Mathematics

19 pages, 311 KiB

Open AccessArticle

by José Luis Usó-Doménech, Josué Antonio Nescolarde-Selva and Hugh Gash

Mathematics 2018, 6(11), 272; https://doi.org/10.3390/math6110272 - 21 Nov 2018

Cited by 2 | Viewed by 8466

There is a fairly widespread belief that the problem of existence is not an essential issue for logic. Logic, though formal, must deal with the problem of existence. However, logic should be limited to describing “formal existence” or “existence of [...] Read more.

There is a fairly widespread belief that the problem of existence is not an essential issue for logic. Logic, though formal, must deal with the problem of existence. However, logic should be limited to describing “formal existence” or “existence of a formal system”. However, the logical problem of existence and how to treat and resolve this problem differ completely from the corresponding metaphysical problem. It is possible to deduce that formal existence is nothing other than belonging to the universe of discourse, so proposing a solution to the logical problem of existence in an epistemological, rather than a metaphysical, context. In this paper, we conclude, from a formal point of view, no universe of discourse is given in advance; any universe of discourse that satisfies the necessary conditions can be used. The extended epistemological belief that there is a universe of discourse defined rigorously, which would be the true and should be “the universe of discourse of logic”, cannot be justified. Full article

11 pages, 261 KiB

Open AccessArticle

The Extremal Graphs of Some Topological Indices with Given Vertex k-Partiteness

by Fang Gao, Xiaoxin Li, Kai Zhou and Jia-Bao Liu

Mathematics 2018, 6(11), 271; https://doi.org/10.3390/math6110271 - 21 Nov 2018

Cited by 1 | Viewed by 2216

Abstract

The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the [...] Read more.

The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general zeroth-order Randić index, and the modified-Wiener index among graphs of order n with vertex k-partiteness not more than

m

. Full article

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

13 pages, 639 KiB

Open AccessArticle

Efficient Solutions of Interval Programming Problems with Inexact Parameters and Second Order Cone Constraints

by Ali Sadeghi, Mansour Saraj and Nezam Mahdavi Amiri

Mathematics 2018, 6(11), 270; https://doi.org/10.3390/math6110270 - 20 Nov 2018

Cited by 1 | Viewed by 2569

Abstract

In this article, a methodology is developed to solve an interval and a fractional interval programming problem by converting into a non-interval form for second order cone constraints, with the objective function and constraints being interval valued functions. We investigate the parametric and [...] Read more.

In this article, a methodology is developed to solve an interval and a fractional interval programming problem by converting into a non-interval form for second order cone constraints, with the objective function and constraints being interval valued functions. We investigate the parametric and non-parametric forms of the interval valued functions along with their convexity properties. Two approaches are developed to obtain efficient and properly efficient solutions. Furthermore, the efficient solutions or Pareto optimal solutions of fractional and non-fractional programming problems over

R_{+}^{n} ⋃ {0}

are also discussed. The main idea of the present article is to introduce a new concept for efficiency, called efficient space, caused by the lower and upper bounds of the respective intervals of the objective function which are shown in different figures. Finally, some numerical examples are worked through to illustrate the methodology and affirm the validity of the obtained results. Full article

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16 pages, 1161 KiB

Open AccessArticle

Identifying the Informational/Signal Dimension in Principal Component Analysis

by Sergio Camiz and Valério D. Pillar

Mathematics 2018, 6(11), 269; https://doi.org/10.3390/math6110269 - 20 Nov 2018

Cited by 12 | Viewed by 3470

Abstract

The identification of a reduced dimensional representation of the data is among the main issues of exploratory multidimensional data analysis and several solutions had been proposed in the literature according to the method. Principal Component Analysis (PCA) is the method that [...] Read more.

