On the exponent average of integer sequences

Let ω(m) denote the number of distinct prime factors of the integerm, let Ω(m) be the number of prime factors ofm counted with multiplicities. The exponent averageA(m) is defined byA(m)=Ω(m)/ω(m) form>1, andA(1)=1. If (m _n) is a sequence of positive integers, we can study the asymptotic exponent average lim _n→∞ A(m_n) (if it exists) resp. lim sup and lim inf. In this article, we consider exponent averages for general sequences, and particularly for sequences of binomial coefficients as well as the divisor function. One of the many results on binomial coefficients is that $$\mathop {\lim }\limits_{n \to \infty } A\left( {\left( {\begin{array}{*{20}c} {2n} \\ n \\ \end{array} } \right)} \right) = 1,$$ which shows that these binomial coefficients are almost squarefree. For the divisor functiond(n), we prove for instance $$\mathop {\lim \sup }\limits_{n \to \infty } \frac{{A(d(n))\log \log n}}{{\log n}} = 1.$$      

Nonlinear deterministic structures and the randomness of protein sequences

To clarify the randomness of protein sequences, we make a detailed analysis of a set of typical protein sequences representing each structural classes by using nonlinear prediction method. No deterministic structures are found in these protein sequences and this implies that they behave as random sequences. We also give an explanation to the controversial results obtained in previous investigations.     

Nonlinear generalized master equations and accounting for initial correlations

We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can thus be followed from the initial reversible stage of the evolution to the irreversible kinetic stage.     

On the average number of records for sequences of not identically distributed random variables

The scheme of n series of independent random variables X ₁₁, X ₂₁, …, X _k1, X ₁₂, X ₂₂, …, X _k2, …, X _1n , X _2n , …, X _kn is considered. Each of these successive series X _1m , X _2m , …, X _km , m = 1, 2, …, n consists of k variables with continuous distribution functions F ₁, F ₂, …, F _k , which are the same for all series. Let N(nk) be the number of upper records of the given nk random variables, and EN(nk) be the corresponding expected value. For EN(nk) exact upper and lower estimates are obtained. Examples are given of the sets of distribution functions for which these estimates are attained.     

Strategies and flexibility in mental calculation

This article focuses on issues around flexibility in mental calculation. When calculating a subtraction or multiplication mentally, it is desirable for students to use a method that suits the problem—to be flexible—but is it fair to describe this as “strategic flexibility”? What exactly is it that happens when someone ‘decides’ to tackle a mental calculation in a particular way? In the article, I make some distinctions among strategies and strategic flexibility in mental calculation to explore and try to clarify the terms and associated ideas. At the heart of the article is an argument that the model of strategic choice is not able to give an adequate account of strategic flexibility in all aspects of mental calculation, and an alternative is proposed.     

On the operations of sequences in rings and binomial type sequences

Given a commutative ring with identity R, many different and interesting operations can be defined over the set \(H_R\) of sequences of elements in R. These operations can also give \(H_R\) the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well-known transforms of sequences (such as the Stirling transform) are special cases of these automorphisms. Moreover, we introduce a novel isomorphism between \(H_R\) equipped with the componentwise sum and the set of the sequences starting with 1 equipped with the binomial convolution product. Finally, thanks to this isomorphism, we find a new method for characterizing and generating all the binomial type sequences.

     

The average distance and the independence number

We prove that in every connected graph the independence number is at least as large as the average distance between vertices.     

A nonnormal look at polychoric correlations: modeling the change in correlations before and after discretization

Two algorithms for establishing a connection between correlations before and after ordinalization under a wide spectrum of nonnormal underlying bivariate distributions are developed by extending the iteratively found normal-based results via the power polynomials. These algorithms are designed to compute the polychoric correlation when the ordinal correlation is specified, and vice versa, along with the distributional properties of latent, continuous variables that are subsequently ordinalized through thresholds dictated by the marginal proportions. The method has broad applicability in the simulation and random number generation world where modeling the relationships between these correlation types is of interest.     

On the flexibility of demand and production rate

In the manufacturing practice, cycle time is usually optimized in order to plan the batch size, and production time among other parameters. In certain situations, the production rate is decreased in order to have lower inventory levels or to deal with a shelf life constraint. This technical note examines the increase/decrease in the demand level along with a discussion concerning flexibility of the production rate. A generalized problem is also formulated in the context of the costs that are incurred in order to maintain certain demand level.     

On the DEA total weight flexibility and the aggregation in cross-efficiency evaluations

This paper discusses the DEA total weight flexibility in the context of the cross-efficiency evaluation. The DMUs in DEA are often assessed with unrealistic weighting schemes in their attempt to achieve the best ratings in their self-evaluation. We claim here that in a peer-appraisal like the cross-efficiency evaluation the cross-efficiencies provided by such weights cannot play the same role as those obtained with more reasonable weights. To address this issue, we propose to calculate the cross-efficiency scores by means of a weighted average of cross-efficiencies, instead of with the usual arithmetic mean, so the aggregation weights reflect the disequilibrium in the profiles of DEA weights that are used. Thus, the cross-efficiencies provided by profiles with large differences in their weights, especially those obtained with zero weights, would be attached lower aggregation weights (less importance) than those provided by more balanced profiles of weights.     

Definition and properties of the input-weighted average lead-time

In this paper a mathematical definition of the input-weighted average lead-time for a general continuous inventory function is introduced. It is theoretically shown that the input-weighted average lead-time is equal to the ratio of average inventory to average input, or equivalently, average inventory to average output.     

Valuing the flexibility of investing in security process innovations

In this paper, we develop a decision model of a firm’s optimal strategy for investment in security process innovations (SPIs) when confronted with a sequence of malicious attacks. The model incorporates real options as a methodology to capture the flexibility embedded in such investment decisions. SPIs, when seamlessly integrated with the organization’s overall business dynamics, induce organizational learning and provide the flexibility of switching to more suitable technologies as the environment of malicious attacks changes. The theoretical contribution of this paper is a mathematical model of the invest-to-learn and switching options generated upon early investment in flexible SPIs. The practical significance of the paper is the application of a binomial lattice model to approximate the continuous-time model, resulting in an easy to use decision aid for managers.