oblivious communication channels and their capacity let be c x1 xn sub 0 1 n be an n n binary error correcting code not necessarily linear let e 0 1 n be an error vector a codeword xepsic is said to be disturbed by the error e if the closest codeword to xopluse is no longer x let ae be the subset of codewords in c that are disturbed by e in this work we study the size of ae in random codes c i e codes in which each codeword is chosen uniformly and independently at random from 0 1 n using recent results of vu random structures and algorithms vol 20 no 3 pp 262 316 2002 on the concentration of non lipschitz functions we show that ae is strongly concentrated for a wide range of values of n and e we apply this result in the study of communication channels we refer to as oblivious roughly speaking a channel w y x is said to be oblivious if the error distribution imposed by the channel is independent of the transmitted codeword x a family of channels psi is said to be oblivious if every member w of the family is oblivious in this work we define oblivious and partially oblivious families of not necessarily memoryless channels and analyze their capacity when considering the capacity of a family of channels psi one must address the design of error correcting codes which allow communication under the uncertainty of which channel wepsipsi is actually used the oblivious channels we define have connections to arbitrarily varying channels with state constraints

exponential error bounds for random codes in the arbitrarily varying channel random codes for the arbitrarily varying channel are investigated the code ensemble is restricted to increase only exponentially with codeword length a quantity called key rate is used as a measure of the rate of increase the reliability function for such codes is related to the reliability function for codes with unlimited key rate explicit results are obtained in some examples of practical interest

an exact method for the calculation of certain titchmarsh weyl m functions we consider the second order linear differential equation where q is real valued in the case we calculate the titchmarsh weyl m function associated with

common randomness in information theory and cryptography i secret sharing as the first part of a study of problems involving common randomness at distance locations information theoretic models of secret sharing generating a common random key at two terminals without letting an eavesdropper obtain information about this key are considered the concept of key capacity is defined single letter formulas of key capacity are obtained for several models and bounds to key capacity are derived for other models

quantum capacity under adversarial quantum noise arbitrarily varying quantum channels we investigate entanglement transmission over an unknown channel in the presence of a third party called the adversary which is enabled to choose the channel from a given set of memoryless but non stationary channels without informing the legitimate sender and receiver about the particular choice that he made this channel model is called an arbitrarily varying quantum channel avqc we derive a quantum version of ahlswede s dichotomy for classical arbitrarily varying channels this includes a regularized formula for the common randomness assisted capacity for entanglement transmission of an avqc quite surprisingly and in contrast to the classical analog of the problem involving the maximal and average error probability we find that the capacity for entanglement transmission of an avqc always equals its strong subspace transmission capacity

real gromov witten theory in all genera and real enumerative geometry construction the first part of this work constructs positive genus real gromov witten invariants of realorientable symplectic manifolds of odd complex dimensions the second part studies the orientations on the moduli spaces of real maps used in constructing these invariants the present paper applies the results of the latter to obtain quantitative and qualitative conclusions about the invariants defined in the former after describing large collections of real orientable symplectic manifolds we show that the genus 1 real gromov witten invariants of sufficiently positive almost kahler threefolds are signed counts of real genus 1 curves only and thus provide direct lower bounds for the counts of these curves in such targets we specify real orientations on the real orientable complete intersections in projective spaces the real gromov witten invariants they determine are in a sense canonically determined by the complete intersection itself at least in most cases we also obtain equivariant localization data that computes the real invariants of projective spaces and determines the contributions from many torus fixed loci for other complete intersections our results confirm walcher s predictions for the vanishing of these invariants in certain cases and for the localization data in other cases

interferometric visibility and coherence recently the basic concept of quantum coherence or superposition has gained a lot of renewed attention after baumgratz et al phys rev lett 113 140401 doi 10 1103 physrevlett 113 140401 following aberg http arxiv org abs quant ph 0612146 have proposed a resource theoretic approach to quantify it this has resulted in a large number of papers and preprints exploring various coherence monotones and debating possible forms for the resource theory here we take the view that the operational foundation of coherence in a state be it quantum or otherwise wave mechanical lies in the observation of interference effects our approach here is to consider an idealized multi path interferometer with a suitable detector in such a way that the visibility of the interference pattern provides a quantitative expression of the amount of coherence in a given probe state we present a general framework of deriving coherence measures from visibility and demonstrate it by analysing several concrete visibility parameters recovering some known coherence measures and obtaining some new ones

