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

cost function shaping of the output error criterion identification of an output error model using the prediction error method leads to an optimization problem built on input output data collected from the system to be identified it is often hard to

on trace identities and universal eigenvalue estimates for some partial differential operators in this article we prove and exploit a trace identity for the spectra of schrodinger operators and similar operators this identity leads to universal bounds on the spectra which apply to low lying eigenvalues eigenvalue asymptotics and to partition functions traces of heat operators in many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated special cases corresponding to known inequalities include those of hile and protter and of yang introduction in this article we prove and exploit an identity for the spectra of self adjoint operators h modeled on the dirichlet laplacian or more generally on schrodinger operators of the form p a x 2 v x 1 where p 1i is the usual momentum operator in convenient units and a x is the magnetic vector potential we recover and extend several known inequalities involving sums differences and ratios of eigenvalues let j j 1 denote the ordered eigenvalues of the dirichlet laplacian on a bounded d dimensional domain with zero dirichlet boundary conditions and recall that hile and protter hipr80 proved that d 4 1 n n j 1 j n 1 j 2 thereby extending an earlier inequality of payne polya and weinberger papowe56 in the last few years it has become clear that these and many similar relationships can be realized as special cases of abstract variational bounds involving the interplay among commutators of 2 a cartesian coordinate xj and the corresponding derivative xj for this analysis and various extensions of 2 see ha88 ho90 ha93 hami95 while inequalities of this type have been fairly sharp for low lying eigenvalues they have been mostly disappointing for higher eigenvalues yang ya91 however received by the editors september 28 1995 1991 mathematics subject classification primary 35j10 35j25 58g25

computationally efficient spatial interpolators based on spartan spatial random fields this paper addresses the spatial interpolation of scattered data in d dimensions the problem is approached using the theory of spartan spatial random fields ssrfs focusing on a specific gaussian ssrf i e the fluctuation gradient curvature fgc model a family of spatial interpolators predictors is formulated by maximizing the fgc ssrf probability density function at each prediction point conditioned by the data an analytical expression for the general uniform bandwidth spartan gubs predictor is derived the linear weights of this predictor involve weighted summations of kernel functions over the sample and prediction points approximations for the sums are obtained at the asymptotic limit of a dense sampling network leading to simplified explicit expressions of the weights an asymptotic locally adaptive spartan alas predictor is defined by means of a kernel family that involves a tunable local parameter the relevant equations are fully developed in d 2 using simulated data in two dimensions we show that the alas prediction accuracy is comparable to that of ordinary kriging ok which is an optimal spatial linear predictor solp the numerical complexity of the alas predictor increases linearly with the sample size in contrast with the cubic dependence of ok for large data sets the alas predictor is shown to be orders of magnitude faster than ok at the cost of a slightly higher prediction dispersion the performance of the alas predictor and ok are compared on a data set of rainfall measurements using cross validation measures

general variational approach to the interpolation problem abstract the talmi and gilat variational approach to the interpolation problem in arbitrary dimension is presented together with the corresponding physical model the connection of this approach to some known spline methods is demonstrated and new interpolation functions are derived for one two and three dimensional cases they are designed to be flexible through the use of meaningful parameters and to give good approximations of both the function itself and its derivatives as well

the map and blend scattered data interpolant on a sphere abstract the map and blend technique constructs a function defined over the sphere that interpolates to a discrete sample of measurements at arbitrary locations on the sphere this technique consists of two different mappings of the sphere to planar domains solving two corresponding scattered data interpolation problems on the planar domains and then the interpolant on the sphere is formed by blending these two planar interpolants if the user has software for solving the scattered data interpolation problem on a planar domain only a small programming effort is needed to implement the map and blend interpolant on the sphere although any interpolant to scattered data on a planar domain can be used in our general technique we use the multiquadric radial basis method

estimates on eigenvalues of laplacian in this paper we study eigenvalues of laplacian on either a bounded connected domain in an n dimensional unit sphere sn 1 or a compact homogeneous riemannian manifold or an n dimensional compact minimal submanifold in an n dimensional unit sphere sn 1 we estimate the k 1 th eigenvalue by the first k eigenvalues as a corollary we obtain an estimate of difference between consecutive eigenvlaues our results are sharper than ones of p c yang and yau 25 leung 19 li 20 and harrel ii and stubbe 12 respectively from weyl s asymptotical formula we know that our estimates are optimal in the sense of the order of k for eigenvalues of laplacian on a bounded connected domain in an n dimensional unit sphere sn 1

