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.     

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.     

Comparing parallel Newton''s method with parallel Laguerre''s method

Newton's and Laguerre's methods can be used to concurrently refine all separated zeros of a polynomial P(z). This paper analyses the rate convergence of both procedures, and its implication on the attainable number n of correct figures. In two special cases the number m of iterations required to reach an accuracy η = 10⁻ⁿ is shown to grow as log_λ n, where λ = 3 for Newton's and λ = 4 for Laguerre's. In the general case m is shown to grow linearly with n for both procedures. An assessment of the efficiency of the two methods is also given, by evaluating the computational complexity of operations in circular arithmetic, and the efficiency indices of the two iterative schemes.     

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.     

A note on a nonlinear semigroup for controlled partially observed diffusions

This paper is concerned with the separated control problems for optimal stochastic controls under partial observations. The ‘pathwise’ method of nonlinear filtering is extended to the case when a function h(X_tY_t, of state X_t, and observation Y_t 3plus additive white noise is observed. Markov and continuity properties of the unnormalized conditional distribution measure are obtained and the Nisio nonlinear semigroup is found.     

Algorithm for converting a forest of quadtrees to a binary array

An algorithm for reconstructing a binary array of size N sx N from its forest of quadtree representation is presented. The algorithm traverses each tree of the forest in preorder and maps each ‘black’ node into the spatial domain. The time complexity in mapping is O(log N × B_n + B_p), where B_n is the number of black nodes in the forest and B_p is the number of black pixels in the N × N array. The algorithm has been implemented on an Apple II.     

On the latent roots of λ matrices

In this paper we consider equations defined by (1.3)–(1.2)–(1.4). We describe in detail three algorithms for the approximate determination of |λ_nr|, for |λ₁| resp. for one of the λ_j's. The single steps of the algorithms are the four fundamental operations and the positive value of k^th roots of positive numbers and the main interest of them lies in the fact that (the algorithms themselves and so) their lengths depend only on n, r and the prescribed relative error and not on the entries of the matrices A_ν.     

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 limit characterization for the number of spanning trees of graphs

In this paper we propose a limit characterization of the behaviour of classes of graphs with respect to their number of spanning trees. Let {G_n} be a sequence of graphs G₀,G₁,G₂,… that belong to a particular class. We consider graphs of the form K_n−G_n that result from the complete graph K_n after removing a set of edges that span G_n. We study the spanning tree behaviour of the sequence {K_n−G_n} when n→∞ and the number of edges of G_n scales according to n. More specifically, we define the spanning tree indicator ({G_n}), a quantity that characterizes the spanning tree behaviour of {K_n−G_n}. We derive closed formulas for the spanning tree indicators for certain well-known classes of graphs. Finally, we demonstrate that the indicator can be used to compare the spanning tree behaviour of different classes of graphs (even when their members never happen to have the same number of edges).     

Boundary matching algorithm for connected component labelling using linear quadtrees

A computationally fast top-down recursive algorithm for connected component labelling using linear quadtrees is presented. The input data structure used is a linear quadtree representing only black leaf nodes. The boundary matching approach used ensures that at most two adjacencies of any black leaf node are considered. Neighbour searching is carried out within restricted subsets of the input quadtree. The time and space complexities of the algorithm are O(Bn) and O(B) respectively for a linear quadtree with B black leaves constructed from a binary array of size 2ⁿ × 2ⁿ. Simulations show the algorithm to be twice as fast as an existing algorithm that uses an identical input data structure. The top-down algorithm presented can also be used to efficiently generate a linear quadtree representing all nodes — ‘grey’, ‘black’ and ‘white’ — in preorder when given an input linear quadtree representing only ‘black’ leaf nodes. The boundary matching algorithm is computationally fast and has low static and dynamic storage costs, making it useful for applications where linear quadtrees are held in main memory.     

Asymptotic expansions of Hankel transforms of functions with logarithmic singularities

Asymptotic expansions as λ → +∞ are obtained for the Hankel transform

whereJ_v(t) is the Bessel function of the first kind and v is a fixed complex number. The function \tf(t) is allowed to have an asymptotic expansion near the origin of the form

Here, Re _n ↑ +∞ and β_n is an arbitrary complex number.     

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.     

Simulation of partially observed Markov decision process and dynamic quality improvement

Computer simulation is used to compare three quality policies. The first policy is ‘do-nothing’. The second is an appraisal policy. The third policy includes prevention that leads to process quality improvement. The simulation model is based on a Partially Observed Markov Decision Process (POMDP). The unobserved states of the process depend on the failure rate, λ. The observed process output is the number of conforming and nonconforming products. The process performance is measured by quality costs per unit. The simulation language used is SLAM II. The power of using computer simulation to model the dynamics of process quality improvement is discussed.     

Research on auto-reasoning process planning using a knowledge based semantic net

A process-planning model (PP model) is proposed to convert the geometric features into manufacture machining operations and sequence the machining operations of the part in a feasible and effective order. The process-planning model (PP model) construct a feature framework that makes a mapping from geometric features into machining operations. A semantic net named the Precedence-Relations-Net is established to reflect the precedence relationships among the machining operations. The vectors and the matrixes are employed to construct a mathematical sequencing model. A part is decomposed into several basic geometrical units, namely, U₁, U₂, … , U_N. For each unit U_i, two vectors, named F_i and P_i, represent the features and machining operations of U_i. Finally, a matrix named PP is used to memorize the process plan, and a matrix – PO (performing objects) – represents the object of machining operations.     

