Experimental Mathematics Articles (Project Euclid)
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The latest articles from Experimental Mathematics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Table of Contents, Experiment. Math., vol. 19, no. 1 (201009)
http://projecteuclid.org/euclid.em/1268404800
<p><strong>Source: </strong>Experiment. Math., Volume 19, Number 1.</p>projecteuclid.org/euclid.em/1268404800_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTOn the Periodized Square of L 2 Cardinal Splineshttp://projecteuclid.org/euclid.em/1317924409<strong>E. Giné</strong>, <strong>C. S. Güntürk</strong>, <strong>W. R. Madych</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 2, 177--188.</p><p><strong>Abstract:</strong><br/>
We establish properties of and propose a conjecture concerning
$\sum_m (S(x + m))2$, where $S$ is a piecewise polynomial cardinal
spline in $L^2(\mathbb{R})$.
</p>projecteuclid.org/euclid.em/1317924409_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTGeneralized Gorshkov–Wirsing Polynomials and the Integer Chebyshev Problemhttp://projecteuclid.org/euclid.em/1317924410<strong>Kevin G. Hare</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 2, 189--200.</p><p><strong>Abstract:</strong><br/>
The integer Chebyshev problem is the problem of finding an
integer polynomial of degree n such that the supremum norm
on [0, 1] is minimized. The most common technique used to
find upper bounds is by explicit construction of an example.
This is often (although not always) done by heavy computational
use of the LLL algorithm and simplex method. One of the first
methods developed to find lower bounds employed a sequence
of polynomials known as the Gorshkov–Wirsing polynomials.
This paper studies properties of the Gorshkov–Wirsing polynomials.
It is shown how to construct generalized Gorshkov–Wirsing
polynomials on any interval $[a, b]$, with $a, b ∈ \mathbb{Q}$. An extensive
search for generalized Gorshkov–Wirsing polynomials is carried
out for a large family of $[a, b]$. Using generalized Gorshkov–
Wirsing polynomials, LLL, and the simplex method, upper and
lower bounds for the integer Chebyshev constant on intervals
other than $[0, 1]$ are calculated. These methods are compared
with other existing methods.
</p>projecteuclid.org/euclid.em/1317924410_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTCombinatorial Properties of the $K^3$ Surface: Simplicial Blowups and Slicingshttp://projecteuclid.org/euclid.em/1317924411<strong>Jonathan Spreer</strong>, <strong>Wolfgang Kühnel</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 2, 201--216.</p><p><strong>Abstract:</strong><br/>
The 4-dimensional abstract Kummer variety $K^4$ with 16 nodes
leads to the $K^3$ surface by resolving the 16 singularities. Here we
present a simplicial realization of this minimal resolution. Starting
with a minimal 16-vertex triangulation of $K^4$, we resolve its
16 isolated singularities—step by step—by simplicial blowups.
As a result we obtain a 17-vertex triangulation of the standard
PL $K^3$ surface. A key step is the construction of a triangulated
version of the mapping cylinder of the Hopf map from real projective
3-space onto the 2-sphere with the minimum number of
vertices. Moreover, we study simplicial Morse functions and the
changes of their levels between the critical points. In this way
we obtain slicings through the $K^3$ surface of various topological
types.
</p>projecteuclid.org/euclid.em/1317924411_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTFinite Symplectic Matrix Groupshttp://projecteuclid.org/euclid.em/1317924412<strong>Markus Kirschmer</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 2, 217--228.</p><p><strong>Abstract:</strong><br/>
This paper classifies the maximal finite subgroups of ${\rm Sp}_{2n}(\mathbb{Q})$ for
$1 ≤ n ≤ 11$ up to conjugacy in ${\rm GL}_{2n}(\mathbb{Q})$.
</p>projecteuclid.org/euclid.em/1317924412_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTEdge-Graph Diameter Bounds for Convex Polytopes with Few Facetshttp://projecteuclid.org/euclid.em/1317924417<strong>David Bremner</strong>, <strong>Lars Schewe</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 3, 229--237.</p><p><strong>Abstract:</strong><br/>
We show that the edge graph of a 6-dimensional polytope with
12 facets has diameter at most 6, thus verifying the $d$-step conjecture
of Klee and Walkup in the case $d = 6$. This implies that
for all pairs $(d, n)$ with $n − d ≤ 6$, the diameter of the edge graph
of a $d$-polytope with $n$ facets is bounded by 6, which proves
the Hirsch conjecture for all $n − d ≤ 6$. We prove this result by
establishing this bound for a more general structure, so-called
matroid polytopes, by reduction to a small number of satisfiability
problems.
</p>projecteuclid.org/euclid.em/1317924417_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTOrthogonal Polynomials with Respect to Self-Similar Measureshttp://projecteuclid.org/euclid.em/1317924418<strong>Steven M. Heilman</strong>, <strong>Philip Owrutsky</strong>, <strong>Robert S. Strichartz</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 3, 238--259.</p><p><strong>Abstract:</strong><br/>
We study experimentally systems of orthogonal polynomialswith
respect to self-similar measures. When the support of the measure
is a Cantor set, we observe some interesting properties of the
polynomials, both on the Cantor set and in the gaps of the Cantor
set. We introduce an effective method to visualize the graph of
a function on a Cantor set. We suggest a new perspective, based
on the theory of dynamical systems, for studying families $P_n(x)$
of orthogonal functions as functions of $n$ for fixed values of $x$.
</p>projecteuclid.org/euclid.em/1317924418_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTApproximation of Partitions of Least Perimeter by Γ-Convergence: Around Kelvin’s Conjecturehttp://projecteuclid.org/euclid.em/1317924419<strong>Édouard Oudet</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 3, 260--270.</p><p><strong>Abstract:</strong><br/>
A numerical process to approximate optimal partitions in any
dimension is reported. The key idea of the method is to relax the
problem into a functional framework based on the famous result
of Γ-convergence obtained by Modica and Mortolla.
</p>projecteuclid.org/euclid.em/1317924419_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTOn the Computation of Class Polynomials with “Thetanullwerte” and Its Applications to the Unit Group Computationhttp://projecteuclid.org/euclid.em/1317924420<strong>Franck Leprévost</strong>, <strong>Michael Pohst</strong>, <strong>Osmanbey Uzunkol</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 3, 271--281.</p><p><strong>Abstract:</strong><br/>
The classical class invariants of Weber are introduced as quotients
of Thetanullwerte, enabling the computation of these
invariants more efficiently than as quotients of values of the
Dedekind η-function. We show also how to compute the unit
group of suitable ring class fields by proving the fact that most
of the invariants introduced by Weber are actually units in the
corresponding ring class fields.
