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AppliedMath, Volume 3, Issue 1 (March 2023) – 14 articles

Cover Story (view full-size image): The underpopulation rule is graph dynamics derived by simplifying the set of rules constituting the Game of Life. The number of label configurations met by a graph during the dynamic process defined by this rule is bounded by a polynomial in the size of the graph if the graph is undirected. As a consequence, predicting the label evolution is an easy problem (i.e., a problem in P) in such a case. In this paper, the generalization of the underpopulation rule to signed and directed graphs is studied. It is proved here that the number of label configurations met by a graph during the dynamic process defined by any generalized underpopulation rule is still bounded by a polynomial in the size of the graph if the graph is undirected and structurally balanced, while it is not bounded by any polynomial in the size of the graph if the graph is directed but unsigned unless P = PSpace. View this paper
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Article
A Rule-Based Approach for Mining Creative Thinking Patterns from Big Educational Data
AppliedMath 2023, 3(1), 243-267; https://doi.org/10.3390/appliedmath3010014 - 20 Mar 2023
Viewed by 848
Abstract
Numerous studies have established a correlation between creativity and intrinsic motivation to learn, with creativity defined as the process of generating original and valuable ideas, often by integrating perspectives from different fields. The field of educational technology has shown a growing interest in [...] Read more.
Numerous studies have established a correlation between creativity and intrinsic motivation to learn, with creativity defined as the process of generating original and valuable ideas, often by integrating perspectives from different fields. The field of educational technology has shown a growing interest in leveraging technology to promote creativity in the classroom, with several studies demonstrating the positive impact of creativity on learning outcomes. However, mining creative thinking patterns from educational data remains a challenging task, even with the proliferation of research on adaptive technology for education. This paper presents an initial effort towards formalizing educational knowledge by developing a domain-specific Knowledge Base that identifies key concepts, facts, and assumptions essential for identifying creativity patterns. Our proposed pipeline involves modeling raw educational data, such as assessments and class activities, as a graph to facilitate the contextualization of knowledge. We then leverage a rule-based approach to enable the mining of creative thinking patterns from the contextualized data and knowledge graph. To validate our approach, we evaluate it on real-world datasets and demonstrate how the proposed pipeline can enable instructors to gain insights into students’ creative thinking patterns from their activities and assessment tasks. Full article
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Article
The Harris Extended Bilal Distribution with Applications in Hydrology and Quality Control
AppliedMath 2023, 3(1), 221-242; https://doi.org/10.3390/appliedmath3010013 - 10 Mar 2023
Viewed by 622
Abstract
In this research work, a new three-parameter lifetime distribution is introduced and studied. It is called the Harris extended Bilal distribution due to its construction from a mixture of the famous Bilal and Harris distributions, resulting from a branching process. The basic properties, [...] Read more.
In this research work, a new three-parameter lifetime distribution is introduced and studied. It is called the Harris extended Bilal distribution due to its construction from a mixture of the famous Bilal and Harris distributions, resulting from a branching process. The basic properties, such as the moment generating function, moments, quantile function, and Rényi entropy, are discussed. We show that the hazard rate function has ideal features for modeling increasing, upside-down bathtub, and roller-coaster data sets. In a second part, the Harris extended Bilal model is investigated from a statistical viewpoint. The maximum likelihood estimation is used to estimate the parameters, and a simulation study is carried out. The flexibility of the proposed model in a hydrological data analysis scenario is demonstrated using two practical data sets and compared with important competing models. After that, we establish an acceptance sampling plan that takes advantage of all of the features of the Harris extended Bilal model. The operating characteristic values, the minimum sample size that corresponds to the maximum possible defects, and the minimum ratios of lifetime associated with the producer’s risk are discussed. Full article
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Article
Linear Trees, Lattice Walks, and RNA Arrays
AppliedMath 2023, 3(1), 200-220; https://doi.org/10.3390/appliedmath3010012 - 09 Mar 2023
Viewed by 815
Abstract
The leftmost column entries of RNA arrays I and II count the RNA numbers that are related to RNA secondary structures from molecular biology. RNA secondary structures sometimes have mutations and wobble pairs. Mutations are random changes that occur in a structure, and [...] Read more.
