# Research by New Projects

Below is a list of our new research projects. For more information click on the title of the project.

## Multi-Soliton Solutions and their Properties

**Advisor(s): **Alicia Machuca

**Abstract:** The study of integrable nonlinear partial differential equations is interesting to mathematicians, engineers, and physicists because such equations have physically important solutions that can be expressed in terms of elementary functions. This research project will focus on studying a certain solution formula for multi-soliton solutions of the Kadomtsev-Petviashvili (KP) equation. The solution formula that will be studied is dependent on a matrix quadruplet, (*A, M, B, C*), and can be expressed in terms of matrix exponentials. The student researchers will make use of this solution formula to examine the physical properties of these multi-soliton solutions analytically.

**Number of students needed:** 2

**Name of students you have identified to work on the project: **Rey Andrade-Flores and Angel Paucar

**Research Description:** In addition to the analytical study of certain solutions the student researchers will be expected to use Mathematica to animate certain solutions and to present their final results in a written report. Students will be expected to attend a conference to present their results.

**Student qualifications:** Completion of Math 200 and programming experience is required.

**Hours per week: **10

Start/Stop dates: October 6 - December 12

## Generating Arbitrary Force Profiles via Buoyancy

**Advisor(s): Jeff Jalkio **

**Abstract:** Buoyant forces arise from pressure gradients in fluids and the buoyant force on a partially submerged object is proportional to the volume which is submerged in the fluid. Because of this proportionality, vertical displacements of a partially submerged object will result in variations in buoyant force. This makes it possible to generate arbitrary force profiles by varying the shape of the object being submerged.

**Number of students needed:** 2

**Name of students you have identified to work on the project: **None

**Research Description:** This project has both theoretical and experimental components. After reviewing the theory of buoyancy, the student researchers will develop the necessary relationships between object profile and force and present a general solution for an arbitrary force profile. They will then solve these equations analytically and/or numerically for a number of particular profiles. For the experimental portion of their work, the students will construct an experimental apparatus for measuring force and position, fabricate their buoyant objects using a 3D printer and experimentally verify their calculations.

**Student qualifications:** Completed Math 113, 114, 200 and Physics 111.

**Hours per week: **20

Start/Stop dates: May 25 - Summer

## Title: Introduction to Bayesian Estimation and Copula Models of Dependence

**Advisor(s): Arkady Shemyakin**

**Abstract:** The main goal of the project is to develop exercises and computer problems as supporting materials for the manuscript **“Introduction to Bayesian Estimation and Copula Models of Dependence”** (a textbook for advanced undergraduates and practitioners). Six first chapters of the book contain the basics of Bayesian theory of estimation and copula modeling. Chapters 7 and 8 discuss several applications to insurance (joint first life and last survivor insurance, automobile warranty claims), risk management (probability of default in mortgage portfolios, credit risk models), and finance (joint co-movements of international markets). We will modify existing exercises and develop new ones, and also will develop detailed solutions.

**Number of students needed:** 3

**Name of students you have identified to work on the project: **None

**Research Description:** Testing and developing problem sets and computer exercises supporting the manuscript currently in preparation.

**Student qualifications:** Completed Math 313 and Stat 314. Also desirable (not necessary): Stat 333, working knowledge or R and LaTex.

**Hours per week: **20

Start/Stop dates: May 25 - Summer

## Title: "Characterizing benign and malignant tumors using Applied Mathematicsâ€ť

**Advisor(s): Cheri Shakiban**

**Abstract:** We will use three different methods, one based on signature curves, one based on invariant histograms and the third on fractal dimension to study images which will be provided to us by the American Institute of Cancer Research (http://www.aicr.org/). We will then apply our methods to distinguish between benign and malignant tumors.

**Number of students needed:** 2

**Name of students you have identified to work on the project: Anna Grim and Victor Gonzalez**

**Research Description:** NA

**Student qualifications:** NA

**Hours per week: **20

Start/Stop dates: May 25 - Summer