</p>projecteuclid.org/euclid.em/1317924420_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTSome Dimensions of Spaces of Finite Type Invariants of Virtual Knotshttp://projecteuclid.org/euclid.em/1317924421<strong>Dror Bar-Natan</strong>, <strong>Iva Halacheva</strong>, <strong>Louis Leung</strong>, <strong>Fionntan Roukema</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 3, 282--287.</p><p><strong>Abstract:</strong><br/>
We compute many dimensions of spaces of finite type invariants
of virtual knots (of several kinds) and the dimensions of the
corresponding spaces of “weight systems,” finding everything to
be in agreement with the conjecture that “every weight system
integrates.”
</p>projecteuclid.org/euclid.em/1317924421_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTResolving Toric Varieties with Nash Blowupshttp://projecteuclid.org/euclid.em/1317924422<strong>Atanas Atanasov</strong>, <strong>Christopher Lopez</strong>, <strong>Alexander Perry</strong>, <strong>Nicholas Proudfoot</strong>, <strong>Michael Thaddeus</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 3, 288--303.</p><p><strong>Abstract:</strong><br/>
It is a long-standing questionwhether an arbitrary variety is desingularized
by finitely many normalized Nash blowups. We consider
this question in the case of a toric variety. We interpret the
normalized Nash blowup in polyhedral terms, show how continued
fractions can be used to give an affirmative answer for a toric
surface, and report on a computer investigation in which over a
thousand 3- and 4-dimensional toric varieties were successfully
resolved.
</p>projecteuclid.org/euclid.em/1317924422_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTNumerical Analysis of Nodal Sets for Eigenvalues of Aharonov–Bohm Hamiltonians on the Square with Application to Minimal Partitionshttp://projecteuclid.org/euclid.em/1317924423<strong>V. Bonnaillie-Noël</strong>, <strong>B. Helffer</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 3, 304--322.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to presenting numerical simulations and a
theoretical interpretation of results for determining the minimal
$k$-partitions of a domain $\Omega$ as considered in Helffer et al., Nodal Domains and Spectral Minimal Partitions .
More precisely, using the double-covering approach introduced
by B. Helffer, M. and T. Hoffmann-Ostenhof, and M. Owen
and further developed for questions of isospectrality by the authors
in collaboration with T. Hoffmann-Ostenhof and S. Terracini
in Helffer et al., and Bonnaillie-Noël et al. Aharonov–Bohm Hamiltonians, Isospectrality and Minimal Partitions.” , we analyze
the variation of the eigenvalues of the one-pole Aharonov–
Bohm Hamiltonian on the square and the nodal picture of
the associated eigenfunctions as a function of the pole. This
leads us to discover new candidates for minimal k-partitions
of the square with a specific topological type and without
any symmetric assumption, in contrast to our previous works. This illustrates
also recent results of B. Noris and S. Terracini; see
Noris and Terracini 10, Nodal Sets of Magnetic Schrödinger Operators of Aharonov–Bohm Type and Energy Minimizing Partitions” . This finally supports or disproves conjectures
for the minimal 3- and 5-partitions on the square.
</p>projecteuclid.org/euclid.em/1317924423_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTA Quartic System with Twenty-Six Limit Cycleshttp://projecteuclid.org/euclid.em/1317924424<strong>Tomas Johnson</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 3, 323--328.</p><p><strong>Abstract:</strong><br/>
We construct a planar quartic system and demonstrate that it
has at least 26 limit cycles. The vector field is symmetric and
integrable, but non-Hamiltonian. The proof is based on a verified
computation of zeros of pseudo-Abelian integrals, together with
the symmetry properties.
</p>projecteuclid.org/euclid.em/1317924424_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTAmicable Pairs and Aliquot Cycles for Elliptic Curveshttp://projecteuclid.org/euclid.em/1317924425<strong>Joseph H. Silverman</strong>, <strong>Katherine E. Stange</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 3, 329--357.</p><p><strong>Abstract:</strong><br/>
An amicable pair for an elliptic curve $E/\mathbb{Q}$ is a pair of primes
$(p, q)$ of good reduction for $E \sharp\tilde{E}_p(\mathbb{F}_p) = q$ and
$\sharp\tilde{E}_q(\mathbb{F}q) = p$.
In this paper we study elliptic amicable pairs and analogously
defined longer elliptic aliquot cycles . We show that there exist
elliptic curves with arbitrarily long aliquot cycles, but that
CM elliptic curves (with $j \neq 0$) have no aliquot cycles of length
greater than two.We give conjectural formulas for the frequency
of amicable pairs. For CM curves, the derivation of precise conjectural
formulas involves a detailed analysis of the values of
the Grössencharacter evaluated at primes $\mathfrak{p}$ in $\operatorname{End}(E )$ having the
property that $\sharp\tilde{E}_{\mathfrak{p}}(\mathbb{F}_{\mathfrak{p}})$ is prime. This is especially intricate for the
family of curves with $j = 0$.
</p>projecteuclid.org/euclid.em/1317924425_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTToward a Salmon Conjecturehttp://projecteuclid.org/euclid.em/1317924426<strong>Daniel J. Bates</strong>, <strong>Luke Oeding</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 3, 358--370.</p><p><strong>Abstract:</strong><br/>
Methods from numerical algebraic geometry are applied in combination
with techniques from classical representation theory to
show that the variety of 3 × 3 × 4 tensors of border rank 4 is
cut out by polynomials of degree 6 and 9. Combined with results
of Landsberg and Manivel, this furnishes a computational
solution of an open problem in algebraic statistics, namely, the
set-theoretic version of Allman’s salmon conjecture for 4 × 4 × 4
tensors of border rank 4. A proof without numerical computation
was given recently by Friedland and Gross.
</p>projecteuclid.org/euclid.em/1317924426_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTCorrigendumhttp://projecteuclid.org/euclid.em/1317924427<p><strong>Source: </strong>Experiment. Math., Volume 20, Number 3, 371--.</p>projecteuclid.org/euclid.em/1317924427_Thu, 06 Oct 2011 14:06 EDTThu, 06 Oct 2011 14:06 EDTRational Points on Some Fano Quadratic Bundleshttp://projecteuclid.org/euclid.em/1323367152<strong>Andreas-Stephan Elsenhans</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 4, 373--379.</p><p><strong>Abstract:</strong><br/>
We study the number of rational points of bounded height on
a certain threefold. The accumulating subvarieties are Zariski
dense in this example. The computations support an extension
of a conjecture of Manin to this situation.
