pair correlation densities of inhomogeneous quadratic forms under explicit diophantine conditions on 2 w eprove that the local two point correlations of the sequence given by the values m 2 n 2 with m n 2 are those of a poisson process this partly confirms a conjecture of berry and tabor 2 on spectral statistics of quantized integrable systems and also establishes a particular case of the quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms of signature 2 2 the proof uses theta sums and ratner s classification of measures invariant under unipotent flows

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

on minima of the absolute value of certain random exponential sums let t n x n j 1 e 2 ij 2 x where stands for a random choice of sign with equal probability it is shown here that with high probability min x e 0 1 t n x n provided n is large and t n x n j 1 r j e 2 ij d x where d 2 3 is fixed and with standard normal r j we show that the minima are typically on the order of n d with high probability and for large n

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

stabilized finite element methods based on multiscale enrichment for the stokes problem this work concerns the development of stabilized finite element methods for the stokes problem considering nonstable different or equal order of velocity and pressure interpolations the approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element boundary on the other hand since the test function space is enriched with bubble like functions a petrov galerkin approach is employed we use such a strategy to propose stable variational formulations for continuous piecewise linear in velocity and pressure and for piecewise linear piecewise constant interpolation pairs optimal order convergence results are derived and numerical tests validate the proposed methods

a robust wg finite element method for convection diffusion reaction equations this paper proposes and analyzes a weak galerkin wg finite element method for 2 and 3 dimensional convection diffusion reaction problems on conforming or nonconforming polygon polyhedral meshes the wg method uses piecewise polynomial approximations of degrees k k 0 for both the scalar function and its trace on the inter element boundaries we show that the method is robust in the sense that the derived a priori error estimates is uniform with respect to the coefficients for sufficient smooth true solutions numerical experiments confirm the theoretical results

integral bilinear forms coxeter transformations and coxeter polynomials of finite posets abstract linear algebra technique in the study of linear representations of finite posets is developed in the paper a concept of a quadratic wandering on a class of posets i is introduced and finite posets i are studied by means of the four integral bilinear forms b i b i b i b i z i z i z 1 1 the associated coxeter transformations and the coxeter polynomials in connection with bilinear forms of dynkin diagrams extended dynkin diagrams and irreducible root systems are also studied bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of dynkin diagrams and extended dynkin diagrams are discussed a homological interpretation of the bilinear forms 1 1 is given and z bilinear equivalences between them are discussed by applying well known results of bongartz loupias and zavadskij shkabara we give several characterisations of posets i with the euler form q i x b i x x weakly positive resp with the reduced euler form q i x b i x x weakly positive and posets i with the tits form q i x b i x x weakly positive

on algorithmic coxeter spectral analysis of positive posets abstract following a general framework of coxeter spectral analysis of signed graphs and finite posets i introduced by simson siam j discrete math 27 827 854 2013 we present efficient numerical algorithms for the coxeter spectral study of finite posets i 1 n i that are positive in the sense that the symmetric gram matrix g i 1 2 c i c i t r m n q is positive definite where c i m n z is the incidence matrix of i encoding the relation i in the framework of scientific computing we present a complete coxeter spectral classification of finite positive posets i of size n i 20 it extends one of the main results obtained in gasiorek et al eur j comb 48 127 142 2015 for posets of size n 10 we also show that the connectivity of such posets i is determined by the complex coxeter spectrum specc i c equivalently by the coxeter polynomial cox i t z t od i one of the main results of the paper is a new technique to compute in a polynomial time a z invertible matrix b i m n z such that b i t r c i b i g d i where g d i m n z is a non symmetric gram matrix of a simply laced dynkin diagram d i a n d n e 6 e 7 e 8 associated with a finite positive poset i

pair correlation of torsion points on elliptic curves we prove the existence of the pair correlation measure associated to torsion points on the real locus e r of an elliptic curve e and provide an explicit formula for the limiting pair correlation function

bubble stabilization of finite element methods for the linearized incompressible navier stokes equations abstract in this paper we discuss the stabilization via bubble functions of a finite element method for stationary linearized incompressible navier stokes equations it is shown that residual free bubbles can reproduce supg

convergence analysis of the residual free bubble method for bilinear elements we present an a priori error analysis for the residual free bubble rfb method applied to diffusion convection equations with a dominating and antisymmetric convection operator special attention is paid to the case of piecewise bilinears in mathbb r 2 and the theoretical comparison with the streamline diffusion finite element sdfe equivalently streamline upwind petrov galerkin supg method

residual free bubbles for advection diffusion problems the general error analysis we develop the general a priori error analysis of residual free bubble finite element approximations to linear elliptic convection dominated diffusion problems subject to homogeneous dirichlet boundary condition optimal order error bounds are derived in various norms using piecewise polynomial finite elements of degree greater than or equal to 1

