Primordial Perturbation of Dark Matter as a Novel Probe of Very Early Universe

Dark matter(DM) is the only possible candidate which would be apart from the thermal equilibrium before Big Bang nucleosynthesis(BBN) in accordance with current DM searches. In this work, we report a generic scenario that primordial perturbation of dark matter(PPDM) can be, effectively, generated and encoded with primordial information of very early universe up to the reheating era. We present an analytical solution of the whole evolution of PPDM. A novel and strong constraint on the reheating process imposed by primordial gravitational wave(PGW) is obtained for the first time. It indicates the ratio of PGW to primordial curvature perturbation(PCP) is not only dependent on the slow-roll spectral index but also, strongly, on the decay process of inflaton at reheating. For very generic reheating process, our result provides a natural explanation of the paucity of PGW in current observations.Comment: 5 pages, 3 figures, Typos on Figure.2 and Figure.3 are corrected in new versio

MeV Dark Matter in light of the Small Scale Crisis

The small-scale crisis is one of the most outstanding puzzles in modern cosmology and astrophysics. It may imply a suppression of matter perturbation at small scale. In this work, by taking into account of the gravitational effects from the non-equilibrium production of DM, we propose a new mechanism, which can realize such desired suppression and alleviate the crisis within the framework of cold dark matter (DM) and simple inflation. Moreover, in this new mechanism, we establish a novel relation between the particle mass of DM, $m_\chi$, and the critical scale of suppression, $k_\star^{-1}$. As $k_\star^{-1}$ will be further constrained in future astrophysical observations, $m_\chi$ can be constrained accordingly with this relation. It thus provides a new method in complementary to other existing strategies of determining $m_\chi$. Furthermore, to illustrate our theoretical prediction, we consider a suppression at $k_\star^{-1}=1~\text{kpc}$ that can partially alleviate the small-scale crisis, and obtain $m_\chi=2.2~\text{MeV}$ for realizing such suppression. Then we plot the power spectrum of linear matter perturbation for this case, and illustrate a salient feature of the suppression that can serves a smoking-gun signature of this new mechanism in future observations.Comment: 5 pages, 5 figure

Thermally Producing and Weakly Freezing-out Dark Matter in Bouncing Universe

We investigate the production and freeze-out of dark matter with a constant thermally averaged cross-section in a generic bouncing universe framework. Our result shows that, there is a novel avenue that dark matter is produced thermally and take a weakly freezing-out process, besides two previously known cases, the thermally production & strongly freezing-out case and the non-thermally production & weakly freezing-out case, in which the relic abundance of dark matter are inverse and proportional to its cross-section respectively. We calculated the relic abundance of dark matter for this new case, and find its relic abundance is independent of its cross-section. We also present the cosmological constraints on the cross-section and mass relation of dark matter for this new case.Comment: 6 pages, 4 figure

Thermal Fluctuations of Dark Matter in Bouncing Cosmology

We investigate the statistical nature of the dark matter particles produced in bouncing cosmology, especially, the evolution of its thermal fluctuations. By explicitly deriving and solving the equation of motion of super-horizon mode, we fully determine the evolution of thermal perturbation of dark matter in a generic bouncing background. And we also show that the evolution of super-horizon modes is stable and will not ruin the background evolution of a generic bouncing universe till the Planck scale. Given no super-horizon thermal perturbation of dark matter appears in standard inflation scenario such as WIMP(-less) miracles, such super-horizon thermal perturbation of dark matter generated during the generic bouncing universe scenario may be significant for testing and distinguishing these two scenario in near future.Comment: 19 pages, 2 figure

On the number of a SDRs of a valued (t,n)-family

A system of distinct representatives (SDR) of a family $F = (A_1, \cdots, A_n)$ is a sequence $(x_1, \cdots, x_n)$ of $n$ distinct elements with $x_i \in A_i$ for $1 \le i \le n$. Let $N(F)$ denote the number of SDRs of a family $F$; two SDRs are considered distinct if they are different in at least one component. For a nonnegative integer $t$, a family $F=(A_1,\cdots,A_n)$ is called a $(t,n)$-family if the union of any $k\ge 1$ sets in the family contains at least $k+t$ elements. The famous Hall's Theorem says that $N(F)\ge 1$ if and only if $F$ is a $(0,n)$-family. Denote by $M(t,n)$ the minimum number of SDRs in a $(t,n)$-family. The problem of determining $M(t,n)$ and those families containing exactly $M(t,n)$ SDRs was first raised by Chang [European J. Combin.{\bf 10}(1989), 231-234]. He solved the cases when $0\le t\le 2$ and gave a conjecture for $t\ge 3$. In this paper, we solve the conjecture. In fact, we get a more general result for so-called valued $(t,n)$-family

