Mathematics

15 pages, 858 KiB

Open AccessArticle

Formulation of Pruning Maps with Rhythmic Neural Firing

by Feng-Sheng Tsai, Yi-Li Shih, Chin-Tzong Pang and Sheng-Yi Hsu

Mathematics 2019, 7(12), 1247; https://doi.org/10.3390/math7121247 - 17 Dec 2019

Viewed by 2518

Rhythmic neural firing is thought to underlie the operation of neural function. This triggers the construction of dynamical network models to investigate how the rhythms interact with each other. Recently, an approach concerning neural path pruning has been proposed in a dynamical network [...] Read more.

Rhythmic neural firing is thought to underlie the operation of neural function. This triggers the construction of dynamical network models to investigate how the rhythms interact with each other. Recently, an approach concerning neural path pruning has been proposed in a dynamical network system, in which critical neuronal connections are identified and adjusted according to the pruning maps, enabling neurons to produce rhythmic, oscillatory activity in simulation. Here, we construct a sort of homomorphic functions based on different rhythms of neural firing in network dynamics. Armed with the homomorphic functions, the pruning maps can be simply expressed in terms of interactive rhythms of neural firing and allow a concrete analysis of coupling operators to control network dynamics. Such formulation of pruning maps is applied to probe the consolidation of rhythmic patterns between layers of neurons in feedforward neural networks. Full article

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19 pages, 845 KiB

Open AccessFeature PaperArticle

A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options

by Sergey Smirnov

Mathematics 2019, 7(12), 1246; https://doi.org/10.3390/math7121246 - 17 Dec 2019

Cited by 6 | Viewed by 2814 | Correction

Abstract

This paper considers super-replication in a guaranteed deterministic problem setting with discrete time. The aim of hedging a contingent claim is to ensure the coverage of possible payoffs under the option contract for all admissible scenarios. These scenarios are given by means of [...] Read more.

This paper considers super-replication in a guaranteed deterministic problem setting with discrete time. The aim of hedging a contingent claim is to ensure the coverage of possible payoffs under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts that depend on the history of prices. The increments of the price at each moment in time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game–theoretic interpretation of pricing American options implies that the corresponding Bellman–Isaacs equations hold for both pure and mixed strategies. In the present paper, we study some properties of the least favorable (for the “hedger”) mixed strategies of the “market” and of their supports in the special case of convex payoff functions. Full article

(This article belongs to the Special Issue Mathematical Game Theory 2019)

17 pages, 496 KiB

Open AccessArticle

How Teacher’s Progressiveness in Using Digital Technologies Influences Levels of Pupils’ Metacognitive Knowledge in Mathematics

by Vlastimil Chytrý, Jaroslav Říčan and Janka Medová

Mathematics 2019, 7(12), 1245; https://doi.org/10.3390/math7121245 - 17 Dec 2019

Cited by 2 | Viewed by 3515

Abstract

The low efficiency of using appropriate strategies to solve problems in the classroom environment is not due to the lack of knowledge of how to classify concepts, but rather due to the failure to apply this knowledge strategically. Therefore, it is necessary to [...] Read more.

The low efficiency of using appropriate strategies to solve problems in the classroom environment is not due to the lack of knowledge of how to classify concepts, but rather due to the failure to apply this knowledge strategically. Therefore, it is necessary to find a balance between them, i.e., to let the pupils discuss the problems while supporting the teacher’s intervention. The aim of the presented study was to examine the influence of a teacher’s progressiveness on the level of metacognitive knowledge of the pupil. Altogether, 47 teachers and 278 pupils at grade 5 were participating in the study. It is shown that the approach of teachers to innovation itself has an influence on the pupil. When comparing all five groups of innovators, the difference among the categories of teachers was significant (

p = 0.044

) with the low effect (

d_{c o h e n} = 0.3)

. When considering only the two almost antagonistic poles of teachers, the innovators and the late majority according to Roger’s innovation diffusion theory, this influence was very strong (

p = 0.009

) and with medium effect (

d_{c o h e n} = 0.725

). Our research shows that it is necessary to address the teacher’s innovativeness, affecting the level of metacognitive knowledge of the pupil as an important prediction tool determining school success. Full article

(This article belongs to the Special Issue New trends in Mathematics Learning)

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19 pages, 590 KiB

Open AccessArticle

Fixed Point Theory for Digital k-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces

by Sang-Eon Han

Mathematics 2019, 7(12), 1244; https://doi.org/10.3390/math7121244 - 16 Dec 2019

Cited by 2 | Viewed by 2305

Abstract

The present paper studies the fixed point property (FPP) for closed k-surfaces. We also intensively study Euler characteristics of a closed k-surface and a connected sum of closed k-surfaces. Furthermore, we explore some relationships between the FPP and [...] Read more.

