Mathematics

20 pages, 932 KiB

Open AccessArticle

Some Reverse Degree-Based Topological Indices and Polynomials of Dendrimers

by Wei Gao, Muhammad Younas, Adeel Farooq, Abaid Ur Rehman Virk and Waqas Nazeer

Mathematics 2018, 6(10), 214; https://doi.org/10.3390/math6100214 - 22 Oct 2018

Cited by 46 | Viewed by 4356

Topological indices collect information from the graph of molecule and help to predict properties of the underlying molecule. Zagreb indices are among the most studied topological indices due to their applications in chemistry. In this paper, we compute first and second reverse Zagreb [...] Read more.

Topological indices collect information from the graph of molecule and help to predict properties of the underlying molecule. Zagreb indices are among the most studied topological indices due to their applications in chemistry. In this paper, we compute first and second reverse Zagreb indices, reverse hyper-Zagreb indices and their polynomials of Prophyrin, Propyl ether imine, Zinc Porphyrin and Poly (ethylene amido amine) dendrimers. Full article

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16 pages, 3678 KiB

Open AccessArticle

Target Fusion Detection of LiDAR and Camera Based on the Improved YOLO Algorithm

by Jian Han, Yaping Liao, Junyou Zhang, Shufeng Wang and Sixian Li

Mathematics 2018, 6(10), 213; https://doi.org/10.3390/math6100213 - 19 Oct 2018

Cited by 39 | Viewed by 7972

Abstract

Target detection plays a key role in the safe driving of autonomous vehicles. At present, most studies use single sensor to collect obstacle information, but single sensor cannot deal with the complex urban road environment, and the rate of missed detection is high. [...] Read more.

Target detection plays a key role in the safe driving of autonomous vehicles. At present, most studies use single sensor to collect obstacle information, but single sensor cannot deal with the complex urban road environment, and the rate of missed detection is high. Therefore, this paper presents a detection fusion system with integrating LiDAR and color camera. Based on the original You Only Look Once (YOLO) algorithm, the second detection scheme is proposed to improve the YOLO algorithm for dim targets such as non-motorized vehicles and pedestrians. Many image samples are used to train the YOLO algorithm to obtain the relevant parameters and establish the target detection model. Then, the decision level fusion of sensors is introduced to fuse the color image and the depth image to improve the accuracy of the target detection. Finally, the test samples are used to verify the decision level fusion. The results show that the improved YOLO algorithm and decision level fusion have high accuracy of target detection, can meet the need of real-time, and can reduce the rate of missed detection of dim targets such as non-motor vehicles and pedestrians. Thus, the method in this paper, under the premise of considering accuracy and real-time, has better performance and larger application prospect. Full article

(This article belongs to the Special Issue Mathematics and Engineering)

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Graphical abstract

10 pages, 368 KiB

Open AccessArticle

The Space–Time Kernel-Based Numerical Method for Burgers’ Equations

by Marjan Uddin and Hazrat Ali

Mathematics 2018, 6(10), 212; https://doi.org/10.3390/math6100212 - 18 Oct 2018

Cited by 14 | Viewed by 3479

Abstract

It is well known that major error occur in the time integration instead of the spatial approximation. In this work, anisotropic kernels are used for temporal as well as spatial approximation to construct a numerical scheme for solving nonlinear Burgers’ equations. The time-dependent [...] Read more.

It is well known that major error occur in the time integration instead of the spatial approximation. In this work, anisotropic kernels are used for temporal as well as spatial approximation to construct a numerical scheme for solving nonlinear Burgers’ equations. The time-dependent PDEs are collocated in both space and time first, contrary to spatial discretization, and time stepping procedures for time integration are then applied. Physically one cannot in general expect that the spatial and temporal features of the solution behaves on the same order. Hence, one should have to incorporate anisotropic kernels. The nonlinear Burgers’ equations are converted by nonlinear transformation to linear equations. The spatial discretizations are carried out to construct differentiation matrices. Comparisons with most available numerical methods are made to solve the Burgers’ equations. Full article

(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)

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17 pages, 3375 KiB

Open AccessArticle

The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

by Haoyu Dong, Changna Lu and Hongwei Yang

Mathematics 2018, 6(10), 211; https://doi.org/10.3390/math6100211 - 18 Oct 2018

Cited by 7 | Viewed by 4200

Abstract

We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can [...] Read more.

We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes. Full article

(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)

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14 pages, 245 KiB

Open AccessArticle

Expressing Sums of Finite Products of Chebyshev Polynomials of the Second Kind and of Fibonacci Polynomials by Several Orthogonal Polynomials

by Taekyun Kim, Dae San Kim, Jongkyum Kwon and Dmitry V. Dolgy

Mathematics 2018, 6(10), 210; https://doi.org/10.3390/math6100210 - 17 Oct 2018

Cited by 17 | Viewed by 3119

Abstract

This paper is concerned with representing sums of the finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials in terms of several classical orthogonal polynomials. Indeed, by explicit computations, each of them is expressed as linear combinations of Hermite, [...] Read more.

