Singular expansions

The Foldy-Wouthuysen (FW) transformation [67] offers a decoupling, which in principle is exact, but it is impractical and leads to a singular expansion in 1/c in the important case of a Coulomb potential [68]. Douglas and Kroll (DK) suggested an alternative decoupling procedure based on a series of appro-... 

Iiifomiation about the behaviour of the 3D Ising ferromagnet near the critical point was first obtained from high- and low-temperatnre expansions. The expansion parameter in the high-temperatnre series is tanli K, and the corresponding parameter in the low-temperatnre expansion is exp(-2A ). A 2D square lattice is self-dual in the sense that the bisectors of the line joining the lattice points also fomi a square lattice and the coefficients of the two expansions, for the 2D square lattice system, are identical to within a factor of two. The singularity occurs when... 

Both expansions are exaet and assuming there is only one singularity, identified with the eritieal point, this must oeeiir when... 

Here the coefficients G2, G, and so on, are frinctions ofp and T, presumably expandable in Taylor series around p p and T- T. However, it is frequently overlooked that the derivation is accompanied by the connnent that since. . . the second-order transition point must be some singular point of tlie themiodynamic potential, there is every reason to suppose that such an expansion camiot be carried out up to temis of arbitrary order , but that tliere are grounds to suppose that its singularity is of higher order than that of the temis of the expansion used . The theory developed below was based on this assumption. 

It is a consequence of the assumptions of the lemma that the second term of the right-hand side of (4.36) is either regular or at least "less seriously" singular than the first term on the right the second term is, in particular, regular if / 0. The proposition states that the coefficient behaves asymptotically like the coefficient of x" in the power expansion of... 

Wang, F. and Li, L. (2002) A singularity excluded approximate expansion scheme in relativistic density functional theory. Theoretical Chemistry Accounts, 108, 53-60. 

CFA can also be defined as an expansion of a contingency table X using the generalized latent vectors in A, B and the singular values in A ... 

This equation cannot be solved by expansion in series, as the coefficients of S(p) and its first derivative result in a singularity at p = 0. Because this point is regular, the substitution Sip) = ps (p) is suggested. If the coefficient of p 2 is set equal to zero, the resulting indicial equation is... 

Flow of trains of surfactant-laden gas bubbles through capillaries is an important ingredient of foam transport in porous media. To understand the role of surfactants in bubble flow, we present a regular perturbation expansion in large adsorption rates within the low capillary-number, singular perturbation hydrodynamic theory of Bretherton. Upon addition of soluble surfactant to the continuous liquid phase, the pressure drop across the bubble increases with the elasticity number while the deposited thin film thickness decreases slightly with the elasticity number. Both pressure drop and thin film thickness retain their 2/3 power dependence on the capillary number found by Bretherton for surfactant-free bubbles. Comparison of the proposed theory to available and new experimental... 

Added in Proof.] We do not here discuss the logarithmic singularities which occur in the virial expansion and have recently been reported by I. Oppenheim and K. Kawasaki [Phys. Rev. 139A, 1763 (1965)]. 

A symmetric matrix A, can usually be factored using the common-dimension expansion of the matrix product (Section 2.1.3). This is known as the singular value decomposition (SVD) of the matrix A. Let A, and u, be a pair of associated eigenvalues and eigenvectors. Then equation (2.3.9) can be rewritten, using equation (2.1.21)... 

In reviewing, the development of a useful polymer theory was a prerequisite for the expansion in research, and explosion in use, of these unique materials. The development occured in several distinct phases. It was marked by several landmark events and papers, and it can claim its conception as the product of remarkably few workers. It is doubtful that the army of scientists, technicians, and engineers involved in polymer research, much less the lay recipients of the wealth and benefits of the resultant technology, realize the impact that so few have had on their lives. We are reminded in a time of diminishing influence of the individual, that singular contributions have been, and should continue to be, an important part of science. 

To improve on the wave function one has to accept that the standard multideterminantal expansion [Eq. (13.3)] is unsuitable for near-exact but practical approximations to the electronic wavefunction. The problem is dear from a simple analysis of the electronic Hamiltonian in Eq. (13.2) singularities in the Coulomb potential at the electron coalescence points necessarily lead to irregularities in first and higher derivatives of the exact wave function with respect to the interpartide coordinate, rj 2. The mathematical consequences of Coulomb singularities are known as electron-electron (correlation) and electron-nuclear cusp conditions and were derived by... 

As shown by Chang, Pelissier and Durand (CPD) [41] a regular expansion, however, can be deduced by isolating the Coulomb singularity by infinite summations. Let us rewrite the equation (38), when z — 0... 

As a result, we have Poincare s resonance singularity at = i for i > 0 in the series expansion of in X. The Friedrichs model discussed above may become nonintegrable. 

This has a unique solution because there are two conditions at r = 0. For the function e(r), the conditions are e(r) = d f)ldf = 0 for r = 0. This yields that near r = 0, the beginning of the Taylor expansion of e is e (r) +. The solution for e becomes singular for r = o like 21n( r — ro ), and cannot be extended beyond this value. This is where the transition occurs between values of T close to (—1) and values close to zero. This is expected to be a very narrow range of values of r where one can neglect the variation of r in Eq. (27) and take it as constant. In this range the equation becomes... 

Equation (1) is obtained by using an expansion in E/ 2c - Vc) on the Dirac Fock equation. This expansion is valid even for a singular Coulombic potential near the nucleus, hence the name regular approximation. This is in contrast with the Pauli method, which uses an expansion in (E — V)I2(. Everything is written in terms of the two component ZORA orbitals, instead of using the large and small component Dirac spinors. This is an extra approximation with respect to the original formalism. 

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