Positive Integer Values Research Articles

An Engineer's Approach to a General Algorithm for Finding the Sum of Powers of Natural Numbers

Convenient formulae for finding the sums of kth power of the first n natural numbers may be useful in some engineering applications. The formulae for k = 1, 2, and 3, are commonly found in the literature. In this paper, an attempt is made to develop a general algorithm for finding the sum for any positive integer value of k. The development of the algorithm is entirely an engineering approach, based purely on a simple geometric interpretation and does not involve any deep mathematics. This algorithm may be used to derive the formulae for different values of k. Some of the possible engineering applications of these formulae are also discussed.

Parametric dependence of the Pollicott-Ruelle resonances for sawtooth maps.

The Pollicott-Ruelle resonances for the sawtooth map are investigated. We turn our attention to the parametric dependence of them with respect to the bifurcation parameter K. It is numerically shown that the resonances move in an erratic way if the bifurcation parameter K is supposed to be time. At certain rational values of K, it is observed that some resonances shrink to z=0. In particular, at positive integer values of K which correspond to the Arnold cat map, all resonances except z=1 (i.e., the equilibrium state) shrink to z=0. This peculiar behavior is rigorously proved in the Appendix. In addition, the diffusion coefficient of this map is numerically calculated in a very accurate way by evaluating the leading resonance.

The Falkner–Skan Equation II: Dynamics and the Bifurcations of P- and Q-Orbits

The Falkner–Skan equation is a reversible three-dimensional system of ordinary differential equations with two distinguished straight-line trajectories which form a heteroclinic loop between fixed points at infinity. We showed in the previous paper (1995, J. Differential Equations119, 336–394) that at positive integer values of the parameter λ there are bifurcations creating large sets of periodic and other interesting trajectories. Here we show that all but two of these trajectories are destroyed in another sequence of bifurcations as λ→∞, and by considering topological invariants and orderings on certain manifolds we obtain unusually detailed information about the sequences of bifurcations which can occur.

Robust and nonblocking supervisor for discrete-event systems with model uncertainty under partial observation

Addresses a robust and nonblocking supervisory control problem based on a framework of discrete-event systems with model uncertainty under partial observation. The uncertainty introduced in the paper is associated with internal and unobservable events occurring in systems. The paper presents a systematic method for modeling uncertainty described as a /spl Delta/-transition representing the internal and unobservable events. In particular, at a state with the assigned positive integer value p a system is assumed to experience at most a p-step state transition by the internal and unobservable event. Under the framework for model uncertainty, the paper presents necessary and sufficient conditions for the existence of a robust supervisor with the nonblocking property.

Compound Poisson approximation in total variation

Poisson approximation in total variation can be successfully established in a wide variety of contexts, involving sums of weakly dependent random variables which usually take the value 0, and occasionally the value 1. If the random variables can take other positive integer values, or if there is stronger dependence between them, compound Poisson approximation may be more suitable. Stein's method, which is so effective in the Poisson context, turns out to be much more difficult to apply for compound Poisson approximation, because the solutions of the Stein equation have undesirable properties. In this paper, we prove new bounds on the absolute values of the solutions to the Stein equation and of their first differences, over certain ranges of their arguments. These enable compound Poisson approximation in total variation to be carried out with almost the same efficiency as in the Poisson case. Even for sums of independent random variables, which have been exhaustively studied in the past, new results are obtained, effectively solving a problem discussed by Le Cam (1965, Bernoulli, Bayes, Laplace. Springer, New York, pp. 179–202), in the context of nonnegative integer valued random variables.

On the generalized dimensions for the Fourier spectrum of the Thue-Morse sequence

We present explicit relations for the generalized dimensions of the spectral measure of the Thue-Morse symbolic sequence for positive integer values of q. Each is expressed through the eigenvalue of the corresponding matrix. It is also demonstrated how these dimensions can be recovered from the products of the values of the autocorrelation function.

