extension criterion via two components of vorticity on strong solutions to the 3 d navier stokes equations we shall show that only two components of vorticity play an essential role to determine possibility of extension of the time interval for the local strong solution to the navier stokes equations then we shall apply our extension theorem to regularity criterion on weak solutions due to serrin and beirao da veiga chae choe proved the same criterion as beirao da veiga only by means of the two components of vorticity we deal with the critical case which they excluded our criterion may be regarded as the generalization of the result of beal kato majda from l to bmo

multiple sign changing solutions to semilinear elliptic resonant problems abstract in this paper we use the topological degree theory and the critical groups to investigate the elliptic equation u f x u in and subject to u 0 on and establish a multiple solutions theorem which guarantees that this problem has at least six nontrivial solutions under some resonant conditions if this problem has only finitely many solutions then of these solutions there are two positive solutions two negative solutions and two sign changing solutions

gap probability of the circular unitary ensemble with a fisher hartwig singularity and the coupled painleve v system we consider the circular unitary ensemble with a fisher hartwig singularity of both jump type and root type at z 1 a rescaling of the ensemble at the fisher hartwig singularity leads to the confluent hypergeometric kernel by studying the asymptotics of the toeplitz determinants we show that the probability of there being no eigenvalues in a symmetric arc about the singularity on the unit circle for a random matrix in the ensemble can be explicitly evaluated via an integral of the hamiltonian of the coupled painleve v system in dimension four this leads to a painleve type representation of the confluent hypergeometric kernel determinant moreover the large gap asymptotics including the constant terms are derived by evaluating the total integral of the hamiltonian in particular we reproduce the large gap asymptotics of the confluent hypergeometric kernel determinant obtained by deift krasovsky and vasilevska and the sine kernel determinant as a special case including the constant term conjectured earlier by dyson

the method of lower and upper solutions for fourth order two point boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u iv f x u x u x u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous

multiple solutions for fourth order boundary value problem abstract in this paper we study the existence and multiplicity of nontrivial solutions for the fourth order two point boundary value problems making use of the theory of fixed point index in cone and leray schauder degree under general conditions on nonlinearity we prove that there exist at least six different nontrivial solutions for the fourth order two point boundary value problems furthermore if the nonlinearity is odd we obtain that there exist at least eight different nontrivial solutions

existence and multiplicity of solutions to 2mth order ordinary differential equations abstract in this paper the existence and multiplicity of solutions are obtained for the 2mth order ordinary differential equation two point boundary value problems 1 m u 2 m t i 1 m 1 m i a i u 2 m i t f t u t for all t 0 1 subject to dirichlet neumann mixed and periodic boundary value conditions respectively where f is continuous a i r for all i 1 2 m since these four boundary value problems have some common properties and they can be transformed into the integral equation of form u i 1 m a i t i u t m f u we firstly deal with this nonlinear integral equation by using the strongly monotone operator principle and the critical point theory we establish some conditions on f which are able to guarantee that the integral equation has a unique solution at least one nonzero solution and infinitely many solutions furthermore we apply the abstract results on the integral equation to the above four 2mth order two point boundary problems and successfully resolve the existence and multiplicity of their solutions

positive solutions of fourth order boundary value problems with two parameters abstract in this paper the existence results of positive solutions are obtained for fourth order boundary value problem u 4 u u f t u 0 where f 0 1 r r is continuous r and satisfy

two regularity criteria for the 3d mhd equations this work establishes two regularity criteria for the 3d incompressible mhd equations the first one is in terms of the derivative of the velocity field in one direction while the second one requires suitable boundedness of the derivative of the pressure in one direction

a numerical scheme for multi point special boundary value problems and application to fluid flow through porous layers abstract in this paper we propose a numerical scheme based on finite differences for the numerical solution of nonlinear multi point boundary value problems over adjacent domains in each subdomain the solution is governed by a different equation the solutions are required to be smooth across the interface nodes the approach is based on using finite difference approximation of the derivatives at the interface nodes smoothness across the interface nodes is imposed to produce an algebraic system of nonlinear equations a modified multi dimensional newton s method is proposed for solving the nonlinear system the accuracy of the proposed scheme is validated by examples whose exact solutions are known the proposed scheme is applied to solve for the velocity profile of fluid flow through multilayer porous media

large gap asymptotics for airy kernel determinants with discontinuities we obtain large gap asymptotics for airy kernel fredholm determinants with any number m of discontinuities these m point determinants are generating functions for the airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles our main result is that the m point determinants can be expressed asymptotically as the product of m 1 point determinants multiplied by an explicit constant pre factor which can be interpreted in terms of the covariance of the counting function of the process

