stationary sets for the wave equation in crystallographic domains let w be a crystallographic group in r n generated by reflections and let be the fundamental domain of w we characterize stationary sets for the wave equation in when the initial data is supported in the interior of the stationary sets are the sets of time invariant zeros of nontrivial solutions that are identically zero at t 0 lie show that for these initial data the n 1 dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group w w w this part comes from a corresponding odd symmetry of the initial data in physical language the result is that if the initial source is localized strictly inside of the crystalline then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

nodal sets and morse indices of solutions of super linear elliptic pdes in this paper we develop a sturm liouville type theory for the nodal sets and morse indices of solutions of super linear elliptic pdes with dirichlet boundary condition it shows that there are some relationships between analytic properties e g lp norm vanishing order of the nodal point andn 1 dimensional hausdorff measure of the nodal set of the solutions as the functions on the domain and morse indices of the solutions as the critical points of the functionalj u 12 u 2 f x u dxonh10

correlated wishart ensembles and chaotic time series we study the correlated wishart ensembles in the context of time series analysis we are interested in the statistics of eigenlevels viz variables associated with independent eigenmodes in the system the motivation of this work is to study the effect of time series correlations on the wishart ensembles in this connection we derive the level density and the two point function for the correlated wishart ensembles by using the binary correlation method using our analytic results we analyze spectra of autocovariance matrices derived from single variable stationary time series we consider the stochastic time series of gaussian variables with exponentially decaying correlations and time series of chaotic maps viz the arnold map the standard map and the stadium billiard map in both cases correlated time series are encountered and analyzed under the framework of random matrix theory it is shown that the eigenlevel statistics for the chaotic maps follow closely those of correlated wishart ensembles it is indicated that the presence of collective modes in the spectra of autocovariance matrices is related to the integrability of the system

decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an exterior domain in ℝ3 we consider the equations of motion of compressible viscous and heat conductive gases in an exterior domain in 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem

complex wishart matrices and conductance in mesoscopic systems exact results it is noted that the hypothesis of independent random complex elements for the off diagonal blocks say m2 and m3 of the transfer matrix describing conductance in a mesoscopic wire allows the eigenvalue distribution of the matrix product m3 m3 or m2 m2 to be computed exactly exact expressions in terms of double wronskian and toeplitz determinants are derived for the distribution of the smallest and second smallest eigenvalue of this and similar random matrix products

quantum ergodicity of boundary values of eigenfunctions suppose that n is a bounded piecewise smooth domain we prove that the boundary values cauchy data of eigenfunctions of the laplacian on with various boundary conditions are quantum ergodic if the classical billiard map on the ball bundle b is ergodic our proof is based on the classical observation that the boundary values of an interior eigenfunction 2 is an eigenfunction of an operator fh on the boundary of with h 1 in the case of the neumann boundary condition fh is the boundary integral operator induced by the double layer potential we show that fh is a semiclassical fourier integral operator quantizing the billiard map plus a small remainder the quantum dynamics defined by fh can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of zelditch zworski 1996 novelties include the facts that fh is not unitary and consequently the boundary values are equidistributed by measures which are not invariant under and which depend on the boundary conditions ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by s ozawa 1993 and was almost simultaneously proved by gerard leichtnam 1993 in the case of convex c1 1 domains with continuous tangent planes and with dirichlet boundary conditions our methods seem to be quite different motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form

doubling inequality and nodal sets for solutions of bi laplace equations we investigate the doubling inequality and nodal sets for the solutions of bi laplace equations a polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for laplace eigenfunctions by logunov in addition we derive an implicit upper bound for the nodal sets of solutions we show two types of doubling inequalities for the solutions of bi laplace equations as a consequence the rate of vanishing is given for the solutions

the pointwise estimates of diffusion wave for the navier stokes systems in odd multi dimensions we study the dissipation of solutions of the isentropic navier stokes equations in odd multi dimensions pointwise estimates of the time asymptotic shape of the solutions are obtained and shown to exhibit the generalized huygen s principle our approach is based on the detailed analysis of the green function of the linearized system this is used to study the coupling of nonlinear diffusion waves

