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Mathematics, Volume 6, Issue 4 (April 2018)

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Cover Story (view full-size image) We investigated the problem of a data-driven, model-based economic model predictive control (MPC) [...] Read more.
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Open AccessArticle
An Estimate of the Root Mean Square Error Incurred When Approximating an fL2(ℝ) by a Partial Sum of Its Hermite Series
Mathematics 2018, 6(4), 64; https://doi.org/10.3390/math6040064 - 23 Apr 2018
Viewed by 604
Abstract
Let f be a band-limited function in $L 2 ( R )$ . Fix $T > 0$ , and suppose $f ′$ exists and is integrable on $[ − T , T ]$ . This paper gives a concrete estimate of the error [...] Read more.
Let f be a band-limited function in $L 2 ( R )$ . Fix $T > 0$ , and suppose $f ′$ exists and is integrable on $[ − T , T ]$ . This paper gives a concrete estimate of the error incurred when approximating f in the root mean square by a partial sum of its Hermite series. Specifically, we show, that for $K = 2 n , n ∈ Z + ,$ $1 2 T ∫ − T T [ f ( t ) − ( S K f ) ( t ) ] 2 d t 1 / 2 ≤ 1 + 1 K 1 2 T ∫ | t | > T f ( t ) 2 d t 1 / 2 + 1 2 T ∫ | ω | > N | f ^ ( ω ) | 2 d ω 1 / 2 + 1 K 1 2 T ∫ | t | ≤ T f N ( t ) 2 d t 1 / 2 + 1 π 1 + 1 2 K S a ( K , T ) ,$ in which $S K f$ is the K-th partial sum of the Hermite series of $f , f ^$ is the Fourier transform of f, $N = 2 K + 1 + 2 K + 3 2$ and $f N = ( f ^ χ ( − N , N ) ) ∨ ( t ) = 1 π ∫ − ∞ ∞ sin ( N ( t − s ) ) t − s f ( s ) d s$ . An explicit upper bound is obtained for $S a ( K , T )$ . Full article
(This article belongs to the Special Issue Applied Mathematics and Mechanics 2017)
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Open AccessArticle
A New Descent Algorithm Using the Three-Step Discretization Method for Solving Unconstrained Optimization Problems
Mathematics 2018, 6(4), 63; https://doi.org/10.3390/math6040063 - 23 Apr 2018
Viewed by 926
Abstract
In this paper, three-step Taylor expansion, which is equivalent to third-order Taylor expansion, is used as a mathematical base of the new descent method. At each iteration of this method, three steps are performed. Each step has a similar structure to the steepest [...] Read more.
In this paper, three-step Taylor expansion, which is equivalent to third-order Taylor expansion, is used as a mathematical base of the new descent method. At each iteration of this method, three steps are performed. Each step has a similar structure to the steepest descent method, except that the generalized search direction, step length, and next iterative point are applied. Compared with the steepest descent method, it is shown that the proposed algorithm has higher convergence speed and lower computational cost and storage. Full article
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Open AccessArticle
t-Norm Fuzzy Incidence Graphs
Mathematics 2018, 6(4), 62; https://doi.org/10.3390/math6040062 - 20 Apr 2018
Cited by 1 | Viewed by 1034
Abstract
It is the case that, in certain applications of fuzzy graphs, a t-norm, instead of a minimum, is more suitable. This requires the development of a new theory of fuzzy graphs involving an arbitrary t-norm in the basic definition of a fuzzy graph. [...] Read more.
It is the case that, in certain applications of fuzzy graphs, a t-norm, instead of a minimum, is more suitable. This requires the development of a new theory of fuzzy graphs involving an arbitrary t-norm in the basic definition of a fuzzy graph. There is very little known about this type of fuzzy graph. The purpose of this paper is to further develop this type of fuzzy graph. We concentrate on the relatively new concept of fuzzy incidence graphs. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle
A Developed Artificial Bee Colony Algorithm Based on Cloud Model
Mathematics 2018, 6(4), 61; https://doi.org/10.3390/math6040061 - 18 Apr 2018
Cited by 2 | Viewed by 1128
Abstract
The Artificial Bee Colony (ABC) algorithm is a bionic intelligent optimization method. The cloud model is a kind of uncertainty conversion model between a qualitative concept $T ˜$ that is presented by nature language and its quantitative expression, which integrates probability theory and [...] Read more.
