Intersection theory and the Alesker product

Alesker has introduced the space $\mathcal V^\infty(M)$ of {\it smooth valuations} on a smooth manifold $M$, 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 pppp collisions

We calculate the production of real photons originating from the photoproduction in relativistic $pp$ collisions. The Weizs$\ddot{\mathrm{a}}$cker-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 $V$, 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 $SO(V)$ to general compact groups $G \subset O(V)$ 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 $M$ is a compact submanifold of euclidean space, then the volumes of small tubes about $M$ are given by a polynomial in the radius $r$, with coefficients that are expressible as integrals of certain scalar invariants of the curvature tensor of $M$ 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 $M$. This perspective yields a fundamental new structure in Riemannian geometry, in the form of a certain abstract module over the polynomial algebra $\mathbb R[t]$ 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 $M$ is a complex space form.Comment: Corrected version, to appear in GAF