adaptive mixed finite element method for reissner mindlin plate abstract mixed finite element methods are designed to overcome shear locking phenomena observed in the numerical treatment of reissner mindlin plate models automatic adaptive mesh refining algorithms are an important tool to improve the approximation behavior of the finite element discretization in this paper a reliable and robust residual based a posteriori error estimate is derived which evaluates a t depending residual norm based on results in d arnold r falk r winther math modell numer anal 31 1997 517 557 the localized error indicators suggest an adaptive algorithm for automatic mesh refinement numerical examples prove that the new scheme is efficient

first order least squares method with weakly imposed boundary condition for convection dominated diffusion problems we present and analyze a first order least squares method for convection dominated diffusion problems which provides robust l 2 a priori error estimate for the scalar variable even if the given data f l 2 the novel theoretical approach is to rewrite the method in the framework of discontinuous petrov galerkin dpg method and then show numerical stability by using a key equation discovered by gopalakrishnan and qiu 2014 this new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations we also show that the condition number of the global matrix is independent of the diffusion coefficient a key feature of the method is that there is no stabilization parameter chosen empirically in addition dirichlet boundary condition is weakly imposed numerical experiments verify our theoretical results and in particular show our way of weakly imposing dirichlet boundary condition is essential to the design of least squares methods numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes which are unstructured quasi uniform

spacing between phase shifts in a simple scattering problem we prove a scattering theoretical version of the berry tabor conjecture for an almost every surface in a class of cylindrical surfaces of revolution the large energy limit of the pair correlation measure of the quantum phase shifts is poisson that is it is given by the uniform measure

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

the pair correlation function of fractional parts of polynomials we investigate the pair correlation function of the sequence of fractional parts of n d n 1 2 n where d 2 is an integer and an irrational we conjecture that for badly approximable the normalized spacings between elements of this sequence have poisson statistics as n

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

inflation algorithm for loop free non negative edge bipartite graphs of corank at least two we continue the study of finite connected loop free edge bipartite graphs with m 3m 3 vertices a class of signed graphs we started in simson 2013 48 and m g siorek et al 2016 19 by means of the non symmetric gram matrix g mm z of its symmetric gram matrix g 12 g g tr mm 12z and the gram quadratic form q zm zq zm z in particular we study connected non negative edge bipartite graphs with n r 3n r 3 vertices of corank r 2r 2 in the sense that the symmetric gram matrix g mn r z g mn r z of is positive semi definite of rank n 1n 1 the edge bipartite graphs of corank r 2r 2 are studied up to the weak gram zz congruence z z where z z means that g btr g bg btr g b for some b mn r z b mn r z such that det b 1det b 1 our main result of the paper asserts that given a connected edge bipartite graph with n r 3n r 3 vertices of corank r 2r 2 there exists a suitably chosen sequence t of the inflation operators of the form tab such that the composite operator t reduces to a connected bigraph d n r such that zd n r and d n r is one of the canonical r vertex extensions a n r n 1n 1 d n r n 4n 4 e 6 r e 7 r and e 8 r with n rn r vertices of the simply laced dynkin diagrams an dn e6 e7 e8an dn e6 e7 e8 with n 1n 1 vertices the algorithm constructs also a matrix b mn r z b mn r z with det b 1det b 1 defining the weak gram zz congruence zd n r that is satisfying the equation gd n r btr g b

on the structure of loop free non negative edge bipartite graphs abstract we continue the study of a class of signed graphs called finite connected loop free edge bipartite graphs bigraphs for short started in simson 2013 33 and continued in g siorek et al 2016 14 and simson zaj c 2017 38 in this paper we present an algorithmic approach to the study of non negative bigraphs with n r 2 vertices of corank r 1 that is bigraphs with the symmetric gram matrix g 1 2 g g t r m n r 1 2 z positive semi definite of rank n 1 where g m n r z is the non symmetric gram matrix of one of the main results of this paper are theorem 2 1 theorem 2 17 asserting that every such a bigraph with n r 2 vertices can be obtained from a corank zero i e positive connected loop free bigraph with n 1 vertices by an r point extension procedure u 1 u r u 1 u r along the r 1 roots u 1 u r z n of the positive definite gram form q z n z v v g v t r it is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop free non negative corank r 1 bigraphs with n r 2 vertices from the corank zero bigraphs with n 1 vertices

