Confluent general-factorial difference and derivative operators and polynomial-exponential interpolation

A previous application of the Newton divided difference series of the displacement function E^z = (1 + Δ)^z = e ^Dz, where the operators Δ and D are the variables, to purely exponential interpolation employing general-factorial differences and derivatives, {Pi;^m_i=0 (Δ - S_i)}f(0) and {Pi;^m_i=0 (D - t_i)}f(0), in which the s_i's and t_i's are distinct[1], is here extended to mixed polynomial-exponential interpolation where the s_i's and t_i's are no longer distinct.     

Experimental numerical studies on a supercomputer of natural convection in an enclosure with localized heating

We present particle simulations of natural convection of a symmetrical, nonlinear, three-dimensional cavity flow problem. Qualitative studies are made in an enclosure with localized heating. The assumption is that particles interact locally by means of a compensating Lennard-Jones type force F, whose magnitude is given by −G/r^p + H/r^q.

In this formula, the parameters G, H, p, q depend upon the nature of the interacting particles and r is the distance between two particles. We also consider the system to be under the influence of gravity. Assuming that there are n particles, the equations relating position, velocity and acceleration at time t_k = kΔt, K = 0, 1, 2, …, are solved simultaneously using the “leap-frog” formulas. The basic formulas relating force and acceleration are Newton's dynamical equations F_i,k = m_ia_i,k, I = 1, 2, 3, …, n, where m_i is the mass of the ith particle.

Extensive and varied computations on a CRAY X - MP/24 are described and discussed, and comparisons are made with the results of others.     

Fuzzy plane geometry: triangles

A fuzzy triangle T (with a discrete-valued membership function) can be regarded as a nest of parallel-sided triangles T_i with successively higher membership values. Such a nest is determined by its max projections on any two of its “sides”. The area (perimeter) of T is a weighted sum of the areas (perimeters) of the T_i's. The side lengths and altitudes of T can also be defined as weighted sums obtained from projections; using these definitions, the perimeter of T is the sum of the side lengths, and the side lengths are related to the vertex angles by the Law of Sines, but there is no simple relationship between the area of T and the products of the side lengths and altitudes.     

Computational complexity of distributed detection problems with information constraints

For a system consisting of a set of sensors S = {S₁, S₂, …, S_m} and a set of objects O = {O₁, O₂, …, O_n}, there are information constraints given by a relation R S × O such that (S_i, O_j) R if and only if S_i is capable of detecting O_j. Each (S_i, O_j) R is assigned a confidence factor (a positive real number) which is either explicitly given or can be efficiently computed. Given that a subset of sensors have detected obstacles, the detection problem is to identify a subset H O with the maximum confidence value. The computational complexity of the detection problem, which depends on the nature of the confidence factor and the information constraints, is the main focus of this paper. This problem exhibits a myriad of complexity levels ranging from a worst-case exponential (in n) lower bound in a general case to an O(m + n) time solvability. We show that the following simple versions of a detection problem are computationally intractable: (a) deterministic formulation, where confidence factors are either 0 or 1; (b) uniform formulation where (S_i, O_j) R, for all S_i S, O_j O; (c) decomposable systems under multiplication operation. We then show that the following versions are solvable in polynomial (in n) time: (a) single object detection; (b) probabilistically independent detection; (c) decomposable systems under additive and nonfractional multiplicative measures; and (d) matroid systems.     

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.     

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.     

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.     

Mach band optical ray illusion—Drawing with the second differential

Mach bands running along convex ramp edges in a computer graphics image produce the illusion of optical rays, and a concave ramp edge defines an interface for “reflection.” The positive and negative brightness enhancements (E) observed in the illusion are simulated by convolving the pattern of gradient increments (Δg) computed at the edges with a Gaussian kernel (G): E ∝ − Δg G.     

Infinite words and biprefix codes

Let A be an alphabet and ƒ be a right infinite word on A. If ƒ is not ultimately periodic then there exists an infinite set {v_ii0} of (finite) words on A such that ƒ=v₀v₁…v_i…, {v_ii1} is a biprefix code and v_i≠v_j for positive integers i≠j.     

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.     

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.     