Growth of the product $∏^n_{j=1} (1-x^{a_j})$

We estimate the maximum of $∏^n_{j=1} |1 - x^{a_j}|$ on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when $a_j$ is $j^k$ or when $a_j$ is a quadratic in j that takes on positive integer values, the maximum grows as exp(c

On the random sum with a stopping rule SN > T>

We consider the random sum Sn >+ X>1 + X>2 + ...+ Xn >with a stopping rule N>= min{n> : Sn > t}>where T>is a predetermined threshold. We assume that X>1 ,X>2 ,...>are independent and identically distributed random variables having positive integer values. N>is also a positive integer‐valued random variable but dependent on the sequence X>1 , X>2 ,...>due to the stopping rule. Given T>and the probability distribution of X,>we first derive the probability distributions of N, Sn,>and of N>conditional on Sn.>Then we compute the first two moments of these three taking an instructive approach based on the absorbing Markov chain.

R-Dominating cliques in graphs with hypertree structure

Let G = ( V, E) be an undirected graph and r be a vertex weight function with positive integer values. A subset (clique) D ⊆ V is an r- dominating set ( clique) in G iff for every vertex v ϵ V there is a vertex u ϵ D with dist ( u, v) ⩽ r( v). This paper contains the following results: 1. (i) We give a simple necessary and sufficient condition for the existence of r-dominating cliques in the case of Helly graphs and of chordal graphs. 2. (ii) For Helly graphs with an r-dominating clique the minimum size of an r-dominating clique coincides with the minimum size of any r-dominating set. 3. (iii) We give a linear-time algorithm for finding a minimum r-dominating clique in dually chordal graphs (a generalization of strongly chordal graphs). These results improve and extend earlier results on r-dominating cliques in chordal and strongly chordal graphs.

Infinite complete group presentations

Knuth-Bendix for strings, when applied to a presentation for a group G, often diverges. In this paper we develop a Knuth-Bendix procedure for equational term rewriting which can find an infinite, complete presentation for G in certain cases where previous procedures fail. These presentations yield an efficient solution of the word problem for G in the same way as finite ones, by reducing any word to a unique normal form. Our presentations consist of finitely many families, each of which is parameterized by a single term rewrite rule. The parameters are formal variables which occur in exponents, and take all positive integer values. We give an example of a group which is not FP ∞, hence which has no finite complete rewrite system, and is not automatic, but which has a complete presentation in our sense.

QUANTUM GROUP APPROACH TO A SOLUBLE VERTEX MODEL WITH GENERALIZED ICE RULE

Using the representation of the quantum group SL q(2) by the Weyl operators of the canonical commutation relations in quantum mechanics, we construct and solve a new vertex model on a square lattice. Random variables on horizontal bonds are Ising variables, and those on the vertical bonds take half positive integer values. The vertex is subjected to a generalized form of the so-called “ice rule,” its property is studied in detail and its free energy calculated with the method of quantum inverse scattering. Remarkably, in analogy with the usual six-vertex model, there exists a “free-fermion” limit with a novel rich operator structure. The existing algebraic structure suggests a possible connection with a lattice neutral plasma of charges, via the fermion-boson correspondence.

Close-Proximity Analysis: Another Variation on the Minimum-Spanning-Tree Problem

Close-Proximity Analysis: Another Variation on the Minimum-Spanning-Tree Problem

The Falkner-Skan Equation I. The Creation of Strange Invariant Sets

We study bifurcations in the Falkner-Skan equation. we show that for positive integer values of the parameter there are bifurcations which create an infinity of periodic and other interesting trajectories. These bifurcations provide a new mechanism for the creation of strange invariant sets in ordinary differential equations, and are co-dimension one.

Influence of fractal kinetics on molecular recognition.

Molecular recognition is a central issue for nearly every biological mechanism. The analysis of molecular recognition to date has been conducted within the framework of classical chemical kinetics, in which the kinetic orders of a reaction have positive integer values. However, recent theoretical and experimental advances have shown that the assumptions inherent in this classical framework are invalid under a variety of conditions in which the reaction environment may be considered nonideal. A good example is provided by reactions that are spatially constrained and diffusion limited. Bimolecular reactions confined within two-dimensional membranes, one-dimensional channels or fractal surfaces in general exhibit kinetic orders that are noninteger. An appropriate framework for the study of these nonideal phenomena is provided by the Power-Law formalism, which includes as special cases the Mass-Action formalism of chemical kinetics and the Michaelis-Menten formalism of enzyme kinetics. The Power-Law formalism is an appropriate representation not only for fractal kinetics per se, but also for other nonideal kinetic phenomena, provided the range of variation in concentration is not too large. After defining some elementary concepts of molecular recognition, and showing how these are manifested in classical kinetic terms, this paper contrasts the implications of classical and fractal kinetics in a few simple cases. The principal distinction lies in the ability of fractal kinetics to nonlinearly transform, rather than proportionally transmit, the input S/N ratio. As a consequence, fractal kinetics create a threshold for the input signal below which no recognition occurs and above which amplified recognition takes place.(ABSTRACT TRUNCATED AT 250 WORDS)

