improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates the problem of estimating the eigenvalues and eigenvectors of the covariance matrix associated with a multivariate stochastic process is considered the focus is on finite sample size situations whereby the number of observations is limited and comparable in magnitude to the observation dimension using tools from random matrix theory and assuming a certain eigenvalue splitting condition new estimators of the eigenvalues and eigenvectors of the covariance matrix are derived that are shown to be consistent in a more general asymptotic setting than the traditional one indeed these estimators are proven to be consistent not only when the sample size increases without bound for a fixed observation dimension but also when the observation dimension increases to infinity at the same rate as the sample size numerical evaluations indicate that the estimators have an excellent performance in small sample size scenarios where the observation dimension and the sample size are comparable in magnitude

on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

subspace based and direct algorithms for distorted circular contour estimation circular features are commonly sought in digital image processing the subspace based line detection slide method proposed to estimate the center and the radius of a single circle in this paper we introduce a novel method for estimating several radii while extending the circle estimation to retrieve circular like distorted contours particularly we develop and validate a new model for virtual signal generation by simulating a circular antenna the circle center is estimated by the slide method a variable speed propagation scheme toward the circular antenna yields a linear phase signal therefore a high resolution method provides the radius either the gradient method or the more robust combination of dividing rectangles and spline interpolation can extend this method for free form object segmentation the retrieval of multiple non concentric circles and rotated ellipses is also considered to evaluate the performance of the proposed methods we compare them with a least squares method hough transform and gradient vector flow we apply the proposed method to hand made images while considering some real world images

a random matrix theoretic approach to handling singular covariance estimates in many practical situations we would like to estimate the covariance matrix of a set of variables from an insufficient amount of data more specifically if we have a set of n independent identically distributed measurements of an m dimensional random vector the maximum likelihood estimate is the sample covariance matrix here we consider the case where n m such that this estimate is singular noninvertible and therefore fundamentally bad we present a radically new approach to deal with this situation based on the idea of dimensionality reduction through an ensemble of isotropically random unitary matrices we obtain two estimates cov and invcov which are estimates for the covariance matrix and the inverse covariance matrix respectively both estimates retain the original eigenvectors while altering the eigenvalues we have a closed form analytical expression for cov and invcov in terms of the eigenvector eigenvalue decomposition of the sample covariance we motivate the use of invcov through applications to linear estimation supervised learning and high resolution spectral estimation we also compare the performance of these estimators with other more conventional methods

g 1 approximation of conic sections by quartic bezier curves this paper proposes a new approximation method for conic sections by quartic bezier curves with g 1 end point continuity we give an upper bound on the hausdorff distance between the conic section and the quartic bezier approximation curve and also show that the approximation order is eight moreover using the subdivision scheme at the shoulder point of the conic section the resulting approximation curve has g 2 curvature continuity with the conic section at the endpoints finally we prove that our approximation has a smaller error bound than previous quartic bezier approximations and also present some numerical examples to demonstrate the validity and effectiveness of our method

an approximation method to circular arcs abstract modern cad cam systems are not able to dispose the circles represented by the implicit equations or the parametric equations with trigonometric function parametric rational polynomials are used to approximate the circular arcs a new approximation method by means of quartic bezier curves is presented to effectively approximate the circular arcs it is shown that our method provides the approximation of order eight and yields a smaller error than previous methods by equally spaced subdivision of circular arcs the curvature continuous spline approximation algorithm of the circular arcs is given

optimal parameter values for approximating conic sections by the quartic bzier curves the previous approximation curves of conic section by quartic bzier curves interpolate the conic section at the specified parameter values in this paper by solving the minimax problem we present the optimal parameter values for approximating conic sections by the quartic bzier curves the upper bound on the hausdorff distance between the conic section and the approximation curves is minimized the method of solving the minimax problem is changed to solve a quartic algebraic equation by delicate reasoning