The Artificial Bee Colony (ABC) algorithm is a bionic intelligent optimization method. The cloud model is a kind of uncertainty conversion model between a qualitative concept $T ˜$ that is presented by nature language and its quantitative expression, which integrates probability theory and the fuzzy mathematics. A developed ABC algorithm based on cloud model is proposed to enhance accuracy of the basic ABC algorithm and avoid getting trapped into local optima by introducing a new select mechanism, replacing the onlooker bees’ search formula and changing the scout bees’ updating formula. Experiments on CEC15 show that the new algorithm has a faster convergence speed and higher accuracy than the basic ABC and some cloud model based ABC variants. Full article
(This article belongs to the Special Issue Evolutionary Computation)
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Open AccessFeature PaperArticle
A Novel Distributed Economic Model Predictive Control Approach for Building Air-Conditioning Systems in Microgrids
Mathematics 2018, 6(4), 60; https://doi.org/10.3390/math6040060 - 17 Apr 2018
Cited by 2 | Viewed by 1106
Abstract
With the penetration of grid-connected renewable energy generation, microgrids are facing stability and power quality problems caused by renewable intermittency. To alleviate such problems, demand side management (DSM) of responsive loads, such as building air-conditioning system (BACS), has been proposed and studied. In [...] Read more.
With the penetration of grid-connected renewable energy generation, microgrids are facing stability and power quality problems caused by renewable intermittency. To alleviate such problems, demand side management (DSM) of responsive loads, such as building air-conditioning system (BACS), has been proposed and studied. In recent years, numerous control approaches have been published for proper management of single BACS. The majority of these approaches focus on either the control of BACS for attenuating power fluctuations in the grid or the operating cost minimization on behalf of the residents. These two control objectives are paramount for BACS control in microgrids and can be conflicting. As such, they should be considered together in control design. As individual buildings may have different owners/residents, it is natural to control different BACSs in an autonomous and self-interested manner to minimize the operational costs for the owners/residents. Unfortunately, such “selfish” operation can result in abrupt and large power fluctuations at the point of common coupling (PCC) of the microgrid due to lack of coordination. Consequently, the original objective of mitigating power fluctuations generated by renewable intermittency cannot be achieved. To minimize the operating costs of individual BACSs and simultaneously ensure desirable overall power flow at PCC, this paper proposes a novel distributed control framework based on the dissipativity theory. The proposed method achieves the objective of renewable intermittency mitigation through proper coordination of distributed BACS controllers and is scalable and computationally efficient. Simulation studies are carried out to illustrate the efficacy of the proposed control framework. Full article
(This article belongs to the Special Issue New Directions on Model Predictive Control) Printed Edition available
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Open AccessArticle
Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics
Mathematics 2018, 6(4), 59; https://doi.org/10.3390/math6040059 - 17 Apr 2018
Cited by 5 | Viewed by 1134
Abstract
A telegraph equation is believed to be an appropriate model of population dynamics as it accounts for the directional persistence of individual animal movement. Being motivated by the problem of habitat fragmentation, which is known to be a major threat to biodiversity that [...] Read more.
A telegraph equation is believed to be an appropriate model of population dynamics as it accounts for the directional persistence of individual animal movement. Being motivated by the problem of habitat fragmentation, which is known to be a major threat to biodiversity that causes species extinction worldwide, we consider the reaction–telegraph equation (i.e., telegraph equation combined with the population growth) on a bounded domain with the goal to establish the conditions of species survival. We first show analytically that, in the case of linear growth, the expression for the domain’s critical size coincides with the critical size of the corresponding reaction–diffusion model. We then consider two biologically relevant cases of nonlinear growth, i.e., the logistic growth and the growth with a strong Allee effect. Using extensive numerical simulations, we show that in both cases the critical domain size of the reaction–telegraph equation is larger than the critical domain size of the reaction–diffusion equation. Finally, we discuss possible modifications of the model in order to enhance the positivity of its solutions. Full article
(This article belongs to the Special Issue Progress in Mathematical Ecology) Printed Edition available
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Open AccessArticle
Theoretical Study of the One Self-Regulating Gene in the Modified Wagner Model
Mathematics 2018, 6(4), 58; https://doi.org/10.3390/math6040058 - 09 Apr 2018
Viewed by 725
Abstract
Predicting how a genetic change affects a given character is a major challenge in biology, and being able to tackle this problem relies on our ability to develop realistic models of gene networks. However, such models are rarely tractable mathematically. In this paper, [...] Read more.
