delay independence of mutual information rate of two symbolic sequences introduced by shannon as a rate of actual transmission mutual information rate mir is an extension of mutual information to a pair of dynamical processes we show a delay independence theorem according to which mir is not sensitive to a time shift between the two processes numerical studies of several benchmark situations confirm that this theoretical asymptotic property remains valid for realistic finite sequences estimations based on block entropies and a causal state machine algorithm perform better than an estimation based on a lempel ziv compression algorithm provided that block length and maximum history length respectively can be chosen larger than the delay mir is thus a relevant index for measuring nonlinear correlations between two experimental or simulated sequences when the transmission delay in input output devices or dephasing in coupled systems is variable or unknown

comment on dirac equation in the background of the nutku helicoid metric j math phys 48 092301 2007 the dirac equation written on the boundary of the nutku helicoid space consists of a system of ordinary differential equations we tried to analyze this system and we found that it has a higher singularity than those of the heun equations which give the solutions of the dirac equation in the bulk we also lose an independent integral of motion on the boundary this facts explain why we could not find the solution of the system on the boundary in terms of known functions we make the stability analysis of the helicoid and catenoid cases and end up with an appendix which gives a new example wherein one encounters a form of the heun equation

estimating entropy rates with bayesian confidence intervals the entropy rate quantifies the amount of uncertainty or disorder produced by any dynamical system in a spiking neuron this uncertainty translates into the amount of information potentially encoded and thus the subject of intense theoretical and experimental investigation estimating this quantity in observed experimental data is difficult and requires a judicious selection of probabilistic models balancing between two opposing biases we use a model weighting principle originally developed for lossless data compression following the minimum description length principle this weighting yields a direct estimator of the entropy rate which compared to existing methods exhibits significantly less bias and converges faster in simulation with monte carlo techinques we estimate a bayesian confidence interval for the entropy rate in related work we apply these ideas to estimate the information rates between sensory stimuli and neural responses in experimental data shlens kennel abarbanel chichilnisky in preparation

on algebraic solutions of lame s differential equation in two previous papers 1 21 we reconsidered and improved klein s method for establishing whether a given second order linear differential equation with rational function coefficients over an algebraic curve has a full set of algebraic solutions that procedure being rather complicated we were interested in checking whether when applied to classical examples it would lead to nontrivial new results the hypergeometric equation being in that respect well understood lame s equation see 0 1 was the simplest type of differential equation for which we could hope to get interesting information by our method before explaining in detail the content of this paper it may be useful to the reader if we recall a few facts proved in 1 21 of which constant use will be made throughout this paper let c be a nonsingular algebraic curve over the complex field c with function field k and let l be a second order linear differential operator on c that is

asymptotic bias in information estimates and the exponential bell polynomials we present a new derivation of the asymptotic correction for bias in the estimate of information from a finite sample the new derivation reveals a relationship between information estimates and a sequence of polynomials with combinatorial significance the exponential bell polynomials and helps to provide an understanding of the form and behavior of the asymptotic correction for bias

on the cubic lattice green functions wheatstone physics laboratory king s college university of london strand london wc2r 2ls u k it is proved that k k 4 eta frac 1 2 1 eta frac 1 2 k k where eta is a complex variable which lies in a certain region scr r 2 of the eta plane and k k pm are complete elliptic integrals of the first kind with moduli k pm which are given by k pm 2 equiv k pm 2 eta frac 1 2 pm frac 1 4 eta 4 eta frac 1 2 frac 1 4 2 eta 1 eta frac 1 2 this basic result is then used to express the face centred cubic and simple cubic lattice green functions at the origin in terms of the square of a complete elliptic integral of the first kind several new identities involving the heun function f a b alpha beta gamma delta eta are also derived next it is shown that the three cubic lattice green functions all have parametric representations which involve the green function for the two dimensional honeycomb lattice finally the results are applied to a variety of problems in lattice statistics in particular a new simplified formula for the generating function of staircase polygons on a four dimensional hypercubic lattice is derived

coverage adjusted entropy estimation summary data on neural coding have frequently been analyzed using information theoretic measures these formulations involve the fundamental and generally difficult statistical problem of estimating entropy we review briefly several methods that have been advanced to estimate entropy and highlight a method the coverage adjusted entropy estimator cae due to chao and shen that appeared recently in the environmental statistics literature this method begins with the elementary horvitz thompson estimator developed for sampling from a finite population and adjusts for the potential new species that have not yet been observed in the sample these become the new patterns or words in a spike train that have not yet been observed the adjustment is due to i j good and is called the good turing coverage estimate we provide a new empirical regularization derivation of the coverage adjusted probability estimator which shrinks the maximum likelihood estimate we prove that the cae is consistent and first order optimal with rate op 1 logn in the class of distributions with finite entropy variance and that within the class of distributions with finite qth moment of the log likelihood the good turing coverage estimate and the total probability of unobserved words converge at rate op 1 logn q we then provide a simulation study of the estimator with standard distributions and examples from neuronal data where observations are dependent the results show that with a minor modification the cae performs much better than the mle and is better than the best upper bound estimator due to paninski when the number of possible words m is unknown or infinite copyright q 2007 john wiley sons ltd

