long time behavior of the solution to the mkdv equation with step like initial data we consider the modified korteweg de vries equation on the line the initial data is the pure step function i e q x 0 cr for x 0 and q x 0 cl for x cr 0 are arbitrary real numbers long time behavior of the solution to the mkdv equation in the case cl cr 0 and t was studied recently in kotlyarov and minakov 2010 j math phys 51 093506 where the structure of the compression wave was obtained in the form of a modulated elliptic wave the goal of this paper is to study the asymptotic behavior of the solution of the initial value problem as t i e we study the long time dynamics of the rarefaction wave using the steepest descent method and the so called g function mechanism we deform the original oscillatory matrix riemann hilbert problem to the explicitly solvable model forms and show that the solution of the initial value problem has a different asymptotic behavior in different regions of the xt plane in the regions x 6c2rt the main term of asymptotics of the solution is equal to cl and cr respectively in the region the asymptotics of the solution tends to an influence of the dispersion is also studied the second term of the asymptotics is obtained in the region x 6c2rt where the background constant cr is perturbed by the self similar vanishing wave

riemann hilbert problems and the mkdv equation with step initial data short time behavior of solutions and the nonlinear gibbs type phenomenon we produce a family of matrix riemann hilbert rh problems parametrized by a given scalar function r k the jump matrix of the rh problem depends on r k and external real parameters x and t we prove the unique solvability of such a problem a solution m x t k of this matrix rh problem gives a smooth solution q x t of the modified korteweg de vries mkdv equation qt 6q2qx qxxx 0 for a special choice of the function r k we show that the corresponding initial data q x 0 is a heaviside type function we consider the small time limit of the solution q x t the leading order term is explicitly calculated as an integral of the airy function the first summand is an exact solution to the linear kdv equation pt pxxx 0 with initial data thus this specific solution q x t to the mkdv equation being smooth with respect to and t 0 takes pointwise its limit values and exhibits a gibbs type phenomenon with rapid oscillations near x 0 as t 0

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

universality at the edge of the spectrum for unitary orthogonal and symplectic ensembles of random matrices we prove universality at the edge of the spectrum for unitary 2 orthogonal 1 and symplectic 4 ensembles of random matrices in the scaling limit for a class of weights w x e v x where v is a polynomial v x 2mx2m 2m 0 the precise statement of our results is given in theorem 1 1 and corollaries 1 2 and 1 4 below for the same class of weights a proof of universality in the bulk of the spectrum is given in 12 for the unitary ensembles and in 9 for the orthogonal and symplectic ensembles our starting point in the unitary case is 12 and for the orthogonal and symplectic cases we rely on our recent work 9 which in turn depends on the earlier work of widom 46 and tracy and widom 42 as in 9 the uniform plancherel rotach type asymptotics for the orthogonal polynomials found in 12 plays a central role the formulae in 46 express the correlation kernels for 1 4 as a sum of a christoffel darboux cd term as in the case 2 together with a correction term in the bulk scaling limit 9 the correction term is of lower order and does not contribute to the limiting form of the correlation kernel by contrast in the edge scaling limit considered here the cd term and the correction term contribute to the same order this leads to additional technical difficulties over and above 49 2006 wiley periodicals inc

the small dispersion limit of the korteweg de vries equation iii in parts i and ii we have derived explicit formulas for the distribution limit u of the solution of the kdv equation as the coefficient of uxxx tends to zero this formula contains n parameters 1 n whose values as well as whose number depends on x and t in section 4 we have shown that for t tb n 1 and the value of was determined in section 5 we have shown that the parameters i satisfy a nonlinear system of partial differential equations in part iii section 6 we show that for t large n 3 and we determine the asymptotic behavior of 1 2 3 and of u and u2 for t large the explicit formulas show that u and u2 are o t 1 and o t 2 respectively see formulas 6 2 and 6 24 in section 7 we study initial data whose value tends to zero as x and to 1 as x if one accepts some plausible guesses about the behavior of solutions with such initial data we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution the function s is expressible in terms of complete elliptic integrals a similar formula is derived for u2 in section 8 we indicate how to extend the treatment of this series of papers to multihumped but still negative initial data

