100 research outputs found
Computability Theory
Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science
There are no maximal d.c.e. wtt-degrees
В статье доказывается, что не существует максимальной 2-в.п. wtt-степени в 2-в.п. wtt-степеня
A survey of results on the d-c.e. and n-c.e. degrees
© Springer International Publishing AG 2017.This paper is a survey on the upper semilattices of Turing and enumeration degrees of n-c.e. sets. Questions on the structural properties of these semilattices, and some model-theoretic properties are considered
Computability Theory
Computability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work
Beyond Mean-Field Low-Lying Excitations of Dipolar Bose Gases
We theoretically investigate various beyond mean-field effects on Bose gases
at zero temperature featuring the anisotropic and long-range dipole-dipole
interaction in addition to the isotropic and short-range contact interaction.
Within the realm of the Bogoliubov-de Gennes theory, we consider static
properties and low-lying excitations of both homogeneous and harmonically
trapped dipolar bosonic gases. For the homogeneous system, the condensate
depletion, the ground-state energy, the equation of state, and the speed of
sound are discussed in detail. Making use of the local density approximation,
we extend these results in order to study the properties of a dipolar Bose gas
in a harmonic trap and in the regime of large particle numbers. After deriving
the equations of motion for the general case of a triaxial trap, we analyze the
influence of quantum fluctuations on important properties of the gas, such as
the equilibrium configuration and the low-lying excitations in the case of a
cylinder-symmetric trap. In addition to the monopole and quadrupole oscillation
modes, we also discuss the radial quadrupole mode. We find that the latter
acquires a quantum correction exclusively due to the dipole-dipole interaction.
As a result, we identify the radial quadrupole as a reasonably accessible
source for the signature of dipolar many-body effects and stress the enhancing
character that dipolar interactions have for quantum fluctuations in the other
oscillation modes.Comment: Version published in PR
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