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Teaching Mathematics, Hiking, Swimming, Travelling, and Doing Mathematics (Moment Problem, Applied Analysis, Control Systems, Finance Mathematics, Mathematics Education)


Ph.D. in Mathematics, University of Wyoming (2013)
M.S. in Mathematics, University of Wyoming (2011)
B.S. in Mathematics (Honors), Trinity College (2009)


Curent Courses


College Trig

12:10 - 1:40 pm
LA 302

College Algebra

(Online Course)
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College Trig

12:10 - 1:40 pm
LA 302

Diff Equations

2:00 - 3:30 pm
LA 313

Diff Equations

2:00 - 3:00 pm
LA 313

Diff Equations

2:00 - 3:30 pm
LA 313

Office Hours

3:40 - 5:00 pm

Office Hours

3:10 - 5:00 pm

Office Hours

3:40 - 5:00 pm

My expectations regarding your written assignments, whenever applicable:

Write Clearly:

Show all work, but don't submit your scratch paper. Be neat and write up carefully with problems in order. Simplify. Staple carefully in the upper left corner. I will assign a large portion of your grade in homeworks for writing.

Submit Your Own Work:

I encourage you to work with others. But write your own homework and do not copy other's.

Timely Submission:

I expect you to submit each completed homework assignment on the specified due date, unless there is a good reason for being late. However, I will not accept late homework after I have already returned the graded assignment to the rest of the class.

Courses taught at the Montana State University Billings (Fall 2013 to present):

M 329: Modern Geometry
M 274: Intro Differential Equations
M 273: Multivariable Calculus
M 242: Methods of Proof
M 171: Calculus I
M 143: Finite Mathematics
M 122: College Trigonometry I
M 121: College Algebra
M 110: Mathematical Computations
M 105: Contemporary Mathematics
STAT 217: Intermediate Statistical Concepts
STAT 216: Introduction to Statistics
STAT 141: Introduction to Statistical Concepts

Courses taught at the University of Wyoming (Fall 2009 to Summer 2013):

MATH 4200: Math Analysis I
MATH 2355: Math Apps for Business
MATH 2310: Applied Differential Equations I
MATH 2210: Calculus III
MATH 2205: Calculus II
MATH 2200: Calculus I
MATH 1450: Algebra and Trigonometry
MATH 1405: Trigonometry
MATH 1400: College Algebra


The Department of Mathematics at Montana State University Billings is committed to providing a superior training in mathematics. The undergraduate mathematics programs at MSUB has a quality and scope that is comparable to those in larger colleges and universities. A student majoring in mathematics will go through a wide range of regularly offered courses. In addition, students can take advantage of department members' willingness to supervise independent studies and tutorials in various areas of mutual interest. In particular, a student with interest in an area not offered as a course can study the material as an independent study with faculty in the department.

Degrees Offered

The Department of Mathematics currently offers the following majors and minors.
B.S. in Mathematics
B.S. in Mathematics (Teaching Option). STEM students who are interested in a career teaching in Montana may be eligible for MSUB Noyce Scholarships, which are generous annual and renewable awards of $17,300.
Minor in Mathematics
Teaching Minor in Mathematics
Minor in Computer Science
Minor in Statistics
Math Courses Flowchart

Why major in Mathematics?

    The mathematics major is designed to give students an exposure to the fundamentals of mathematics as well as a solid mathematical foundation. As a mathematics major, students get a rigorous training in both pure and appplied mathematics to acquire a strong proficiency in knowledge and application of mathematical concepts in various fields and professions.

    A student who majors in mathematics can pursue graduate study in mathematics, statistics, or computer science, or careers requiring a strong mathematical background. More importantly, the broad range of mathematics, statistics and computer courses required for a mathematics major combined with the general education requirements enables a student to pursue a post-baccalaureate plan that may not be math-related. Moreover, the liberal-arts setting of the program allows mathematics majors with broad interests to take courses in various disciplines offered on-campus and opt for minor in various fields.

Some Useful External Links


Research Interests:

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


  1. Saroj Aryal, and Baudry Metangmo, A note on higher dimensional generalization of Viskovatov-like method., Integers, 2018. (Under Review)

  2. Saroj Aryal, Hayoung Choi, and Farhad Jafari. Sparse Hamburger moment multisequences. In Problems and recent methods in operator theory, volume 687 of Contem. Math., pages 21-30. Amer. Math. Soc., Providence, RI, 2017

  3. Saroj Aryal, Hayoung Choi, and Farhad Jafari. Hamburger moment sequences and their moment subsequences. Linear Multilinear Algebra, 65(9):1838-1851, 2017

  4. Saroj Aryal, and Rakesh Sah. Market Volatility and market indexed exchange traded funds. Review of Business Research, 14:163-170, 10 2014

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.

    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.

    S-Fractions and Stieltjes Transforms

      Stieljes continued fractions, also known as S-Fractions, are closely related to Stieltjes Transforms of a non-decreasing measure: $$\frac{1}{c_1z+\frac{1}{c_2+\frac{1}{c_3z+\cdots+\frac{1}{c_{2n}+\frac{1}{c_{2n+1}z+\cdots}}}}}=\int_0^{\infty}\frac{d\sigma(x)}{z+x},$$ 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.

    Finance Mathematics: Performance of Stocks Market

      Market Indexed Exchange Traded Funds (ETFs) 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.