low precision a d conversion for maximum information rate in channels with memory analog to digital converters that maximize the information rate between the quantized channel output sequence and the channel input sequence are designed for discrete time channels with intersymbol interference additive noise and for independent and identically distributed signaling optimized scalar quantizers with regions achieve the full information rate of log2 bits per channel use with a transmit alphabet of size at infinite signal to noise ratio these quantizers however are not necessarily uniform quantizers low precision scalar and two dimensional analog to digital converters are designed at finite signal to noise ratio and an upper bound on the information rate is derived simulation results demonstrate the effectiveness of the designed quantizers over conventional quantizers the advantage of the new quantizers is further emphasized by an example of a channel for which a slicer with a single threshold at zero and a carefully optimized channel input with memory fail to achieve a rate of one bit per channel use at high signal to noise ratio in contrast to memoryless binary signaling and an optimized quantizer

subblock constrained codes for real time simultaneous energy and information transfer consider an energy harvesting receiver that uses the same received signal both for decoding information and for harvesting energy which is employed to power its circuitry in the scenario where the receiver has limited battery size a signal with bursty energy content may cause power outage at the receiver since the battery will drain during intervals with low signal energy in this paper we analyze subblock energy constrained codes seccs which ensure that sufficient energy is carried within every subblock duration we consider discrete memoryless channels and characterize the secc capacity and the secc error exponent and provide useful bounds for these values we also study constant subblock composition codes csccs which are a subclass of seccs where all the subblocks in every codeword have the same fixed composition and this subblock composition is chosen to maximize the rate of information transfer while meeting the energy requirement compared with constant composition codes cccs we show that csccs incur a rate loss and that the error exponent for csccs is also related to the error exponent for cccs by the same rate loss term we exploit the symmetry in csccs to obtain a necessary and sufficient condition on the subblock length for avoiding power outage at the receiver furthermore for cscc sequences we present a tight lower bound on the average energy per symbol within a sliding time window we provide numerical examples highlighting the tradeoff between the delivery of sufficient energy to the receiver and achieving high information transfer rates it is observed that the ability to use energy in real time imposes less of penalty compared with the ability to use information in real time

a conditional entropy power inequality for dependent variables we provide a condition under which a version of shannon s entropy power inequality will hold for dependent variables we first provide a fisher information inequality extending that found in the independent case the key ingredients are a conditional expectation representation for the score function of a sum and the de bruijn identity which relates entropy and fisher information

matrix elements and wigner coefficients for uq gl n in this series of papers the fundamental wigner coefficients and the matrix elements of the generators for the quantum group uq gl n are derived the uq gl n uq gl n 1 reduced wigner coefficients and reduced matrix elements are determined algebraically as eigenvalues of certain uq gl n 1 invariants in the quantum group uq gl n the matrix elements of the elementary quantum group generators in the gel fand tsetlin basis are derived below while all fundamental wigner coefficients are derived in the second paper of the series

simple measure for complexity a measure of complexity is proposed based on appropriately defined notions of order and disorder which has a considerable degree of flexibility in its dependence on these concepts the possible functional dependencies which result encompass those of many earlier definitions of complexity the proposed measure is in principle easy to calculate and has the property of an intensive thermodynamic quantity with appropriate choices of parameters it behaves similarly to effective measure complexity for the logistic map it is also a generalization of the normalized complexity of lopez ruiz et al but does not suffer from over universality

tensor operators and projection techniques in infinite dimensional representations of semi simple lie algebras polynomial identities satisfied by the infinitesimal generators of a semi simple lie group are employed to construct projection operators which project a tensor product representation v lambda x v v lambda finite dimensional and irreducible v an infinite dimensional representation admitting an infinitesimal character onto a primary subrepresentation such projection operators project out the generalised shift components of a tensor operator acting on an infinite dimensional representation v in direct analogy with the finite dimensional analysis of bracken and green 1971 applications of these methods to evaluating the matrix elements of the generators of the locally compact groups u p q and o p q in unitary discrete irreducible representations are discussed

