Implementation of Logical Functions in the Game of Life

The Game of Life cellular automaton is a classical example of a massively parallel collision-based computing device. The automaton exhibits mobile patterns, gliders, and generators of the mobile patterns, glider guns, in its evolution. We show how to construct basic logical perations, AND, OR, NOT in space-time configurations of the cellular automaton. Also decomposition of complicated Boolean functions is discussed. Advantages of our technique are demonstrated on an example of binary adder, realized via collision of glider streams

Cactus group and monodromy of Bethe vectors

Cactus group is the fundamental group of the real locus of the Deligne-Mumford moduli space of stable rational curves. This group appears naturally as an analog of the braid group in coboundary monoidal categories. We define an action of the cactus group on the set of Bethe vectors of the Gaudin magnet chain corresponding to arbitrary semisimple Lie algebra $\mathfrak{g}$. Cactus group appears in our construction as a subgroup in the Galois group of Bethe Ansatz equations. Following the idea of Pavel Etingof, we conjecture that this action is isomorphic to the action of the cactus group on the tensor product of crystals coming from the general coboundary category formalism. We prove this conjecture in the case $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_2$ (in fact, for this case the conjecture almost immediately follows from the results of Varchenko on asymptotic solutions of the KZ equation and crystal bases). We also present some conjectures generalizing this result to Bethe vectors of shift of argument subalgebras and relating the cactus group with the Berenstein-Kirillov group of piecewise-linear symmetries of the Gelfand-Tsetlin polytope.Comment: 23 pages, sections 2 and 3 revise

Using the Hadamard and related transforms for simplifying the spectrum of the quantum baker's map

We rationalize the somewhat surprising efficacy of the Hadamard transform in simplifying the eigenstates of the quantum baker's map, a paradigmatic model of quantum chaos. This allows us to construct closely related, but new, transforms that do significantly better, thus nearly solving for many states of the quantum baker's map. These new transforms, which combine the standard Fourier and Hadamard transforms in an interesting manner, are constructed from eigenvectors of the shift permutation operator that are also simultaneous eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal) symmetry.Comment: Version to appear in J. Phys. A. Added discussions; modified title; corrected minor error

Priestley duality for MV-algebras and beyond

We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions