Fast evaluation of radial basis functions: I

This paper describes some new techniques for the rapid evaluation and fitting of radial basic functions. The techniques are based on the hierarchical and multipole expansions recently introduced by several authors for the calculation of many-body potentials. Consider in particular the N term thin-plate spline, s(x) = Σ_j=1^N d_jφ(x−x_j), where φ(u) = |u|²log|u|, in 2-dimensions. The direct evaluation of s at a single extra point requires an extra O(N) operations. This paper shows that, with judicious use of series expansions, the incremental cost of evaluating s(x) to within precision ε, can be cut to O(1+|log ε|) operations. In particular, if A is the interpolation matrix, a_i,j = φ(x_i−x_j, the technique allows computation of the matrix-vector product Ad in O(N), rather than the previously required O(N²) operations, and using only O(N) storage. Fast, storage-efficient, computation of this matrix-vector product makes pre-conditioned conjugate-gradient methods very attractive as solvers of the interpolation equations, Ad = y, when N is large.     

Cardinal interpolation with differences of tempered functions

In this paper, we investigate the existence and uniqueness of cardinal interpolants associated with functions arising from the k^th order iterated discrete Laplacian ^k applied to certain radial basis functions. In particular, we concentrate on determining, for a given radial function Φ, which functions ^kΦ give rise to cardinal interpolation operators which are both bounded and invertible ℓ₂ (Z³). In addition to solving the cardinal interpolation problem (CIP) associated with such functions ^kΦ, our approach provides a unified framework and simpler proofs for the CIP associated with polyharmonic splines and Hardy multiquadrics.     

Intersection of finitely generated congruences over term algebra

We show that it is decidable for any given ground term rewrite systems R and S if there is a ground term rewrite system U such that ↔_U^*=↔_R^*∩↔_S^*. If the answer is yes, then we can effectively construct such a ground term rewrite system U. In other words, for any given finitely generated congruences ρ and τ over the term algebra, it is decidable if ρ∩τ is a finitely generated congruence. If the answer is yes, then we can effectively construct a ground term rewrite system U such that ↔_U^*=ρ∩τ.     

On canonical number systems

Let be such that d1,p_d=1,p₀2 and . We are proving in this note a new criterion for the pair to be a canonical number system. This enables us to prove that if p₂,…,p_d−1,∑_i=1^dp_i0 and p₀>2∑_i=1^d|p_i|, then is a canonical number system.     

Timoshenko beam finite element with four dof and convergence of O(h)

In situations where transverse shear deformations and rotary inertia in beams are important, elements based on the Timoshenko beam theory are useful. Among the two-noded, four DOF elements derived from the minimum total potential energy principle, the HTK. element proposed by Hughes et al. using linear displacement functions for both w and θ and the T1CC4 element proposed by Tessler et al. using quadratic displacement function for w and linear displacement function for θ are well known in the literature. The convergence of the HTK element in the thin beam situation has been too poor due to shear locking but by using selective integration this element can be shown to be equivalent to the T1CC4 element which has a rate of convergence of O(h²). In this paper a five DOF element with w and θ at the end nodes and θ at the middle node and based on the cubic displacement function for w and the quadratic displacement function for θ is first developed. Statically condensing the middle rotational DOF, the well-known (4 × 4) stiffness matrix using the φ-factor defined as φ = 12EI/kGAL² and hitherto obtained only through a flexibility approach or closed-form solution of the governing equations of the Timoshenko beam theory is derived. This element based on cubic displacement function for w has rate of convergence of O(h⁴), is completely free of shear locking and performs equally well in thin as well as thick beam situations.     

Existence of multiple positive solutions for functional differential equations

In this paper, the existence of at least three positive solutions for the boundary value problem (BVP) of second-order functional differential equation with the form y″(t) + f(t, y^t) = 0, for t ε [0,1], y(t) -βy′(t) =η(t), for t ε [−τ,0], −γy(t) + Δy′(t) = ζ(t), for t ε [1, 1 + a], is studied. Moreover, we investigate the existence of at least three partially symmetric positive solutions for the above BVP with Δ = βγ.     

