the measurable kesten theorem we give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d regular graphs it follows that the a finite d regular ramanujan graph g contains a negligible number of cycles of size less than c log log g we prove that infinite d regular ramanujan unimodular random graphs are trees through benjamini schramm convergence this leads to the following rigidity result if most eigenvalues of a d regular finite graph g fall in the alon boppana region then the eigenvalue distribution of g is close to the spectral measure of the d regular tree in particular g contains few short cycles in contrast we show that d regular unimodular random graphs with maximal growth are not necessarily trees institute of mathematical statistics 2016

control of flow separation over a forward facing step by model reduction abstract design and implementation of reduced order optimal controller for flow separation are investigated the reduced order controller design is based on proper orthogonal decomposition pod and galerkin projection detailed finite element simulations are performed and pod is applied to the resulting data to extract the most energetic eigenmodes these global eigenmodes are then used in conjunction with the galerkin projection to obtain a reduced order low dimensional model reduced order models are not only attractive for real time control computation but also crucial for detailed stability and bifurcation analysis we investigate the design of optimal controller for flow separation in a channel with this model where actuation is performed on a small part of the boundary two different types of surface actuation are considered tangential blowing and suction through a single slot best locations for actuation and adaptive procedure for reduced order model are explored the proposed approaches are evaluated by investigating the control of flow separation over a forward facing step channel our methods are found to be efficient and fast and our methods demonstrate a significant reduction in computational time and feasibility dramatic separation delays are observed on all cases it is found that the tangential blowing is more efficient in mitigating flow separation and reducing wake spread

a collocation quadrature based sturm liouville problem solver we present a computational method for solving a class of boundary value problems in sturm liouville form the algorithms are based on global polynomial collocation methods and produce discrete representations of the eigenfunctions error control is performed by evaluating the eigenvalue problem residuals generated when the eigenfunctions are interpolated to a finer discretization grid eigenfunctions that produce residuals exceeding an infinity norm bound are discarded because the computational approach involves the generation of quadrature weights and arrays for discrete differentiation operations our computational methods provide a convenient framework for solving boundary value problems by eigenfunction expansion and other projection methods

strong and weak constraint variational assimilations for reduced order fluid flow modeling in this work we propose and evaluate two variational data assimilation techniques for the estimation of low order surrogate experimental dynamical models for fluid flows both methods are built from optimal control recipes and rely on proper orthogonal decomposition and a galerkin projection of the navier stokes equation the techniques proposed differ in the control variables they involve the first one introduces a weak dynamical model defined only up to an additional uncertainty time dependent function whereas the second one handles a strong dynamical constraint in which the dynamical system s coefficients constitute the control variables both choices correspond to different approximations of the relation between the reduced basis on which is expressed the motion field and the basis components that have been neglected in the reduced order model construction the techniques have been assessed on numerical data and for real experimental conditions with noisy particle image velocimetry data

counting and equidistribution for quaternion algebras we aim at studying automorphic forms of bounded analytic conductor in the totally definite quaternion algebra setting we prove the equidistribution of the universal family with respect to an explicit and geometrically meaningful measure it leads to answering the sato tate conjectures in this case and contains the counting law of the universal family with a power savings error term

central limit theorems for elliptic curves and modular forms with smooth weight functions abstract in 11 the second and third named authors established a central limit theorem for the error term in the sato tate law for families of modular forms this method was adapted to families of elliptic curves in 3 by the first and second named authors in this context a central limit theorem was established only under a strong hypothesis going beyond the riemann hypothesis in the present paper we consider a smoothed version of the sato tate conjecture which allows us to overcome several limitations in particular for the smoothed version we are able to establish a central limit theorem for much smaller families of modular forms and we succeed in proving a theorem of this type for families of elliptic curves under the riemann hypothesis for l functions associated to hecke eigenforms for the full modular group

dpiv driven flow simulation a new computational paradigm we present a new approach to simulating unsteady fluid flows with only very few degrees of freedom by employing directly eigenmodes extracted from digital particle image velocimetry experimental data in particular we formulate standard galerkin and nonlinear galerkin approximations of the incompressible navier stokes equations using hierarchical empirical eigenfunctions extracted from an ensemble of flow snapshots we demonstrate that standard galerkin approaches produce simulations capable of capturing the short term dynamics of the flow but nonlinear galerkin projections are more effective in capturing both the short and long term dynamics leading to bounded solutions these findings are documented by applying these approaches to flow past a stationary circular cylinder at reynolds number 610

