bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

quantization of edge currents for continuous magnetic operators abstract for a magnetic hamiltonian on a half plane given by the sum of the landau operator with dirichlet boundary conditions and a random potential a quantization theorem for the edge currents is proven this shows that the concept of edge channels also makes sense in presence of disorder moreover gaussian bounds on the heat kernel and its covariant derivatives are obtained

gap labelling and the pressure on the boundary in quantum systems described by covariant families of 1 particle schrodinger operators on half spaces the pressure on the boundary per unit energy is topologically quantised if the fermi energy lies in a gap of the bulk spectrum its relation with the integrated density of states can be expressed in an integrated version of streda s formula this leads also to a gap labelling theorem for systems with constant magnetic field the proof uses non commutative topology

bulk universality of general β ensembles with non convex potential we prove the bulk universality of the ensembles with non convex regular analytic potentials for any 0 this removes the convexity assumption appeared in the earlier work p bourgade l erdos and h t yau universality of general ensembles preprint arxiv 0907 5605 2011 the convexity condition enabled us to use the logarithmic sobolev inequality to estimate events with small probability the new idea is to introduce a convexified measure so that the local statistics are preserved under this convexification

convergence of line search methods for unconstrained optimization line search methods are traditional and successful methods for solving unconstrained optimization problems its convergence has attracted more attention in recent years in this paper we analyze the general results on convergence of line search methods with seven line search rules it is clarified that the search direction plays a main role in these methods and that step size guarantees the global convergence in some cases it is also proved that many line search methods have same convergence property these convergence results can enable us to design powerful effective and stable algorithms in practice finally a class of special line search methods is investigated

line search algorithms for locally lipschitz functions on riemannian manifolds this paper presents line search algorithms for finding extrema of locally lipschitz functions defined on riemannian manifolds to this end we generalize the so called wolfe conditions for nonsmooth functions on riemannian manifolds using varepsilon subgradient oriented descent directions and the wolfe conditions we propose a nonsmooth riemannian line search algorithm and establish the convergence of our algorithm to a stationary point moreover we extend the classical bfgs algorithm to nonsmooth functions on riemannian manifolds numerical experiments illustrate the effectiveness and efficiency of the proposed algorithm

boundary maps for c crossed products with r with an application to the quantum hall effect the boundary map in k theory arising from the wiener hopf extension of a crossed product algebra with r is the connes thom isomorphism in this article the wiener hopf extension is combined with the heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces and cyclic cohomol ogy it then follows directly from a non commutative stokes theorem that this map is dual w r t connes pairing of cyclic cohomology with k theory as an application we prove equality of quantized bulk and edge conductivities for the integer quantum hall effect described by continuous magnetic schrodinger operators 1 motivation and main result in a commonly used approach to study aperiodic solids particles in the bulk of the medium are described by covariant families of one particle schrodinger operators h whereis the probability space of configurations furnished with an ergodic action of space translations crossed product algebras provide a natural framework for such families be86 in particular their bounded functions are represented by elements of a c crossed product the so called bulk algebra the non commutative topology of the c algebra is a useful tool to construct topological invariants resulting from pairings between k group elements and higher traces some of these invariants may be physi cally interpreted as topologically quantised quantities the quantised hall conductivity is such an example the physics near a boundary of the solid can also be described by a c algebra the so called edge algebra the bulk algebra being essentially a crossed product of the edge algebra with r or with z in the tight binding approximation krs02 both algebras are tied together in the wiener hopf extension or respectively the toeplitz extension this extension gives rise to boundary maps between the k groups and the higher traces of the bulk and edge algebra which allow one to equate bulk and edge in variants this topological relation and its physical interpretations is our main objective we discuss one prominent physical example of this the quantum hall effect where the

