phenomenes de symetrie dans des formes lineaires en polyzetas we give two generalizations in arbitrary depth of the symmetry phenomenon used by ball rivoal to prove that infinitely many values of riemann zeta function at odd integers are irrational these generalizations concern multiple series of hypergeometric type which can be written as linear forms in some specific multiple zeta values the proof makes use of the regularization procedure for multiple zeta values with logarithmic divergence

conical zeta values and their double subdivision relations abstract we introduce the concept of a conical zeta value as a geometric generalization of a multiple zeta value in the context of convex cones the quasi shuffle and shuffle relations of multiple zeta values are generalized to open cone subdivision and closed cone subdivision relations respectively for conical zeta values in order to achieve the closed cone subdivision relation we also interpret linear relations among fractions as subdivisions of decorated closed cones as a generalization of the double shuffle relation of multiple zeta values we give the double subdivision relation of conical zeta values and formulate the extended double subdivision relation conjecture for conical zeta values

rigorous properties and uncertainty like relationships on product complexity measures application to atomic systems most of research on complexities and the corresponding conclusions have been obtained by numerically quantifying their values and little attention has been paid to their theoretical properties and the exact meaning within an statistical framework valid for any arbitrary n dimensional distribution in the present work different product complexities constructed as a product of two relevant information quantities associated to the concepts of uncertainty and localization are investigated different results on product complexities such as rigorous bounds uncertainty like inequalities relationships among themselves and also statistical interpretations are given for the sake of completeness some of these analytical results on product complexities are numerically checked for the one particle densities of atomic systems in both conjugated spaces

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

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

intensive entropic non triviality measure abstract we discuss a way of characterizing probability distributions complementing that provided by the celebrated notion of information measure with reference to a measure of complexity that we call a nontriviality measure our starting point is the lmc measure of complexity advanced by lopez ruiz et al phys lett a 209 1995 321 and its analysis by anteneodo and plastino phys lett a 223 1997 348 an improvement of some of their troublesome characteristics is thereby achieved basically we replace the euclidean distance to equilibrium by the jensen shannon divergence the resulting measure turns out to be i an intensive quantity and ii allows one to distinguish between different degrees of periodicity we apply the cured measure to the logistic map so as to clearly exhibit its advantages

gait analysis and the bootstrap this paper is about random coefficient trigonometric regression models and their use in gait analysis here gait analysis means free speed walking on a level surface our study focuses on bootstrap based prediction regions for the angular rotation curves of test children when the relevant training data are gathered from normal children of comparable ages considerations that led to our choice of model and use of the bootstrap are given prediction regions and empirical bootstrap distributions are displayed as is their application to several test cases also included is a study of the almost sure asymptotic behavior of the theoretical bootstrap probability of the prediction regions

euler s constant euler s work and modern developments this paper has two parts the first part surveys euler s work on the constant 0 57721 bearing his name together with some of his related work on the gamma function values of the zeta function and divergent series the second part describes various mathematical developments involving euler s constant as well as another constant the euler gompertz constant these developments include connections with arithmetic functions and the riemann hypothesis and with sieve methods random permutations and random matrix products it also includes recent results on diophantine approximation and transcendence related to euler s constant

a statistical complexity measure with nonextensive entropy and quasi multiplicativity the properties of a statistical complexity measure that are characterized by nonextensivity in entropy have been investigated which is of so called disequilibrium type considering the composition law for two systems with different nonextensivities quasi multiplicativity a nontrivial relation between the nonextensive parameters and the fluctuating bit number in information theory has been mentioned to see the time evolution of the nonextensive complexity measure we examine systems having a lognormal distribution the underlying dynamics for which is known to obey a random multiplicative process in the presence of a boundary constraint

generalized constrained multiple correspondence analysis a comprehensive approach for imposing both row and column constraints on multivariate discrete data is proposed that may be called generalized constrained multiple correspondence analysis gcmca in this method each set of discrete data is first decomposed into several submatrices according to its row and column constraints and then multiple correspondence analysis mca is applied to the decomposed submatrices to explore relationships among them this method subsumes existing constrained and unconstrained mca methods as special cases and also generalizes various kinds of linearly constrained correspondence analysis methods an example is given to illustrate the proposed method

thermodynamic depth of causal states objective complexity via minimal representations thermodynamic depth is an appealing but flawed structural complexity measure it depends on a set of macroscopic states for a system but neither its original introduction by lloyd and pagels nor any follow up work has considered how to select these states depth therefore is at root arbitrary computational mechanics an alternative approach to structural complexity provides a definition for a system close quote s minimal necessary causal states and a procedure for finding them we show that the rate of increase in thermodynamic depth or ital dive is the system close quote s reverse time shannon entropy rate and so depth only measures degrees of macroscopic randomness not structure to fix this we redefine the depth in terms of the causal state representation emdash epsilon machines emdash and show that this representation gives the minimum dive consistent with accurate prediction thus epsilon machines are optimally shallow copyright ital 1999 ital the american physical society

