an approximation to δ couplings on graphs we discuss a general parametrization for vertices of quantum graphs and show in particular how the s and couplings at an n edge vertex can be approximated by means of n 1 couplings of the type provided the latter are properly scaled

why do block length and delay behave differently if feedback is present for output symmetric discrete memoryless channels dmcs at even moderately high rates fixed block length communication systems show no improvements in their error exponents with feedback this paper studies systems with fixed end to end delay and shows that feedback generally provides dramatic gains in the error exponents a new upper bound the uncertainty focusing bound is given on the probability of symbol error in a fixed delay communication system with feedback this bound turns out to have a form similar to viterbi s bound used for the block error probability of convolutional codes as a function of the fixed constraint length the uncertainty focusing bound is shown to be asymptotically achievable with noiseless feedback for erasure channels as well as for any output symmetric dmc that has strictly positive zero error capacity furthermore it can be achieved in a delay universal anytime fashion even if the feedback itself is delayed by a small amount finally it is shown that for end to end delay it is generally possible at high rates to beat the sphere packing bound for general dmcs thereby providing a counterexample to a conjecture of pinsker

approximation of quantum graph vertex couplings by scaled schrodinger operators on thin branched manifolds we discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds we show that if a neumann type laplacian on such manifolds is amended by suitable potentials the resulting schrodinger operators can approximate non trivial vertex couplings the latter include not only the couplings but also those with wavefunctions discontinuous at the vertex we work out the example of the symmetric couplings and make a conjecture that the same method can be applied to all couplings invariant with respect to the time reversal we conclude with a result that certain vertex couplings cannot be approximated by a pure laplacian

on circuit size complexity and the low hierarchy in np let a be a set having polynomial size circuits if a is also known to be in np then we may conclude that the graph of the polynomial size circuits for a is actually in pi 2 p using this observation we show that sets in np which have polynomial size ciruits are in l 3 p the third level of the low hierarchy in np by a similar technique we are able to show that some other intuitively low sets in np are in l 2 p and even in a certain refinement of l 2 p as a consequence sparse sets are not strong nondeterministic polynomial time turing complete in np unless the polynomial time hierarchy collapses to delta 2 p

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

spectral statistics for the dirac operator on graphs we determine conditions for the quantization of graphs using the dirac operator for both two and four component spinors according to the bohigas giannoni schmit conjecture for such systems with time reversal symmetry the energy level statistics are expected in the semiclassical limit to correspond to those of random matrices from the gaussian symplectic ensemble this is confirmed by numerical investigation the scattering matrix used to formulate the quantization condition is found to be independent of the type of spinor we derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

coding for the feedback gel fand pinsker channel and the feedforward wyner ziv source we consider both channel coding and source coding with perfect past feedback feedforward in the presence of side information it is first observed that feedback does not increase the capacity of the gel fand pinsker channel nor does feedforward improve the achievable rate distortion performance in the wyner ziv problem we then focus on the gaussian case showing that as in the absence of side information feedback feedforward allows to efficiently attain the respective performance limits in particular we derive schemes via variations on that of schalkwijk and kailath these variants which are as simple as their origin and require no binning are shown to achieve respectively the capacity of costa s channel and the wyner ziv rate distortion function finally we consider the finite alphabet setting and derive schemes for both the channel and the source coding problems that attain the fundamental limits using variations on schemes of ahlswede and ooi and wornell and of martinian and wornell respectively

recurrent multiple repetition coding for channels with feedback we consider multiple repetition strategies with fixed delay decoding for discrete memoryless channels with noiseless feedback existing binary schemes by schalkwijk and zigangirov are analyzed and their results are extended the general error exponents are computed and presented by elegant expressions in the strictly symmetric case an important class of precoded sequences so called flip sequences is found and their degrading effect on the error exponent is investigated this effect is shown negligible when the repetition parameters are chosen such that the transmission rate is maximized even when signalling at channel capacity the error exponent is shown to be strictly positive

the free dirac operator on compact and noncompact graphs the free dirac operator defined on composite one dimensional structures consisting of finitely many half lines and intervals is investigated the influence of the connection points between the constituents is modeled by transition conditions for the wave functions or equivalently by different self adjoint extensions of the dirac operator general relations between the parameters of the extensions and the eigenvalues resp the scattering coefficients are derived and then applied to the cases of a bundle of half lines a point defect a branching line and an eye shaped structure

