﻿ Saroj Aryal, Assistant Professor

### Research

#### Research Interests

Moment Problem, Applied Analysis, Control Systems, Finance Mathematics, Continued Fractions, Mathematics Education

#### Current Research Projects

Listed below are some of the research projects I am currently working on. Students who are interested in working in a math research, feel free to come by my office and we talk about various possibilities. In addition, if you are interested in something else in various areas of pure and applied mathematics and statistics, you are always welcome come to talk with me and we can find a project to work on together.

1. Sparse Moment Problem

Moments of a geometric object conveys a lot of its topological information such as how round it is, where it is located, what direction it is tapered, where its mass is centered, etc. For simple objects like an ellipsoid, just a few terms in the moment sequence are enough to identify it. However, only infinitely many terms can uniquely identify more complicated shapes like a polygon or a quadrature domain. While numerous applications of moment problems with a complete set of moments have been identified, they are mostly limited to theoretical observations. For practical implementation of moment problems, it is vital to be able to deal with missing moment data since data obtained from physical experiments and phenomena are often corrupt or incomplete. We call a positive subsequence of a moment sequence a sub-moment sequence and the moment problem of sub-moment sequence a sub-moment problem. The main goal of this research is to give a complete characterization of the sub-moment problem and to develop a deep connection between the measures arising from the sub-moment problem and the original moment problem. Recent advances in image reconstruction from sparse MRI data, partial data transmission and sparse signals, make this problem timely and of broad interest.

The Hamburger Theorem states that a sequence $\{s_k\}_{k=0}^{\infty}$ can be represented as $\displaystyle moments, s_k=\int_{\mathbb R}x^kd\sigma(x),\, k=0,1,2,...$, where $\sigma$ is positive measure, if and only if for any finite set of real numbers $\xi_i$, we have $\displaystyle\sum_{i,j=0}^m\xi_i\xi_js_{i+j}\geq 0, \, m=0, 1, 2....$

We have established some results that fully charaterize submoment sequences of a moment sequence in one- and multi-dimensions. This completely answers the first question above. Current work in progress is aimed at answering the second question. That is, we want to establish a connection between a moment solution of a submoment sequence and its original moment solution.

2. S-Fractions and Stieltjes Transforms

Stieljes continued fractions, also known as S-Fractions, are closely related to Stieltjes Transforms of a non-decreasing measure: $$\cfrac{1}{c_1z+\cfrac{1}{c_2+\cfrac{1}{c_3z+\cdots+\cfrac{1}{c_{2n}+\cfrac{1}{c_{2n+1}z+\cdots}}}}}=\int_0^{\infty}\frac{d\sigma(x)}{z+x},$$ where $\displaystyle c_k>0, k=0,1,2,...,z\in\mathbb C\setminus\mathbb R$ and $\sigma$ is a non-decreasing measure. We study how perturbation of terms of an S-fraction affects the measure associated with the corresponding Stieltjes transform. We further investigate the effects of perturbation of Stieltjes moments.

3. Finance Mathematics: Performance of Stocks Market

Market Indexed Exchange Traded Funds (ETF’s) have become popular in recent times. Our primary goal is to establish a methods for investors to obtain best resturns through statistical analysis of past data as well as mathematical evaluations. We examine the reasons for the demand in these instruments. Further, we want to determine relation between volume of trading and volatility index. and between the tracking error of these funds and volatility.

4. Mathematics and Statistics Education

We are interested in finding methods for optimal use of technology for enhancing student learning outcomes in college mathematics and statistics.

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