Predicting how a genetic change affects a given character is a major challenge in biology, and being able to tackle this problem relies on our ability to develop realistic models of gene networks. However, such models are rarely tractable mathematically. In this paper, we propose a mathematical analysis of the sigmoid variant of the Wagner gene-network model. By considering the simplest case, that is, one unique self-regulating gene, we show that numerical simulations are not the only tool available to study such models: theoretical studies can be done too, by mathematical analysis of discrete dynamical systems. It is first shown that the particular sigmoid function can be theoretically investigated. Secondly, we provide an illustration of how to apply such investigations in the case of the dynamical system representing the one self-regulating gene. In this context, we focused on the composite function $f a ( m . x )$ where $f a$ is the parametric sigmoid function and m is a scalar not in ${ 0 , 1 }$ and we have proven that the number of fixed-point can be deduced theoretically, according to the values of a and m. Full article
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Open AccessArticle
Quasirecognition by Prime Graph of the Groups 2D2n(q) Where q < 105
Mathematics 2018, 6(4), 57; https://doi.org/10.3390/math6040057 - 09 Apr 2018
Viewed by 909
Abstract
Let G be a finite group. The prime graph $Γ ( G )$ of G is defined as follows: The set of vertices of $Γ ( G )$ is the set of prime divisors of $| G |$ and two distinct vertices p [...] Read more.
Let G be a finite group. The prime graph $Γ ( G )$ of G is defined as follows: The set of vertices of $Γ ( G )$ is the set of prime divisors of $| G |$ and two distinct vertices p and $p ′$ are connected in $Γ ( G )$ , whenever G contains an element of order $p p ′$ . A non-abelian simple group P is called recognizable by prime graph if for any finite group G with $Γ ( G ) = Γ ( P )$ , G has a composition factor isomorphic to P. It is been proved that finite simple groups $2 D n ( q )$ , where $n ≠ 4 k$ , are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups $2 D 2 k ( q )$ , where $k ≥ 9$ and q is a prime power less than $10 5$ . Full article
Open AccessArticle
Primes and the Lambert W function
Mathematics 2018, 6(4), 56; https://doi.org/10.3390/math6040056 - 08 Apr 2018
Cited by 4 | Viewed by 1149
Abstract
The Lambert W function, implicitly defined by $W ( x ) e W ( x ) = x$ , is a relatively “new” special function that has recently been the subject of an extended upsurge in interest and applications. In this note, I [...] Read more.
The Lambert W function, implicitly defined by $W ( x ) e W ( x ) = x$ , is a relatively “new” special function that has recently been the subject of an extended upsurge in interest and applications. In this note, I point out that the Lambert W function can also be used to gain a new perspective on the distribution of the prime numbers. Full article
Open AccessArticle
On Small Deviation Asymptotics In L2 of Some Mixed Gaussian Processes
Mathematics 2018, 6(4), 55; https://doi.org/10.3390/math6040055 - 05 Apr 2018
Cited by 2 | Viewed by 749
Abstract
We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes. The simplest example here is the linear combination of the Wiener process and the Brownian bridge. We get the precise final result in this case [...] Read more.
We study the exact small deviation asymptotics with respect to the Hilbert norm for some mixed Gaussian processes. The simplest example here is the linear combination of the Wiener process and the Brownian bridge. We get the precise final result in this case and in some examples of more complicated processes of similar structure. The proof is based on Karhunen–Loève expansion together with spectral asymptotics of differential operators and complex analysis methods. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
Open AccessFeature PaperArticle
g-Convex Weight Sequences
Mathematics 2018, 6(4), 54; https://doi.org/10.3390/math6040054 - 05 Apr 2018
Viewed by 819
Abstract
In this paper, we introduce the notion of g-convex weight sequence (gcws) for connected graphs based on the concept of g-convexity and g-weight. g-weight is a natural generalization of the notion of branch weight for trees. We investigate the various questions [...] Read more.
In this paper, we introduce the notion of g-convex weight sequence (gcws) for connected graphs based on the concept of g-convexity and g-weight. g-weight is a natural generalization of the notion of branch weight for trees. We investigate the various questions of realization of an integer sequence as a g-convex weight sequence for trees and some special classes of graphs such as complete graphs, windmill and degenerate windmill graphs and wheels. Full article
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Open AccessArticle
Neutrosophic Triplet G-Module
Mathematics 2018, 6(4), 53; https://doi.org/10.3390/math6040053 - 05 Apr 2018
Cited by 7 | Viewed by 798
Abstract
In this study, the neutrosophic triplet G-module is introduced and the properties of neutrosophic triplet G-module are studied. Furthermore, reducible, irreducible, and completely reducible neutrosophic triplet G-modules are defined, and relationships of these structures with each other are examined. Also, it is shown [...] Read more.