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

the xi function despite the fact that spectral and inverse spectral properties of one dimensional schrsdinger operators h d2 dx2 have been extensively studied for seventy five years there remain large areas where our knowledge is limited for example while the inverse theory for operators on l2 co co is well understood in case v is periodic 12 24 25 35 39 42 49 it is not understood in case limjx o v x co and h has discrete spectrum our goal here is to introduce a special function x a on r associated to h which we believe will be a valuable tool in the spectral and inverse spectral theory in a sense we will make precise it complements the weyl m functions m x a a main application of which we will make here concerns a generalization of the trace formula for schrsdinger operators to general v s recall the well known trace formula for periodic potentials let v x v x t 1 then by floquet theory see e g 10 37 44

optimal feedback communication via posterior matching in this paper we introduce a fundamental principle for optimal communication over general memoryless channels in the presence of noiseless feedback termed posterior matching using this principle we devise a simple sequential generic feedback transmission scheme suitable for a large class of memoryless channels and input distributions achieving any rate below the corresponding mutual information this provides a unified framework for optimal feedback communication in which the horstein scheme bsc and the schalkwijk kailath scheme awgn channel are special cases thus as a corollary we prove that the horstein scheme indeed attains the bsc capacity settling a longstanding conjecture we further provide closed form expressions for the error probability of the scheme over a range of rates and derive the achievable rates in a mismatch setting where the scheme is designed according to the wrong channel model several illustrative examples of the posterior matching scheme for specific channels are given and the corresponding error probability expressions are evaluated the proof techniques employed utilize novel relations between information rates and contraction properties of iterated function systems

an achievable rate region for the multiple access channel with feedback an achievable rate region r 1 leq i x 1 y x 2 u r 2 leq i x 2 y x 1 u r 1 r 2 leq i x 1 x 2 y where p u x l x 2 y p u p x l u p x 2 u p y x l x 2 is established for the multiple access channel with feedback time sharing of these achievable rates yields the rate region of this paper this region generally exceeds the achievable rate region without feedback and exceeds the rate point found by gaarder and wolf for the binary erasure multiple access channel with feedback the presence of feedback allows the independent transmitters to understand each other s intended transmissions before the receiver has sufficient information to achieve the desired decoding this allows the transmitters to cooperate in the transmission of information that resolves the residual uncertainty of the receiver at the same time independent information from the transmitters is superimposed on the cooperative correction information the proof involves list codes and block markov encoding

the error exponent of variable length codes over markov channels with feedback the error exponent of markov channels with feedback is studied in the variable length block coding setting burnashev s classic result is extended to finite state ergodic markov channels for these channels a single letter characterization of the reliability function is presented under the assumption of full causal output feedback and full causal observation of the channel state both at the transmitter and at the receiver side tools from stochastic control theory are used in order to treat channels with intersymbol interference isi specifically the convex analytic approach to markov decision processes is adopted in order to handle problems with stopping time horizons induced by variable length coding schemes

multiple access channels with states causally known at transmitters it has been recently shown by lapidoth and steinberg that strictly causal state information can be beneficial in multiple access channels macs specifically it was proved that the capacity region of a two user mac with independent states each known strictly causally to one encoder can be enlarged by letting the encoders send compressed past state information to the decoder in this study a generalization of the said strategy is proposed whereby the encoders compress also the past transmitted codewords along with the past state sequences the proposed scheme uses a combination of long message encoding compression of the past state sequences and codewords without binning and joint decoding over all transmission blocks the proposed strategy has been recently shown by lapidoth and steinberg to strictly improve upon the original one capacity results are then derived for a class of channels that include two user modulo additive state dependent macs moreover the proposed scheme is extended to state dependent macs with an arbitrary number of users finally output feedback is introduced and an example is provided to illustrate the interplay between feedback and availability of strictly causal state information in enlarging the capacity region

error exponent for gaussian channels with partial sequential feedback this paper studies the error exponent of block coding over an additive white gaussian noise channel where a fraction f of the channel output symbols are revealed to the transmitter through noiseless feedback if the code rate exceeds fc where c is the channel capacity then the probability of decoding error cannot decay faster than exponentially with block length however if the code rate is below fc the error probability can decrease faster than exponentially with the block length as with full feedback f 1 this is achieved by combining a feedback code and a forward error control code and jointly decoding them at the receiver this scheme can attain higher reliability than rate splitting in which feedback and forward codes independently encode separate source messages

traces and inverse nodal problems for dirac type integro differential operators on a graph abstract in this paper dirac type integro differential operators on the star graph are studied first we derive the asymptotic expressions for the large eigenvalues second the regularized trace formulae are obtained for these operators we also provide asymptotic expressions for the nodal points and give a constructive procedure for solving an inverse nodal problem

