divergence statistics for testing uniform association in cross classifications in this paper we consider the problem of testing uniform association in cross classifications having ordered categories taking as test statistic one in the family proposed by conde and salicru j conde m salicru uniform association in contingency tables associated to csiszar divergence statistics and probability letters 37 1998 149 154 we consider two approximations to the null distribution of the test statistics in this family an estimation of the asymptotic null distribution and a bootstrap estimator we prove that both approximations are asymptotically equivalent to study their finite sample performance we carried out two simulation experiments whose results are presented from the simulations it can be concluded that the bootstrap estimator behaves much better than the estimated asymptotic null distribution

estimation efficiency under privacy constraints we investigate the problem of estimating a random variable y under a privacy constraint dictated by another correlated random variable x when x and y are discrete we express the underlying privacy utility tradeoff in terms of the privacy constrained guessing probability mathcal h p xy varepsilon and the maximum probability mathsf p mathsf c y z of correctly guessing y given an auxiliary random variable z where the maximization is taken over all p z y ensuring that mathsf p mathsf c x z leq varepsilon for a given privacy threshold varepsilon geq 0 we prove that mathcal h p xy cdot is concave and piecewise linear which allows us to derive its expression in closed form for any varepsilon when x and y are binary in the non binary case we derive mathcal h p xy varepsilon in the high utility regime i e for sufficiently large but nontrivial values of varepsilon under the assumption that y and z have the same alphabets we also analyze the privacy constrained guessing probability for two scenarios in which x y and z are binary vectors when x and y are continuous random variables we formulate the corresponding privacy utility tradeoff in terms of mathsf sensr p xy varepsilon the smallest normalized minimum mean squared error mmse incurred in estimating y from a gaussian perturbation z here the minimization is taken over a family of gaussian perturbations z for which the mmse of f x given z is within a factor 1 varepsilon from the variance of f x for any non constant real valued function f we derive tight upper and lower bounds for mathsf sensr when y is gaussian for general absolutely continuous random variables we obtain a tight lower bound for mathsf sensr p xy varepsilon in the high privacy regime i e for small varepsilon

sharp lower bounds on the extractable randomness from non uniform sources extraction of uniform randomness from noisy non uniform sources is an important primitive in many security applications e g pseudo random number generators privacy preserving biometrics and key storage based on physical unclonable functions generic extraction methods exist using universal hash functions there is a trade off between the length of the extracted bit string and the uniformity of the string in the literature there are proven lower bounds on this length as a function of the desired uniformity the best known bound involves a quantity known as smooth min entropy unfortunately there exist at least three definitions of smooth entropy in this paper we compare three of these definitions and we derive improved lower bounds on the extractable randomness we also investigate the use of almost universal hash functions which are slightly worse at extracting randomness than universal hash functions but are preferable in practice because they require far less resources in devices we show that using them has negligible effect on the extractable randomness

sandwiched renyi divergence satisfies data processing inequality sandwiched quantum renyi divergence has been recently defined in the independent works of wilde et al strong converse for the classical capacity of entanglement breaking channels preprint arxiv 1306 1586 2013 and muller lennert et al on quantum renyi entropies a new definition some properties and several conjectures preprint arxiv 1306 3142v1 2013 this new quantum divergence has already found applications in quantum information theory here we further investigate properties of this new quantum divergence in particular we show that sandwiched renyi divergence satisfies the data processing inequality for all values of 1 moreover we prove that holevo information a variant of holevo information defined in terms of sandwiched renyi divergence is super additive our results are based on holder s inequality the riesz thorin theorem and ideas from the theory of complex interpolation we also employ sion s minimax theorem

quantum achievability proof via collision relative entropy in this paper we provide a simple framework for deriving one shot achievable bounds for some problems in quantum information theory our framework is based on the joint convexity of the exponential of the collision relative entropy and is a partial quantum generalization of the technique of yassaee et al from classical information theory based on this framework we derive one shot achievable bounds for the problems of communication over classical quantum channels quantum hypothesis testing and classical data compression with quantum side information we argue that our one shot achievable bounds are strong enough to give the asymptotic achievable rates of these problems even up to the second order

