eigenvalues of elliptic boundary value problems with an indefinite weight function on etudie les problemes aux valeurs propres elliptiques autoadjoints au r x u dans un ensemble ouvert r k a est un operateur positif d ordre 2m et le poids r est une fonction qui change de signe dans et peut etre discontinue on determine le comportement asymptotique des valeurs propres

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

eigencurves for two parameter sturm liouville equations this paper concerns two parameter sturm liouville problems of the form p x y q x y lambda r x mu y quad a leqslant x leqslant b with self adjoint boundary conditions at a and b the set of lambda mu in bf r 2 for which there exists a nontrivial y satisfying the differential equation and the boundary conditions turns out to be a countable union of graphs of analytic functions our focus is on these graphs which are termed eigencurves in the literature although eigencurves have been used in a variety of ways for about a century they seem comparatively underdeveloped in their own right our plan is to give motivation for the topic elementary properties of eigencurves illustrations on a simple example first studied by richardson in 1918 and since then by several authors and some natural questions which may whet the reader s appetite some of these questions lead to new types of inverse sturm liouville problems

schauder bases in quotients of subspaces of l 2 x introduction questions and problems related to schauder basis are among the most classical ones in the banach space theory and have been studied from the earliest days the fact that every infinite dimensional banach space has an infinite dimensional subspace with a basis has been observed already in banach s book ba at about the same time the existence of bases in classical banach spaces such as c 0 1 and lp 0 1 was established an importance of the concept of a basis lies in the fact that it provides a natural method of approximation of vectors and operators in the space a question related to the approximation property in banach spaces was asked by mazur in the scottish book scb this question and the problem of the existence of a basis in every banach space stayed open for about 40 years although not because of the lack of interest cf e g sin in 1972 p enflo constructed a banach space which fails the approximation property and hence does not admit a schauder basis his discovery triggered a flurry of activities resulting in constructions of subspaces failing the approximation property in all classical banach spaces not isomorphic to a hilbert space such subspaces have been also found in all banach spaces sufficiently far from the hilbert space cf e g l t l l t 2 and references therein techniques developed until that moment did not provide quantitative finitedimensional estimates for basis constants it took another 10 years before a new probabilistic argument was introduced by e d gluskin g l and consequently the existence of n dimensional banach spaces with basis constant tending to infinity was proved independently by gluskin g 2 and s j szarek sz l at last a circle got closed a method of constructing infinite dimensional examples out of finite dimensional ones was proposed by j bourgain b l and developed by szarek sz 3 to obtain an 2 sum of finite dimensional banach spaces which has no basis but obviously admits a so called finite dimensional decomposition in this paper we prove a striking isomorphic characterization of a hilbert space in terms of a schauder basis first observe that the class of banach spaces isomorphic to a hilbert space is closed under operations of passing to closed

an exotic quasidiagonal operator let h be a separable infinite dimensional hilbert space we construct a block diagonal hence quasidiagonal operator t on h i e t tn for some tn s called blocks acting on finite dimensional mutually orthogonal subspaces of h which is not a norm limit of a sequence of algebraically m normal operators or equivalently block diagonal operators each of which has uniformly bounded sizes of blocks the construction is based on random techniques

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

aspects of nonnormality for iterative methods abstract recently new optimal krylov subspace methods have been discovered for normal matrices in light of this novel ways to quantify nonnormality are considered in connection with various families of matrices we use as a criterion how for a given matrix these iterative methods introduced can be employed via e g inexpensive matrix factorizations the unitary orbit of the set of binormal matrices provides a natural extension of normal matrices its elements yield polynomially normal matrices of moderate degree in this context several matrix nearness problems arise

graphical krein signature theory and evans krein functions two concepts evidently very different in nature have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory i the krein signature of an eigenvalue a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half plane leading to dynamical instability under perturbation of the system and ii the evans function an analytic function detecting the location of eigenvalues one might expect these two concepts to be related but unfortunately examples demonstrate that there is no way in general to deduce the krein signature of an eigenvalue from the evans function for example by studying derivatives of the latter the purpose of this paper is to recall and popularize a simple graphical interpretation of the krein signature well known in the spectral theory of polynomial operator pencils once esta