the capacity region of the gaussian multiple input multiple output broadcast channel the gaussian multiple input multiple output mimo broadcast channel bc is considered the dirty paper coding dpc rate region is shown to coincide with the capacity region to that end a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality to show that a superposition of gaussian codes is optimal for the degraded vector broadcast channel and that dpc is optimal for the nondegraded case furthermore the capacity region is characterized under a wide range of input constraints accounting as special cases for the total power and the per antenna power constraints

wigner coefficients for a semisimple lie group and the matrix elements of the o n generators a purely algebraic infinitesimal method for obtaining multiplicity free wigner coefficients is presented the method is applied to obtain analytic expressions for the complete matrix elements of all o n generators moreover the structure of these matrix elements in terms of reduced matrix elements wigner coefficients and reduced wigner coefficients is made explicit by comparison with the wigner eckart theorem explicit analytic expressions are obtained for the fundamental wigner coefficients of o n finally the results are presented in a form which is directly analogous with the corresponding results for u n

bounds on the information rate of intertransition time restricted binary signaling over an awgn channel upper and lower bounds on the capacity of a continuous time additive white gaussian noise awgn channel with bilevel spl plusmn spl radic p input signals subjected to a minimum inter transition time t sub min constraint are derived the channel model and input constraints reflect basic features of certain magnetic recording systems the upper bounds are based on duncan s relation between the average mutual information in an awgn regime and the mean square error mse of an optimal causal estimator evaluation or upper bounding the mse of suboptimal causal estimators yields the desired upper bounds the lower bound is found by invoking the extended mrs gerber s lemma and asymptotic properties of the entropy of max entropic bipolar d k codes asymptotic results indicate that at low snr pt sub min n sub 0 with n sub 0 designating the noise one sided power spectral density the capacity tends to p n sub 0 nats per second nats s i e it equals the capacity in the simplest average power limited case at high snr the capacity in the simplest average power limited case at high snr the capacity behaves asymptotically as t sub min sup 1 ln snr ln snr nats s demonstrating the degradation relatively to t sub avg sup 1 lnsnr which is the asymptotic known behavior of the capacity with a bilevel average intertransition time t sub avg restricted channel input additional lower bounds are obtained by considering specific signaling formats such as pulsewidth modulation the effect of mild channel filtering on the lower bounds on capacity is also addressed and novel techniques to lower bound the capacity in this case are introduced

offdiagonal complexity a computationally quick complexity measure for graphs and networks abstract a vast variety of biological social and economical networks shows topologies drastically differing from random graphs yet the quantitative characterization remains unsatisfactory from a conceptual point of view motivated from the discussion of small scale free networks a biased link distribution entropy is defined which takes an extremum for a power law distribution this approach is extended to the node node link cross distribution whose nondiagonal elements characterize the graph structure beyond link distribution cluster coefficient and average path length from here a simple and computationally cheap complexity measure can be defined this offdiagonal complexity odc is proposed as a novel measure to characterize the complexity of an undirected graph or network while both for regular lattices and fully connected networks odc is zero it takes a moderately low value for a random graph and shows high values for apparently complex structures as scale free networks and hierarchical trees the odc approach is applied to the helicobacter pylori protein interaction network and randomly rewired surrogates

on the rate of convergence in the entropic central limit theorem we study the rate at which entropy is produced by linear combinations of independent random variables which satisfy a spectral gap condition

power law distributions resulting from finite resources abstract an elementary stochastic model termed the normalization model is put forward which does demonstrate that power laws generically occur in systems with finite resources the model is capable to exhibit power law distributions with arbitrary power law exponents nevertheless for a large fraction of the parameter space power law exponents near unity are obtained as an application of the normalization mechanism we consider a network growth saturation model this model extends the scale free network model sf to include the fact of finite resources in the network growth saturation model the scale free property holds only for the growth period within the stationary regime we obtain power law distributions of the weight of the edges among the vertices we conjecture that this pattern will be found in the internet if it reaches the steady state

representation theory of the symplectic groups i certain aspects of symplectic group representation theory are investigated in particular it is shown that every irreducible representation of sp n admits a relatively large class of states referred to herein as canonical states which possess properties analogous to the gelfand tsetlin states appearing in the theory of the orthogonal and unitary groups the properties of canonical states are investigated and some matrix element formulas are derived

