Open Systems in Classical Mechanics

Span categories provide a framework for formalizing mathematical models of open systems in classical mechanics. The categories appearing in classical mechanics do not have pullbacks, which requires the use of generalized span categories. We introduce categories \LagSy and \HamSy that respectively provide a categorical framework for the Lagrangian and Hamiltonian descriptions of open classical mechanical systems. The morphisms of \LagSy and \HamSy correspond to such open systems, and composition of morphisms models the construction of systems from subsystems. The Legendre transformation gives a functor from \LagSy to \HamSy that translates from the Lagrangian to the Hamiltonian perspective.Comment: 31 page

Airy functions over local fields

Airy integrals are very classical but in recent years they have been generalized to higher dimensions and these generalizations have proved to be very useful in studying the topology of the moduli spaces of curves. We study a natural generalization of these integrals when the ground field is a non-archimedean local field such as the field of p-adic numbers. We prove that the p-adic Airy integrals are locally constant functions of moderate growth and present evidence that the Airy integrals associated to compact p-adic Lie groups also have these properties.Comment: Minor change

On Infinitesimal Generators and Feynman-Kac Integrals of Adelic Diffusion

A prime $p$, an exponent, and a diffusion constant together specify a $p$-adic diffusion equation and a measure on the Skorokhod space of $p$-adic valued paths. The product, $P$, taken over the prime numbers of these measures with a fixed exponent is a probability measure on the product of the $p$-adic path spaces. Bounds on the exit probabilities for $p$-adic paths imply that the adelic paths have full measure in the product space if and only if the sum, $\sigma$, of the diffusion constants is finite. Finiteness of $\sigma$ implies that there is an adelic Vladimirov operator, $\Delta_{\mathbb A}$, and an associated diffusion equation whose fundamental solution gives rise to the measure induced by $P$ on an adelic Skorohod space. All moments of the random variable that counts the number of components of an adelic path that have journeyed outside of the ring of integers within a fixed time are finite. Given a simple adelic potential $V$, we obtain a path integral representation for the dynamical semigroup associated to the adelic Schr\"{o}dinger operator $\Delta_{\mathbb A} + V$.Comment: 33 page

Constructing Span Categories From Categories Without Pullbacks

Span categories provide an abstract framework for formalizing mathematical models of certain systems. The mathematical descriptions of some systems, such as classical mechanical systems, require categories that do not have pullbacks and this limits the utility of span categories as a formal framework. Given categories $\mathscr{C}$ and $\mathscr{C}^\prime$, we introduce the notion of span tightness of a functor $\mathcal F$ from $\mathscr{C}$ to $\mathscr{C}^\prime$ as well as the notion of an $\mathcal F$-pullback of a cospan in $\mathscr{C}$. If $\mathcal F$ is span tight, then we can form a generalized span category ${\rm Span}(\mathscr{C},\mathcal F)$ and circumvent the technical difficulty of $\mathscr{C}$ failing to have pullbacks. Composition in ${\rm Span}(\mathscr{C},\mathcal F)$ uses $\mathcal F$-pullbacks rather than pullbacks and in this way differs from the category ${\rm Span}(\mathscr{C})$ but reduces to it when both $\mathscr{C}$ has pullbacks and $\mathcal F$ is the identity functor.Comment: 20 pages, 14 figure

Brownian Motion in the pp-Adic Integers is a Limit of Discrete Time Random Walks

Vladimirov defined an operator on balls in $\mathbb Q_p$, the $p$-adic numbers, that is analogous to the Laplace operator in the real setting. Kochubei later provided a probabilistic interpretation of the operator. This Vladimirov-Kochubei operator generates a real-time diffusion process in the ring of $p$-adic integers, a Brownian motion in $\mathbb Z_p$. The current work shows that this process is a limit of discrete time random walks. It motivates the construction of the Vladimirov-Kochubei operator, provides further intuition about the properties of ultrametric diffusion, and gives an example of the weak convergence of stochastic processes in a profinite group.20 page

Buffon's Problem determines Gaussian Curvature in three Geometries

The classical Buffon problem requires a precise presentation in order to be meaningful. We reinterpret the classical problem in the planar setting with a needle whose length is equal to the grating width and find analogs of this problem in the settings of the sphere and the Poincar\'e disk. We show that the probability that the needle intersects the grating in these non euclidean settings tends to the probability of the intersection in the planar setting as the length of the needle tends to zero. Finally, we calculate the Gaussian curvature of the spaces from probability deficits related to the generalized Buffon problem, obtaining a result similar to the Bertrand-Diguet-Puiseux Theorem.Comment: 21 pages, 2 figure

Brownian Motion in a Vector Space over a Local Field is a Scaling Limit

For any natural number $d$, the Vladimirov-Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a $d$-dimensional vector space $V$ over a local field $K$. Just as the Laplace operator on $L^2(\mathbb R^d)$ is the infinitesimal generator of Brownian motion with state space $\mathbb R^d$, the Vladimirov-Taibleson operator on $L^2(V)$ is the infinitesimal generator of real-time Brownian motion with state space $V$. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted $V$ to be the $p$-adic numbers.Comment: 23 page

CellProfiler plugins -- an easy image analysis platform integration for containers and Python tools

CellProfiler is a widely used software for creating reproducible, reusable image analysis workflows without needing to code. In addition to the >90 modules that make up the main CellProfiler program, CellProfiler has a plugins system that allows for creation of new modules which integrate with other Python tools or tools that are packaged in software containers. The CellProfiler-plugins repository contains a number of these CellProfiler modules, especially modules that are experimental and/or dependency-heavy. Here, we present an upgraded CellProfiler-plugins repository with examples of accessing containerized tools, improved documentation, and added citation/reference tools to facilitate the use and contribution of the community.Comment: 17 pages, 2 figures, 1 tabl