Trivial solution

A tircial solution to this equation is x = 0. For a non-trivial solution, we require that the deterniinant A - AI equals zero. One way to determine the eigenvalues and their associated eigenvectors is thus to expend the determinant to give a polynomial equation in A. Ko." our 3x3 symmetric matrix this gives ... 

The use of selectively reduced integration to obtain accurate non-trivial solutions for incompressible flow problems by the continuous penalty method is not robust and failure may occur. An alternative method called the discrete penalty technique was therefore developed. In this technique separate discretizations for the equation of motion and the penalty relationship (3.6) are first obtained and then the pressure in the equation of motion is substituted using these discretized forms. Finite elements used in conjunction with the discrete penalty scheme must provide appropriate interpolation orders for velocity and pressure to satisfy the BB condition. This is in contrast to the continuous penalty method in which the satisfaction of the stability condition is achieved indirectly through... 

The set of eigenvalue-eigenveetor equations has non-trivial (v(k) = 0 is "trivial") solutions if... 

In addition to the trivial solutions, there is a /S-periodic upside-down barrier trajectory called instanton, or bounce [Langer 1969 Callan and Coleman 1977 Polyakov 1977]. At jS oo the instanton dwells mostly in the vicinity of the point x = 0, attending the barrier region (near x ) only during some finite time (fig. 20). When jS is raised, the instanton amplitude... 

Unlike the trivial solution x = 0, the instanton, as well as the solution x(t) = x, is not the minimum of the action S[x(t)], but a saddle point, because there is at least one direction in the space of functions x(t), i.e. towards the absolute minimum x(t) = 0, in which the action decreases. Hence if we were to try to use the approximation of steepest descents in the path integral (3.13), we would get divergences from these two saddle points. This is not surprising, because the partition function corresponding to the unbounded Hamiltonian does diverge. 

This formula, however, tacitly supposes that the instanton period depends monotonically on its amplitude so that the zero-amplitude vibrations in the upside-down barrier possess the smallest possible period 2nla>. This is obvious for sufficiently nonpathological one-dimensional potentials, but in two dimensions this is not necessarily the case. Benderskii et al. [1993] have found that there are certain cases of strongly bent two-dimensional PES when the instanton period has a minimum at a finite amplitude. Therefore, the cross-over temperature, formally defined as the lowest temperature at which the instanton still exists, turns out to be higher than that predicted by (4.7). At 7 > Tc the trivial solution Q= Q Q is the saddle-point coordinate) emerges instead of instanton, the action equals S = pV (where F " is the barrier height at the saddle point) and the Arrhenius dependence k oc exp( — F ") holds. 

This solution for (X) in which all the unknowns are zero is called the trivial solution. A nontrivial solution to Equation (A.27) exists, therefore, only when matrix [A] is singular, that is, when A = 0. 

These equations are satisfied if 7a = 0, 7z = 0, but this trivial solution is of no interest. To ensure a nontrivial solution, it is sufficient to require that the determinant of the coefficients of 7a and 7z be equal to zero, namely. 

Deterministic Randomness. On the one hand, equation 4.8 is a trivial linear difference equation possessing an equally as trivial solution for each initial point Xq Xn = 2"a o (mod 1). Once an initial point is chosen, the future iterates are determined uniquely. As such, this simple system is an intrinsically deterministic one. On the other hand, look again at the binary decimal expansion of a randomly selected a o- This expansion can also be thought of as a particular semi-infinite sequence of coin tosses. 

Note that the system (11.55) is valid for small deviations of the interface from Xf when mx f < 1 and exp( ix[) 1. Estimations show that the term in the thermal balance equation on the interfece is small in comparison with the term Aca3i and ALa32. Moreover, since Pg.l( g//ilg) < 1 and (/Jl/Zilg) < it is possible to neglect the second term in the expressions for coefficients an and 01.2,2 and assume that 0 31 = (ml - OgPg.l g), 32 = (ml - ol l)- Then the non-trivial solution of... 

