conormal problem of higher order parabolic systems with time irregular coefficients the paper is a comprehensive study of lp and schauder estimates for higher order divergence type parabolic systems with discontinuous coefficients on a half space and cylindrical domains with the conormal derivative boundary conditions for the lp estimates we assume that the leading coefficients are only bounded and measurable in the t variable and have vanishing mean oscillations vmox with respect to x we also prove the schauder estimates in two situations the coefficients are holder continuous only in the x variable the coefficients are holder continuous in the t variable as well on the lateral boundary

schur recursions error formulas and convergence of rational estimators for stationary stochastic sequences an exact and approximate realization theory for estimation and model filters of second order stationary stochastic sequences is presented the properties of j lossless matrices as a unifying framework are used necessary and sufficient conditions for the exact realization of an estimation filter and a model filter as a submatrix of a j lossless system are deduced an extension of the so called schur algorithm yields an approximate j lossless realization based on partial past information about the process the geometric properties of such partial realizations and their convergence are studied finally connections with the nevanlinna pick problem are made and how the techniques presented constitute a generalization of many aspects of the levinson szego theory of partial realizations is shown as a consequence generalized recursive formulas for reproducing kernels and christoffel darboux formulas are obtained in this paper the scalar case is considered the matrix case will be considered in a separate publication

mixed boundary value problems for elliptic and parabolic differential equations of second order on considere les problemes aux valeurs limites mixtes suivants lu f 1 dans mu f 2 sur u f 3 sur avec l operateur elliptique et pu f 1 dans 0 t mu f 2 sur 0 t u f 3 sur 0 t 0 pour des operateurs p paraboliques pour des operateurs obliques generaux m et pour des domaines ou et se rencontrent a des angles arbitraires on montre l existence de solutions classiques quand les angles sont non nuls et f 3 est continue de holder

localization for the anderson model on a strip with singular potentials abstract we prove exponential localization in the anderson model on a one dimensional strip for any potential whose probability distribution is not concentrated on a single point and possesses some finite moment in particular we obtain localization for bernoulli potentials on a strip

moment problems and orthogonal functions abstract the caratheodory coefficient problem for an infinite sequence c n can be formulated as follows find a caratheodory function f z i e an analytic function mapping the unit disk d z z into the right half plane p w re w 0 whose taylor coefficients at a fixed points in d for convenience the origin are the c n this problem is equivalent to the trigonometric moment problem find a measure on such that e i n d c n for n 0 1 2 these problems are closely related to the theory of szego polynomials i e orthogonal polynomials on the unit circle t z z 1 let n be an arbitrary given sequence of not necessarily distinct points in d and let w n be a given sequence of numbers the nevanlinna pick problem for this situation is to find a caratheodory function f z with the interpolation property f n w n for all n when points n are repeated the interpolation property involves the appropriate number of taylor coefficients also this problem is connected with the problem of finding a measure generating moments the nevanlinna pick problem is related to a theory of rational functions which are orthogonal on t with poles at the points 1 n this relationship is analogous to and generalizes the relationship between the caratheodory coefficient problem or trigonometric moment problem and polynomials orthogonal on t the situation when points n are given on the boundary t or d is also briefly discussed

global c regularity for second order quasilinear elliptic euqations in divergence form on considere des equations elliptiques quasilineaires d ordre 2 sous forme divergence sous des conditions aux limites de dirihlet

eigenvalues and eigenfunctions of metric measure manifolds in this paper we study the eigenvalues and eigenfunctions of metric measure manifolds we prove that any eigenfunction is c 1 alpha at its critical points and c infty elsewhere moreover the eigenfunction corresponding to the first eigenvalue in the dirichlet problem does not change sign we also discuss the first eigenvalue the sobolev constants and their relationship with the isoperimetric constants 2000 mathematics subject classification 47j05 47j10 53c60 58e05 58c40