sparse principal component analysis via axis aligned random projections we introduce a new method for sparse principal component analysis based on the aggregation of eigenvector information from carefully selected axis aligned random projections of the sample covariance matrix unlike most alternative approaches our algorithm is non iterative so it is not vulnerable to a bad choice of initialization we provide theoretical guarantees under which our principal subspace estimator can attain the minimax optimal rate of convergence in polynomial time in addition our theory provides a more refined understanding of the statistical and computational trade off in the problem of sparse principal component estimation revealing a subtle interplay between the effective sample size and the number of random projections that are required to achieve the minimax optimal rate numerical studies provide further insight into the procedure and confirm its highly competitive finite sample performance

constrained polynomial approximation of quadric surfaces quadric surfaces are one of the geometric elements most commonly used for shape expression and mechanical accessory cartography in three dimensions since quadric surfaces cannot be represented exactly using polynomials it is not suitable to deal with most problems encountered in engineering application for this reason we present a new method for polynomial approximation of quadric surfaces the approximation tensor product surface obtained by the proposed method has no limit on its degree i e arbitrary degree it has c k h k h 0 1 2 continuity at four corners with the original quadric surface meanwhile the method has an optimal approximation result in the l2 norm and it can also be applied to piecewise continuous quadric surface patches or a quadric surface patch combined with surface subdivision scheme and the resulting piecewise approximation patches are globally c0 continuous finally error estimation for the approximation method is given and numerical examples are presented to validate the feasibility and effectiveness of our method

besov spaces with variable smoothness and integrability abstract in this article we introduce besov spaces with variable smoothness and integrability indices we prove independence of the choice of basis functions as well as several other basic properties we also give sobolev type embeddings and show that our scale contains variable order holder zygmund spaces as special cases we provide an alternative characterization of the besov space using approximations by analytic functions

approximation of conic sections by curvature continuous quartic bezier curves in this paper we propose two approximation methods of conic section by quartic bezier curves these are the extensions of the quartic bezier approximations of circular arcs presented in ahn and kim 1997 1 and kim and ahn 2007 10 to conic cases we also give the error bounds of the hausdorff distances between the conic section and the approximation curves and show that the error bounds have the approximation order eight our methods yield quartic g 2 curvature continuous spline approximations of conic sections using the subdivision scheme stated in floater 1995 1997 5 11 we illustrate our results by some numerical examples

using the matrix pencil method to solve phase unwrapping this correspondence presents a phase unwrapping pu algorithm based on the matrix pencil mp method a brief discussion is given on the relationship between pu and instantaneous frequency estimation the pu algorithm and its properties are presented in detail results show a significant improvement in pu with respect to previous algorithms

doubly constrained robust capon beamformer the standard capon beamformer scb is known to have better resolution and much better interference rejection capability than the standard data independent beamformer when the array steering vector is accurately known however the major problem of the scb is that it lacks robustness in the presence of array steering vector errors in this paper we will first provide a complete analysis of a norm constrained capon beamforming nccb approach which uses a norm constraint on the weight vector to improve the robustness against array steering vector errors and noise our analysis of nccb is thorough and sheds more light on the choice of the norm constraint than what was commonly known we also provide a natural extension of the scb which has been obtained via covariance matrix fitting to the case of uncertain steering vectors by enforcing a double constraint on the array steering vector viz a constant norm constraint and a spherical uncertainty set constraint which we refer to as the doubly constrained robust capon beamformer dcrcb nccb and dcrcb can both be efficiently computed at a comparable cost with that of the scb performance comparisons of nccb dcrcb and several other adaptive beamformers via a number of numerical examples are also presented

capon algorithm mean squared error threshold snr prediction and probability of resolution below a specific threshold signal to noise ratio snr the mean squared error mse performance of signal parameter estimates derived from the capon algorithm degrades swiftly prediction of this threshold snr point is of practical significance for robust system design and analysis the exact pairwise error probabilities for the capon and bartlett algorithm derived herein are given by simple finite sums involving no numerical integration include finite sample effects and hold for an arbitrary colored data covariance via an adaptation of an interval error based method these error probabilities along with the local error mse predictions of vaidyanathan and buckley facilitate accurate prediction of the capon threshold region mse performance for an arbitrary number of well separated sources circumventing the need for numerous monte carlo simulations a large sample closed form approximation for the capon threshold snr is provided for uniform linear arrays a new exact two point measure of the probability of resolution for the capon algorithm that includes the deleterious effects of signal model mismatch is a serendipitous byproduct of this analysis that predicts the snrs required for closely spaced sources to be mutually resolvable by the capon algorithm last a general strategy is provided for obtaining accurate mse predictions that account for signal model mismatch

