curiosities of arithmetic gases statistical mechanical systems with an exponential density of states are considered the arithmetic analog of parafermions of arbitrary order is constructed and a formula for boson parafermion equivalence is obtained using properties of the riemann zeta function interactions nontrivial mixing among arithmetic gases using the concept of twisted convolutions are also introduced examples of exactly solvable models are discussed in detail

primes in short intervals contrary to what would be predicted on the basis of cramer s model con cerning the distribution of prime numbers we develop evidence that the distribution of x h x for 0 x n is approximately normal with mean h and variance h log n h when n h n 1

on the zeros of the ramanujan t dirichlet series in the critical strip we describe computations which show that each of the first 12069 zeros of the ramanujan r dirichlet series of the form it in the region 0 t 6397 is simple and lies on the line 6 the failures of gram s law in this region are also noted the first 5018 zeros and 2228 successive zeros beginning with the 20001st zero are also calculated the distribution of the normalized spacing of the zeros is examined and it appears to be that of the eigenvalues of random matrices these comptuations are done with a berry keating formula for the r dirichlet series and evaluated using mathematica

random matrix theory and the zeros of ζ s we study the density of the roots of the derivative of the characteristic polynomial z u z of an n n random unitary matrix with distribution given by haar measure on the unitary group based on previous random matrix theory models of the riemann zeta function s this is expected to be an accurate description for the horizontal distribution of the zeros of s to the right of the critical line we show that as n the fraction of the roots of z u z that lie in the region 1 x n 1 z 1 tends to a limit function we derive asymptotic expressions for this function in the limits x and x 0 and compare them with numerical experiments

basic theory of f bounded quantification abstract system f bounded is a second order typed lambda calculus where the basic features of object oriented languages can be naturally modelled f bounded extends the better known system f in a way that provides an immediate solution for the treatment of the so called binary methods although more powerful than f and also quite natural system f bounded has only been superficially studied from a foundational perspective and many of its essential properties have been conjectured but never proved in the literature the aim of this paper is to give a solid foundation to f bounded by addressing and proving the key properties of the system in particular transitivity elimination completeness of the type checking semi algorithm the subject reduction property for reduction conservativity with respect to system f and antisymmetry of a full subsystem are considered and various possible formulations for system f bounded are compared finally a semantic interpretation of system f bounded is presented based on partial equivalence relations

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

level spacings distribution for large random matrices gaussian fluctuations we study the level spacings distribution for eigenvalues of large n x n matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1 defining by 77n s the number of nearest neighbors spacings greater than s 0 smaller than s 0 we prove functional limit theorem for the process 77n s e n s n1 2 giving weak convergence of this distribution to some gaussian random process on 0 x the limiting gaussian random process is universal for all classical compact groups it is holder continuous with any exponent less than 1 2 similar results can be obtained for the n level spacings distribution

weighted estimators for proportional hazards regression with missing covariates missing covariate data are common in epidemiologic studies and disease prevention trials in this article regression parameter estimation in the cox proportional hazards model is considered when certain covariates are observed for all study subjects and other covariate data are collected only for a subset the article presents both simple weighted and kernel assisted fully augmented weighted estimators that use the partially incomplete data nonparametrically we use nonparametric methods to estimate selection probabilities in the simple weighted estimating functions we also use nonparametric kernel smoothing techniques to estimate certain conditional expectations in fully augmented weighted estimating functions the proposed methods are nonparametric in the sense that they require neither a model for the missing data mechanism nor specification of the conditional distribution of missing covariates given observed covariates these estimators allow the missing data mechanism to depend on outcome variables

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

bias and efficiency of multiple imputation compared with complete case analysis for missing covariate values when missing data occur in one or more covariates in a regression model multiple imputation mi is widely advocated as an improvement over complete case analysis cc we use theoretical arguments and simulation studies to compare these methods with mi implemented under a missing at random assumption when data are missing completely at random both methods have negligible bias and mi is more efficient than cc across a wide range of scenarios for other missing data mechanisms bias arises in one or both methods in our simulation setting cc is biased towards the null when data are missing at random however when missingness is independent of the outcome given the covariates cc has negligible bias and mi is biased away from the null with more general missing data mechanisms bias tends to be smaller for mi than for cc since mi is not always better than cc for missing covariate problems the choice of method should take into account what is known about the missing data mechanism in a particular substantive application importantly the choice of method should not be based on comparison of standard errors we propose new ways to understand empirical differences between mi and cc which may provide insights into the appropriateness of the assumptions underlying each method and we propose a new index for assessing the likely gain in precision from mi the fraction of incomplete cases among the observed values of a covariate fico

