Progress in spin dynamics solid-state nuclear magnetic resonance with the application of Floquet–Magnus expansion to chemical shift anisotropy

The purpose of this article is to present an historical overview of theoretical approaches used for describing spin dynamics under static or rotating experiments in solid state nuclear magnetic resonance. The article gives a brief historical overview for major theories in nuclear magnetic resonance and the promising theories. We present the first application of Floquet–Magnus expansion to chemical shift anisotropy when irradiated by BABA pulse sequence.     

The large time behavior of solutions of a diffusion equation involving a nonlocal convection term

We study the existence, uniqueness, and asymtotic behaviour of non-negative solutions to a parabolic diffusion equation involving a nonlocal con-vective term on a bounded domain ? contained in Rⁿ. We apply the In variance Principle of LaSalle and Hale to prove that the solution tends to zero as t → ∞     

Generalization of the Bardeen-Cooper-Schrieffer method for pair interactions

We consider the transition to temperature-dependent equations for ultrasecond-quantized problems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 584–592, March, 2008.     

On the isolated singularities of the solutions of the Gaussian curvature equation for nonnegative curvature

The precise asymptotic behaviour of the solutions to the two-dimensional curvature equation Δu=k(z)e2u with e2u∈L¹ for bounded nonnegative curvature functions −k(z) near isolated singularities is obtained.     

On the index of the boundary value problem for a homogeneous Hamiltonian system of differential equations

The index of the homogeneous self-adjoint boundary value problem for the Hamiltonian systems of ordinary differential equations is introduced. It is assumed that the system has a nontrivial solution. The relationship between the index of an eigenvalue of the nonlinear eigenvalue problem and the index of the corresponding homogeneous problem is established. Properties of the index of the problem and those of the eigenvalue are examined.     

Critical exponents of Manhattan Hamiltonian walks in two dimensions,from Potts andO(n) models

We consider a set of Hamiltonian circuits filling a Manhattan lattice, i.e., a square lattice with alternating traffic regulation. We show that the generating function (with fugacityz) of this set is identical to the critical partition function of aq-state Potts model on an unoriented square lattice withq ^1/2 =z. The set of critical exponents governing correlations of Hamiltonian circuits is derived using a Coulomb gas technique. These exponents are also found to be those of an O(n) vector model in the low-temperature phase withn =q ^1/2 =z. The critical exponents in the limitz = 0 are then those of spanning trees (q= 0) and of dense polymers (n=0,T < T_c), corresponding to a conformal theory with central chargeC = –2. This shows that the Manhattan orientation and the Hamiltonian constraint of filling all the lattice are irrelevant for the infrared critical properties of Hamiltonian walks.     

Can the electron magnetic resonance (EMR) techniques measure the crystal (ligand) field parameters?

In this paper, the question posed in the title is critically examined on the basis of the available literature evidence implying the positive answer. The distinction between, on the one hand, the actual crystal field (CF) or equivalently ligand field (LF) related quantities and, on the other hand, the actual zero-field splitting (ZFS) or equivalently fine structure (FS) quantities, is elucidated. The origin and possible roots of the incorrect terminology consisting in mixing up the two physically distinct quantities at different levels are examined. Aspects concerning Hamiltonians, parameters, energy level splitting, and nature of the operators involved are taken into account. Problems with the various notations for the operators and parameters used in the electron magnetic resonance (EMR) area are also identified and reviewed. A large number of cases of incorrect terminology and other inconsistencies identified in the course of a comprehensive literature survey are analyzed and systematically classified. Implications of the confusion in question, which go beyond the simple semantic issues, are discussed. The results of the survey reveal that the two most serious categories of this confusion lead to misinterpretation of the experimental EMR data. Several examples of serious misinterpretations found in the books, reviews, and original papers are discussed. The incorrect terminology contributes also to misleading keyword classifications of papers in journals as well as references in scientific literature databases. Thus, the database searches may produce unreliable outcomes. Examples of such outcomes are also shown. It is concluded that, in order to prevent further proliferation of the incorrect terminology and thus to increase reliability of the published EMR data, a concerted effort within the EMR community is indispensable. Various ways in this regard at the international level are suggested.     

Extremal k-edge-hamiltonian hypergraphs

An r-uniform hypergraph is k-edge-hamiltonian iff it still contains a hamiltonian-chain after deleting any k edges of the hypergraph. What is the minimum number of edges in such a hypergraph? We give lower and upper bounds for this question for several values of r and k.     

Numerical Energy Conservation for Multi-Frequency Oscillatory Differential Equations

The long-time near-conservation of the total and oscillatory energies of numerical integrators for Hamiltonian systems with highly oscillatory solutions is studied in this paper. The numerical methods considered are symmetric trigonometric integrators and the St?rmer–Verlet method. Previously obtained results for systems with a single high frequency are extended to the multi-frequency case, and new insight into the long-time behaviour of numerical solutions is gained for resonant frequencies. The results are obtained using modulated multi-frequency Fourier expansions and the Hamiltonian-like structure of the modulation system. A brief discussion of conservation properties in the continuous problem is also included. AMS subject classification (2000) 65L05, 65P10