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.     

Nonlinear robust stabilization of multi-parameter families of systems

We show that given any family of asymptotically stabilizable LTI systems depending continuously on a parameter that lies in some subset [a₁,b₁]××[a_p,b_p] of , there exists a C⁰ time-varying state feedback law v(t,x) (resp. a C⁰ time-invariant feedback law v(x)) which robustly globally exponentially stabilizes (resp. which robustly stabilizes, not asymptotically) the family. Further, if these systems are obtained by linearizing some nonlinear systems, then v(t,x) locally exponentially stabilizes these nonlinear systems. Finally, v(t,x) globally exponentially stabilizes any time-varying system which switches “slowly enough” between the given LTI systems.     

Thin element applications of a new formulation for C structural elements

A new formulation was proposed recently for the removal of the shear and membrane locking mechanisms from the finite element equations of the structural C⁰ shell, plate and beam elements. The use of full integration with the proposed formulation does not allow development of the zero energy modes or the softening effects, usually associated with the use of the technique of reduced integration in C⁰ plate and shell element applications. In the present paper a beneficial side effect of the new formulation is presented with regard to the development of the purely machine dependent locking. Questions concerning the introduction of softening effects by the new formulation in some flat C⁰ plate/shell element applications are addressed.     

On extended P4-reducible and extended P4-sparse graphs

A graph G was defined in [16] as P₄-reducible, if no vertex in G belongs to more than one chordless path on four vertices or P₄. A graph G is defined in [15] as P₄-sparse if no set of five vertices induces more than one P₄, in G. P₄-sparse graphs generalize both P₄-reducible and the well known class of p₄-free graphs or cographs. In an extended abstract in [11] the first author introduced a method using the modular decomposition tree of a graph as the framework for the resolution of algorithmic problems. This method was applied to the study of P₄-sparse and extended P₄-sparse graphs.

In this paper, we begin by presenting the complete information about the method used in [11]. We propose a unique tree representation of P₄-sparse and a unique tree representation of P₄-reducible graphs leading to a simple linear recognition algorithm for both classes of graphs. In this way we simplify and unify the solutions for these problems, presented in [16–19]. The tree representation of an n-vertex P₄-sparse or a P₄-reducible graph is the key for obtaining O(n) time algorithms for the weighted version of classical optimization problems solved in [20]. These problems are NP-complete on general graphs.

Finally, by relaxing the restriction concerning the exclusion of the C₅ cycles from P₄-sparse and P₄-reducible graphs, we introduce the class of the extended P4-sparse and the class of the extendedP₄-reducible graphs. We then show that a minimal amount of additional work suffices for extending most of our algorithms to these new classes of graphs.     

Undecidable properties of monoids with word problem solvable in linear time

Let C₁ be the class of finitely presented monoids with word problem solvable in linear time. Let P be a Markov property of monoids related to class C₁ in some sense. It is undecidable given a monoid in C₁ whether it satisfies P. Let C and C′ be classes of finitely presented monoids with word problem solvable in some time-bounds. If C contains C₁ and C′ properly contains C, then it is undecidable given a monoid in C′ whether it belongs to C.     

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.     

On renaming a set of clauses as a Horn set

Let S = {C₁, …, C_m} be a set of clauses in the propositional calculus and let n denote the number of variables appearing these clauses. We present and O(mn) time algorithm to test whether S can be renamed as a Horn set.     

Biquartic C-surface splines over irregular meshes

C¹-surface splines define tangent continuous surfaces from control points in the manner of tensor-product (B-)splines, but allow a wider class of control meshes capable of outlining arbitrary free-form surfaces with or without boundary. In particular, irregular meshes with non-quadrilateral cells and more or fewer than four cells meeting at a point can be input and are treated in the same conceptual frame work as tensor-product B-splines; that is, the mesh points serve as control points of a smooth piecewise polynomial surface representation that is local and evaluates by averaging. Biquartic surface splines extend and complement the definition of C¹-surface splines in a previous paper (Peters, J SLAM J. Numer. Anal. Vol 32 No 2 (1993) 645–666) improving continuity and shape properties in the case where the user chooses to model entirely with four-sided patches. While tangent continuity is guaranteed, it is shown that no polynomial, symmetry-preserving construction with adjustable blends can guarantee its surfaces to lie in the local convex hull of the control mesh for very sharp blends where three patches join. Biquartic C¹-surface splines do as well as possible by guaranteeing the property whenever more than three patches join and whenever the blend exceeds a certain small threshold.     

On the sensitivity of camera calibration

The orientation position errors of an object's coordinate frame are determined when the offset of image centre and lens distortion are not included in the calibration process. The orientation and position errors are [(u₀)² + (v₀)²]^0.5/f and [(u²₀+v²₀)T²_z + (u²₀T²_z + v²₀T_y²)]^0.5/f, respectively, where f is the focal length, (u₀, v₀) is the offset of image centre and (T_x T_y T_z) is the position of an object. We also obtain the following conclusions: (a) The offset of image centre has little effect on the determinations of the position and orientation of a coordinate frame; (b) the lens distortion will not dramatically change the position and orientation of a coordinate frame; (c) the scale factor has a great effect on the position of a coordinate frame, and on the accuracy of measurement; (d) the offset of image centre is more sensitive than the lens distortion on the determinations of the position and orientation of a coordinate frame. Finally, some experimental results are given to demonstrate the theoretical analysis given in this paper.     