regularity conditions in the clt for linear eigenvalue statistics of wigner matrices abstract we show that the variance of centred linear statistics of eigenvalues of gue matrices remains bounded for large n for some classes of test functions less regular than lipschitz functions this observation is suggested by the limiting form of the variance which has previously been computed explicitly but it does not seem to appear in the literature we combine this fact with comparison techniques following tao vu and erdos yau et al and a littlewood paley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the holder class c 1 2 in the case of matrices of gaussian convolution type we also give a variance bound which implies the clt for test functions in the sobolev space h 1 and c 1 for general wigner matrices satisfying moment conditions if the additional assumption of the test function being supported away from the edge of the spectrum is made we prove the clt for test functions of regularity h 1 2 l and h 1 2 for gue and johansson matrices respectively previous results on the clt impose the existence and continuity of at least one classical derivative

a trust region algorithm with a worst case iteration complexity of mathcal o epsilon 3 2 o 3 2 for nonconvex optimization we propose a trust region algorithm for solving nonconvex smooth optimization problems for any overline epsilon in 0 infty 0 the algorithm requires at most mathcal o epsilon 3 2 o 3 2 iterations function evaluations and derivative evaluations to drive the norm of the gradient of the objective function below any epsilon in 0 overline epsilon 0 this improves upon the mathcal o epsilon 2 o 2 bound known to hold for some other trust region algorithms and matches the mathcal o epsilon 3 2 o 3 2 bound for the recently proposed adaptive regularisation framework using cubics also known as the arc algorithm our algorithm entitled trace follows a trust region framework but employs modified step acceptance criteria and a novel trust region update mechanism that allow the algorithm to achieve such a worst case global complexity bound importantly we prove that our algorithm also attains global and fast local convergence guarantees under similar assumptions as for other trust region algorithms we also prove a worst case upper bound on the number of iterations function evaluations and derivative evaluations that the algorithm requires to obtain an approximate second order stationary point

direct search methods then and now we discuss direct search methods for unconstrained optimization we give a modern perspective on this classical family of derivative free algorithms focusing on the development of direct search methods during their golden age from 1960 to 1971 we discuss how direct search methods are characterized by the absence of the construction of a model of the objective we then consider a number of the classical direct search methods and discuss what research in the intervening years has uncovered about these algorithms in particular while the original direct search methods were consciously based on straightforward heuristics more recent analysis has shown that in most but not all cases these heuristics actually suffice to ensure global convergence of at least one subsequence of the sequence of iterates to a first order stationary point of the objective function

a reduced space line search method for unconstrained optimization via random descent directions abstract in this paper we propose an iterative method based on reduced space approximations for unconstrained optimization problems the method works as follows among iterations samples are taken about the current solution by using for instance a normal distribution for all samples gradients are computed approximated in order to build reduced spaces onto which descent directions of cost functions are estimated by using such directions intermediate solutions are updated the overall process is repeated until some stopping criterion is satisfied the convergence of the proposed method is theoretically proven by using classic assumptions in the line search context experimental tests are performed by using well known benchmark optimization problems and a non linear data assimilation problem the results reveal that as the number of sample points increase gradient norms go faster towards zero and even more in the data assimilation context error norms are decreased by several order of magnitudes with regard to prior errors when the assimilation step is performed by means of the proposed formulation

gap labelling conjecture with nonzero magnetic field abstract given a constant magnetic field on euclidean space r p determined by a skew symmetric p p matrix and a z p invariant probability measure on the disorder set which is by hypothesis a cantor set where the action is assumed to be minimal the corresponding integrated density of states of any self adjoint operator affiliated to the twisted crossed product algebra c z p where is the multiplier on z p associated to takes on values on spectral gaps in the magnetic gap labelling group the magnetic frequency group is defined as an explicit countable subgroup of r involving pfaffians of and its sub matrices we conjecture that the magnetic gap labelling group is a subgroup of the magnetic frequency group we give evidence for the validity of our conjecture in 2d 3d the jordan block diagonal case and the periodic case in all dimensions

edge universality of beta ensembles we prove the edge universality of the beta ensembles for any beta ge 1 provided that the limiting spectrum is supported on a single interval and the external potential is fancyscript c 4 and regular we also prove that the edge universality holds for generalized wigner matrices for all symmetry classes moreover our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class fancyscript c 4