left definite sturm liouville problems abstract left definite regular self adjoint sturm liouville problems with either separated or coupled boundary conditions are studied we give an elementary proof of the existence of eigenvalues for these problems for any fixed equation we establish a sequence of inequalities among the eigenvalues for different boundary conditions and estimate the range of each eigenvalue as a function on the space of boundary conditions some of our results here yield an algorithm for numerically computing the eigenvalues of a left definite problem with an arbitrary coupled boundary condition our inequalities imply that the well known asymptotic formula for the eigenvalues in the separated case also holds in the coupled case moreover we study the continuous and differentiable dependence of the eigenvalues of the general left definite problem on all the parameters in its differential equation and boundary condition

entropy numbers of embeddings of schatten classes abstract let 0 p q and denote by s p n and s q n the corresponding finite dimensional schatten classes we prove optimal bounds up to constants only depending on p and q for the entropy numbers of natural embeddings between s p n and s q n this complements the known results in the classical setting of natural embeddings between finite dimensional l p spaces due to schutt edmunds triebel triebel and guedon litvak kuhn we present a rather short proof that uses all the known techniques as well as a constructive proof of the upper bound in the range n n n 2 that allows deeper structural insight and is therefore interesting in its own right our main result can also be used to provide an alternative proof of recent lower bounds in the area of low rank matrix recovery

asymptotics of eigencurves for second order ordinary differential equations i abstract the regular two parameter sturm liouville equation py qy f r y is studied for l1 coefficients with p r 0 for each fixed number n of internal zeros of the eigenfunctions y n is analytic in necessary and sufficient conditions which are in fact independent of n are given for lim n gl to exist as or asymptotic expansions for n are derived in cases of existence and non existence of lim n

inverse spectral problems for left definite sturm liouville equations with indefinite weight abstract inverse sturm liouville problems with indefinite weight are considered theorems analogous to those of both hochstadt and gelfand levitan are proved as a sample result under suitable symmetry conditions we show that half of one spectrum suffices to determine the potential function uniquely

on the existence and uniqueness of complex structure and spaces with few operators we construct a 2 n dimensional real normed space whose banach mazur distance to the set of spaces admitting complex structure is of order nl 2 and two complex n dimensional normed spaces which are isometric as real spaces but whose complex banach masur distance is of order n both orders of magnitude are the largest possible we also construct finite dimensional spaces with the property that all well bounded operators on them are rather small in the sense of some ideal norm perturbations of multiples of identity we also state some metatheorem which can be used to produce spaces with various pathological properties 1 introduction and the main results the main purpose of this paper is to prove the following results theorem 1 1 given n there exists a 2n dimensional real normed space x such that whenever y is an n dimensional complex normed space and yr is y treated as a real space then the banach mazur distance d x yr c4 where c is a numerical constant given a complex normed space y denote by y the space obtained from y by replacing the multiplication by scalar x y xy by k y a y xy and preserving the rest of the structure addition norm we then have theorem 1 2 given n there ejcists an n dimensional space y over c such that d y y cn where c is a numerical constant here d denotes distance of complex spaces the orders of magnitude o 4 and o n in theorems 1 1 and 1 2 are of course the largest possible in the first case the n dimensional complex hilbert space always gives the upper estimate in the second d yl y2 n if dimyl dim y2 n in both the real and complex case a weaker version of theorem 1 1 with estimate o 5 n log n was shown recently by p mankiewicz 8 a version of theorem 1 2 with estimate o n log n was shown independently of this paper and approximately at the same time by j bourgain 1 who also settled the infinite dimensional problem constructing a complex banach space y such that y and y are not isomorphic with all the consequences see corollary 1 3 below similar problems were considered recently by n j kalton 5 who constructed an analogous example received by the editors november 1 1984 and in revised form march 4 1985 1980 mathematics subject classification primary 46b20 51a99 secondary 46l99 47b10 47d15 52a22 52a40 60e15 supported in part by nsf dms 8401906 c 1986 american mathematical society 0002 9947 86 1 00 25 per page