In this study, the neutrosophic triplet G-module is introduced and the properties of neutrosophic triplet G-module are studied. Furthermore, reducible, irreducible, and completely reducible neutrosophic triplet G-modules are defined, and relationships of these structures with each other are examined. Also, it is shown that the neutrosophic triplet G-module is different from the G-module. Full article
Open AccessFeature PaperArticle
On Angles and Pseudo-Angles in Minkowskian Planes
Mathematics 2018, 6(4), 52; https://doi.org/10.3390/math6040052 - 03 Apr 2018
Viewed by 1039
Abstract
The main purpose of the present paper is to well define Minkowskian angles and pseudo-angles between the two null directions and between a null direction and any non-null direction, respectively. Moreover, in a kind of way that will be tried to be made [...] Read more.
The main purpose of the present paper is to well define Minkowskian angles and pseudo-angles between the two null directions and between a null direction and any non-null direction, respectively. Moreover, in a kind of way that will be tried to be made clear at the end of the paper, these new sorts of angles and pseudo-angles can similarly to the previously known angles be seen as (combinations of) Minkowskian lengths of arcs on a Minkowskian unit circle together with Minkowskian pseudo-lengths of parts of the straight null lines. Full article
(This article belongs to the Special Issue Differential Geometry)
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Open AccessFeature PaperArticle
Data Driven Economic Model Predictive Control
Mathematics 2018, 6(4), 51; https://doi.org/10.3390/math6040051 - 02 Apr 2018
Cited by 4 | Viewed by 1212
Abstract
This manuscript addresses the problem of data driven model based economic model predictive control (MPC) design. To this end, first, a data-driven Lyapunov-based MPC is designed, and shown to be capable of stabilizing a system at an unstable equilibrium point. The data driven [...] Read more.
This manuscript addresses the problem of data driven model based economic model predictive control (MPC) design. To this end, first, a data-driven Lyapunov-based MPC is designed, and shown to be capable of stabilizing a system at an unstable equilibrium point. The data driven Lyapunov-based MPC utilizes a linear time invariant (LTI) model cognizant of the fact that the training data, owing to the unstable nature of the equilibrium point, has to be obtained from closed-loop operation or experiments. Simulation results are first presented demonstrating closed-loop stability under the proposed data-driven Lyapunov-based MPC. The underlying data-driven model is then utilized as the basis to design an economic MPC. The economic improvements yielded by the proposed method are illustrated through simulations on a nonlinear chemical process system example. Full article
(This article belongs to the Special Issue New Directions on Model Predictive Control) Printed Edition available
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Open AccessArticle
Credibility Measure for Intuitionistic Fuzzy Variables
Mathematics 2018, 6(4), 50; https://doi.org/10.3390/math6040050 - 02 Apr 2018
Cited by 2 | Viewed by 788
Abstract
Credibility measures in vague environments are to quantify the approximate chance of occurrence of fuzzy events. This paper presents a novel definition about credibility for intuitionistic fuzzy variables. We axiomatize this credibility measure and to clarify, give some examples. Based on the notion [...] Read more.
Credibility measures in vague environments are to quantify the approximate chance of occurrence of fuzzy events. This paper presents a novel definition about credibility for intuitionistic fuzzy variables. We axiomatize this credibility measure and to clarify, give some examples. Based on the notion of these concepts, we provide expected values, entropy, and general formulae for the central moments and discuss them through examples. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle
Large Deviation Results and Applications to the Generalized Cramér Model
Mathematics 2018, 6(4), 49; https://doi.org/10.3390/math6040049 - 02 Apr 2018
Viewed by 834
Abstract
In this paper, we prove large deviation results for some sequences of weighted sums of random variables. These sequences have applications to the probabilistic generalized Cramér model for products of primes in arithmetic progressions; they could lead to new conjectures concerning the (non-random) [...] Read more.
In this paper, we prove large deviation results for some sequences of weighted sums of random variables. These sequences have applications to the probabilistic generalized Cramér model for products of primes in arithmetic progressions; they could lead to new conjectures concerning the (non-random) set of products of primes in arithmetic progressions, a relevant topic in number theory. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
Open AccessArticle
A Numerical Method for Solving a Class of Nonlinear Second Order Fractional Volterra Integro-Differntial Type of Singularly Perturbed Problems
Mathematics 2018, 6(4), 48; https://doi.org/10.3390/math6040048 - 27 Mar 2018
Viewed by 1318
Abstract
In this paper, we study a class of fractional nonlinear second order Volterra integro-differential type of singularly perturbed problems with fractional order. We divide the problem into two subproblems. The first subproblems is the reduced problem when $ϵ = 0 .$ The second [...] Read more.