approximation of circular arcs and offset curves by bezier curves of high degree in this paper we present an exact error analysis for circle approximation by bezier curve the approximation method is a special case of floater s conic approximation method comput aided geom design 8 1991 135 using this we propose an approximation method of the offset curve of given plane bezier curve by bezier curve of the same degree

asymptotic achievability of the cramer rao bound for noisy compressive sampling we consider a model of the form y ax n where x isin cm is sparse with at most l nonzero coefficients in unknown locations y isin cn is the observation vector a isin cn times m is the measurement matrix and n isin cn is the gaussian noise we develop a cramer rao bound on the mean squared estimation error of the nonzero elements of x corresponding to the genie aided estimator gae which is provided with the locations of the nonzero elements of x intuitively the mean squared estimation error of any estimator without the knowledge of the locations of the nonzero elements of x is no less than that of the gae assuming that l n is fixed we establish the existence of an estimator that asymptotically achieves the cramer rao bound without any knowledge of the locations of the nonzero elements of x as n rarr infin for a a random gaussian matrix whose elements are drawn i i d according to n 0 1

sobolev s inequalities and vanishing integrability for riesz potentials of functions in the generalized lebesgue space lp logl q abstract our aim in this paper is to deal with sobolev s inequalities for riesz potentials of functions belonging to l p log l q to do so we study the boundedness of hardy littlewood maximal functions and apply the hedberg s trick as an application we treat vanishing integrability for riesz potentials

circle approximation by g2 bezier curves of degree n with 2n 1 extreme points abstract in this paper we present a new circle approximation method using g 2 bezier curve of degree n the signed error function of which is an alternant or equioscillating with 2 n 1 extreme points for n 2 3 4 the new circle approximation curve is obtained in explicit form as is the hausdorff distance between the curve and a circular arc compared with previous g 2 quartic circle approximation methods the proposed g 2 quartic approximation yields the smallest hausdorff distance

a general framework for the optimal approximation of circular arcs by parametric polynomial curves abstract we propose a general framework for a geometric approximation of circular arcs by parametric polynomial curves the approach is based on a constrained uniform approximation of an error function by scalar polynomials the system of nonlinear equations for the unknown control points of the approximating polynomial given in the bezier form is derived and a detailed analysis provided for some low degree cases which were not studied yet at least for these cases the solutions can be in principal written in a closed form and provide the best known approximants according to the simplified radial distance a general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given

curvatures of the quadratic rational bezier curves abstract we find necessary and sufficient conditions for the curvature of a quadratic rational bezier curve to be monotone in 0 1 to have a unique local minimum to have a unique local maximum and to have both extrema in 0 1 and we also visualize them in figures as an application we present a necessary and sufficient condition for the offset curve to be regular and to have the same tangent direction with the given quadratic rational bezier curve and give a simple algorithm to find it

a separable cross entropy approach to power spectral estimation an approach to power spectrum estimation that is based on a separable cross entropy modeling procedure is presented the authors start with a model of a multichannel multidimensional stationary gaussian random process that is sampled on a nonuniform grid an approximate separable model in which selected frequency samples of the process are modeled as independent random variables is then fitted to it two cross entropy like criteria are used to select optimal separable approximations one of them yields a spectral estimation algorithm that is a generalized version of capon s maximum likelihood method and the other is similar to classical windowing methods they discuss different strategies for designing bandpass filters for use with the cross entropy approach