The present paper studies the fixed point property (FPP) for closed k-surfaces. We also intensively study Euler characteristics of a closed k-surface and a connected sum of closed k-surfaces. Furthermore, we explore some relationships between the FPP and Euler characteristics of closed k-surfaces. After explaining how to define the Euler characteristic of a closed k-surface more precisely, we confirm a certain consistency of the Euler characteristic of a closed k-surface and a continuous analog of it. In proceeding with this work, for a simple closed k-surface in

Z^{3}

, say

S_{k}

, we can see that both the minimal 26-adjacency neighborhood of a point

x \in S_{k}

, denoted by

M_{k} (x)

, and the geometric realization of it in

R^{3}

, denoted by

D_{k} (x)

, play important roles in both digital surface theory and fixed point theory. Moreover, we prove that the simple closed 18-surfaces

M S S_{18}

and

M S S_{18}^{'}

do not have the almost fixed point property (AFPP). Consequently, we conclude that the triviality or the non-triviality of the Euler characteristics of simple closed k-surfaces have no relationships with the FPP in digital topology. Using this fact, we correct many errors in many papers written by L. Boxer et al. Full article

(This article belongs to the Special Issue Fixed Point Theory and Related Nonlinear Problems with Applications)

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13 pages, 260 KiB

Open AccessArticle

Constructing Some Logical Algebras with Hoops

by M. Aaly Kologani, Seok-Zun Song, R. A. Borzooei and Young Bae Jun

Mathematics 2019, 7(12), 1243; https://doi.org/10.3390/math7121243 - 16 Dec 2019

Cited by 3 | Viewed by 3360

Abstract

In any logical algebraic structures, by using of different kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, in this paper by considering a hoop algebra or a hoop, that is introduced by Bosbach, the notion [...] Read more.

In any logical algebraic structures, by using of different kinds of filters, one can construct various kinds of other logical algebraic structures. With this inspirations, in this paper by considering a hoop algebra or a hoop, that is introduced by Bosbach, the notion of co-filter on hoops is introduced and related properties are investigated. Then by using of co-filter, a congruence relation on hoops is defined, and the associated quotient structure is studied. Thus Brouwerian semilattices, Heyting algebras, Wajsberg hoops, Hilbert algebras and BL-algebras are obtained. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics)

13 pages, 925 KiB

Open AccessFeature PaperArticle

A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional

by Hyun Geun Lee, Jaemin Shin and June-Yub Lee

Mathematics 2019, 7(12), 1242; https://doi.org/10.3390/math7121242 - 16 Dec 2019

Cited by 3 | Viewed by 2925

Abstract

Various Cahn–Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new [...] Read more.

Various Cahn–Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new convex splitting and a constrained Convex Splitting (cCS) scheme based on the splitting. We show analytically that the cCS scheme is mass conserving and satisfies the partition of unity constraint at the next time level. It is uniquely solvable and energy stable. Furthermore, we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method to develop a high-order (up to third-order) cCS scheme for the multi-component CH system. We also show analytically that the high-order cCS scheme is unconditionally energy stable. Numerical experiments with ternary and quaternary systems are presented, demonstrating the accuracy, energy stability, and capability of the proposed high-order cCS scheme. Full article

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16 pages, 502 KiB

Open AccessArticle

Exploring the Link between Academic Dishonesty and Economic Delinquency: A Partial Least Squares Path Modeling Approach

by Elena Druică, Călin Vâlsan, Rodica Ianole-Călin, Răzvan Mihail-Papuc and Irena Munteanu

Mathematics 2019, 7(12), 1241; https://doi.org/10.3390/math7121241 - 15 Dec 2019

Cited by 16 | Viewed by 4970

Abstract

This paper advances the study of the relationship between the attitude towards academic dishonesty and other types of dishonest and even fraudulent behavior, such as tax evasion and piracy. It proposes a model in which the attitudes towards two types of cheating and [...] Read more.

