a rigid local system with monodromy group the big conway group 2 and two others with monodromy group the suzuki group 6

shape optimization of shell structures based on nurbs description using automatic differentiation the consolidation of the link among four fields in computational mathematics and mechanics is the main objective of this work surfaces based on nurbs non uniform rational b spline mathematical optimization the finite element method fem in structural analysis and automatic differentiation ad are applied in shape optimization of shells this problem is performed taking into account the fact that material and mechanical characteristics influence both the structural shape and the thickness variation in order to obtain the best performance with respect to a specific criterium some techniques were implemented to modify the shell geometry conserving the same parameterization without a new finite element mesh generation the shape modification is carried out using an optimization code based on the data obtained by a finite element analysis and gradients evaluation in this work the optimization procedure is performed using an sqp sequential quadratic programming algorithm where the variables are the control points in homogeneous coordinates the knot vectors and the thickness the functional value is determined by the fem and gradients are evaluated using ad some examples are analyzed and discussed as a consequence of the shape optimization shells with high structural performance and aesthetically beautiful shapes can be obtained copyright 2011 john wiley sons ltd

zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

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

improvements of the weil bound for artin schreier curves for the artin schreier curve y q y f x defined over a finite field mathbb f q of q elements the celebrated weil bound for the number of mathbb f q r rational points can be sharp especially in super singular cases and when r is divisible in this paper we show how the weil bound can be significantly improved using ideas from moment l functions and katz s work on l adic monodromy calculations roughly speaking we show that in favorable cases which happens quite often one can remove an extra sqrt q factor in the error term

ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

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

rationality of trace and norm l functions for a given l adic sheaf f on a commutative algebraic group g over a finite field k and an integer r 1 we define the rth local norm l function of f at a point t g k and prove its rationality this function gives information on the sum of the local frobenius traces of f over the points of g kr where kr is the extension of degree r of k with norm t for g the 1 dimensional affine line or the torus these sums can in turn be used to estimate the number of rational points on curves or the absolute value of exponential sums which are invariant under a large group of translations or homotheties

on the self fractional fourier functions it is shown that a self fractional fourier function for an angle where n and m are indivisible integers n m is also a self fractional fourier function for angles j 1 2 this allows us to define a self fractional fourier function of order m an eigenvalue of a fractional fourier operator for angle is equal to where l is an integer

the fractional fourier transform and time frequency representations the functional fourier transform frft which is a generalization of the classical fourier transform was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community to which it may however be potentially useful the frft depends on a parameter spl alpha and can be interpreted as a rotation by an angle spl alpha in the time frequency plane an frft with spl alpha spl pi 2 corresponds to the classical fourier transform and an frft with spl alpha 0 corresponds to the identity operator on the other hand the angles of successively performed frfts simply add up as do the angles of successive rotations the frft of a signal can also be interpreted as a decomposition of the signal in terms of chirps the authors briefly introduce the frft and a number of its properties and then present some new results the interpretation as a rotation in the time frequency plane and the frft s relationships with time frequency representations such as the wigner distribution the ambiguity function the short time fourier transform and the spectrogram these relationships have a very simple and natural form and support the frft s interpretation as a rotation operator examples of frfts of some simple signals are given an example of the application of the frft is also given

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

distribution of lattice points visible from the origin let be a region in the plane which contains the origin is star shaped with respect to the origin and has a piecewise c 1 boundary for each integer q 1 we consider the integer lattice points from which are visible from the origin and prove that the 1 st consecutive spacing distribution of the angles formed with the origin exists this is a probability measure supported on an interval m with m 0 its repartition function is explicitly expressed as the convolution between the square of the distance from origin function and a certain kernel

determinantal processes with number variance saturation consider dyson s hermitian brownian motion model after a finite time s where the process is started at n equidistant points on the real line these n points after time s form a determinantal process and has a limit as n this limting determinantal process has the interesting feature that it shows number variance saturation the variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity number variance saturation is also seen for example in the zeros of the riemann function 21 2 the process can also be constructed using non intersecting paths and we consider several variants of this construction one construction leads to a model which shows a transition from a non universal behaviour with number variance saturation to a universal sine kernel behaviour as we go up the line

