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 process of analyzing hypergeometric representations, we found---experimentally and strikingly---that the $C_{n,k}$ almost certainly satisfy certain interindicial relations including discrete $k$-recurrences. Using generating functions, differential theory, complex analysis, and Wilf--Zeilberger algorithms we are able to prove some central cases of these relations.

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.

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.

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.

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.

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.

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 groups of simple groups. Our results are based on major amounts of computation. We make substantial use of systems for computational group theory and, in particular, of computer implementations of coset enumeration.

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

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.

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 dependence of $\FracMod{1}{\g}{d}$ on $d$ is complicated and beautiful, depending on the congruence class of $d$ mod $n$ and on generating curves that resemble a double helix within a given congruence class. We also summands in the direct sum decomposition as a function on the space of all $\g$-modules in a fixed dimension, and plot its histogram. This is related to the concept (used in quantum information theory) of noiseless subsystem. We identify a simple function that is conjectured to be the asymptotic form of the aforementioned histogram, and verify numerically that this is correct for $\su_n$.