two sided generalized hyperbolic qr factorization and its perturbation analysis abstract we first propose the two sided generalized hyperbolic qr factorization of a matrix pair which is a generalization of the generalized qr factorization more explicitly two orthogonal matrices in the generalized qr factorization are replaced with two corresponding j orthogonal matrices where j is a signature matrix then we consider the perturbation analysis of the new matrix factorization some first order normwise perturbation bounds are derived using the refined matrix equation approach and the updated matrix vector equation approach the corresponding perturbation bounds for the generalized qr factorization are also obtained as the special case in comparison these bounds can be much tighter than the previous ones derived from barrlund 1994 a barrlund perturbation bounds for the generalized qr factorization linear algebra appl 207 1994 251 271

perturbation bound of singular linear systems a classical perturbation bound in numerical linear algebra is extended to more general cases of singular linear algebraic systems and weighted linear least squares problem componentwise perturbation analysis of linear least squares problem is also derived

distance for degree raising and reduction of triangular bezier surfaces the problem of degree reduction and degree raising of triangular bezier surfaces is considered the l 2 and l 2 measures of distance combined with the least squares method are used to get a formula for the bezier points the methods use the matrix representations of the degree reduction and degree raising

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

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

on the early history of the singular value decomposition this paper surveys the contributions of five mathematicians eugenio beltrami 1835 1899 camille jordan 1838 1921 james joseph sylvester 1814 1897 erhard schmidt 1876 1959 and hermann weyl 1885 1955 who were responsible for establishing the existence of the singular value decomposition and developing its theory

the representation and approximation for the weighted moore penrose inverse in hilbert space we present a unified representation theorem of the weighted moore penrose inverse in hilbert space specific expressions and computational procedures for the weighted moore penrose inverse in hilbert space can be uniformly derived

optimal multi degree reduction of triangular bezier surfaces with corners continuity in the norm l2 this paper derives an approximation algorithm for multi degree reduction of a degree n triangular bezier surface with corners continuity in the norm l 2 the new idea is to use orthonormality of triangular jacobi polynomials and the transformation relationship between bivariate jacobi and bernstein polynomials this algorithm has a very simple and explicit expression in matrix form i e the reduced matrix depends only on the degrees of the surfaces before and after degree reduction and the approximation error of this degree reduced surface is minimum and can get a precise expression before processing of degree reduction combined with surface subdivision the piecewise degree reduced patches possess global c 0 continuity finally several numerical examples are presented to validate the effectiveness of this algorithm

two sided hyperbolic svd abstract in this paper we propose the two sided hyperbolic svd 2hsvd for square matrices i e a u v where u and v are j unitary j diag 1 and is a real diagonal matrix of double hyperbolic singular values we show that with some natural conditions such decomposition exists without the use of hyperexchange matrices in other words u and v are really j unitary with regard to j and not some matrix j which is permutationally similar to matrix j we provide full characterization of 2hsvd and completely relate it to the semidefinite j polar decomposition

on level 2 condition number for the weighted moore penrose inverse in this paper we present characterizations for the level 2 condition number of the weighted moore penrose inverse i e cond m n a cond m n 2 a cond m n a 1 where cond m n a is the condition number of the weighted moore penrose inverse of a rectangular matrix and cond m n 2 a is the level 2 condition number of this problem this paper extends the result by cucker diao and wei f cucker h diao y wei on the level 2 condition number for moore penrose inversion 2005 unpublished report and improves the results by wei and wang y wei d wang condition numbers and perturbation of weighted moore penrose inverse and weighted linear least squares problem appl math comput 145 2003 45 58

the hyperbolic schur decomposition we propose a hyperbolic counterpart of the schur decomposition with the emphasis on the preservation of structures related to some given hyperbolic scalar product we give results regarding the existence of such a decomposition and research the properties of its block triangular factor for various structured matrices

