Uniqueness of solution

It may be noted that an elastic material for which potentials of this sort exist is called a hyperelastic material. Hyperelasticity ensures the existence and uniqueness of solutions to intial/boundary value problems for an elastic material undergoing small deformations, and also implies that all acoustic wave speeds in the material are real and positive. 

It should be noted that the normality conditions, arising from the work assumption applied to inelastic loading, ensure the existence and uniqueness of solutions to initial/boundary value problems for inelastic materials undergoing small deformations. Uniqueness of solutions is not always desirable, however. Inelastic deformations often lead to instabilities such as localized deformations. It is quite possible that the work assumption, which is essentially a stability postulate, is too strong in these cases. Normality is a necessary condition for the work assumption. Instabilities, while they may occur in real deformations, are therefore likely to be associated with loss of normality and violation of the work assumption. 

Interest in developing and refining the mathematical methods of operations research has become intensified and sophisticated. Attention is generally given to a priori upper bounds on the number of solutions of a problem, the existence and uniqueness of solutions,... 

The method has been thoroughly analyzed in the electron microscopy application by Ferwerda and colleagues [see references cited by Van Toom and Ferwerda (1977)]. Convergence properties, uniqueness of solutions, and... 

Existence and uniqueness of solutions to the b.v.p. analogous to (2.2.1) has been proved in numerous contexts (see, e.g., [2]—[6]) and can be easily inferred for (2.2.1). We shall not do it here. Instead we shall assume the existence and uniqueness for (2.2.1) and similar formulations and, based on this assumption, we shall discuss some simple properties of the appropriate solutions. These properties will follow from those of the solution of the one-dimensional Poisson-Boltzmann equation, combined with two elementary comparison theorems for the nonlinear Poisson equation. These theorems follow from the Green s function representation for the solution of the nonlinear Poisson equation with a monotonic right-hand side (or from the maximum principle arguments [20]) and may be stated as follows. 

Existence and uniqueness of solutions to (4.2.12) and to the appropriate algebraic equations for ji, B have to be studied separately in any specific electro-diffusional set-up. 

Such purely mathematical problems as the existence and uniqueness of solutions of parabolic partial differential equations subject to free boundary conditions will not be discussed. These questions have been fully answered in recent years by the contributions of Evans (E2), Friedman (Fo, F6, F7), Kyner (K8, K9), Miranker (M8), Miranker and Keller (M9), Rubinstein (R7, R8, R9), Sestini (S5), and others, principally by application of fixed-point theorems and Green s function techniques. Readers concerned with these aspects should consult these authors for further references. 

Any governing model equations have to be supplemented by initial and boundary conditions, all together called side conditions. Their definition means imposing certain conditions on the dependent variable and/or functions of it (e.g. its derivative) on the boundary (in time and space) for uniqueness of solution. A proper choice of side conditions is crucial and usually represents a significant portion of the computational effort. Simply speaking, boundary conditions are the mathematical description of the different situations that occur at the boundary of the chosen domain that produce different results within the same physical system (same governing equations). A proper and accurate specification of the boundary conditions is necessary to produce relevant results from the calculation. Once the mathematical expressions of all boundary conditions are defined the so-called properly-posed problem is reached. Moreover, it must be noted that in fuel cell modeling there are various... 

The common underlying principle in the approaches for characterizing the solvability of a DAE system is to obtain, either explicitly, or implicitly, a local representation of an equivalent ODE system, for which available results on existence and uniqueness of solutions are applicable. The derivation of the underlying ODE system involves the repeated differentiation of the algebraic constraints of the DAE, and it is this differentiation process that leads to the concept of a DAE index that is widely used in the literature. For the semi-explicit DAE systems (A. 10) that are of interest to us here, the index has the following definition. 

In order to understand the problem of finding TS with three or more DOFs, it is useful to address the question of dimensionalities, in configuration and phase space. In classical, Hamiltonian dynamics, transition states are grounded on the idea that certain surfaces (more precisely, certain manifolds) act as barriers in phase space. It is possible to devise barriers in phase space, since in phase space, in contrast to configuration space, two trajectories never cross [uniqueness of solutions of ODEs, see Eq. (4)]. In order to construct a barrier in phase space, the first step is to construct a manifold if that is made of a set of trajectories [8]. 

Gusarov, A. L., 1981, On uniqueness of solution of inverse magnetotelluric problem for two-dimensional media (in Russian) Mathematical Models in Geophysics, Moscow State University, 31-61. 

Our treatment of vectorf ields has been very informal. In particular, we have taken a cavalier attitude toward questions of existence and uniqueness of solutions to... 

Chastened by this example, we state a theorem that provides sufficient conditions for existence and uniqueness of solutions to x = /(x). Existence and Uniqueness Theorem Consider the initial value problem X = fix), X(0) = Xg. ... 

In practice, such vector fields arise when we have a first-order system 0 = /(0), where /(0) is a real-valued, 27t-periodic function. That is, f d + 2it) = /(0) for all real 0. Moreover, we assume (as usual) that f(Q) is smooth enough to guarantee existence and uniqueness of solutions. Although this system could be regarded as a special case of a vector field on the line, it is usually clearer to think of it as a vector field on the circle (as in Example 4,1,1). This means that we don t distin-... 