This paper is concerned with representing sums of the finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials in terms of several classical orthogonal polynomials. Indeed, by explicit computations, each of them is expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, which involve the hypergeometric functions

_{1} F_{1}

and

_{2} F_{1}

. Full article

26 pages, 374 KiB

Open AccessArticle

Prioritized Linguistic Interval-Valued Aggregation Operators and Their Applications in Group Decision-Making Problems

by Kamal Kumar and Harish Garg

Mathematics 2018, 6(10), 209; https://doi.org/10.3390/math6100209 - 17 Oct 2018

Cited by 17 | Viewed by 2660

Abstract

The linguistic interval-valued intuitionistic fuzzy (LIVIF) set is an efficient tool to represent data in the form of interval membership degrees in a qualitative rather than a quantitative manner. The LIVIF set combines the features of interval-valued intuitionistic fuzzy sets (IFSs) and the [...] Read more.

The linguistic interval-valued intuitionistic fuzzy (LIVIF) set is an efficient tool to represent data in the form of interval membership degrees in a qualitative rather than a quantitative manner. The LIVIF set combines the features of interval-valued intuitionistic fuzzy sets (IFSs) and the linguistic variables (LV) and hence provides more freedom to decision-makers. Under this environment, the main objective of this manuscript is to propose some new aggregation operators by capturing the prioritized relationship between the objects. For this, different weighted averaging and geometric aggregation operators are proposed in which preferences related to each object are expressed in terms of LIVIF numbers. Desirable properties of the proposed operators are studied. Further, a group decision-making (DM) approach is presented to solve the multi-attribute DM problems, and its efficiency has been verified with an illustrative example. Full article

(This article belongs to the Special Issue Nonlinear Analysis Using Fuzzy Mathematics)

19 pages, 291 KiB

Open AccessArticle

Fixed Point Results on Δ-Symmetric Quasi-Metric Space via Simulation Function with an Application to Ulam Stability

by Badr Alqahtani, Andreea Fulga and Erdal Karapınar

Mathematics 2018, 6(10), 208; https://doi.org/10.3390/math6100208 - 17 Oct 2018

Cited by 45 | Viewed by 3195

Abstract

In this paper, in the setting of

Δ

-symmetric quasi-metric spaces, the existence and uniqueness of a fixed point of certain operators are scrutinized carefully by using simulation functions. The most interesting side of such operators is that they do not form a [...] Read more.

In this paper, in the setting of

Δ

-symmetric quasi-metric spaces, the existence and uniqueness of a fixed point of certain operators are scrutinized carefully by using simulation functions. The most interesting side of such operators is that they do not form a contraction. As an application, in the same framework, the Ulam stability of such operators is investigated. We also propose some examples to illustrate our results. Full article

(This article belongs to the Special Issue Stability Problems)

13 pages, 294 KiB

Open AccessArticle

Some Results on the Deficiencies of Some Differential-Difference Polynomials of Meromorphic Function

by Hong-Yan Xu, Xiu-Min Zheng and Hua Wang

Mathematics 2018, 6(10), 207; https://doi.org/10.3390/math6100207 - 16 Oct 2018

Viewed by 2275

Abstract

For a transcendental meromorphic function

f (z)

, the main aim of this paper is to investigate the properties on the zeros and deficiencies of some differential-difference polynomials. Some results about the deficiencies of some differential-difference polynomials concerning Nevanlinna deficiency and [...] Read more.

For a transcendental meromorphic function

f (z)

, the main aim of this paper is to investigate the properties on the zeros and deficiencies of some differential-difference polynomials. Some results about the deficiencies of some differential-difference polynomials concerning Nevanlinna deficiency and Valiron deficiency are obtained, which are a generalization of and improvement on previous theorems given by Liu, Lan and Zheng, etc. Full article

11 pages, 460 KiB

Open AccessArticle

The Double Roman Domination Numbers of Generalized Petersen Graphs P(n, 2)

by Huiqin Jiang, Pu Wu, Zehui Shao, Yongsheng Rao and Jia-Bao Liu

Mathematics 2018, 6(10), 206; https://doi.org/10.3390/math6100206 - 16 Oct 2018

Cited by 13 | Viewed by 3042

Abstract

A double Roman dominating function (DRDF) f on a given graph G is a mapping from

V (G)

to

{0, 1, 2, 3}

in such a way that a vertex u for which [...] Read more.

