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Full-Text Articles in Physical Sciences and Mathematics
Construction Of Spline Type Orthogonal Scaling Functions And Wavelets, Tian-Xiao He, Tung Nguyen, '15
Construction Of Spline Type Orthogonal Scaling Functions And Wavelets, Tian-Xiao He, Tung Nguyen, '15
Tian-Xiao He
No abstract provided.
Enumeration Problems For A Linear Congruence Equation, Tian-Xiao He, Wun-Seng Chou, Peter Shiue
Enumeration Problems For A Linear Congruence Equation, Tian-Xiao He, Wun-Seng Chou, Peter Shiue
Tian-Xiao He
Let m ≥ 2 and r ≥ 1 be integers and let c Є Zm = {0, 1, …,m ─ 1}. In this paper, we give an upper bound and a lower bound for the number of unordered solutions x1, …, xn Є Zm of the congruence x1 + x2 + ••• + xr ≡ c mod m. Exact formulae are also given when m or r is prime. This solution number involves the Catalan number or generalized Catalan number in some special cases. Moreover, the enumeration problem has interrelationship with the restricted integer partition.
Polynomials That Have Golden Ratio Zeros, Tian-Xiao He, Jack Maier, Kurt Vanness
Polynomials That Have Golden Ratio Zeros, Tian-Xiao He, Jack Maier, Kurt Vanness
Tian-Xiao He
When the golden ratio and its conjugate are zeros to a polynomial, two of the coefficients are functions of the Fibonacci sequence in terms of the other coefficients, which characterize the polynomial completely. These functions are used to derive some Fn, Ln, and golden ratio identities. In many cases, this is generalized to the Lucas sequences Un and Vn, with an associated quadratic root pair. Horadam sequences are produced in the series of linear and constant coefficients of the series of polynomials Having ra and rb zeros when all of the other coefficients are equal.