Paired-domination in inflated graphs

The inflation G_I of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique K_d. A set S of vertices in a graph G is a paired dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired domination number γ_p(G) is the minimum cardinality of a paired dominating set of G. In this paper, we show that if a graph G has a minimum degree δ(G)2, then n(G)γ_p(G_I)4m(G)/[δ(G)+1], and the equality γ_p(G_I) = n(G) holds if and only if G has a perfect matching. In addition, we present a linear time algorithm to compute a minimum paired-dominating set for an inflation tree.     

3-D silicon vector sensor based on a novel parallel-field Hall microdevice

A single-chip 3-D silicon vector sensor for magnetic-field measurement is presented. This triaxial device functionally integrates in a common transducer region two novel five-contacts parallel-field Hall elements for in-plane components B_x and B_y and four cross-coupled orthogonal triple Hall elements for the B_z component, perpendicular to the chip surface. The effective sensor volume is 150 μm×150 μm×100 μm. The main advantage of this 3-D vector magnetometer is the strongly reduced (with about 45%) channel cross-sensitivities. This is achieved by amperometric output mode of operation using an original read-out circuitry, keeping equal voltage conditions on the top of the chip with and without magnetic field. The sensitivities per channel reaches S_A(B_x)=S_A(B_y)=356 μA/T and S_A(B_z)=260 μA/T at a supply current 10 mA, the measured equivalent offsets of the three channel outputs are 25/35 mT and the low frequency noise is mainly a 1/f. The lowest detected magnetic induction of the three output channels is about 15/20 μT in the range 1≤f≤ 10² Hz and the frequency response of the 3-D device together with the read-out electronics is at least 30 kHz.     

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.     

ANTS: Agents on Networks, Trees, and Subgraphs

Efficient exploration of large networks is a central issue in data mining and network maintenance applications. In most existing work there is a distinction between the active ‘searcher’ which both executes the algorithm and holds the memory and the passive ‘searched graph’ over which the searcher has no control at all. Large dynamic networks like the Internet, where the nodes are powerful computers and the links have narrow bandwidth and are heavily-loaded, call for a different paradigm, in which a noncentralized group of one or more lightweight autonomous agents traverse the network in a completely distributed and parallelizable way. Potential advantages of such a paradigm would be fault tolerance against network and agent failures, and reduced load on the busy nodes due to the small amount of memory and computing resources required by the agent in each node. Algorithms for network covering based on this paradigm could be used in today’s Internet as a support for data mining and network control algorithms. Recently, a vertex ant walk ( ) method has been suggested [I.A. Wagner, M. Lindenbaum, A.M. Bruckstein, Ann. Math. Artificial Intelligence 24 (1998) 211–223] for searching an undirected, connected graph by an a(ge)nt that walks along the edges of the graph, occasionally leaving ‘pheromone’ traces at nodes, and using those traces to guide its exploration. It was shown there that the ant can cover a static graph within time nd, where n is the number of vertices and d the diameter of the graph. In this work we further investigate the performance of the method on dynamic graphs, where edges may appear or disappear during the search process. In particular we prove that (a) if a certain spanning subgraph S is stable during the period of covering, then the method is guaranteed to cover the graph within time nd_s, where d_s is the diameter of S, and (b) if a failure occurs on each edge with probability p, then the expected cover time is bounded from above by nd((logΔ/log(1/p))+((1+p)/(1−p))), where Δ is the maximum vertex degree in the graph. We also show that (c) if G is a static tree then it is covered within time 2n.     

C- and C-continuous spline-interpolation of a regular triangular net of points

The investigations focus on the construction of a C^k-continuous (k=0,1,2) interpolating spline-surface for a given data set consisting of points P_ijk arranged in a regular triangular net and corresponding barycentric parameter triples (u_i,v_j,w_k). We try to generalize an algorithm by A.W. Overhauser who solved the analogous problem for the case of a univariate data set. As a straightforward generalization does not work out we adapt the Overhauser-construction. We use some blending of basic surfaces with uniquely determined basic functions. This yields a spline-surface with a polynomial parametric representation which display C¹- or C²-continuity along the common curve of two adjacent sub-patches. Local control of the emerging spline surface is provided which means moving one data point P changes only some of the sub-patches around P and does not affect regions lying far away.     

The application of explicit Nyström methods to singular second order differential equations

An important class of singular second order initial value problems is y″ + (2/x)y′+f(x,y) = 0, 0 < x < x_f, y(0) = a, y′(0) = 0; this class includes, for example, the well-known singular equations of Emden and Liouville. The purpos of this paper is to show the interesting result that explicit Nyström methods, existing for the direct integration of special second order regular initial value problems, can be used for the integration of this class of singular initial value problems and the methods show their proper respective orders of convergence. This is justified mathematically and demonstrated computationally.