This paper advances the study of the relationship between the attitude towards academic dishonesty and other types of dishonest and even fraudulent behavior, such as tax evasion and piracy. It proposes a model in which the attitudes towards two types of cheating and fraud are systematically analyzed in connection with a complex set of latent construct determinants and control variables. It attempts to predict the tolerance towards tax evasion and social insurance fraud and piracy, using academic cheating as the main predictor. The proposed model surveys 504 student respondents, uses a partial least squares—path modeling analysis, and employs two subsets of latent constructs to account for context and disposition. The relationship between the outcome variable and the subset of predictors that account for context is mediated by yet another latent construct—Preoccupation about Money—that has been shown to strongly influence people’s attitude towards a whole range of social and economic behaviors. The results show academic dishonesty is a statistically significant predictor of an entire range of unethical and fraudulent behavior acceptance, and confirm the role played by both contextual and dispositional variables; moreover, they show that dispositional and contextual variables tend to be segregated according to how they impact the outcome. They also show that money priming does not act as a mediator, in spite of its stand-alone impact on the outcome variables. The most important result, however, is that the effect size of the main predictor is large. The contribution of this paper is two-fold: it advances a line of research previously sidestepped, and it proposes a comprehensive and robust model with a view to establish a hierarchy of significance and effect size in predicting deviance and fraud. Most of all, this research highlights the central role played by academic dishonesty in predicting the acceptance of any type of dishonest behavior, be it in the workplace, at home, or when discharging one’s responsibilities as a citizen. The results presented here give important clues as to where to start intervening in order to discourage the acceptance of deviance and fraud. Educators, university professors, and academic administrators should be at the forefront of targeted campaigns and policies aimed at fighting and reducing academic dishonesty. Full article

(This article belongs to the Special Issue Quantitative Methods for Economics and Finance)

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15 pages, 269 KiB

Open AccessArticle

On the Study of Fixed Points for a New Class of α-Admissible Mappings

by Mohamed Abdalla Darwish, Mohamed Jleli, Donal O’Regan and Bessem Samet

Mathematics 2019, 7(12), 1240; https://doi.org/10.3390/math7121240 - 14 Dec 2019

Cited by 1 | Viewed by 1968

Abstract

In this paper, we discuss the existence of fixed points for new classes of mappings. Some examples are presented to illustrate our results. Full article

(This article belongs to the Section Mathematics and Computer Science)

31 pages, 1249 KiB

Open AccessArticle

Moving Information Horizon Approach for Dynamic Game Models

by Ovanes Petrosian, Lihong Shi, Yin Li and Hongwei Gao

Mathematics 2019, 7(12), 1239; https://doi.org/10.3390/math7121239 - 14 Dec 2019

Cited by 7 | Viewed by 3166

Abstract

In the paper, a new class of dynamic game models with a moving information horizon or dynamic updating is studied. In this class of games, players do not have full information about the game structure (motion equations, payoff functions) on the interval on [...] Read more.

In the paper, a new class of dynamic game models with a moving information horizon or dynamic updating is studied. In this class of games, players do not have full information about the game structure (motion equations, payoff functions) on the interval on which the game is defined. It is supposed that the players at each stage of the dynamic game have only truncated information about the game structure defined by the information horizon. Cooperative and noncooperative settings are considered in the paper. Results are illustrated using the oligopoly advertising game model, and comparison between the solution in the initial game model and in the game model with moving information horizon is presented. Simulation results are presented. Full article

(This article belongs to the Special Issue Mathematical Game Theory 2019)

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15 pages, 280 KiB

Open AccessArticle

A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms

by Pablo Alegre, Joaquín Barrera and Alfonso Carriazo

Mathematics 2019, 7(12), 1238; https://doi.org/10.3390/math7121238 - 13 Dec 2019

Cited by 1 | Viewed by 2147

Abstract

The Maslov form is a closed form for a Lagrangian submanifold of

C^{m}

, and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the [...] Read more.

The Maslov form is a closed form for a Lagrangian submanifold of

C^{m}

, and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal. Full article

(This article belongs to the Special Issue Inequalities in Geometry and Applications)

36 pages, 9358 KiB

Open AccessArticle

Application of Differential Evolution Algorithm Based on Mixed Penalty Function Screening Criterion in Imbalanced Data Integration Classification

by Yuelin Gao, Kaiguang Wang, Chenyang Gao, Yulong Shen and Teng Li

Mathematics 2019, 7(12), 1237; https://doi.org/10.3390/math7121237 - 13 Dec 2019

Cited by 5 | Viewed by 2346

Abstract

There are some processing problems of imbalanced data such as imbalanced data sets being difficult to integrate efficiently. This paper proposes and constructs a mixed penalty function data integration screening criterion, and proposes Differential Evolution Integration Algorithm Based on Mixed Penalty Function Screening [...] Read more.

