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

generalized chirp like polyphase sequences with optimum correlation properties a new general class of polyphase sequences with ideal periodic autocorrelation function is presented the new class of sequences is based on the application of zadoff chu polyphase sequences of length n sm sup 2 where s and m are any positive integers it is shown that the generalized chirp like sequences of odd length have the optimum crosscorrelation function under certain conditions finally recently proposed generalized p4 codes are derived as a special case of the generalized chirp like sequence

optimal families of perfect polyphase sequences from the array structure of fermat quotient sequences we show that a p ary polyphase sequence of period p 2 from the fermat quotients is perfect that is its periodic autocorrelation is zero for all non trivial phase shifts we call this fermat quotient sequence we propose a collection of optimal families of perfect polyphase sequences using the fermat quotient sequences in the sense of the sarwate bound that is the cross correlation of two members in a family is upper bounded by p to investigate some relation between fermat quotient sequences and frank zadoff sequences and to construct optimal families including these sequences we introduce generators of p ary polyphase sequences of period p 2 using their p times p array structures we call an optimal generator to be the generator of some p ary polyphase sequences which are perfect and which gives an optimal family by the proposed construction finally we propose an algebraic construction for optimal generators as another main result a lot of optimal families of size p 1 can be constructed from these optimal generators some of which are known to be from the fermat quotient sequences or from the frank zadoff sequences but some families are new for p geq 11 the relation between the fermat quotient sequences and the frank zadoff sequences is determined as a by product

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 polynomial factorial diophantine equations we study equations of the form p x n and show that for some classes of polynomials p the equation has only finitely many solutions this is the case say if p is irreducible of degree greater than 1 or has an irreducible factor of relatively large degree this is also the case if the factorization of p contains some large power s of irreducible s for example we can show that the equation x r x 1 n has only finitely many solutions for r 4 but not that this is the case for 1 r 3 although it undoubtedly should be we also study the equation p x h n where h n is one of several other highly divisible sequences proving again that for various classes of polynomials these equations have only finitely many solutions

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

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

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

theory and applications of q ary interleaved sequences a new class of q ary sequences called interleaved sequence is introduced their periods shift equivalence linear spans and autocorrelation functions are derived the interleaved sequences include a large number of popular sequences such as multiplexed sequences clock controlled sequences kasami 1966 sequences gmw sequences geometric sequences and no 1989 sequences a special class of the interleaved sequences is constructed by mapping gf q sup m sequences into gf q sequences in terms of different bases of gf q sup m over gf q as an application of the theory of interleaved sequences some new families of binary pseudo random sequences are constructed which have large linear spans optimal periodic cross autocorrelation functions balance and the rapidly hopped properties a complete comparison of the new family of sequences with the gold sequence family the kasami small and large set sequence families the bent function sequence family and the no sequence family is discussed this shows that the new sequence families have important advantages for use in spread spectrum multiple access communication systems

a construction of odd length generators for optimal families of perfect sequences in this paper we give a construction of optimal families of n ary perfect sequences of period n 2 where n is a positive odd integer for this we re define perfect generators and optimal generators of any length n which were originally defined only for odd prime lengths by park song kim and golomb in 2016 but investigate the necessary and sufficient condition for these generators for arbitrary length n based on this we propose a construction of odd length optimal generators by using odd prime length optimal generators for a fixed odd integer n and its odd prime factor p the proposed construction guarantees at least n p p 1 phi n p phi p phi p 1 phi n 2 inequivalent optimal generators of length n in the sense of constant multiples cyclic shifts and or decimations here phi cdot is euler s totient function from an optimal generator one can construct lots of different n ary optimal families of period n 2 all of which contain p text min 1 perfect sequences where p text min is the least positive prime factor of n

a finite element level set redistancing method based on gradient recovery we introduce a new redistancing method for level set functions this method applies in a finite element setting and uses a gradient recovery technique based on the recovered gradient a quasi normal field on the zero level of the finite element level set function is defined and from this an approximate signed distance function is determined for this redistancing method rigorous error bounds are derived for example the distance between the original zero level and the zero level after redistancing can be shown to be bounded by ch k 1 if finite elements of degree k are used in the discretization

