相关论文: The Chung-Graham Expansion
Recently a permutation on Dyck paths, related to the chip firing game, was introduced and studied by Barnabei et al.. It is called $\gamma$-operator, and uses symmetries and reflections to relate Dyck paths having the same length. A…
We study the generalized continued fraction expansions of complex numbers in term of elements from Euclidean subrings, especially Gaussian or Eisenstein integers, in a general framework as pursued in [3] and [1]. We introduce a common…
In 1973, J.-R. Chen showed that every large even integer is a sum of a prime and a product of at most two primes. In this paper, the author indicates and fixes the issues in a simplified proof of this result given by Pan et al.
Categorical quantum mechanics (CQM) and the theory of quantum groups rely heavily on the use of structures that have both an algebraic and co-algebraic component, making them well-suited for manipulation using diagrammatic techniques.…
Using the classic two's complement notation of signed integers, the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers. We introduce a Fibonacci-equivalent of…
The method of preliminary group classification is rigorously defined, enhanced and related to the theory of group classification of differential equations. Typical weaknesses in papers on this method are discussed and strategies to overcome…
Chung, Diaconis, and Graham considered random processes of the form X_{n+1}=a_n X_n+b_n (mod p) where p is odd, X_0=0, a_n=2 always, and b_n are i.i.d. for n=0,1,2,... . In this paper, we show that if P(b_n=-1)=P{b_n=0)=P(b_n=1)=1/3, then…
We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85-th birthday.
Various approaches to the numerical representation of the Incomplete Gamma Function F_m(z) for complex arguments z and small integer indexes m are compared with respect to numerical fitness (accuracy and speed). We consider power series,…
We study methods of inducing metrics on unital completely positive maps by employing seminorms arising in noncommutative geometry. Our main approach relies on the development of an infinite-dimensional $C^*$-algebraic analogue of the…
We present a multidimensional generalization of Zeckendorf's Theorem (any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers) to a large family of linear recurrences. This extends work of Anderson and…
We show that a number of conditions on oriented graphs, all of which are satisfied with high probability by randomly oriented graphs, are equivalent. These equivalences are similar to those given by Chung, Graham and Wilson in the case of…
The Fibonomial coefficients are well-known analogues of the classical binomial coefficients. In 2009, Sagan and Savage introduced a combinatorial interpretation for these coefficients, based on tiling a rectangular grid. More recently,…
Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive…
We show that an iterative method for computing the center of mass (CM) of q units of mass, placed on a unit interval [0, 1] along the x- axis, give rise to a simple procedure for expanding rational numbers less than unity in powers of r/s <…
A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…
Let $\Gamma$ be a countable discrete group. Given any sequence $(f_n)_{n\geq 1}$ of $\ell^p$-normalized functions ($p\in [1,2)$), consider the associated positive definite matrix coefficients $\langle f_n, \rho(\cdot) f_n\rangle$ of the…
We investigate the special class of formulas made up of arbitrary but finite com- binations of addition, multiplication, and exponentiation gates. The inputs to these formulas are restricted to the integral unit 1. In connection with such…
An interesting characterization of the Fibonacci numbers is that, if we write them as $F_1 = 1$, $F_2 = 2$, $F_3 = 3$, $F_4 = 5, ...$, then every positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. This is…
By means of the derivative operator and Chu-Vandermonde convolution, four families of summation formulas involving harmonic numbers with even or odd indexes are established.