improved bounds on lossless source coding and guessing moments via renyi measures this paper provides upper and lower bounds on the optimal guessing moments of a random variable taking values on a finite set when side information may be available these moments quantify the number of guesses required for correctly identifying the unknown object and similarly to arikan s bounds they are expressed in terms of the arimoto renyi conditional entropy although arikan s bounds are asymptotically tight the improvement of the bounds in this paper is significant in the non asymptotic regime relationships between moments of the optimal guessing function and the map error probability are also established characterizing the exact locus of their attainable values the bounds on optimal guessing moments serve to improve non asymptotic bounds on the cumulant generating function of the codeword lengths for fixed to variable optimal lossless source coding without prefix constraints non asymptotic bounds on the reliability function of discrete memoryless sources are derived as well relying on these techniques lower bounds on the cumulant generating function of the codeword lengths are derived by means of the smooth renyi entropy for source codes that allow decoding errors

renyi resolvability and its applications to the wiretap channel the conventional channel resolvability problem refers to the determination of the minimum rate required for an input process so that the output distribution approximates a target distribution in either the total variation distance or the relative entropy in contrast to previous works in this paper we use the normalized or unnormalized renyi divergence with the renyi parameter in 0 2 cup infty to measure the level of approximation we also provide asymptotic expressions for normalized renyi divergence when the renyi parameter is larger than or equal to 1 as well as lower and upper bounds for the case when the same parameter is smaller than 1 we characterize the renyi resolvability which is defined as the minimum rate required to ensure that the renyi divergence vanishes asymptotically the renyi resolvabilities are the same for both the normalized and unnormalized divergence cases in addition when the renyi parameter smaller than 1 consistent with the traditional case where the renyi parameter is equal to 1 the renyi resolvability equals the minimum mutual information over all input distributions that induce the target output distribution when the renyi parameter is larger than 1 the renyi resolvability is in general larger than the mutual information the optimal renyi divergence is proven to vanish at least exponentially fast for both of these two cases as long as the code rate is larger than the renyi resolvability the optimal exponential rate of decay for i i d random codes is also characterized exactly we apply these results to the wiretap channel and completely characterize the optimal tradeoff between the rates of the secret and non secret messages when the leakage measure is given by the unnormalized renyi divergence this tradeoff differs from the conventional setting when the leakage is measured by the traditional mutual information

arimoto renyi conditional entropy and bayesian m ary hypothesis testing this paper gives upper and lower bounds on the minimum error probability of bayesian m ary hypothesis testing in terms of the arimoto renyi conditional entropy of an arbitrary order alpha the improved tightness of these bounds over their specialized versions with the shannon conditional entropy alpha 1 is demonstrated in particular in the case where m is finite we show how to generalize fano s inequality under both the conventional and list decision settings as a counterpart to the generalized fano s inequality allowing m to be infinite a lower bound on the arimoto renyi conditional entropy is derived as a function of the minimum error probability explicit upper and lower bounds on the minimum error probability are obtained as a function of the arimoto renyi conditional entropy for both positive and negative alpha furthermore we give upper bounds on the minimum error probability as functions of the renyi divergence in the setup of discrete memoryless channels we analyze the exponentially vanishing decay of the arimoto renyi conditional entropy of the transmitted codeword given the channel output when averaged over a random coding ensemble

renyi s divergence and entropy rates for finite alphabet markov sources in this work we examine the existence and the computation of the renyi divergence rate lim sub n spl rarr spl infin 1 n d sub spl alpha p sup n spl par q sup n between two time invariant finite alphabet markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p sup n and q sup n respectively this yields a generalization of a result of nemetz 1974 where he assumed that the initial probabilities under p sup n and q sup n are strictly positive the main tools used to obtain the renyi divergence rate are the theory of nonnegative matrices and perron frobenius theory we also provide numerical examples and investigate the limits of the renyi divergence rate as spl alpha spl rarr 1 and as spl alpha spl darr 0 similarly we provide a formula for the renyi entropy rate lim sub n spl rarr spl infin 1 n h sub spl alpha p sup n of markov sources and examine its limits as spl alpha spl rarr 1 and as spl alpha spl darr 0 finally we briefly provide an application to source coding

