canonical p dimension of algebraic groups we describe a way to compute the p relative version of the berhuy reichstein canonical dimension for an arbitrary split semisimple algebraic group over an arbitrary field of an arbitrary characteristic p is any prime integer the canonical p dimension is computed for all split simple groups of classical types

lieb thirring type inequalities and gagliardo nirenberg inequalities for systems this paper is devoted to inequalities of lieb thirring type let v be a nonnegative potential such that the corresponding schrodinger operator has an unbounded sequence of eigenvalues i v i n we prove that there exists a positive constant c such that if d 2 then x i n i v c z rd v d 2 dx

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

lower bounds of eigenvalues for a class of bi subelliptic operators let be a bounded open domain in rnrn with smooth boundary and x x1 x2 xm x x1 x2 xm be a system of real smooth vector fields defined on with the boundary which is non characteristic for x if x satisfies the hormander s condition then the vector fields are finitely degenerate and the sum of square operators x i 1mxi2 is a subelliptic operator let k k be the k th eigenvalue for the bi subelliptic operator x2 on in this paper we introduce the generalized metivier s condition and study the lower bounds of dirichlet eigenvalues for the operator x2 on some finitely degenerate systems of vector fields x which satisfy the hormander s condition or the generalized metivier s condition by using the subelliptic estimates we shall give a explicit lower bound estimates of k k which is polynomial increasing in k with the order relating to the hormander index or the generalized metivier index

a lower bound for the number of negative eigenvalues of schrodinger operators we prove a lower bound for the number of negative eigenvalues for a schr o dinger operator on a riemannian manifold via the integral of the potential

new bounds on the lieb thirring constants improved estimates on the constants l d for 1 2 3 2 d n in the inequalities for the eigenvalue moments of schrodinger operators are established

symmetric operations in algebraic cobordism abstract in this article we describe certain new cohomological operations in algebraic cobordisms these operations give the natural obstructions for the cobordism class to be represented by the embedding also they permit to work with algebraic cobordisms and chow groups in a more subtle way than the landweber novikov operations related to 2 torsion effects we describe applications to the computation of the algebraic cobordisms of a pfister quadrics and to the problem of rationality of cycles

multidimensional lower bounds for the eigenvalues of stokes and dirichlet laplacian operators this article is concerned with the improvements of certain eigenvalue inequalities of stokes operator and dirichlet laplacian related to the berezin li yau type inequalities the formulas proved extend the earlier works of melas a lower bound for sums of eigenvalues of the laplacian proc am math soc 131 2 631 636 2002 10 1090 s0002 9939 02 06834 x on dirichlet laplacian and of ilyin lower bounds for the spectrum of the laplace and stokes operators discrete contin dyn syst 28 1 131 146 2010 10 3934 dcds 2010 28 131 on stokes operator for any dimension d 2 and they are asymptotically sharp as are the earlier inequalities of berezin li yau melas and ilyin

morse index of some min max critical points i application to multiplicity results on donne un resultat general sur des bornes inferieures pour des indices de morse de points critiques obtenus par des principes de min max en combinant cette information avec une inegalite semiclassique on obtient des estimations pointues sur la croissance de certaines valeurs critiques a partir desquelles on deduit de nouveaux resultats de multiplicite pour des solutions d equations elliptiques semi lineaires d ordre 2

lieb thirring inequality with semiclassical constant and gradient error term abstract in 1975 lieb and thirring derived a semiclassical lower bound on the kinetic energy for fermions which agrees with the thomas fermi approximation up to a constant factor whenever the optimal constant in their bound coincides with the semiclassical one is a long standing open question we prove an improved bound with the semiclassical constant and a gradient error term which is of lower order

the eigenvalue problem and infinitely many sign changing solutions for an elliptic equation with critical hardy constant abstract by li yau s idea and a more precise hardy s inequality we obtain a new lower bound of eigenvalues for an elliptic equation with critical hardy constant then using the lower bounds we prove that there are infinitely many sign changing solutions for a class of elliptic equation with critical hardy constant which extends the results in schechter and zou 2005 9 to the critical hardy constant case

