Mathematics

20 pages, 541 KiB

Open AccessArticle

Enhancing Strong Neighbor-Based Optimization for Distributed Model Predictive Control Systems

by Shan Gao, Yi Zheng and Shaoyuan Li

Mathematics 2018, 6(5), 86; https://doi.org/10.3390/math6050086 - 22 May 2018

Cited by 10 | Viewed by 2887

This paper considers a class of large-scale systems which is composed of many interacting subsystems, and each of them is controlled by an individual controller. For this type of system, to improve the optimization performance of the entire closed-loop system in a distributed [...] Read more.

This paper considers a class of large-scale systems which is composed of many interacting subsystems, and each of them is controlled by an individual controller. For this type of system, to improve the optimization performance of the entire closed-loop system in a distributed framework without the entire system’s information or too-complicated network information, connectivity is always an important topic. To achieve this purpose, a distributed model predictive control (DMPC) design method is proposed in this paper, where each local model predictive control (MPC) considers the optimization performance of its strong coupling subsystems and communicates with them. A method to determine the strength of the coupling relationship based on the closed-loop system’s performance and subsystem network connectivity is proposed for the selection of each subsystem’s neighbors. Finally, through integrating the steady-state calculation, the designed DMPC is able to guarantee the recursive feasibility and asymptotic stability of the closed-loop system in the cases of both tracking set point and stabilizing system to zeroes. Simulation results show the efficiency of the proposed DMPC. Full article

(This article belongs to the Special Issue New Directions on Model Predictive Control)

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13 pages, 339 KiB

Open AccessFeature PaperArticle

Some Notes about Inference for the Lognormal Diffusion Process with Exogenous Factors

by Patricia Román-Román, Juan José Serrano-Pérez and Francisco Torres-Ruiz

Mathematics 2018, 6(5), 85; https://doi.org/10.3390/math6050085 - 21 May 2018

Cited by 17 | Viewed by 3640

Abstract

Different versions of the lognormal diffusion process with exogenous factors have been used in recent years to model and study the behavior of phenomena following a given growth curve. In each case considered, the estimation of the model has been addressed, generally by [...] Read more.

Different versions of the lognormal diffusion process with exogenous factors have been used in recent years to model and study the behavior of phenomena following a given growth curve. In each case considered, the estimation of the model has been addressed, generally by maximum likelihood (ML), as has been the study of several characteristics associated with the type of curve considered. For this process, a unified version of the ML estimation problem is presented, including how to obtain estimation errors and asymptotic confidence intervals for parametric functions when no explicit expression is available for the estimators of the parameters of the model. The Gompertz-type diffusion process is used here to illustrate the application of the methodology. Full article

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

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12 pages, 295 KiB

Open AccessArticle

Comparison of Differential Operators with Lie Derivative of Three-Dimensional Real Hypersurfaces in Non-Flat Complex Space Forms

by George Kaimakamis, Konstantina Panagiotidou and Juan De Dios Pérez

Mathematics 2018, 6(5), 84; https://doi.org/10.3390/math6050084 - 20 May 2018

Cited by 1 | Viewed by 2977

Abstract

In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field

P = ϕ A - A ϕ

is given and three-dimensional real hypersurfaces in non-flat complex space forms whose [...] Read more.

In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field

P = ϕ A - A ϕ

is given and three-dimensional real hypersurfaces in non-flat complex space forms whose tensor field

P

satisfies geometric conditions are classified. Full article

(This article belongs to the Special Issue Differential Geometry)

11 pages, 211 KiB

Open AccessArticle

Generalized Hyers-Ulam Stability of Trigonometric Functional Equations

by Elhoucien Elqorachi and Michael Th. Rassias

Mathematics 2018, 6(5), 83; https://doi.org/10.3390/math6050083 - 18 May 2018

Cited by 12 | Viewed by 2747

Abstract

In the present paper we study the generalized Hyers–Ulam stability of the generalized trigonometric functional equations [...] Read more.

In the present paper we study the generalized Hyers–Ulam stability of the generalized trigonometric functional equations

f (x y) + μ (y) f (x σ (y)) = 2 f (x) g (y) + 2 h (y), x, y \in S;

f (x y) + μ (y) f (x σ (y)) = 2 f (y) g (x) + 2 h (x), x, y \in S,

where S is a semigroup,

σ

:

S ⟶ S

is a involutive morphism, and

μ

:

S ⟶ C

is a multiplicative function such that

μ (x σ (x)) = 1

for all

x \in S .