</p>projecteuclid.org/euclid.em/1323367152_Thu, 08 Dec 2011 12:59 ESTThu, 08 Dec 2011 12:59 ESTSome Experiments with Integral Apollonian Circle Packingshttp://projecteuclid.org/euclid.em/1323367153<strong>Elena Fuchs</strong>, <strong>Katherine Sanden</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 4, 380--399.</p><p><strong>Abstract:</strong><br/>
Bounded Apollonian circle packings (ACPs) are constructed by
repeatedly inscribing circles into the triangular interstices of a
Descartes configuration of four mutually tangent circles, one of
which is internally tangent to the other three. If the original four
circles have integer curvature, all of the circles in the packing will
have integer curvature as well. In "Letter to Lagarias," Sarnak proves that
there are infinitely many circles of prime curvature and infinitely
many pairs of tangent circles of prime curvature in a primitive
integral ACP. (A primitive integral ACP is one in which no integer
greater than 1 divides the curvatures of all of the circles in the
packing.) In this paper, we give a heuristic backed up by numerical
data for the number of circles of prime curvature less than x
and the number of "kissing primes," or pairs of circles of prime
curvature less than x , in a primitive integral ACP.We also provide
experimental evidence toward a local-to-global principle for the
curvatures in a primitive integral ACP.
</p>projecteuclid.org/euclid.em/1323367153_Thu, 08 Dec 2011 12:59 ESTThu, 08 Dec 2011 12:59 ESTFrequencies of Successive Pairs of Prime Residueshttp://projecteuclid.org/euclid.em/1323367154<strong>Avner Ash</strong>, <strong>Laura Beltis</strong>, <strong>Robert Gross</strong>, <strong>Warren Sinnott</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 4, 400--411.</p><p><strong>Abstract:</strong><br/>
We consider statistical properties of the sequence of ordered
pairs obtained by taking the sequence of prime numbers and
reducing modulo $m$. Using an inclusion/exclusion argument and
a cutoff of an infinite product suggested by Pólya, we obtain a
heuristic formula for the "probability" that a pair of consecutive
prime numbers of size approximately x will be congruent to
$(a, a + d) modulo m$. We demonstrate some symmetries of our
formula. We test our formula and some of its consequences
against data for x in various ranges.
</p>projecteuclid.org/euclid.em/1323367154_Thu, 08 Dec 2011 12:59 ESTThu, 08 Dec 2011 12:59 ESTSmall Subgroups of ${\sf SL}(3, \mathbb{Z})$http://projecteuclid.org/euclid.em/1323367155<strong>D. D. Long</strong>, <strong>A. W. Reid</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 4, 412--425.</p><p><strong>Abstract:</strong><br/>
Motivated in part by various questions of Serre, Labourie and
Lubotzky, we consider the question of representing the fundamental
group of the figure eight knot complement into ${\sf SL}(3, \mathbb{Z})$.
We explore questions of faithfulness and finite index for such
representations.
</p>projecteuclid.org/euclid.em/1323367155_Thu, 08 Dec 2011 12:59 ESTThu, 08 Dec 2011 12:59 ESTNumerical Evidence for the Equivariant Birch and Swinnerton-Dyer Conjecturehttp://projecteuclid.org/euclid.em/1323367156<strong>Werner Bley</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 4, 426--456.</p><p><strong>Abstract:</strong><br/>
Let $E/ \mathbb{Q}$ be an elliptic curve and $K/ \mathbb{Q}$
a finite Galois extension
with group $G$. We write $E_K$ for the base change of $E$ and consider
the equivariant Tamagawa number conjecture for the pair
$(h^1(E_K )(1), \mathbb{Z}[G])$. This conjecture is an equivariant refinement of
the Birch and Swinnerton-Dyer conjecture for $E/K$. For almost
all primes $l$, we derive an explicit formulation of the conjecture
that makes it amenable to numerical verifications. We use
this to provide convincing numerical evidence in favor of the
conjecture.
</p>projecteuclid.org/euclid.em/1323367156_Thu, 08 Dec 2011 12:59 ESTThu, 08 Dec 2011 12:59 ESTEquivalence Classes for the μ-Coefficient of Kazhdan–Lusztig Polynomials in $S_n$http://projecteuclid.org/euclid.em/1323367157<strong>Gregory S. Warrington</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 4, 457--466.</p><p><strong>Abstract:</strong><br/>
We study equivalence classes relating to the Kazhdan–Lusztig
$μ(x,w)$ coefficients in order to help explain the scarcity of distinct
values. Each class is conjectured to contain a “crosshatch”
pair. We also compute the values attained by $μ(x,w)$ for the
permutation groups $S_10$ and $S_11$.
</p>projecteuclid.org/euclid.em/1323367157_Thu, 08 Dec 2011 12:59 ESTThu, 08 Dec 2011 12:59 ESTCensus of the Complex Hyperbolic Sporadic Triangle Groupshttp://projecteuclid.org/euclid.em/1323367158<strong>Martin Deraux</strong>, <strong>John R. Parker</strong>, <strong>Julien Paupert</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 4, 467--386.</p><p><strong>Abstract:</strong><br/>
The goal of this paper is to give a conjectural census of complex
hyperbolic sporadic triangle groups. We prove that only finitely
many of these sporadic groups are lattices.
We also give a conjectural list of all lattices among sporadic
groups, and for each group in the list we give a conjectural
group presentation, as well as a list of cusps and generators
for their stabilizers. We describe strong evidence for these conjectural
statements, showing that their validity depends on the
solution of reasonably small systems of quadratic inequalities in
four variables.
</p>projecteuclid.org/euclid.em/1323367158_Thu, 08 Dec 2011 12:59 ESTThu, 08 Dec 2011 12:59 ESTOn the Integral Cohomology of Bianchi Groupshttp://projecteuclid.org/euclid.em/1323367159<strong>Mehmet Haluk Şengün</strong><p><strong>Source: </strong>Experiment. Math., Volume 20, Number 4, 487--505.</p><p><strong>Abstract:</strong><br/>
Extensive and systematic machine computations are carried out
to investigate the integral cohomology of the Euclidean Bianchi
groups and their congruence subgroups. The collected data give
insight into several aspects, including the asymptotic behavior
of the torsion in the first homology. Along with the experimental
work, some basic properties of the integral cohomology are
recorded with an eye toward the liftability issue of Hecke eigenvalue
systems.