The identification of a reduced dimensional representation of the data is among the main issues of exploratory multidimensional data analysis and several solutions had been proposed in the literature according to the method. Principal Component Analysis (PCA) is the method that has received the largest attention thus far and several identification methods—the so-called stopping rules—have been proposed, giving very different results in practice, and some comparative study has been carried out. Some inconsistencies in the previous studies led us to try to fix the distinction between signal from noise in PCA—and its limits—and propose a new testing method. This consists in the production of simulated data according to a predefined eigenvalues structure, including zero-eigenvalues. From random populations built according to several such structures, reduced-size samples were extracted and to them different levels of random normal noise were added. This controlled introduction of noise allows a clear distinction between expected signal and noise, the latter relegated to the non-zero eigenvalues in the samples corresponding to zero ones in the population. With this new method, we tested the performance of ten different stopping rules. Of every method, for every structure and every noise, both power (the ability to correctly identify the expected dimension) and type-I error (the detection of a dimension composed only by noise) have been measured, by counting the relative frequencies in which the smallest non-zero eigenvalue in the population was recognized as signal in the samples and that in which the largest zero-eigenvalue was recognized as noise, respectively. This way, the behaviour of the examined methods is clear and their comparison/evaluation is possible. The reported results show that both the generalization of the Bartlett’s test by Rencher and the Bootstrap method by Pillar result much better than all others: both are accounted for reasonable power, decreasing with noise, and very good type-I error. Thus, more than the others, these methods deserve being adopted. Full article

(This article belongs to the Special Issue New Paradigms and Trends in Quantitative Ecology)

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19 pages, 308 KiB

Open AccessArticle

On Domain of Nörlund Matrix

by Kuddusi Kayaduman and Fevzi Yaşar

Mathematics 2018, 6(11), 268; https://doi.org/10.3390/math6110268 - 20 Nov 2018

Cited by 3 | Viewed by 2020

Abstract

In 1978, the domain of the Nörlund matrix on the classical sequence spaces l_p and l_∞ was introduced by Wang, where 1 ≤ p < ∞. Tuğ and Başar studied the matrix domain of Nörlund mean on the sequence spaces f [...] Read more.

In 1978, the domain of the Nörlund matrix on the classical sequence spaces l_p and l_∞ was introduced by Wang, where 1 ≤ p < ∞. Tuğ and Başar studied the matrix domain of Nörlund mean on the sequence spaces f₀ and f in 2016. Additionally, Tuğ defined and investigated a new sequence space as the domain of the Nörlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space

b s (N^{t})

and

c s (N^{t})

and examined the domain of the Nörlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their α-, β-, γ-duals, and characterized their matrix transformations on this space and into this space. Full article

19 pages, 280 KiB

Open AccessArticle

Some PPF Dependent Fixed Point Theorems for Generalized α-F-Contractions in Banach Spaces and Applications

by Yeol Je Cho, Shin Min Kang and Peyman Salimi

Mathematics 2018, 6(11), 267; https://doi.org/10.3390/math6110267 - 19 Nov 2018

Cited by 1 | Viewed by 2431

Abstract

In this paper, we introduce the concepts of an

α

-admissible nonself-mapping, an

α

-F-contractive nonself-mapping, a weak

α

-F-contractive nonself-mapping, and a generalized

α

-F-contractive nonself-mapping and prove some

P P F

(past-present-future)-dependent fixed point theorems [...] Read more.

In this paper, we introduce the concepts of an

α

-admissible nonself-mapping, an

α

-F-contractive nonself-mapping, a weak

α

-F-contractive nonself-mapping, and a generalized

α

-F-contractive nonself-mapping and prove some

P P F

(past-present-future)-dependent fixed point theorems for the proposed contractive nonself-mappings in certain Razumikhin classes. By using our results, we derive some

P P F

-dependent fixed point theorems for an

α

-F-contractive nonself-mapping endowed with a graph or a partial order. Finally, we give some applications to illustrate the main results. Full article

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

8 pages, 747 KiB

Open AccessArticle

On a Length Problem for Univalent Functions

by Mamoru Nunokawa, Janusz Sokół and Nak Eun Cho

Mathematics 2018, 6(11), 266; https://doi.org/10.3390/math6110266 - 19 Nov 2018

Cited by 2 | Viewed by 2394

Abstract

Let g be an analytic function with the normalization in the open unit disk. Let

L (r)

be the length of

g ({z : | z | = r})

. In this paper we present a correspondence [...] Read more.

