Extremal probabilities for Gaussian quadratic forms

 Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q)=1. We prove that for an arbitrary x>0, inf _Q P(Q≤x)=P(χ² _n /n≤x), where χ _n ² is a chi-square distributed rv with n=n(x) degrees of freedom, n(x) is a non-increasing function of x, n=1 iff x>x(1)=1.5364…, n=2 iff x[x(2),x(1)], where x(2)=1.2989…, etc., n(x)≤rank(Q). A similar statement is not true for the supremum: if 1<x<2 and Z ₁ ,Z ₂ are independent standard Gaussian rv's, then sup_0≤λ≤1/2 P{λZ ₁ ² +(1−λ)Z ₂ ² ≤x} is taken not at λ=0 or at λ=1/2 but at 0<λ=λ(x)<1/2, where λ(x) is a continuous, increasing function from λ(1)=0 to λ(2)=1/2, e.g. λ(1.5)=.15…. Applications of our theorems include asymptotic quantiles of U and V-statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes. Received: 24 June 2002 / Revised version: 26 January 2003 Published online: 15 April 2003 Research supported by NSA Grant MDA904-02-1-0091 Mathematics Subject Classification (2000): Primary 60E15, 60G15; Secondary 62G10     

Linearization of third-order ordinary differential equations by point and contact transformations

We present here the solution of the problem on linearization of third-order ordinary differential equations by means of point and contact transformations. We provide, in explicit forms, the necessary and sufficient conditions for linearization, the equations for determining the linearizing point and contact transformations as well as the coefficients of the resulting linear equations.     

Invariant Lagrangians and a new method of integration of nonlinear equations

A method for solving the inverse variational problem for differential equations admitting a Lie group is presented. The method is used for determining invariant Lagrangians and integration of second-order nonlinear differential equations admitting two-dimensional noncommutative Lie algebras. The method of integration suggested here is quite different from Lie's classical method of integration of second-order ordinary differential equations based on canonical forms of two-dimensional Lie algebras. The new method reveals existence and significance of one-parameter families of singular solutions to nonlinear equations of second order.     

Platinum (II) phosphine complexes with acetylene ligands containing 1,4,5,8-naphthalenediimide: Synthesis, crystal structure and electrochemistry

A new acetylene ligand modified with the electron-accepting 1,4,5,8-naphthalenediimide group and its Pt(II) phosphine complexes are reported. The Pt(II) complexes incorporate optical properties of the imide group, and display two reversible imide-centred reduction processes.     

In situ muSR and NMR investigation of methanol dissociation on carbon-supported nanoscaled Pt-Ru catalyst

In situ muSR and nuclear magnetic resonance (NMR) experiments on the investigation of methanol decomposition on Pt-Ru nanoscaled catalyst supported on Vulcan XC-72 carbon black (Pt-Ru/Vulcan XC-72) are presented. Avoided-level-crossing muon spin resonance measurements reveal the resonance line at magnetic field 2.018 T, we assign to transient CH₂?OMu? formaldehyde muonated radical. Results obtained allow us to estimate the contribution of chemical decomposition reaction process for the formaldehyde—carbon oxide—hydrogen conversion on Pt-Ru/Vulcan XC-72 catalytic material. NMR spectrum of the methanol covered Pt-Ru/Vulcan XC-72 consists of three broad resonance lines. Two of them are assigned to OH and CH₃ groups of methanol bound to the carbon black surface. Following the existing literature, which indicates that polymerization of formaldehyde readily takes place when the monomer is adsorbed on some active surface, the third one is assigned to the resonance signal of oligomeric species of formaldehyde. Taking into account the results of our experiments on electrochemical hydrogenation of carbon nanotubes, we do not exclude that the hydrogen arising as a result of methanol decomposition would also contribute to this signal. The high surface area of amorphous carbon nanoparticles of Vulcan XC-72 is favourable to adsorb the hydrogen on their surface.     

Crown Ethers with Exocyclic Organophosphorus Groups

Abstract

In this report is adduced all data on synthesis, structure, chemical and complexingproperties, practical aspects of using the crown ethers with exocyclic organophosphorusgroups. The first articles on this class compounds was published in 1974. After 1988 [1] was published more than 100 papers which are discussed in this report. Also in this paper is included the author's published and non published data on syntheses, structure, chemical properties, toxic and biological activity the exocyclic organophosphorus derivatives of various aza-crown ethers (1–3) [2–4].     

Group theoretical nature of conservation theorems

The purpose of this paper is to establish a group theoretical foundation of conservation theorems for arbitrary systems of partial differential equations. It is shown that any conservation law for differential equations is derivable from the invariance properties of the equations with respect to a group of Lei-Bäcklund tangent transformations.