Nonnegativity

Thus loops, utility paths, and stream splits offer the degrees of freedom for manipulating the network cost. The problem is one of multivariable nonlinear optimization. The constraints are only those of feasible heat transfer positive temperature difference and nonnegative heat duty for each exchanger. Furthermore, if stream splits exist, then positive bremch flow rates are additional constraints. 

Normalization is the process of finding a multiplicative constant for the wave function such that the integral of / over all space is 1.0. All space in this calculation is nonnegative because r cannot be less than 0. 

Thus, we have obtained that the right-hand side of (1.40) is always nonnegative, which gives (1.39). To derive a complete system of relations describing the interaction between the punch and the plate we should add to (1.38), (1.39) the constitutive law equations of Sections 1.1.3 and 1.1.4. 

The right-hand side is nonnegative here in view of (1.61), therefore (A) > 0. Lemma 1.1 is proved. 

The second term of the right-hand side of this relation converges to zero in view of the weak convergence of the third term is nonnegative. Hence... 

By (1.97), the right-hand side is nonnegative, which proves the assertion. Lemma 1.4. The following estimate takes place ... 

The right-hand side is nonnegative here, hence in view of the monotony of 0(t) the relation... 

We consider an equilibrium problem for a shell with a crack. The faces of the crack are assumed to satisfy a nonpenetration condition, which is an inequality imposed on the horizontal shell displacements. The properties of the solution are analysed - in particular, the smoothness of the stress field in the vicinity of the crack. The character of the contact between the crack faces is described in terms of a suitable nonnegative measure. The stability of the solution is investigated for small perturbations to the crack geometry. The results presented were obtained in (Khludnev, 1996b). 

The nonpenetration condition considered in this section leads to new effects such as the appearance of interaction forces between crack faces. It is of interest to establish the highest regularity of the solution up to the crack faces and thus to analyse the smoothness of the interaction forces. The regularity of the solution stated in this section entails the components of the strain and stress tensors to belong to in the vicinity of the crack and the interaction forces to belong to T. If the crack shape is not regular, i.e. 0 1), the interaction forces can be characterized by the nonnegative... 

Lemma 2.5. There exists a nonnegative measure jx G such that for... 

By the inequality (2.252), valid for all functions % possessing the property XN > 0 in this definition of L is correct. Similar to (Landkof, 1966) the functional L can be extended on Co l,p)- The extended functional is linear and positive, and hence it is continuous. This implies that there exists a nonnegative measure p G such that for all (f G Co l,p) a... 

Because g is nonnegative, one can see that ff is a coercive, strongly convex and lower semicontinuous functional. Therefore, there exists a unique solution w G H Qc) of the problem (3.204) or (3.205) (see Section 1.2). 

In so doing we have omitted the nonnegative term 5 p a)ij,aij — a)f). Since satisfies (5.11), and because the coefficients Uijki,bijki have the positive definiteness property (5.10), the inequality (5.20) results in the estimate... 

In this case the boundary conditions (5.81) are included in (5.84). At the first step we get a priori estimates. Assume that the solutions of (5.79)-(5.82) are smooth enough. Multiply (5.79), (5.80) by Vi, Oij — ij, respectively, and integrate over fl. Taking into account that the penalty term is nonnegative this provides the inequality... 

Integrate by parts in the fifth and sixth terms of the left-hand side of (5.152) taking into account the boundary conditions (5.149)-(5.151) and the Green formula like (5.138) for the domain fic- The penalty term is nonnegative and satisfy the equation (5.144). Hence the uniform in the s,5 estimate follows. 

In so doing we have omitted the nonnegative term containing the penalty operator. Using the formula (5.181), the integration by parts can be done in the third and the fifth terms of the left-hand side of (5.189). Also, note that Mij satisfy equation (5.175). Integration of (5.189) in t from 0 to t results in the inequality... 

Differentiate with respect to t the equations (5.185), (5.186) and multiply by Vjifiij — Mij, respectively. The nonnegative term (see Lions, 1969)... 

In this subsection we construct a nonnegative measure characterizing the work of interacting forces. The measure is defined on the Borel subsets of I. The space of continuous functions defined on I with compact supports is denoted by Co(I). 

Theorem 5.12. A nonnegative measure can he defined on the a-algebra of Borel subsets of I such that the representation... 

Now we can differentiate the equations (5.334)-(5.337) with respect to t and multiply by Vt, Wt, nt, rrit, respectively. The penalty terms are nonnegative and, therefore, they can be neglected. As a result, the following differential inequality is derived ... 

As the goal is to minimize the objective function, releasing the constraint into the feasible region must not decrease the objective function. Using the shadow price argument above, it is evident that the multipher must be nonnegative (Ref. 177). 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

The following work assumption is now made. The external work done on any arbitrarily chosen region of a body undergoing a finite smooth closed cycle of homogeneous deformation %(ti, f ) is nonnegative... 

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