Enumerating Colorings, Tensions and Flows in Cell Complexes

We study quasipolynomials enumerating proper colorings, nowhere-zero tensions, and nowhere-zero flows in an arbitrary CW-complex $X$, generalizing the chromatic, tension and flow polynomials of a graph. Our colorings, tensions and flows may be either modular (with values in $\mathbb{Z}/k\mathbb{Z}$ for some $k$) or integral (with values in $\{-k+1,\dots,k-1\}$). We obtain deletion-contraction recurrences and closed formulas for the chromatic, tension and flow quasipolynomials, assuming certain unimodularity conditions. We use geometric methods, specifically Ehrhart theory and inside-out polytopes, to obtain reciprocity theorems for all of the aforementioned quasipolynomials, giving combinatorial interpretations of their values at negative integers as well as formulas for the numbers of acyclic and totally cyclic orientations of $X$.Comment: 28 pages, 3 figures. Final version, to appear in J. Combin. Theory Series

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

Low-level processing for real-time image analysis

A system that detects object outlines in television images in real time is described. A high-speed pipeline processor transforms the raw image into an edge map and a microprocessor, which is integrated into the system, clusters the edges, and represents them as chain codes. Image statistics, useful for higher level tasks such as pattern recognition, are computed by the microprocessor. Peak intensity and peak gradient values are extracted within a programmable window and are used for iris and focus control. The algorithms implemented in hardware and the pipeline processor architecture are described. The strategy for partitioning functions in the pipeline was chosen to make the implementation modular. The microprocessor interface allows flexible and adaptive control of the feature extraction process. The software algorithms for clustering edge segments, creating chain codes, and computing image statistics are also discussed. A strategy for real time image analysis that uses this system is given

An intriguing hyperelliptic Shimura curve quotient of genus 16

Let $F$ be the maximal totally real subfield of $\mathbf{Q}(\zeta_{32})$, the cyclotomic field of $32$nd roots of unity. Let $D$ be the quaternion algebra over $F$ ramified exactly at the unique prime above $2$ and 7 of the real places of $F$. Let $\mathcal{O}$ be a maximal order in $D$, and $X_0^D(1)$ the Shimura curve attached to $\mathcal{O}$. Let $C = X_0^D(1)/\langle w_D \rangle$, where $w_D$ is the unique Atkin-Lehner involution on $X_0^D(1)$. We show that the curve $C$ has several striking features. First, it is a hyperelliptic curve of genus $16$, whose hyperelliptic involution is exceptional. Second, there are $34$ Weierstrass points on $C$, and exactly half of these points are CM points; they are defined over the Hilbert class field of the unique CM extension $E/F$ of class number $17$ contained in $\mathbf{Q}(\zeta_{64})$, the cyclotomic field of $64$th roots of unity. Third, the normal closure of the field of $2$-torsion of the Jacobian of $C$ is the Harbater field $N$, the unique Galois number field $N/\mathbf{Q}$ unramified outside $2$ and $\infty$, with Galois group $\mathrm{Gal}(N/\mathbf{Q})\simeq F_{17} = \mathbf{Z}/17\mathbf{Z} \rtimes (\mathbf{Z}/17\mathbf{Z})^\times$. In fact, the Jacobian $\mathrm{Jac}(X_0^D(1))$ has the remarkable property that each of its simple factors has a $2$-torsion field whose normal closure is the field $N$. Finally, and perhaps the most striking fact about $C$ is that it is also hyperelliptic over $\mathbf{Q}$

Faster Algorithms for Sparse ILP and Hypergraph Multi-Packing/Multi-Cover Problems

In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in $P \cap Z^n$, assuming that $P$ is a polyhedron, defined by systems $A x \leq b$ or $Ax = b,\, x \geq 0$ with a sparse matrix $A$. We develop algorithms for these problems that outperform state of the art ILP and counting algorithms on sparse instances with bounded elements. We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximal Matching problems