The leftmost column entries of RNA arrays I and II count the RNA numbers that are related to RNA secondary structures from molecular biology. RNA secondary structures sometimes have mutations and wobble pairs. Mutations are random changes that occur in a structure, and wobble pairs are known as non-Watson–Crick base pairs. We used topics from RNA combinatorics and Riordan array theory to establish connections among combinatorial objects related to linear trees, lattice walks, and RNA arrays. In this paper, we establish interesting new explicit bijections (one-to-one correspondences) involving certain subclasses of linear trees, lattice walks, and RNA secondary structures. We provide an interesting generalized lattice walk interpretation of RNA array I. In addition, we provide a combinatorial interpretation of RNA array II as RNA secondary structures with n bases and k base-point mutations where ω of the structures contain wobble base pairs. We also establish an explicit bijection between RNA structures with mutations and wobble bases and a certain subclass of lattice walks. Full article
(This article belongs to the Special Issue Feature Papers in AppliedMath)
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Article
Analyzing Health Data Breaches: A Visual Analytics Approach
AppliedMath 2023, 3(1), 175-199; https://doi.org/10.3390/appliedmath3010011 - 09 Mar 2023
Viewed by 1085
Abstract
This research studies the occurrence of data breaches in healthcare provider settings regarding patient data. Using visual analytics and data visualization tools, we study the distribution of healthcare breaches by state. We review the main causes and types of breaches, as well as [...] Read more.
This research studies the occurrence of data breaches in healthcare provider settings regarding patient data. Using visual analytics and data visualization tools, we study the distribution of healthcare breaches by state. We review the main causes and types of breaches, as well as their impact on both providers and patients. The research shows a range of data breach victims. Network servers are the most popular location for common breaches, such as hacking and information technology (IT) incidents, unauthorized access, theft, loss, and improper disposal. We offer proactive recommendations to prepare for a breach. These include, but are not limited to, regulatory compliance, implementing policies and procedures, and monitoring network servers. Unfortunately, the results indicate that the probability of data breaches will continue to rise. Full article
(This article belongs to the Special Issue Feature Papers in AppliedMath)
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Article
A Collection of New Trigonometric- and Hyperbolic-FGM-Type Copulas
AppliedMath 2023, 3(1), 147-174; https://doi.org/10.3390/appliedmath3010010 - 03 Mar 2023
Cited by 2 | Viewed by 695
Abstract
Copula analysis was created to explain the dependence of two or more quantitative variables. Due to the need for in-depth data analysis involving complex variable relationships, there is always a need for new copula models with original features. As a modern example, for [...] Read more.
Copula analysis was created to explain the dependence of two or more quantitative variables. Due to the need for in-depth data analysis involving complex variable relationships, there is always a need for new copula models with original features. As a modern example, for the analysis of circular or periodic data types, trigonometric copulas are particularly attractive and recommended. This is, however, an underexploited topic. In this article, we propose a new collection of eight trigonometric and hyperbolic copulas, four based on the sine function and the others on the tangent function, all derived from the construction of the famous Farlie–Gumbel–Morgenstern copula. In addition to their original trigonometric and hyperbolic functionalities, the proposed copulas have the feature of depending on three parameters with complementary roles: one is a dependence parameter; one is a shape parameter; and the last can be viewed as an angle parameter. In our main findings, for each of the eight copulas, we determine a wide range of admissible values for these parameters. Subsequently, the capabilities, features, and functions of the new copulas are thoroughly examined. The shapes of the main functions of some copulas are illustrated graphically. Theoretically, symmetry in general, stochastic dominance, quadrant dependence, tail dependence, Archimedean nature, correlation measures, and inference on the parameters are investigated. Some copula shapes are illustrated with the help of figures. On the other hand, some two-dimensional inequalities are established and may be of separate interest. Full article
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Article
Renormalization in Quantum Brain Dynamics
AppliedMath 2023, 3(1), 117-146; https://doi.org/10.3390/appliedmath3010009 - 22 Feb 2023
Viewed by 990
Abstract
We show renormalization in Quantum Brain Dynamics (QBD) in 3+1 dimensions, namely Quantum Electrodynamics with water rotational dipole fields. First, we introduce the Lagrangian density for QBD involving terms of water rotational dipole fields, photon fields and their interactions. Next, we [...] Read more.
