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AppliedMath, Volume 2, Issue 1 (March 2022) – 9 articles

Cover Story (view full-size image): This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how solely the frictional abrasion of a stone on a planar beach might lead to the oval shapes that have been observed empirically. View this paper
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11 pages, 260 KiB  
Article
Resurrecting the Prospect of Supplementary Variableswith the Principle of Local Realism
by Frank Lad
AppliedMath 2022, 2(1), 159-169; https://doi.org/10.3390/appliedmath2010009 - 16 Mar 2022
Cited by 1 | Viewed by 1470
Abstract
I produce a coherent mathematical formulation of the supplementary variables structure for Aspect’s experimental test of Bell’s inequality as devised by Clauser, Horne, Shimony, and Holt, a formalization which has been widely considered to be impossible. Contrary to Aspect’s understanding, it is made [...] Read more.
I produce a coherent mathematical formulation of the supplementary variables structure for Aspect’s experimental test of Bell’s inequality as devised by Clauser, Horne, Shimony, and Holt, a formalization which has been widely considered to be impossible. Contrary to Aspect’s understanding, it is made clear that a supplementary variable formulation can represent any tendered probability distribution whatsoever. This includes both the QM distribution and the “naive distribution”, which he had suggested as a foil. It has long been known that quantum theory does not support a complete distribution for the components of the thought experiment that underlies the inequality. However, further than that, here I identify precisely the bounding polytope of distributions that do cohere with both its explicit premises and with the prospect of supplementary variables. In this context, it is found once again that every distribution within this polytope respects the conditions of Bell’s inequality, and that the famous evaluation of the gedankenexpectation defying it as 22 is mistaken. The argument is relevant to all subsequent embellishments of experimental methodology post Aspect, designed to block seven declared possible loopholes. The probabilistic prognostications of quantum theory are not denied, nor are the experimental observations. However, their inferential implications have been misrepresented. Full article
16 pages, 868 KiB  
Article
One-Dimensional Matter Waves as a Multi-State Bit
by Jacopo Giacomelli
AppliedMath 2022, 2(1), 143-158; https://doi.org/10.3390/appliedmath2010008 - 1 Mar 2022
Viewed by 2184
Abstract
We design a simple technique to control the position of a localized matter wave. Our system is composed of two counter-phased periodic potentials and a third optical lattice, which can be either periodic or disordered. The only control needed on the system is [...] Read more.
We design a simple technique to control the position of a localized matter wave. Our system is composed of two counter-phased periodic potentials and a third optical lattice, which can be either periodic or disordered. The only control needed on the system is a three-state switch that allows the sudden selection of the desired potential. The method is proposed as a possible new alternative to achieving the realization of a multi-state bit. We show that this framework is robust, and that the multi-state bit behavior can be observed under weak assumptions. Given the current degree of development of matter wave control in optical lattices, we believe that the proposed device would be easily reproducible in a laboratory, allowing for testing and industrial applications. Full article
(This article belongs to the Special Issue Feature Papers in AppliedMath)
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12 pages, 307 KiB  
Article
Difference in Coulomb Electrostatic Energy for Localized versus Delocalized Electrons and Electron Pairs—Exact Results Based on Cubic Charge Distributions
by Hanno Essén and Johan C.-E. Stén
AppliedMath 2022, 2(1), 131-142; https://doi.org/10.3390/appliedmath2010007 - 18 Feb 2022
Viewed by 2065
Abstract
Wigner showed that a sufficiently thin electron gas will condense into a crystal of localized electrons. Here, we show, using a model based on cubic charge distributions that gives exact results, that the Coulomb repulsion energy of localized charge distributions is lower than [...] Read more.
