Summer Al Hamdani

316 Van Vleck Hall,

480 Lincoln Dr,

Madison, WI 53705

I am a mathematics Ph.D. student at University of Wisconsin-Madison. My doctoral minor is in statistics. My primary interests of the mathematical variety generally revolve around analysis and applied math/statistics. Nowadays, I am more of a casual reader of medical physics - my interests are in imaging and diagnostic modalities.

Before starting the doctoral program, I completed my undergraduate studies in Mathematics and Biomedical Physics along with a Master’s in Mathematics at California State University, Fresno (go Bulldogs!). My Master’s thesis studied the zero distribution of Chebyshev polynomials of the second kind with binomial-type coefficients. Afterwards, I briefly worked in institutional research and data science. Further details can be found on my CV.

The best way to reach me is by email: alhamdani at math dot wisc dot edu. I usually try to respond within 24 ± 12 hours on business days.

news

Jan 22, 2023	I updated this webpage! How exciting.

selected publications

Zeros of a binomial combination of Chebyshev polynomials

Summer Al Hamdani, and Khang Tran

International Journal of Number Theory, 2022

Abs arXiv HTML

For 0 < α< 1, we study the zeros of the sequence of polynomials { P_m(z) }_m=0^∞generated by the reciprocal of (1 − t)^α(1 − 2zt + t^2), expanded as a power series in t. Equivalently, this sequence is obtained from a linear combination of Chebyshev polynomials whose coefficients have a binomial form. We show that the number of zeros of P_m(z) outside the interval (−1, 1) is bounded by a constant independent of m.
Zero distribution of binomial combinations of Chebyshev polynomials of the second kind

Summer Al Hamdani

Master’s thesis, California State University, Fresno, May 2021

Abs PDF

For α∈\mathbbR such that 0 < α≤1, we can construct a sequence of real polynomials { P_m(z) }_m=0^∞that is generated by 1/(1-t)^α(1-2zt+t^2) = \sum_m=0^∞P_m(z)t^m . The sequence { P_m(z) }_m=0^∞is a binomial combination of the well-known Chebyshev polynomials of the second kind, which have all real zeros on the interval (−1, 1). We prove that there exists a constant C (independent of m) such that the number of zeros of P_m(z) outside of the interval (−1, 1) is at most C for all m ∈\mathbbN.