The NLC-width and clique-width for powers of graphs of bounded tree-width

The $k$-power graph of a graph $G$ is a graph with the same vertex set as $G$, in that two vertices are adjacent if and only if, there is a path between them in $G$ of length at most $k$. A $k$-tree-power graph is the $k$-power graph of a tree, a $k$-leaf-power graph is the subgraph of some $k$-tree-power graph induced by the leaves of the tree.We show that (1) every $k$-tree-power graph has NLC-width at most $k+2$ and clique-width at most $k+2+max\{?\frac{k}{2}??1,0\}$, (2) every $k$-leaf-power graph has NLC-width at most $k$ and clique-width at most $k+max\{?\frac{k}{2}??2,0\}$, and (3) every $k$-power graph of a graph of tree-width $l$ has NLC-width at most ${(k+1)}^{l+1}?1$, and clique-width at most $2?{(k+1)}^{l+1}?2$.     

An asymptotically optimal online algorithm to minimize the total completion time on two multipurpose machines with unit processing times

In the majority of works on online scheduling on multipurpose machines the objective is to minimize the makespan. We, in contrast, consider the objective of minimizing the total completion time. For this purpose, we analyze an online-list scheduling problem of $n$ jobs with unit processing times on a set of two machines working in parallel. Each job belongs to one of two sets of job types. Jobs belonging to the first set can be processed on either of the two machines while jobs belonging to the second set can only be processed on the second machine. We present an online algorithm with a competitive ratio of ${\rho }_{LB}+O\left(\frac{1}{n}\right)$, where ${\rho }_{LB}$ is a lower bound on the competitive ratio of any online algorithm and is equal to $1+{\left(\frac{?\alpha +\sqrt{4{\alpha }^3?{\alpha }^2+2\alpha ?1}}{2{\alpha }^2+1}\right)}^2$ where $\alpha =\frac{1}{3}+\frac{1}{6}{(116?6\sqrt{78})}^{1/3}+\frac{{(58+3\sqrt{78})}^{1/3}}{3{\left(2\right)}^{2/3}}\approx 1.918$. This result implies that our online algorithm is asymptotically optimal.     

Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs

Suppose there exists a Hadamard 2-$(m,\frac{m?1}{2},\frac{m?3}{4})$ design having skew incidence matrix. If there exists a conference graph on $2m?1$ vertices, then there exists a regular Hadamard matrix of order $4m^2$. A conference graph on $2m+3$ vertices yields a regular Hadamard matrix of order $4{(m+1)}^2$.     

Energy dissipation in the Smagorinsky model of turbulence

The Smagorinsky model often severely over-dissipates flows and, consistently, previous estimates of its energy dissipation rate blow up as $Re\to \infty$. This report estimates time averaged model dissipation, $〈{\epsilon }_S〉$, under periodic boundary conditions as$〈{\epsilon }_S〉\le 2\frac{U^3}{L}+{\mathrm{Re}}^{-1}\frac{U^3}{L}+\frac{32}{27}{C}_S^2{\left(\frac{\delta }{L}\right)}^2\frac{U^3}{L}\text{,}$ where $U,L$ are global velocity and length scales and $C_S\simeq 0.1,\delta <1$ are model parameters. Thus, in the absence of boundary layers, the Smagorinsky model does not over dissipate.     

The Klein–Gordon equation in the Anti-de Sitter cosmology

This paper deals with the Klein–Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass μ is larger than $-\frac{1}{4}$, the Cauchy problem is well-posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza–Klein tower and we deduce a uniform decay as ${|t|}^{-\frac{3}{2}}$. We investigate the case $\mu =\frac{{\nu }^2-1}{2}$, $\nu \in {\mathbb{N}}^{⁎}$, which encompasses the gravitational fluctuations, $\nu =4$, and the electromagnetic waves, $\nu =2$. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as ${|t|}^{-2-\sqrt{\mu +\frac{1}{4}}}$, and we get global $L^p$ estimates of Strichartz type. When ν is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We address the cosmological model of the negative-tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza–Klein tower. We establish some $L^2-L^{\infty }$ estimates in suitable weighted Sobolev spaces.