duality under dependency inversion and noether theory for second order lagrangians abstract we consider the duality which is obtained by interchanging dependent and independent variables in a lagrangian this duality is studied here for a second order lagrangian which could describe e g a euler bernoulli beam more generally a one dimensional elastic body of higher grade however the essential theory follows purely formally for a general lagrangian next the corresponding noether theory of transformation groups and invariants is developed making use of duality as far as possible and showing the limits of duality theory particular emphasis is laid on the difference between relations equally valid on any curve and those valid only on the solution curve stressing the difference between true and apparent invariants

short time critical dynamics of the ising model studied within the n fold algorithm abstract the short time behaviour of the critical two dimensional ising model is studied for the bortz kalos lebowitz n fold way algorithm bkl algorithm for square lattices of linear sizes l 110 128 140 170 and 200 we calculate the short time critical behaviour for the bkl algorithm and also for the heat bath algorithm comparison of these results shows that although power law scaling form emerges from the early times of the dynamics there are noticeable differences which do not produce negligible effects on the estimates of critical exponents the observed universality of short time critical behaviour supplements our fundamental knowledge of critical phenomena however short time dynamics seem to be related only approximately with equilibrium critical properties therefore any attempt to estimate equilibrium critical exponents from the short time regime would be superfluous except for the purpose of comparison because these exponents have been determined very precisely by different methods

a second eigenvalue bound for the dirichlet schrodinger operator let i v be the ith eigenvalue of the schrodinger operator with dirichlet boundary conditions on a bounded domain omega subset mathbb r n and with the positive potential v following the spirit of the payne polya weinberger conjecture and under some convexity assumptions on the spherically rearranged potential v we prove that 2 v 2 s1 v here s1 denotes the ball centered at the origin that satisfies the condition 1 v 1 s1 v

uniform concentration compactness for sobolev spaces on variable domains abstract we present a new method for proving existence results in shape optimization problems involving the eigenvalues of the dirichlet laplace operator this method brings together the convergence theory and the concentration compactness principle given a sequence of open sets a n n n in r n not necessarily bounded but of uniformly bounded measure we prove a concentration compactness result in l l 2 r n for the sequence of resolvent operators r a n n n where r a n l 2 r n h 1 0 a n r a n 1

shape recognition using eigenvalues of the dirichlet laplacian the eigenvalues of the dirichlet laplacian are used to generate three different sets of features for shape recognition and classification in binary images the generated features are rotation translation and size invariant the features are also shown to be tolerant of noise and boundary deformation these features are used to classify hand drawn synthetic and natural shapes with correct classification rates ranging from 88 9 to 99 2 the classification was done using few features only two features in some cases and simple feedforward neural networks or minimum euclidian distance

a numerical study of the spectral gap we present a numerical study of the spectral gap of the dirichlet laplacian k 2 k 1 k of a planar convex region k besides providing supporting numerical evidence for the long standing gap conjecture that k 3 2 d2 k where d k denotes the diameter of k our study suggests new types of bounds and several conjectures regarding the dependence of the gap not only on the diameter but also on the perimeter and the area one of these conjectures is a stronger version of the gap conjecture mentioned above a similar study is carried out for the quotient of the first two dirichlet eigenvalues

on the range of the first two dirichlet and neumann eigenvalues of the laplacian in this paper we study the set of points in the plane defined by x y lambda 1 omega lambda 2 omega omega 1 where lambda 1 omega lambda 2 omega are either the two first eigenvalues of the dirichlet laplacian or the two first non trivial eigenvalues of the neumann laplacian we consider the case of general open sets together with the case of convex open domains we give some qualitative properties of these sets show some pictures obtained through numerical computations and state several open problems

inequalities for eigenvalues of fourth order elliptic operators in divergence form on riemannian manifolds abstract in this paper we study eigenvalue of linear fourth order elliptic operators in divergence form with dirichlet boundary condition on a bounded domain in a compact riemannian manifolds with boundary possibly empty and provide a general inequality for them as an application by using this inequality we study eigenvalues of this operator on compact domains of complete submanifolds in a euclidean space

isoperimetric bounds for higher eigenvalue ratios for the n dimensional fixed membrane problem we give several results which extend our recent proof of the payne polya weinberger conjecture to ratios of higher eigenvalues in particular we show that for a bounded domain n the eigenvalues of its dirichlet laplacian obey where m denotes the m th eigenvalue and j p k denotes the k th positive zero of the bessel function j p x certain extensions of this result are given the most general being the bound where k 2 and l m denotes the number of nodal domains of an m th eigenfunction our results imply certain further conjectures of payne polya and weinberger concerning 3 2 and 4 3 in addition we find a resonably good bound on 4 1 we also briefly discuss extensions to schrodinger operators and other elliptic eigenvalue problems