on the capacity of a direct detection photon channel with intertransition constrained binary input the classical directed detection photon channel is modeled by an output nu sub t observed signal describing the photon arrival poisson count process with intensity rate lambda sub t lambda sub 0 where lambda sub t photons s is the channel input information carrying intensity and lambda sub 0 photons s is the dark current intensity upper and lower bounds on the capacity of this channel are presented for two level binary inputs taking on the extreme value lambda sub t in 0 a where a denotes the peak power satisfying an average power constraint e lambda sub t

a binary analog to the entropy power inequality let x sub n y sub n be independent stationary binary random sequences with entropy h x h y respectively let h zeta zeta log zeta 1 zeta log 1 zeta 0 or sigma x sigma y where sigma z h sup 1 h z and alpha beta alpha 1 beta beta 1 alpha when y sub n are independent identically distributed this reduces to mrs gerber s lemma from a d wyner and j ziv 1973

simulation based computation of information rates for channels with memory the information rate of finite state source channel models can be accurately estimated by sampling both a long channel input sequence and the corresponding channel output sequence followed by a forward sum product recursion on the joint source channel trellis this method is extended to compute upper and lower bounds on the information rate of very general channels with memory by means of finite state approximations further upper and lower bounds can be computed by reduced state methods

a new entropy power inequality a strengthened version of shannon s entropy power inequality for the case where one of the random vectors involved is gaussian is proved in particular it is shown that if independent gaussian noise is added to an arbitrary multivariate random variable the entropy power of the resulting random variable is a concave function of the variance power of the added noise the strengthened inequality is shown to hold for the class of stable distributions

on renyi entropy power inequalities this paper gives improved renyi entropy power inequalities r epis consider a sum s n sum k 1 n x k of n independent continuous random vectors taking values on mathbb r d and let alpha in 1 infty an r epi provides a lower bound on the order alpha renyi entropy power of s n that up to a multiplicative constant which may depend in general on n alpha d is equal to the sum of the order alpha renyi entropy powers of the n random vectors x k k 1 n for alpha 1 the r epi coincides with the well known entropy power inequality by shannon the first improved r epi is obtained by tightening the recent r epi by bobkov and chistyakov which relies on the sharpened young s inequality a further improvement of the r epi also relies on convex optimization and results on rank one modification of a real valued diagonal matrix

on divergence power inequalities expressions for entropy power inequality epi shannon type divergence power inequalities dpis in two cases time discrete and time continuous of stationary random processes are given the new expressions connect the divergence rate of the sum of independent processes the individual divergence rate of each process and their power spectral densities all divergences are between a process and a gaussian process with same second order statistics and are assumed to be finite a new proof of the shannon epi based on the relationship between divergence and causal minimum mean square error cmmse in gaussian channels with large signal to noise ratio is also shown

information theoretic description of networks we present a new information theoretic approach for network characterizations it is developed to describe the general type of networks with n nodes and l directed and weighted links i e it also works for the simpler undirected and unweighted networks the new information theoretic measures for network characterizations are based on a transmitter receiver analogy of effluxes and influxes based on these measures we classify networks as either complex or non complex and as either democracy or dictatorship networks directed networks in particular are furthermore classified as either information spreading and information collecting networks the complexity classification is based on the information theoretic network complexity measure medium articulation ma it is proven that special networks with a medium number of links l n1 5 show the theoretical maximum complexity ma logn 2 2 a network is complex if its ma is larger than the average ma of appropriately randomized networks ma mar a network is of the democracy type if its redundancy r h b a it is an information collecting network otherwise an information spreading network finally different real networks directed and undirected weighted and unweighted are classified according to our general scheme

multiplicity free wigner coefficients for semisimple lie groups i the u n pattern calculus this is the first paper in a series of two dedicated to a new algebraic determination of the multiplicity free reduced wigner coefficients for the lie groups u n and o n the approach employed enables a direct nonrecursive derivation of reduced wigner coefficients the absolute value squared of the reduced wigner coefficients is expressed as a rational polynomial function numerator polynomial divided by denominator polynomial whose weyl group symmetries are seen to fall out naturally in our approach from the transformation properties of polynomial functions determined by casimir invariants a unified treatment of the composition laws of reduced wigner coefficients and the evaluation of their associated denominator polynomials is presented which applies to both u n and o n an explicit formula for the numerator polynomials of u n is also derived the numerator polynomials for the orthogonal groups will be given in the second paper of the series

