Mirko Degli Esposti

Classical limit of the Quantized Hyperbolic Toral Automorphisms

The canonical quantization of any hyperbolic symplectomorphismA of the 2-torus yields a periodic unitary operator on aN-dimenional Hilbert space,N=1/h. We prove that this quantum system becomes ergodic and mixing at the classical limit (N→∞,N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly speread in phase space.

On the origin of long-range correlations in texts

The complexity of human interactions with social and natural phenomena is mirrored in the way we describe our experiences through natural language. In order to retain and convey such a high dimensional information, the statistical properties of our linguistic output has to be highly correlated in time. An example are the robust observations, still largely not understood, of correlations on arbitrary long scales in literary texts. In this paper we explain how long-range correlations flow from highly structured linguistic levels down to the building blocks of a text (words, letters, etc..). By combining calculations and data analysis we show that correlations take form of a bursty sequence of events once we approach the semantically relevant topics of the text. The mechanisms we identify are fairly general and can be equally applied to other hierarchical settings.

Markov constraints for generating lyrics with style

We address the issue of generating texts in the style of an existing author, that also satisfy structural constraints imposed by the genre of the text. We focus on song lyrics, for which structural constraints are well-defined: rhyme and meter. Although Markov processes are known to be suitable for representing style, they are difficult to control in order to satisfy non-local properties, such as structural constraints, that require long distance modeling. We show that the framework of Constrained Markov Processes allows us to precisely generate texts that are consistent with a corpus, while being controllable in terms of rhymes and meter, a result that no other technique, to our knowledge, could achieve to date. Controlled Markov processes consist in reformulating Markov processes in the context of constraint satisfaction. We describe how to represent stylistic and structural properties in terms of constraints in this framework and we provide an evaluation of our method by comparing it to both pure Markov and pure constraint-based approaches. We show how this approach can be used for the semi-automatic generation of lyrics in the style of a popular author that has the same structure as an existing song.

Word length n-grams for text re-use detection

The automatic detection of shared content in written documents –which includes text reuse and its unacknowledged commitment, plagiarism– has become an important problem in Information Retrieval. This task requires exhaustive comparison of texts in order to determine how similar they are. However, such comparison is impossible in those cases where the amount of documents is too high. Therefore, we have designed a model for the proper pre-selection of closely related documents in order to perform the exhaustive comparison afterwards. We use a similarity measure based on word-level n-grams, which proved to be quite effective in many applications As this approach becomes normally impracticable for real-world large datasets, we propose a method based on a preliminary word-length encoding of texts, substituting a word by its length, providing three important advantages: (i) being the alphabet of the documents reduced to nine symbols, the space needed to store n-gram lists is reduced; (ii) computation times are decreased; and (iii) length n-grams can be represented in a trie, allowing a more flexible and fast comparison. We experimentally show, on the basis of the perplexity measure, that the noise introduced by the length encoding does not decrease importantly the expressiveness of the text. The method is then tested on two large datasets of co-derivatives and simulated plagiarism.

An example of mathematical authorship attribution

In this paper we discuss a novel mathematical approach to authorship attribution which we implemented recently to face a concrete problem of author recognition. The fundamental ideas for our methods came from statistical mechanics and information theory. We combine two approaches. Both of them use similarity measures between couples of texts as indicators of stylistic closeness: the first one is based on the comparison of frequencies of fixed length substrings (n-grams) throughout the texts; the second one relies on a suitable use of compression algorithms as relative entropy approximators, in the spirit of the so-called Ziv–Merhav theorem. The two methods were separately developed and then combined, together with a suitable and theoretically founded ranking analysis, to produce an original authorship attribution procedure that yielded very successful results on the specific problem to which it was applied. This ranking analysis could be of interest also in other application fields.

Distribution of closed orbits for linear automorphisms of tori

In this paper we study the distribution properties of periodic orbits for the linear hyperbolic automorphisms of the d-torus. We first obtain an explicit expression of the dynamical zeta function and prove general equidistribution results similar to those obtained for axiom A flows. We then study in detail some families of periodic orbits living on invariant prime lattices: they have the property that the integral of any character along any single orbit can be reduced to a number theoretic exponential sum over a finite field. This fact enables us to obtain explicit estimates on their asymptotic distributional properties.

The mathematical aspects of quantum maps

to Dynamical Systems.- Number Theoretic Background.- Mathematical Aspects of Quantum Maps.- Numerical Aspects of Eigenvalue and Eigenfunction Computations for Chaotic Quantum Systems.- From Normal to Anomalous Deterministic Diffusion.

Quantization of the classical Lie algorithm in the Bargmann representation

Abstract For any polynomial perturbation of a d -dimensional system of non-resonant harmonic oscillators it is proved, writing the classical Hamiltonian in complex coordinates and the corresponding Schrodinger operator in the Bargmann representation, that the classical Lie algorithm generating the canonical perturbation expansion can be “exactly” quantized to yield the Rayleigh-Schrodinger series near any unperturbed bound state.

Mathematical Aspects of Quantum Maps

In this contribution we will review the basic mathematical aspects of quantum maps. Here is a brief outline of the topics covered in this contribution. Most of the material comes from [13], [12], [9] and also [24], [23]. 1. The space of the states and quantization of observables 2. Quantum dynamics over the torus 3. Quantized cat maps 4. The quantum baker’s map and the sawtooth maps 5. Equidistribution of eigenfunctions 6. The period of Amod N and an introduction to the Hecke operators 7. Value distribution of eigenfunctions 8. Equidistribution of Eigenfunctions for discontinuous maps

An infinite step billiard

A class of non-compact billiards is introduced, namely the infinite step billiards, i.e. systems of a point particle moving freely in the domain , with elastic reflections on the boundary; here , and . After describing some generic ergodic features of these dynamical systems, we turn to a more detailed study of the example . Playing an important role in this case are the so-called escape orbits, that is, orbits going to monotonically in the X-velocity. A fairly complete description of them is given. This enables us to prove some results concerning the topology of the dynamics on the billiard.