analysis of remaining uncertainties and exponents under various conditional renyi entropies we analyze the asymptotics of the normalized remaining uncertainty of a source when a compressed or hashed version of it and correlated side information is observed for this system commonly known as slepian wolf source coding we establish the optimal minimum rate of compression of the source to ensure that the remaining uncertainties vanish we also study the exponential rate of decay of the remaining uncertainty to zero when the rate is above the optimal rate of compression in this paper we consider various classes of random universal hash functions instead of measuring remaining uncertainties using traditional shannon information measures we do so using two forms of the conditional renyi entropy among other techniques we employ new one shot bounds and the moments of type class enumerator method see merhav for these evaluations we show that these asymptotic results are generalizations of the strong converse exponent and the error exponent of the slepian wolf problem under maximum a posteriori decoding

the decorated teichmuller space of punctured surfaces a principal 5 bundle over the usual teichmuller space of ans times punctured surface is introduced the bundle is mapping class group equivariant and admits an invariant foliation several coordinatizations of the total space of the bundle are developed there is furthermore a natural cell decomposition of the bundle finally we compute the coordinate action of the mapping class group on the total space the total space is found to have a rich equivariant geometric structure we sketch some connections with arithmetic groups diophantine approximations and certain problems in plane euclidean geometry furthermore these investigations lead to an explicit scheme of integration over the moduli spaces and to the construction of a universal teichmuller space which we hope will provide a formalism for understanding some connections between the teichmuller theory the kp hierarchy and the virasoro algebra these latter applications are pursued elsewhere

gelfand yaglom perez theorem for generalized relative entropy functionals the measure theoretic definition of kullback leibler relative entropy or simply kl entropy plays a basic role in defining various classical information measures on general spaces entropy mutual information and conditional forms of entropy can be expressed in terms of kl entropy and hence properties of their measure theoretic analogs will follow from those of measure theoretic kl entropy these measure theoretic definitions are key to extending the ergodic theorems of information theory to non discrete cases a fundamental theorem in this respect is the gelfand yaglom perez gyp theorem m s pinsker information and information stability of random variables and process 1960 holden day san francisco ca english ed 1964 translated and edited by amiel feinstein theorem 2 4 2 which states that measure theoretic relative entropy equals the supremum of relative entropies over all measurable partitions this paper states and proves the gyp theorem for renyi relative entropy of order greater than one consequently the result can be easily extended to tsallis relative entropy

renyi divergence measures for commonly used univariate continuous distributions abstract probabilistic distances also called divergences which in some sense assess how close two probability distributions are from one another have been widely employed in probability statistics information theory and related fields of particular importance due to their generality and applicability are the renyi divergence measures this paper presents closed form expressions for the renyi and kullback leibler divergences for nineteen commonly used univariate continuous distributions as well as those for multivariate gaussian and dirichlet distributions in addition a table summarizing four of the most important information measure rates for zero mean stationary gaussian processes namely renyi entropy differential shannon entropy renyi divergence and kullback leibler divergence is presented lastly a connection between the renyi divergence and the variance of the log likelihood ratio of two distributions is established thereby extending a previous result by song j stat plan infer 93 2001 on the relation between renyi entropy and the log likelihood function a table with the corresponding variance expressions for the univariate distributions considered here is also included

reciprocity relations and generalized classic entropic quantifiers that lack trace form abstract in this effort we show that the legendre reciprocity relations thermodynamics essential formal feature are respected by general classic entropic functionals even if they are not of trace form nature in contrast with shannon s or tsallis cases further with reference to the maximum entropy maxent variational process we encounter important cases relevant to physical applications currently discussed in the research literature in which the associated reciprocity relations exhibit anomalies we show that these anomalies can be cured by carefully discriminating between apparently equivalent entropic forms

