Lipschitz condition

Equation 24 is a necessary condition for the presence of the internal discontinuity. If it is satisfied, for some g(u) and y, then Equation 11 fails to meet Lipschitz condition (see, for example, (4) for discussion of this as well as some other facts from the theory of differential equations) and continuous solutions cannot exist. Therefore, Equation 24 is also a sufficient condition for the presence of the internal discontinuity. Thus, we have proved the following ... 

To tackle the question of the growth of local error, we still must make an important assumption on namely that it satisfies a Lipschitz condition of the form... 

To complete the convergence proof for Verlet s method, we would still need to verify the second assumption. This requires the assumption that the force field F satisfy a Lipschitz condition ... 

Under these assumptions, we now find the conditions which ensure the existence of an invariant toroidal manifold tt y = H((p) of the system (2.1) whose derivatives satisfy the Lipschitz conditions. To find this torus, we use, as before, an iteration process. 

Then there exists n s [0, iol such that for all e [0,. O] the system of equations (3.1) possesses the invariant toroidal set (fi)-I = (9> H) satiffying the Lipschitz condition with respect to 9 and such that... 

To insure the uniqueness of the solution some additional restriction, such as a Lipschitz condition, is necessary. 

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. 

Curvature condition (1). This is a weighted Lipschitz condition for F, writ-... 

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. 

Remark 2 The property of stability can be interpreted as a Lipschitz continuity condition on the perturbation function v(y). 

The objective function /( ) and the inequality constraint g(x) are convex since f(x) is separable quadratic (sum of quadratic terms, each of which is a linear function of xi, x2,X3, respectively) and g(x) is linear. The equality constraint h(x) is linear. The primal problem is also stable since v(0) is finite and the additional stability condition (Lipschitz continuity-like) is satisfied since f(x) is well behaved and the constraints are linear. Hence, the conditions of the strong duality theorem are satisfied. This is why... 

Remark 2 The e-convergence test are sufficient for e-improvement but they are not necessary. An additional condition, which is an inverse Lipschitz assumption, needs to be introduced so as to prove the necessity. This additional condition states that for points of a certain distance apart the value of the feasibility cut should differ by at least some amount. This is stated in the following theorem of Holmberg (1990). 

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 ... 

Initial value problems with ordinary differential equations (ODE) have well-defined conditions (based on Lipschitz continuity of the time derivatives) that guarantee unique solutions. Conditions for unique solutions of DAEs (Equations 14.2 and 14.3) are less well defined. One way to guarantee existence and uniqueness of DAE solutions is to confirm that the DAE can be converted (at least implicitly) to an initial value ODE. A general analysis of these DAE properties can be found in [5] and is beyond the scope of this chapter. On the other hand, for a workable analysis, one needs to ensure a regularity condition on the DAE characterized by its index. 

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