10,843 research outputs found
Resolutions of small sets of fat points
We investigate the minimal graded free resolutions of ideals of at most n+1
fat points in general position in P^n. Our main theorem is that these ideals
are componentwise linear. This result yields a number of corollaries, including
the multiplicity conjecture of Herzog, Huneke, and Srinivasan in this case. On
the computational side, using an iterated mapping cone process, we compute
formulas for the graded Betti numbers of ideals associated to two fat points in
P^n, verifying a conjecture of Fatabbi, and at most n+1 general double points
in P^n.Comment: 15 pages; very minor revisions plus some additional references; to
appear in JPA
Interplay between curvature and Planck-scale effects in astrophysics and cosmology
Several recent studies have considered the implications for astrophysics and
cosmology of some possible nonclassical properties of spacetime at the Planck
scale. The new effects, such as a Planck-scale-modified energy-momentum
(dispersion) relation, are often inferred from the analysis of some quantum
versions of Minkowski spacetime, and therefore the relevant estimates depend
heavily on the assumption that there could not be significant interplay between
Planck-scale and curvature effects. We here scrutinize this assumption, using
as guidance a quantum version of de Sitter spacetime with known Inonu-Wigner
contraction to a quantum Minkowski spacetime. And we show that, contrary to
common (but unsupported) beliefs, the interplay between Planck-scale and
curvature effects can be significant. Within our illustrative example, in the
Minkowski limit the quantum-geometry deformation parameter is indeed given by
the Planck scale, while in the de Sitter picture the parameter of quantization
of geometry depends both on the Planck scale and the curvature scalar. For the
much-studied case of Planck-scale effects that intervene in the observation of
gamma-ray bursts we can estimate the implications of "quantum spacetime
curvature" within robust simplifying assumptions. For cosmology at the present
stage of the development of the relevant mathematics one cannot go beyond
semiheuristic reasoning, and we here propose a candidate approximate
description of a quantum FRW geometry, obtained by patching together pieces
(with different spacetime curvature) of our quantum de Sitter. This
semiheuristic picture, in spite of its limitations, provides rather robust
evidence that in the early Universe the interplay between Planck-scale and
curvature effects could have been particularly significant.Comment: 26 pages
Introduction to Iltis: An Interactive, Web-Based System for Teaching Logic
Logic is a foundation for many modern areas of computer science. In
artificial intelligence, as a basis of database query languages, as well as in
formal software and hardware verification --- modelling scenarios using logical
formalisms and inferring new knowledge are important skills for going-to-be
computer scientists. The Iltis project aims at providing a web-based,
interactive system that supports teaching logical methods. In particular the
system shall (a) support to learn to model knowledge and to infer new knowledge
using propositional logic, modal logic and first-order logic, and (b) provide
immediate feedback and support to students. This article presents a
prototypical system that currently supports the above tasks for propositional
logic. First impressions on its use in a second year logic course for computer
science students are reported
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