20,823 research outputs found
On volumes of hyperideal tetrahedra with constrained edge lengths
Hyperideal tetrahedra are the fundamental building blocks of hyperbolic
3-manifolds with geodesic boundary. The study of their geometric properties (in
particular, of their volume) has applications also in other areas of
low-dimensional topology, like the computation of quantum invariants of
3-manifolds and the use of variational methods in the study of circle packings
on surfaces.
The Schl\"afli formula neatly describes the behaviour of the volume of
hyperideal tetrahedra with respect to dihedral angles, while the dependence of
volume on edge lengths is worse understood. In this paper we prove that, for
every , where is an explicit constant, regular hyperideal
tetrahedra of edge length maximize the volume among hyperideal
tetrahedra whose edge lengths are all not smaller than .
This result provides a fundamental step in the computation of the ideal
simplicial volume of an infinite family of hyperbolic 3-manifolds with geodesic
boundary.Comment: 20 pages, 2 figures, Some minor changes, To appear in Periodica
Mathematica Hungaric
Vector-Valued Modular Forms from the Mumford Form, Schottky-Igusa Form, Product of Thetanullwerte and the Amazing Klein Formula
Vector-valued Siegel modular forms are the natural generalization of the
classical elliptic modular forms as seen by studying the cohomology of the
universal abelian variety. We show that for g>=4, a new class of vector-valued
modular forms, defined on the Teichmuller space, naturally appears from the
Mumford forms, a question directly related to the Schottky problem. In this
framework we show that the discriminant of the quadric associated to the
complex curves of genus 4 is proportional to the square root of the products of
Thetanullwerte \chi_{68}, which is a proof of the recently rediscovered Klein
`amazing formula'. Furthermore, it turns out that the coefficients of such a
quadric are derivatives of the Schottky-Igusa form evaluated at the Jacobian
locus, implying new theta relations involving the latter, \chi_{68} and the
theta series corresponding to the even unimodular lattices E_8\oplus E_8 and
D_{16}^+. We also find, for g=4, a functional relation between the singular
component of the theta divisor and the Riemann period matrix.Comment: 17 pages. Final version in Proc. Amer. Math. So
The Singular Locus of the Theta Divisor and Quadrics through a Canonical Curve
A section K on a genus g canonical curve C is identified as the key tool to
prove new results on the geometry of the singular locus Theta_s of the theta
divisor. The K divisor is characterized by the condition of linear dependence
of a set of quadrics containing C and naturally associated to a degree g
effective divisor on C. K counts the number of intersections of special
varieties on the Jacobian torus defined in terms of Theta_s. It also identifies
sections of line bundles on the moduli space of algebraic curves, closely
related to the Mumford isomorphism, whose zero loci characterize special
varieties in the framework of the Andreotti-Mayer approach to the Schottky
problem, a result which also reproduces the only previously known case g=4.
This new approach, based on the combinatorics of determinantal relations for
two-fold products of holomorphic abelian differentials, sheds light on basic
structures, and leads to the explicit expressions, in terms of theta functions,
of the canonical basis of the abelian holomorphic differentials and of the
constant defining the Mumford form. Furthermore, the metric on the moduli space
of canonical curves, induced by the Siegel metric, which is shown to be
equivalent to the Kodaira-Spencer map of the square of the Bergman reproducing
kernel, is explicitly expressed in terms of the Riemann period matrix only, a
result previously known for the trivial cases g=2 and g=3. Finally, the induced
Siegel volume form is expressed in terms of the Mumford form.Comment: 88+1 page
CNNs found to jump around more skillfully than RNNs: Compositional generalization in seq2seq convolutional networks
Lake and Baroni (2018) introduced the SCAN dataset probing the ability of
seq2seq models to capture compositional generalizations, such as inferring the
meaning of "jump around" 0-shot from the component words. Recurrent networks
(RNNs) were found to completely fail the most challenging generalization cases.
We test here a convolutional network (CNN) on these tasks, reporting hugely
improved performance with respect to RNNs. Despite the big improvement, the CNN
has however not induced systematic rules, suggesting that the difference
between compositional and non-compositional behaviour is not clear-cut.Comment: accepted as a short paper at ACL 201
Higher genus superstring amplitudes from the geometry of moduli spaces
We show that the higher genus 4-point superstring amplitude is strongly
constrained by the geometry of moduli space of Riemann surfaces. A detailed
analysis leads to a natural proposal which satisfies several conditions. The
result is based on the recently derived Siegel induced metric on the moduli
space of Riemann surfaces and on combinatorial products of determinants of
holomorphic abelian differentials.Comment: 23 pp. A change in citing, to appear in Nucl. Phys.
TDOA--based localization in two dimensions: the bifurcation curve
In this paper, we complete the study of the geometry of the TDOA map that
encodes the noiseless model for the localization of a source from the range
differences between three receivers in a plane, by computing the Cartesian
equation of the bifurcation curve in terms of the positions of the receivers.
From that equation, we can compute its real asymptotic lines. The present
manuscript completes the analysis of [Inverse Problems, Vol. 30, Number 3,
Pages 035004]. Our result is useful to check if a source belongs or is closed
to the bifurcation curve, where the localization in a noisy scenario is
ambiguous.Comment: 11 pages, 3 figures, to appear in Fundamenta Informatica
Asymptotically idempotent aggregation operators for trust management in multi-agent systems
The study of trust management in
multi-agent system, especially distributed,
has grown over the last
years. Trust is a complex subject
that has no general consensus in literature,
but has emerged the importance
of reasoning about it computationally.
Reputation systems takes
into consideration the history of an
entity’s actions/behavior in order to
compute trust, collecting and aggregating
ratings from members in a
community. In this scenario the aggregation
problem becomes fundamental,
in particular depending on
the environment. In this paper we
describe a technique based on a class
of asymptotically idempotent aggregation
operators, suitable particulary
for distributed anonymous environments
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