Search Results (22)

This paper studies a terminating (“modified”) Appell function ${\mathcal{F}}_1^{\ast }(\alpha ,\beta ,{\beta }^{\prime },\beta +{\beta }^{\prime };x,y),$ with $\alpha \in {\mathbb{Z}}_{\ge 1}$ and $\beta ,{\beta }^{\prime }\in {\mathbb{Z}}_{\le -1}$, together with the associated terminating Gauss function ${}_{2}F_1^{\ast }\left(\begin{array}{c}\alpha ,\beta \\ \beta +{\beta }^{\prime }\end{array};z\right).$ The reduction formula for the Appell function ${\mathcal{F}}_1^{\ast }(\alpha ,\beta ,{\beta }^{\prime },\beta +{\beta }^{\prime };x,y)$ to ${\displaystyle \frac{1}{{(1-y)}^{\alpha }}{}_{2}F_1\left(\begin{array}{c}\alpha ,\beta \\ \beta +{\beta }^{\prime }\end{array};\frac{x-y}{1-y}\right)}$ breaks down when $(\beta ,{\beta }^{\prime })$ are integers less than or equal to $-1$ and it needs to be substituted with a revised identity that includes an explicit additional correction term ${\mathcal{CORR}}^{(\alpha ,\beta ,{\beta }^{\prime })}(x,y)$. This correction term is initially computed for specific cases (particularly for $\alpha =1,2,3$) and subsequently formulated and verified generally through mathematical induction on $\alpha$. The final expression demonstrates a structured pattern reminiscent of binomial/Pascal coefficients and leads to various corollaries, including simplified boundary scenarios (such as when $\alpha =1$). Full article

The purpose of this article is to highlight the connections between two seemingly distinct domains: random walks and the distribution of angular-momentum projections in quantum physics (the magnetic quantum numbers m). It is well known that there is indeed a deep mathematical link between the two, via the vector composition of angular momenta and rotational symmetry. Random walks are considered in the framework of an interpretation of the probability of microstates in statistical physics. The ideas presented in this work aim to illustrate the relevance of this perspective for modeling angular momentum in atomic physics. Full article

Pascal’s Triangle, renowned for its geometric elegance and profound applications across combinatorics, algebra, and probability, has fascinated mathematicians for centuries. While its origins can be traced to Chinese, Persian, and European mathematical traditions, the study of its higher-dimensional analogues remains notably underexplored. This paper offers a systematic and self-contained study of Pascal Pyramids and Pascal Simplexes with their proofs. It encompasses both classical results (such as multinomial identities) and novel contributions (including boundary and scaling properties), as well as fresh perspectives (such as graph-theoretic interpretations) that are rarely documented in the existing literature. Full article

Pascal’s triangle is a classical mathematical construct, historically studied for centuries, that organises binomial coefficients in a triangular array and serves as a cornerstone in combinatorics, algebra, and number theory. Herein, we propose to model it with Petri nets, a formal specification technique derived from discrete event systems. A minimal Petri net is created that generates Pascal’s triangle under a simple arithmetic rule. Token counts in each place coincide with binomial coefficients, providing a direct combinatorial interpretation. Two other classical structures emerge from this model: by colouring tokens depending on their parity, the Sierpiński triangle appears; by routing tokens randomly at each branching, the binomial distribution arises, converging to a Gaussian limit as depth increases. As a result, a single Petri construction unifies three mathematical objects: Pascal’s Triangle, Sierpiński’s Triangle, and the Gaussian distribution. This connection illustrates the invaluable potential of Petri nets as unifying tools for modelling discrete mathematical structures and beyond. Full article

This work addresses closed-form expressions for the distributions $P\left(M\right)$ of the magnetic quantum numbers M and $Q\left(J\right)$ of total angular momentum J for non-equivalent fermions in single-j orbits. Such quantities play an important role in both nuclear and atomic physics, through the shell models. Using irreducible representations of the rotation group, different kinds of formulas are presented, involving multinomial coefficients, generalized Pascal triangle coefficients, or hypergeometric functions. Special cases are discussed, and the connections between $P\left(M\right)$ (and therefore $Q\left(J\right)$) and mathematical functions such as elementary symmetric, cyclotomic, and Jacobi polynomials are outlined. Full article