A double Roman dominating function (DRDF) f on a given graph G is a mapping from

V (G)

to

{0, 1, 2, 3}

in such a way that a vertex u for which

f (u) = 0

has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which

f (u) = 1

has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value

w (f) = \sum_{u \in V (G)} f (u)

. The minimum weight of a DRDF on a graph G is called the double Roman domination number

γ_{d R} (G)

of G. In this paper, we determine the exact value of the double Roman domination number of the generalized Petersen graphs

P (n, 2)

by using a discharging approach. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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18 pages, 2146 KiB

Open AccessArticle

Modeling Mindsets with Kalman Filter

by Takashi Yamauchi

Mathematics 2018, 6(10), 205; https://doi.org/10.3390/math6100205 - 16 Oct 2018

Cited by 3 | Viewed by 3626

Abstract

Mathematical models have played an essential role in interface design. This study focused on “mindsets”—people’s tacit beliefs about attributes—and investigated the extent to which: (1) mindsets can be extracted in a motion trajectory in target selection, and (2) a dynamic state-space model, such [...] Read more.

Mathematical models have played an essential role in interface design. This study focused on “mindsets”—people’s tacit beliefs about attributes—and investigated the extent to which: (1) mindsets can be extracted in a motion trajectory in target selection, and (2) a dynamic state-space model, such as the Kalman filter, helps quantify mindsets. Participants were experimentally manipulated to hold fixed or growth mindsets in a “mock” memory test, and later performed a concept-learning task in which the movement of the computer cursor was recorded in every trial. By inspecting motion trajectories of the cursor, we observed clear disparities in the impact of mindsets; participants who were induced with a fixed mindset moved the cursor faster as compared to those who were induced with a growth mindset. To examine further the mechanism of this influence, we fitted a Kalman filter model to the trajectory data; we found that system-level error-covariance in the Kalman filter model could effectively separate motion trajectories gleaned from the two mindset conditions. Taken together, results from the experiment suggest that people’s mindsets can be captured in motor trajectories in target selection and the Kalman filter helps quantify mindsets. It is argued that people’s personality, attitude, and mindset are embodied in motor behavior underlying target selection and these psychological variables can be studied mathematically with a feedback control system. Full article

(This article belongs to the Special Issue Human-Computer Interaction: New Horizons)

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13 pages, 590 KiB

Open AccessFeature PaperArticle

Stochastic Comparisons and Dynamic Information of Random Lifetimes in a Replacement Model

by Antonio Di Crescenzo and Patrizia Di Gironimo

Mathematics 2018, 6(10), 204; https://doi.org/10.3390/math6100204 - 16 Oct 2018

Cited by 6 | Viewed by 2886

Abstract

We consider a suitable replacement model for random lifetimes, in which at a fixed time an item is replaced by another one having the same age but different lifetime distribution. We focus first on stochastic comparisons between the involved random lifetimes, in order [...] Read more.

We consider a suitable replacement model for random lifetimes, in which at a fixed time an item is replaced by another one having the same age but different lifetime distribution. We focus first on stochastic comparisons between the involved random lifetimes, in order to assess conditions leading to an improvement of the system. Attention is also given to the relative ratio of improvement, which is proposed as a suitable index finalized to measure the goodness of the replacement procedure. Finally, we provide various results on the dynamic differential entropy of the lifetime of the improved system. Full article

(This article belongs to the Special Issue Stochastic Processes with Applications)

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25 pages, 1760 KiB

Open AccessArticle

High-Order Finite-Element Framework for the Efficient Simulation of Multifluid Flows

by Thibaut Metivet, Vincent Chabannes, Mourad Ismail and Christophe Prud’homme

Mathematics 2018, 6(10), 203; https://doi.org/10.3390/math6100203 - 15 Oct 2018

Cited by 5 | Viewed by 3241

Abstract

In this paper, we present a comprehensive framework for the simulation of Multifluid flows based on the implicit level-set representation of interfaces and on an efficient solving strategy of the Navier-Stokes equations. The mathematical framework relies on a modular coupling approach between the [...] Read more.

In this paper, we present a comprehensive framework for the simulation of Multifluid flows based on the implicit level-set representation of interfaces and on an efficient solving strategy of the Navier-Stokes equations. The mathematical framework relies on a modular coupling approach between the level-set advection and the fluid equations. The space discretization is performed with possibly high-order stable finite elements while the time discretization features implicit Backward Differentation Formulae of arbitrary order. This framework has been implemented within the Feel++ library, and features seamless distributed parallelism with fast assembly procedures for the algebraic systems and efficient preconditioning strategies for their resolution. We also present simulation results for a three-dimensional Multifluid benchmark, and highlight the importance of using high-order finite elements for the level-set discretization for problems involving the geometry of the interfaces, such as the curvature or its derivatives. Full article

(This article belongs to the Special Issue Modern Finite Element Methods)

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14 pages, 2291 KiB

Open AccessArticle

The Opening Capability for Security against Privacy Infringements in the Smart Grid Environment

by Sungwook Eom and Jun-Ho Huh

Mathematics 2018, 6(10), 202; https://doi.org/10.3390/math6100202 - 14 Oct 2018

Cited by 10 | Viewed by 3526

Abstract

It is now known that more information can be leaked into the smart grid environment than into the existing environment. In particular, specific information such as energy consumption data can be exposed via smart devices. Such a phenomenon can incur considerable risks due [...] Read more.

