Brian Deer

Tewa Kpulun

For our computational project Brian and I will be focusing our efforts on Neural Networks and the Brain. We will be using the Ising model and the Monte Carlo method to model a network of neurons and investigate pattern recognition. With this, we will be able to learn more about content addressable memory, an important component of human memory. Each neuron can be in either two states, spin up or spin down where spin up corresponds to a neuron being active and spin down corresponds to a neuron being inactive. With this comparison, we are able to apply the Ising model to understand the behavior of pattern recognition within the brain.

We will be looking at a very simplified version of the human brain, which we will refer to as the neural network. Our main topics will be how many patterns can the neural network store, the amount of input information that is needed for a successful retrieval of a stored pattern, how long those retrievals would take for various levels of input information, and if the size of the neural network affects it’s overall performance.

An acoustical plane wave, much like a light source, displays diffraction patterns based on the object it encounters (slits, shapes, objects, etc.). A sphere is one such object of particular interest, as the diffraction pattern produced by a sphere can be used to deconstruct sound signals on the surface of the sphere (a signal which would otherwise be nonsense because of the diffraction). Spheres have been increasingly used in recent years as the basic geometry of microphone arrays, and they offer many advantages over, for example, linear microphone arrays. The project will therefore focus on modelling and visually displaying diffraction of an acoustic wave due to a sphere, and will (hopefully) delve into spherical harmonic decomposition of the sound field on the sphere.

The advantage of using a computer to find a numerical solution rather than an analytical solution is quite clear when considering the relationship between Jupiter, the Sun, and the Earth. Rather, it is impossible to achieve an analytical solution for most celestial mechanics problems involving 3 or more bodies, thus numerical approaches are the only alternative.

Our project begins with modeling the Sun/Earth/Jupiter system with the Sun fixed at the origin. The three bodies are related via the inverse square law, and the positions of Jupiter and the Earth are calculated using their respective equations of motion (the sum of the forces from the other two bodies). Once a working model of the planetary orbits is established, it is worth investigating the effects of Jupiter on the Earth’s orbit when Jupiter’s mass is increased. We know that the current orbit is stable, but at what point does it become unstable?

If the mass of Jupiter is increased by roughly a thousand fold, it actually becomes comparable to the Sun’s mass. In this case, the model can be expanded to approximate a true 3-body system, with the origin at the center of mass of the system instead of fixed at the sun. The advantage of this model is that it allows us to observe how all three planets are effected by this change in mass. How large an effect, and at what point the orbits becomes unstable, remain open to investigation.

The Monte Carlo method has extended into many different aspects of the world, including, but not limited to physics and finance. The Ising model models ferromagnetism using the concept that an electron can either be spin up or spin down. A practical way to solve the Ising model is to use the Monte Carlo method, which calculates probabilities for a discrete number of iterations. Elias Kim and I want to investigate how the Ising model and the Monte Carlo method aid economists in making decisions on portfolios and risk analysis. The Ising model, which uses complex interactions between electrons and energy, can be translated to the complex parameters associated with stock volatility. To begin we will use MatLab to model the simple Ising model in chapter 8 of Computational Physics, by Nicholas J. Giordano and Hisao Nakanishi. Ultimately, we hope to compare our results in MatLab to studies that have already been published.

Using the Euler cromer method, we will investigate the movement of solar entities in orbit. First, we will write a program that numerically finds the position and velocity of the earth as it orbits the sun. Second, we will write a program that numerically finds position and velocity of the moon as it orbits the earth as the earth orbits the sun. Third, we will write a program that numerically finds the position and velocity of a large star such as the sun as a massive planet orbits it, or rather as they orbit and affect each other. To calculate these movements we will use basic equations of motion and Kepler’s laws.

For our project, we are interested in exploring realistic projectile motion as it pertains to the human body.  While air resistance and spin often produce negligible effects for the objects in introductory mechanics, normally modeled as point particles, they can produce significant effects on objects with larger surface area and mass, such as the human body.

In addition to this, with athletic competitions often being decided by seconds in races or mere inches in jumping events, minimizing adverse effects can ensure victory.  We are interested in modeling the human body in motion in MatLAB in Olympic jumping events, contorting limbs in obscure positions and seeing how it affects the ultimate outcome of the event.