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

testing for chaos in deterministic systems with noise abstract recently we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal lyapunov exponent in this paper we investigate the capability of the test to cope with moderate amounts of noisy data comparisons are made between an improved version of our test and both the tangent space method and direct method for computing the maximal lyapunov exponent the evidence of numerical experiments ranging from the logistic map to an eight dimensional lorenz system of differential equations the lorenz 96 system suggests that our method is superior to tangent space methods and that it compares very favourably with direct methods

transverse laplacians for substitution tilings pearson and bellissard recently built a spectral triple the data of riemannian noncommutative geometry for ultrametric cantor sets they derived a family of laplace beltrami like operators on those sets motivated by the applications to specific examples we revisit their work for the transversals of tiling spaces which are particular self similar cantor sets we use bratteli diagrams to encode the self similarity and cuntz krieger algebras to implement it we show that the abscissa of convergence of the function of the spectral triple gives indications on the exponent of complexity of the tiling we determine completely the spectrum of the laplace beltrami operators give an explicit method of calculation for their eigenvalues compute their weyl asymptotics and a seeley equivalent for their heat kernels

fixed energy universality for generalized wigner matrices we prove the wigner dyson mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and hermitian wigner matrices previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for hermitian matrices if the energy parameter is fixed we develop a homogenization theory of the dyson brownian motion and show that microscopic universality follows from mesoscopic statistics 2015 wiley periodicals inc

dyson brownian motion for general beta and potential at the edge in this paper we compare the solutions of the dyson brownian motion for general beta and potential v and the associated mckean vlasov equation near the edge under suitable conditions on the initial data and the potential v we obtain optimal rigidity estimates of particle locations near the edge after a short time t mathrm o 1 our argument uses the method of characteristics along with a careful estimate involving an equation of the edge with the rigidity estimates as an input we prove a central limit theorem for mesoscopic statistics near the edge which as far as we know has been done for the first time in this paper additionally combining our results with landon and yau edge statistics of dyson brownian motion arxiv 1712 03881 2017 we give a proof of the local ergodicity of the dyson brownian motion for general beta and potential at the edge i e we show the distribution of extreme particles converges to the tracy widom beta distribution in a short time

spaces of tilings finite telescopic approximations and gap labeling the continuous hull of a repetitive tiling t in d with the finite pattern condition fpc inherits a minimal d lamination structure with flat leaves and a transversal open image in new window which is a cantor set this class of tiling includes the penrose the amman benkker ones in 2d as well as the icosahedral tilings in 3d we show that the continuous hull with its canonical d action can be seen as the projective limit of a suitable sequence of branched oriented and flat compact d manifolds as a consequence the longitudinal cohomology and the k theory of the corresponding c algebra open image in new window are obtained as direct limits of cohomology and k theory of ordinary manifolds moreover the space of invariant finite positive measures can be identified with a cone in the d th homology group canonically associated with the orientation of d at last the gap labeling theorem holds given an invariant ergodic probability measure on the hull the corresponding integrated density of states ids of any selfadjoint operators affiliated to open image in new window takes on values on spectral gaps in the module generated by the occurrence probabilities of finite patches in the tiling

fixed energy universality of dyson brownian motion abstract we consider dyson brownian motion for classical values of with deterministic initial data v we prove that the local eigenvalue statistics coincide with the goe gue in the fixed energy sense after time t 1 n if the density of states of v is bounded above and below down to scales t in a window of size l t our results imply that fixed energy universality holds for essentially any random matrix ensemble for which averaged energy universality was previously known our methodology builds on the homogenization theory developed in 16 which reduces the microscopic problem to a mesoscopic problem as an auxiliary result we prove a mesoscopic central limit theorem for linear statistics of various classes of test functions for classical dyson brownian motion