</p>projecteuclid.org/euclid.em/1323367159_Thu, 08 Dec 2011 12:59 ESTThu, 08 Dec 2011 12:59 EST<link>http://projecteuclid.org/euclid.em/1338430809</link><description><strong>Reza Rezaeian Farashahi</strong>, <strong>Igor E. Shparlinksi</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 1, 1--10.</p><p><strong>Abstract:</strong><br/>
We obtain explicit formulas for the number of nonisomorphic
elliptic curves with a given group structure (considered as an abstract
abelian group) and the number of distinct group structures
of all elliptic curves over a finite field. We use these formulas
to derive some asymptotic estimates and tight upper and lower
bounds for various counting functions related to classification
of elliptic curves according to their group structure. Finally, we
present results of some numerical tests that exhibit several interesting
phenomena in the distribution of group structures.
</p></description><guid isPermaLink="false">projecteuclid.org/euclid.em/1338430809_Wed, 30 May 2012 22:20 EDT</guid><pubDate>Wed, 30 May 2012 22:20 EDT</pubDate></item><item><title>On Group Structures Realized by Elliptic Curves over Arbitrary Finite Fieldshttp://projecteuclid.org/euclid.em/1338430810<strong>William D. Banks</strong>, <strong>Francesco Pappalardi</strong>, <strong>Igor E. Shparlinski</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 1, 11--25.</p><p><strong>Abstract:</strong><br/>
We study the collection of group structures that can be realized
as a group of rational points on an elliptic curve over a finite field
(such groups are well known to be of rank at most two). We also
study various subsets of this collection that correspond to curves
over prime fields or to curves with a prescribed torsion. Some
of our results are rigorous and are based on recent advances
in analytic number theory; some are conditional under certain
widely believed conjectures; and others are purely heuristic in
nature.
</p>projecteuclid.org/euclid.em/1338430810_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTSelf-Intersection Numbers of Curves in the Doubly Punctured Planehttp://projecteuclid.org/euclid.em/1338430811<strong>Moira Chas</strong>, <strong>Anthony Phillips</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 1, 26--37.</p><p><strong>Abstract:</strong><br/>
We address the problem of computing bounds for the selfintersection
number (the minimum number of generic self intersection
points) of members of a free homotopy class of
curves in the doubly punctured plane as a function of their
combinatorial length $L$ ; this is the number of letters required
for a minimal description of the class in terms of a set of standard
generators of the fundamental group and their inverses.
We prove that the self-intersection number is bounded above
by $L^2/4 + L/2 − 1$, and that when $L$ is even, this bound is
sharp; in that case, there are exactly four distinct classes attaining
that bound. For odd L we conjecture a smaller upper bound,
$(L^2 − 1)/4$, and establish it in certain cases in which we show
that it is sharp. Furthermore, for the doubly punctured plane,
these self-intersection numbers are bounded below, by $L/2 − 1$
if $L$ is even, and by $(L − 1)/2$ if $L$ is odd. These bounds are sharp.
</p>projecteuclid.org/euclid.em/1338430811_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTThe Chebotarev Invariant of a Finite Grouphttp://projecteuclid.org/euclid.em/1338430812<strong>Emmanuel Kowalski</strong>, <strong>David Zywina</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 1, 38--56.</p><p><strong>Abstract:</strong><br/>
We consider invariants of a finite group related to the number of
random (independent, uniformly distributed) conjugacy classes
that are required to generate it. These invariants are intuitively
related to problems of Galois theory.We find group-theoretic expressions
for them and investigate their values both theoretically
and numerically.
</p>projecteuclid.org/euclid.em/1338430812_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTSmooth Structures on Eschenburg Spaces: Numerical Computationshttp://projecteuclid.org/euclid.em/1338430813<strong>Leo T. Butler</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 1, 57--64.</p><p><strong>Abstract:</strong><br/>
This paper numerically computes the topological and smooth
invariants of Eschenburg–Kruggel spaces with small fourth cohomology
group, following Kruggel’s determination of the Kreck–Stolz invariants of Eschenburg spaces satisfying condition C. It is shown that
each topological Eschenburg–Kruggel space with small fourth
cohomology group has each of its 28 oriented smooth structures
represented by an Eschenburg–Kruggel space. Our investigations
also suggest that there is an action of $\mathbb{Z}_{12}$ on the set of homotopy
classes of Eschenburg–Kruggel spaces, the nature of which
remains to be understood.
The calculations are done in C++ with the GNU GMP arbitrary precision
library and Jon Wilkening’s C++ wrapper.
</p>projecteuclid.org/euclid.em/1338430813_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTMosaic Supercongruences of Ramanujan Typehttp://projecteuclid.org/euclid.em/1338430814<strong>Jesús Guillera</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 1, 65--68.</p><p><strong>Abstract:</strong><br/>
In this article, we present analogues of supercongruences of Ramanujan
type observed by L. Van Hamme and W. Zudilin. Our
congruences are inspired by Ramanujan-type series that involve
quadratic algebraic numbers.
</p>projecteuclid.org/euclid.em/1338430814_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTCertified Numerical Homotopy Trackinghttp://projecteuclid.org/euclid.em/1338430815<strong>Carlos Beltrán</strong>, <strong>Anton Leykin</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 1, 69--83.</p><p><strong>Abstract:</strong><br/>
Given a homotopy connecting two polynomial systems, we provide
a rigorous algorithm for tracking a regular homotopy path
connecting an approximate zero of the start system to an approximate
zero of the target system. Our method uses recent
results on the complexity of homotopy continuation rooted in
the alpha theory of Smale. Experimental results obtained with
an implementation in the numerical algebraic geometry package
Macaulay2 demonstrate the practicality of the algorithm. In
particular, we confirm the theoretical results for random linear
homotopies and illustrate the plausibility of a conjecture by Shub
and Smale on a good initial pair.
</p>projecteuclid.org/euclid.em/1338430815_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTThe Sato–Tate Distribution and the Values of Fourier Coefficients of Modular Newformshttp://projecteuclid.org/euclid.em/1338430816<strong>Josep González</strong>, <strong>Jorge Jiménez-Urroz</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 1, 84--102.</p><p><strong>Abstract:</strong><br/>
The Sato–Tate conjecture has been recently settled in great generality.
One natural question now concerns the rate of convergence
of the distribution of the Fourier coefficients of modular
newforms to the Sato–Tate distribution. In this paper, we address
this issue, imposing congruence conditions on the primes and on
the Fourier coefficients as well. Assuming a proper error term in
the convergence to a conjectural limiting distribution, supported
by experimental data, we prove the Lang–Trotter conjecture, and
in the direction of Lehmer’s conjecture, we prove that $\tau (p) = 0$
has at most finitely many solutions. In fact, we propose a conjecture,
much more general than Lehmer’s, about the vanishing
of Fourier coefficients of any modular newform.