symbolic dynamics and periodic orbits for the cardioid billiard the periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics the corresponding partition is mapped to a topologically well ordered symbol plane in the symbol plane the pruning front is obtained from orbits running either into or through the cusp we show that all periodic orbits correspond to maxima of the lagrangian and give a complete list up to code length 15 the symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines we show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned all these orbits are related to finite orbits starting and ending in the cusp we obtain an analytical estimate of the kolmogorov sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits furthermore the statistical properties of periodic orbits are investigated

representations and computations of 2 3 we investigate full rank representations of 2 3 and 2 4 inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space in certain particular cases we obtain new results or generalizations related with 2 3 and 2 4 inverses in a hilbert unitary space direct and iterative methods for computing 2 3 and 2 4 inverses in an indefinite inner product space are developed numerical examples with different metric matrices are presented

good degree reduction of bezier curves using jacobi polynomials abstract the constrained chebyshev polynomial is the error function of the best degree reduction of polynomial with c1 continuity in this paper we propose the constrained jacobi polynomial as an alternative error function for good degree reduction although the degree reduction is not the best approximation it is more useful than the constrained chebyshev polynomial since its coefficients are represented explicitly but the coefficients of the constrained chebyshev polynomial are not we present the uniform error bounds of the constrained jacobi polynomials and subdivision scheme for degree reduction within given tolerance we also apply our method to an example and compare its result to that of the best degree reduction

some topics on weighted moore penrose inverse weighted least squares and weighted regularized tikhonov problems it is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of linear systems the classical normwise relative condition number formulas are given by higham linear algebra appl 214 1995 193 here we consider the condition number formulas for the weighted moore penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems ax b moreover we consider the nearness to rank deficiency finally we give some decomposition tools and their application in the regularized weighted tikhonov problems

saturation of number variance in embedded random matrix ensembles we study fluctuation properties of embedded random matrix ensembles of noninteracting particles for ensemble of two noninteracting particle systems we find that unlike the spectra of classical random matrices correlation functions are nonstationary in the locally stationary region of spectra we study the number variance and the spacing distributions the spacing distributions follow the poisson statistics which is a key behavior of uncorrelated spectra the number variance varies linearly as in the poisson case for short correlation lengths but a kind of regularization occurs for large correlation lengths and the number variance approaches saturation values these results are known in the study of integrable systems but are being demonstrated for the first time in random matrix theory we conjecture that the interacting particle cases which exhibit the characteristics of classical random matrices for short correlation lengths will also show saturation effects for large correlation lengths

degree reduction of composite bezier curves this paper deals with the problem of multi degree reduction of a composite bezier curve with the parametric continuity constraints at the endpoints of the segments we present a novel method which is based on the idea of using constrained dual bernstein polynomials to compute the control points of the reduced composite curve in contrast to other methods ours minimizes the l2 error for the whole composite curve instead of minimizing the l2 errors for each segment separately as a result an additional optimization is possible examples show that the new method gives much better results than multiple application of the degree reduction of a single bezier curve

recurrent neural networks for computing weighted moore penrose inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 three recurrent neural networks are presented for computing the weighted moore penrose inverse of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of weighted moore penrose inverse under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a weighted moore penrose inverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a weighted moore penrose inverse and a lagrangian matrix with constant connections

iterative process for g2 multi degree reduction of bezier curves 2 continuity abstract in this paper the issue of multi degree reduction of bezier curves with c 1 and g 2 continu ity at the end points of the curve is considered an iterative method which is the first of this type is derived it is shown that this algorithm converges and can be applied itera tively to get the required accuracy some examples and figures are given to demonstrate the efficiency of this method

a note on iterative process for g2 multi degree reduction of bezier curves abstract in the paper a rababah s mann iterative process for g 2 multi degree reduction of bezier curves applied mathematics and computation 217 2011 8126 8133 rababah and mann proposed an iterative method for multi degree reduction of bezier curves with c 1 and g 2 continuity at the endpoints in this paper we provide a theoretical proof for the existence of the unique solution in the first step of the iterative process while the proof in their paper applies only in some special cases also we give a complete convergence proof for the iterative method we solve the problem by using convex quadratic optimization

a note on the existence of the hyperbolic singular value decomposition abstract in this note we present a new matrix decomposition for a matrix pair a b with a hermitian the hyperbolic svd comes as a special case of the decomposition with a set to be the signature matrix

perturbation bounds for constrained and weighted least squares problems we derive perturbation bounds for the constrained and weighted linear least squares ls problems both the full rank and rank deficient cases are considered the analysis generalizes some results of earlier works