modulated elliptic wave and asymptotic solitons in a shock problem to the modified korteweg de vries equation we study the long time asymptotic behavior of the cauchy problem for the modified korteweg de vries equation with an initial function of the step type this function rapidly tends to zero as and to some positive constant c as in 1989 khruslov and kotlyarov have found khruslov and kotlyarov 1989 inverse problems 5 1075 88 that for a large time the solution breaks up into a train of asymptotic solitons located in the domain is a constant the number n of these solitons grows unboundedly as in 2010 kotlyarov and minakov have studied temporary asymptotics of the solution of the cauchy problem on the whole line kotlyarov and minakov 2010 j math phys 51 093506 and have found that in the domain this solution is described by a modulated elliptic wave we consider here the modulated elliptic wave in the domain our main result shows that the modulated elliptic wave also breaks up into solitons which are similar to the asymptotic solitons in khruslov and kotlyarov 1989 inverse problems 5 1075 88 but differ from them in phase it means that the modulated elliptic wave does not represent the asymptotics of the solution in the domain the correct asymptotic behavior of the solution is given by the train of asymptotic solitons given in khruslov and kotlyarov 1989 inverse problems 5 1075 88 however in the asymptotic regime as in the region we can watch precisely a pair of solitons with numbers n one of them is the asymptotic soliton while the other soliton is generated from the elliptic wave their phases become closer to each other for a large n i e these solitons are also close to each other this result gives the answer on a very important question about matching of the asymptotic formulas in the mentioned region where the both formulas are well defined thus we have here a new and previously unknown mechanism 5 35 of matching of the asymptotics of the solution in the adjacent regions

the stability of the quadratic functional equation on amenable groups all the literature on the stability of the quadratic functional equation focus on the case where the relevant domain is an abelian group or a normed space in this note we solve the stability problem of the quadratic functional equation on amenable groups

projected sequential quadratic programming methods in this paper we introduce and analyze a class of optimization methods called projected sequential quadratic programming sqp methods for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on parts of the variables such problems frequently arise in the numerical solution of optimal control problems projected sqp methods combine the ideas of projected newton methods and sqp methods they use the simple projection onto the set defined by the bound constraints and maintain feasibility with respect to these constraints the iterates are computed using an extension of sqp methods and require only the solution of the linearized equality constraint global convergence of these methods is enforced using a constrained merit function and an armijo like line search we discuss global and local convergence properties of these methods the identification of active indices and we present numerical examples for an optimal control problem governed by a nonlinear h

mesoscopic colonization of a spectral band we consider the unitary matrix model in the limit where the size of the matrices becomes infinite and in the critical situation when a new spectral band is about to emerge in previous works the number of expected eigenvalues in the neighborhood of the band was fixed and finite a situation that was termed birth of a cut or first colonization we now consider the transitional regime where this microscopic population in the new band grows without bounds but at a slower rate than the size of the matrix the local population in the new band organizes in a mesoscopic regime in between the macroscopic behavior of the full system and the previously studied microscopic one the mesoscopic colony may form a finite number of new bands with a maximum number dictated by the degree of criticality of the original potential we describe the delicate scaling limit that realizes and controls the mesoscopic colony the method we use is the steepest descent analysis of the riemann hilbert problem that is satisfied by the associated orthogonal polynomials

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

on the convergence theory of trust region based algorithms for equality constrained optimization in a recent paper dennis el alem and maciel proved global convergence to a stationary point for a general trust region based algorithm for equality constrained optimization this general algorithm is based on appropriate choices of trust region subproblems and seems particularly suitable for large problems this paper shows global convergence to a point satisfying the second order necessary optimality conditions for the same general trust region based algorithm the results given here can be seen as a generalization of the convergence results for trust regions methods for unconstrained optimization obtained by more and sorensen the behavior of the trust radius and the local rate of convergence are analyzed some interesting facts concerning the trust region subproblem for the linearized constraints the quasi normal component of the step and the hard case are presented it is shown how these results can be applied to a class of discretized optimal control problems