nonlinear finite element computation of the equilibrium stability and motion of the extensional beam and ring abstract discrete gauss integration of the element total potential energy is applied to the formation of a cubic cubic c1 extensible completely nonlinear curved beam finite element the versatility accuracy effectiveness and robustness of the element and the newton raphson technique used to solve the nonlinear algebraic stiffness equation set up with it is numerically demonstrated by computations of the nonlinear equilibrium stability and motion of beams and rings

low lying zeros of cubic dirichlet l functions and the ratios conjecture abstract we compute the one level density for the family of cubic dirichlet l functions when the support of the fourier transform of the associated test function is in 1 1 we also establish the ratios conjecture prediction for the one level density for this family and confirm that it agrees with the one level density we obtain

ramanujan hardy littlewood riesz phenomena for hecke forms abstract we generalize a result of ramanujan hardy and littlewood to the setting of primitive hecke forms which interestingly exhibits faster convergence than in the initial case of the riemann zeta function we also provide a criterion in the spirit of riesz for the riemann hypothesis for the associated l functions

free fermions and the classical compact groups there is a close connection between the ground state of non interacting fermions in a box with classical absorbing reflecting and periodic boundary conditions and the eigenvalue statistics of the classical compact groups the associated determinantal point processes can be extended in two natural directions i we consider the full family of admissible quantum boundary conditions i e self adjoint extensions for the laplacian on a bounded interval and the corresponding projection correlation kernels ii we construct the grand canonical extensions at finite temperature of the projection kernels interpolating from poisson to random matrix eigenvalue statistics the scaling limits in the bulk and at the edges are studied in a unified framework and the question of universality is addressed whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is a priori not obvious we complete the picture by constructing a finite temperature extension of the haar measure on the classical compact groups the eigenvalue statistics of the resulting grand canonical matrix models of random size corresponds exactly to the grand canonical measure of free fermions with classical boundary conditions

a general procedure for the direct computation of turning and bifurcation points we address the formulation and numerical implementation of general purpose algorithme suitable for the direct computation of stability points in non linear solid mechanics the effectiveness of the resulting algorithm is illustrated by numerical simulation that include bifurcation and loss of stability in elastoplasticity non linear rods and non linear three dimensional shells

application of the implicitly updated arnoldi method with a complex shift and invert strategy in mhd the implicitly updated arnoldi method introduced by sorensen with an internal qr iteration is a very useful eigenvalue solver for nonsymmetric eigenvalue problems to make this method rigorous in finding internal eigenvalues a complex shift and invert strategy is used therefore a complex variant of the method has been constructed and the method has been compared with a lanczos method as implemented by cullum et al for a practical problem in magnetohydrodynamics

statistics for ordinary artin schreier covers and other rank strata we study the distribution of the number of points and of the zeroes of the zeta function in different p rank strata of artin schreier covers over fq when q is fixed and the genus goes to infinity the p rank strata considered include the ordinary family the whole family and the family of covers with p rank equal to p 1 while the zeta zeroes always approach the standard gaussian distribution the number of points over fq has a distribution that varies with the specific family

jensen polynomials for the riemann zeta function and other sequences in 1927 polya proved that the riemann hypothesis is equivalent to the hyperbolicity of jensen polynomials for the riemann zeta function s at its point of symmetry this hyperbolicity has been proved for degrees d 3 we obtain an asymptotic formula for the central derivatives 2 n 1 2 that is accurate to all orders which allows us to prove the hyperbolicity of all but finitely many of the jensen polynomials of each degree moreover we establish hyperbolicity for all d 8 these results follow from a general theorem which models such polynomials by hermite polynomials in the case of the riemann zeta function this proves the gaussian unitary ensemble random matrix model prediction in derivative aspect the general theorem also allows us to prove a conjecture of chen jia and wang on the partition function

ramanujan fourier series the wiener khintchine formula and the distribution of prime pairs the wiener khintchine formula plays a central role in statistical mechanics it is shown here that the problem of prime pairs is related to autocorrelation and hence to a wiener khintchine formula experimental evidence is given for this c 1999 elsevier science b v all rights reserved