Let g be an analytic function with the normalization in the open unit disk. Let

L (r)

be the length of

g ({z : | z | = r})

. In this paper we present a correspondence between g and

L (r)

for the case when g is not necessary univalent. Furthermore, some other results related to the length of analytic functions are also discussed. Full article

(This article belongs to the Special Issue Fractional-Order Integral and Derivative Operators and Their Applications)

16 pages, 273 KiB

Open AccessArticle

Scheduling and Planning in Service Systems with Goal Programming: Literature Review

by Şeyda Gür and Tamer Eren

Mathematics 2018, 6(11), 265; https://doi.org/10.3390/math6110265 - 19 Nov 2018

Cited by 36 | Viewed by 5175

Abstract

Background: People want to be able to evaluate different kinds of information in a good way. There are various methods that they develop in such situations. Among the optimization methods, the goal programming method is often used when there are multiple objectives that [...] Read more.

Background: People want to be able to evaluate different kinds of information in a good way. There are various methods that they develop in such situations. Among the optimization methods, the goal programming method is often used when there are multiple objectives that decision makers want to accomplish. Because scheduling and planning problems have multiple objectives that are desired to be achieved, the goal programming method helps the researcher in contradictory situations between these goals. Methods: This study includes, examines, and analyzes recent research on service scheduling and planning. In the literature, service scheduling and planning studies have been examined using goal programming method from past to today. Results: The studies are detailed according to the type of goal programming, according to scheduling types, the purpose used in the studies, and the methods integrated with the goal programming. There are 142 studies in Emerald, Science Direct, Jstor, Springer, Taylor and Francis, Google Scholar, etc. databases that are examined in detail. For readers, diversification has been made to facilitate the identification of these studies and a detailed overview has been presented. Conclusion: As a result of the study, studies with the goal programming in the literature have been seen. The readers’ perspectives for planning and scheduling are discussed. Full article

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

19 pages, 360 KiB

Open AccessFeature PaperArticle

Tsallis Entropy of Fuzzy Dynamical Systems

by Dagmar Markechová

Mathematics 2018, 6(11), 264; https://doi.org/10.3390/math6110264 - 18 Nov 2018

Viewed by 1839

Abstract

This article deals with the mathematical modeling of Tsallis entropy in fuzzy dynamical systems. At first, the concepts of Tsallis entropy and Tsallis conditional entropy of order

q,

where

q

is a positive real number not equal to 1, of fuzzy partitions [...] Read more.

This article deals with the mathematical modeling of Tsallis entropy in fuzzy dynamical systems. At first, the concepts of Tsallis entropy and Tsallis conditional entropy of order

q,

where

q

is a positive real number not equal to 1, of fuzzy partitions are introduced and their mathematical behavior is described. As an important result, we showed that the Tsallis entropy of fuzzy partitions of order

q > 1

satisfies the property of sub-additivity. This property permits the definition of the Tsallis entropy of order

q > 1

of a fuzzy dynamical system. It was shown that Tsallis entropy is an invariant under isomorphisms of fuzzy dynamical systems; thus, we acquired a tool for distinguishing some non-isomorphic fuzzy dynamical systems. Finally, we formulated a version of the Kolmogorov–Sinai theorem on generators for the case of the Tsallis entropy of a fuzzy dynamical system. The obtained results extend the results provided by Markechová and Riečan in Entropy, 2016, 18, 157, which are particularized to the case of logical entropy. Full article

12 pages, 280 KiB

Open AccessArticle

The Coefficients of Powers of Bazilević Functions

by Nak Eun Cho, Virendra Kumar and Ji Hyang Park

Mathematics 2018, 6(11), 263; https://doi.org/10.3390/math6110263 - 18 Nov 2018

Cited by 2 | Viewed by 2260

Abstract

In the present work, a sharp bound on the modulus of the initial coefficients for powers of strongly Bazilević functions is obtained. As an application of these results, certain conditions are investigated under which the Littlewood-Paley conjecture holds for strongly Bazilević functions for [...] Read more.

In the present work, a sharp bound on the modulus of the initial coefficients for powers of strongly Bazilević functions is obtained. As an application of these results, certain conditions are investigated under which the Littlewood-Paley conjecture holds for strongly Bazilević functions for large values of the parameters involved therein. Further, sharp estimate on the generalized Fekete-Szegö functional is also derived. Relevant connections of our results with the existing ones are also made. Full article

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

12 pages, 771 KiB

Open AccessArticle

Random Best Proximity Points for α-Admissible Mappings via Simulation Functions

by Chayut Kongban, Poom Kumam and Juan Martínez-Moreno

Mathematics 2018, 6(11), 262; https://doi.org/10.3390/math6110262 - 18 Nov 2018

Cited by 1 | Viewed by 2991

Abstract

In this paper, we introduce a new concept of random

α

-proximal admissible and random

α

-

Z

-contraction. Then we establish random best proximity point theorems for such mapping in complete separable metric spaces. [...] Read more.

