366 research outputs found
Spectral analysis on infinite Sierpinski fractafolds
A fractafold, a space that is locally modeled on a specified fractal, is the
fractal equivalent of a manifold. For compact fractafolds based on the
Sierpinski gasket, it was shown by the first author how to compute the discrete
spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian.
A similar problem was solved by the second author for the case of infinite
blowups of a Sierpinski gasket, where spectrum is pure point of infinite
multiplicity. Both works used the method of spectral decimations to obtain
explicit description of the eigenvalues and eigenfunctions. In this paper we
combine the ideas from these earlier works to obtain a description of the
spectral resolution of the Laplacian for noncompact fractafolds. Our main
abstract results enable us to obtain a completely explicit description of the
spectral resolution of the fractafold Laplacian. For some specific examples we
turn the spectral resolution into a "Plancherel formula". We also present such
a formula for the graph Laplacian on the 3-regular tree, which appears to be a
new result of independent interest. In the end we discuss periodic fractafolds
and fractal fields
A Non-Riemannian Metric on Space-Time Emergent From Scalar Quantum Field Theory
We show that the two-point function
\sigma(x,x')=\sqrt{} of a scalar quantum field theory
is a metric (i.e., a symmetric positive function satisfying the triangle
inequality) on space-time (with imaginary time). It is very different from the
Euclidean metric |x-x'| at large distances, yet agrees with it at short
distances. For example, space-time has finite diameter which is not universal.
The Lipschitz equivalence class of the metric is independent of the cutoff.
\sigma(x,x') is not the length of the geodesic in any Riemannian metric.
Nevertheless, it is possible to embed space-time in a higher dimensional space
so that \sigma(x,x') is the length of the geodesic in the ambient space.
\sigma(x,x') should be useful in constructing the continuum limit of quantum
field theory with fundamental scalar particles
Nontangential limits and Fatou-type theorems on post-critically finite self-similar sets
In this paper we study the boundary limit properties of harmonic functions on
, the solutions to the Poisson equation where is a p.c.f. set
and its Laplacian given by a regular harmonic structure. In
particular, we prove the existence of nontangential limits of the corresponding
Poisson integrals, and the analogous results of the classical Fatou theorems
for bounded and nontangentially bounded harmonic functions.Comment: 22 page
Optimal frequency for microfluidic mixing across a fluid interface
A new analytical tool for determining the optimum frequency for a micromixing strategy to mix two fluids across their interface is presented. The frequency dependence of the flux is characterized in terms of a Fourier transform related to the apparatus geometry. Illustrative microfluidic mixing examples based on electromagnetic forcing and fluid pumping strategies are presented.Sanjeeva Balasuriy
The structure of the space of affine Kaehler curvature tensors as a complex module
We use results of Matzeu and Nikcevic to decompose the space of affine
Kaehler curvature tensors as a direct sum of irreducible modules in the complex
settin
Harmonic analysis of iterated function systems with overlap
In this paper we extend previous work on IFSs without overlap. Our method
involves systems of operators generalizing the more familiar Cuntz relations
from operator algebra theory, and from subband filter operators in signal
processing.Comment: 37 page
Convergence of the Magnus series
The Magnus series is an infinite series which arises in the study of linear
ordinary differential equations. If the series converges, then the matrix
exponential of the sum equals the fundamental solution of the differential
equation. The question considered in this paper is: When does the series
converge? The main result establishes a sufficient condition for convergence,
which improves on several earlier results.Comment: 11 pages; v2: added justification for conjecture, minor
clarifications and correction
Leibnizian, Galilean and Newtonian structures of spacetime
The following three geometrical structures on a manifold are studied in
detail: (1) Leibnizian: a non-vanishing 1-form plus a Riemannian
metric \h on its annhilator vector bundle. In particular, the possible
dimensions of the automorphism group of a Leibnizian G-structure are
characterized. (2) Galilean: Leibnizian structure endowed with an affine
connection (gauge field) which parallelizes and \h. Fixed
any vector field of observers Z (), an explicit Koszul--type
formula which reconstruct bijectively all the possible 's from the
gravitational and vorticity fields
(plus eventually the torsion) is provided. (3) Newtonian: Galilean structure
with \h flat and a field of observers Z which is inertial (its flow preserves
the Leibnizian structure and ). Classical concepts in Newtonian
theory are revisited and discussed.Comment: Minor errata corrected, to appear in J. Math. Phys.; 22 pages
including a table, Late
Generalized and weighted Strichartz estimates
In this paper, we explore the relations between different kinds of Strichartz
estimates and give new estimates in Euclidean space . In
particular, we prove the generalized and weighted Strichartz estimates for a
large class of dispersive operators including the Schr\"odinger and wave
equation. As a sample application of these new estimates, we are able to prove
the Strauss conjecture with low regularity for dimension 2 and 3.Comment: Final version, to appear in the Communications on Pure and Applied
Analysis. 33 pages. 2 more references adde
Quantization of Dirac fields in static spacetime
On a static spacetime, the solutions of the Dirac equation are generated by a
time-independent Hamiltonian. We study this Hamiltonian and characterize the
split into positive and negative energy. We use it to find explicit expressions
for advanced and retarded fundamental solutions and for the propagator.
Finally, we use a fermion Fock space based on the positive/negative energy
split to define a Dirac quantum field operator whose commutator is the
propagator.Comment: LaTex2e, 17 page
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