27,456 research outputs found
Polynomial Distributions and Transformations
Polynomials are common algebraic structures, which are often used to
approximate functions including probability distributions. This paper proposes
to directly define polynomial distributions in order to describe stochastic
properties of systems rather than to assume polynomials for only approximating
known or empirically estimated distributions. Polynomial distributions offer a
great modeling flexibility, and often, also mathematical tractability. However,
unlike canonical distributions, polynomial functions may have non-negative
values in the interval of support for some parameter values, the number of
their parameters is usually much larger than for canonical distributions, and
the interval of support must be finite. In particular, polynomial distributions
are defined here assuming three forms of polynomial function. The
transformation of polynomial distributions and fitting a histogram to a
polynomial distribution are considered. The key properties of polynomial
distributions are derived in closed-form. A piecewise polynomial distribution
construction is devised to ensure that it is non-negative over the support
interval. Finally, the problems of estimating parameters of polynomial
distributions and generating polynomially distributed samples are also studied.Comment: 21 pages, no figure
Tropical Theta Functions and Log Calabi-Yau Surfaces
We generalize the standard combinatorial techniques of toric geometry to the
study of log Calabi-Yau surfaces. The character and cocharacter lattices are
replaced by certain integral linear manifolds described by Gross, Hacking, and
Keel, and monomials on toric varieties are replaced with the canonical theta
functions which GHK defined using ideas from mirror symmetry. We describe the
tropicalizations of theta functions and use them to generalize the dual pairing
between the character and cocharacter lattices. We use this to describe
generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and
finite Fourier series expansions. We hope that these techniques will generalize
to higher-rank cluster varieties.Comment: 40 pages, 2 figures. The final publication is available at Springer
via http://dx.doi.org/10.1007/s00029-015-0221-y, Selecta Math. (2016
Arithmetic geometry of toric varieties. Metrics, measures and heights
We show that the height of a toric variety with respect to a toric metrized
line bundle can be expressed as the integral over a polytope of a certain
adelic family of concave functions. To state and prove this result, we study
the Arakelov geometry of toric varieties. In particular, we consider models
over a discrete valuation ring, metrized line bundles, and their associated
measures and heights. We show that these notions can be translated in terms of
convex analysis, and are closely related to objects like polyhedral complexes,
concave functions, real Monge-Amp\`ere measures, and Legendre-Fenchel duality.
We also present a closed formula for the integral over a polytope of a function
of one variable composed with a linear form. This allows us to compute the
height of toric varieties with respect to some interesting metrics arising from
polytopes. We also compute the height of toric projective curves with respect
to the Fubini-Study metric, and of some toric bundles.Comment: Revised version, 230 pages, 3 figure
Background independent quantizations: the scalar field II
We are concerned with the issue of quantization of a scalar field in a
diffeomorphism invariant manner. We apply the method used in Loop Quantum
Gravity. It relies on the specific choice of scalar field variables referred to
as the polymer variables. The quantization, in our formulation, amounts to
introducing the `quantum' polymer *-star algebra and looking for positive
linear functionals, called states. Assumed in our paper homeomorphism
invariance allows to derive the complete class of the states. They are
determined by the homeomorphism invariant states defined on the CW-complex
*-algebra. The corresponding GNS representations of the polymer *-algebra and
their self-adjoint extensions are derived, the equivalence classes are found
and invariant subspaces characterized. In the preceding letter (the part I) we
outlined those results. Here, we present the technical details.Comment: 51 pages, LaTeX, no figures, revised versio
- …