average ranks of elliptic curves tension between data and conjecture rational points on elliptic curves are the gems of arithmetic they are to diophantine geometry what units in rings of integers are to algebraic number theory what algebraic cycles are to algebraic geometry a rational point in just the right context at one place in the theory can inhibit and control thanks to ideas of kolyvagin kol88 the existence of rational points and other mathematical structures elsewhere despite all that we know about these objects the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today we have a network of heuristics and conjectures regarding rational points and we have massive data accumulated to exhibit instances of the phenomena generally we would expect that our data support our conjectures and if not we lose faith in our conjectures but here there is a somewhat more surprising interrelation between data and conjecture they are not exactly in open conflict one with the other but they are no great comfort to each other either we discuss various aspects of this story including recent heuristics and data that attempt to resolve this mystery we shall try to convince the reader that despite seeming discrepancy data and conjecture are in fact in harmony

l functions with large analytic rank and abelian varieties with large algebraic rank over function fields the goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of l functions allows one to produce many examples of l functions over function fields vanishing to high order at the center point of their functional equation conjectures of birch and swinnerton dyer bloch and beilinson relate the orders of vanishing of some of these l functions to mordell weil groups and other groups of algebraic cycles for certain abelian varieties of high analytic rank we are also able to prove the conjecture of birch and swinnerton dyer thus establishing the existence of large mordell weil groups in those cases in the rest of this section we state the main results of the paper

low lying zeros of l functions with orthogonal symmetry we investigate the moments of a smooth counting function of the zeros near the central point of l functions of weight k cuspidal newforms of prime level n we split by the sign of the functional equations and show that for test functions whose fourier transform is supported in 1 n 1 n as n the first n centered moments are gaussian by extending the support to 1 n 1 1 n 1 we see non gaussian behavior in particular the odd centered moments are nonzero for such test functions if we do not split by sign we obtain gaussian behavior for support in 2 n 2 n if 2k n the nth centered moments agree with random matrix theory in this extended range providing additional support for the katz sarnak conjectures the proof requires calculating multidimensional integrals of the nondiagonal terms in the bessel kloosterman expansion of the petersson formula we convert these multidimensional integrals to one dimensional integrals already considered in the work of iwaniec luo and sarnak ils and derive a new and more tractable expression for the nth centered moments for such test functions this new formula facilitates comparisons between number theory and random matrix theory for test functions supported in 1 n 1 1 n 1 by simplifying the combinatorial arguments as an application we obtain bounds for the percentage of such cusp forms with a given order of vanishing at the central point

computing l functions with large conductor an algorithm is given to efficiently compute l functions with large conductor in a restricted range of the critical strip examples are included for about 24000 dihedral galois representations with conductor near 10 7 the data shows good agreement with a symplectic random matrix model

the approximate solution of defining equations for generalized turning points a set of defining equations for turning points is shown to produce a test function that can be computed iteratively to double accuracy an inexact newton method is then derived that gives a method for the efficient calculation of turning points applications are given to discretized boundary value problems using conjugate gradients

sato tate theorem for families and low lying zeros of automorphic l functions we consider certain families of automorphic representations over number fields arising from the principle of functoriality of langlands let g be a reductive group over a number field f which admits discrete series representations at infinity let l g widehat g rtimes mathrm gal bar f f be the associated l group and r l g rightarrow mathrm gl d mathbb c a continuous homomorphism which is irreducible and does not factor through mathrm gal bar f f the families under consideration consist of discrete automorphic representations of g mathbb a f of given weight and level and we let either the weight or the level grow to infinity we establish a quantitative plancherel and a quantitative sato tate equidistribution theorem for the satake parameters of these families this generalizes earlier results in the subject notably of sarnak prog math 70 321 331 1987 and serre j am math soc 10 1 75 102 1997 as an application we study the distribution of the low lying zeros of the associated family of l functions l s pi r assuming from the principle of functoriality that these l functions are automorphic we find that the distribution of the 1 level densities coincides with the distribution of the 1 level densities of eigenvalues of one of the unitary symplectic and orthogonal ensembles in accordance with the katz sarnak heuristics we provide a criterion based on the frobenius schur indicator to determine this symmetry type if r is not isomorphic to its dual r vee then the symmetry type is unitary otherwise there is a bilinear form on mathbb c d which realizes the isomorphism between r and r vee if the bilinear form is symmetric resp alternating then r is real resp quaternionic and the symmetry type is symplectic resp orthogonal

