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AppliedMath, Volume 2, Issue 3 (September 2022) – 10 articles

Cover Story (view full-size image): The infinite sequence of triangular numbers, {Tn}={n(n+1)/2}, takes on interesting structure upon performing the modulo by any positive integer m. The key structural feature is 2m periodicity which allows for the properties of this infinite sequence to be studied via a 2m-member sequence of residue classes. Additional structural features are present and can be explored using various representations including with network graphs. The cover image shows a graph representation of the triangular numbers modulo several different values of m. Graphs for low composite numbers exhibit web-like structure. Graphs for high composites get rather complicated, but one can still see a web-like overall structure. View this paper
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11 pages, 298 KiB  
Article
The Wiener–Hopf Equation with Probability Kernel and Submultiplicative Asymptotics of the Inhomogeneous Term
by Mikhail Sgibnev
AppliedMath 2022, 2(3), 501-511; https://doi.org/10.3390/appliedmath2030029 - 19 Sep 2022
Cited by 1 | Viewed by 1248
Abstract
We consider the inhomogeneous Wiener–Hopf equation whose kernel is a nonarithmetic probability distribution with positive mean. The inhomogeneous term behaves like a submultiplicative function. We establish asymptotic properties of the solution to which the successive approximations converge. These properties depend on the asymptotics [...] Read more.
We consider the inhomogeneous Wiener–Hopf equation whose kernel is a nonarithmetic probability distribution with positive mean. The inhomogeneous term behaves like a submultiplicative function. We establish asymptotic properties of the solution to which the successive approximations converge. These properties depend on the asymptotics of the submultiplicative function. Full article
21 pages, 5300 KiB  
Article
The Mathematics of Catastrophe
by Ted Gyle Lewis
AppliedMath 2022, 2(3), 480-500; https://doi.org/10.3390/appliedmath2030028 - 14 Sep 2022
Cited by 1 | Viewed by 2150
Abstract
A mathematical description of catastrophe in complex systems modeled as a network is presented with emphasis on network topology and its relationship to risk and resilience. We present mathematical formulas for computing risk, resilience, and likelihood of faults in nodes/links of network models [...] Read more.
A mathematical description of catastrophe in complex systems modeled as a network is presented with emphasis on network topology and its relationship to risk and resilience. We present mathematical formulas for computing risk, resilience, and likelihood of faults in nodes/links of network models of complex systems and illustrate the application of the formulas to simulation of catastrophic failure. This model is not related to nonlinear “Catastrophe theory” by René Thom, E.C. Zeeman and others. Instead, we present a strictly probabilistic network model for estimating risk and resilience—two useful metrics used in practice. We propose a mathematical model of exceedance probability, risk, and resilience and show that these properties depend wholly on vulnerability, consequence, and properties of the network representation of the complex system. We use simulation of the network under simulated stress causing one or more nodes/links to fail, to extract properties of risk and resilience. In this paper two types of stress are considered: viral cascades and flow cascades. One unified definition of risk, MPL, is proposed, and three kinds of resilience illustrated—viral cascading, blocking node/link, and flow resilience. The principal contributions of this work are new equations for risk and resilience and measures of resilience based on vulnerability of individual nodes/links and network topology expressed in terms of spectral radius, bushy, and branchy metrics. We apply the model to a variety of networks—hypothetical and real—and show that network topology needs to be included in any definition of network risk and resilience. In addition, we show how simulations can identify likely future faults due to viral and flow cascades. Simulations of this nature are useful to the practitioner. Full article
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14 pages, 549 KiB  
Article
Mathematical Models for Population Growth with Variable Carrying Capacity: Analytical Solutions
by M. Rodrigo and D. Zulkarnaen
AppliedMath 2022, 2(3), 466-479; https://doi.org/10.3390/appliedmath2030027 - 29 Aug 2022
Cited by 3 | Viewed by 3372
Abstract
A general population model with variable carrying capacity consisting of a coupled system of nonlinear ordinary differential equations is proposed, and a procedure for obtaining analytical solutions for three broad classes of models is provided. A particular case is when the population and [...] Read more.