some information theoretic saddlepoints in order to study communication in the presence of jamming or vice versa one can formulate a two person zero sum game with mutual information as the payoff function since mutual information is convex concave in just the right way one often finds saddlepoint strategies which are simultaneously optimal for both players in this paper we shall discuss these saddlepoints for four distinct cases additive noise average power constraints and any combination of hard soft input output quantization one of these cases soft soft has been previously studied shannon dobrushin blachman but the other three appear to be new

minimum ms e gerber s lemma mrs gerber s lemma lower bounds the entropy at the output of a binary symmetric channel in terms of the entropy of the input process in this paper we lower bound the output entropy via a different measure of input uncertainty pertaining to the minimum mean squared error prediction cost of the input process we show that in many cases our bound is tighter than the one obtained from mrs gerber s lemma as an application we evaluate the bound for binary hidden markov processes and obtain new estimates for the entropy rate

the pattern calculus for tensor operators in quantum groups an explicit algebraic evaluation is given for all q tensor operators with unit norm belonging to the quantum group uq u n and having extremal operator shift patterns acting on arbitrary uq u n irreps these rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns a more conceptual derivation of these results is discussed using fundamental properties of the q 6j coefficients

markov processes asymptotically achieve the capacity of finite state intersymbol interference channels recent progress in capacity evaluation has made it possible to compute a sequence of lower bounds on the capacity of a finite state intersymbol interference isi channel by finding a sequence of optimized markov input processes with increasing order for which the state of the process is the previous input symbols in this correspondence we prove that as the order goes to infinity the sequence of optimized markov sources asymptotically achieves the capacity of the channel the conclusion is extended to two dimensional finite state isi channels the binary symmetric channel bsc with constrained inputs and general indecomposable finite state channels with a mild constraint

on the capacity of binary and gaussian channels with run length limited inputs bounds on the capacity of binary symmetric channels and additive gaussian channels with run length limited two level binary bipolar inputs are presented and their tightness is demonstrated for some cases stationary input sequences which do not degrade capacity are considered when deriving the bounds lower bounds on the magnetic recording density for a simple continuous time recording model incorporating a minimal intertransition constraint are evaluated for soft and hard decisions a superiority of about 1 5 db in signal to noise ratio is observed for the soft decision scheme

extension of an entropy property for binary input memoryless symmetric channels the channel output entropy property introduced by a d wyner and j ziv ibid vol it 19 p 769 762 nov 1973 for a binary symmetric channel is extended to arbitrary memoryless symmetric channels with binary inputs and discrete or continuous outputs this yields lower bounds on the achievable information rates of these channels under constrained binary inputs using the interpretation of entropy as a measure of order and randomness the authors deduce that output sequences of memoryless symmetric channels induced by binary inputs are of a higher degree of randomness if the redundancy of the input binary sequence is spread in memory rather than in one dimensional asymmetry it is of interest to characterize the general class of schemes for which this interpretation holds

casimir invariants and characteristic identities for gl a full set of higher order casimir invariants for the lie algebra gl is constructed and shown to be well defined in the category ofs generated by the highest weight unitarizable irreducible representations with only a finite number of nonzero weight components moreover the eigenvalues of these casimir invariants are determined explicitly in terms of the highest weight characteristic identities satisfied by certain infinite matrices with entries from gl are also determined and generalize those previously obtained for gl n by bracken and green a j bracken and h s green j math phys 12 2099 1971 h s green ibid 12 2106 1971

on runlength codes several results on binary d k codes are given first a novel derivation for the capacity of these codes based on information theoretic principles is given based on this result the spectrum of a d k code is computed finally the problem of computing the capacity of the binary symmetric channel under the condition that the input sequences satisfy the d k constraint is considered lower bounds on the capacity of such a channel are derived