This paper presents a novel Q-value-based formability assessment for optimizing deep drawing processes. The Q-value, derived from thinning limit diagrams (TLDs), uses offset thinning and wrinkling limit curves to define severity levels. It is calculated by summing the product of Pascal’s triangle weighting factors and normalized element counts within each severity level. The effectiveness of this Q-value assessment was demonstrated using experimentally validated finite element analysis (FEA) to optimize blank size, tool geometry, and drawbead design (male bead height and contra-bead radius) for a deep-drawn AA5754-O automotive fuel tank. Validation of FEA results with experimental thickness measurements showed that the Barlat and Lian 1989 yield criterion provided higher accuracy than Hill’s 1948 model. An optimal condition, determined using the Q-value, consists of a 430 mm × 525 mm blank formed by a redesigned tool cooperated with optimized semi-circular drawbead geometries, achieving experimental significant formability improvements by minimizing wrinkling and thinning. During optimization, this study revealed a significant interaction between blank width and length, which influenced formability. Side-wall wrinkles were attributed to insufficient tool support for the blank during forming and were relieved through tool redesign. Furthermore, increasing the male drawbead height effectively reduced wrinkling but led to increased thinning, whereas increasing the contra-bead radius had the opposite effect. Full article

The combinatorial problem in this paper is motivated by a variant of the famous traveling salesman problem where the salesman must return to the starting point after each delivery. The total length of a delivery route is related to a metric known as closeness centrality. The closeness centrality of a vertex v in a graph G was defined in 1950 by Bavelas to be $CC\left(v\right)={\displaystyle \frac{\left|V\right(G\left)\right|-1}{SD\left(v\right)}}$, where $SD\left(v\right)$ is the sum of the distances from v to each of the other vertices (which is one-half of the total distance in the delivery route). We provide a real-world example involving the Metro Atlanta Rapid Transit Authority rail network and identify stations whose $SD$ values are nearly identical, meaning they have a similar proximity to other stations in the network. We then consider theoretical aspects involving asymmetric trees. For integer values of k, we considered the asymmetric tree with paths of lengths $k,2k,\dots ,nk$ that are incident to a center vertex. We investigated trees with different values of k, and for $k=1$ and $k=2$, we established necessary and sufficient conditions for the existence of two vertices with identical $SD$ values, which has a surprising connection to the triangular numbers. Additionally, we investigated asymmetric trees with paths incident to two vertices and found a sufficient condition for vertices to have equal $SD$ values. This leads to new combinatorial proofs of identities arising from Pascal’s triangle. Full article

Paratemporal methods based on tree expansion have proven to be effective in efficiently generating the trajectories of stochastic systems. However, combinatorial explosion of branching arising from multiple choice points presents a major hurdle that must be overcome to implement such techniques. In this paper, we tackle this scalability problem by developing a systems theory-based framework covering both conventional and proposed tree expansion algorithms for speeding up discrete event system stochastic simulations while preserving the desired accuracy. An example is discussed to illustrate the tree expansion framework in which a discrete event system specification (DEVS) Markov stochastic model takes the form of a tree isomorphic to a free monoid over the branching alphabet. We derive the computation times for baseline, non-merging, and merging tree expansion algorithms to compute the distribution of output values at any given depth. The results show the remarkable reduction from exponential to polynomial dependence on depth effectuated by node merging. We relate these results to the similarly reduced computation time of binomial coefficients underlying Pascal’s triangle. Finally, we discuss the application of tree expansion to estimating temporal distributions in stochastic simulations involving serial and parallel compositions with potential real-world use cases. Full article