a coxeter type classification of one peak principal posets abstract we present a complete coxeter spectral classification of finite one peak posets i t that are principal in the sense that their symmetric gram matrix g i 1 2 c i c i t r m i q is positive semi definite of rank i 1 where c i m i z is the incidence matrix of i equivalently posets t with the quadratic tits form q t z t 1 z in the sense of drozd non negative and the kernel ker q t v z t 1 q t v 0 z t 1 infinite cyclic we continue the study started in m g siorek d simson 2012 11 we show that up to the strong gram z congruence and the weak gram z congruence every one peak principal poset is uniquely determined by its complex coxeter spectrum specc i c moreover we calculate the exact number of all such posets of a given size up to isomorphism and present their hasse quivers we finish the paper with conjecture on the exact number of principal positive posets according to our knowledge no formula is known for the number of non isomorphic posets in a general case

on the a posteriori error analysis for linear fokker planck models in convection dominated diffusion problems abstract this work is aimed at the derivation of reliable and efficient a posteriori error estimates for convection dominated diffusion problems motivated by a linear fokker planck problem appearing in computational neuroscience we obtain computable error bounds of functional type for the static and time dependent case and for different boundary conditions mixed and pure neumann boundary conditions finally we present a set of various numerical examples including discussions on mesh adaptivity and space time discretisation the numerical results confirm the reliability and efficiency of the error estimates derived

quasi optimality of the supg method for the one dimensional advection diffusion problem in this paper we propose quasi optimal error estimates in various norms for the streamline upwind petrov galerkin supg method applied to the linear one dimensional advection diffusion problem we follow the classical argument due to babuska and brezzi therefore the goal of this work is the proof of the inf sup and of the continuity conditions for the bilinear stabilized variational form with respect to suitable norms these norms are suggested by our previous work g sangalli technical report 1221 ian cnr pavia italy 2001 numer math to appear in which we analyze the continuous multidimensional advection diffusion operator we obtain these results by means of function space interpolation

a petrov galerkin enriched method a mass conservative finite element method for the darcy equation starting from the non stable p1 p0 discretization we build enhanced methods for the darcy equation which are stable and locally mass conservative the methods are derived in a petrov galerkin framework where both velocity and pressure trial spaces are enriched with multiscale functions these functions solve local problems correcting the residuals of the strong equations in each element and interior edge which leads to a velocity space enhanced with functions belonging to the lowest order raviart thomas space several numerical tests validate the methods

stable discretizations of convection diffusion problems via computable negative order inner products a new functional framework for consistently stabilizing discrete approximations to convection diffusion problems was recently proposed by the authors the key ideas are the evaluation of the residual in an inner product of the type h 1 2 unlike classical supg methods which use elemental weighted l2 inner products and the realization of this inner product via explicitly computable multilevel decompositions of function spaces such as those given by wavelets or hierarchical finite elements in this paper we first provide further motivations for our approach next we carry on a detailed analysis of the method which covers all regimes convection dominated and diffusion dominated a consistent part of the analysis justifies the use of easily computable truncated forms of the stabilizing inner product numerical results in close agreement with the theory are given at the end of the paper

the energy level spacing for two harmonic oscillators with golden mean ratio of frequencies the limit distribution of energy level spacing is studied for the system of two harmonic oscillators with generic ratio of frequencies it is proved that for any fixed generic ratio no limit distribution exists but for random with any absolutely continuous distributionp d on 0 1 a universal random limit distribution of the energy level spacing exists some properties of the random limit distribution are discussed

applications of the pseudo residual free bubbles to the stabilization of convection diffusion reaction problems it is known that the enrichment of the polynomial finite element space of degree 1 by bubble functions results in a stabilized scheme of the supg type for the convection diffusion reaction problems in particular the residual free bubbles rfb can assure stabilized methods but they are usually difficult to compute unless the configuration is simple therefore it is important to devise numerical algorithms that provide cheap approximations to the rfb functions contributing a good stabilizing effect to the numerical method overall here we propose a stabilization technique based on the rfb method and particularly designed to treat the most interesting case of small diffusion we replace the rfb functions by their cheap yet efficient approximations which retain the same qualitative behavior the approximate bubbles are computed on a suitable sub grid the choice of whose nodes are critical and determined by minimizing the residual of a local problem with respect to l 1 norm the resulting numerical method has similar stability features with the rfb method for the whole range of problem parameters this fact is also confirmed by numerical experiments we also note that the location of the sub grid nodes suggested by the strategy herein coincides with the one in brezzi et al math models methods appl sci 13 445 461 2003

applications of the pseudo residual free bubbles to the stabilization of convection diffusion problems abstract residual free bubbles have been recently introduced in order to compute optimal values for the stabilization methods a la hughes franca however unless in very special situations one dimensional problems limit cases etc they require the actual solution of pde problems the bubble problems in each element thus they are very difficult to be used in practice in this paper we present for the special case of convection dominated elliptic problems a cheap way to compute approximately the solution of the bubble problem in each element this provides as a consequence a cheap way to compute good approximations for the optimal values of the stabilization parameters

