From moments of sum to moments of product

We provide an identity that relates the moment of a product of random variables to the moments of different linear combinations of the random variables. Applying this identity, we obtain new formulae for the expectation of the product of normally distributed random variables and the product of quadratic forms in normally distributed random variables. In addition, we generalize the formulae to the case of multivariate elliptically distributed random variables. Unlike existing formulae in the literature, our new formulae are extremely efficient for computational purposes.     

Two-photon spectroscopy of SnO2 under hydrostatic pressure

The pressure shift of S excitons in the rutile-type semiconductor tin oxide (SnO₂) is measured by two-photon absorption. From these data the pressure coefficients of the band gap (62.0 meV/GPa) and of the exciton binding energy (0.87 meV/GPa) are determined.     

NMR investigation of chlorophosphorylation products of N‐vinylazoles

The structure of novel phosphorus‐containing N‐vinylazoles prepared by action of phosphorus pentachloride has been studied by multinuclear ¹H, ¹³C, ¹⁵N, ³¹P and two‐dimensional (2D) NMR spectroscopy. N‐vinyl‐substituted 1,2‐diazoles and 1,2,3‐triazoles have undergone phosphorylation, exclusively, on double bond. N‐vinylazoles‐based hexa‐coordinated phosphorus compounds have been synthesized for the first time. ³¹P NMR spectroscopy provides the most convenient and unambiguous method for the investigation of E—Z‐isomeric structures of phosphorylated enamines. Copyright © 2008 John Wiley & Sons, Ltd.     

Identifications in modular group algebras

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by D_n(G)=G∩(1+ωⁿ(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the D_n(G). We then extend a theorem of Jennings that provides a basis for the quotient ωⁿ(G)/ωⁿ⁺¹(G) in terms of a basis of the restricted Lie algebra associated with the D_n(G). We shall use these theorems to prove the main results of this paper. For G a finite p-group and n a positive integer, we prove that G∩(1+ω(G)ωⁿ(S))=D_n+1(S) and G∩(1+ω²(G)ωⁿ(S))=D_n+2(S)D_n+1(S∩D₂(G)). The analogous results for integral group rings of free groups have been previously obtained by Gruenberg, Hurley, and Sehgal.     

On classical electrodynamics of point particles and mass renormalization: Some preliminary results

Apparently, no rigorous results exist for the dynamics of a classical point particle interacting with the electromagnetic field, as described by the standard Maxwell-Lorentz equations. Some results are given here for the corresponding linearized system (dipole approximation) in the presence of a mechanical linear restoring force. We consider a regularization of the system (Pauli-Fierz model), and explicitly solve the Cauchy problem in terms of normal modes. Then we study the limit of the particle's motion as the regularization is removed. We prove that the particle's motion corresponding to smooth initial data for the field has a well-defined limit if mass is renormalized, while the motion is trivial (i.e. the particle does not move at all) if mass is not renormalized. Moreover, the limit particle's motion corresponding to an interesting class of initial data satifies exactly the Abraham-Lorentz-Dirac equation. Finally, for generic initial data the limit motion is runaway.     

On Weakly Periodic-Like Rings and Commutativity

A well-known theorm of Jacobson asserts that a ring R with the property that for every x in R there exists an integer n(x) > 1 such that x^n(x) = x is necessarily commutative. With this as motivation, we define an N₀-ring to be a ring which satisfies a weaker hypothesis than the “x^n(x) = x” condition in Jacobson’s Theorem. We consider commutativity of N₀-rings, usually with the additional hypothesis that the ground ring is also weakly periodic-like.     

A posteriori error estimators for obstacle problems – another look

We show that a posteriori estimators for the obstacle problem are easily obtained from the theory for linear equations. The theory would be even simpler if the Lagrange multiplier does not have a nonconforming contribution as it has in actual finite element computations.     

Computing the spectrum of non-self-adjoint Sturm–Liouville problems with parameter-dependent boundary conditions

This paper deals with the computation of the eigenvalues of non-self-adjoint Sturm–Liouville problems with parameter-dependent boundary conditions using the regularized sampling method.     

Connection factors in the Schrödinger equation with a polynomial potential

Given a second order differential equation with two singular points, namely the origin and infinity, the connection factors allow to split a power series solution into formal solutions with known asymptotic behavior. A procedure is suggested to obtain those factors, as quotients of Wronskians of the mentioned solutions, in the case of a Schrödinger equation with a polynomial potential. Application of the procedure to particular cases, whose connection factors are already known, allows us to obtain new relations for quotients and products of gamma functions.