrasts sharply with the asymptotically flat case the main idea consists in combining inverse mean curvature flow with work done by shi tam regarding boundary behavior of compact manifolds assuming the penrose inequality holds we also derive a nontrivial inequality for functions on s

second order asymptotics of visible mixed quantum source coding via universal codes the simplest example of a quantum information source with memory is a mixed source which emits signals entirely from one of two memoryless quantum sources with given a priori probabilities considering a mixed source consisting of a general one parameter family of memoryless sources we derive the second order asymptotic rate for fixed length visible source coding furthermore we specialize our main result to a mixed source consisting of two memoryless sources our results provide the first example of the second order asymptotics for a quantum information processing task employing a resource with memory for the case of a classical mixed source using a finite alphabet our results reduce to those obtained by nomura and han to prove the achievability part of our main result we introduce universal quantum source codes achieving the second order asymptotic rates these are obtained by an extension of hayashi s construction of their classical counterparts

two way source coding with a fidelity criterion let x i y i i 1 2 cdots be a sequence of independent identically distributed bivariate random variables and consider the following communication situation the x component of the process is observed at some location say a while the y component is observed at a different location say b the x y component of the process has to be reproduced at location b a it is desired to find the amount of information that should be exchanged between the two locations so that the distortions incurred will not exceed some predetermined tolerance a single letter characterization of a certain region cal k subset cal r 4 of rates and distortions is given this region contains all and only the achievable rates and distortions for the special case where block coding is used and the information is conveyed through a one way link that can switch direction only k times per block

existence and characterization of regions minimizing perimeter under a volume constraint inside euclidean cones we study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a euclidean cone our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half space this allows us to give some criteria ensuring existence of isoperimetric regions for instance local convexity of the cone at some boundary point we also characterize which are the stable regions in a convex cone i e second order minima of perimeter under a volume constraint from this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone

capacity results for the discrete memoryless network a discrete memoryless network dmn is a memoryless multiterminal channel with discrete inputs and outputs a sequence of inner bounds to the dmn capacity region is derived by using code trees capacity expressions are given for three classes of dmns 1 a single letter expression for a class with a common output 2 a two letter expression for a binary symmetric broadcast channel bc with partial feedback and 3 a finite letter expression for push to talk dmns the first result is a consequence of a new capacity outer bound for common output dmns the third result demonstrates that the common practice of using a time sharing random variable does not include all time sharing possibilities namely time sharing of channels several techniques for improving the bounds are developed 1 causally conditioned entropy and directed information simplify the inner bounds 2 code trellises serve as simple code trees 3 superposition coding and binning with code trees improves rates numerical computations show that the last technique enlarges the best known rate regions for a multiple access channel mac and a bc both with feedback in addition to the rate bounds a sequence of inner bounds to the dmn reliability function is derived a numerical example for a two way channel illustrates the behavior of the error exponents

multiterminal source encoding with one distortion criterion the authors unify earlier investigations concerning the encoding of two correlated sources x sub k y sub k by means of separate encoders decoding is done by a single decoder which receives the outputs from both encoders the reconstruction of x sub k is required to be perfect in the usual shannon sense the authors determine the admissible rate region r d where d is the distortion of the reconstruction of y sub k the binary hamming case is investigated explicitly

foliations by stable spheres with constant mean curvature for isolated systems with general asymptotics we prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the regge teitelboim condition near infinity it is known that the hamiltonian center of mass is well defined for manifolds satisfying this condition we also show that the foliation is asymptotically concentric and its geometric center is the center of mass the construction of the foliation generalizes the results of huisken yau ye and metzger where strongly asymptotically flat manifolds and their small perturbations were studied

