Lipschitz constant

In a recent study Wang and Hofmann (1999) have stressed the importance of nonisothermal rate data. From a simple theoretical analysis they conclude that kinetic and transport data obtained under isothermal conditions in a laboratory reactor cannot logically be used to simulate any other type of reactor. This is because of the behavior of the Lipschitz constant L, which is a measure of the sensitivity of the reaction to different models. It tells us how any two models would diverge at the end of a reactor under different thermal conditions of operation. It is therefore a useful criterion for selecting the best model. It has been shown that L is different for different reactor models ... 

Proof First observe that due to smoothness and the assumptions on , we have a global Lipschitz constant... 

Lipschitz constant. In fact, the relation (2.14) implies that... 

Lawson JD (1967) Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J Numer Anal 4 372-380... 

Assutnc that f has a Lipschitz constant L. Thcn the difference between the two solutions can be bounded by... 

A number of inequalities on the so called concentration of measure phenomenon ([Talagrand (1996)], [Ledoux (2001)], [Villani (2003)]) are available for product probability spaces. We say that the IID sequence of integrable random variables satisfies a concentration inequality if there exists a continuous function [0, oo) — [0, oo), limt oo (t) = 0, such that for every n and every Lipschitz convex function G K — R with Lipschitz constant 1, that is... 

There exists a constant M > 0 such that f(f, x,8) < M. f(f, X, a) is Lipschitz continuous with respect to x and a with Lipschitz constants and kf , respectively. 

The first and second terms on the right-hand side of (A.6) can be estimated using the Lipschitz constants Xx and X ... 

A priori estimates in the case of a variable operator A. So far we have established stability in Ha under the agreement that operator A is constant, that is, independent of t. In the case when A(t) = A t) > 0 depends on t, this obstacle necessitates imposing the Lipschitz continuity of the operator A(t) in the variable t... 

Let s consider a more practical scheme named as proportional placement (PP) based on offset lag, i.e., 9=fp to)+aLp, where a is constant placement coefficient less than 1, and Lp is the first neighbor s offset lag. Since the first neighbor must have been a new peer when it entered the system, we can refer to a very familiar formula Xti =bx +c, which is a contraction mapping when the Lipschitz condition satisfies bspecific initial offset. 

It is easy to show that if // > 1, the derivative of/(w) is zero, so that it is a constant. This case is not of great interest, so it is always assumed that /i < 1. For // = 1, the Hdlder condition is termed the Lipschitz condition and is obeyed by any differentiable function, and others not in this class. For // < 1, the condition implies continuity in the ordinary sense. The case // = 0, which is excluded, is consistent with discontinuity. A function obeying this condition at a point, or on a line, will be described as obeying the H(ji) condition on that set, if n is specified or otherwise just the H condition. 

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