Interval orders and reverse mathematics

We study the reverse mathematics of interval orders. We establish the logical strength of the implications between various definitions of the notion of interval order. We also consider the strength of different versions of the characterization theorem for interval orders: a partial order is an interval order if and only if it does not contain $2 \oplus 2$. We also study proper interval orders and their characterization theorem: a partial order is a proper interval order if and only if it contains neither $2 \oplus 2$ nor $3 \oplus 1$.Comment: 21 pages; to appear in Notre Dame Journal of Formal Logic; minor changes from the previous versio

Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension

We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is $\Pi^0_2$-complete and that the set of Cauchy problems which locally have a unique solution is $\Sigma^0_3$-complete. We prove that the set of Cauchy problems which have a global solution is $\Sigma^0_4$-complete and that the set of ordinary differential equation which have a global solution for every initial condition is $\Pi^0_3$-complete. We prove that the set of Cauchy problems for which both uniqueness and globality hold is $\Pi^0_2$-complete

The Veblen functions for computability theorists

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) "If X is a well-ordering, then so is epsilon_X", and (2) "If X is a well-ordering, then so is phi(alpha,X)", where alpha is a fixed computable ordinal and phi the two-placed Veblen function. For the former statement, we show that omega iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ACA_0^+ over RCA_0. To prove the latter statement we need to use omega^alpha iterations of the Turing jump, and we show that the statement is equivalent to Pi^0_{omega^alpha}-CA_0. Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement "if X is a well-ordering, then so is phi(X,0)" is equivalent to ATR_0 over RCA_0.Comment: 26 pages, 3 figures, to appear in Journal of Symbolic Logi

Time-scale analysis non-local diffusion systems, applied to disease models

The objective of the present paper is to use the well known Ross-Macdonald models as a prototype, incorporating spatial movements, identifying different times scales and proving a singular perturbation result using a system of local and non-local diffusion. This results can be applied to the prototype model, where the vector has a fast dynamics, local in space, and the host has a slow dynamics, non-local in space

Forme di ‘scrittura quotidiana’ a Roma e nel mondo romano tra la fine dell’età repubblicana e la prima età imperiale

In questo contributo si è cercato di verificare la possibilità che nel mondo antico, anche in contesti per i quali si può presupporre che poche fossero le persone davvero capaci di leggere e scrivere, il ricorso alla scrittura fosse comunque presupposto e generalizzato. Gli esempi addotti suggeriscono che lo sviluppo delle relazioni economiche nel mondo romano (a partire dal III secolo a.C.) abbiano determinato le condizioni per un fenomeno di questo genere. Altamente significativo delle peculiarità della società romana appare il caso dello schiavo ‘letterato’, che ha la possibilità di aver parte negli affari del padrone.In this paper we tried to explore the possibility that in the ancient world, where many people are supposed not to be able to read or write, the use of written languages was not something restricted to a small, high-status group. The examples here discussed suggest that in Rome, with the development of economic relations (from the third century B.C.), a wide range of people could participate in the use of writing in some fashion. A good example is the ‘literate slave’ who stands a chance of taking part in the owner’s business

Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries

In this work we study the behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating thin region with reaction terms concentrated in a neighborhood of the oscillatory boundary. Our main result is concerned with the upper and lower semicontinuity of the set of solutions. We show that the solutions of our perturbed equation can be approximated with ones of a one-dimensional equation, which also captures the effects of all relevant physical processes that take place in the original problem