Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

For both localized and periodic initial data, we prove local existence in classical energy space $H^s, s>\frac{3}{2}$, for a class of dispersive equations $u_{t}+(n(u))_{x}+Lu_{x}=0$ with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators $L$ whose symbol is of temperate growth, and $n(\cdot)$ in local Sobolev space $H^{s+2}_{\mathrm{loc}}(\mathbb{R})$. In particular, the results include non-smooth and exponentially growing nonlinearities. Our proof is based on a combination of semi-group methods and a new composition result for Besov spaces. In particular, we extend a previous result for Nemytskii operators on Besov spaces on $\mathbb{R}$ to the periodic setting by using the difference-derivative characterization of Besov spaces

Power Partial Isometry Index and Ascent of a Finite Matrix

We give a complete characterization of nonnegative integers $j$ and $k$ and a positive integer $n$ for which there is an $n$-by-$n$ matrix with its power partial isometry index equal to $j$ and its ascent equal to $k$. Recall that the power partial isometry index $p(A)$ of a matrix $A$ is the supremum, possibly infinity, of nonnegative integers $j$ such that $I, A, A^2, \ldots, A^j$ are all partial isometries while the ascent $a(A)$ of $A$ is the smallest integer $k\ge 0$ for which $\ker A^k$ equals $\ker A^{k+1}$. It was known before that, for any matrix $A$, either $p(A)\le\min\{a(A), n-1\}$ or $p(A)=\infty$. In this paper, we prove more precisely that there is an $n$-by-$n$ matrix $A$ such that $p(A)=j$ and $a(A)=k$ if and only if one of the following conditions holds: (a) $j=k\le n-1$, (b) $j\le k-1$ and $j+k\le n-1$, and (c) $j\le k-2$ and $j+k=n$. This answers a question we asked in a previous paper.Comment: 11 page

Numerical Ranges of KMS Matrices

A KMS matrix is one of the form J_n(a)=[{array}{ccccc} 0 & a & a^2 &... & a^{n-1} & 0 & a & \ddots & \vdots & & \ddots & \ddots & a^2 & & & \ddots & a 0 & & & & 0{array}] for $n\ge 1$ and $a$ in $\mathbb{C}$. Among other things, we prove the following properties of its numerical range: (1) $W(J_n(a))$ is a circular disc if and only if $n=2$ and $a\neq 0$, (2) its boundary $\partial W(J_n(a))$ contains a line segment if and only if $n\ge 3$ and $|a|=1$, and (3) the intersection of the boundaries $\partial W(J_n(a))$ and $\partial W(J_n(a)[j])$ is either the singleton \{\min\sigma(\re J_n(a))\} if $n$ is odd, $j=(n+1)/2$ and $|a|>1$, or the empty set $\emptyset$ if otherwise, where, for any $n$-by-$n$ matrix $A$, $A[j]$ denotes its $j$th principal submatrix obtained by deleting its $j$th row and $j$th column ($1\le j\le n$), \re A its real part $(A+A^*)/2$, and $\sigma(A)$ its spectrum.Comment: 35 page

Throughput and Robustness Guaranteed Beam Tracking for mmWave Wireless Networks

With the increasing demand of ultra-high-speed wireless communications and the existing low frequency band (e.g., sub-6GHz) becomes more and more crowded, millimeter-wave (mmWave) with large spectra available is considered as the most promising frequency band for future wireless communications. Since the mmWave suffers a serious path-loss, beamforming techniques shall be adopted to concentrate the transmit power and receive region on a narrow beam for achieving long distance communications. However, the mobility of users will bring frequent beam handoff, which will decrease the quality of experience (QoE). Therefore, efficient beam tracking mechanism should be carefully researched. However, the existing beam tracking mechanisms concentrate on system throughput maximization without considering beam handoff and link robustness. This paper proposes a throughput and robustness guaranteed beam tracking mechanism for mobile mmWave communication systems which takes account of both system throughput and handoff probability. Simulation results show that the proposed throughput and robustness guaranteed beam tracking mechanism can provide better performance than the other beam tracking mechanisms.Comment: Accepted by IEEE/CIC ICCC 201

Evolutionary dynamics of cooperation on interdependent networks with Prisoner's Dilemma and Snowdrift Game

The world in which we are living is a huge network of networks and should be described by interdependent networks. The interdependence between networks significantly affects the evolutionary dynamics of cooperation on them. Meanwhile, due to the diversity and complexity of social and biological systems, players on different networks may not interact with each other by the same way, which should be described by multiple models in evolutionary game theory, such as the Prisoner's Dilemma and Snowdrift Game. We therefore study the evolutionary dynamics of cooperation on two interdependent networks playing different games respectively. We clearly evidence that, with the increment of network interdependence, the evolution of cooperation is dramatically promoted on the network playing Prisoner's Dilemma. The cooperation level of the network playing Snowdrift Game reduces correspondingly, although it is almost invisible. In particular, there exists an optimal intermediate region of network interdependence maximizing the growth rate of the evolution of cooperation on the network playing Prisoner's Dilemma. Remarkably, players contacting with other network have advantage in the evolution of cooperation than the others on the same network.Comment: 6 pages, 6 figure