relationships between adaptive minimum variance beamforming and optimal source localization for many years the popular minimum variance mv adaptive beamformer has been well known for not having been derived as a maximum likelihood ml estimator this paper demonstrates that by use of a judicious decomposition of the signal and noise the log likelihood function of source location is in fact directly proportional to the adaptive mv beamformer output power in the proposed model the measurement consists of an unknown temporal signal whose spatial wavefront is known as a function of its unknown location which is embedded in complex gaussian noise with unknown but positive definite covariance further in cases where the available observation time is insufficient a constrained ml estimator is derived here that is closely related to mv beamforming with a diagonally loaded data covariance matrix estimate the performance of the constrained ml estimator compares favorably with robust mv techniques giving slightly better root mean square error rmse angle of arrival estimation of a plane wave signal in interference more importantly however the fact that such optimal ml techniques are closely related to conventional robust mv methods such as diagonal loading lends theoretical justification to the use of these practical approaches

levinson s algorithm wold s decomposition and spectral estimation levinson s algorithm is developed in the context of mean square estimation and is applied to a variety of topics related to wiener filtering and spectral estimation the study includes the innovations approach to prediction theory wold s decomposition lattice filters autoregressive processes the method of maximum entropy and the general class of extrapolating spectra

power spectral density bounds corresp the determination of a power density spectrum from a finite set of correlation samples is an ill posed problem furthermore it is not possible even to bound the values that consistent power density spectra can take on at a particular point a more reasonable problem is to try to determine the total spectral power in some frequency interval although this power cannot be determined exactly upper and lower bounds on its possible values can be determined this observation leads to a unified treatment of certain classical and modern spectral estimation techniques and to new interpretations for two data adaptive spectral estimators maximum likelihood method mlm and data adaptive spectral estimator dase according to these new interpretations mlm and dase provide upper bounds on spectral power in a specified frequency region subject to the assumption that the spectral density is constant in that region these methods make no use of an extendibility constraint that can be used to obtain tight upper bounds as well as nontrivial lower bounds on power cybenko has studied a related problem of bounding windowed power for an arbitrary window with no assumptions about the form of the spectral density a new type of classical resolution limit for these bounds is derived and a numerical example is presented

some new g1 quartic parametric approximants of circular arcs abstract in this paper a method for approximation of circular arc by quartic bezier curve is presented interpolation by geometrically continuous g 1 parametric polynomials is considered and necessary and sufficient conditions for the existence of the solution are established optimal asymptotic approximation order eight is also confirmed for the obtained approximant application of our new solution to approximation of conic sections is presented and some numerical examples provided

a new interpretation of capon s maximum likelihood method of frequency wavenumber spectral estimation the maximum likelihood method of frequency wavenumber spectral estimation mlm is a high resolution alternative to conventional fourier analysis this paper provides a new interpretation for mlm the point of view taken is that at each wavenumber we attempt to estimate the power of a discrete planewave at that wavenumber a problem complicated by the presence of interfering energy at other wavenumbers noise with an unknown spectral density matrix it is shown that subject to some assumptions the problem of choosing the variance of the planewave complex amplitude that is the plane wave power and the noise covariance matrix jointly to maximize the likelihood function for the observations has no unique solution there is a continuous range of values for the estimates all of which have the same maximum likelihood the lower bound on the planewave power estimate being zero and the upper bound being the mlm power estimate some important differences between the problem of estimating the planewave complex amplitude for each independent data segment and the problem of estimating the variance of the planewave complex amplitude are indicated

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x

existence and multiplicity of weak solutions for elliptic dirichlet problems with variable exponent abstract under appropriate growth conditions on the nonlinearity the existence of multiple solutions for nonlinear elliptic dirichlet problems with variable exponent is established the approach is based on variational methods some applications and examples illustrate the obtained results

explicit g 1 approximation of conic sections using bezier curves of arbitrary degree in this paper we propose an explicit and effective method for g 1 approximation of conic sections with arbitrary degree polynomial curves the geometric constraints can be expressed by two variables then we construct an objective function as the alternative approximation error function in the l 2 norm which is a quadratic function in these two variables to minimize the objective function is equivalent to solving a system of linear equations with two variables since its coefficient matrix is invertible we can explicitly obtain the unique solution and then derive the explicit g 1 approximation of conic sections by bezier curves of arbitrary degree the proposed method has an optimal approximation in the l 2 norm i e the l 2 distance error reaches minimum and also can process the conic section with center angle larger than without subdivision scheme finally numerical examples demonstrate the effectiveness of our method