As an application, we establish the generalized Hyers–Ulam stability theorem on amenable monoids and when

σ

is an involutive automorphism of S. Full article

(This article belongs to the Special Issue Stability Problems)

27 pages, 7669 KiB

Open AccessArticle

Linearization of the Kingman Coalescent

by Paul F. Slade

Mathematics 2018, 6(5), 82; https://doi.org/10.3390/math6050082 - 14 May 2018

Cited by 3 | Viewed by 4181

Abstract

Kingman’s coalescent process is a mathematical model of genealogy in which only pairwise common ancestry may occur. Inter-arrival times between successive coalescence events have a negative exponential distribution whose rate equals the combinatorial term

(\begin{matrix} n \\ 2 \end{matrix})

where n denotes the number [...] Read more.

Kingman’s coalescent process is a mathematical model of genealogy in which only pairwise common ancestry may occur. Inter-arrival times between successive coalescence events have a negative exponential distribution whose rate equals the combinatorial term

(\begin{matrix} n \\ 2 \end{matrix})

where n denotes the number of lineages present in the genealogy. These two standard constraints of Kingman’s coalescent, obtained in the limit of a large population size, approximate the exact ancestral process of Wright-Fisher or Moran models under appropriate parameterization. Calculation of coalescence event probabilities with higher accuracy quantifies the dependence of sample and population sizes that adhere to Kingman’s coalescent process. The convention that probabilities of leading order

N^{- 2}

are negligible provided

n ≪ N

is examined at key stages of the mathematical derivation. Empirically, expected genealogical parity of the single-pair restricted Wright-Fisher haploid model exceeds 99% where

n \leq \frac{1}{2} \sqrt[3]{N}

; similarly, per expected interval where

n \leq \frac{1}{2} \sqrt{N / 6}

. The fractional cubic root criterion is practicable, since although it corresponds to perfect parity and to an extent confounds identifiability it also accords with manageable conditional probabilities of multi-coalescence. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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23 pages, 646 KiB

Open AccessArticle

A Time-Non-Homogeneous Double-Ended Queue with Failures and Repairs and Its Continuous Approximation

by Antonio Di Crescenzo, Virginia Giorno, Balasubramanian Krishna Kumar and Amelia G. Nobile

Mathematics 2018, 6(5), 81; https://doi.org/10.3390/math6050081 - 11 May 2018

Cited by 27 | Viewed by 3494

Abstract

We consider a time-non-homogeneous double-ended queue subject to catastrophes and repairs. The catastrophes occur according to a non-homogeneous Poisson process and lead the system into a state of failure. Instantaneously, the system is put under repair, such that repair time is governed by [...] Read more.

We consider a time-non-homogeneous double-ended queue subject to catastrophes and repairs. The catastrophes occur according to a non-homogeneous Poisson process and lead the system into a state of failure. Instantaneously, the system is put under repair, such that repair time is governed by a time-varying intensity function. We analyze the transient and the asymptotic behavior of the queueing system. Moreover, we derive a heavy-traffic approximation that allows approximating the state of the systems by a time-non-homogeneous Wiener process subject to jumps to a spurious state (due to catastrophes) and random returns to the zero state (due to repairs). Special attention is devoted to the case of periodic catastrophe and repair intensity functions. The first-passage-time problem through constant levels is also treated both for the queueing model and the approximating diffusion process. Finally, the goodness of the diffusive approximating procedure is discussed. Full article

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

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17 pages, 545 KiB

Open AccessFeature PaperArticle

On the Bounds for a Two-Dimensional Birth-Death Process with Catastrophes

by Anna Sinitcina, Yacov Satin, Alexander Zeifman, Galina Shilova, Alexander Sipin, Ksenia Kiseleva, Tatyana Panfilova, Anastasia Kryukova, Irina Gudkova and Elena Fokicheva

Mathematics 2018, 6(5), 80; https://doi.org/10.3390/math6050080 - 11 May 2018

Cited by 2 | Viewed by 3134

Abstract

The model of a two-dimensional birth-death process with possible catastrophes is studied. The upper bounds on the rate of convergence in some weighted norms and the corresponding perturbation bounds are obtained. In addition, we consider the detailed description of two examples with 1-periodic [...] Read more.