computation of eigenvalues and solutions of regular sturm liouville problems using haar wavelets the paper presents a novel method for the computation of eigenvalues and solutions of sturm liouville eigenvalue problems sleps using truncated haar wavelet series this is an extension of the technique proposed by hsiao to solve discretized version of variational problems via haar wavelets the proposed method aims to cover a wider class of problems by applying it to historically important and a very useful class of boundary value problems thereby enhancing its applicability to demonstrate the effectiveness and efficiency of the method various celebrated sturm liouville problems are analyzed for their eigenvalues and solutions also eigensystems are investigated for their asymptotic and oscillatory behavior the proposed scheme unlike the conventional numerical schemes such as rayleigh quotient and rayleigh ritz approximation gives eigenpairs simultaneously and provides upper and lower estimates of the smallest eigenvalue and it is found to have quadratic convergence with increase in resolution

poincare inequality in mean value for gaussian polytopes let k n g 1 g n be the absolute convex hull of n independent standard gaussian random points in mathbb r n with n n we prove that for any 1 lipschitz function f mathbb r n rightarrow mathbb r the polytope k n satisfies the following poincare inequality in mean value mathbb e omega int limits k n omega left f x frac 1 textup vol n left k n omega right int limits k n omega f y dy right 2 dx leq frac c n mathbb e omega int limits k n omega x 2dx where c 0 is an absolute constant this poincare inequality is the one suggested by a conjecture of kannan lovasz and simonovits for general convex bodies moreover we prove in mean value that the volume of the polytope k n is concentrated in a subexponential way within a thin euclidean shell with the optimal dependence of the dimension n an important tool of the proofs is a representation of the law of g 1 g n conditioned by the event that the convex hull of g 1 g n is a n 1 face of k n as an application we also get an estimate of the number of n 1 faces of the polytope k n valid for every n n

on the hasse principle for quartic hypersurfaces we establish the hasse principle for smooth projective quartic hypersurfaces of dimension greater than or equal to 28 defined over q

a new form of the circle method and its application to quadratic forms if the coefficients r n satisfy suitable arithmetic conditions the behaviour of f will be determined by an appropriate rational approximation a q to with small values of q usually producing large values of f when lies in an interval a q a q with q small a major arc one hopes to estimate f asymptotically while if the corresponding q is large for the minor arcs one hopes that f will be small at least on average one can distinguish two different types of application of the circle method the first type following the above pattern uses one of a variety of mean value techniques hua s inequality for example to give an upper bound for the minor arc contribution results on waring s problem are generally proved this way for example the second type of application on which we shall be focusing our attention avoids the minor arcs completely and treats f asymptotically for all here one cannot take advantage of the mean value ideas used in the first type of problem the savings come instead from the availability of good estimates for complete exponential sums kloosterman s work 12 on quaternary quadratic forms and the papers of the author 5 and hooley 8 and 9 on cubic forms are of this latter type we should point out that such applications do not include any problems in which f is the generating function for a polynomial of degree 4 or more there are

on the geometry of random convex sets between polytopes and zonotopes abstract in this work we study a class of random convex sets that interpolate between polytopes and zonotopes these sets arise from considering a q t h moment q 1 of an average of order statistics of 1 dimensional marginals of a sequence of n n independent random vectors in r n we consider the random model of isotropic log concave distributions as well as the uniform distribution on an l p n sphere 1 p with respect to the cone probability measure and study the geometry of these sets in terms of the support function and mean width we provide asymptotic formulas for the expectation of these geometric functionals which are sharp up to absolute constants our model includes and generalizes the standard one for random polytopes

