Functional calculus for non-commuting operators with real spectra via an iterated Cauchy formula

We define a smooth functional calculus for a non-commuting tuple of (unbounded) operators A_j on a Banach space with real spectra and resolvents with temperate growth, by means of an iterated Cauchy formula. The construction is also extended to tuples of more general operators allowing smooth functional calculii. We also discuss the relation to the case with commuting operators.     

Bianchi type-V model with a perfect fluid and A-term

A self-consistent system of gravitational field with a binary mixture of perfect fluid and dark energy given by a cosmological constant has been considered in Bianchi Type-V universe. The perfect fluid is chosen to be obeying either the equation of state p=γρ with γ ε |0,1| or a van der Waals equation of state. The role of A-term in the evolution of the Bianchi Type-V universe has been studied.     

New aspects of thermal analysis,Part I. Resolution of DSC and means for its optimization

The definitions of the temperature resolution, the so-called resolution of DSC instruments given in literature are discussed. A new definition of the resolution for DSC instruments is presented and outlined. The main characteristic introduced in this new definition is a minimum between two caloric events as a prerequisite of an existing resolution. Possible candidates of test substances have been evaluated. The oligomer n-hexatriacontane is revealing an interesting phenomenon, namely a lambda transition which is in the peak temperature only 2 K lower than the melting temperature of 76°C. The substance was selected as an ideal test substance for the quantification of the resolution of DSC instruments. The lambda transition is a second order process which may reach under certain conditions the saturation of the occurring molecular dislocation within 0.2 K, and after saturation the heat flow rate drops sharply down. Investigations concerning the main characteristics of n-hexatriacontane in respect to the temperatures of transition (lambda transition and melting), to the involved enthalpies, and to the resolution factors were performed as functions of the sample mass and the heating rate. The importance of relevant evaluation procedures increasing the resolution factors of DSC curves are discussed and these procedures are integrated into the testing of the resolution. The necessity for widening the experimental scope from instruments to evaluation procedures is forced by the existence of instruments with built-in signal treatments based on electronic devices and software procedures. A comparison with literature data is outlined for all of the mentioned characteristic values of n-hexatriacontane. This revised version was published online in August 2006 with corrections to the Cover Date.     

A law of the iterated logarithm for processes with independent increments

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Generalized solutions of a stochastic partial differential equation

We discuss the Cauchy problem of a certain stochastic parabolic partial differential equation arising in the nonlinear filtering theory, where the initial data and the nonhomogeneous noise term of the equation are given by Schwartz distributions. The generalized (distributional) solution is represented by a partial (conditional) generalized expectation ofT(t)° _0,t ^–1 , whereT(t) is a stochastic process with values in distributions and _s,t is a stochastic flow generated by a certain stochastic differential equation. The representation is used for getting estimates of the solution with respect to Sobolev norms.Further, by applying the partial Malliavin calculus of Kusuoka-Stroock, we show that any generalized solution is aC -function under a condition similar to Hörmander's hypoellipticity condition.     

Rate conservation laws: A survey

We survey the rate conservation law, RCL for short, arising in queues and related stochastic models. RCL was recognized as one of the fundamental principles to get relationships between time and embedded averages such as the extended Little's formulaH=G, but we show that it has other applications. For example, RCL is one of the important techniques for deriving equilibrium equations for stochastic processes. It is shown that the various techniques, including Mecke's formula for a stationary random measure, can be formulated as RCL. For this purpose, we start with a new definition of the rate with respect to a random measure, and generalize RCL by using it. We further introduce the notion of quasi-expectation, which is a certain extension of the ordinary expectation, and derive RCL applicable to the sample average results. It means that the sample average formulas such asH=G can be obtained as the stationary RCL in the quasi-expectation framework. We also survey several extensions of RCL and discuss examples. Throughout the paper, we would like to emphasize how results can be easily obtained by using a simple principle, RCL.     

The law of the solution to a nonlinear hyperbolicSPDE

Let { _s,t ,(s,t ₊ ² } be a white noise on ₊ ² . We consider the hyperbolic stochastic partial differential equation {ie863-3} The purpose of this paper is to study the law of the solution to this equation. We analyze the existence and smoothness of the density using the tools of Malliavin Calculus. Finally we prove a large deviation principle on the space of continuous functions, for the family of probabilities obtained by perturbation of the noise in the equation.This work has been partially supported by the grant of the DGICYT No. PB 930052 and the EU Science project CT 910459.     

Derivative of the exponential mapping for infinite dimensional lie groups

It is proved that for infinite dimensional Lie groups in the sense of the differential calculus of Frölicher and Kriegl the derivative of the exponential mappings is given by the formula d(exp)(X)Y=d_exp(X)(e) ₀ ¹ Ad_exp(–tX) Y dt, where stands for the left translation ande is the neutral element.This work was supported by the Alexander von Humboldt-Stiftung.     

Reaction of 4,4′-dimethyldiphenyl ether with phosphorus trichloride in the presence of anhydrous aluminum chloride

The reaction of 4,4-dimethyldiphenyl ether with phosphorus trichloride in the presence of anhydrous aluminum chloride was studied. This reaction affords 2,8-dimethyl-10H-10⁵-phenoxaphosphine 10-oxide as virtually the only product. In air, the latter in an alkaline solution is quantitatively transformed into 10-hydroxy-2,8-dimethyl-10H-10⁵-phenoxaphosphine 10-oxide.__________Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 12, pp. 2762–2765, December, 2004.