special correspondences and chow traces of landweber novikov operations in paper vi07 a vishik using the techniques of symmetric operations in algebraic cobordism proved that changing the base field by the function field of a smooth projective quadric doesn t change the property of being rational for cycles of small codimension this fact which he calls main tool lemma plays the crucial role in his construction of fields with u invariant 2 1 in the present paper we prove m t l for the class of varieties introduced by m rost in the context of the proof of bloch kato conjecture namely for varieties which possess a special correspondence see ro06 definition 5 1 as in the vishik s proof the main technical tools are algebraic cobordism of m levine and f morel lm07 generalized rost degree formula and divisibility of chow traces of certain landweber novikov operations therefore we always assume that our base field k has characteristic 0 we use the following notation by k we denote the algebraic closure of k and by xk the respective base change x k k of a variety x by k we denote the function field of x given a prime p by ch x we denote the chow ring of x modulo p torsion taken with z p coefficients i e ch x ch x p where ch x ch x p tors we say that a cycle from ch xk is defined over k if it belongs to the image of the restriction map ch x ch xk the following notion will be central in this paper

the thermodynamic limit of quantum coulomb systems part ii applications abstract in a previous paper c hainzl m lewin j p solovej the thermodynamic limit of quantum coulomb systems part i general theory adv math 2009 doi 10 1016 j aim 2008 12 010 this issue we have developed a general theory of thermodynamic limits we apply it here to three different coulomb quantum systems for which we prove the convergence of the free energy per unit volume the first system is the crystal for which the nuclei are classical particles arranged periodically in space and only the electrons are quantum particles we recover and generalize a previous result of fefferman in the second example both the nuclei and the electrons are quantum particles submitted to a periodic magnetic field we thereby extend a seminal result of lieb and lebowitz finally in our last example we take again classical nuclei but optimize their position to our knowledge such a system was never treated before the verification of the assumptions introduced in c hainzl m lewin j p solovej the thermodynamic limit of quantum coulomb systems part i general theory adv math 2009 doi 10 1016 j aim 2008 12 010 this issue uses several tools which have been introduced before in the study of large quantum systems in particular an electrostatic inequality of graf and schenker is one main ingredient of our new approach

convergence of finite element solutions of stochastic partial integro differential equations driven by white noise numerical approximation of a stochastic partial integro differential equation driven by a space time white noise is studied by truncating a series representation of the noise with finite element method for spatial discretization and convolution quadrature for time discretization sharp order convergence of the numerical solutions is proved up to a logarithmic factor numerical examples are provided to support the theoretical analysis

on lieb thirring inequalities for higher order operators with critical and subcritical powers let i h l v denote the negative eigenvalues of the operatorh l u l u v 0 x d onl 2 d we prove the two sided estimate open image in new window we discuss bounds on the riesz means open image in new window

a note on the fractal dimension of attractors of dissipative dynamical systems inotherwords thehausdorandfractaldimensionshavethesameupperbound turningtotheapplicationswerstnotethatinalltheexamplesofpartialdierentialequationsconsideredin 8 thefunctionf d isconcave secondly inthetwoparticularexamples areaction diusionsystemandtwo dimensionalnavier stokessystem weobtainf d explicitlyand consequently weobtainexplicitupperboundsofthefractaldimensionoftheattractorintermsofthephysicalparameters finally we observe that theorem 2 1 can be obtained by a careful analysis ofthecorrespondingresultsin 8 however weprovetheorem2 1generalizingtotheinnite dimensionalcasethemethodof 2 wherethekaplan yorkeestimateforthefractaldimensionofthelorenzattractorhasbeenobtained 2 main estimatesupposethatasemigroups

on some sharp spectral inequalities for schrodinger operators on semiaxis in this paper we obtain sharp lieb thirring inequalities for a schrodinger operator on semiaxis with a matrix potential and show how they can be used to other related problems among them are spectral inequalities on star graphs and spectral inequalities for schrodinger operators on half spaces with robin boundary conditions

le spectre de certaines mosaiques poissoniennes du plan et l enveloppe convexe du pont brownien the spectral empirical function of a homogeneous isotropic poisson mosaic process is a functional of the perimeter of the convex hull of planar brownian bridge some geometrical identities follow

gevrey regularity for navier stokes equations under lions boundary conditions abstract the navier stokes system is considered in a compact riemannian manifold gevrey class regularity is proven under lions boundary conditions in 2d for the rectangle cylinder and hemisphere and in 3d for the rectangle the cases of the 2d sphere and 2d and 3d torus are also revisited