On some dynamical subsets of the rauzy fractal

Let ( ,(+1)_n) be the adic system associated to the substitution: 1 → 12,…,(n − 1) → 1n, n → 1. In Sirvent (1996) it was shown that there exist a subset Cⁿ of and a map h_n: C → Cⁿ such that the dynamical system (C, h_n) is semiconjugate to ( ). In this paper we compute the Hausdorff and Billingsley dimensions of the geometrical realizations of the set Cⁿ on the (n− l)-dimensional torus. We also show that the dynamical system (Cⁿ,h_n) cannot be realized on the (n − 1)-torus.     

Heaps and heapsort on secondary storage

A heap structure designed for secondary storage is suggested that tries to make the best use of the available buffer space in primary memory. The heap is a complete multi-way tree, with multi-page blocks of records as nodes, satisfying a generalized heap property. A special feature of the tree is that the nodes may be partially filled, as in B-trees. The structure is complemented with priority-queue operations insert and delete-max. When handling a sequence of S operations, the number of page transfers performed is shown to be O(∑_{i = 1}^S(1/P) log_(M/P)(N_i/P)), where P denotes the number of records fitting into a page, M the capacity of the buffer space in records, and N_i, the number of records in the heap prior to the ith operation (assuming P 1 and S> M c · P, where c is a small positive constant). The number of comparisons required when handling the sequence is O(∑_{i = 1}^S log₂ N_i). Using the suggested data structure we obtain an optimal external heapsort that performs O((N/P) log_(M/P)(N/P)) page transfers and O(N log₂ N) comparisons in the worst case when sorting N records.     

A modified C Coons'' patch

A modified C² Coons' patch, free of any compatibility constraints, has recently been developed in [Barnhill '83]. In this short note we show how it can be modified and simplified to produce a computationally more efficient C² transfinite interpolation scheme over rectangles.     

Burstiness in broadband integrated networks

Since broadband integrated networks have to cope with a wide range of bit rates, the notion of burstiness which expresses the irregularity of a flow, has been recognized as a vital question for such networks. In burstiness characterization encountered in the literature, special attention is given to the squared coefficient of variation of interarrival time (Cv²) in a cell arrival process. In order to observe the impact of a bursty flow on a queue, we introduce in this paper a new class of arrival process, the n-stage Markov Modulated Bernoulli Process, MMBP_n, for short, and its peculiar case, the n-stage Hyper-Bernoulli process, denoted by HBP_n. We numerically solve the MMBP_n/D/1/K and we compute in particular the rejection probability and the mean waiting time. For that purpose, a relation between the stationary queue length distribution and arrival time distribution is established. This relation adapts the GASTA equality to the arrival process under consideration. We then discuss the relevance of Cv² for burstiness characterization through an example: the HBP₂/D/1/K queue. We show that Cv² becomes significant only when local overload occurs, i.e., when the arrival rate is momentarily greater than the server rate. The results are then applied to two basic ATM problems: traffic characterization and buffer dimensioning using bursty inputs.     

Algorithms of discretization of algebraic spatial curves on homogeneous cubical grids

In this paper new methods of discretization (integer approximation) of algebraic spatial curves in the form of intersecting surfaces P(x, y, z) = 0 and Q(x, y, z) = 0 are analyzed.

The use of homogeneous cubical grids G(h³) to discretize a curve is the essence of the method. Two new algorithms of discretization (on 6-connected grid G_6c(h³) and 26-connected grid G₂₆(h³)) are presented based on the method above. Implementation of the algorithms for algebraic spatial curves is suggested. The elaborated algorithms are adjusted for application in computer graphics and numerical control of machine tools.     

An iterative method for computing multivariate C piecewise polynomial interpolants

This paper describes a successive overrelaxation (SOR) method for computing a multivariate C¹ piecewise polynomial interpolant. Given a collection of points in R^m together with a triangulation of those points, the scheme described requires only the values of the function to be interpolated at the given points. The result is a C¹ interpolant whose restriction to each of the triangles in the triangulation is a polynomial of degree n.     

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.     

Preemptive LCFS scheduling in hard real-time applications

We investigate the real-time behaviour of a (discrete time) single server system with preemptive LCFS task scheduling. The main result deals with the probability distribution of a random variable SRD(T), which describes the time the system operates without violating a fixed task service time deadline T. The tree approach, used for the derivation of our results, is also suitable for revisiting problems in queueing theory.

Relying on a simple general probability model, asymptotic formulas concerning all moments of SRD(T) are determined; for instance, the expectation of SRD(T) is proved to grow exponentially in T, i.e., E[SRD(T)]C^TT^3/2 for some > 1.     

Minimizing the size of an identifying or locating-dominating code in a graph is NP-hard

Let G=(V,E) be an undirected graph and C a subset of vertices. If the sets B_r(v)∩C, vV (respectively, vVC), are all nonempty and different, where B_r(v) denotes the set of all points within distance r from v, we call C an r-identifying code (respectively, an r-locating-dominating code). We prove that, given a graph G and an integer k, the decision problem of the existence of an r-identifying code, or of an r-locating-dominating code, of size at most k in G, is NP-complete for any r.     

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.     

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.