stein s method and characters of compact lie groups stein s method is used to study the trace of a random element from a compact lie group or symmetric space central limit theorems are proved using very little information character values on a single element and the decomposition of the square of the trace into irreducible components this is illustrated for lie groups of classical type and dyson s circular ensembles the approach in this paper will be useful for the study of higher dimensional characters where normal approximations need not hold

universality of the edge distribution of eigenvalues of wigner random matrices with polynomially decaying distributions of entries we consider an ensemble of wigner symmetric random matrices a n a ij i j 1 n with matrix elements a ij being i i d symmetrically distributed random variables open image in new window we assume that open image in new window and that

the size of selmer groups for the congruent number problem where np is the number of solutions of the congruence y x d2x mod p then ld s has an analytic continuation as an entire function on the complex plane the conjecture of birch and swinnerton dyer then states in particular that the rank r d of ed is equal to the order r d of ld s at s 1 this being the so called analytic rank while we cannot at present find r d in all cases we can at least determine whether or not ld 1 0 and hence conjecturally whether or not r d 0 moreover one has a functional equation for ld s which relates its values at s and 2 s via a sign change ed 1 one may deduce that 1 r d ed it would then follow from the conjecture of birch and swinnerton dyer that the rank is positive whenever ed 1 according to the calculations of birch and stephens 2 one has

moments of the logarithmic derivative of characteristic polynomials from so n and usp 2n we study moments of the logarithmic derivative of characteristic polynomials of orthogonal and symplectic random matrices in particular we compute the asymptotics for large matrix size n of these moments evaluated at points that are approaching 1 this follows the work of bailey et al j math phys 60 8 083509 2019 where they computed these asymptotics in the case of unitary random matrices

gaps between zeros of gl 2 l functions abstract let l s f be an l function associated to a primitive holomorphic or maass cusp form f on gl 2 over q combining mean value estimates of montgomery and vaughan with a method of ramachandra we prove a formula for the mixed second moments of derivatives of l 1 2 i t f and via a method of hall use it to show that there are infinitely many gaps between consecutive zeros of l s f along the critical line that are at least 3 1 732 times the average spacing using general pair correlation results due to murty and perelli in conjunction with a technique of montgomery we also prove the existence of small gaps between zeros of any primitive l function of the selberg class in particular when f is a primitive holomorphic cusp form on gl 2 over q we prove that there are infinitely many gaps between consecutive zeros of l s f along the critical line that are at most 0 823 times the average spacing

the lowest eigenvalue of jacobi random matrix ensembles and painleve vi we present two complementary methods each applicable in a different range to evaluate the distribution of the lowest eigenvalue of random matrices in a jacobi ensemble the first method solves an associated painleve vi nonlinear differential equation numerically with suitable initial conditions that we determine the second method proceeds via constructing the power series expansion of the painleve vi function our results are applied in a forthcoming paper in which we model the distribution of the first zero above the central point of elliptic curve l function families of finite conductor and of conjecturally orthogonal symmetry

self fractional fourier functions and selection of modes it is shown that any function from can be represented as a linear combination of m self fractional fourier functions of order m which are orthogonal to each other each of them contains a selection of hermite gauss modes of the generator function the physical meaning of the synthesis of self fractional fourier functions is discussed

on the numerical evaluation of fredholm determinants some significant quantities in mathematics and physics are most naturally expressed as the fredholm determinant of an integral operator most notably many of the distribution functions in random matrix theory though their numerical values are of interest there is no systematic numerical treatment of fredholm determinants to be found in the literature instead the few numerical evaluations that are available rely on eigenfunction expansions of the operator if expressible in terms of special functions or on alternative numerically more straightforwardly accessible analytic expressions e g in terms of painleve transcendents that have masterfully been derived in some cases in this paper we close the gap in the literature by studying projection methods and above all a simple easily implementable general method for the numerical evaluation of fredholm determinants that is derived from the classical nystrom method for the solution of fredholm equations of the second kind using gauss legendre or clenshaw curtis as the underlying quadrature rule we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel in particular we get exponential convergence for analytic kernels which are typical in random matrix theory the application of the method to the distribution functions of the gaussian unitary ensemble gue in the bulk scaling limit and the edge scaling limit is discussed in detail after extending the method to systems of integral operators we evaluate the two point correlation functions of the more recently studied airy and airy 1 processes