It is now known that more information can be leaked into the smart grid environment than into the existing environment. In particular, specific information such as energy consumption data can be exposed via smart devices. Such a phenomenon can incur considerable risks due to the fact that both the amount and the concreteness of information increase when more types of information are combined. As such, this study aimed to develop an anonymous signature technique along with a signature authentication technique to prevent infringements of privacy in the smart grid environment, and they were tested and verified at the testbed used in a previous study. To reinforce the security of the smart grid, a password and anonymous authentication algorithm which can be applied not only to extendable test sites but also to power plants, including nuclear power stations, was developed. The group signature scheme is an anonymous signature schemes where the authenticator verifies the group signature to determine whether the signer is a member of a certain group but he/she would not know which member actually signed in. However, in this scheme, the identity of the signer can be revealed through an “opener” in special circumstances involving accidents, incidents, or disputes. Since the opener can always identify the signer without his/her consent in such cases, the signer would be concerned about letting the opener find out his/her anonymous activities. Thus, an anonymous signature scheme where the signer issues a token when entering his/her signature to allow the opener to confirm his/her identity only from that token is proposed in this study. Full article

(This article belongs to the Special Issue Mathematics and Engineering)

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45 pages, 436 KiB

Open AccessArticle

Approaches to Multiple Attribute Decision Making with Interval-Valued 2-Tuple Linguistic Pythagorean Fuzzy Information

by Jie Wang, Guiwu Wei and Hui Gao

Mathematics 2018, 6(10), 201; https://doi.org/10.3390/math6100201 - 13 Oct 2018

Cited by 76 | Viewed by 4133

Abstract

The Maclaurin symmetric mean (MSM) operator is a classical mean type aggregation operator used in modern information fusion theory, which is suitable to aggregate numerical values. The prominent characteristic of the MSM operator is that it can capture the interrelationship among multi-input arguments. [...] Read more.

The Maclaurin symmetric mean (MSM) operator is a classical mean type aggregation operator used in modern information fusion theory, which is suitable to aggregate numerical values. The prominent characteristic of the MSM operator is that it can capture the interrelationship among multi-input arguments. Motivated by the ideal characteristic of the MSM operator, in this paper, we expand the MSM operator, generalized MSM (GMSM), and dual MSM (DMSM) operator with interval-valued 2-tuple linguistic Pythagorean fuzzy numbers (IV2TLPFNs) to propose the interval-valued 2-tuple linguistic Pythagorean fuzzy MSM (IV2TLPFMSM) operator, interval-valued 2-tuple linguistic Pythagorean fuzzy weighted MSM (IV2TLPFWMSM) operator, interval-valued 2-tuple linguistic Pythagorean fuzzy GMSM (IN2TLPFGMSM) operator, interval-valued 2-tuple linguistic Pythagorean fuzzy weighted GMSM (IV2TLPFWGMSM) operator, interval-valued 2-tuple linguistic Pythagorean fuzzy DMSM (IN2TLPFDMSM) operator, Interval-valued 2-tuple linguistic Pythagorean fuzzy weighted DMSM (IV2TLPFWDMSM) operator. Then the multiple attribute decision making (MADM) methods are developed with these three operators. Finally, an example of green supplier selection is used to show the proposed methods. Full article

(This article belongs to the Special Issue Recent Advances in Game and Decision Theory: Structures, Models, Applications and Software Implementation)

17 pages, 381 KiB

Open AccessArticle

High-Order Energy and Linear Momentum Conserving Methods for the Klein-Gordon Equation

by He Yang

Mathematics 2018, 6(10), 200; https://doi.org/10.3390/math6100200 - 12 Oct 2018

Cited by 4 | Viewed by 2622

Abstract

The Klein-Gordon equation is a model for free particle wave function in relativistic quantum mechanics. Many numerical methods have been proposed to solve the Klein-Gordon equation. However, efficient high-order numerical methods that preserve energy and linear momentum of the equation have not been [...] Read more.