quadratic growth of convergence radii for eigenvalues of two parameter sturm liouville equations abstract the n th eigenvalue of the equation y r x y 0 a x b subject to self adjoint boundary conditions admits a power series expansion into powers of for sufficiently small it is proved that the convergence radii of these series grow like n 2 as n tends to infinity provided r x is analytic on a b applications to the airy and mathieu equation are given

a krein space approach to symmetric ordinary differential operators with an indefinite weight function on considere les proprietes spectrales du probleme differentiel l f 1 n p 0 f n n 1 n 1 p 1 f n 1 n 1 p n f rf sur un intervalle fini ou infini a b avec des coefficients reels localement sommables 1 p 0 p 1 p n r sous l hypothese que p 0 0 et que la fonction poids r change de signe sur a b

eigencurves of non definite sturm liouville problems for the p laplacian abstract the eigencurves are studied for a weighted quasi linear sturm liouville type problem of the form p y p 1 r q s sgn y y p 1 on 0 1 with sturmian type boundary conditions p being the p laplacian topics include holder classes c k and asymptotics including existence of oblique asymptotes an example illustrates precision of our holder exponents when p 2

counting rational points on algebraic varieties for any n 2 let z pn be a geometrically integral algebraic variety of degree d this article is concerned with the number nz b of q rational points on z which have height at most b for any e 0 we establish the estimate nz b od e n bdimz e provided that d e as indicated the implied constant depends at most on d e and n

singular left definite sturm liouville problems abstract we study singular left definite sturm liouville problems with an indefinite weight function the existence of eigenvalues is established based on the existence of eigenvalues of corresponding right definite problems furthermore for each singular left definite problem with limit circle non oscillatory endpoints we construct a regular left definite problem with the same eigenvalues and use it to obtain properties of eigenvalues and eigenfunctions inequalities among eigenvalues recently established for regular left definite problems are extended to the singular case

restricted strong convexity and weighted matrix completion optimal bounds with noise we consider the matrix completion problem under a form of row column weighted entrywise sampling including the case of uniformentrywise sampling as a special case we analyze the associated random observation operator and prove that with high probability it satisfies a form of restricted strong convexity with respect to weighted frobenius norm using this property we obtain as corollaries a number of error bounds on matrix completion in the weighted frobenius norm under noisy sampling and for both exact and near low rank matrices our results are based on measures of the spikiness and low rankness of matrices that are less restrictive than the incoherence conditions imposed in previous work our technique involves an m estimator that includes controls on both the rank and spikiness of the solution and we establish non asymptotic error bounds in weighted frobenius norm for recovering matrices lying with lq balls of bounded spikiness using information theoretic methods we show that no algorithm can achieve better estimates up to a logarithmic factor over these same sets showing that our conditions on matrices and associated rates are essentially optimal

spectral inversion of an indefinite sturm liouville problem due to richardson we study an indefinite sturm liouville problem due to richardson whose complicated eigenvalue dependence on a parameter has been a puzzle for decades in atomic physics a process exists that inverts the usual schr dinger situation of an energy eigenvalue depending on a coupling parameter into the so called sturmian problem where the coupling parameter becomes the eigenvalue which then depends on the energy we observe that the richardson equation is of the sturmian type this means that the richardson and its related schr dinger eigenvalue functions are inverses of each other and that the richardson spectrum is therefore no longer a puzzle

cubic hypersurfaces and a version of the circle method for number fields a version of the hardy littlewood circle method is developed for number fields k q and is used to show that non singular projective cubic hypersurfaces over k always have a k rational point when they have dimension at least 8

oscillation theory for sturm liouville problems with indefinite coefficients oscillation and related results are given for the problem under separated end conditions assuming 1 p q and r l 1 i attention focuses on the two cases i p 0 with r indefinite and ii p indefinite with r 0

rational points on quartic hypersurfaces let x be a projective non singular quartic hypersurface of dimension 39 or more which is defined over we show that x is non empty provided that x is non empty and x has p adic points for every prime p 2009 walter de gruyter

