A Hamiltonian, explicit algorithm with spectral accuracy for the ‘good’ Boussinesq system

We construct an explicit pseudo-spectral method for the numerical solution of the soliton-producing ‘good’ Boussinesq system w_t = u_xxx + u_x + (u²)_x, u_t = w_x. The new scheme preserves a discrete Poisson structure similar to that of the continuous system. The scheme is shown to converge with spectral spatial accuracy. A numerical illustration is given.     

On the robust adaptive control of a contour-following robotic system

To demonstrate the force sensing and control, a possible model of a contour-following system is represented by a fourth-order linear continuous-time time-invariant system, in which stiffness k_t, robot natural frequency, ω_n, linear accommodation gain k_x, and angular accommodation gain k_φ are all constants obtained by measurement and experiment. This model works well for following a short contour. To satisfactorily follow a longer contour, k_t, ω_n, k_x and k_φ can be treated as unknown constants or time-varying variables. When they are considered as unknown constants, a robust model reference adaptive controller can be used to achieve both stability and tracking without having to know or find the true values of those constants, given bounded input or output disturbances and stable unmodeled dynamics. If ω_n is assumed to be a given constant but K_t, k_x, and k_φ are assumed to be unknown variables, then one has a linear time-varying plant and other types of model reference adaptive controllers have to be used to achieve the same purpose. In this paper, the schemes from various robust model reference adaptive control design will be studied and comparison and suggestions will also be made based on the simulation results for the contour-following robotic system mentioned above.     

Properties of Noncentral Dirichlet Distributions

Let X₁,…, X_r+1 be independent random variables, X_iGa (a_i, θ, δ_i), i = 1,…, r + 1. Define and V_i = X_i/X_r+1, i = 1,…, r. Then, (U₁,…, U_r) and (V₁,…, V_r) follow noncentral Dirichlet Type 1 and Type 2 distributions, respectively. In this article several properties of these distributions and their connections with the uniform, the noncentral multivariate-F and the noncentral multivariate-t distributions are discussed.     

1-Inverses for polynomial matrices of non-constant rank

Let p₁, … p_t be polynomials in ⁿ with a variety V of common zeros contained in a suitable open set U. Explicit formulas are provided to construct rational functions λ₁, … λ_s such that Σ_i=1^sp_iλ_i 1, and such that the singularities of the λ_i are contained in U. This result is applied to compute rational functions-valued 1-inverses of matrices with polynomial coefficients, which do not have constant rank, while retaining control over the location of the singularities of the rational functions themselves.     

Computing multidimensional Delaunay tessellations

We recall some properties of Voronoi and Delaunay tessellations in any numbers of dimensions. We then propose a solution to the following problem: Given the Delaunay tessellation of n d-dimensional data points X₁,…, X_n, the proble is to insert a new data point X and to update the tessellation accordingly. The solution proposed achieves minimum space-complexity.     

Order-configuration functions: Mathematical characterizations and applications to digital signal and image processing

We call a function f in n variables an order-configuration function if for any x₁,…, x_n such that x_i1 … x_in we have f(x₁,…, x_n) = x_t, where t is determined by the n-tuple (i₁,…, i_n) corresponding to that ordering. Equivalently, it is a function built as a minimum of maxima, or a maximum of minima. Well-known examples are the minimum, the maximum, the median, and more generally rank functions, or the composition of rank functions. Such types of functions are often used in nonlinear processing of digital signals or images (for example in the median or separable median filter, min-max filters, rank filters, etc.). In this paper we study the mathematical properties of order-configuration functions and of a wider class of functions that we call order-subconfiguration functions. We give several characterization theorems for them. We show through various examples how our concepts can be used in the design of digital signal filters or image transformations based on order-configuration functions.     

A complete algebraic solvability test for the nonstrict Lyapunov inequality

For arbitrary equally sized square complex matrices A and Q (Q Hermitian), the paper provides a complete algebraic test for verifying the existence of a Hermitian solution X of the nonstrict Lyapunov inequality A^*X + XA + Q 0. If existing, we exhibit how to construct a solution. Our approach involves the validation problem for the linear matrix inequality Σ_j=1^k (A_j^*X_jB_j + B_j^*X_j^*A_j) + Q> 0 in X_j, for which we provide an algebraic solvability test and a construct solutions if the kernels of A_j or, dually, those of B_j form an isotonic sequence.     

Long time behaviour of the solution of a system of equations from new theory of shock dynamics

The new theory of shock dynamics gives a system of n + 2 equations, the solution of which determines the propagation of a discontinuity in the initial condition. For the model equation u_t + u u_x = 0, the long time behaviour of the solution is investigated for various confined initial data. The suitability of using Padé approximants for the solution is examined. Special exact solutions are obtained.     