Wigner showed that a sufficiently thin electron gas will condense into a crystal of localized electrons. Here, we show, using a model based on cubic charge distributions that gives exact results, that the Coulomb repulsion energy of localized charge distributions is lower than that of delocalized distributions in spite of the fact that the total overall charge distribution is the same. Assuming a simple cubic geometry, we obtain an explicit result for the energy reduction. This reduction results from the exclusion of self-interactions of the electrons. The corresponding results for electron pairs are also discussed. Full article
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13 pages, 341 KiB  
Article
Numerical Solutions of the Hattendorff Differential Equation for Multi-State Markov Insurance Models
by Nathan Ritchey and Rajeev Rajaram
AppliedMath 2022, 2(1), 118-130; https://doi.org/10.3390/appliedmath2010006 - 11 Feb 2022
Viewed by 2123
Abstract
We use the representation of a continuous time Hattendorff differential equation and Matlab to compute 2σt(j), the solution of a backwards in time differential equation that describes the evolution of the variance of [...] Read more.
We use the representation of a continuous time Hattendorff differential equation and Matlab to compute 2σt(j), the solution of a backwards in time differential equation that describes the evolution of the variance of Lt(j), the loss at time t random variable for a multi-state Markovian process, given that the state at time t is j. We demonstrate this process by solving examples of several instances of a multi-state model which a practitioner can use as a guide to solve and analyze specific multi-state models. Numerical solutions to compute the variance 2σt(j) enable practitioners and academic researchers to test and simulate various state-space scenarios, with possible transitions to and from temporary disabilities, to permanent disabilities, to and from good health, and eventually to a deceased state. The solution method presented in this paper allows researchers and practitioners to easily compute the evolution of the variance of loss without having to resort to detailed programming. Full article
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14 pages, 343 KiB  
Article
Rotating Binaries
by Anant Gupta, Idriss J. Aberkane, Sourangshu Ghosh, Adrian Abold, Alexander Rahn and Eldar Sultanow
AppliedMath 2022, 2(1), 104-117; https://doi.org/10.3390/appliedmath2010005 - 3 Feb 2022
Viewed by 4322
Abstract
This paper investigates the behavior of rotating binaries. A rotation by r digits to the left of a binary number B exhibits in particular cases the divisibility lN1(B)·r+1, where l is the [...] Read more.
This paper investigates the behavior of rotating binaries. A rotation by r digits to the left of a binary number B exhibits in particular cases the divisibility lN1(B)·r+1, where l is the bit-length of B and N1(B) is the Hamming weight of B, that is the number of ones in B. The integer r is called the left-rotational distance. We investigate the connection between this rotational distance, the length, and the Hamming weight of binary numbers. Moreover, we follow the question under which circumstances the above-mentioned divisibility is true. We have found out and will demonstrate that this divisibility occurs for kn+c cycles. Full article
50 pages, 9859 KiB  
Article
A Phase-Field Perspective on Mereotopology
by Georg J. Schmitz
AppliedMath 2022, 2(1), 54-103; https://doi.org/10.3390/appliedmath2010004 - 17 Jan 2022
Cited by 2 | Viewed by 3406
Abstract
Mereotopology is a concept rooted in analytical philosophy. The phase-field concept is based on mathematical physics and finds applications in materials engineering. The two concepts seem to be disjoint at a first glance. While mereotopology qualitatively describes static relations between things, such as [...] Read more.
Mereotopology is a concept rooted in analytical philosophy. The phase-field concept is based on mathematical physics and finds applications in materials engineering. The two concepts seem to be disjoint at a first glance. While mereotopology qualitatively describes static relations between things, such as x isConnected y (topology) or x isPartOf y (mereology) by first order logic and Boolean algebra, the phase-field concept describes the geometric shape of things and its dynamic evolution by drawing on a scalar field. The geometric shape of any thing is defined by its boundaries to one or more neighboring things. The notion and description of boundaries thus provides a bridge between mereotopology and the phase-field concept. The present article aims to relate phase-field expressions describing boundaries and especially triple junctions to their Boolean counterparts in mereotopology and contact algebra. An introductory overview on mereotopology is followed by an introduction to the phase-field concept already indicating its first relations to mereotopology. Mereotopological axioms and definitions are then discussed in detail from a phase-field perspective. A dedicated section introduces and discusses further notions of the isConnected relation emerging from the phase-field perspective like isSpatiallyConnected, isTemporallyConnected, isPhysicallyConnected, isPathConnected, and wasConnected. Such relations introduce dynamics and thus physics into mereotopology, as transitions from isDisconnected to isPartOf can be described. Full article
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15 pages, 288 KiB  
Article
Berry–Esseen Bounds of the Quasi Maximum Likelihood Estimators for the Discretely Observed Diffusions
by Jaya P. N. Bishwal
AppliedMath 2022, 2(1), 39-53; https://doi.org/10.3390/appliedmath2010003 - 8 Jan 2022
Viewed by 2029
Abstract
For stationary ergodic diffusions satisfying nonlinear homogeneous Itô stochastic differential equations, this paper obtains the Berry–Esseen bounds on the rates of convergence to normality of the distributions of the quasi maximum likelihood estimators based on stochastic Taylor approximation, under some regularity conditions, when [...] Read more.
