6,218 research outputs found
Saturation and Irredundancy for Spin(8)
We explicitly calculate the triangle inequalities for the group PSO(8).
Therefore we explicitly solve the eigenvalues of sum problem for this group
(equivalently describing the side-lengths of geodesic triangles in the
corresponding symmetric space for the Weyl chamber-valued metric). We then
apply some computer programs to verify two basic questions/conjectures. First,
we verify that the above system of inequalities is irredundant. Then, we verify
the ``saturation conjecture'' for the decomposition of tensor products of
finite-dimensional irreducible representations of Spin(8). Namely, we show that
for any triple of dominant weights a, b, c such that a+b+c is in the root
lattice, and any positive integer N, the tensor product of the irreducible
representations V(a) and V(b) contains V(c) if and only if the tensor product
of V(Na) and V(Nb) contains V(Nc).Comment: 22 pages, 2 figure
The algebro-geometric study of range maps
Localizing a radiant source is a widespread problem to many scientific and
technological research areas. E.g. localization based on range measurements
stays at the core of technologies like radar, sonar and wireless sensors
networks. In this manuscript we study in depth the model for source
localization based on range measurements obtained from the source signal, from
the point of view of algebraic geometry. In the case of three receivers, we
find unexpected connections between this problem and the geometry of Kummer's
and Cayley's surfaces. Our work gives new insights also on the localization
based on range differences.Comment: 38 pages, 18 figure
The Metric Nearness Problem
Metric nearness refers to the problem of optimally restoring metric properties to
distance measurements that happen to be nonmetric due to measurement errors or otherwise. Metric
data can be important in various settings, for example, in clustering, classification, metric-based
indexing, query processing, and graph theoretic approximation algorithms. This paper formulates
and solves the metric nearness problem: Given a set of pairwise dissimilarities, find a “nearest” set
of distances that satisfy the properties of a metric—principally the triangle inequality. For solving
this problem, the paper develops efficient triangle fixing algorithms that are based on an iterative
projection method. An intriguing aspect of the metric nearness problem is that a special case turns
out to be equivalent to the all pairs shortest paths problem. The paper exploits this equivalence and
develops a new algorithm for the latter problem using a primal-dual method. Applications to graph
clustering are provided as an illustration. We include experiments that demonstrate the computational
superiority of triangle fixing over general purpose convex programming software. Finally, we
conclude by suggesting various useful extensions and generalizations to metric nearness
Numerical evolution of squeezed and non-Gaussian states in loop quantum cosmology
In recent years, numerical simulations with Gaussian initial states have
demonstrated the existence of a quantum bounce in loop quantum cosmology in
various models. A key issue pertaining to the robustness of the bounce and the
associated physics is to understand the quantum evolution for more general
initial states which may depart significantly from Gaussianity and may have no
well defined peakedness properties. The analysis of such states, including
squeezed and highly non-Gaussian states, has been computationally challenging
until now. In this manuscript, we overcome these challenges by using the
Chimera scheme for the spatially flat, homogeneous and isotropic model sourced
with a massless scalar field. We demonstrate that the quantum bounce in this
model occurs even for states which are highly squeezed or are non-Gaussian with
multiple peaks and with little resemblance to semi-classical states. The
existence of the bounce is found to be robust, being independent of the
properties of the states. The evolution of squeezed and non-Gaussian states
turns out to be qualitatively similar to that of Gaussian states, and satisfies
strong constraints on the growth of the relative fluctuations across the
bounce. We also compare the results from the effective dynamics and find that,
although it captures the qualitative aspects of the evolution for squeezed and
highly non-Gaussian states, it always underestimates the bounce volume. We show
that various properties of the evolution, such as the energy density at the
bounce, are in excellent agreement with the predictions from an exactly
solvable loop quantum cosmological model for arbitrary states.Comment: 26 pages, 16 figures. v2: Discussion of the main results expande
On the Conjectures Regarding the 4-Point Atiyah Determinant
For the case of 4 points in Euclidean space, we present a computer aided
proof of Conjectures II and III made by Atiyah and Sutcliffe regarding Atiyah's
determinant along with an elegant factorization of the square of the imaginary
part of Atiyah's determinant
Relative equilibria of four identical satellites
We consider the Newtonian 5-body problem in the plane, where 4 bodies have
the same mass m, which is small compared to the mass M of the remaining body.
We consider the (normalized) relative equilibria in this system, and follow
them to the limit when m/M -> 0. In some cases two small bodies will coalesce
at the limit. We call the other equilibria the relative equilibria of four
separate identical satellites. We prove rigorously that there are only three
such equilibria, all already known after the numerical researches in [SaY]. Our
main contribution is to prove that any equilibrium configuration possesses a
symmetry, a statement indicated in [CLO2] as the missing key to proving that
there is no other equilibrium.Comment: 16 pages, 2 figure
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