There are some processing problems of imbalanced data such as imbalanced data sets being difficult to integrate efficiently. This paper proposes and constructs a mixed penalty function data integration screening criterion, and proposes Differential Evolution Integration Algorithm Based on Mixed Penalty Function Screening Criteria (DE-MPFSC algorithm). In addition, the theoretical validity and the convergence of the DE-MPFSC algorithm are analyzed and proven by establishing the Markov sequence and Markov evolution process model of the DE-MPFSC algorithm. In this paper, the entanglement degree and enanglement degree error are introduced to analyze the DE-MPFSC algorithm. Finally, the effectiveness and stability of the DE-MPFSC algorithm are verified by UCI machine learning datasets. The test results show that the DE-MPFSC algorithm can effectively improve the effectiveness and application of imbalanced data classification and integration, improve the internal classification of imbalanced data and improve the efficiency of data integration. Full article

(This article belongs to the Special Issue Evolutionary Computation & Swarm Intelligence)

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25 pages, 326 KiB

Open AccessArticle

Classifying Evolution Algebras of Dimensions Two and Three

by M. Eugenia Celorrio and M. Victoria Velasco

Mathematics 2019, 7(12), 1236; https://doi.org/10.3390/math7121236 - 13 Dec 2019

Cited by 11 | Viewed by 2679

Abstract

We classified evolution algebras of dimensions two and three. Evolution algebras of dimensions three were classified recently obtaining 116 non-isomorphic types of algebras. Herein, with a new approach, we classify these algebras into 14 non-isomorphic types of algebra, so that this new classification [...] Read more.

We classified evolution algebras of dimensions two and three. Evolution algebras of dimensions three were classified recently obtaining 116 non-isomorphic types of algebras. Herein, with a new approach, we classify these algebras into 14 non-isomorphic types of algebra, so that this new classification is easier to deal with. Full article

25 pages, 2522 KiB

Open AccessArticle

Digital Supply Chain through Dynamic Inventory and Smart Contracts

by Pietro De Giovanni

Mathematics 2019, 7(12), 1235; https://doi.org/10.3390/math7121235 - 13 Dec 2019

Cited by 25 | Viewed by 5144

Abstract

This paper develops a digital supply chain game, modeling marketing and operation interactions between members. The main novelty of the paper concerns a comparison between static and dynamic solutions of the supply chain game achieved when moving from traditional to digital platforms. Therefore, [...] Read more.

This paper develops a digital supply chain game, modeling marketing and operation interactions between members. The main novelty of the paper concerns a comparison between static and dynamic solutions of the supply chain game achieved when moving from traditional to digital platforms. Therefore, this study proposes centralized and decentralized versions of the game, comparing their solutions under static and dynamic settings. Moreover, it investigates the decentralized supply chain by evaluating two smart contracts: Revenue sharing and wholesale price contracts. In both cases, the firms use an artificial intelligence system to determine the optimal contract parameters. Numerical and qualitative analyses are used for comparing configurations (centralized, decentralized), settings (static, dynamic), and contract schemes (revenue sharing contract, wholesale price contract). The findings identify the conditions under which smart revenue sharing mechanisms are worth applying. Full article

(This article belongs to the Special Issue Applications of Operational Research and Mathematical Models in Management)

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17 pages, 303 KiB

Open AccessArticle

Convergence Theorems of Variational Inequality for Asymptotically Nonexpansive Nonself Mapping in CAT(0) Spaces

by Kyung Soo Kim

Mathematics 2019, 7(12), 1234; https://doi.org/10.3390/math7121234 - 12 Dec 2019

Cited by 4 | Viewed by 1879

Abstract

The aim of this manuscript is to get the strong convergence theorems of the Moudafi’s viscosity approximation methods for an asymptotically nonexpansive nonself mapping in

C A T (0)

spaces. Full article

9 pages, 264 KiB

Open AccessArticle

Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues

by Masoumeh Farkhondeh, Mohammad Habibi, Doost Ali Mojdeh and Yongsheng Rao

Mathematics 2019, 7(12), 1233; https://doi.org/10.3390/math7121233 - 12 Dec 2019

Cited by 1 | Viewed by 2405

Abstract

If G is a graph, its Laplacian is the difference between the diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs

G_{1}

and

G_{2}

is a graph [...] Read more.

