On a Surface Associated with Pascal’s Triangle

Beiu, Valeriu; Dăuş, Leonard; Jianu, Marilena; Mihai, Adela; Iordache, Mihai

doi:10.3390/sym14020411

Cited by 3 publications

(2 citation statements)

References 23 publications

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“…which can be seen in Fig. 4, and explore level curves (see [28]) on this surface (for preliminary results see [7]).…”

Section: Pascal's Surfacementioning

confidence: 89%

“…The idea of starting from the well-known discrete Pascal's triangle and generalizing it to a continuous Pascal's surface is not new [32,33,34,35,47,63,69,70]. Still, the novelty of this paper steams from our aim to: (i) use Pascal's surface for a particular application (network reliability), and (ii) explore level curves on Pascal's surface (see also [7]).…”

Section: Introduction and Motivationsmentioning

confidence: 99%

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Approximating the Level Curves on Pascal’s Surface

Dăuş

Jianu

Nagy

et al. 2022

INT J COMPUT COMMUN, Int. J. Comput. Commun. Control

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It is well-known that in general the algorithms for determining the reliability polynomial associated to a two-terminal network are computationally demanding, and even just bounding the coefficients can be taxing. Obviously, reliability polynomials can be expressed in Bernstein form, hence all the coefficients of such polynomials are fractions of the binomial coefficients. That is why we have very recently envisaged using an extension of the classical discrete Pascal’s triangle (which comprises all the binomial coefficients) to a continuous version/surface. The fact that this continuous Pascal’s surface has real values in between the binomial coefficients makes it appealing as being a mathematical concept encompassing all the coefficients of all the reliability polynomials (which are integers, as resulting from counting processes) and more. This means that, the coefficients of any reliability polynomial can be represented as discrete steps (on level curves of integer values) on Pascal’s surface. The equation of this surface was formulated by means of the gamma function, for which quite a few approximation formulas are known. Therefore, we have started by reviewing many of those results, and have used a selection of those approximations for the level curves problem on Pascal’s surface. Towards the end, we present fresh simulations supporting the claim that some of these could be quite useful, as being both (reasonably) easy to calculate as well as fairly accurate.

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“…which can be seen in Fig. 4, and explore level curves (see [28]) on this surface (for preliminary results see [7]).…”

Section: Pascal's Surfacementioning

confidence: 89%