Boundedness

In the same section, we also see that the source of the appropriate analytic behavior of the wave function is outside its defining equation (the Schibdinger equation), and is in general the consequence of either some very basic consideration or of the way that experiments are conducted. The analytic behavior in question can be in the frequency or in the time domain and leads in either case to a Kramers-Kronig type of reciprocal relations. We propose that behind these relations there may be an equation of restriction, but while in the former case (where the variable is the frequency) the equation of resh iction expresses causality (no effect before cause), for the latter case (when the variable is the time), the restriction is in several instances the basic requirement of lower boundedness of energies in (no-relativistic) spectra [39,40]. In a previous work, it has been shown that analyticity plays further roles in these reciprocal relations, in that it ensures that time causality is not violated in the conjugate relations and that (ordinary) gauge invariance is observed [40]. 

The equation of restriction can embody causality, lower boundedness of energies in the spectrum, positive wavenumber in the outgoing wave (all these in nonrelativistic physics) and interactions inside the light cone only, conditions of mass speciality, and so on in relativistic physics. In the case of interest in this... 

A boundedness and semicontinuity of the operator / follow from the same properties of I and P. The verification of the equivalence of conditions... 

The angular brackets ( , ) denote the integration over flc- In virtue of the linearity, boundedness, and coercivity of the form a(-, ), there exists a unique solution to (2.165). 

On the other hand, bearing in mind the boundedness of the set U, we can assume... 

Remark. The specific choice of bijki as the inverse of the Uijki for the elliptic regularization appears to be natural, since in the case of pure elastic (with K = [I/ (R)] , respectively p a) = 0), the boundary condition (5.16) reduces to (5.9). However, the proof of Theorem 5.1 works with any other choice of bijki as long as requirements of symmetry, boundedness and coercivity are met. 

A derivation of the next estimate requires the boundedness of the penalty term in L Q), i.e. uniformly in e,6... 

In this case the condition u(a ,0) = Ug x) and the boundary conditions are approximated exactly. For instance, one of the schemes arising in Section 1.2 is good enough for the difference approximation of the initial equation. No doubt, we preassumed not only the existence and continuity of the derivatives involved in the equation on the boundary of the domain in view (at. r = 0 or f = 0), but also the existence and boundedness of the third derivatives of a solution for raising the order of approximation of boundary and initial conditions. 

When = 0 we impose the boundedness condition j (0) < oo being equivalent to the requirements... 

Sufficiency. The stability with respect to the initial data means the boundedness of the resolving operator... 

Lemma 2 Let scheme (32) be given with a constant operator C. The condition of p-stabihty of this scheme is equivalent to the boundedness of the norm of the transition operator... 

In the second case we obtain a linear equation related to y and then use the elimination method for solving it. The uniform convergence with the rate 0 t + h ) takes place under the extra restrictions concerning the boundedness of the derivatives d k /du, d k /dx du, d k /dx. ... 

This means that the system of differential equations (55)-(56) generates an approximation of order 1 in a summarized sense to the Cauchy problem (51) under the extra restrictions on the existence and boundedness of the derivative A t)cPu/dt in some suitable norm. 

With the identification of the TS trajectory, we have taken the crucial step that enables us to carry over the constructions of the geometric TST into time-dependent settings. We now have at our disposal an invariant object that is analogous to the fixed point in an autonomous system in that it never leaves the barrier region. However, although this dynamical boundedness is characteristic of the saddle point and the NHIMs, what makes them important for TST are the invariant manifolds that are attached to them. It remains to be shown that the TS trajectory can take over their role in this respect. In doing so, we follow the two main steps of time-independent TST first describe the dynamics in the linear approximation, then verify that important features remain qualitatively intact in the full nonlinear system. 

The original Hohenberg-Kohn theorem was directly applicable to complete systems [14], The first adaptation of the Hohenberg-Kohn theorem to a part of a system involved special conditions the subsystem considered was a part of a finite and bounded entity regarded as a hypothetical system [21], The boundedness condition, in fact, the presence of a boundary beyond which the hypothetical system did not extend, was a feature not fully compatible with quantum mechanics, where no such boundaries can exist for any system of electron density, such as a molecular electron density. As a consequence of the Heisenberg uncertainty relation, molecular electron densities cannot have boundaries, and in a rigorous sense, no finite volume, however large, can contain a complete molecule. 

System (1) is of the same type as other systems which have recently appeared in the literature on allelopathic competitions (Hsu and Waltman, 1998, Braselton and Waltman, 2001, Fergola et al., 2004). Usually the mathematical analysis of such systems, first requires one to check if their solutions satisfy the properties of global existence in the future, positivity, boundedness and uniqueness. Subsequently, it often happens that, due to the difficulties of integration of these systems, we look for special biologically meaningful solutions such as steady-states or periodic... 

The division between constraints and desirable properties is somewhat vague. For example, die boundedness property (ii) is in fact an essential property of any mixing model used for reacting scalars. It could thus rightly be considered as a constraint, and not just a desirable property. 

Realizability and boundedness of all variables are assured. In particular, since the chemical source term is treated exactly, mass and element conservation is guaranteed at the notional-particle level. 

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