## Recursos de colección

1. #### The Propositional Theory of Closure

McCluskey, A. E.; McIntyre, D. W.; Watson, W. S.
We study the simplest fragment of topological theory: those statements that can be expressed using one set variable, interior and closure operators, and inclusion. We introduce a formal system that is simple enough to be implemented on a computer and exhaustively studied and yet rich enough to be sound and complete for the fragment of theory under consideration. This fragment is rich enough to capture concepts such as regular open sets, extremal disconnectedness, partition topologies, and the nodec property.

2. #### Sparse Representation for Cyclotomic Fields

Fieker, Claus
Currently, all major implementations of cyclotomic fields as well as number fields are based on a dense model in which elements are represented either as dense polynomials in the generator of the field or as coefficient vectors with respect to a fixed basis. While this representation allows for the asymptotically fastest arithmetic for general elements, it is unsuitable for fields of degree greater than $10^4$ that arise in certain applications such as character theory for finite groups. We propose instead a sparse representation for cyclotomic fields that is particularly tailored to representation theory. We implemented our ideas in MAGMA and...

3. #### Constructing Hyperbolic Polyhedra Using Newton's Method

Roeder, Roland K. W.
We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton's method. Under the restriction that the dihedral angles must be nonobtuse, Andreev's theorem provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyhedron realizing a given combinatorial structure $C$. Andreev's theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Our construction uses Newton's method and a homotopy to explicitly follow the existence proof presented by Andreev, providing both a very clear illustration of a proof of Andreev's theorem and a convenient way to construct three-dimensional...

4. #### Lenstra's Constant and Extreme Forms in Number Fields

Coulangeon, R.; Icaza, M. I.; O'Ryan, M.
In this paper we compute $\gamma_{K,2$ for $K=\mathbb{Q}(\rho)$, where $\rho$ is the real root of the polynomial $x^3 -x^2 +1 =0$. We refine some techniques introduced in Baeza, et al. to construct all possible sets of minimal vectors for perfect forms. These refinements include a relation between minimal vectors and the Lenstra constant. This construction gives rise to results that can be applied in several other cases.

5. #### On the Singularization of the Two-Dimensional Jacobi--Perron Algorithm

Schratzberger, B.
We present a constructive method to convert the (two-dimensional) Jacobi--Perron evolution of $(x_1,x_2) \in [0,1]^2$ into the corresponding evolution of Podsypanin, and vice versa. A similar approach allows us to extend the result to Brun's algorithm. The method, based on the techniques of singularization and insertion, is built up in steps. From experiments, we assume that each step terminates after finitely many states almost everywhere.

6. #### An Explicit Formula for the Arithmetic--Geometric Mean in Genus 3

Lehavi, D.; Ritzenthaler, C.
The arithmetic--geometric mean algorithm for calculating elliptic integrals of the first type was introduced by Gauss. The analogous algorithm for abelian integrals of genus $2$ was introduced by Richelot (1837) and Humbert (1901). We present the analogous algorithm for abelian integrals of genus 3.

7. #### Elliptic Curves as Attractors in ${\mathbb P}^2$, Part 1: Dynamics

Bonifant, Araceli; Dabija, Marius; Milnor, John
We study rational maps of the real or complex projective plane of degree two or more, concentrating on those that map a genus-one curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich variety of interesting dynamical behaviors that are perhaps familiar to the applied dynamics community but not to specialists in several complex variables. For example, we describe smooth attractors with riddled or intermingled attracting basins, and we observe blowout'' bifurcations when the transverse Lyapunov exponent for the invariant curve changes sign. In the complex case, we prove that the genus-one curve (a...

9. #### A Hardy-Ramanujan Formula for Lie Algebras

Ritter, Gordon
We study certain combinatoric aspects of the set of all unitary representations of a finite-dimensional semisimple Lie algebra $\g$. We interpret the Hardy--Ramanujan--Rademacher formula for the integer partition function as a statement about $\su_2$, and explore in some detail the generalization to other Lie algebras. We conjecture that the number $\Mod{\g}{d}$ of $\g$-modules in dimension $d$ is given by $(\alpha/d) \exp(\beta d^\gamma)$ for $d \gg 1$, which (if true) has profound consequences for other combinatorial invariants of $\g$-modules. In particular, the fraction $\FracMod{1}{\g}{d}$ of $d$-dimensional\linebreak $\g$-modules that have a one-dimensional submodule is determined by the generating function for $\Mod{\g}{d}$. The...

10. #### Equality of Polynomial and Field Discriminants

Ash, Avner; Brakenhoff, Jos; Zarrabi, Theodore
We give a conjecture concerning when the discriminant of an irreducible monic integral polynomial equals the discriminant of the field defined by adjoining one of its roots to $\Q$. We discuss computational evidence for it. An appendix by the second author gives a conjecture concerning when the discriminant of an irreducible monic integral polynomial is square-free and some computational evidence for it.

