A new formalism for the dynamic modelling of cables

This article proposes a new formalism for the dynamic modelling of cables that can even be applied when they are submitted to cross flow of water or air. An important application is the case of umbilical cables used in remotely operated vehicles. The primary basis for the formulation is to assume that the continuous flexibility is represented by a discrete approach, consisting of rigid links connected by elastic joints, allowing movement in three dimensions. Each elastic joint allows three independent movements, called elevation, azimuth and torsion (twist). A significant contribution of the proposed formalism is the development of a compact equation that allows obtaining the Lagrangian of the system directly and automatically, regardless of the number of links chosen to form a chain of rigid bodies connected by flexible joints to represent the continuous flexibility of the cable. This formulation allows the construction of an algorithm for obtaining the equations of the dynamic model of flexible cables.     

Hollow-fiber glass-reinforced plastics with increased specific bending stiffness

It is shown that the use of hollow glass fibers makes possible a considerable increase in the specific bending stiffness of a glass-reinforced plastic as compared with solid-fiber material. At the same time, the thermophysical and dielectric properties are significantly improved.Mekhanika Polimerov, Vol. 2, No. 3, pp. 380–382, 1966     

On the equivalence between some discrete and continuous optimization problems

The simplex algorithm for linear programming is based on the well-known equivalence between the problem of maximizing a linear functionf on a polyhedronP and the problem of maximizingf over the setV _P of all vertices ofP. The equivalence between these two problems is also exploited by some methods for maximizing a convex or quasi-convex function on a polyhedron.In this paper we determine some very general conditions under which the problem of maximizingf overP is equivalent, in some sense, to the problem of maximizingf overV _P . In particular, we show that these two problems are equivalent whenf is convex or quasi-convex on all the line segments contained inP and parallel to some edge ofP.In the case whereP is a box our results extend a well-known result of Rosenberg for 0–1 problems. Furthermore, whenP is a box or a simplex, we determine some classes of functions that can be maximized in polynomial time overP.This paper has been partially written while the author was visiting the Rutgers Center for Operations Research (RUTCOR). The support of the Air Force grants AFORS-89-0512 and AFORS-90-0008 is gratefully acknowledged.     

Prediction of Euclidean distances with discrete and continuous outcomes

The objective of this paper is first to predict generalized Euclidean distances in the context of discrete and quantitative variables and then to derive their statistical properties. We first consider the simultaneous modelling of discrete and continuous random variables with covariates and obtain the likelihood. We derive an important property useful for its practical maximization. We then study the prediction of any Euclidean distances and its statistical proprieties, especially for the Mahalanobis distance. The quality of distance estimation is analyzed through simulations. This results are applied to our motivating example: the official distinction procedure of rapeseed varieties.     

A comparison of uniform and discrete polynomial approximation

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On discrete and continuous wirtinger inequalities

We shall present several generalizations of discrete Wirtinger's inequality, and establish their continuous analogs.     

A discrete filled function algorithm embedded with continuous approximation for solving max-cut problems

In this paper, a discrete filled function algorithm embedded with continuous approximation is proposed to solve max-cut problems. A new discrete filled function is defined for max-cut problems, and properties of the function are studied. In the process of finding an approximation to the global solution of a max-cut problem, a continuation optimization algorithm is employed to find local solutions of a continuous relaxation of the max-cut problem, and then global searches are performed by minimizing the proposed filled function. Unlike general filled function methods, characteristics of max-cut problems are used. The parameters in the proposed filled function need not to be adjusted and are exactly the same for all max-cut problems that greatly increases the efficiency of the filled function method. Numerical results and comparisons on some well known max-cut test problems show that the proposed algorithm is efficient to get approximate global solutions of max-cut problems.     

On discrete and continuous quotient Riesz spaces

Order properties of quotient Riesz spaces E/N(f) by null ideals N(f) are investigated. We show relationships between properties of a Riesz space E and its order dual E ^~ and properties of quotients E/N(f) where f runs over some subspaces of E ^~. A characterization of metrizable locally convex topological Riesz spaces whose all quotients (by proper closed ideals) are discrete is also given.     

Random evolutions in discrete and continuous time

This paper points out a connection between random evolutions and products of random matrices. This connection is useful in predicting the long-run growth rate of a single-type, continuously changing population in randomly varying environments using only observations at discrete points in time. A scalar Markov random evolution is specified by the n×n irreducible intensity matrix or infinitesimal generator Q = (q_ij) of a time-homogeneous Markov chain and by n finite real growth rates (scalars) s_i. The scalar Markov random evolution is the quantity M^C(t) = exp(Σⁿ_j=1s_jg^C_j (t)), where g^C_j(t) is the occupancy times in state j up to time t. The scalar Markov product of random matrices induced by this scalar Markov random evolution is the quantity M^D(t) = exp(Σⁿ_j=1s_jg^D_j (t)), where g^D_j(t) is the occupancy time in state j up to and including t of the discrete-time Markov chain with stochastic one-step transition matrix P = e^Q. We show that lim_t→∞(1/t)E(logM^D(t))=lim_t→∞(1/t)E(logM^C(t)) but that in general lim_t→∞(1/t)logE(M^C(t)) ≠ lim_t→∞(1/t)logE(M^D(t)). Thus the mean Malthusian parameter of population biologists is invariant with respect to the choice of continuous or discrete time, but the rate of growth of average population size is not. By contrast with a single-type population, in multitype populations whose growth is governed by non-commuting operators, the mean Malthusian parameter may be destined for a less prominent role as a measure of long-run growth.     

