Fractional Gaussian Fields (FGF’s) are a family of stochastic processes that can be defined on any compact manifold M and model the behavior of random oscillations on M. FGF’s can be used to model for the behavior of bosons in particle physics, but they also find use in exploiting the connection between random matrix theory and number theory to prove the Riemann Hypothesis. For these reasons, studying the extreme values of these fields is of great interest to a variety of subjects, and yet computing the distributions of the maxima and minima of the FGF are challenging, open problems. We define discrete fractional gaussian fields on manifolds M that converge to the continuous fractional field, and by numerically simulating the discrete fractional gaussian fields, we generate conjectures for the behavior of the extreme values of the continuous field.

Loading...

Taking too long?

Reload document

| Open in new tab

In this poster, we begin by defining the Fractional Gaussian Field (FGF) X_s on S1 with parameter s. For s>1/4, we index our Gaussian random variables by points on the circle S1, and for s\leq 1/4 we index these random variables by smooth, real-valued functions on S1. Using methods from Fourier analysis and probability, we prove that these definitions make sense (i.e. that the series defining the FGF pointwise converges almost surely). We then prove that the sample paths of X_s are almost surely Lipschitz when s>3/4, Holder with exponent 1/k for s>1/2k, and k times continuously differentiable for s>(k+1)/2. We then prove the analogous results on the d dimensional torus Td.

The theory of FGF’s on manifolds is currently a topic of interest to cosmologists studying cosmic microwave background radiation and particle physicists studying Liouville quantum gravity. We outline a procedure for defining the FGF on an arbitrary compact Riemannian manifold (M,g), using tools from functional analysis and probability. Our current interest is in regularity conditions for paths of FGF on the sphere S2, which are more difficult to study due to the sphere’s positive Ricci curvature.

Loading...

Taking too long?

Reload document

| Open in new tab

Petersen and Tenner’s depth measures the distance of a permutation from its symmetric group’s identity and is known to be bound below by the average of two other measures of distance, length and reflection length. Permutations where this bound is an equality (i.e., permutations that are shallow) are known to not be characterizable by the containment and avoidance of patterns. However, using Hadjicostas and Monico’s recursive definition for shallowness, we show that the subset of shallow permutations which avoid 4231 can be characterized in this way. Such permutations are the ones which avoid 4231, 3412, 34521, 54123, 365214, 541632, 7652143, and 5476321.

Loading...

Taking too long?

Reload document

| Open in new tab

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.

Loading...

Taking too long?

Reload document

| Open in new tab

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.

Loading...

Taking too long?

Reload document

| Open in new tab

Bulk_and_Shear_Relativistic_Fluids  Data

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.

Loading...

Taking too long?

Reload document

| Open in new tab

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. 

Loading...

Taking too long?

Reload document

| Open in new tab

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, ψ1 is sub-multiplicative and 1/ψ1 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.

Loading...

Taking too long?

Reload document

| Open in new tab

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.

Loading...

Taking too long?

Reload document

| Open in new tab