estimates for eigenvalues on riemannian manifolds abstract in this paper we investigate eigenvalues of the dirichlet eigenvalue problem of laplacian on a bounded domain in an n dimensional complete riemannian manifold m when m is an n dimensional euclidean space r n the conjecture of polya is well known the kth eigenvalue k of the dirichlet eigenvalue problem of laplacian satisfies k 4 2 n vol 2 n k 2 n for k 1 2 li and yau p li s t yau on the schrodinger equation and the eigenvalue problem comm math phys 88 1983 309 318 cf lieb e lieb the number of bound states of one body schrodinger operators and the weyl problem in proc sympos pure math vol 36 1980 pp 241 252 have given a partial solution for the conjecture of polya that is they have proved 1 k i 1 k i n n 2 4 2 n vol 2 n k 2 n for k 1 2 which is sharp in the sense of average in this paper we consider a general setting for complete riemannian manifolds we establish an analog of the li and yau s inequality for eigenvalues of the dirichlet eigenvalue problem of laplacian on a bounded domain in a complete riemannian manifold furthermore we obtain a universal inequality for eigenvalues of the dirichlet eigenvalue problem of laplacian on a bounded domain in a hyperbolic space h n 1 from it we prove that when the bounded domain tends to h n 1 all eigenvalues tend to n 1 2 4

liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities abstract we extend our recent results on the classification of stable solutions of the equation u f u in r n where f 0 is a general convex non decreasing function

weyl s type estimates on the eigenvalues of critical schrodinger operators using improved hardy sobolev inequalities motivated by the work karachalios n i 2008 lett math phys 83 189 99 we present explicit asymptotic estimates on the eigenvalues of the critical schrodinger operator involving inverse square potential based on improved hardy sobolev type inequalities

improved regression calibration the likelihood for generalized linear models with covariate measurement error cannot in general be expressed in closed form which makes maximum likelihood estimation taxing a popular alternative is regression calibration which is computationally efficient at the cost of inconsistent estimation we propose an improved regression calibration approach a general pseudo maximum likelihood estimation method based on a conveniently decomposed form of the likelihood it is both consistent and computationally efficient and produces point estimates and estimated standard errors which are practically identical to those obtained by maximum likelihood simulations suggest that improved regression calibration which is easy to implement in standard software works well in a range of situations

on the bolza problem abstract we consider the following boundary value problem x v x 0 x 0 x 0 x t x 1 where t 0 x 0 x 1 r n are given we prove that this problem has infinitely many solutions provided v is even and superquadratic

multiplicity results for some perturbed elliptic problems in unbounded domains with non homogeneous boundary conditions abstract we study the following nonlinear elliptic problem u m u g x u f x in r n b r u h on b r u 0 as x with n 3 m r under suitable conditions on the radial functions g f and h multiplicity results are proved in the perturbed case f 0 and h 0 by using bolle s perturbation method and suitable growth estimates on min max critical levels

estimation by data augmentation in regression models with continuous and discrete covariates measured with error estimation methods are considered for regression models which have both misclassified discrete covariates and continuous covariates measured with error adjusted parameter estimates are obtained using the method of data augmentation where the true values of the covariates measured with error are regarded as missing data validation data on the covariates are assumed to be available the distinction between internal and external validation data is emphasized and its effects on the analysis are examined the method is illustrated with simulated data 1997 by john wiley sons ltd

dirichlet and neumann eigenvalue problems on domains in euclidean spaces abstract we obtain here some inequalities for the eigenvalues of dirichlet and neumann value problems for general classes of operators or system of operators acting in l 2 or l 2 c m r d d 1

steklov eigenvalues and quasiconformal maps of simply connected planar domains we investigate isoperimetric upper bounds for sums of consecutive steklov eigenvalues of planar domains the normalization involves the perimeter and scale invariant geometric factors which measure deviation of the domain from roundness we prove sharp upper bounds for both starlike and simply connected domains for a large collection of spectral functionals including partial sums of the zeta function and heat trace the proofs rely on a special class of quasiconformal mappings

bayesian analysis of misclassified binary data from a matched case control study with a validation sub study bayesian methods are proposed for analysing matched case control studies in which a binary exposure variable is sometimes measured with error but whose correct values have been validated for a random sample of the matched case control sets three models are considered model 1 makes few assumptions other than randomness and independence between matched sets while models 2 and 3 are logistic models with model 3 making additional distributional assumptions about the variation between matched sets with models 1 and 2 the data are examined in two stages the first stage analyses data from the validation sample and is easy to perform the second stage analyses the main body of data and requires mcmc methods all relevant information is transferred between the stages by using the posterior distributions from the first stage as the prior distributions for the second stage with model 3 a hierarchical structure is used to model the relationship between the exposure probabilities of the matched sets which gives the potential to extract more information from the data all the methods that are proposed are generalized to studies in which there is more than one control for each case the bayesian methods and a maximum likelihood method are applied to a data set for which the exposure of every patient was measured using both an imperfect measure that is subject to misclassification and a much better measure whose classifications may be treated as correct to test methods the latter information was suppressed for all but a random sample of matched sets