The Klein-Gordon equation is a model for free particle wave function in relativistic quantum mechanics. Many numerical methods have been proposed to solve the Klein-Gordon equation. However, efficient high-order numerical methods that preserve energy and linear momentum of the equation have not been considered. In this paper, we propose high-order numerical methods to solve the Klein-Gordon equation, present the energy and linear momentum conservation properties of our numerical schemes, and show the optimal error estimates and superconvergence property. We also verify the performance of our numerical schemes by some numerical examples. Full article

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16 pages, 2142 KiB

Open AccessArticle

Rheological Behavior of Rockmass Supported with Rockbolts Based on Viscoelastic Analysis Method

by Gang Wang, Wei Han, Chuanzheng Liu, Ke Wang and Hengjie Luan

Mathematics 2018, 6(10), 199; https://doi.org/10.3390/math6100199 - 11 Oct 2018

Cited by 1 | Viewed by 2608

Abstract

In this paper, we present the viscoelastic solutions for rockmass supported with discretely mechanically or frictionally coupled (DMFC) rockbolts to reveal the coupling rheological mechanisms. The analytical solutions are first acquired by applying the Laplace inverse transforms. The effect of different viscosity coefficients [...] Read more.

In this paper, we present the viscoelastic solutions for rockmass supported with discretely mechanically or frictionally coupled (DMFC) rockbolts to reveal the coupling rheological mechanisms. The analytical solutions are first acquired by applying the Laplace inverse transforms. The effect of different viscosity coefficients and supporting parameters on the coupling model rheological behavior are then investigated. It is concluded that the variation of the rockbolt axial force or rock mass stress and displacement have a close relationship with rheological parameters and support parameters. In addition, the variations of mechanical states of rockbolts and rock mass are closely related to the rheological model. Full article

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12 pages, 245 KiB

Open AccessArticle

System of Extended General Variational Inequalities for Relaxed Cocoercive Mappings in Hilbert Space

by Kyung Soo Kim

Mathematics 2018, 6(10), 198; https://doi.org/10.3390/math6100198 - 11 Oct 2018

Cited by 4 | Viewed by 2126

Abstract

In this manuscript, we study a system of extended general variational inequalities (SEGVI) with several nonlinear operators, more precisely, six relaxed

(α, r)

-cocoercive mappings. Using the projection method, we show that a system of extended general variational inequalities is [...] Read more.

In this manuscript, we study a system of extended general variational inequalities (SEGVI) with several nonlinear operators, more precisely, six relaxed

(α, r)

-cocoercive mappings. Using the projection method, we show that a system of extended general variational inequalities is equivalent to the nonlinear projection equations. This alternative equivalent problem is used to consider the existence and convergence (or approximate solvability) of a solution of a system of extended general variational inequalities under suitable conditions. Full article

17 pages, 552 KiB

Open AccessFeature PaperArticle

Network Reliability Modeling Based on a Geometric Counting Process

by Somayeh Zarezadeh, Somayeh Ashrafi and Majid Asadi

Mathematics 2018, 6(10), 197; https://doi.org/10.3390/math6100197 - 11 Oct 2018

Cited by 7 | Viewed by 3250

Abstract

In this paper, we investigate the reliability and stochastic properties of an n-component network under the assumption that the components of the network fail according to a counting process called a geometric counting process (GCP). The paper has two parts. In the [...] Read more.

In this paper, we investigate the reliability and stochastic properties of an n-component network under the assumption that the components of the network fail according to a counting process called a geometric counting process (GCP). The paper has two parts. In the first part, we consider a two-state network (with states up and down) and we assume that its components are subjected to failure based on a GCP. Some mixture representations for the network reliability are obtained in terms of signature of the network and the reliability function of the arrival times of the GCP. Several aging and stochastic properties of the network are investigated. The reliabilities of two different networks subjected to the same or different GCPs are compared based on the stochastic order between their signature vectors. The residual lifetime of the network is also assessed where the components fail based on a GCP. The second part of the paper is concerned with three-state networks. We consider a network made up of n components which starts operating at time

t = 0

. It is assumed that, at any time

t > 0

, the network can be in one of three states up, partial performance or down. The components of the network are subjected to failure on the basis of a GCP, which leads to change of network states. Under these scenarios, we obtain several stochastic and dependency characteristics of the network lifetime. Some illustrative examples and plots are also provided throughout the article. Full article

(This article belongs to the Special Issue Stochastic Processes with Applications)

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11 pages, 1078 KiB

Open AccessArticle

Adaptive System for Steering a Ship Along the Desired Route

by Piotr Borkowski

Mathematics 2018, 6(10), 196; https://doi.org/10.3390/math6100196 - 10 Oct 2018

Cited by 17 | Viewed by 3604

Abstract

An adaptive ship steering system along a preset track is an example of an intelligent system. An optimal linear quadratic regulator (LQR) regulator with a symmetric indicator of control quality was adopted as the control algorithm. The model identification was based on the [...] Read more.