further considerations on residual free bubbles for advective diffusive equations abstract we further consider the galerkin method for advective diffusive equations in two dimensions the finite dimensional space employed is of piecewise polynomials enriched with residual free bubbles rfb we show that in general this method does not coincide with the supg method unless the piecewise polynomials are spanned by linear functions furthermore a simple stability analysis argument displays the effect of the rfb on the reduced space of piecewise polynomials which in some situations is not equivalent to streamline diffusion for bilinears

the residual free bubble finite element method on anisotropic partitions the subject of this work is the analysis and implementation of stabilized finite element methods on anisotropic meshes we develop the anisotropic a priori error analysis of the residual free bubble rfb method applied to elliptic convection dominated convection diffusion problems in two dimensions with finite element spaces of type q k k geq 1 in the case of p 1 finite elements relying on the equivalence of the rfb method to classical stabilized finite element methods we propose a new rule justified through the analysis of the rfb method for selecting the stabilization parameter in classical stabilized methods on two dimensional anisotropic triangulations

variational multiscale analysis the fine scale green s function projection optimization localization and stabilized methods we derive an explicit formula for the fine scale green s function arising in variational multiscale analysis the formula is expressed in terms of the classical green s function and a projector which defines the decomposition of the solution into coarse and fine scales the theory is presented in an abstract operator format and subsequently specialized for the advection diffusion equation it is shown that different projectors lead to fine scale green s functions with very different properties for example in the advection dominated case the projector induced by the h 1 0 seminorm produces a fine scale green s function which is highly attenuated and localized these are very desirable properties in a multiscale method and ones that are not shared by the l 2 projector by design the coarse scale solution attains optimality in the norm associated with the projector this property combined with a localized fine scale green s function indicates the possibility of effective methods with local character for dominantly hyperbolic problems the constructs lead to a new class of stabilized methods and the relationship between h 1 0 optimality and the streamline upwind petrov galerkin supg method is described

enhanced rfb method the residual free bubble method rfb is a parameter free stable finite element method that has been successfully applied to a wide range of boundary value problems exhibiting multiple scale behaviour if some local features of the solution are known a priori the approximation properties of the rfb finite element space can be improved by enriching it on selected edges of the partition by edge bubbles that are supported on pairs of neighbouring elements motivated by this idea we define and analyse an enhanced residual free bubble method for the solution of convection dominated convection diffusion problems in 2 d our a priori analysis highlights the limitations of the rfb method and the improved global approximation properties of the new method the theoretical results are supported by detailed numerical experiments

a robust a posteriori estimator for the residual free bubbles method applied to advection diffusion problems we develop the a posteriori error analysis for the rfb method applied to the linear advection diffusion problem the numerical error measured in suitable norms is estimated in terms of the numerical residual the robustness is investiged in the sense that we prove uniform equivalence between a norm of the numerical residual and a particular norm of the error

coupling stabilized finite element methods with finite difference time integration for advection diffusion reaction problems abstract we present some numerical schemes for the unsteady advection diffusion reaction linear problem in one space dimension we investigate two possible different ways of combining the discretization in time and in space where the sequence of the discretizations is interchanged discretization in time is performed by using the crank nicolson finite difference scheme while for the space discretization we consider three classical stabilized finite element schemes and the more recent link cutting bubble strategy proposed in f brezzi g hauke l d marini g sangalli link cutting bubbles for the stabilization of convection diffusion reaction problems math models methods appl sci 13 2003 445 461 numerical experiments are presented to assess and valuate the capabilities of the proposed methods an l 1 error analysis of the link cutting bubble strategy for solving the steady problem is included

spectral statistics on zoll surfaces the subject of this article is a family of partly conjectural spectral invariants of a laplace operator with a discrete spectrum j known as its k level correlation functions k and level spacings distribution the 2 level or pair correlation function 2 for instance measures the asymptotic number of spacings i j between normalized eigenvalues i as the eigenvalues tend to infinity the normalization is so that the mean level spacing i 1 i is one these invariants have been studied by many physicists including wigner dyson and berry and are conjectured to have certain universal forms depending on the qualitative behaviour of the corresponding classical geodesic flow