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

trace identities for commutators with applications to the distribution of eigenvalues we prove trace identities for commutators of operators which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic schrodinger operators and schrodinger operators on immersed manifolds in particular we prove bounds on the eigenvalue lambda n 1 in terms of the lower spectrum bounds on ratios of means of eigenvalues and universal monotonicity properties of eigenvalue moments which imply sharp versions of lieb thirring inequalities in the case of a schrodinger operator on an immersed manifold of dimension d we derive a version of reilly s inequality bounding the eigenvalue lambda n 1 of the laplace beltrami operator by a universal constant times vertical bar vertical bar h vertical bar n 2 infinity 2 d

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

eigenvalue inequalities for klein gordon operators we consider the pseudodifferential operators hm associated by the prescriptions of quantum mechanics to the klein gordon hamiltonian p 2 m2 when restricted to a bounded open domain rd when the mass m is 0 the operator h0 coincides with the generator of the cauchy stochastic process with a killing condition on the operator h0 is sometimes called the fractional laplacian with power 12 cf r banuelos t kulczycki eigenvalue gaps for the cauchy process and a poincare inequality j funct anal 211 2 2004 355 423 r banuelos t kulczycki the cauchy process and the steklov problem j funct anal 234 2006 199 225 e giere the fractional laplacian in applications http www eckhard giere de math publications review pdf we prove several universal inequalities for the eigenvalues 0 1 2 of hm and their means k 1k l 1k l among the inequalities proved are k cst k 1 d for an explicit optimal semiclassical constant depending only on the dimension d for any dimension d 2 and any k k 1 d 1d 1 k furthermore when d 2 and k 2j k j d21 d d 1 kj 1d finally we present some analogous estimates allowing for an operator including an external potential energy field i e hm v x for v x in certain function classes

useful redundancy in parameter and time delay estimation for continuous time models in this paper we propose an algorithm to estimate the parameters including time delay of continuous time systems based on instrumental variable identification methods to overcome the multiple local minima of the cost function associated with the estimation of a time delay system we utilize the useful redundancy technique specifically the cost function is filtered through a set of low pass filters to improve convexity with the useful redundancy technique exploited to achieve convergence to the global minimum of the optimization problem numerical examples are presented to demonstrate the effectiveness of the proposed algorithm

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

brief paper some properties of the output error method the output error identification method is studied in various respects the stationary points of the associated loss function are investigated sufficient conditions for a unique local minimum are given the loss functions can be minimized using a quasilinearization algorithm such an algorithm will give good local convergence it is however shown that global convergence does not always occur the output error method is also compared with some other estimation methods from the accuracy point of view it is proved that a prediction error method will give better accuracy an instrumental variable technique may give better or worse accuracy depending on the actual noise correlation

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

differential inequalities for riesz means and weyl type bounds for eigenvalues we derive differential inequalities and difference inequalities for riesz means of eigenvalues of the dirichlet laplacian r z x k z k

smooth interpolation of scattered data by local thin plate splines abstract an algorithm and the corresponding computer program for solution of the scattered data interpolation problem is described given points x k y k f k k 1 n a locally defined function f x y which has the property f x k y k f k k 1 n is constructed the algorithm is based on a weighted sum of locally defined thin plate splines and yields an interpolation function which is differentiable the program is available from the author