gram matrix of bernstein basis this note presents explicit expressions for the inverses of the gram matrix of the bernstein basis and its principal submatrices by taking the advantages of the transformations between the bernstein basis and the constrained dual bernstein basis using the symmetry property fast calculation of these matrices and their inverses is achieved significant improvements are obtained for applications including polynomial approximation of functions and degree reduction of bezier curves

on polynomial approximation of circular arcs and helices we present a simple method for polynomial approximation of circular arcs and helices by expressing the trigonometric functions using the two point taylor expansion we obtain the degree 2n 1 polynomial for the approximation problem in an efficient way which is very convenient to increase the degree of polynomial by adding new terms an upper bound on the approximation error is available so that we can obtain the lowest degree polynomial curve that can approximate a circular arc or helix segment within any user prescribed error tolerance

joint spatial division and multiplexing the large scale array regime we propose joint spatial division and multiplexing jsdm an approach to multiuser mimo downlink that exploits the structure of the correlation of the channel vectors in order to allow for a large number of antennas at the base station while requiring reduced dimensional channel state information at the transmitter csit jsdm achieves significant savings both in the downlink training and in the csit uplink feedback thus making the use of large antenna arrays at the base station potentially suitable also for frequency division duplexing fdd systems for which uplink downlink channel reciprocity cannot be exploited in the proposed scheme the multiuser mimo downlink precoder is obtained by concatenating a prebeamforming matrix which depends only on the channel second order statistics with a classical multiuser precoder based on the instantaneous knowledge of the resulting reduced dimensional effective channel matrix we prove a simple condition under which jsdm incurs no loss of optimality with respect to the full csit case for linear uniformly spaced arrays we show that such condition is approached in the large number of antennas limit for this case we use szego s asymptotic theory of toeplitz matrices to show that a dft based prebeamforming matrix is near optimal requiring only coarse information about the users angles of arrival and angular spread finally we extend these ideas to the case of a 2 d base station antenna array with 3 d beamforming including multiple beams in the elevation angle direction we provide guidelines for the prebeamforming optimization and calculate the system spectral efficiency under proportional fairness and max min fairness criteria showing extremely attractive performance our numerical results are obtained via asymptotic random matrix theory avoiding lengthy monte carlo simulations and providing accurate results for realistic finite number of antennas and users

bezier representation of the constrained dual bernstein polynomials abstract explicit formulae for the bezier coefficients of the constrained dual bernstein basis polynomials are derived in terms of the hahn orthogonal polynomials using difference properties of the latter polynomials efficient recursive scheme is obtained to compute these coefficients applications of this result to some problems of cagd is discussed

the best g 1 cubic and g 2 quartic bezier approximations of circular arcs we obtain cubic and quartic bezier approximations of circular arcs that respectively satisfy g 1 and g 2 end point interpolation conditions we identify the necessary and sufficient conditions for such approximations to be the best in the sense that they have the minimum hausdorff distance to the circular arc we then establish the existence and uniqueness of these best approximations and present practical methods to calculate them which are verified by examples

random matrix derived shrinkage of spectral precision matrices there has been much research on shrinkage methods for real valued covariance matrices and their inverses precision matrices in spectral analysis of p vector valued time series complex valued spectral matrices and precision matrices arise and good shrinkage methods are often required most notably when the estimated complex valued spectral matrix is singular as an improvement on the ledoit wolf lw type of spectral matrix estimator we use random matrix theory to derive a rao blackwell estimator for a spectral matrix its inverse being a rao blackwellized estimator for the spectral precision matrix a random matrix method has previously been proposed for complex valued precision matrices it was implemented by very costly simulations we formulate a fast completely analytic approach moreover we derive a way of selecting an important parameter using predictive risk methodology we show that both the rao blackwell estimator and the random matrix estimator of the precision matrix can substantially outperform the inverse of the lw estimator in a time series setting our new methodology is applied to eeg derived time series data where it is seen to work well and deliver substantial improvements for precision matrix estimation

capacity for potentials of functions in musielak orlicz spaces abstract we define a capacity for potentials of functions in musielak orlicz spaces basic properties of such capacity are studied we also estimate the capacity of balls and give some applications of the estimates