nearest neighbor distances on a circle multidimensional case we study the distances called spacings between pairs of neighboring energy levels for the quantum harmonic oscillator specifically we consider all energy levels falling between e and e 1 and study how the spacings between these levels change for various choices of e particularly when e goes to infinity primarily we study the case in which the spring constant is a badly approximable vector we first give the proof by boshernitzan dyson that the number of distinct spacings has a uniform bound independent of e then if the spring constant has components forming a basis of an algebraic number field we show that when normalized up to a unit the spacings are from a finite set moreover in the specific case that the field has one fundamental unit the probability distribution of these spacings behaves quasiperiodically in loge we conclude by studying the spacings in the case that the spring constant is not badly approximable providing examples for which the number of distinct spacings is unbounded

local spacings along curves we investigate local spacing problems along curves via smooth maps moreover we provide explicit formulas for the nearest neighbor spacing distribution function of torsion points on elliptic curves over r and of rational points on the unit circle

local analysis of hdg methods for convection dominated diffusion problems in this paper we establish local error analysis of hybridizable discontinuous galerkin hdg methods for convection dominated diffusion equation in two types of subdomains away from the layers the first subdomain is of o log 1 h h 1 2 away from the interior layers and of o log 1 h h away from the boundary layers and the second one is of o away from the outflow of the boundary our local error bound for the first subdomain can be improved by a factor of log 1 h we use weighted estimates to prove the first result by constructing new weight functions and the inf sup condition of the bilinear form to prove the second one

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

bubble stabilization of linear finite element methods for nonlinear evolutionary convection diffusion equations abstract a spatial stabilization via bubble functions of linear finite element methods for nonlinear evolutionary convection diffusion equations is discussed the method of lines with supg discretization in space leads to numerical schemes that are not only difficult to implement when considering nonlinear evolutionary equations but also do not produce satisfactory results the method we propose can be seen as an alternative to this kind of methods once the numerical approximation belonging to a linear finite element space enriched with bubble functions is obtained the bubble part is discarded the linear part is shown to give a stabilized approximation to the solution being approached the bubble functions are deduced using a linear steady convection diffusion model in such a way that the linear part of the approximation to the linear steady model after static condensation of bubbles reproduces the supg method in the convection dominated regime however for the nonlinear evolutionary equations we consider in the paper the method we propose is not equivalent to the supg method some numerical experiments are provided in the paper to show the efficiency of the procedure

analysis of the advection diffusion operator using fractional order norms in this paper we obtain a family of optimal estimates for the linear advection diffusion operator more precisely we define norms on the domain of the operator and norms on its image such that it behaves as an isomorphism it stays bounded as well as its inverse does uniformly with respect to the diffusion parameter the analysis makes use of the interpolation theory between function spaces one motivation of the present work is our interest in the theoretical properties of stable numerical methods for this kind of problem we will only give some hints here and we will take a deeper look in a further paper

estimation of higher sobolev norm from lower order approximation given an accurate piecewise constant flux approximation q of a smooth but unknown exact flux p how can one compute a sobolev norm of p over a small domain omega this question arises in a duality argument of goal oriented finite element a posteriori error analysis two equivalent practical estimators eta m and eta mathcal e are presented for an approximation q of p h 1 omega and model situations are addressed where q is asymptotically an upper and lower bound of p h 1 omega

spectral statistics for quantized skew translations on the torus we study the spectral statistics for quantized skew translations on the torus which are ergodic but not mixing for irrational parameters it is shown explicitly that in this case the level spacing distribution and other common spectral statistics like the number variance do not exist in the semiclassical limit

a priori error analysis of residual free bubbles for advection diffusion problems we develop an a priori error analysis of a finite element approximation to the elliptic advection diffusion equation eps delta u conv cdot nabla u f subject to a homogeneous dirichlet boundary condition based on the use of residual free bubble functions an optimal order error bound is derived in the so called stability norm biggl eps nabla v 2 l 2 omega sum t h t conv cdot nabla v 2 l 2 t biggr 1 2 where ht denotes the diameter of element t in the subdivision of the computational domain

level repulsion in the spectrum of two dimensional harmonic oscillators the asymptotic energy limit of the density p s of spacings between adjacent levels of the two dimensional harmonic oscillator tdho spectrum is studied it is shown that in any integer segment m m 1 containing approximately m levels of the tdho spectrum m n p s has the form wi s si where i takes on at most three values for large m p s displays strong level repulsion for irrational but it does not settle on a stationary form nor does its average over m this is in marked contrast with the behaviour of generic integrable systems for which the poisson statistics p s exp s is known to apply

a small scale density of states formula let m g be a c compact riemann manifold with classical hamiltonian h c t m assume that the corresponding quantization p1 op h is quantum completely integrable we establish an microlocal weyl law on short spectral intervals of size 2 e e 0 for various families of operators p1u u i containing p1 both in the mean and pointwise a e for u i the microlocalization refers to a small tubular neighbourhood of a non degenerate stable periodic bicharacteristic t m 0

