'Cheng, Dr. Leslie' Archive

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**August 4, 2011**
Abstract: Hoang Ha Mentor: Leslie Cheng This summer, I am doing research on financial mathematics with Professor Leslie Cheng as my advisor. Financial mathematics is a field of applied math that uses mathematical techniques and models to make optimal decisions in the financial market. Since this subject is new to me, I will first obtain […]

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**July 26, 2011**
Abstract: Nuzhat Binte Arif Mentor: Professor Leslie Cheng My topic of interest for research this summer is Fourier Analysis. Fourier Analysis, named after French mathematician and physicist Joseph Fourier, is the mathematics of transforming complex waves into simple sine-cosine curves so that they can be better understood and easily analyzed. Fourier Analysis on Euclidean spaces […]

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**July 25, 2011**
Abstract: Hao Jiang Mentor: Leslie Cheng An option is a contract that permits the owner to purchase or sell an asset at a fixed price until a specific date. To have a strong market in options, we need to be able to efficiently evaluate the approximate worth of the options. There are various solutions to […]

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**July 13, 2011**
Abstract: Tong Wu Mentor: Leslie Cheng This summer I am working with Professor Leslie Cheng. My summer research project involves financial mathematics. Financial mathematics is the branch of applied math that analyzes the financial market. For example, we can use our models and formulas to calculate the fair price of financial instruments, such as options […]

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**June 24, 2010**
Harmonic analysis, or fourier transform, is the branch of mathematics that studies functions by deconstructing functions into simple building blocks. The materials learned in this branch can be applied to many different fields besides mathematics, including engineering, physics, applied mathematics, chemistry, signal processing, astronomy, optics, and more. Out of the possible fields, I chose to study the use of harmonic analysis in its application on medical imaging devices, such as X-rays and electrocardiographs (ECG).

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**June 24, 2010**
I will study applications to telecommunication. Also, I will learn how to write in LaTex. This program allows writers to create large and complex projects. Unlike Microsoft Word, the writer is not distracted with the presentation of his or her work. In LaTex, the writer simply writes a command in order to format the project to his or her desire.

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**June 23, 2010**
Abstract: Kaushiki Dunusinghe Mentor: Dr. Cheng My summer research will be in the eld of harmonic analysis. Har- monic analysis is an area of mathematics which studies the representation of signals or functions as the superposition of basic waves. Basic waves are called harmonics”, and hence the name harmonic analysis”. The mathematical study of harmonic […]

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**May 28, 2010**
This summer, I will be doing research with Professor Leslie Cheng as my adviser. My research will focus on financial mathematics and harmonic analysis. Because I am an A.B./M.A. student in mathematics, this experience may later provide me with a topic to explore for my master’s thesis.

The first part of my research focuses on financial mathematics. Financial mathematics is concerned with calculating how much one should pay for financial securities like stocks, bonds, and options, and finding what combinations of the securities make the best investment. An option is the right to buy or sell an asset or a stock. I will focus on lookback options, a type of exotic option that pays the option buyer the maximum value of the stock over time at a specified expiration date. By completing background reading from previous financial mathematics lectures, I will try to modify the Black-Scholes Formula for European call options.

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**May 12, 2010**
My summer research is in the area of harmonic analysis, the branch of mathematics which studies the representation of signals or functions as the superposition of basic waves. The basic waves are called “harmonics”, hence the name “harmonic analysis”.

In this branch of mathematics, there is great concern about examining mathematical objects by decomposing them into many simpler building blocks and studying these simpler components. This would make the study of the original object easier and more comprehensible in most cases. For instance, to study a function (a musical sound, for example) we could break it down into a sum of the pure harmonics, sine and cosine, which will allow us to study these simple graphs, to better understand our original function’s behavior. Harmonic analysis has become a vast subject with applications in areas as diverse as acoustics, optics, electrical engineering, quantum mechanics, and neuroscience.

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**May 11, 2010**
The field of harmonic analysis is the branch of mathematics that studies the representation of signals or functions as the superposition of basic waves. The basic waves are called “harmonics”, hence the name “harmonic analysis.” In the past two centuries, it has become a vast subject with applications in areas as diverse as signal processing, quantum mechanics, and neuroscience.

Harmonic analysis has a long and rich history dating back to eighteenth century studies of the wave equation pioneered by Fourier. The succeeding 150 years saw a blossoming of Fourier analysis into a powerful set of tools, including the Fourier transform, in mathematical physics, engineering, applied partial differential equations, and pure mathematics. Because of its efficient numerical approximations, the Fourier transform forms the foundation for most image and signal processing algorithms. The 1930’s and 1940’s were a relatively quiet time for Fourier analysis, but, beginning in the 1950’s, the focus of Fourier analysis became singular integrals pioneered by Calderón and Zygmund. Singular integrals are mathematical objects that look infinite but when properly interpreted are finite and well behaved.