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

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

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

On the unsplittable minimal zero-sum sequences over finite cyclic groups of prime order

Let $p > 155$ be a prime and let $G$ be a cyclic group of order $p$. Let $S$ be a minimal zero-sum sequence with elements over $G$, i.e., the sum of elements in $S$ is zero, but no proper nontrivial subsequence of $S$ has sum zero. We call $S$ is unsplittable, if there do not exist $g$ in $S$ and $x,y \in G$ such that $g=x+y$ and $Sg^{-1}xy$ is also a minimal zero-sum sequence. In this paper we show that if $S$ is an unsplittable minimal zero-sum sequence of length $|S|= \frac{p-1}{2}$, then $S=g^{\frac{p-11}{2}}(\frac{p+3}{2}g)^4(\frac{p-1}{2}g)$ or $g^{\frac{p-7}{2}}(\frac{p+5}{2}g)^2(\frac{p-3}{2}g)$. Furthermore, if $S$ is a minimal zero-sum sequence with $|S| \ge \frac{p-1}{2}$, then \ind(S) \leq 2.Comment: 11 page

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

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 $N\geq 1$, $M(t)=am\left( t\right) +b$, $m\in C(\mathbb{R}^{+})$ and $f(x,u)=g(x,u)+h(x)u^{q-1}$. We require that $f$ 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 RN\mathbb{R}^{N}

We are concerned with a class of Kirchhoff type equations in $\mathbb{R}^{N}$ 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 $N\geq 1,$ $\lambda>0$ is a parameter, $M(t)=am(t)+b$ with $a,b>0$ and $m\in C(\mathbb{R}^{+},\mathbb{R}^{+})$, $V\in C(\mathbb{R}^{N},\mathbb{R}^{+})$ and $f\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R})$ satisfying $\lim_{|u|\rightarrow \infty }f(x,u) /|u|^{k-1}=q(x)$ uniformly in $x\in \mathbb{R}^{N}$ for any $2<k<2^{\ast}$($2^{\ast}=\infty$ for $N=1,2$ and $2^{\ast}=2N/(N-2)$ for $N\geq 3$). Unlike most other papers on this problem, we are more interested in the effects of the functions $m$ and $q$ 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