examples of heun and mathieu functions as solutions of wave equations in curved spaces we give examples of where the heun function exists as solutions of wave equations encountered in general relativity as a new example we find that while the dirac equation written in the background of nutku helicoid metric yields mathieu functions as its solutions in four spacetime dimensions the trivial generalization to five dimensions results in the double confluent heun function we reduce this solution to the mathieu function with some transformations

estimation of the entropy based on its polynomial representation estimating entropy from empirical samples of finite size is of central importance for information theory as well as the analysis of complex statistical systems yet this delicate task is marred by intrinsic statistical bias here we decompose the entropy function into a polynomial approximation function and a remainder function the approximation function is based on a taylor expansion of the logarithm given n observations we give an unbiased linear estimate of the first n power series terms based on counting sets of k coincidences for the remainder function we use nonlinear bayesian estimation with a nearly flat prior distribution on the entropy that was developed by nemenman shafee and bialek our simulations show that the combined entropy estimator has reduced bias in comparison to other available estimators

integrability of the one dimensional schrodinger equation we present a definition of integrability for the one dimensional schrodinger equation which encompasses all known integrable systems i e systems for which the spectrum can be explicitly computed for this we introduce the class of rigid functions built as liouvillian functions but containing all solutions of rigid differential operators in the sense of katz and a notion of natural of boundary conditions we then make a complete classification of rational integrable potentials many new integrable cases are found some of them physically interesting

high order fuchsian equations for the square lattice ising model χ 6 this paper deals with the six particle contribution to the magnetic susceptibility of the square lattice ising model we have generated modulo a prime series coefficients for the length of the series is sufficient to produce the corresponding fuchsian linear differential equation modulo a prime we obtain the fuchsian linear differential equation that annihilates the depleted series the factorization of the corresponding differential operator is performed using a method of factorization modulo a prime introduced in a previous paper the depleted differential operator is shown to have a structure similar to the corresponding operator for it splits into factors of smaller orders with the left most factor of order 6 being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral e the right most factor has a direct sum structure and using series calculated modulo several primes all the factors in the direct sum have been reconstructed in exact arithmetics

globally nilpotent differential operators and the square ising model we recall various multiple integrals with one parameter related to the isotropic square ising model and corresponding respectively to the n particle contributions of the magnetic susceptibility to the lattice form factors to the two point correlation functions and to their extensions the univariate analytic functions defined by these integrals are holonomic and even g functions they satisfy fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties we recall the explicit forms found in previous work of these fuchsian equations as well as their russian doll and direct sum structures these differential operators are selected fuchsian linear differential operators and their remarkable properties have a deep geometrical origin they are all globally nilpotent or sometimes even have zero p curvature we also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known focusing on the factorized parts of all these operators we find out that the global nilpotence of the factors resp p curvature nullity corresponds to a set of selected structures of algebraic geometry elliptic curves modular curves curves of genus five six and even a remarkable weight 1 modular form emerging in the three particle contribution 3 of the magnetic susceptibility of the square ising model noticeably this associated weight 1 modular form is also seen in the factors of the differential operator for another n fold integral of the ising class 3 h for the staircase polygons counting and in ap ry s study of 3 g functions naturally occur as solutions of globally nilpotent operators in the case where we do not have g functions but hamburger functions one irregular singularity at 0 or that correspond to the confluence of singularities in the scaling limit the p curvature is also found to verify new structures associated with simple deformations of the nilpotent property

binless strategies for estimation of information from neural data we present an approach to estimate information carried by experimentally observed neural spike trains elicited by known stimuli this approach makes use of an embedding of the observed spike trains into a set of vector spaces and entropy estimates based on the nearest neighbor euclidean distances within these vector spaces l f kozachenko and n n leonenko probl peredachi inf 23 9 1987 using numerical examples we show that this approach can be dramatically more efficient than standard bin based approaches such as the direct method s p strong r koberle r r de ruyter van steveninck and w bialek phys rev lett 80 197 1998 for amounts of data typically available from laboratory experiments

symmetry and the chazy equation there are three different actions of the unimodular lie group sl 2 c on a two dimensional space in every case we show how an ordinary differential equation admitting sl 2 as a symmetry group can be reduced in order by three and the solution recovered from that of the reduced equation via a pair of quadratures and the solution to a linear second order equation a particular example is the chazy equation whose general solution can be expressed as a ratio of two solutions to a hypergeometric equation the reduction method leads to an alternative formula in terms of solutions to the lame equation resulting in a surprising transformation between the lame and hypergeometric equations finally we discuss the painleve analysis of the singularities of solutions to the chazy equation

algebraic solutions of the lame equation revisited a minor error in the necessary conditions for the algebraic form of the lam e equation to have a finite projective monodromy group and hence for it to have only algebraic solutions is pointed out see f baldassarri on algebraic solutions of lams differential equation j differential equations41 1 1981 44 58 it is shown that if the group is the octahedral group then the degree parameter of the equation may differ by from an integer this possibility was missed the omission affects a recent result on the monodromy of the weierstrass form of the lamequation see r c churchill two generator subgroups of and the hypergeometric riemann and lam e equations j symbolic computa tion 28 4 5 1999 521 545 the weierstrass form which is a differential equation on an elliptic curve may have after all an octahedral projective monodromy group

