A basic framework is derived for the development of a higher-dimensional discrete function theory in a Clifford algebra context. The concept of a discrete monogenic function is introduced as a proper generalization of the discrete holomorphic, or monodiffric, functions introduced by Isaacs in the 1950s. A concrete model is provided for the definition of the corresponding discrete Dirac operator.

We discuss a comparison of the entropy of pseudo-Anosov maps and the volume of their mapping tori. Recent study of the Weil--Petersson geometry of Teichmüller space tells us that the entropy and volume admit linear inequalities for both directions under some bounded geometry condition. Based on experiments, we present various observations on the relation between minimal entropies and volumes, and on bounding constants for the entropy over the volume from below. We also provide explicit bounding constants for a punctured torus case.

We study several families of planar quadratic diffeomorphisms near a Bogdanov-Takens bifurcation. For each family, the skeleton of the associated bifurcation diagram can be deduced from the interpolating flow. However, a zone of chaos confined between two lines of homoclinic bifurcation that are exponentially close to one another is observed. The goal of this paper is to test numerically an accurate asymptotic expansion for the width of this chaotic zone for different families.

We determine the smallest possible regulator $R(P,Q)$ for a rank-2 subgroup $\mathbb{Z}P\oplus\mathbb{Z}Q$ of an elliptic curve $E$ over $\mathbb{C}(t)$ of discriminant degree $12n$ for $n=1$ (a rational elliptic surface) and $n=2$ (a $K3$ elliptic surface), exhibiting equations for all $(E,P,Q)$ attaining the minimum. The minimum $R(P,Q) = 1/36$ for a rational elliptic surface was known, but a formula for $(E,P,Q)$ was not, nor was the fact that this is the minimum for an elliptic curve of discriminant degree 12 over a function field of any genus. For a $K3$ surface, both the minimal regulator $R(P,Q)=1/100$ and the explicit equations are new. We also prove that 1/100 is the minimum for an elliptic curve of discriminant degree 24 over a function field of any genus. The optimal $(E,P,Q)$ are uniquely characterized by having $mP$ and $m'Q$ integral for $m\leq M$ and $m'\leq M'$, where $(M,M') = (3,3)$ for $n=1$ and $(M,M') = (6,3)$ for $n=2$. In each case $MM'$ is maximal. We use the connection with integral points to find explicit equations for the curves. As an application we use the $K3$ surface to produce, in a new way, the elliptic curves $E/\mathbb{Q}$ with nontorsion points of smallest known canonical height. These examples appeared previously in Noam D. Elkies. “Nontorsion Points of Low Height on Elliptic Curves over $mathbb{Q}$.”.

We present a new method to approximate the Neumann spectrum of a Laplacian on a fractal $K$ in the plane as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of domains that approximate $K$ from the outside. The method allows a numerical approximation of eigenvalues and eigenfunctions for lower portions of the spectrum. We present experimental evidence that the method works by looking at examples in which the spectrum of the fractal Laplacian is known (the unit interval and the Sierpiński gasket). We also present a speculative description of the spectrum on the standard Sierpiński carpet, where existence of a self-similar Laplacian is known, and also on nonsymmetric and random carpets and the octagasket, where existence of a self-similar Laplacian is not known. At present we have no explanation as to why the method should work. Nevertheless, we are able to prove some new results about the structure of the spectrum involving ``miniaturization'' of eigenfunctions that we discovered by examining the experimental results obtained using our method.

A Lissajous knot is one that can be parameterized as

$K(t)=\left (\cos (n_x t+\phi_x), \cos (n_y t+\phi_y), \cos (n_z t+\phi_z)\right )$,

where the frequencies $n_x$, $n_y$, and $n_z$ are relatively prime integers and the phase shifts $\phi_x$, $\phi_y$, and $\phi_z$ are real numbers. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems that allow us to place bounds on the number of Lissajous knot types with given frequencies and to efficiently sample all possible Lissajous knots with a given set of frequencies. In particular, we systematically tabulate all Lissajous knots with small frequencies and as a result substantially enlarge the tables of known Lissajous knots.

A Fourier-$(i, j, k)$ knot is similar to a Lissajous knot except that the $x$, $y$, and $z$ coordinates are now each described by a sum of $i$, $j$, and $k$ cosine functions, respectively. According to Lamm, every knot is a Fourier-$(1,1,k)$ knot for some $k$. By randomly searching the set of Fourier-$(1,1,2)$ knots we find that all 2-bridge knots with up to 14 crossings are either Lissajous or Fourier-$(1,1,2)$ knots. We show that all twist knots are Fourier-$(1,1,2)$ knots and give evidence suggesting that all torus knots are Fourier-$(1,1,2)$ knots.

As a result of our computer search, several knots with relatively small crossing numbers are identified as potential counterexamples to interesting conjectures.

We prove that the moduli space $C(d)$ of plane curves of degree $d$ (with respect to projective equivalence) is rational except possibly if $d= 6, 7, 8, 11, 12, 14, 15, 16, 18, 20, 23, 24, 26, 32, 48$.