The Airy transform and associated polynomials

The Airy transform is an ideally suited tool to treat problems in classical and quantum optics. Even though the relevant mathematical aspects have been thoroughly investigated, the possibilities it offers are wide and some features, such as the link with special functions and polynomials, still contain unexplored aspects. In this note we will show that the so called Airy polynomials are essentially the third order Hermite polynomials. We will also prove that this identification opens the possibility of developing new conjectures on the properties of this family of polynomials.     

Bessel functions of third order and the distribution of cyclic planar motions with three directions

The exact distribution of a cyclic planar motion with three directions is explicitly derived in terms of Bessel functions of order three (suitably combined). The absolutely continuous part of the distribution is proved to satisfy suitable boundary conditions and some of its properties are analyzed. The transformations converting the governing equations of order three is presented and its solutions (used here) derived by applying the Frobenius method.     

Entanglement as a Semantic Resource

The characteristic holistic features of the quantum theoretic formalism and the intriguing notion of entanglement can be applied to a field that is far from microphysics: logical semantics. Quantum computational logics are new forms of quantum logic that have been suggested by the theory of quantum logical gates in quantum computation. In the standard semantics of these logics, sentences denote quantum information quantities: systems of qubits (quregisters) or, more generally, mixtures of quregisters (qumixes), while logical connectives are interpreted as special quantum logical gates (which have a characteristic reversible and dynamic behavior). In this framework, states of knowledge may be entangled, in such a way that our information about the whole determines our information about the parts; and the procedure cannot be, generally, inverted. In spite of its appealing properties, the standard version of the quantum computational semantics is strongly “Hilbert-space dependent”. This certainly represents a shortcoming for all applications, where real and complex numbers do not generally play any significant role (as happens, for instance, in the case of natural and of artistic languages). We propose an abstract version of quantum computational semantics, where abstract qumixes, quregisters and registers are identified with some special objects (not necessarily living in a Hilbert space), while gates are reversible functions that transform qumixes into qumixes. In this framework, one can give an abstract definition of the notions of superposition and of entangled pieces of information, quite independently of any numerical values. We investigate three different forms of abstract holistic quantum computational logic.     

The dynamics of the weakly perturbed motion of a liquid-filled gyroscope and the control problem

The Cauchy problem for the motion of a dynamically symmetrical rigid body with a cavity, filled with an ideal liquid, which is perturbed from uniform rotation, is considered in a linear formulation. The problem of the simultanious solution of the equations of hydrodynamics and the mechanics of a rigid body is reduced to the solution of an eigenvalue problem which depends solely on the geometry of the cavity and the subsequent integration of a system of differential equations.     

Embedding a novel objective function in a two-phased local search for robust vertex coloring

In this paper, we propose a two-phased local search for vertex coloring. The algorithm alternately executes two closely interacting functionalities, i.e., a stochastic and a deterministic local search. The stochastic phase is basically based on biased random sampling that, according to a probability matrix storing the probability a vertex can be assigned to a color, iteratively constructs feasible colorings. The deterministic phase, instead, consists in assigning sequentially, according to a given ordering, each vertex to the color which causes the lowest increase of the solution penalty, and then, when the schedule is constructed, swap operations are executed to improve the performance. The interaction between the two phases is implemented by tunnelling information of what happened during a phase to the successive ones. Beyond the algorithm scheme, the novelty of the approach stems from the fact that the objective function is not minimizing the number of colors but a new penalty function. The proposed approach is tested on known benchmarks for the studied problem available on the public domain. From a comparison to the state of the art it appears that the proposed approach is robust and is able to achieve best known results.     

A note on Chebyshev polynomials

New families of generating functions and identities concerning the Chebyshev polynomials are derived. It is shown that the proposed method allows the derivation of sum rules involving products of Chebyshev polynomials and addition theorems. The possiblity of extending the results to include gnerating functions involving products of Chebyshev and other polynomials is finally analyzed.

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Sunto Si derivano nuove famiglie di funzioni generatrici e di identità relative ai polinomi di Chebyshev. Si dimostra che il metodo proposto permette la derivazione di regole di somma relative a prodotti di polinomi di Chebyshev e teoremi di addizione. La possibilità di estendere i risultati includendos funzioni generatrici di prodotti di polinomi di Chebyshev ed altri polinomi è infine analizzata.

     

Scalarization and Kuhn-Tucker-Like Conditions for Weak Vector Generalized Quasivariational Inequalities

We consider a weak vector generalized quasivariational inequality. By introducing a method of scalarization which does not require any assumption on the data and by using previous results of the authors concerning scalar generalized quasivariational inequalities, we present Kuhn-Tucker-like conditions for this problem in the case in which the set-valued operator of the constraints is defined by a finite number of inequalities