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

transport maps for beta matrix models and universality we construct approximate transport maps for non critical beta matrix models that is maps so that the push forward of a non critical beta matrix model with a given potential is a non critical beta matrix model with another potential up to a small error in the total variation distance one of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension in addition we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models

the asymptotics of monotone subsequences of involutions we compute the limiting distributions of the lengths of the longest monotone subsequences of random signed involutions with or without conditions on the number of fixed points and negated points as the sizes of the involutions tend to infinity the resulting distributions are depending on the number of fixed points 1 the tracy widom distributions for the largest eigenvalues of random goe gue gse matrices 2 the normal distribution or 3 new classes of distributions which interpolate between pairs of the tracy widom distributions we also consider the second rows of the corresponding young diagrams in each case the convergence of moments is also shown the proof is based on the algebraic work of j baik and e rains in 7 which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the riemann hilbert analysis for the orthogonal polynomials by p deift k johansson and baik in 3

sum rules via large deviations in this paper a sum rule means a relationship between a functional defined on a subset of all probability measures on r involving the reverse kullback leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction the first sum rule is the szeg o verblunsky theorem see simon 2011 theorem 1 8 6 p 29 and concerns the case of trigonometrical polynomials on the torus more recently killip and simon killip and simon 2003 have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution indeed this sum rule includes a contribution of the atomic part of the measure away from the support of the circular law in this paper we recover this sum rules by using only large deviations tools on random matrices furthermore we obtain new up to our knowledge magic sum rules for the reverse kullback leibler divergence with respect to the pastur marchenko or kesten mckay distributions as in the semicircular case these formulas generally include a contribution of the atomic part appearing away from the support of the reference measure from the point of view of large deviations we obtain these formulas by showing a large deviation principle for the point mass measure concentrated on random eigenvalues of classical ensembles and combine it with a large deviation principle due to gamboa and rouault 2011 for the recursion coefficients

smoothing sqp methods for solving degenerate nonsmooth constrained optimization problems with applications to bilevel programs we consider a degenerate nonsmooth and nonconvex optimization problem for which the standard constraint qualification such as the generalized mangasarian fromovitz constraint qualification gmfcq may not hold we use smoothing functions with the gradient consistency property to approximate the nonsmooth functions and introduce a smoothing sequential quadratic programming sqp algorithm under the l infty penalty framework we show that any accumulation point of a selected subsequence of the iteration sequence generated by the smoothing sqp algorithm is a clarke stationary point provided that the sequence of multipliers and the sequence of penalty parameters are bounded furthermore we propose a new condition called the weakly generalized mangasarian fromovitz constraint qualification wgmfcq that is weaker than the gmfcq we show that the extended version of the wgmfcq guarantees the boundedness of the sequence of multipliers and the sequence of penalty parameters and thus guarantees the global c

discrete painleve equations and their appearance in quantum gravity we discuss an algorithmic approach for both deriving discrete analogues of painleve equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations as a concrete example we show that a discrete analogue of painleve i can be used to characterize similarity solutions of the kac moerbeke equation it turns out that these similarity solutions also satisfy a special case of painleve iv equation in addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions as an example we use the wkb method in the presence of two turning points of the third order to parametrize at the continuous limit the solution of painleve i in terms of the solution of discrete painleve i finally we show that these results are useful for investigating the partition function of the matrix model in 2d quantum gravity associated with the measure exp t1z2 t2z4 t3z6

differential systems for biorthogonal polynomials appearing in 2 matrix models and the associated riemann hilbert problem we consider biorthogonal polynomials that arise in the study of a generalization of two matrix hermitian models with two polynomial potentials v1 x v2 y of any degree with arbitrary complex coefficients finite consecutive subsequences of biorthogonal polynomials windows of lengths equal to the degrees of the potentials v1 and v2 satisfy systems of ode s with polynomial coefficients as well as pde s deformation equations with respect to the coefficients of the potentials and recursion relations connecting consecutive windows a compatible sequence of fundamental systems of solutions is constructed for these equations the stokes sectorial asymptotics of these fundamental systems are derived through saddle point integration and the riemann hilbert problem characterizing the differential equations is deduced