For stationary ergodic diffusions satisfying nonlinear homogeneous Itô stochastic differential equations, this paper obtains the Berry–Esseen bounds on the rates of convergence to normality of the distributions of the quasi maximum likelihood estimators based on stochastic Taylor approximation, under some regularity conditions, when the diffusion is observed at equally spaced dense time points over a long time interval, the high-frequency regime. It shows that the higher-order stochastic Taylor approximation-based estimators perform better than the basic Euler approximation in the sense of having smaller asymptotic variance. Full article
23 pages, 5211 KiB  
Article
On the Oval Shapes of Beach Stones
by Theodore P. Hill
AppliedMath 2022, 2(1), 16-38; https://doi.org/10.3390/appliedmath2010002 - 7 Jan 2022
Viewed by 5517
Abstract
This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how frictional abrasion alone of a stone on a planar beach might lead to the oval shapes observed empirically. The underlying [...] Read more.
This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how frictional abrasion alone of a stone on a planar beach might lead to the oval shapes observed empirically. The underlying idea in this theory is the intuitive observation that the rate of ablation at a point on the surface of the stone is proportional to the product of the curvature of the stone at that point and the likelihood the stone is in contact with the beach at that point. Specifically, key roles in this new model are played by both the random wave process and the global (non-local) shape of the stone, i.e., its shape away from the point of contact with the beach. The underlying physical mechanism for this process is the conversion of energy from the wave process into the potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, which is both non-linear and non-local. On the other hand, preliminary numerical experiments are presented in both the deterministic continuous-time setting using standard curve-shortening algorithms and a stochastic discrete-time polyhedral-slicing setting using Monte Carlo simulation. Full article
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15 pages, 13314 KiB  
Article
Non-Linear Analysis of River System Dynamics Using Recurrence Quantification Analysis
by Athanasios Fragkou, Avraam Charakopoulos, Theodoros Karakasidis and Antonios Liakopoulos
AppliedMath 2022, 2(1), 1-15; https://doi.org/10.3390/appliedmath2010001 - 6 Jan 2022
Cited by 2 | Viewed by 3005
Abstract
Understanding the underlying processes and extracting detailed characteristics of rivers is critical and has not yet been fully developed. The purpose of this study was to examine the performance of non-linear time series methods on environmental data. Specifically, we performed an analysis of [...] Read more.
Understanding the underlying processes and extracting detailed characteristics of rivers is critical and has not yet been fully developed. The purpose of this study was to examine the performance of non-linear time series methods on environmental data. Specifically, we performed an analysis of water level measurements, extracted from sensors, located on specified stations along the Nestos River (Greece), with Recurrence Plots (RP) and Recurrence Quantification Analysis (RQA) methods. A more detailed inspection with the sliding windows (epoqs) method was applied on the Recurrence Rate, Average Diagonal Line and Trapping Time parameters, with results showing phase transitions providing useful information about the dynamics of the system. The suggested method seems to be promising for the detection of the dynamical transitions that can characterize distinct time windows of the time series and reveals information about the changes in state within the whole time series. The results will be useful for designing the energy policy investments of producers and also will be helpful for dam management assessment as well as government energy policy. Full article
(This article belongs to the Special Issue Feature Papers in AppliedMath)
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