Through computer enumeration with the aid of topological results, we catalogue all 18 closed nonorientable $\ppirr$ 3-manifolds that can be formed from eight or fewer tetrahedra. In addition, we give an overview as to how the 100 resulting minimal triangulations are constructed. Observations and conjectures are drawn from the census data, and future potential for the nonorientable census is discussed. Some preliminary 9-tetrahedron results are also included.

A symmetric subset of the reals is one that remains invariant under some reflection $x\mapsto c-x$. We consider, for any $0<\e \le 1$, the largest real number $\De$ such that every subset of $[0,1]$ with measure greater than $\e$ contains a symmetric subset with measure $\De$. In this paper we establish upper and lower bounds for $\De$ of the same order of magnitude: For example, we prove that $\De=2\e-1$ for $\frac{11}{16}\le\e\le1$ and that $0.59\e^2<\De<0.8\e^2$ for $0<\e\le\frac{11}{16}$.

This continuous problem is intimately connected with a corresponding discrete problem. A set $S$ of integers is called a $\Bg$ set if for any given $m$ there are at most $g$ ordered pairs $(s_1,s_2)\in S \times S$ with $s_1+s_2=m$; in the case $g=2$, these are better known as Sidon sets. Our lower bound on $\De$ implies that every $\Bg$ set contained in $\{1,2,\dotsc,n\}$ has cardinality less than $1.30036\sqrt{gn}$. This improves a result of Green for $g\ge 30$. Conversely, we use a probabilistic construction of $\Bg$ sets to establish an upper bound on $\De$ for small $\e$

The minimum dilatation of pseudo-Anosov 5-braids is shown to be the largest zero $\lambda_5 \approx 1.72208$ of $x^4 - x^3 - x^2 - x + 1$, which is attained by $\sigma_1\sigma_2\sigma_3\sigma_4\sigma_1\sigma_2$.

We propose a rigorous computational method to prove the uniform hyperbolicity of discrete dynamical systems. Applying the method to the real Hénon family, we prove the existence of many regions of hyperbolic parameters in the parameter plane of the family.

We give two examples of algebras of differential operators associated with families of matrix-valued orthogonal polynomials arising from representations of $\SU(N+1)$. The first gives a commutative algebra and the second a noncommutative one.

A base-$b$ pseudoprime (psp) is a composite $N$ satisfying $b^{N-1}\equiv 1\pmod N$. We use computer searches to count odd base-3 psp near $10^n$ for $n$ up to 19. The counts indicate that a good approximation to the probability of a random odd number near $z$ being a psp is $P(z)=z^{-0.59}$. Integrating $P$ yields a psp-counting function, $Q(x)=(x^{0.41})/0.82$, which gives estimated counts close to known actual counts up to 10$^{\mbox{\ASF 19}}$, although these estimates are probably not valid for all $x$.

A table comparing pseudoprime counts up to 10$^{\mbox{\ASF 11}}$ for bases 2, 3, 5, 7, 11, 13, 17, is included.

We study one of the central open questions in one-dimensional renormalization theory---the conjectural universality of golden-mean Siegel disks. We present an approach to the problem based on cylinder renormalization proposed by the second author. Numerical implementation of this approach relies on the constructive measurable Riemann mapping theorem proved by the first author. Our numerical study yields convincing evidence to support the hyperbolicity conjecture in this setting.

In this paper, we show that the Newton polytope of an observation $Y$ from a two-dimensional hidden Markov model\linebreak (2D HMM) lies in a three-dimensional subspace of its ambient eight-dimensional space, whose vertices correspond to the most likely explanations (``hidden'' states) for $Y$ given the model. For each Newton polytope, there exists a set of ``essential'' vertices, which form a skeleton for the polytope. All observations in the same orbit (identical under translations, rotations, and transpositions) have the same Newton polytope. Our main conjecture is that the maximal number of vertices of any Newton polytope is of order $n^2$

We present an algorithm for computing the dimensions of higher secant varieties of minimal orbits. Experiments with this algorithm lead to many conjectures on secant dimensions, especially of Grassmannians and Segre products. For these two classes of minimal orbits we give a short proof of the relation---known from the work of Ehrenborg, Catalisano--Geramita--Gimigliano, and Sturmfels--Sullivant---between the existence of certain codes and nondefectiveness of certain higher secant varieties.

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 $20$