existence and asymptotics of eigenvalues of indefinite systems of sturm liouville and dirac type abstract existence and asymptotic behavior of eigenvalues of indefinite sturm liouville and dirac systems with integrable coefficients are investigated using prufer angle analysis the results generalize those previously obtained by atkinson mingarelli gohberg and krein and are based on a new theorem that determines the asymptotic behavior of the solution of a riccati type equation containing a large parameter

spectral analysis of singular ordinary differential operators with indefinite weights abstract in this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions we prove a general perturbation result on the local spectral properties of selfadjoint operators in krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares this result is applied to singular indefinite sturm liouville operators and higher order singular ordinary differential operators with indefinite weight functions

multiparameter sturm liouville problems with eigenparameter dependent boundary conditions abstract a system of ordinary differential equations y j q j y j k 1 n k r jk y j j 1 n 0 1 with real valued and continuous coefficient functions q j r jk is studied on 0 1 subject to boundary conditions y j 0 y j 0 cot j and b j y j 1 d j y j 1 e t j c j y j 1 a j y j 1 0 2 for j 1 n here e t e 1 e 2 e n is an arbitrary n n matrix of real numbers and j a j d j b j c j 0 a point 1 n t c n satisfying 0 1 and 0 2 is called an eigenvalue of the system results are given on the existence and location of the eigenvalues and completeness and oscillation of the eigenfunctions

on the existence of eigenvalues of the schrodinger operatorh λw in a gap of σ h the authors prove a number of results asserting the existence of eigenvalues of the schrodinger operatorh w in a gap of h whereh v andv andw are bounded the existence of these eigenvalues is an important element in the theory of the colour of crystals the basic theorems are proved in v stronger results forv 1 are also presented

spectral properties of two parameter eigenvalue problems ii we study the self adjoint eigenvalue problem w x 0 in hilbert space for one equation in two parameters here is bounded below with compact resolvent for each 1 2 we give necessary and sufficient conditions for the existence of so that holds with w 0 and we investigate the geometry of the set z 0 of such we also discuss higher order solution sets z i where the ith eigenvalue of w vanishes deriving various asymptotic results in a unified fashion

non real eigenvalues of indefinite sturm liouville problems abstract the present paper deals with non real eigenvalues of regular indefinite sturm liouville problems a priori bounds and sufficient conditions of the existence for non real eigenvalues are obtained under mild integrable conditions of coefficients

on coupled multiparameter nonlinear elliptic systems on considere le systeme de problemes non lineaires aux valeurs limites de dirichlet lu x f u x v x lv x g u x v x x domaine borne de r n l est un operateur aux derivees partielles lineaire fortement et uniformement elliptique et sont des parametres reels f g r 2 r des applications continues telles que f 0 0 0 g 0 0 on presente une analyse detaillee de l ensemble des solutions du point de vue de la theorie de la bifurcation a plusieurs parametres

lagrange s four squares theorem with one prime and threealmost prime variables it is conjectured that every sufficiently large integer n equiv 4 pmod 24 should be a sum of the squares of 4 primes the best approximation to this in the literature is the result of brudern and fouvry j reine angew math 454 1994 59 96 who showed that every sufficiently large integer n equiv 4 pmod 24 is a sum of the squares of 4 almost primes each of which has at most 34 prime factors the present paper proves such a result with the square of one prime and 3 almost primes which in this case have at most 101 prime factors each the work of brudern and fouvry was based on kloosterman s approach to representations by quaternary forms but this does not lend itself to situations in which one of the variables is restricted to be a prime instead the present paper works with an almost all result for the representation of numbers m as sums of 3 squares to use this approach one has to take m of the form n p 2 and such numbers are too sparse for the standard theory it is therefore necessary to use an amplification procedure which emphasizes those integers m for which n m is a square all this machinery is coupled with kloosterman s version of the circle method there are considerable technical complications in which bounds for the kloosterman sum play a key role at one point in the argument a saving has to be extracted from a non trivial averaging over the denominators of the farey arcs this is an instance of the second kloosterman refinement