51/ algebra properties

We think there is a need to investigate the algebraic properties underlying the various graph and matrix representations in order to discover the properties which are common to every stereoisomerization, to every ligand partition and perhaps, to every coordination number. This attempt is necessary if we want to understand the maximum number of phenomena with a minimum number of concepts. 

The matrix gab is symmetric and non-singular and has other purely algebraic properties that can be demonstrated by first defining the matrices... 

Equation (64) justifies the application of flux-balance analysis even in the face of (i) fast short-term fluctuations and (ii) periodic long term for example, circadian variability. The steady state balance condition restricts the feasible steady-state flux distributions to the flux cone P = v° G IRr IVv0 = 0. The reduction of the admissible flux space, with some of its algebraic properties already summarized in Section III.B, is exploited by several computational approaches, most notably Flux Balance Analysis (FBA) [61, 71, 235] and elementary flux modes (EFMs) [96, 236 238],... 

Schuster, S. Hilgetag, C. Woods, J. H. Fell, D. A. Reaction routes in biochemical reaction systems algebraic properties, validated calculation procedure and example from nucleotide metabolism. J Math Biol 2002,45 153-181. 

The representation of an EMB(A) by its fee-matrix B = (b ), an n x n symmetric matrix with positive integer entries and well-defined algebraic properties, corresponds to the act of translating an object of chemistry into a genuine mathematical object. This implies that the chemistry of an EMB(A) corresponds to the algebraic properties of the matrix B. 

By the word infinitesimals we refer to the nilpotent quantities, whose second or higher powers are zero. Some of their algebraic properties are quite different from the the properties of the finite quantities [56]. For example, et, — 0 does not necessarily imply e — 0 or % = 0, if e and are infinitesimals. Also, the infinitesimals do not possess inverses that is, e-1 is not defined. 

A way to overcome the difficulties in the definition of the Hermitian phase operator has been proposed by Pegg and Barnett [40,45]. Their method is based on a contraction of the infinite-dimensional Hilbert-Fock space of photon states Within this method, the quantum phase variable is determined first in a finite 5-dimensional subspace of //, where the polar decomposition is allowed. The formal limit, v oc is taken only after the averages of the operators, describing the physical quantities, have been calculated. Let us stress that any restriction of dimension of the Hilbert-Fock space of photons is equivalent to an effective violation of the algebraic properties of the photon operators and therefore can lead to an inadequate picture of quantum fluctuations [46]. 

We also note that, in contrast to the Pegg-Bamett formalism [45], we consider an extended space of states, including the Hilbert-Fock state of photons as well as the space of atomic states [36,46,53,54]. The quantum phase of radiation is defined, in this case, by mapping of corresponding operators from the atomic space of states to the whole Hilbert-Fock space of photons. This procedure does not lead to any violation of the algebraic properties of multipole photons and therefore gives an adequate picture of quantum phase fluctuations [46],... 

It is seen that (157) has the operator structure and algebraic properties similar to those of (136)—(137). At r 0, the set (157) exactly coincides with (136)— (137). Due to the commutation relations (155), the operators (157) have the same algebraic properties as do (136)—(137) at any given point. In particular, we can construct the local representation of the radiation phase operators in the same way as in Section IV, using the operator (158) instead of (63). By construction, this gives us the 57/(2) quantum phase of spin or polarization with the properties described in Section IV.C. 

It follows from the algebraic properties of the above sum that almost all eigenvalues f2rk lie between a pair of the original frequencies corh, so that the bands of the phonons and of the phonon-like modes coincide. 

The Dirac matrices are defined abstractly by their algebraic properties. Using the notation 04 = 0, the defining properties are the anticommutation relations... 

For particles with spin-1/2 we would expect (on the basis of nonrelativistic quantum mechanics) that spinors with two components would be sufficient. But the Dirac spinors have to be (at least) four-dimensional. A mathematical reason lies in the nature of the algebraic properties that have to be satisfied by the Dirac matrices a and 0 if the Dirac equation should satisfy the relativistic energy-momentum relation in the sense described above, see (6). 

This requirement does not fix the Dirac matrices uniquely, and thus the whole Dirac theory and all systematic approximations to it could equally well be formulated in terms of general four-dimensional quaternions, which are independent of a special representation and rely only on the algebraic properties of the Clifford algebra [8-10]. Such an implementation of the Dirac theory is known to speed up diagonalisation procedures significantly, and has successfully been employed in modem four-component relativistic program packages like Dirac... 

But the R-matrix has some very interesting and useful formal, algebraic properties in addition to this physical interpretation. 

Many of the algebraic properties of the single-particle Green s function, in particular Dyson s equation, are transferable to two-particle propagators if they are constructed starting from an orthonormal set of primary states. In this paper we will construct propagators Q(u)) with the help of the extended... 

Note that the terms G and J are proportional to the single-particle density Pij — (V l W) of secondary reference state extended states A,B). The choice of this reference wavefunction is arbitrary for the algebraic properties of the extended wave-function and, in particular, for the derivation of the Dyson equation. Here it obviously introduces differences and the freedom of choice can be used to change the static self energy. The only condition that tp) has to comply with in order to have the full formalism at hand, is to be cm eigenfunction of the Fock-space Hamiltonian H (c. f. Sec. IIC). The two obvious choices for y )... 

There are some further algebraic properties of R n) that are simple to establish. 

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