Moments of conjugacy classes of binary words

For each nonempty binary word w=c₁c₂c_q, where c_i{0,1}, the nonnegative integer ∑_i=1^q (q+1−i)c_i is called the moment of w and is denoted by M(w). Let [w] denote the conjugacy class of w. Define M([w])={M(u): u[w]}, N(w)={M(u)−M(w): u[w]} and δ(w)=max{M(u)−M(v): u,v[w]}. Using these objects, we obtain equivalent conditions for a binary word to be an -word (respectively, a power of an -word). For instance, we prove that the following statements are equivalent for any binary word w with |w|2: (a) w is an -word, (b) δ(w)=|w|−1, (c) w is a cyclic balanced primitive word, (d) M([w]) is a set of |w| consecutive positive integers, (e) N(w) is a set of |w| consecutive integers and 0N(w), (f) w is primitive and [w]St.     

The evolution of the gingerbread man

Two versions of the evolution of the Gingerbread Man for the iteration z → z^x + c are presented. A previously published version shows an asymmetric evolution as positive real x increases, while a new version shows a symmetric evolution, and several interesting features. Fractal “growth” is described that occurs along certain “shock” lines with this new evolution. A floating “object” appears while approaching certain positive even integer values of x. Finally, a few comments are made about images created from negative values of x.     

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.     

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 nearest neighbour error rates

Let A_k,l,E_k,l denote the acceptance and error rates in the (k,l) case of the nearest neighbour decision rule. We show that if q>k and A_q,m = A_k,l then E_q,m ≤ E_k,l.     

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.     

Entropy maximisation and open queueing networks with priorities and blocking

A review is carried out on the characterisation and algorithmic implementation of an extended product-form approximation, based on the principle of maximum entropy (ME), for a wide class of arbitrary finite capacity open queueing network models (QNMs) with service and space priorities. A single server finite capacity GE/GE/1/N queue with R (R>1) distinct priority classes, compound Poisson arrival processes (CPPs) with geometrically distributed batches and generalised exponential (GE) service times is analysed via entropy maximisation, subject to suitable GE-type queueing theoretic constraints, under preemptive resume (PR) and head-of-line (HOL) scheduling rules combined with complete buffer sharing (CBS) and partial buffer sharing (PBS) management schemes stipulating a sequence of buffer thresholds {N=(N₁,…,N_R),0<N_i≤N_i−1,i=2,…,R}. The GE/GE/1/N queue is utilised, in conjunction with GE-type first two moment flow approximation formulae, as a cost-effective building block towards the establishment of a generic ME queue-by-queue decomposition algorithm for arbitrary open QNMs with space and service priorities under repetitive service blocking with random destination (RS-RD). Typical numerical results are included to illustrate the credibility of the ME algorithm against simulation for various network topologies and define experimentally pessimistic GE-type performance bounds. Remarks on the extensions of the ME algorithm to other types of blocking mechanisms, such as repetitive service blocking with fixed destination (RS-FD) and blocking-after-service (BAS), are included.     

Sojourn time distribution in the M/M/1 queue with discriminatory processor-sharing

In this paper, we consider a queue with multiple K job classes, Poisson arrivals, exponentially distributed required service times in which a single processor serves according to the DPS discipline. More precisely, if there are n_i class i jobs in the system, i=1,…,K, each class j job receives a fraction _j/∑_i=1^K_in_i of the processor capacity. For this queue, we obtain a system of equations for joint transforms of the sojourn time and the number of jobs. Using this system of equations we find the moments of the sojourn time as a solution of linear simultaneous equations, which solves an open problem.     

Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues

In many calculations, spectral discretization in space is coupled with a standard ordinary differential equation formula in time. To analyze the stability of such a combination, one would like simply to test whether the eigenvalues of the spatial discretization operator (appropriately scaled by the time step k) lie in the stability region for the o.d.e. formula, but it is well known that this kind of analysis is in general invalid. In the present paper we rehabilitate the use of stability regions by proving that a discrete linear multistep ‘method of lines’ approximation to a partial differential equation is Lax-stable, within a small algebraic factor, if and only if all of the -pseudo-eigenvalues of the spatial discretization operator lie within O() of the stability region as → 0. An -pseudo-eigenvalue of a matrix A is any number that is an eigenvalue of some matrix A + E with E ; our arguments make use of resolvents and are closely related to the Kreiss matrix theorem. As an application of our general result, we show that an explicit N-point Chebyshev collocation approximation of u_t = −xu_x on [−1, 1] is Lax-stable if and only if the time step satisfies k = O(N⁻²), although eigenvalue analysis would suggest a much weaker restriction of the form k CN⁻¹.