Piaget’s theory of cognitive development stands as a cornerstone in understanding how individuals perceive, reason, and comprehend the world around them. We have summarized the fundamental principles of Piaget’s theory, exploring stages of child development. Through assimilation and accommodation, individuals continually adapt their mental schemas to incorporate new information as well as new experiences. Furthermore, we have emphasized the significance of play, curiosity, and social interaction in fostering cognitive advancement and give examples in the context of learning mathematics. While Piaget’s theory has faced critiques and modifications over time, its enduring legacy continues to inform educational practices and psychological research.

Poster Presentation

Session 3

2:45pm – 4:00pm
Grand Salon

Mathematics

In this presentation, I am investigating Octonions (O) as rank three tensors, their multiplication, and their various constructions Octonions are an Algebra that have the following properties: The are non-commutative (ab ̸ = ba) nor are they anti-commutative (ab ̸ = -ba). They also are non-associative as (a(bc)̸ =(ab)c). They do however abide by the basis square multiplication of: e2_1 = e2_2 = e2_3 = e2_4 = e2_5 = e2_6 = e2_7 = −1 = −(e0)2 where e0 is the real basis and e1, e2, e3, e4, e5, e6, e7 are the imaginary basis, also known as the 7 vector portion of the octonion.

Octonions are an extension of quaternions with additional imaginary basis e_4 with the associated multiplication giving 8 basis. They pose an important role at the current edge of particle physics research being the mathematical system proposed to describe supersymmetry.

A common issue with octonions is that they are not associative. They are commonly attempted to be transformed into matrices which in themselves are associative. This is a contradiction but is considered necessary by many physicists. This will also be investigated as it is currently under great debate.

Poster Presentation

Session 3

2:45pm – 4:00pm
Grand Salon

Mathematics

This research presents a method for analyzing quasiperiodic videos by leveraging the principles of topological decoupling. By unraveling the intricate spatial patterns inherent in these videos, we appeal to persistent homology to identify and track persistent topological features over time. Our findings demonstrate the efficacy of topological decoupling in extracting meaningful spatiotemporal features,which can be harnessed for various applications including de-noising, anomaly detection, and video synthesis. This ork bridges the fields of topology and video processing, providing valuable insights into the nature of quasiperiodic phenomena and unlocking new avenues for interpreting and manipulating complex visual patterns. This research explores the behavior of oscillators within the video context. Specifically, the focus is on a video consisting of two oscillators that continuously move back and forth.This study not only enhances our understanding of the dynamical systems depicted in quasiperiodic videos but also paves the way for innovative approaches in video editing and motion control, offering promising implications for the field of computational video analysis.

Poster Presentation

Session 3

2:45pm – 4:00pm
Grand Salon

Mathematics

“Laser-driven acceleration of protons presents exciting opportunities for future advancements in ultra-intense laser science and particle research. Optimizing laser parameters provides more control over the configuration of the laser system, producing a proton beam with an energy spectrum with desired characteristics. Invertible Neural Networks (INNs) are a recent development in deep learning and offer a promising approach to optimize these parameters. INNs are a class of normalizing flows with a computationally-efficient Jacobian determinant. This enables bidirectional training, reconstruction of input data from latent representations, and model interpretability. INNs have the potential to solve the inverse problem of achieving desired ion energy distributions by determining optimal laser parameters, and generate high-fidelity synthetic data through inverse transformations.

As a proof of concept, we developed and trained an INN on MNIST data to learn a model capable of forward and backward predictions. Our INN can construct unique images of handwritten digits, and accurately classify new images without retraining.”

Poster Presentation

Session 3

2:45pm – 4:00pm
Grand Salon

Mathematics

We are exploring the concept of teaching limits in calculus. Currently, test results show that many students have problems with this topic and since the limits are a prerequisite knowledge for understanding differentiation, integration, sequences, we aim to improve student learning and the related assessment of their performance. Such that we will be exploring the concept of limits in calculus by demonstrating through examples on how limits bring forward the knowledge for understanding differentiation, integration, and sequences. We also will be explaining the importance of why teachers introduce a very difficult topic such as limits in calculus before introducing any other topic like differentiation, integration, and sequences. Where ultimately we want to show that limits are the backbone of calculus such that if students fail to master the concepts of limits, then they will find it hard to go forward to the next calculus topics. Such that our main goal is to design activities helping students to understand these concepts better.