A general population model with variable carrying capacity consisting of a coupled system of nonlinear ordinary differential equations is proposed, and a procedure for obtaining analytical solutions for three broad classes of models is provided. A particular case is when the population and carrying capacity per capita growth rates are proportional. As an example, a generalised Thornley–France model is given. Further examples are given when the growth rates are not proportional. A criterion when inflexion may occur is also provided, and results of numerical simulations are presented. Full article
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9 pages, 286 KiB  
Article
Ranks with Respect to a Projective Variety and a Cost-Function
by Edoardo Ballico
AppliedMath 2022, 2(3), 457-465; https://doi.org/10.3390/appliedmath2030026 - 2 Aug 2022
Viewed by 1129
Abstract
Let XPr be an integral and non-degenerate variety. A “cost-function” (for the Zariski topology, the semialgebraic one, or the Euclidean one) is a semicontinuous function w:=[1,+)+ such that [...] Read more.
Let XPr be an integral and non-degenerate variety. A “cost-function” (for the Zariski topology, the semialgebraic one, or the Euclidean one) is a semicontinuous function w:=[1,+)+ such that w(a)=1 for a non-empty open subset of X. For any qPr, the rank rX,w(q) of q with respect to (X,w) is the minimum of all aSw(a), where S is a finite subset of X spanning q. We have rX,w(q)<+ for all q. We discuss this definition and classify extremal cases of pairs (X,q). We give upper bounds for all rX,w(q) (twice the generic rank) not depending on w. This notion is the generalization of the case in which the cost-function w is the constant function 1. In this case, the rank is a well-studied notion that covers the tensor rank of tensors of arbitrary formats (PARAFAC or CP decomposition) and the additive decomposition of forms. We also adapt to cost-functions the rank 1 decomposition of real tensors in which we allow pairs of complex conjugate rank 1 tensors. Full article
11 pages, 388 KiB  
Article
Local Normal Approximations and Probability Metric Bounds for the Matrix-Variate T Distribution and Its Application to Hotelling’s T Statistic
by Frédéric Ouimet
AppliedMath 2022, 2(3), 446-456; https://doi.org/10.3390/appliedmath2030025 - 1 Aug 2022
Viewed by 1360
Abstract
In this paper, we develop local expansions for the ratio of the centered matrix-variate T density to the centered matrix-variate normal density with the same covariances. The approximations are used to derive upper bounds on several probability metrics (such as the total variation [...] Read more.
In this paper, we develop local expansions for the ratio of the centered matrix-variate T density to the centered matrix-variate normal density with the same covariances. The approximations are used to derive upper bounds on several probability metrics (such as the total variation and Hellinger distance) between the corresponding induced measures. This work extends some previous results for the univariate Student distribution to the matrix-variate setting. Full article
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13 pages, 300 KiB  
Article
A Note on the Appearance of the Simplest Antilinear ODE in Several Physical Contexts
by Dmitry Ponomarev
AppliedMath 2022, 2(3), 433-445; https://doi.org/10.3390/appliedmath2030024 - 19 Jul 2022
Viewed by 1278
Abstract
We review several one-dimensional problems such as those involving linear Schrödinger equation, variable-coefficient Helmholtz equation, Zakharov–Shabat system and Kubelka–Munk equations. We show that they all can be reduced to solving one simple antilinear ordinary differential equation [...] Read more.
We review several one-dimensional problems such as those involving linear Schrödinger equation, variable-coefficient Helmholtz equation, Zakharov–Shabat system and Kubelka–Munk equations. We show that they all can be reduced to solving one simple antilinear ordinary differential equation ux=fxux¯ or its nonhomogeneous version ux=fxux¯+gx, x0,x0R. We point out some of the advantages of the proposed reformulation and call for further investigation of the obtained ODE. Full article
(This article belongs to the Special Issue Feature Papers in AppliedMath)
40 pages, 7655 KiB  
Article
Continued Fractions and Probability Estimations in Shor’s Algorithm: A Detailed and Self-Contained Treatise
by Johanna Barzen and Frank Leymann
AppliedMath 2022, 2(3), 393-432; https://doi.org/10.3390/appliedmath2030023 - 18 Jul 2022
Cited by 2 | Viewed by 2240
Abstract
Shor’s algorithm for prime factorization is a hybrid algorithm consisting of a quantum part and a classical part. The main focus of the classical part is a continued fraction analysis. The presentation of this is often short, pointing to text books on number [...] Read more.