extreme gaps between eigenvalues of random matrices this paper studies the extreme gaps between eigenvalues of random matrices we give the joint limiting law of the smallest gaps for haar distributed unitary matrices and matrices from the gaussian unitary ensemble in particular the kth smallest gap normalized by a factor n 4 3 has a limiting density proportional to x3k 1e x3 concerning the largest gaps normalized by n logn they converge in lp to a constant for all p 0 these results are compared with the extreme gaps between zeros of the riemann zeta function

topological recursion in enumerative geometry and random matrices we review the method of symplectic invariants recently introduced to solve matrix models loop equations in the so called topological expansion and further extended beyond the context of matrix models for any given spectral curve one defines a sequence of differential forms and a sequence of complex numbers fg called symplectic invariants we recall the definition of fg s and we explain their main properties in particular symplectic invariance integrability modularity as well as their limits and their deformations then we give several examples of applications in particular matrix models enumeration of discrete surfaces maps algebraic geometry and topological strings and non intersecting brownian motions

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

superlinear convergence of affine scaling interior point newton methods for infinite dimensional nonlinear problems with pointwise bounds we develop and analyze a superlinearly convergent affine scaling interior point newton method for infinite dimensional problems with pointwise bounds in lp space the problem formulation is motivated by optimal control problems with lp controls and pointwise control constraints the finite dimensional convergence theory by coleman and li siam j optim 6 1996 pp 418 445 makes essential use of the equivalence of norms and the exact identifiability of the active constraints close to an optimizer with strict complementarity since these features are not available in our infinite dimensional framework algorithmic changes are necessary to ensure fast local convergence the main building block is a newton like iteration for an affine scaling formulation of the kkt condition we demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsizes even arbitrarily close to a nondegenerate solution using a pointwise projection instead we prove superlinear convergence under a weak strict complementarity condition and convergence with mbox q rate 1 under a slightly stronger condition if a smoothing step is available we discuss how the algorithm can be embedded in the class of globally convergent trust region interior point methods recently developed by m heinkenschloss and the authors numerical results for the control of a heating process confirm our theoretical findings

gaussian fluctuation for the number of particles in airy bessel sine and other determinantal random point fields we prove the central limit theorem clt for the number of eigenvalues near the spectrum edge for certain hermitian ensembles of random matrices to derive our results we use a general theorem essentially due to costin and lebowitz concerning the gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions as another corollary of the costin lebowitz theorem we prove the clt for the empirical distribution function of the eigenvalues of random matrices from classical compact groups

strong asymptotics of the orthogonal polynomials with respect to a measure supported on the plane we consider the orthogonal polynomials pn z with respect to the measure z a 2nce n z 2da z over the whole complex plane we obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where n grows to infinity with n the asymptotics are described in terms of three probability measures associated with the problem the first measure is the limit of the counting measure of zeros of the polynomials which is captured by the g function much in the spirit of ordinary orthogonal polynomials on the real line the second measure is the equilibrium measure that minimizes a certain logarithmic potential energy supported on a region k of the complex plane the third measure is the harmonic measure of kc with a pole at this appears as the limit of the probability measure given up to the normalization constant by the squared modulus of the nth orthogonal polynomial times the orthogonality measure i e pn z 2 z a 2nce n z 2da z the compact region k that is the support of the second measure undergoes a topological transition under the variation of the parameter t n n in a double scaling limit near the critical point given by tc a a 2c we observe the hastings mcleod solution to painleve ii in the asymptotics of the orthogonal polynomials 2014 wiley periodicals inc

generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields the equilibrium measure in the presence of an external field plays a role in a number of areas in analysis for example in random matrix theory the limiting mean density of eigenvalues is precisely the density of the equilibrium measure typical behavior for the equilibrium measure is 1 it is positive on the interior of a finite number of intervals 2 it vanishes like a square root at endpoints and 3 outside the support there is strict inequality in the euler lagrange variational conditions if these conditions hold then the limiting local eigenvalue statistics is loosely described by a bulk in which there is universal behavior involving the sine kernel and edge effects in which there is a universal behavior involving the airy kernel through techniques from potential theory and integrable systems we show that this regular behavior is generic for equilibrium measures associated with real analytic external fields in particular we show that for any one parameter family of external fields v c the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c we discuss applications of our results to random matrices orthogonal polynomials and integrable systems 2000 john wiley sons inc

