# Brown Bag Seminar

A mathematical model of the vascular conducted response

The vascular conducted response (VCR) is an electrical signal propagated along the walls of small blood vessels independent of blood flow direction or perivascular nerves. VCRs are thought to be essential in signaling changes in oxygen demand of local tissue to distal upstream arterioles, allowing for coordinated adjustments in vessel diameter along the entire flow pathway. However, the properties and driving mechanisms of VCRs are not well understood. Recent research suggests potassium inward rectifier (Kir) channels play a significant role in extending conduction distance. Here, we present a mathematical model of the VCR investigating the effects of Kir expression on signal behavior. This work is supervised by Dr. Timothy Secomb.

## When

## Where

*Math, 402*

The role of thermal noise in the laminar-turbulent transition of hypersonic boundary layers

The transition of hypersonic boundary layers from laminar to turbulent has important practical implications for the design of hypersonic aircraft. Despite significant progress in recent decades, the phenomenon is still not well understood, particularly for atmospheric flight conditions. Thermal noise, which has origin at the molecular level, has been proposed as a possible explanation for transition in cases where the level of external disturbances is low. In this talk, a method is developed for computing the amplitude of disturbances induced by thermal noise in high-speed compressible boundary layers, beginning from the equations of fluctuating hydrodynamics. The approach relies on a coupling of linear stability theory (LST) with receptivity theory and asymptotic methods. Extensions to the theory in cases for which the stability spectrum is particularly complicated are discussed.

## When

## Where

*Math, 402*

Asymptotic stability of boundary layers in hypersonic chemically reacting mixtures subject to perturbations induced by thermal noise

In fluid dynamics, the boundary layer is a thin region of fluid flow near fluid-solid interfaces where viscosity plays a significant role. These boundary layers can usually be categorized as being either smooth (laminar) or turbulent. The transition of this boundary layer from a laminar state to a turbulent state is generally not well understood yet nonetheless crucial in many engineering applications. This particular work is motivated by the need to predict the location of this boundary layer transition over the body of hypersonic aircrafts where the fluid flow is both hypersonic (moving over 5 times the speed of sound) and chemically reacting. In this context, there has been recent interest in how the unavoidable thermal noise responsible for Brownian motion can play a role in the onset of turbulence for certain flight conditions. In this talk a brief introduction to boundary layer stability theory will be presented, a particular stability problem will be motivated and stated, this will then be followed by some results.

## When

## Where

*Math, 402*

SIAM Lunch with Alumnus and Professor John Gemmer, Department of Mathematics, Wake Forest University

Come and learn about:

- Wisdom and perspective on academic careers
- How to set yourself up well while in graduate school to land academic postdocs/jobs
- How an alumnus of the Applied Math Program has built a successful career in the years since graduating

A bit about him… John Gemmer is an alumnus of the Program in Applied Mathematics here at Arizona, obtaining his PhD in 2012. His is now an Assistant Professor in the Department of Mathematics and Statistics at Wake Forest University, after postdoctoral fellowships at the Arizona Center for Mathematical Sciences here and at Brown University. He has since built a successful academic career teaching and mentoring students, while developing a strong research program in applying analysis, variational calculus, asymptotics, differential geometry, and numerical simulations to study phenomena in the physical and biological sciences. Meanwhile, he's inspired a few first/second-year students in our Program to come here!

## When

## Where

*Math, 402*

Using Quskit, DiNT and implementing the Hellman Feynman theorem for the computation of molecular dynamics. (Argonne National Lab research experience)

This past summer I was privileged enough to work at Argonne national lab. There I worked on computational chemistry and my main project was on molecular dynamics using Qiskit. Qiskit is a powerful computational framework developed by IBM for the purpose for quantum computing. It gives the user access to quantum simulators and Quantum computers across the globe. Using DiNT(Direct Nonadiabatic trajectories),a program designed for classical molecular dynamics, and Qiskit we develope a hybrid program to compute the trajectories of diatomic molecules. I shall share some of my results and experiences during my time at Argonne.

## When

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*Math, 402*

External Funding for Graduate Students

Please join us for an information session on the types of funding opportunities available to you as a graduate student. We will discuss how to search for funding, specific opportunities, and give you tips for applying successfully.

## When

## Where

*Math, 402*

The EM algorithm, Deep Learning and the Softmax Layer

The EM Algorithm as described by Dempster et. al. provides excellent approaches to very difficult inference problems. Recent approaches in deep learning, such as variational autoencoders, give a gradient based approach to similar problems. While the above are *unsupervised* algorithms, it is the case that the softmax layer commonly used in *supervised* classification shares many features with the EM algorithm. We will discuss the connections between these algorithms.

## When

## Where

*Math, 402*

Drive-Based Motivation for Coordination of Limit Cycle Behaviors

Constructing autonomous systems capable of high-level behaviors often involves reducing these behaviors to a collection of low-level tasks. This requires developing a method for switching among possible tasks. Recent work has developed continuous dynamical systems that have an internal drive state to select the desired task. In one particular result, authors considered a scenario where individual behaviors were encoded in control vector fields with unique, globally stable equilibria. A further level of complexity arises when one seeks to create a system that switches between tasks encoded as globally attracting sets with recurrent behaviors, rather than as point attractors. This presentation outlines the problem using the recently-developed drive-based dynamical framework. First we generalize the formulation of tasks as one part attracting set and one part recurrent behavior on said attracting set. Then as a proof-of-concept we demonstrate the existence of an attracting set consisting of orbits that repeatedly flow between two canonical limit cycles (e.g., Hopf oscillators). Finally we give some general results for the case of arbitrary disjoint limit cycles.

## When

## Where

*Math, 402*

A new type of supercritical collapse for intense long-wavelength ultrashort laser pulses

The study of nonlinear light-matter interactions was born in the 1960s when the first powerful cw laser sources came online. Soon after, researchers began developing intense pulsed laser sources with durations of only a few femtosecond (fs) (10^(-15) s) capable of delivering Terawatt and even Petawatt powers. Recently interest has shifted from visible and near-infrared sources to long-wavelength-infrared (LWIR) fs-scale pulses. In this talk we introduce a canonical description for such few-cycle pulses in the limit of weak dispersion and relatively strong nonlinear effects. Our model predicts a new type of carrier-wave resolved supercritical collapse, leading to extreme spatiotemporal confinement of the pulse's electric field, and whose dynamics persist in physically-more-complete computational models.

## When

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*Math, 402*

Boundary layer potentials with some applications from imaging

**Philip Hoskins, Department of Mathematics, University of Arizona**

The use of integral equations to solve boundary value problems has a long history, beginning with the study of gravitation and electrostatics in the late 18th and early 19th centuries. It was shown that solutions of the Neumann and Dirichlet problems for Laplace's equation can be represented by boundary layer potentials, which are certain integrals taken over the boundary of the domain of interest. Similar techniques can be used to study the solutions of other basic PDEs from mathematical physics. While these techniques are considered outdated for the treatment of PDE, they are still important today in many applications. We will start with an accessible introduction to boundary layer integrals and then discuss their applications to some medical imaging techniques, such as magnetoacoustic imaging and photoacoustic tomography.

## When

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*Math, 402*