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

cylindrical algebraic decomposition i the basic algorithm given a set of r variate integral polynomials a cylindrical algebraic decomposition cad of euclidean r space e r partitions e r into connected subsets compatible with the zeros of the polynomials by compatible with the zeros of the polynomials we mean that on each subset of e r each of the polynomials either vanishes everywhere or nowhere for example consider the bivariate polynomial y 4 2 y 3 y 2 3 x 2 y 2 x 4

necessary and sufficient conditions for rational quartic representation of conic sections conic section is one of the geometric elements most commonly used for shape expression and mechanical accessory cartography a rational quadratic bezier curve is just a conic section it cannot represent an elliptic segment whose center angle is not less than p however conics represented in rational quartic format when compared to rational quadratic format enjoy better properties such as being able to represent conics up to 2 p but not including 2 p without resorting to negative weights and possessing better parameterization therefore it is actually worth studying the necessary and sufficient conditions for the rational quartic bezier representation of conics this paper attributes the rational quartic conic sections to two special kinds that is degree reducible and improperly parameterized on this basis the necessary and sufficient conditions for the rational quartic bezier representation of conics are derived they are divided into two parts bezier control points and weights these conditions can be used to judge whether a rational quartic bezier curve is a conic section or for a given conic section present positions of the control points and values of the weights of the conic section in form of a rational quartic bezier curve many examples are given to show the use of our results

voronoi diagram and medial axis algorithm for planar domains with curved boundaries i theoretical foundations in this first installment of a two part paper the underlying theory for an algorithm that computes the voronoi diagram and medial axis of a planar domain bounded by free form polynomial or rational curve segments is presented an incremental approach to computing the voronoi diagram is used wherein a single boundary segment is added to an existing boundary segment set at each step the introduction of each new segment entails modifying the voronoi regions of the existing boundary segments and constructing the voronoi region of the new segment we accomplish this by i computing the bisector of the new segment with each of the current boundary segments ii updating the voronoi regions of the current boundary segments by partitioning them with these bisectors and iii constructing the voronoi region of the new segment as a union of regions obtained from the partitioning in ii when all boundary segments are included and their voronoi regions have been constructed the voronoi diagram of the boundary is obtained as the union of the voronoi polygons for each boundary segment to construct the medial axis of a planar domain we first compute the voronoi diagram of its boundary the medial axis is then obtained from the voronoi diagram by i removing certain edges of the voronoi diagram that do not belong to the medial axis and ii adding certain edges that do belong to the medial axis but are absent from the voronoi diagram unambiguous characterizations for edges in both these categories are given details of algorithms based on this theory are deferred to the second installment of this two part paper

approximating conic sections by constrained bezier curves of arbitrary degree in this paper an algorithm for approximating conic sections by constrained bezier curves of arbitrary degree is proposed first using the eigenvalues of recurrence equations and the method of undetermined coefficients some exact integral formulas for the product of two bernstein basis functions and the denominator of rational quadratic form expressing conic section are given then using the least squares method a matrix based representation of the control points of the optimal bezier approximation curve is deduced this algorithm yields an explicit arbitrary degree bezier approximation of conic sections which has function value and derivatives at the endpoints that match the function value and the derivatives of the conic section up to second order and is optimal in the l 2 norm to reduce error the method can be combined with a curve subdivision scheme computational examples are presented to validate the feasibility and effectiveness of the algorithm for a whole curve or its part generated by a subdivision