lower bounds on achievable rates for limited bitshift correcting codes limited codes runlength limited charge constrained and so on capable of correcting shifts of their symbols are considered the error correction ability is characterized by the minimal bitshift distance d sub bs of a code for a given spl delta d sub bs n where n is the code length the achievable code rate r sub spl alpha is lower bounded we prove the existence of codes of rate r spl ges r sub spl alpha

estimating the entropy of a signal with applications we present a new estimator of the entropy of continuous signals we model the unknown probability density of data in the form of an ar spectrum density and use regularized long ar models to identify the ar parameters we then derive both an analytical expression and a practical procedure for estimating the entropy from sample data we indicate how to incorporate recursive and adaptive features in the procedure we evaluate and compare the new estimator with other estimators based on histograms kernel density models and order statistics finally we give several examples of applications an adaptive version of our entropy estimator is applied to detection of law changes blind deconvolution and source separation

graph distance measures based on topological indices revisited graph distance measures based on topological indices have been already explored by dehmer et al also inequalities for those graph distance measures have been proved in this paper we continue studying such comparative graph measures based on the well known wiener index graph energy and randic index respectively we prove extremal properties of the graph distance measures for some special classes of graphs to demonstrate useful properties of the measures we also discuss numerical results to conclude the paper we state some open problems

invariants and reduced matrix elements associated with the lie superalgebra gl m n we construct explicit formulae for the eigenvalues of certain invariants of the lie superalgebra gl m n using characteristic identities we discuss how such eigenvalues are related to reduced wigner coefficients and the reduced matrix elements of generators and thus provide a first step to a new algebraic derivation of matrix element formulae for all generators of the algebra

on the matrix elements of the u n generators a straightforward derivation of the matrix elements of the u n generators is presented using algebraic infinitesimal techniques an expression for the general fundamental wigner coefficients of the group is obtained as a polynomial in the group generators this enables generalized matrix elements to be defined without explicit reference to basis states such considerations are important for treating groups such as sp 2n whose basis states are not known

noisy constrained capacity for bsc channels we study the classical problem of noisy constrained capacity in the case of the binary symmetric channel bsc namely the capacity of a bsc whose input is a sequence from a constrained set as stated by fan while calculation of the noise free capacity of constrained sequences is well known the computation of the capacity of a constraint in the presence of noise has been an unsolved problem in the half century since shannon s landmark paper we first express the constrained capacity of a binary symmetric channel with d k constrained input as a limit of the top lyapunov exponents of certain matrix random processes then we compute asymptotic approximations of the noisy constrained capacity for cases where the noise parameter e is small in particular we show that when k 2d the error term excess of capacity beyond the noise free capacity is o e whereas it is o eloge when k 2d in both cases we compute the coefficient of the error term in the course of establishing these findings we also extend our previous results on the entropy of a hidden markov process to higher order finite memory processes these conclusions are proved by a combination of analytic and combinatorial methods

the rate distortion function for the quadratic gaussian ceo problem a new multiterminal source coding problem called the ceo problem was presented and investigated by berger zhang and viswanathan recently viswanathan and berger have investigated an extension of the ceo problem to gaussian sources and call it the quadratic gaussian ceo problem they considered this problem from a statistical viewpoint deriving some interesting results in this paper we consider the quadratic gaussian ceo problem from a standpoint of multiterminal rate distortion theory we regard the ceo problem as a certain multiterminal remote source coding problem with infinitely many separate encoders whose observations are conditionally independent if the remote source is given this viewpoint leads us to a complete solution to the problem we determine the tradeoff between the total amount of rate and squared distortion deriving an explicit formula of the rate distortion function the derived function has the form of a sum of two nonnegative functions one is a classical rate distortion function for single gaussian source and the other is a new rate distortion function which dominates the performance of the system for a relatively small distortion it follows immediately from our result that the conjecture of viswanathan and berger on the asymptotic behavior of minimum squared distortion for large rates is true