linear types and non size increasing polynomial time computation we propose a linear type system with recursion operators for inductive datatypes which ensures that all definable functions are polynomial time computable the system improves upon previous such systems in that recursive definitions can be arbitrarily nested in particular no predicativity or modality restrictions are made

self organized criticality attributed to a central limit like convergence effect abstract self organized criticality is a hypothesis used to explain the origin of 1 f noise and other scaling behaviors despite being proposed nearly 30 years ago no consensus exists as to its exact definition or mathematical mechanism s recently a model for 1 f noise was proposed based on a family of statistical distributions known as the tweedie exponential dispersion models these distributions are characterized by an inherent scale invariance that manifests as a variance to mean power law called fluctuation scaling they also serve as foci of convergence in a limit theorem on independent and identically distributed distributions fluctuation scaling can be modeled by self similar stochastic processes that relate the variance to mean power law to 1 f noise through their correlation structure a hypothesis is proposed whereby the effects of self organized criticality are mathematically modeled by the tweedie distributions and their convergence behavior as applied to self similar stochastic processes sandpile model fluctuations are shown to manifest 1 f noise fluctuation scaling and to conform to the tweedie compound poisson distribution the tweedie models and their convergence theorem allow for a mechanistic explanation of 1 f noise and fluctuation scaling in phenomena conventionally attributed to self organized criticality thus providing a paradigm shift in our understanding of these phenomena

light types for polynomial time computation in lambda calculus we present a polymorphic type system for lambda calculus ensuring that well typed programs can be executed in polynomial time dual light affine logic dlal dlal has a simple type language with a linear and an intuitionistic type arrow and one modality it corresponds to a fragment of light affine logic lal we show that contrarily to lal dlal ensures good properties on lambda terms and not only on proof nets subject reduction is satisfied and a well typed term admits a polynomial bound on the length of any of its beta reduction sequences we also give a translation of lal into dlal and deduce from it that all polynomial time functions can be represented in dlal

strong normalization of mlf via a calculus of coercions ml f is a type system extending ml with first class polymorphism as in system f the main goal of the present paper is to show that ml f enjoys strong normalization i e it has no infinite reduction paths the proof of this result is achieved in several steps we first focus on xml f the church style version of ml f and show that it can be translated into a calculus of coercions terms are mapped into terms and instantiations into coercions this coercion calculus can be seen as a decorated version of system f so that the simulation result entails strong normalization of xml f through the same property of system f we then transfer the result to all other versions of ml f using the fact that they can be compiled into xml f and showing there is a bisimulation between the two we conclude by discussing what results and issues are encountered when using the candidates of reducibility approach to the same problem

a proof of strong normalization for f 2 f ω and beyond abstract we present an evaluation technique for proving strong normalization sn we use the technique to give sn proofs for f2 f bounded quantification subtypes and f the evaluation technique derives sn as a corollary of the soundness of the typing rules under an appropriate evaluation semantics the evaluation semantics yields simpler type sets than those used in the earlier sn proofs the type sets discussed here form a complete lattice under classical union and intersection the simplified type sets allow a simplified treatment of a variety of type constructors

on the pair correlation of zeros of the riemann zeta function in 1972 montgomery 20 21 introduced a new method for studying the zeros of the riemann zeta function one of his main accomplishments was to determine partially the pair correlation of zeros and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros perhaps more importantly he conjectured on number theoretic grounds an asymptotic formula for the pair correlation of zeros and found that the form of this correlation exactly agreed with the gaussian unitary ensemble gue model for random hermitian matrices which had been studied earlier by physicists he was therefore able to formulate a general n correlation conjecture for zeros during the 1980s odlyzko 23 24 performed extensive numerical calculations of the correlations for zeros in ranges up to the 10th zero and found excellent agreement with the gue model more recently hejhal 17 was able to prove the same partial result for triple correlation as montgomery proved for pair correlation and rudnick and sarnak 25 have done the same for n correlation rudnick and sarnak extended their results to a large class of l functions and also showed that the riemann hypothesis rh is not needed for smoother forms of the asymptotic result very recently bogomolny and keating 1 2 have used a prime twin type conjecture to derive heuristically the n correlation conjecture beyond the range where the results of rudnick and sarnak apply the conclusion of all this work is to rmly establish but not prove the gue distribution for zeros of many zeta functions there is a dual relationship between zeros of the riemann zeta function and prime numbers following montgomery s work it was realized that information on pair correlation of zeros could be used to obtain information on primes this connection was developed by gallagher and mueller 10 and heath brown 15 later goldston and montgomery 14 found an equivalence under the riemann hypothesis between the pair correlation of zeros and the variance for the number of primes in short intervals this equivalence arises out of the explicit formula via a parseval relation together with a tauberian theorem from this work one sees that to obtain new information on primes from zeros will require some new insight on the zeros and while the connections to statistical physics mentioned above are a possible source of this insight so far no progress has been made on this fundamental problem the motivation for this paper is the observation that there are additional tools