mac with action dependent state information at one encoder the growing interest in action dependent channels motivates us to extend the study of action dependent settings which until now focused on point to point models to multiple access channels macs in this paper we consider a two user state dependent mac in which one of the encoders called the informed cognitive encoder is allowed to take an action that affects the formation of the channel states two independent messages are to be sent through the channel 1 a common message known to both encoders and 2 a private message known only to the informed encoder in addition the informed encoder has access to the sequence of channel states in a noncausal manner our framework generalizes the previously evaluated settings of state dependent point to point channels with actions and macs with common messages we derive a single letter characterization of the capacity region for this setting using this general result we obtain and compute the capacity region for the gaussian action dependent mac the special methods used in solving the gaussian case are then applied to obtain the capacity of the gaussian action dependent point to point channel a problem left open and solved now finally we establish some dualities between action dependent channel coding and source coding problems in particular we obtain a duality equivalence between the considered mac setting and the rate distortion model known as successive refinement with actions this is done by developing a set of simple duality principles that enables us to successfully evaluate the outcome of one problem given the outcome of the other

a minkowski inequality for hypersurfaces in the anti de sitter schwarzschild manifold we prove a sharp inequality for hypersurfaces in the n dimensional anti de sitter schwarzschild manifold for general n 3 this inequality generalizes the classical minkowski inequality for surfaces in the three dimensional euclidean space and has a natural interpretation in terms of the penrose inequality for collapsing null shells of dust the proof relies on a new monotonicity formula for inverse mean curvature flow and uses a geometric inequality established by the first author in 3 c 2015 wiley periodicals inc

quantum data processing we prove a data processing inequality for quantum communication channels which states that processing a received quantum state may never increase the mutual information between input and output states

on the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3 manifolds in this note we study constant mean curvature surfaces in asymptotically flat 3 manifolds we prove that outside a given compact subset in an asymptotically flat 3 manifold with positive mass stable spheres of given constant mean curvature are unique therefore we are able to conclude that the foliation of stable spheres of constant mean curvature in an asymptotically flat 3 manifold with positive mass outside a given compact subset is unique

on the reversible extraction of classical information from a quantum source consider a source of pure quantum states with von neumann entropy s by the quantum source coding theorem arbitrarily long strings of signals may be encoded asymptotically into s qubits per signal the schumacher limit in such a way that entire strings may be recovered with arbitrarily high fidelity suppose that classical storage is free while quantum storage is expensive and suppose that the states of do not fall into two or more orthogonal subspaces we show that if can be compressed with arbitrarily high fidelity into a qubits per signal plus any amount of auxiliary classical storage then a must still be at least as large as the schumacher limit s of thus no part of the quantum information content of can be faithfully replaced by classical information if the states do fall into orthogonal subspaces then a may be less than s but only by an amount not exceeding the amount of classical information specifying the subspace for a signal from the source

compressibility of quantum mixed state signals we present a formula that determines the optimal number of qubits per message that allows asymptotically faithful compression of the quantum information carried by an ensemble of mixed states the set of mixed states determines a decomposition of the hilbert space into the redundant part and the irreducible part after removing the redundancy the optimal compression rate is shown to be given by the von neumann entropy of the reduced ensemble

three positive solutions for a nonlinear nth order m point boundary value problem abstract in this paper we consider the existence of at least three positive solutions for the nonlinear n th order m point boundary value problem u n t f t u 0 t 0 1 u 0 0 u 0 u n 2 0 0 u 1 i 1 m 2 k i u i where n 2 k i 0 i 1 2 m 2 0 1 2 m 2 1 the associated green s function for the n th order m point boundary value problem is first given and growth conditions are imposed on the nonlinearity f which yield the existence of multiple positive solutions by using the leggett williams fixed point theorem

communicating probability distributions a rate distortion problem is solved that is motivated by a quantum data compression problem the goal is to send information about a source string x so that a receiver can construct a second string y for which the joint empirical probability distribution of x and y is close to some desired distribution the problem differs from the usual rate distortion problems in that one must consider both remote sources and distortion functions that are not averages of per letter distortion functions