monotonicity thinning and discrete versions of the entropy power inequality we consider the entropy of sums of independent discrete random variables in analogy with shannon s entropy power inequality where equality holds for normals in our case infinite divisibility suggests that equality should hold for poisson variables we show that some natural analogues of the epi do not in fact hold but propose an alternative formulation which does always hold the key to many proofs of shannon s epi is the behavior of entropy on scaling of continuous random variables we believe that renyi s operation of thinning discrete random variables plays a similar role to scaling and give a sharp bound on how the entropy of ultra log concave random variables behaves on thinning in the spirit of the monotonicity results established by artstein ball barthe and naor we prove a stronger version of concavity of entropy which implies a strengthened form of our discrete epi

solution of shannon s problem on the monotonicity of entropy it is shown that if x1 x2 are independent and identically distributed square integrable random variables then the entropy of the normalized sum ent x1 xn over n is an increasing function of n this resolves an old problem which goes back to 6 7 5 the result also has a version for non identically distributed random variables or random vectors

reduced wigner coefficients for uq gl n this is the second paper in a series concerned with the racah wigner calculus for the quantum group uq gl n in this approach all reduced wigner coefficients are determined algebraically by the eigenvalues of certain generalized casimir invariants here the method is applied to evaluate all fundamental wigner coefficients and reduced wigner coefficients for uq gl n

a computational approach to construct a multivariate complete graph invariant in this paper we present a computational approach for finding complete graph invariants specifically we generate exhaustive sets of connected non isomorphic graphs with 9 and 10 vertices and demonstrate that a 97 dimensional multivariate graph invariant is capable to distinguish each of the non isomorphic graphs furthermore in order to tame the computational complexity of the problem caused by the vast number of graphs e g involving over 10 million networks with 10 vertices we suggest a low dimensional iterative procedure that is based on highly discriminative individual graph invariants we show that also this computational approach leads to a perfect discrimination overall our numerical results prove the existence of such graph invariants for networks with 9 and 10 vertices furthermore we show that our iterative approach has a polynomial time complexity

multiplicity free wigner coefficients for semisimple lie groups ii a pattern calculus for o n in this paper a direct algebraic derivation of a large class of multiplicity free reduced wigner coefficients is presented for the orthogonal groups in particular all elementary reduced wigner coefficients including those for the fundamental spinor and tensor representations are obtained the results are presented in a form directly analogous to the corresponding results obtained for the unitary groups

bounds on the capacity of the bit shift magnetic recording channel a simple statistical channel model is suggested to account for single position bit shifts that have been identified as one of the major impairments in magnetic recording the authors investigate bounds on the capacity of this channel where the channel inputs are the d k codes commonly used in magnetic recordings for d or 2 this channel is conveniently described in terms of phrase lengths where a bit shift causes a phrase either to shrink or to expand the inherent correlation present in consecutive shift affected phrases introduces memory into the channel model sequences of nondecreasing lower bounds and nonincreasing upper bounds on the capacity are evaluated and investigated for a variety of parameters lower bounds on the zero error capacity are also studied the channel model is extended to a concatenated scheme of a bit shift channel connected in tandem with the binary symmetric channel thus capturing both major error generating mechanisms the bit shifts as well as randomly generated errors lower and upper bounds on the capacity of this concatenated channel model with d k input sequences are derived and discussed

the capacity cost function of discrete additive noise channels with and without feedback we consider modulo q additive noise channels where the noise process is a stationary irreducible and aperiodic markov chain of order k we begin by investigating the capacity cost function c spl beta of such additive noise channels without feedback we establish a tight upper bound to c spl beta which holds for general not necessarily markovian stationary q ary noise processes this bound constitutes the counterpart of the wyner ziv lower bound to the rate distortion function of stationary sources with memory we also provide two simple lower bounds to c spl beta which along with the upper bound can be easily calculated using the blahut algorithm for the computation of channel capacity numerical results indicate that these bounds form a tight envelope on c spl beta we next examine the effect of output feedback on the capacity cost function of these channels and establish a lower bound to the capacity cost function with feedback c sub fb spl beta we show both analytically and numerically that for a particular feedback encoding strategy and a class of markov noise sources the lower bound to c sub fb spl beta is strictly greater than c spl beta this demonstrates that feedback can increase the capacity cost function of discrete channels with memory