streamline upwind petrov galerkin method supg vs residual free bubbles rfb abstract the purpose of this paper is to present the residual free bubbles rfb method and to show its relationship with the streamline upwind petrov galerkin method supg the paper is organized as follows after the introduction we show how the failure of classical finite element methods for convection dominated convection diffusion problem can be explained by energy considerations then we describe the residual free bubbles ideas and in the last section we show the equivalence of rfb and supg in a simple case

bubble and multiscale stabilization of bilinear finite element methods for transient advection diffusion equations on rectangular grids in this paper a petrov galerkin type stabilization for a time dependent advection diffusion equation is considered we first enrich the bilinear test space with bubble functions and the bilinear trial space with a special combination of bubble and multiscale functions for the steady state advection diffusion equation it is known that solving the residual equation obtained by the bubble elimination procedure is as difficult as solving the steady case of the original problem which makes the enriched methods quite costly for two dimensional problems in this study we instead utilize their cheap and efficient approximations in each rectangular element then we suggest a recipe for a proper adaptation of these functions combined with the generalized euler time integration to the unsteady problem some numerical experiments for typical model problems are presented to illustrate the accuracy of our method

on the correlations of directions in the euclidean plane let r v x y q denote the repartition of the v level correlation measure of the finite set of directions p x y p where p x y is the fixed point x y 0 1 2 and p is an integer lattice point in the square q q 2 we show that the average of the pair correlation repartition r 2 x y q over x y in a fixed disc d 0 converges as q oo more precisely we prove for every a r and 0 1 10 the estimate formula math we also prove that for each individual point x y 0 1 2 the 6 level correlation r 6 x y q diverges at any point a r 5 as q and we give an explicit lower bound for the rate of divergence

degeneracy moments for the square billiard we calculate the semiclassical asymptotics of the moments of a function that characterizes the accidental energy level degeneracies of the square billiard the results quantify the deviations from generic poisson statistics in this spectrum

analysis of a multiscale discontinuous galerkin method for convection diffusion problems we study a multiscale discontinuous galerkin method introduced in t j r hughes g scovazzi p bochev and a buffa comput meth appl mech engrg 195 2006 pp 2761 2787 that reduces the computational complexity of the discontinuous galerkin method seemingly without adversely affecting the quality of results for a stabilized variant we are able to obtain the same error estimates for the convection diffusion equation as for the usual discontinuous galerkin method we assess the stability of the unstabilized case numerically and find that the inf sup constant is positive bounded uniformly away from zero and very similar to that for the usual discontinuous galerkin method

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

stability of a streamline diffusion finite element method for turning point problems a one dimensional singularly perturbed problem with a boundary turning point is considered in this paper let v h be the linear finite element space on a suitable grid t h a variant of streamline diffusion finite element method is proved to be almost uniform stable in the sense that the numerical approximation u h satisfies u u email protected

structure and a coxeter dynkin type classification of corank two non negative posets inspired by famous results of drozd 1974 13 nazarova ro ter 1975 42 and bondarenko stepochkina 2005 8 on the importance of finite posets i with the non negative tits quadratic form q i zi z in the representation theory we study the finite non negative posets i of corank two that is such posets i that the symmetric gram matrix gi 12 ci citr mi q of i is positive semi definite of corank two where ci mi z is the incidence matrix of i a structure of such posets is described in the paper it is shown that every such a poset i can be obtained in a natural way from a corank zero poset j and a pair of roots of j following main ideas of the coxeter spectral analysis of posets we associate with any connected corank two poset i the incidence quadratic form qi zi z the coxeter matrix coxi ci ci tr the coxeter polynomial coxi t det t e coxi z t its complex coxeter spectrum specci the coxeter number ci n the reduced coxeter number c i n the incidence defect homomorphism i zi kerqi zi and a simply laced dynkin diagram dyni in particular we study the question when the coxeter spectrum specci of i together with dyni determines the incidence matrix ci of i hence the poset i uniquely up to a z congruence by applying linear algebra techniques and a combinatorial computer aided analysis of posets we prove that the question has an affirmative answer for connected corank two posets i with at most 15 elements a complete list of such posets is presented and their coxeter type invariants are computed it follows that i 6 there is only one such a poset i of cardinality 6 and 8 such posets of cardinality 7 up to the poset duality i iop the existence of a i mesh translation quiver ri kerqi i on the set ri kerqi is discussed where ri v zi qi v 1 is the set of roots of qi