on the deficiency indices and self adjointness of symmetric hamiltonian systems abstract the main purpose of this paper is to investigate the formal deficiency indices n of a symmetric first order system jf bf h f on an interval i where i r or i r here j b h are n n matrix valued functions and the hamiltonian h 0 may be singular even everywhere we obtain two results for such a system to have minimal numbers n 0 if i r resp n n if i r and a criterion for their maximality n 2n for i r as well as the quasi regularity this covers the kac krein and de branges trans amer math soc 99 1961 118 theorems on 2 2 canonical systems and some results from kogan and rofe beketov proc roy soc edinburgh sect a 74 1974 75 5 some conditions for a canonical system to have intermediate formal deficiency indices are presented too we also obtain a generalization of the well known titchmarsh sears theorem for second order sturm liouville type equations this contains results due to lidskii and krein as special cases we present two approaches to the above problems one dealing with formal deficiency indices and one dealing with ordinary deficiency indices our main non formal approach is based on the investigation of a symmetric linear relation smin which is naturally associated to a first order system this approach works in the framework of extension theory and therefore we investigate in detail the domain d s min of s min in particular we prove the so called regularity theorem for d s min as a byproduct of the regularity result we obtain very short proofs of generalizations of the main results of the paper by kogan and rofe beketov 1974 75

on the convergence of rational functions orthogonal on the unit circle rational functions orthogonal on the unit circle with prescribed poles are studied we establish some convergence theorems for the orthogonal rational functions we generalize some of the results for szego polynomials to rational orthogonal functions

positive solutions of the dirichlet problem for the prescribed mean curvature equation abstract we discuss existence and multiplicity of positive solutions of the prescribed mean curvature problem div u 1 u 2 f x u in u 0 on in a general bounded domain r n depending on the behavior at zero or at infinity of f x s or of its potential f x s 0 s f x t d t our main effort here is to describe in a way as exhaustive as possible all configurations of the limits of f x s s 2 at zero and of f x s s at infinity which yield the existence of one two three or infinitely many positive solutions either strong or weak or bounded variation solutions are considered our approach is variational and combines critical point theory the lower and upper solutions method and elliptic regularization

szego functions and multipoint pade approximation abstract rational functions analogous to szego polynomials are introduced and their application in multipoint pade approximation is discussed

generalized szego theory in frequency analysis abstract a sequence of positive borel measures with infinite support on the unit circle is given converging in the weak star topology to a measure with support consisting of a finite number n 0 of mass points the sequences of monic rational functions orthogonal with respect to inner products determined by the measures are studied with the focus being on the asymptotic behavior of the zeros the main result is twofold for a fixed n n 0 n 0 of the zeros interesting zeros tend to the mass points of the limiting measure while the remaining n n 0 zeros uninteresting zeros may not converge at all however by considering subsequences limits may be obtained and the remaining zeros and their limits will be located in a closed disk z r

orthogonal rational functions with poles on the unit circle let n be a sequence of not necessarily distinct points on the unit circle t z c z 1 set ln span 1 1 1 1 n l n 0ln where we have used the notation n nk 1 z k let m be a positive linear functional defined on the space l l with m r real for functions that are real on t define r s m r z s 1 z for r s l in particular if m is given as m r r ei d for some measure then r s r ei s ei d let the orthogonal system n be obtained from 1 n by orthogonalization three term recurrence relations quadrature formulas moment theory and interpolation properties connected with the functional m and the system n are discussed

regularity of a minimal surface at its free boundary in this paper we shall consider a stationary minimal surface x whose free or at least partially free boundary lies on a given supporting surface s it is well known that if the supporting surface s has no boundary then x possesses the same smoothness as s does if on the other hand the boundary of s is nonvoid one can at most expect c t 1 2 regularity of x at its free boundary c f 1 5 6 this c 1 1 2 boundary regularity was proved in the paper 6 by s hildebrandt and j c c nitsche and in the paper 5 by m griiter s hildebrandt and j c c nitsche under the condition that the supporting surface is of class c 3 for some ae 0 1 now combining the ideas of d kinderlehrer 7 and m giaquinta e giusti 4 and modifying the argument in 6 concerning the conformality relation we are able to show the c 1 1 2 regularity under a c2 assumption which in turn seems to be the weakest even if the minimal surface under consideration are minima of the area we also treat the regularity problem of general stationary surfaces under the c 1 1 condition and of minima of the dirichlet integral under a c 1 condition thus without conformality assumption