termination of system f bounded a complete proof abstract systemf bounded is a second order typed calculus with subtyping which has been defined to carry out foundational studies about the type systems of object oriented languages the almost recursive nature of the essential feature of this system makes one wonder whether it retains the strong normalization property with respect to first and second order reduction of systemf we prove that this is the case the proof is carried out to the last detail to allow the reader to be convinced of its correctness

the bakerian lecture 1987 quantum chaology bounded or driven classical systems often exhibit chaos exponential instability that persists but their quantum counterparts do not nevertheless there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic and this motivates the following definition definition quantum chaology is the study of semiclassical but nonclassical behaviour characteristic of systems whose classical motion exhibits chaos this is illustrated by the statistics of energy levels on scales comparable with the mean level spacing of order h n for n freedoms these fall into universality classes for classically chaotic systems the statistics are those of random matrices real symmetric or complex hermitian depending on the presence or absence of time reversal symmetry for classically regular ones the statistics are poisson on larger scales of order h i e classically small but semiclassically large universality breaks down these phenomena are being explained by representing spectra in terms of classical closed orbits universal spectral behaviour has its origin in very long orbits non universal behaviour depends only on short ones

extreme gaps between eigenvalues of random matrices this paper studies the extreme gaps between eigenvalues of random matrices we give the joint limiting law of the smallest gaps for haar distributed unitary matrices and matrices from the gaussian unitary ensemble in particular the kth smallest gap normalized by a factor n 4 3 has a limiting density proportional to x3k 1e x3 concerning the largest gaps normalized by n logn they converge in lp to a constant for all p 0 these results are compared with the extreme gaps between zeros of the riemann zeta function

paths based criteria and application to linear logic subsystems characterizing polynomial time abstract several variants of linear logic have been proposed to characterize complexity classes in the proofs as programs correspondence light linear logic lll ensures a polynomial bound on reduction time and characterizes in this way polynomial time p t i m e in this paper we study the complexity of linear logic proof nets and propose three semantic criteria based on context semantics stratification dependence control and nesting stratification alone entails an elementary time bound the three criteria entail together a polynomial time bound these criteria can be used to prove the complexity soundness of several existing variants of linear logic we define a decidable syntactic subsystem of linear logic s d n l l we prove that the proof nets of sdnll satisfy the three criteria which implies that s d n l l is sound for p t i m e several previous subsystems of linear logic characterizing polynomial time lll m l 4 maximal system of ms are embedded in s d n l l proving its p t i m e completeness

on the zeros of the riemann zeta function in the critical strip iv etude de la localisation des zeros complexes de la fonction zeta de riemann verification numerique de l hypothese de riemann pour les 1 500 000 001 premiers zeros de la forme it dans la bande critique avec 0 t 545 439 823 215 tous ces zeros ayant pour partie reelle 1 2

fast algorithms for multiple evaluations of the riemann zeta function the best previously know algorithm for evaluating the riemann zeta function it with bounded and t large to moderate accuracy was based on the rienmann siegel formula and required on the order of t 1 2 operations for each value that was computed new algorithms are presented in this paper which enable one to compute any single value of it with fixed and t t t t 1 2 in o te operations on numbers of o log t bits for any e 0 for example these algorithms lead to methods for numerically verifying the riemann hypothesis for the first n zeros in what is expected to be o n 1 e operations

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

light logics and optimal reduction completeness and complexity typing of lambda terms in elementary and light affine logic eal and lal respectively has been studied for two different reasons on the one hand the evaluation of typed terms using lal eal respectively proof nets admits a guaranteed polynomial elementary respectively bound on the other hand these terms can also be evaluated by optimal reduction using the abstract version of lamping s algorithm the first reduction is global while the second one is local and asynchronous we prove that for lal eal respectively typed terms lamping s abstract algorithm also admits a polynomial elementary respectively bound we also give a proof of its soundness and completeness for eal and lal with type fixpoints by using a simple geometry of interaction model context semantics

computational strategies for the riemann zeta function we provide a compendium of evaluation methods for the riemann zeta function presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new we concentrate primarily on practical computational issues such issues depending on the domain of the argument the desired speed of computation and the incidence of what we call value recycling c 2000 elsevier science b v all rights reserved

