5,487 research outputs found
Many Normal Measures
We characterize the situation of having many normal measures on a measurable
cardinal. We show the plausibility of having many normal measures on each
compact cardinal.Comment: 12 page
Conformal Dilatonic Cosmology
Gravitation and the standard model of particle physics are incorporated
within a single conformal scalar-tensor theory, where the scalar field is
complex. The Higgs field has a dynamical expectation value, as has the Planck
mass, but the relative strengths of the fundamental interactions are unchanged.
Initial cosmic singularity and the horizon problem are avoided, and spatial
flatness is natural. There were no primordial phase transitions; consequently,
no topological defects were produced. Quantum excitations of the dilaton phase
induced a slightly red-tilted spectrum of gaussian and adiabatic scalar
perturbations, but no analogous primordial gravitational waves were generated.
Subsequent cosmological epochs through nucleosynthesis are as in standard
cosmology. A generalized Schwarzschild-de Sitter metric, augmented with a
linear potential term, describes the exterior of stars and galaxies, such that
there is no need for dark matter on galactic scales.Comment: 5 page
Bipartite graphs and monochromatic squares
We prove that consistently every bipartite graph of size
contains either a clique or an independent subset of
size for every , where is a
successor cardinal
Weak diamond and Galvin's property
We prove that the Devlin-Shelah weak diamond implies Galvin's property. On
the other hand, Galvin's property is consistent with the negation of the weak
diamond, and even with Martin's axiom. We show that the proper forcing axiom
implies a relative to the negation of Galvin's property for
Scale Invariant Gravitation and Unambiguous Interpretation of Physical Theories
Our conventional system of physical units is based on local or microscopic
{\it dimensional} quantities which are {\it defined}, for convenience or
otherwise aesthetic reasons, to be spacetime-independent. A more general choice
of units may entail variation of fundamental physical quantities (`constants')
in spacetime. The theory of gravitation generally does not satisfy conformal
symmetry, i.e. it is not invariant to local changes of the unit of length.
Consequently, the {\it dimensionless} action associated with the
Einstein-Hilbert action () of gravitation, , is
not invariant to local changes of the length unit; clearly an unsatisfactory
feature for a dimensionless quantity. Here we amend the phase by adding extra
terms that account for spacetime variation of the physical `constants' in
arbitrary unit systems. In such a unit system, all dimensional quantities are
implicitly spacetime-dependent; this is achieved by a conformal transformation
of the metric augmented by appropriate metric-dependent rescalings of the
dimensional quantities. The resulting modified dimensionless action is
scale-invariant, i.e. independent of the unit system, as desired. The deep
connection between gravitation, dimensionless physical quantities, and quantum
mechanics, is elucidated and the implicit ambiguity in interpretations of
dimensional quantities is underlined.Comment: 5 pages, submitted, title changed, presentation improve
Conformal Higgs Gravity
It is shown that gravitation naturally emerges from the standard model of
particle physics if local scale invariance is imposed in the context of a
single conformal (Weyl-symmetric) theory. Gravitation is then
conformally-related to the standard model via a conformal transformation,
merely a function of the number of fermionic particles dominating the energy
density associated with the ground state of the physical system. Doing so
resolves major puzzles afflicting the standard models of particle physics and
cosmology, clearly indicating these to be artifacts stemming from universally
employing the system of units selected here and now. In addition to the three
known fundamental interactions mediated by gauge bosons, a scalar-tensor
interaction is also accommodated by the theory; its inertial and gravitational
sectors are characterized by whether contributions to the Weyl tensor vanish or
are finite, respectively. In this approach both inertia and gravity are viewed
as collective phenomena, with characteristic gravitational Planck scale devoid
of fundamental meaning; consequently, mass hierarchy and Higgs mass instability
concerns are avoided altogether. Only standard model particles gravitate; dark
matter and dark energy have an inertial origin, and since the Higgs field does
not interact with photons it is an ideal candidate for explaining the dark
sector of cosmology. On cosmological scales the dynamical vacuum-like Higgs
self-coupling accounts for dark energy, and its observed proximity at present
to the energy density of nonrelativistic matter is merely a consistency
requirement. Spatially varying vacuum expectation value of the Higgs field
could likely account for the apparent cold dark matter on both galactic and
cosmological scales.Comment: Significantly revised and extended versio
Cosmology in Conformal Dilatonic Gravity
Gravitation is described in the context of a dilatonic theory that is
conformally related to general relativity. All dimensionless ratios of
fundamental dimensional quantities, e.g. particle masses and the Planck mass,
as well as the relative strengths of the fundamental interactions, are fixed
constants. An interplay between the positive energy density associated with
relativistic matter (and possibly with negative spatial curvature) and the
negative energy associated with dynamical dilaton phase results in a
non-singular, flat cosmological model with no horizon, and -- as a direct
consequence of absence of phase transitions in the early universe -- with no
production of topological defects. The (logarithmic) time-derivative of the
field modulus is degenerate with the Hubble function, and all cosmological
epochs of the standard model are unchanged except at the very early universe.