eigenmodes of isospectral drums recently it was proved that there exist nonisometric planar regions that have identical laplace spectra that is one cannot hear the shape of a drum the simplest isospectral regions known are bounded by polygons with reentrant corners while the isospectrality can be proven mathematically analytical techniques are unable to produce the eigenvalues themselves furthermore standard numerical methods for computing the eigenvalues such as adaptive finite elements are highly inefficient physical experiments have been performed to measure the spectra but the accuracy and flexibility of this method are limited we describe an algorithm due to descloux and tolley comput methods appl mech engrg 39 1983 pp 37 53 that blends singular finite elements with domain decomposition and show that with a modification that doubles its accuracy this algorithm can be used to compute efficiently the eigenvalues for polygonal regions we present results accurate to 12 digits for the most famous pair of isospectral drums as well as results for another pair

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

on conservation laws of continuum mechanics abstract a unified variational formulation is advanced leading to field equations and conservation laws in general mechanical continua the formulation is applicable to dynamic processes in any medium which admits a lagrangian as a result of the procedure employed physical or canonical momenta emerge on the same level as material momenta the simplicity and transparency of the relations derived is a consequence of strict separation of descriptions in physical and material space which precludes the use of displacement as a field variable one of the results shows that a distinction between field equations or equations of motion and conservation laws is no longer essential and all basic relations are in fact balance laws

class of monte carlo algorithms for dynamic problems leads to an adaptive method we introduce a class of monte carlo algorithms that solve a dynamic problem defined by the transition rates and the initial state of a discrete system this class contains the method of bortz kalos and lebowitz bkl j comp phys 17 10 1975 as a limit we show that introducing a constant time step in a metropolis algorithm leads to an approximation of the solution in which the system relaxation times are underestimated this can be corrected if the time step is an adequate stochastic variable thus we are able to define kinetic metropolis algorithms and generalize them in a case of nonconstant numbers of attempt configurations the algorithm class allows us to introduce a useful method in which the calculation of transition rates are exploited for the next step in an adaptive way this method corresponds to a kinetic metropolis algorithm when the rejection probability is reasonable and becomes similar to the bkl method otherwise we describe and compare four different algorithms applied to a physical example about the diffusion in lattice gases

range of the first two eigenvalues of the laplacian for each planar domain d of unit area the first two dirichlet eigenvalues of on d determine a point 1 d 2 d in the 1 2 plane as d varies over all such domains this point varies over a set r which we determine its boundary consists of two semi infinite straight lines and a curve connecting their endpoints this curve is found numerially we also show how to minimize the n th eigenvalue when the minimizing domain is diconnected for n 3 we show that the minimizing domain is connected and that 3 is a local minimum for d a circular disc

is it possible to tune a drum p a was partially supported by fct portugal through the program investigador fct with reference if 00177 2013 and the scientific project ptdc matcal 4334 2014

on the variation with respect to a parameter of zeros of bessel and q bessel functions abstract we apply what is essentially the hellmann feynman theorem in a discrete setting to derive representations for the derivatives with respect to a parameter of the positive zeros of a family of entire functions these representations prove that the zeros are monotone functions of the parameters involved this family includes the bessel functions jv x and their basic analog jv 2 x q as a by product we show that j v 1 2 v 1 increases with v when v 1 jv 1 being the smallest positive zero of jv x we also use the lommel polynomials to derive improved lower bounds for the smallest positive zero of jv x

spatial two level interacting particle simulations and information theory based error quantification we propose a hierarchy of two level kinetic monte carlo methods for sampling high dimensional stochastic lattice particle dynamics with complex interactions the method is based on the efficient coupling of different spatial resolution levels taking advantage of the low sampling cost in a coarse space and developing local reconstruction strategies from coarse grained dynamics furthermore a natural extension to a multilevel kinetic coarse grained monte carlo is presented microscopic reconstruction corrects possibly significant errors introduced through coarse graining leading to the controlled error approximation of the sampled stochastic process in this manner the proposed algorithm overcomes known shortcomings of coarse graining of particle systems with complex interactions such as combined long and short range particle interactions and or complex lattice geometries specifically we provide error analysis for the approximation of long time stationary dynamics in terms of relative entropy measu

a lower bound to the spectral threshold in curved tubes we consider the laplacian in curved tubes of arbitrary cross section rotating together with the frenet frame along curves in euclidean spaces of arbitrary dimension subject to dirichlet boundary conditions on the cylindrical surface and neumann conditions at the ends of the tube we prove that the spectral threshold of the laplacian is estimated from below by the lowest eigenvalue of the dirichlet laplacian in a torus determined by the geometry of the tube