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

approaching word length distribution via level spectra abstract treating a text after the removal of paragraphs and punctuations as a spectrum of blanks the distributions of the length of words of ten languages are analyzed using models from the statistical theory of spectra it is found that the ten languages can be classified into two families one with words that follow a wigner like distribution while the words of the other obey a poisson like distribution

an abstract approach to stratification in linear logic we study the notion of stratification as used in subsystems of linear logic with low complexity bounds on the cut elimination procedure the so called light subsystems from an abstract point of view introducing a logical system in which stratification is handled by a separate modality this modality which is a generalization of the paragraph modality of girard s light linear logic arises from a general categorical construction applicable to all models of linear logic we thus learn that stratification may be formulated independently of exponential modalities when it is forced to be connected to exponential modalities it yields interesting complexity properties in particular from our analysis stem three alternative reformulations of baillot and mazza s linear logic by levels one geometric one interactive and one semantic

on the histogram as a density estimator l 2 theory let f be a probability density on an interval i finite or infinite i includes its finite endpoints if any and f vanishes outside of i let x1 x k be independent random variables with common density f the empirical histogram for the x s is often used to estimate f to define this object choose a reference point xosi and a cell width h let nj be the number of x s falling in the j th class interval

on the expressivity of elementary linear logic elementary linear logic is a simple variant of linear logic due to girard and which characterizes in the proofs as programs approach the class of elementary functions that is to say functions computable in time bounded by a tower of exponentials of fixed height other systems like light and soft linear logics have then been defined to characterize in a similar way the more interesting complexity class of polynomial time functions but at the price of either a more complicated syntax or of more sophisticated encodings these logical systems can serve as the basis of type systems for calculus ensuring polynomial time complexity bounds on well typed terms this paper aims at reviving interest in elementary linear logic by showing that despite its simplicity it can capture smaller complexity classes than that of elementary functions for that we carry a detailed analysis of its normalization procedure and study the complexity of functions represented by a given type we then show that by considering a slight variant of this system with type fixpoints and free weakening elementary affine logic with fixpoints we can characterize the complexity of functions of type w k 2 b where w and b are respectively types for binary words and booleans the key point is a sharper study of the normalization bounds we characterize in this way the class p of polynomial time predicates and more generally the hierarchy of classes k exp for k 0 where k exp is the union of dtime 2 k n i for i 1

multiple imputation for interval estimation from simple random samples with ignorable nonresponse abstract several multiple imputation techniques are described for simple random samples with ignorable nonresponse on a scalar outcome variable the methods are compared using both analytic and monte carlo results concerning coverages of the resulting intervals for the population mean using m 2 imputations per missing value gives accurate coverages in common cases and is clearly superior to single imputation m 1 in all cases the performances of the methods for various m can be predicted well by linear interpolation in 1 m 1 between the results for m 2 and m as a rough guide to assure coverages of interval estimates within 2 of the nominal level when using the preferred methods the number of imputations per missing value should increase from 2 to 3 as the nonresponse rate increases from 10 to 60

on the distribution of spacings between zeros of the zeta function etude numerique de la distribution des espacements des zeros de la fonction zeta de riemann observation de certains phenomenes inatendus

disproof of the mertens conjecture the mertens conjecture states that m x 1 where m x n x n and n is the mo bius function this conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the riemann hypothesis this paper disproves the mertens conjecture by showing that x lim sup m x x 2 1 06 the disproof relies on extensive computations with the zeros of the zeta function and does not provide an explicit counterexample disproof of the mertens conjecture a m odlyzko at t bell laboratories murray hill new jersey 07974 usa and h j j te riele centre for mathematics and computer science kruislaan 413 1098 sj amsterdam the netherlands

expected estimating equations for missing data measurement error and misclassification with application to longitudinal nonignorable missing data missing data measurement error and misclassification are three important problems in many research fields such as epidemiological studies it is well known that missing data and measurement error in covariates may lead to biased estimation misclassification may be considered as a special type of measurement error for categorical data nevertheless we treat misclassification as a different problem from measurement error because statistical models for them are different indeed in the literature methods for these three problems were generally proposed separately given that statistical modeling for them are very different the problem is more challenging in a longitudinal study with nonignorable missing data in this article we consider estimation in generalized linear models under these three incomplete data models we propose a general approach based on expected estimating equations eees to solve these three incomplete data problems in a unified fashion this eee approach can be easily implemented and its asymptotic covariance can be obtained by sandwich estimation intensive simulation studies are performed under various incomplete data settings the proposed method is applied to a longitudinal study of oral bone density in relation to body bone density

