We analyse the dynamics of the Friedmann-Robertson-Walker and Bianchi I cosmologies when considering a viscous fluid. By examining the role of the regulator introduced by Lichnerowicz to avoid the causality issue in Eckart theory, we formulate a dynamical system which we solve analytically in both cosmological models.

# Author: Anastasiia

# “Viscous Relativistic Hydrodynamics” by Jordan Marajh (Queen Mary University of London)

What would happen if you threw honey really hard at a black hole as opposed to water?

Mathematicians and physicists of the field studying Einstein’s Field Equations in hopes of explaining physical phenomena know so much about ideal fluids. However, the community requires research into viscous fluids of different types, i.e. admitting different types of viscosity.

We encounter those of bulk and shear viscosity in the hopes of either expanding the viscous terms of the stress-energy tensor and/or changing the physical background. Analysing the dynamical systems is a large focus point for the qualitative understanding of the fluids.

# “Tensor-Based Approaches to fMRI Classification” by Katherine Keegan (Mary Baldwin University)

To analyze the abundance of multidimensional data, tensor-based frameworks have been developed. Traditional matrix-based frameworks extract the most relevant features of vectorized data using the matrix-SVD. However, we may lose crucial high-dimensional relationships in this process. To facilitate efficient multidimensional feature extraction, we propose a projection-based classification algorithm using the t-SVDM, a tensor-based extension of the matrix-SVD. We apply our algorithm to the StarPlus functional Magnetic Resonance Imaging (fMRI) dataset. Through our numerical experiments, we conclude that there exists a more accurate tensor-based approach to fMRI classification than the best possible equivalent matrix-based approach. Our research showcases the potential of tensor-based classification frameworks, and justifies further research into the usage of tensors for numerous other classification tasks.

# “Decategorification of HFK_n(L)” by Larry Gu (USC)

Using a definition of Euler characteristic for fractionally-graded complexes based on roots of unity, we show that the Euler characteristics of Dowlin’s “sl(n)-like” Heegaard Floer knot invariants HFK_n recover both Alexander polynomial evaluations and sl(n) polynomial evaluations at certain roots of unity for links in S^3. We show that the equality of these evaluations can be viewed as the decategorified content of the conjectured spectral sequences relating sl(n) homology and HFK_n. This is joint work with Professor Andy Manion.

# “Lyapunov-type inequalities for third order nonlinear equations” by Brian Behrens (UConn)

We derive Lyapunov-type inequalities for general third-order nonlinear equations involving multiple *ψ*-Laplacian operators of the form

$$(\psi_{2}((\psi_{1}(u’))’))’ + q(x)f(u) = 0,$$

where

$$\psi_{2}\text{ and }\psi_{1}$$ are odd, increasing functions, *ψ*1 is sub-multiplicative and 1/*ψ*1 is convex, and *f* is a continuous function which satisfies a sign condition. Our results utilize q+ and q-, as opposed to |q| which appears in most results in the literature. Additionally, these new inequalities cover previously obtained results and notably do not require the use of Holder’s or the Cauchy-Schwartz inequality, unlike other proofs. Furthermore, using the obtained inequalities, we obtain a lower bound for the number of zeroes and properties of oscillatory solutions. We conclude with a constraint on the location of the maximum of a solution.

# “Properties of Reduced Convex Hulls” by Ben Arora (UConn)

A set S is convex if, for any two points in S, the line segment connecting them is also in S. Given a set of points P, we define their convex hull as the smallest convex set that contains P.

Sometimes, we want to shrink the convex hull of a set of points in a non-uniform manner. This gives rise to the more general concept of a “reduced convex hull.” Such objects originated in binary classification problems in machine learning, and have been studied further in the field of computational geometry. We will explore various properties of reduced convex hulls, and how these properties behave as the number of points in our set gets large.

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When posters are available please leave your questions/comments/feedback for the corresponding post. If you are the author, you should monitor the comments during the assigned time on Saturday and reply to posted comments.

# “Computational modeling of wind effects on a tall building by the finite element method” by Bakdauren Narbayev (Nazarbayev University)

Pedestrian wind comfort plays important role in the urban environment. In our work, we consider a model obtained using the Computational Fluid Dynamics (CFD) around the tall building. Our focus is the Tower of Abu Dhabi Plaza in Nur-Sultan city (Kazakhstan), which will be the tallest building of Central Asia with a height of 382 m. We investigated the effect of the wind velocity for pedestrians solving the incompressible time-dependent Navier-Stokes equations in the deal.II library by the Finite Element Method (FEM). We present numerical simulation results for various scenarios. It has been found that the velocity profile can vary in the domain that creates different pedestrian comfort conditions including the extreme category at places dedicated to the pedestrian walking.

# “Dynamical Characterization & Analysis of the Optimization Algorithms: Linearized Bregman and Iterative Shrinkage Thresholding Algorithm” by Yihua Xu (Georgia Tech)

Sparse recovery optimization algorithms are utilized in machine learning, imaging, and parameter fitting in problems, as well as a multitude of other fields. Compressive sensing, a prominent field in mathematics this past decade, has motivated the revival of sparse recovery algorithms with ?-1 norm minimization. Although small underdetermined problems are substantially well understood, large, inconsistent, nearly sparse systems have not been investigated with as much detail. In this dynamical study, two commonly used sparse recovery optimization algorithms, Linearized Bregman and Iterative Shrinkage Thresholding Algorithm are compared. The dependence of their dynamical behaviors on the threshold hyper-parameter and different entry sizes in the solution suggests complementary advantages and disadvantages. These results prompted the creation of a hybrid method which benefits from favorable characteristics from both optimization algorithms such as less chatter and quick convergence. The Hybrid method is proposed, analyzed, and evaluated as outperforming and superior to both linearized Bregman and Iterative Shrinkage Thresholding Algorithm, principally due to the Hybrid’s versatility when processing diverse entry sizes.

# “Agmon-Type Decay of Eigenfunctions for Schrodinger Operators with a Non-compactly Supported Classical Region” by Hengrui Zhu (Oberlin College)

A classical result by Agmon implies that an eigenfunction of a Schrodinger operator on R^d decays exponentially if the classical region associated with that eigenstate is compactly supported. A Schrodinger operator is the sum of the negative Laplacian and a multiplication operator by a real valued function, known as the potential. The classical region is defined as the subset of R^d for which the potential is less than the eigenvalue.

In this work, we extend the result to Schrodinger operators with an eigenvalue such that the classical region is not compactly supported. We show that assuming integrability of the classical region with respect to an increasing weight function implies L^2-decay of the eigenfunction with respect to the same weight. Here, decay is measured in the Agmon metric which takes into account anisotropies of the potential.