stability and accuracy of adapted finite element methods for singularly perturbed problems the stability and accuracy of a standard finite element method fem and a new streamline diffusion finite element method sdfem are studied in this paper for a one dimensional singularly perturbed connvection diffusion problem discretized on arbitrary grids both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity often in the form of boundary layers of the solution surprisingly the accuracy of the standard fem is shown to depend crucially on the uniformity of the grid away from the singularity in other words the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but in contrast it is robust with respect to perturbation of properly adapted grid inside the boundary layer motivated by this discovery a new sdfem is developed based on a special choice of the stabilization bubble function the new method is shown to have an optimal maximum norm stability and approximation property in the sense that u u n infty leq c inf v n in v n u v n infty where un is the sdfem approximation in linear finite element space vn of the exact solution u finally several optimal convergence results for the standard fem and the new sdfem are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered

one peak posets with positive quadratic tits form their mesh translation quivers of roots and programming in maple and python by applying linear algebra and computer algebra tools we study finite posets with positive quadratic tits form our study is motivated by applications of matrix representations of posets in representation theory where a matrix representation of a partially ordered set t p1 pn with a partial order means a block matrix m m1 m2 mn over a field k of size d d1 dn determined up to all elementary row transformations elementary column transformations within each of the substrips m1 m2 mn and additions of linear combinations of columns of mi to columns of mj if pi pj drozd 10 proves that t has only a finite number of direct sum indecomposable representations if and only if its quadratic tits form q x1 xn x x12 xn2 x 2 pi pjxixj x x1 xn is weakly positive i e q a1 an a 0 for all non zero vectors a1 an a with integral non negative coefficients in this case there exists an indecomposable representation m of size d d1 dn if and only if d1 dn d is a root of q i e q d1 dn d 1 bondarenko and stepochkina 8 give a list of posets t with positive tits form consisting of four infinite series and 108 posets up to duality in the paper we construct this list in an alternative way by applying computational algorithms implemented in maple and python moreover given any poset t of the list we show that a the coxeter polynomial coxi t of the poset i t obtained from t by adding a unique maximal element called a peak is a coxeter polynomial f i t of a simply laced dynkin diagram i an dn e6 e7 e8 and b the set of orbits of the set rq of integral roots of q admits a mesh translation quiver structure rq of a cylinder shape where is the coxeter transformation of i t in the sense of drozd 10

on the stability of residual free bubbles for convection diffusion problemsand their approximation by a two level finite element method we consider the galerkin finite element method for partial diffferential equations in two dimensions where the finite dimensional space used consists of piecewise isoparametric polynomials enriched with bubble functions writing l for the differential operator we show that for elliptic convection diffusion problems the component of the bubble enrichment that stabilizes the method is essentially equivalent to a petrov galerkin method with an l spline exponentially fitted trial space and piecewise polynomial test space the remaining component of the bubble influences the accuracy of the method a stability inequality recently obtained by brezzi franca and russo for a limiting case of bubbles applied to convection diffusion problems is shown to be slightly weaker than the standard stability inequality that is obtained for the sdfem supg method thereby demonstrating that the bubble approach is in general slightly less stable than the streamline diffusion method when the trial functions are piecewise linear we show that residual free bubbles are as stable as sdfem supg and we extend this stability inequality to include positive mesh peclet numbers in the convection dominated regime approximate computations of the residual free bubbles are performed using a two level finite element method

global and local error analysis for the residual free bubbles method applied to advection dominated problems we prove the stability and a priori global and local error analysis for the residual free bubbles rfb finite element method applied to advection dominated advection diffusion problems

on the choice of a stabilizing subgrid for convection diffusion problems abstract supg and residual free bubbles are closely related methods that have been used with success to stabilize a certain number of problems including advection dominated flows in recent times a slightly different idea has been proposed to choose a suitable subgrid in each element and then solving standard galerkin on the augmented grid for this however the correct location of the subgrid node s plays a crucial role here for the model problem of linear advection diffusion equations we propose a simple criterion to choose a single internal node such that the corresponding plain galerkin scheme on the augmented grid provides the same a priori error estimates that are typically obtained with supg or rfb methods

stability and error analysis of mixed finite volume methods for advection dominated problems abstract we consider a convection diffusion reaction problem and we analyze a stabilized mixed finite volume scheme introduced in 1 the scheme is presented in the format of discontinuous galerkin methods and error bounds are given proving o h convergence in the l 2 norm for the scalar variable which is approximated with piecewise constant elements

systems of subspaces of a unitary space abstract for a finite poset p p 1 p t we study systems u 1 u t u of subspaces of a unitary space u such that u i u j if p i p j two systems u 1 u t u and v 1 v t v are said to be isometric if there exists an isometry u v such that u i v i we classify such systems up to isometry if p is a semichain we prove that the problem of their classification is unitarily wild if p is not a semichain a classification problem is called unitarily wild if it contains the problem of classifying linear operators on a unitary space which is hopeless in a certain sense

