17,413 research outputs found
Independence in computable algebra
We give a sufficient condition for an algebraic structure to have a
computable presentation with a computable basis and a computable presentation
with no computable basis. We apply the condition to differentially closed, real
closed, and difference closed fields with the relevant notions of independence.
To cover these classes of structures we introduce a new technique of safe
extensions that was not necessary for the previously known results of this
kind. We will then apply our techniques to derive new corollaries on the number
of computable presentations of these structures. The condition also implies
classical and new results on vector spaces, algebraically closed fields,
torsion-free abelian groups and Archimedean ordered abelian groups.Comment: 24 page
Towards a Model Theory for Transseries
The differential field of transseries extends the field of real Laurent
series, and occurs in various context: asymptotic expansions, analytic vector
fields, o-minimal structures, to name a few. We give an overview of the
algebraic and model-theoretic aspects of this differential field, and report on
our efforts to understand its first-order theory.Comment: Notre Dame J. Form. Log., to appear; 33 p
Loops and Knots as Topoi of Substance. Spinoza Revisited
The relationship between modern philosophy and physics is discussed. It is
shown that the latter develops some need for a modernized metaphysics which
shows up as an ultima philosophia of considerable heuristic value, rather than
as the prima philosophia in the Aristotelian sense as it had been intended, in
the first place. It is shown then, that it is the philosophy of Spinoza in
fact, that can still serve as a paradigm for such an approach. In particular,
Spinoza's concept of infinite substance is compared with the philosophical
implications of the foundational aspects of modern physical theory. Various
connotations of sub-stance are discussed within pre-geometric theories,
especially with a view to the role of spin networks within quantum gravity. It
is found to be useful to intro-duce a separation into physics then, so as to
differ between foundational and empirical theories, respectively. This leads to
a straightforward connection bet-ween foundational theories and speculative
philosophy on the one hand, and between empirical theories and sceptical
philosophy on the other. This might help in the end, to clarify some recent
problems, such as the absence of time and causality at a fundamental level. It
is implied that recent results relating to topos theory might open the way
towards eventually deriving logic from physics, and also towards a possible
transition from logic to hermeneutic.Comment: 42 page
Towards the QFT on Curved Spacetime Limit of QGR. I: A General Scheme
In this article and a companion paper we address the question of how one
might obtain the semiclassical limit of ordinary matter quantum fields (QFT)
propagating on curved spacetimes (CST) from full fledged Quantum General
Relativity (QGR), starting from first principles. We stress that we do not
claim to have a satisfactory answer to this question, rather our intention is
to ignite a discussion by displaying the problems that have to be solved when
carrying out such a program. In the present paper we propose a scheme that one
might follow in order to arrive at such a limit. We discuss the technical and
conceptual problems that arise in doing so and how they can be solved in
principle. As to be expected, completely new issues arise due to the fact that
QGR is a background independent theory. For instance, fundamentally the notion
of a photon involves not only the Maxwell quantum field but also the metric
operator - in a sense, there is no photon vacuum state but a "photon vacuum
operator"! While in this first paper we focus on conceptual and abstract
aspects, for instance the definition of (fundamental) n-particle states (e.g.
photons), in the second paper we perform detailed calculations including, among
other things, coherent state expectation values and propagation on random
lattices. These calculations serve as an illustration of how far one can get
with present mathematical techniques. Although they result in detailed
predictions for the size of first quantum corrections such as the gamma-ray
burst effect, these predictions should not be taken too seriously because a)
the calculations are carried out at the kinematical level only and b) while we
can classify the amount of freedom in our constructions, the analysis of the
physical significance of possible choices has just begun.Comment: LaTeX, 47 p., 3 figure
Time in quantum gravity
Quantum gravity--the marriage of quantum physics with general relativity--is
bound to contain deep and important lessons for the nature of physical time.
Some of these lessons shall be canvassed here, particularly as they arise from
quantum general relativity and string theory and related approaches. Of
particular interest is the question of which of the intuitive aspects of time
will turn out to be fundamental, and which 'emergent' in some sense.Comment: 18 pages, 1 figur
GenEvA (II): A phase space generator from a reweighted parton shower
We introduce a new efficient algorithm for phase space generation. A parton
shower is used to distribute events across all of multiplicity, flavor, and
phase space, and these events can then be reweighted to any desired analytic
distribution. To verify this method, we reproduce the e+e- -> n jets tree-level
result of traditional matrix element tools. We also show how to improve
tree-level matrix elements automatically with leading-logarithmic resummation.
This algorithm is particularly useful in the context of a new framework for
event generation called GenEvA. In a companion paper [arXiv:0801.4026], we show
how the GenEvA framework can address contemporary issues in event generation.Comment: 54 pages, 20 figures, v2: corrected typos, added reference
A Century of Gravity: 1901--2000 (plus some 2001)
This lecture consists of two parts. The first is a (totally unsystematic)
survey of some of the high points in the evolution of gravity and its
successors, primarily in the course of the past century. The second summarizes
some new work on surprising properties of higher spin fields in
cosmological backgrounds: the presence of \L gives rise to discrete sets of
massive models endowed with gauge invariances, that divide the (m^2, \L)
plane into unitary and non-unitary phases. The unitary region common to
fermions and bosons shrinks to flat space ( \L \to 0 ) as their spins
increase.Comment: 12 pages, 1 eps Fig. Invited Lecture at 2001: A Spacetime Odyssey,
Ann Arbor, May 200
Cyclic Orbit Codes
In network coding a constant dimension code consists of a set of
k-dimensional subspaces of F_q^n. Orbit codes are constant dimension codes
which are defined as orbits of a subgroup of the general linear group, acting
on the set of all subspaces of F_q^n. If the acting group is cyclic, the
corresponding orbit codes are called cyclic orbit codes. In this paper we give
a classification of cyclic orbit codes and propose a decoding procedure for a
particular subclass of cyclic orbit codes.Comment: submitted to IEEE Transactions on Information Theor
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