global convergence conditions in maximum likelihood estimation maximum likelihood estimation has been widely applied in system identification because of consistency its asymptotic efficiency and sufficiency however gradient based optimisation of the likelihood function might end up in local convergence in this article we derive various new non local minimum conditions in both open and closed loop system when the noise distribution is a gaussian process here we consider different model structures in particular ararmax bj and oe models

universal inequalities for the eigenvalues of schrodinger operators on submanifolds we establish inequalities for the eigenvalues of schr o dinger operators on compact submanifolds possibly with nonempty boundary of euclidean spaces of spheres and of real complex and quaternionic projective spaces which are related to inequalities for the laplacian on euclidean domains due to payne polya and weinberger and to yang but which depend in an explicit way on the mean curvature in later sections we prove similar results for schr o dinger operators on homogeneous riemannian spaces and more generally on any riemannian manifold that admits an eigenmap into a sphere as well as for the kohn laplacian on subdomains of the heisenberg group among the consequences of this analysis are an extension of reilly s inequality bounding any eigenvalue of the laplacian in terms of the mean curvature and spectral criteria for the immersibility of manifolds in homogeneous spaces

on eigenvalues of a system of elliptic equations and of the biharmonic operator abstract let be a bounded domain in an n dimensional euclidean space r n we study eigenvalues of an eigenvalue problem of a system of elliptic equations u grad div u u in u 0 estimates for eigenvalues of the above eigenvalue problem are obtained furthermore we obtain an upper bound on the k 1 th eigenvalue k 1 we also obtain sharp lower bound for the first eigenvalue of two kinds of eigenvalue problems of the biharmonic operator on compact manifolds with boundary and positive ricci curvature

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

upper bounds for the neumann eigenvalues on a bounded domain in euclidean space abstract let k be the the kth eigenvalue for the neumann boundary value problem with respect to the laplace operator on a bounded domain with piecewise smooth boundary in rn weyl s asymptotic formula states that k c n k v 2 n we aim to give an upper bound for partial sums jk 1 j of eigenvalues our bound depends only on the volume of and is to some extent asymptotically correct for any domain because of the analogy between the problems of obtaining lower bounds for dirichlet eigenvalues and upper bounds for neumann eigenvalues cf g polya proc london math soc 3 11 1961 419 433 we can adapt the technique used by li and yau commun math phys 88 1983 309 318 for the task of estimating dirichlet eigenvalues

input design as a tool to improve the convergence of pem the prediction error method pem is related to an optimization problem built on input output data collected from the system to be identified it is often hard to find the global solution of this optimization problem because the corresponding objective function presents local minima and or the search space is constrained to a nonconvex set the shape of the cost function and hence the difficulty in solving the optimization problem depends directly on the experimental conditions more specifically on the spectrum of the input output data collected from the system therefore it seems plausible to improve the convergence to the global minimum by properly choosing the spectrum of the input in this paper we address this problem we present a condition for convergence to the global minimum of the cost function and propose its inclusion in the input design we present the application of the proposed approach to case studies where the algorithms tend to get trapped in nonglobal minima

a method for interpolating scattered data based upon a minimum norm network a method for interpolating scattered data is described ctiven xi yi zi i 1 n a bivariate function s with continuous first order partial derivatives is defined which has the property that s xi yi zi i 1 n the method is based upon a triangulation of the domain and a curve network which has certain minimum pseudonorm properties algorithms and examples are included

warping digital images using thin plate splines this note is to direct attention brack to thin plate spline because evaluation algorithms which became generaly available early in 1992 have dramatically reduced their computational complexity thin plate splines make very effective warping functions but special attention must be paid when calculating their coefficients and when evaluating them many times