spectral statistics of erdős renyi graphs i local semicircle law we consider the ensemble of adjacency matrices of erd s renyi random graphs that is graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one we prove that as long as pn with a speed at least logarithmic in n the density of eigenvalues of the erd s renyi ensemble is given by the wigner semicircle law for spectral windows of length larger than n 1 up to logarithmic corrections as a consequence all eigenvectors are proved to be completely delocalized in the sense that the l norms of the l2 normalized eigenvectors are at most of order n 1 2 with a very high probability the estimates in this paper will be used in the companion paper spectral statistics of erd s renyi graphs ii eigenvalue spacing and the extreme eigenvalues 2011 preprint to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pn n2 3

beta ensembles stochastic airy spectrum and a diffusion we prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low lying eigenvalues of the random schrodinger operator d2 dx2 x 2 bx restricted to the positive half line where b x is white noise in doing so we extend the definition of the tracy widom distributions to all 0 and also analyze their tails last in a parallel development we provide a second characterization of these laws in terms of a one dimensional diffusion the proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit in particular we show how tracy widom laws arise from a functional central limit theorem

a derivative free trust region framework for variational data assimilation this study develops a hybrid ensemble variational approach for solving data assimilation problems the method called tr 4d enkf is based on a trust region framework and consists of three computational steps first an ensemble of model runs is propagated forward in time and snapshots of the state are stored next a sequence of basis vectors is built and a low dimensional representation of the data assimilation system is obtained by projecting the model state onto the space spanned by the ensemble perturbations finally the low dimensional optimization problem is solved in the reduced space using a trust region approach the size of the trust region is updated according to the relative decrease of the reduced order surrogate cost function the analysis state is projected back onto the full space and the process is repeated with the current analysis serving as a new background a heuristic approach based on the trust region size is proposed in order to adjust the background error statistics from one iteration to the next experimental simulations are carried out using the atmospheric general circulation model speedy the results show that tr 4d enkf is an efficient computational approach and is more accurate than the current state of the art 4d enkf implementations such as the pod 4d enkf and the iterative subspace minimization methods

the wigner semi circle law and eigenvalues of matrix valued diffusions a system of stochastic differential equations for the eigenvalues of a symmetric matrix whose components are independent ornstein uhlenbeck processes is derived this corresponds to a diffusion model of an interacting particles system with linear drift towards the origin and electrostatic inter particle repulsion the associated empirical distribution of particles is shown to converge weakly as the number of particles tends to infinity to a limiting measure valued process which may be characterized as the weak solution of a deterministic ode the wigner semi circle density is found to be one of the equilibrium points of this limiting equation

sparse quasi newton updates with positive definite matrix completion quasi newton methods are powerful techniques for solving unconstrained minimization problems variable metric methods which include the bfgs and dfp methods generate dense positive definite approximations and therefore are not applicable to large scale problems to overcome this difficulty a sparse quasi newton update with positive definite matrix completion that exploits the sparsity pattern e i j nabla 2 f x ij neq 0 rm for some x in r n of the hessian is proposed the proposed method first calculates a partial approximate hessian h qn ij i j in f where f supseteq e using an existing quasi newton update formula such as the bfgs or dfp methods next a full matrix h k 1 which is a maximum determinant positive definite matrix completion of h qn ij i j in f is obtained if the sparsity pattern e or its extension f has a property related to a chordal graph then the matrix h k 1 can be expressed as products of some sparse matrices the time and space requirements of the proposed method are lower than those of the bfgs or the dfp methods in particular when the hessian matrix is tridiagonal the complexities become o n the proposed method is shown to have superlinear convergence under the usual assumptions

higher order analogues of tracy widom distributions via the lax method we study the distribution of the largest eigenvalue in hermitian one matrix models when the spectral density acquires an extra number of k 1 zeros at the edge the distributions are directly expressed through the norms of orthogonal polynomials on a semi infinite interval as an alternative to using fredholm determinants they satisfy nonlinear recurrence relations which we show form a lax pair making contact to the string literature in the early 1990s the technique of pseudo differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the painleve xxxiv hierarchy these are the higher order analogues of the tracy widom distribution using known backlund transformations we show how to simplify earlier equivalent results that are derived from fredholm determinant theory valid for even k in terms of the painleve ii hierarchy