Poster Presentation

Session 3

2:45pm – 4:00pm
Grand Salon

Mathematics

Partial differential equations are often used to model the behavior of systems in many different fields of study. For instance is physics they can used to model particle behavior, corrosion models for biological quantities as well as electricity and magnetism.
Here for this topic, the significance of this topic/study is based upon nonlocal gradient operators where non local differential operators are very important in the studies of nonlocal models especially nonlocal diffusion models which apply to many different areas in research.
For the background it is composed of partial differential equations in 3-D dimensions. Also it is  where the variational setting is the nonlocal Dirichlet energies within the energy densities which are quadratic in nonlocal gradients. For the gradient dependent second order equations where the influence on the solution can arise in different ways on one hand there are semi-linear equations and the idea of drift or transport.
We will present the difference between local and nonlocal heat kernels with respect to linear and nonlinear models. Also we want to distinguish the difference between the homogenous versus nonhomogeneous equations in particular their applications .

Poster Presentation

Session 3

2:45pm – 4:00pm
Grand Salon

Mathematics

In this presentation I am going through the various Algebras in Physics and their uses and rationalizations for why they are used. The Algebras covered in this presentation are R, C, H, Cl+ 3,0, O, and Exterior Algebras. Discussing wave functions, Quantum Mechanics, Rotations, Cross Product as wedge products, Spinors, and Supersymmetry.

I will be exploring representations in physics and identifying consistent patterns within numerical systems that would solve a particular property of the equation.

The real numbers are the basic continuous number system most capable of being identified. It is any number that can have a decimal and has only one basis. The next would be the complex numbers, denoted as a + bi. A real number:a and a complex numbe:b with the imaginary generator i where i^2 = -1. A lesser known but very common algebra would be the quaternions. they come in the form a +bi + cj +dk where i^ = j^2 = k^2 = ijk = -1

Poster Presentation

Session 2

1:00pm – 2:15pm
Grand Salon

Mathematics

This project is looking to show the benefits of further research of peridynamics on biological materials by using data analytics. By looking into the medical costs of specific ailments, researching surveys on quality of life, I look to find significant figures that justify my claim that further and more aggressive research should be invested into the topic. I will be presenting an existing model of peridynamics and show what its current applications are used for. Furthermore, I will present challenges that researchers are and will be facing when trying to apply a model to biological materials. However, if researchers develop and remodel the framework of peridynamics that was originally formulated for engineered materials, the potential benefits of understanding, predicting, and ultimately controlling the mechanical behavior of biological materials is vast. It can lead to improvements of medical care effectiveness, decrease the costs of care, and thus, increase quality of life.

Poster Presentation

Session 2

1:00pm – 2:15pm
Grand Salon

Mathematics

Whereas there have been analytical proofs for Apery’s constant (the sum of the reciprocals of cubes), there have been no geometrical proofs. In our search for such a proof, we constructed a new type of triangular spiral and associated it with the Apery’s constant equation. Through an infinite process, we stacked right-triangles with specific values on top of each other, obtaining a growing spiral. We extended this construction to many series of similar form, namely, real values of the Riemann-Zeta function. There is a well known closed form for the Riemann-Zeta function values at even integers, but not at odd integers. The spirals we constructed apply to both even and odd integers. We investigated the properties of these spirals through drawings and computer programming. In addition to spirals, we built interesting angles associated with these Riemann-Zeta values. Our work may contribute to the pursuit of a closed form expression of Apery’s constant, which has been a dream of many mathematicians for nearly three centuries.

Poster Presentation

Session 2

1:00pm – 2:15pm
Grand Salon

Mathematics

Linear programming, also known as linear optimization, plays a pivotal role in resource allocation and decision-making across various fields. In this study, we concentrate on the powerful simplex method that is currently widely used and may involve many variables. This method is an example of optimization problem for functions (representing resources) with domains restricted by linear constrains. The shape of the domain is polygonal (or looks like a simplex in several dimensions). The process navigates through multiple variables to identify the most efficient utilization of resources in achieving desired outcomes. We aim to elucidate the mechanics of the simplex method and showcase examples of real-world applications. We provide visualization in the case of two variables. For more variables the method used matrices and row reduction to find the optimal answer (hence computers can solve these types of problems). By exploring its functionality and practical implications, we unveil the versatility and effectiveness of the simplex method as a tool for optimization in many different environments.

Poster Presentation

Session 2

1:00pm – 2:15pm
Grand Salon

Mathematics