</p>projecteuclid.org/euclid.em/1338430816_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTA Local Version of Szpiro’s Conjecturehttp://projecteuclid.org/euclid.em/1338430824<strong>Michael A. Bennett</strong>, <strong>Soroosh Yazdani</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 2, 103--116.</p><p><strong>Abstract:</strong><br/>
Szpiro’s conjecture asserts the existence of an absolute constant
$K \gt 6$ such that if $E$ is an elliptic curve over $\mathbb{Q}$, the
minimal discriminant $\Delta(E)$ of $E$ is bounded above in modulus
by the $K$th power of the conductor $N(E)$ of $E$ . An immediate
consequence of this is the existence of an absolute upper
bound on $\min\{v_p(\Delta(E )) : p |\Delta(E )\}$. In this paper, we will prove
this local version of Szpiro’s conjecture under the (admittedly
strong) additional hypotheses that $N(E)$ is divisible by a “large”
prime $p$ and that $E$ possesses a nontrivial rational isogeny. We
will also formulate a related conjecture that if true, we prove
to be sharp. Our construction of families of curves for which
$\min{v_p(\Delta(E)) : p | \Delta(E )} \ge 6$ provides an alternative proof of a
result of Masser on the sharpness of Szpiro’s conjecture.We close
the paper by reporting on recent computations of examples of
curves with large Szpiro ratio.
</p>projecteuclid.org/euclid.em/1338430824_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTComputation of Harmonic Weak Maass Formshttp://projecteuclid.org/euclid.em/1338430825<strong>Jan H. Bruinier</strong>, <strong>Fredrik Strömberg</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 2, 117--131.</p><p><strong>Abstract:</strong><br/>
Harmonic weak Maass forms of half-integral weight have been
the subject of much recent work. They are closely related to
Ramanujan’s mock theta functions, and their theta lifts give rise
to Arakelov Green functions, and their coefficients are often related
to central values and derivatives of Hecke L-functions. We
present an algorithm to compute harmonic weak Maass forms
numerically, based on the automorphymethod due to Hejhal and
Stark. As explicit examples we consider harmonic weak Maass
forms of weight 1/2 associated to the elliptic curves 11a1, 37a1,
37b1. We have made extensive numerical computations, and
the data we obtained are presented in this paper. We expect
that experiments based on our data will lead to a better understanding
of the arithmetic properties of the Fourier coefficients
of harmonic weak Maass forms of half-integral weight.
</p>projecteuclid.org/euclid.em/1338430825_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTOn Repeated Values of the Riemann Zeta Function on the Critical Linehttp://projecteuclid.org/euclid.em/1338430826<strong>William D. Banks</strong>, <strong>Sarah Kang</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 2, 132--140.</p><p><strong>Abstract:</strong><br/>
Let $\zeta (s)$ be the Riemann zeta function. In this paper, we study repeated
values of $\zeta (s)$ on the critical line, and we give evidence to
support our conjecture that for every nonzero complex number
$z$, the equation $\zeta (1/2 + i t) = z$ has at most two solutions $t \in R$.
We prove a number of related results, some of which are unconditional,
and some of which depend on the truth of the Riemann
hypothesis. We also propose some related conjectures that are
implied by Montgomery’s pair correlation conjecture.
</p>projecteuclid.org/euclid.em/1338430826_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTComputational Approaches to Poisson Traces Associated to Finite Subgroups of ${\rm Sp}_2(\mathbb{C})$http://projecteuclid.org/euclid.em/1338430827<strong>Pavel Etingof</strong>, <strong>Sherry Gong</strong>, <strong>Aldo Pacchiano</strong>, <strong>Qingchun Ren</strong>, <strong>Travis Schedler</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 2, 141--170.</p><p><strong>Abstract:</strong><br/>
We reduce the computation of Poisson traces on quotients of
symplectic vector spaces by finite subgroups of symplectic automorphisms
to a finite one by proving several results that bound
the degrees of such traces as well as the dimension in each
degree. This applies more generally to traces on all polynomial
functions that are invariant under invariant Hamiltonian flow.We
implement these approaches by computer together with direct
computation for infinite families of groups, focusing on complex
reflection and abelian subgroups of ${\rm GL}_2(\mathbb{C}) \lt {\rm Sp}_4(\mathbb{C})$, Coxeter
groups of rank $\le 3$ and types $A_4$, $B_4 = C_4$, and $D_4$, and subgroups
of ${\rm SL}_2(\mathbb{C})$.
</p>projecteuclid.org/euclid.em/1338430827_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTSemistable Vector Bundles and Tannaka Duality from a Computational Point of Viewhttp://projecteuclid.org/euclid.em/1338430828<strong>Almar Kaid</strong>, <strong>Ralf Kasprowitz</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 2, 171--188.</p><p><strong>Abstract:</strong><br/>
We develop a semistability algorithm for vector bundles that are
given as a kernel of a surjective morphism between splitting bundles
on the projective space $\mathbb{P}^N$ over an algebraically closed field
$K$ . This class of bundles is a generalization of syzygy bundles.
We show how to implement this algorithm in a computer algebra
system. Further, we give applications, mainly concerning the
computation of Tannaka dual groups of stable vector bundles of
degree 0 on $\mathbb{P}^N$ and on certain smooth complete intersection
curves. We also use our algorithm to close a case left open in
a recent work of L. Costa, P. Macias Marques, and R. M. Miró-
Roig regarding the stability of the syzygy bundle of general forms.
Finally, we apply our algorithm to provide a computational approach
to tight closure. All algorithms are implemented in the
computer algebra system CoCoA.
</p>projecteuclid.org/euclid.em/1338430828_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTA Markov Chain on the Symmetric Group That Is Schubert Positive?http://projecteuclid.org/euclid.em/1338430829<strong>Thomas Lam</strong>, <strong>Lauren Williams</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 2, 189--192.</p><p><strong>Abstract:</strong><br/>
We study a multivariate Markov chain on the symmetric group
with remarkable enumerative properties. We conjecture that the
stationary distribution of this Markov chain can be expressed
in terms of positive sums of Schubert polynomials. This Markov
chain is a multivariate generalization of a Markov chain introduced
by the first author in the study of random affine Weyl
group elements.
