Rado's theorem for rings and modules

We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary domain or a noetherian ring. The crucial idea is to study partition regularity for general modules rather than only for rings. Contrary to previous techniques, our approach is independent of the characteristic of the coefficient ring.Comment: 19 page

Which weakly ramified group actions admit a universal formal deformation?

Consider a formal (mixed-characteristic) deformation functor D of a representation of a finite group G as automorphisms of a power series ring k[[t]] over a perfect field k of positive characteristic. Assume that the action of G is weakly ramified, i.e., the second ramification group is trivial. Examples of such representations are provided by a group action on an ordinary curve: the action of a ramification group on the completed local ring of any point on such a curve is weakly ramified. We prove that the only such D that are not pro-representable occur if k has characteristic two and G is of order two or isomorphic to a Klein group. Furthermore, we show that only the first of those has a non-pro-representable equicharacteristic deformation functor.Comment: 16 pages; further minor correction

Simultaneous pp-orderings and minimising volumes in number fields

In the paper "On the interpolation of integer-valued polynomials" (Journal of Number Theory 133 (2013), pp. 4224--4232.) V. Volkov and F. Petrov consider the problem of existence of the so-called $n$-universal sets (related to simultaneous $p$-orderings of Bhargava) in the ring of Gaussian integers. We extend their results to arbitrary imaginary quadratic number fields and prove an existence theorem that provides a strong counterexample to a conjecture of Volkov-Petrov on minimal cardinality of $n$-universal sets. Along the way, we discover a link with Euler-Kronecker constants and prove a lower bound on Euler-Kronecker constants which is of the same order of magnitude as the one obtained by Ihara.Comment: new version, substantial corrections in section 6, will appear in Journal of Number Theor

Uniqueness of solutions of parabolic semilinear nonlocal-boundary problems

AbstractThe purpose of the paper is to give two theorems about the uniqueness of solutions of parabolic semilinear nonlocal-boundary problems. The paper is a continuation of previous papers by Byszewski and the generalization of some results from [R. Rabczuk, “Introduction to Differential Inequalities,” PWN, Warsaw, 1976 [Polish]; J. Chabrowski, On nonlocal problems for parabolic equations, Nagoya Math. J. 93 (1984), 109–131], The theorems obtained in this paper can be applied in the theories of diffusion and heat conduction with better effects than the analogous theorems about parabolic initial-boundary problems and than the analogous theorems about parabolic periodic-boundary problems

Dynamically affine maps in positive characteristic

We study fixed points of iterates of dynamically affine maps (a generalisation of Latt\`es maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function of the dynamical system: it is either rational or non-holonomic, depending on specific characteristics of the map. We also study the algebraicity of the so-called tame zeta function, the generating function for periodic points of order coprime to $p$. We then verify these hypotheses for dynamically affine maps on the projective line, generalising previous work of Bridy, and, in arbitrary dimension, for maps on Kummer varieties arising from multiplication by integers on abelian varieties.Comment: Lois van der Meijden co-authored Appendix B. 31 p

Colonizing Chaco Canyon: Mapping Antiquity in the Territorial Southwest

The 1849 Navajo Expedition was the first official US military mapping of Navajoland after the Mexican Cession, and has been recognized by historians as the first sustained window into the region and its people. Lieutenant James H. Simpson of the US Topographical Corps of Engineers was ordered to accompany the punitive expedition to document the route. Captivated by the stone ruins of Chaco Canyon, Simpson made a side excursion to record and map the structures, and contributed to the way Chaco is interpreted and imagined to this day. In this paper, I follow Lieutenant Simpson\u27s survey party, tracing their discovery and mapping of Chaco Canyon. Through an analysis of Simpson\u27s map and journal, I argue that the mapping effort served to fix Chaco in a new geography of antiquity that redrew the history and future of the nation, and attempted to discipline unfamiliar peoples and landscapes into the national body. This mapping constructed Chaco as a national resource, fixing its significance in both prehistory and the moment of its discovery. Tracing the particular ways this knowledge was produced through the discovery and mapping of Chaco shows how both colonial cartographies and notions of antiquity are foundational to the ongoing project of settler colonialism