Conditional law of large numbers and entropy of Markov processes

The importance of a law of large numbers is well known. In this paper, conditional law of large numbers for Markov processes is proved, which can be used in computing quantities related to sub-Markov sequences. A variational interpretation for this limit is given, which shows that typically the quantities of interests tend to minimize a certain entropy. A quasi-compact operator argument is involved.

About a complex operator exponential function of a complex operator argument main property

Operator functions e^A, sin B, cos B of the operator argument from the Banach algebra of bounded linear operators acting from E to E are considered in the Banach space E . For trigonometric operator functions sin B, cos B, formulas for the sine and cosine of the sum of the arguments are derived that are similar to the scalar case. In the proof of these formulas, the composition of ranges with operator terms in the form of Cauchy is used. The basic operator trigonometric identity is given. For a complex operator exponential function e^Z of an operator argument Z from the Banach algebra of complex operators, using the formulas for the cosine and sine of the sum, the main property of the exponential function is proved. Operator functions e^At , sin Bt, cos Bt, e^Zt of a real argument t∈(-∞;∞) are considered. The facts stated for the operator functions of the operator argument are transferred to these functions. In particular, the group property of the operator exponent e^Zt is given. The rule of differentiation of the function e^Zt is indicated. It is noted that the operator functions of the real argument t listed above are used in constructing a general solution of a linear n th order differential equation with constant bounded operator coefficients in a Banach space.

La combinatoire des termes. Exemple : nectar de fruits

Terminology is henceforth based on textual analysis, although its methodology remains paradigmatic. Concordances need to be interpreted, and a theory of combinations is needed. In this article, Harris’ (1968, 1976, 1988) « operator-argument » theory is used, though words, concepts and terms are strictly distinguished. Specialised texts are made up of words as are every text, but some of them denote special concepts, and these are terms. The case of nectar de fruits, as used in EU regulations and FAO Codex alimentarius and on corporate websites, is used to show the relevance of combinatory analysis of concepts and of terms in establishing the ontology of the fruit industry.

HYPER-OPERATORS FOR NONCOMMUTATIVE CALCULUS IN QUANTUM PHYSICS

In quantum physics, one has to deal with linear operator valued functions of a linear operator argument. The derivatives and higher derivatives of such functions are no longer linear operators on state vector spaces. They are hyper-operators of various kinds. Two kinds of notations for hyper-operators are presented. We show that by using them, one can easily write down these hyper-operators, and the calculation on hyper-operators can be reduced to the calculation on ordinary operators. The hyper-operators are shown to be a powerful tool for noncommutative calculus in quantum physics.