Graph-different permutations

We strengthen and put in a broader perspective previous results of the first two authors on colliding permutations. The key to the present approach is a new non-asymptotic invariant for graphs.Comment: 1+14 page

Transitive decomposition of symmetry groups for the nn-body problem

Periodic and quasi-periodic orbits of the $n$-body problem are critical points of the action functional constrained to the Sobolev space of symmetric loops. Variational methods yield collisionless orbits provided the group of symmetries fulfills certain conditions (such as the \emph{rotating circle property}). Here we generalize such conditions to more general group types and show how to constructively classify all groups satisfying such hypothesis, by a decomposition into irreducible transitive components. As examples we show approximate trajectories of some of the resulting symmetric minimizers

Exact solutions of exactly integrable quantum chains by a matrix product ansatz

Most of the exact solutions of quantum one-dimensional Hamiltonians are obtained thanks to the success of the Bethe ansatz on its several formulations. According to this ansatz the amplitudes of the eigenfunctions of the Hamiltonian are given by a sum of permutations of appropriate plane waves. In this paper, alternatively, we present a matrix product ansatz that asserts that those amplitudes are given in terms of a matrix product. The eigenvalue equation for the Hamiltonian define the algebraic properties of the matrices defining the amplitudes. The existence of a consistent algebra imply the exact integrability of the model. The matrix product ansatz we propose allow an unified and simple formulation of several exact integrable Hamiltonians. In order to introduce and illustrate this ansatz we present the exact solutions of several quantum chains with one and two global conservation laws and periodic boundaries such as the XXZ chain, spin-1 Fateev-Zamolodchikov model, Izergin-Korepin model, Sutherland model, t-J model, Hubbard model, etc. Formulation of the matrix product ansatz for quantum chains with open ends is also possible. As an illustration we present the exact solution of an extended XXZ chain with $z$-magnetic fields at the surface and arbitrary hard-core exclusion among the spins.Comment: 57 pages, no figure

Interlocked permutations

The zero-error capacity of channels with a countably infinite input alphabet formally generalises Shannon's classical problem about the capacity of discrete memoryless channels. We solve the problem for three particular channels. Our results are purely combinatorial and in line with previous work of the third author about permutation capacity.Comment: 8 page

The combinatorics of the colliding bullets problem

The finite colliding bullets problem is the following simple problem: consider a gun, whose barrel remains in a fixed direction; let $(V_i)_{1\le i\le n}$ be an i.i.d.\ family of random variables with uniform distribution on $[0,1]$; shoot $n$ bullets one after another at times $1,2,\dots, n$, where the $i$th bullet has speed $V_i$. When two bullets collide, they both annihilate. We give the distribution of the number of surviving bullets, and in some generalisation of this model. While the distribution is relatively simple (and we found a number of bold claims online), our proof is surprisingly intricate and mixes combinatorial and geometric arguments; we argue that any rigorous argument must very likely be rather elaborate.Comment: 29 page

Moments of vicious walkers and M\"obius graph expansions

A system of Brownian motions in one-dimension all started from the origin and conditioned never to collide with each other in a given finite time-interval $(0, T]$ is studied. The spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue-statistics of random Hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the Gaussian unitary ensemble (GUE) statistics to the Gaussian orthogonal ensemble (GOE) statistics as the time $t$ is going on from 0 to $T$. In the present paper, we characterize this GUE-to-GOE transition by presenting the graphical expansion formula for the moments of positions of vicious walkers. In the GUE limit $t \to 0$, only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus. Following the time evolution of the vicious walkers, however, the graphs with twisted ribbons, called M\"obius graphs, increase their contribution to our expansion formula, and we have to deal with the topology of non-orientable surfaces. Application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the M\"obius expansion using the Stirling numbers of the first kind.Comment: REVTeX4, 11 pages, 1 figure. v.2: calculations of the Green function and references added. v.3: minor additions and corrections made for publication in Phys.Rev.

The Lifshitz-Khalatnikov Kasner index parametrization and the Weyl Tensor

The scale invariant Petrov classification of the Weyl tensor is linked to the scale invariant combination of the Kasner index constraints, and the Lifshitz-Khalatnikov Kasner index parametrization scheme turns out to be a natural way of adapting to this symmetry, while hiding the permutation symmetry that is instead made manifest by the Misner parametrization scheme. While not so interesting for the Kasner spacetime by itself, it gives a geometrical meaning to the famous Kasner map transitioning between Kasner epochs and Kasner eras, equivalently bouncing between curvature walls, in the BLK-Mixmaster dynamics exhibited by spatially homogeneous cosmologies approaching the initial cosmological singularity and the inhomogeneous generalization of this dynamics.Comment: 16 page Latex cimento.cls formatted document with 6 EPS figures annd 2 PicTeX figures; to appear in the Proceedings of the First Italian-Pakistan Workshop on Relativistic Astrophysics which will be published as a special issue of Nuovo Cimento

Colliding Wave Solutions, Duality, and Diagonal Embedding of General Relativity in Two-Dimensional Heterotic String Theory

The non-linear sigma model of the dimensionally reduced Einstein (-Maxwell) theory is diagonally embedded into that of the two-dimensional heterotic string theory. Consequently, the embedded string backgrounds satisfy the (electro-magnetic) Ernst equation. In the pure Einstein theory, the Matzner-Misner SL(2,{\bf R}) transformation can be viewed as a change of conformal structure of the compactified flat two-torus, and in particular its integral subgroup SL(2,{\bf Z}) acts as the modular transformation. The Ehlers SL(2,{\bf R}) and SL(2,{\bf Z}) similarly act on another torus whose conformal structure is induced through the Kramer-Neugebauer involution. Either of the Matzner-Misner and the Ehlers SL(2,{\bf Z}) can be embedded to a special T-duality, and if the former is chosen, then the Ehlers SL(2,{\bf Z}) is shown to act as the S-duality on the four-dimensional sector. As an application we obtain some new colliding string wave solutions by using this embedding as well as the inverse scattering method.Comment: 32 pages, revte