On generalized entropy measures and pathways

Product probability property, known in the literature as statistical independence, is examined first. Then generalized entropies are introduced, all of which give generalizations to Shannon entropy. It is shown that the nature of the recursivity postulate automatically determines the logarithmic functional form for Shannon entropy. Due to the logarithmic nature, Shannon entropy naturally gives rise to additivity, when applied to situations having product probability property. It is argued that the natural process is non-additivity, important, for example, in statistical mechanics [C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys. 52 (1988) 479-487; E.G.D. Cohen, Boltzmann and Einstein: statistics and dynamics—an unsolved problem, Pramana 64 (2005) 635-643.], even in product probability property situations and additivity can hold due to the involvement of a recursivity postulate leading to a logarithmic function. Generalized entropies are introduced and some of their properties are examined. Situations are examined where a generalized entropy of order α leads to pathway models, exponential and power law behavior and related differential equations. Connection of this entropy to Kerridge's measure of “inaccuracy” is also explored.     

Subdifferential representation of homogeneous functions and extension of smoothness in Banach spaces

In this paper, we mainly consider proximal subdifferentials of lower semicontinuous functions defined on real Hilbert space and Clarke＇s subdifferentials of locally Lipschitzian functions defined on Banach space respectively, and obtain the generalized Euler identity of homogenous functions. Then, by introducing a multifunction F, we extend the smoothness of sphere and differentiability of norm function in Banach space.     

Intersective polynomials and the primes

Intersective polynomials are polynomials in Z[x] having roots every modulus. For example, P₁(n)=n² and P₂(n)=n²−1 are intersective polynomials, but P₃(n)=n²+1 is not. The purpose of this note is to deduce, using results of Green and Tao (2006) [8] and Lucier (2006) [16], that for any intersective polynomial h, inside any subset of positive relative density of the primes, we can find distinct primes p₁,p₂ such that p₁−p₂=h(n) for some integer n. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number p such that p+2 is the product of at most 2 primes).     

Convergence of compressible Euler-Poisson system to incompressible Euler equations

In this paper, we study the quasi-neutral limit of compressible Euler-Poisson equations in plasma physics in the torus T^d. For well prepared initial data the convergence of solutions of compressible Euler-Poisson equations to the solutions of incompressible Euler equations is justified rigorously by an elaborate energy methods based on studies on an λ-weighted Lyapunov-type functional. One main ingredient of establishing uniformly a priori estimates with respect to λ is to use the curl-div decomposition of the gradient.     

The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers

In a recent paper Dattoli and Srivastava [3], by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli-Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form.     

On the fourth-order difference equation for the associated Meixner polynomials

Three equivalent forms of the fourth-order difference equation obeyed by the associated Meixner polynomials (with a nonnegative real association parameter) are derived from a refinement of a recent result due to Letessier et al. (1996).     

On the Capability of the Generalized Gamma Function to Represent Spray Drop‐Size Distribution

The three‐parameter, Generalized Gamma function solution of a recent MEF formulation used to derive liquid spray drop‐size distribution, is applied to sprays resulting from three different atomization processes. The objectives of these applications are to determine the sign of the parameters for which this function reports a more reliable fit and to further understand the parameter stability problem reported elsewhere. It is found that the lack of stability of the parameters is related to a characteristic feature of the mathematical function and appears for a series of spray drop‐size distributions with constant shape. For each situation analyzed in the present study, the Generalized Gamma function provides a very good fit with parameters that are either constant or correlated to the working conditions. As far as the sign of the parameters is concerned, the results show that the best formulation is a function of the spray and that it is impossible to know, a priori, which parameter sign will report the best fit. Finally, for one situation, it is found that the Generalized Gamma function allows extrapolation of drop sizes outside the measured values. All of the results converge to conclude that the three‐parameter Generalized Gamma function, which is identical to the well‐known Nukiyama‐Tanasawa distribution, accumulates valuable attributes to represent liquid spray drop‐size distributions.     

On log-normal convolutions: An analytical–numerical method with applications to economic capital determination

We put forward an efficient algorithm for approximating the sums of independent and log-normally distributed random variables. Namely, by combining tools from probability theory and numerical analysis, we are able to compute the cumulative distribution functions of the just-mentioned sums to a high precision and in a relatively short computing time. We illustrate the effectiveness of the new method in the contexts of the individual and collective risk models, aggregate economic capital determination, and economic capital allocation.     

A remark on congruences for coefficients of Faber polynomials

Faber polynomials appear when weight zero Hecke operators act on the modular j-invariant. They are polynomials in j with rational integral coefficents. Using the theory of p-adic modular forms we establish some congruences and divisibilities for these coefficients. 2000 Mathematics Subject Classification Primary—11F33 Supported by NSF grant DMS-0501225.