304 research outputs found
Identities for hyperelliptic P-functions of genus one, two and three in covariant form
We give a covariant treatment of the quadratic differential identities
satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of
genera 1, 2 and 3
Dense Packings of Superdisks and the Role of Symmetry
We construct the densest known two-dimensional packings of superdisks in the
plane whose shapes are defined by |x^(2p) + y^(2p)| <= 1, which contains both
convex-shaped particles (p > 0.5, with the circular-disk case p = 1) and
concave-shaped particles (0 < p < 0.5). The packings of the convex cases with p
1 generated by a recently developed event-driven molecular dynamics (MD)
simulation algorithm [Donev, Torquato and Stillinger, J. Comput. Phys. 202
(2005) 737] suggest exact constructions of the densest known packings. We find
that the packing density (covering fraction of the particles) increases
dramatically as the particle shape moves away from the "circular-disk" point (p
= 1). In particular, we find that the maximal packing densities of superdisks
for certain p 6 = 1 are achieved by one of the two families of Bravais lattice
packings, which provides additional numerical evidence for Minkowski's
conjecture concerning the critical determinant of the region occupied by a
superdisk. Moreover, our analysis on the generated packings reveals that the
broken rotational symmetry of superdisks influences the packing characteristics
in a non-trivial way. We also propose an analytical method to construct dense
packings of concave superdisks based on our observations of the structural
properties of packings of convex superdisks.Comment: 15 pages, 8 figure
Borel-Cantelli sequences
A sequence in is called Borel-Cantelli (BC) if
for all non-increasing sequences of positive real numbers with
the set
has full Lebesgue measure. (To put it informally, BC
sequences are sequences for which a natural converse to the Borel-Cantelli
Theorem holds).
The notion of BC sequences is motivated by the Monotone Shrinking Target
Property for dynamical systems, but our approach is from a geometric rather
than dynamical perspective. A sufficient condition, a necessary condition and a
necessary and sufficient condition for a sequence to be BC are established. A
number of examples of BC and not BC sequences are presented.
The property of a sequence to be BC is a delicate diophantine property. For
example, the orbits of a pseudo-Anosoff IET (interval exchange transformation)
are BC while the orbits of a "generic" IET are not.
The notion of BC sequences is extended to more general spaces.Comment: 20 pages. Some proofs clarifie
Testing Hardy nonlocality proof with genuine energy-time entanglement
We show two experimental realizations of Hardy ladder test of quantum
nonlocality using energy-time correlated photons, following the scheme proposed
by A. Cabello \emph{et al.} [Phys. Rev. Lett. \textbf{102}, 040401 (2009)].
Unlike, previous energy-time Bell experiments, these tests require precise
tailored nonmaximally entangled states. One of them is equivalent to the
two-setting two-outcome Bell test requiring a minimum detection efficiency. The
reported experiments are still affected by the locality and detection
loopholes, but are free of the post-selection loophole of previous energy-time
and time-bin Bell tests.Comment: 5 pages, revtex4, 6 figure
Optimal Packings of Superballs
Dense hard-particle packings are intimately related to the structure of
low-temperature phases of matter and are useful models of heterogeneous
materials and granular media. Most studies of the densest packings in three
dimensions have considered spherical shapes, and it is only more recently that
nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs
(whose shapes are defined by |x1|^2p + |x2|^2p + |x3|^2p <= 1) provide a
versatile family of convex particles (p >= 0.5) with both cubic- and
octahedral-like shapes as well as concave particles (0 < p < 0.5) with
octahedral-like shapes. In this paper, we provide analytical constructions for
the densest known superball packings for all convex and concave cases. The
candidate maximally dense packings are certain families of Bravais lattice
packings. The maximal packing density as a function of p is nonanalytic at the
sphere-point (p = 1) and increases dramatically as p moves away from unity. The
packing characteristics determined by the broken rotational symmetry of
superballs are similar to but richer than their two-dimensional "superdisk"
counterparts, and are distinctly different from that of ellipsoid packings. Our
candidate optimal superball packings provide a starting point to quantify the
equilibrium phase behavior of superball systems, which should deepen our
understanding of the statistical thermodynamics of nonspherical-particle
systems.Comment: 28 pages, 16 figure
Multidimensional continued fractions, dynamical renormalization and KAM theory
The disadvantage of `traditional' multidimensional continued fraction
algorithms is that it is not known whether they provide simultaneous rational
approximations for generic vectors. Following ideas of Dani, Lagarias and
Kleinbock-Margulis we describe a simple algorithm based on the dynamics of
flows on the homogeneous space SL(2,Z)\SL(2,R) (the space of lattices of
covolume one) that indeed yields best possible approximations to any irrational
vector. The algorithm is ideally suited for a number of dynamical applications
that involve small divisor problems. We explicitely construct renormalization
schemes for (a) the linearization of vector fields on tori of arbitrary
dimension and (b) the construction of invariant tori for Hamiltonian systems.Comment: 51 page
Complete intersections: Moduli, Torelli, and good reduction
We study the arithmetic of complete intersections in projective space over
number fields. Our main results include arithmetic Torelli theorems and
versions of the Shafarevich conjecture, as proved for curves and abelian
varieties by Faltings. For example, we prove an analogue of the Shafarevich
conjecture for cubic and quartic threefolds and intersections of two quadrics.Comment: 37 pages. Typo's fixed. Expanded Section 2.
Ranks of twists of elliptic curves and Hilbert's Tenth Problem
In this paper we investigate the 2-Selmer rank in families of quadratic
twists of elliptic curves over arbitrary number fields. We give sufficient
conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer
rank, and we give lower bounds for the number of twists (with bounded
conductor) that have a given 2-Selmer rank. As a consequence, under appropriate
hypotheses we can find many twists with trivial Mordell-Weil group, and
(assuming the Shafarevich-Tate conjecture) many others with infinite cyclic
Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our
results that if the Shafarevich-Tate conjecture holds, then Hilbert's Tenth
Problem has a negative answer over the ring of integers of every number field.Comment: Minor changes. To appear in Inventiones mathematica
Patterns in rational base number systems
Number systems with a rational number as base have gained interest
in recent years. In particular, relations to Mahler's 3/2-problem as well as
the Josephus problem have been established. In the present paper we show that
the patterns of digits in the representations of positive integers in such a
number system are uniformly distributed. We study the sum-of-digits function of
number systems with rational base and use representations w.r.t. this
base to construct normal numbers in base in the spirit of Champernowne. The
main challenge in our proofs comes from the fact that the language of the
representations of integers in these number systems is not context-free. The
intricacy of this language makes it impossible to prove our results along
classical lines. In particular, we use self-affine tiles that are defined in
certain subrings of the ad\'ele ring and Fourier
analysis in . With help of these tools we are able to
reformulate our results as estimation problems for character sums
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