AppliedMath

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

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

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

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

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

We use a payment pattern of the type $\{1^k,2^k,3^k,\dots \}$ 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 $\{1^r,2^r,3^r,\dots \}$, 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

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

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

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

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 $2\sqrt{2}$ 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

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

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

We use the representation of a continuous time Hattendorff differential equation and Matlab to compute ${}^2{\sigma }_t^{\left(j\right)}$, the solution of a backwards in time differential equation that describes the evolution of the variance of $L_t^{\left(j\right)}$, 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 ${}^2{\sigma }_t^{\left(j\right)}$ 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

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 $l\mid N_1\left(B\right)·r+1$, where l is the bit-length of B and $N_1\left(B\right)$ 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

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

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

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

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

In finance, the most efficient portfolio is the tangency portfolio, which is formed by the intersection point of the efficient frontier and the capital market line. This paper defines and explores a time-varying tangency portfolio under nonlinear constraints (TV-TPNC) problem as a nonlinear programming (NLP) problem. Because meta-heuristics are commonly used to solve NLP problems, a semi-integer beetle antennae search (SIBAS) algorithm is proposed for solving cardinality constrained NLP problems and, hence, to solve the TV-TPNC problem. The main results of numerical applications in real-world datasets demonstrate that our method is a splendid substitute for other evolutionary methods. Full article