differentiability of implicit functions beyond the implicit function theorem abstract the implicit function theorem ift can be used to deduce the differentiability of an implicit mapping s u y given by the equation e y u 0 however the ift is not applicable when different norms are necessary for the differentiation of e w r t y and the invertibility of the partial derivative e y y u we prove theorems ensuring the twice differentiability of the mapping s which can be applied in this case we highlight the application of our results to quasilinear partial differential equations whose principal part depends nonlinearly on the gradient of the state y

random dirac operators with time reversal symmetry quasi one dimensional stochastic dirac operators with an odd number of channels time reversal symmetry but otherwise efficiently coupled randomness are shown to have one conducting channel and absolutely continuous spectrum of multiplicity two this follows by adapting the criteria of guivarch raugi and goldsheid margulis to the analysis of random products of matrices in the group so 2l and then a version of kotani theory for these operators absence of singular spectrum can be shown by adapting an argument of jaksic last if the potential contains random dirac peaks with absolutely continuous distribution

convergence of orthogonal rational functions outside of the unit circle rational functions orthogonal on the unit circle with prescribed poles lying outside the unit circle are studied we study the asymptotic behaviors for the orthogonal rational functions outside of the unit circle

delocalization in random polymer models a random polymer model is a one dimensional jacobi matrix randomly composed of two finite building blocks if the two associated transfer matrices commute the corresponding energy is called critical such critical energies appear in physical models an example being the widely studied random dimer model it is proven that the lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport showing that it is almost surely overdiffusive even though the models are known to have pure point spectrum with exponentially localized eigenstates for almost every configuration of the polymers furthermore the level spacing is shown to be regular at the critical energy

the random phase property and the lyapunov spectrum for disordered multi channel systems a random phase property establishing in the weak coupling limit a link between quasi one dimensional random schrodinger operators and full random matrix theory is advocated briefly summarized it states that the random transfer matrices placed into a normal system of coordinates act on the isotropic frames and lead to a markov process with a unique invariant measure which is of geometric nature on the elliptic part of the transfer matrices this measure is invariant under the unitaries in the hermitian symplectic group of the universality class under study while the random phase property can up to now only be proved in special models or in a restricted sense we provide strong numerical evidence that it holds in the anderson model of localization a main outcome of the random phase property is a perturbative calculation of the lyapunov exponents which shows that the lyapunov spectrum is equidistant and that the localization lengths for large systems in the unitary orthogonal and symplectic ensemble differ by a factor 2 each in an anderson ando model on a tubular geometry with magnetic field and spin orbit coupling the normal system of coordinates is calculated and this is used to derive explicit energy dependent formulas for the lyapunov spectrum

foreword to the proceedings of the orthoquad 2014 conference orthoquad 2014 was an international symposium on orthogonality quadrature and related topics held on january 20 24 2014 in puerto de la cruz tenerife spain it was held in memory of professor pablo gonzalez vera 1955 2012

the conormal derivative problem for elliptic equations of variational type on considere des solutions du probleme qu 0 dans nu 0 sur ou est un domaine borne dans r n et q et n sont definis par qu x div a x u x du x b x u x du x x nu x a x u x du x x x x

rates of convergence of multipoint rational approximants and quadrature formulas on the unit circle abstract in this paper multipoint rational approximants to the riesz herglotz transform of a borel measure supported on are considered we give estimates for the rate of convergence of these approximants these estimates are obtained from the asymptotic behaviour of a sequence of rational functions orthogonal with respect to and having prescribed poles 1 k where k 1 is a sequence of complex numbers compactly contained in the open unit disk as an application we also give the rate of convergence of rational szego quadrature formulas for integrals with respect to

