Computing isomorphism numbers of F-crystals using the level torsions
Journal of Number Theory, 2012
In this paper, we use the level torions to compute isomorphism numbers of various F-crystals.
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publications
Subtle invariants of F-crystals
Journal of Ramanujan Mathematical Society, 2014
In this paper, we prove Vasiu’s conjecture that the isomorphism number of an F-crystal is equal to its level torsion.
Minimal F-crystals and isomorphism numbers of isosimple F-crystals
Mathematische Nachrichten, 2017
In this paper, we generalize minimal p‐divisible groups to minimal F-crystals. Using minimal heights, we give an estimate of isomorphism number of isosimple F-crystals.
Finding a unique solution to Radon-Kaczmarz puzzles
Pi Mu Epsilon Journal, 2018
In this paper, we determine the size of a Radon-Kaczmarz Puzzle so that it has a unique solution when give a fixed set of clues.
Dimensions of automorphism group schemes of finite level truncation of F-cyclic F-crystals
Transactions of the American Mathematical Society, 2021
In this paper, we develop a combinatorial formual to calculate the dimension of the automorphism group of F-cyclic F-crystal over an algebraically close dfield at some finite level. This is a joint work with Zeyu Ding.
The Number Line through Guided Inquiry
MAA Press: An Imprint of the American Mathematical Society, 2022
This is a real analysis textbook for future secondary teachers and general undergraudate mathematics major. This is a joint work with David M. Clark.
Arithmetic Partial Derivative
Journal of Integer Sequences, 2022
In this paper, we investigate the dynamical system of higher order of arithmetic partial derivatives. We show that there are infinitely many numbers with exactly n anti-partial derivatives for any nonnegative integer n.
A Generalization of Arithmetic Derivative to p-adic Fields and Number Fields
Notes on Number Theory and Discrete Mathematics, 2024
In this paper, we generalize the arithmetic partial derivative to local fields like the p-adic fields, and the arithmetic derivative to global fields like the number fields.
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teaching
Guidelines for letters of recommendation
General advice, Mathematics Department, Utica University, 2022
This page contains advice for people asking for a letter of recommendation from me (or someone like me).
MAT 143: Mathematical Analysis for Business and Economics
Undergraduate course, Mathematics Department, Utica University, 2025
Welcome to the home page of Dr. Xiao Xiao’s Mathematical Analysis for Business and Economics course at Utica University. You can find all the information for this course on this page. Please check this page frequently for announcements and assignments.
MAT 334 Introduction to Abstract Algebra
Undergraduate course, Mathematics Department, Utica University, 2025
Welcome to the home page of Prof. Xiao Xiao’s MAT 334 Introduction to Abstract Algebra course at Utica University. You can find all the information for this course on this page.
MAT 351 Euclidean and Non-Euclidean Geometry
Undergraduate course, Mathematics Department, Utica University, 2025
Welcome to the home page of Prof. Xiao Xiao’s MAT 351 Euclidean and Non-Euclidean Geometry course at Utica University. You can find all the information for this course on this page.