89,634 research outputs found
Approximate roots of a valuation and the Pierce-Birkhoff Conjecture
This paper is a step in our program for proving the Piece-Birkhoff Conjecture
for regular rings of any dimension (this would contain, in particular, the
classical Pierce-Birkhoff conjecture which deals with polynomial rings over a
real closed field). We first recall the Connectedness and the Definable
Connectedness conjectures, both of which imply the Pierce - Birkhoff
conjecture. Then we introduce the notion of a system of approximate roots of a
valuation v on a ring A (that is, a collection Q of elements of A such that
every v-ideal is generated by products of elements of Q). We use approximate
roots to give explicit formulae for sets in the real spectrum of A which we
strongly believe to satisfy the conclusion of the Definable Connectedness
conjecture. We prove this claim in the special case of dimension 2. This proves
the Pierce-Birkhoff conjecture for arbitrary regular 2-dimensional rings
Quantum correlations in Newtonian space and time: arbitrarily fast communication or nonlocality
We investigate possible explanations of quantum correlations that satisfy the
principle of continuity, which states that everything propagates gradually and
continuously through space and time. In particular, following [J.D. Bancal et
al, Nature Physics 2012], we show that any combination of local common causes
and direct causes satisfying this principle, i.e. propagating at any finite
speed, leads to signalling. This is true even if the common and direct causes
are allowed to propagate at a supraluminal-but-finite speed defined in a
Newtonian-like privileged universal reference frame. Consequently, either there
is supraluminal communication or the conclusion that Nature is nonlocal (i.e.
discontinuous) is unavoidable.Comment: It is an honor to dedicate this article to Yakir Aharonov, the master
of quantum paradoxes. Version 2 contains some more references and a clarified
conclusio
Problem of Time in Quantum Gravity
The Problem of Time occurs because the `time' of GR and of ordinary Quantum
Theory are mutually incompatible notions. This is problematic in trying to
replace these two branches of physics with a single framework in situations in
which the conditions of both apply, e.g. in black holes or in the very early
universe. Emphasis in this Review is on the Problem of Time being multi-faceted
and on the nature of each of the eight principal facets. Namely, the Frozen
Formalism Problem, Configurational Relationalism Problem (formerly Sandwich
Problem), Foliation Dependence Problem, Constraint Closure Problem (formerly
Functional Evolution Problem), Multiple Choice Problem, Global Problem of Time,
Problem of Beables (alias Problem of Observables) and Spacetime
Reconstruction/Replacement Problem. Strategizing in this Review is not just
centred about the Frozen Formalism Problem facet, but rather about each of the
eight facets. Particular emphasis is placed upon A) relationalism as an
underpinning of the facets and as a selector of particular strategies
(especially a modification of Barbour relationalism, though also with some
consideration of Rovelli relationalism). B) Classifying approaches by the full
ordering in which they embrace constrain, quantize, find time/history and find
observables, rather than only by partial orderings such as "Dirac-quantize". C)
Foliation (in)dependence and Spacetime Reconstruction for a wide range of
physical theories, strategizing centred about the Problem of Beables, the
Patching Approach to the Global Problem of Time, and the role of the
question-types considered in physics. D) The Halliwell- and
Gambini-Porto-Pullin-type combined Strategies in the context of semiclassical
quantum cosmology.Comment: Invited Review: 26 pages including 2 Figures. This v2 has a number of
minor improvements and correction
The Invisible Thin Red Line
The aim of this paper is to argue that the adoption of an unrestricted principle of bivalence is compatible with a metaphysics that (i) denies that the future is real, (ii) adopts nomological indeterminism, and (iii) exploits a branching structure to provide a semantics for future contingent claims. To this end, we elaborate what we call Flow Fragmentalism, a view inspired by Kit Fine (2005)âs non-standard tense realism, according to which reality is divided up into maximally coherent collections of tensed facts. In this way, we show how to reconcile a genuinely A-theoretic branching-time model with the idea that there is a branch corresponding to the thin red line, that is, the branch that will turn out to be the actual future history of the world
Connected Coalition Formation and Voting Power in the Council of the European Union: An Endogenous Policy Approach. EIPA Working Paper 99/W/05
Resorting to political economy approaches, this paper attempts to associate the industrial structure in the European Union (EU) to the coalition formation process between European member states. Using a well-known measure of relative voting power, the (normalized) Banzhaf power index, we relax the common assumption that coalitions form randomly. Instead, we adopt the standard interest group model and look at the structure of European industry, mainly in terms of industrial concentration in the EU, as an indicator of its lobbying influence on domestic politics and governmentsâ preferences. This, in turn, influences the political stance, and thus the coalition building process, of the different member states in the Council. We derive estimates on membersâ relative influence within the Council for different policy areas in the broader framework of industry and trade, on the basis of both weighted votes and likely patterns of coalition-formation in the Council
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