Efficient winning strategies in random-turn Maker-Breaker games

We consider random-turn positional games, introduced by Peres, Schramm, Sheffield and Wilson in 2007. A $p$-random-turn positional game is a two-player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is $p$). We analyze the random-turn version of several classical Maker-Breaker games such as the game Box (introduced by Chv\'atal and Erd\H os in 1987), the Hamilton cycle game and the $k$-vertex-connectivity game (both played on the edge set of $K_n$). For each of these games we provide each of the players with a (randomized) efficient strategy which typically ensures his win in the asymptotic order of the minimum value of $p$ for which he typically wins the game, assuming optimal strategies of both players.Comment: 20 page

Linearly Dichroic Plasmonic Lens and Hetero-Chiral Structures

We present theoretical and experimental study of plasmonic Hetero-Chiral structures, comprised of constituents with opposite chirality. We devise, simulate and experimentally demonstrate different schemes featuring selective surface plasmon polariton focusing of orthogonal polarization states and standing plasmonic vortex fields.Comment: 9 pages, 6 figure

Packing, counting and covering Hamilton cycles in random directed graphs

A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so-called Posá ‘rotation-extension’ technique for the undirected analogue. Let D(n, p) denote the random digraph on vertex set [n], obtained by adding each directed edge independently with probability p. Here we present a general and a very simple method, using known results, to attack problems of packing and counting Hamilton cycles in random directed graphs, for every edge-probability p > logC(n)/n. Our results are asymptotically optimal with respect to all parameters and apply equally well to the undirected case

The F2\mathbb{F}_2-Rank and Size of Graphs

We consider the extremal family of graphs of order $2^n$ in which no two vertices have identical neighbourhoods, yet the adjacency matrix has rank only $n$ over the field of two elements. A previous result from algebraic geometry shows that such graphs exist for all even $n$ and do not exist for odd $n$. In this paper we provide a new combinatorial proof for this result, offering greater insight to the structure of graphs with these properties. We introduce a new graph product closely related to the Kronecker product, followed by a construction for such graphs for any even $n$. Moreover, we show that this is an infinite family of strongly-regular quasi-random graphs whose signed adjacency matrices are symmetric Hadamard matrices. Conversely, we provide a combinatorial proof that for all odd $n$, no twin-free graphs of minimal $\mathbb{F}_2$-rank exist, and that the next best-possible rank $(n+1)$ is attainable, which is tight.Comment: Added comparison to the results of Godsil and Royle. We thank Sam Adriaensen for bringing them to our attentio

Efficient winning strategies in random-turn Maker-Breaker games

We consider random-turn positional games, introduced by Peres, Schramm, Sheffield and Wilson in 2007. A p-random-turn positional game is a two-player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is p). We analyze the random-turn version of several classical MakerBreaker games such as the game Box (introduced by Chvátal and Erdős in 1987), the Hamilton cycle game and the k-vertex-connectivity game (both played on the edge set of K n ). For each of these games we provide each of the players with a (randomized) efficient strategy which typically ensures his win in the asymptotic order of the minimum value of p for which he typically wins the game, assuming optimal strategies of both players

On the Precarious Path of Reverse Neuro-Engineering

In this perspective we provide an example for the limits of reverse engineering in neuroscience. We demonstrate that application of reverse engineering to the study of the design principle of a functional neuro-system with a known mechanism, may result in a perfectly valid but wrong induction of the system's design principle. If in the very simple setup we bring here (static environment, primitive task and practically unlimited access to every piece of relevant information), it is difficult to induce a design principle, what are our chances of exposing biological design principles when more realistic conditions are examined? Implications to the way we do Biology are discussed