A subjective Bayesian approach to the theory of queues II — Inference and information in M/M/1 queues

This is a sequel to Part I of A Subjective Bayesian Approach to the Theory of Queues. The focus here is on inference and a use of Shannon's measure of information for assessing the amount of information conveyed by the various types of data from queues. The notation and terminology used here is established in Part I.     

Remarks on commuting exponentials in Banach algebras, II

Suppose that and are elements of a complex unital Banach algebra such that the spectrum of is -congruence-free and . We show that then is the sum of nilpotent elements. If denotes the spectral radius of , then we show that the additional assumption implies that

     

Modelling population growth vialaguerre-type exponentials

We use the Laguerre-type exponentials, i.e., eigenfunctions of the Laguerre-type derivatives, in order to construct new models for population growth. Relevant modifications of the classical exponential, logistic, and Volterra-Lotka models are investigated.     

Iterated exponential inequalities: a classroom-tailored approach

We explore the connection between the notion of critical point for a function of one variable and various inequalities for iterated exponentials defined on the positive semiline of real numbers.     

A remark on commuting operator exponentials

In a previous paper the author proved that for square matrices with algebraic entries expexpexpexp if and only if . This result is extended here to bounded operators on an arbitrary Banach space.

     

Spectrality of certain self‐affine measures on the generalized spatial Sierpinski gasket

The self‐affine measure corresponding to a upper or lower triangle expanding matrix M and the digit set in the space is supported on the generalized spatial Sierpinski gasket, where are the standard basis of unit column vectors in . We consider in this paper the existence of orthogonal exponentials on the Hilbert space , i.e., the spectrality of . Such a property is directly connected with the entries of M and is not completely determined. For this generalized spatial Sierpinski gasket, we present a method to deal with the spectrality or non‐spectrality of . As an application, the spectral property of a class of such self‐affine measures are clarified. The results here generalize the corresponding results in a simple manner.     

Spectral Self-Affine Measures on the Generalized Three Sierpinski Gasket

The self-affine measure μM,Dassociated with an iterated function system{φd(x)=M~(-1)(x + d)}_(d∈D) is uniquely determined. It only depends upon an expanding matrix M and a finite digit set D. In the present paper we give some sufficient conditions for finite and infinite families of orthogonal exponentials. Such research is necessary to further understanding the non-spectral and spectral of μM,D. As an application,we show that the L~2(μM,D) space has infinite families of orthogonal exponentials on the generalized three Sierpinski gasket. We then consider the spectra of a class of self-affine measures which extends several known conclusions in a simple manner.     

Spectrality of a class of self-affine measures with decomposable digit sets

The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight.The spectral and non-spectral problems on the selfaffine measures have some surprising connections with a number of areas in mathematics,and have been received much attention in recent years.In the present paper,we shall determine the spectrality and non-spectrality of a class of self-affine measures with decomposable digit sets.We present a method to deal with such case,and clarify the spectrality and non-spectrality of a class of self-affine measures by applying this method.     

Optimization of Bilinear Systems Using a Higher-Order Variational Method

This paper derives some optimization results for bilinear systems using a higher-order method by characterizing them over matrix Lie groups. In the derivation of the results, first a bilinear system is transformed to a left-invariant system on matrix Lie groups. Then, the product of exponential representation is used to express this system in canonical form. Next, the conditions for optimality are obtained by the principles of variational calculus. It is demonstrated that closed-form analytical solutions exist for classes of bilinear systems whose Lie algebra are nilpotent.     

Estimate of an Entire Function in the Union of Two Angles

We obtain an estimate of an entire function expressed as the limit of a sequence of polynomials in exponentials in the union of two angles via an estimate of the function and its derivatives on a set lying on a horizontal strip.