strong converse for the classical capacity of entanglement breaking and hadamard channels via a sandwiched renyi relative entropy a strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel along with a corresponding achievability statement for rates below the capacity such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication here we show that such a strong converse theorem holds for the classical capacity of all entanglement breaking channels and all hadamard channels the complementary channels of the former these results follow by bounding the success probability in terms of a sandwiched renyi relative entropy by showing that this quantity is subadditive for all entanglement breaking and hadamard channels and by relating this quantity to the holevo capacity prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results

on the entropy of a hidden markov process we study the entropy rate of a hidden markov process hmp defined by observing the output of a binary symmetric channel whose input is a first order binary markov process despite the simplicity of the models involved the characterization of this entropy is a long standing open problem by presenting the probability of a sequence under the model as a product of random matrices one can see that the entropy rate sought is equal to a top lyapunov exponent of the product this offers an explanation for the elusiveness of explicit expressions for the hmp entropy rate as lyapunov exponents are notoriously difficult to compute consequently we focus on asymptotic estimates and apply the same product of random matrices to derive an explicit expression for a taylor approximation of the entropy rate with respect to the parameter of the binary symmetric channel the accuracy of the approximation is validated against empirical simulation results we also extend our results to higher order markov processes and to renyi entropies of any order

the ziv zakai renyi bound for joint source channel coding shannon s capacity and rate distortion function combined with the separation principle provide tight bounds for the minimum possible distortion in joint source channel coding these bounds however are usually achievable only in the limit of a large block length in their 1973 paper ziv and zakai introduced a family of alternative capacity and rate distortion functions based on functionals satisfying the data processing inequality which potentially give tighter bounds for systems with a small block length there is a considerable freedom as to how to choose those functionals and the ways of finding the best possible functionals yielding the best bounds for a given source channel combination are not specified we examine recently conjectured high snr asymptotic expressions for the ziv zakai bounds based on the renyi divergence functional we derive nonasymptotic bounds on the ziv zakai renyi rate distortion function and capacity for a broad class of sources and additive noise channels which hold for arbitrary snr and prove the conjectured asymptotic expressions in the limit of a small distortion high snr the results lead to new bounds on the best achievable distortion in finite dimensional joint source channel coding examples are presented where the new bounds achieve significant improvement upon shannon s original bounds

α z renyi relative entropies we consider a two parameter family of renyi relative entropies d z that are quantum generalisations of the classical renyi divergence d p q this family includes many known relative entropies or divergences such as the quantum relative entropy the recently defined quantum renyi divergences as well as the quantum renyi relative entropies all its members satisfy the quantum generalizations of renyi s axioms for a divergence we consider the range of the parameters z for which the data processing inequality holds we also investigate a variety of limiting cases for the two parameters obtaining explicit formulas for each one of them

the polya information divergence extensions of sanov s theorem and the conditional limit theorem colt are established for a multicolor polya eggenberger pe urn sampling scheme giving the polya information divergence and a polya extension to the maximum relative entropy maxent method polya maxent includes the standard maxent as well as its variants used in bose einstein fermi dirac and intermediate acharya swamy statistics as special cases in the pe setting standard maxent is in general asymptotically inconsistent

entropy of spatial network ensembles we analyze complexity in spatial network ensembles through the lens of graph entropy mathematically we model a spatial network as a soft random geometric graph i e a graph with two sources of randomness namely nodes located randomly in space and links formed independently between pairs of nodes with probability given by a specified function the pair connection function of their mutual distance we consider the general case where randomness arises in node positions as well as pairwise connections i e for a given pair distance the corresponding edge state is a random variable classical random geometric graph and exponential graph models can be recovered in certain limits we derive a simple bound for the entropy of a spatial network ensemble and calculate the conditional entropy of an ensemble given the node location distribution for hard and soft probabilistic pair connection functions under this formalism we derive the connection function that yields maximum entropy under general constraints finally we apply our analytical framework to study two practical examples ad hoc wireless networks and the us flight network through the study of these examples we illustrate that both exhibit properties that are indicative of nearly maximally entropic ensembles

differentials on graph complexes abstract we study the cohomology of complexes of ordinary non decorated graphs introduced by m kontsevich we construct spectral sequences converging to zero whose first page contains the graph cohomology in particular these spectral sequences may be used to show the existence of an infinite series of previously unknown and provably non trivial cohomology classes and put constraints on the structure of the graph cohomology as a whole