a comparison of multiple imputation and fully augmented weighted estimators for cox regression with missing covariates several approaches exist for handling missing covariates in the cox proportional hazards model the multiple imputation mi is relatively easy to implement with various software available and results in consistent estimates if the imputation model is correct on the other hand the fully augmented weighted estimators fawes recover a substantial proportion of the efficiency and have the doubly robust property in this paper we compare the fawes and the mi through a comprehensive simulation study for the mi we consider the multiple imputation by chained equation and focus on two imputation methods bayesian linear regression imputation and predictive mean matching simulation results show that the imputation methods can be rather sensitive to model misspecification and may have large bias when the censoring time depends on the missing covariates in contrast the fawes allow the censoring time to depend on the missing covariates and are remarkably robust as long as getting either the conditional expectations or the selection probability correct due to the doubly robust property the comparison suggests that the fawes show the potential for being a competitive and attractive tool for tackling the analysis of survival data with missing covariates

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

energy level statistics of the hadamard gutzwiller ensemble abstract the statistical properties of the quantal energy levels of the hadamard gutzwiller ensemble whose classical members belong to the class of systems with hard chaos are investigated based on a sample of 4500 energy levels it is shown that the short range statistics as nearest neighbour spacing distributions are governed by the goe predictions of random matrix theory which was first surmised by wigner and by landau and smorodinsky for nuclear level statistics this result strengthens the hypothesis that quantum systems with chaotic classical counterpart display level repulsion as predicted by random matrix theory however the level statistics describing correlations over greater level distances deviate from the goe predictions which is explained as a simple consequrnce of the fact that the spectral rigidity 3 l introduced by dyson and mehta saturates non universally at a finite value for l in complete agreement with the semiclassical theory developed by berry

type inference for light affine logic via constraints on words light affine logic lal is a system due to girard and asperti capturing the complexity class p in a proof theoretical approach based on linear logic lal provides a typing for lambda calculus which guarantees that a well typed program is executable in polynomial time on any input we prove that the lal type inference problem for lambda calculus is decidable for propositional lal to establish this result we reformulate the type assignment system into an equivalent one which makes use of subtyping and is more flexible we then use a reduction to a satisfiability problem for a system of inequations on words over a binary alphabet for which we provide a decision procedure

the zeta function on the critical line numerical evidence for moments and random matrix theory models results of extensive computations of moments of the riemann zeta function on the critical line are presented calculated values are compared with predictions motivated by random matrix theory the results can help in deciding between those and competing predictions it is shown that for high moments and at large heights the variability of moment values over adjacent intervals is substantial even when those intervals are long as long as a block containing 109 zeros near zero number 1023 more than anything else the variability illustrates the limits of what one can learn about the zeta function from numerical evidence it is shown the rate of decline of extreme values of the moments is modelled relatively well by power laws also some long range correlations in the values of the second moment as well as asymptotic oscillations in the values of the shifted fourth moment are found the computations described here relied on several representations of the zeta function the numerical comparison of their effectiveness that is presented is of independent interest for future large scale computations

on an interpretation of safe recursion in light affine logic we introduce a subalgebra bc of bellantoni and cook s safe recursion function algebra bc functions of the subalgebra have safe arguments that are non contractible i e non duplicable we propose a definition of safe and normal variables in light affine logic lal and show that bc is the largest subalgebra that is interpretable in lal relative to that definition though bc itself is not pf complete there are extensions of it by additional schemes for defining functions with safe arguments that are and are still interpretable in lal and so preserve pf closure we focus on one such which is bc augmented by a definition by cases construct and a restricted form of definition by recursion scheme over safe arguments as a corollary we obtain a new proof of the pf completeness of lal

time delay resonances riemann zeros and chaos in a model quantum scattering system the quantum treatment of an intrinsically chaotic model scattering system originally studied by gutzwiller 1983 is extended to include the time delay and to make explicit the zeros of the riemann zeta function in the mathematical expressions for the scattering matrix s and the time delay the system consists of a particle moving on a two dimensional surface of constant negative curvature the scattering in this unusual system is dominated by resonances associated with poles of s in the complex momentum plane the real parts of these poles are one half of the imaginary parts of the riemann zeros the resonances have a constant width but their average spacing varies with the momentum the focal point is the manifestation of chaotic behaviour in the time delay features considered in this regard include the momentum dependence of the time delay in regions of overlapping and isolated resonances the decomposition of the time delay into an average dynamical component and a fluctuating chaotic component and characterisation of the fluctuating component by its autocorrelation function