When the IRP is traced, successive points are obtained following the energy gradient. Because there is no external force or torque, the path is irrotational and leaves the center of mass fixed. Sets of points coming from separate geometry optimizations (as in the case of the DC model) introduce the additional problem of their relative orientation. In fact, the distance in MW coordinates between adjacent points is altered by the rotation or translation of their respeetive referenee axes. The problem of translation has the trivial solution of centering the referenee axes at the eenter of mass of the system. On the other hand for non planar systems, the problem of rotations does not have an analytical solution and must be solved by numeiieal minimization of the distanee between sueeessive points as a funetion of the Euler angles of the system [16,24]. 

A necessary condition for obtaining non-trivial solutions for u. and v in eq. (31.6a) is that the determinants of the coefficient matrices must be zero, which results into the so-called characteristic equation ... 

The procedure is schematically shown in Fig. 34.29. Equation (34.10) represents a homogeneous system of equations with a trivial solution r, = 0. Because component / is absent in the concentration vector, this component does not contribute to the matrix T °. As a consequence the rank of T is one less than its number of rows. A non-trivial solution therefore can be calculated. The value of one element of r, is arbitrarily chosen and the other elements are calculated by a simple regression [17]. Because the solution depends on the initially chosen value, the size (scale) of the true factors remains undetermined. By repeating this procedure for all columns c, (t = 1 to p), one obtains all columns of R, the entire rotation matrix. 

The trivial solution (w = -Wo ) is not interesting since it defines the same boundary line. A non-trivial solution is found by the following procedure ... 

These simultaneous linear homogeneous equations determine c and C2 and have a non-trivial solution if the determinant of the coefficients of c, C2 vanishes... 

This determinant or its equivalent algebraic expansion is known as the secular equation. In equation (9.12) the parameters Ci correspond to the unknown quantities Xi in equation (9.13) and the terms (//, — WSu) correspond to the coefficients au- Thus, a non-trivial solution for the N parameters c, exists only if the determinant with elements Hu — Su) vanishes... 

If Nv < Nr, Nd < 0, and the problem is over defined only a trivial solution is possible. 

If C m is zero, we have the trivial solution C = 0. It is obvious from Eq. (2-2) immediately. For a more interesting situation in which C is nonzero, or for C to deviate from the initial C0,... 

Given a prediction of the liquid-phase activity coefficients, from say the NRTL or UNIQUAC equations, then Equations 4.69 and 4.70 can be solved simultaneously for x and x . There are a number of solutions to these equations, including a trivial solution corresponding with x[ = x[. For a solution to be meaningful ... 

This ensures liquid-liquid equilibrium. Trivial solutions whereby x = xf need to be avoided. The results are shown in Figure 4.8 to be x[ = 0.59, xf = 0.98. The system forms a hetero azeotrope. 

It is a common problem to solve a set of homogeneous equations of the form Ax = 0. If the matrix is non-singular the only solutions are the trivial ones, x = x2 = = xn = 0. It follows that the set of homogeneous equations has non-trivial solutions only if A = 0. This means that the matrix has no inverse and a new strategy is required in order to get a solution. 

Non-trivial solutions are obtained only if the determinant of coefficients vanishes. Reduced to a two-dimensional problem, e.g. in K(z, t), the secular equation becomes... 

For neutral atoms, N = Z, (23) requires x ixo) = 0> so that x vanishes at the same point as x- Since this condition cannot be satisfied for a finite value xo by non-trivial solutions, the point xo must be at infinity. The solution X(x) for a neutral atom must hence be asymptotic to the x-axis, x(°°) = 0. There is no boundary to the neutral atom in the Thomas-Fermi model. 

The simultaneous equations (37) and (38) have non-trivial solutions only if the determinant of coefficients vanishes giving rise to the secular equation... 

The integrals of the H and S matrices are generally referred to as matrix elements. The condition for a non-trivial solution is that the secular determinant should vanish,... 

In order to get non-trivial solutions the determinant of Ap needs to be zero ... 

Obviously, the trivial solution v,=0 (/ = L > n) does not fit our needs and we must search for solutions as a constrained problem in which the solution vector is of constant, yet arbitrary, length. In other words, we become interested in the vector with some criterion of best direction regardless of its magnitude, which we may conveniently take as unity. Let us lump the C/ coefficients into the m x n matrix A and the n coefficients Vj into the vector x , hence... 

This represents a homogeneous system of equations. There is, as always with homogeneous equations, a trivial solution t fO. 

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