We show renormalization in Quantum Brain Dynamics (QBD) in 3+1 dimensions, namely Quantum Electrodynamics with water rotational dipole fields. First, we introduce the Lagrangian density for QBD involving terms of water rotational dipole fields, photon fields and their interactions. Next, we show Feynman diagrams with 1-loop self-energy and vertex function in dipole coupling expansion in QBD. The counter-terms are derived from the coupling expansion of the water dipole moment. Our approach will be applied to numerical simulations of Kadanoff–Baym equations for water dipoles and photons to describe the breakdown of the rotational symmetry of dipoles, namely memory formation processes. It will also be extended to the renormalization group method for QBD with running parameters in multi-scales. Full article
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Article
A Multilevel Monte Carlo Approach for a Stochastic Optimal Control Problem Based on the Gradient Projection Method
AppliedMath 2023, 3(1), 98-116; https://doi.org/10.3390/appliedmath3010008 - 13 Feb 2023
Viewed by 825
Abstract
A multilevel Monte Carlo (MLMC) method is applied to simulate a stochastic optimal problem based on the gradient projection method. In the numerical simulation of the stochastic optimal control problem, the approximation of expected value is involved, and the MLMC method is used [...] Read more.
A multilevel Monte Carlo (MLMC) method is applied to simulate a stochastic optimal problem based on the gradient projection method. In the numerical simulation of the stochastic optimal control problem, the approximation of expected value is involved, and the MLMC method is used to address it. The computational cost of the MLMC method and the convergence analysis of the MLMC gradient projection algorithm are presented. Two numerical examples are carried out to verify the effectiveness of our method. Full article
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Article
A Fast Algorithm for the Eigenvalue Bounds of a Class of Symmetric Tridiagonal Interval Matrices
AppliedMath 2023, 3(1), 90-97; https://doi.org/10.3390/appliedmath3010007 - 03 Feb 2023
Viewed by 779
Abstract
The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we improve the theoretical results presented in a previous paper “A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices” and provide [...] Read more.
The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we improve the theoretical results presented in a previous paper “A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices” and provide a fast algorithm to find the upper and lower bounds of the interval eigenvalues of a class of symmetric tridiagonal interval matrices. Full article
Editorial
Acknowledgment to the Reviewers of AppliedMath in 2022
AppliedMath 2023, 3(1), 88-89; https://doi.org/10.3390/appliedmath3010006 - 20 Jan 2023
Viewed by 618
Abstract
High-quality academic publishing is built on rigorous peer review [...] Full article
Article
Coquaternions, Metric Invariants of Biologic Systems and Malignant Transformations
AppliedMath 2023, 3(1), 60-87; https://doi.org/10.3390/appliedmath3010005 - 16 Jan 2023
Viewed by 964
Abstract
Different hypotheses of carcinogenesis have been proposed based on local genetic factors and physiologic mechanisms. It is assumed that changes in the metric invariants of a biologic system (BS) determine the general mechanisms of cancer development. Numerous pieces of data demonstrate the existence [...] Read more.
Different hypotheses of carcinogenesis have been proposed based on local genetic factors and physiologic mechanisms. It is assumed that changes in the metric invariants of a biologic system (BS) determine the general mechanisms of cancer development. Numerous pieces of data demonstrate the existence of three invariant feedback patterns of BS: negative feedback (NFB), positive feedback (PFB) and reciprocal links (RL). These base patterns represent basis elements of a Lie algebra sl(2,R) and an imaginary part of coquaternion. Considering coquaternion as a model of a functional core of a BS, in this work a new geometric approach has been introduced. Based on this approach, conditions of the system are identified with the points of three families of hypersurfaces in R42: hyperboloids of one sheet, hyperboloids of two sheets and double cones. The obtained results also demonstrated the correspondence of an indefinite metric of coquaternion quadratic form with negative and positive entropy contributions of the base elements to the energy level of the system. From that, it can be further concluded that the anabolic states of the system will correspond to the points of a hyperboloid of one sheet, whereas catabolic conditions correspond to the points of a hyperboloid of two sheets. Equilibrium states will lie in a double cone. Physiologically anabolic and catabolic states dominate intermittently oscillating around the equilibrium. Deterioration of base elements increases positive entropy and causes domination of catabolic states, which is the main metabolic determinant of cancer. Based on these observations and the geometric representation of a BS’s behavior, it was shown that conditions related to cancer metabolic malfunction will have a tendency to remain inside the double cone. Full article
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Article
Analytical Approximation of the Jackknife Linking Error in Item Response Models Utilizing a Taylor Expansion of the Log-Likelihood Function
AppliedMath 2023, 3(1), 49-59; https://doi.org/10.3390/appliedmath3010004 - 05 Jan 2023
Viewed by 1079
Abstract
Linking errors in item response models quantify the dependence on the chosen items in means, standard deviations, or other distribution parameters. The jackknife approach is frequently employed in the computation of the linking error. However, this jackknife linking error could be computationally tedious [...] Read more.