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

the effect of convolving families of l functions on the underlying group symmetries let fn and gm be families of primitive automorphic l functions for gln aq and glm aq respectively such that as n m the statistical behavior 1 level density of the low lying zeros of l functions in fn resp gm agrees with that of the eigenvalues near 1 of matrices in g1 resp g2 as the size of the matrices tend to infinity where each gi is one of the classical compact groups unitary u symplectic sp or orthogonal o so even so odd assuming that the convolved families of l functions fn gm are automorphic we study their 1 level density we also study convolved families of the form f gm for a fixed f under natural assumptions on the families which hold in many cases we can associate to each family l of l functions a symmetry constant cl equal to 0 resp 1 or 1 if the corresponding low lying zero statistics agree with those of the unitary resp symplectic or orthogonal group our main result is that cf g cf cg the symmetry type of the convolved family is the product of the symmetry types of the two families a similar statement holds for the convolved families f gm we provide examples built from dirichlet l functions and holomorphic modular forms and their symmetric powers an interesting special case is to convolve two families of elliptic curves with positive rank in this case the symmetry group of the convolution is independent of the ranks in accordance with the general principle of multiplicativity of the symmetry constants but the ranks persist before taking the limit n m as lower order terms

mr2071847 2005e 03139 terui kazushige light affine set theory a naive set theory of polynomial time english summary studia logica 77 2004 no 1 9 40

detecting squarefree numbers we present an algorithm based on the explicit formula for l functions and conditional on grh for proving that a given integer is squarefree with little or no knowledge of its factorization we analyze the algorithm both theoretically and practically and use it to prove that several rsa challenge numbers are not squarefull f a non zero element of k x then it is well known that f is squarefree if and only if gcd f f 0 1 and the latter condition may be checked quickly in deterministic polynomial time by the euclidean algorithm it is a long standing question in algorithmic number theory whether there is a correspondingly simple procedure to test if a given integer is squarefree in particular can one determine whether n 2 z is squarefree more rapidly than by factoring it in this paper we describe an algorithm conditional on the generalized riemann hy pothesis grh for proving an integer squarefree with little or no knowledge of its factor ization and analyze the complexity of the algorithm both theoretically and practically in particular we present some heuristic evidence based on random matrix theory and other probabilistic calculations that our algorithm runs in deterministic subexponential time o exp logn 2 3 o 1 although this is poorer than the performance expected of the cur rent best known factoring algorithms our method is able to give partial results that one does not obtain from a failed attempt at factoring in particular we show the following see

on fluctuations of riemann s zeta zeros it is shown that the normalized fluctuations of riemann s zeta zeros around their predicted locations follow the gaussian law it is also shown that fluctuations of two zeros gamma k and gamma k x with x sim left log k right beta beta 0 for large k follow the two variate gaussian distribution with correlation left 1 beta right

multinomial logistic regression with missing outcome data an application to cancer subtypes many diseases such as cancer and heart diseases are heterogeneous and it is of great interest to study the disease risk specific to the subtypes in relation to genetic and environmental risk factors however due to logistic and cost reasons the subtype information for the disease is missing for some subjects in this article we investigate methods for multinomial logistic regression with missing outcome data including a bootstrap hot deck multiple imputation bhmi simple inverse probability weighted sipw augmented inverse probability weighted aipw and expected estimating equation eee estimators these methods are important approaches for missing data regression the bhmi modifies the standard hot deck multiple imputation method such that it can provide valid confidence interval estimation under the situation when the covariates are discrete the sipw aipw and eee estimators are numerically identical when the covariates are continuous nonparametric smoothers can be applied to estimate the selection probabilities and the estimating scores these methods perform similarly extensive simulations show that all of these methods yield unbiased estimators while the complete case cc analysis can be biased if the missingness depends on the observed data our simulations also demonstrate that these methods can gain substantial efficiency compared with the cc analysis the methods are applied to a colorectal cancer study in which cancer subtype data are missing among some study individuals

soft linear logic and polynomial time we present a subsystem of second order linear logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms and vice versa