correcting for measurement error in binary and continuous variables using replicates measurement error in exposures and confounders leads to bias in regression coefficients it is possible to adjust for this bias if true values or independent replicates are observed on a subsample we extend a method suitable for quantitative variables to the situation where both binary and quantitative variables are present binary variables with independent replicates introduce two extra problems i the error is correlated with the true value and ii the measurement error probabilities are unidentified if only two replicates are available we show that under plausible assumptions adjustment for error in binary confounders does not need to address these problems the regression coefficient for a binary exposure is overadjusted if methods for continuous variables are used correct adjustment is possible either if three replicates are available or if further assumptions can be made otherwise bounds can be put on the correctly adjusted value and these bounds are reasonably close together if the exposure has prevalence near 0 5

bounds on eigenvalues of dirichlet laplacian in this paper we investigate an eigenvalue problem of dirichlet laplacian on a bounded domain in an n dimensional euclidean space rn if k 1 is the k 1 th eigenvalue of dirichlet laplacian on then we prove that for n 41 and k geq 41 lambda k 1 leq k frac2n lambda 1 and for any n and k lambda k 1 leq c 0 n k k frac2n lambda 1 with c 0 n k leq j 2 n 2 1 j 2 n 2 1 1 where jp k denotes the k th positive zero of the standard bessel function jp x of the first kind of order p from the asymptotic formula of weyl and the partial solution of the conjecture of polya we know that our estimates are optimal in the sense of order of k

wesd weighted spectral distance for measuring shape dissimilarity this paper presents a new distance for measuring shape dissimilarity between objects recent publications introduced the use of eigenvalues of the laplace operator as compact shape descriptors here we revisit the eigenvalues to define a proper distance called weighted spectral distance wesd for quantifying shape dissimilarity the definition of wesd is derived through analyzing the heat trace this analysis provides the proposed distance with an intuitive meaning and mathematically links it to the intrinsic geometry of objects we analyze the resulting distance definition present and prove its important theoretical properties some of these properties include 1 wesd is defined over the entire sequence of eigenvalues yet it is guaranteed to converge 2 it is a pseudometric 3 it is accurately approximated with a finite number of eigenvalues and 4 it can be mapped to the 0 1 interval last experiments conducted on synthetic and real objects are presented these experiments highlight the practical benefits of wesd for applications in vision and medical image analysis

global bounds of schrodinger heat kernels with negative potentials abstract we obtain global in time bounds for the heat kernel g of the schrodinger operator l v the potential v satisfies v x c d x b near infinity with b 0 the result can be described as follows suppose l is positive and b 2 then g g x t y 0 has a global upper bound which is a standard gaussian times a polynomial function of x y t however when l is not nonnegative and negative eigenvalues exist g g x t 0 0 is bounded below by a standard gaussian times ectwithc 0 in other words the growth rate of g x t 0 0 is comparable with the heat kernel of c regardless how fast v decays near infinity

sums of laplace eigenvalues rotations and tight frames in higher dimensions the sum of the first n 1 eigenvalues of the laplacian is shown to be maximal among simplexes for the regular simplex the regular tetrahedron in three dimensions maximal among parallelepipeds for the hypercube and maximal among ellipsoids for the ball provided the volume and moment of inertia of an auxiliary body are suitably normalized this result holds for dirichlet robin and neumann eigenvalues additionally the cubical torus is shown to be maximal among flat tori the proof involves euclidean tight frames generated by orbits of the rotation group of the extremal domain among general convex domains the ball is conjectured to maximize sums of neumann eigenvalues provided the volume and moment of inertia of the polar dual are suitably normalized

