Notes on MATH 109, group theory.

Lectures

• SU-MATH109 SEP272023

PSets

These links are dead.

• SU-MATH109 Problem Set 1

Course logistics

• midterm: November 1st, final: December 14th, 8:30-11:30
• WIM assignment: December 8th, start of class (no late submissions)
• PSets: 8 in total, posted on Wednesdays at 8A, due following Tuesday at 8A

“Stuffing some stuff into buckets”

How many ways are there to sort \(n\) distinct objects to \(r\) buckets?

\begin{equation} r^{n} \end{equation}

grouping with entirely indistinct objects

You can simply reframe the grouping problem as permutation of the objects with \(r-1\) dividers along with your old \(n\) objects.

i.e.: sort this thing —

So:

\begin{equation} \frac{(n+r-1)!}{n! (r-1)!} \end{equation}

The Guilded Age is a period in history between 1877 and 1900. This period deepened divide in racism, deepened the split between poor and rich, and the fluidity of American social classes became more set in this time.

Links to Organize

• Imperialism
• New American South

Why is the “Guilded Age” “Guilded”?

Guilded: Outside Lined with Gold, Inside Contains Metal and is Less Valuable.

The Guilded Age consists of three different sections:

One-Liner

UAV navigation through leveraging updrafts, handling their unpredictability with POMDPs and Receeding Horizon.

Novelty

• Developed new method for low-cost POMDP online solving
• Cool bird.

Notable Methods

two main steps

• explore: determine thermal parameters
• exploit: plan a trajectory to exploit the thermal

formulation

• \(\mathcal{S}\): \(s^{u} \in \mathbb{R}^{6}\), the joint state of the UAV (2D location wrt fixed point + air speech + heading, bank, roll, altitude), and \(s^{th} \in \mathbb{R}^{2}\),the thermal status (thermal center x and y relative to UAV)
• \(\mathcal{A}\): discretized arc trajectory segments by bank angles \(\phi_{1 \dots n}\), which executes for a fixed \(T_{A}\) seconds
• \(\mathcal{T}\): Gaussian of \(s^{u}\) over the dynamics of the UAV, and over fixed noise covariance \(Q\)
• \(\mathcal{R}\): \(h_{s’}-h_{s}\), the change in altitude….
• \(\mathcal{O}\): senor readings
• \(O(a, s’, o)\): fixed noise covariance \(R\)
• \(b_0\): product of two Gaussian of the UAV’s position and the belief about the underlying thermals
• \(update(b,a,o)\): EKF

modeling assumptions:

DOI: 10.3389/fcomp.2021.642517

One-Liner

Used WLS data to augment CTP from ADReSS Challenge and trained it on a BERT with good results.

Novelty

• Used WLS data with CTP task to augment ADReSS DementiaBank data

Notable Methods

WLS data is not labeled, so authors used Semantic Verbal Fluency tests that come with WLS to make a presumed conservative diagnoses. Therefore, control data is more interesting:

Key Figs

Table 2

Data-aug of ADReSS Challenge data with WSL controls (no presumed AD) trained with a BERT. As expected the conservative control data results in better ferf