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AppliedMath, Volume 2, Issue 2 (June 2022) – 10 articles

Cover Story (view full-size image): Brain-inspired artificial neural networks trained via gradient descent have revolutionized machine learning, but it remains unknown how biological neural networks learn. In this work, we propose a reinforcement learning methodology to generate and apply biologically feasible learning rules in artificial neural networks. Surprisingly, a simple application of the methodology quickly yielded a learning rule that was directly applicable to much larger networks than initially trained for. With slow and consistent convergence, the new learning rule produced networks with performance on par with those trained via gradient descent. These results imply a bright future for our methodology in creating and understanding robust, decentralized, and intelligent systems that go beyond gradient descent. View this paper
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14 pages, 458 KiB  
Article
Verifying Measures of Quantum Entropy
by Giancarlo Pastor and Jae-Oh Woo
AppliedMath 2022, 2(2), 312-325; https://doi.org/10.3390/appliedmath2020019 - 17 Jun 2022
Viewed by 1736
Abstract
This paper introduces a new measure of quantum entropy, called the effective quantum entropy (EQE). The EQE is an extension, to the quantum setting, of a recently derived classical generalized entropy. We present a thorough verification of its properties. As its predecessor, the [...] Read more.
This paper introduces a new measure of quantum entropy, called the effective quantum entropy (EQE). The EQE is an extension, to the quantum setting, of a recently derived classical generalized entropy. We present a thorough verification of its properties. As its predecessor, the EQE is a semi-strict quasi-concave function; it would be capable of generating many of the various measures of quantum entropy that are useful in practice. Thereafter, we construct a consistent estimator for our proposed measure and empirically test its estimation error, under different system dimensions and number of measurements. Overall, we build the grounds of the EQE, which will facilitate the analyses and verification of the next innovative quantum technologies. Full article
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28 pages, 1295 KiB  
Article
Measuring Dependencies between Variables of a Dynamical System Using Fuzzy Affiliations
by Niklas Wulkow
AppliedMath 2022, 2(2), 284-311; https://doi.org/10.3390/appliedmath2020018 - 16 Jun 2022
Viewed by 1581
Abstract
A statistical, data-driven method is presented that quantifies influences between variables of a dynamical system. The method is based on finding a suitable representation of points by fuzzy affiliations with respect to landmark points using the Scalable Probabilistic Approximation algorithm. This is followed [...] Read more.
A statistical, data-driven method is presented that quantifies influences between variables of a dynamical system. The method is based on finding a suitable representation of points by fuzzy affiliations with respect to landmark points using the Scalable Probabilistic Approximation algorithm. This is followed by the construction of a linear mapping between these affiliations for different variables and forward in time. This linear mapping, or matrix, can be directly interpreted in light of unidirectional dependencies, and relevant properties of it are quantified. These quantifications, given by the sum of singular values and the average row variance of the matrix, then serve as measures for the influences between variables of the dynamics. The validity of the method is demonstrated with theoretical results and on several numerical examples, covering deterministic, stochastic, and delayed types of dynamics. Moreover, the method is applied to a non-classical example given by real-world basketball player movement, which exhibits highly random movement and comes without a physical intuition, contrary to many examples from, e.g., life sciences. Full article
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15 pages, 405 KiB  
Article
On the Exact Solution of Nonlocal Euler–Bernoulli Beam Equations via a Direct Approach for Volterra-Fredholm Integro-Differential Equations
by Efthimios Providas
AppliedMath 2022, 2(2), 269-283; https://doi.org/10.3390/appliedmath2020017 - 10 Jun 2022
Cited by 3 | Viewed by 1901
Abstract
First, we develop a direct operator method for solving boundary value problems for a class of nth order linear Volterra–Fredholm integro-differential equations of convolution type. The proposed technique is based on the assumption that the Volterra integro-differential operator is bijective and its [...] Read more.
First, we develop a direct operator method for solving boundary value problems for a class of nth order linear Volterra–Fredholm integro-differential equations of convolution type. The proposed technique is based on the assumption that the Volterra integro-differential operator is bijective and its inverse is known in closed form. Existence and uniqueness criteria are established and the exact solution is derived. We then apply this method to construct the closed form solution of the fourth order equilibrium equations for the bending of Euler–Bernoulli beams in the context of Eringen’s nonlocal theory of elasticity (two phase integral model) under a transverse distributed load and simply supported boundary conditions. An easy to use algorithm for obtaining the exact solution in a symbolic algebra system is also given. Full article
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8 pages, 294 KiB  
Article
Computer-Algebra-Software-Assisted Calculus Instruction, Not Calculus for Dummies: Bespoke Applications Necessitate Theory
by Daniel A. Griffith
AppliedMath 2022, 2(2), 261-268; https://doi.org/10.3390/appliedmath2020016 - 7 Jun 2022
Viewed by 1414
Abstract
Today, calculus frequently is taught with artificial intelligence in the form of computer algebra systems. Although these software packages may reduce tedium associated with the mechanics of calculus, they may be less effective if not supplemented by the accompanying teaching of calculus theory. [...] Read more.