The model of a two-dimensional birth-death process with possible catastrophes is studied. The upper bounds on the rate of convergence in some weighted norms and the corresponding perturbation bounds are obtained. In addition, we consider the detailed description of two examples with 1-periodic intensities and various types of death (service) rates. The bounds on the rate of convergence and the behavior of the corresponding mathematical expectations are obtained for each example. Full article

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

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19 pages, 264 KiB

Open AccessArticle

Global Behavior of Certain Nonautonomous Linearizable Three Term Difference Equations

by E. J. Janowski and M. R. S. Kulenović

Mathematics 2018, 6(5), 79; https://doi.org/10.3390/math6050079 - 09 May 2018

Viewed by 2711

Abstract

We investigate the nonautonomous difference equation with real initial conditions and coefficients

g_{i}, i = 0, 1

which are in general functions of n and/or the state variables

x_{n}, x_{n - 1}, \dots

, and satisfy [...] Read more.

We investigate the nonautonomous difference equation with real initial conditions and coefficients

g_{i}, i = 0, 1

which are in general functions of n and/or the state variables

x_{n}, x_{n - 1}, \dots

, and satisfy

g_{0} + g_{1} = 1

. We also obtain some global results about the behavior of solutions of the nonautonomous non-homogeneous difference equation where

g_{i}, i = 0, 1, 2

are functions of n and/or the state variables

x_{n}, x_{n - 1}, \dots

, with

g_{0} + g_{1} = 1

. Our results are based on the explicit formulas for solutions. We illustrate our results by numerous examples. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications)

18 pages, 321 KiB

Open AccessArticle

On Some Sufficiency-Type Stability and Linear State-Feedback Stabilization Conditions for a Class of Multirate Discrete-Time Systems

by M. De la Sen

Mathematics 2018, 6(5), 78; https://doi.org/10.3390/math6050078 - 09 May 2018

Cited by 1 | Viewed by 2771

Abstract

This paper presents and discusses the stability of a discrete multirate sampling system whose sets of sampling rates (or sampling periods) are the integer multiple of those operating on all the preceding substates. Each of such substates is associated with a particular sampling [...] Read more.

This paper presents and discusses the stability of a discrete multirate sampling system whose sets of sampling rates (or sampling periods) are the integer multiple of those operating on all the preceding substates. Each of such substates is associated with a particular sampling rate. The sufficiency-type stability conditions are derived based on simple conditions on the norm, spectral radius and numerical radius of the matrix of the dynamics of a system parameterized at the largest sampling period. Full article

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

27 pages, 2768 KiB

Open AccessFeature PaperArticle

The “Lévy or Diffusion” Controversy: How Important Is the Movement Pattern in the Context of Trapping?

by Danish A. Ahmed, Sergei V. Petrovskii and Paulo F. C. Tilles

Mathematics 2018, 6(5), 77; https://doi.org/10.3390/math6050077 - 09 May 2018

Cited by 7 | Viewed by 5452

Abstract

Many empirical and theoretical studies indicate that Brownian motion and diffusion models as its mean field counterpart provide appropriate modeling techniques for individual insect movement. However, this traditional approach has been challenged, and conflicting evidence suggests that an alternative movement pattern such as [...] Read more.

Many empirical and theoretical studies indicate that Brownian motion and diffusion models as its mean field counterpart provide appropriate modeling techniques for individual insect movement. However, this traditional approach has been challenged, and conflicting evidence suggests that an alternative movement pattern such as Lévy walks can provide a better description. Lévy walks differ from Brownian motion since they allow for a higher frequency of large steps, resulting in a faster movement. Identification of the ‘correct’ movement model that would consistently provide the best fit for movement data is challenging and has become a highly controversial issue. In this paper, we show that this controversy may be superficial rather than real if the issue is considered in the context of trapping or, more generally, survival probabilities. In particular, we show that almost identical trap counts are reproduced for inherently different movement models (such as the Brownian motion and the Lévy walk) under certain conditions of equivalence. This apparently suggests that the whole ‘Levy or diffusion’ debate is rather senseless unless it is placed into a specific ecological context, e.g., pest monitoring programs. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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10 pages, 284 KiB

Open AccessArticle

On the Semigroup Whose Elements Are Subgraphs of a Complete Graph

by Yanisa Chaiya, Chollawat Pookpienlert, Nuttawoot Nupo and Sayan Panma

Mathematics 2018, 6(5), 76; https://doi.org/10.3390/math6050076 - 09 May 2018

Viewed by 2837

Abstract

Let

K_{n}

be a complete graph on n vertices. Denote by

S K_{n}

the set of all subgraphs of

K_{n}

. For each

G, H \in S K_{n}

, the ring sum of G and H is a [...] Read more.

