Global Polynomials Research Articles

Intersection homology and Alexander modules of hypersurface complements

Let V be a degree d , reduced hypersurface in \mathbb{CP}^{n+1} , n \geq 1 , and fix a generic hyperplane, H . Denote by \mathcal{U} the (affine) hypersurface complement, \mathbb{CP}^{n+1}-V \cup H , and let \mathcal{U}^c be the infinite cyclic covering of \mathcal{U} corresponding to the kernel of the total linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules H_i(\mathcal{U}^c;\mathbb{Q}) of the hypersurface complement and show that, if i \leq n , these are torsion over the ring of rational Laurent polynomials. We also obtain obstructions on the associated global polynomials: their zeros are roots of unity of order d and are entirely determined by the local topological information encoded by the link pairs of singular strata of a stratification of the pair (\mathbb{CP}^{n+1},V) . As an application, we give obstructions on the eigenvalues of monodromy operators associated to

Nonlinear speech analysis using models for chaotic systems

In this paper, we use concepts and methods from chaotic systems to model and analyze nonlinear dynamics in speech signals. The modeling is done not on the scalar speech signal, but on its reconstructed multidimensional attractor by embedding the scalar signal into a phase space. We have analyzed and compared a variety of nonlinear models for approximating the dynamics of complex systems using a small record of their observed output. These models include approximations based on global or local polynomials as well as approximations inspired from machine learning such as radial basis function networks, fuzzy-logic systems and support vector machines. Our focus has been on facilitating the application of the methods of chaotic signal analysis even when only a short time series is available, like phonemes in speech utterances. This introduced an increased degree of difficulty that was dealt with by resorting to sophisticated function approximation models that are appropriate for short data sets. Using these models enabled us to compute for short time series of speech sounds useful features like Lyapunov exponents that are used to assist in the characterization of chaotic systems. Several experimental insights are reported on the possible applications of such nonlinear models and features.

Treatment of discontinuity in the reproducing kernel element method

AbstractA discontinuous reproducing kernel element approximation is proposed in the case where weak discontinuity exists over an interface in the physical domain. The proposed method can effectively take care of the discontinuity of the derivative by truncating the window function and global partition polynomials. This new approximation keeps the advantage of both finite element methods and meshfree methods as in the reproducing kernel element method. The approximation has the interpolation property if the support of the window function is contained in the union of the elements associated with the corresponding node; therefore, the continuity of the primitive variables at nodes on the interface is ensured. Furthermore, it is smooth on each subregion (or each material) separated by the interface. The major advantage of the method is its simplicity in implementation and it is computationally efficient compared to other methods treating discontinuity. The convergence of the numerical solution is validated through calculations of some material discontinuity problems. Copyright © 2005 John Wiley & Sons, Ltd.

A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes

In this paper we propose a Lagrange multiplier method for the finite element solution of multi-domain elliptic partial differential equations using non-matching meshes. The interface Lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes (e.g, we will consider global polynomials as multiplier). The ideas are illustrated using Poisson’s equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.

Reproducing kernel element method Part III: Generalized enrichment and applications

In this part of the work, a notion of generalized enrichment is proposed to construct the global partition polynomials or to enrich global partition polynomial basis with extra terms corresponding to the higher order derivatives of primary variable. This is accomplished by either multiplying enrichment functions with the original global partition polynomials, or increasing the order of global partition polynomials in the same mesh. Without refining mesh, high order consistency in interpolation hierarchy with generalized Kronecker delta property can be straightforwardly achieved in quadrilateral and triangular mesh in 2D by the proposed scheme. Comparing with the traditional finite element methods, the construction proposed here has more flexibility and only needs minimal degrees of freedom. The optimal element with high reproducing capacity and overall minimal degrees of freedom can be constructed by the generalized enrichment procedure. Two optimal elements in two dimensional space have been constructed: T10P3I 4 3 triangular element satisfies third order consistency condition with only 10 degrees of freedom, and Q15P4I 4 3 quadrilateral element satisfies fourth order consistency condition with 15 degrees of freedom. The performance of interpolation hierarchy is evaluated through solving some bench-mark problems for thin (Kirchhoff) plates.

Reproducing kernel element method Part II: Globally conforming Im/ Cn hierarchies

In this part of the work, a minimal degrees of freedom, arbitrary smooth, globally compatible, I m / C n interpolation hierarchy is constructed in the framework of reproducing kernel element method (RKEM) for arbitrary multiple dimensional domains. This is the first interpolation hierarchical structure that has been constructed with both minimal degrees of freedom and higher order smoothness or continuity over multi-dimensional domain. The proposed hierarchical structure possesses the generalized Kronecker property, i.e. ∂ αΨ (β) I/ ∂x α(x J)=δ IJδ αβ, |α|,|β|⩽m. This contribution is the latest breakthrough of an outstanding problem––construction of a minimal degrees of freedom, globally conforming, I m / C n finite element interpolation fields on an arbitrary mesh or subdivision of multiple dimension. The newly constructed globally conforming interpolant is a hybrid of a set of C ∞ global partition polynomials with a highly smooth ( C n ) compactly supported meshfree partition of unity. Examples of compatible RKEM hierarchical interpolations are illustrated, and they are used in a Galerkin procedure to solve differential equations.