We demonstrate that both linear order perturbation theory and the spherical
collapse model are equivalent to those in the standard model, up to
modifications caused by the phase of the (complex) scalar field and its
perturbations. Consequently, our alternative theory automatically passes the
main classical cosmological tests. Quantum excitations of the phase of the
scalar field generate a slightly red-tilted spectrum of adiabatic and gaussian
scalar perturbations on the largest scales. However, this framework does not
provide a similar mechanism for producing primordial gravitational waves on
these scales. A spherically symmetric vacuum solution that approximately
describes the exterior of gravitationally bound systems (e.g., stars and
galaxies) by a modified Schwarzschild-de Sitter metric, augmented with an
additional linear potential term, could possibly explain galactic rotation
curves and strong gravitational lensing with no recourse to dark matter.Comment: 28 pages, submitte
Simple wedge points
Let V be a finite set of points in the plane, not contained in a line. Assume
|V| = n is an odd number, and |L \cap V| \leq 3 for every line L which is
spanned by V. We prove that every simple line L_{a,b} in V creates a simple
wedge (i.e., a triple {a, b, c} \subseteq V such that L_{a,b} and L_{a,c} are
simple lines). We also show that both restrictions on V (namely |V| is odd and
|L \cap V| \leq 3) are needed. We conjecture, further, that if |V | = n is an
odd number then V contains a simple wedge, even if V is not 3-bounded. We
introduce a method for proving this, which gives (in this paper) partial
results.Comment: 10 page
Eisenstein Series and Breakdown of Semiclassical Correspondence
We consider certain Lagrangian states associated to unstable horocycles on
the modular surface , and show that for
sufficiently large logarithmic times, expectation values for the wave
propagated states diverge from the classical transport along geodesics. This is
due to the fact that these states "escape to the cusp" very quickly, at
logarithmic times, while the geodesic flow continues to equidistribute on the
surface. The proof relies crucially on the analysis of expectation values for
Eisenstein series initiated by Luo-Sarnak and Jakobson, based on subconvexity
estimates for relevant -functions--- in other words, this is a very special
case in which we can analyze long time propagation explicitly with tools from
arithmetic.Comment: 11 page
The Rational and Computational Scope of Probabilistic Rule-Based Expert Systems
Belief updating schemes in artificial intelligence may be viewed as three
dimensional languages, consisting of a syntax (e.g. probabilities or certainty
factors), a calculus (e.g. Bayesian or CF combination rules), and a semantics
(i.e. cognitive interpretations of competing formalisms). This paper studies
the rational scope of those languages on the syntax and calculus grounds. In
particular, the paper presents an endomorphism theorem which highlights the
limitations imposed by the conditional independence assumptions implicit in the
CF calculus. Implications of the theorem to the relationship between the CF and
the Bayesian languages and the Dempster-Shafer theory of evidence are
presented. The paper concludes with a discussion of some implications on
rule-based knowledge engineering in uncertain domains.Comment: Appears in Proceedings of the Second Conference on Uncertainty in
Artificial Intelligence (UAI1986
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