reduced order modeling of the upper tropical pacific ocean model using proper orthogonal decomposition the proper orthogonal decomposition pod is shown to be an efficient model reduction technique for simulating physical processes governed by partial differential equations in this paper we make an initial effort to investigate problems related to pod reduced modeling of a large scale upper ocean circulation in the tropic pacific domain we construct different pod models with different choices of snapshots and different number of pod basis functions the results from these different pod models are compared with that of the original model the main findings are 1 the large scale seasonal variability of the tropic pacific obtained by the original model is well captured by a low dimensional system of order 22 which is constructed using 20 snapshots and 7 leading pod basis functions 2 the rms errors for the upper ocean layer thickness of the pod model of order 22 are less than 1m that is less than 1 of the average thickness and the correlation between the upper ocean layer thickness with that from the pod model is around 0 99 3 retaining modes that capture 99 energy is necessary in order to construct pod models yielding a high accuracy

transcendence of hecke operators in the big hecke algebra we prove a transcendence result of hecke operator t l over the ring generated by diamond operators over in a non cm component of the cyclotomic ordinary big p adic hecke algebra and we discuss its conjectural implication of a modulo p version to the vanishing of the cyclotomic invariant 2014

the energy operator and new scaling relations for the incompressible navier stokes equations the incompressible navier stokes equations have been widely studied due to their vast ability to model hydrodynamic phenomena the nonlinear character of these equations produces a hierarchical system of dynamical scales that offer challenging barriers to computation simulation and scale elimination modeling here the incompressible navier stokes equations are decomposed and studied under the context of global eigensystems that have a direct relation to the dynamical character of the scales of motion in particular the self adjoint linear reynolds orr operator is shown to correspond to the energy operator of this nonlinear system of equations and has the distinguishing property of having an eigensystem that orders the time derivative of the integral kinetic energy for the nonlinear navier stokes operator the eigensystem of the reynolds orr operator is shown to yield a dynamical range of scales that can be characterized by an energy containing range an inertial range and a dissipation range it i

approximate inertial manifolds for the kuramoto sivashinsky equation analysis and computations abstract we evaluate several alternative methods for the approximation of inertial manifolds for the one dimensional kuramoto sivashinsky equation kse a method motivated by the dynamics originally developed for the navier stokes equation is adapted for the kse rigorous error estimates are obtained and compared to those of other methods introduced in the literature formal relationships between these other methods and the one introduced here are established numerical bifurcation diagrams of the various approximate inertial forms for the kse are presented we discuss the correspondence between the rigorous error estimates and the accuracy of the computational results these methods can be adapted to other dissipative partial differential equations

on approximate inertial manifolds to the navier stokes equations abstract recently the theory of inertial manifolds has shown that the long time behavior the dynamics of certain dissipative partial differential equations can be fully discribed by that of a finite ordinary differential system although we are still unable to prove existence of inertial manifolds to the navier stokes equations we present here a nonlinear finite dimensional analytic manifold that approximates closely the global attractor in the two dimensional case and certain bounded invariant sets in the three dimensional case this approximate manifold and others allow us to introduce modified galerkin approximations

nonlinear feedback control of parabolic partial differential equation systems with time dependent spatial domains partial differential equations pdes with time dependent spatial domains whose dynamics can be partitioned into slow and fast ones initially a nonlinear model reduction scheme similar to the one introduced in christofides and daoutidis j math anal appl 216 1997 398 420 which is based on combinations of galerkin s method with the concept of approximate inertial manifold is employed for the derivation of low order ordinary differential equation ode systems that yield solutions which are close up to a desired accuracy to the ones of the pde system for almost all times then these ode systems are used as the basis for the explicit construction of nonlinear time varying output feedback controllers via geometric control methods the controllers guarantee stability and enforce the output of the closed loop parabolic pde system to follow up to a desired accuracy a prespecified response for almost all times provided that the separation of the slow and fast dynamics is sufficiently large differences in the nature of the model reduction and control problems between parabolic pde systems with fixed and moving spatial domains are identified and discussed the proposed control method is used to stabilize an unstable steady state of a diffusion reaction process whose spatial domain changes with time it is shown to lead to a significant reduction on the order of the stabilizing nonlinear output feedback controller and outperform a nonlinear controller synthesis method that does not account for the variation of the spatial domain q 1999 academic press