Global unitary fixing and matrix-valued correlations in matrix models

We consider the partition function for a matrix model with a global unitary invariant energy function. We show that the averages over the partition function of global unitary invariant trace polynomials of the matrix variables are the same when calculated with any choice of a global unitary fixing, while averages of such polynomials without a trace define matrix-valued correlation functions, that depend on the choice of unitary fixing. The unitary fixing is formulated within the standard Faddeev–Popov framework, in which the squared Vandermonde determinant emerges as a factor of the complete Faddeev–Popov determinant. We give the ghost representation for the FP determinant, and the corresponding BRST invariance of the unitary-fixed partition function. The formalism is relevant for deriving Ward identities obeyed by matrix-valued correlation functions.

The Finite Element Method for Computing the Stationary Distribution of an SRBM in a Hypercube with Applications to Finite Buffer Queueing Networks

This paper proposes an algorithm, referred to as BNAfm (Brownian network analyzer with finite element method), for computing the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in a hypercube. The SRBM serves as an approximate model of queueing networks with finite buffers. Our BNAfm algorithm is based on the finite element method and an extension of a generic algorithm developed by Dai and Harrison [14]. It uses piecewise polynomials to form an approximate subspace of an infinite-dimensional functional space. The BNAfm algorithm is shown to produce good estimates for stationary probabilities, in addition to stationary moments. This is in contrast to the BNAsm algorithm (Brownian network analyzer with spectral method) of Dai and Harrison [14], which uses global polynomials to form the approximate subspace and which sometimes fails to produce meaningful estimates of these stationary probabilities. Extensive computational experiences from our implementation are reported, which may be useful for future numerical research on SRBMs. A three-station tandem network with finite buffers is presented to illustrate the effectiveness of the Brownian approximation model and our BNAfm algorithm.

Global polynomial approximation for Symm's equation on polygons

Summary. To solve 1D linear integral equations on bounded intervals with nonsmooth input functions and solutions, we have recently proposed a quite general procedure, that is essentially based on the introduction of a nonlinear smoothing change of variable into the integral equation and on the approximation of the transformed solution by global algebraic polynomials. In particular, the new procedure has been applied to weakly singular equations of the second kind and to solve the generalized air foil equation for an airfoil with a flap. In these cases we have obtained arbitrarily high orders of convergence through the solution of very-well conditioned linear systems. In this paper, to enlarge the domain of applicability of our technique, we show how the above procedure can be successfully used also to solve the classical Symm's equation on a piecewise smooth curve. The collocation method we propose, applied to the transformed equation and based on Chebyshev polynomials of the first kind, has shown to be stable and convergent. A comparison with some recent numerical methods using splines or trigonometric polynomials shows that our method is highly competitive.

A Spectral Patching Method for Direct Trajectory Optimization

A new direct method for trajectory optimization is presented in this paper. This method is based on using global orthogonal polynomials instead of the traditional spline-integration methods to discretize the dynamical constraints. The states and controls are approximated by interpolating polynomials with unknown coefficients as the values of the functions at the collocation points. The collocation points are the Legendre-Gauss-Lobatto points which provide the smallest error in interpolation in the least-squares sense. They also provide an expression for the Lagrange interpolating polynomials used as trial functions in terms of Legendre polynomials. In order to achieve a greater freedom in the distribution of the collocation points, we present a patching method wherein the time domain may be arbitrarily divided into a few subintervals. The state equations over these subintervals are discretized by the spectral collocation method. Then, the solutions on these subintervals are “patched” by a continuity condition. In this patching method, by judiciously combining the ideas of the traditional collocation methods with global spectral methods, we achieve advantages of both methods while using neither. Results for the simplified Moon-landing problem (selected in honor of Professor Battin) show that the spectral patching method works well for just two subintervals.

GLOBAL NONLINEAR POLYNOMIAL MODELS: STRUCTURE, TERM CLUSTERS AND FIXED POINTS

In this paper, it is shown that the number and location of the fixed points of global nonlinear polynomials can be specified in terms of the term clusters and cluster coefficients of the respective models. It is also shown that if the structure (that is, the model basis) of a nonlinear polynomial model of degree ℓ includes all possible terms then such a model will always have ℓ nontrivial fixed points. In modeling problems, where the estimated polynomials are to reproduce fundamental invariants of the original system, this situation will not always be welcome and therefore suggests that the structure of polynomial models should be carefully chosen. Ways of achieving this are briefly mentioned.

