This is a preliminary investigation of the geometry and dynamics of rational maps with only two critical points.

We give a numerical method for approximating critical points of the Ginzburg-Landau functional, and present test results in the form of plots of the corresponding electron densities, magnetic fields, and currents. Our domains include a rectangle, a rectangle with a rectangular hole in the center, and a rectangle with two rectangular holes. In each case, we found several critical points. The plots reveal interesting patterns, including the existence of counter-currents (adjacent currents in opposite directions).

Rubinstein and Sarnak investigated systems of inequalities of the form $\pi(x;q,a_1)>\cdots>\pi(x;q,a_r)$, where $\pi(x;q,b)$ denotes the number of primes up to x that are congruent to b mod q. They showed, under standard hypotheses on the zeros of Dirichlet L-functions mod q, that the set of positive real numbers x for which these inequalities hold has positive (logarithmic) density $\delta_{q;a_1,\dots,a_r}>0$. They also discovered the surprising fact that a certain distribution associated with these densities is not symmetric under permutations of the residue classes $a_j$ in general, even if the $a_j$ are all squares or all nonsquares mod q (a condition necessary to avoid obvious biases of the type first observed by Chebyshev). This asymmetry suggests, contrary to prior expectations, that the densities $\delta_{q;a_1,\dots,a_r}$ themselves vary under permutations of the $a_j$.

Here we derive (under the hypotheses used by Rubinstein and Sarnak) a general formula for the densities $\delta_{q;a_1,\dots,a_r}$, and we use this formula to calculate many of these densities when $q\le12$ and $r\le4$. For the special moduli q=8 and q=12, and for $\{a_1,a_2,a_3\}$ a permutation of the nonsquares {3,5,7} mod 8 and {5,7,11} mod12, respectively, we rigorously bound the error in our calculations, thus verifying that these densities are indeed asymmetric under permutation of the $a_j$. We also determine several situations in which the densities $\delta_{q;a_1,\dots,a_r}$ remain unchanged under certain permutations of the $a_j$, and some situations in which they are provably different.

We present a numerical test for determining whether a given set of numbers is the set of Fourier coefficients of a Maass form, without knowing its eigenvalue. Our method extends directly to consideration of holomorphic newforms. The test is applied to show that the Kloosterman sums $\pm S(1,1;p)\big/\hskip-1pt\sqrt p$ are not the coefficients of a Maass form with small level and eigenvalue. Source code and the calculated Kloosterman sums are available electronically.

We show that the unboundedness of the ranks of the quadratic twists of an elliptic curve is equivalent to the divergence of certain infinite series.

In a companion paper, Rubin and Silverberg relate the question of unboundedness of rank in families of quadratic twists of elliptic curves to the convergence or divergence of certain series. Here we give a computational application of their ideas on counting the rational points in such families; namely, to find curves of high rank in families of quadratic twists. We also observe that the algorithm seems to find as many curves of positive even rank as it does curves of odd rank. Results are given in the case of the congruent number elliptic curves, which are the quadratic twists of the curve $y^2 = x^3 - x$; for this family, the highest rank found is 6.

We use the Dorfmeister-Pedit-Wu construction to use three new classes of immersed CMC cylinders, each of which includes surfaces with umbilics. The first class consists of cylinders with one end asymptotic to a Delaunay surface. The second class presents surfaces with a closed planar geodesic. In the third class each surface has a closed curve of points with a common tangent plane. An appendix, by the third author, describes the DPW potentials that appear to give CMC punctured spheres with a k Delaunay ends (k-noids): the evidence is experiemental at present. These can have both unduloidal and nodoidal ends.

We consider some more or less classical hypersurfaces in projective space, known to be birational to a quotient of the unit ball in the corresponding dimension by an arithmetic subgroup. We are interested in understanding the intersection of each such hypersurface with its Hessian from the point of view of arithmetic groups. In addition to unifying certain results found previously in the literature, we compute for four of these hypersurfaces the Hessian as well as its intersection with the hypersurface.

Character-theoretic methods using Clifford algebras are developed to describe the quadratic forms on a vector space V that are invariant under a finite subgroup G of $\GL(V)$ such that the order of the commutator factor group of G is odd.

Let D be a positive integer such that D and $D{-}1$ are not perfect squares; denote by $X_0$, $Y_0$, $X_1$, $Y_1$ the least positive integers such that $X_0^2 - (D{-}1) Y_0^2 = 1$ and $X_1^2 - D Y_1^2 = 1$; and put $\rho(D) = \log X_1 / \log X_0$. We prove here that $\rho(D)$ can be arbitrarily large. Indeed, we exhibit an infinite family of values of D for which $\rho(D) \gg D^{1/6}/ \log D$. We also provide some heuristic reasoning which suggests that there exists an infinitude of values of D for which $\rho(D) \gg \sqrt{D} \log \log D / \log D$, and that this is the best possible result under the Extended Riemann Hypothesis. Finally, we present some numerical evidence in support of this heuristic.