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

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

spectral fluctuations of billiards with mixed dynamics from time series to superstatistics a statistical analysis of the eigenfrequencies of two sets of superconducting microwave billiards one with mushroomlike shape and the other from the family of the limacon billiards is presented these billiards have mixed regular chaotic dynamics but different structures in their classical phase spaces the spectrum of each billiard is represented as a time series where the level order plays the role of time two most important findings follow from the time series analysis first the spectra can be characterized by two distinct relaxation lengths this is a prerequisite for the validity of the superstatistical approach which is based on the folding of two distribution functions second the shape of the resulting probability density function of the so called superstatistical parameter is reasonably approximated by an inverse chi2 distribution this distribution is used to compute nearest neighbor spacing distributions and compare them with those of the resonance frequencies of billiards with mixed dynamics within the framework of superstatistics the obtained spacing distribution is found to present a good description of the experimental ones and is of the same or even better quality as a number of other spacing distributions including the one from berry and robnik however in contrast to other approaches toward a theoretical description of spectral properties of systems with mixed dynamics superstatistics also provides a description of properties of the eigenfunctions in terms of a superstatistical generalization of the porter thomas distribution indeed the inverse chi2 parameter distribution is found suitable for the analysis of experimental resonance strengths in the limacon billiards within the framework of superstatistics

explicit algorithms for multiwise merging of bezier curves this paper presents a novel scheme called c r s multiwise merging for merging multiple segments of bezier curves using a single bezier curve it is considered as an extension of the existing pairwise merging to avoid the limitations caused by recursively applying pairwise merging to the multiple case an explicit algorithm is developed to obtain the merged curve which preserves c r and c s continuity at the endpoints and is optimal in the sense that the l 2 or l 2 distance is minimized as an application we develop explicit algorithms for g 1 multiwise merging always producing better results than c 1 multiwise merging

best polynomial degree reduction on q lattices with applications to q orthogonal polynomials we study the best polynomial degree reduction with respect to the q l2 norm we study a finite analogue with respect to finite q lattices we present applications to q orthogonal polynomials we show that a weighted least squares approximation of q bezier coefficients provides the best polynomial degree reduction in the q l2 norm we also provide a finite analogue of this result with respect to finite q lattices and we present applications of these results to q orthogonal polynomials

classes of normal matrices in indefinite inner products abstract we study classes of matrices defined by various normality properties with respect to an indefinite complex inner product the relationships between many such properties all of them equivalent to the normality in case of a definite inner product are described in particular a canonical form is developed for the class of matrices that are polynomials of a self adjoint matrix

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

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

spectral statistics in the quantized cardioid billiard the spectral statistics of the strongly chaotic cardioid billiard is studied the analysis is based on the first 11 000 quantal energy levels for both odd and even symmetry it is found that the level spacing distribution is in good agreement with the gaussian orthogonal ensemble distribution of random matrix theory in the cases of the number variance and rigidity we observe agreement with the random matrix model for short range correlations only whereas for long range correlations both statistics saturate in agreement with semiclassical expectations furthermore the conjecture that for classically chaotic systems the normalized mode fluctuations have a universal gaussian distribution with unit variance is tested and found to be in very good agreement for both symmetry classes by means of the gutzwiller trace formula the trace of the cosine modulated heat kernel is studied since the billiard boundary is focusing there are conjugate points that give rise to zeros instead of exclusively at gaussian peaks at the locations of the periodic orbits