on the conditional renyi entropy the renyi entropy of general order unifies the well known shannon entropy with several other entropy notions like the min entropy or collision entropy in contrast to the shannon entropy there seems to be no commonly accepted definition for the conditional renyi entropy several versions have been proposed and used in the literature in this paper we reconsider the definition for the conditional renyi entropy of general order as proposed by arimoto in the seventies we show that this particular notion satisfies several natural properties in particular we show that it satisfies monotonicity under conditioning meaning that conditioning can only reduce the entropy and a weak form of chain rule which implies that the decrease in entropy due to conditioning is bounded by the number of bits one conditions on none of the other suggestions for the conditional renyi entropy satisfies both these properties finally we show a natural interpretation of the conditional renyi entropy in terms of unconditional renyi divergence and we show consistency with a recently proposed notion of conditional renyi entropy in the quantum setting

on renyi information for ergodic diffusion processes in this paper we derive explicit formulas of the renyi information shannon entropy and song measure for the invariant density of one dimensional ergodic diffusion processes in particular the diffusion models considered include the hyperbolic the generalized inverse gaussian the pearson the exponential family and a new class of skew t diffusions

on the estimation of differential entropy from data located on embedded manifolds estimation of the differential entropy from observations of a random variable is of great importance for a wide range of signal processing applications such as source coding pattern recognition hypothesis testing and blind source separation in this paper we present a method for estimation of the shannon differential entropy that accounts for embedded manifolds the method is based on high rate quantization theory and forms an extension of the classical nearest neighbor entropy estimator the estimator is consistent in the mean square sense and an upper bound on the rate of convergence of the estimator is given because of the close connection between compression and shannon entropy the proposed method has an advantage over methods estimating the renyi entropy through experiments on uniformly distributed data on known manifolds and real world speech data we show the accuracy and usefulness of our proposed method

strong correlations between the exponent α and the particle number for a renyi monoatomic gas in gibbs statistical mechanics appealing to the 1902 gibbs formalism for classical statistical mechanics sm the first sm axiomatic theory ever that successfully explained equilibrium thermodynamics we show that already at the classical level there is a strong correlation between renyi s exponent and the number of particles for very simple systems no reference to heat baths is needed for such a purpose

multiple edges in m kontsevich s graph complexes and computations of the dimensions and euler characteristics we provide a generating function for the graded dimensions of m kontsevich s graph complexes of ordinary graphs this generating function can be used to compute the euler characteristic in each loop order furthermore we show that graphs with multiple edges can be omitted from these graph complexes

on some entropy functionals derived from renyi information divergence we consider the maximum entropy problems associated with renyi q entropy subject to two kinds of constraints on expected values the constraints considered are a constraint on the standard expectation and a constraint on the generalized expectation as encountered in nonextensive statistics the optimum maximum entropy probability distributions which can exhibit a power law behaviour are derived and characterized the renyi entropy of the optimum distributions can be viewed as a function of the constraint this defines two families of entropy functionals in the space of possible expected values general properties of these functionals including nonnegativity minimum convexity are documented their relationships as well as numerical aspects are also discussed finally we work out some specific cases for the reference measure q x and recover in a limit case some well known entropies

on quantum renyi entropies a new generalization and some properties the renyi entropies constitute a family of information measures that generalizes the well known shannon entropy inheriting many of its properties they appear in the form of unconditional and conditional entropies relative entropies or mutual information and have found many applications in information theory and beyond various generalizations of renyi entropies to the quantum setting have been proposed most prominently petz s quasi entropies and renner s conditional min max and collision entropy however these quantum extensions are incompatible and thus unsatisfactory we propose a new quantum generalization of the family of renyi entropies that contains the von neumann entropy min entropy collision entropy and the max entropy as special cases thus encompassing most quantum entropies in use today we show several natural properties for this definition including data processing inequalities a duality relation and an entropic uncertainty relation