Many economists work on problems where they cannot be sure how different factors affect each other. In the context of physics, the Ising model is used to study the interactions between bodies that make up complex systems. A number of internal and external factors influence the way these bodies interact with each other to affect the whole system. There are several ways to solve Ising models, but one of the most pragmatic is the Monte Carlo method, which calculates the probability of certain factors occurring over and over again at varying probabilities until the system reaches an optimal level.  Adam Warner and I would like study the ways these techniques aid problems done by economists. First, we will use MatLab to model a simple Ising system in the context of a physics problem relating either to magnetism or phase transition. From there, we will each look at different ways that Monte Carlo methods and Ising Models apply to financial models. This analysis will include a survey of utility-based decision making, market booms and crashes, and the volatility of stock pricing. We hope to compare the results we obtain through our MatLab programs to established studies done by financial institutions.

Audio signal processing is taking an audio recording, represented as a wave function of amplitude against time, and mathematically manipulating it, often in order to improve the quality.

Transposing the pitch of an audio recording is easy if you allow for changes in the length of the file. If you shorten the time of the wavefunction the frequency increases, and similarly if you lengthen the time of the audio file you decrease the frequency of the wavefunction. However, it is much more computationally intensive to transpose an audio file without changing its length. Fourier analysis is used to change the wavefunction from the time domain to the frequency domain, changing the frequencies represented, then using the inverse Fourier transform to return it to the time domain without lengthening the time of the file. You cannot simply linearly shift all of the frequencies represented in the Fourier transform because the harmonic ratios must remain intact.

This can be used for many reasons, entertainment being common, but there are scientific reasons as well; such as studying infrasonic and ultrasonic waves. The study of ultrasonic waves brings about some issues with computers. To capture ultrasound waves the sampling rate must be considered in order to avoid aliasing of frequencies above the nyquist frequency.

We are going to represent ultrasonic sound waves in the audible range.

by: Juan Vasquez & Robert Sciortino

For my computational physics project I would like to investigate the relationship between the precession and the eccentricity of a planet’s orbit due to general relativity as detailed in chapter 4, section 3 of Computational Physics, by Nicholas J. Giordano and Hisao Nakanishi. Precession is the phenomenon of the rotation of the orientation of axes of a planet’s (elliptical) orbit with respect to time caused by the gravitational forces exerted by other planets. However, general relativity predicts deviations from the inverse-square law, which (for example) allows the Sun to contribute an additional 43 arcseconds per century to the precession of Mercury (as the distance between the two is small enough for the deviations to have an effect). The rate of precession examined in this project will only be that which is caused by the model planet’s host star. This will entail the construction of a planetary motion program using Newton’s law of gravitation and the Euler-Chromer method; which will later be adjusted to allow for the addition of a variable precession rate. The value of the planet’s perihelion will be held constant at the value for Mercury’s perihelion, but the eccentricity and size of the orbits will be allowed to vary. Mass of the planet will be held constant, at the same value as for Mercury.
If I have additional time left over I could also add another planet (such as Jupiter) to my code to make a three-body simulation of the “Mercury-like” planet.

Our project will be aimed at studying the behavior of a sample of N molecules when they are heated by a pulsed laser source. For this project, we will be using Newton’s second law, statistical mechanics, and the information on molecular dynamics simulations in Chapter 9 of Giordano and Nakanishi’s Computational Physics.

In our project we will tackle molecular heating simulations at different levels of difficulty. We will start simply by simulating gas molecules in a box and move on to more complex topics that involve heating and then finally pulsed heating of the gases. From these simulations we will study how long it will take for the gas particles to come to thermal equilibrium and how the speed distributions of the gas particles look when the system is in equilibrium.

After gases, we will try to create similar simulations to study the properties of crystalline solids. If we have time, we will also study the Fermi-Pasta-Ulam problem.
We decided upon this project as this is closely related to what we are studying in our thermal physics class right now and what we studied in classical mechanics in the past. Both branches of science are interesting and very challenging and we hope that undertaking this project will give us a better understanding of the concepts we have learnt so far.