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

on the distribution of imaginary parts of zeros of the riemann zeta function ii we continue our investigation of the distribution of the fractional parts of where is a fixed non zero real number and runs over the imaginary parts of the non trivial zeros of the riemann zeta function we establish some connections to montgomery s pair correlation function and the distribution of primes in short intervals we also discuss analogous results for a more general l function

a front tracking method for computation of interfacial flows with soluble surfactants a finite difference front tracking method is developed for computations of interfacial flows with soluble surfactants the method is designed to solve the evolution equations of the interfacial and bulk surfactant concentrations together with the incompressible navier stokes equations using a non linear equation of state that relates interfacial surface tension to surfactant concentration at the interface the method is validated for simple test cases and the computational results are found to be in a good agreement with the analytical solutions the method is then applied to study the cleavage of drop by surfactant a problem proposed as a model for cytokinesis h p greenspan on the dynamics of cell cleavage j theor biol 65 1 1977 79 h p greenspan on fluid mechanical simulations of cell division and movement j theor biol 70 1 1978 125 finally the method is used to model the effects of soluble surfactants on the motion of buoyancy driven bubbles in a circular tube and the results are found to be in a good agreement with available experimental data

a new general construction for generalized bent functions a simple yet general method of constructing bent functions is presented for certain applications it is of interest to minimize the size of the image of a bent function in this study a tight lower bound on the number of elements in the image is established under certain conditions

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

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

finite element discretization of the navier stokes equations with a free capillary surface the instationary navier stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions a stable finite element discretization is presented the key idea is the treatment of the curvature terms by a variational formulation in the context of a discontinuous in time space time element discretization stability in weak energy norms can be proved numerical examples in 2 and 3 space dimensions are given

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

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

on the decimations of frank sequences sets of decimated frank 1962 sequences with optimal correlation properties were first considered by alltop 1984 by regarding decimated frank sequences as special cases of the generalized frank sequences proposed by suehiro and hatori 1988 the set size of decimated frank sequences can be doubled without sacrificing the optimality of the correlation properties besides the proposed sequences form representatives of generalized frank sequences of different classes they thus provide an efficient way to generate signature sequences for spread spectrum multiple access systems with dynamic signature assignments

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

modulatable orthogonal sequences and their application to ssma systems a set of n 1 orthogonal sequences of period n sup 2 is proposed where n is a natural number each orthogonal sequence proposed can be modulated by n complex numbers of absolute value 1 so the modulated sequence is also orthogonal when n is an odd prime number the absolute value of the cross correlation function between any two of the n 1 orthogonal sequences is constant and satisfies the mathematical lower bound this property of the cross correlation function is not changed when each of the two orthogonal sequences is modulated by n complex numbers of absolute value 1 two spread spectrum multiple access ssma systems using these sequences are proposed one system is an asynchronous ssma system using the proposed sequences unmodulated the cochannel interference peak between any two channels in this system realizes the mathematical lower bound for an asynchronous ssma system using a set of orthogonal sequences the other system is a synchronous ssma system without cochannel interference which uses the modulated form of the proposed sequences

