We study the analytic function $\eta$ defined on the unit disk by $\eta(\lambda)=\lim_{n\to +\infty} P_\lambda^{\circ n}(-\lambda/2)/\lambda^n,$ where $P_\lambda$ is the quadratic polynomial $z\mapsto \lambda z+z^2$.

For quadratics with period-one negative continued fraction expansions,

X\theta =\frac{1}{ a-{\dfrac{\mathstrut 1}{a-{\dfrac{\mathstrut 1}{a- \cdots }}}}},

we show that the inhomogeneous Lagrange spectrum,

\bL (\theta) :=\bigl\{ \liminf\nolimits_{|n|\rightarrow \infty} |n|@\|n\theta -\gamma\| : \gamma \in \funnyR,\; \gamma \not\in \funnyZ+\theta \funnyZ\bigr\},

contains an inhomogeneous Hall's ray $[0,c(\theta)]$ with $$c(\theta)=\tfrac{1}{4}\bigl(1-O(a^{-1/2})\bigr)\hbox{.}

We describe gaps in the spectrum showing that this is essentially best possible. Pictures of computed spectra are included. Investigating such pictures led us to these results.

Let $M$ and $P$ be Seifert 3-manifolds. Is there a degree one map $f\colon M \rightarrow P\,$? The problem was completely solved by Hayat-Legrand, Wang, and Zieschang for all cases except when $P$ is the Poincaré homology sphere. We investigate the remaining case by elaborating and implementing a computer algorithm that calculates the degree. As a result, we get an explicit experimental expression for the degree through numerical invariants of the induced homomorphism $f_{\#}\colon \pi_1 (M) \rightarrow \pi_1(P)$.

Let $T:\CH1 \times \CH2 \times \CH3 \ra \C $ be a trilinear form, where $\CH1$, $\CH2$, $\CH3$ are separable Hilbert spaces. In the hypothesis that at least two of the three spaces are finite \dimn/al we show that the norm square $\lam=\Norm T{}2$ is a root of a certain algebraic equation, usually of very high degree, which we baptize the millennial equation, because it is an analogue of the secular equation in the bilinear case. More generally, as indicated in the title, we can consider singular values of a trilinear form and their squares too satisfy the same equation. We work out the binary case (all three spaces are two dimensional). Even in this case the situation is complex, so, in the absence of any genuine results, we content ourselves with advancing a number of conjectures suggested by computer experiments. Finally, we connect the singular values of a trilinear form with the critical values of an associated family of a one parameter family of bilinear forms. Also here we have to offer mainly only experimental evidence.

The pentagram map is defined on the space of convex $n$-gons (considered up to projective equivalence) by drawing the diagonals that join second-nearest-neighbors in an $n$-gon and taking the new $n$-gon formed by the intersections. We prove that this map is recurrent; thus, for almost any starting polygon, repeated application of the pentagram map will show a near copy of the starting polygon appear infinitely often under various perspectives.

We give a symbolic algorithm to check whether certain holomorphic foliations of the complex projective plane have an algebraic solution. This algorithm is then used to produce new examples of foliations without algebraic solutions.

We announce the creation of a database of invariant rings. This database contains a large number of invariant rings of finite groups, mostly in the modular case. It gives information on generators and structural properties of the invariant rings. The main purpose is to provide a tool for researchers in invariant theory.

We develop a method to compute the Hermite-Humbert constants $\gam_{K,n}$ of a real quadratic number field $K$, the analogue of the classical Hermite constant $\gam_n$ when $\funnyQ$ is replaced by a quadratic extension. In the case $n=2$, the problem is equivalent to the determination of lowest points of fundamental domains in $\H^2$ for the Hilbert modular group over $K$, that had been studied experimentally by H. Cohn. We establish the results he conjectured for the fields $ \funnyQ@(\sqrt{2})$, $\funnyQ@(\sqrt{3})$ and $\funnyQ@(\sqrt{5})$. The method relies on the characterization of extreme forms in terms of perfection and eutaxy given by the second author in an earlier paper.

Let $E$ be an elliptic curve defined over a small finite field $\funnyF_q$, and let $p$ be a prime number. We give a conjectural formula for the probability that the order of the quotient group $E(\funnyF_{q^p})/E(\funnyF_q)$ is prime, and compare it with experimental data. The motivation for this study comes from public key cryptography.

