Kakeya sets over non-archimedean local rings

In a recent paper of Ellenberg, Oberlin, and Tao, the authors asked whether there are Besicovitch phenomena in F_q[[t]]^n. In this paper, we answer their question in the affirmative by explicitly constructing a Kakeya set in F_q[[t]]^n of measure 0. Furthermore, we prove that any Kakeya set in F_q[[t]]^2 or Z_p^2 is of Minkowski dimension 2.Comment: 10 page

Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories

In this paper, we use a geometric technique developed by Gonz\'alez-Prieto, Logares, Mu\~noz, and Newstead to study the $G$-representation variety of surface groups $\mathfrak{X}_G(\Sigma_g)$ of arbitrary genus for $G$ being the group of upper triangular matrices of fixed rank. Explicitly, we compute the virtual classes in the Grothendieck ring of varieties of the $G$-representation variety and the moduli space of $G$-representations of surface groups for $G$ being the group of complex upper triangular matrices of rank $2$, $3$ and $4$ via constructing a topological quantum field theory. Furthermore, we show that in the case of upper triangular matrices the character map from the moduli space of $G$-representations to the $G$-character variety is not an isomorphism

Algebraic 3D Graphic Statics: reciprocal constructions

The recently developed 3D graphic statics (3DGS) lacks a rigorous mathematical definition relating the geometrical and topological properties of the reciprocal polyhedral diagrams as well as a precise method for the geometric construction of these diagrams. This paper provides a fundamental algebraic formulation for 3DGS by developing equilibrium equations around the edges of the primal diagram and satisfying the equations by the closeness of the polygons constructed by the edges of the corresponding faces in the dual/reciprocal diagram. The research provides multiple numerical methods for solving the equilibrium equations and explains the advantage of using each technique. The approach of this paper can be used for compression-and-tension combined form-finding and analysis as it allows constructing both the form and force diagram based on the interpretation of the input diagram. Besides, the paper expands on the geometric/static degrees of (in)determinacies of the diagrams using the algebraic formulation and shows how these properties can be used for the constrained manipulation of the polyhedrons in an interactive environment without breaking the reciprocity between the two

Power map permutations and symmetric differences in finite groups

Let $G$ be a finite group. For all $a \in \Z$, such that $(a,|G|)=1$, the function $\rho_a: G \to G$ sending $g$ to $g^a$ defines a permutation of the elements of $G$. Motivated by a recent generalization of Zolotarev's proof of classic quadratic reciprocity, due to Duke and Hopkins, we study the signature of the permutation $\rho_a$. By introducing the group of conjugacy equivariant maps and the symmetric difference method on groups, we exhibit an integer $d_{G}$ such that $\text{sgn}(\rho_a)=(\frac{d_G}{a})$ for all $G$ in a large class of groups, containing all finite nilpotent and odd order groups.Comment: Electronic version of an article to be published as, Journal of Algebra and its Applications, 2011, DOI No: 10.1142/S0219498811005051, \c{opyright} copyright World Scientific Publishing Company, http://www.worldscinet.com/jaa/jaa.shtm

Explicit computations of Hida families via overconvergent modular symbols

In [Pollack-Stevens 2011], efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of $p$-adic $L$-functions and have further been applied to compute rational points on elliptic curves (e.g. [Darmon-Pollack 2006, Trifkovi\'c 2006]). In this paper, we generalize these algorithms to the case of families of overconvergent modular symbols. As a consequence, we can compute $p$-adic families of Hecke-eigenvalues, two-variable $p$-adic $L$-functions, $L$-invariants, as well as the shape and structure of ordinary Hida-Hecke algebras.Comment: 51 pages. To appear in Research in Number Theory. This version has added some comments and clarifications, a new example, and further explanations of the previous example

Finitely Presentable Higher-Dimensional Automata and the Irrationality of Process Replication

Higher-dimensional automata (HDA) are a formalism to model the behaviour of concurrent systems. They are similar to ordinary automata but allow transitions in higher dimensions, effectively enabling multiple actions to happen simultaneously. For ordinary automata, there is a correspondence between regular languages and finite automata. However, regular languages are inherently sequential and one may ask how such a correspondence carries over to HDA, in which several actions can happen at the same time. It has been shown by Fahrenberg et al. that finite HDA correspond with interfaced interval pomset languages generated by sequential and parallel composition and non-empty iteration. In this paper, we seek to extend the correspondence to process replication, also known as parallel Kleene closure. This correspondence cannot be with finite HDA and we instead focus here on locally compact and finitely branching HDA. In the course of this, we extend the notion of interval ipomset languages to arbitrary HDA, show that the category of HDA is locally finitely presentable with compact objects being finite HDA, and we prove language preservation results of colimits. We then define parallel composition as a tensor product of HDA and show that the repeated parallel composition can be expressed as locally compact and as finitely branching HDA, but also that the latter requires infinitely many initial states.Comment: 25 pages, 3 figure