spatiotemporal analysis of complex signals theory and applications we present a space time description of regular and complex phenomena which consists of a decomposition of a spatiotemporal signal into orthogonal temporal modes that we call chronos and orthogonal spatial modes that we call topos this permits the introduction of several characteristics of the signal three characteristic energies and entropies one temporal one spatial and one global and a characteristic dimension although the technique is general we concentrate on its applications to hydrodynamic problems specifically the transition to turbulence we consider two cases of application a coupled map lattice as a dynamical system model for spatiotemporal complexity and the open flow instability on a rotating disk in the latter we show a direct relation between the global entropy and the different instabilities that the flow undergoes as reynolds number increases

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

an explicit trace formula of jacquet zagier type for hilbert modular forms abstract we give an exact formula of the average of adjoint l functions of holomorphic hilbert cusp forms with a fixed weight and a square free level which is a generalization of zagier s formula known for the case of elliptic cusp forms on sl 2 z as an application we prove that the satake parameters of hilbert cusp forms with a fixed weight and with growing square free levels are equidistributed in an ensemble constructed by values of the adjoint l functions

a combinatorial theorem on the cartan invariants of the schur algebra s b abstract we prove an identity for the cartan invariants of the schur algebra s b using combinatorial properties of row semistandard tableaux the problem arose in connection with the construction of projective resolutions of the irreducible s b modules

unsteady two dimensional flows in complex geometries comparative bifurcation studies with global eigenfunction expansions we present a bifurcation study of the incompressible navier stokes equations in a model complex geometry a spatially periodic array of cylinders in a channel the dynamics of the flow include a hopf bifurcation from steady to oscillatory flow at an approximate reynolds number r of 350 and the appearance of a second frequency at approximately r simeq 890 the multiple frequency dynamics include a substantial increase in spatial and temporal scales with reynolds number as compared with the simple limit cycle oscillation present close to r 350 numerical bifurcation studies of the dynamics are performed using three forms of global eigenfunction expansions the first basis set is derived through principal factor analysis karhunen loeve expansion of snapshots from accurate direct spectral element numerical solutions of the navier stokes equations the second set is obtained from the eigenfunctions of the stokes operator for this geometry finally eigenfunctions are derived from a singular stokes operator i e the stokes operator modified to include a variable coefficient which vanishes at the domain boundaries truncated systems of sim 100 odes are obtained through projection of the navier stokes equations onto the basis sets and a comparative study of the resulting dynamical models is performed

error estimates of proper orthogonal decomposition eigenvectors and galerkin projection for a general dynamical system arising in fluid models we study estimates for proper orthogonal decomposition eigenvectors and eigenvalues as well as error estimates between the exact solution of a 2d navier stokes model and the numerical approach when the proper orthogonal decomposition method is considered these estimates are also extended when bifurcation diagram are calculated using the so called p pod or spod methods with a new cut off criterion to minimize noisy modes produced by the p pod method

effective multiplicity one on gl n and narrow zero free regions for rankin selberg l functions we establish zero free regions tapering as an inverse power of the analytic conductor for rankin selberg l functions on gln gln such zero free regions are equivalent to commensurate lower bounds on the edge of the critical strip and in the case of l s on the residue at s 1 as an application we show that a cuspidal automorphic representation on gln is determined by a finite number of its dirichlet series coefficients and that this number grows at most polynomially in the analytic conductor introduction let a be the ring of adeles over a number field f and let and be two cuspidal representations of gln a with restricted tensor product decompositions v v and v