</p>projecteuclid.org/euclid.em/1338430829_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTExtended Torelli Map to the Igusa Blowup in Genus 6, 7, and 8http://projecteuclid.org/euclid.em/1338430830<strong>Valery Alexeev</strong>, <strong>Ryan Livingston</strong>, <strong>Joseph Tenini</strong>, <strong>Maxim Arap</strong>, <strong>Xiaoyan Hu</strong>, <strong>Lauren Huckaba</strong>, <strong>Patrick McFaddin</strong>, <strong>Stacy Musgrave</strong>, <strong>Jaeho Shin</strong>, <strong>Catherine Ulrich</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 2, 193--203.</p><p><strong>Abstract:</strong><br/>
It was conjectured in Yukihiko Namikawa, “On the Canonical Holomorphic
Map from the Moduli Space of Stable Curves
to the Igusa Monoidal Transform,” that the Torelli map $M_g \to
A_g$ associating to a curve its Jacobian extends to a regular map
from the Deligne–Mumford moduli space of stable curves $\bar{M}_g$ to
the (normalization of the) Igusa blowup $\bar{A}^{\rm cent}_g$. A counterexample
in genus $g = 9$ was found in Valery Alexeev and Adrian Brunyate,
“Extending Torelli Map to Toroidal Compactifications
of Siegel Space.” Here, we prove that the extended map is regular for all $g \le 8$, thus
completely solving the problem in every genus.
</p>projecteuclid.org/euclid.em/1338430830_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTSums of Three Squareful Numbershttp://projecteuclid.org/euclid.em/1338430831<strong>T. D. Browning</strong>, <strong>K. Van Valckenborgh</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 2, 204--211.</p><p><strong>Abstract:</strong><br/>
We investigate the frequency of positive squareful numbers
$x, y, z \le B$ for which $x + y = z$ and present a conjecture concerning
its asymptotic behavior.
</p>projecteuclid.org/euclid.em/1338430831_Wed, 30 May 2012 22:20 EDTWed, 30 May 2012 22:20 EDTCritical Values of Higher Derivatives of Twisted Elliptic $L$-Functionshttp://projecteuclid.org/euclid.em/1347541272<strong>Jack Fearnley</strong>, <strong>Hershy Kisilevsky</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 3, 213--222.</p><p><strong>Abstract:</strong><br/>
Let $L (E /\mathbb{Q} , s)$ be the $L$-function of an elliptic curve $E$ defined
over the rational field $\mathbb{Q}$. Assuming the Birch–Swinnerton-Dyer
conjectures, we examine special values of the $r$th derivatives,
$L^{(r)}(E , 1, \chi)$, of twists by Dirichlet characters of $L (E /\mathbb{Q} , s)$ when
$L (E , 1, \chi) = • • • = L^{(r−1)} (E , 1, \chi) = 0$.
</p>projecteuclid.org/euclid.em/1347541272_Thu, 13 Sep 2012 09:01 EDTThu, 13 Sep 2012 09:01 EDTRamanujan-like Series for $1/\pi^2$ and String Theoryhttp://projecteuclid.org/euclid.em/1347541273<strong>Gert Almkvist</strong>, <strong>Jesús Guillera</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 3, 223--234.</p><p><strong>Abstract:</strong><br/>
Using the machinery from the theory of Calabi–Yau differential
equations, we find formulas for $1/\pi^2$ of hypergeometric and
nonhypergeometric types.
</p>projecteuclid.org/euclid.em/1347541273_Thu, 13 Sep 2012 09:01 EDTThu, 13 Sep 2012 09:01 EDTThe Riemann Zeta Function on Arithmetic Progressionshttp://projecteuclid.org/euclid.em/1347541274<strong>J örn Steuding</strong>, <strong>Elias Wegert</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 3, 235--240.</p><p><strong>Abstract:</strong><br/>
We prove asymptotic formulas for the first discrete moment of the
Riemann zeta function on certain vertical arithmetic progressions
inside the critical strip. The results give some heuristic arguments
for a stochastic periodicity that we observed in the phase portrait
of the zeta function.
</p>projecteuclid.org/euclid.em/1347541274_Thu, 13 Sep 2012 09:01 EDTThu, 13 Sep 2012 09:01 EDTThe Noncommutative A-Polynomial of $(−2, 3, n)$ Pretzel Knotshttp://projecteuclid.org/euclid.em/1347541275<strong>Stavros Garoufalidis</strong>, <strong>Christoph Koutschan</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 3, 241--251.</p><p><strong>Abstract:</strong><br/>
We study $q$-holonomic sequences that arise as the colored
Jones polynomial of knots in 3-space. The minimal-order recurrence
for such a sequence is called the (noncommutative)
A-polynomial of a knot. Using the method of guessing , we
obtain this polynomial explicitly for the $K_p = (−2, 3, 3 + 2p)$
pretzel knots for $p= −5, \dots , 5$. This is a particularly interesting
family, since the pairs $(K_p,−K_{−p})$ are geometrically similar
(in particular, scissors congruent) with similar character varieties.
Our computation of the noncommutative $A$-polynomial complements
the computation of the $A$-polynomial of the pretzel knots
done by the first author and Mattman, supports the AJ conjecture
for knots with reducible $A$-polynomial, and numerically computes
the Kashaev invariant of pretzel knots in linear time. In a
later publication, we will use the numerical computation of the
Kashaev invariant to numerically verify the volume conjecture
for the above-mentioned pretzel knots.
</p>projecteuclid.org/euclid.em/1347541275_Thu, 13 Sep 2012 09:01 EDTThu, 13 Sep 2012 09:01 EDTThe Secant Conjecture in the Real Schubert Calculushttp://projecteuclid.org/euclid.em/1347541276<strong>Luis D. García-Puente</strong>, <strong>Nickolas Hein</strong>, <strong>Christopher Hillar</strong>, <strong>Abraham Martín del Campo</strong>, <strong>James Ruffo</strong>, <strong>Frank Sottile</strong>, <strong>Zach Teitler</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 3, 252--265.</p><p><strong>Abstract:</strong><br/>
We formulate the secant conjecture, which is a generalization
of the Shapiro conjecture for Grassmannians. It asserts that an
intersection of Schubert varieties in a Grassmannian is transverse
with all points real if the flags defining the Schubert varieties
are secant along disjoint intervals of a rational normal curve.
We present theoretical evidence for this conjecture as well as
computational evidence obtained in over one terahertz-year of
computing, and we discuss some of the phenomena we observed
in our data.
</p>projecteuclid.org/euclid.em/1347541276_Thu, 13 Sep 2012 09:01 EDTThu, 13 Sep 2012 09:01 EDTZero Cells of the Siegel–Gottschling Fundamental Domain of Degree 2http://projecteuclid.org/euclid.em/1347541277<strong>Takahiro Hayata</strong>, <strong>Takayuki Oda</strong>, <strong>Tomoki Yatougo</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 3, 266--279.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{F}_n$ be a fundamental domain of the Siegel upper half-space of
degree $n$ with respect to the Siegel modular group $\operatorname{Sp}(n, \mathbb{Z})$. According
to Siegel himself, $\mathcal{F}_n$ is determined by only finitely many
polynomial inequalities. In case of degree $n = 2$, Gottschling determined
the minimal set of inequalities. The boundary of $\mathcal{F}_2$ is
of great concern in the literature not only from a homological
point of view but also from the geometry of numbers. In this paper
we compute the vertices of $\mathcal{F}_2$ under the condition that the
defining ideal is zero-dimensional (“0-cells”). We also discuss
an equivalence relation among 0-cells.
