1.
Computing Gröbner fans of toric ideals - Huber, Birkett; Thomas, Rekha R.
The monomial initial ideals of a graded polynomial ideal are in bijection with the vertices of a convex polytope known as the state polytope of the ideal. The Gröbner fan of the ideal is the normal fan of its state polytope. In this paper we present a software system called TiGERS (Toric Gröbner bases Enumeration by Reverse Search) for computing the Gröbner fan of a toric ideal by enumerating the edge graph of its state polytope. The key contributions are an inexpensive algorithm for local change of Gröbner bases in toric ideals and the identification of a reverse search tree...
2.
Two short presentations for Lyons' sporadic simple group - Gebhardt, Volker
Using an algorithm developed by the author, two new presentations for R. Lyons' sporadic simple group $\Ly$ are established, which contain fewer relations and are shorter than previously known ones.
3.
Integral geometry and real zeros of Thue-Morse polynomials - Doche, Christophe; France, Michel Mendès
We study the average number of intersecting points of a given curve with random hyperplanes in an $n$-dimensional Euclidean space. As noticed by A. Edelman and E. Kostlan, this problem is closely linked to finding the average number of real zeros of random polynomials. They showed that a real polynomial of degree $n$ has on average $\frac{2}{\pi}\log n +O(1)$ real zeros (M. Kac's theorem).
¶ This result leads us to the following problem: given a real sequence $(\alpha_k)_{k\in\N }$, study the average $$\frac{1}{N}\sum_{n=0}^{N-1} \rho(f_{n}),$$ where $\rho(f_n)$ is the number of real zeros of $f_n(X)=\alpha_0+\alpha_1X+\cdots+\alpha_nX^n$. We give theoretical results for the Thue-Morse...
4.
Computing Hecke eigenvalues below the cohomological dimension - Gunnells, Paul E.
Let $\funnyGamma$ be a torsion-free finite-index subgroup of $\SL_{n} (\Z )$ or $\GL_{n} (\Z )$, and let $\nu $ be the cohomological dimension of $\funnyGamma $. We present an algorithm to compute the eigenvalues of the Hecke operators on $H^{\nu -1} (\funnyGamma ;\Z )$, for n= 2, 3, and 4. In addition, we describe a modification of the modular symbol algorithm of Ash and Rudolph for computing Hecke eigenvalues on $H^{\nu } (\funnyGamma ;\Z )$.
5.
Nested squares and evaluations of integer products - Dilcher, Karl
The identity $$\medmuskip 0mu minus 2mu \bigl((x^2-85)^2@-@@4176\bigr)^2-2880^2=(x^2-@
1^2)\*(x^2-@ 7^2)\*(x^2-@ 11^2)\*(x^2-@ 13^2),$$ discovered by R. E. Crandall, allows the evaluation of a product
of 8 integers by a succession of 3 squares and 3 subtractions. The question arises whether there exist formulas like Crandall's with more than 3 nested squares. It will be shown that this is not the case; however, there are infinitely many formulas of length 3.
6.
Treating the exceptional cases of the MeatAxe - Ivanyos, Gábor; Lux, Klaus
We show that the Holt-Rees extension of the standard MeatAxe procedure finds submodules of modules over finite algebras with positive probability in more cases than originally claimed. For the case when the Holt-Rees method fails we propose a further, but still simple and efficient extension.
7.
Conjugacy classes of the hyperelliptic mapping class group of genus 2 and 3 - Ahara, Kazushi; Takasawa, Mitsuhiko
We present tables of conjugacy classes of the hyperelliptic mapping class group of genus 2 and 3, and some theorems on the Sp representation, the Jones representation, and Meyer's function.
8.
Infinite regular hexagon sequences on a triangle - Smith, Alvy Ray
The well-known dual pair of Napoleon equilateral triangles intrinsic to each triangle is extended to infinite sequences of them, shown to be special cases of infinite regular hexagon sequences on each triangle. A set of hexagon-to-hexagon transformations, the hex operators, is defined for this purpose, a set forming an abelian monoid under function composition. The sequences result from arbitrary strings of hex operators applied to a particular truncation of a given triangle to a hexagon. The deep structure of the sequence constructions reveals surprising infinite
sequences of nonconcentric, symmetric equilateral triangle pairs parallel to one of the sequences of hexagons and...
