Pelegrí Viader

Riesz-Nágy Singular Functions Revisited

In 1952 F. Riesz and Sz.Nagy published an example of a monotonic continuous function whose derivative is zero almost everywhere, that is to say, a singular function. Besides, the function was strictly increasing. Their example was built as the limit of a sequence of deformations of the identity function. As an easy consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In this paper we revisit the Riesz-N´agy family of functions and we relate it to a system for real number representation which we call (t, t-1)–expansions. With the help of these real number expansions we generalize the family. The singularity of the functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity.

A total order in [0,1] defined through a 'next' operator

A ‘next’ operator, σ, is built on the set R1 =(0, 1] − {1 − 1 / e} defining a partial order that, with the help of the axiom of choice, can be extended to a total order in R1. In addition, the orbits {σn(α)}n ∈ Z are all dense in R1 and are constituted by elements of the same arithmetical character: if α is an algebraic irrational of degree k, all the elements in αs orbit are algebraic of degree k; if α is transcendental, all are transcendental. Moreover, the asymptotic distribution function of the sequence formed by the elements in any of the half-orbits is a continuous, strictly increasing, singular function very similar to the well-known Minkowskis ?(⋅) function.

On actually computable bijections between ℕ and Q

We present three different ways of getting an actually computable enumeration of Q+ in the sense of being able to know exactly which rational occupies a given position and vice versa. The first enumeration is based on the Pierce expansion model for representing real numbers. The other two are based on regular continued fractions.

A New Singular Function

Abstract A new continuous strictly increasing singular function is described with the help of the ternary and binary systems for real number representation; in this, our function is similar to Cantors function, but in other aspects it is quite unusual. We are able to determine a condition to identify many points for which the derivative vanishes or is infinite; for other singular functions constructed with the help of a system of representation of real numbers, this condition depends on some metrical properties of the growth of averages of the sum of all the digits of the representation, but in the case of this new function, it depends on the frequency of occurrence of the digit 2 in the usual ternary expansion of a number.

A non-quasi-competitive Cournot oligopoly with stability

This paper presents a classical Cournot oligopoly model with some peculiar features: it is non--quasi--competitive as price under N-poly is greater than monopoly price; Cournot equilibrium exists and is unique with each new entry; the successive equilibria after new entries are stable under the adjustment mechanism that assumes that actual output of each seller is adjusted proportionally to the difference between actual output and profit maximizing output. Moreover, the model tends to perfect competition as N goes to infinity, reaching the monopoly price again.

On a Problem of Alfred Renyi

Alfred Renyi, in a paper of 1962, A new approach to the theory of Engels series, proposed a problem related to the growth of the elements of an Engels series. In this paper, we reformulate and solve Renyis problem for both, Engels series and Pierce expansions.

Fermat and the Quadrature of the Folium of Descartes

(2004). Fermat and the Quadrature of the Folium of Descartes. The American Mathematical Monthly: Vol. 111, No. 3, pp. 216-229.

Non-Quasi-Competitiveness and Stability in Cournot's Model

This paper retakes previous work of the authors, about the relationship between non{quasi{competitiveness (the increase in price caused by an increase in the number of oligopolists) and stability of the equilibrium in the classical Cournot oligopoly model. Though it has been widely accepted in the literature that the loss of quasi{competitiveness is linked, in the long run as new rms entered the market, to instability of the model, the authors in their previous work put forward a model in which a situation of monopoly changed to duopoly losing quasi{competitiveness but maintaining the stability of the equilibrium. That model could not, at the time, be extended to any number of oligopolists. The present paper exhibits such an extension. An oligopoly model is shown in which the loss of quasi{competiti- veness resists the presence in the market of as many rms as one wishes and where the successive Cournots equilibrium points are unique and asymptotically stable. In this way, for the rst time, the conjecture that non{quasi{competitiveness and instability were equivalent in the long run, is proved false.