This paper proposes a novel code for optical code division multiple access (OCDMA) systems, called the three-dimensional (3D) spectral/temporal/spatial single weight zero cross-correlation (3D-SWZCC) code. The proposed code could potentially be used in the next generation of passive optical networks (NG-PONs) to provide a 3D-SWZCC-OCDMA-NG-PON system. The developed code has a high capacity and a zero cross-correlation property that completely suppresses the multiple access interference (MAI) effects that are a main drawback for OCDMA systems. Previously, a two-dimensional (2D) SWZCC code was proposed for two-dimensional OCDMA (2D-OCDMA) systems. It works by devoting the first and second components to spectral and spatial encodings, respectively. However, the proposed code aims to carry out encoding domains in spectral, time, and spatial aspects for the first, second, and third components, respectively. One-dimensional, 2D, and 3D systems can support up to 68, 157, and 454 active users with total code lengths equal to 68, 171, and 273, respectively. Numerical results reveal that the 3D-SWZCC code outperforms codes from previous studies, including 3D codes such as perfect difference (PD), PD/multi-diagonal (PD/MD), dynamic cyclic shift/MD (DCS/MD), and Pascal’s triangle zero cross-correlation (PTZCC), according to various metrics. The system function is provided by exhibiting the architecture of the transmitter and receiver in the PON context, where the proposed code demonstrates its effectiveness in meeting optical communication requirements based on 3D-OCDMA-PON by producing a high quality factor (Q) of 18.8 and low bit error rate (BER) of 3.48 × 10⁻²⁹ over a long distance that can reach 30 Km for a data rate of 0.622 Gbps. Full article

In this paper, the numerical solutions of the backward and forward non-homogeneous wave problems are derived to address the nonlocal boundary conditions. When boundary conditions are not set on the boundaries, numerical instability occurs, and the solution may have a significant boundary error. For this reason, it is challenging to solve such nonlinear problems by conventional numerical methods. First, we derive a nonlocal boundary shape function (NLBSF) from incorporating the Pascal triangle as free functions; hence, the new, two-parameter Pascal bases are created to automatically satisfy the specified conditions for the solution. To satisfy the wave equation in the domain by the collocation method, the solution of the forward nonlocal wave problem can be quickly obtained with high precision. For the backward nonlocal wave problem, we construct the corresponding NLBSF and Pascal bases, which exactly implement two final time conditions, a left-boundary condition and a nonlocal boundary condition; in addition, the numerical method for the backward nonlocal wave problem under two-side, nonlocal boundary conditions is also developed. Nine numerical examples, including forward and backward problems, are tested, demonstrating that this scheme is more effective and stable. Even for boundary conditions with a large noise at final time, the solution recovered in the entire domain for the backward nonlocal wave problem is accurate and stable. The accuracy and efficiency of the method are validated by comparing the estimation results with the existing literature. Full article

An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous extension of Pascal’s triangle including: Gauss curvatures, mean curvatures, geodesics, and level curves, as well as their symmetries. Full article

The paper promotes the notion of computational experiment supported by a multi-tool digital environment as a means of the development of new mathematical knowledge in the context of education. The main study of the paper deals with the issues of teaching this knowledge to secondary teacher candidates within a graduate capstone mathematics education course. The interplay of mathematics and education is considered through the lens of using technology to enhance one’s mathematical background by advancing ideas from mostly known to genuinely unknown. In this paper, the knowns consist of Fibonacci numbers, Pascal’s triangle, and continued fractions; among the unknowns are Fibonacci-like polynomials and generalized golden ratios in the form of cycles of various lengths. The paper discusses the interplay of pragmatic and epistemic uses of digital tools by the learners of mathematics. The data for the study were collected over the years through solicited comments by teacher candidates enrolled in the capstone course. The main results indicate the candidates’ appreciation of the need for deep mathematical knowledge as an instrument of the modern-day pedagogy aimed at making high schoolers interested in the subject matter. Full article

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the symmetric matrix of initial conditions. Full article

In this paper we use a graph interpretation of distance Fibonacci polynomials to get a new generalization of Lucas polynomials in the distance sense. We give a direct formula, a generating function and we prove some identities for generalized Lucas polynomials. We present Pascal-like triangles with left-justified rows filled with coefficients of these polynomials, in which one can observe some symmetric patterns. Using a general Q-matrix and a symmetric matrix of initial conditions we also define matrix generators for generalized Lucas polynomials. Full article

In this paper we introduce and study $(2,k)$-distance Fibonacci polynomials which are natural extensions of $(2,k)$-Fibonacci numbers. We give some properties of these polynomials—among others, a graph interpretation and matrix generators. Moreover, we present some connections of $(2,k)$-distance Fibonacci polynomials with Pascal’s triangle. Full article