a sharp bound for the ratio of the first two dirichlet eigenvalues of a domain in a hemisphere of ⁿ for a domain q contained in a hemisphere of the n dimensional sphere sn we prove the optimal result a2 ai q a2 al 0 for the ratio of its first two dirichlet eigenvalues where q the symmetric rearrangement of q in sn is a geodesic ball in n having the same n volume as q we also show that a2 a1 for geodesic balls of geodesic radius 01 less than or equal to 7r 2 is an increasing function of 01 which runs between the value jn 2 1 in 2 1 1 2 for 1 0 this is the euclidean value and 2 n l n for 01 7r 2 here jv k denotes the kth positive zero of the bessel function j t this result generalizes the payne polya weinberger conjecture which applies to bounded domains in euclidean space and which we had proved earlier our method makes use of symmetric rearrangement of functions and various technical properties of special functions we also prove that among all domains contained in a hemisphere of sn and having a fixed value of a1 the one with the maximal value of a2 is the geodesic ball of the appropriate radius this is a stronger but slightly less accessible isoperimetric result than that for a2 1i various other results for a1 and a2 of geodesic balls in sn are proved in the course of our work

the method of fundamental solutions applied to some inverse eigenproblems in this work we address the application of the method of fundamental solution mfs as a forward solver in some shape optimization problems in two and three dimensional domains it is well known kac s problem that a set of eigenvalues does not determine uniquely the shape of the domain moreover even the existence problem is not well defined due to the ashbaugh benguria inequality although these results constitute counterexamples in the general problem of shape determination from the eigenfrequencies we can address simpler questions in shape determination using the mfs for instance we apply the mfs to build domains that include a specific finite set of eigenvalues or that have an eigenmode that verifies some prescribed conditions as a particular case an eigenmode that defines a certain nodal line

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

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

ratios of bessel functions and roots of αjν x xj ν x 0 abstract derivatives with respect to order and argument x of the ratio j x j 1 x of bessel functions are studied for all real values of and x our results generalize and sharpen previously known results and allow the deduction of a more complete description including monotonicity and multiplicity of the positive roots of the equation j x xj x 0 for all real and

fourier transform null variety and laplacian s eigenvalues we consider a quantity the distance to the origin from the null variety of the fourier transform of the characteristic function of we conjecture rstly that is maximized among all convex balanced domains r d of a xed volume by a ball and also that is bounded above by the square root of the second dirichlet eigenvalue of we prove some weaker versions of these conjectures in dimension two as well as their validity for domains asymptotically close to a disk and also discuss further links between and the eigenvalues of the laplacians

on the optimization of the principal eigenvalue for single centre point interaction operators in a bounded region we investigate relations between spectral properties of a single centre point interaction hamiltonian describing a particle confined to a bounded domain with a dirichlet boundary and the geometry of for this class of operators krein s formula yields an explicit representation of the resolvent in terms of the integral kernel of the unperturbed one d z 1 we use a moving plane analysis to characterize the behaviour of the ground state energy of the hamiltonian with respect to the point interaction position and the shape of in particular we establish some conditions showing how to place the interaction to optimize the principal eigenvalue

on the inverse spectral problem for euclidean triangles we consider the inverse spectral problem for the laplace operator on triangles with dirichlet boundary conditions providing numerical evidence to the effect that the eigenvalue triplet 1 2 3 is sufficient to determine a triangle uniquely on the other hand we show that other combinations such as 1 2 4 will not be enough and that there will exist at least two triangles with the same values on these triplets

optimization of spectral functions of dirichlet laplacian eigenvalues we consider the shape optimization of spectral functions of dirichlet laplacian eigenvalues over the set of star shaped symmetric bounded planar regions with smooth boundary the regions are represented using fourier cosine coefficients and the optimization problem is solved numerically using a quasi newton method the method is applied to maximizing two particular nonsmooth spectral functions the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue both of which are generalizations of the payne polya weinberger ratio the optimal values and attaining regions for n 13 are presented and interpreted as a study of the range of the dirichlet laplacian eigenvalues for both spectral functions and each n the optimal attaining region has multiplicity two nth eigenvalue

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

optimization of sums and quotients of dirichlet laplacian eigenvalues we study some shape optimization problems related to sums and quotients of dirichlet laplacian eigenvalues l n for planar domains we show how to minimize a sum l k l k 1 w k 1 2 when the minimizing domain is disconnected in particular we prove that the optimizers in the cases k 1 and k 2 are connected we develop a numerical method for solving shape optimization eigenvalue problems which is applied to determine the first fourteen optimizers for sums of consecutive dirichlet eigenvalues and quotients of type l k l 1 k 2 3 this last problem was already studied by osting using a different numerical method and we obtain similar results

reviving the method of particular solutions fox henrici and moler made famous a method of particular solutions for computing eigenvalues and eigenmodes of the laplacian in planar regions such as polygons we explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties the crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant similar methods may be used to improve other mesh free algorithms for a variety of computational problems