strong unique continuation for higher order elliptic equations with gevrey coefficients article history we address the strong unique continuation problem for higher order elliptic partial differential equations in 2d with gevrey co efficients we provide a quantitative estimate of unique continu ation observability estimate and prove that the solutions satisfy the strong unique continuation property for ranges of the gevrey exponent strictly including non analytic gevrey classes as an ap plication we obtain a new upper bound on the hausdorff length of the nodal sets of solutions with a polynomial dependence on the coefficients published by elsevier inc

the nodal line of the second eigenfunction of the robin laplacian in r2 can be closed abstract we construct a multiply connected domain in r 2 for which the second eigenfunction of the laplacian with robin boundary conditions has an interior nodal line in the process we adapt a bound of donnelly fefferman type to obtain a uniform estimate on the size of the nodal sets of a sequence of solutions to a certain class of elliptic equations in the interior of a sequence of domains which does not depend directly on any boundary behaviour this also gives a new proof of the nodal line property of the example in the dirichlet case

on some decay properties of stokes semigroup of compressible viscous fluid flow in a 2 dimensional exterior domain in this paper we proved the local energy decay and some lp lq decay properties of solutions to the initial boundary value problem for the stokes equations of compressible viscous fluid flow in a 2 dimensional exterior domain kobayashi 1997 19 and kobayashi and shibata 1999 21 treated the same problem in a 3 dimensional exterior domain and our results obtained in this paper are an extension of results in kobayashi 1997 19 and kobayashi and shibata 1999 21 to the 2 dimensional case the fundamental solution to the resolvent equations for the stokes equations in r2 has a logarithmical singularity at 0 being a resolvent parameter while it is continuous up to 0 in r3 this difference requires us a new idea to prove the local energy decay estimate once getting the local energy decay estimate the required lp lq decay estimates in the exterior domain are obtained by combining the local energy estimate and the lp lq estimates in r2 by a cut off technique

optimal lp lq convergence rates for the compressible navier stokes equations with potential force in this paper we are concerned with the optimal lp lq convergence rates for the compressible navier stokes equations with a potential external force in the whole space under the smallness assumption on both the initial perturbation and the external force in some sobolev spaces the optimal convergence rates of the solution in lq norm with 2 q 6 and its first order derivative in l2 norm are obtained when the initial perturbation is bounded in lp with 1 p 6 5 the proof is based on the energy estimates on the solution to the nonlinear problem and some lp lq estimates on the semigroup generated by the corresponding linearized operator

the morphology of nodal lines random waves versus percolation in this paper we investigate the properties of nodal structures in random wave fields and in particular we scrutinize their recently proposed connection with short range percolation models we propose a measure which shows the difference between monochromatic random waves which are characterized by long range correlations and gaussian fields with short range correlations which are naturally assumed to be better modelled by percolation theory we also study the relevance of the quantities which we compute to the probability that nodal lines are in the vicinity of a given reference line

recursion for the smallest eigenvalue density of β wishart laguerre ensemble the statistics of the smallest eigenvalue of wishart laguerre ensemble is important from several perspectives the smallest eigenvalue density is typically expressible in terms of determinants or pfaffians these results are of utmost significance in understanding the spectral behavior of wishart laguerre ensembles and among other things unveil the underlying universality aspects in the asymptotic limits however obtaining exact and explicit expressions by expanding determinants or pfaffians becomes impractical if large dimension matrices are involved for the real matrices beta 1 edelman has provided an efficient recurrence scheme to work out exact and explicit results for the smallest eigenvalue density which does not involve determinants or matrices very recently an analogous recurrence scheme has been obtained for the complex matrices beta 2 in the present work we extend this to beta wishart laguerre ensembles for the case when exponent alpha in the associated laguerre weight function lambda alpha e beta lambda 2 is a non negative integer while beta is positive real this also gives access to the smallest eigenvalue density of fixed trace beta wishart laguerre ensemble as well as moments for both cases moreover comparison with earlier results for the smallest eigenvalue density in terms of certain hypergeometric function of matrix argument results in an effective way of evaluating these explicitly exact evaluations for large values of n the matrix dimension and alpha also enable us to compare with tracy widom density and large deviation results of katzav and castillo we also use our result to obtain the density of the largest of the proper delay times which are eigenvalues of the wigner smith matrix and are relevant to the problem of quantum chaotic scattering

