On Maximal Unbordered Factors

Given a string $S$ of length $n$, its maximal unbordered factor is the longest factor which does not have a border. In this work we investigate the relationship between $n$ and the length of the maximal unbordered factor of $S$. We prove that for the alphabet of size $\sigma \ge 5$ the expected length of the maximal unbordered factor of a string of length~$n$ is at least $0.99 n$ (for sufficiently large values of $n$). As an application of this result, we propose a new algorithm for computing the maximal unbordered factor of a string.Comment: Accepted to the 26th Annual Symposium on Combinatorial Pattern Matching (CPM 2015

Three Laisses from the Franco-Italian \u3cem\u3eSong of Roland\u3c/em\u3e

These three laisses, that is, decasyllabic verse-paragraphs of various lengths whose lines end with the same assonance, are from the thirteenth-century Franco-Italian version of The Song of Roland, known as the V4 manuscript of St. Mark\u27s Church, Venice. They illustrate creative thirteenth-century innovations, in a slightly different language, added to the much better known twelfth-century Anglo-Norman French Chanson de Roland, known as the Oxford version. “La Chanson de Roland”: The French Corpus, Vol. 1, Part 2, The Venice 4 Version, 2005, Brepols, 2005, pp. 97, 100-1, 246-7

Conserved quantities in non-abelian monopole fields

Van Holten's covariant Hamiltonian framework is used to find conserved quantities for an isospin-carrying particle in a non-Abelian monopole-like field. For a Wu-Yang monopole we find the most general scalar potential such that the combined system admits a conserved Runge-Lenz vector. It generalizes the fine-tuned inverse-square plus Coulomb potential, found before by McIntosh and Cisneros, and by Zwanziger, for a charged particle in the field of a Dirac monopole. Following Feh\'er, the result is interpreted as describing motion in the asymptotic field of a self-dual Prasad-Sommerfield monopole. In the effective non-Abelian field for nuclear motion in a diatomic molecule due to Moody, Shapere and Wilczek, a conserved angular momentum is constructed, despite the non-conservation of the electric charge. No Runge-Lenz vector has been found.Comment: 8 pages, RevTex no figures. An error corrected and a new Section adde

Some Spacetimes with Higher Rank Killing-Stackel Tensors

By applying the lightlike Eisenhart lift to several known examples of low-dimensional integrable systems admitting integrals of motion of higher-order in momenta, we obtain four- and higher-dimensional Lorentzian spacetimes with irreducible higher-rank Killing tensors. Such metrics, we believe, are first examples of spacetimes admitting higher-rank Killing tensors. Included in our examples is a four-dimensional supersymmetric pp-wave spacetime, whose geodesic flow is superintegrable. The Killing tensors satisfy a non-trivial Poisson-Schouten-Nijenhuis algebra. We discuss the extension to the quantum regime

A suffix tree or not a suffix tree?

In this paper we study the structure of suffix trees. Given an unlabeled tree τ on n nodes and suffix links of its internal nodes, we ask the question ”Is τ a suffix tree?”, i.e., is there a string S whose suffix tree has the same topological structure as τ? We place no restrictions on S, in particular we do not require that S ends with a unique symbol. This corresponds to considering the more general definition of implicit or extended suffix trees. Such general suffix trees have many applications and are for example needed to allow efficient updates when suffix trees are built online. Deciding if τ is a suffix tree is not an easy task, because, with no restrictions on the final symbol, we cannot guess the length of a string that realizes τ from the number of leaves. And without an upper bound on the length of such a string, it is not even clear how to solve the problem by an exhaustive search. In this paper, we prove that τ is a suffix tree if and only if it is realized by a string S of length n−1, and we give a linear-time algorithm for inferring S when the first letter on each edge is known. This generalizes the work of I et al. [Discrete Appl. Math. 163, 2014]

Effect of physical aging on the low-frequency vibrational density of states of a glassy polymer

The effects of the physical aging on the vibrational density of states (VDOS) of a polymeric glass is studied. The VDOS of a poly(methyl methacrylate) glass at low-energy (<15 meV), was determined from inelastic neutron scattering at low-temperature for two different physical thermodynamical states. One sample was annealed during a long time at temperature lower than Tg, and another was quenched from a temperature higher than Tg. It was found that the VDOS around the boson peak, relatively to the one at higher energy, decreases with the annealing at lower temperature than Tg, i.e., with the physical aging.Comment: To be published in Europhys. Let

Simple model for the vibrations of embedded elastically cubic nanocrystals

The purpose of this work is to calculate the vibrational modes of an elastically anisotropic sphere embedded in an isotropic matrix. This has important application to understanding the spectra of low-frequency Raman scattering from nanoparticles embedded in a glass matrix. First some low frequency vibrational modes of a free cubically elastic sphere are found to be nearly independent of one combination of elastic constants. This is then exploited to obtain an isotropic approximation for these modes which enables to take into account the surrounding isotropic matrix. This method is then used to quantatively explain recent spectra of gold and copper nanocrystals in glasses.Comment: 6 pages, 5 figure

Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view

The "metric" structure of nonrelativistic spacetimes consists of a one-form (the absolute clock) whose kernel is endowed with a positive-definite metric. Contrarily to the relativistic case, the metric structure and the torsion do not determine a unique Galilean (i.e. compatible) connection. This subtlety is intimately related to the fact that the timelike part of the torsion is proportional to the exterior derivative of the absolute clock. When the latter is not closed, torsionfreeness and metric-compatibility are thus mutually exclusive. We will explore generalisations of Galilean connections along the two corresponding alternative roads in a series of papers. In the present one, we focus on compatible connections and investigate the equivalence problem (i.e. the search for the necessary data allowing to uniquely determine connections) in the torsionfree and torsional cases. More precisely, we characterise the affine structure of the spaces of such connections and display the associated model vector spaces. In contrast with the relativistic case, the metric structure does not single out a privileged origin for the space of metric-compatible connections. In our construction, the role of the Levi-Civita connection is played by a whole class of privileged origins, the so-called torsional Newton-Cartan (TNC) geometries recently investigated in the literature. Finally, we discuss a generalisation of Newtonian connections to the torsional case.Comment: 79 pages, 7 figures; v2: added material on affine structure of connection space, former Section 4 postponed to 3rd paper of the serie

On the Number of Unbordered Factors

We illustrate a general technique for enumerating factors of k-automatic sequences by proving a conjecture on the number f(n) of unbordered factors of the Thue-Morse sequence. We show that f(n) = 4 and that f(n) = n infinitely often. We also give examples of automatic sequences having exactly 2 unbordered factors of every length