renyi s information transfer between financial time series a b s t r a c t in this paper we quantify the statistical coherence between financial time series by means of the renyi entropy with the help of campbell s coding theorem we show that the renyi entropy selectively emphasizes only certain sectors of the underlying empirical distribution while strongly suppressing others this accentuation is controlled with renyi s parameter q to tackle the issue of the information flow between time series we formulate the concept of renyi s transfer entropy as a measure of information that is transferred only between certain parts of underlying distributions this is particularly pertinent in financial time series where the knowledge of marginal events such as spikes or sudden jumps is of a crucial importance we apply the renyian information flow to stock market time series from 11 world stock indices as sampled at a daily rate in the time period 02 01 1990 31 12 2009 corresponding heat maps and net information flows are represented graphically a detailed discussion of the transfer entropy between the dax and s p500 indices based on minute tick data gathered in the period 02 04 2008 11 09 2009 is also provided our analysis shows that the bivariate information flow between world markets is strongly asymmetric with a distinct information surplus flowing from the asia pacific region to both european and us markets an important yet less dramatic excess of information also flows from europe to the us this is particularly clearly seen from a careful analysis of renyi information flow between the dax and s p500 indices

dimensional regularization of renyi s statistical mechanics abstract we show that typical renyi s statistical mechanics quantifiers exhibit poles we are referring to the partition function z and the mean energy u renyi s entropy is characterized by a real parameter the poles emerge in a numerable set of rational numbers belonging to the line physical effects of these poles are studied by appeal to dimensional regularization as usual interesting effects are found as for instance gravitational ones in particular negative specific heats

the asymptotics of posterior entropy and error probability for bayesian estimation we consider the bayesian parameter estimation problem where the value of a finitary parameter x should be decided on the basis of i i d sample y sup n of size n in this context the amount of missing information on x after observing y sup n may be evaluated by the posterior entropy which is often called the equivocation or the conditional entropy of x given y sup n while it is well known that the minimum possible probability of error in estimating x is achieved by the maximum a posteriori probability map estimator in this work the focus is on the asymptotic relation between the posterior entropy and the map error probability as the sample size n becomes sufficiently large it is shown that if the sample size n is large enough the posterior entropy as well as the map error probability decay with n to zero at the identical exponential rate and that the maximum achievable exponent for this decay is determined by the minimum chernoff information over all the possible pairs of distinct parameter values

disequilibrium complexity the schottky effect and q entropies in paramagnetism abstract we investigate connections between statistical quantifiers and paramagnetism more concretely we apply the notions of i disequilibrium and ii statistical complexity to a paramagnetic system of non coupled dipoles interesting insights are thereby obtained in particular we encounter a kind of criticality not associated to the temperature but to the disequilibrium

some properties of renyi entropy and renyi entropy rate in this paper we define the conditional renyi entropy and show that the so called chain rule holds for the renyi entropy then we introduce a relation for the rate of renyi entropy and use it to derive the rate of the renyi entropy for an irreducible aperiodic markov chain we also show that the bound for the renyi entropy rate is simply the shannon entropy rate

m kontsevich s graph complex and the grothendieck teichmuller lie algebra we show that the zeroth cohomology of m kontsevich s graph complex is isomorphic to the grothendieck teichmuller lie algebra mathfrak grt 1 the map is explicitly described this result has applications to deformation quantization and duflo theory we also compute the homotopy derivations of the gerstenhaber operad they are parameterized by mathfrak grt 1 up to one class or two depending on the definitions more generally the homotopy derivations of the non unital e n operads may be expressed through the cohomology of a suitable graph complex our methods also give a second proof of a result of h furusho stating that the pentagon equation for mathfrak grt 1 elements implies the hexagon equation

perturbative series and the moduli space of riemann surfaces on utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires f g ou f est une surface et g une epine de f