two sided bounds on minimum error quantum measurement on the reversibility of quantum dynamics and on maximum overlap using directional iterates in a unified framework we estimate the following quantities of interest in quantum information theory 1 the minimum error distinguishability of arbitrary ensembles of mixed quantum states 2 the approximate reversibility of quantum dynamics in terms of entanglement fidelity this is referred to as channel adapted quantum error recovery when applied to the composition of an encoding operation and a noise channel 3 the maximum overlap between a bipartite pure quantum state and a bipartite mixed state that may be achieved by applying a local quantum operation to one part of the mixed state and 4 the conditional min entropy of bipartite quantum states a refined version of the author s techniques j tyson j math phys 50 032016 2009 for bounding the first quantity is employed to give two sided estimates of the remaining three quantities we obtain a closed form approximate reversal channel using a state dependent kraus decomposition our reversal may be interpreted as a quadratically

duality of a source coding problem and the semi deterministic broadcast channel with rate limited cooperation the wyner ahlswede korner wak empirical coordination problem where the encoders cooperate via a finite capacity one sided link is considered the coordination capacity region is derived by combining several source coding techniques such as wyner ziv coding binning and superposition coding furthermore a semi deterministic sd broadcast channel bc with one sided decoder cooperation is considered duality principles relating the two problems are presented and the capacity region for the sd bc setting is derived the direct part follows from an achievable region for a general bc that is tight for the sd scenario a converse is established by using telescoping identities the sd bc is shown to be operationally equivalent to a class of relay bcs and the correspondence between their capacity regions is established the capacity region of the sd bc is transformed into an equivalent region that is shown to be dual to the admissible region of the wak problem in the sense that the information measures defining the corner points of both regions coincide achievability and converse proofs for the equivalent region are provided for the converse we use a probabilistic construction of auxiliary random variables that depends on the distribution induced by the codebook several examples illustrate the results

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

second order asymptotics for source coding dense coding and pure state entanglement conversions we introduce two variants of the information spectrum relative entropy defined by tomamichel and hayashi which have the particular advantage of satisfying the data processing inequality i e monotonicity under quantum operations this property allows us to obtain one shot bounds for various information processing tasks in terms of these quantities moreover these relative entropies have a second order asymptotic expansion which in turn yields tight second order asymptotics for optimal rates of these tasks in the independent and identically distributed setting the tasks studied in this paper are fixed length quantum source coding noisy dense coding entanglement concentration pure state entanglement dilution and transmission of information through a classical quantum channel in the latter case we retrieve the second order asymptotics obtained by tomamichel and tan our results also yield the known second order asymptotics of fixed length classical source coding derived by hayashi the second order asymptotics of entanglement concentration and dilution provide a refinement of the inefficiency of these protocols a quantity which in the case of entanglement dilution was studied by harrow and lo we prove how the discrepancy between the optimal rates of these two processes in the second order implies the irreversibility of entanglement concentration established by kumagai and hayashi in addition the spectral divergence rates of the information spectrum approach isa can be retrieved from our relative entropies in the asymptotic limit this enables us to directly obtain the more general results of the isa from our one shot bounds

positive solutions for second order m point boundary value problems by using the schauder fixed point theorem and analysis method we establish the existence of solutions for the m point boundary value problem u t a t f u 0 u 0 0 u 1 i 1m 2kiu i b where b ki 0 i 1 2 m 2 0 b

stationary quantum source coding in this article the quantum version of the source coding theorem is obtained for a completely ergodic source this result extends schumacher s quantum noiseless coding theorem for memoryless sources the control of the memory effects requires some earlier results of hiai and petz on high probability subspaces our result is equivalently considered as a compression theorem for noiseless stationary channels

the mass of asymptotically hyperbolic manifolds motivated by certain problems in general relativity and riemannian geometry we study manifolds which are asymptotic to the hyperbolic space in a certain sense it is shown that an invariant the so called total mass can be defined unambiguously a positive mass theorem is established by using the spinor method