An element for static, vibration and buckling analysis of thick laminated plates

A conforming finite element formulation of the equations governing composite multilayered plates using Reddy's higher-order theory is presented. The element has eight degrees of freedom, u₀, v₀, w, ∂w/∂x, ∂w/∂y, ∂²w/∂x∂y, γ_x, γ_y, per node. The transverse displacement of the present element is described by a modified bicubic displacement function while the in-plane displacements and shear-rotations are interpolated quadraticly. The element is evaluated for its accuracy in the analysis of static, vibration, and buckling of anisotropic rectangular plates with different lamination schemes and boundary conditions. The conforming finite element described here for the higher-order theory gives fairly accurate results for displacements, stresses, buckling loads, and natural frequencies.     

Efficient enumeration of all minimal separators in a graph

This paper presents an efficient algorithm for enumerating all minimal a-b separators separating given non-adjacent vertices a and b in an undirected connected simple graph G = (V, E), Our algorithm requires O(n³R_ab) time, which improves the known result of O(n⁴R_ab) time for solving this problem, where ¦V¦= n and R_ab is the number of minimal a-b separators. The algorithm can be generalized for enumerating all minimal A-B separators that separate non-adjacent vertex sets A, B < V, and it requires O(n²(n − n_A − n_b)R_AB) time in this case, where n_a = ¦A¦, n_B = ¦B¦ and r_AB is the number of all minimal A−B separators. Using the algorithm above as a routine, an efficient algorithm for enumerating all minimal separators of G separating G into at least two connected components is constructed. The algorithm runs in time O(n³R⁺_Σ + n⁴R_Σ), which improves the known result of O(n⁶R_Σ) time, where R_σ is the number of all minimal separators of G and R_ΣR⁺_Σ = ∑_1i, v_j) ER_vivj n − 1)/2 − m)R_Σ. Efficient parallelization of these algorithms is also discussed. It is shown that the first algorithm requires at most O((n/log n)R_ab) time and the second one runs in time O((n/log n)R⁺_Σ+n log nR_Σ) on a CREW PRAM with O(n³) processors.     

Mean behaviour and distribution properties of multiplicative functions

Mean-value theorems with sharp quantitative remainder term estimates and theorems on the distribution of values are proved for a large class of multiplicative functions. This class is characterized by the existence of an abscissa of absolute convergence ≤ 1 for the Dirichlet series of the quotient , where is the Dirichlet series associated with ƒ and ζ(s) denotes the Riemann zeta function. In particular, the mean behaviour of ƒ on certain sequences a, namely, that of Mersenne numbers 2ⁿ −1 and that of shifted primes p−1 with natural n and prime p, respectively, is studied and cluster points of the image sequence f(a) of a are determined. The strength of the results is demonstrated by new results for some specific functions.     

Similarity solutions for surface-tension driven flows in a slot with an insulated bottom

We study a two-point boundary values problem (TPBVP) T + Q(Aƒƒ″−(ƒ′)²)=β, ƒ(0)=ƒ(1)=ƒ′(1)=ƒPrime;(0)+1=0, 1A<∞ which arises from a study of similarity transformation for surface-tension drive flows of low Prandtl number fluids in a slot with an insulated bottom. Numerical solutions of the (TPBVP) are found for cases of assumed linear (A = 2) and quadratic (A = 1) radiation of temperature along the free surface. The corresponding bifurcation diagrams in terms of Q and β indicate that multiple solutions occur at some Q when A = 1 and unique solution exists for each Q when A = 2. Some existing properties of the TPBVP when A = 1 and 2, are also verified mathematically.     

Hermite interpolation with simple planar PH curves by speed reparametrization

We introduce a new method of solving C¹ Hermite interpolation problems, which makes it possible to use a wider range of PH curves with potentially better shapes. By characterizing PH curves by roots of their hodographs in the complex representation, we introduce PH curves of type K(t−c)²ⁿ⁺¹+d. Next, we introduce a speed reparametrization. Finally, we show that, for C¹ Hermite data, we can use PH curves of type K(t−c)²ⁿ⁺¹+d or strongly regular PH quintics satisfying the G¹ reduction of C¹ data, and use these curves to solve the original C¹ Hermite interpolation problem.     