the entropy power inequality and mrs gerber s lemma for groups of order 2 n shannon s entropy power inequality can be viewed as characterizing the minimum differential entropy achievable by the sum of two independent random variables with fixed differential entropies the entropy power inequality has played a key role in resolving a number of problems in information theory it is therefore interesting to examine the existence of a similar inequality for discrete random variables in this paper we obtain an entropy power inequality for random variables taking values in a group of order 2 n i e for such a group g we explicitly characterize the function f g x y giving the minimum entropy of the sum of two independent g valued random variables with respective entropies x and y random variables achieving the extremum in this inequality are thus the analogs of gaussians in this case and these are also determined it turns out that f g x y is convex in x for fixed y and by symmetry convex in y for fixed x this is a generalization to groups of order 2 n of the result known as mrs gerber s lemma

on the symmetry properties of outer product reduction the symmetry properties of outer product coupling coefficients for the symmetric group and of clebsch gordan coefficients for the unitary group and of the corresponding isoscalar factors are discussed a simple rule is given for determination of a phase factor of outer reduction symmetry of sn and u n the rule is provided with a relationship between the phase factor of a young diagram with its subdiagrams

invariants and matrix elements of the quantum group u_q gl n mathbb c revisited in a previous paper the generator matrix elements and dual vector reduced wigner coefficients rwcs were evaluated via the polynomial identities satisfied by a certain matrix constructed from the r matrix r and its twisted counterpart r t t circle r here we provide an alternative evaluation utilising the r matrix r over tilde r t 1 this provides a new direct derivation of the vector rwcs obtained indirectly in earlier work via a symmetry relation this approach has the advantage that it generalises to the lie superalgebra case which will be investigated elsewhere

a randomized algorithm for the capacity of finite state channels inspired by ideas from the field of stochastic approximation we propose a randomized algorithm to compute the capacity of a finite state channel with a markovian input when the mutual information rate of the channel is concave with respect to the chosen parameterization the proposed algorithm proves to be convergent to the capacity of the channel almost surely with the derived convergence rate we also discuss the convergence behavior of the algorithm without the concavity assumption

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

a note on the shannon capacity of run length limited codes corresp it is proven that 100 percent efficient fixed rate codes for run length limited rll d k and rll charge constrained d k c channels are possible in only two eases namely d k c 0 1 1 and 1 3 3 specifically the binary shannon capacity of rll d k constrained systems is shown to be irrational for all values of d k 0 leq d for rll charge constrained systems with parameters d k c the binary capacity is irrational for all values of d k c 0 leq d except 0 1 1 and 1 3 3 which both have binary capacity 1 2

bounds on information combining with quantum side information bounds on information combining are entropic inequalities that determine how the information entropy of a set of random variables can change when these are combined in certain prescribed ways such bounds play an important role in classical information theory particularly in coding and shannon theory entropy power inequalities are special instances of them the arguably most elementary kind of information combining is the addition of two binary random variables a cnot gate and the resulting quantities play an important role in belief propagation and polar coding we investigate this problem in the setting where quantum side information is available which has been recognized as a hard setting for entropy power inequalities our main technical result is a non trivial and close to optimal lower bound on the combined entropy which can be seen as an almost optimal quantum mrs gerber s lemma our proof uses three main ingredients 1 a new bound on the concavity of von neumann entropy which is tight in the regime of low pairwise state fidelities 2 the quantitative improvement of strong subadditivity due to fawzi renner in which we manage to handle the minimization over recovery maps and 3 recent duality results on classical quantum channels due to renes et al we furthermore present conjectures on the optimal lower and upper bounds under quantum side information supported by interesting analytical observations and strong numerical evidence we finally apply our bounds to polar coding for binary input classical quantum channels and show the following three results 1 even non stationary channels polarize under the polar transform 2 the blocklength required to approach the symmetric capacity scales at most sub exponentially in the gap to capacity and 3 under the aforementioned lower bound conjecture a blocklength polynomial in the gap suffices

