20,616 research outputs found
Bounds on the maximum multiplicity of some common geometric graphs
We obtain new lower and upper bounds for the maximum multiplicity of some
weighted and, respectively, non-weighted common geometric graphs drawn on n
points in the plane in general position (with no three points collinear):
perfect matchings, spanning trees, spanning cycles (tours), and triangulations.
(i) We present a new lower bound construction for the maximum number of
triangulations a set of n points in general position can have. In particular,
we show that a generalized double chain formed by two almost convex chains
admits {\Omega}(8.65^n) different triangulations. This improves the bound
{\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by
Aichholzer et al.
(ii) We present a new lower bound of {\Omega}(12.00^n) for the number of
non-crossing spanning trees of the double chain composed of two convex chains.
The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years.
(iii) Using a recent upper bound of 30^n for the number of triangulations,
due to Sharir and Sheffer, we show that n points in the plane in general
position admit at most O(68.62^n) non-crossing spanning cycles.
(iv) We derive lower bounds for the number of maximum and minimum weighted
geometric graphs (matchings, spanning trees, and tours). We show that the
number of shortest non-crossing tours can be exponential in n. Likewise, we
show that both the number of longest non-crossing tours and the number of
longest non-crossing perfect matchings can be exponential in n. Moreover, we
show that there are sets of n points in convex position with an exponential
number of longest non-crossing spanning trees. For points in convex position we
obtain tight bounds for the number of longest and shortest tours. We give a
combinatorial characterization of the longest tours, which leads to an O(nlog
n) time algorithm for computing them
A ferrofluid based neural network: design of an analogue associative memory
We analyse an associative memory based on a ferrofluid, consisting of a
system of magnetic nano-particles suspended in a carrier fluid of variable
viscosity subject to patterns of magnetic fields from an array of input and
output magnetic pads. The association relies on forming patterns in the
ferrofluid during a trainingdphase, in which the magnetic dipoles are free to
move and rotate to minimize the total energy of the system. Once equilibrated
in energy for a given input-output magnetic field pattern-pair the particles
are fully or partially immobilized by cooling the carrier liquid. Thus produced
particle distributions control the memory states, which are read out
magnetically using spin-valve sensors incorporated in the output pads. The
actual memory consists of spin distributions that is dynamic in nature,
realized only in response to the input patterns that the system has been
trained for. Two training algorithms for storing multiple patterns are
investigated. Using Monte Carlo simulations of the physical system we
demonstrate that the device is capable of storing and recalling two sets of
images, each with an accuracy approaching 100%.Comment: submitted to Neural Network
A tensor network study of the complete ground state phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square lattice
Using infinite projected entangled pair states, we study the ground state
phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square
lattice directly in the thermodynamic limit. We find an unexpected partially
nematic partially magnetic phase in between the antiferroquadrupolar and
ferromagnetic regions. Furthermore, we describe all observed phases and discuss
the nature of the phase transitions involved.Comment: 27 pages, 15 figures; v3: adjusted sections 1 and 3, and added a
paragraph to section 5.2.
Entanglement entropy for the long range Ising chain
We consider the Ising model in a transverse field with long-range
antiferromagnetic interactions that decay as a power law with their distance.
We study both the phase diagram and the entanglement properties as a function
of the exponent of the interaction. The phase diagram can be used as a guide
for future experiments with trapped ions. We find two gapped phases, one
dominated by the transverse field, exhibiting quasi long range order, and one
dominated by the long range interaction, with long range N\'eel ordered ground
states. We determine the location of the quantum critical points separating
those two phases. We determine their critical exponents and central-charges. In
the phase with quasi long range order the ground states exhibit exotic
corrections to the area law for the entanglement entropy coexisting with gapped
entanglement spectra.Comment: 5 pages, all comments welcom
Conversion from linear to circular polarization in FPGA
Context: Radio astronomical receivers are now expanding their frequency range
to cover large (octave) fractional bandwidths for sensitivity and spectral
flexibility, which makes the design of good analogue circular polarizers
challenging. Better polarization purity requires a flatter phase response over
increasingly wide bandwidth, which is most easily achieved with digital
techniques. They offer the ability to form circular polarization with perfect
polarization purity over arbitrarily wide fractional bandwidths, due to the
ease of introducing a perfect quadrature phase shift. Further, the rapid
improvements in field programmable gate arrays provide the high processing
power, low cost, portability and reconfigurability needed to make practical the
implementation of the formation of circular polarization digitally. Aims: Here
we explore the performance of a circular polarizer implemented with digital
techniques. Methods: We designed a digital circular polarizer in which the
intermediate frequency signals from a receiver with native linear polarizations
were sampled and converted to circular polarization. The frequency-dependent
instrumental phase difference and gain scaling factors were determined using an
injected noise signal and applied to the two linear polarizations to equalize
the transfer characteristics of the two polarization channels. This
equalization was performed in 512 frequency channels over a 512 MHz bandwidth.
