Theorem on Reducibility

The main result, obtained when solving the problem formulated above, is given by the following theorem. 

Theorem 5.1. (Martinyuk and Perestyuk, 1974). Suppose that the system (1.1) satisfies the following conditions  

Then one cm always find a sufficiently small positive constmt Mq, such that for 

This theorem implies that the fundamental matrix of (1.1) has the form 

Since A is the diagonal matrix with real, diffoent, and nonzero elements Xj,the equation (2.9) is always solvable moreover, the function t/j( p) is analytic in the region 

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We have a A/J 0 and I A I 83 —> 0 as s — °o, and thus, in order to complete the proof of the theorem on reducibility, it remains to show that the number sequence A )(0) converges and that the sequence of functions d> 0, 0) (and of their derivatives up to the -th order) converges uniformly (see, Bogolyubov, Mitropolsky, and Samoilenko (1969)). This, in fact, completes the proof of the theorem, because by means of the change of variables... 

Then, in the Old Ages (1940 or 1951-1967) some ingenious people became aware that, in the case of two-body interactions, it is the two-particle reduced density matrix (2-RDM) that carries in a compact way all the relevant information about the system (energy, correlations, etc.). Early insight by Husimi (1940) and challenges by Charles Coulson were completed by a clear realization and formulation of the A-representability problem by John Coleman in 1951 (for the history, see his book [1] and Chapters 1 and 17 of the present book). Then a series of theorems on A-representability followed, by John Coleman and many... 

M. Rosina, Some theorems on uniqueness and reconstruction of higher-order density matrices, in Many-Electron Densities and Reduced Density Matrices (J. Cioslowski, ed.), Kluwer Academic/ Plenum Pubhshers, New York, 2000, pp. 19—32. 

Here a and (3 are some constants depending on the system size N, as discussed in the following section. The proof of the theorem is reduced to evaluate the remainder terms after successive canonical perturbations, and it is possible to make the remainder exponentially small. 

Use the GTF product. Use the Gaussian product theorem to reduce the two basis-function products to linear combinations of GTFs centred on just two points... 

The historic development has been accounted for in many reviews (see Refs. [9,13]). Important theorems regarding fermionic behaviour was developed by Yang [14], Coleman [15] and Sasaki [16], for a recent review on reduced density matrices and the famous N-representability problem, see Ref. [17]. 

Before we start proving the theorem on the reducibility of the system (1.1), let us formulate and prove some auxiliary statements which will be necessary in what follows. 

This statement is known as Tihonov s theorem. The equalities show that in the limit (as /a— 0) the solution z(t, /x), y t, /x) of (2.1), (2.2) tends to the solution of the reduced problem (2.3), (2.9). That is why the above theorem is called the theorem on passage to the limit. Notice that the transition to the limit for y takes place for all t in the interval Moreover, this limiting process is uniform. Transition to the limit for z takes place for any t except t = 0. This is clear since z(0, p.) = z 7 z(0). The limiting process for z will be uniform outside a small neighborhood of the initial point. We call this neighborhood the boundary initial) layer. 

One of the main conditions in the theorem on the passage to the limit is the condition for the existence of an isolated root z =

singularly perturbed equations, and particularly in most problems of chemical kinetics, this condition is not satisfied because the reduced equation does not have an isolated root. Instead, it has a family of solutions depending on one or several parameters. This case will be called the critical case. 

For heteronuclear decoupling, the TPPM (Two Pulse Phase Modulation) sequence " is now widely employed in solid-state NMR. It was extended to PMFM and FMPM by Gan and Ernst, who also found that the phase inversion in TPPM brings about a secondary averaging that further reduces the heteronuclear coupling. Eden and Levitt revisited the problem of heteronuclear decoupling based on the symmetry consideration of the Hamiltonian. Based on the property of the lower orders of the interaction Hamiltonians, a general theorem on the pulse sequence was conjectured that allows one to predict which terms are eliminated and which are symmetry-allowed without... 

In Sec. 13.5 we consider the bifurcation of the homoclinic loop of a saddle without any restrictions on the dimensions of its stable and imstable manifolds. We prove a theorem which gives the conditions for the birth of a single periodic orbit from the loop [134], and also formulate (without proof) a theorem on complex dynamics in a neighborhood of a homoclinic loop to a saddle-focus. Here, we show how the non-local center manifold theorem (Chap. 6 of Part I) can be used for simple saddles to reduce our analysis to known results (Theorem 13.6). 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

The volumetric properties of fluids are represented not only by equations of state but also by generalized correlations. The most popular generalized correlations are based on a three-parameter theorem of corresponding states which asserts that the compressibiHty factor is a universal function of reduced temperature, reduced pressure, and a parameter CO, called the acentric factor ... 

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... 

The integral on the right hand side represents the moment of inertia of the pendulum with respect to the axis passing through the center of gravity, and Equation (3.55) describes the well-known theorem of mechanics. Bearing in mind that, we already introduced the reduced length /, (Equation (3.54)), let us assume... 

Rather than giving the general expression for the Hellmann-Feynman theorem, we focus on the equation for a general diatomic molecule, because from it we can leam how p influences the stability of a bond. We take the intemuclear axis as the z axis. By symmetry, the x and y components of the forces on the two nuclei in a diatomic are zero. The force on a nucleus a therefore reduces to the z component only, Fz A, which is given by... 

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