An adaptive ship steering system along a preset track is an example of an intelligent system. An optimal linear quadratic regulator (LQR) regulator with a symmetric indicator of control quality was adopted as the control algorithm. The model identification was based on the continuous version of the least squares method. A significant part of the article presents the proof of the stability of the proposed system. The results of the calculation experiments are provided to confirm the effective and correct working of the system. Full article

(This article belongs to the Special Issue Mathematics on Automation Control Systems)

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9 pages, 245 KiB

Open AccessFeature PaperArticle

Some Results on S-Contractions of Type E

by Andreea Fulga and Erdal Karapınar

Mathematics 2018, 6(10), 195; https://doi.org/10.3390/math6100195 - 9 Oct 2018

Cited by 6 | Viewed by 2431

Abstract

In this manuscript, we consider the compositions of simulation functions and E-contraction in the setting of a complete metric space. We investigate the existence and uniqueness of a fixed point for this composite form. We give some illustrative examples and provide an [...] Read more.

In this manuscript, we consider the compositions of simulation functions and E-contraction in the setting of a complete metric space. We investigate the existence and uniqueness of a fixed point for this composite form. We give some illustrative examples and provide an application. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point Theory and Its Applications)

7 pages, 239 KiB

Open AccessArticle

Controlled Metric Type Spaces and the Related Contraction Principle

by Nabil Mlaiki, Hassen Aydi, Nizar Souayah and Thabet Abdeljawad

Mathematics 2018, 6(10), 194; https://doi.org/10.3390/math6100194 - 8 Oct 2018

Cited by 175 | Viewed by 7114

Abstract

In this article, we introduce a new extension of b-metric spaces, called controlled metric type spaces, by employing a control function

α (x, y)

of the right-hand side of the b-triangle inequality. Namely, the triangle inequality in the [...] Read more.

In this article, we introduce a new extension of b-metric spaces, called controlled metric type spaces, by employing a control function

α (x, y)

of the right-hand side of the b-triangle inequality. Namely, the triangle inequality in the new defined extension will have the form,

d (x, y) \leq α (x, z) d (x, z) + α (z, y) d (z, y), for all x, y, z \in X .

Examples of controlled metric type spaces that are not extended b-metric spaces in the sense of Kamran et al. are given to show that our extension is different. A Banach contraction principle on controlled metric type spaces and an example are given to illustrate the usefulness of the structure of the new extension. Full article

(This article belongs to the Special Issue Recent Advances in Fixed Point Theory and Its Applications)

9 pages, 232 KiB

Open AccessArticle

On Some Inequalities Involving Liouville–Caputo Fractional Derivatives and Applications to Special Means of Real Numbers

by Bessem Samet and Hassen Aydi

Mathematics 2018, 6(10), 193; https://doi.org/10.3390/math6100193 - 8 Oct 2018

Cited by 16 | Viewed by 2480

Abstract

We are concerned with the class of functions

f \in C^{1} ([a, b]; R)

,

a, b \in R

,

a < b

, such that

|^{c} D_{a}^{α} f|

is convex or [...] Read more.

We are concerned with the class of functions

f \in C^{1} ([a, b]; R)

,

a, b \in R

,

a < b

, such that

|^{c} D_{a}^{α} f|

is convex or

|^{c} D_{b}^{α} f|

is convex, where

0 < α < 1

,

^{c} D_{a}^{α} f

is the left-side Liouville–Caputo fractional derivative of order

α

of f and

^{c} D_{b}^{α} f

is the right-side Liouville–Caputo fractional derivative of order

α

of f. Some extensions of Dragomir–Agarwal inequality to this class of functions are obtained. A parallel development is made for the class of functions

f \in C^{1} ([a, b]; R)

such that

|^{c} D_{a}^{α} f|

is concave or

|^{c} D_{b}^{α} f|

is concave. Next, an application to special means of real numbers is provided. Full article

13 pages, 341 KiB

Open AccessArticle

Tunneling Time in Attosecond Experiments and Time Operator in Quantum Mechanics

by Ossama Kullie

Mathematics 2018, 6(10), 192; https://doi.org/10.3390/math6100192 - 8 Oct 2018

Cited by 1 | Viewed by 3528

Abstract

Attosecond science is of a fundamental interest in physics. The measurement of the tunneling time in attosecond experiments, offers a fruitful opportunity to understand the role of time in quantum mechanics (QM). We discuss in this paper our tunneling time model in relation [...] Read more.

