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

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

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

Correctors for the Neumann problem in thin domains with locally periodic oscillatory structure

In this paper we are concerned with convergence of solutions of the Poisson equation with Neumann boundary conditions in a two-dimensional thin domain exhibiting highly oscillatory behavior in part of its boundary. We deal with the resonant case in which the height, amplitude and period of the oscillations are all of the same order which is given by a small parameter $\epsilon > 0$. Applying an appropriate corrector approach we get strong convergence when we replace the original solutions by a kind of first-order expansion through the Multiple-Scale Method.Comment: to appear in Quarterly of Applied Mathematic

Epimorphisms between linear orders

We study the relation on linear orders induced by order preserving surjections. In particular we show that its restriction to countable orders is a bqo.Comment: 15 pages; in version 2 we corrected some typos and rewrote the paragraphs introducing the results of subsection 3.3 (statements and proofs are unchanged

Invariantly universal analytic quasi-orders

We introduce the notion of an invariantly universal pair (S,E) where S is an analytic quasi-order and E \subseteq S is an analytic equivalence relation. This means that for any analytic quasi-order R there is a Borel set B invariant under E such that R is Borel bireducible with the restriction of S to B. We prove a general result giving a sufficient condition for invariant universality, and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread. In fact it occurs for a great number of complete analytic quasi-orders, arising in different areas of mathematics, when they are paired with natural equivalence relations.Comment: 31 pages, 1 figure, to appear in Transactions of the American Mathematical Societ