Non-increasing Function Of Time Research Articles

Дискретная замкнутая одночастичная цепочка контуров

A discrete dynamical system called a closed chain of contours is considered. This system belongs to the class of the contour networks introduced by A. P. Buslaev. The closed chain contains N contours. There are 2m cells and a particle at each contour. There are two points on any contour called a node such that each of these points is common for this contour and one of two adjacent contours located on the left and right. The nodes divide each contour into equal parts. At any time t = 0,1, 2,... any particle moves onto a cell forward in the prescribed direction. If two particles simultaneously try to cross the same node, then only the particle of the left contour moves. The time function is introduced, that is equal to 0 or 1. This function is called the potential delay of the particle. For t ≥ m, the equality of this function to 1 implies that the time before the delay of the particle is not greater than m. The sum of all particles potential delays is called the potential of delays. From a certain moment, the states of the system are periodically repeated (limit cycles). Suppose the number of transitions of a particle on the limit cycle is equal to S(T) and the period is equal to T. The ratio S(T) to T is called the average velocity of the particle. The following theorem have been proved. 1) The delay potential is a non-increasing function of time, and the delay potential does not change in any limit cycle, and the value of the delay potential is equal to a non-negative integer and does not exceed 2N/3. 2) If the average velocity of particles is less than 1 for a limit cycle, then the period of the cycle (this period may not be minimal) is equal to (m + 1)N. 3) The average velocity of particles is equal to v = 1 - H/((m + 1)N), where H is the potential of delays on the limit cycle. 4) For any m, there exists a value N such that there exists a limit cycle with H > 0 and, therefore, v < 1.

From local to global asymptotic stabilizability for weakly contractive control systems

A nonlinear control system is said to be weakly contractive in the control if the flow that it generates is non-expanding (in the sense that the distance between two trajectories is a non-increasing function of time) for some fixed Riemannian metric independent of the control. We prove in this paper that for such systems, local asymptotic stabilizability implies global asymptotic stabilizability by means of a dynamic state feedback. We link this result and the so-called Jurdjevic and Quinn approach.

The Random Gas of Hard Spheres

The inconsistency between the time-reversible Liouville equation and time-irreversible Boltzmann equation has been pointed out by Loschmidt. To avoid Loschmidt’s objection, here we propose a new dynamical system to model the motion of atoms of gas, with their interactions triggered by a random point process. Despite being random, this model can approximate the collision dynamics of rigid spheres via adjustable parameters. We compute the exact statistical steady state of the system, and determine the form of its marginal distributions for a large number of spheres. We find that the Kullback–Leibler entropy (a generalization of the conventional Boltzmann entropy) of the full system of random gas spheres is a non-increasing function of time. Unlike the conventional hard sphere model, the proposed random gas system results in a variant of the Enskog equation, which is known to be a more accurate model of dense gas than the Boltzmann equation. We examine the hydrodynamic limit of the derived Enskog equation for spheres of constant mass density, and find that the corresponding Enskog–Euler and Enskog–Navier–Stokes equations acquire additional effects in both the advective and viscous terms.

On Chu's disturbance energy

Chu [On the energy transfer to small disturbances in fluid flow (part I), Acta Mechanica 1 (1965) 215–234] proposed a positive definite energy norm for characterizing the level of fluctuation in a disturbance. In the absence of heat transfer at the boundaries, work done by boundary or body forces, heat and material sources of energy, this norm is a monotone, non-increasing function of time. In this paper, we show that Chu's disturbance energy defines an inner product, with respect to which the conservation equations of fluid motion linearized about a uniform base flow are self-adjoint. This ensures that the eigenvectors of the linearized operator are orthogonal to each other, and the property that the energy norm is a non-increasing function of time in the absence of physical sources of energy follows as an immediate consequence. Examples from numerical simulations of Euler equations are presented to highlight the importance of choosing an energy norm that is consistent with the underlying physics. We demonstrate that the disturbance energy as measured by Chu's norm does not exhibit spurious transient growth in the absence of physical sources of energy and hence is suitable for analyzing thermoacoustic instability.

Markovian Approximation for the Nosé–Hoover Method and H-Theorem

A Langevin equation with state-dependent random force is considered. When the Helmholtz free energy is a nonincreasing function of time (the H-theorem), a generalized Einstein relation is obtained. A stochastic process of the Nose–Hoover method is discussed on the basis of the Markovian approximation. It is found that the generalized Einstein relation holds for the Fokker–Planck equation associated with the stochastic Nose–Hoover equation. The present result indicates that the Nose–Hoover dynamics coarse-grained with time satisfies the H-theorem and therefore the canonical distribution is guaranteed for the subsystem.

A Note on the Mean Residual Life Function of a Parallel System

Abstract One of the most important types of system structures is the parallel structure. In the present article, we propose a definition for the mean residual life function of a parallel system and obtain some of its properties. The proposed definition measures the mean residual life function of a parallel system consisting of n identical and independent components under the condition that n - i, i = 0, 2, …, n - 1, components of the system are working and other components of the system have already failed. It is shown that, for the case where the components of the system have increasing hazard rate, the mean residual life function of the system is a nonincreasing function of time. Finally, we will obtain an upper bound for the proposed mean residual life function.

