37 research outputs found
Computability on quasi-Polish spaces
International audienceWe investigate the effectivizations of several equivalent definitions of quasi-Polish spaces and study which characterizations hold effectively. Being a computable effectively open image of the Baire space is a robust notion that admits several characterizations. We show that some natural effectivizations of quasi-metric spaces are strictly stronger
Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Dynamical Systems
In this paper we study a class of dynamical systems defined by Pfaffian maps. It is a sub-class of o-minimal dynamical systems which capture rich
continuous dynamics and yet can be studied using finite bisimulations.
The existence of finite bisimulations for o-minimal dynamical and hybrid systems has been shown by several authors; see e.g. Brihaye et al (2004), Davoren (1999), Lafferriere et al (2000).
The next natural question to investigate is how the sizes of such bisimulations can be bounded. The first step in this direction was done by Korovina et al (2004) where a double exponential upper bound was shown for Pfaffian dynamical and hybrid systems. In the present paper we improve this bound to a single exponential upper bound. Moreover we show that this bound is tight in general, by exhibiting a parameterized class of systems on which the exponential bound is attained.
The bounds provide a basis for designing efficient algorithms for computing
bisimulations, solving reachability and motion planning problems
Relativistic theory of inverse beta-decay of polarized neutron in strong magnetic field
The relativistic theory of the inverse beta-decay of polarized neutron, , in strong magnetic field is developed. For the proton
wave function we use the exact solution of the Dirac equation in the magnetic
filed that enables us to account exactly for effects of the proton momentum
quantization in the magnetic field and also for the proton recoil motion. The
effect of nucleons anomalous magnetic moments in strong magnetic fields is also
discussed. We examine the cross section for different energies and directions
of propagation of the initial neutrino accounting for neutrons polarization. It
is shown that in the super-strong magnetic field the totally polarized neutron
matter is transparent for neutrinos propagating antiparallel to the direction
of polarization. The developed relativistic approach can be used for
calculations of cross sections of the other URCA processes in strong magnetic
fields.Comment: 41 pages in LaTex including 11 figures in PostScript, discussion on
nucleons AMM interaction with magnetic field is adde
Black-hole concept of a point-like nucleus with supercritical charge
The Dirac equation for an electron in the central Coulomb field of a
point-like nucleus with the charge greater than 137 is considered. This
singular problem, to which the fall-down onto the centre is inherent, is
addressed using a new approach, based on a black-hole concept of the singular
centre and capable of producing cut-off-free results. To this end the Dirac
equation is presented as a generalized eigenvalue boundary problem of a
self-adjoint operator. The eigenfunctions make complete sets, orthogonal with a
singular measure, and describe particles, asymptotically free and
delta-function-normalizable both at infinity and near the singular centre
. The barrier transmission coefficient for these particles responsible for
the effects of electron absorption and spontaneous electron-positron pair
production is found analytically as a function of electron energy and charge of
the nucleus. The singular threshold behaviour of the corresponding amplitudes
substitutes for the resonance behaviour, typical of the conventional theory,
which appeals to a finite-size nucleus.Comment: 22 pages, 5 figures, LATEX requires IOPAR
Recent Advances in Σ-definability over Continuous Data Types
The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas
Σ_K–constraints for Hybrid Systems
In this paper we introduce and study computational aspects
of Σ_K-constraints which are powerful enough to represent computable continuous data, but also simple enough to be an approach to approximate constraint solving for a large class of quantified continuous constraints.
We illustrate how Σ_K-constraints can be used for reasoning
about hybrid systems
Semantic Characterisations of Second-order Computability Over the Real Numbers
We propose semantic characterisations of second-order computability over the reals based on -denability theory. Notions of computability for operators and real-valued functionals dened on the class of continuous functions are introduced via domain theory. We consider the reals with and without equality and prove theorems which connect computable operators and real-valued functionals with validity of nite -formulas.