If G is a graph, its Laplacian is the difference between the diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs

G_{1}

and

G_{2}

is a graph

G = G_{1} ⊙_{u v} G_{2}

with

V (G) = V (G_{1}) \cup V (G_{2})

and

E (G) = E (G_{1}) \cup E (G_{2}) \cup {e = u v}

where

u \in V (G_{1})

and

v \in V (G_{2})

. In this paper, we study some structural conditions ensuring the presence of 2 in the Laplacian spectrum of bicyclic graphs of type

G_{1} ⊙_{u v} G_{2}

. We also provide a condition under which a bicyclic graph with a perfect matching has a Laplacian eigenvalue 2. Moreover, we characterize the broken sun graphs and the one-edge connection of two broken sun graphs by their Laplacian eigenvalue 2. Full article

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21 pages, 4213 KiB

Open AccessArticle

Cost-Based Optimum Design of Reinforced Concrete Retaining Walls Considering Different Methods of Bearing Capacity Computation

by Neda Moayyeri, Sadjad Gharehbaghi and Vagelis Plevris

Mathematics 2019, 7(12), 1232; https://doi.org/10.3390/math7121232 - 12 Dec 2019

Cited by 29 | Viewed by 20500

Abstract

This paper investigates the effect of computing the bearing capacity through different methods on the optimum construction cost of reinforced concrete retaining walls (RCRWs). Three well-known methods of Meyerhof, Hansen, and Vesic are used for the computation of the bearing capacity. In order [...] Read more.

This paper investigates the effect of computing the bearing capacity through different methods on the optimum construction cost of reinforced concrete retaining walls (RCRWs). Three well-known methods of Meyerhof, Hansen, and Vesic are used for the computation of the bearing capacity. In order to model and design the RCRWs, a code is developed in MATLAB. To reach a design with minimum construction cost, the design procedure is structured in the framework of an optimization problem in which the initial construction cost of the RCRW is the objective function to be minimized. The design criteria (both geotechnical and structural limitations) are considered constraints of the optimization problem. The geometrical dimensions of the wall and the amount of steel reinforcement are used as the design variables. To find the optimum solution, the particle swarm optimization (PSO) algorithm is employed. Three numerical examples with different wall heights are used to capture the effect of using different methods of bearing capacity on the optimal construction cost of the RCRWs. The results demonstrate that, in most cases, the final design based on the Meyerhof method corresponds to a lower construction cost. The research findings also reveal that the difference among the optimum costs of the methods is decreased by increasing the wall height. Full article

(This article belongs to the Special Issue Mathematical Physics II)

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12 pages, 250 KiB

Open AccessFeature PaperArticle

A Characterization of Polynomial Density on Curves via Matrix Algebra

by Carmen Escribano, Raquel Gonzalo and Emilio Torrano

Mathematics 2019, 7(12), 1231; https://doi.org/10.3390/math7121231 - 12 Dec 2019

Cited by 1 | Viewed by 2037

Abstract

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces

L^{2} (μ)

, with

μ

a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the [...] Read more.

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces

L^{2} (μ)

, with

μ

a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure

μ

. To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes,

γ (M)

and

λ (M)

, associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index

γ

and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index

γ

that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices. Full article

(This article belongs to the Special Issue Recent Trends on Orthogonal Polynomials: Approximation Theory and Applications)

13 pages, 348 KiB

Open AccessArticle

Fractional Integrations of a Generalized Mittag-Leffler Type Function and Its Application

by Kottakkaran Sooppy Nisar

Mathematics 2019, 7(12), 1230; https://doi.org/10.3390/math7121230 - 12 Dec 2019

Cited by 6 | Viewed by 2602

Abstract

A generalized form of the Mittag-Leffler function denoted by

_{p} E_{q; δ}^{λ, μ; ν} (x)

is established and studied in this paper. The fractional integrals involving the newly defined function are investigated. As an application, the solutions of [...] Read more.

A generalized form of the Mittag-Leffler function denoted by

_{p} E_{q; δ}^{λ, μ; ν} (x)

is established and studied in this paper. The fractional integrals involving the newly defined function are investigated. As an application, the solutions of a generalized fractional kinetic equation containing this function are derived and the nature of the solution is studied with the help of graphical analysis. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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24 pages, 493 KiB

Open AccessArticle

A Clustering System for Dynamic Data Streams Based on Metaheuristic Optimisation

by Jia Ming Yeoh, Fabio Caraffini, Elmina Homapour, Valentino Santucci and Alfredo Milani

Mathematics 2019, 7(12), 1229; https://doi.org/10.3390/math7121229 - 12 Dec 2019

Cited by 24 | Viewed by 3833

Abstract

This article presents the Optimised Stream clustering algorithm (OpStream), a novel approach to cluster dynamic data streams. The proposed system displays desirable features, such as a low number of parameters and good scalability capabilities to both high-dimensional data and numbers of clusters in [...] Read more.

