7,896 research outputs found
Falconer distance problem, additive energy and Cartesian products
A celebrated result due to Wolff says if is a compact subset of , then the Lebesgue measure of the distance set is positive if the Hausdorff dimension of is greater than
. In this paper we improve the barrier by a small
exponent for Cartesian products. In higher dimensions, also in the context of
Cartesian products, we reduce Erdogan's exponent to
. The proof uses a combination of Fourier analysis and
additive comibinatorics.Comment: 9 page
Pinned distance problem, slicing measures and local smoothing estimates
We improve the Peres-Schlag result on pinned distances in sets of a given
Hausdorff dimension. In particular, for Euclidean distances, with we prove that for any , there
exists a probability measure on such that for -a.e. ,
(1) if ;
(2) has positive Lebesgue measure if ;
(3) has non-empty interior if .
We also show that in the case when , for -a.e. , has
positive Lebesgue measure. This describes dimensions of slicing subsets of ,
sliced by spheres centered at .
In our proof, local smoothing estimates of Fourier integral operators (FIO)
plays a crucial role. In turn, we obtain results on sharpness of local
smoothing estimates by constructing geometric counterexamples
Equilateral triangles in subsets of of large Hausdorff dimension
We prove that subsets of , of large enough Hausdorff
dimensions contain vertices of an equilateral triangle. It is known that
additional hypotheses are needed to assure the existence of equilateral
triangles in two dimensions (see \cite{CLP14}). We show that no extra
conditions are needed in dimensions four and higher. The three dimensional case
remains open.
Some interesting parallels exist between the triangle problem in Euclidean
space and its counter-part in vector spaces over finite fields. We shall
outline these similarities in hopes of eventually achieving a comprehensive
understanding of this phenomenon in the setting of locally compact abelian
groups.Comment: 10 pages, no picture
Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisymmetric objects
We present a high-order accurate boundary-based solver for three-dimensional
(3D) frequency-domain scattering from a doubly-periodic grating of smooth
axisymmetric sound-hard or transmission obstacles. We build the one-obstacle
solution operator using separation into P azimuthal modes via the FFT, the
method of fundamental solutions (with N proxy points lying on a curve), and
dense direct least-squares solves; the effort is O(N^3P) with a small constant.
Periodizing then combines fast multipole summation of nearest neighbors with an
auxiliary global Helmholtz basis expansion to represent the distant
contributions, and enforcing quasi-periodicity and radiation conditions on the
unit cell walls. Eliminating the auxiliary coefficients, and preconditioning
with the one-obstacle solution operator, leaves a well-conditioned square
linear system that is solved iteratively. The solution time per incident wave
is then O(NP) at fixed frequency. Our scheme avoids singular quadratures,
periodic Green's functions, and lattice sums, and its convergence rate is
unaffected by resonances within obstacles. We include numerical examples such
as scattering from a grating of period 13 {\lambda} x 13{\lambda} of
highly-resonant sound-hard "cups" each needing NP = 64800 surface unknowns, to
10-digit accuracy, in half an hour on a desktop.Comment: 22 pages, 9 figures, submitted to Journal of Computational Physic
Universality of the shear viscosity in supergravity
Kovtun, Son and Starinets proposed a bound on the shear viscosity of any
fluid in terms of its entropy density. We argue that this bound is always
saturated for gauge theories at large 't Hooft coupling, which admit
holographically dual supergravity description.Comment: 14 pages, Late
Tunneling Spectroscopy of a Spiral Luttinger Liquid in Contact with Superconductors
One-dimensional wires with Rashba spin-orbit coupling, magnetic field, and
strong electron-electron interactions are described by a spiral Luttinger
liquid model. We develop a theory to investigate the tunneling density of
states into a spiral Luttinger liquid under the proximity effect with
superconductors. This approach provides a way to disentangle the delicate
interplay between superconducting correlations and strong electron
interactions. If the wire-superconductor boundary is dominated by Andreev
reflection, we find that in the vicinity of the interface the zero-bias
tunneling anomaly reveals a power law enhancement with the unusual exponent.
Far away from the interface strong correlations inherent to the Luttinger
liquid prevail and restore conventional suppression of the tunneling density of
states at the Fermi level, which acquire, however, a Friedel-like oscillatory
envelope with the period renormalized by the strength of the interaction.Comment: 7 pages, 4 figure
Holographic Software for Quantum Networks
We introduce a pictorial approach to quantum information, called holographic
software. Our software captures both algebraic and topological aspects of
quantum networks. It yields a bi-directional dictionary to translate between a
topological approach and an algebraic approach. Using our software, we give a
topological simulation for quantum networks. The string Fourier transform (SFT)
is our basic tool to transform product states into states with maximal
entanglement entropy. We obtain a pictorial interpretation of Fourier
transformation, of measurements, and of local transformations, including the
-qudit Pauli matrices and their representation by Jordan-Wigner
transformations.
We use our software to discover interesting new protocols for multipartite
communication. In summary, we build a bridge linking the theory of planar para
algebras with quantum information.Comment: 48 pages. Accepted for publication in SCIENCE CHINA Mathematic
Qudit Isotopy
We explore a general diagrammatic framework to understand qudits and their
braiding, especially in its relation to entanglement. This involves
understanding the role of isotopy in interpreting diagrams that implement
entangling gates as well as some standard quantum information protocols. We
give qudit Pauli operators and comment on their structure, both from an
algebraic and from a diagrammatic point of view. We explain alternative models
for diagrammatic interpretations of qudits and their transformations. We use
our diagrammatic approach to define an entanglement-relay protocol for
long-distance entanglement. Our approach rests on algebraic and topological
relations discovered in the study of planar para algebras. In summary, this
work provides bridges between the new theory of planar para algebras and
quantum information, especially in questions involving entanglement
Modeling Morphology of Social Network Cascades
Cascades represent an important phenomenon across various disciplines such as
sociology, economy, psychology, political science, marketing, and epidemiology.
An important property of cascades is their morphology, which encompasses the
structure, shape, and size. However, cascade morphology has not been rigorously
characterized and modeled in prior literature. In this paper, we propose a
Multi-order Markov Model for the Morphology of Cascades () that can
represent and quantitatively characterize the morphology of cascades with
arbitrary structures, shapes, and sizes. can be used in a variety of
applications to classify different types of cascades. To demonstrate this, we
apply it to an unexplored but important problem in online social networks --
cascade size prediction. Our evaluations using real-world Twitter data show
that based cascade size prediction scheme outperforms the baseline
scheme based on cascade graph features such as edge growth rate, degree
distribution, clustering, and diameter. based cascade size prediction
scheme consistently achieves more than 90% classification accuracy under
different experimental scenarios.Comment: 12 pages, technical repor
Compressed Teleportation
In a previous paper we introduced holographic software for quantum networks,
inspired by work on planar para algebras. This software suggests the definition
of a compressed transformation. Here we utilize the software to find a CT
protocol to teleport compressed transformations. This protocol serves multiple
parties with multiple persons.Comment: 3 page
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