Matrices functions

V.L. Druskin and L.A. Knizhnerman Krylov subspace approximation of eigen-pairs and matrix functions in exact and computer arithmetics. Num. Lin. Alg. Appl., 2 (1995) 205-217... 

Abstract. We present novel time integration schemes for Newtonian dynamics whose fastest oscillations are nearly harmonic, for constrained Newtonian dynamics including the Car-Parrinello equations of ab initio molecular dynamics, and for mixed quantum-classical molecular dynamics. The methods attain favorable properties by using matrix-function vector products which are computed via Lanczos method. This permits to take longer time steps than in standard integrators. 

Of course, to make the scheme (4) practical, we must be able to compute the products of matrix functions a At A) and

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To form the only non-zero matrix elements of Hrot within the J, M, K> basis, one can use the following properties of the rotation-matrix functions (see, for example, Zare s book on Angular Momentum) ... 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

A square matrix of order n can be considered a vector in a space of n2 dimensions, and if the matrix function j... defines a vector norm in 2-space, it will be called a generalized matrix norm. Thus a generalized matrix norm satisfies... 

A matrix function v(...) will be called a mvltiplioative (strictly, submultiplicative) matrix norm, or simply a matrix norm, if it is a generalized matrix norm, and satisfies also... 

A. A. Martynyuk, Stability by Liapunov s Matrix Function Method with Applications... 

For many synthetic copolymers, it becomes possible to calculate all desired statistical characteristics of their primary structure, provided the sequence is described by a Markov chain. Although stochastic process 31 in the case of proteinlike copolymers is not a Markov chain, an exhaustive statistic description of their chemical structure can be performed by means of an auxiliary stochastic process 3iib whose states correspond to labeled monomeric units. As a label for unit M , it was suggested [23] to use its distance r from the center of the globule. The state of this stationary stochastic process 31 is a pair of numbers, (a, r), the first of which belongs to a discrete set while the second one corresponds to a continuous set. Stochastic process ib is remarkable for being stationary and Markovian. The probability of the transition from state a, r ) to state (/i, r") for the process of conventional movement along a heteropolymer macromolecule is described by the matrix-function of transition intensities... 

The procedure followed here is clearly that of matrix multiplication and Q can be any matrix whatever, for the proof does not require it to be a coordinate. It follows almost immediately that if F is a matrix function of two other matrices P and Q, then... 

The spreadsheet in Figure 4-62 is heavily matrix based (see Chapter 2, for an introduction to basic matrix functions in Excel). It is the only way to keep the structure reasonably simple. The matrix C in cells A21 C31 is computed in the usual way, see equation (4.63) the parameters required to compute the concentration matrix are in cells Q4 S4, they include the initial concentration for species A and the two rate constants that are to be fitted. In cells E 16 018 the computation of the best absorptivity matrix A for any given concentration matrix C, is done as a matrix equation, as demonstrated in The Pseudo-Inverse in Excel (p.146). Similarly the matrix Ycaic in cells E21 031 is written as the matrix product CA. Even the calculation of the square sum of the residuals in cell R7 is written in a compact way, using the Excel function SUMXMY2, especially designed for this purpose. We refer to... 

Matrix elements are scalar-valued matrix functions of the exponent matrices Lk- Therefore, the appropriate mathematical tool for finding derivatives is the matrix differential calculus [116, 118]. Using this, the derivations are nontrivial but straightforward. We will only present the final results of the derivations. The reader wishing to derive these formulas, or other matrix derivatives, is referred to the Ref. 116 and references therein. 

Equation (95) reveals an interesting link between g-density functional theory and g-matrix functional theory. Consider rewriting Eq. (95) in a form analogous to Eq. (83),... 

A. Nagy, Density-matrix functional theory. Phys. Rev. A 66, 022505 (2002). 

A. Nagy and C. Amovilli, Effective potential in density matrix functional theory. J. Chem. Phys. 121, 6640-6648 (2004). 

Chloroquine is the drug of choice for preventing and treating acute forms of malaria caused by P. vivax, P. malariae, P ovale, as well as sensitive forms of P. falciparum. The mechanism of its action is not completely clear, although there are several hypotheses explaining its antimalarial activity. Chloroquine and its analogs inhibit synthesis of nucleic acids of the parasite by affecting the matrix function of DNA. This happens by preliminary... 

The left and right eigenvectors are equal only when the populations at all spin sites are the same. As a result of the invariance of modal matrices to matrix functions, the dynamic matrix L can be written as... 

Let us discuss briefly properties of the condensate matrix function /. According to the definitions of the Green s functions g the functions fa(x) are related to following correlation functions... 

Matrix Function group on the beads Group to be coupled Products"... 

Equation (52) may be solved for the off-shell reaction K-matrix functions by requiring to be an eigenfunction of the complete projected Hamiltonian with eigenvalue E ... 

Let us search for a satisfactory model to transform our verbal portrait of a natural catastrophe into notions and indicators subject to formalized description and transformation. With this aim in view, we select m elements of subsystems at the lowest level in the N U H system, the interaction between which we determine using the matrix function A = a,/, where ait is an indicator of the level of dependence of the relationships between subsystems i and j. Then, the I(t) parameter can be estimated as the sum ... 

Since an exact solution is assumed outside ri, the Schrodinger matrix functional is... 

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