In this paper, we study a class of fractional nonlinear second order Volterra integro-differential type of singularly perturbed problems with fractional order. We divide the problem into two subproblems. The first subproblems is the reduced problem when $ϵ = 0 .$ The second subproblems is fractional Volterra integro-differential problem. We use the finite difference method to solve the first problem and the reproducing kernel method to solve the second problem. In addition, we use the pade’ approximation. The results show that the proposed analytical method can achieve excellent results in predicting the solutions of such problems. Theoretical results are presented. Numerical results are presented to show the efficiency of the proposed method. Full article
(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)
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Open AccessArticle
Hesitant Probabilistic Fuzzy Linguistic Sets with Applications in Multi-Criteria Group Decision Making Problems
Mathematics 2018, 6(4), 47; https://doi.org/10.3390/math6040047 - 26 Mar 2018
Cited by 6 | Viewed by 1496
Abstract
Uncertainties due to randomness and fuzziness comprehensively exist in control and decision support systems. In the present study, we introduce notion of occurring probability of possible values into hesitant fuzzy linguistic element (HFLE) and define hesitant probabilistic fuzzy linguistic set (HPFLS) for ill [...] Read more.
Uncertainties due to randomness and fuzziness comprehensively exist in control and decision support systems. In the present study, we introduce notion of occurring probability of possible values into hesitant fuzzy linguistic element (HFLE) and define hesitant probabilistic fuzzy linguistic set (HPFLS) for ill structured and complex decision making problem. HPFLS provides a single framework where both stochastic and non-stochastic uncertainties can be efficiently handled along with hesitation. We have also proposed expected mean, variance, score and accuracy function and basic operations for HPFLS. Weighted and ordered weighted aggregation operators for HPFLS are also defined in the present study for its applications in multi-criteria group decision making (MCGDM) problems. We propose a MCGDM method with HPFL information which is illustrated by an example. A real case study is also taken in the present study to rank State Bank of India, InfoTech Enterprises, I.T.C., H.D.F.C. Bank, Tata Steel, Tata Motors and Bajaj Finance using real data. Proposed HPFLS-based MCGDM method is also compared with two HFL-based decision making methods. Full article
(This article belongs to the Special Issue Fuzzy Mathematics) Printed Edition available
Open AccessArticle
Study on the Development of Neutrosophic Triplet Ring and Neutrosophic Triplet Field
Mathematics 2018, 6(4), 46; https://doi.org/10.3390/math6040046 - 23 Mar 2018
Cited by 9 | Viewed by 679 | Correction
Abstract
Rings and fields are significant algebraic structures in algebra and both of them are based on the group structure. In this paper, we attempt to extend the notion of a neutrosophic triplet group to a neutrosophic triplet ring and a neutrosophic triplet field. [...] Read more.
Rings and fields are significant algebraic structures in algebra and both of them are based on the group structure. In this paper, we attempt to extend the notion of a neutrosophic triplet group to a neutrosophic triplet ring and a neutrosophic triplet field. We introduce a neutrosophic triplet ring and study some of its basic properties. Further, we define the zero divisor, neutrosophic triplet subring, neutrosophic triplet ideal, nilpotent integral neutrosophic triplet domain, and neutrosophic triplet ring homomorphism. Finally, we introduce a neutrosophic triplet field. Full article
Open AccessArticle
Asynchronous Iterations of Parareal Algorithm for Option Pricing Models
Mathematics 2018, 6(4), 45; https://doi.org/10.3390/math6040045 - 21 Mar 2018
Cited by 4 | Viewed by 1023
Abstract
Spatial domain decomposition methods have been largely investigated in the last decades, while time domain decomposition seems to be contrary to intuition and so is not as popular as the former. However, many attractive methods have been proposed, especially the parareal algorithm, which [...] Read more.
Spatial domain decomposition methods have been largely investigated in the last decades, while time domain decomposition seems to be contrary to intuition and so is not as popular as the former. However, many attractive methods have been proposed, especially the parareal algorithm, which showed both theoretical and experimental efficiency in the context of parallel computing. In this paper, we present an original model of asynchronous variant based on the parareal scheme, applied to the European option pricing problem. Some numerical experiments are given to illustrate the convergence performance and computational efficiency of such a method. Full article
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