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

nonlocal operators of order near zero abstract we study dirichlet forms defined by nonintegrable levy kernels whose singularity at the origin can be weaker than that of any fractional laplacian we show some properties of the associated sobolev type spaces in a bounded domain such as symmetrization estimates hardy inequalities compact inclusion in l 2 or the inclusion in some lorentz space we then apply those properties to study the associated nonlocal operator l and the dirichlet and neumann problems related to the equations l u f x and l u f u in

refined eigenvalue bounds on the dirichlet fractional laplacian the purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the dirichlet fractional laplacian operator 2 restricted to a bounded domain d with d 2 1 2 and d 3 0 2 our main result yields a sharper lower bound in the sense of weyl asymptotics for the berezin li yau type inequality improving the previous result s yildirim yolcu and t yolcu discrete contin dyn syst 35 5 2209 2225 2015 furthermore we give a result improving the bounds for analogous elliptic operators y c kim e print arxiv 1406 5407

qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area we prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed riemannian manifolds in terms of their index and area restricting to the case where the hypersurface has dimension less than seven in particular we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index the total curvature along the sequence is quantised in terms of the total curvature of some limit surface plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in euclidean space all of which are finite thus we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary

effective limiting absorption principles and applications the limiting absorption principle asserts that if h is a suitable schrodinger operator and f lives in a suitable weighted l2 space namely h 0 1 2 sigma for some sigma 0 then the functions r lambda i varepsilon f h lambda i varepsilon 1 f converge in another weighted l2 space h 0 1 2 sigma to the unique solution u of the helmholtz equation h lambda u f which obeys the sommerfeld outgoing radiation condition in this paper we investigate more quantitative or effective versions of this principle for the schrodinger operator on asymptotically conic manifolds with short range potentials and in particular consider estimates of the form r lambda i varepsilon f h 0 1 2 sigma leq c lambda h f h 0 1 2 sigma we are particularly interested in the exact nature of the dependence of the constants c lambda h on both lambda and h it turns out that the answer to this question is quite subtle with distinctions being made between low energies lambda ll 1 medium energies lambda sim 1 and large energies lambda gg 1 and there is also a non trivial distinction between qualitative estimates on a single operator h possibly obeying some spectral condition such as non resonance or a geometric condition such as non trapping and quantitative estimates which hold uniformly for all operators h in a certain class using elementary methods integration by parts and ode techniques we give some sharp answers to these questions as applications of these estimates we present a global in time local smoothing estimate and pointwise decay estimates for the associated time dependent schrodinger equation as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation under some additional assumptions on the operator h

bounds on the number of bound states for potentials with critical decay at infinity let v be a potential whose negative part v decays like c x 2 at infinity if c is not too large then the schrodinger operator hv v on rn n 3 has only a finite number of bound states although the associated classical phase space volume is infinite optimal conditions are derived for the absence of bound states and a family of bounds on the total number of bound states for the operator hv of the form n0 v c n rn v c x 2 x x0 2 n dn x the basic tool used to obtain results is a family of sharp sobolev inequalities for the operator a c c x 2

asymptotic behaviour of extremal averages of laplacian eigenvalues we study the convergence of extrema of averages of eigenvalues of the dirichlet laplacian on domains in mathbb r n under both measure and surface measure restrictions in the former case we prove that the sequence of averages to the power n 2 is sub additive and determine the first term in its asymptotics in the high frequency limit in the latter case we show that the sequence of minimisers converges to the ball as the frequency goes to infinity similar results hold for neumann boundary conditions

a bound for the eigenvalue counting function for krein von neumann and friedrichs extensions for an arbitrary open nonempty bounded set rn n n and sufficiently smooth coefficients a b q we consider the closed strictly positive higher order differential operator a 2m a b q in l2 defined on w02m 2 associated with the differential expression 2m a b q j k 1n i j bj aj k i k bk q m m n and its krein von neumann extension ak 2m a b q in l2 denoting by n ak 2m a b q 0 the eigenvalue counting function corresponding to the strictly positive eigenvalues of ak 2m a b q we derive the bound n ak 2m a b q cvn 2 n 1 2m2m n n 2m n 2m 0 where c c a b q 0 with c in 0 0 is connected to the eigenfunction expansion of the self adjoint operator a 2m a b q in l2 rn defined on w2m 2 rn corresponding to 2m a b q here vn n 2 n 2 2 denotes the euclidean volume of the unit ball in rn our method of proof relies on variational considerations exploiting the fundamental link between the krein von neumann extension and an underlying abstract buckling problem and on the distorted fourier transform defined in terms of the eigenfunction transform of a 2 a b q in l2 rn we also consider the analogous bound for the eigenvalue counting function for the friedrichs extension af 2m a b q in l2 of a 2m a b q no assumptions on the boundary of are made

a positive density analogue of the lieb thirring inequality the lieb thirring inequalities give a bound on the negative eigenvalues of a schrodinger operator in terms of an l p norm of the potential these are dual to bounds on the h 1 norms of a system of orthonormal functions here we extend these bounds to analogous inequalities for perturbations of the fermi sea of noninteracting particles i e for perturbations of the continuous spectrum of the laplacian by local potentials

