A New Condition for Blow-up Solutions to Discrete Semilinear Heat Equations on Networks

The purpose of this paper is to introduce a new condition $$\hbox{(C)$\hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{2}+\gamma,\,\,u>0$}$$ for some $\alpha, \beta, \gamma>0$ with $0<\beta\leq\frac{\left(\alpha-2\right)\lambda_{0}}{2}$, where $\lambda_{0}$ is the first eigenvalue of discrete Laplacian $\Delta_{\omega}$, with which we obtain blow-up solutions to discrete semilinear heat equations \begin{equation*} \begin{cases} u_{t}\left(x,t\right)=\Delta_{\omega}u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in S\times\left(0,+\infty\right),\\ u\left(x,t\right)=0, & \left(x,t\right)\in\partial S\times\left[0,+\infty\right),\\ u\left(x,0\right)=u_{0}\geq0(nontrivial), & x\in\overline{S} \end{cases} \end{equation*} on a discrete network $S$. In fact, it will be seen that the condition (C) improves the conditions known so far.Comment: 19 page

Latency-Optimal Uplink Scheduling Policy in Training-based Large-Scale Antenna Systems

In this paper, an uplink scheduling policy problem to minimize the network latency, defined as the air-time to serve all of users with a quality-of-service (QoS), under an energy constraint is considered in a training-based large-scale antenna systems (LSAS) employing a simple linear receiver. An optimal algorithm providing the exact latency-optimal uplink scheduling policy is proposed with a polynomial-time complexity. Via numerical simulations, it is shown that the proposed scheduling policy can provide several times lower network latency over the conventional ones in realistic environments. In addition, the proposed scheduling policy and its network latency are analyzed asymptotically to provide better insights on the system behavior. Four operating regimes are classified according to the average received signal quality, $\rho$, and the number of BS antennas, $M$. It turns out that orthogonal pilots are optimal only in the regime $\rho\gg1$ and $M\ll \log^2\rho$. In other regimes ($\rho\ll 1$ or $M\gg \log^2\rho$), it turns out that non-orthogonal pilots become optimal. More rigorously, the use of non-orthogonal pilots can reduce the network latency by a factor of $\Theta(M)$ when $\rho\ll 1$ or by a factor of $\Theta(\sqrt{M}/\log M)$ when $\rho\gg 1$ and $M\gg \log\rho$, which would be a critical guideline for designing 5G future cellular systems.Comment: submitted to IEEE Transactions on Information Theor

A New Condition for the Concavity Method of Blow-up Solutions to p-Laplacian Parabolic Equations

In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equations \begin{equation} \begin{cases} u_{t}\left(x,t\right)=\mbox{div}(|\nabla u\left(x,t\right)|^{p-2}\nabla u(x,t))+f(u(x,t)), & \left(x,t\right)\in \Omega\times\left(0,+\infty\right), \newline u\left(x,t\right)=0, & \left(x,t\right)\in\partial \Omega\times\left[0,+\infty\right), \newline u\left(x,0\right)=u_{0}\geq0, & x\in\overline{\Omega}, \end{cases} \end{equation} where $p\geq2$ and $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ $(N\geq1)$ with smooth boundary $\partial\Omega$. The main contribution of this work is to introduce a new condition \mbox{$(C_{p})$$\hspace{1cm} \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{p}+\gamma,\,\,u>0$} for some $\alpha, \beta, \gamma>0$ with $0<\beta\leq\frac{\left(\alpha-p\right)\lambda_{1, p}}{p}$, where $\lambda_{1, p}$ is the first eigenvalue of p-Laplacian $\Delta_{p}$, and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition $(C_{p})$ improves the conditions ever known so far.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1706.0349

A New Condition for the Concavity Method of Blow-up Solutions to Semilinear Heat Equations

In this paper, we consider the semilinear heat equations under Dirichlet boundary condition u_{t}\left(x,t\right)=\Delta u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in \Omega\times\left(0,+\infty\right), u\left(x,t\right)=0, & \left(x,t\right)\in\partial \Omega\times\left[0,+\infty\right), u\left(x,0\right)=u_{0}\geq0, & x\in\overline{\Omega}, where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$ $(N\geq1)$ with smooth boundary $\partial\Omega$. The main contribution of our work is to introduce a new condition $$(C) \alpha \int_{0}^{u}f(s)ds \leq uf(u)+\beta u^{2}+\gamma,\,\,u>0$$ for some $\alpha, \beta, \gamma>0$ with $0<\beta\leq\frac{\left(\alpha-2\right)\lambda_{0}}{2}$, where $\lambda_{0}$ is the first eigenvalue of Laplacian $\Delta$, and we use the concavity method to obtain the blow-up solutions to the semilinear heat equations. In fact, it will be seen that the condition (C) improves the conditions known so far.Comment: 7 page

Bond-site duality and phase transition nature of explosive percolations on a two-dimensional lattice

To establish the bond-site duality of explosive percolations in 2 dimension, the site and bond explosive percolation models are carefully defined on a square lattice. By studying the cluster distribution function and the behavior of the second largest cluster, it is shown that the duality in which the transition is discontinuous exists for the pairs of the site model and the corresponding bond model which relatively enhances the intra-bond occupation. In contrast the intra-bond-suppressed models which have no corresponding site models undergo the continuous transition and satisfy the normal scaling ansatz as ordinary percolation.Comment: 5 pages, 2 figure

Explosive site percolation with a product rule

We study the site percolation under Achlioptas process (AP) with a product rule in a $2-dimensional$ (2D) square lattice. From the measurement of the cluster size distribution, $P_s$, we find that $P_s$ has a very robust power-law regime followed by a stable hump near the transition threshold. Based on the careful analysis on the $P_s$ distribution, we show that the transition should be discontinuous. The existence of the hysteresis loop in order parameter also verifies that the transition is discontinuous in 2D. Moreover we also show that the transition nature from the product rule is not the same as that from a sum rule in 2D

Partitions weighted by the parity of the crank

A partition statistic ` crank' gives combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula, Ramanujan type congruences, and q-series identities that the number of partitions with even crank $M_e(n)$ minus the number of partitions with odd crank $M_o(n)$ satisfies. For example, we show that $M_e(5n+4)-M_o(5n+4)\equiv 0 \pmod 5.$ We also determine the exact values of $M_e(n)-M_o(n)$ in case of partitions into distinct parts, which are at most two and zero for infinitely many $n$

Automatic Detection and Decoding of Photogrammetric Coded Targets

Close-range Photogrammetry is widely used in many industries because of the cost effectiveness and efficiency of the technique. In this research, we introduce an automated coded target detection method which can be used to enhance the efficiency of the Photogrammetry.Comment: 3 pages, 4 figures, Electronics, Information and Communications (ICEIC), 2014 International Conference o

(2+1) dimensional black holes in warped product scheme

Exploiting a multiply warped product manifold scheme, we study the interior solutions of the Banados-Teitelboim-Zanelli black holes and the exterior solutions of the de Sitter black holes in the (2+1) dimensions.Comment: 13 page

Warped products and Reissner-Nordstrom metric

We study a multiply warped products manifold associated with the Reissner-Nordstrom metric to investigate the physical properties inside the black hole event horizons. It is shown that, different from the uncharged Schwarzschild metric, the Ricci curvature components inside the Reissner-Nordstrom black hole horizons are nonvanishing, while the Einstein scalar curvature vanishes even in the interior of the charged metric. Introducing a perfect fluid inside the Reissor-Nordstrom black hole, it is also shown that the charge plays effective roles of decreasing the mass-energy density and the pressure of the fluid inside the black hole.Comment: 7 page