Fast Algorithm for Partial Covers in Words

A factor $u$ of a word $w$ is a cover of $w$ if every position in $w$ lies within some occurrence of $u$ in $w$. A word $w$ covered by $u$ thus generalizes the idea of a repetition, that is, a word composed of exact concatenations of $u$. In this article we introduce a new notion of $\alpha$-partial cover, which can be viewed as a relaxed variant of cover, that is, a factor covering at least $\alpha$ positions in $w$. We develop a data structure of $O(n)$ size (where $n=|w|$) that can be constructed in $O(n\log n)$ time which we apply to compute all shortest $\alpha$-partial covers for a given $\alpha$. We also employ it for an $O(n\log n)$-time algorithm computing a shortest $\alpha$-partial cover for each $\alpha=1,2,\ldots,n$

On quantum mechanics with a magnetic field on R^n and on a torus T^n, and their relation

We show in elementary terms the equivalence in a general gauge of a U(1)-gauge theory of a scalar charged particle on a torus T^n = R^n/L to the analogous theory on R^n constrained by quasiperiodicity under translations in the lattice L. The latter theory provides a global description of the former: the quasiperiodic wavefunctions defined on R^n play the role of sections of the associated hermitean line bundle E on T^n, since also E admits a global description as a quotient. The components of the covariant derivatives corresponding to a constant (necessarily integral) magnetic field B = dA generate a Lie algebra g_Q and together with the periodic functions the algebra of observables O_Q . The non-abelian part of g_Q is a Heisenberg Lie algebra with the electric charge operator Q as the central generator; the corresponding Lie group G_Q acts on the Hilbert space as the translation group up to phase factors. Also the space of sections of E is mapped into itself by g in G_Q . We identify the socalled magnetic translation group as a subgroup of the observables' group Y_Q . We determine the unitary irreducible representations of O_Q, Y_Q corresponding to integer charges and for each of them an associated orthonormal basis explicitly in configuration space. We also clarify how in the n = 2m case a holomorphic structure and Theta functions arise on the associated complex torus. These results apply equally well to the physics of charged scalar particles on R^n and on T^n in the presence of periodic magnetic field B and scalar potential. They are also necessary preliminary steps for the application to these theories of the deformation procedure induced by Drinfel'd twists.Comment: Latex2e file, 22 pages. Final version appeared in IJT

k-Approximate Quasiperiodicity under Hamming and Edit Distance

Quasiperiodicity in strings was introduced almost 30 years ago as an extension of string periodicity. The basic notions of quasiperiodicity are cover and seed. A cover of a text T is a string whose occurrences in T cover all positions of T. A seed of text T is a cover of a superstring of T. In various applications exact quasiperiodicity is still not sufficient due to the presence of errors. We consider approximate notions of quasiperiodicity, for which we allow approximate occurrences in T with a small Hamming, Levenshtein or weighted edit distance. In previous work Sip et al. (2002) and Christodoulakis et al. (2005) showed that computing approximate covers and seeds, respectively, under weighted edit distance is NP-hard. They, therefore, considered restricted approximate covers and seeds which need to be factors of the original string T and presented polynomial-time algorithms for computing them. Further algorithms, considering approximate occurrences with Hamming distance bounded by k, were given in several contributions by Guth et al. They also studied relaxed approximate quasiperiods that do not need to cover all positions of T. In case of large data the exponents in polynomial time complexity play a crucial role. We present more efficient algorithms for computing restricted approximate covers and seeds. In particular, we improve upon the complexities of many of the aforementioned algorithms, also for relaxed quasiperiods. Our solutions are especially efficient if the number (or total cost) of allowed errors is bounded. We also show NP-hardness of computing non-restricted approximate covers and seeds under Hamming distance. Approximate covers were studied in three recent contributions at CPM over the last three years. However, these works consider a different definition of an approximate cover of T, that is, the shortest exact cover of a string T\u27 with the smallest Hamming distance from T

Calogero-Moser models with noncommutative spin interactions

We construct integrable generalizations of the elliptic Calogero-Sutherland-Moser model of particles with spin, involving noncommutative spin interactions. The spin coupling potential is a modular function and, generically, breaks the global spin symmetry of the model down to a product of U(1) phase symmetries. Previously known models are recovered as special cases.Comment: Version to appear in Phys. Rev. Let

String Covering: A Survey

The study of strings is an important combinatorial field that precedes the digital computer. Strings can be very long, trillions of letters, so it is important to find compact representations. Here we first survey various forms of one potential compaction methodology, the cover of a given string x, initially proposed in a simple form in 1990, but increasingly of interest as more sophisticated variants have been discovered. We then consider covering by a seed; that is, a cover of a superstring of x. We conclude with many proposals for research directions that could make significant contributions to string processing in future