Almost automorphic solutions to logistic equations with discrete and continuous delay

We obtain sufficient conditions for the existence and uniqueness of a positive compact almost automorphic solution to a logistic equation with discrete and continuous delay. Moreover, we provide a counterexample to some results in literature which deal with the uniqueness of almost periodic solutions to logistic type equations.     

Estimation of states and parameters in continuous non-linear systems with discrete observations

We consider a class of continuous non-linear systems defined by the ordinary differential equation x = f(x, t) + g(x, t)u, where u is an unknown input representing noise or disturbances. The object is to estimate states and parameters in these systems by means of a fixed number of discrete observations y_i = h(x(t_i), t_i) + v_i, 1 ? i ? m, where the v_i represents unknown errors in the measurements y_i. No statistical assumptions are made concerning the nature of the unknown input u or the unknown measurement errors v_i. A weighted least squares criterion is defined as a measure of the optimal estimate. A result concerning the existence of solutions of the differential equation which minimize the criterion is presented. The necessary conditions for an optimal estimate, a set of Euler-Lagrange equations and multi-point discontinuous non-linear boundary conditions, are given. The multi-point problem is converted to an equivalent continuous two-point boundary value problem of larger dimension in the case in which the observations are assumed to be linear functions of the state. A pair of equivalent quasilinearization algorithms is defined for the two-point system and the multi-point system. Quadratic convergence for these algorithms is proved.     

Global aspects of the continuous and discrete Newton method: A case study

Newton's method has recently become one of the paradigms in the revival of Julia set theory and complex dynamical systems. This paper, to a large extent experimental in nature, investigates Newton's method for some particular model problems as a real dynamical system of several simultaneous equations guided by the Julia set theory.     

Oscillatory behavior in microbial continuous culture with discrete time delay

By introducing discrete time delay into the model for producing 1,3-propanediol by microbial continuous fermentation, we consider the stability and Hopf bifurcation of the delay differential system. Through numerical simulations, we get the rule of branch value changing with parameter and draw the pictures of periodic solutions and phase diagrams with specified parameters. The effect of time delay suggests that the system can qualitatively describe oscillatory phenomena occurring in the experiment.     

Singular discrete and continuous mixed boundary value problems

For each $n\in \mathbb{N}$, $n\ge 2$, we prove the existence of a solution $(u_0,\dots ,u_n)\in {\mathbb{R}}^{n+1}$ of the singular discrete problem $\frac{1}{h^2}{\Delta }^2{u}_{k?1}+f(t_k,u_k)=0\text{,}k=1,\dots ,n?1\text{,}$$\Delta u_0=0\text{,}{u}_n=0\text{,}$ where $u_k>0$ for $k=0,\dots ,n?1$. Here $T\in (0,\infty )$, $h=\frac{T}{n}$, $t_k=hk$, $f(t,x):[0,T]\times (0,\infty )\to \mathbb{R}$ is continuous and has a singularity at $x=0$. We prove that for $n\to \infty$ the sequence of solutions of the above discrete problems converges to a solution $y$ of the corresponding continuous boundary value problem $y^{″}\left(t\right)+f(t,y\left(t\right))=0\text{,}$$y^{\prime }\left(0\right)=0\text{,}y\left(T\right)=0\text{.}$     

Difference estimates for continuous and discrete operator semigroups

For suitable bounded operator semigroups (e ^tA ) _t≥0 in a Banach space, we characterize the estimate ‖Ae ^tA ‖≤c/F(t) for large t, where F is a function satisfying a sublinear growth condition. The characterizations are by holomorphy estimates on the semigroup, and by estimates on powers of the resolvent. We give similar characterizations of the difference estimate ‖T ⁿ −T ⁿ⁺¹‖≤c/F(n) for a power-bounded linear operator T, when F(n) grows faster than n ^1/2 for large n.     

Weak stability and weak discrete convergence of continuous mappings

Summary The present paper establishes new equivalence theorems concerning weak discrete convergence, weak stability and consistency for sequences of continuous mappings as well as associated estimates. The method of increasing precisions yields an algorithm for the numerical solution of weakly discretely convergent problems. The general results are illustrated by an application to double sequences and difference quotients.     

A participative and facilitative conceptual modelling framework for discrete event simulation studies in healthcare

Existing approaches to conceptual modelling (CM) in discrete-event simulation do not formally support the participation of a group of stakeholders. Simulation in healthcare can benefit from stakeholder participation as it makes possible to share multiple views and tacit knowledge from different parts of the system. We put forward a framework tailored to healthcare that supports the interaction of simulation modellers with a group of stakeholders to arrive at a common conceptual model. The framework incorporates two facilitated workshops. It consists of a package including: three key stages and sub-stages; activities and guidance; tools and prescribed outputs. The CM framework is tested in a real case study of an obesity system. The benefits of using this framework in healthcare studies and more widely in simulation are discussed. The paper also considers how the framework meets the CM requirements.