Today, calculus frequently is taught with artificial intelligence in the form of computer algebra systems. Although these software packages may reduce tedium associated with the mechanics of calculus, they may be less effective if not supplemented by the accompanying teaching of calculus theory. This paper presents two examples from spatial statistics in which computer software in an unsupervised auto-execution mode fails, or can fail, to yield correct calculus results. Accordingly, it emphasizes the need to teach calculus theory when using software packages such as Mathematica and Maple. Full article
14 pages, 479 KiB  
Article
From Modelling Turbulence to General Systems Modelling
by Alexander Y. Klimenko
AppliedMath 2022, 2(2), 247-260; https://doi.org/10.3390/appliedmath2020015 - 26 May 2022
Cited by 1 | Viewed by 1383
Abstract
Complex adaptive and evolutionary systems can, at least in principle, be modelled in ways that are similar to modelling of complex mechanical (or physical) systems. While quantitative modelling of turbulent reacting flows has been developed over many decades due to availability of experimental [...] Read more.
Complex adaptive and evolutionary systems can, at least in principle, be modelled in ways that are similar to modelling of complex mechanical (or physical) systems. While quantitative modelling of turbulent reacting flows has been developed over many decades due to availability of experimental data, modelling of complex evolutionary systems is still in its infancy and has huge potential for further development. This work analyses recent trends, points to the similarity of modelling approaches used in seemingly different areas, and suggests a basic classification for such approaches. Availability of data in the modern computerised world allows us to use tools previously developed in physics and applied mathematics in new domains of scientific inquiry that previously were not amendable by quantitative evaluation and modelling, while raising concerns about the associated ethical and legal issues. While the utility of big data has been repeatedly demonstrated in various practical applications, these applications, as far as we can judge, do not involve the scientific goal of conceptual modelling of emergent collective behaviour in complex evolutionary systems. Full article
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13 pages, 641 KiB  
Article
Improved Exploration in Reinforcement Learning Environments with Low-Discrepancy Action Selection
by Stephen W. Carden, Jedidiah O. Lindborg and Zheni Utic
AppliedMath 2022, 2(2), 234-246; https://doi.org/10.3390/appliedmath2020014 - 16 May 2022
Viewed by 1925
Abstract
Reinforcement learning (RL) is a subdomain of machine learning concerned with achieving optimal behavior by interacting with an unknown and potentially stochastic environment. The exploration strategy for choosing actions is an important component for enabling the decision agent to discover how to obtain [...] Read more.
Reinforcement learning (RL) is a subdomain of machine learning concerned with achieving optimal behavior by interacting with an unknown and potentially stochastic environment. The exploration strategy for choosing actions is an important component for enabling the decision agent to discover how to obtain high rewards. If constructed well, it may reduce the learning time of the decision agent. Exploration in discrete problems has been well studied, but there are fewer strategies applicable to continuous dynamics. In this paper, we propose a Low-Discrepancy Action Selection (LDAS) process, a novel exploration strategy for environments with continuous states and actions. This algorithm focuses on prioritizing unknown regions of the state-action space with the intention of finding ideal actions faster than pseudo-random action selection. Results of experimentation with three benchmark environments elucidate the situations in which LDAS is superior and introduce a metric for quantifying the quality of exploration. Full article
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22 pages, 335 KiB  
Article
Polynomial Annuities
by Rajeev Rajaram and Nathan Ritchey
AppliedMath 2022, 2(2), 212-233; https://doi.org/10.3390/appliedmath2020013 - 5 May 2022
Viewed by 1706
Abstract
We use a payment pattern of the type {1k,2k,3k,} to generalize the standard level payment and increasing annuity to polynomial payment patterns. We derive explicit formulas for the present value of an [...] Read more.