approximating tensor product bezier surfaces with tangent plane continuity we present a simple method for degree reduction of tensor product bezier surfaces with tangent plane continuity in l 2 norm continuity constraints at the four corners of surfaces are considered so that the boundary curves preserve endpoints continuity of any order a we obtain matrix representations for the control points of the degree reduced surfaces by the least squares method a simple optimization scheme that minimizes the perturbations of some related control points is proposed and the surface patches after adjustment are c continuous in the interior and g 1 continuous at the common boundaries we show that this scheme is applicable to surface patches defined on chessboard like domains

existence of the hyperbolic singular value decomposition abstract the hyperbolic singular value decomposition is defined on a general n m matrix and an m m signature matrix pair it is employed in finding the eigenstructure of any matrix that is expressed as the difference of two matrix outer products such differences arise in signal processing applications in the context of the covariance differencing the hyperbolic svd applies in problems where the conventional svd cannot be employed the existence of the hyperbolic singular value decomposition is here extended to the most general case where neither the general matrix nor the matrix product is assumed full rank

quantum surface of section method demonstration of semiclassical berry robnik energy level spacing distribution in a generic two dimensional hamiltonian system the recently developed quantum surface of section method is applied to a search for extremely high lying energy levels in a simple but generic hamiltonian system between integrability and chaos namely the semiseparable two dimensional oscillator using the stretch of 13445 consecutive levels with the sequential number around 1 8 107 eighteen million we have clearly demonstrated the validity of the semiclassical berry robnik level spacing distribution while at 1000 times smaller sequential quantum numbers we find the very persistent quasi universal phenomenon of power law level repulsion which is globally very well described by the phenomenological brody distribution this is the first statistically significant numerical confirmation of the berry robnik surmise in an autonomous two dimensional hamiltonian system

positive real matrices in indefinite inner product spaces and invariant maximal semidefinite subspaces in this paper positive real matrices in indefinite inner product spaces are studied this class of matrices is intimately connected to dissipative matrices in the complex case first the complex case and then the real case are treated an explicit construction is given for invariant semidefinite subspaces

constrained multi degree reduction of triangular bezier surfaces using dual bernstein polynomials we propose a novel approach to the problem of multi degree reduction of bezier triangular patches with prescribed boundary control points we observe that the solution can be given in terms of bivariate dual discrete bernstein polynomials the algorithm is very efficient thanks to using the recursive properties of these polynomials the complexity of the method is o n 2m 2 n and m being the degrees of the input and output bezier surfaces respectively if the approximation with appropriate boundary constraints is performed for each patch of several smoothly joined triangular bezier surfaces the result is a composite surface of global c r continuity with a prescribed order r some illustrative examples are given

a simple matrix form for degree reduction of bezier curves using chebyshev bernstein basis transformations we use the matrices of transformations between chebyshev and bernstein basis and the matrices of degree elevation and reduction of chebyshev polynomials to present a simple and efficient method for r times degree elevation and optimal r times degree reduction of bezier curves with respect to the weighted l2 norm for the interval 0 1 using the weight function w x 1 4x 4x2 the error of the degree reduction scheme is given and the degree reduction with continuity conditions is also considered

distribution of resonance strengths in microwave billiards of mixed and chaotic dynamics a new measure for statistical properties of the wave function components of quantum systems the distribution of the product of two partial widths is introduced it is tested with data obtained in analog experiments with microwave billiards where the product of two partial widths equals the resonance strengths in the microwave spectra the billiards are from the family of the limacons one with chaotic and two with mixed classical dynamics for completely chaotic systems the partial widths generically obey a porter thomas distribution we show that in this case the distribution of their product equals a k0 distribution while we find deviations of the experimental strength distribution from the k0 distribution for the billiards with mixed dynamics the distributions agree perfectly for the chaotic billiard when taking into account the experimental threshold of detection in the theoretical description hence the strength distribution provides another stringent test for the connection between statistical properties of systems with classical chaotic dynamics and random matrix theory