A Note on the Mean Residual Life Function of a Parallel System

One of the most important types of system structures is the parallel structure. In the present article, we propose a definition for the mean residual life function of a parallel system and obtain some of its properties. The proposed definition measures the mean residual life function of a parallel system consisting of n identical and independent components under the condition that n - i, i = 0, 2, …, n - 1, components of the system are working and other components of the system have already failed. It is shown that, for the case where the components of the system have increasing hazard rate, the mean residual life function of the system is a nonincreasing function of time. Finally, we will obtain an upper bound for the proposed mean residual life function.

Strategies for maximizing seller's profit under unknown buyer's valuations

Suppose there is a seller that has an unlimited number of units of a single product for sale. The seller at each moment of time posts a price for his/her product. Based on the posted price, at each moment of time, a buyer decides whether or not to buy a unit of that product from the seller. The only information about the buyer available to the seller is the seller's sales history. Further, we assume that the maximal unit price the buyer is willing to pay does not change over time. The question then is how should the seller price his/her product to maximize profits? To address this question, we use the notion of loss functions. Intuitively, a loss function is a measure, at each moment of time, of the lost opportunity to make a profit. In particular, we provide a polynomial-time algorithm that finds a pricing algorithm (strategy) for the seller that minimizes the average (total) losses over time. Further, we provide efficient algorithms for finding pricing strategies for maximizing the cumulative seller's profits when the sales period is limited, there is a limited supply of the product for sale, and the buyer's valuation is a non-increasing function of time, where those functions are known to the seller. We also present preliminary results on pricing strategies that minimize the maximum loss, and we show show that there is no strategy minimizing both the total loss and the maximum loss at the same time.

Generalized monotone schemes, discrete paths of extrema, and discrete entropy conditions

Solutions of conservation laws satisfy the monotonicity property: the number of local extrema is a non-increasing function of time, and local maximum/minimum values decrease/increase monotonically in time. This paper investigates this property from a numerical standpoint. We introduce a class of fully discrete in space and time, high order accurate, difference schemes, called generalized monotone schemes. Convergence toward the entropy solution is proven via a new technique of proof, assuming that the initial data has a finite number of extremum values only, and the flux-function is strictly convex. We define discrete paths of extrema by tracking local extremum values in the approximate solution. In the course of the analysis we establish the pointwise convergence of the trace of the solution along a path of extremum. As a corollary, we obtain a proof of convergence for a MUSCL-type scheme that is second order accurate away from sonic points and extrema.

Pushing Disks Together—The Continuous-Motion Case

If disks are moved so that each center—center distance does not increase, must the area of their union also be nonincreasing? We show that the answer is yes, assuming that there is a continuous motion such that each center—center distance is a nonincreasing function of time. This generalizes a previous result on unit disks. Our proof relies on a recent construction of Edelsbrunner and on new isoperimetric inequalities of independent interest. We go on to show analogous results for the intersection and for holes between disks.

An approximate solution of the integral equation of renewal theory

In this study, an approximation to the solution of the renewal integral equation is constructed. Performance of the new method is evaluated and it is shown that the approximation provides an upper bound for the renewal function when the hazard function is a non-increasing function of time.

An approximate solution of the integral equation of renewal theory

In this study, an approximation to the solution of the renewal integral equation is constructed. Performance of the new method is evaluated and it is shown that the approximation provides an upper bound for the renewal function when the hazard function is a non-increasing function of time.

A time-dependent stopping problem with application to live organ transplants.

We consider a time-dependent stopping problem and its application to the decision-making process associated with transplanting a live organ. "Offers" (e.g., kidneys for transplant) become available from time to time. The values of the offers constitute a sequence of independent identically distributed positive random variables. When an offer arrives, a decision is made whether to accept it. If it is accepted, the process terminates. Otherwise, the offer is lost and the process continues until the next arrival, or until a moment when the process terminates by itself. Self-termination depends on an underlying lifetime distribution (which in the application corresponds to that of the candidate for a transplant). When the underlying process has an increasing failure rate, and the arrivals form a renewal process, we show that the control-limit type policy that maximizes the expected reward is a nonincreasing function of time. For non-homogeneous Poisson arrivals, we derive a first-order differential equation for the control-limit function. This equation is explicitly solved for the case of discrete-valued offers, homogeneous Poisson arrivals, and Gamma distributed lifetime. We use the solution to analyze a detailed numerical example based on actual kidney transplant data.

On the Method of Characteristics in Rate‐Type Viscoelasticity

AbstractThis paper deals with the standard method of characteristics for a system of partial differential equations encountered in viscoelasticity (and also in viscoplasticity, electric transmission lines, etc.). The second law of thermodynamics implies the existence of a nonlinear functional called the total energy, which for the exact solutions of the considered problems is a nonincreasing function of time. A sufficient condition on the integration step is given such that the total energy of the numerical solution be also a non‐increasing function of the number of time integration steps. An example is given showing that, if this condition is violated then the numerical solution diverges in time.