This article presents the Optimised Stream clustering algorithm (OpStream), a novel approach to cluster dynamic data streams. The proposed system displays desirable features, such as a low number of parameters and good scalability capabilities to both high-dimensional data and numbers of clusters in the dataset, and it is based on a hybrid structure using deterministic clustering methods and stochastic optimisation approaches to optimally centre the clusters. Similar to other state-of-the-art methods available in the literature, it uses “microclusters” and other established techniques, such as density based clustering. Unlike other methods, it makes use of metaheuristic optimisation to maximise performances during the initialisation phase, which precedes the classic online phase. Experimental results show that OpStream outperforms the state-of-the-art methods in several cases, and it is always competitive against other comparison algorithms regardless of the chosen optimisation method. Three variants of OpStream, each coming with a different optimisation algorithm, are presented in this study. A thorough sensitive analysis is performed by using the best variant to point out OpStream’s robustness to noise and resiliency to parameter changes. Full article

(This article belongs to the Special Issue Evolutionary Computation & Swarm Intelligence)

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11 pages, 262 KiB

Open AccessArticle

Countably Expansiveness for Continuous Dynamical Systems

by Manseob Lee and Jumi Oh

Mathematics 2019, 7(12), 1228; https://doi.org/10.3390/math7121228 - 12 Dec 2019

Cited by 1 | Viewed by 1980

Abstract

Expansiveness is very closely related to the stability theory of the dynamical systems. It is natural to consider various types of expansiveness such as countably-expansive, measure expansive, N-expansive, and so on. In this article, we introduce the new concept of countably expansiveness [...] Read more.

Expansiveness is very closely related to the stability theory of the dynamical systems. It is natural to consider various types of expansiveness such as countably-expansive, measure expansive, N-expansive, and so on. In this article, we introduce the new concept of countably expansiveness for continuous dynamical systems on a compact connected smooth manifold M by using the dense set D of M, which is different from the weak expansive flows. We establish some examples having the countably expansive property, and we prove that if a vector field X of M is

C^{1}

stably countably expansive then it is quasi-Anosov. Full article

(This article belongs to the Section Mathematics and Computer Science)

13 pages, 253 KiB

Open AccessArticle

Fractions and Pythagorean Tuning—An Interdisciplinary Study in Secondary Education

by Rocío Chao-Fernández, Dorinda Mato-Vázquez and Aurelio Chao-Fernández

Mathematics 2019, 7(12), 1227; https://doi.org/10.3390/math7121227 - 12 Dec 2019

Cited by 2 | Viewed by 3425

Abstract

Formal education is experiencing a series of reforms that favor the integration of the contents of different areas in the teaching and learning of the different educational stages. The present study examined the use of an interdisciplinary music and mathematics experience in Secondary [...] Read more.

Formal education is experiencing a series of reforms that favor the integration of the contents of different areas in the teaching and learning of the different educational stages. The present study examined the use of an interdisciplinary music and mathematics experience in Secondary Education in Galicia (Spain) in the 2016/17 academic year. A descriptive–exploratory design was used, through a Likert questionnaire applied to 197 students with a diagnostic test and a reference test, and a study of multiple cases was carried out in which information was collected through classroom observations. The results show improvements in the understanding of mathematical and musical concepts, and attitudes and procedures so we can argue that the use of interdisciplinary activities have favored the development of teaching–learning opportunities in mathematical and musical training. Full article

(This article belongs to the Special Issue Mathematical Modeling and Simulation in Science and Engineering Education)

14 pages, 813 KiB

Open AccessEditor’s ChoiceArticle

Formative Transcendence of Flipped Learning in Mathematics Students of Secondary Education

by Jesús López Belmonte, Arturo Fuentes Cabrera, Juan Antonio López Núñez and Santiago Pozo Sánchez

Mathematics 2019, 7(12), 1226; https://doi.org/10.3390/math7121226 - 12 Dec 2019

Cited by 43 | Viewed by 7630

Abstract

Educational technology is achieving great potential in the formative processes of today’s society. Flipped learning is considered as a pedagogical innovation derived from the technological influence in learning spaces. The general objective of the research is to analyze the effectiveness of flipped learning [...] Read more.

Educational technology is achieving great potential in the formative processes of today’s society. Flipped learning is considered as a pedagogical innovation derived from the technological influence in learning spaces. The general objective of the research is to analyze the effectiveness of flipped learning on a traditional teaching and learning approach in the subject of Mathematics. To achieve this objective, an experimental design of a descriptive and correlational type has been followed through a quantitative research method. Two study groups have been set up. In the control group, the contents have been imparted from a traditional perspective, and in the experimental group, innovation has been applied through the use of flipped learning. The sample of participants has been chosen by means of intentional sampling and reached the figure of 60 students in the 4th year of Secondary Education at an educational center in Ceuta (Spain). A questionnaire has been used for data collection. The results reflect that the application of flipped learning has obtained better assessment in established attitudinal and mathematical indicators. It is concluded that with the use of flipped learning, motivation and skills are increased in the analysis and representation of graphs. Full article

