38 research outputs found
An Algorithmic Characterization of Polynomial Functions over
In this paper we consider polynomial representability of functions defined
over , where is a prime and is a positive integer. Our aim is
to provide an algorithmic characterization that (i) answers the decision
problem: to determine whether a given function over is polynomially
representable or not, and (ii) finds the polynomial if it is polynomially
representable. The previous characterizations given by Kempner (1921) and
Carlitz (1964) are existential in nature and only lead to an exhaustive search
method, i.e., algorithm with complexity exponential in size of the input. Our
characterization leads to an algorithm whose running time is linear in size of
input. We also extend our result to the multivariate case
On Rudimentarity, Primitive Recursivity and Representability
It is quite well-known from Kurt G¨odel’s (1931) ground-breaking Incompleteness Theorem that rudimentary relations (i.e., those definable by bounded formulae) are primitive recursive, and that primitive recursive functions are representable in sufficiently strong arithmetical theories. It is also known, though perhaps not as well-known as the former one, that some primitive recursive relations are not rudimentary. We present a simple and elementary proof of this fact in the first part of the paper. In the second part, we review some possible notions of representability of functions studied in the literature, and give a new proof of the equivalence of the weak representability with the (strong) representability of functions in sufficiently strong arithmetical theories
To the theory of generalized Stieltjes transforms
We identify measures arising in the representations of products of
generalized Stieltjes transforms as generalized Stieltjes transforms, provide
optimal estimates for the size of those measures, and address a similar issue
for generalized Cauchy transforms. In the latter case, in two particular
settings, we give criteria ensuring that the measures are positive. On this
way, we also obtain new, applicable conditions for representability of
functions as generalized Stieltjes transforms, thus providing a partial answer
to a problem posed by Sokal and shedding a light at spectral multipliers
emerged recently in probabilistic studies. As a byproduct of our approach, we
improve several known results on Stieltjes and Hilbert transforms.Comment: 55 page
Inconsistency, paraconsistency and ω-inconsistency
In this paper I'll explore the relation between ω-inconsistency and plain inconsistency, in the context of theories that intend to capture semantic concepts. In particular, I'll focus on two very well known inconsistent but non-trivial theories of truth: LP and STTT. Both have the interesting feature of being able to handle semantic and arithmetic concepts, maintaining the standard model. However, it can be easily shown that both theories are ω-inconsistent. Although usually a theory of truth is generally expected to be ω-consistent, all conceptual concerns don't apply to inconsistent theories. Finally, I'll explore if it's possible to have an inconsistent, but ω-consistent theory of truth, restricting my analysis to substructural theories.Fil: Da Re, Bruno. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Instituto de Investigaciones Filosóficas - Sadaf; Argentin
Ethics and economics in Karl Menger: how did social sciences cope with Hilbertism
This paper deals with the contributions made to the social sciences by the mathematician Karl Menger (1902-1985), the son of the more famous economist, Carl Menger. Mathematician and a logician, he focused on whether it was possible to explain the social order in formal terms.1 He stressed the need to find the appropriate means with which to treat them, avoiding recourse to historical descriptions, which are unable to yield social laws. He applied Hilbertism to economics and ethics in order to build an axiomatic and formalized model of the individual behavior and the dynamics of social groups.
Starting the Dismantling of Classical Mathematics
This paper uses the relevant logic, MCQ, of meaning containment to explore mathematics without various classical theses, in particular, without the law of excluded middle