mesoscopic eigenvalue density correlations of wigner matrices we investigate to what extent the microscopic wigner gaudin mehta dyson wgmd or sine kernel statistics of random matrix theory remain valid on mesoscopic scales to that end we compute the connected two point spectral correlation function of a wigner matrix at two mesoscopically separated points in the mesoscopic regime density correlations are much weaker than in the microscopic regime our result is an explicit formula for the two point function this formula implies that the wgmd statistics are valid to leading order on all mesoscopic scales that in the real symmetric case there are subleading corrections matching precisely the wgmd statistics while in the complex hermitian case these subleading corrections are absent we also uncover non universal subleading correlations which dominate over the universal ones beyond a certain intermediate mesoscopic scale the proof is based on a hierarchy of schwinger dyson equations for a sufficiently large class of polynomials in the entries of the green function the hierarchy is indexed by a tree whose depth is controlled using stopping rules a key ingredient in the derivation of the stopping rules is a new estimate on the density of states which we prove to have bounded derivatives of all order on all mesoscopic scales

an analogue of the thom isomorphism for crossed products of a c algebra by an action of r abstract in this paper we show the existence and uniqueness of a natural isomorphism oj of kj a with kj 1 a r j z 2 where a r is a c dynamical r system k is the functor of topological k theory and a r is the crossed product of a by the action of r the pimsner voiculescu exact sequence is obtained as a corollary we show that given an invariant trace on a with dual trace gt one has gto 1 u 1 2 i u u for any unitary u in the domain of the derivation of a associated to the action finally we show that the crossed product of c s3 continuous functions on the 3 sphere by a minimal diffeomorphism is a simple c algebra with no nontrivial idempotent

interacting brownian particles and the wigner law in this paper we study interacting diffusing particles governed by the stochastic differential equationsdx j t n db j t d jon x 1 x n dt j 1 2 n here theb jare independent brownian motions in d and o n x 1 x n n i j v x i x j n i u x 1 the potentialv has a singularity at 0 strong enough to keep the particles apart and the potentialu serves to keep the particles from escaping to infinity our interest is in the behaviour as the number of particles increases without limit which we study through the empirical measure process we prove tightness of these processes in the case ofd 1 v x log x u x x 2 2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure valued process exists this process is deterministic and converges to the wigner law ast some information on the rates of convergence is derived and the case of a cauchy initial distribution is analysed completely

rotation numbers boundary forces and gap labelling we review the johnson moser rotation number and the k0 theoretical gap labelling of bellissard for one dimensional schr odinger operators we compare them with two further gap labels one being related to the motion of dirichlet eigenvalues the other being a k1 theoretical gap label we argue that the latter provides a natural generalization of the johnson moser rotation number to higher dimensions

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

bulk universality for deformed wigner matrices we consider n n random matrices of the form h w v where w is a real symmetric or complex hermitian wigner matrix and v is a random or deterministic real diagonal matrix whose entries are independent of w we assume subexponential decay for the matrix entries of w and we choose v so that the eigenvalues of w and v are typically of the same order for a large class of diagonal matrices v we show that the local statistics in the bulk of the spectrum are universal in the limit of large n

rigidity and a mesoscopic central limit theorem for dyson brownian motion for general beta β and potentials we study dyson brownian motion with general potential v and for general beta ge 1 for short times t o 1 and under suitable conditions on v we obtain a local law and corresponding rigidity estimates on the particle locations that is with overwhelming probability the particles are close to their classical locations with an almost optimal error estimate under the condition that the density of states of the initial data is bounded below and above down to the scale eta ll t ll 1 we prove a mesoscopic central limit theorem for linear statistics at all scales eta with n 1 ll eta ll t