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

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

finite element discretization error analysis of a general interfacial stress functional the finite element discretization of a variable interfacial stress tensor sigma gamma x is considered variable interfacial tension is included as a special case interfacial stress for example occurs in two phase flow problems in the weak formulation the interfacial stress gives rise to a functional which is supported on the interface gamma a finite element discretization of this functional is presented and analyzed the discretization admits almost independent meshes for the approximation of the interface and the approximation of the flow variables the main result is an mathcal o h k 1 2 error bound for the interfacial stress functional in a natural norm if the discrete interface is an mathcal o h k 1 approximation of gamma the bound is shown to be sharp in numerical experiments

analysis of an xfem discretization for stokes interface problems we consider a stationary stokes interface problem in the discretization the interface is not aligned with the triangulation for the discretization we use the p 1 extended finite element space p 1 xfem for the pressure and the standard conforming p 2 finite element space for the velocity since this pair is not necessarily lbb stable a consistent stabilization term known from the literature is added for the discrete bilinear form an inf sup stability result is derived which is uniform with respect to h mesh size parameter the viscosity quotient mu 1 mu 2 and the position of the interface in the triangulation based on this discretization error bounds are derived an optimal preconditioner for the stiffness matrix corresponding to this pair p 1 xfe for pressure and p 2 fe for velocity is presented the preconditioner has block diagonal form with a multigrid preconditioner for the velocity block and a new schur complement preconditioner optimality of this block preconditioner

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

on the distribution of imaginary parts of zeros of the riemann zeta function ii mathematics subject classification 2000 primary 11m26 secondary 11k38 we continue our investigation of the distribution of the fractional parts of whereis a fixed non zero real number and runs over the imaginary parts of the non trivial zeros of the riemann zeta function we establish some connections to mont gomery s pair correlation function and the distribution of primes in short intervals we also discuss analogous results for a more general l function

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 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

robust preconditioning for xfem applied to time dependent stokes problems we consider a quasi stationary stokes interface problem with a reaction term proportional to tau 1 delta t geq 0 as obtained by a time discretization of a time dependent stokes problem the mesh used for space discretization is not aligned with the interface we use the p 1 extended finite element space for the pressure approximation and the standard conforming p 2 finite element space for the velocity approximation a pressure stabilization term known from the literature is added since the fe pair is not lbb stable for the stabilized discrete bilinear form we derive a new inf sup stability result a new schur complement preconditioner is proposed and analyzed we present an analysis which proves robustness of the preconditioner with respect to h tau with tau in 0 c 0 cup c 1 h 2 infty and the position of the interface numerical results are included which indicate that the preconditioner is robust for the whole parameter range tau geq 0 and also with respect to the viscos

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

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

prime polynomials in short intervals and in arithmetic progressions in this paper we establish function field versions of two classical conjectures on prime numbers the first says that the number of primes in intervals x x x is about x logx the second says that the number of primes p x in the arithmetic progression p a modd for d x1 is about x d where is the euler totient function more precisely for short intervals we prove let k be a fixed integer then q i f i f k q holds uniformly for all prime powers q degree k monic polynomials f fq t and 0 f q where 0 is either 1k or 2k if p k k 1 or 3k if further p 2 and degf 1 here i f g fq t deg f g degf and q i f denotes the number of prime polynomials in i f we show that this estimation fails in the neglected cases for arithmetic progressions we prove let k be a fixed integer then q k d f q k d q holds uniformly for all relatively prime polynomials d f fq t satisfying d qk 1 0 where 0 is either 3k or 4k if p 2 and f d is a constant here q k is the number of degree k prime polynomials and q k d f is the number of such polynomials in the arithmetic progression p f modd we also generalize these results to arbitrary factorization types

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

finite element discretization error analysis of a surface tension force in two phase incompressible flows we consider a standard model for a stationary two phase incompressible flow with surface tension in the variational formulation of the model a linear functional which describes the surface tension force occurs this functional depends on the location and the curvature of the interface in a finite element discretization method the functional has to be approximated for an approximation method based on a laplace beltrami representation of the curvature we derive sharp bounds for the approximation error a new modified approximation method with a significantly smaller error is introduced

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

decimations of the frank heimiller sequences the frank heimiller sequence of period n2with n phases is constructed by writing the n by n discrete fourier transform matrix by rows the periodic autocorrelation sidelobes for such a sequence are all zero small families of n phase sequences with optimal periodic cross correlations can be formed by judicious selection of decimations of the frank heimiller sequence

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality