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 $\aleph_1$

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 ($S_{EH}$) of gravitation, $\phi_{EH}=S_{EH}/\hbar$, 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

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 $\kappa^+\times\kappa^+$ contains either a clique or an independent subset of size $\tau\times\tau$ for every $\tau\in\kappa^+$, where $\kappa$ is a successor cardinal

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 $PSL(2,\mathbb{Z})\backslash\mathbb{H}$, 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 $L$-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

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