space mapping for optimal control of partial differential equations solving optimal control problems for nonlinear partial differential equations represents a significant numerical challenge due to the tremendous size and possible model difficulties e g nonlinearities of the discretized problems in this paper a novel space mapping technique for solving the aforementioned problem class is introduced analyzed and tested the advantage of the space mapping approach compared to classical multigrid techniques lies in the flexibility of not only using grid coarsening as a model reduction but also employing perhaps less nonlinear surrogates the space mapping is based on a regularization approach which in contrast to other space mapping techniques results in a smooth mapping and thus avoids certain irregular situations at kinks a new broyden update formula for the sensitivities of the space map is also introduced this quasi newton update is motivated by the usual secant condition combined with a secant condition resulting from differentiating the space mapping surrogate the overall algorithm employs a trust region framework for global convergence issues involved in the computations are highlighted and a report on a few illustrative numerical tests is given

on semiclassical zero dispersion limit solutions of the focusing nonlinear schrodinger equation we calculate the leading order term of the solution of the focusing nonlinear cubic schrodinger equation nls in the semiclassical limit for a certain one parameter family of initial conditions this family contains both solitons and pure radiation in the pure radiation case our result is valid for all times t 0 we utilize the riemann hilbert problem formulation of the inverse scattering problem to obtain the leading order term of the solution error estimates are provided 2004 wiley periodicals inc

the behavior of the weyl function in the zero dispersion kdv limit the moment formulas that globally characterize the zero dispersion limit of the korteweg devries kdv equation are known to be expressed in terms of the solution of a maximization problem here we establish a direct relation between this maximizer and the zero dispersion limit of the logarithm of the jost functions associated with the inverse spectral transform all the kdv conserved densities are encoded in the spatial derivative of these functions known as weyl functions we show the weyl functions are densities of measures that converge in the weak sense to a limiting measure this limiting measure encodes all of the weak limits of the kdv conserved densities moreover we establish the weak limit of spectral measures associated with the dirichlet problem

harmonic analysis in phase space and finite weyl heisenberg ensembles weyl heisenberg ensembles are translation invariant determinantal point processes on mathbb r 2d associated with the schrodinger representation of the heisenberg group and include as examples the ginibre ensemble and the polyanalytic ensembles which model the higher landau levels in physics we introduce finite versions of the weyl heisenberg ensembles and show that they behave analogously to the finite ginibre ensembles more specifically guided by the observation that the ginibre ensemble with n points is asymptotically close to the restriction of the infinite ginibre ensemble to the disk of area n we define finite wh ensembles as adequate finite approximations of the restriction of infinite wh ensembles to a given domain omega we provide a precise rate for the convergence of the corresponding one point intensities to the indicator function of omega as omega is dilated and the process is rescaled proportionally thermodynamic regime the construction and analysis rely neither on explicit formulas nor on the asymptotics for orthogonal polynomials but rather on phase space methods second we apply our construction to study the pure finite ginibre type polyanalytic ensembles which model finite particle systems in a single landau level and are defined in terms of complex hermite polynomials on a technical level we show that finite wh ensembles provide an approximate model for finite polyanalytic ginibre ensembles and we quantify the corresponding deviation by means of this asymptotic description we derive estimates for the rate of convergence of the one point intensity of polyanalytic ginibre ensembles in the thermodynamic limit

local analysis of a new multipliers method abstract in this paper we introduce a penalty function and a corresponding multipliers method for the solution of a class of nonlinear programming problems where the equality constraints have a particular structure the class models optimal control and engineering design problems with bounds on the state and control variables and has wide applicability the multipliers method updates multipliers corresponding to inequality constraints maintaining their nonnegativity instead of dealing with multipliers associated with equality constraints the basic local convergence properties of the method are proved and a dual framework is introduced we also analyze the properties of the penalized problem related with the penalty function

ulam s stability of a generalization of the frechet functional equation abstract we prove some general stability and hyperstability results for a generalization of the well known frechet equation stemming from the characterization of the inner product spaces due to jordan and von neumann the main result yields several stability outcomes for various other well known functional equations thus we obtain in particular some new inequalities characterizing the inner product spaces as the main tool we use a fixed point theorem for some function spaces

