Nonsymmetrical distance between probability distributions,entropy and the theorem of pythagoras

The Kullback-Leibler information I[Q¦p¦ for discrimination in favor of the probability distribution Q against P is considered as a nonsymmetrical analog of one half of the square of the distance between the points Q and P. For the n-dimensional planes we take the exponential families. We shall prove a nonsymmetrical analogue of the theorem of Pythagoras in the formulation: The squared length of an oblique line equals the sum of the squared lengths of the perpendicular and the projection of the oblique line, and also an analog of the cosine theorem and the like.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 323–332, September, 1968.     

Moment information and entropy evaluation for probability densities

How much information does the sequence of integer moments carry about the corresponding unknown absolutely continuous distribution? We prove that a reliable evaluation of the corresponding Shannon entropy can be done by exploiting some known theoretical results on the entropy convergence, uniquely involving exact moments without solving the underlying moment problem. All the procedure essentially rests on the solution of linear systems, with nearly singular matrices, and hence it requires both calculations in high precision and a pre-conditioning technique. Numerical examples are provided to support the theoretical results.     

Characteristics of queueing systems observed at events and the connection between stochastic intensity and palm probability

This article explores the link between the concepts of stochastic intensity and Palm probability and gives a new proof and useful extensions to the so-called PASTA property of queueing theory.     

The connection between normalizable and spectral operators

A necessary and sufficient condition for an operator to be normalizable is given in terms of Dunford’s spectral theory. This paper is a part of the author’s Ph.D. thesis to be submitted to the Hebrew University. The author wishes to express his indebtedness to Professor S. R. Foguel for his guidance and kind encouragement.     

On the relation between topological entropy and entropy dimension

Mathematical Notes - For the Lipschitz mapping of a metric compact set into itself, there is a classical upper bound on topological entropy, namely, the product of the entropy dimension of the...     

Existence and stability of a discrete probability distribution by maximum entropy approach

The discrete maximum entropy (ME) probability distribution which can take on a finite number of values and whose first moments are assigned, is considered. The necessary and sufficient conditions for the existence of a maximum entropy solution are identical to the general ones for the finite moment problem. The entropy decreasing by adding one more moment is studied. Unstability of the distribution recovering is proved when an increasing number of moments is used.     

Relative entropy between quantum ensembles

Relative entropy between two quantum states, which quantifies to what extent the quantum states can be distinguished via whatever methods allowed by quantum mechanics, is a central and fundamental quantity in quantum information theory. However, in both theoretical analysis (such as selective measurements) and practical situations (such as random experiments), one is often encountered with quantum ensembles, which are families of quantum states with certain prior probability distributions. How can we quantify the quantumness and distinguishability of quantum ensembles? In this paper, by use of a probabilistic coupling technique, we propose a notion of relative entropy between quantum ensembles, which is a natural generalization of the relative entropy between quantum states. This generalization enjoys most of the basic and important properties of the original relative entropy. As an application, we use the notion of relative entropy between quantum ensembles to define a measure for quantumness of quantum ensembles. This quantity may be useful in quantum cryptography since in certain circumstances it is desirable to encode messages in quantum ensembles which are the most quantum, thus the most sensitive to eavesdropping. By use of this measure of quantumness, we demonstrate that a set consisting of two pure states is the most quantum when the states are 45° apart.     

Some relations between entropy and approximation numbers

A general result is obtained which relates the entropy numbers of compact maps on Hilbert space to its approximation numbers. Compared with previous works in this area, it is particularly convenient for dealing with the cases where the approximation numbers decay rapidly. A nice estimation between entropy and approximation numbers for noncompact maps is given. Partially supported by the National 863 Project of China and the National Natural Science Foundation of China (Grant No. 19771003).     

The gap between probability and prevalence: Loneliness in vector spaces

The best available definition of a subset of an infinite dimensional, complete, metric vector space being ``small' is Christensen's Haar zero sets, equivalently, Hunt, Sauer, and Yorke's shy sets. The complement of a shy set is a prevalent set. There is a gap between prevalence and likelihood. For any probability on , there is a shy set with . Further, when is locally convex, any i.i.d. sequence with law repeatedly visits neighborhoods of only a shy set of points if the neighborhoods shrink to at any rate.

     

A connection between multi-linear and Volterra systems

The Volterra system is a non-linear system with the structure of the Volterra series. The Volterra series is attractive from the system-theoretic point of view, since it enables to obtain the output of a class of non-linear systems in terms of the input explicitly rather than involving input-output coupling terms and allows substantial simplifications for the numerical simulation. The Volterra system allows to derive the stability condition as well, i.e. obtain a bound on the output for a given bound on the input function, especially for the bilinear system. The bilinear system possesses the Volterra series. This paper derives the Volterra series formalism from the multi-linear system involving the coupling term attributed to (k-1)th order non-linearity and output function, where 1<k. The bilinear system becomes a special case of the non-linear problem of concern here. The Volterra series formalism of this paper is derived using the discrete counterpart of the phase space analysis for the non-linear non-autonomous system. The main result of the paper, i.e the Volterra series formalism of the multi-linear system of concern here, is somewhat more general, since the Volterra series representations for bilinear and tri-linear systems, etc. can be obtained as its special cases.     

A connection between Fourier and Schur multipliers

We construct a continuous functionf on the circle such thatf is a Fourier multiplier inl _p , 1<p<, but (x, y)=f(x–y) is not a Schur multiplier inS _p ,p2.     

The connection between dielectric susceptibility and weight by volume of vitreous plastics

Experimental results of measurements of dielectric susceptibilities of vitreous plastics by use of SHF electromagnetic waves are presented. A connection is shown between the dielectric susceptibility over the specimen as it depends on the technological pressing process.V. I. Ul'yanov (Lenin) Leningrad Electrotechnical Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1133–1135, November–December, 1972.