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

transient analysis of electro osmotic transport by a reduced order modelling approach transient behaviour of electro osmotic transport in typical electrokinetic channels is studied in this paper the time needed for the electro osmotic flow to reach steady state exhibits multiple time scales depending on whether the flow is governed by either a viscous force electrokinetic force or by a combination of both when an intersection is present in the electrokinetic channel such as in a cross or a t channel the flow in the main channel and in the intersection gets to steady state at different times a weighted karhunen loeve kl decomposition method is proposed in this paper to generate the global basis function for reduced order simulation the key idea in a weighted kl approach is that instead of minimizing a least squares measure of error between the linear subspace spanned by the basis functions and the observation space we minimize the weighted error between the two spaces the global basis functions in a weighted kl approach can be generated by computing the singular value decomposition svd of the matrix containing the weighted snapshots we show that the weighted kl decomposition based reduced order model is computationally more efficient and can capture the multiple time scales encountered in electro osmotic transport much more effectively compared to the classical kl decomposition based reduced order model copyright 2003 john wiley sons ltd

asymptotic behavior of supercuspidal representations and sato tate equidistribution for families abstract we establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes for the tame supercuspidals constructed by j k yu we prove the limit multiplicity property with error terms thereby we obtain a sato tate equidistribution for the hecke eigenvalues of these families the main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p adic group tend to zero uniformly for every noncentral semisimple element

eigenvalue spacings for quantized cat maps according to one of the basic conjectures in quantum chaos the eigenvalues of a quantized chaotic hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices we study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms cat maps our numerical experiments indicate that for generic choices of cat maps the unfolded consecutive spacing distribution in the irreducible components of the nth quantization given by the n dimensional weil representation approaches the goe gse law of random matrix theory for certain special arithmetic transformations related to the ramanujan graphs of lubotzky phillips and sarnak the experiments indicate that the unfolded consecutive spacing distribution follows poisson statistics we provide a sharp estimate in that direction

integral moments of l functions we give a new heuristic for all of the main terms in the integral moments of various families of primitive l functions the results agree with previous conjectures for the leading order terms our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary orthogonal or symplectic matrices where the moments are defined by the appropriate group averages this lends support to the idea that arithmetical l functions have a spectral interpretation and that their value distributions can be modelled using random matrix theory numerical examples show good agreement with our conjectures

nonlinear control of incompressible fluid flow application to burgers equation and 2d channel flow abstract this paper proposes a methodology for the synthesis of nonlinear finite dimensional feedback controllers for incompressible newtonian fluid flows described by two dimensional navier stokes equations combination of galerkin s method with approximate inertial manifolds is employed for the derivation of low order ordinary differential equation ode systems that accurately describe the dominant dynamics of the flow these ode systems are subsequently used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed loop system to follow the reference input asymptotically the method is successfully used to synthesize nonlinear finite dimensional output feedback controllers for the burgers equation and the two dimensional channel flow that enhance the convergence rate to the spatially uniform steady state and the parabolic velocity profile respectively the performance of the proposed controllers is successfully tested through simulations and is shown to be superior to the one of linear controllers

approximation of attractors large eddy simulations and multiscale methods recent advances in the mathematical theory of the navier stokes equations have produced new insight in the mathematical theory of turbulence in particular the study of the attractor for the navier stokes equations produced the first connection between two approaches to turbulence that seemed far apart namely the conventional approach of kolmogorov and the dynamical systems theory approach similarly the study of the approximation of the attractor in connection with the newly introduced concept of approximate inertial manifolds has produced a new approach to large eddy simulations and the study of the interaction of small and large eddies in turbulent flows our aim in this article is to survey and describe some of the new results concerning the functional properties of the navier stokes equations and to discuss their relevance to turbulence

positive polynomial constraints for pod based model predictive controllers this paper presents an application of positive polynomials to the reduction of the number of temperature constraints of a proper orthogonal decomposition pod based predictive controller for a non isothermal tubular reactor the objective of the controller is to maintain the reactor at a desired operating condition in spite of disturbances in the feed flow while keeping the maximum temperature low enough to avoid the formation of undesired byproducts the controller is based on a model derived by means of pod which reduces the high dimensionality of the discretized system used to approximate the partial differential equations that model the reactor however pod does not lead to a reduction in the number of temperature constraints which is typically very large if we use univariate polynomials to approximate part of the basis vectors derived with the pod technique it is possible to apply the theory of positive polynomials to find good approximations of the temperature constraints by linear matrix inequalities and to get a reduction in their number this is the approach that is followed in this paper the simulation results show that the predictive controller presented a good behavior and that it dealt with the temperature constraints very well