Shor’s algorithm for prime factorization is a hybrid algorithm consisting of a quantum part and a classical part. The main focus of the classical part is a continued fraction analysis. The presentation of this is often short, pointing to text books on number theory. In this contribution, we present the relevant results and proofs from the theory of continued fractions in detail (even in more detail than in text books), filling the gap to allow a complete comprehension of Shor’s algorithm. Similarly, we provide a detailed computation of the estimation of the probability that convergents will provide the period required for determining a prime factor. Full article
(This article belongs to the Special Issue Applications of Number Theory to the Sciences and Mathematics)
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14 pages, 322 KiB  
Article
Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis
by Tohru Morita
AppliedMath 2022, 2(3), 379-392; https://doi.org/10.3390/appliedmath2030022 - 15 Jul 2022
Viewed by 1280
Abstract
Discussions are presented by Morita and Sato on the problem of obtaining the particular solution of an inhomogeneous differential equation with polynomial coefficients in terms of the Green’s function. In a paper, the problem is treated in distribution theory, and in another paper, [...] Read more.
Discussions are presented by Morita and Sato on the problem of obtaining the particular solution of an inhomogeneous differential equation with polynomial coefficients in terms of the Green’s function. In a paper, the problem is treated in distribution theory, and in another paper, the formulation is given on the basis of nonstandard analysis, where fractional derivative of degree, which is a complex number added by an infinitesimal number, is used. In the present paper, a simple recipe based on nonstandard analysis, which is closely related with distribution theory, is presented, where in place of Heaviside’s step function H(t) and Dirac’s delta function δ(t) in distribution theory, functions Hϵ(t):=1Γ(1+ϵ)tϵH(t) and δϵ(t):=ddtHϵ(t)=1Γ(ϵ)tϵ1H(t) for a positive infinitesimal number ϵ, are used. As an example, it is applied to Kummer’s differential equation. Full article
20 pages, 1294 KiB  
Article
Dualities and Asymptotic Mixtures Using Functional-Order Differentiation
by Aris Alexopoulos
AppliedMath 2022, 2(3), 359-378; https://doi.org/10.3390/appliedmath2030021 - 6 Jul 2022
Viewed by 1928
Abstract
New definitions for fractional integro-differential operators are presented and referred to as delayed fractional operators. It is shown that delayed fractional derivatives give rise to the notion of functional order differentiation. Functional differentiation can be used to establish dualities and asymptotic mixtures between [...] Read more.
New definitions for fractional integro-differential operators are presented and referred to as delayed fractional operators. It is shown that delayed fractional derivatives give rise to the notion of functional order differentiation. Functional differentiation can be used to establish dualities and asymptotic mixtures between unrelated theories, something that conventional fractional or integer operators cannot do. In this paper, dualities and asymptotic mixtures are established between arbitrary functions, probability densities, the Gibbs–Shannon entropy and Hellinger distance, as well as higher-dimensional particle geometries in quantum mechanics. Full article
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33 pages, 4737 KiB  
Article
Structure of Triangular Numbers Modulo m
by Darin J. Ulness
AppliedMath 2022, 2(3), 326-358; https://doi.org/10.3390/appliedmath2030020 - 1 Jul 2022
Viewed by 2107
Abstract
This work focuses on the structure and properties of the triangular numbers modulo m. The most important aspect of the structure of these numbers is their periodic nature. It is proven that the triangular numbers modulo m forms a 2m-cycle [...] Read more.
This work focuses on the structure and properties of the triangular numbers modulo m. The most important aspect of the structure of these numbers is their periodic nature. It is proven that the triangular numbers modulo m forms a 2m-cycle for any m. Additional structural features and properties of this system are presented and discussed. This discussion is aided by various representations of these sequences, such as network graphs, and through discrete Fourier transformation. The concept of saturation is developed and explored, as are monoid sets and the roles of perfect squares and nonsquares. The triangular numbers modulo m has self-similarity and scaling features which are discussed as well. Full article
(This article belongs to the Special Issue Applications of Number Theory to the Sciences and Mathematics)
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