Attosecond science is of a fundamental interest in physics. The measurement of the tunneling time in attosecond experiments, offers a fruitful opportunity to understand the role of time in quantum mechanics (QM). We discuss in this paper our tunneling time model in relation to two time operator definitions introduced by Bauer and Aharonov–Bohm. We found that both definitions can be generalized to the same type of time operator. Moreover, we found that the introduction of a phenomenological parameter by Bauer to fit the experimental data is unnecessary. The issue is resolved with our tunneling model by considering the correct barrier width, which avoids a misleading interpretation of the experimental data. Our analysis shows that the use of the so-called classical barrier width, to be precise, is incorrect. Full article

(This article belongs to the Special Issue Time and Time Dependence in Quantum Mechanics)

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16 pages, 924 KiB

Open AccessArticle

On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

by Shahid Imran, Muhammad Kamran Siddiqui, Muhammad Imran and Muhammad Hussain

Mathematics 2018, 6(10), 191; https://doi.org/10.3390/math6100191 - 8 Oct 2018

Cited by 14 | Viewed by 4175

Abstract

Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex [...] Read more.

Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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19 pages, 392 KiB

Open AccessArticle

Price and Treatment Decisions in Epidemics: A Differential Game Approach

by Andrea Di Liddo

Mathematics 2018, 6(10), 190; https://doi.org/10.3390/math6100190 - 2 Oct 2018

Cited by 2 | Viewed by 2385

Abstract

We consider a pharmaceutical company that sells a drug that is useful in the treatment of an infectious disease. A public authority buys the drug to heal at least a portion of the infected population. The authority has an overall budget for all [...] Read more.

We consider a pharmaceutical company that sells a drug that is useful in the treatment of an infectious disease. A public authority buys the drug to heal at least a portion of the infected population. The authority has an overall budget for all health care costs in the country and can only allocate a (small) part of the budget to the purchase of the drug. The government chooses the amount of drug to be purchased in order to minimize both the number of infectious people and the perceived cost of the operation along a given time horizon. This cost can be modeled through a linear or quadratic function of the monetary cost (as generally happens in the literature) or through a specific function (blow-up) that makes the budget constraint endogenous. The pharmaceutical company chooses the price of the drug in order to maximize its profit and knowing the budget constraints of the buyer. The resulting differential game is studied by supposing the simplest possible dynamics for the population. Two different games are proposed and their solutions are discussed: a cooperative game in which the two players bargain for the price of the drug and the quantity is purchased with the aim of maximizing the overall payoff and a competitive game in which the seller announces a price strategy to the buyer and binds to it; the buyer reacts by choosing the quantity to be purchased. In the case of linear or quadratic costs, the solution provided (for budget levels is not high enough) that the government spends the entire budget to purchase the drug. This drawback does not occur when the blow-up cost function is used. Full article

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

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16 pages, 5988 KiB

Open AccessArticle

Urban-Tissue Optimization through Evolutionary Computation

by Diego Navarro-Mateu, Mohammed Makki and Ana Cocho-Bermejo

Mathematics 2018, 6(10), 189; https://doi.org/10.3390/math6100189 - 2 Oct 2018

Cited by 16 | Viewed by 6382

Abstract

The experiments analyzed in this paper focus their research on the use of Evolutionary Computation (EC) applied to a parametrized urban tissue. Through the application of EC, it is possible to develop a design under a single model that addresses multiple conflicting objectives. [...] Read more.

The experiments analyzed in this paper focus their research on the use of Evolutionary Computation (EC) applied to a parametrized urban tissue. Through the application of EC, it is possible to develop a design under a single model that addresses multiple conflicting objectives. The experiments presented are based on Cerdà’s master plan in Barcelona, specifically on the iconic Eixample block which is grouped into a 4 × 4 urban Superblock. The proposal aims to reach the existing high density of the city while reclaiming the block relations proposed by Cerdà’s original plan. Generating and ranking multiple individuals in a population through several generations ensures a flexible solution rather than a single “optimal” one. Final results in the Pareto front show a successful and diverse set of solutions that approximate Cerdà’s and the existing Barcelona’s Eixample states. Further analysis proposes different methodologies and considerations to choose appropriate individuals within the front depending on design requirements. Full article

(This article belongs to the Special Issue Evolutionary Computation)

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9 pages, 272 KiB

Open AccessArticle

New Types of F_c-Contractions and the Fixed-Circle Problem

by Nihal Taş, Nihal Yılmaz Özgür and Nabil Mlaiki

Mathematics 2018, 6(10), 188; https://doi.org/10.3390/math6100188 - 2 Oct 2018

Cited by 47 | Viewed by 3408

Abstract

In this paper we investigate some fixed-circle theorems using Ćirić’s technique (resp. Hardy-Rogers’ technique, Reich’s technique and Chatterjea’s technique) on a metric space. To do this, we define new types of

F_{c}

-contractions such as Ćirić type, Hardy-Rogers type, Reich type and [...] Read more.

