55,148 research outputs found
Intersection theory and the Alesker product
Alesker has introduced the space of {\it smooth
valuations} on a smooth manifold , and shown that it admits a natural
commutative multiplication. Although Alesker's original construction is highly
technical, from a moral perspective this product is simply an artifact of the
operation of intersection of two sets. Subsequently Alesker and Bernig gave an
expression for the product in terms of differential forms. We show how the
Alesker-Bernig formula arises naturally from the intersection interpretation,
and apply this insight to give a new formula for the product of a general
valuation with a valuation that is expressed in terms of intersections with a
sufficiently rich family of smooth polyhedra.Comment: further revisons, now 23 page
Assessment method for photo-induced waveguides
A method to probe the guiding characteristics of waveguides formed in
real-time is proposed and evaluated. It is based on the analysis of the time
dependent light distribution observed at the exit face of the waveguide while
progressively altering its index profile and probed by a large diameter optical
beam. A beam propagation method is used to model the observed dynamics. The
technique is applied to retrieve the properties of soliton-induced waveguides
Evolution of Cooperation in Public Goods Games with Stochastic Opting-Out
This paper investigates the evolution of strategic play where players drawn
from a finite well-mixed population are offered the opportunity to play in a
public goods game. All players accept the offer. However, due to the
possibility of unforeseen circumstances, each player has a fixed probability of
being unable to participate in the game, unlike similar models which assume
voluntary participation. We first study how prescribed stochastic opting-out
affects cooperation in finite populations. Moreover, in the model, cooperation
is favored by natural selection over both neutral drift and defection if return
on investment exceeds a threshold value defined solely by the population size,
game size, and a player's probability of opting-out. Ultimately, increasing the
probability that each player is unable to fulfill her promise of participating
in the public goods game facilitates natural selection of cooperators. We also
use adaptive dynamics to study the coevolution of cooperation and opting-out
behavior. However, given rare mutations minutely different from the original
population, an analysis based on adaptive dynamics suggests that the over time
the population will tend towards complete defection and non-participation, and
subsequently, from there, participating cooperators will stand a chance to
emerge by neutral drift. Nevertheless, increasing the probability of
non-participation decreases the rate at which the population tends towards
defection when participating. Our work sheds light on understanding how
stochastic opting-out emerges in the first place and its role in the evolution
of cooperation.Comment: 30 pages, 4 figures. This is one of the student project papers arsing
from the Mathematics REU program at Dartmouth 2017 Summer. See
https://math.dartmouth.edu/~reu/ for more info. Comments are always welcom
Real photons produced from photoproduction in collisions
We calculate the production of real photons originating from the
photoproduction in relativistic collisions. The
Weizscker-Williams approximation in the photoproduction is
considered. Numerical results agree with the experimental data from
Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC). We find
that the modification of the photoproduction is more prominent in large
transverse momentum region.Comment: 2 figure
Convolution of convex valuations
We show that the natural "convolution" on the space of smooth, even,
translation-invariant convex valuations on a euclidean space , obtained by
intertwining the product and the duality transform of S. Alesker, may be
expressed in terms of Minkowski sum. Furthermore the resulting product extends
naturally to odd valuations as well. Based on this technical result we give an
application to integral geometry, generalizing Hadwiger's additive kinematic
formula for to general compact groups acting
transitively on the sphere: it turns out that these formulas are in a natural
sense dual to the usual (intersection) kinematic formulas.Comment: 18 pages; Thm. 1.4. added; references updated; other minor changes;
to appear in Geom. Dedicat
Riemannian curvature measures
A famous theorem of Weyl states that if is a compact submanifold of
euclidean space, then the volumes of small tubes about are given by a
polynomial in the radius , with coefficients that are expressible as
integrals of certain scalar invariants of the curvature tensor of with
respect to the induced metric. It is natural to interpret this phenomenon in
terms of curvature measures and smooth valuations, in the sense of Alesker,
canonically associated to the Riemannian structure of . This perspective
yields a fundamental new structure in Riemannian geometry, in the form of a
certain abstract module over the polynomial algebra that
reflects the behavior of Alesker multiplication. This module encodes a key
piece of the array of kinematic formulas of any Riemannian manifold on which a
group of isometries acts transitively on the sphere bundle. We illustrate this
principle in precise terms in the case where is a complex space form.Comment: Corrected version, to appear in GAF
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