Contraction theorem

This approximation can be used only if An is positive, which will be true in most cases. Equation 6.59 coincides with 6.57 for 17 close to one, and with 6.58 for ij close to zero. Equation 6.59 can be solved as an implicit relationship. The value of 17 can be easily obtained by using the contraction theorem. This is illustrated in Table 6.7 for>4nj =An = 1. 

Table 6.7 Illustration of the contraction theorem for the calculation of an approximate value of the effectiveness factor rj defined by Equation 6.59...

We will also use the theorem on contraction mappings. A mapping S y —> y is called a contraction mapping if it is Lipschitz continuous,... 

Theorem 1.21. If S is a contraction mapping in a Hilbert space V then there exists a fixed point u such that Su = u and solutions u of the equation... 

The proof is by induction. It is clearly true for two factors since then it reduces to the definition of the contraction symbol. Furthermore, it is sufficient to prove the theorem under the assumption that Z is a creation operator and that all the operators UV XY are destruction operators. If UV- - -XY are all destruction operators and Z is a creation operator, we may then add any number of creation operators to the left of all factors on both sides of Eq. (10-196) within the N product, without impairing the validity of our theorem, since the contraction between two creation operators gives zero. If on the other hand Z is a destruction operator and UV - - - are creation operators, then Eq. (10-196) reduces to a trivial identity... 

Since DN(UV-- -XY)Z = N(DUV- - -XY)Z, the theorem is proved for n + 1 factors. This lemma can be generalized by multiplying both sides of Eq. (10-196) by an arbitrary number of contracted factors, and using Eq. (10-195) to bring these factors within the N products. Wick s theorem now states that a T product can be decomposed into a unique sum of normal products as follows ... 

As a specific application of Wick s theorem and the above contraction... 

The contractions that are encountered in applying Wick s theorem in the computation of the -matrix are given by... 

Conditional probability, 267 density function, 152 Condon, E. U., 404 Configuration space amplitude, 501 Heisenberg operator, 507 operators, 507, 514, 543 Conservation laws for light particles (leptons), 539 for heavy particles (baryons), 539 Continuous memoryless channels, 239 Contraction symbol for two time-labelled operators, 608 Control of flow, 265 Converse to coding theorem, 215 Convex downward function, 210 Convex upward function, 209 Cook, L. F 724... 

The theorem also applies to a heterogeneous system, such as a liquid in presence of its saturated vapour, or in presence of the solid. In the former case, vapour is liquefied by compression and gives out its latent heat. Under isothermal conditions this would escape as fast as produced, but if the heat is compelled to remain in the system, it raises the temperature and thereby increases the pressure. If, on the other hand, a mixture of ice and water is compressed, ice melts and the mass is cooled by abstraction of heat. If heat is allowed to enter from outside, so as to restore the original temperature, more ice melts, and the pressure falls by reason of the contraction. 

In the absence of nuclear energy sources, a star contracts on a thermal timescale and radiates energy at the expense of gravitational potential energy. Since, by the Virial Theorem, the total energy... 

D. R. Alcoba, Equivalence theorems between the solutions of the fourth-order modified contracted Schrodinger equation and those of the Schrodinger equation. Phys. Rev. A 65, 032519 (2002). 

For products of a operators one finds again a generalized Wick theorem like Eqs. (10) and (11), but in addition to particle contractions we also have hole contractions and combined contractions, even full contractions, that is, simple numbers ... 

Note that these expressions as such are never needed explicitly. What one does need are the contraction rules (i.e., the generalized Wick theorem) derived from these expressions. These will be given in the next section. 

Generalized normal ordering is intimately linked to the cumulants Xk of the k-particle density matrices (for short, density cumulants). The contractions in the sense of the generalized Wick theorem involve the... 

Recall that in his Theorems 3 and 4 Hans Kummer [3] defined a contraction operator, L, which maps a linear operator on A-space onto an operator on p-space and an expansion operator, E, which maps an operator on p-space onto an operator on A-space. Note that the contraction and expansion operators are super operators in the sense that they act not on spaces of wavefunctions but on linear spaces consisting of linear operators on wavefunction spaces. If the two-particle reduced Hamiltonian is defined as... 

This is sometimes called the similarity theorem. The horizontal stretching of a function in one domain results in horizontal contraction and amplitude growth in the other. In fact, these changes occur in such a way that the area under the curve in the other domain remains constant. 

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