Contraction mappings

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... 

Harriman (20) has shown that this map is "onto" i.e. any element of comes from at least one element of S. Note that this jH-operty does not rule out the possibility that an element of can also come from operators not in. This "onto" property should be compared to the case that arises in the N-representability problem (28) where not every positive two-particle operator comes from a state in fj so the contraction map in that case does not have the onto property. 

The FORDO is defined by a linear contraction map,, given by the following... 

C N Contraction map from A-particle Trace Class Operators to p-particle Trace Class operators. 

Two different approaehes to this problem will be described in this work. They are based in quite different philosophies, but both are aimed at determining the RDM without a previous knowledge of the WF. Another common feature of these two approaches is that they both employ the discrete Matrix representation of the Contraction Mapping (MCM) [17,18]. Applying this MCM is the alternative, in discrete form, to integrating with respect to a set of electron variables and it is a much simpler tool to use. 

Three distinct sets of linear mappings for the partial 3-positivity matrices in Eqs. (31)-(36) are important (i) the contraction mappings, which relate the lifted metric matrices to the 2-positive matrices in Eqs. (27)-(29) (ii) the linear interconversion mappings from rearranging creation and annihilation operators to interrelate the lifted metric matrices and (iii) antisymmetry (or symmetry) conditions, which enforce the permutation of the creation operators for fermions (or bosons). Note that the correct permutation of the annihilation operators is automatically enforced from the permutation of the creation operators in (iii) by the Hermiticity of the matrices. 

Properties of the 2-RDM and the V-Representability Ih-oblem The Matrix Contracting Mapping The Contracted Schrodinger Equation... 

As mentioned in Section I, Cho [13], Cohen and Frishberg [14, 15], and Nakatsuji [16] integrated the Schrodinger equation and obtained an equation that they called the density equation. This equation was at the time also studied by Schlosser [44] for the 1-TRDM. In 1986 Valdemoro [17] applied a contracting mapping to the matrix representation of the Schrodinger equation and obtained the contracted Schrodinger equation (CSE). In 1986, at the Coleman Symposium where the CSE was first reported, Lowdin asked whether there was a connection between the CSE and the Nakatsuji s density equation. It came out that both... 

This quantity averages the Shannon entropies conditional on the Gamma and lognormal models, with weights given by their posterior probabilities. In Appendix B, we show that the average entropy is a concave function on the space of probability distributions which is monotone under contractive maps (Sebastiani and... 

The following convergence theorem (sometimes called the contraction mapping theorem) will provide this information. 

This can be shown graphically in Fig. A.2 with plots of functions x and f(x). The solution a is simply the intersection of these two functions. It is recognized that if the slope of the function f(x) is less than that of the function x (i.e., df/dx < 1) in the neighborhood of a, we see in Fig. A.2 that any point in that neighborhood will map itself into the same domain (see the direction of arrows in Fig. A.2). This explains why the theory was given the name of contraction mapping. 

Figure A2 Graphical representation of one-dimensional contraction mapping.

There is no guarantee of convergence (the contracting mapping theorem must be applied, but it is conservative). 

By applying the principle of contracting mappings, one can easily prove that the equation (2.15) has a solution continuous in t and x provided that... 

By virtue of the principle of contracting mappings the equation (3.12) has a unique periodic solution z-zjf) satisfying the inequality... 

Since 5>0, d[fip),f(q)]contraction mapping (see Appendix) and there must be a unique stable limit cycle in the four regions. In Fig. 14, we give this construction for the parameters used to compute Fig. 12. Although there is good agreement between the dynamics in the piecewise linear and the continuous equations, no proof of stable limit cycle oscillations has been found for Eq. (48) or (50). However, there has been a recent proof for the existence of nonlocal periodic solutions of Eq. (45) using fixed-point methods. ... 

The iterative scheme in Figure 2.7 left is a contraction mapping it approaches the equilibrium after every iteration. 

In words, the application of a contraction mapping to any two points strictly reduces (i.e., a = 1 does not work) the distance between these points. The norm in the definition can be any norm, i.e., the mapping can be a contraction in one norm and not a contraction in another norm. 

One can think of a contraction mapping in terms of iterative play player 1 selects some strategy, then player 2 selects a strategy based on the decision by player 1, etc. If the best response mapping is a contraction, the NE obtained as a result of such iterative play is stable but the opposite is not necessarily true, i.e., no matter where the game starts, the final outcome is the same. See also Moulin (1986) for an extensive treatment of stable equilibria. 

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