Let

K_{n}

be a complete graph on n vertices. Denote by

S K_{n}

the set of all subgraphs of

K_{n}

. For each

G, H \in S K_{n}

, the ring sum of G and H is a graph whose vertex set is

V (G) \cup V (H)

and whose edges are that of either G or H, but not of both. Then

S K_{n}

is a semigroup under the ring sum. In this paper, we study Green’s relations on

S K_{n}

and characterize ideals, minimal ideals, maximal ideals, and principal ideals of

S K_{n}

. Moreover, maximal subsemigroups and a class of maximal congruences are investigated. Furthermore, we prescribe the natural order on

S K_{n}

and consider minimal elements, maximal elements and covering elements of

S K_{n}

under this order. Full article

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11 pages, 250 KiB

Open AccessArticle

L²-Harmonic Forms on Incomplete Riemannian Manifolds with Positive Ricci Curvature

by Junya Takahashi

Mathematics 2018, 6(5), 75; https://doi.org/10.3390/math6050075 - 09 May 2018

Viewed by 3117

Abstract

We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial

L^{2}

-harmonic forms and on which the

L^{2}

-Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds. [...] Read more.

We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial

L^{2}

-harmonic forms and on which the

L^{2}

-Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds. Full article

(This article belongs to the Special Issue Differential Geometry)

16 pages, 297 KiB

Open AccessArticle

Neutrosophic Permeable Values and Energetic Subsets with Applications in BCK/BCI-Algebras

by Young Bae Jun, Florentin Smarandache, Seok-Zun Song and Hashem Bordbar

Mathematics 2018, 6(5), 74; https://doi.org/10.3390/math6050074 - 07 May 2018

Cited by 5 | Viewed by 2120

Abstract

The concept of a

(\in,

\in)

-neutrosophic ideal is introduced, and its characterizations are established. The notions of neutrosophic permeable values are introduced, and related properties are investigated. Conditions for the neutrosophic level sets to be energetic, right stable, and [...] Read more.

The concept of a

(\in,

\in)

-neutrosophic ideal is introduced, and its characterizations are established. The notions of neutrosophic permeable values are introduced, and related properties are investigated. Conditions for the neutrosophic level sets to be energetic, right stable, and right vanished are discussed. Relations between neutrosophic permeable S- and I-values are considered. Full article

9 pages, 249 KiB

Open AccessArticle

Upper Bound Design for the Lipschitz Constant of the F_G(ν,q)-Entropy Operator

by Yuri S. Popkov

Mathematics 2018, 6(5), 73; https://doi.org/10.3390/math6050073 - 07 May 2018

Cited by 2 | Viewed by 2077

Abstract

This paper develops an upper bound design method of the Lipschitz constant for the generalized Fermi–Dirac information entropy operator with a polyhedral admissible set. We introduce the concept of a normal operator from this class in which the constraint matrix has normalized columns. [...] Read more.

This paper develops an upper bound design method of the Lipschitz constant for the generalized Fermi–Dirac information entropy operator with a polyhedral admissible set. We introduce the concept of a normal operator from this class in which the constraint matrix has normalized columns. Next, we establish a connection between the normal and original operator. Finally, we demonstrate that the normal operator is majorized by the linear one and find numerical characteristics of this majorant. Full article

24 pages, 2475 KiB

Open AccessFeature PaperArticle

Numerical Methods for a Two-Species Competition-Diffusion Model with Free Boundaries

by Shuang Liu and Xinfeng Liu

Mathematics 2018, 6(5), 72; https://doi.org/10.3390/math6050072 - 03 May 2018

Cited by 11 | Viewed by 3500

Abstract

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of [...] Read more.

