7,144 research outputs found
Closed geodesics and billiards on quadrics related to elliptic KdV solutions
We consider algebraic geometrical properties of the integrable billiard on a
quadric Q with elastic impacts along another quadric confocal to Q. These
properties are in sharp contrast with those of the ellipsoidal Birkhoff
billiards. Namely, generic complex invariant manifolds are not Abelian
varieties, and the billiard map is no more algebraic. A Poncelet-like theorem
for such system is known. We give explicit sufficient conditions both for
closed geodesics and periodic billiard orbits on Q and discuss their relation
with the elliptic KdV solutions and elliptic Calogero systemComment: 23 pages, Latex, 1 figure Postscrip
Development of an unsteady aerodynamic analysis for finite-deflection subsonic cascades
An unsteady potential flow analysis, which accounts for the effects of blade geometry and steady turning, was developed to predict aerodynamic forces and moments associated with free vibration or flutter phenomena in the fan, compressor, or turbine stages of modern jet engines. Based on the assumption of small amplitude blade motions, the unsteady flow is governed by linear equations with variable coefficients which depend on the underlying steady low. These equations were approximated using difference expressions determined from an implicit least squares development and applicable on arbitrary grids. The resulting linear system of algebraic equations is block tridiagonal, which permits an efficient, direct (i.e., noniterative) solution. The solution procedure was extended to treat blades with rounded or blunt edges at incidence relative to the inlet flow
Three-dimensional quasi-periodic shifted Green function throughout the spectrum--including Wood anomalies
This work presents an efficient method for evaluation of wave scattering by
doubly periodic diffraction gratings at or near "Wood anomaly frequencies". At
these frequencies, one or more grazing Rayleigh waves exist, and the lattice
sum for the quasi-periodic Green function ceases to exist. We present a
modification of this sum by adding two types of terms to it. The first type
adds linear combinations of "shifted" Green functions, ensuring that the
spatial singularities introduced by these terms are located below the grating
and therefore outside of the physical domain. With suitable coefficient choices
these terms annihilate the growing contributions in the original lattice sum
and yield algebraic convergence. Convergence of arbitrarily high order can be
obtained by including sufficiently many shifts. The second type of added terms
are quasi-periodic plane wave solutions of the Helmholtz equation which
reinstate certain necessary grazing modes without leading to blow-up at Wood
anomalies. Using the new quasi-periodic Green function, we establish, for the
first time, that the Dirichlet problem of scattering by a smooth doubly
periodic scattering surface at a Wood frequency is uniquely solvable. We also
present an efficient high-order numerical method based on the this new Green
function for the problem of scattering by doubly periodic three-dimensional
surfaces at and around Wood frequencies. We believe this is the first solver in
existence that is applicable to Wood-frequency doubly periodic scattering
problems. We demonstrate the proposed approach by means of applications to
problems of acoustic scattering by doubly periodic gratings at various
frequencies, including frequencies away from, at, and near Wood anomalies
Phase Structure and Compactness
In order to study the influence of compactness on low-energy properties, we
compare the phase structures of the compact and non-compact two-dimensional
multi-frequency sine-Gordon models. It is shown that the high-energy scaling of
the compact and non-compact models coincides, but their low-energy behaviors
differ. The critical frequency at which the sine-Gordon model
undergoes a topological phase transition is found to be unaffected by the
compactness of the field since it is determined by high-energy scaling laws.