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

a second proof of the payne polya weinberger conjecture without using product representations or elaborate comparisons of zeros we prove the two key properties of the bessel function ratiojp 1jp 1 1x jpjp 1x that we used to prove the payne polya weinberger conjecture in these new proofs we use only differential equations and the rayleigh ritz method for estimating lowest eigenvalues the new proofs admit generalization to other related problems where our previous proofs fail

inverse spectral problem for analytic domains ii ℤ2 symmetric domains this paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit the algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary and on a feynman diagrammatic analysis of the stationary phase expansion of the oscillatory integral the algorithm is particularly effective for euclidean plane domains possessing a 2 symmetry which reverses the orientation of a bouncing ball orbit it is also very effective for domains with dihedral symmetries for simply connected analytic euclidean plane domains in either symmetry class we prove that the domain is determined within the class by either its dirichlet or neumann spectrum this improves and generalizes the best prior inverse result that simply connected analytic plane domains with two symmetries are spectrally determined within that class

the boundary collocation method with meshless concept for acoustic eigenanalysis of two dimensional cavities using radial basis function abstract in this paper a meshless method for the acoustic eigenfrequencies using radial basis function rbf is proposed the coefficients of influence matrices are easily determined by the two point functions in determining the diagonal elements of the influence matrices two techniques limiting approach and invariant method are employed based on the rbf in the imaginary part kernel the method results in spurious eigenvalues which can be separated by using the singular value decomposition svd technique in conjunction with the fredholm alternative theorem to understand why the spurious eigenvalues occur analytical study in the discrete system by discretizing the circular boundary is conducted by using circulants by using the svd updating terms and documents the true and spurious eigensolutions can be extracted out respectively several examples are demonstrated to see the validity of the present method

on the buckling eigenvalue problem we prove a density result which allows us to justify the application of the method of fundamental solutions to solve the buckling eigenvalue problem of a plate we address an example of an analytic convex domain for which the first eigenfunction does change the sign and present a large scale numerical study with polygons providing numerical evidence to some new conjectures

generalizations of noether s theorem in classical mechanics in this paper a review is presented of various approaches to the generalization of the version of noether s theorem which is presented in most textbooks on classical mechanics its motivation is the controversy still persisting around the possible scope of a noether type theorem allowing for velocity dependent transformations our analysis is centered around the one factor common to all known treatments namely the structure of the related first integral we first discuss the most general framework in which a function of the above mentioned structure constitutes a first integral of a given lagrangian system and show that one cannot really talk about an interrelationship between symmetries and first integrals there we then compare different proposed generalizations of noether s theorem by describing the nature of the restrictions which characterize them when they are situated within the broadest framework we prove a seemingly new equivalence result between the two main approaches that of invaria

on low eigenvalues of the laplacian with mixed boundary conditions abstract by means of the so called symmetrization we study the eigenvalue problem for the laplace operator with mixed boundary conditions we obtain various bounds for combinations of the low eigenvalues and some sharp comparison results for the first eigenfunction in terms of bessel functions

isogeometric shape optimization of vibrating membranes we consider a model problem of isogeometric shape optimization of vibrating membranes whose shapes are allowed to vary freely the main obstacle we face is the need for robust and inexpensive extension of a b spline parametrization from the boundary of a domain onto its interior a task which has to be performed in every optimization iteration we experiment with two numerical methods one is based on the idea of constructing a quasi conformal mapping whereas the other is based on a spring based mesh model for carrying out this task which turn out to work sufficiently well in the present situation we perform a number of numerical experiments with our isogeometric shape optimization algorithm and present smooth optimized membrane shapes our conclusion is that isogeometric analysis fits well with shape optimization

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

minimizing the second eigenvalue of the laplace operator with dirichlet boundary conditions in this paper we are interested in the minimization of the second eigenvalue of the laplacian with dirichlet boundary conditions amongst convex plane domains with given area the natural candidate to be the optimum was the stadium a convex hull of two identical tangent disks we refute this conjecture nevertheless we prove the existence of a minimizer we also study some qualitative properties of the minimizer regularity geometric properties

two new weyl type bounds for the dirichlet laplacian in this paper we prove two new weyl type upper estimates for the eigenvalues of the dirichlet laplacian as a consequence we obtain the following lower bounds for its counting function for 1 one has n 2 n 2 1 h n 1 n 2 1 n 2 and n n 2 n 4 n 2 1 h n 1 4 n 1 n 2 1 n 2 where h n 2n j 2 n 2 1 1j 2 n 2 jn 2 1 1 is a constant which depends on n the dimension of the underlying space and bessel functions and their zeros