Discontinuous quasivariational-like inequalities

In this note, we deal with the following problem: given X Rⁿ, a multification gG : X → 2^X, two (single-valued) maps f : X → Rⁿ, η : X × X → Rⁿ, find a point x* X such that x* Γ (x*) and f(x*), η(x,x*) ≥ 0 for all x Γ(x*). We prove an existence theorem in which, in particular, the function f is not supposed to be continuous.     

Similarity-based ranking and query processing in multimedia databases

Since media-based evaluation yields similarity values, results to a multimedia database query, Q(Y₁,…,Y_n), is defined as an ordered list S_Q of n-tuples of the form X₁,…,X_n. The query Q itself is composed of a set of fuzzy and crisp predicates, constants, variables, and conjunction, disjunction, and negation operators. Since many multimedia applications require partial matches, S_Q includes results which do not satisfy all predicates. Due to the ranking and partial match requirements, traditional query processing techniques do not apply to multimedia databases. In this paper, we first focus on the problem of “given a multimedia query which consists of multiple fuzzy and crisp predicates, providing the user with a meaningful final ranking”. More specifically, we study the problem of merging similarity values in queries with multiple fuzzy predicates. We describe the essential multimedia retrieval semantics, compare these with the known approaches, and propose a semantics which captures the requirements of multimedia retrieval problem. We then build on these results in answering the related problem of “given a multimedia query which consists of multiple fuzzy and crisp predicates, finding an efficient way to process the query.” We develop an algorithm to efficiently process queries with unordered fuzzy predicates (sub-queries). Although this algorithm can work with different fuzzy semantics, it benefits from the statistical properties of the semantics proposed in this paper. We also present experimental results for evaluating the proposed algorithm in terms of quality of results and search space reduction.     

Analytic numerical solution of coupled semi-infinite diffusion problems

In this paper, we consider coupled semi-infinite diffusion problems of the form u_t(x, t)− A² u_xx(x,t) = 0, x> 0, t> 0, subject to u(0,t)=B and u(x,0)=0, where A is a matrix in , and u(x,t), and B are vectors in . Using the Fourier sine transform, an explicit exact solution of the problem is proposed. Given an admissible error and a domain D(x₀,t₀)={(x,t);0≤x≤x₀, t≥t₀ > 0, an analytic approximate solution is constructed so that the error with respect to the exact solution is uniformly upper bounded by in D(x₀, t₀).     

Numerical inverse isoparametric mapping in remeshing and nodal quantity contouring

In finite element analysis, isoparametric mapping defined as [(ξ, η) → (x, y): X = N_iξ_i] is widely used. It is a one-to-one mapping and its construction is especially elegant for elements of a variable number of nodes showing its versatile applicability to model curved boundaries. In certain analyses, such as remeshing in dynamic analyses or contouring and others, the inverse of this mapping is inevitably valuable, but its determination is not so straightforward. To avoid solving a system of nonlinear equations, generally an iterative technique of order N² in a two-dimensional mesh is often resorted to. In the paper, this technique is improved by systematically bisecting along a predefined line that reduces the iterations to only order N. Its applications in remeshing and nodal quantity contouring are demonstrated and a possible extension for stress contouring is also discussed. The FORTRAN subroutines for the technique proposed are also given.     

Theory analysis on FSLDA and ULDA

It is first revealed that the Fisher criterion ratio of each FSLDA discriminant vector must not be less than that of corresponding ULDA discriminant vector. So, the phenomenon in Yang et al. (Pattern Recognition 35 (2002) 2665) is not strange but certain, and must be available in all experiments! In addition, it is also first illustrated that in fact ULDA discriminant vectors are the S_t− orthogonal eigenvectors of a generalized eigenequation. As a result, the algorithms to obtain S_t− orthogonal eigenvectors of the generalized eigenequation are equivalent to the ULDA algorithm. Consequently, it is possible to work out ULDA discriminant vectors more efficiently.     

Analytic-numerical solutions with a priori error bounds for time-dependent mixed partial differential problems

The aim of this paper is double. First, we point out that the hypothesis D(t₁)D(t₂) = D(t₂)D(t₁) imposed in [1] can be removed. Second, a constructive method for obtaining analytic-numerical solutions with a prefixed accuracy in a bounded domain Ω(t₀,t₁) = [0,p] × [t₀,t₁], for mixed problems of the type u_t(x,t) − D(t)u_xx(x,t) = 0, 0 < x < p, t> 0, subject to u(0,t) = u(p,t) = 0 and u(x,0) = F(x) is proposed. Here, u(x,t) and F(x) are r-component vectors, D(t) is a C^{r × r} valued analytic function and there exists a positive number δ such that every eigenvalue z of (1/2) (D(t) + D(t)^H) is bigger than δ. An illustrative example is included.     