Linking errors in item response models quantify the dependence on the chosen items in means, standard deviations, or other distribution parameters. The jackknife approach is frequently employed in the computation of the linking error. However, this jackknife linking error could be computationally tedious if many items were involved. In this article, we provide an analytical approximation of the jackknife linking error. The newly proposed approach turns out to be computationally much less demanding. Moreover, the new linking error approach performed satisfactorily for datasets with at least 20 items. Full article
Article
Effects of the Queue Discipline on System Performance
AppliedMath 2023, 3(1), 37-48; https://doi.org/10.3390/appliedmath3010003 - 03 Jan 2023
Viewed by 1283
Abstract
Queue systems are essential in the modelling of transport systems. Increasing requirements from the beneficiaries of logistic services have led to a broadening of offerings. Consequently, models need to consider transport entities with priorities being assigned in relation to the costs corresponding to [...] Read more.
Queue systems are essential in the modelling of transport systems. Increasing requirements from the beneficiaries of logistic services have led to a broadening of offerings. Consequently, models need to consider transport entities with priorities being assigned in relation to the costs corresponding to different classes of customers and/or processes. Waiting lines and queue disciplines substantially affect queue system performance. This paper aims to identify a solution for decreasing the waiting time, the total time in the system, and, overall, the cost linked to queueing delays. The influence of queue discipline on the waiting time and the total time in the system is analysed for several cases: (i) service for priority classes at the same rate of service with and without interruptions, and (ii) service for several priority classes with different service rates. The presented analysis is appropriate for increasing the performance of services dedicated to freight for two priority classes. It demonstrates how priority service can increase system performance by reducing the time in the system for customers with high costs. In addition, in the considered settings, the total time in the system is reduced for all customers, which leads to resource savings for system infrastructures. Full article
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Article
Game of Life-like Opinion Dynamics: Generalizing the Underpopulation Rule
AppliedMath 2023, 3(1), 10-36; https://doi.org/10.3390/appliedmath3010002 - 28 Dec 2022
Viewed by 817
Abstract
Graph dynamics for a node-labeled graph is a set of updating rules describing how the labels of each node in the graph change in time as a function of the global set of labels. The underpopulation rule is graph dynamics derived by simplifying [...] Read more.
Graph dynamics for a node-labeled graph is a set of updating rules describing how the labels of each node in the graph change in time as a function of the global set of labels. The underpopulation rule is graph dynamics derived by simplifying the set of rules constituting the Game of Life. It is known that the number of label configurations met by a graph during the dynamic process defined by such rule is bounded by a polynomial in the size of the graph if the graph is undirected. As a consequence, predicting the labels evolution is an easy problem (i.e., a problem in P) in such a case. In this paper, the generalization of the underpopulation rule to signed and directed graphs is studied. It is here proved that the number of label configurations met by a graph during the dynamic process defined by any so generalized underpopulation rule is still bounded by a polynomial in the size of the graph if the graph is undirected and structurally balanced, while it is not bounded by any polynomial in the size of the graph if the graph is directed although unsigned unless P = PSpace. Full article
(This article belongs to the Special Issue Feature Papers in AppliedMath)
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Article
Chaotic Behavior of the Zakharov-Kuznetsov Equation with Dual-Power Law and Triple-Power Law Nonlinearity
by and
AppliedMath 2023, 3(1), 1-9; https://doi.org/10.3390/appliedmath3010001 - 26 Dec 2022
Viewed by 972
Abstract
The main idea of this paper is to study the chaotic behavior of Zakharov–Kuznetsov equation with perturbation. By taking the traveling wave transformation, we transform the perturbed Zakharov–Kuznetsov equation with dual-power law and triple-power law nonlinearity into planar dynamic systems, and then analyze [...] Read more.
The main idea of this paper is to study the chaotic behavior of Zakharov–Kuznetsov equation with perturbation. By taking the traveling wave transformation, we transform the perturbed Zakharov–Kuznetsov equation with dual-power law and triple-power law nonlinearity into planar dynamic systems, and then analyze how the external perturbed terms affect the chaotic behavior. We emphasize here that there is no chaotic phenomenon for the non-perturbed ZK equation, thus it is only caused by the external perturbed terms. Full article
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