minimum rates of approximate sufficient statistics given a sufficient statistic for a parametric family of distributions one can estimate the parameter without access to the data however the memory or code size for storing the sufficient statistic may nonetheless still be prohibitive indeed for n independent samples drawn from a k nomial distribution with d k 1 degrees of freedom the length of the code scales as d log n o 1 in many applications we may not have a useful notion of sufficient statistics e g when the parametric family is not an exponential family and we may also not need to reconstruct the generating distribution exactly by adopting a shannon theoretic approach in which we allow a small error in estimating the generating distribution we construct various approximate sufficient statistics and show that the code length can be reduced to d 2 log n o 1 we consider errors measured according to the relative entropy and variational distance criteria for the code constructions we leverage rissanen s minimum description length principle which yields a non vanishing error measured according to the relative entropy for the converse parts we use clarke and barron s formula for the relative entropy of a parameterized distribution and the corresponding mixture distribution however this method only yields a weak converse for the variational distance we develop new techniques to achieve vanishing errors and we also prove strong converses the latter means that even if the code is allowed to have a non vanishing error its length must still be at least d 2 log n

a theory of network equivalence part i point to point channels a family of equivalence tools for bounding network capacities is introduced given a network n with node set v the capacity of n is a set of non negative vectors with elements corresponding to all possible multicast connections in n a vector is in the capacity region for n if and only if it is possible to simultaneously and reliably establish all multicast connections across n at the given rates any other demand type with independent messages is a special case of this multiple multicast problem and is therefore included in the given rate region in part i we show that the capacity of a network n is unchanged if any independent memoryless point to point channel in n is replaced by a noiseless bit pipe with throughput equal to the removed channel s capacity it follows that the capacity of a network comprised entirely of such point to point channels equals the capacity of an error free network that replaces each channel by a noiseless bit pipe of the corresponding capacity a related separation result was known previously for a single multicast connection over an acyclic network of independent memoryless point to point channels our result treats general connections e g a collection of simultaneous unicasts and allows cyclic or acyclic networks

large isoperimetric regions in asymptotically hyperbolic manifolds we show the existence of isoperimetric regions of sufficiently large volumes in general asymptotically hyperbolic three manifolds furthermore we show that large coordinate spheres are uniquely isoperimetric in manifolds that are schwarzschild anti desitter at infinity these results have important repercussions for our understanding of spacelike hypersurfaces in lorentzian space times which are asymptotic to null infinity in fact as an application of our results we verify the asymptotically hyperbolic penrose inequality in the special case of the existence of connected isoperimetric regions of all volumes

effective versions of the positive mass theorem the study of stable minimal surfaces in riemannian 3 manifolds m g with non negative scalar curvature has a rich history in this paper we prove rigidity of such surfaces when m g is asymptotically flat and has horizon boundary as a consequence we obtain an effective version of the positive mass theorem in terms of isoperimetric or more generally closed volume preserving stable cmc surfaces that is appealing from both a physical and a purely geometric point of view we also include a proof of the following conjecture of schoen an asymptotically flat riemannian 3 manifold with non negative scalar curvature that contains an unbounded area minimizing surface is isometric to flat mathbb r 3

large outlying stable constant mean curvature spheres in initial data sets we give examples of asymptotically flat three manifolds m g which admit arbitrarily large constant mean curvature spheres that are far away from the center of the manifold this resolves a question raised by huisken and yau invent math 124 281 311 1996 on the other hand we show that such surfaces cannot exist when m g has nonnegative scalar curvature this result depends on an intricate relationship between the scalar curvature of the initial data set and the isoperimetric ratio of large stable constant mean curvature surfaces

three positive solutions for second order m point boundary value problems for the second order m point boundary value problemu t f t u 0 0 0 i 1 2 m 2 0 x 1 x 2 x m 2 1 growth conditions are imposed on f which yield the existence of at least three positive solutions by using the leggett williams fixed point theorem the associated green s function for the m point boundary value problem is also given

