The k-nacci triangle and applications Kuhapatanakul, Kantaphon; Anantakitpaisal, Pornpawee
Cogent mathematics,
20/1/1/, Letnik:
4, Številka:
1
Journal Article
Recenzirano
Odprti dostop
A generalization of the classical Fibonacci numbers
is the k-generalized Fibonacci numbers
for
whose first k terms are
and each term afterward is the sum of the preceding k terms. In this article, we ...first introduce the k-nacci triangle to derive an explicit formula of the nth k-generalized Fibonacci number. Second, we also introduce the k-generalized Pascal triangle for deriving the formula of the k-generalized Fibonacci numbers.
This article defines the so called Generalized Matrix Variate Jensen-Logistic distribution. The relevant applications of this class of distributions in Configuration Shape Theory consist of a more ...efficient computation, supported by the corresponding inference. This demands the solution of two important problems: (1) the development of analytical and efficient formulae for their k-th derivatives and (2) the use of the derivatives to transform the configuration density into a polynomial density under some special matrix Kummer relation, indexed in this case by the Jensen-Logistic kernel. In this article, we solve these problems by deriving a simple formula for the k-th derivative of the density function, avoiding the usual partition theory framework and using a generalization of Pascal triangles. Then we apply the results by obtaining the associated Jensen-Logistic Kummer relations and the configuration polynomial density in the setting of Statistical Shape Theory.
The discrete Pascal transform (DPT) is a relatively recently introduced spectral transform based on the concept of the Pascal triangle which has been known for centuries. It is used in digital image ...processing, digital filtering, pattern recognition, watermarking, and related areas. Its applicability is limited by the O(N 2 ) asymptotical time complexity of best current algorithms for its computation, where N is the size of the function to be processed. In this paper, we propose a method for the efficient computation of the DPT in O(N log N) time, based on the factorization of its transform matrix into a product of three matrices with special structure - two diagonal matrices and a Toeplitz matrix. The Toeplitz matrix is further embedded into a circulant matrix of order 2N. The diagonalization of the circulant matrix by the Fourier matrix permits the use of the fast Fourier transform (FFT) for performing the computations, leading to an algorithm with the overall computational complexity of O(N log N). Since the entries in the Toeplitz matrix have very different magnitudes, the numerical stability of this algorithm is also discussed. We also consider the issues in implementing the proposed algorithm for highly-parallel computation on graphics processing units (GPUs). The experiments show that computing the DPT using the proposed algorithm processed on GPUs is orders of magnitude faster than the best current approach. As a result, the proposed method can significantly extend the practical applicability of the discrete Pascal transform.
This paper makes a theoretical comparative analysis of the Reed-Muller-Fourier Transform, Pascal matrices based on the Pascal triangle, and the Discrete Pascal Transform. The Reed-Muller-Fourier ...Transform was not originated by a Pascal matrix, however it happens to show a strong family resemblance with it, sharing several basic properties. Its area of application is the multiple-valued switching theory, mainly to obtain polynomial expressions from the value vector of multiple-valued functions. The Discrete Pascal Transform was introduced over a decade later, based on an ad hoc modification of a Pascal matrix, for applications on picture processing. It is however shown that a Discrete Pascal Transform of size p, taken modulo p equals the special Reed-Muller-Fourier Transform for the same p and n = 1. The Sierpinski fractal is close related to the Pascal matrix. Data structures based on the Sierpinski triangle have been successfully used to solve special problems in switching theory. Some of them will be addressed in the paper.
The Fibonacci partition triangles Fahr, Philipp; Ringel, Claus Michael
Advances in mathematics,
07/2012, Letnik:
230, Številka:
4-6
Journal Article
Recenzirano
Odprti dostop
In two previous papers we have presented partition formulas for the Fibonacci numbers motivated by the appearance of the Fibonacci numbers in the representation theory of the 3-Kronecker quiver and ...its universal cover, the 3-regular tree. Here we show that the basic information can be rearranged in two triangles. They are quite similar to the Pascal triangle of the binomial coefficients, but in contrast to the additivity rule for the Pascal triangle, we now deal with additivity along “hooks”, or, equivalently, with additive functions for valued translation quivers. As for the Pascal triangle, we see that the numbers in these Fibonacci partition triangles are given by evaluating polynomials. We show that the two triangles can be obtained from each other by looking at differences of numbers, it is sufficient to take differences along arrows and knight’s moves.
We extend the concept of a binomial coefficient to all integer values of its parameters. Our approach is purely algebraic, but we show that it is equivalent to the evaluation of binomial coefficients ...by means of the
Γ
-function. In particular, we prove that the traditional rule of “negation” is wrong and should be substituted by a slightly more complex rule. We also show that the “cross product” rule remains valid for the extended definition.
In this paper, we partially solve an open problem, due to J.C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer
n, that are Steinhaus triangles containing ...all the elements of
Z
/
n
Z
with the same multiplicity. For every odd number
n, we build an orbit in
Z
/
n
Z
, by the linear cellular automaton generating the Pascal triangle modulo
n, which contains infinitely many balanced Steinhaus triangles. This orbit, in
Z
/
n
Z
, is obtained from an integer sequence called the universal sequence. We show that there exist balanced Steinhaus triangles for at least 2/3 of the admissible sizes, in the case where
n is an odd prime power. Other balanced Steinhaus figures, such as Steinhaus trapezoids, generalized Pascal triangles, Pascal trapezoids or lozenges, also appear in the orbit of the universal sequence modulo
n odd. We prove the existence of balanced generalized Pascal triangles for at least 2/3 of the admissible sizes, in the case where
n is an odd prime power, and the existence of balanced lozenges for all admissible sizes, in the case where
n is a square-free odd number.
Two running themes of this paper are as follows: (1) there is an underlying unity—which in fact is ‘identity’ of the substance—of all major world religions, and (2) different modes of Universe and ...the unification of cognitions therein are expressions of answer to various metaphysical questions. The present endeavour in this way keeps the unity of human society—a step towards realizing ‘
vasudhāeva kuṭumbakam
’—‘Whole of Earth is a family’—as its ultimate goal. This paper envisages the realization of this objective through arriving at the common metaphysical structure of religions that constitute the core of humanity’s beliefs. The endeavour in this sense is essential for a globalized world since a globalized world of twenty-first century will be awfully susceptible to cataclysmic possibilities in the absence of knowledge of existence and understanding of such a common metaphysical structure, unlike as in the non-globalized world of nineteenth or even that of twentieth century, for example.
PDF