arbitrarily varying channels with constrained inputs and states random coding theorems are proved for discrete memoryless arbitrarily varying channels avcs with constraints on the transmitted codewords and channel state sequences two types of constraints are considered peak i e required for each n length sequence almost surely and average over the message set or over an ensemble for peak constraints on the codewords and on the channel state sequences the avc is shown to have a strong random coding capacity if the codewords and or the channel state sequences are constrained in the average sense the avcs do not possess strong capacities only epsilon capacities are shown to exist

arbitrarily small amounts of correlation for arbitrarily varying quantum channels as our main result show that in order to achieve the randomness assisted message and entanglement transmission capacities of a finite arbitrarily varying quantum channel it is not necessary that sender and receiver share asymptotically perfect common randomness rather it is sufficient that they each have access to an unlimited amount of uses of one part of a correlated bipartite source this access might be restricted to an arbitrary small nonzero fraction per channel use without changing the main result we investigate the notion of common randomness it turns out that this is a very costly resource generically it cannot be obtained just by local processing of a bipartite source this result underlines the importance of our main result also the asymptotic equivalence of the maximal and average error criterion for classical message transmission over finite arbitrarily varying quantum channels is proven at last we prove a simplified symmetrizability condition for finite arbitrarily varying quantum channels

arbitrarily varying channels with general alphabets and states previous results of csiszar and narayan 1988 1989 1991 on the capacity of discrete arbitrarily varying channels with input and state constraints are extended to the case of arbitrary alphabets and state sets for channels with scalar or vector inputs and additive interference consisting of a deterministic part and noise both arbitrarily varying subject to power constraints the capacity for deterministic codes is shown to be equal to the random coding capacity if the input power exceeds the power of the deterministic interference and zero otherwise explicit capacity formulas are also given

additive bounds of minimum output entropies for unital channels and an exact qubit formula we find an upper bound for the minimum output entropy of a unital quantum channel and obtain an exact formula for general qubit channels our techniques incorporate the renyi entropies particularly with renyi parameter alpha 2 moreover since our upper bound is additive under tensor product we get as a corollary an upper bound for the classical capacity of unital quantum channels interestingly our upper bound for the classical capacity depends only on the operator norm of matrix representations of channels on the space of traceless hermitian operators and is tight in the sense that it gives the precise quantity of classical capacity of the werner holevo channel as an example we study quantum channels with operator sum representation that is made of the discrete weyl operators generalized pauli operators and explain how our formula works in this case finally we find new examples for which the minimum output renyi 2 entropy is additive

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

capacity of the gaussian arbitrarily varying channel the gaussian arbitrarily varying channel with input constraint gamma and state constraint lambda admits input sequences x x sub 1 x sub n of real numbers with sigma x sub i sup 2

the common randomness capacity of a pair of independent discrete memoryless channels we study the following problem two agents alice and bob are connected to each other by independent discrete memoryless channels they wish to generate common randomness i e agree on a common random variable by communicating interactively over the two channels assuming that alice and bob are allowed access to independent external random sources at rates in bits per step of communication of h sub a and h sub b respectively we show that they can generate common randomness at a rate of max min h sub a h w q i p v min h sub b h v p i q w bits per step by exploiting the noise on the two channels here v is the channel from alice to bob and w is the channel from bob to alice the maximum is over all probability distributions p and q on the input alphabets of v and w respectively we also prove a strong converse which establishes the above rate as the highest attainable in this situation

a comparison theorem for gromov witten invariants in the symplectic category abstract we exploit the geometric approach to the virtual fundamental class due to fukaya ono and li tian to compare gromov witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order various special cases of the comparison theorem in this paper have long been used in the algebraic category some of them have also appeared in the symplectic setting combined with the inherent flexibility of the symplectic category the main theorem leads to a confirmation of pandharipande s gopakumar vafa prediction for gw invariants of fano classes in 6 dimensional symplectic manifolds the proof of the main theorem uses deformations of the cauchy riemann equation that respect the submanifold and carleman similarity principle for solutions of perturbed cauchy riemann equations in a forthcoming paper we apply a similar approach to relative gromov witten invariants and the absolute relative correspondence in genus 0

on error exponents for arbitrarily varying channels the minimum probability of error achievable by random codes on the arbitrarily varying channel avc is investigated new exponential error bounds are found and applied to the avc with and without input and state constraints also considered is a simple subclass of random codes called randomly modulated codes in which encoding and decoding operations are separate from code randomization a universal coding theorem is proved which shows the existence of randomly modulated codes that achieve the same error bounds as fully random codes for all avcs