distribution of the smallest eigenvalue in complex and real correlated wishart ensembles for the correlated gaussian wishart ensemble we compute the distribution of the smallest eigenvalue and a related gap probability we obtain exact results for the complex 2 and for the real case 1 for a particular set of empirical correlation matrices we find universality in the spectral density for both real and complex ensembles and all kinds of rectangularity we calculate the asymptotic and universal results for the gap probability and the distribution of the smallest eigenvalue we use the supersymmetry method in particular the generalized hubbard stratonovich transformation and superbosonization

laplace operators with eigenfunctions whose nodal set is a knot we prove that given any knot in a compact 3 manifold m there exists a riemannian metric on m such that there is a complex valued eigenfunction u of the laplacian corresponding to the first nontrivial eigenvalue whose nodal set u 1 0 has a connected component given by higher dimensional analogs of this result will also be considered

nodal domains of the equilateral triangle billiard we characterize the eigenfunctions of an equilateral triangle billiard in terms of its nodal domains the number of nodal domains has a quadratic form in terms of the quantum numbers with a non trivial number theoretic factor the patterns of the eigenfunctions follow a group theoretic connection in a way that makes them predictable as one goes from one state to another extensive numerical investigations bring out the distribution functions of the mode number and signed areas the statistics of the boundary intersections is also treated analytically finally the distribution functions of the nodal loop count and the nodal counting function are shown to contain information about the classical periodic orbits using the semiclassical trace formula we believe that the results belong generically to non separable systems thus extending the previous works which are concentrated on separable and chaotic systems

on large time behavior of solutions to the compressible navier stokes equations in the half space in r3 the navier stokes equation for compressible viscous fluid is considered on the half space in r3 under the zero dirichlet boundary condition for the momentum with initial data near an arbitrarily given equilibrium of positive constant density and zero momentum time decay properties in l2 norms for solutions of the linearized problem are investigated to obtain the rate of convergence in l2 norms of solutions to the equilibrium when initial data are sufficiently close to the equilibrium in some lower bounds are derived for solutions to the linearized problem one of which indicates a nonlinear phenomenon not appearing in the case of the cauchy problem on the whole space

volume estimates on the critical sets of solutions to elliptic pdes in this paper we study solutions to elliptic linear equations l u i aij x ju bi x iu c x u 0 either on n or a riemannian manifold under the assumption that the coefficient functions aij are lipschitz bounded we focus our attention on the critical set c u x u 0 and the singular set c u x u 0 and more importantly on effective versions of these currently with just the lipschitz regularity of the coefficients the strongest results in the literature say that the singular set is n 2 dimensional however at this point it has not even been shown that hn 2 c unless the coefficients are smooth fundamentally this is due to the need of an regularity theorem that requires higher smoothness of the coefficients as the frequency increases we introduce new techniques for estimating the critical and singular set which avoids the need of any such regularity consequently we prove that if the frequency of u is bounded by then we have the estimates hn 2 c u c 2 and hn 2 s u c 2 depending on whether the equation is critical or not more importantly we prove corresponding estimates for the effective critical and singular sets even under the assumption of real analytic coefficients these results are much sharper than those currently in the literature we also give applications of the technique to give estimates on the volume of the nodal set of solutions and estimates for the corresponding eigenvalue problem 2017 wiley periodicals inc

number of nodal domains of eigenfunctions on non positively curved surfaces with concave boundary it is an open problem in general to prove that there exists a sequence of delta g eigenfunctions varphi j k on a riemannian manifold m g for which the number n varphi j k of nodal domains tends to infinity with the eigenvalue our main result is that n varphi j k rightarrow infty along a subsequence of eigenvalues of density 1 if m g is a non positively curved surface with concave boundary i e a generalized sinai or lorentz billiard unlike the recent closely related work of ghosh reznikov sarnak and of the authors on the nodal domain counting problem the surfaces need not have any symmetries

critical correlations in a z invariant inhomogeneous ising model abstract we study the critical spin spin correlation function in the z invariant inhomogeneous ising model which includes the rectangular triangular honeycomb and checkerboard lattices as special cases using quadratic difference equations we derive various explicit representations for the correlation function confirming and simplifying also a miraculous result of baxter

