5,014 research outputs found
On three parameters of invisibility graphs
The invisibility graph I(X) of a set X ⊆ Rd is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and
only if the straight-line segment connecting the two corresponding points is not fully contained in X . We consider the following three parameters of a set X : the clique number ω(I(X)), the chromatic number χ(I(X)) and the inimum number γ(X) of convex subsets of X that cover X. We settle a conjecture of Matousek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3.
We also find sets X in R5 with χ(I(X)) = 2, but γ(X) arbitrarily large.Czech Science FoundationMinistry of Education, Youth and Sports of the Czech RepublicEuropean Science FoundationOrszágos Tudományos Kutatási Alapprogramok (OTKA)Centre Interfacultaire BernoulliSwiss National Science Foundatio
Active Invisibility Cloaks in One Dimension
We outline a general method of constructing finite-range cloaking potentials
which render a given finite-range real or complex potential
unidirectionally reflectionless or invisible at a wavenumber of our
choice. We give explicit analytic expressions for three classes of cloaking
potentials which achieve this goal while preserving some or all of the other
scattering properties of . The cloaking potentials we construct are the
sum of up to three constituent unidirectionally invisible potentials. We also
discuss their application in making bidirectionally invisible at ,
and demonstrate the application of our method to obtain anti-reflection and
invisibility cloaks for a Bragg reflector.Comment: 7 pages, 4 figures, expanded version including a discussion of
absorbing cloaking potentials, to appear in Phys. Rev.
Invisibility and PT-symmetry
For a general complex scattering potential defined on a real line, we show
that the equations governing invisibility of the potential are invariant under
the combined action of parity and time-reversal (PT) transformation. We
determine the PT-symmetric an well as non-PT-symmetric invisible configurations
of an easily realizable exactly solvable model that consists of a two-layer
planar slab consisting of optically active material. Our analysis shows that
although PT-symmetry is neither necessary nor sufficient for the invisibility
of a scattering potential, it plays an important role in the characterization
of the invisible configurations. A byproduct of our investigation is the
discovery of certain configurations of our model that are effectively
reflectionless in a spectral range as wide as several hundred nanometers.Comment: 11 pages, 3 figures, revised version, accepted for publication in
Phys.Rev.
Perturbative Unidirectional Invisibility
We outline a general perturbative method of evaluating scattering features of
finite-range complex potentials and use it to examine complex perturbations of
a rectangular barrier potential. In optics, these correspond to modulated
refractive index profiles of the form , where is real,
is complex-valued, and . We give a comprehensive
description of the phenomenon of unidirectional invisibility for such media,
proving five general theorems on its realization in -symmetric and
non--symmetric material. In particular, we establish the
impossibility of unidirectional invisibility for -symmetric samples
whose refractive index has a constant real part and show how a simple scaling
transformation of a unidirectionally invisible -symmetric index
profile with may be used to generate a hierarchy of unidirectionally
invisible -symmetric index profiles with . The results
pertaining unidirectional invisibility for open up the way for the
experimental studies of this phenomenon in a variety of active material. As an
application of our general results, we show that a medium with , and real, and can support unidirectional
invisibility only for . We then construct unidirectionally invisible
index profiles of the form , with
complex, real, , and .Comment: 15 pages, 3 figure
Unidirectional Invisibility and PT-Symmetry with Graphene
We investigate the reflectionlessness and invisibility properties in the
transverse electric (TE) mode solution of a linear homogeneous optical system
which comprises the -symmetric structures covered by graphene
sheets. We derive analytic expressions, indicate roles of each parameter
governing optical system with graphene and justify that optimal conditions of
these parameters give rise to broadband and wide angle invisibility. Presence
of graphene turns out to shift the invisible wavelength range and to reduce the
required gain amount considerably, based on its chemical potential and
temperature. We substantiate that our results yield broadband reflectionless
and invisible configurations for realistic materials of small refractive
indices, usually around , and of small thickness sizes with graphene
sheets of rather small temperatures and chemical potentials. Finally, we
demonstrate that pure -symmetric graphene yields invisibility at
small temperatures and chemical potentials.Comment: 20 pages, 1 table 17 figure
Reconstruction of Planar Domains from Partial Integral Measurements
We consider the problem of reconstruction of planar domains from their
moments. Specifically, we consider domains with boundary which can be
represented by a union of a finite number of pieces whose graphs are solutions
of a linear differential equation with polynomial coefficients. This includes
domains with piecewise-algebraic and, in particular, piecewise-polynomial
boundaries. Our approach is based on one-dimensional reconstruction method of
[Bat]* and a kind of "separation of variables" which reduces the planar problem
to two one-dimensional problems, one of them parametric. Several explicit
examples of reconstruction are given.
Another main topic of the paper concerns "invisible sets" for various types
of incomplete moment measurements. We suggest a certain point of view which
stresses remarkable similarity between several apparently unrelated problems.
In particular, we discuss zero quadrature domains (invisible for harmonic
polynomials), invisibility for powers of a given polynomial, and invisibility
for complex moments (Wermer's theorem and further developments). The common
property we would like to stress is a "rigidity" and symmetry of the invisible
objects.
* D.Batenkov, Moment inversion of piecewise D-finite functions, Inverse
Problems 25 (2009) 105001Comment: Proceedings of Complex Analysis and Dynamical Systems V, 201
Forecasting Financial Extremes: A Network Degree Measure of Super-exponential Growth
Investors in stock market are usually greedy during bull markets and scared
during bear markets. The greed or fear spreads across investors quickly. This
is known as the herding effect, and often leads to a fast movement of stock
prices. During such market regimes, stock prices change at a super-exponential
rate and are normally followed by a trend reversal that corrects the previous
over reaction. In this paper, we construct an indicator to measure the
magnitude of the super-exponential growth of stock prices, by measuring the
degree of the price network, generated from the price time series. Twelve major
international stock indices have been investigated. Error diagram tests show
that this new indicator has strong predictive power for financial extremes,
both peaks and troughs. By varying the parameters used to construct the error
diagram, we show the predictive power is very robust. The new indicator has a
better performance than the LPPL pattern recognition indicator.Comment: 16 pages, 6 figure
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