‎The concept of self-similarity on subsets of algebraic varieties‎ ‎is defined by considering algebraic endomorphisms of the variety‎ ‎as `similarity' maps‎. ‎Self-similar fractals are subsets of algebraic varieties‎ ‎which can be written as a finite and disjoint union of‎ ‎`similar' copies‎. ‎Fractals provide a framework in which‎, ‎one can‎ ‎unite some results and conjectures in Diophantine g...

We show that the order three algebraic differential equation over Q satisfied by the analytic j-function defines a non-א0-categorical strongly minimal set with trivial forking geometry relative to the theory of differentially closed fields of characteristic zero answering a long-standing open problem about the existence of such sets. The theorem follows from Pila’s modular Ax-Lindemann-Weierstr...

In this paper we use the Cayley-Bacharach theorem of classical algebraic geometry to construct several universal algebras on algebraic curves using divisors and complete intersection cycles and study the equational identities valid for these synthetic constructions. These results are not necessarily new; in fact, all of them may be “easily” provable by resorting to such powerful tools as the Ri...

I. The Complex Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 II. Elementary Properties and Examples of Analytic Fns. . . . . . . . . . . . . . . . 3 Differentiability and analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . ...

This paper examines existence, continuity and characterization of optimal paths under “recursive” preferences. Current utility is a fixed (aggregator) function of current consumption and future utility. For suitable aggregators, a useful refinement of the Contraction Mapping Theorem generates the utility function, as in Lucas and Stokey. A broader class of aggregators is handled via a limiting ...

We first provide an overview of the basic results in the geometry of elliptic curves, introducing the Picard Group, Weierstrass Equations, and Isogenies. This is followed by a discussion of the structure of m-torsion points on an elliptic curve, introducing such tools as the Weil pairing and the l-adic Tate module. The paper culminates in a theorem counting the rational points on an elliptic cu...