The Fibonacci partition triangles Fahr, Philipp; Ringel, Claus Michael
Advances in mathematics (New York. 1965),
07/2012, Letnik:
230, Številka:
4-6
Journal Article
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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.
The recent worldwide explosion of the financial market originating also from the Black–Scholes equation proved how fundamental could be Brownian motion to real life. Brownian motion is deeply rooted ...into discrete spaces that are well represented by a Tartaglia–Pascal triangle (TPt). Furthermore, this mapping can be extended to the case of the Schrödinger equation: one of the key equations of quantum mechanics. The connection arises from the asymptotic formula for the binomial coefficients and the normal probability density function. This paper shows how this mapping between the discrete spaces, represented through some forms of TPt, extends to Brownian motion in different geometries. One of the well-known cases is the heat equation; another one is the Black–Scholes equation that derives from geometric Brownian motion. It is shown that the TPt becomes a periodic structure for the Brownian motion on a circle. For the geometric Brownian motion, we get a scale deformed TPt the main effect being scale deformations into the corresponding Newton binomial formula. In the asymptotic limit, one recovers the known formula for the sum on a row of the TPt. This approach unveils discrete structures underlying Brownian motion on different geometries revealing a possible conjecture that, for a given stochastic motion, it is always possible to associate a discrete map such that a TPt is obtained. In a general case, outcomes of the elements of the triangle become real numbers.
We interpret the reciprocation process in
K
x
as a fixed point problem related to contractive functions for certain adequate ultrametric spaces. This allows us to give a dynamical interpretation ...of certain arithmetical triangles introduced herein. Later we recognize, as a special case of our construction, the so-called Riordan group which is a device used in combinatorics. In this manner we give a new and alternative way to construct the proper Riordan arrays. Our point of view allows us to give a natural metric on the Riordan group turning this group into a topological group. This construction allows us to recognize a countable descending chain of normal subgroups.
In a recent paper (Farina et al. in Signal Image Video Process 1–16,
2011
), it was shown a clean connection between the solution of the classical heat equation and the Tartaglia/Pascal/Yang-Hui ...triangle. When the time variable in the heat equation is substituted with the imaginary time, the heat equation becomes the Schrödinger equation of the quantum mechanics. So, a conjecture was put forward about a connection between the solution of the Schrödinger equation and a suitable generalization of the Tartaglia triangle. This paper proves that this conjecture is true and shows a new—as far as the authors are aware—result concerning the generalization of the classical Tartaglia triangle by introducing the “complex-valued Tartaglia triangle.” A “complex-valued Tartaglia triangle” is just the square root of an ordinary Tartaglia triangle, with a suitable phase factor calculated via a discretized version of the ordinary continuous case of the Schrödinger equation. So, taking the square of this complex-valued Tartaglia triangle gives back exactly the probability distribution of a discrete random walk. We also discuss about potential connections between the theories of stochastic processes and quantum mechanics: a connection debated since the inception of the theories and still lively hot today.
The aim of this paper is to provide a historical perspective of Tartaglia-Pascal’s triangle with its relations to physics, finance, and statistical signal processing. We start by introducing ...Tartaglia’s triangle and its numerous properties. We then consider its relationship with a number of topics: the Newton binomial, probability theory (in particular with the Gaussian probability density function, pdf), the Fibonacci sequence, the heat equation, the Schrödinger equation, the Black–Scholes equation of mathematical finance and stochastic filtering theory. Thus, the main contribution of this paper is to present a systematic review of the triangle properties, its connection to statistical theory, and its numerous applications. The paper has mostly a scientific-educational character and is addressed to a wide circle of readers. Sections 7 and 8 are more technical; thus, they may be of interest to more expert readers.
In view of the several publications on the application of the Finite Element Method (FEM) to compute regional gravity anomaly involving only 8 nodes on the periphery of a rectangular map, we present ...an interactive FORTRAN program, FEAODD.FOR, for wider applicability of the technique. A brief description of the theory of FEM is presented for the sake of completeness. The efficacy of the program has been demonstrated by analyzing the gravity anomaly over Salt dome, South Houston, USA using two differently oriented rectangular blocks and over chromite deposits, Camaguey, Cuba. The analyses over two sets of data reveal that the outline of the ore body/structure matches well with the maxima of the residuals. Further, the data analyses over South Houston, USA, have revealed that though the broad regional trend remains the same for both the blocks, the magnitudes of the residual anomalies differ approximately by 25% of the magnitude as obtained from previous studies.
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