Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm

Linear rank-width is a linearized variation of rank-width, and it is deeply related to matroid path-width. In this paper, we show that the linear rank-width of every $n$-vertex distance-hereditary graph, equivalently a graph of rank-width at most $1$, can be computed in time $\mathcal{O}(n^2\cdot \log_2 n)$, and a linear layout witnessing the linear rank-width can be computed with the same time complexity. As a corollary, we show that the path-width of every $n$-element matroid of branch-width at most $2$ can be computed in time $\mathcal{O}(n^2\cdot \log_2 n)$, provided that the matroid is given by an independent set oracle. To establish this result, we present a characterization of the linear rank-width of distance-hereditary graphs in terms of their canonical split decompositions. This characterization is similar to the known characterization of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex separation and search number of a graph. Inf. Comput., 113(1):50--79, 1994]. However, different from forests, it is non-trivial to relate substructures of the canonical split decomposition of a graph with some substructures of the given graph. We introduce a notion of `limbs' of canonical split decompositions, which correspond to certain vertex-minors of the original graph, for the right characterization.Comment: 28 pages, 3 figures, 2 table. A preliminary version appeared in the proceedings of WG'1

Separation of variables for the classical and quantum Neumann model

The method of separation of variables is shown to apply to both the classical and quantum Neumann model. In the classical case this nicely yields the linearization of the flow on the Jacobian of the spectral curve. In the quantum case the Schr\"odinger equation separates into one--dimensional equations belonging to the class of generalized Lam\'e differential equations.Comment: 16 page

Kowalevski's analysis of the swinging Atwood's machine

We study the Kowalevski expansions near singularities of the swinging Atwood's machine. We show that there is a infinite number of mass ratios $M/m$ where such expansions exist with the maximal number of arbitrary constants. These expansions are of the so--called weak Painlev\'e type. However, in view of these expansions, it is not possible to distinguish between integrable and non integrable cases.Comment: 30 page

Formal Design of Asynchronous Fault Detection and Identification Components using Temporal Epistemic Logic

Autonomous critical systems, such as satellites and space rovers, must be able to detect the occurrence of faults in order to ensure correct operation. This task is carried out by Fault Detection and Identification (FDI) components, that are embedded in those systems and are in charge of detecting faults in an automated and timely manner by reading data from sensors and triggering predefined alarms. The design of effective FDI components is an extremely hard problem, also due to the lack of a complete theoretical foundation, and of precise specification and validation techniques. In this paper, we present the first formal approach to the design of FDI components for discrete event systems, both in a synchronous and asynchronous setting. We propose a logical language for the specification of FDI requirements that accounts for a wide class of practical cases, and includes novel aspects such as maximality and trace-diagnosability. The language is equipped with a clear semantics based on temporal epistemic logic, and is proved to enjoy suitable properties. We discuss how to validate the requirements and how to verify that a given FDI component satisfies them. We propose an algorithm for the synthesis of correct-by-construction FDI components, and report on the applicability of the design approach on an industrial case-study coming from aerospace.Comment: 33 pages, 20 figure

Continuum limit of the Volterra model, separation of variables and non standard realizations of the Virasoro Poisson bracket

The classical Volterra model, equipped with the Faddeev-Takhtadjan Poisson bracket provides a lattice version of the Virasoro algebra. The Volterra model being integrable, we can express the dynamical variables in terms of the so called separated variables. Taking the continuum limit of these formulae, we obtain the Virasoro generators written as determinants of infinite matrices, the elements of which are constructed with a set of points lying on an infinite genus Riemann surface. The coordinates of these points are separated variables for an infinite set of Poisson commuting quantities including $L\_0$. The scaling limit of the eigenvector can also be calculated explicitly, so that the associated Schroedinger equation is in fact exactly solvable.Comment: Latex, 43 pages Synchronized with the to be published versio

Probability distribution of the maximum of a smooth temporal signal

We present an approximate calculation for the distribution of the maximum of a smooth stationary temporal signal X(t). As an application, we compute the persistence exponent associated to the probability that the process remains below a non-zero level M. When X(t) is a Gaussian process, our results are expressed explicitly in terms of the two-time correlation function, f(t)=.Comment: Final version (1 major typo corrected; better introduction). Accepted in Phys. Rev. Let

Topological characteristics of oil and gas reservoirs and their applications

We demonstrate applications of topological characteristics of oil and gas reservoirs considered as three-dimensional bodies to geological modeling.Comment: 12 page

Photon Splitting in a Very Strong Magnetic Field

Photon splitting in a very strong magnetic field is analyzed for energy $\omega < 2m$. The amplitude obtained on the base of operator-diagram technique is used. It is shown that in a magnetic field much higher than critical one the splitting amplitude is independent on the field. Our calculation is in a good agreement with previous results of Adler and in a strong contradiction with recent paper of Mentzel et al.Comment: 5 pages,Revtex , 4 figure