A distribution map for the one-median location problem on a network

In this paper we consider the Weber (one-median) location problem on a network. The weights at the nodes are drawn from a multivariate normal distribution. We find the probability that the optimal location is at each of the nodes. This set of probabilities is the distribution map. The methodology is illustrated on two networks of 20 nodes each.     

An improved lower bound for the expected number of real zeros of a random algebraic polynomials

In this note we estimate the lower bound of the average number of real zeros of a random algebraic polynomials when the random coefficients are standard normal random variables     

A single-period inventory placement problem for a supply chain with the expected profit objective

Consider the expected profit maximizing inventory placement problem in an N-stage, supply chain facing a stochastic demand for a single planning period for a specialty item with a very short selling season. Each stage is a stocking point holding some form of inventory (e.g., raw materials, subassemblies, product returns or finished products) that after a suitable transformation can satisfy customer demand. Stocking decisions are made before demand occurs. Because of delays, only a known fraction of demand at a stage will wait for shipments. Unsatisfied demand is lost. The revenue, salvage value, ordering, shipping, processing, and lost sales costs are proportional. There are fixed costs for utilizing stages for stock storage. After characterizing an optimal solution, we propose an algorithm for its computation. For the zero fixed cost case, the computations can be done on a spreadsheet given normal demands. For the nonnegative fixed cost case, we develop an effective branch and bound algorithm.     

A formula for the lyapunov numbers of a stochastic flow. application to a perturbation theorem

We present a formula for the Lyapunov exponents of the flow of a nonlinear stochastic system. (These exponents characterise the asymptotic behaviour of the derivative flow, and negative exponents are associated with clustering of the flow). This formula is analogous to that of Khas'minskii, who deals with a linear system. We use this fojoruila to show that if we have an ordinary dynamical system which is Lyapunov stable (i.e. all the exponents are negative) then so are certain stochastic perturbations of it.     

A method for maximizing the reliability coefficient of a communications network

The most common idea of network reliability in the literature is a numerical parameter calledoverall network reliability, which is the probability that the network will be in a successful state in which all nodes can mutually communicate. Most papers concentrate on the problem of calculating the overall network reliability which is known to be an NP hard problem. In the present paper, the question asked is how to find a method for determining a reliable subnetwork of a given network. Givenn terminals and one central computer, the problem is to construct a network that links each terminal to the central computer, subject to the following conditions: (1) each link must be economically feasible; (2) the minimum number of links should be used; and (3) the reliability coefficient should be maximized. We argue that the network satisfying condition (2) is a spanning arborescence of the network defined by condition (1). We define the idea of thereliability coefficient of a spanning arborescence of a network, which is the probability that a node at average distance from the root of the arborescence can communicate with the root. We show how this coefficient can be calculated exactly when there are no degree constraints on nodes of the spanning arborescence, or approximately when such degree constraints are present. Computational experience for networks consisting of up to 900 terminals is given.This report was prepared as part of the activities of the Management Science Research Group, Carnegie-Mellon University, under Contract No. N00014-82-K-0329 NR 047–048 with the U.S. Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

A recursive pricing formula for a path-dependent option under the constant elasticity of variance diffusion

In this paper, we consider a path-dependent option in finance under the constant elasticity of variance diffusion. We use a perturbation argument and the probabilistic representation (the Feynman–Kac theorem) of a partial differential equation to obtain a complete asymptotic expansion of the option price in a recursive manner based on the Black–Scholes formula and prove rigorously the existence of the expansion with a convergence error.     

A numerical method for solving stochastic programming problems with moment constraints on a distribution function

The stochastic programming problem is considered in the case of a distribution function with partially known random parameters. A minimax approach is taken, and a numerical method is proposed for problems when information on the distribution function can be expressed in the form of finitely many moment constraints. Convergence is proved and results of numerical experiments are reported.     

A low-Mach number method for the numerical simulation of complex flows

A new numerical procedure which considers a modification to the artificial acoustic stiffness correction method (AASCM) is here presented, to perform simulations of low Mach number flows with the compressible Navier–Stokes equations. An extra term is added to the energy fluxes instead of using an energy source correction term as in the original model. This new scheme re-scales the speed of sound to values similar to the flow velocity, enabling the use of larger time steps and leading to a more stable numerical method. The new method is validated performing Large Eddy Simulations on test problems. The effect of a crucial numerical parameter alpha is evaluated as well as the robustness of the method to variations of the Mach number. Numerical results are compared to the existing experimental data showing that the new method achieves good agreement increasing the time-step, and therefore accelerating the computation for low-Mach convective flows.     

A direct method to calculate the energy evolution of a turbulent flow

The classical configuration of the power spectrum of a homogeneous turbulent flow provides a functional relation between energy and enstrophy which may be used to find the evolution of energy in the intermediate stages of finite time and viscosity to which the usual asymptotic arguments do not apply. The method has a minimal computational cost compared with the direct numerical integration of the whole Navier–Stokes system. It is applied here to study the energy evolution both theoretically and numerically, obtaining the minima of energy compatible with the power spectra as well as the rate of decay of the energy towards them. The results may be useful to study the compatibility of the Kolmogorov and Kraichnan spectra with the observed energy evolution of a turbulent flow.     

Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter

We study least energy solutions of a quasilinear Schrödinger equation with a small parameter. We prove that the ground state is nondegenerate and unique up to translations and phase shifts using bifurcation theory.     

A note on distance-regular graphs with a small number of vertices compared to the valency

In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α>2$\alpha >2$, there are finitely many distance-regular graphs Γ$\Gamma$ with valency k$k$, diameter D≥3$D\ge 3$ and v$v$ vertices satisfying v≤αk$v\le \alpha k$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k≥3$k\ge 3$, diameter D≥3$D\ge 3$ and _c2≥εk$c_2\ge \epsilon k$ for a given 0<ε<1$0<\epsilon <1$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite).     

Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance

In this paper, we extend the Cramér-Lundberg risk model perturbed by diffusion to incorporate the jumps of surplus investment return. Under the assumption that the jump of surplus investment return follows a compound Poisson process with Laplace distributed jump sizes, we obtain the explicit closed-form expression of the resulting Gerber-Shiu expected discounted penalty (EDP) function through the Wiener-Hopf factorization technique instead of the integro-differential equation approach. Especially, when the claim distribution is of Phase-type, the expression of the EDP function is simplified even further as a compact matrix-type form. Finally, the financial applications include pricing barrier option and perpetual American put option and determining the optimal capital structure of a firm with endogenous default.     

A variance reduction technique for use with the extrapolated Euler method for numerical solution of stochastic differential equations

A simple, effective technique is described and tested for reducing the variation in estimated expectations of functions of functions of solutions of stochastic differential equations. The technique is implemented with extrapolated Euler method for numerical solution of stochastic differential equations     

A method for analysis of linear dynamic systems driven by stationary non-Gaussian noise with applications to turbulence-induced random vibration

A method is developed for approximating the properties of the state of a linear dynamic system driven by a broad class of non-Gaussian noise, namely, by polynomials of filtered Gaussian processes. The method involves four steps. First, the mean and correlation functions of the state of the system are calculated from those of the input noise. Second, higher order moments of the state are calculated based on Itô’s formula for continuous semimartingales. It is shown that equations governing these moments are closed, so that moment of any order of the state can be calculated exactly. Third, a conceptually simple technique, which resembles the Galerkin method for solving differential equations, is proposed for constructing approximations for the marginal distribution of the state from its moments. Fourth, translation models are calibrated to representations of the marginal distributions of the state as well as its second moment properties. The resulting models can then be utilized to estimate properties of the state, such as the mean rate at which the state exits a safe set. The implementation of the proposed method is demonstrated by numerous examples, including the turbulence-induced random vibration of a flexible plate.     

Asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small puncture

Let X(t) be a positive recurrent diffusion process corresponding to an operator L on a domain D⊆R^d with oblique reflection at ∂D if D≠R^d. For each x∈D, we define a volume-preserving norm that depends on the diffusion matrix a(x). We calculate the asymptotic behavior as ε→0 of the expected hitting time of the ε-ball centered at x and of the principal eigenvalue for L in the exterior domain formed by deleting the ball, with the oblique derivative boundary condition at ∂D and the Dirichlet boundary condition on the boundary of the ball. This operator is non-self-adjoint in general. The behavior is described in terms of the invariant probability density at x and Det(a(x)). In the case of normally reflected Brownian motion, the results become isoperimetric-type equalities.     

A note on the solution to a common thermal network problem encountered in heat-transfer analysis of spacecraft

A widely used discretization method for modeling thermal systems is the thermal network approach. The network approach is derived from energy balance equations and is equivalent to a particular finite difference discretization of the underlying heat-transfer equation. The steady-state problems that arise in the analysis of spacecraft systems using network models are frequently dominated by radiative transfer, which introduces quartic nonlinearities in the equations. Although these systems are routinely encountered, there has not appeared any detailed analysis of these equations. Questions of existence and uniqueness of solutions and numerical methods for solving the systems have never been addressed in any generality. In this paper, general existence and uniqueness properties of the network equations are established. Globally convergent methods for solving the systems are developed and insight into the relative success of existing methods is given. Numerical examples are presented illustrating the methods. The perspective adopted here is also useful in interdisciplinary applications. A simple example involving thermal control is used to demonstrate this.     

A density result for the variation of a material with respect to small inclusions

We consider the family of materials obtained, via homogenization, by replacing a small portion, of size ?, of a fixed material by other materials. In a previous paper we have obtained a subset of the set of ‘derivatives’ of this family with respect to ? in $?=0$. In the present Note we prove that this set is, in fact, dense. This result can be applied, for example, to obtain optimality conditions for composite materials. To cite this article: J. Casado-Díaz et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

A class of bivariate negative binomial distributions with different index parameters in the marginals

In this paper, we consider a new class of bivariate negative binomial distributions having marginal distributions with different index parameters. This feature is useful in statistical modelling and simulation studies, where different marginal distributions and a specified correlation are required. This feature also makes it more flexible than the existing bivariate generalizations of the negative binomial distribution, which have a common index parameter in the marginal distributions. Various interesting properties, such as canonical expansions and quadrant dependence, are obtained. Potential application of the proposed class of bivariate negative binomial distributions, as a bivariate mixed Poisson distribution, and computer generation of samples are examined. Numerical examples as well as goodness-of-fit to simulated and real data are also given here in order to illustrate the application of this family of bivariate negative binomial distributions.