circular sector area preserving approximation of circular arcs by geometrically smooth parametric polynomials abstract the quality of the approximation of circular arcs by parametric polynomials is usually measured by the hausdorff distance it is sometimes important that a parametric polynomial approximant additionally preserves some particular geometric property in this paper we study the circular sector area preserving parametric polynomial approximants of circular arcs a general approach to this problem is considered and corresponding nonlinear equations are derived for the approximants possessing the maximal order of geometric smoothness a scalar nonlinear equation is analyzed in detail for the parabolic the cubic and the quartic case the existence of the admissible solution is confirmed moreover the uniqueness of the solution with the optimal approximation order with respect to the radial distance is proved theoretical results are confirmed by numerical examples

some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size this paper analyzes whether standard covariance matrix tests work when dimensionality is large and in particular larger than sample size in the latter case the singularity of the sample covariance matrix makes likelihood ratio tests degenerate but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well defined we study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together with their ratio converging to a finite nonzero limit we find that the existing test for sphericity is robust against high dimensionality but not the test for equality of the covariance matrix to a given matrix for the latter test we develop a new correction to the existing test statistic that makes it robust against high dimensionality

circle approximation using ln bezier curves of even degree and its application abstract we present an approximation method of circular arcs using linear normal ln bezier curves of even degree four and higher our method achieves g m continuity for endpoint interpolation of a circular arc by a ln bezier curve of degree 2m for m 2 3 we also present the exact hausdorff distance between the circular arc and the approximating ln bezier curve we show that the ln curve has an approximation order of 2 m 2 for m 2 3 our approximation method can be applied to offset approximation so obtaining a rational bezier curve as an offset approximant we derive an algorithm for offset approximation based on the ln circle approximation and illustrate our method with some numerical examples

voronoi diagram and medial axis algorithm for planar domains with curved boundaries ii detailed algorithm description abstract details of algorithms to construct the voronoi diagrams and medial axes of planars domain bounded by free form polynomial or rational curve segments are presented based on theoretical foundations given in the first installment ramamurthy and farouki j comput appl math 1999 102 119 141 of this two part paper in particular we focus on key topological and computational issues that arise in these constructions the topological issues include i the data structures needed to represent various geometrical entities bisectors voronoi regions etc and ii the boolean operations i e union intersection and difference on planar sets required by the algorithm specifically representations for the voronoi polygons of boundary segments and for individual voronoi diagram or medial axis edges are proposed since these edges may be segments of a nonrational algebraic curves curve curve bisectors b rational curves point curve bisectors or c straight lines point point bisectors data structures tailored to each of these geometrical entities are introduced the computational issues addressed include the curve intersection algorithms required in the boolean operations and iterative schemes used to precisely locate bifurcation or n prong points n 3 of the voronoi diagram and medial axis a selection of computed voronoi diagram and medial axis examples is included to illustrate the capabilities of the algorithm

cylindrical algebraic decomposition ii an adjacency algorithm for the plane in part i of the present paper we defined cylindrical algebraic decompositions cad s and described an algorithm for cad construction in part ii we give an algorithm that provides information about the topological structure of a cad of the plane informally two disjoint cells in e r r 1 are adjacent if they touch each other formally they are adjacent if their union is connected in a picture of a cad eg fig 2 of part i it is obvious to the eye which pairs of cells are adjacent however the cad algorithm of part i does not actually produce this information nor did the original version of that algorithm in collins 1975

on the behavior of a capon type spectral density estimator in this paper we propose a capon type estimator for the spectrum of a stationary time series this estimator may be viewed as an alternative to classical periodogram based estimators its advantage is that it copes with the leakage effect by using implicity automatic adaptive windowing we show its asymptotic equivalence to a random variable which is a quadratic form in the observations thus obtaining the asymptotic normality of the capon estimator we also study its asymptotic bias and variance

sequence of gn ln polynomial curves approximating circular arcs abstract in this paper we derive a sequence of linear normal ln curves b 2 n of degree 2 n which are g n endpoint interpolations of a circular arc and have approximation order 2 n 2 this is an extension of the circle approximation method by ln bezier curves given in ahn and hoffmann 2014 to all even degrees we also extend the circle approximation to an ellipse approximation by g n ln curves of degree 2 n an upper bound of the hausdorff distance between the ellipse and its ln approximation is obtained we illustrate our results through an ln approximation of convolution curves of ellipses and a spline curve

