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Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are(More)
We study algebraic dynamical systems (and, more generally, σ-varieties) Φ : A n C → A n C given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of " clusters " from which one may easily read off(More)
We show that a connected group interpretable in a compact complex manifold (a meromorphic group) is definably an extension of a complex torus by a linear algebraic group, generalizing results in [4]. A special case of this result, as well as one of the ingredients in the proof, is that a strongly minimal modular meromorphic group is a complex torus,(More)
The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the Ax-Kochen-Ershov principle is proven for a theory of valued D-fields of residual characteristic zero. The model theory of differential and difference fields has been extensively studied (see for example [7, 3]) and(More)
We give axiomatizations and prove quantifier elimination theorems for first-order theories of unramified valued fields with an automorphism having a close interaction with the valuation. We achieve an analogue of the classical Ostrowski theory of pseudoconvergence. In the outstanding case of Witt vectors with their Frobenius map, we use the ∂-ring formalism(More)
Let (K; v) be a complete discretely valued eld of characteristic zero with an algebraically closed residue eld of positive characteristic. Let : K ! K be a continuous automorphism of K inducing a Frobenius auto-morphism on the residue eld. We prove quantiier-elimination for (K; v;) in a language with angular component maps and in a language with predicates(More)
In this thesis we introduce a general notion of a D-ring generalizing that of a differential or difference ring. In Chapter 3, this notion is specialized to consider valued fields D-fields: valued fields K having an operator D : K → K and a fixed element e ∈ K satisfying D(x + y) = Dx + Dy, D(1) = 0, D(xy) = xDy + yDx + eDxDy, v(e) ≥ 0, and v(Dx) ≥ v(x).(More)
Communication, i.e., moving data, between levels of a memory hierarchy or between parallel processors on a network, can greatly dominate the cost of computation, so algorithms that minimize communication can run much faster (and use less energy) than algorithms that do not. Motivated by this, attainable communication lower bounds were established in [12,(More)
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