Let {\small $K$} be an algebraic function field over a finite field. Let {\small $L$} be an extension field of {\small $K$} of degree at least 3. Let {\small $R$} be a finite set of valuationsof {\small $K$} and denote by {\small $S$} the set of extensions of valuations of {\small $R$} to {\small $L$}. Denote by {\small $O_{K,R},O_{L,S}$} the ring of {\small $R$}-integers of {\small $K$} and {\small $S$}-integers of {\small $L$}, respectively. Assume that {\small $\alpha\in O_{L,S}$} with {\small $L=K(\alpha)$}, let {\small $0\neq \mu\in O_{K,R}$}, and consider the solutions {\small $(x,y)\in O_{K,R}$} of the Thue equation

{\small \[ N_{L/K}(x-\alpha y)=\mu.\]}

We give an efficient method for calculating the {\small $R$}-integral solutions of the above equation. The method is different from that in our previous paper and is much more efficient in many cases.

We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is {\small $\#P$}-hard in general, we show that when the rank of the Lie algebra is assumed fixed, then there is a polynomial-time algorithm, based on counting lattice points in polytopes. In fact, for Lie algebras of type {\small $A$}, there is an algorithm, based on the ellipsoid algorithm, to decide when the coefficients are nonzero in polynomial time for arbitrary rank. Our experiments show that the lattice point algorithm is superior in practice to the standard techniques for computing multiplicities when the weights have large entries but small rank. Using an implementation of this algorithm, we provide experimental evidence for two conjectured generalizations of the saturation property of Littlewood-Richardson coefficients. One of these conjectures seems to be valid for types {\small $B$}, {\small $C$}, and {\small $D$}.

Let {\small $R$} be a complete discrete valuation ring with finite residue field, and let {\small $r_n$} be the probability that a random monic polynomial over {\small $R$} of degree {\small $n$} factors over {\small $R$} into linear factors. We study {\small $r_n$} in detail. Among other things, we show that {\small $r_n$} satisfies an interesting recursion, make a conjecture on the asymptotic behavior of {\small $r_n$} as {\small $n$} goes to infinity, and prove the conjecture in the case that the residue field has two elements.

In this paper, we find bounds on the smallest or base eigenvalue for the Laplacian on the noncompact surfaces within a certain parameterized class. We attempt to fit a curve to the resulting data. These eigenvalues are related to the Hausdorff dimension of certain fractals and to some problems in number theory.

In 1934 Hardy, Littlewood, and Pólya generalized Hilbert's inequality to the case in which the parameters are not conjugate. Determination of the best constant in this generalization is still an unsolved problem. An experimental approach is presented that yields numerical values that agree with theory in the cases in which an exact answer is known. The results may be a guide to a further theoretical approach.

We present a computer-oriented method of producing pictures of Bers embeddings of the Teichmüller space of once-punctured tori. The coordinate plane is chosen in such a way that the accessory parameter is hidden in the relative position of the origin. Our algorithm consists of two steps. For each point in the coordinate plane, we first compute the corresponding monodromy representation by numerical integration along certain loops. Then we decide whether the representation is discrete by applying Jørgensen's theory on the quasi-Fuchsian space of once-punctured tori.

We summarize how OPTi draws the parameter space. Each point in the picture of a parameter space corresponds to a group, and the program colors the point according to whether the group is discrete or indiscrete. Applying Jørgensen's inequality to certain sets of generators, OPTi first tries to decide indiscreteness of the group. If the process fails for generators up to a certain depth, the program then tries to construct the Ford region. When it succeeds in constructing the Ford region, Poincaré's polyhedron theorem guarantees the discreteness of the group.

Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of $L$-functions at the center of the critical strip are used to motivate a series of conjectures concerning the value distribution of the Fourier coefficients of half-integral-weight modular forms related to these $L$-functions. Our conjectures may be viewed as being analogous to the Sato-Tate conjecture for integral-weight modular forms. Numerical evidence is presented in support of them.

We give several conjectures that allow us to derive many series for {\small $ 1/ \pi$} and {\small $ 1/ \pi^2$}. These series include Ramanujan's series, as well as those associated with the Domb numbers and Apéry numbers. We have checked the conjectures numerically in many examples with a precision of two hundred digits.

This article furthers the study of nonlinear elliptic partial difference equations (PdE) on graphs. We seek solutions {\small $u:V\to\R$} to the semilinear elliptic partial difference equation {\small $-Lu + f(u) = 0$} on a graph {\small $G=(V,E)$}, where {\small $L$} is the (negative) Laplacian on the graph {\small $G$}. We extend techniques used to prove existence theorems and derive numerical algorithms for the partial differential equation (PDE) {\small $\Delta u + f(u) = 0$}. In particular, we prove the existence of sign-changing solutions and solutions with symmetry in the superlinear case. Developing variants of the mountain pass, modified mountain pass, and gradient Newton-Galerkin algorithms for our discrete nonlinear problem, we compute and describe many solutions. Letting {\small $f=f(\lambda,u)$}, we construct bifurcation diagrams and relate the results to the developed theory.

The main results of A. Zorich and I. Dynnikov regarding plane sections of periodic surfaces are extended to the piecewise linear case. As an application, the stereographic map of a truncated octahedron, extended to all of {\small $\Rt$} by periodicity, is analyzed numerically.