Subspace selection algorithms to be used with the nonlinear projection methods in solving systems of nonlinear equations

The nonlinear projection methods are minimization procedures for solving systems of nonlinear equations. They permit reevaluation of n_k, 1 ≤ n_k ≤ n, components of the approximate solution vector at each iteration step where n is the dimension of the system. At iteration step k, the reduction in the norm of the residue vector depends upon the n_k components which are reevaluated. These n_k components are obtained by solving a linear system.

We present two algorithms for determining the components to be modified at each iteration of the nonlinear projection method and compare the use of these algorithms to Newton's method. The computational examples demonstrate that Newton's method, which reevaluates all components of the approximate solution vector at each iteration, can be accelerated by using the projection techniques.     

Global reparametrization for curve approximation

For approximation of a set of points by a parametric curve X(t) the choice of parameters t_j is essential. None of the parametrization strategies is optimal. To obtain good approximation results, reparametrization (parameter correction) of the points is necessary. In general, these reparametrization methods work only locally. We present a global reparametrization method which leads to dramatically better results.     

Nonlinear finite element for modeling reinforced concrete columns in three-dimensional dynamic analysis

A new finite element is proposed for slender, flexure-dominated reinforced concrete columns subjected to cyclic biaxial bending with axial load, and its implementation into a program for the nonlinear static or dynamic analysis of structures in three-dimensions, is described. The element belongs to the class of distributed inelasticity discrete models for the nonlinear dynamic response analysis of frame structures to earthquake ground motions. The element tangent flexibility matrix is constructed at each time step by Gauss-Lobatto integration of the section tangent flexibility matrix along the member length. The tangent flexibility matrix of the cross-section relates the increment of the vector of the three normal stress resultants N, M_y, M_z, to the vector increment of the section deformation measures. ε_o, _y, _z, and is constructed on the basis of the bounding surface of the cross-section, which is defined as the locus of points in the space of the normalized N, M_y, M_z, which correspond to ultimate strength. The bounding surface concept enables the model to produce realistic predictions for the nonlinear response of the cross-section to any arbitrary loading path in the space N-M_y-M_z.The bounding surface is introduced and utilized in a very flexible manner, enabling a variety of cross-sectional shapes to be treated in a unified way. As this flexibility is at the expense of computational simplicity and memory size requirements, emphasis is placed on algorithmic techniques to facilitate numerical implementation and to increase computational efficiency.     

Nonlinear thermal and mechanical analysis of edge effects in angle-ply laminates

A method has been developed for the thermal analysis of materially nonlinear Carbon Fibre Reinforced Plastic (CFRP) laminated plates using a quasi-three-dimensional iso-parametric finite element. The variation of the linear expansitivity of CFRP lamina with temperature in the transverse direction is included. The nonlinear initial thermal stresses resulting from thermal cooling from the stress-free temperature of 132.22 to 20°C in the [0/90]_s and [±45]_s laminates were found. These two types of laminates were then subject to a uniform applied strain ε_x until failure was detected inside the laminate according to a maximum strain failure criterion. This nonlinear analysis was based on an initial stress iteration method formulated in a previous paper. The [±45]_s laminate analysis was carried out with and without resin layers between the laminae modelled. All results obtained were compared with those of previous investigators. It was found that with initial thermal stresses included the [0/90]_s laminate failed at an earlier stage than without initial thermal stresses included. The [±45], laminate with initial thermal stresses included failed at a later stage than that without initial thermal effects and the inclusion of resin layers delayed the failure even further.     

Simultaneous H control of a finite collection of linear plants with a single nonlinear digital controller

This paper considers the problem of simultaneous H^∞ control of a finite collection of linear time-invariant systems via a nonlinear digital output feedback controller. The main result is given in terms of the existence of suitable solutions to Riccati algebraic equations and a dynamic programming equation. Our main result shows that if the simultaneous H^∞ control problem for k linear time-invariant plants of orders n₁,n₂,…,n_k can be solved, then this problem can be solved via a nonlinear time-invariant controller of order nn₁+n₂++n_k.     

On computing the entropy of cellular automata

We study the topological entropy of a particular class of dynamical systems: cellular automata. The topological entropy of a dynamical system (X,F) is a measure of the complexity of the dynamics of F over the space X. The problem of computing (or even approximating) the topological entropy of a given cellular automata is algorithmically undecidable (Ergodic Theory Dynamical Systems 12 (1992) 255). In this paper, we show how to compute the entropy of two important classes of cellular automata namely, linear and positively expansive cellular automata. In particular, we prove a closed formula for the topological entropy of D-dimensional (D1) linear cellular automata over the ring Z_m (m2) and we provide an algorithm for computing the topological entropy of positively expansive cellular automata.