the capacity of a vector gaussian arbitrarily varying channel the random coding capacity of a vector gaussian arbitrarily varying channel vgavc is determined along with a simple general method for computing this capacity the vgavc is a discrete time memoryless vector channel with an input power constraint and additive gaussian noise that is further corrupted by an additive jamming signal the statistics of this jamming signal are unknown and can be arbitrary subject only to a power constraint

strong converse exponents for a quantum channel discrimination problem and quantum feedback assisted communication this paper studies the difficulty of discriminating between an arbitrary quantum channel and a replacer channel that discards its input and replaces it with a fixed state the results obtained here generalize those known in the theory of quantum hypothesis testing for binary state discrimination we show that in this particular setting the most general adaptive discrimination strategies provide no asymptotic advantage over non adaptive tensor power strategies this conclusion follows by proving a quantum stein s lemma for this channel discrimination setting showing that a constant bound on the type i error leads to the type ii error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels the strong converse part of the lemma states that any attempt to make the type ii error decay to zero at a rate faster than the channel relative entropy implies that the type i error necessarily converges to one we then refine this latter result by identifying the optimal strong converse exponent for this task as a consequence of these results we can establish a strong converse theorem for the quantum feedback assisted capacity of a channel sharpening a result due to bowen furthermore our channel discrimination result demonstrates the asymptotic optimality of a non adaptive tensor power strategy in the setting of quantum illumination as was used in prior work on the topic the sandwiched renyi relative entropy is a key tool in our analysis finally by combining our results with recent results of hayashi and tomamichel we find a novel operational interpretation of the mutual information of a quantum channel mathcal n as the optimal type ii error exponent when discriminating between a large number of independent instances of mathcal n and an arbitrary worst case replacer channel chosen from the set of all replacer channels

capacity and decoding rules for classes of arbitrarily varying channels the capacity of an arbitrarily varying channel avc is considered for deterministic codes with the average probability of error criterion and typically subject to at state constraint first sufficient conditions are provided that enable relatively simple decoding rules such as typicality maximum mutual information and minimum distance to attain capacity then the possibly noisy or channels and group adder channels are studied in detail for the former the capacity is explicitly determined and shown to be attainable by minimum distance decoding next for a large class of addictive avcs in addition to providing an intuitively suggestive simplification of the general avc capacity formula it is proven that capacity can be attained by a universal decoding rule finally the effect of random state selections on capacity is studied the merits and limitations of a previous mutual information game approach are also discussed

state constraints for the multiple access arbitrarily varying channel a discrete multiple access arbitrarily varying channel with two transmitters and one receiver is considered it has recently been shown that if such a channel is symmetrizable then the capacity region has an empty interior it is shown that if the channel is subject to a suitable state constraint then the capacity region can contain certain open rectangles and thereby possess a nonempty interior even if the channel is symmetrizable a new weak converse under a state constraint is also proved these results are then used to establish the capacity region of the two user adder channel under state constraint 1 2

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

universal coding for classical quantum channel we construct a universal code for a stationary and memoryless classical quantum channel as a quantum version of the universal coding by csiszar and korner our code is constructed utilizing a combination of irreducible representations a decoder introduced through the quantum information spectrum and the packing lemma

enumeration of genus two curves with a fixed complex structure in ℙ 2 and ℙ 3 the main concrete result of this paper is enumeration of genus two curves with complex structure fixed in p and p along the way rational curves with certain simple singularities are counted as well while the methods described can be used to count positive genus curves in some other cases the most powerful direct applications of the machinery developed are to enumeration of rational curves with a very large class of singularities in projective spaces

the moduli space of stable vector bundles over a real algebraic curve we study the spaces of stable real and quaternionic vector bundles on a real algebraic curve the basic relationship is established with unitary representations of an extension of mathbb z 2 by the fundamental group by comparison with the space of real or quaternionic connections some of the basic topological invariants of these spaces are calculated

on the deterministic code capacity region of an arbitrarily varying multiple access channel under list decoding we study the capacity region cl of an arbitrarily varying multiple access channel avmac for deterministic codes with decoding into a list of a fixed size l and for the average error probability criterion motivated by known results in the study of fixed size list decoding for a point to point arbitrarily varying channel we define for every avmac whose capacity region for random codes has a nonempty interior a nonnegative integer called its symmetrizability it is shown that for every l cl has an empty interior and for every l 1 2 cl equals the nondegenerate capacity region of the avmac for random codes with a known single letter characterization for a binary avmac with a nondegenerate random code capacity region it is shown that the symmetrizability is always finite

