RealLife: the continuum limit of Larger Than Life cellular automata

Let A:={0,1}. A `cellular automaton' (CA) is a shift-commuting transformation of A^{Z^D} determined by a local rule. Likewise, a `Euclidean automaton' is a shift-commuting transformation of A^{R^D} determined by a local rule. `Larger than Life' (LtL) CA are long-range generalizations of J.H. Conway's Game of Life CA, proposed by K.M. Evans. We prove a conjecture of Evans: as their radius grows to infinity, LtL CA converge to a `continuum limit' Euclidean automaton, which we call `RealLife'. We also show that the `life forms' (fixed points, periodic orbits, and propagating structures) of LtL CA converge to life forms of RealLife. Finally we prove a number of existence results for fixed points of RealLife.Comment: 22 pages, 3 figures. Final Versio

Projective geometries arising from Elekes-Szab\'o problems

We generalise the Elekes-Szab\'o theorem to arbitrary arity and dimension and characterise the complex algebraic varieties without power saving. The characterisation involves certain algebraic subgroups of commutative algebraic groups endowed with an extra structure arising from a skew field of endomorphisms. We also extend the Erd\H{o}s-Szemer\'edi sum-product phenomenon to elliptic curves. Our approach is based on Hrushovski's framework of pseudo-finite dimensions and the abelian group configuration theorem.Comment: 48 pages. Minor improvements in presentation. To appear in ASEN

Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic

We show that algebraic analogues of universal group covers, surjective group homomorphisms from a $\mathbb{Q}$-vector space to $F^{\times}$ with "standard kernel", are determined up to isomorphism of the algebraic structure by the characteristic and transcendence degree of $F$ and, in positive characteristic, the restriction of the cover to finite fields. This extends the main result of "Covers of the Multiplicative Group of an Algebraically Closed Field of Characteristic Zero" (B. Zilber, JLMS 2007), and our proof fills a hole in the proof given there.Comment: Version accepted by the Bull. London Math. So

A Note on Divisible Points of Curves

Let $C$ be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3. We partially answer a question posed by Levin on points on $C$ for which a non-trivial power lies again on $C$. Our results have connections to Zilber's Conjecture on Intersections with Tori and yield to methods arising in transcendence theory and the theory of o-minimal structures.Comment: Published version, but with an error fixed in the formula for the function on page

Visualizing Change Through Interactive Photography: Transforming Identities, Transforming Research

In September of 1991 while I was still in the first critical months of establishing my credibility among the people with whom I wanted to work, two Hmong children asked me to teach them "to make pictures" after watching me photograph an ear of corn. My garden plot was adjacent to their families' plot at the Laotian Garden Farm in east Visalia, an agricultural town in California's San Joaquin Valley. My immediate "yes" flagged the beginning of an ongoing endeavor that I, very briefly, thought would be a temporary diversion from my real work: the gathering of hard data from adults. Eventually I came to understand that what happened between us -- what they learned from me and what I learned from them -- struck at the very heart of the research

Don Antonio de Alcedo\u27s Spanish Slant on the Upper Country

Don Antonio de Alcedo’s 1786-89 Diccionario geografico-historico de las Indias Occidentale o America includes entries about countries, provinces, cities, villages, and hamlets; about every aspect of political, cultural and natural history; and about products from cochineal to zarzaparilla, from Alaska to Tierra del Fuego. As a Spanish military officer he naturally focused most of this monumental five-volumework on New Spain, from the North American southwest to Patagonia and Chile. This article covers his descriptions of Lake Superior, Nipigon, Michipicoten, and Michilimackinac

Blaney Park Resort, Part I: Birth and Transformation

For decades, Blaney Park Resort in Michigan’s Upper Peninsula drew visitors from all over the Midwest to an unparalleled array of activities. From nearby and faraway, tourists and local residents came by train, small plane, and vehicle to Blaney Park. U. P. residents remember Blaney as the scene of formal receptions, informal socializing, dining, dancing and drinking, and a whopper smorgasbord. From horseback riding, swimming, golf, archery and snow sports; to hunting, fishing and trapping; to elbow-bending and toe-tapping, Blaney had something for everyone

Some Definability Results in Abstract Kummer Theory

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.Comment: 21 pages; minor notational fixe