moore penrose invertible normal and hermitian elements in rings in this paper we present several new characterizations of normal and hermitian elements in rings with involution in purely algebraic terms and considerably simplify proofs of already existing characterizations

degree reduction of bezier curves by l1 approximation with endpoint interpolation abstract we consider the one degree reduction problem with endpoint interpolation in the l1 norm we obtain the best one degree reduction of bezier curve of the degree n 5 with endpoint interpolation by using perfect splines for the general degree n we propose a good one degree reduction by use of an appropriate transform of the tchebycheff polynomials un x of the second kind of degree n by use of the good one degree reduction subdivision algorithm is given to get one degree reduced bezier curve within a given tolerance some numerical experiments are also given

vector least squares solutions for coupled singular matrix equations the weighted least squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition in this paper a family of algorithms are applied to solve these problems based on the kronecker structures subsequently we construct a computationally efficient solutions of coupled restricted singular matrix equations furthermore the need to compute the weighted drazin and weighted moore penrose inverses and the use of tian s work and lev ari s results are due to appearance in the solutions of these problems the several special cases of these problems are also considered which includes the well known coupled sylvester matrix equations finally we recover the iterative methods to the weighted case in order to obtain the minimum d norm g vector least squares solutions for the coupled sylvester matrix equations and the results lead to the least squares solutions and invertible solutions as a special case

modified jacobi bernstein basis transformation and its application to multi degree reduction of bezier curves this paper reports new modified jacobi polynomials mjps we derive the basis transformation between mjps and bernstein polynomials and vice versa this transformation is merging the perfect least square performance of the new polynomials together with the geometrical insight of bernstein polynomials the mjps with indexes corresponding to the number of endpoints constraints are the natural basis functions for least square approximation of bezier curves using mjps leads us to deal with the constrained jacobi polynomials and the unconstrained jacobi polynomials as orthogonal polynomials the mjps are automatically satisfying the homogeneous boundary conditions thereby the main advantage of using mjps in multi degree reduction of bezier curves on computer aided geometric design cagd is that the constraints in cagd are also satisfied and that decreases the steps of multi degree reduction algorithm several numerical results for the multi degree reduction of bezier curves on cagd are given we establish and prove the basis transformation between the modified jacobi polynomials mjps and bernstein polynomials and vice versa this transformation is merging the perfect least square performance of the mjps with the geometrical insight of the bernstein form the mjps with indexes corresponding to the number of endpoint constraints are the natural basis functions for least square approximation of bezier curves several numerical results are considered

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

behaviour of boundary functions for quantum billiards we study the behaviour of the normal derivative of eigenfunctions of the helmholtz equation inside billiards with dirichlet boundary condition these boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest therefore they form a reduced representation of the quantum system analogous to the poincare section of the classical system for the normal derivatives we introduce an equivalent to the standard green function and derive an integral equation on the boundary based on this integral equation we compute the first two terms of the mean asymptotic behaviour of the boundary functions for large energies the first term is universal and independent of the shape of the billiard the second one is proportional to the curvature of the boundary the asymptotic behaviour is compared with numerical results for the stadium billiard different limacon billiards and the circle billiard and good agreement is found furthermore we derive an asymptotic completeness relation for the boundary functions

extension and generalization properties of the weighted minkowski inverse in a minkowski space for an arbitrary matrix the weighted minkowski inverse a m n m n m related to the positive definite matrices m m m and n m n of an arbitrary matrix a m m n including singular and rectangular is one of the important generalized inverses for solving matrix equations in minkowski space in this paper the results are introduced in the following three ways first we establish some new and attractive properties of the weighted minkowski inverse a m n in a minkowski space second new representations and conditions for the continuity of the weighted minkowski inverse a m n in a minkowski space are discussed finally some illustrated counterexamples are also studied to show that some well known properties of the weighted moore penrose inverse a m n in a hilbert space h are not valid in the minkowski inverse a m n in a minkowski space