9.
Counting crystallographic groups in low dimensions - Plesken, Wilhelm; Schulz, Tilman
We present the results of our computations concerning the space groups of dimension 5 and 6. We find 222 018 and 28 927 922 isomorphism types of these groups, respectively. Some overall statistics on the number of $\funnyQ$-classes and $\funnyZ$-classes in dimensions up to six are provided. The computations were done with the package CARAT, which can parametrize, construct and identify all crystallographic groups up to dimension 6.
10.
On the dimensions of certain incommensurably constructed sets - Veerman, J. J. P.; Stoi?, B. D.
It is known that the Hausdorff dimension of the invariant set $\Lambda_t$ of an iterated function system ${\cal F}_t$ on $\R^n$ depending smoothly on a parameter $t$ varies lower-semicontinuously, but not necessarily continuously. For a specific family of systems we investigate numerically the conjecture that discontinuities in the dimension only arise when in some iterate of the iterated function system two or more branches coincide. This happens in a dense set of codimension one. All other points are conjectured to be points of continuity.
11.
Polynomials with height 1 and prescribed vanishing at 1 - Borwein, Peter; Mossinghoff, Michael J.
We study the minimal degree d(m) of a polynomial with all coefficients in $\{-1,0,1\}$ and a zero of order m at 1. We determine d(m) for $m\leq10$ and compute all the extremal polynomials. We also determine the minimal degree for $m=11$ and $m=12$ among certain symmetric polynomials, and we find explicit examples with small degree for $m\leq21$. Each of the extremal examples is a pure product polynomial. The method uses algebraic number theory and combinatorial computations and relies on showing that a polynomial with bounded degree, restricted coefficients, and a zero of high order at 1 automatically vanishes at several...
12.
Symplectic packings in cotangent bundles of tori - Maley, F. Miller; Mastrangeli, Jean; Traynor, Lisa
Finding optimal packings of a symplectic manifold with symplectic embeddings of balls is a well known problem. In the following, an alternate symplectic packing problem is explored where the target and domains are 2n-dimensional manifolds which have first homology group equal to $\funnyZ^n$ and the embeddings induce isomorphisms of first homology. When the target and domains are $\funnyT^n \times V$ and $\funnyT^n \times U$ in the cotangent bundle of the torus, all such symplectic packings give rise to packings of $V$ by copies of $U$ under $\GL(n,\funnyZ)$ and translations. For arbitrary dimensions, symplectic packing invariants are computed when packing a...
13.
The irreducible six-dimensional complex representations of ${\rm Aut}(F\sb 2)$ that are nontrivial on $F\sb 2$ - ?okovi?, Dragomir .
Denote by $\Phi_2$ the automorphism group of the free group $F_2$ on two generators. We classify the irreducible 6-dimensional complex representations of $\Phi_2$ whose restriction to $F_2$ is nontrivial. J. Dyer, E. Formanek, and E. Grossman have shown how the Bürau representation of the braid group $B_4$ gives rise to a one-parameter family of irreducible 6-dimensional representations of $\Phi_2$. The faithfulness question for these and some other closely related representations of $\Phi_2$ is open. Our classification shows that all other 6-dimensional representations of $\Phi_2$ are not faithful.
14.
On monochromatic arithmetic progressions having odd step - Jungic, Veselin
We perform an experiment concerning certain sets of 2-colorings and the existence of monochromatic arithmetic progressions of odd step. We use the results of the experiment to prove more general statements.
15.
Rational knots and a theorem of Kanenobu - Stoimenow, Alexander
We give examples where Kanenobu's necessary condition for the rationality of a knot is not sufficient, and show that such examples are atypical.
16.
On rational maps with two critical points - Milnor, John
This is a preliminary investigation of the geometry and dynamics of rational maps with only two critical points.
17.
Critical points of the Ginzburg-Landau functional on multiply-connected domains - Neuberger, J. W.; Renka, R. J.
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).
18.
Biases in the Shanks-Rényi prime number race - Feuerverger, Andrey; Martin, Greg
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...
19.
A test for identifying Fourier coefficients of automorphic forms and application to Kloosterman sums - Booker, Andrew R.
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.
20.
Ranks of elliptic curves in families of quadratic twists - Rubin, Karl; Silverberg, Alice
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.
1.
