45,625 research outputs found
Topological Symmetries of R^3, II
If a fintie group G acts topologically and faithfully on R^3, then G is a
subgroup of O(3
Dynamic process of free space excitation of asymmetry resonant microcavity
The underlying physics and detailed dynamical processes of the free space
beam excitation to the asymmetry resonant microcavity are studied numerically.
Taking the well-studied quadrupole deformed microcavity as an example, we use a
Gaussian beam to excite the high-Q mode. The simulation provides a powerful
platform to study the underlying physics. The transmission spectrum and
intracavity energy can be obtained directly. Irregular transmission spectrum
was observed, showing asymmetric Fano-type lineshapes which could be attributed
to interference between the different light paths. Then excitation efficiencies
about the aim distance of the incident Gaussian beam and the rotation angle of
the cavity were studied, showing great consistence with the reversal of
emission efficiencies. By projecting the position dependent excitation
efficiency to the phase space, the correspondence between the excitation and
emission was demonstrated. In addition, we compared the Husimi distributions of
the excitation processes and provided more direct evidences of the dynamical
tunneling process in the excitation process
An Optimization Method of Asymmetric Resonant Cavities for Unidirectional Emission
In this paper, we studied the repeatability and accuracy of the ray
simulation for one kind of Asymmetric Resonant Cavities (ARCs)
Half-Quadrupole-Half-Circle shaped cavity, and confirmed the robustness of the
directionality about the shape errors. Based on these, we proposed a
hill-climbing algorithm to optimize the ARCs for unidirectional emission.
Different evaluation functions of directionality were tested and we suggested
using the function of energy contained in a certain angle for highly collimated
and unidirect ional emission. By this method, we optimized the ARCs to obtain
about 0.46 of the total radiated energy in divergence angle of 40 degree in the
far field. This optimization method is very powerful for the shape engineering
of ARCs and could be applied in future studies of ARCs with specific emission
properties
Chevalley's theorem for affine Nash groups
We formulate and prove Chevalley's theorem in the setting of affine Nash
groups. As a consequence, we show that the semi-direct product of two almost
linear Nash groups is still an almost linear Nash group
Local Coefficients Revisited
Two simple "simplicial approximation" tricks are invoked to prove basic
results involving (co)-homology with local coefficients
Propagation dynamics of Fisher-KPP equation with time delay and free boundaries
Incorporating free boundary into time-delayed reaction-diffusion equations
yields a compatible condition that guarantees the well-posedness of the initial
value problem. With the KPP type nonlinearity we then establish a
vanishing-spreading dichotomy result. Further, when the spreading happens, we
show that the spreading speed and spreading profile are nonlinearly determined
by a delay-induced nonlocal semi-wave problem. It turns out that time delay
slows down the spreading speed.Comment: 38 pages, 0 figure
On the unsplittable minimal zero-sum sequences over finite cyclic groups of prime order
Let be a prime and let be a cyclic group of order . Let
be a minimal zero-sum sequence with elements over , i.e., the sum of
elements in is zero, but no proper nontrivial subsequence of has sum
zero. We call is unsplittable, if there do not exist in and such that and is also a minimal zero-sum sequence.
In this paper we show that if is an unsplittable minimal zero-sum sequence
of length , then
or
. Furthermore, if is a
minimal zero-sum sequence with , then \ind(S) \leq 2.Comment: 11 page
On the indefinite Kirchhoff type problems with local sublinearity and linearity
The purpose of this paper is to study the indefinite Kirchhoff type problem:
\begin{equation*} \left\{ \begin{array}{ll} M\left(
\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+u^{2})dx\right) \left[ -\Delta u+u\right]
=f(x,u) & \text{in }\mathbb{R}^{N}, \\ 0\leq u\in H^{1}\left(
\mathbb{R}^{N}\right), & \end{array} \right. \end{equation*} where ,
, and . We require that is \textquotedblleft
local\textquotedblright\ sublinear at the origin and \textquotedblleft
local\textquotedblright\ linear at infinite. Using the mountain pass theorem
and Ekeland variational principle, the existence and multiplicity of nontrivial
solutions are obtained. In particular, the criterion of existence of three
nontrivial solutions is established
The effect of nonlocal term on the superlinear Kirchhoff type equations in
We are concerned with a class of Kirchhoff type equations in
as follows: \begin{equation*} \left\{ \begin{array}{ll} -M\left(
\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right) \Delta u+\lambda V\left( x\right)
u=f(x,u) & \text{in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), &
\end{array}% \right. \end{equation*}% where is a
parameter, with and , and
satisfying
uniformly in for any ( for and
for ). Unlike most other papers on this problem,
we are more interested in the effects of the functions and on the
number and behavior of solutions. By using minimax method as well as
Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity
of positive solutions for the above problem
The effect of heterogeneity on flocking behavior and systemic risk
The goal of this paper is to study organized flocking behavior and systemic
risk in heterogeneous mean-field interacting diffusions. We illustrate in a
number of case studies the effect of heterogeneity in the behavior of systemic
risk in the system, i.e., the risk that several agents default simultaneously
as a result of interconnections. We also investigate the effect of
heterogeneity on the "flocking behavior" of different agents, i.e., when agents
with different dynamics end up following very similar paths and follow closely
the mean behavior of the system. Using Laplace asymptotics, we derive an
asymptotic formula for the tail of the loss distribution as the number of
agents grows to infinity. This characterizes the tail of the loss distribution
and the effect of the heterogeneity of the network on the tail loss
probability
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