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of two species due to the interaction of their free boundaries. On the one hand, extremely small time steps are usually needed due to the stiffness of the system. On the other hand, it is always difficult to efficiently and accurately handle the moving boundaries especially with competition of two species. To overcome these numerical difficulties, we introduce a front tracking method coupled with an implicit solver for the 1D model. For the general 2D model, we use a level set approach to handle the moving boundaries to efficiently treat complicated topological changes. Several numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches. Full article

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

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20 pages, 5567 KiB

Open AccessArticle

Two-Level Finite Element Approximation for Oseen Viscoelastic Fluid Flow

by Nasrin Jahan Nasu, Md. Abdullah Al Mahbub, Shahid Hussain and Haibiao Zheng

Mathematics 2018, 6(5), 71; https://doi.org/10.3390/math6050071 - 03 May 2018

Cited by 5 | Viewed by 3174

Abstract

In this paper, a two-level finite element method for Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is presented. With the newly proposed algorithm, solving a large system of the constitutive equations will not be much more complex than the solution [...] Read more.

In this paper, a two-level finite element method for Oseen viscoelastic fluid flow obeying an Oldroyd-B type constitutive law is presented. With the newly proposed algorithm, solving a large system of the constitutive equations will not be much more complex than the solution of one linearized equation. The viscoelastic fluid flow constitutive equation consists of nonlinear terms, which are linearized by taking a known velocity

b (x)

, and transforms into the Oseen viscoelastic fluid flow model. Since Oseen viscoelastic fluid flow is already linear, we use a two-level method with a new technique. The two-level approach is consistent and efficient to study the coupled system which contains nonlinear terms. In the first step, the solution on the coarse grid is derived, and the result is used to determine the solution on the fine mesh in the second step. The decoupling algorithm takes two steps to solve a linear system on the fine mesh. The stability of the algorithm is derived for the temporal discretization and obtains the desired error bound. Two numerical experiments are executed to show the accuracy of the theoretical analysis. The approximations of the stress tensor, velocity vector, and pressure field are

P 1

-discontinuous,

P 2

-continuous and

P 1

-continuous finite elements respectively. Full article

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

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12 pages, 1173 KiB

Open AccessFeature PaperArticle

On Short-Term Loan Interest Rate Models: A First Passage Time Approach

by Giuseppina Albano and Virginia Giorno

Mathematics 2018, 6(5), 70; https://doi.org/10.3390/math6050070 - 03 May 2018

Cited by 5 | Viewed by 3449

Abstract

In this paper, we consider a stochastic diffusion process able to model the interest rate evolving with respect to time and propose a first passage time (FPT) approach through a boundary, defined as the “alert threshold”, in order to evaluate the risk of [...] Read more.

In this paper, we consider a stochastic diffusion process able to model the interest rate evolving with respect to time and propose a first passage time (FPT) approach through a boundary, defined as the “alert threshold”, in order to evaluate the risk of a proposed loan. Above this alert threshold, the rate is considered at the risk of usury, so new monetary policies have been adopted. Moreover, the mean FPT can be used as an indicator of the “goodness” of a loan; i.e., when an applicant is to choose between two loan offers, s/he will choose the one with a higher mean exit time from the alert boundary. An application to real data is considered by analyzing the Italian average effect global rate by means of two widely used models in finance, the Ornstein-Uhlenbeck (Vasicek) and Feller (Cox-Ingersoll-Ross) models. Full article

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

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19 pages, 1361 KiB

Open AccessArticle

Safeness Index-Based Economic Model Predictive Control of Stochastic Nonlinear Systems

by Zhe Wu, Helen Durand and Panagiotis D. Christofides

Mathematics 2018, 6(5), 69; https://doi.org/10.3390/math6050069 - 03 May 2018

Cited by 9 | Viewed by 3046

Abstract

Process operational safety plays an important role in designing control systems for chemical processes. Motivated by this, in this work, we develop a process Safeness Index-based economic model predictive control system for a broad class of stochastic nonlinear systems with input constraints. A [...] Read more.

Process operational safety plays an important role in designing control systems for chemical processes. Motivated by this, in this work, we develop a process Safeness Index-based economic model predictive control system for a broad class of stochastic nonlinear systems with input constraints. A stochastic Lyapunov-based controller is first utilized to characterize a region of the state-space surrounding the origin, starting from which the origin is rendered asymptotically stable in probability. Using this stability region characterization and a process Safeness Index function that characterizes the region in state-space in which it is safe to operate the process, an economic model predictive control system is then developed using Lyapunov-based constraints to ensure economic optimality, as well as process operational safety and closed-loop stability in probability. A chemical process example is used to demonstrate the applicability and effectiveness of the proposed approach. Full article

(This article belongs to the Special Issue New Directions on Model Predictive Control)

