Mathematical Theorems

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

A Practical Hint20. In order to most accurately determine PI in Eq. (7.21), a mathematical theorem concerning convolution of a function with a shape function are helpful. The measured primary beam profile of the Kratky camera... 

The second term on the right hand side of Equation 3.16 introduces complications because it couples x[ and xj. The first term, on the other hand is easily solved because it involves no coupling. The resolution of the difficulty introduced by the second term is to take advantage of the symmetry of the fy matrix. Note that each f is an element of a symmetric matrix and the second derivatives fy are independent of the order of differentiation. There is a well known mathematical theorem on the diagonalization of symmetric matrices which states (as applied to Equation 3.16) that when we introduce a new coordinate Q ... 

The procedure has been tested primarily on realistic distillation column models. This choice was deliberate because most industrial processes have similar gain, deadtime, and lag transfer functions. Undoubtedly some pathological transfer functions can be found that the procedure cannot handle. But we are interested in a practical engineering tool, not elegant, rigorous all-inclusive mathematical theorems. 

A mathematical theorem defining the form of the polynomial corresponding to (a + b)", where n is a positive integer. The coefficients for each term of such a polynomial are given by Pascal s triangle. See Pascal s Triangle... 

The entire field of density functional theory rests on two fundamental mathematical theorems proved by Kohn and Hohenberg and the derivation of a... 

The method discussed so far originates from an analysis of n-electrons quantum states based on general mathematical theorems such as Kato s [18]. The particle model is not an element of the theory. In this sense, no direct relation to orbital methods including diabatic ones can be found at the present level of development. The diabaticity is related to a global conservation of the nodal planes. And the calculations reported here have been monitored to keep up to this level. 

Physically this may be visualized as a set of discrete states x with probability pn embedded in a continuous range. If P(x) consists of delta functions alone, i.e., if P(x) = 0, it can also be considered as a probability distribution pn on the discrete set of states x . A mathematical theorem asserts that any distribution on — oo < x < oo can be written in the form (1.3), apart from a third term, which, however, is of rather pathological form and does not appear to occur in physical problems. ... 

Cj is constant), is both the upper bound [32, 33, 39] and the lower [37] estimate of the exact solution. Equation (5.2.1) results from a strict mathematical theorem [32, 33] and is proved by its coincidence for d = 1 with the exact solution for both continuous [31, 37] and lattice [43] statements of the problem. 

Proof. A term from logic and mathematics describing an argument from premise to conclusion using strictly logical principles. In mathematics, theorems or propositions are established by... 

However, it must be noted explicitly that the converse is not true and that the structure of the J-fractions (3.10) and (3.11) is similar but more general than Eqs. (3.12) and (3.13) thus caution must be used in applying mathematical theorems valid for S-fractions or their even contractions to the memory J-fractions. Note, for instance, that (3.11) allows the possibility a = 0, fej 0, while (3.13) does not allow 03 = 0. 02 3 0-However, the close formal relationship between S-fractions and J-fractions allows one in a number of situations to consider properties and theorems for some particular form and to transfer appropriately the results to other situations. 

It is a mathematical theorem that the eigenvalue of a matrix (Equation 76), with the largest absolute value, is smaller than the largest of 2 hij and also smaller than the largest of 2 hij. We will call such a sum... 

The general mathematical theorem states that any density distribution p(r)... 

In addition, we should remark that invariant tori we often found in numerical simulations are not truly KAM tori guaranteed rigorously in the mathematical theorem. The perturbation strength s is so small and any chaotic orbits cannot be detected in phase space if we perform numerical simulations under the original condition of the KAM theorem as for the smallness of . 

For instance, as for the recurrence phenomenon discovered in the FPU nonlinear lattices [25], whose explicit Hamiltonian will be presented in Section IV, there may exist a gap between the results of numerical simulations and the mathematical theorem, but a rigorous result certainly plays a significant role and theoretical arguments based on it can be deduced [26],... 

The mathematical theorems needed in order to derive the governing model equations are defined in this appendix. 

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