p adic multiple zeta values ii tannakian interpretations we establish a tannakian formalism of p adic multiple polylogarithms and p adic multiple zeta values introduced in our previous paper via a comparison isomorphism between a de rham fundamental torsor and a rigid fundamental torsor of the projective line minus three points and also discuss its hodge and e tale analogues as an application we give a way to erase log poles of p adic multiple polylogarithms and introduce overconvergent p adic multiple polylogarithms which might be p adic multiple analogue of zagier s single valued complex polylogarithms

entropy and complexity analysis of dirac delta like quantum potentials abstract the dirac delta like quantum mechanical potentials are frequently used to describe and interpret numerous phenomena in many scientific fields including atomic and molecular physics condensed matter and quantum computation the entropy and complexity properties of potentials with one and two dirac delta functions are here analytically calculated and numerically discussed in both position and momentum spaces we have studied the information theoretic lengths of fisher renyi and shannon types as well as the cramer rao fisher shannon and lmc shape complexities of the lowest lying stationary states of one delta and twin delta they allow us to grasp and quantify different facets of the spreading of the charge and momentum of the system far beyond the celebrated standard deviation

brain electrical activity analysis using wavelet based informational tools ii tsallis non extensivity and complexity measures the processing of information by the brain is reflected in dynamical changes of the electrical activity in i time ii frequency and iii space therefore the concomitant studies require methods capable of describing the qualitative variation of the signal in both time and frequency the present work complements the efforts described in rosso et al physica a 313 2002 587 devoted to non extensive information measures a la tsallis here we show that the notion of complexity using 1 appropriate information theory tools adapted to a non extensive scenario and 2 scalp eeg data provides one with valuable insights into the dynamics of neural activity in particular using different complexity measures we detect the presence of states characterized by both order and maximal complexity

evaluation of the multiple zeta values zeta 2 dots 2 3 2 dots 2 a formula is given for the special multiple zeta values occurring in the title as rational linear combinations of products m p 2n with m odd the existence of such a formula had been proved using motivic arguments by francis brown but the explicit formula more precisely certain 2 adic properties of its coefficients were needed for his proof in cite1 of the conjecture that all periods of mixed tate motives over z are q 2pi 1 linear combinations of multiple zeta values the formula is proved indirectly by computing the generating functions of both sides in closed form one as the product of a sine function and a 3 f 2 hypergeometric function and one as a sum of 14 products of sine functions and digamma functions and then showing that both are entire functions of exponential growth and that they agree at sufficiently many points to force their equality we also show that the space spanned by the multiple zeta values in question coincides with the space of double zeta values of odd weight and find a relation between this space and the space of cusp forms on the full modular group

new properties of multiple harmonic sums modulo p and p analogues of leshchiner s series in this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences 1 a c 1 b 2 a c 2 b and prove a number of congruences for these sums modulo a prime p the congruences obtained allow us to find nice p analogues of leshchiner s series for zeta values and to refine a result due to m hoffman and j zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo p moreover we are also able to provide a new proof of zagier s formula for zeta 2 a 3 2 b based on a finite identity for partial sums of the zeta star series

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

existence conditions and spreading properties of extreme entropy d dimensional distributions abstract the extremization of the information theoretic measures fisher information shannon entropy tsallis entropy which complementary describe the spreading of the physical states of natural systems gives rise to fundamental equations of motion and or conservation laws at times the associated extreme entropy distributions are known for some given constraints usually moments or radial expectation values in this work first we give the existence conditions of the maxent probability distributions in a d dimensional scenario where two moments not necessarily of consecutive order are known then we find general relations which involve four elements the extremized entropy the other two information theoretic measures and the variance of the extremum density in scenarios with different dimensionalities and moment constraints

a statistical measure of complexity with nonextensive entropy abstract a statistical measure of complexity utilising the concept of entropy or information is proposed our way in this study is to use a nonextensive entropy instead of an extensive additive shannon entropy in the definition but can be characterised as a difference between the qth order renyi entropy and the second one furthermore we devise a conditional joint and mutual complexity measure as a coherent possibility the behavior of the measure for the logistic map shows that it is more sensitive to nonextensivity at the transition point ac 3 8284 than any other values when 0