on the estimation of transfer functions regularizations and gaussian processes revisited intrigued by some recent results on impulse response estimation by kernel and nonparametric techniques we revisit the old problem of transfer function estimation from input output measurements we formulate a classical regularization approach focused on finite impulse response fir models and find that regularization is necessary to cope with the high variance problem this basic regularized least squares approach is then a focal point for interpreting other techniques like bayesian inference and gaussian process regression the main issue is how to determine a suitable regularization matrix bayesian prior or kernel several regularization matrices are provided and numerically evaluated on a data bank of test systems and data sets our findings based on the data bank are as follows the classical regularization approach with carefully chosen regularization matrices shows slightly better accuracy and clearly better robustness in estimating the impulse response than the standard approach the prediction error method maximum likelihood pem ml approach if the goal is to estimate a model of given order as well as possible a low order model is often better estimated by the pem ml approach and a higher order model is often better estimated by model reduction on a high order regularized fir model estimated with careful regularization moreover an optimal regularization matrix that minimizes the mean square error matrix is derived and studied the importance of this result lies in that it gives the theoretical upper bound on the accuracy that can be achieved for this classical regularization approach

universal bounds for eigenvalues of the polyharmonic operators we study eigenvalues of polyharmonic operators on compact riemannian manifolds with boundary possibly empty in particular we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a euclidean space this inequality controls the kth eigenvalue by the lower eigenvalues independently of the particular geometry of the domain our inequality is sharper than the known payne polya weinberg type inequality and also covers the important yang inequality on eigenvalues of the dirichlet laplacian we also prove universal inequalities for the lower order eigenvalues of the polyharmonic operator on compact domains in a euclidean space which in the case of the biharmonic operator and the buckling problem strengthen the estimates obtained by ashbaugh finally we prove universal inequalities for eigenvalues of polyharmonic operators of any order on compact domains in the sphere

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

universal bounds for eigenvalues of the biharmonic operator abstract in this paper we study the eigenvalues of the clamped plate problem 2 u u in d u d u d 0 where d is a bounded connected domain in an n dimensional complete minimal submanifold of a unit m sphere s m 1 or of an m dimensional euclidean space r m let 0 1 2 k be the eigenvalues of the above problem we obtain universal bounds on k 1 in terms the first k eigenvalues independent of the domains for example when d is contained in an n dimensional complete minimal submanifold of s m 1 we show that k 1 1 k i 1 k i 1 k n i 1 k k 1 i 1 2 2 n 4 i 1 2 n 2 1 2 i 1 k k 1 i 1 2 4 i 1 2 n 2 1 2 from which one can obtain a more explicit upper bound on k 1 in terms of 1 k see corollary 1 when d is contained in a complete n dimensional minimal submanifold of r m we prove the inequality k 1 1 k i 1 k k 8 n 2 n 2 1 2 1 k i 1 k i k 1 i 1 2 which generalizes the main theorem in cheng and yang 2006 10 that states that the same estimate holds when d is a connected and bounded domain in r n

commutators spectral trace identities and universal estimates for eigenvalues using simple commutator relations we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self adjoint operator acting in a hilbert space applications involve abstract universal estimates for the eigenvalue gaps as particular examples we present simple proofs of the classical universal estimates for eigenvalues of the dirichlet laplacian as well as of some known and new results for other differential operators and systems we also suggest an extension of the methods to the case of non self adjoint operators

domain independent upper bounds for eigenvalues of elliptic operators let q c ir m be a bounded open set aq its boundary and a the laplacian on rm consider the elliptic differential equation 1 au au in q u 0 on aq it is known that the eigenvalues i of 1 satisfy n m n 2 i l n 1l 1 provided that n l an in this paper we abstract the method used by hile and protter 2 to establish 2 and apply the method to a variety of second order elliptic problems in particular to all constant coefficient problems we then consider a variety of higher order problems and establish an extension of 2 for problem 1 where the laplacian is replaced by a more general operator in a hilbert space