11. #### On the Integral Carathéodory Property

Bruns, Winfried
In this note we document the existence of a finitely generated rational cone that is not covered by its unimodular Hilbert subcones, but satisfies the integral Carathéodory property. We explain the algorithms that decide these properties and describe our experimental approach that led to the discovery of the examples

12. #### On the Efficiency of the Simple Groups of Order Less Than a Million and Their Covers

Campbell, Colin M.; Havas, George; Ramsay, Colin; Robertson, Edmund F.
There is much interest in finding short presentations for the finite simple groups. In a previous paper we produced nice efficient presentations for all small simple groups and for their covering groups. Here we extend those results from simple groups of order less than100,000 up to order one million, but we leave one simple group and one covering group for which the efficiency question remains unresolved. We give presentations that are better than what was previously available, in terms of length and in terms of computational properties, in the process answering two previously unresolved problems about the efficiency of covering...

13. #### Schwarz Reflection Geometry II: Local and Global Behavior of the Exponential Map

Calini, Annalisa; Langer, Joel
A local normal form is obtained for geodesics in the space $\Lambda=\{\Gamma\}$ of analytic Jordan curves in the extended complex plane with symmetric space multiplication $\Gamma_1\cdot\Gamma_2$ defined by Schwarzian reflection of $\Gamma_2$ in $\Gamma_1$. Local geometric features of $(\Lambda, \cdot)$ will be seen to reflect primarily the structure of the Witt algebra, while issues of global behavior of the exponential map will be viewed in the context of conformal mapping theory.

14. #### The $D_4$ Root System Is Not Universally Optimal

Cohn, Henry; Conway, John H.; Elkies, Noam D.; Kumar, Abhinav
We prove that the $D_4$ root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code. We further conjecture that there is no universally optimal spherical code of 24 points in $S^3$, based on numerical computations suggesting that every 5-design consisting of 24 points in $S^3$ is in a 3-parameter family (which we describe explicitly, based on a construction due to Sali) of deformations of the $D_4$ root system.

15. #### Finding All Elliptic Curves with Good Reduction Outside a Given Set of Primes

Cremona, J. E.; Lingham, M. P.
We describe an algorithm for determining elliptic curves defined over a given number field with a given set of primes of bad reduction. Examples are given over $\Q$ and over various quadratic fields.

16. #### Jointly Periodic Points in Cellular Automata: Computer Explorations and Conjectures

Boyle, Mike; Lee, Bryant
We develop a rather elaborate computer program to investigate the jointly periodic points of one-dimensional cellular automata. The experimental results and mathematical context lead to questions, conjectures, and a contextual theorem.

17. #### Tropical Polytopes and Cellular Resolutions

Develin, Mike; Yu, Josephine
Tropical polytopes are images of polytopes in an affine space over the Puiseux series field under the degree map. This viewpoint gives rise to a family of cellular resolutions of monomial ideals that generalize the hull complex of Bayer and Sturmfels, instances of which improve upon the hull resolution in the sense of being smaller. We also suggest a new definition of a face of a tropical polytope, which has nicer properties than previous definitions; we give examples and provide many conjectures and directions for further research in this area.

18. #### Hypergeometric Forms for Ising-Class Integrals

Bailey, D. H.; Borwein, D.; Borwein, J. M.; Crandall, R. E.
We apply experimental-mathematical principles to analyze the integrals C_{n,k} and:= \frac{1}{n!} \int_0^{\infty} \cdots \int_0^{\infty} \frac{dx_1 \, dx_2 \cdots \, dx_n}{(\cosh x_1 + \dots + \cosh x_n)^{k+1}. ¶ These are generalizations of a previous integral $C_n := C_{n,1}$ relevant to the Ising theory of solid-state physics. We find representations of the $C_{n,k}$ in terms of Meijer $G$-functions and nested Barnes integrals. Our investigations began by computing 500-digit numerical values of $C_{n,k}$ for all integers $n, k$, where $n \in [2, 12]$ and $k \in [0,25]$. We found that some $C_{n,k}$ enjoy exact evaluations involving Dirichlet $L$-functions or the Riemann zeta function. In the...

A cuspidal curve is a curve whose singularities are all cusps, i.e., unibranched singularities. This article describes computations that lead to the following conjecture: A rational cuspidal plane curve of degree greater than or equal to six has at most three cusps. The curves with precisely three cusps occur in three series. Assuming the Flenner--Zaidenberg rigidity conjecture, the above conjecture is verified up to degree $20$