multiple series connected to hoffman s conjecture on multiple zeta values abstract recent results of zlobin and cresson fischler rivoal allow one to decompose any suitable p uple series of hypergeometric type into a linear combination over the rationals of multiple zeta values of depth at most p in some cases only the multiple zeta values with 2 s and 3 s are involved as in hoffman s conjecture in this text we study the depth p part of this linear combination namely the contribution of the multiple zeta values of depth exactly p we prove that it satisfies some symmetry property as soon as the p uple series does and make some conjectures on the depth p 1 part of the linear combination when p 3 our result generalizes the property that very well poised univariate hypergeometric series involve only zeta values of a given parity which is crucial in the proof by rivoal and ball rivoal that 2 n 1 is irrational for infinitely many n 1 the main feature of the proof is an algebraic approach based on representations of z 2 z p s p

shuffle products for multiple zeta values and partial fraction decompositions of zeta functions of root systems the shuffle product plays an important role in the study of multiple zeta values mzvs this is expressed in terms of multiple integrals and also as a product in a certain non commutative polynomial algebra over the rationals in two indeterminates in this paper we give a new interpretation of the shuffle product in fact we prove that the procedure of shuffle products essentially coincides with that of partial fraction decompositions of mzvs of root systems as an application we give a proof of extended double shuffle relations without using drinfel d integral expressions for mzvs furthermore our argument enables us to give some functional relations which include double shuffle relations

minimum description length induction bayesianism and kolmogorov complexity the relationship between the bayesian approach and the minimum description length approach is established we sharpen and clarify the general modeling principles minimum description length mdl and minimum message length mml abstracted as the ideal mdl principle and defined from bayes s rule by means of kolmogorov complexity the basic condition under which the ideal principle should be applied is encapsulated as the fundamental inequality which in broad terms states that the principle is valid when the data are random relative to every contemplated hypothesis and also these hypotheses are random relative to the universal prior the ideal principle states that the prior probability associated with the hypothesis should be given by the algorithmic universal probability and the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized if we restrict the model class to finite sets then application of the ideal principle turns into kolmogorov s minimal sufficient statistic in general we show that data compression is almost always the best strategy both in model selection and prediction

functional multiple set canonical correlation analysis we propose functional multiple set canonical correlation analysis for exploring associations among multiple sets of functions the proposed method includes functional canonical correlation analysis as a special case when only two sets of functions are considered as in classical multiple set canonical correlation analysis computationally the method solves a matrix eigen analysis problem through the adoption of a basis expansion approach to approximating data and weight functions we apply the proposed method to functional magnetic resonance imaging fmri data to identify networks of neural activity that are commonly activated across subjects while carrying out a working memory task

canonical correlation analysis when the data are curves summary it is not immediately straightforward to extend canonical correlation analysis to the context of functional data analysis where the data are themselves curves or functions the obvious approach breaks down and it is necessary to use a method involving smoothing in some way such a method is introduced and discussed with reference to a data set on human gait the breakdown of the unsmoothed method is illustrated in a practical context and is demonstrated theoretically a consistency theorem for the smoothed method is proved in an increasing number of problems in a wide range of fields the data observed are not the univariate or multivariate observations of classical statistics but are func tions observed continuously ramsay and dalzell 1991 give the name functional data analysis to the analysis of data of this kind in most cases the observations will be functions of time or a closely related variable but there are clearly applications where the functions are surfaces observed over two or three dimensional space the motivating example for the present paper is a set of data on children s gait collected by the motion analysis laboratory at children s hospital san diego california see olshen et al 1989 for full details for each of a number of children several angles made by the child s joints knee hip etc are observed during the child s gait cycle one aim of the study is to gain understanding of the gait cycle of a normal child to make comparisons with children suffering from walking diffi culties these data motivated rice and silverman 1991 to discuss the extension of principal component analysis to the functional setting and to explain how smoothing can be incorporated into the analysis in a natural way in a substantial complemen tary paper ramsay and dalzell 1991 discuss different approaches to regression and principal component analysis in functional data analysis illustrating their work by a meteorological example the insights of functional data analysis may be helpful in chemometrics for instance a standard chemometric problem is the analysis of spectra observed in chromatography and these are to all intents and purposes functional observations another obvious area of potential relevance is the analysis of growth curve data the focus in the present paper will be on canonical correlation analysis cca in the gait example one might ask how variability in the knee angle cycle is related to

scaling properties of composite information measures and shape complexity for hydrogenic atoms in parallel magnetic and electric fields abstract the scaling properties of various composite information theoretic measures shannon and renyi entropy sums fisher and onicescu information products tsallis entropy ratio fisher shannon product and shape complexity are studied in position and momentum spaces for the non relativistic hydrogenic atoms in the presence of parallel magnetic and electric fields such measures are found to be invariant at the fixed values of the scaling parameters given by s 1 b 3 4 0 2 z 2 m 2 e 3 and s 2 f 4 4 0 3 z 3 e 5 m 2 numerical results which support the validity of the scaling properties are shown by choosing the representative example of the position space shape complexity physical significance of the resulting scaling behavior is discussed