approximation of circular arcs by bezier curves abstract for the circular arc of angle 0 we present the explicit form of the best gc 3 quartic approximation and the best gc 2 quartic approximations of various types and give the explicit form of the hausdorff distance between the circular arc and the approximate bezier curves for each case we also show the existence of the gc 4 quintic approximations to the arc and find the explicit form of the best gc 3 quintic approximation in certain constraints and their distances from the arc all approximations we construct in this paper have the optimal order of approximation twice of the degree of approximate bezier curves

compact embeddings in the generalized sobolev space w01 p g and existence of solutions for nonlinear elliptic problems abstract our aim in this paper is to deal with the compact embedding in the generalized sobolev space w 0 1 p g with a variable exponent satisfying the log holder condition as an application we find a nontrivial weak solution of the nonlinear elliptic problem div u x p x 2 u x f x u x in g u x 0 on g which is an extension of fan zhang x fan g h zhang existence of solutions for p x laplacian dirichlet problem nonlinear anal 52 2003 pp 1843 1852 theorem 4 7

besov type spaces with variable smoothness and integrability abstract in this article the authors introduce besov type spaces with variable smoothness and integrability the authors then establish their characterizations respectively in terms of transforms in the sense of frazier and jawerth smooth atoms or peetre maximal functions as well as a sobolev type embedding as an application of their atomic characterization the authors obtain a trace theorem of these variable besov type spaces

cs decomposition based bayesian subspace estimation in numerous applications it is required to estimate the principal subspace of the data possibly from a very limited number of samples additionally it often occurs that some rough knowledge about this subspace is available and could be used to improve subspace estimation accuracy in this case this is the problem we address herein and in order to solve it a bayesian approach is proposed the main idea consists of using the cs decomposition of the semi orthogonal matrix whose columns span the subspace of interest this parametrization is intuitively appealing and allows for non informative prior distributions of the matrices involved in the cs decomposition and very mild assumptions about the angles between the actual subspace and the prior subspace the posterior distributions are derived and a gibbs sampling scheme is presented to obtain the minimum mean square distance estimator of the subspace of interest numerical simulations and an application to real hyperspectral data assess the validity and the performances of the estimator

on a relation between maximum likelihood classification and minimum relative entropy classification corresp maximum likelihood ml and minimum relative entropy mre minimum cross entropy classification of samples from an unknown probability density when the hypotheses comprise an exponential family are considered it is shown that ml and mre lead to the same classification nde and the result is illustrated in terms of a method for estimating covariance matrices recently developed by burg luenberger and wenger mre classification applies to the general case in which it cannot be assumed that the samples were generated by one of the hypothesis densities the common use of ml in this case is technically incorrect but the equivalence of mre and ml provides a theoretical justification

existence results for some anisotropic dirichlet problems abstract this paper concerns with a class of elliptic anisotropic dirichlet problems depending of one real parameter on bounded euclidean domains our approach is based on variational and topological methods more concretely along the paper we show the existence of at least two weak solutions for the treated problem by using a direct consequence of the celebrated pucci and serrin theorem in addition to a local minimum result for differentiable functionals due to ricceri this abstract approach has been developed for equations on carnot groups see 15

evaluating the boundary and covering degree of planar minkowski sums and other geometrical convolutions algorithms are developed based on topological principles to evaluate the boundary and internal structure of the minkowski sum of two planar curves a graph isotopic to the envelope curve is constructed by computing its characteristic points the edges of this graph are in one to one correspondence with a set of monotone envelope segments a simple formula allows a degree to be assigned to each face defined by the graph indicating the number of times its points are covered by the minkowski sum the boundary can then be identified with the set of edges that separate faces of zero and non zero degree and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy for applications that require only the minkowski sum boundary the algorithm minimizes geometrical computations on the internal envelope edges that do not contribute to the final boundary in other applications this internal structure is of interest and the algorithm provides comprehensive information on the covering degree for different regions within the minkowski sum extensions of the algorithm to the computation of minkowski sums in r 3 and other forms of geometrical convolution are briefly discussed