pair correlation of sums of rationals with bounded height for each positive integer q let f q denote the farey sequence of order q we prove the existence of the pair correlation measure associated to the sum f q f q modulo 1 as q tends to infinity and compute the corresponding limiting pair correlation function

random matrices and l functions in recent years there has been a growing interest in connections between the statistical properties of number theoretical l functions and random matrix theory we review the history of these connections some of the major achievements and a number of applications

the square free sieve and the rank of elliptic curves let e be an elliptic curve over q a celebrated theorem of mordell asserts that e q the abelian group of rational points of e is finitely generated by the rank of e we mean the rank of e q thus the rank of e is positive if and only if e possesses an infinity of rational points relatively few general qualitative assertions can be made about the rank as e varies how large can the rank get although we expect that there are elliptic curves over q with arbitrarily high ranks this is presently unknown what is the average size of the rank recently brumer and mcguinness have reported on their study of 310 716 elliptic curves of prime conductor less than 108 where they have found that 20 06 of those curves have even rank 2 cf bm also see forthcoming work of brumer where subject to a number of standard conjectures he shows that 2 3 is an upper bound for the average rank for all elliptic curves over q ordered in terms of their faltings height what is the behavior of the rank over the family of twists of a given elliptic curve here one has three natural kinds of families of twists 1 quadratic twists one can take any elliptic curve and systematically twist it by all quadratic characters this is the type of family of elliptic curves we are concerned with in this paper specifically if e is an elliptic curve over q given by the weierstrass equation y2 x a x b and d is any squarefree integer the quadratic twist of e by d ed is given by the equation

an adjoint variable method for sensitivity analysis of non linear elastic systems summary an adjoint variable method for design sensitivity analysis of non linear elastic systems is presented the method uses domain parameterization and a mutual form of the hu washizu energy principle and extends results reported in a recent work for linear elastic systems to non linear elasticity non linearities due to finite deformations and non linear hyperelastic constitutive models are considered in contrast to other methods for non linear sensitivity analysis the present formulation can be applied with force displacement or mixed approximate solution methods the mutual energy expression used in the adjoint sensitivity derivation is developed from a non linear extension of the hu washizu energy functional and yields a linear governing equation for the adjoint system this has important ramifications for the computational cost of a sensitivity analyses of non linear systems excluding the cost of determining the response of the system the cost of a sensitivity analysis for a non linear system is essentially the same as that for a linear system finite element implementation of the resulting sensitivity expressions is discussed and two numerical examples are presented the first example involves large deformations of a mooney rivlin body while the second involves design sensitivity analysis for mixed solution methods

design sensitivity analysis of non linear structures in regular and critical states abstract a general design sensitivity formulation is introduced for non linear elastic structures reaching critical equilibrium states a discretized formulation is considered first and the sensitivity analysis is discussed for regular and critical states including analysis of post critical behavior next a variational approach is presented for beam and surface structures for which generalized stress and strain operator formulation is applied the formulae for sensitivity of bifurcation loads are derived in terms of adjoint fields which are shown to be equivalent to post buckling fields in the case of symmetric bifurcation

numerical computations concerning the erh this paper describes a computation which established the erh to height 10000 for all primitive dirichlet l series with conductor q 13 and to height 2500 for all q 72 all composite q 112 and other moduli the computations were based on euler maclaurin summation care was taken to obtain mathematically rigorous results the zeros were first located within 10 12 then rigorously separated using an interval arithmetic package a generalized turing criterion was used to show there were no zeros off the critical line statistics about the spacings between zeros were compiled to test the pair correlation conjecture and gue hypothesis

advances in random matrix theory zeta functions and sphere packing over four hundred years ago sir walter raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles these investigations aroused the curiosity of the astronomer johannes kepler and led to a problem that has gone centuries without a solution why is the familiar cannonball stack the most efficient arrangement possible here we discuss the solution that hales found in 1998 almost every part of the 282 page proof relies on long computer verifications random matrix theory was developed by physicists to describe the spectra of complex nuclei in particular the statistical fluctuations of the eigenvalues the energy levels follow certain universal laws based on symmetry types we describe these and then discuss the remarkable appearance of these laws for zeros of the riemann zeta function which is the generating function for prime numbers and is the last special function from the last century that is not understood today explaining this phenomenon is a central problem these topics are distinct so we present them separately with their own introductory remarks