publications

2022

1. Zeros of a binomial combination of Chebyshev polynomials

   Summer Al Hamdani, and Khang Tran

   International Journal of Number Theory, 2022

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   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.

2021

1. Zero distribution of binomial combinations of Chebyshev polynomials of the second kind

   Summer Al Hamdani

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

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   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.

2018

1. Improvements and reproducibility of an optimal grazing-incidence position method to L-shell x-ray fluorescence measurements of lead in bone and soft tissue phantoms

   Mihai Raul Gherase, and Summer Al-Hamdani

   Biomedical Physics and Engineering Express, Nov 2018

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   L-shell x-ray fluorescence (LXRF) is a non-invasive approach to lead (Pb) concentration measurements in human bone. The method is based on the detection of the characteristic x-ray photons of Pb at 10.5 and 12.6 keV and experimental studies were designed to perform in vivo human bone Pb measurements. In later studies, however, the initial LXRF methodology was shown to have poor accuracy and precision. In a recent publication, we investigated an optimal grazing-incidence position (OGIP) approach using a submillimeter x-ray beam from an integrated x-ray tube and polycapillary x-ray lens table-top system. The OGIP method effectively reduced the x-ray scatter and produced a Pb detection limit of  5 μg/g for a 2 mm soft tissue phantom thickness. In this study, the OGIP methodology was improved by using 10 s x-ray spectra acquisitions at sequential positions 0.5 mm apart. The measured Sr Kα peak height versus position data was used to spectroscopically identify the bone phantom and the OGIP. The data was fitted with the analytical convolution between a Gaussian and an exponential decay. The position corresponding to the maximum of the fitted convolution function was then selected as the OGIP. Four phantom sets were used. A cylindrical plaster-of-Paris bone phantom doped with Pb in a concentration of 74 μg/g was used as a bare bone phantom or with one of the three overlying polyoxymethylene cylindrical shell soft tissue phantoms of 1, 2, and 3 mm thickness. The reproducibility of the OGIP method was assessed in five independent trials using each of the four phantom sets. The coefficient of variation (COV) percentage values of the Sr Kα peak height measurements were below 5%. The new procedure decreased by more than threefold the duration and radiation dose of the earlier approach.

2. A microbeam grazing-incidence approach to L-shell x-ray fluorescence measurements of lead concentration in bone and soft tissue phantoms

   Mihai Raul Gherase, and Summer Al-Hamdani

   Physiological Measurement, Mar 2018

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   Objective: L-shell x-ray fluorescence (LXRF) is a non-invasive approach to lead (Pb) concentration measurements in the human bone. The first studies were published in the early 1980s. In the same period the K-shell x-ray fluorescence (KXRF) method using a Cd-109 radionuclide source was developed and later improved and refined. Lower sensitivity and calibration difficulties associated with the LXRF method led the KXRF to be the most adopted method for in vivo human bone Pb studies. In the present study a microbeam-based grazing-incidence approach to Pb LXRF measurements was investigated. Approach: The microbeam produced by an integrated x-ray tube and polycapillary x-ray lens (PXL) unit was used to excite cylindrical plaster-of-Paris (poP) bone phantoms doped with Pb in seven concentrations: 0, 8, 16, 29, 44, 59, and 74 µg g−1. Two 1 mm- and 3 mm-thick cylindrical shell soft tissue phantoms were made out of polyoxymethylene (POM) plastic. Three bone-soft tissue phantom sets corresponding to the 0, 1, and 3 mm POM thickness values resulted. Each phantom was placed between the microbeam and the detector; its position was controlled using a positioning stage. Small steps (0.1–0.5 mm) and short 30 s x-ray spectra acquisitions were used to find the optimal phantom position according to the maximum observed Sr Kα peak height. At the optimal geometry, five 180 s x-ray spectra were acquired for each phantom set. Calibration lines were obtained using the fitted peak heights of the two observed Pb Lα and Pb Lβ peaks. Main results: The lowest detection limit (DL) values were (2.9  ±  0.2), (4.9  ±  0.3), and (23  ±  3) µg g−1, respectively. The order of magnitude of the absorbed radiation dose in the POM plastic for the 180 s irradiation was estimated to be  <1 mGy. Significance: The results are superior to a relatively recently published LXRF phantom study and show promise for future designs of in vivo LXRF measurements.

3. On Classical Multiplier Sequences

   Summer Al-Hamdani, and Alexandra Leon

   The PUMP Journal of Undergraduate Research, Jan 2018

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   We provide the detailed proofs of two recent claims of Csordas and Forgács asserting that two particular Bessel-type functions generate classical multiplier sequences whose generic terms are Cauchy-products of Laguerre polynomials and hypergeometric func- tions, respectively. The way these sequences are generated involves functions from the Laguerre-Pólya class. We address the question whether these functions are unique in any way as generators of a given classical multiplier sequence.