in search of minimal random braid configurations a geometric braid consists of n curves stretching between two parallel planes a group structure may be defined on a geometric braid yielding an algebraic braid braids provide information on the structure of knots and links and have many practical applications for example in fluid mechanics and astrophysics a well known problem in braid theory consists of finding minimal crossing number braids no efficient algorithm which solves this problem using group theory seems to be possible for n 3 here we investigate several different approaches to obtaining minimal configurations we employ three different relaxation techniques and compare them with each other and with an algebraic heuristic algorithm in terms of minimization of energy and crossing number and time efficiency by energy we mean total string length of the braid it is found that more than half of the crossings of a sufficiently large braid in terms of crossing number and number of strings are redundant we analyse the different methods and say in what circumstances which method is to be favoured and conclude that minimum braid energy and minimum braid crossing number are substantially different measures of topological complexity for braids

a new approach to the word and conjugacy problems in the braid groups abstract a new presentation of the n string braid group b n is studied using it a new solution to the word problem in b n is obtained which retains most of the desirable features of the garside thurston solution and at the same time makes possible certain computational improvements we also give a related solution to the conjugacy problem but the improvements in its complexity are not clear at this writing

minimum crossing numbers for 3 braids given a braid on n strings find an algorithm which generates an artin braid word b of minimal length this is an important unsolved problem a solution would give us the most economical way of notating and drawing braids the length of an artin word equals the number of crossings seen in a braid diagram minimum crossing numbers provide a measure of complexity for braids this paper presents an algorithm for n 3 a three dimensional configuration space for 3 braids will also be defined and analysed

troublesome aspects of the renyi maxent treatment we study in great detail the possible existence of a renyi associated thermodynamics with negative results in particular we uncover a hidden relation in renyi s variational problem maxent this relation connects the two associated lagrange multipliers canonical ensemble with the mean energy u and the renyi parameter as a consequence of such relation we obtain anomalous renyi maxent thermodynamic results

monotonicity of a relative renyi entropy we show that a recent definition of relative renyi entropy is monotone under completely positive trace preserving maps this proves a recent conjecture of muller lennert et al on quantum renyi entropies a new definition some properties j math phys 54 122203 2013 e print arxiv 1306 3142v1 see also e print arxiv 1306 3142

fuzzy extractors how to generate strong keys from biometrics and other noisy data we provide formal definitions and efficient secure techniques for turning noisy information into keys usable for any cryptographic application and in particular reliably and securely authenticating biometric data our techniques apply not just to biometric information but to any keying material that unlike traditional cryptographic keys is 1 not reproducible precisely and 2 not distributed uniformly we propose two primitives a fuzzy extractor reliably extracts nearly uniform randomness r from its input the extraction is error tolerant in the sense that r will be the same even if the input changes as long as it remains reasonably close to the original thus r can be used as a key in a cryptographic application a secure sketch produces public information about its input w that does not reveal w and yet allows exact recovery of w given another value that is close to w thus it can be used to reliably reproduce error prone biometric inputs without incurring the security risk inherent in storing them we define the primitives to be both formally secure and versatile generalizing much prior work in addition we provide nearly optimal constructions of both primitives for various measures of closeness of input data such as hamming distance edit distance and set difference

conditional entropies and their relation to entanglement criteria we discuss conditional renyi and tsallis entropies for bipartite quantum systems of finite dimension we investigate the relation between the positivity of conditional entropies and entanglement properties it is in particular shown that any state having a negative conditional entropy with respect to any value of the entropic parameter is distillable since it violates the reduction criterion moreover we show that the entanglement of werner states in odd dimensions can neither be detected by entropic criteria nor by any other spectral criterion

conditional renyi entropies there is no generally accepted definition of conditional renyi entropy the unconditional renyi entropy depends on a parameter which for the case of min entropy takes the value even for this particular case there are several proposals for the definition of conditional entropy this paper describes three general definitions of conditional renyi entropy that were found or suggested in the literature their properties are studied and their values as a function of are compared the particular case of min entropy is widely used in cryptography as a security parameter this case is studied in some detail