an explicit method for g 3 merging of two bezier curves this paper presents an explicit method for the g 3 merging problem of two bezier curves the main idea is to express the l 2 distance as a quadratic function of some parameters provided by g 3 continuity conditions an efficient non iterative algorithm is proposed to obtain the optimal merged curve when the l 2 distance is minimized the uniqueness of the global minimum is also proven this method can be applied to two adjacent curves with different degrees and has the ability to obtain satisfactory merging results by using curves of lower degree the efficiency and accuracy of the proposed explicit method are illustrated through several comparative examples

semiclassical trace formulas for noninteracting identical particles we extend the gutzwiller trace formula to systems of noninteracting identical particles the standard relation for isolated orbits does not apply since the energy of each particle is separately conserved causing the periodic orbits to occur in continuous families the identical nature of the particles also introduces discrete permutational symmetries we exploit the formalism of creagh and littlejohn phys rev a 44 836 1991 who have studied semiclassical dynamics in the presence of continuous symmetries to derive many body trace formulas for the full and symmetry reduced densities of states numerical studies of the three particle cardioid billiard are used to explicitly illustrate and test the results of the theory

explicit g2 constrained degree reduction of bezier curves by quadratic optimization in this paper we revisit g 2 constrained degree reduction of bezier curves which has been solved in our previous work by using iterative methods we propose an explicit and effective method for g 1 constrained degree reduction and c 1g 2 constrained degree reduction our main idea is to express the distance function defined in the l 2 norm as a strictly convex quadratic function of two variables which becomes a quadratic optimization problem we can explicitly obtain the unique solution by solving two linear equations such that the distance function is minimized the existence of the unique solution is also proved

multi degree reduction of tensor product bezier surfaces with general boundary constraints abstract we propose an efficient approach to the problem of multi degree reduction of rectangular bezier patches with prescribed boundary control points we observe that the solution can be given in terms of constrained bivariate dual bernstein polynomials the complexity of the method is o mn 1 n 2 with m min m1 m2 where n1 n2 and m1 m2 is the degree of the input and output bezier surface respectively if the approximation with appropriate boundary constraints is performed for each patch of several smoothly joined rectangular bezier surfaces the result is a composite surface of global cr continuity with a prescribed r 0 in the detailed discussion we restrict ourselves to r 0 1 which is the most important case in practical application some illustrative examples are given

bezier curves based on lupas p q analogue of bernstein functions in cagd abstract this paper deals with the extension of rational lupas bernstein functions lupas bezier curves and surfaces involving p q integers as shape parameters for all p 0 and q 0 two different techniques such as de casteljau s algorithm and korovkin s type approximation based on p q integers are used a two parameter family for lupas p q bernstein functions is constructed and their degree elevation and reduction properties have been studied for lupas p q bezier curves some of their basic properties as well as degree elevation and de casteljau algorithm have been discussed the new curves have some properties similar to lupas q bezier curves similarly the corresponding tensor product for lupas bezier surfaces over the rectangular domain u v 0 1 0 1 depending on four parameters is constructed the de casteljau algorithm and degree evaluation properties of the surfaces for these generalizations and some fundamental properties are discussed we get lupas q bezier surfaces for u v 0 1 0 1 when we set the parameter p 1 p 2 1 with the help of rational lupas bernstein functions p q lupas bernstein operators are constructed based on korovkin s type approximation it has been shown that the sequence of p q analogue of lupas bernstein operators l p n q n n f x converges uniformly to f x c 0 1 if and only if 0 q n p n 1 such that lim n q n 1 lim n p n 1 and lim n p n n 1 lim n q n n 1 on the other hand for any p 0 fixed and p 1 the sequence l p q n f x converges uniformly to f x c 0 1 if and only if f x a x b for some a b r furthermore in comparison to q bezier curves and surfaces based on lupas q bernstein rational functions our generalization gives us more flexibility in controlling the shapes of curves and surfaces