rate of convergence of non stationary flow to the steady flow of compressible viscous fluid we consider a compressible viscous fluid affected by external forces of general form which are small and smooth enough in suitable norms in r 3 in shibata and tanaka y shibata k tanaka on the steady flow of compressible viscous fluid and its stability with respect to initial disturbance j math soc japan 55 2003 797 826 we proved the unique existence and some regularity of the steady flow and its globally in time stability with respect to a small initial disturbance in the h 3 framework in this paper we investigate the rate of the convergence of the non stationary flow to the corresponding steady flow when the initial data are small enough in the h 3 and also belong to l 6 5

large time behavior of the semigroup on lp spaces associated with the linearized compressible navier stokes equation in a cylindrical domain abstract large time behavior of solutions to the linearized compressible navier stokes equation around the motionless state in a cylindrical domain is investigated the l p decay estimates of the associated semigroup are established for all 1 p it is also shown that the time asymptotic leading part of the semigroup is given by a one dimensional heat semigroup

nodal domains statistics a criterion for quantum chaos we consider the distribution of the properly normalized numbers of nodal domains of wave functions in 2d quantum billiards we show that these distributions distinguish clearly between systems with integrable separable or chaotic underlying classical dynamics and for each case the limiting distribution is universal system independent thus a new criterion for quantum chaos is provided by the statistics of the wave functions which complements the well established criterion based on spectral statistics

janossy densities i determinantal ensembles we derive an elementary formula for janossy densities for determinantal point processes with a finite rank projection type kernel in particular for 2 polynomial ensembles of random matrices we show that the janossy densities on an interval i can be expressed in terms of the christoffel darboux kernel for the orthogonal polynomials on the complement of i

nodal sets for solutions of elliptic equations on etudie sur un domaine connexe r n l ensemble zero u 1 0 d une solution u d une equation elliptique a ij d i d j u b j d j u cu 0 ou a ij b j c sont bornes et a ij est continu on demontre que la mesure de hausdorff a n 1 dimensions de u 1 0 est finie dans un voisinage d un point x 0 ou u a un ordre fini d annulation

asymptotic behavior of solutions of the compressible navier stokes equations on the half space asymptotic behavior of solutions of the compressible navier stokes equations on the half space r n n 2 is considered around a given constant equilibrium a solution formula for the linearized problem is derived and l p estimates for solutions of the linearized problem are obtained for 2 p it is shown that as in the case of the cauchy problem the leading part of the solution of the linearized problem is decomposed into two parts one that behaves like diffusion waves and the other one purely diffusively there however are some aspects different from the cauchy problem especially in considering spatial derivatives it is also shown that the solution of the linearized problem approaches for large times the solution of the nonstationary stokes problem in some l p spaces and as a result a solution formula for the nonstationary stokes problem is obtained large time behavior of solutions of the nonlinear problem is then investigated in l p spaces for 2 p by applying the results on the linearized analysis and the weighted energy method the results indicate that there may be some nonlinear interaction phenomena not appearing in the cauchy problem

the nodal set of solutions to some elliptic problems sublinear equations and unstable two phase membrane problem abstract we are concerned with the nodal set of solutions to equations of the form u u q 1 u q 1 in b 1 where 0 q 1 2 b 1 b 1 0 is the unit ball in r n n 2 and u max u 0 u max u 0 are the positive and the negative part of u respectively this class includes the unstable two phase membrane problem q 1 as well as sublinear equations for 1 q 2 we prove the following main results a the finiteness of the vanishing order at every point and the complete characterization of the order spectrum b a weak non degeneracy property c regularity of the nodal set of any solution the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having hausdorff dimension at most n 2 locally finite when n 2 d a partial stratification theorem ultimately the main features of the nodal set are strictly related with those of the solutions to linear or superlinear equations with two remarkable differences first of all the admissible vanishing orders can not exceed the critical value 2 2 q at threshold we find a multiplicity of homogeneous solutions yielding the non validity of any estimate of the n 1 dimensional measure of the nodal set of a solution in terms of the vanishing order as a byproduct we also prove the strong unique continuation property for the unstable obstacle problem corresponding to the case 0 the proofs are based on monotonicity formulae for a 2 parameter family of weiss type functionals blow up arguments and the classification of homogeneous solutions

