Fixed signs

Remark Quite often, the Dirichlet problem is approximated by the method based on the difference approximation at the near-boundary nodes of the Laplace operator on an irregular pattern, with the use of formulae (14) instead of (16) at the nodes x G However, in some cases the difference operator so constructed does not possess several important properties intrinsic to the initial differential equation, namely, the self-adjointness and the property of having fixed sign, For this reason iterative methods are of little use in studying grid equations and will be excluded from further consideration. 

For p > 4 the difference operator (15) lacks the property of having fixed sign (ellipticity). In each such case it is recommended to refer to another operator A, which preserves the ellipticity property for any p ... 

TEST SIGN BIT OF MICROSOFT MANTTISSA 1= -NEG MANTISSA-GO FIX SIGN IN APU FP FORMAT MS MANTISSA BIT ALWAYS SET IN APU FP FORMAT GET HL,BC,DE... 

Formally, a Liapounov function is a function y(ai, a2. . . a ) of fixed sign that is everywhere positive (or negative) and its time derivative... 

A large local value of h r) means a large resistance to change of the shape of the local electron density. It involves fluctuations in nuclear positions, e.g. in vibrations. The averaging in Equation (6.28) is over an ensemble. Since the covariance of p, and V does not necessarily have a fixed sign, we cannot say that hir) is a maximum or minimum. 

Fixed signs Unyielding, determined, focused, opposed to change... 

The important role of random fields in the disordered systems is confirmed by the form of their phase diagram, schematically represented in Fig. 1.11 [26]. It is seen that the boundaries of phases (paraphase (P), phase with long-range order (F), short-range order (SG) and mixed phase (FSG)) are defined by the ratio of mean value of random field Jo (mean field) to its dispersion J. In the disordered ferroics the nonzero mean field originates from fixed sign part of dipole-dipole or spin-spin interaction while alternative sign part defines the dispersion of the random field. 

Let us discuss these questions in more details. In magnetic systems the fixed sign interaction is defined by exchange interaction of spins that can be written in the form [25] ... 

In ionic lattices these shifted (off-central) impurities are known to be the electric dipoles. The indirect interaction of such dipoles via soft phonon mode of a dielectric host exists in the incipient ferroelectrics like KTaOs, SrTiOs, PbTe. It leads to following fixed sign interaction ... 

If the reaction is irreversible, fj has a fixed sign and vanishes only at equilibrium, it follows then from Eq.(2.4.14)—(2.4.16) that is monotonic. As Vq increases, the bounds of the integral in Eq. (2.8.20) increase as Tq, but the volume of integration increases as Tq. Thus, fj must be negligible except in a surface layer with thickness (2 ). The reaction... 

In the case of a reversible reaction, fj has no fixed sign, need not be monotonic, and sustained oscillations cannot be ruled out except by introducing entropy considerations. Suppose that there exists a positive definite form (t(u), vanishing only on the equilibrium manifold and satisfying a conservation equation of the type... 

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