In this paper we investigate some fixed-circle theorems using Ćirić’s technique (resp. Hardy-Rogers’ technique, Reich’s technique and Chatterjea’s technique) on a metric space. To do this, we define new types of

F_{c}

-contractions such as Ćirić type, Hardy-Rogers type, Reich type and Chatterjea type. Two illustrative examples are presented to show the effectiveness of our results. Also, it is given an application of a Ćirić type

F_{c}

-contraction to discontinuous self-mappings which have fixed circles. Full article

29 pages, 12692 KiB

Open AccessFeature PaperArticle

About Revisiting Domain Decomposition Methods for Poroelasticity

by Horacio Florez

Mathematics 2018, 6(10), 187; https://doi.org/10.3390/math6100187 - 2 Oct 2018

Cited by 1 | Viewed by 3377

Abstract

In this paper, we revisit well-established domain decomposition (DD) schemes to perform realistic simulations of coupled flow and poroelasticity problems on parallel computers. We define distinct solution schemes to take into account different transmission conditions among subdomain boundaries. Indeed, we examine two different [...] Read more.

In this paper, we revisit well-established domain decomposition (DD) schemes to perform realistic simulations of coupled flow and poroelasticity problems on parallel computers. We define distinct solution schemes to take into account different transmission conditions among subdomain boundaries. Indeed, we examine two different approaches, i.e., Dirichlet-Neumann (DN) and the mortar finite element method (MFEM), and we recognize their advantages and disadvantages. The MFEM significantly lessens the computational cost of reservoir compaction and subsidence calculations by dodging the conforming Cartesian grids that arise from the pay-zone onto its vicinity. There is a manifest necessity of producing non-matching interfaces between the reservoir and its neighborhood. We thus employ MFEM over nonuniform rational B-splines (NURBS) surfaces to stick these non-conforming subdomain parts. We then decouple the mortar saddle-point problem (SPP) using the Dirichlet-Neumann domain decomposition (DNDD) scheme. We confirm that this procedure is proper for calculations at the field level. We also carry comprehensive comparisons between the conventional and non-matching solutions to prove the method’s accuracy. Examples encompass linking finite element codes for slightly compressible single-phase and poroelasticity. We have used this program to a category of problems ranking from near-borehole applications to whole field subsidence estimations. Full article

(This article belongs to the Special Issue Modern Finite Element Methods)

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9 pages, 268 KiB

Open AccessArticle

Non-Cash Risk Measure on Nonconvex Sets

by Chang Cong and Peibiao Zhao

Mathematics 2018, 6(10), 186; https://doi.org/10.3390/math6100186 - 1 Oct 2018

Cited by 6 | Viewed by 2674

Abstract

Monetary risk measures defined on a convex set are interpreted as the smallest amount of external cash that must be added to a portfolio to make the portfolio being acceptable. In the present paper, the authors introduce a new concept: non-cash risk measure, [...] Read more.

Monetary risk measures defined on a convex set are interpreted as the smallest amount of external cash that must be added to a portfolio to make the portfolio being acceptable. In the present paper, the authors introduce a new concept: non-cash risk measure, which does as a nonconvex risk measure work in a nonconvex set. In addition, the authors arrive at a convex extension of the non-cash risk measure, and offer the relationship between the non-cash risk measure and its extension. Full article

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11 pages, 6896 KiB

Open AccessArticle

A Surface of Section for Hydrogen in Crossed Electric and Magnetic Fields

by Korana Burke and Kevin Mitchell

Mathematics 2018, 6(10), 185; https://doi.org/10.3390/math6100185 - 29 Sep 2018

Viewed by 2711

Abstract

A well defined global surface of section (SOS) is a necessary first step in many studies of various dynamical systems. Starting with a surface of section, one is able to more easily find periodic orbits as well as other geometric structures that govern [...] Read more.

A well defined global surface of section (SOS) is a necessary first step in many studies of various dynamical systems. Starting with a surface of section, one is able to more easily find periodic orbits as well as other geometric structures that govern the nonlinear dynamics of the system in question. In some cases, a global surface of section is relatively easily defined, but in other cases the definition is not trivial, and may not even exist. This is the case for the electron dynamics of a hydrogen atom in crossed electric and magnetic fields. In this paper, we demonstrate how one can define a surface of section and associated return map that may fail to be globally well defined, but for which the dynamics is well defined and continuous over a region that is sufficiently large to include the heteroclinic tangle and thus offers a sound geometric approach to studying the nonlinear dynamics. Full article

(This article belongs to the Special Issue Applied Analysis of Ordinary Differential Equations 2018)

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Mathematics, Volume 6, Issue 10 (October 2018) – 42 articles

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