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

statistical properties of spectral fluctuations for a quantum system with infinitely many components extending the idea formulated in makino phys rev e 67 066205 2003 that is based on the berry robnik approach m v berry and m robnik j phys a 17 2413 1984 we investigate the statistical properties of a two point spectral correlation for a classically integrable quantum system the eigenenergy sequence of this system is regarded as a superposition of infinitely many independent components in the semiclassical limit we derive the level number variance lnv in the limit of infinitely many components and discuss its deviations from poisson statistics the slope of the limiting lnv is found to be larger than that of poisson statistics when the individual components have a certain accumulation this property agrees with the result from the semiclassical periodic orbit theory that is applied to a system with degenerate torus actions d biswas m azam and s v lawande phys rev a 43 5694 1991

spectral form factors of rectangle billiards the berry tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a poisson process we prove that in the case of a rectangle billiard with random ratio of sides the sum behaves for random and n large like a random walk in the complex plane with a non gaussian limit distribution the expectation value of the distribution is zero its variance which is essentially the average pair correlation function is one in accordance with the berry tabor conjecture but all higher moments 4 diverge the proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces the berry tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space

correlations for pairs of closed geodesics in this article we consider natural counting problems for closed geodesics on negatively curved surfaces we present asymptotic estimates for pairs of closed geodesics the differences of whose lengths lie in a prescribed family of shrinking intervals related pair correlation problems have been studied in both quantum chaos and number theory

stabilized finite element approximation of transient incompressible flows using orthogonal subscales abstract in this paper we present a stabilized finite element method to solve the transient navier stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales the latter are approximately accounted for so as to end up with a stable finite element problem which in particular allows to deal with convection dominated flows and the use of equal velocity pressure interpolations three main issues are addressed the first is a method to estimate the behavior of the stabilization parameters based on a fourier analysis of the problem for the subscales secondly the way to deal with transient problems discretized using a finite difference scheme is discussed finally the treatment of the nonlinear term is also analyzed a very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space in the transient case this simplifies considerably the numerical scheme

resonant harmonic oscillators and eigenvalue multiplicity explicit formulae are worked out for the eigenvalue multiplicity of a system of n independent quantum harmonic oscillators in the general case of 1 or s or n 1 resonance relations among the frequencies omega 1 omega n as a particular case we prove that even though the quantum numbers are always less than the degrees of freedom the eigenvalues are in general intrinsically degenerate only in the completely resonant case s n 1

spatial approximation of the radiation transport equation using a subgrid scale finite element method in this paper we present stabilized finite element methods to discretize in space the monochromatic radiation transport equation these methods are based on the decomposition of the unknowns into resolvable and subgrid scales with an approximation for the latter that yields a problem to be solved for the former this approach allows us to design the algorithmic parameters on which the method depends which we do here when the discrete ordinates method is used for the directional approximation we concentrate on two stabilized methods namely the classical supg technique and the orthogonal subscale stabilization a numerical analysis of the spatial approximation for both formulations is performed which shows that they have a similar behavior they are both stable and optimally convergent in the same mesh dependent norm a comparison with the behavior of the galerkin method for which a non standard numerical analysis is done is also presented

level spacings for integrable quantum maps in genus zero we study the pair correlation function for a variety of completely integrable quantum maps in one degree of freedom for simplicity we assume that the classical phase space m is the riemann sphere p 1 and that the classical map is a fixed time map of a hamilton flow the quantization is then a unitary n n matrix u t n and its pair correlation measure gives the distribution of spacings between eigenvalues in an interval of length comparable to the mean level spacing 1 n the physicists conjecture berry tabor conjecture is that as should converge to the pair correlation function of a poisson process for any 2 parameter family of hamiltonians of the form with we prove that this conjecture is correct for almost all along the subsequence of planck constants in the addendum to this paper z addendum we further show that for polynomial phases the a e convergence to poisson holds along the full sequence of planck constants for the cesaro means of

rate of convergence of finite difference approximations for degenerate ordinary differential equations in this paper we study finite difference approximations for the following linear stationary convection diffusion equations 1 2 2 x u x b x u x u x f x x r where is allowed to be degenerate we first propose a new weighted finite difference scheme motivated by approximating the diffusion process associated with the equation in the strong sense we show that under certain conditions this scheme converges with the first order rate and that such a rate is sharp to the best of our knowledge this is the first sharp result in the literature moreover by using the connection between our scheme and the standard upwind finite difference scheme we get the rate of convergence of the latter which is also new

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100