However, the compact two-frequency sine-Gordon model has first and second order
phase transitions determined by the low-energy scaling: we show that these are
absent in the non-compact model.Comment: 21 pages, 5 figures, minor changes, final version, accepted for
publication in JHE
Fast multi-dimensional scattered data approximation with Neumann boundary conditions
An important problem in applications is the approximation of a function
from a finite set of randomly scattered data . A common and powerful
approach is to construct a trigonometric least squares approximation based on
the set of exponentials . This leads to fast numerical
algorithms, but suffers from disturbing boundary effects due to the underlying
periodicity assumption on the data, an assumption that is rarely satisfied in
practice. To overcome this drawback we impose Neumann boundary conditions on
the data. This implies the use of cosine polynomials as basis
functions. We show that scattered data approximation using cosine polynomials
leads to a least squares problem involving certain Toeplitz+Hankel matrices. We
derive estimates on the condition number of these matrices. Unlike other
Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context
cannot be diagonalized by the discrete cosine transform, but they still allow a
fast matrix-vector multiplication via DCT which gives rise to fast conjugate
gradient type algorithms. We show how the results can be generalized to higher
dimensions. Finally we demonstrate the performance of the proposed method by
applying it to a two-dimensional geophysical scattered data problem
Optical signatures of nonlocal plasmons in graphene
We theoretically investigate under which conditions nonlocal plasmon response
in monolayer graphene can be detected. To this purpose, we study optical
scattering off graphene plasmon resonances coupled using a subwavelength
dielectric grating. We compute the graphene conductivity using the Random Phase
Approximation (RPA) obtaining a nonlocal conductivity and we calculate the
optical scattering of the graphene-grating structure. We then compare this with
the scattering amplitudes obtained if graphene is modeled by the local RPA
conductivity commonly used in the literature. We find that the graphene plasmon
wavelength calculated from the local model may deviate up to from the
more accurate nonlocal model in the small-wavelength (large-) regime. We
also find substantial differences in the scattering amplitudes obtained from
the two models. However, these differences in response are pronounced only for
small grating periods and low temperatures compared to the Fermi temperature.Comment: Accepted for publication in Physical Review B. 15 pages, 9 figure
Quantum Logical States and Operators for Josephson-like Systems
We give a formal algebraic description of Josephson-type quantum dynamical
systems, i.e., Hamiltonian systems with a cos theta-like potential term. The
two-boson Heisenberg algebra plays for such systems the role that the h(1)
algebra does for the harmonic oscillator. A single Josephson junction is
selected as a representative of Josephson systems. We construct both logical
states (codewords) and logical (gate) operators in the superconductive regime.
The codewords are the even and odd coherent states of the two-boson algebra:
they are shift-resistant and robust, due to squeezing. The logical operators
acting on the qubit codewords are expressed in terms of operators in the
enveloping of the two-boson algebra. Such a scheme appears to be relevant for
quantum information applications.Comment: 12 pages in RevTex. In press, Journal of Physics A/Letter
Lattice potentials and fermions in holographic non Fermi-liquids: hybridizing local quantum criticality
We study lattice effects in strongly coupled systems of fermions at a finite
density described by a holographic dual consisting of fermions in
Anti-de-Sitter space in the presence of a Reissner-Nordstrom black hole. The
lattice effect is encoded by a periodic modulation of the chemical potential
with a wavelength of order of the intrinsic length scales of the system. This
corresponds with a highly complicated "band structure" problem in AdS, which we
only manage to solve in the weak potential limit. The "domain wall" fermions in
AdS encoding for the Fermi surfaces in the boundary field theory diffract as
usually against the periodic lattice, giving rise to band gaps. However, the
deep infrared of the field theory as encoded by the near horizon AdS2 geometry
in the bulk reacts in a surprising way to the weak potential. The hybridization
of the fermions bulk dualizes into a linear combination of CFT1 "local quantum
critical" propagators in the bulk, characterized by momentum dependent
exponents displaced by lattice Umklapp vectors. This has the consequence that
the metals showing quasi-Fermi surfaces cannot be localized in band insulators.
In the AdS2 metal regime, where the conformal dimension of the fermionic
operator is large and no Fermi surfaces are present at low T/\mu, the lattice
gives rise to a characteristic dependence of the energy scaling as a function
of momentum. We predict crossovers from a high energy standard momentum AdS2
scaling to a low energy regime where exponents found associated with momenta
"backscattered" to a lower Brillioun zone in the extended zone scheme. We
comment on how these findings can be used as a unique fingerprint for the
detection of AdS2 like "pseudogap metals" in the laboratory.Comment: 42 pages, 5 figures; v2, minor correction, to appear in JHE
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