enveloping algebra annihilators and projection techniques for finite dimensional cyclic modules of a semisimple lie algebra some results on the structure of finite dimensional cyclic modules for a semisimple lie algebra are presented cyclic modules arise naturally in constructing symmetry adapted states of a system using projection projecting out states with definite symmetry from an arbitrary state is related to the properties of the cyclic module generated by an important example of a cyclic module is the tensor product of two irreducible modules v v which is cyclically generated by the vector v v where v resp v is the minimal resp maximal weight vector of v resp v for this particular case we determine the explicit form of the annihilator in the universal enveloping algebra of the cyclic vector v v it is hoped that this result may add new insight into the clebsch gordan multiplicity problem as an application of this result projection operators are constructed which project from an arbitrary vector of weight a maximal weight vector of weight

relationship between sn and u n isoscalar factors and higher order u n invariants general relationships expressing u n and sn coupling and transformation isoscalar factors in terms of u n racah and 9 coefficients are derived the absolute values of u n racah coefficients involving at most k column irreducible representations are shown to be identical with su k racah coefficients in particular an explicit relationship is established between the u n and su 2 approaches to the many electron correlation problem

entropy and random vectors barron 1 produced a proof of the central limit theorem for real valued iid random variables in the sense of convergence in relative entropy here we establish a similar result for independent real valued random vectors not necessarily identically distributed the main developments required are a generalisation of de bruijn s identity and various inequalities proposed in ref 2

an asymptotic expression for the information and capacity of a multidimensional channel with weak input signals an asymptotic expression is derived for the shannon mutual information between the input and output signals for a relatively large class of continuous alphabet memoryless channels in the case of weak input signals when the input space is multidimensional the authors extend a result of ibragimov and khas minskii 1972 from the one dimensional to the n dimensional case the asymptotic expression obtained relates the shannon 1948 mutual information function and the fisher information matrix this expression is used to derive an asymptotic expression for the capacity of continuous alphabet memoryless channels with vector valued weak input signals this asymptotic capacity involves the largest eigenvalue of the fisher information matrix evaluated at the zero input signal

information theoretic proofs of entropy power inequalities while most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information up to now shannon s entropy power inequality epi is an exception existing information theoretic proofs of the epi hinge on representations of differential entropy using either fisher information or minimum mean square error mmse which are derived from de bruijn s identity in this paper we first present an unified view of these proofs showing that they share two essential ingredients 1 a data processing argument applied to a covariance preserving linear transformation 2 an integration over a path of a continuous gaussian perturbation using these ingredients we develop a new and brief proof of the epi through a mutual information inequality which replaces stam and blachman s fisher information inequality fii and an inequality for mmse by guo shamai and verdu used in earlier proofs the result has the advantage of being very simple in that it relies only on the basic properties of mutual information these ideas are then generalized to various extended versions of the epi zamir and feder s generalized epi for linear transformations of the random variables takano and johnson s epi for dependent variables liu and viswanath s covariance constrained epi and costa s concavity inequality for the entropy power

what is a complex graph abstract many papers published in recent years show that real world graphs g n m n nodes m edges are more or less complex in the sense that different topological features deviate from random graphs here we narrow the definition of graph complexity and argue that a complex graph contains many different subgraphs we present different measures that quantify this complexity for instance c 1 e the relative number of non isomorphic one edge deleted subgraphs i e deck size however because these different subgraph measures are computationally demanding we also study simpler complexity measures focussing on slightly different aspects of graph complexity we consider heuristically defined product measures the products of two quantities which are zero in the extreme cases of a path and clique and entropy measures quantifying the diversity of different topological features the previously defined network graph complexity measures medium articulation and offdiagonal complexity odc belong to these two classes we study odc measures in some detail and compare it with our new measures for all measures the most complex graph g c max has a medium number of edges between the edge numbers of the minimum and the maximum connected graph n 1 m g c max n n 1 2 interestingly for some measures c this number scales exactly with the geometric mean of the extremes m g c max n 2 n 1 n 1 5 all graph complexity measures are characterized with the help of different example graphs for all measures the corresponding time complexity is given finally we discuss the complexity of 33 real world graphs of different biological social and economic systems with the six computationally most simple measures including odc the complexities of the real graphs are compared with average complexities of two different random graph versions complete random graphs just fixed n m and rewired graphs with fixed node degrees