Blended Hermite interpolants

In (Röschel, l997) B-spline technique was used for blending of Lagrange interpolants. In this paper we generalize this idea replacing Lagrange by Hermite interpolants. The generated subspline b(t) interpolates the Hermite input data consisting of parameter values t_i and corresponding derivatives a_i,j, j=0,…,_i−1, and is called blended Hermite interpolant (BHI). It has local control, is connected in affinely invariant way with the input and consists of integral (polynomial) segments of degree 2·k−1, where k−1max{_i}−1 denotes the degree of the B-spline basis functions used for the blending. This method automatically generates one of the possible interpolating subsplines of class C^k−1 with the advantage that no additional input data is necessary.     

Bivariate hermite subdivision

A subdivision scheme for constructing smooth surfaces interpolating scattered data in R³ is proposed. It is also possible to impose derivative constraints in these points. In the case of functional data, i.e., data are given in a properly triangulated set of points {(x_i, y_i)}_i=1^N from which none of the pairs (x_i,y_i) and (x_j,y_j) with i≠j coincide, it is proved that the resulting surface (function) is C¹. The method is based on the construction of a sequence of continuous splines of degree 3. Another subdivision method, based on constructing a sequence of splines of degree 5 which are once differentiable, yields a function which is C² if the data are not ‘too irregular’. Finally the approximation properties of the methods are investigated.     

Uniform and optimal schemes for stiff initial-value problems

We formulate a class of difference schemes for stiff initial-value problems, with a small parameter ε multiplying the first derivative. We derive necessary conditions for uniform convergence with respect to the small parameter ε, that is the solution of the difference scheme u_i^h satisfies |u_i^h−u(x_i)| Ch, where C is independent of h and ε. We also derive sufficient conditions for uniform convergence and show that a subclass of schemes is also optimal in the sense that |u_i^h−u(x_i)| C min (h, ε). Finally, we show that this class contains higher-order schemes.     

A new routing algorithm for multirate rearrangeable Clos networks

In this paper, we study the problem of finding routing algorithms on the multirate rearrangeable Clos networks which use as few number of middle-stage switches as possible. We propose a new routing algorithm called the “grouping algorithm”. This is a simple algorithm which uses fewer middle-stage switches than all known strategies, given that the number of input-stage switches and output-stage switches are relatively small compared to the size of input and output switches. In particular, the grouping algorithm implies that m = 2n+(n−1)/2^k is a sufficient number of middle-stage switches for the symmetric three-stage Clos network C(n,m,r) to be multirate rearrangeable, where k is any positive integer and rn/(2^k−1).     

Discrete-time system modelling in Lp with orthonormal basis functions

In this paper, model sets for linear time-invariant systems spanned by fixed pole orthonormal bases are investigated. The obtained model sets are shown to be complete in L_p(T) (1<p<∞), the Lebesque spaces of functions on the unit circle T, and in C(T), the space of periodic continuous functions on T. The L_p norm error bounds for estimating systems in L_p(T) by the partial sums of the Fourier series formed by the orthonormal functions are computed for the case 1<p<∞. Some inequalities on the mean growth of the Fourier series are also derived. These results have application in estimation and model reduction.     

Minimizing the distance to one evader while chasing another

To approach a simple game Δ² of P and E = {E₁, E₂} with no a priori evaders' role assignment and the payoff equal to the distance to one evader at an instant of catching another, we introduce a concept of casting and study the games Δ_1,2 and Δ_2,1 for preassigned and Δ_p² for open-loop casting procedures. Since Δ_p² is reduced to Δ_1,2 or Δ_2,1 which, in turn, are distinguished only by their notations, we focus attention mainly on Δ_1,2. According to the tenet of transition, Δ_1,2 is divided into a concatenation of Δ_1,2^b (basic) and Δ_1,2^a (auxiliary) games that model the problem before and after the first instant of E₁ capture. The games Δ_1,2^a, Δ_1,2^b, Δ_1,2 are studied one after another with use of the Isaacs' approach extended by Berkowitz, Breakwell, Bernhard et al.     

Existence and uniqueness of unmixed solutions of the discrete-time algebraic Riccati equation

A solution X of a discrete-time algebraic Riccati equation is called unmixed if the corresponding closed-loop matrix Φ(X) has the property that the common roots of det(sI−Φ(X)) and det(I−sΦ(X)^*) (if any) are on the unit circle. A necessary and sufficient condition is given for existence and uniqueness of an unmixed solution such that the eigenvalues of Φ(X) lie in a prescribed subset of .