We give explicit examples of infinite families of elliptic curves $E$ over $\funnyQ$ with (nonconstant) quadratic twists over $\funnyQ(t)$ of rank at least $2$ and $3$. We recover some results announced by Mestre, as well as some additional families. Suppose $D$ is a squarefree integer and let $r_E(D)$ denote the rank of the quadratic twist of $E$ by $D$. We apply results of Stewart and Top to our examples to obtain results of the form

{\#\{D : |D|<x, \, r_E(D) \ge 2\} \gg x^{1/3},

{\#\{D : |D|<x, \, r_E(D) \ge 3\} \gg x^{1/6}}

for all sufficiently large $x$.

We present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomorphisms, suitable for Jeff Weeks's program SnapPea. The package is an extension of the software described in Brinkmann [2000]. It consists of two programs. \jmt\ computes triangulations and prints them in a human-readable format. \jsnap\ converts this format into SnapPea's triangulation file format and may be of independent interest because it allows for quick and easy generation of input for SnapPea. As an application, we obtain a new solution to the restricted conjugacy problem in the mapping class group.

We investigate, through numerical experiments, the asymptotic behavior of the length $H_d(n)$ of a maximal chain (longest totally ordered subset) of a set of $n$ points drawn from a uniform distribution on the $d$-dimensional unit cube {\bf V}\math{_D=[0,1]^d}. For \math{d\ge2}, it is known that \math{c_d(n)=H_d(n)/n^{1/d}} converges in probability to a constant \math{c_d<e}, with \math{\lim_{d\rightarrow\infty} c_d=e}. For \math{d=2}, the problem has been extensively studied, and it is known that \math{c_2=2}; \math{c_d} is not currently known for any \math{d\ge3}. Straightforward Monte Carlo simulations to obtain \math{c_d} have already been proposed, and shown to be beyond the scope of current computational resources. In this paper, we present a computational approach which yields feasible experiments that lead to estimates for \math{c_d}. We prove that \math{H_d(n)} can be estimated by considering only those chains close to the diagonal of the cube. A new conjecture regarding the asymptotic behavior of \math{c_d(n)} leads to even more efficient experiments. We present experimental support for our conjecture, and the new estimates of \math{c_d} obtained from our experiments, for \math{d\in\{3,4,5,6\}}.

Max Koecher's axiomatic treatment of self-dual cones is generalized to pairs of dual cones. This leads to a powerful algorithm, à la Voronoi, to calculate the normalizer $N_{\GL_n(\funnyZ )}(G)$ and to decide conjugacy in $\GL_n(\funnyZ )$ for finite unimodular groups $G\leq \GL_n(\funnyZ )$.

Consider the operator

\frakd_Q(f)=\frac {d^k}{dx^k}(Q(x)f(x)),

where $Q(x)$ is some fixed polynomial of degree $k$. One can easily see that $\frakd_Q$ has exactly one polynomial eigenfunction $p_n(x)$ in each degree $n\ge 0$ and its eigenvalue $\la_{n,k}$ equals $(n+k)!/{n!}$. A more intriguing fact is that all zeros of $p_{n}(x)$ lie in the convex hull of the set of zeros to $Q(x)$. In particular, if $Q(x)$ has only real zeros then each $p_{n}(x)$ enjoys the same property. We formulate a number of conjectures on different properties of $p_{n}(x)$ based on computer experiments as, for example, the interlacing property, a formula for the asymptotic distribution of zeros etc. These polynomial eigenfunctions might be thought of as a generalization of the classical Gegenbauer polynomials with half-integer superscript, this case arising when our $Q(x)$ is an integer power of $x^2-1$.

We investigate in greater detail a sampling formula given by the first author for functions whose spectrum lies in a Cantor set $K$ of a special type introduced by Jorgensen and Pedersen, where the sampling set is extremely thin, and the sampling function is quite different from the usual sinc function. We obtain new properties of the sampling function, and we give approximate descriptions of both local and global behavior of functions with spectrum in $K$. Some experimental results are described, and more can be found at http://mathlab.cit.cornell.edu/tillman.