64,904 research outputs found
O-Minimal Invariants for Linear Loops
The termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Such deceptively simple questions also relate to a number of deep open problems, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this paper, we introduce the class of o-minimal invariants, which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture
Non-minimal -type corrections to holographic superconductor
We study -dimensional holographic superconductors in the presence of
non-minimally coupled electromagnetic field to gravity by considering an
arbitrary linear combination of -type invariants with three parameters.
Our analytical analysis shows that the non-minimal couplings affect the
condensate and the critical temperature.Comment: Accepted for publication in Modern Physics Letters
Linearity defects of modules over commutative rings
This article concerns linear parts of minimal resolutions of finitely
generated modules over commutative local, or graded rings. The focus is on the
linearity defect of a module, which marks the point after which the linear part
of its minimal resolution is acyclic. The results established track the change
in this invariant under some standard operations in commutative algebra. As one
of the applications, it is proved that a local ring is Koszul if and only if it
admits a Koszul module that is Cohen-Macaulay of minimal degree. An injective
analogue of the linearity defect is introduced and studied. The main results
express this new invariant in terms of linearity defects of free resolutions,
and relate it to other ring theoretic and homological invariants of the module.Comment: 23 pages, minor modification
Higher derivative couplings and massive supergravity in three dimensions
We develop geometric superspace settings to construct arbitrary higher
derivative couplings (including R^n terms) in three-dimensional supergravity
theories with N=1,2,3 by realising them as conformal supergravity coupled to
certain compensators. For all known off-shell supergravity formulations, we
construct supersymmetric invariants with up to and including four derivatives.
As a warming-up exercise, we first give a new and completely geometric
derivation of such invariants in N=1 supergravity. Upon reduction to
components, they agree with those given in arXiv:0907.4658 and arXiv:1005.3952.
We then carry out a similar construction in the case of N=2 supergravity for
which there exist two minimal formulations that differ by the choice of
compensating multiplet: (i) a chiral scalar multipet; (ii) a vector multiplet.
For these formulations all four derivative invariants are constructed in
completely general and gauge independent form. For a general supergravity model
(in the N=1 and minimal N=2 cases) with curvature-squared and lower order
terms, we derive the superfield equations of motion, linearise them about
maximally supersymmetric backgrounds and obtain restrictions on the parameters
that lead to models for massive supergravity. We use the non-minimal
formulation for N = 2 supergravity (which corresponds to a complex linear
compensator) to construct a novel consistent theory of massive supergravity. In
the case of N = 3 supergravity, we employ the off-shell formulation with a
vector multiplet as compensator to construct for the first time various higher
derivative invariants. These invariants may be used to derive models for N = 3
massive supergravity. As a bi-product of our analysis, we also present
superfield equations for massive higher spin multiplets in (1,0), (1,1) and
(2,0) anti-de Sitter superspaces.Comment: 84 pages; V3: references added, minor modifications, published
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