Circular polarization was formed by quadrature phase shifting and summing the
equalized linear polarization signals. Results: We obtained polarization purity
of -25 dB corresponding to a D-term of 0.06 over the whole bandwidth.
Conclusions: This technique enables construction of broad-band radio astronomy
receivers with native linear polarization to form circular polarization for
VLBI.Comment: 11 pages 8 figure
Spin ice thin films: Large-N theory and Monte Carlo simulations
We explore the physics of highly frustrated magnets in confined geometries,
focusing on the Coulomb phase of pyrochlore spin ices. As a specific example,
we investigate thin films of nearest-neighbor spin ice, using a combination of
analytic large-N techniques and Monte Carlo simulations. In the simplest film
geometry, with surfaces perpendicular to the [001] crystallographic direction,
we observe pinch points in the spin-spin correlations characteristic of a
two-dimensional Coulomb phase. We then consider the consequences of crystal
symmetry breaking on the surfaces of the film through the inclusion of orphan
bonds. We find that when these bonds are ferromagnetic, the Coulomb phase is
destroyed by the presence of fluctuating surface magnetic charges, leading to a
classical Z_2 spin liquid. Building on this understanding, we discuss other
film geometries with surfaces perpendicular to the [110] or the [111]
direction. We generically predict the appearance of surface magnetic charges
and discuss their implications for the physics of such films, including the
possibility of an unusual Z_3 classical spin liquid. Finally, we comment on
open questions and promising avenues for future research.Comment: 17 pages, 11 figures. Minor improvements, typos correcte
The Iray Light Transport Simulation and Rendering System
While ray tracing has become increasingly common and path tracing is well
understood by now, a major challenge lies in crafting an easy-to-use and
efficient system implementing these technologies. Following a purely
physically-based paradigm while still allowing for artistic workflows, the Iray
light transport simulation and rendering system allows for rendering complex
scenes by the push of a button and thus makes accurate light transport
simulation widely available. In this document we discuss the challenges and
implementation choices that follow from our primary design decisions,
demonstrating that such a rendering system can be made a practical, scalable,
and efficient real-world application that has been adopted by various companies
across many fields and is in use by many industry professionals today
Linear And Nonlinear Arabesques: A Study Of Closed Chains Of Negative 2-Element Circuits
In this paper we consider a family of dynamical systems that we call
"arabesques", defined as closed chains of 2-element negative circuits. An
-dimensional arabesque system has 2-element circuits, but in addition,
it displays by construction, two -element circuits which are both positive
vs one positive and one negative, depending on the parity (even or odd) of the
dimension . In view of the absence of diagonal terms in their Jacobian
matrices, all these dynamical systems are conservative and consequently, they
can not possess any attractor. First, we analyze a linear variant of them which
we call "arabesque 0" or for short "A0". For increasing dimensions, the
trajectories are increasingly complex open tori. Next, we inserted a single
cubic nonlinearity that does not affect the signs of its circuits (that we call
"arabesque 1" or for short "A1"). These systems have three steady states,
whatever the dimension is, in agreement with the order of the nonlinearity. All
three are unstable, as there can not be any attractor in their state-space. The
3D variant (that we call for short "A1\_3D") has been analyzed in some detail
and found to display a complex mixed set of quasi-periodic and chaotic
trajectories. Inserting cubic nonlinearities (one per equation) in the same
way as above, we generate systems "A2\_D". A2\_3D behaves essentially as
A1\_3D, in agreement with the fact that the signs of the circuits remain
identical. A2\_4D, as well as other arabesque systems with even dimension, has
two positive -circuits and nine steady states. Finally, we investigate and
compare the complex dynamics of this family of systems in terms of their
symmetries.Comment: 22 pages, 12 figures, accepted for publication at Int. J. Bif. Chao
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