reduced genus one gromov witten invariants in a previous paper we described a natural closed subset m 0 1 k x a j of the moduli space m1 k x a j of stable genus one j holomorphic maps into a symplectic manifold x in this paper we generalize the definition of the main component to moduli spaces of perturbed in a restricted way j holomorphic maps this generalization implies that m 0 1 k x a j just like m1 k x a j carries a virtual fundamental class and can be used to define symplectic invariants these truly genus one invariants constitute part of the standard genus one gromovwitten invariants which arise from the entire moduli space m1 k x a j the new invariants are more geometric and can be used to compute the genus one gw invariants of complete intersections as shown in a separate paper

the common randomness capacity of a network of discrete memoryless channels we generalize our previous results on generating common randomness at two terminals to a situation where any finite number of agents interconnected by an arbitrary network of independent point to point discrete memoryless channels wish to generate common randomness by interactive communication over the network our main result is an exact characterization of the common randomness capacity of such a network i e the maximum number of bits of randomness that all the agents can agree on per step of communication as a by product we also obtain a purely combinatorial result viz a characterization of the incidence vectors of the spanning arborescences rooted at a specified vertex in a digraph and having exactly one edge exiting the root as precisely the extreme points of a certain unbounded convex polyhedron described by a system of linear inequalities

common randomness in information theory and cryptography ii cr capacity for pt i see ibid vol 39 p 1121 1993 the common randomness cr capacity of a two terminal model is defined as the maximum rate of common randomness that the terminals can generate using resources specified by the given model we determine cr capacity for several models including those whose statistics depend on unknown parameters the cr capacity is shown to be achievable robustly by common randomness of nearly uniform distribution no matter what the unknown parameters are our cr capacity results are relevant for the problem of identification capacity and also yield a new result on the regular transmission capacity of arbitrarily varying channels with feedback

the capacity of the arbitrarily varying channel revisited positivity constraints a well known result of r ahlswede 1970 asserts that the deterministic code capacity of an arbitrarily varying channel avc under the average error probability criterion either equals its random code capacity or else is zero a necessary and sufficient condition is identified for deciding between these alternative namely the capacity is zero if and only if the avc is symmetrizable the capacity of the avc is determined with constraints on the transmitted codewords as well as on the channel state sequences and it is demonstrated that it may be positive but less than the corresponding random code capacity a special case of the results resolves a weakened version of a fundamental problem of coding theory

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

communication over individual channels a communication problem in considered where no mathematical model is specified for the channel the achievable rates are determined as a function of the channel input and output sequences known a posteriori without assuming any a priori relation between them for discrete channels the empirical mutual information between the input and output sequences is shown to be achievable while for continuous channels the achievable rate is based on the empirical correlation between the sequences a rate adaptive scheme employing feedback which achieves these rates asymptotically with a guaranteed reliability without prior knowledge of the channel behavior is presented

spinor states of real rational curves in real algebraic convex 3 manifolds and enumerative invariants let x be a real algebraic convex 3 manifold whose real part is equipped with a pin i structure we show that every irreducible real rational curve with non empty real part has a canonical spinor state belonging to f 1g the main result is then that the algebraic count of the number of real irreducible rational curves in a given numerical equivalence class passing through the appropriate number of points does not depend on the choice of the real conflguration of points provided that these curves are counted with respect to their spinor states these invariants provide lower bounds for the total number of such real rational curves independantly of the choice of the real conflguration of points

on random coding error exponents of watermarking systems watermarking codes are analyzed from an information theoretic viewpoint as a game between an information hider and an active attacker while the information hider embeds a secret message watermark in a covertext message typically text image sound or video stream within a certain distortion level the attacker processes the resulting watermarked message within limited additional distortion in attempt to invalidate the watermark for the case where the covertext source is memoryless or more generally where there exists some transformation that makes it memoryless we provide a single letter characterization of the maximin game of the random coding error exponent associated with the average probability of erroneously decoding the watermark this single letter characterization is in effect because if the information hider utilizes a memoryless channel to generate random codewords for every covertext message the causal attacker will maximize the damage by implementing a memoryless channel as well partial results for the dual minimax game and the conditions for the existence of a saddle point are also presented