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

asymptotic analysis of the density of states in random matrix models associated with a slowly decaying weight the asymptotic behavior of polynomials that are orthogonal with respect to a slowly decaying weight is very different from the asymptotic behavior of polynomials that are orthogonal with respect to a freud type weight while the latter has been extensively studied much less is known about the former following an earlier investigation into the zero behavior we study here the asymptotics of the density of states in a unitary ensemble of random matrices with a slowly decaying weight this measure is also naturally connected with the orthogonal polynomials it is shown that after suitable rescaling the weak limit is the same as the weak limit of the rescaled zeros

asymptotics of step like solutions for the camassa holm equation abstract we study the long time asymptotics of solution of the cauchy problem for the camassa holm equation with a step like initial datum by using the nonlinear steepest descent method and the so called g function approach we show that the camassa holm equation exhibits a rich structure of sharply separated regions in the x t half plane with qualitatively different asymptotics which can be described in terms of a modulated finite gap elliptic function and a finite number of solitons

a feasible trust region sequential quadratic programming algorithm an algorithm for smooth nonlinear constrained optimization problems is described in which a sequence of feasible iterates is generated by solving a trust region sequential quadratic programming sqp subproblem at each iteration and by perturbing the resulting step to retain feasibility of each iterate by retaining feasibility the algorithm avoids several complications of other trust region sqp approaches the objective function can be used as a merit function and the sqp subproblems are feasible for all choices of the trust region radius global convergence properties are analyzed under various assumptions on the approximate hessian under additional assumptions superlinear convergence to points satisfying second order sufficient conditions is proved

comments on the core of the direct method for proving hyers ulam stability of functional equations in this short paper the core of the direct method for proving stability of functional equations is described in a clear way and in a quite general form

on universality for orthogonal ensembles of random matrices we prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum our starting point is the tracy widom formula for the matrix reproducing kernel the key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive toeplitz matrix and a two diagonal skew symmetric toeplitz matrix

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

long time asymptotics of the nonlinear schrodinger equation shock problem the long time asymptotics of two colliding plane waves governed by the focusing nonlinear schrodinger equation are analyzed via the inverse scattering method we find three asymptotic regions in space time a region with the original wave modified by a phase perturbation a residual region with a one phase wave and an intermediate transition region with a modulated two phase wave the leading order terms for the three regions are computed with error estimates using the steepest descent method for riemann hilbert problems the nondecaying initial data requires a new adaptation of this method a new breaking mechanism involving a complex conjugate pair of branch points emerging from the real axis is observed between the residual and transition regions also the effect of the collision is felt in the plane wave state well beyond the shock front at large times 2007 wiley periodicals inc

a fixed point approach to the stability of functional equations in non archimedean metric spaces abstract in this note we prove a simple fixed point theorem for a special class of complete metric spaces namely complete non archimedean metric spaces which are connected with some problems coming from quantum physics p adic strings and superstrings we also show that this theorem is a very efficient and convenient tool for proving the hyers ulam stability of a quite wide class of functional equations in a single variable

inverse scattering transform for the defocusing nonlinear schrodinger equation with fully asymmetric non zero boundary conditions abstract we formulate the inverse scattering transform ist for the defocusing nonlinear schrodinger nls equation with fully asymmetric non zero boundary conditions i e when the limiting values of the solution at space infinities have different non zero moduli the theory is formulated without making use of riemann surfaces and instead by dealing explicitly with the branched nature of the eigenvalues of the associated scattering problem for the direct problem we give explicit single valued definitions of the jost eigenfunctions and scattering coefficients over the whole complex plane and we characterize their discontinuous behavior across the branch cut arising from the square root behavior of the corresponding eigenvalues we pose the inverse problem as a riemann hilbert problem on an open contour and we reduce the problem to a standard set of linear integral equations finally for comparison purposes we present the single sheet branch cut formulation of the inverse scattering transform for the initial value problem with symmetric equimodular non zero boundary conditions as well as for the initial value problem with one sided non zero boundary conditions and we also briefly describe the formulation of the inverse scattering transform when a different choice is made for the location of the branch cuts