We use a payment pattern of the type {1k,2k,3k,} to generalize the standard level payment and increasing annuity to polynomial payment patterns. We derive explicit formulas for the present value of an n-year polynomial annuity, the present value of an m-monthly n-year polynomial annuity, and the present value of an n-year continuous polynomial annuity. We also use the idea to extend the annuities to payment patterns derived from analytic functions, as well as to payment patterns of the type {1r,2r,3r,}, with r being an arbitrary real number. In the process, we develop possible approximations to k! and for the gamma function evaluated at real numbers. Full article
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16 pages, 308 KiB  
Article
Kaczmarz Anomaly in Tomography Problems
by Achiya Dax
AppliedMath 2022, 2(2), 196-211; https://doi.org/10.3390/appliedmath2020012 - 25 Apr 2022
Cited by 1 | Viewed by 1352
Abstract
The Kaczmarz method is an important tool for solving large sparse linear systems that arise in computerized tomography. The Kaczmarz anomaly phenomenon has been observed recently when solving certain types of random systems. This raises the question of whether a similar anomaly occurs [...] Read more.
The Kaczmarz method is an important tool for solving large sparse linear systems that arise in computerized tomography. The Kaczmarz anomaly phenomenon has been observed recently when solving certain types of random systems. This raises the question of whether a similar anomaly occurs in tomography problems. The aim of the paper is to answer this question, to examine the extent of the phenomenon and to explain its reasons. Another tested issue is the ability of random row shuffles to sharpen the anomaly and to accelerate the rate of convergence. The results add important insight into the nature of the Kaczmarz method. Full article
11 pages, 892 KiB  
Article
Gradient-Free Neural Network Training via Synaptic-Level Reinforcement Learning
by Aman Bhargava, Mohammad R. Rezaei and Milad Lankarany
AppliedMath 2022, 2(2), 185-195; https://doi.org/10.3390/appliedmath2020011 - 12 Apr 2022
Cited by 1 | Viewed by 2485
Abstract
An ongoing challenge in neural information processing is the following question: how do neurons adjust their connectivity to improve network-level task performance over time (i.e., actualize learning)? It is widely believed that there is a consistent, synaptic-level learning mechanism in specific brain regions, [...] Read more.
An ongoing challenge in neural information processing is the following question: how do neurons adjust their connectivity to improve network-level task performance over time (i.e., actualize learning)? It is widely believed that there is a consistent, synaptic-level learning mechanism in specific brain regions, such as the basal ganglia, that actualizes learning. However, the exact nature of this mechanism remains unclear. Here, we investigate the use of universal synaptic-level algorithms in training connectionist models. Specifically, we propose an algorithm based on reinforcement learning (RL) to generate and apply a simple biologically-inspired synaptic-level learning policy for neural networks. In this algorithm, the action space for each synapse in the network consists of a small increase, decrease, or null action on the connection strength. To test our algorithm, we applied it to a multilayer perceptron (MLP) neural network model. This algorithm yields a static synaptic learning policy that enables the simultaneous training of over 20,000 parameters (i.e., synapses) and consistent learning convergence when applied to simulated decision boundary matching and optical character recognition tasks. The trained networks yield character-recognition performance comparable to identically shaped networks trained with gradient descent. The approach has two significant advantages in comparison to traditional gradient-descent-based optimization methods. First, the robustness of our novel method and its lack of reliance on gradient computations opens the door to new techniques for training difficult-to-differentiate artificial neural networks, such as spiking neural networks (SNNs) and recurrent neural networks (RNNs). Second, the method’s simplicity provides a unique opportunity for further development of local information-driven multiagent connectionist models for machine intelligence analogous to cellular automata. Full article
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15 pages, 576 KiB  
Article
Accumulators and Bookmaker’s Capital with Perturbed Stochastic Processes
by Dominic Cortis and Muhsin Tamturk
AppliedMath 2022, 2(2), 170-184; https://doi.org/10.3390/appliedmath2020010 - 1 Apr 2022
Viewed by 1754
Abstract
The sports betting industry has been growing at a phenomenal rate and has many similarities to the financial market in that a payout is made contingent on an outcome of an event. Despite this, there has been little to no mathematical focus on [...] Read more.
The sports betting industry has been growing at a phenomenal rate and has many similarities to the financial market in that a payout is made contingent on an outcome of an event. Despite this, there has been little to no mathematical focus on the potential ruin of bookmakers. In this paper, the expected profit of a bookmaker and probability of multiple soccer matches are observed via Dirac notations and Feynman’s path calculations. Furthermore, we take the unforeseen circumstances into account by subjecting the betting process to more uncertainty. A perturbed betting process, set by modifying the conventional stochastic process, is handled to scale and manage this uncertainty. Full article
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