sign changing solutions to second order integral boundary value problems abstract in this paper by using the fixed point index theory and leray schauder degree theory we consider the existence and multiplicity of sign changing solutions to nonlinear second order integral boundary value problem u t f u t for all t 0 1 subject to u 0 0 and u 1 g 0 1 u s d s where f g c r r we obtain some new existence results concerning sign changing solutions by computing hardly eigenvalues and the algebraic multiplicities of the associated linear problem if f and g satisfy certain conditions then this problem has at least six different nontrivial solutions two positive solutions two negative solutions and two sign changing solutions moreover if f and g are also odd then the problem has at least eight different nontrivial solutions which are two positive two negative and four sign changing solutions

deformations of the tracy widom distribution in random matrix theory the tracy widom tw distribution describes the behavior of the largest eigenvalue we consider here two models in which tw undergoes transformations in the first one disorder is introduced in the gaussian ensembles by superimposing an external source of randomness a competition between tw and a normal gaussian distribution results depending on the spreading of the disorder the second model consists of removing at random a fraction of correlated eigenvalues of a random matrix the usual formalism of fredholm determinants extends naturally a continuous transition from tw to the weilbull distribution characteristic of extreme values of an uncorrelated sequence is obtained

nearest neighbour spacing distributions of the β hermite ensemble of random matrices the evolution with of the distributions of the spacing s between nearest neighbour levels of unfolded spectra of random matrices from the hermite ensemble he is investigated by monte carlo simulations the random matrices from the he are real symmetric and tridiagonal where which can take any positive value is the reciprocal of the temperature in the classical electrostatic interpretation of eigenvalues the distribution of eigenvalues coincide with those of the three classical gaussian ensembles for 1 2 4 the use of the he ensemble results in an incomparable speed up and efficiency of numerical simulations of all spectral characteristics of large random matrices generalized gamma distributions are shown to be excellent approximations of the nearest neighbor spacing nns distributions for any while being still simple they account both for the level repulsion in s when s 0 and for the whole shape of the nns distributions in the range of s which is accessible to experiment or to most numerical simulations the exact nns distribution of the goe 1 is in particular significantly better described by a generalized gamma distribution than it is by the wigner surmise while the best generalized gamma approximation coincides essentially with the wigner surmise for 2 they describe too the evolution of the level repulsion between that of a poisson distribution and that of a goe distribution when increases from 0 to 1 the distribution of ln s related to the electrostatic interaction energy between neighbouring charges is accordingly well approximated by a generalized gumbel distribution for any 0 the distributions of the minimum nn spacing between eigenvalues of matrices from the he obtained both from as calculated eigenvalues and from unfolded eigenvalues are brody distributions which are classically used to characterize the spectral fluctuations of various physical systems

approaching word length distribution via level spectra abstract treating a text after the removal of paragraphs and punctuations as a spectrum of blanks the distributions of the length of words of ten languages are analyzed using models from the statistical theory of spectra it is found that the ten languages can be classified into two families one with words that follow a wigner like distribution while the words of the other obey a poisson like distribution

multiple sign changing solutions to the sturm liouville boundary value problem with resonance abstract in this paper we study the sturm liouville boundary value problem p x u q x u f x u x 0 1 subject to u 0 u 0 u 1 u 1 0 by constructing a new sobolev space h 1 0 1 we discuss the existence of multiple solutions especially the existence of multiple sign changing solutions to this problem when the nonlinear f is resonant both at 0 and by combining the methods of the morse theory the topological degree and the fixed point index we establish a multiple solutions theorem which guarantees that the problem has at least six nontrivial solutions if this problem has only finitely many solutions then of these solutions there are two positive solutions two negative solutions and two sign changing solutions

a class of fourth order singular boundary value problems by constructing lower and upper solutions and with the maximal theorem the author gives a necessary and sufficient condition for the existence of c 2 0 1 as well as c 3 0 1 positive solutions of singular boundary value problems for a class of fourth order sublinear differential equations

multiple solutions of some boundary value problems with parameters abstract in this paper we study the existence and multiplicity of nontrivial solutions for the following second order dirichlet nonlinear boundary value problem with odd order derivative u t a u t b u t f t u t for all t 0 1 with u 0 u 1 0 where a b r 1 f c 1 0 1 r 1 r 1 by using the morse theory we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution two nontrivial solutions and infinitely many solutions separately

incomplete determinantal processes from random matrix to poisson statistics we study linear statistics of a class of determinantal processes which interpolate between poisson and gue ginibre statistics in dimension 1 or 2 these processes are obtained by performing an independent bernoulli percolation on the particle configuration of a log gas confined in a general potential we show that depending on the expected number of deleted particles there is a universal transition for mesoscopic linear statistics namely at small scales the point process behave according to random matrix theory while at large scales it needs to be renormalized because the variance of any linear statistic diverges the crossover is explicitly characterized as the superposition of a h 1 or h 1 2 correlated gaussian noise depending on the dimension and an independent poisson process the proof consists in computing the limits of the cumulants of linear statistics using the asymptotics of the correlation kernel of the process