Dimensional scaling for Regge trajectories

Using dimensional scaling, we were able to obtain a systematic expansion for Regge trajectories in 1/ k , where k=(D-)/2 and D is the number of spatial dimensions. Bound states for the power-law potential were obtained from Regge trajectories by requiring the angular momentum quantum number to take on positive integer values. For scattering states, we calculated the positions of Regge poles for the Lennard-Jones (6,4) and (12,6) potentials. The results to first order in 1/k were in good agreement with both semiclassical and quantum calculations. The same expansion was used to obtain the positions of Regge poles for complex optical potentials

Shepp statistic for Markov chains application to a long-run average cost criterion

This paper is a generalization to Markov chains of the work of Shepp [6] in the i.i.d case. Shepp studies the limiting values of the averagesTn= (Sn+f(n)–Sn)/f(n) whereSn=X0+X1+ · ·· +Xn,X0= 0,n= 1, 2, ···, is a sum of mutually independent and identically distributed random variables. The functionftakes positive integer values and non-decreasingly tends to infinity. Here we take a class of functionsfin central positionf(n) = [clogn],c> 0,n= 1, 2, ···. There are many refinements of the functionfin the i.i.d case [1], [2]. Here we consider the more general case whereX1,· ··,Xnis an irreducible and recurrent Markov chain. The state space of the chain is either compact or countable.

Shepp statistic for Markov chains application to a long-run average cost criterion

This paper is a generalization to Markov chains of the work of Shepp [6] in the i.i.d case. Shepp studies the limiting values of the averages Tn = (Sn + f(n) – Sn )/f(n) where Sn = X0 + X 1+ · ·· + Xn , X 0 = 0, n = 1, 2, ···, is a sum of mutually independent and identically distributed random variables. The function f takes positive integer values and non-decreasingly tends to infinity. Here we take a class of functions f in central position f(n) = [c log n], c > 0, n = 1, 2, ···. There are many refinements of the function f in the i.i.d case [1], [2]. Here we consider the more general case where X 1 , · ··, Xn is an irreducible and recurrent Markov chain. The state space of the chain is either compact or countable.

P-adic interpolation of dedekind sums

In this article we give an explicit representation of p-adic Dedekind sums and their reciprocity laws by using p-adic measure theory. We then study the consequences of the p-adic reciprocity law for particular positive integer values in which case we can recover a reciprocity law for Dedekind sums attached to particular Dirichlet characters. This gives a proof different from that of Nagasaka.

Closed-Form Solutions of Higher Order Duration Measures

This paper derives formulas for higher order duration measures, including D(1) (i.e. Macaulay duration), D(2) (i.e., slope duration), D(3) (curvature duration), etc. We develop a general iterative method to obtain formulas for any higher order measure D(m), for an arbitrary positive integer value for m. The higher order duration measures can be used to hedge against virtually any type of interest rate shift. Our algorithm can be easily adapted for spreadsheet applications.

A linear code for the sawtooth and cat maps

The sawtooth maps are a one-parameter set of piecewise linear area preserving maps on the torus. For positive integer values of the parameter K they are automorphisms of the torus, known as the cat maps. We present a symbolic dynamics for these maps in which the symbols are integers. This code is related to a practical problem of the stabilisation of a system which is perturbed by impulses. The code is linear in the sense that an orbit and its code are linearly related, so it is not difficult to obtain a good approximation to one from the other in practice. A stationary stochastic process for generating the code is given explicitly. The theory uses Green function methods, which are also used to study ordered periodic orbits and cantori. The problems of using a similar code for arbitrary area preserving twist maps on the torus are briefly discussed.