Definite positive

For transition state searches, none of the above updates is particularly appropriate as a positive definite Hessian is not desired. A more usefiil update in this case is the Powell update [16] ... 

The EF algoritlnn [ ] is based on the work of Cerjan and Miller [ ] and, in particular, Simons and coworkers [70,1Y. It is closely related to the augmented Hessian (rational fiinction) approach[25]. We have seen in section B3.5.2.5 that this is equivalent to addmg a constant level shift (damping factor) to the diagonal elements of the approximate Hessian H. An appropriate level shift effectively makes the Hessian positive definite, suitable for minimization. 

Here, M is a constant, symmetric positive definite mass matrix. We assume without loss of generality that M is simply the identity matrix I. Otherwise, this is achieved by the familiar transformation... 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

X (-l/2), NOTE X must be real symmetrie, and positive definite. 

L°° Tl) depend only on x and possess the usual properties of symmetry and positive definiteness. As a matter of convenience we choose arbitrary fixed functions G and M - G L Q) satisfying the conditions... 

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... 

If ti satisfies necessaiy conditions [Eq. (3-80)], the second term disappears in this last line. Sufficient conditions for the point to be a local minimum are that the matrix of second partial derivatives F is positive definite. This matrix is symmetric, so all of its eigenvalues are real to be positive definite, they must all be greater than zero. 

This linear quadratic program will have a unique solution if B i) is kept positive definite. Efncient solution methods exist for solving it (Refs. 119 and 123). 

Since the deformation tensor F is nonsingular, it may be decomposed uniquely into a proper orthogonal tensor R and a positive-definite symmetric tensor U by the polar decomposition theorem... 

In this section, well-known properties of second-order positive-definite symmetric tensors and functions involving them will be cited without proof. The principal values and principal vectors (m = 1, 2, 3) of a symmetric second-order tensor A are given by... 

Associated with /4 is a quadric surface, which for positive-definite symmetric A is an ellipsoid, whose intercepts with the principal directions are the principal values. When expanded, (A.72) becomes... 

The title of the book requires that a definition of plastics materials be given. This is however very difficult. For the purpose of this book I eventually used as a working definition Those materials which are considered to be plastics materials by common acceptance . Not a positive definition but one which is probably less capable of being criticised than any other definition I have seen. Perhaps a rather more useful definition but one which requires clarification is... 

The Ajj are the sum of the product of the individual laminae Qy and the laminae thjcknesses. Thus, the only ways to obtain a zero individual Ay are for all Qy to be zero or for some Qy to be negative and some positive so that their products with their respective thicknesses sum torero. From the expressions for Uie transformed larnina stiffnesses, Qy, in Equation (2.80), apparently Qii, Q- 2, Q22 66 positive-definite... 

Both entropic and coulombic contributions are bounded from below and it can be verified that the second variation of is positive definite so that the above equations correspond to a minimum [27]. Using conditions in the bulk we can eliminate //, from the equations. Then we get the Boltzmann equation in which the electric potential verifies the Poisson equation by construction. Hence is equivalent within MFA to the... 

Although I used the example of ethene, where n =2, the same consideration, apply to ZDO calculations on all conjugated molecules. All overlap matrices are real symmetric, positive definite and so have eigenvalues > 0. 

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