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

relativized polynomial time hierarchies having exactly k levels it is proved that for every integer k geqq 0 there is an oracle a k relative to which the polynomial time hierarchy collapses so that it has exactly k levels furthermore sets b k and c k may be constructed so that relative to b k the polynomial time hierarchy has exactly k levels and the class pspace coincides with the polynomial time hierarchy and relative to c k the polynomial time hierarchy has exactly k levels and the class pspace is different from the polynomial time hierarchy

the entropy of a randomly stopped sequence a wald like equation is proved for the entropy of a randomly stopped sequence of independent identically distributed discrete random variables x sub 1 x sub 2 with a nonanticipating stopping time n the authors first define a general stopping time and the associated stopped sequence and then present the two main theorems for the entropy of a stopped sequence the formal proofs of the lemmas necessary for the proof of the theorems are given the randomness in the stopped sequence x sup n is the expected number of calls for x times the entropy per call plus the residual randomness in the stopping time conditioned on the unstopped sequence x sup infinity

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

separating the low and high hierarchies by oracles abstract the relativized low and high hierarchies within np are considered an oracle a is constructed such that the low and high hierarchies relative to a are infinite and for each k an oracle ak is constructed such that the low and high hierarchies relative to ak have exactly k levels

inverse nodal problems for dirac type integro differential operators abstract the inverse nodal problem for dirac differential operator perturbated by a volterra integral operator is studied we prove that dense subset of the nodal points determines the coefficients of differential part and gives partial information on the coefficients of integral part of the operator we also provide an algorithm to reconstruct the coefficients of the problem by using the nodal points

strong nondeterministic polynomial time reducibilities abstract building on the work of adleman and manders 1 we define a new class of reducibilities the strong nondeterministic polynomial time reducibilities we show that several reducibilities in this class differ over the recursive sets and assuming np conp that they each determine different classes of np hard sets comparisons in strength are also made between the strong nondeterministic polynomial time reducibilities and the classes of deterministic and non deterministic polynomial time reducibilities in ladner lynch and selman 6

joint source channel coding with feedback this paper quantifies the fundamental limits of variable length transmission of a general possibly analog source over a memoryless channel with noiseless feedback under a distortion constraint we consider excess distortion average distortion and guaranteed distortion d semifaithful codes in contrast to the asymptotic fundamental limit a general conclusion is that allowing variable length codes and feedback leads to a sizable improvement in the fundamental delay distortion tradeoff in addition we investigate the minimum energy required to reproduce k source samples with a given fidelity after transmission over a memoryless gaussian channel and we show that the required minimum energy is reduced with feedback and an average rather than maximal power constraint

fast iterative coding techniques for feedback channels a class of capacity achieving low complexity high reliability variable rate coding schemes is developed for communication over discrete memoryless channels with noiseless feedback algorithms for encoding and decoding that require computations growing linearly with the number of channel inputs used are developed the error exponent associated with the scheme is shown to be optimal and implies that capacity is achievable simulations are performed and support the analytically predicted high performance and low complexity

extrinsic jensen shannon divergence applications to variable length coding this paper considers the problem of variable length coding over a discrete memoryless channel with noiseless feedback this paper provides a stochastic control view of the problem whose solution is analyzed via a newly proposed symmetrized divergence termed extrinsic jensen shannon ejs divergence it is shown that strictly positive lower bounds on ejs divergence provide nonasymptotic upper bounds on the expected code length this paper presents strictly positive lower bounds on ejs divergence and hence nonasymptotic upper bounds on the expected code length for the following two coding schemes 1 variable length posterior matching and 2 maxejs coding scheme that is based on a greedy maximization of the ejs divergence as an asymptotic corollary of the main results this paper also provides a rate reliability test variable length coding schemes that satisfy the condition s of the test for parameters r and e are guaranteed to achieve a rate r and an error exponent e the results are specialized for posterior matching and maxejs to obtain deterministic one phase coding schemes achieving capacity and optimal error exponent for the special case of symmetric binary input channels simpler deterministic schemes of optimal performance are proposed and analyzed