minimal entropy of states emerging from noisy quantum channels we consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel which is modeled by the action of a completely positive trace preserving or stochastic map we provide strong support for the conjecture that this minimal entropy is additive namely that the minimum entropy can be achieved when product states are transmitted explicitly we prove that for a tensor product of two unital stochastic maps on qubit states using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states we give a separate argument based on the geometry of the image of the set of density matrices under stochastic maps which suggests that the minimal entropy conjecture holds for nonunital as well as for unital maps we also show that the maximal norm of the output states is multiplicative for most product maps on n qubit states including all those for which at least one map is unital for the class of unital channels on c sup 2 we show that additivity of minimal entropy implies that the holevo see ieee trans inform theory vol 44 p 269 73 1998 and russian math surv p 1295 1331 1999 capacity of the channel is additive over two inputs achievable with orthogonal states and equal to the shannon capacity this implies that superadditivity of the capacity is possible only for nonunital channels

list decoding for arbitrarily varying multiple access channel revisited list configuration and symmetrizability in a recent work s nitinawarat obtained a lower bound and an upper bound for the minimum list size in list decoding for an arbitrarily varying multiple access channel avmac for which the interior of the capacity region of deterministic list codes is nonempty in the same paper he proved that for a binary avmac the minimum list size is finite if and only if the interior of the capacity region of random correlated codes is nonempty the goal of this paper is to close the gap between the two bounds for the minimum list size we find a necessary and sufficient condition for the list codes for an avmac to have a capacity region with a nonempty interior in terms of bipartite graphs therefore we determine the minimum list size moreover we prove that for any avmac the minimum list size is finite if and only if the interior of the capacity region of random correlated codes is nonempty

universal quantum state merging we determine the optimal entanglement rate of quantum state merging when assuming that the state is unknown except for its membership in a certain set of states we find that merging is possible at the lowest rate allowed by the individual states additionally we establish a lower bound for the classical cost of state merging under state uncertainty to this end we give an elementary proof for the cost in case of a perfectly known state which makes no use of the resource framework as applications of our main result we determine the capacity for one way entanglement distillation if the source is not perfectly known moreover we give another achievability proof for the entanglement generation capacity over compound quantum channels

universal communication over arbitrarily varying channels consider the problem of universally communicating over an arbitrarily varying channel i e a channel comprised of an unknown arbitrary sequence of memoryless channels it is shown that there is a communication system using feedback and common randomness that asymptotically attains with high probability the capacity of the time averaged channel universally for every sequence of channels this attainable rate is optimal under certain conditions while no prior knowledge of the channel sequence is assumed the capacity of the time averaged channel meets or exceeds the traditional arbitrarily varying channel avc capacity for every memoryless avc defined over the same alphabets and therefore the system universally attains the random code avc capacity without knowledge of the avc parameters the presented system combines rateless coding with a universal prediction scheme for the input prior distribution from which the codebook is randomly drawn because at each point in time the future of the channel sequence is unknown to the communicators the adaptation of the input behavior by universally predicting the prior plays a major role in the result

on the capacity region of the discrete additive multiple access arbitrarily varying channel the discrete additive multiple access arbitrarily varying channel avc with two senders and one receiver is considered necessary and sufficient conditions are given for its deterministic code average probability of error capacity region under a state constraint to have a nonempty interior in the case that no state constraint is present the capacity region is characterized exactly in the case of the noiseless mod 2 adder avc using state constraint function l s s and subject to a state constraint l less than or equal to 0 13616917 the capacity region is shown to be a 45 degree triangle whose legs have length 1 h l where h denotes the binary entropy function

one shot classical quantum capacity and hypothesis testing the one shot classical capacity of a quantum channel quantifies the amount of classical information that can be transmitted through a single use of the channel such that the error probability is below a certain threshold in this work we show that this capacity is well approximated by a relative entropy type measure defined via hypothesis testing combined with a quantum version of stein s lemma our results give a conceptually simple proof of the well known holevo schumacher westmoreland theorem for the capacity of memoryless channels more generally we obtain tight capacity formulas for arbitrary not necessarily memoryless channels

correlated sources help transmission over an arbitrarily varying channel it is well known that the deterministic code capacity for the average error probability criterion of an arbitrarily varying channel avc either equals its random code capacity or zero here it is shown that if two components of a correlated source are additionally available to the sender and receiver respectively the capacity always equals its random code capacity