a riemann hilbert problem for skew orthogonal polynomials we find a local d 1 x d 1 riemann hilbert problem characterizing the skew orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d our riemann hilbert problem is similar to a local dxd riemann hilbert problem found by kuijlaars and mclaughlin characterizing the bi orthogonal polynomials this gives more motivation for finding methods to compute asymptotics of high order riemann hilbert problems and brings us closer to finding full asymptotic expansions of the skew orthogonal polynomials

higher order analogues of the tracy widom distribution and the painleve ii hierarchy we study fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points the kernels are natural higher order analogues of the airy kernel and

limit correlation functions for fixed trace random matrix ensembles universal limits for the eigenvalue correlation functions in the bulk of the spectrum are shown for a class of nondeterminantal random matrices known as the fixed trace or the hilbert schmidt ensemble these universal limits have been proved before for determinantal hermitian matrix ensembles and for some special classes of the wigner random matrices

the law of large numbers for the maximum of almost gaussian log correlated fields coming from random matrices we compute the leading asymptotics as n rightarrow infty of the maximum of the field q n q log q a n q in mathbb c for any unitarily invariant hermitian random matrix a n associated to a non critical real analytic potential hence we verify the leading order in a conjecture of fyodorov and simm nonlinearity 29 2837 2016 arxiv 1503 07110 math ph formulated for the gue the method relies on a classical upper bound and a more sophisticated lower bound based on a variant of the second moment method which exploits the hyperbolic branching structure of the field q n q q in mathbb h specifically we compare q n to an idealized gaussian field by means of exponential moments in principle this method could also be applied to random fields coming from other point processes provided that one can compute certain mixed exponential moments for unitarily invariant ensembles we show that these assumptions follow from the fyodorov strahov formula fyodorov and strahov in j phys a 36 12 3203 3213 2003 https doi org 10 1088 0305 4470 36 12 320 and asymptotics of orthogonal polynomials derived in deift et al commun pure appl math 52 11 1335 1425 1999 https doi org 10 1002 sici 1097 0312 199911 52 11 3 0 co 2 1

the inverse scattering transform for the focusing nonlinear schrodinger equation with asymmetric boundary conditions the inverse scattering transform ist as a tool to solve the initial value problem for the focusing nonlinear schrodinger nls equation with non zero boundary values ql r t al re 2ial r2t i l r as x is presented in the fully asymmetric case for both asymptotic amplitudes and phases i e with al ar and l r the direct problem is shown to be well defined for nls solutions q x t such that q x t ql r t l1 1 r with respect to x for all t 0 and the corresponding analyticity properties of eigenfunctions and scattering data are established the inverse scattering problem is formulated both via left and right marchenko integral equations and as a riemann hilbert problem on a single sheet of the scattering variables l r k2 al r2 where k is the usual complex scattering parameter in the ist the time evolution of the scattering coefficients is then derived showing that unlike the case of solutions with equal amplitudes as x here both reflection and transmission coefficients have

the k tacnode process the tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems for dyson brownian bridges the tacnode process describes the grazing collision of two packets of walkers we consider such a dyson sea on the unit circle with drift for any k in mathbb z we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which k particles are expected to wrap around the circle we derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized hastings mcleod solutions to the inhomogeneous painleve ii equation the method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight

variations on the drygas equation and its stability abstract we study the stability of the drygas functional equation g x y g x y 1 2 g x g y g y 1 mixing the direct method of the proof with the method of the invariant means dropping the assumption about the domain to be an abelian group we assume that the function we are dealing with is central i e g x y g y x is approximatively central i e g x y g y x satisfies the kannappan condition i e g x y z g x z y or that the group is amenable