indices for testing neural codes one of the most critical challenges in systems neuroscience is determining the neural code a principled framework for addressing this can be found in information theory with this approach one can determine whether a proposed code can account for the stimulus response relationship specifically one can compare the transmitted information between the stimulus and the hypothesized neural code with the transmitted information between the stimulus and the behavioral response if the former is smaller than the latter i e if the code cannot account for the behavior the code can be ruled out the information theoretic index most widely used in this context is shannon s mutual information the shannon test however is not ideal for this purpose while the codes it will rule out are truly nonviable there will be some nonviable codes that it will fail to rule out here we describe a wide range of alternative indices that can be used for ruling codes out the range includes a continuum from shannon information to measures of the performance of a bayesian decoder we analyze the relationship of these indices to each other and their complementary strengths and weaknesses for addressing this problem

applications of fuchsian differential equations to free boundary problems in a couple of recent papers free boundary problems and a class of conformal mappings involving curvilinear quadrilaterals were analysed primarily using transform methods in both the free boundary and conformal mapping problems there is an underlying relation with fuchsian differential equations and an alternative conceptually simpler solution technique for these problems would use solutions of these fuchsian equations thus it was conjectured in those papers that a class of fuchsian differential equations commonly known as heun9s equation has in some special but relatively important cases degenerate solutions these involve hypergeometric functions this complementary solution method is in many cases more convenient than the transform approach the purpose of this paper is to explore the connections with the fuchsian equations directly and use the solutions to solve free boundary problems the specific examples treated here come from a quasi steady approximation to solidification problems and are not without interest in their own right there are few analytical solutions for solidification problems in geometries of practical interest and the solutions found here should be of use in that regard

entropy estimation in turing s perspective a new nonparametric estimator of shannon s entropy on a countable alphabet is proposed and analyzed against the well known plug in estimator the proposed estimator is developed based on turing s formula which recovers distributional characteristics on the subset of the alphabet not covered by a size n sample the fundamental switch in perspective brings about substantial gain in estimation accuracy for every distribution with finite entropy in general a uniform variance upper bound is established for the entire class of distributions with finite entropy that decays at a rate of o ln n n compared to o ln n 2 n for the plug in in a wide range of subclasses the variance of the proposed estimator converges at a rate of o 1 n and this rate of convergence carries over to the convergence rates in mean squared errors in many subclasses specifically for any finite alphabet the proposed estimator has a bias decaying exponentially in n several new bias adjusted estimators are also discussed

the upward bias in measures of information derived from limited data samples extracting information measures from limited experimental samples such as those normally available when using data recorded in vivo from mammalian cortical neurons is known to be plagued by a systematic error which tends to bias the estimate upward we calculate here the average of the bias under certain conditions as an asymptotic expansion in the inverse of the size of the data sample the result agrees with numerical simulations and is applicable as an additive correction term to measurements obtained under such conditions moreover we discuss the implications for measurements obtained through other usual procedures

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data

on kummer s twenty four solutions of the hypergeometric differential equation the p adic analyticity of the boyarsky matrix associated with the hypergeometric function f a b c x has been investigated in an earlier article the transformation of this matrix under translation of a b c by z3 was determined at that time this article gives the transformation of this matrix under the extended kummer group the p adic implications of quadratic and higher degree transformations remain open the object of this article is to explain the relation between kummer s twenty four solutions of this equation 1 ax i a 12 c a b ab y 0 and the p adic theory of this equation for simplicity of exposition let us assume that a b c satisfy the condition 2 a b c a c b all lie outside z the unique solution of 1 analytic at a 0 is given by the classical formula 3 f a b c e a s b s ms s o c s where a s f a s f a s e n s 0 the euler integral formula f a b c n 17 c jb f 4 bs c a f b r c b l a b c wa b c x b x 1 1 x cb1 l x a d valid for re c re b 0 a restriction which may be removed 7 p 60 equation 13 by changing the path of integration shows that we are concerned with the periods of a certain differential as is well known 5 6 11 p 107 12 the kummer list may be obtained by transformation of this integral representation indeed there received by the editors september 1 1982 and in revised form september 28 1983 1980 mathematics subject classification primary 12b40 34a30 1984 american mathematical society 0002 9947 84 1 00 25 per page

on reducing the heun equation to the hypergeometric equation abstract the reductions of the heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified heun to hypergeometric reductions are similar to classical hypergeometric identities but the conditions for the existence of a reduction involve features of the heun equation that the hypergeometric equation does not possess namely its cross ratio and accessory parameters the reductions include quadratic and cubic transformations which may be performed only if the singular points of the heun equation form a harmonic or an equianharmonic quadruple respectively and several higher degree transformations this result corrects and extends a theorem in a previous paper which found only the quadratic transformations siam j math anal 10 3 1979 655