multiterminal source coding under logarithmic loss we consider the classical two encoder multiterminal source coding problem where distortion is measured under logarithmic loss we provide a single letter description of the achievable rate distortion region for all discrete memoryless sources with finite alphabets by doing so we also give the rate distortion region for the m encoder ceo problem also under logarithmic loss several applications and examples are given

on large volume preserving stable cmc surfaces in initial data sets let m g be a complete 3 dimensional asymptotically flat manifold with everywhere positive scalar curvature we prove that given a compact subset k subset m all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid k this complements the results of g huisken and s t yau and of j qing and g tian on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to schwarzschild with mass m gt 0 the analysis in g huisken and s t yau and in j qing and g tian takes place in the asymptotic regime of m here we adapt ideas from the minimal surface proof of the positive mass theorem by r schoen and s t yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of m that is far from euclidean

operational duality between lossy compression and channel coding we explore the duality between lossy compression and channel coding in the operational sense whether a capacity achieving encoder decoder sequence achieves the rate distortion function of the dual problem when the channel decoder encoder is the source compressor decompressor resp and vice versa we show that if used as a lossy compressor the maximum likelihood channel decoder of a randomly chosen capacity achieving codebook achieves the rate distortion function almost surely however operational duality does not hold for every capacity achieving encoder decoder sequence or rate distortion achieving compressor decompressor sequence we show that there exist optimal channel coding lossy compression schemes which fail when used for the dual lossy compression channel coding resp problem

compressing quantum mixed state sources by sending classical information we consider visible compression for discrete memoryless sources of mixed quantum states when only classical information can be sent from alice to bob we assume that bob knows the source statistics and that alice and bob have access to the same source of random numbers we put in an information theoretic framework some previous results on visible compression for sources of states with commuting density operators and remove the commutativity requirement we derive a general achievable compression rate which is for the noncommutative case still higher than the known lower bound we also present several related problems of classical information theory and show how they can be used to answer some questions of the mixed state compression problem

proof of the riemannian penrose inequality using the positive mass theorem we prove the riemannian penrose conjecture an important case of a con jecture 41 made by roger penrose in 1973 by defining a new flow of metrics this flow of metrics stays inside the class of asymptotically flat riemannian 3 manifolds with nonnegative scalar curvature which contain minimal spheres in particular if we consider a riemannian 3 manifold as a totally geodesic submanifold of a space time in the context of general relativity then outermost minimal spheres with total area a correspond to apparent horizons of black holes contributing a massa 16 scalar cur vature corresponds to local energy density at each point and the rate at which the metric becomes flat at infinity corresponds to total mass also called the adm mass the riemannian penrose conjecture then states that the total mass of an asymptotically flat 3 manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes the flow of metrics we define continuously evolves the original 3 metric to a schwarzschild 3 metric which represents a spherically symmetric black hole in vacuum we define the flow such that the area of the minimal spheres which flow outward and hence the mass contributed by the black holes in each of the metrics in the flow is constant and then use the positive mass theorem to show that the total mass of the metrics is nonincreasing then since the total mass equals the mass of the black hole in a schwarzschild metric the riemannian penrose conjecture follows we also refer the reader to the beautiful work of huisken and ilmanen 30 who used inverse mean curvature flows of surfaces to prove that the total mass is at least the mass contributed by the largest black hole in sections 1 and 2 we motivate the problem discuss important quantities like total mass and horizons of black holes and state the positive mass theorem and the penrose conjecture for riemannian 3 manifolds in section 3 we give the proof of the riemannian penrose

common randomness and distributed control a counterexample when agents collaborate to perform a control task it is of interest to characterize the set of joint probability distributions they can achieve on their joint action space when they are passively provided with external common randomness we give a simple counterexample to a natural conjecture about this class of joint distributions