Dynamical effects of overparametrization in nonlinear models

This paper is concemed with dynamical reconstruction for nonlinear systems. The effects of the driving function and of the complexity of a given representation on the bifurcation patter are investigated. It is shown that the use of different driving functions to excite the system may yield models with different bifurcation patterns. The complexity of the reconstructions considered is quantified by the embedding dimension and the number of estimated parameters. In this respect it appears that models which reproduce the original bifurcation behaviour are of limited complexity and that excessively complex models tend to induce ghost bifurcations and spurious dynamical regimes. Moreover, some results suggest that the effects of overparametrization on the global dynamical behaviour of a nonlinear model may be more deleterious than the presence of moderate noise levels. In order to precisely quantify the complexity of the reconstructions, global polynomials are used although the results are believed to apply to a much wider class of representations including neural networks.

Dynamical effects of overparametrization in nonlinear models

Abstract This paper is concemed with dynamical reconstruction for nonlinear systems. The effects of the driving function and of the complexity of a given representation on the bifurcation patter are investigated. It is shown that the use of different driving functions to excite the system may yield models with different bifurcation patterns. The complexity of the reconstructions considered is quantified by the embedding dimension and the number of estimated parameters. In this respect it appears that models which reproduce the original bifurcation behaviour are of limited complexity and that excessively complex models tend to induce ghost bifurcations and spurious dynamical regimes. Moreover, some results suggest that the effects of overparametrization on the global dynamical behaviour of a nonlinear model may be more deleterious than the presence of moderate noise levels. In order to precisely quantify the complexity of the reconstructions, global polynomials are used although the results are believed to apply to a much wider class of representations including neural networks.

Chaotic time series. Part II. System Identification and Prediction

This paper is the second in a series of two, and describes the current state of the art in modelling and prediction of chaotic time series. Sampled data from deterministic non-linear systems may look stochastic when analysed with linear methods. However, the deterministic structure may be uncovered and non-linear models constructed that allow improved prediction. We give the background for such methods from a geometrical point of view, and briefly describe the following types of methods: global polynomials, local polynomials, multi layer perceptrons and semi-local methods including radial basis functions. Some illustrative examples from known chaotic systems are presented, emphasising the increase in prediction error with time. We compare some of the algorithms with respect to prediction accuracy and storage requirements, and list applications of these methods to real data from widely different areas.

Approximate three-dimensional analysis of rectangular thick laminated plates: Bending, vibration and buckling

A global-local approach is proposed to analyse thick laminated plates. This approach treats a thick laminated plate as a three-dimensional inhomogeneous anisotropic elastic body. The cross-section of a laminated plate is first discretized into conventional eight-node elements. The interpolation function along the span of the plate is defined by the cubic B 3-spline function. The displacement functions can be expressed as the product of the usual isoparametric shape functions and the spline function. A set of global polynomials of an appropriate order is selected to transform the nodal variables of the cross-section to a much smaller set of generalized parameters associated with the polynomials. These parameters can be obtained by means of the standard Rayleigh-Ritz technique. The total number of unknowns involved is drastically reduced with a minor sacrifice of accuracy. The six components of stresses, the fundamental natural frequencies and the critical buckling loads can be determined with acceptable accuracy. Numerical examples are given to demonstrate the accuracy and effectiveness of the global-local procedures.

Cell averaging Chebyshev methods for hyperbolic problems

This paper describes a cell averaging method for the Chebyshev approximations of first order hyperbolic equations in conservation form. We present formulas for transforming between pointwise data at the collocation points and cell averaged quantities, and vice-versa. This step, trivial for the finite difference and Fourier methods, is nontrivial for the global polynomials used in Spectral methods. We then prove that the cell averaging methods presented are stable for linear scalar hyperbolic equations and present numerical simulations of shock-density wave interaction using the new cell averaging Chebyshev methods.

Spectral methods with sparse matrices

Spectral methods employ global polynomials for approximation. Hence they give very accurate approximations for smooth solutions. Unfortunately, for Dirichlet problems the matrices involved are dense and have condition numbers growing asO(N 4) for polynomials of degree ?N in each variable. We propose a new spectral method for the Helmholtz equation with a symmetric and sparse matrix whose condition number grows only asO(N 2). Certain "algebraic spectral multigrid methods" can be efficiently used for solving the resulting system. Numerical results are presented which show that we have probably found the most effective solver for spectral systems.

A Partial Characterization of Poised Hermite–Birkhoff Interpolation Problems

A Partial Characterization of Poised Hermite–Birkhoff Interpolation Problems