(This article belongs to the Special Issue Futuring Mathematics Education—Core Teaching Practices to Prepare Students for the 21st Century: An International Perspective)

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18 pages, 315 KiB

Open AccessArticle

Inequalities by Means of Generalized Proportional Fractional Integral Operators with Respect to Another Function

by Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom and Yu-Ming Chu

Mathematics 2019, 7(12), 1225; https://doi.org/10.3390/math7121225 - 11 Dec 2019

Cited by 100 | Viewed by 4123

Abstract

In this article, we define a new fractional technique which is known as generalized proportional fractional (GPF) integral in the sense of another function

Ψ .

The authors prove several inequalities for newly defined GPF-integral with respect to another function

Ψ .

Our [...] Read more.

In this article, we define a new fractional technique which is known as generalized proportional fractional (GPF) integral in the sense of another function

Ψ .

The authors prove several inequalities for newly defined GPF-integral with respect to another function

Ψ .

Our consequences will give noted outcomes for a suitable variation to the GPF-integral in the sense of another function

Ψ

and the proportionality index

ς

. Furthermore, we present the application of the novel operator with several integral inequalities. A few new properties are exhibited, and the numerical approximation of these new operators is introduced with certain utilities to real-world problems. Full article

15 pages, 309 KiB

Open AccessArticle

New Generalized Mizoguchi-Takahashi’s Fixed Point Theorems for Essential Distances and e⁰-Metrics

by Binghua Jiang, Huaping Huang and Wei-Shih Du

Mathematics 2019, 7(12), 1224; https://doi.org/10.3390/math7121224 - 11 Dec 2019

Cited by 2 | Viewed by 2233

Abstract

In this paper, we present some new generalizations of Mizoguchi-Takahashi’s fixed point theorem which also improve and extend Du-Hung’s fixed point theorem. Some new examples illustrating our results are also given. By applying our new results, some new fixed point theorems for essential [...] Read more.

In this paper, we present some new generalizations of Mizoguchi-Takahashi’s fixed point theorem which also improve and extend Du-Hung’s fixed point theorem. Some new examples illustrating our results are also given. By applying our new results, some new fixed point theorems for essential distances and e⁰-metrics were established. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)

16 pages, 60309 KiB

Open AccessArticle

A Numerical Study on the Crack Development Behavior of Rock-Like Material Containing Two Intersecting Flaws

by Bing Dai, Ying Chen, Guoyan Zhao, Weizhang Liang and Hao Wu

Mathematics 2019, 7(12), 1223; https://doi.org/10.3390/math7121223 - 11 Dec 2019

Cited by 12 | Viewed by 3346

Abstract

It is quite often that rocks contain intersecting cracks. Therefore, crack behavior cannot be completely studied by only considering several isolated, single flaws. To investigate the crack behavior of rock or rock-like material containing intersecting flaws under uniaxial loading, numerical simulations were carried [...] Read more.

It is quite often that rocks contain intersecting cracks. Therefore, crack behavior cannot be completely studied by only considering several isolated, single flaws. To investigate the crack behavior of rock or rock-like material containing intersecting flaws under uniaxial loading, numerical simulations were carried out using parallel bonded-particle models containing two intersecting flaws with different inclination angles (varying β) and different intersection angles (varying αα). The crack propagation processes are analyzed and two typical patterns of linkage are observed between two intersecting flaws: (1) One-tip-linkage that contains three subtypes: Coalescence position near the tip; coalescence position at the flaw, but far away from the tip; coalescence position outside the flaw at a certain distance from the tip; and (2) two-tip-linkage with two subtypes: Straight linkage and arc linkage. The geometries of flaws influence the coalescence type. Moreover, the effects of intersection angle α and inclination angle β on the peak stress, the stress of crack initiation, and the stress of crack coalescence are also investigated in detail. Full article

(This article belongs to the Special Issue Mathematical Physics II)

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14 pages, 273 KiB

Open AccessArticle

Topological Properties of E-Metric Spaces with Applications to Fixed Point Theory

by Huaping Huang

Mathematics 2019, 7(12), 1222; https://doi.org/10.3390/math7121222 - 11 Dec 2019

Cited by 10 | Viewed by 3197

Abstract

The purpose of this paper is to present some topological properties in E-metric spaces such as the properties of e-sequences, the decision conditions of e-Cauchy sequences, the characteristics of non-normal cones, and so on. Moreover, the theorem of nested closed-balls [...] Read more.