application of the τ function theory of painleve equations to random matrices pv piii the lue jue and cue with denoting an average with respect to the eigenvalue pdf for the laguerre unitary ensemble the object of our study is e n i a x l i l 0 i l for i 0 s and i s where x i l 1 for l i and x i l 0 other wise using okamoto s development of the theory of the painleve v equation it is shown that e n i a is a function associated with the hamiltonian therein and so can be characterized as the solution of a certain second order second degree differential equation or in terms of the solution of certain difference equations the cases 0 and 2 are of particular interest because they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue in the case i s e n i a is simply related to an average in the jacobi unitary ensemble and this in turn is simply related to certain averages over the orthogonal group the unitary symplectic group and the circular unitary ensemble the latter integrals are of interest for their combinatorial content also considered are the hard edge and soft edge scaled limits of e n i a in particular in the hard edge scaled limit it is shown that the limiting quantity e hard 0 s a can be evaluated as a function associated with the hamiltonian in okamoto s theory of the painleve iii equation

eigenfunctions with prescribed nodal sets in this paper we consider the problem of prescribing the nodal set of low energy eigenfunctions of the laplacian our main result is that given any separating closed hypersurface in a compact n manifold m there is a riemannian metric on m such that the nodal set of its first nontrivial eigenfunction is we present a number of variations on this result which enable us to show in particular that the first nontrivial eigenfunction can have as many non degenerate critical points as one wishes

uniqueness of global positive solution branches of nonlinear elliptic problems has been studied in many publications the first global description was given in 19 nonetheless there were open questions some of which are answered in this paper we motivate some of these questions now for a suitable class of smooth fimctions f ir ir problem 1 1 has infinitely many solutions for 2 1 this was shown in 15 via the method of suband supersolutions which also gives an order of these solutions 0 0 which is not excluded in 15 there is also a global branch continuum c of solutions 2 u bifurcating from the trivial solution line 2 0 at 2 21 where 1 21 f 0 is the principal eigenvalue of the linearization

the distribution and moments of the smallest eigenvalue of a random matrix of wishart type abstract given a random rectangular m n matrix with elements from a normal distribution what is the distribution of the smallest singular value to pose an equivalent question in the language of multivariate statistics what is the distribution of the smallest eigenvalue of a matrix from the central wishart distribution in the null case we give new results giving the distribution as a simple recursion this includes the more difficult case when n m is an even integer without resorting to zonal polynomials and hypergeometric functions of matrix arguments with the recursion one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials exponentials and at worst ordinary hypergeometric functions we further elaborate on the special cases when n m 0 1 2 and 3 and give a numerical table of the expected values for 2 m 25 and 0 n m 25

nodal intersections for random eigenfunctions on the torus we investigate the number of nodal intersections of random gaussian laplace eigenfunctions on the standard two dimensional flat torus arithmetic random waves with a fixed smooth reference curve with nonvanishing curvature the expected intersection number is universally proportional to the length of the reference curve times the wavenumber independent of the geometry our main result prescribes the asymptotic behavior of the nodal intersections variance for smooth curves in the high energy limit remarkably it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber and the geometry of the given curve in particular this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability

correlation functions of the bc calogero sutherland model the bc type calogero sutherland model csm is an integrable extension of the ordinary a type csm that possesses a reflection symmetry point the bc cs mi sr elated to the chiral classes of random matrix ensembles rmes in exactly the same way as the a csm is related to the dyson classes we first develo pt he fermionic replica model formalism suitable to treat all chiral rmes by exploiting generalized colour flavour transformation we then extend the method to find the exact asymptotics of the bc csm density profile consistency of our result with the c 1g aussian conformal field theory description is verified the emerging friedel oscillations structure and sum rules are discussed in details we also compute the distribution of the particle nearest to the reflection point

a remark on a curvature gap for minimal surfaces in the ball we extend to higher codimension earlier characterization of the equatorial disk and the critical catenoid by a pinching condition on the length of their second fundamental form among free boundary minimal surfaces in the three dimensional euclidean ball due to l ambrozio and i nunes

variance calculations and the bessel kernel in the laguerre ensembleof n xn hermitian matrices it is of interest both theoretically and for applications to quantum transport problems to compute the variance of a linear statistic denoted varnf asn furthermore this statistic often contains an additional parameter a for which the limit is most interesting and most difficult to compute numerically we derive exact expressions for both limn varnf and lim limn varnf

