A study on some generalized multiplicative and generalized additive arithmetic functions

In this paper by an arithmetic function we shall mean a real-valued function on the set of positive integers. We recall the definitions of some common arithmetic convolutions. We also recall the definitions of a multiplicative function, a generalized multiplicative function (or briefly a GM-function), an additive function and a generalized additive function (or briefly a GA-function). We shall study in details some properties of GM-functions as well as GA-functions using some particular arithmetic convolutions namely the Narkiewicz’s A-product and the author’s B-product. We conclude our discussion with some examples.

Empirical lognormality of biological variation: implications for the ‘zero-force evolutionary law’

ABSTRACTThe zero-force evolutionary law (ZFEL) of McShea et al. states that independently evolving entities, with no forces or constraints acting on them, will tend to accumulate differences and therefore diverge from each other. McShea et al. quantified the law by assuming normality on an additive arithmetic scale and reflecting negative differences as absolute values, systematically augmenting perceived divergence. The appropriate analytical framework is not additive but proportional, where logarithmic transformation is required to achieve normality. Logarithms and logarithmic differences can be negative but the proportions they represent cannot be negative. Reformulation of ZFEL in a proportional or geometric reference frame indicates that when entities evolve randomly and independently, differences smaller than any initial difference are balanced by differences larger than the initial difference. Total variance increases with each step of a random walk, but there is no statistical expectation of divergence between random-walk lineages.

Sur la concentration de certaines fonctions additives

Improving on estimates of Erdős, Halász and Ruzsa, we provide new upper and lower bounds for the concentration function of the limit law of certain additive arithmetic functions under hypotheses involving only their average behaviour on the primes. In particular we partially confirm a conjecture of Erdős and Kátai. The upper bound is derived via a reappraisal of the method of Diamond and Rhoads, resting upon the theory of functions with bounded mean oscillation.

About the equivalent replaceability of the double induction axiom

In this paper the first order predicate calculus with the axioms of additive arithmetic is investigated. The conditions of the equivalent replaceability of a double induction axiom is presented.

About the some conditions of the replaceability of the double induction

The provability of the axiom of double induction (ADI) with the open induction formula in the additive arithmetic is investigated. The system of additional axioms and theirs provability by ADI is presented.