on the capacity of channels with unknown interference the process of communicating in the presence of interference that is unknown or hostile is modeled as a two person zero sum game with the communicator and the jammer as the players the objective function considered is the rate of reliable communication the communicator s strategies are encoders and distributions on a set of quantizers the jammer s strategies are distributions on the noise power subject to certain constraints various conditions are considered on the jammer s strategy set and on the communicator s knowledge for the case where the decoder is uninformed of the actual quantizer chosen it is shown that from the communicator s perspective the worst case jamming strategy is a distribution concentrated on a finite number of points thereby converting a functional optimization problem into a nonlinear programming problem moreover the worst case distributions can be characterized by means of necessary and sufficient conditions which are easy to verify for the case where the decoder is informed of the actual quantizer chosen the existence of saddle point strategies is demonstrated the analysis is also seen to be valid for a number of situations where the jammer is adaptive

rateless codes for avc models the arbitrarily varying channel avc is a channel model whose state is selected maliciously by an adversary fixed blocklength coding assumes a worst case bound on the adversary s capabilities which leads to pessimistic results this paper defines a variable length perspective on this problem for which achievable rates are shown that depend on the realized actions of the adversary specifically rateless codes are constructed which require a limited amount of common randomness these codes are constructed for two kinds of avc models in the first the channel state cannot depend on the channel input and in the second it can as a by product the randomized coding capacity of the avc with state depending on the transmitted codeword is found and shown to be achievable with a small amount of common randomness the results for this model are proved using a randomized strategy based on list decoding

list decoding for the arbitrarily varying channel under state constraints list decoding for arbitrarily varying channels avcs under state constraints is investigated it is shown that rates within of the randomized coding capacity of avcs with input dependent state can be achieved under maximal error with list decoding using lists of size o 1 under the average error criterion an achievable rate and converse bound are given for lists of size l these bounds are based on two different notions of symmetrizability and do not coincide in general an example is given which shows that for list size l the capacity may be positive but strictly smaller than the randomized coding capacity in contrast to the situation without constraints

pseudocycles and integral homology we describe a natural isomorphism between the set of equivalence classes or pseudocycles and the integral homology groups of a smooth manifold our arguments generalize to settings well suited for applications in enumerative algebraic geometry and for construction of the virtual fundamental class in the gromov witten theory

upper bounds on the capacity of binary channels with causal adversaries in this paper we consider the communication of information in the presence of a causal adversarial jammer in the setting under study a sender wishes to communicate a message to a receiver by transmitting a codeword x x1 xn bit by bit over a communication channel the sender and the receiver do not share common randomness the adversarial jammer can view the transmitted bits xi one at a time and can change up to a p fraction of them however the decisions of the jammer must be made in a causal manner namely for each bit xi the jammer s decision on whether to corrupt it or not must depend only on xj for j i this is in contrast to the classical adversarial jamming situations in which the jammer has no knowledge of x or knows x completely in this study we present upper bounds that hold under both the average and maximal probability of error criteria on the capacity which hold for both deterministic and stochastic encoding schemes

polytope codes against adversaries in networks this paper investigates a network coding problem wherein an adversary controls a subset of nodes in the network of limited quantity but unknown location this problem is shown to be more difficult than that of an adversary controlling a given number of edges in the network in that linear codes are insufficient to solve the node problem the class of polytope codes is introduced polytope codes are constant composition codes operating over bounded polytopes in integer vector fields the polytope structure creates additional complexity but it induces properties on marginal distributions of code vectors so that validities of codewords can be checked by internal nodes of the network it is shown that polytope codes achieve a cut set bound for a class of planar networks it is also shown that this cut set bound is not always tight and a tighter bound is given for an example network

achieving the empirical capacity using feedback memoryless additive models we address the problem of universal communications over an unknown channel with an instantaneous noiseless feedback and show how rates corresponding to the empirical behavior of the channel can be attained although no rate can be guaranteed in advance first we consider a discrete modulo additive channel with alphabet x where the noise sequence zn is arbitrary and unknown and may causally depend on the transmitted and received sequences and on the encoder s message possibly in an adversarial fashion although the classical capacity of this channel is zero we show that rates approaching the empirical capacity log x h emp zn can be universally attained where h emp zn is the empirical entropy of zn for the more general setting where the channel can map its input to an output in an arbitrary unknown fashion subject only to causality we model the empirical channel actions as the modulo addition of a realized noise sequence and show that the same result applies if common randomness is available the results are proved constructively by providing a simple sequential transmission scheme approaching the empirical capacity