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

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

universal inequalities for eigenvalues of the buckling problem on spherical domains in this paper we study the eigenvalues of the buckling problem on domains in a unit sphere we obtain universal bounds on the k 1 th eigenvalue in terms of the first k eigenvalues independent of the domains

image surface approximation with irregular samples a method to approximate image surfaces using irregular samples is proposed experimental results show that a high compression ratio can be achieved for simple or complicated images a method accounting for the three dimensional effects is proposed as a way of selecting the irregular samples from the image approximation spline knots are chosen along the contours and the knot replacement is sensitive to the curvature along contours and the density of contours in the near neighborhood in general the proposed method was shown to reduce the reconstruction error both qualitatively and quantitatively and can be used to model the image surface for the purpose of data compression and representation this approach retains key features in images and throws away redundant data

large signal to noise ratio quantification in mle for ararmax models it has been shown that closed loop linear system identification by indirect method can be generally transferred to open loop ararmax autoregressive autoregressive moving average with exogenous input estimation for such models the gradient related optimisation with large enough signal to noise ratio snr can avoid the potential local convergence in maximum likelihood estimation to ease the application of this condition the threshold snr needs to be quantified in this paper we build the amplitude coefficient which is an equivalence to the snr and prove the finiteness of the threshold amplitude coefficient within the stability region the quantification of threshold is achieved by the minimisation of an elaborately designed multi variable cost function which unifies all the restrictions on the amplitude coefficient the corresponding algorithm based on two sets of physically realisable system input output data details the minimisation and also points out how to use the gradient related method to est

on sums of graph eigenvalues we use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite simple combinatorial graphs these include estimates for the adjacency matrix of a graph and for both the standard combinatorial laplacian and the renormalized laplacian we also provide upper bounds for sums of squares of eigenvalues of these three matrices among our results we generalize an inequality of fiedler for the extreme eigenvalues of the graph laplacian to a bound on the sums of the smallest or largest k such eigenvalues k n furthermore if lambda j are the eigenvalues of the graph laplacian h delta in increasing order on a finite graph with vertical bar nu vertical bar vertices and vertical bar epsilon vertical bar edges which is isomorphic to a subgraph of the v dimensional infinite cubic lattice then the spectral sums obey a weyl type upper bound a simplification of which reads sigma k 1 j 1 lambda j pi 2 vertical bar epsilon vertical bar 3 k vertical bar nu vertical bar 1 2 v for each k vertical bar nu vertical bar this and related estimates for sigma k 1 j 1 lambda 2 j provide a family of necessary conditions for the embeddability of the graph in a lattice of dimension nu or less c 2014 elsevier inc all rights reserved

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

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

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

universal bounds for eigenvalues of the biharmonic operator on riemannian manifolds in this paper we consider eigenvalues of the dirichlet biharmonic operator on compact riemannian manifolds with boundary possibly empty and prove a general inequality for them by using this inequality we study eigenvalues of the dirichlet biharmonic operator on compact domains in a euclidean space or a minimal submanifold of it and a unit sphere we obtain universal bounds on the k 1 th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains the estimate for the k 1 th eigenvalue of bounded domains in a euclidean space improves an important inequality obtained recently by cheng and yang

berezin li yau inequalities on domains on the sphere abstract we prove berezin li yau inequalities for the dirichlet and neumann eigenvalues on domains on the sphere s d 1 a sharp explicit bound for the sums of the neumann eigenvalues is obtained for all dimensions d in the case of s 2 we also obtain sharp lower bounds with correction terms for the vector laplacian and the stokes operator

universal inequalities for the eigenvalues of laplace and schrodinger operators on submanifolds we establish inequalities for the eigenvalues of schrodinger oper ators on compact submanifolds possibly with nonempty boundary of euclidean spaces of spheres and of real complex and quaternionic projective spaces which are related to inequalities for the laplacian on euclidean domains due to payne polya and weinberger and to yang but which depend in an explicit way on the mean curvature in later sections we prove similar results for schrodinger operators on homogeneous riemannian spaces and more generally on any rie mannian manifold that admits an eigenmap into a sphere as well as for the kohn laplacian on subdomains of the heisenberg group among the consequences of this analysis are an extension of reilly s inequality bounding any eigenvalue of the laplacian in terms of the mean curvature and spectral criteria for the immersibility of manifolds in homogeneous spaces