eigenvalues of the laplacian on regular polygons and polygons resulting from their disection abstract the eigenvalue problem is considered for the laplacian on regular polygons with either dirichlet or neumann boundary conditions which will be related to the unit circle by a conformal mapping the polygonal problem is then equivalent to a weighted eigenvalue problem on the circle with the same boundary conditions upper bounds are found for the eigenvalues by the rayleigh ritz method where the trial functions are the eigenfunctions of the unweighted problem on the circle these are products of bessel and trigonometrical functions and so the required integrals simplify greatly with a new recursion formula used to generate some bessel function integrals numerical results are given for the case of the hexagon with dirichlet conditions consideration of symmetry classes makes computations more efficient and gives as a byproduct the eigenvalues of a number of polygons such as trapezoids and diamonds which result from disecting the hexagon comparisons of the hexagon results are made with previous work

the cramer rao bound for estimating a sparse parameter vector the goal of this contribution is to characterize the best achievable mean squared error mse in estimating a sparse deterministic parameter from measurements corrupted by gaussian noise to this end an appropriate definition of bias in the sparse setting is developed and the constrained cramer rao bound crb is obtained this bound is shown to equal the crb of an estimator with knowledge of the support set for almost all feasible parameter values consequently in the unbiased case our bound is identical to the mse of the oracle estimator combined with the fact that the crb is achieved at high signal to noise ratios signal to noise ratio snrs by the maximum likelihood technique our result provides a new interpretation for the common practice of using the oracle estimator as a gold standard against which practical approaches are compared

eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

dynamical spectral rigidity among mathbb z _2 symmetric strictly convex domains close to a circle appendix b coauthored with h hezari

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

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

an analogue of the prime number theorem for closed orbits of axiom a flows for an axiom a flow restricted to a basic set we extend the zeta function to an open set containing w s h where h is the topological entropy this enables us to give an asymptotic formula for the number of closed orbits by adapting the wiener ikehara proof of the prime number theorem

the rkhs approach to minimum variance estimation revisited variance bounds sufficient statistics and exponential families the mathematical theory of reproducing kernel hilbert spaces rkhss provides powerful tools for minimum variance estimation mve problems here we extend the classical rkhs based analysis of mve in several directions we develop a geometric formulation of five known lower bounds on the estimator variance barankin bound cramer rao bound constrained cramer rao bound bhattacharyya bound and hammersley chapman robbins bound in terms of orthogonal projections onto a subspace of the rkhs associated with a given mve problem we show that under mild conditions the barankin bound the tightest possible lower bound on the estimator variance is a lower semicontinuous function of the parameter vector we also show that the rkhs associated with an mve problem remains unchanged if the observation is replaced by a sufficient statistic finally for mve problems conforming to an exponential family of distributions we derive novel closed form lower bounds on the estimator variance and show that a reduction of the parameter set leaves the minimum achievable variance unchanged

the covering spectrum of a compact length space we define a new spectrum for compact length spaces and riemannian manifolds called the covering spectrum which roughly measures the size of the one dimensional holes in the space more specifically the covering spectrum is a set of real numbers 0 which identify the distinct covers of the space we investigate the relationship between this covering spectrum the length spectrum the marked length spectrum and the laplace spectrum we analyze the behavior of the covering spectrum under gromov hausdorff convergence and study its gap phenomenon

capacity of reproducing kernel spaces in learning theory the capacity of reproducing kernel hilbert spaces rkhs plays an essential role in the analysis of learning theory covering numbers and packing numbers of balls of these reproducing kernel spaces are important measurements of this capacity we first present lower bound estimates for the packing numbers by means of nodal functions then we show that if a mercer kernel is c sup s for some s 0 being not an even integer the rkhs associated with this kernel can be embedded into c sup s 2 this gives upper bound estimates for the covering number concerning sobolev smooth kernels examples and applications to v sub spl gamma dimension and tikhonov 1977 regularization are presented to illustrate the upper and lower bound estimates

density distribution for isospectral circular membranes a special conformal mapping is used to yield the areal density distribution for inhomogeneous circular membranes which are isospectral with each other and with the congruent circular homogeneous membrane for fixed boundary and for free boundary implications of this nonuniqueness for the inverse problem in two dimensions are discussed

counting isospectral manifolds abstract given a simple lie group h of real rank at least 2 we show that the maximum cardinality of a set of isospectral non isometric h locally symmetric spaces of volume at most x grows at least as fast as x c log x log log x 2 where c c h is a positive constant in contrast with the real rank 1 case this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of belolipetsky and lubotzky 2 our proof uses sunada s method results of 2 and some deep results from number theory we also discuss an open number theoretical problem which would imply an even faster growth estimate

transverse vibrations of composite membranes of arbitrary boundary shape this paper deals with an approximate method based on the technique of conformal mapping and a variational formulation which allows for a straightforward solution of the title problem for a certain type of composite membranes numerical results are presented for membranes of regular polygonal shape whose inner circular core possesses a density 1 while the remaining is characterized by 2 the fundamental eigenvalues are also verified by the fourier expansion collocation method and by a finite element algorithm