on the genus one gromov witten invariants of complete intersections we state and prove a long elusive relation between genus one gromov witten of a complete intersection and twisted gromov witten invariants of the ambient projective space as shown in a previous paper certain naturally arising cones of holomorphic vector bundle sections over the main component m 0 1 k p d of the moduli space of stable genus one holomorphic maps into p have a well defined euler class in this paper we extend this result to moduli spaces of perturbed in a restricted way j holomorphic maps this extension is used to show that these cones are the correct genus one analogues of the vector bundles relating genus zero gromov witten invariants of a complete intersection to those of the ambient projective space a relationship for highergenus invariants is conjectured as well

a channel under simultaneous jamming and eavesdropping attack correlated random coding capacities under strong secrecy criteria we give a complete characterization of the correlated random coding secrecy capacity of arbitrarily varying wiretap channels avwcs we apply two alternative strong secrecy criteria which both lead to the same multi letter formula the difference of these criteria lies in the treatment of correlated randomness they coincide in the case of uncorrelated codes on the basis of the derived formula we show that the correlated random coding secrecy capacity is continuous as a function of the avwc in contrast to the discontinuous uncorrelated coding secrecy capacity in the proof of the secrecy capacity formula for correlated random codes we apply an auxiliary channel which is compound from the sender to the intended receiver and arbitrarily varying from the sender to the eavesdropper

resonances for dirac operators on the half line abstract we consider the 1d dirac operator on the half line with compactly supported potentials we study resonances as the poles of scattering matrix or equivalently as the zeros of modified fredholm determinant we obtain the following properties of the resonances 1 asymptotics of counting function 2 estimates on the resonances and the forbidden domain

on the deterministic code capacity of the multiple access arbitrarily varying channel a method that allows one to decide whether or not the capacity region of a multiple access arbitrarily varying channel avc has a nonempty interior is discussed using the method of types and an approach different from j h jahn ibid vol 27 no 3 p 212 226 1981 this problem is partially solved the notion of symmetrizability for the two user avc as an extension of the same notion for the single user avc is considered it is shown that if a multiple access avc is symmetrizable then its capacity region has an empty interior for the two user avc this means that at least one and perhaps both users cannot reliably transmit information across the channel more importantly it is shown that if the channel is suitably nonsymmetrizable then the capacity region has a nonempty interior and both users can reliably transmit information across the channel the proofs rely heavily on a complicated decoding rule conditions under which simpler multiple message decoding techniques might suffice are therefore examined in particular conditions under which the universal maximal mutual information decoding rule will be effective are given

identification without randomization in the theory of identification via noisy channels randomization in the encoding has a dramatic effect on the optimal code size namely it grows double exponentially in the blocklength whereas in the theory of transmission it has the familiar exponential growth we consider now instead of the discrete memoryless channel dmc more robust channels such as the familiar compound cc and arbitrarily varying channels avc they can be viewed as models for jamming situations we make the pessimistic assumption that the jammer knows the input sequence before he acts this forces communicators to use the maximal error concept and also makes randomization in the encoding superfluous now for a dmc w by a simple observation made by ahlswede and dueck 1989 in the absence of randomization the identification capacity say c sub nri w equals the logarithm of the number of different row vectors in w we generalize this to compound channels a formidable problem arises if the dmc w is replaced by the avc w in fact for 0 1 matrices only in w we are exactly as for transmission led to the equivalent zero error capacity of shannon but for general w the identification capacity c sub nri w is quite different from the transmission capacity c w an observation is that the separation codes of ahlswede 1989 are also relevant here we present a lower bound on c sub nri w it implies for instance for w sub 0 1 sup 1 0 sub spl delta 1 spl delta 1 0 spl delta spl isin 0 1 2 that c sub nri w 1 which is obviously tight it exceeds c w which is known to exceed 1 h spl delta where h is the binary entropy function we observe that a separation code with worst case average list size l which we call an nra code can be partitioned into l 2 sup ne transmission codes this gives a nonsingle letter characterization of the capacity of avc with maximal probability of error in terms of the capacity of codes with list decoding we also prove that randomization in the decoding does not increase c sub i w and c sub nri w finally we draw attention to related work on source coding