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11 pages, 253 KiB

Open AccessArticle

Non-Unique Fixed Point Results in Extended B-Metric Space

by Badr Alqahtani, Andreea Fulga and Erdal Karapınar

Mathematics 2018, 6(5), 68; https://doi.org/10.3390/math6050068 - 02 May 2018

Cited by 41 | Viewed by 4421

Abstract

In this paper, we investigate the existence of fixed points that are not necessarily unique in the setting of extended b-metric space. We state some examples to illustrate our results. Full article

(This article belongs to the Special Issue Fixed Point Theory)

26 pages, 1796 KiB

Open AccessArticle

NC-Cross Entropy Based MADM Strategy in Neutrosophic Cubic Set Environment

by Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Florentin Smarandache and Tapan Kumar Roy

Mathematics 2018, 6(5), 67; https://doi.org/10.3390/math6050067 - 29 Apr 2018

Cited by 14 | Viewed by 3005

Abstract

The objective of the paper is to introduce a new cross entropy measure in a neutrosophic cubic set (NCS) environment, which we call NC-cross entropy measure. We prove its basic properties. We also propose weighted NC-cross entropy and investigate its basic properties. We [...] Read more.

The objective of the paper is to introduce a new cross entropy measure in a neutrosophic cubic set (NCS) environment, which we call NC-cross entropy measure. We prove its basic properties. We also propose weighted NC-cross entropy and investigate its basic properties. We develop a novel multi attribute decision-making (MADM) strategy based on a weighted NC-cross entropy measure. To show the feasibility and applicability of the proposed multi attribute decision-making strategy, we solve an illustrative example of the multi attribute decision-making problem. Full article

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24 pages, 2278 KiB

Open AccessArticle

Effects of Viral and Cytokine Delays on Dynamics of Autoimmunity

by Farzad Fatehi, Yuliya N. Kyrychko and Konstantin B. Blyuss

Mathematics 2018, 6(5), 66; https://doi.org/10.3390/math6050066 - 28 Apr 2018

Cited by 7 | Viewed by 3498

Abstract

A major contribution to the onset and development of autoimmune disease is known to come from infections. An important practical problem is identifying the precise mechanism by which the breakdown of immune tolerance as a result of immune response to infection leads to [...] Read more.

A major contribution to the onset and development of autoimmune disease is known to come from infections. An important practical problem is identifying the precise mechanism by which the breakdown of immune tolerance as a result of immune response to infection leads to autoimmunity. In this paper, we develop a mathematical model of immune response to a viral infection, which includes T cells with different activation thresholds, regulatory T cells (Tregs), and a cytokine mediating immune dynamics. Particular emphasis is made on the role of time delays associated with the processes of infection and mounting the immune response. Stability analysis of various steady states of the model allows us to identify parameter regions associated with different types of immune behaviour, such as, normal clearance of infection, chronic infection, and autoimmune dynamics. Numerical simulations are used to illustrate different dynamical regimes, and to identify basins of attraction of different dynamical states. An important result of the analysis is that not only the parameters of the system, but also the initial level of infection and the initial state of the immune system determine the progress and outcome of the dynamics. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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19 pages, 962 KiB

Open AccessFeature PaperArticle

Economic Model Predictive Control with Zone Tracking

by Su Liu and Jinfeng Liu

Mathematics 2018, 6(5), 65; https://doi.org/10.3390/math6050065 - 25 Apr 2018

Cited by 14 | Viewed by 3259

Abstract

In this work, we propose a framework for economic model predictive control (EMPC) with zone tracking. A zone tracking stage cost is incorporated into the existing EMPC framework to form a multi-objective optimization problem. We provide sufficient conditions for asymptotic stability of the [...] Read more.

In this work, we propose a framework for economic model predictive control (EMPC) with zone tracking. A zone tracking stage cost is incorporated into the existing EMPC framework to form a multi-objective optimization problem. We provide sufficient conditions for asymptotic stability of the optimal steady state and characterize the exact penalty for the zone tracking cost which prioritizes zone tracking objective over economic objective. Moreover, an algorithm to modify the target zone based on the economic performance and reachability of the optimal steady state is proposed. The modified target zone effectively decouples the dynamic zone tracking and economic objectives and simplifies parameter tuning. Full article

(This article belongs to the Special Issue New Directions on Model Predictive Control)

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Mathematics, Volume 6, Issue 5 (May 2018) – 22 articles

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