the energy and helicity of knotted magnetic flux tubes magnetic relaxation of a magnetic field embedded in a perfectly conducting incompressible fluid to minimum energy magnetostatic equilibrium states is considered it is supposed that the magnetic field is confined to a single flux tube which may be knotted a local non orthogonal coordinate system zero framed with respect to the knot is introduced and the field is decomposed into toroidal and poloidal ingredients with respect to this system the helicity of the field is then determined this vanishes for a field that is either purely toroidal or purely poloidal the magnetic energy functional is calculated under the simplifying assumptions that the tube is axially uniform and of circular cross section the case of a tube with helical axis is first considered and new results concerning kink mode instability and associated bifurcations are obtained the case of flux tubes in the form of torus knots is then considered and the ground state energy function m h where h is an internal twist parameter is obtained as expected m h which is a topological invariant of the knot increases with increasing knot complexity the function m h provides an upper bound on the corresponding function m h that applies when the above constraints on tube structure are removed the technique is applicable to any knot admitting a parametric representation on condition that points of vanishing curvature are excluded

a pseudorandom generator from any one way function pseudorandom generators are fundamental to many theoretical and applied aspects of computing we show how to construct a pseudorandom generator from any one way function since it is easy to construct a one way function from a pseudorandom generator this result shows that there is a pseudorandom generator if and only if there is a one way function

tunable measures for information leakage and applications to privacy utility tradeoffs we introduce a tunable measure for information leakage called maximal alpha leakage this measure quantifies the maximal gain of an adversary in inferring any potentially random function of a dataset from a release of the data the inferential capability of the adversary is in turn quantified by a class of adversarial loss functions that we introduce as alpha loss alpha in 1 infty cup infty the choice of alpha determines the specific adversarial action and ranges from refining a belief about any function of the data for alpha 1 to guessing the most likely value for alpha infty while refining the alpha text th moment of the belief for alpha in between maximal alpha leakage then quantifies the adversarial gain under alpha loss over all possible functions of the data in particular for the extremal values of alpha 1 and alpha infty maximal alpha leakage simplifies to mutual information and maximal leakage respectively for alpha in 1 infty this measure is shown to be the arimoto channel capacity of order alpha we show that maximal alpha leakage satisfies data processing inequalities and a sub additivity property thereby allowing for a weak composition result building upon these properties we use maximal alpha leakage as the privacy measure and study the problem of data publishing with privacy guarantees wherein the utility of the released data is ensured via a hard distortion constraint unlike average distortion hard distortion provides a deterministic guarantee of fidelity we show that under a hard distortion constraint for alpha 1 the optimal mechanism is independent of alpha and therefore the resulting optimal tradeoff is the same for all values of alpha 1 finally the tunability of maximal alpha leakage as a privacy measure is also illustrated for binary data with average hamming distortion as the utility measure

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

operational interpretation of renyi information measures via composite hypothesis testing against product and markov distributions we revisit the problem of asymmetric binary hypothesis testing against a composite alternative hypothesis we introduce a general framework to treat such problems when the alternative hypothesis adheres to certain axioms in this case we find the threshold rate the optimal error and strong converse exponents at large deviations from the threshold and the second order asymptotics at small deviations from the threshold we apply our results to find the operational interpretations of various renyi information measures in case the alternative hypothesis is comprised of bipartite product distributions we find that the optimal error and strong converse exponents are determined by the variations of renyi mutual information in case the alternative hypothesis consists of tripartite distributions satisfying the markov property we find that the optimal exponents are determined by the variations of renyi conditional mutual information in either case the relevant notion of renyi mutual information depends on the precise choice of the alternative hypothesis as such this paper also strengthens the view that different definitions of renyi mutual information conditional entropy and conditional mutual information are adequate depending on the context in which the measures are used

homotopy graph complex for configuration and knot spaces we prove that the primitive part of the sinha homology spectral sequence e2 term for the space of long knots is rationally isomorphic to the homotopy epsilon 2 term we also define natural graph complexes computing the rational homotopy of configuration and of knot spaces

exponential decreasing rate of leaked information in universal random privacy amplification we derive a new upper bound for eve s information in secret key generation from a common random number without communication this bound improves on bennett s bound based on the renyi entropy of order 2 because the bound obtained here uses the renyi entropy of order 1 s for s 0 1 this bound is applied to a wire tap channel then we derive an exponential upper bound for eve s information our exponent is compared with hayashi s exponent for the additive case the bound obtained here is better the result is applied to secret key agreement by public discussion

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