application of chebyshev ii bernstein basis transformations to degree reduction of bezier curves a polynomial curve on 0 1 can be expressed in terms of bernstein polynomials and chebyshev polynomials of the second kind we derive the transformation matrices that map the bernstein and chebyshev coefficients into each other and examine the stability of this linear map in the p 1 and norms the condition number of the chebyshev bernstein transformation matrix grows at a significantly slower rate with n than in the power bernstein case and the rate is very close somewhat faster to the legendre bernstein case using the transformation matrices we present a method for the best multi degree reduction with respect to the t t 2 weighted square norm for the unconstrained case which is further developed to provide a good approximation to the best multi degree reduction with constraints of endpoints continuity of orders r s r s 0 this method has a quadratic complexity and may be ill conditioned when it is applied to the curves of high degree we estimate the posterior l 1 error bounds for degree reduction

best one sided approximation of polynomials under l 1 norm in this paper we develop an analytic solution for the best one sided approximation of polynomials under l1 norm that is we find two polynomials with lower degree which bound the given polynomial such that the areas between the bounding polynomials and the given polynomial attain minimum the key ingredient of our technique is a characterization for one sided approximations based on orthogonal polynomials this result is applied in the degree reduction of interval polynomial bezier curves in computer aided design

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

a simple proof of the product theorem for ep matrices abstract we give a simple proof of a recent theorem of hartwig and katz on the product of ep matrices

new universal aspects of diffusion in strongly chaotic systems we study some new universal aspects of diffusion in chaotic systems especially those having very large lyapunov coefficients on the chaotic indecomposable topologically transitive component we do this by discretizing the chaotic component on the surface of section sos in a large number n of simplectically equally big cells in the sense of equal relative invariant ergodic measure normalized so that the total measure of the chaotic component is unity by iterating the transition of the chaotic orbit through the sos where j counts the number of iterations discrete time and assuming complete lack of correlation even between consecutive crossings which can be justified due to the very large lyapunov exponents we show the universal approach of the relative measure of the occupied cells denoted by to the asymptotic value of unity in the following way so that in the limit of big n we have for j n fixed the exponential law this analytic result is verified numerically in a variety of specific systems for a plane billiard robnik 1983 for a 3d billiard prosen 1997 for an ergodic logistic map tent map for a standard map k 400 and for the hydrogen atom in a strong magnetic field the agreement is almost perfect except in the latter two systems for some long time deviations on very small scales however for henon heiles system and for the standard map k 3 the deviations are noticeable though not very big only about 1 we have tested the random number generators press et al 1986 and confirmed that some are almost perfect ran0 and ran3 whilst two of them ran1 and ran2 exhibit big deviations

the representation and approximation for the weighted moore penrose inverse we present a unified representation theorem for the weighted moore penrose inverse specific expressions and computational procedures for the weighted moore penrose inverse can be uniformly derived

singular value decomposition in minkowski space abstract the well known singular value decomposition theorem states that every matrix m can be written in the form m qdr where d is diagonal and q and r are orthogonal we explore the analogous decomposition where q and r are required to be orthogonal with respect to the minkowski rather than the euclidean metric

a unified approach for degree reduction of polynomials in the bernstein basis part i real polynomials the polynomials determined in the bernstein bezier basis enjoy considerable popularity and the algorithms for reducing their degree are of practical importance in computer aided design applications on the other hand the conversion between the bernstein and the power basis is ill conditioned thus only the degree reduction algorithms which may be carried out without using this conversion are of practical value our unified approach enables us to describe all the algorithms of this kind known in the literature to construct a number of new ones which are better conditioned and cheaper than some of the currently used ones and to study the errors of all of them in a simple homogeneous way all these algorithms can be applied componentwise to a vector valued polynomial representing a bezier curve consider the values of the derivatives whose orders vary successively from 1 to a given number i or k at the start and end point respectively of this curve the current algorithms allow us to preserve these points and values for i equal to k the new ones do that also without the latter constraint