Decentralized Primary Frequency Control in Power Networks

We augment existing generator-side primary frequency control with load-side control that are local, ubiquitous, and continuous. The mechanisms on both the generator and the load sides are decentralized in that their control decisions are functions of locally measurable frequency deviations. These local algorithms interact over the network through nonlinear power flows. We design the local frequency feedback control so that any equilibrium point of the closed-loop system is the solution to an optimization problem that minimizes the total generation cost and user disutility subject to power balance across entire network. With Lyapunov method we derive a sufficient condition ensuring an equilibrium point of the closed-loop system is asymptotically stable. Simulation demonstrates improvement in both the transient and steady-state performance over the traditional control only on the generators, even when the total control capacity remains the same.Comment: 7 pages, 2 figures. Submitted to CDC 201

Path covering number and L(2,1)-labeling number of graphs

A {\it path covering} of a graph $G$ is a set of vertex disjoint paths of $G$ containing all the vertices of $G$. The {\it path covering number} of $G$, denoted by $P(G)$, is the minimum number of paths in a path covering of $G$. An {\sl $k$-L(2,1)-labeling} of a graph $G$ is a mapping $f$ from $V(G)$ to the set ${0,1,...,k}$ such that $|f(u)-f(v)|\ge 2$ if $d_G(u,v)=1$ and $|f(u)-f(v)|\ge 1$ if $d_G(u,v)=2$. The {\sl L(2,1)-labeling number $\lambda (G)$} of $G$ is the smallest number $k$ such that $G$ has a $k$-L(2,1)-labeling. The purpose of this paper is to study path covering number and L(2,1)-labeling number of graphs. Our main work extends most of results in [On island sequences of labelings with a condition at distance two, Discrete Applied Maths 158 (2010), 1-7] and can answer an open problem in [On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005), 208-223]

A Note on Roman \{2\}-domination problem in graphs

For a graph $G=(V,E)$, a Roman $\{2\}$-dominating function (R2DF)$f:V\rightarrow \{0,1,2\}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either there exists a neighbor $u\in N(v)$, with $f(u)=2$, or at least two neighbors $x,y\in N(v)$ having $f(x)=f(y)=1$. The weight of a R2DF is the sum $f(V)=\sum_{v\in V}{f(v)}$, and the minimum weight of a R2DF is the Roman $\{2\}$-domination number $\gamma_{\{R2\}}(G)$. A R2DF is independent if the set of vertices having positive function values is an independent set. The independent Roman $\{2\}$-domination number $i_{\{R2\}}(G)$ is the minimum weight of an independent Roman $\{2\}$-dominating function on $G$. In this paper, we show that the decision problem associated with $\gamma_{\{R2\}}(G)$ is NP-complete even when restricted to split graphs. We design a linear time algorithm for computing the value of $i_{\{R2\}}(T)$ for any tree $T$. This answers an open problem raised by Rahmouni and Chellali [Independent Roman $\{2\}$-domination in graphs, Discrete Applied Mathematics 236 (2018), 408-414]. Chellali, Haynes, Hedetniemi and McRae \cite{chellali2016roman} have showed that Roman $\{2\}$-domination number can be computed for the class of trees in linear time. As a generalization, we present a linear time algorithm for solving the Roman $\{2\}$-domination problem in block graphs

The Scale-invariant Power Spectrum of Primordial Curvature Perturbation in CSTB Cosmos

We investigate the spectrum of cosmological perturbations in a bounce cosmos modeled by a scalar field coupled to the string tachyon field (CSTB cosmos). By explicit computation of its primordial spectral index we show the power spectrum of curvature perturbations, generated during the tachyon matter dominated contraction phase, to be nearly scale invariant. We propose a unified space of parameters for a systematic study of inflationary/bouncing cosmologies. We find that CSTB cosmos is dual--in Wands's sense--to the slow-roll inflation model as can be easily seen from this unified parameter space. Guaranteed by the dynamical attractor behavior of CSTB Cosmos, this scale invariance is free of the fine-tuning problem, in contrast to the slow-roll inflation model.Comment: 19 pages, 2 figure

Distributed Automatic Load-Frequency Control with Optimality in Power Systems

With the increasing penetration of renewable energy resources, power systems face new challenges in balancing power supply and demand and maintaining the nominal frequency. This paper studies load control to handle these challenges. In particular, a fully distributed automatic load control (ALC) algorithm, which only needs local measurement and local communication, is proposed. We prove that the load control algorithm globally converges to an optimal operating point which minimizes the total disutility of users, restores the nominal frequency and the scheduled tie-line power flows, and respects the load capacity limits and the thermal constraints of transmission lines. It is further shown that the asymptotic convergence still holds even when inaccurate system parameters are used in the control algorithm. In addition, the global exponential convergence of the reduced ALC algorithm without considering the capacity limits is proved and leveraged to study the dynamical tracking performance and robustness of the algorithm. Lastly, the effectiveness, optimality, and robustness of the proposed algorithm are demonstrated via numerical simulations.Comment: 16 page