</p>projecteuclid.org/euclid.em/1347541277_Thu, 13 Sep 2012 09:01 EDTThu, 13 Sep 2012 09:01 EDTThe $T$-Graph of a Multigraded Hilbert Schemehttp://projecteuclid.org/euclid.em/1347541278<strong>Milena Hering</strong>, <strong>Diane Maclagan</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 3, 280--297.</p><p><strong>Abstract:</strong><br/>
The $T$-graph of a multigraded Hilbert scheme records the zeroand
one-dimensional orbits of the $T = (K^*)^n$ action on the
Hilbert scheme induced from the $T$-action on $\mathbb{A}^n$. It has vertices
the $T$-fixed points, and edges the one-dimensional $T$-orbits.
We give a combinatorial necessary condition for the existence
of an edge between two vertices in this graph. For the Hilbert
scheme of points in the plane, we give an explicit combinatorial
description of the equations defining the scheme parameterizing
all one-dimensional torus orbits whose closures contain two
given monomial ideals. For this Hilbert scheme we show that
the $T$-graph depends on the ground field, resolving a question
of Altmann and Sturmfels.
</p>projecteuclid.org/euclid.em/1347541278_Thu, 13 Sep 2012 09:01 EDTThu, 13 Sep 2012 09:01 EDTA Note on Beauville $p$-Groupshttp://projecteuclid.org/euclid.em/1347541279<strong>Nathan Barker</strong>, <strong>Nigel Boston</strong>, <strong>Ben Fairbairn</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 3, 298--306.</p><p><strong>Abstract:</strong><br/>
We examine which $p$-groups of order $\le p^6$ are Beauville. We
completely classify them for groups of order $\le p^4$. We also show
that the proportion of 2-generated groups of order $p^5$ that are
Beauville tends to 1 as $p$ tends to infinity; this is not true, however,
for groups of order $p^6$. For each prime $p$ we determine the
smallest nonabelian Beauville $p$-group.
</p>projecteuclid.org/euclid.em/1347541279_Thu, 13 Sep 2012 09:01 EDTThu, 13 Sep 2012 09:01 EDTConjectures and Experiments Concerning the Moments of $L (1/2, \chi_d)$http://projecteuclid.org/euclid.em/1347541280<strong>Matthew W. Alderson</strong>, <strong>Michael O. Rubinstein</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 3, 307--328.</p><p><strong>Abstract:</strong><br/>
We report on some extensive computations and experiments
concerning the moments of quadratic Dirichlet $L$-functions at
the critical point. We computed the values of $L (1/2, \chi_d)$ for
$−5 × 10^{10} \lt d \lt 1.3 × 10^{10}$ in order to numerically test conjectures
concerning the moments $\sum_{|d|\lt X} L (1/2, \chi_d)^k$. Specifically,
we tested the full asymptotics for the moments conjectured by
Conrey, Farmer, Keating, Rubinstein, and Snaith, as well as the
conjectures of Diaconu, Goldfeld, Hoffstein, and Zhang concerning
additional lower-order terms in the moments. We also
describe the algorithms used for this large-scale computation.
</p>projecteuclid.org/euclid.em/1347541280_Thu, 13 Sep 2012 09:01 EDTThu, 13 Sep 2012 09:01 EDTTwisted Alexander Polynomials of Hyperbolic Knotshttp://projecteuclid.org/euclid.em/1356038817<strong>Nathan M. Dunfield</strong>, <strong>Stefan Friedl</strong>, <strong>Nicholas Jackson</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 4, 329--352.</p><p><strong>Abstract:</strong><br/>
We study a twisted Alexander polynomial naturally associated
to a hyperbolic knot in an integer homology 3-sphere via a lift of
the holonomy representation to $\mathrm{SL}(2, \mathbb{C})$. It is an unambiguous
symmetric Laurent polynomial whose coefficients lie in the trace
field of the knot. It contains information about genus, fibering,
and chirality, and moreover, is powerful enough to sometimes
detect mutation.
We calculated this invariant numerically for all $313\, 209$ hyperbolic
knots in $S^3$ with at most 15 crossings, and found that in
all cases it gave a sharp bound on the genus of the knot and
determined both fibering and chirality.
We also study how such twisted Alexander polynomials vary
as one moves around in an irreducible component $X_0$ of the
$\mathrm{SL}(2, \mathbb{C})$-character variety of the knot group. We show how to
understand all of these polynomials at once in terms of a polynomial
whose coefficients lie in the function field of $X_0$. We use
this to help explain some of the patterns observed for knots in
$S^3$, and explore a potential relationship between this universal
polynomial and the Culler–Shalen theory of surfaces associated
to ideal points.
</p>projecteuclid.org/euclid.em/1356038817_Thu, 20 Dec 2012 16:27 ESTThu, 20 Dec 2012 16:27 ESTDiscriminants, Symmetrized Graph Monomials, and Sums of Squareshttp://projecteuclid.org/euclid.em/1356038818<strong>Per Alexandersson</strong>, <strong>Boris Shapiro</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 4, 353--361.</p><p><strong>Abstract:</strong><br/>
In 1878, motivated by the requirements of the invariant theory
of binary forms, J. J. Sylvester constructed, for every graph
with possible multiple edges but without loops, its symmetrized
graph monomial, which is a polynomial in the vertex labels of
the original graph. We pose the question for which graphs this
polynomial is nonnegative or a sum of squares. This problem is
motivated by a recent conjecture of F. Sottile and E. Mukhin on
the discriminant of the derivative of a univariate polynomial and
by an interesting example of P. and A. Lax of a graph with four
edges whose symmetrized graph monomial is nonnegative but
not a sum of squares.We present detailed information about symmetrized
graph monomials for graphs with four and six edges,
obtained by computer calculations.