tracy widom distributions in critical unitary random matrix ensembles and the coupled painleve ii system we study fredholm determinants of the painleve ii and painleve xxxiv kernels in certain critical unitary random matrix ensembles these determinants describe special gap probabilities of eigenvalues we obtain tracy widom formulas for the fredholm determinants which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled painleve ii system in dimension four moreover the large gap asymptotics for these fredholm determinants are derived where the constant terms are given explicitly in terms of the riemann zeta function

truncated linear statistics associated with the eigenvalues of random matrices ii partial sums over proper time delays for chaotic quantum dots invariant ensembles of random matrices are characterized by the distribution of their eigenvalues lambda 1 ldots lambda n we study the distribution of truncated linear statistics of the form tilde l sum i 1 p f lambda i with p n this problem has been considered by us in a previous paper when the p eigenvalues are further constrained to be the largest ones or the smallest in this second paper we consider the same problem without this restriction which leads to a rather different analysis we introduce a new ensemble which is related but not equivalent to the thinned ensembles introduced by bohigas and pato this question is motivated by the study of partial sums of proper time delays in chaotic quantum dots which are characteristic times of the scattering process using the coulomb gas technique we derive the large deviation function for tilde l large deviations of linear statistics l sum i 1 n f lambda i are usually dominated by the energy of the coulomb gas which scales as sim n 2 implying that the relative fluctuations are of order 1 n for the truncated linear statistics considered here there is a whole region including the typical fluctuations region where the energy of the coulomb gas is frozen and the large deviation function is purely controlled by an entropic effect because the entropy scales as sim n the relative fluctuations are of order 1 sqrt n our analysis relies on the mapping on a problem of p fictitious non interacting fermions in n energy levels which can exhibit both positive and negative effective absolute temperatures we determine the large deviation function characterizing the distribution of the truncated linear statistics and show that for the case considered here f lambda 1 lambda the corresponding phase diagram is separated in three different phases

the method of lower and upper solutions for a bending of an elastic beam equation abstract in this paper by combining a new maximum principle and the theory of the two parameter linear eigenvalue problem we develop the monotone method in the presence of lower and upper solutions for the problem u 4 x f x u x u x 0 x 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous

randomly incomplete spectra and intermediate statistics by randomly removing a fraction of levels from a given spectrum a model is constructed that describes a crossover from this spectrum to a poisson spectrum the formalism is applied to the transitions towards poisson from random matrix theory rmt spectra and picket fence spectra it is shown that the fredholm determinant formalism of rmt extends naturally to describe incomplete rmt spectra

existence and uniqueness results for the bending of an elastic beam equation at resonance etude d un probleme non lineaire aux valeurs limites pour la flexion d une poutre elastique en appui simple aux deux extremites dans le cas de resonance theoremes d existence et d unicite

on the analysis of incomplete spectra in random matrix theory through an extension of the jimbo miwa ueno differential abstract several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by jimbo miwa and ueno jmu in the early 1980s 45 recent advances in the theory of tau functions 41 based on earlier works of b malgrange and m bertola have allowed to extend the original jimbo miwa ueno differential form to a 1 form closed on the full space of extended monodromy data of the underlying lax pairs this in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions including the exact computation of all relevant constant factors we use this method to efficiently compute the tail asymptotics of soft edge hard edge and bulk scaled distribution and gap functions in the complex wishart ensemble provided each eigenvalue particle has been removed independently with probability 1 0 1

positive solutions of some nonlocal fourth order boundary value problem by the use of the krasnosel skii s fixed point theorem the existence of one or two positive solutions for the nonlocal fourth order boundary value problemu 4 t email protected t email protected t u t u t 00 email protected c 0 1 x 0 x 0 0

positive solutions of fourth order two point boundary value problems in this paper by using the krasnoselskii fixed point theorem we study the existence of one or multiple positive solution of the fourth order two point boundary value problem y 4 t f t y t y t y 0 y 1 y 0 y 1 0 we also give some examples to illustrate our results

large deformations of the tracy widom distribution i non oscillatory asymptotics we analyze the left tail asymptotics of deformed tracy widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability 1 gamma in 0 1 as gamma varies a transition from tracy widom statistics gamma 1 to classical weibull statistics gamma 0 was observed in the physics literature by bohigas et al phys rev e 79 031117 2009 we provide a description of this transition by rigorously computing the leading order left tail asymptotics of the thinned goe gue and gse tracy widom distributions in this paper we obtain the asymptotic behavior in the non oscillatory region with gamma in 0 1 fixed for the goe gue and gse distributions and gamma uparrow 1 at a controlled rate for the gue distribution this is the first step in an ongoing program to completely describe the transition between tracy widom and weibull statistics as a corollary to our results we obtain a new total integral formula involving the ablowitz segur solution to the second painleve equation