lower bounds for parametric estimation with constraints a chapman robbins form of the barankin bound is used to derive a multiparameter cramer rao cr type lower bound on estimator error covariance when the parameter theta in r sup n is constrained to lie in a subset of the parameter space a simple form for the constrained cr bound is obtained when the constraint set theta sub c can be expressed as a smooth functional inequality constraint it is shown that the constrained cr bound is identical to the unconstrained cr bound at the regular points of theta sub c i e where no equality constraints are active on the other hand at those points theta in theta sub c where pure equality constraints are active the full rank fisher information matrix in the unconstrained cr bound must be replaced by a rank reduced fisher information matrix obtained as a projection of the full rank fisher matrix onto the tangent hyperplane of the full rank fisher matrix onto the tangent hyperplane of the constraint set at theta a necessary and sufficient condition involving the forms of the constraint and the likelihood function is given for the bound to be achievable and examples for which the bound is achieved are presented in addition to providing a useful generalization of the cr bound the results permit analysis of the gain in achievable mse performance due to the imposition of particular constraints on the parameter space without the need for a global reparameterization

representability of algebras finite over their centers abstract any associative left noetherian algebra over a field which is finitely generated as an algebra over a central subring is representable as a special case we give a short proof of the well known theorem that any algebra over a field which also is finite as a module over a central affine algebra is representable we give a counterexample to some natural generalizations as well as a conjectured generic counterexample together with specific positive results for irreducible algebras and finitely presented algebras also based on joint work of the second author with amitsur posthumous we show that any semiprimary pi algebra with radical squared 0 is weakly representable

the nilpotency of the radical in a finitely generated p i ring abstract let r x 1 x k be a p i ring satisfying a monic polynomial identity one of its coefficients is 1 where is a central noetherian subring it is proven that n r the nil radical of r is nilpotent as a corollary by taking f a field we settle affirmatively the open problem posed in c procesi rings with polynomial identities p 186 marcel dekker new york 1973 we prove the jacobson radical of a finitely generated p i algebra is nilpotent

on the ground state of regular polygonal billiards the helmholtz equation for regular polygons is investigated by means of the conformal mapping from the circle which provides an expansion parameter for the approximate evaluation of the lowest eigenvalue and the corresponding eigenvector

isospectral shapes with neumann and alternating boundary conditions the isospectrality of a well known pair of shapes constructed from two arrangements of seven congruent right isosceles triangles with the neumann boundary condition is verified numerically to high precision equally strong numerical evidence for isospectrality is presented for the eigenvalues of this standard pair in new boundary configurations with alternating dirichlet and neumann boundary conditions along successive edges good agreement with theory is obtained for the corresponding spectral staircase functions strong numerical evidence is also presented for isospectrality in an example of a different pair of shapes whose basic building block triangle is not isosceles some possible confirmatory experiments involving fluids are suggested

derivative reproducing properties for kernel methods in learning theory the regularity of functions from reproducing kernel hilbert spaces rkhss is studied in the setting of learning theory we provide a reproducing property for partial derivatives up to order s when the mercer kernel is c 2 s for such a kernel on a general domain we show that the rkhs can be embedded into the function space c s these observations yield a representer theorem for regularized learning algorithms involving data for function values and gradients examples of hermite learning and semi supervised learning penalized by gradients on data are considered

length spectra and degeneration of flat metrics in this paper we consider flat metrics semi translation structures on surfaces of finite type there are two main results the first is a complete description of when a set of simple closed curves is spectrally rigid that is when the length vector determines a metric among the class of flat metrics secondly we give an embedding into the space of geodesic currents and use this to obtain a compactification for the space of flat metrics the geometric interpretation is that flat metrics degenerate to mixed structures on the surface part flat metric and part measured foliation

free energy as a geometric invariant the free energy plays a fundamental role in statistical and condensed matter physics a related notion of free energy plays an important role in the study of hyperbolic dynamical systems in this paper we introduce and study a natural notion of free energy for surfaces with variable negative curvature this geometric free energy encodes a new type of marked length spectrum of closed geodesics which lies between the well known marked length spectrum marked by the corresponding element of the fundamental group and the unmarked length spectrum we prove that the free energy parametrizes the boundary of the domain of convergence of a poincare series which also encodes this spectrum we also show that this new length spectrum or equivalently the geometric free energy is not an isometry invariant in the final section we use tools from multifractal analysis to effect a fine asymptotic comparison of word length and geodesic length of closed geodesics we hope that our geometric understanding of free energy will provide new insight into this fundamental physical and dynamical quantity

a bound on mean square estimate error a lower bound on mean square estimate error is derived as an instance of the covariance inequality by concatenating the generating matrices for the bhattacharyya and barankin bounds it represents a generalization of the bhattacharyya barankin cramer rao hammersley chapman robbins kiefer and mcaulay hofstetter bounds in that all of these bounds may be derived as special cases the bound is applicable to biased estimates of functions of a multidimensional parameter termed the hybrid bhattacharyya barankin bound it may be written as the sum of the mth order bhattacharyya bound and a nonnegative term similar in form to the rth order hammersley chapman robbins bound it is intended for use when small error bounds such as the cramer rao bound may not be tight unlike many large error bounds it provides a smooth transition between the small error and large error regions as an example application bounds are placed on the variance of unbiased position estimates derived from passive array measurements here the hybrid bhattacharyya barankin bound enters the large error region at a larger snr and in the large error region is noticeably greater than either the hammersley chapman robbins or the bhattacharyya bounds

isospectral membranes a connection between shape and density a connection is established via conformal maps between vibrating membranes that are isospectral with respect to shape and those that are isospectral with respect to density in particular inhomogeneous circular membranes are constructed that are isospectral to polygonal membranes of uniform density via the schwarz christoffel mapping although some corners of the polygons lead to singularities in the constructed densities the densities are shown to be integrable