In other words, existence and uniqueness of solutions are guaranteed if f is continuously differentiable. The proof of the theorem is similar to that for the case n = 1, and can be found in most texts on differential equations. Stronger versions of the theorem are available, but this one suffices for most applications. 

From now on, we ll assume that all our vector fields are smooth enough to ensure the existence and uniqueness of solutions, starting from any point in phase space. 

Another important outcome of these considerations is the following. The uniqueness of solutions of the Newton equations of motion implies that phase point trajectories do not cross. If we follow the motions of phase points that started at a given volume element in phase space we will therefore see all these points evolving in time into an equivalent volume element, not necessarily of the same geometrical shape. The number of points in this new volume is the same as the original one, and Eq. (1.107) implies that also their density is the same. Therefore, the new volume (again, not necessarily the shape) is the same as the original one. If we think of this set of points as molecules of some multidimensional fluid, the nature of the time evolution implies that this fluid is totally incompressible. Equation (1.107) is the mathematical expression of this incompressibility property. 

Consider an equilibrium thennodynamic ensemble, say a set of atomic systems characterized by the macroscopic variables T (temperature), Q (volume), andTV (number of particles). Each system in this ensemble contains N atoms whose positions and momenta are assigned according to the distribution function (5.2) subjected to the volume restriction. At some given time each system in this ensemble is in a particular microscopic state that coiTesponds to a point (r, p- ) in phase space. As the system evolves in time such a point moves according to the Newton equations of motion, defining what we call a phase space trajectory (see Section 1.2.2). The ensemble coiTesponds to a set of such trajectories, defined by their starting point and by the Newton equations. Due to the uniqueness of solutions of the Newton s equations, these trajectories do not intersect with themselves or with each other. 

Although the problem defined by (3-95) and (3-96) is time dependent, it is linear in uJ and confined to the bounded spatial domain, 0 < r < 1. Thus it can be solved by the method of separation of variables. In this method we first find a set of eigensolutions that satisfy the DE (3-95) and the boundary condition at r = 1 then we determine the particular sum of those eigensolutions that also satisfies the initial condition at 7 = 0. The problem (3-95) and (3-96) comprises one example of the general class of so-called Sturm-Louiville problems for which an extensive theory is available that ensures the existence and uniqueness of solutions constructed by means of eigenfunction expansions by the method of separation of variables.14 It is assumed that the reader is familiar with the basic technique, and the solution of (3-95) and (3-96) is simply outlined without detailed proofs. We begin with the basic hypothesis that a solution of (3-95) exists in the separable form... 

Existence and Uniqueness of Solutions to Holland s Equations fora Case of Multicolumn... 

Both DAEs and ODEs can be considered special cases of this structure. Brenan et al. [1] provide further reading on existence and uniqueness of solutions to these models, which are considerably more complex issues than in the case of simple ODEs, Initial conditions are required for dxjdt as well as x in this model,... 

Now that it is clear that identiliability is concerned with the question of uniqueness of solutions for the basic parameters from the observation function of a given experiment, we have to introduce the various types of identihability that have been defined in the literature. 

Assuming that all the variables in the physical problem are independent of the azimuthal coordinate (axisymmetry), the projections of Table 11.3 model equations along the three coordinate axes are given in Table 11.4 together with the boundary conditions chosen at the four peripheral boundaries of the porous medium z = 0, z = L, r=0,r= R. We will assume without proof, uniqueness of solution for the system of equations describing this ferrohydrodynamic model. 

One of the most important properties of a typical classical molecular Hamiltonian system is the existence and uniqueness of solutions started from a generic initial condition. The traditional study of existence and uniqueness for systems of nonlinear ordinary differential equations gives local results only, but for Hamiltonian systems considered in molecular dynamics the solutions are typically globally defined. The uniqueness of solutions is easily verified in the usual way (as for the local result for uniqueness of solutions). The key point is that, with the energy constraint, solutions typically remain bounded for all time. 

Although various attempts have been made to modify the Thomas-Fermi method, all of these attempts have failed to make the method reliable, because it has no physical background establishing the uniqueness of solutions and the existence of density functionals, and it also cannot reproduce even chemical bonds qualitatively. As a consequence, this method had been forgotten until the mid-1960s. 

Gavalas (1968) was an early pioneer in the treatment of the deterministic models of chemical reaction kinetics. His book deals with homogeneous systems and systems with diffusion as well. Basing himself upon recent results in nonlinear functional analysis he treats such fundamental questions as stoichiometry, existence and uniqueness of solutions and the number and stability of equilibrium states. Up to that time this treatise might be considered the best (although brief and concise) summary of the topic. 

Similar to first-order equations, the issue of existence and uniqueness of a solution to second order equations must be dealt with. Below is a theorem [1,4] that addresses the existence and uniqueness of solutions of second-order differential equations. 

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