On Index Coding and Graph Homomorphism

In this work, we study the problem of index coding from graph homomorphism perspective. We show that the minimum broadcast rate of an index coding problem for different variations of the problem such as non-linear, scalar, and vector index code, can be upper bounded by the minimum broadcast rate of another index coding problem when there exists a homomorphism from the complement of the side information graph of the first problem to that of the second problem. As a result, we show that several upper bounds on scalar and vector index code problem are special cases of one of our main theorems. For the linear scalar index coding problem, it has been shown in [1] that the binary linear index of a graph is equal to a graph theoretical parameter called minrank of the graph. For undirected graphs, in [2] it is shown that $\mathrm{minrank}(G) = k$ if and only if there exists a homomorphism from $\bar{G}$ to a predefined graph $\bar{G}_k$. Combining these two results, it follows that for undirected graphs, all the digraphs with linear index of at most k coincide with the graphs $G$ for which there exists a homomorphism from $\bar{G}$ to $\bar{G}_k$. In this paper, we give a direct proof to this result that works for digraphs as well. We show how to use this classification result to generate lower bounds on scalar and vector index. In particular, we provide a lower bound for the scalar index of a digraph in terms of the chromatic number of its complement. Using our framework, we show that by changing the field size, linear index of a digraph can be at most increased by a factor that is independent from the number of the nodes.Comment: 5 pages, to appear in "IEEE Information Theory Workshop", 201

Linearized Holographic Isotropization at Finite Coupling

We study holographic isotropization of an anisotropic homogeneous non-Abelian strongly coupled plasma in the presence of Gauss-Bonnet corrections. It was verified before that one can linearize Einstein's equations around the final black hole background and simplify the complicated setup. Using this approach, we study the expectation value of the boundary stress tensor. Although we consider small values of the Gauss-Bonnet coupling constant, it is found that finite coupling leads to significant increasing of the thermalization time. By including higher order corrections in linearization, we extend the results to study the effect of the Gauss-Bonnet coupling on the entropy production on the event horizon.Comment: V2 and v3 are the same! version 4 is ne

Surfing in the phase space of spin-orbit coupling in binary asteroid systems

For a satellite with an irregular shape, which is the common shape among asteroids, the well-known spin-orbit resonance problem could be changed to a spin-orbit coupling problem since a decoupled model does not accurately capture the dynamics of the system. In this paper, having provided a definition for close binary asteroid systems, we explore the structure of the phase space in a classical Hamiltonian model for spin-orbit coupling in a binary system. To map out the geography of resonances analytically and the cartography of resonances numerically, we reformulate a fourth-order gravitational potential function, in Poincare variables, via Stokes coefficients. For a binary system with a near-circular orbit, isolating the Hamiltonian near each resonance yields the pendulum model. Analysis of the results shows the geographical information, including the location and width of resonances, is modified due to the prominent role of the semi-major axis in the spin-orbit coupling model but not structurally altered. However, this resulted in modified Chirikov criterion to predict onset of large-scale chaos. For a binary system with arbitrary closed orbit, we thoroughly surf in the phase space via cartography of resonances created by fast Lyapunov indicator (FLI) maps. The numerical study confirms the analytical results, provides insight into the spin-orbit coupling, and shows some bifurcations in the secondary resonances which can occur due to material transfer. Also, we take the (65803) Didymos binary asteroid as a case to show analytical and numerical results.Comment: 16 pages, 16 figures, accepted for publication in MNRA

On AVCs with Quadratic Constraints

In this work we study an Arbitrarily Varying Channel (AVC) with quadratic power constraints on the transmitter and a so-called "oblivious" jammer (along with additional AWGN) under a maximum probability of error criterion, and no private randomness between the transmitter and the receiver. This is in contrast to similar AVC models under the average probability of error criterion considered in [1], and models wherein common randomness is allowed [2] -- these distinctions are important in some communication scenarios outlined below. We consider the regime where the jammer's power constraint is smaller than the transmitter's power constraint (in the other regime it is known no positive rate is possible). For this regime we show the existence of stochastic codes (with no common randomness between the transmitter and receiver) that enables reliable communication at the same rate as when the jammer is replaced with AWGN with the same power constraint. This matches known information-theoretic outer bounds. In addition to being a stronger result than that in [1] (enabling recovery of the results therein), our proof techniques are also somewhat more direct, and hence may be of independent interest.Comment: A shorter version of this work will be send to ISIT13, Istanbul. 8 pages, 3 figure