counting nodal lines which touch the boundary of an analytic domain we consider the zeros on the boundary of a neumann eigenfunction of a real analytic plane domain we prove that the number of its boundary zeros is o where 2 we also prove that the number of boundary critical points of either a neumann or dirichlet eigenfunction is o it follows that the number of nodal lines of components of the nodal set which touch the boundary is of order this upper bound is of the same order of magnitude as the length of the total nodal line but is the square root of the courant bound on the number of nodal components in the interior more generally the results are proved for piecewise analytic domains

lower bounds for nodal sets of dirichlet and neumann eigenfunctions let phi be a dirichlet or neumann eigenfunction of the laplace beltrami operator on a compact riemannian manifold with boundary we prove lower bounds for the size of the nodal set phi 0

a global existence result for the compressible navier stokes equations in the critical l p framework the present paper is dedicated to the global well posedness issue for the barotropic compressible navier stokes system in the whole space mathbb r d with d 2 we aim at extending the work by danchin inventiones mathematicae 141 3 579 614 2000 to a critical framework which is not related to the energy space for small perturbations of a stable equilibrium state in the sense of suitable lp type besov norms we establish the global existence as a consequence like for incompressible flows one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true in passing we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows

spectrally accurate quadratures for evaluation of layer potentials close to the boundary for the 2d stokes and laplace equations dense particulate flow simulations using integral equation methods demand accurate evaluation of stokes layer potentials on arbitrarily close interfaces in this paper we generalize techniques for close evaluation of laplace double layer potentials in j helsing and r ojala j comput phys 227 2008 pp 2899 2921 we create a globally compensated trapezoid rule quadrature for the laplace single layer potential on the interior and exterior of smooth curves this exploits a complex representation a product quadrature in the style of kress for the sawtooth function careful attention to branch cuts and second kind barycentric type formulae for cauchy integrals and their derivatives upon this we build accurate single and double layer stokes potential evaluators by expressing them in terms of laplace potentials we test their convergence for vesicle vesicle interactions for an extensive set of laplace and stokes problems and when applying the system matrix in a boundary value problem solver

visualization of special eigenmode shapes of a vibrating elliptical membrane analysis of eigenfunctions and eigenvalues of the laplacian is important in the understanding of distributed parameter vibration systems and quantum mechanics the elliptical domain is advantageous for such work because it allows the explicit representation of eigenfunctions as products of the mathieu functions in ann phys 9 1960 pp 24 75 kelley and rubinow used wave propagation geometrical optics and wkb methods to show that two special types of eigenmodes whispering gallery and bouncing ball modes exist on a general convex domain and they illustrated the case of an elliptical domain in this paper we develop a numerical package of the mathieu and modified mathieu functions to actually construct profiles of a sequence of eigenfunctions in order to visualize such special eigenmodes and others in the process we have also been able to observe a new type of transition state named focusing modes herein which seem to have a complementary behavior to the bouncing ball modes numerical eige

recurrence equations for the computation of correlations in the 1 r 2 quantum many body system in a previous paper the two particle distribution function and one particle density matrix for the quantum many body system with the 1 r2 pair potential have been expressed as limiting cases of selberg correlation integrals recurrence equations are derived which allow rapid evaluation of these multidimensional integrals the exact results for the two particle distribution are compared with the harmonic approximation

distribution of the smallest eigenvalue in the correlated wishart model wishart random matrix theory is of major importance for the analysis of correlated time series the distribution of the smallest eigenvalue for wishart correlation matrices is particularly interesting in many applications in the complex and in the real case we calculate it exactly for arbitrary empirical eigenvalues i e for fully correlated gaussian wishart ensembles to this end we derive certain dualities of matrix models in ordinary space we thereby completely avoid the otherwise unsurmountable problem of computing a highly nontrivial group integral our results are compact and much easier to handle than previous ones furthermore we obtain a new universality for the distribution of the smallest eigenvalue on the proper local scale

