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Project Euclid (Hosted at Cornell University Library) (192.979 recursos)
Experimental Mathematics
Experimental Mathematics
Dzhumadil’daev, Askar; Zusmanovich, Pasha
Begue, Matthew; Kelleher, Daniel J.; Nelson, Aaron; Panzo, Hugo; Pellico, Ryan; Teplyaev, Alexander
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...
Bogart, Tristram; Hemmecke, Ray; Petrovíc, Sonja
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...
Böhm, Janko; Eisenbud, David; Nitsche, Max J.
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...
Bailey, David H.; Borwein, Jonathan M.; Calude, Cristian S.; Dinneen, Michael J.; Dumitrescu, Monica; Yee, Alex
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.
Baig, Salman; Hall, Chris
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...
Alexandersson, Per; Shapiro, Boris
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...
Dunfield, Nathan M.; Friedl, Stefan; Jackson, Nicholas
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...
Alderson, Matthew W.; Rubinstein, Michael O.
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.
Barker, Nathan; Boston, Nigel; Fairbairn, Ben
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.
Hering, Milena; Maclagan, Diane
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.
Hayata, Takahiro; Oda, Takayuki; Yatougo, Tomoki
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.
García-Puente, Luis D.; Hein, Nickolas; Hillar, Christopher; del Campo, Abraham Martín; Ruffo, James; Sottile, Frank; Teitler, Zach
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.
Garoufalidis, Stavros; Koutschan, Christoph
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...
Steuding, J örn; Wegert, Elias
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.
Almkvist, Gert; Guillera, Jesús
Using the machinery from the theory of Calabi–Yau differential
equations, we find formulas for $1/\pi^2$ of hypergeometric and
nonhypergeometric types.
Fearnley, Jack; Kisilevsky, Hershy
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
$L (E , 1, \chi) = • • • = L^{(r−1)} (E , 1, \chi) = 0$.
Browning, T. D.; Van Valckenborgh, K.
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.
Alexeev, Valery; Livingston, Ryan; Tenini, Joseph; Arap, Maxim; Hu, Xiaoyan; Huckaba, Lauren; McFaddin, Patrick; Musgrave, Stacy; Shin, Jaeho; Ulrich, Catherine
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.
Lam, Thomas; Williams, Lauren
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.