renormalization of multiple zeta values abstract multiple zeta values mzvs in the usual sense are the special values of multiple variable zeta functions at positive integers their extensive studies are important in both mathematics and physics with broad connections and applications in contrast very little is known about the special values of multiple zeta functions at non positive integers since the values are usually undefined we define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory and following the hopf algebra approach of connes and kreimer this definition of renormalized mzvs agrees with the convergent mzvs and extends the work of ihara kaneko zagier on renormalization of mzvs with positive arguments we further show that the important quasi shuffle stuffle relation for usual mzvs remains true for the renormalized mzvs

the ponzano regge model and parametric representation we give a parametric representation of the effective noncommutative field theory derived from a kappa deformation of the ponzano regge model and define a generalized kirchhoff polynomial with kappa correction terms obtained in a kappa linear approximation we then consider the corresponding graph hypersurfaces and the question of how the presence of the correction term affects their motivic nature we look in particular at the tetrahedron graph which is the basic case of relevance to quantum gravity with the help of computer calculations we verify that the number of points over finite fields of the corresponding hypersurface does not fit polynomials with integer coefficients hence the hypersurface of the tetrahedron is not polynomially countable this shows that the correction term can change significantly the motivic properties of the hypersurfaces with respect to the classical case

uncertainty relationships in many body systems rigorous relationships among radial and logarithmic expectation values of an n body system are obtained by means of information theoretic methods especially interesting are the uncertainty expressions in terms of the product r n pb p which generalize the well known relation r2 p2 2 9n2 4 additionally an inequality involving the uncertainty in the logarithmic radius and the logarithmic momentum is shown for illustration the aocmcy of the inequalities found are numerically analysed in a hameefock framework for atomic systems finally it is pointed out how these inequalities are improved in atoms by raking into account other physical quantities or by considering some semiempirical results on atomic information entropies the one particle density p r is a fundamental quantity in the study of the physical and chemical properties of many fermion systems parr and yang 1989 since hohenberg and kohn 1964 formulated their theorem about the existence of a universal functional e pl for the energy of the many fermion system much effort has been paid in the study of the function p

features of the extension of a statistical measure of complexity to continuous systems we discuss some aspects of the extension to continuous systems of a statistical measure of complexity introduced by l opez ruiz mancini and calbet phys lett a 209 321 1995 in general the extension of a magnitude from the discrete to the continuous case is not a trivial process and requires some kind of choice in the present study several possibilities appear available one of them is examined in detail some interesting properties desirable for any magnitude of complexity are discovered on this particular extension

the algebra of multitangent functions abstract multizeta values are numbers appearing in many different contexts unfortunately their arithmetics remains mostly out of reach in this article we define a functional analogue of the algebra of multizeta values namely the algebra of multitangent functions which are 1 periodic functions defined by a process formally similar to multizeta values we introduce here the fundamental notions of reduction into monotangent functions projection onto multitangent functions and that of trifactorization giving a way of writing a multitangent function in terms of hurwitz multizeta functions this explains why the multitangent algebra is a functional analogue of the algebra of multizeta values we then discuss the most important algebraic and analytic properties of these functions and their consequences on multizeta values as well as their regularization in the divergent case each property of multitangents has a pendant on the side of multizeta values this allows us to propose new conjectures which have been checked up to the weight 18

mixed tate motives over z we prove that the category of mixed tate motives over z is spanned by the motivic fundamental group of p 1 minus three points we prove a conjecture by m hoffman which states that every multiple zeta value is a q linear combination of n 1 n r where n i 2 3

square lattice ising model susceptibility series expansion method and differential equation for χ 3 in a previous paper 2004 j phys a math gen 37 9651 68 we have given the fuchsian linear differential equation satisfied by 3 the three particle contribution to the susceptibility of the isotropic square lattice ising model this paper gives the details of the calculations with some useful tricks and tools allowing one to obtain long series in polynomial time the method is based on series expansion in the variables that appear in the n 1 dimensional integrals representing the n particle contribution to the isotropic square lattice ising model susceptibility the integration rules are straightforward due to remarkable formulae we derived for these variables we obtain without any numerical approximation 3 as a fully integrated series in the variable w s 2 1 s2 where s sh 2k with k j kt the conventional ising model coupling constant we also give some perspectives and comments on these results

explicit double shuffle relations and a generalization of eulerʼs decomposition formula abstract we give an explicit formula for the shuffle relation in a general double shuffle framework that specializes to double shuffle relations of multiple zeta values and multiple polylogarithms as an application we generalize the well known decomposition formula of euler that expresses the product of two riemann zeta values as a sum of double zeta values to a formula that expresses the product of two multiple polylogarithm values as a sum of other multiple polylogarithm values