estimates on the eigenvalues of the clamped plate problem on domains in euclidean spaces the purpose of this article is two fold first we obtain some certain bounds for the sums of positive and negative powers of the eigenvalues of the clamped plate problem of the dirichlet bi laplacian operator 2 d restricted to a bounded domain d rd with d 2 second we establish lower bounds for the sums of eigenvalues of 2 d sharper than the bounds recently obtained by cheng and wei a lower bound for eigenvalues of a clamped plate problem calculus var partial differ equ 42 3 4 579 590 2011 10 1007 s00526 011 0399 6 all these estimates are sharp in the sense of weyl asymptotics

a bayesian approach to prospective binary outcome studies with misclassification in a binary risk factor misclassification in a binary exposure variable within an unmatched prospective study may lead to a biased estimate of the disease exposure relationship it usually gives falsely small credible intervals because uncertainty in the recorded exposure is not taken into account when there are several other perfectly measured covariates interrelationships may introduce further potential for bias bayesian methods are proposed for analysing binary outcome studies in which an exposure variable is sometimes misclassified but its correct values have been validated for a random subsample of the subjects this bayesian approach can model relationships between explanatory variables and between exploratory variables and the probabilities of misclassification three logistic regressions are used to relate disease to true exposure misclassified exposure to true exposure and true exposure to other covariates credible intervals may be used to make decisions about whether certain parameters are unnecessary and hence whether the model can be reduced in complexity in the disease exposure model for parameters representing coefficients related to perfectly measured covariates the precision of posterior estimates is only slightly lower than would be found from data with no misclassification for the risk factor which has misclassification the estimates of model coefficients obtained are much less biased than those with misclassification ignored

universal bounds for eigenvalues of a buckling problem in this paper we investigate an eigenvalue problem for a biharmonic operator on a bounded domain in an n dimensional euclidean space which is also called a buckling problem we introduce a new method to construct nice trial functions and we derive a universal inequality for higher eigenvalues of the buckling problem by making use of the trial functions thus we give an affirmative answer for the problem on universal bounds for eigenvalues of the buckling problem which was proposed by payne polya and weinberger in 14 and this problem has been mentioned again by ashbaugh in 1

sums of laplace eigenvalues rotationally symmetric maximizers in the plane abstract the sum of the first n 1 eigenvalues of the laplacian is shown to be maximal among triangles for the equilateral triangle maximal among parallelograms for the square and maximal among ellipses for the disk provided the ratio area 3 moment of inertia for the domain is fixed this result holds for both dirichlet and neumann eigenvalues and similar conclusions are derived for robin boundary conditions and schrodinger eigenvalues of potentials that grow at infinity a key ingredient in the method is the tight frame property of the roots of unity for general convex plane domains the disk is conjectured to maximize sums of neumann eigenvalues

index energy estimates for yang mills connections and einstein metrics we prove a conformally invariant estimate for the index of schrodinger operators acting on vector bundles over four manifolds related to the classical cwikel lieb rozenblum estimate applied to yang mills connections we obtain a bound for the index in terms of its energy which is conformally invariant and captures the sharp growth rate furthermore we derive an index estimate for einstein metrics in terms of the topology and the einstein hilbert energy lastly we derive conformally invariant estimates for the betti numbers of an oriented four manifold with positive scalar curvature

infinitely many sign changing solutions for a class of biharmonic equation without symmetry abstract in this paper via a perturbation theory we prove the existence of infinitely many sign changing solutions for a class of biharmonic equations

motivic decomposition of a compactification of a merkurjev suslin variety we provide a motivic decomposition of a twisted form of a smooth hyperplane section of gr 3 6 this twisted form is a norm variety corresponding to a symbol in the milnor k theory k3 3 as an application we construct a torsion element in the chow group of this variety msc2000 20g15 14f43

optimal spectral rectangles and lattice ellipses we consider the problem of minimizing the k th eigenvalue of rectangles with unit area and dirichlet boundary conditions this problem corresponds to finding the ellipse centred at the origin with axes on the horizontal and vertical axes with the smallest area containing k integer lattice points in the first quadrant we show that as k goes to infinity the optimal rectangle approaches the square and correspondingly the optimal ellipse approaches the circle we also provide a computational method for determining optimal rectangles for any k and relate the rate of convergence to the square with the conjectured error term for gauss s circle problem