classical capacity of classical quantum arbitrarily varying channels we prove that the average error classical capacity c w of a classicalquantum arbitrarily varying channel cq avc w equals 0 or else the random code capacity c ahlswede s dichotomy we also establish a necessary and sufficient condition for c w 0

on the deterministic code capacity of the two user discrete memoryless arbitrarily varying general broadcast channel with degraded message sets an inner bound on the deterministic code capacity region of the two user discrete memoryless arbitrarily varying general broadcast channel avgbc was characterized by jahn assuming that the common message capacity is nonzero however he did not indicate how one could decide whether the latter capacity is positive csiszaacuter and narayan s result for the single user arbitrarily varying channel avc establishes the missing part in jahn s characterization nevertheless being based on ahlswede s elimination technique jahn s characterization is not applicable for symmetrizable channels under state constraint here the various notions of symmetrizability for the two user broadcast avc are defined sufficient non symmetrizability condition that renders the common message capacity of the avgbc positive is identified using an approach different from jahn s the decoding rules we use establish an achievable region under state and input constraints for the family of degraded message sets codes over the avgbc

asymptotic phase asymptotic modulus and titchmarsh weyl coefficient for a dirac system abstract this paper studies a series representation for the jost function associated with a dirac system under suitable hypotheses the series yields the asymptotic form of the titchmarsh weyl coefficient valid not only in sectors of the upper halfplane but also the real axis the asymptotic form of the spectral function higher derivatives of the jost function and the limiting form of the asymptotic phase are also considered extensions of certain scattering formulas known in the schrodinger case are given for the dirac system

invariants of real symplectic 4 manifolds and lower bounds in real enumerative geometry we first build the moduli spaces of real rational pseudo holomorphic curves in a given real symplectic 4 manifold then following the approach of gromov and witten 3 19 11 we define invariants under deformation of real symplectic 4 manifolds these invariants provide lower bounds for the number of real rational j holomorphic curves which realize a given homology class and pass through a given real configuration of points

capacity and random coding exponents for channel coding with side information capacity formulas and random coding exponents are derived for a generalized family of gel fand pinsker coding problems these exponents yield asymptotic upper bounds on the achievable log probability of error in our model information is to be reliably transmitted through a noisy channel with finite input and output alphabets and random state sequence and the channel is selected by a hypothetical adversary partial information about the state sequence is available to the encoder adversary and decoder the design of the transmitter is subject to a cost constraint two families of channels are considered 1 compound discrete memoryless channels cdmc and 2 channels with arbitrary memory subject to an additive cost constraint or more generally to a hard constraint on the conditional type of the channel output given the input both problems are closely connected the random coding exponent is achieved using a stacked binning scheme and a maximum penalized mutual information decoder which may be thought of as an empirical generalized maximum a posteriori decoder for channels with arbitrary memory the random coding exponents are larger than their cdmc counterparts applications of this study include watermarking data hiding communication in presence of partially known interferers and problems such as broadcast channels all of which involve the fundamental idea of binning

inverse scattering on the line for a dirac system the whole line version of the gelfand levitan marchenko glm equation for a dirac system is studied a new derivation of the glm equation is given under weaker hypotheses than frolov s earlier treatment sov math dokl 13 1468 1972 and the complete inversion is carried out in some explicit cases in which a spectral gap is present previous calculations of this type are restricted either to a scalar potential or degenerate gap applications are discussed in connection with optical couplers and soliton equations

on the structure of certain natural cones over moduli spaces of genus one holomorphic maps we show that certain naturally arising cones over the main component of a moduli space of j0 holomorphic maps into pn have a well defined euler class we also prove that this is the case if the standard complex structure j0 on pn is replaced by a nearby almost complex structure j the genus zero analogue of the cone considered in this paper is a vector bundle the genus zero gromov witten invariant of a projective complete intersection can be viewed as the euler class of such a vector bundle as shown in a separate paper this is also the case for the genus one part of the genus one gw invariant the remaining part is a multiple of the genus zero gw invariant

lower bounds and asymptotics of real double hurwitz numbers we study the real counterpart of double hurwitz numbers called real double hurwitz numbers here we establish a lower bound for these numbers with respect to their dependence on the distribution of branch points we use it to prove under certain conditions existence of real hurwitz covers as well as logarithmic equivalence of real and classical hurwitz numbers the lower bound is based on the tropical computation of real hurwitz numbers in markwig and rau math z 281 1 2 501 522 2015