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

stochastic schrodinger operators and jacobi matrices on the strip we discuss stochastic schrodinger operators and jacobi matrices with wave functions taking values in l so there are 2l lyaponov exponents 1 l 0 l 1 2l 1 our results include the fact that if 1 0 on a set positive measure thenv is deterministic and one that says that e exactly 2j s are zero is the essential support of the a c spectrum of multiplicity 2j

a density problem for orthogonal rational functions abstract let n n 1 be a sequence of points in the open unit disk in the complex plane and let b 0 1 and b n z k 0 n k k k z 1 k z n 1 2 k k 1 when k 0 we put l span b n n 0 1 2 and we consider the following moment problem given a positive definite hermitian inner product on l l find a nondecreasing function on or a positive borel measure on such that f g f e i g e i d for f g l we give a necessary and sufficient condition called n extremality on a solution of the moment problem in order that l is dense in l2

best possible estimates of solutions to the robin boundary value problem for linear elliptic nondivergence second order equations in a neighborhood of the conical point abstract we investigate the behavior of strong solutions to the robin boundary value problem for linear elliptic nondivergence second order equations in a neighborhood of the boundary conical point we establish precise exponent of the solution decreasing rate

l p estimates for parabolic equations in reifenberg domains abstract the inhomogeneous neumann problem for parabolic equations in divergence form is studied an optimal regularity requirement on the domain for the l p theory is investigated assuming that the principal coefficients are supposed to be in the john nirenberg space with small bmo semi norms and that the domain is a reifenberg flat domain

elliptic equations in divergence form with partially bmo coefficients the solvability in sobolev spaces is proved for divergence form second order elliptic equations in the whole space a half space and a bounded lipschitz domain for equations in the whole space or a half space the leading coefficients aij are assumed to be only measurable in one direction and have locally small bmo semi norms in the other directions for equations in a bounded domain additionally we assume that aij have small bmo semi norms in a neighborhood of the boundary of the domain we give a unified approach of both the dirichlet boundary problem and the conormal derivative problem we also investigate elliptic equations in sobolev spaces with mixed norms under the same assumptions on the coefficients

local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations we consider solutions and subsolutions and supersolutions of the boundary value problem au x u du diju u x u du 0 in 2 s x diu y x u g x on3fi for a lipschitz domain s a positive definite matrix valued function u j and a vector field s which points uniformly into q without making any continuity assumptions on the known functions we prove harnack and holder estimates for u near 3i2 in addition we bound the l norm of u near 3i2 in terms of an appropriate lp norm and the known functions our approach is based on that for the corresponding interior estimates of trudinger this paper is concerned with analogs for solutions of oblique boundary value problems of the results in 10 our results will extend previous work of lieberman and trudinger 5 and of nadirashvili 9 specifically we consider second order differential operators q of the form

error estimates for optimal control problems of a class of quasilinear equations arising in variable viscosity fluid flow we consider optimal control problems of quasilinear elliptic equations with gradient coefficients arising in variable viscosity fluid flow the state equation is monotone and the controls are of distributed type we prove that the control to state operator is twice frechet differentiable for this class of equations a finite element approximation is studied and an estimate of optimal order h is obtained for the control the result makes use of the distributed structure of the controls together with a regularity estimate for elliptic equations with holder coefficients and a second order sufficient optimality condition the paper ends with a numerical experiment where the approximation order is computationally tested

multiplicity bound of singular spectrum for higher rank anderson models abstract in this work we prove a bound on the multiplicity of the singular spectrum for a certain class of anderson hamiltonians the operator in consideration is the form h n z d n p n on the hilbert space l 2 z d where is discrete laplacian p n are projection onto l 2 x z d n i l i x i n i 1 l i for some l 1 l d n and n n are i i d real bounded random variables following an absolutely continuous distribution we prove that the multiplicity of the singular spectrum is bounded above by 2 d d independent of l i i 1 d when l i 1 2 n 3 n for all i and g c d l i 1 l j 1 1 for i j the singular spectrum is also simple