</p>projecteuclid.org/euclid.em/1356038818_Thu, 20 Dec 2012 16:27 ESTThu, 20 Dec 2012 16:27 ESTExperimental Data for Goldfeld’s Conjecture over Function Fieldshttp://projecteuclid.org/euclid.em/1356038819<strong>Salman Baig</strong>, <strong>Chris Hall</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 4, 362--374.</p><p><strong>Abstract:</strong><br/>
This paper presents empirical evidence supporting Goldfeld’s
conjecture on the average analytic rank of a family of quadratic
twists of a fixed elliptic curve in the function field setting. In particular,
we consider representatives of the four classes of nonisogenous
elliptic curves over $\mathbb{F}_q(t) with $(q, 6) = 1$ possessing
two places of multiplicative reduction and one place of additive
reduction. The case of $q = 5$ provides the largest data set as
well as the most convincing evidence that the average analytic
rank converges to 1/2, which we also show is a lower bound
following an argument of Kowalski. The data were generated via
explicit computation of the $L$-function of these elliptic curves,
and we present the key results necessary to implement an algorithm
to efficiently compute the $L$-function of nonisotrivial
elliptic curves over $\mathbb{F}_q(t) by realizing such a curve as a quadratic
twist of a pullback of a "versal" elliptic curve. We also provide
a reference for our open-source library ELLFF, which provides
all the necessary functionality to compute such $L$-functions, and
additional data on analytic rank distributions as they pertain to
the density conjecture.
</p>projecteuclid.org/euclid.em/1356038819_Thu, 20 Dec 2012 16:27 ESTThu, 20 Dec 2012 16:27 ESTAn Empirical Approach to the Normality of πhttp://projecteuclid.org/euclid.em/1356038820<strong>David H. Bailey</strong>, <strong>Jonathan M. Borwein</strong>, <strong>Cristian S. Calude</strong>, <strong>Michael J. Dinneen</strong>, <strong>Monica Dumitrescu</strong>, <strong>Alex Yee</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 4, 375--384.</p><p><strong>Abstract:</strong><br/>
Using the results of several extremely large recent computations, we tested positively the normality of a prefix
of roughly four trillion hexadecimal digits of $\pi$. This result was
used by a Poisson process model of normality of $\pi$: in this model,
it is extraordinarily unlikely that $\pi$ is not asymptotically normal
base 16, given the normality of its initial segment.
</p>projecteuclid.org/euclid.em/1356038820_Thu, 20 Dec 2012 16:27 ESTThu, 20 Dec 2012 16:27 ESTDecomposition of Semigroup Algebrashttp://projecteuclid.org/euclid.em/1356038821<strong>Janko Böhm</strong>, <strong>David Eisenbud</strong>, <strong>Max J. Nitsche</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 4, 385--394.</p><p><strong>Abstract:</strong><br/>
Let $A \supseteq B$ be cancellative abelian semigroups, and let $R$ be an
integral domain. We show that the semigroup ring $R[B]$ can be
decomposed, as an $R[A]$-module, into a direct sum of $R[A]$-submodules of the quotient ring of $R[A]$. In the case of a finite
extension of positive affine semigroup rings, we obtain an algorithm
computing the decomposition. When $R[A]$ is a polynomial
ring over a field, we explain how to compute many ring-theoretic
properties of $R[B]$ in terms of this decomposition. In particular,
we obtain a fast algorithm to compute the Castelnuovo–Mumford
regularity of homogeneous semigroup rings. As an application
we confirm the Eisenbud–Goto conjecture in a range of new
cases. Our algorithms are implemented in the MACAULAY2 package
MONOMIAL ALGEBRAS.
</p>projecteuclid.org/euclid.em/1356038821_Thu, 20 Dec 2012 16:27 ESTThu, 20 Dec 2012 16:27 ESTUniversal Gröbner Bases of Colored Partition Identitieshttp://projecteuclid.org/euclid.em/1356038822<strong>Tristram Bogart</strong>, <strong>Ray Hemmecke</strong>, <strong>Sonja Petrovíc</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 4, 395--401.</p><p><strong>Abstract:</strong><br/>
Associated to any toric ideal are two special generating sets:
the universal Gröbner basis and the Graver basis, which encode
polyhedral and combinatorial properties of the ideal, or equivalently,
its defining matrix. If the two sets coincide, then the
complexity of the Graver bases of the higher Lawrence liftings of
the toric matrices is bounded.
While a general classification of all matrices for which both sets
agree is far from known, we identify all such matrices within
two families of nonunimodular matrices, namely, those defining
rational normal scrolls and those encoding homogeneous
primitive colored partition identities. This also allows us to show
that higher Lawrence liftings of matrices with fixed Gröbner and
Graver complexities do not preserve equality of the two bases.
The proof of our classification combines computations with the
theoretical tool of Graver complexity of a pair of matrices.
</p>projecteuclid.org/euclid.em/1356038822_Thu, 20 Dec 2012 16:27 ESTThu, 20 Dec 2012 16:27 ESTRandom Walks on Barycentric Subdivisions and the Strichartz Hexacarpethttp://projecteuclid.org/euclid.em/1356038823<strong>Matthew Begue</strong>, <strong>Daniel J. Kelleher</strong>, <strong>Aaron Nelson</strong>, <strong>Hugo Panzo</strong>, <strong>Ryan Pellico</strong>, <strong>Alexander Teplyaev</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 4, 402--417.</p><p><strong>Abstract:</strong><br/>
We investigate simple random walks on graphs generated by
repeated barycentric subdivisions of a triangle. We use these
random walks to study the diffusion on the self-similar fractal
known as the Strichartz hexacarpet, which is generated as the
limit space of these graphs. We make this connection rigorous
by establishing a graph isomorphism between the hexacarpet
approximations and graphs produced by repeated barycentric
subdivisions of the triangle. This includes a discussion of various
numerical calculations performed on these graphs and their
implications to the diffusion on the limiting space. In particular,
we prove that equilateral barycentric subdivisions—a metric
space generated by replacing the metric on each 2-simplex of
the subdivided triangle with that of a scaled Euclidean equilateral
triangle—converge to a self-similar geodesic metric space
of dimension log(6)/ log(2), or about 2.58. Our numerical experiments
give evidence to a conjecture that the simple random
walks on the equilateral barycentric subdivisions converge to a
continuous diffusion process on the Strichartz hexacarpet corresponding
to a different spectral dimension (estimated numerically
to be about 1.74).
</p>projecteuclid.org/euclid.em/1356038823_Thu, 20 Dec 2012 16:27 ESTThu, 20 Dec 2012 16:27 ESTCorrigendum to: "The Alternative Operad Is Not Koszul" by Askar Dzhumadil’daev and Pasha Zusmanovichhttp://projecteuclid.org/euclid.em/1356038824<strong>Askar Dzhumadil’daev</strong>, <strong>Pasha Zusmanovich</strong><p><strong>Source: </strong>Experiment. Math., Volume 21, Number 4, 418--.</p>projecteuclid.org/euclid.em/1356038824_Thu, 20 Dec 2012 16:27 ESTThu, 20 Dec 2012 16:27 EST