A Note on Superspirals of Confluent Type

Inoguchi, Jun-ichi; Ziatdinov, Rushan; Miura, Kenjiro T.

doi:10.3390/math8050762

Cited by 5 publications

(5 citation statements)

References 9 publications

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“…The confluent hyper-geometric equation is an important differential equation that is used in many areas of classical and quantum physics, chemistry and engineering [9]. This equation also arises in optics [10,11], classical electrodynamics [9,11], classical waves [12,13], diffusion [14], fluid flow [15], heat transfer [16], general relativity [17,18], semi-classical quantum mechanics [19], quantum chemistry [20,21], graphic design [22], finance and many other areas. The solutions of the confluent hyper-geometric equation depend in an essential way on whether or not and are integers.…”

Section: Original Research Articlementioning

confidence: 99%

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Logistics Financial Function of Fractal Dispersion of Hausdorff Measure Prior to Crash Market Signal

Adindu-Dick

2022

JAMCS

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The world of finance works better through logistics. We consider the logistics function in large market crashes corresponding to values of packing dimension of Rn or \(\alpha\) (max) by analyzing the fractal dispersion of Hausdorff prior to market signal with constraint of a zero heat capacity. We also advocate a stochastic Ito’s iterated procedure for locating optimal market crises signal relative to the Heat equation to give an early warning. The resultant differential equation is solve to obtain the recurrence formula.

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Section: Original Research Articlementioning

confidence: 99%

“…Hence, is a singular point. Starting with to see if it is regular, we study the following limits: (22) .…”

Section: Applicationsmentioning

confidence: 99%

Logistics Financial Function of Fractal Dispersion of Hausdorff Measure Prior to Crash Market Signal

Adindu-Dick

2022

JAMCS

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“…Many well-known spirals [42], including a clothoid, are special cases of this class of curves. The most generalized class of curves with a monotonic curvature function, called superspirals, was introduced in [27] and studied via similarity geometry in recent works [40][41]. Equations of these curves are expressed through Gaussian hypergeometric functions and are numerically integrated by adaptive integration methods such as the Gauss-Kronrod method.…”

Section: Natural Beauty Of Spiral Curvesmentioning

confidence: 99%

Multi-Criteria Assessment of Shape Quality in CAD Systems of the Future

Muftejev

Ziatdinov

Nabiyev

2020

Proceedings of the 30th International Conference on Computer Graphics and Machine Vision (GraphiCon 2020). Part 2

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Unlike many other works, where authors are usually focused on one or two quality criteria, the current manuscript, which is a generalization of the article [35] published in Russian, offers a multi-criteria approach to the assessment of the shape quality of curves that constitute component parts of the surfaces used for the computer modelling of object shapes in various types of design. Based on the analysis of point particle motion along a curved path, requirements for the quality of functional curves are proposed: a high order of smoothness, a minimum number of curvature extrema, minimization of the maximum value of curvature and its variation rate, minimization of the potential energy of the curve, and aesthetic analysis from the standpoint of the laws of technical aesthetics. The authors do not set themselves the task of giving a simple and precise mathematical definition of such curves. On the contrary, this category can include various curves that meet certain quality criteria, the refinement and addition of which is possible in the near future. Engineering practice shows that quality criteria can change over time, which does not diminish the need to develop multi-criteria methods for assessing the quality of geometric shapes. Technical issues faced during edge rounding in 3D models that affect the quality of industrial design product shape have been reviewed as an example of the imperfection of existing CAD systems.

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“…Moreover, we note that there is a very nice discussion of Landau levels that also employs confluent hypergeometric functions [18]. The confluent hypergeometric equation also arises in optics [15,[19][20][21][22], classical electrodynamics [6,19,23], classical waves [7,24,25], diffusion [26], fluid flow [27], heat transfer [28], general relativity [29][30][31][32], semiclassical quantum mechanics [33], quantum chemistry [34,35], graphic design [36], and many other areas. The solutions of the confluent hypergeometric equation depend in an essential way on whether or not 𝑎, 𝑏, and 𝑎 − 𝑏 are integers and the standard references (see below) do not present these solutions, with appropriate qualifications, in a user-friendly way.…”

Section: Introductionmentioning

confidence: 99%

A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions

Mathews¹,

Esrick²,

Teoh³

et al. 2022

Condens. Matter Phys.

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The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and M ≡ z1-bM(1+a-b, 2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation.

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A Note on Superspirals of Confluent Type

Cited by 5 publications

References 9 publications

Logistics Financial Function of Fractal Dispersion of Hausdorff Measure Prior to Crash Market Signal

Logistics Financial Function of Fractal Dispersion of Hausdorff Measure Prior to Crash Market Signal

Multi-Criteria Assessment of Shape Quality in CAD Systems of the Future

A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions

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