The purpose of this paper is to present some topological properties in E-metric spaces such as the properties of e-sequences, the decision conditions of e-Cauchy sequences, the characteristics of non-normal cones, and so on. Moreover, the theorem of nested closed-balls in such spaces is displayed. In addition, some principal applications to fixed point theory are also given. Full article

(This article belongs to the Special Issue Fixed Point Theory and Dynamical Systems with Applications)

14 pages, 416 KiB

Open AccessArticle

A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems

by Raudys R. Capdevila, Alicia Cordero and Juan R. Torregrosa

Mathematics 2019, 7(12), 1221; https://doi.org/10.3390/math7121221 - 11 Dec 2019

Cited by 7 | Viewed by 2714

Abstract

In this work, a new class of iterative methods for solving nonlinear equations is presented and also its extension for nonlinear systems of equations. This family is developed by using a scalar and matrix weight function procedure, respectively, getting sixth-order of convergence in [...] Read more.

In this work, a new class of iterative methods for solving nonlinear equations is presented and also its extension for nonlinear systems of equations. This family is developed by using a scalar and matrix weight function procedure, respectively, getting sixth-order of convergence in both cases. Several numerical examples are given to illustrate the efficiency and performance of the proposed methods. Full article

(This article belongs to the Special Issue Selected Papers from Iterative Processes for Solving Nonlinear Problems: Convergence and Stability of ICIAM 2019 and MME&HB 2019)

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2 pages, 195 KiB

Open AccessCorrection

Correction: Alsaedi, A., et al. Existence and Stability Results for a Fractional Order Differential Equation with Non-Conjugate Riemann–Stieltjes Integro-Multipoint Boundary Conditions. Mathematics 2019, 7, 249

by Bashir Ahmad, Ymnah Alruwaily, Ahmed Alsaedi and Sotiris K. Ntouyas

Mathematics 2019, 7(12), 1220; https://doi.org/10.3390/math7121220 - 10 Dec 2019

Viewed by 1695

Abstract

In [1], the authors wish to make the following corrections [...] Full article

13 pages, 948 KiB

Open AccessFeature PaperArticle

The Enhancement of Academic Performance in Online Environments

by Francisco I. Chicharro, Elena Giménez and Íñigo Sarría

Mathematics 2019, 7(12), 1219; https://doi.org/10.3390/math7121219 - 10 Dec 2019

Cited by 5 | Viewed by 6605

Abstract

Distance education has been gaining popularity for the last years. The proficiency in online environments of both learners and teachers explains the success of this methodology. An evaluation of graduate students’ performance within numerical analysis is discussed. In order to improve the marks [...] Read more.

Distance education has been gaining popularity for the last years. The proficiency in online environments of both learners and teachers explains the success of this methodology. An evaluation of graduate students’ performance within numerical analysis is discussed. In order to improve the marks obtained by the students, specific actions have been performed over the years and data from different classes has been analyzed using statistical tools. The results show that the actions proposed seemed to help the students in their learning process. Full article

(This article belongs to the Special Issue New trends in Mathematics Learning)

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21 pages, 3022 KiB

Open AccessArticle

Periodic Solutions for a Four-Dimensional Coupled Polynomial System with N-Degree Homogeneous Nonlinearities

by Yuanyuan Tian and Jing Li

Mathematics 2019, 7(12), 1218; https://doi.org/10.3390/math7121218 - 10 Dec 2019

Cited by 1 | Viewed by 1977

Abstract

This paper studies the periodic solutions of a four-dimensional coupled polynomial system with N-degree homogeneous nonlinearities of which the unperturbed linear system has a center singular point in generalization resonance

1 : n

at the origin. Considering arbitrary positive integers

n

and [...] Read more.

This paper studies the periodic solutions of a four-dimensional coupled polynomial system with N-degree homogeneous nonlinearities of which the unperturbed linear system has a center singular point in generalization resonance

1 : n

at the origin. Considering arbitrary positive integers

n

and

N

with

n \leq N

and

N \geq 2

, the new explicit expression of displacement function for the four-dimensional system is detected by introducing the technique on power trigonometric integrals. Then some precise and detailed results in comparison with the existing works, including the existence condition, the exact number, and the parameter control conditions of periodic solutions, are obtained, which can provide a new theoretical description and mechanism explanation for the phenomena of emergence and disappearance of periodic solutions. Results obtained in this paper improve certain existing results under some parameter conditions and can be extensively used in engineering applications. To verify the applicability and availability of the new theoretical results, as an application, the periodic solutions of a circular mesh antenna model are obtained by theoretical method and numerical simulations. Full article

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Mathematics, Volume 7, Issue 12 (December 2019) – 114 articles

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