increasing subsequences and the hard to soft edge transition in matrix ensembles our interest is in the cumulative probabilities pr l t l for the maximum length of increasing subsequences in poissonized ensembles of random permutations random fixed point free involutions and reversed random fixed point free involutions it is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary orthogonal and symplectic symmetry respectively the gap probabilities can be written as a sum over correlations for certain determinantal point processes from these expressions a proof can be given that the limiting form of pr l t l in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary orthogonal and symplectic symmetry respectively thereby reclaiming theorems due to baik deift johansson and baik rains

wishart and anti wishart random matrices we provide a compact exact representation for the distribution of the matrix elements of the wishart type random matrices for any finite number of rows and columns of without any large n approximations in particular we treat the case when the wishart type random matrix contains redundant non random information which is a new result this representation is of interest for a procedure for reconstructing the redundant information hidden in wishart matrices with potential applications to numerous models based on biological social and artificial intelligence networks

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives

order statistics and ginibre s ensembles the moduli of the eigenvalues at the edge of ginibre s complex and quaternion gaussian random matrix ensembles are shown to respond to a limit theorem identical in nature to that for independent identically distributed sequences this is a companion work to ref 15 in which the limit law for the scaled spectral radius of these ensembles was identified

some estimates of solutions for the equations of motion of compressible viscous fluid in the three dimensional exterior domain abstract we consider the equations of motion of compressible viscous fluid in an exterior domain in r 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem in particular we shall give l 1 estimates which implies the diffusion wave of compressible navier stokes flow

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

eigenvalue statistics for product complex wishart matrices the eigenvalue statistics for complex wishart matrices where is equal to the product of r complex gaussian matrices and the inverse of s complex gaussian matrices are considered in the case r s the exact form of the global density is computed the averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function for finite n the eigenvalue probability density function is computed and is shown to be an example of a biorthogonal ensemble a double contour integral form of the corresponding correlation kernel is derived which allows the hard edge scaled limit to be computed the limiting kernel is given in terms of certain meijer g functions and is identical to that found in the recent work of kuijlaars and zhang in the case s 0 properties of the kernel and corresponding correlation functions are discussed

nodal sets of solutions of equations involving magnetic schrodinger operator in three dimensions it is well known that the complexity of the nodal set of a function mainly comes from the singular set on which both the function and the gradient vanish the singular set of a real valued solution of a linear elliptic equation has been well investigated for a complex valued solution of a linear equation involving a magnetic schrodinger operator the structure of the nodal set has not been well investigated yet excepted in the two dimensional case in this paper we extend the arguments of garafalo and lin indiana univ math j 35 245 268 1986 and of han indiana univ math j 43 983 1002 1994 to show that the singular set of such a solution in a three dimensional domain is countably 1 rectifiable the functions considered in this paper include the order parameter in the ginzburg landau theory of superconductivity and the eigenfunctions of the magnetic schrodinger operator

the geometric complexity of special lagrangian t2 cones we prove a number of results on the geometric complexity of special lagrangian slg t2 cones in 3 every slg t2 cone has a fundamental integer invariant its spectral curve genus we prove that the spectral curve genus of an slg t2 cone gives a lower bound for its geometric complexity i e the area the stability index and the legendrian index of any slg t2 cone are all bounded below by explicit linearly growing functions of the spectral curve genus we prove that the cone on the clifford torus which has spectral curve genus zero in s5 is the unique slg t2 cone with the smallest possible legendrian index and hence that it is the unique stable slg t2 cone this leads to a classification of all rigid index 1 slg cone types in dimension three for cones with spectral curve genus two we give refined lower bounds for the area the legendrian index and the stability index one consequence of these bounds is that there exist s1 invariant slg torus cones of arbitrarily large area legendrian and stability indices we explain some consequences of our results for the programme due to joyce to understand the most common three dimensional isolated singularities of generic families of slg submanifolds in almost calabi yau manifolds

complex zeros of real ergodic eigenfunctions we determine the limit distribution as of complex zeros for holomorphic continuations to grauert tubes of real eigenfunctions of the laplacian on a real analytic compact riemannian manifold m g with ergodic geodesic flow if phi j k is an ergodic sequence of eigenfunctions we prove the weak limit formula frac 1 lambda j z phi j k mathbb c to frac i pi partial bar partial xi g where z phi j k mathbb c is the current of integration over the complex zeros and where overline partial is with respect to the adapted complex structure of lempert szoke and guillemin stenzel