spectral structure of anderson type hamiltonians we study self adjoint operators of the form h h 0 n n where the n s are a family of orthonormal vectors and the n s are independently distributed random variables with absolutely continuous probability distributions we prove a general structural theorem saying that for each pair n m if the cyclic subspaces corresponding to the vectors n and m are not completely orthogonal then the restrictions of h to these subspaces are unitarily equivalent with probability one this has some consequences for the spectral theory of such operators in particular we show that well behaved absolutely continuous spectrum of anderson type hamiltonians must be pure and use this to prove the purity of absolutely continuous spectrum in some concrete cases

a central limit theorem for products of random matrices and goe statistics for the anderson model on long boxes we consider products of random matrices that are small independent identically distributed perturbations of a fixed matrix mathcal t 0 focusing on the eigenvalues of mathcal t 0 of a particular size we obtain a limit to a sde in a critical scaling previous results required mathcal t 0 to be a conjugated unitary matrix so it could not have eigenvalues of different modulus from the result we can also obtain a limit sde for the markov process given by the action of the random products on the flag manifold applying the result to random schrodinger operators we can improve some results by valko and virag showing goe statistics for the rescaled eigenvalue process of a sequence of anderson models on long boxes in particular we solve a problem posed in their work

strong and weak convergence of orthogonal systems of rational functions on the unit circle abstract pan k strong and weak convergence of orthogonal systems of rational functions on the unit circle journal of computational and applied mathematics 46 1993 427 436 we study weak and strong convergence of rational functions that are orthonormal on the unit circle and have all their poles in a given sequence of points outside the unit disk the erd s measures have been characterized as well

asymptotic properties of zeros of orthogonal rational functions abstract a sequence x m m 0 1 2 of real numbers gives rise to an absolutely continuous measure n t on the unit circle given by d t n d 12 n 1 m 0 x m t m e im 2 where n is a natural number t 0 1 each measure n t determines a sequence n n t of monic orthogonal rational functions with prescribed poles outside the closed unit disk in particular these rational functions may be polynomials a measure 0 with support consisting of the points 1 n0 on the unit circle is given let n n0 under a suitable condition on weak convergence of the measures and with a proper ordering of the zeros zk n n t of n n t it can be shown that lim t 1 lim t z k n n t k k 1 n 0 and that there exists a positive number n z k n n t n for k n0 1 n for all n and all t 0 1 the theory is applied to the problem of determining unknown frequencies in a trigonometric signal x m ij iajeimwj m 0 1 2

regularity of solutions of nonlinear degenerate parabolic systems soit le systeme parabolique u i t div u p 2 u i 0 i m avec p 1 ou u i u i x t grad x et x r n ouvert pour toute solution de ce systeme avec max 1 2n n 2 p u est continu dans t

the conormal derivative problem for equations of variational type in nonsmooth domains it is well known that elliptic boundary value problems in smooth domains have smooth solutions but if the domain is say c 1 the solutions need not be lipschitz recently korevaar has identified a class of lipschitz domains in which solutions of the capillary problem are lipschitz assuming the contact angle relates correctly to the geometry of the domain lipschitz bounds for more general boundary value problems in the same class of domains are proved applications to variational inequalities are also considered

extensions of szego s theory of rational functions orthogonal on the unit circle rational functions orthogonal on the unit circle with prescribed poles lying outside the unit circle are studied we study the asymptotic behaviors for the orthogonal rational functions when the measure does not satisfy the szego condition

boundary regularity for linear and quasilinear variational inequalities on montre a l aide d une extension d une methode de jensen que les derivees secondes des solutions de divers problemes d obstacles lineaires sont bornees sous des hypotheses de regularite des donnees plus faibles que celles de jensen

gradient estimates for singular fully nonlinear elliptic equations abstract we study a class of singular fully nonlinear elliptic equations under suitable natural conditions with the model equation being u b d u d u c d u 2 f for lipschitz functions b 0 c and f where d denotes distance to the boundary of the domain and is a suitable extension of the interior unit normal we show that such equations have a unique globally c 1 solution without any a priori prescription of boundary data