critical points of green s functions on complete manifolds we prove that the number of critical points of a li tam green s function on a complete open riemannian surface of finite type admits a topological upper bound given by the first betti number of the surface in higher dimensions we show that there are no topological upper bounds on the number of critical points by constructing for each nonnegative integer n a riemannian manifold diffeomorphic to r n n 3 whose minimal green s function has at least n non degenerate critical points variations on the method of proof of the latter result yield contractible n manifolds whose minimal green s functions have level sets diffeomorphic to any fixed codimension 1 compact submanifold of r n

sharp norm estimates of layer potentials and operators at high frequency abstract in this paper we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency 2 in the high frequency regime the key problem is the dependence of mapping norms on the parameter for single layer potentials we find that the l 2 l 2 norms decay in the rate of decay depends on the curvature of the norm is 3 4 in piecewise smooth domains and 5 6 if the boundary is positively curved the double layer potential however displays uniform l 2 l 2 bounds independent of curvature by various examples we show that all our estimates on layer potentials are sharp appendix a by galkowski gives bounds l 2 l 2 for the single and double layer operators at high frequency that are sharp modulo log in this case both the single and double layer operator bounds depend upon the curvature of the boundary

intersection bounds for nodal sets of planar neumann eigenfunctions with interior analytic curves let r be a bounded piecewise smooth domain and be a neumann or dirichlet eigenfunction with eigenvalue and nodal set n x x 0 let h be an interior c curve consider the intersection number n h h n we first prove that for general piecewise analytic domains and under an appropriate goodness condition on h see theorem 1 1 n h oh as then using theorem 1 1 we prove in theorem 1 2 that the bound in is satisfied in the case of quantum ergodic qe sequences of interior eigenfunctions provided is convex and h has strictly positive geodesic curvature

upper bounds of nodal sets for eigenfunctions of eigenvalue problems the aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains the examples include biharmonic steklov eigenvalue problems buckling eigenvalue problems and champed plate eigenvalue problems the geometric measure of nodal sets are derived from doubling inequalities and growth estimates for eigenfunctions it is done through analytic estimates of morrey nirenberg and carleman estimates

exact results and universal asymptotics in the laguerre random matrix ensemble for various values of the parameters in the laguerre random matrix ensemble the distribution of the smallest eigenvalue and the scaled n level distribution function are calculated exactly in terms of generalized hypergeometric functions in certain cases these functions are expressed as multidimensional integrals from which asymptotic formulas are calculated and predictions of universal behavior verified

on the geometry of the level sets of bounded static potentials in this paper we present a new approach to the study of asymptotically flat static metrics arising in general relativity in the case where the static potential is bounded we introduce new quantities which are proven to be monotone along the level set flow of the potential function we then show how to use these properties to detect the rotational symmetry of the static solutions deriving a number of sharp inequalities among these we prove the validity without any dimensional restriction of the riemannian penrose inequality as well as of a reversed version of it in the class of asymptotically flat static metrics with connected horizon as a consequence of our analysis a simple proof of the classical 3 dimensional black hole uniqueness theorem is recovered and some geometric conditions are discussed under which the same statement holds in higher dimensions

kullback leibler distance as a measure of the information filtered from multivariate data we show that the kullback leibler distance is a good measure of the statistical uncertainty of correlation matrices estimated by using a finite set of data for correlation matrices of multivariate gaussian variables we analytically determine the expected values of the kullback leibler distance of a sample correlation matrix from a reference model and we show that the expected values are known also when the specific model is unknown we propose to make use of the kullback leibler distance to estimate the information extracted from a correlation matrix by correlation filtering procedures we also show how to use this distance to measure the stability of filtering procedures with respect to statistical uncertainty we explain the effectiveness of our method by comparing four filtering procedures two of them being based on spectral analysis and the other two on hierarchical clustering we compare these techniques as applied both to simulations of factor models and empirical data we investigate the ability of these filtering procedures in recovering the correlation matrix of models from simulations we discuss such ability in terms of both the heterogeneity of model parameters and the length of data series we also show that the two spectral techniques are typically more informative about the sample correlation matrix than techniques based on hierarchical clustering whereas the latter are more stable with respect to statistical uncertainty

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m