In this paper, we introduce a new concept of random

α

-proximal admissible and random

α

-

Z

-contraction. Then we establish random best proximity point theorems for such mapping in complete separable metric spaces. Full article

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

11 pages, 260 KiB

Open AccessArticle

Common Fixed Points for Mappings under Contractive Conditions of (α,β,ψ)-Admissibility Type

by Wasfi Shatanawi

Mathematics 2018, 6(11), 261; https://doi.org/10.3390/math6110261 - 18 Nov 2018

Cited by 19 | Viewed by 2497

Abstract

In this paper, we introduce the notion of

(α, β, ψ)

-contraction for a pair of mappings

(S, T)

defined on a set X. We use our new notion to create and prove a common [...] Read more.

In this paper, we introduce the notion of

(α, β, ψ)

-contraction for a pair of mappings

(S, T)

defined on a set X. We use our new notion to create and prove a common fixed point theorem for two mappings defined on a metric space

(X, d)

under a set of conditions. Furthermore, we employ our main result to get another new result. Our results are modifications of many existing results in the literature. An example is included in order to show the authenticity of our main result. Full article

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

8 pages, 249 KiB

Open AccessArticle

Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

by Janak Raj Sharma, Ioannis K. Argyros and Sunil Kumar

Mathematics 2018, 6(11), 260; https://doi.org/10.3390/math6110260 - 17 Nov 2018

Cited by 6 | Viewed by 2173

Abstract

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since [...] Read more.

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

10 pages, 248 KiB

Open AccessArticle

Inequalities on Sasakian Statistical Manifolds in Terms of Casorati Curvatures

by Chul Woo Lee and Jae Won Lee

Mathematics 2018, 6(11), 259; https://doi.org/10.3390/math6110259 - 17 Nov 2018

Cited by 4 | Viewed by 2219

Abstract

A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and

\nabla^{*}

in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded [...] Read more.

A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and

\nabla^{*}

in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded above from Casorati curvatures on C-totally real (Legendrian and slant) submanifolds of a Sasakian statistical manifold of constant

φ

-sectional curvature. In addition, we give examples to show that the total space is a sphere. Full article

(This article belongs to the Special Issue Differential Geometry)

16 pages, 359 KiB

Open AccessArticle

Determinant Forms, Difference Equations and Zeros of the q-Hermite-Appell Polynomials

by Subuhi Khan and Tabinda Nahid

Mathematics 2018, 6(11), 258; https://doi.org/10.3390/math6110258 - 17 Nov 2018

Cited by 13 | Viewed by 2383

Abstract

The present paper intends to introduce the hybrid form of q-special polynomials, namely q-Hermite-Appell polynomials by means of generating function and series definition. Some significant properties of q-Hermite-Appell polynomials such as determinant definitions, q-recurrence relations and q-difference equations [...] Read more.

The present paper intends to introduce the hybrid form of q-special polynomials, namely q-Hermite-Appell polynomials by means of generating function and series definition. Some significant properties of q-Hermite-Appell polynomials such as determinant definitions, q-recurrence relations and q-difference equations are established. Examples providing the corresponding results for certain members belonging to this q-Hermite-Appell family are considered. In addition, graphs of certain q-special polynomials are demonstrated using computer experiment. Thereafter, distribution of zeros of these q-special polynomials is displayed. Full article

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

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19 pages, 1085 KiB

Open AccessArticle

Strong Convergence Theorems for Fixed Point Problems for Nonexpansive Mappings and Zero Point Problems for Accretive Operators Using Viscosity Implicit Midpoint Rules in Banach Spaces

by Huancheng Zhang, Yunhua Qu and Yongfu Su

Mathematics 2018, 6(11), 257; https://doi.org/10.3390/math6110257 - 16 Nov 2018

Cited by 3 | Viewed by 2245

Abstract

This paper uses the viscosity implicit midpoint rule to find common points of the fixed point set of a nonexpansive mapping and the zero point set of an accretive operator in Banach space. Under certain conditions, this paper obtains the strong convergence results [...] Read more.

This paper uses the viscosity implicit midpoint rule to find common points of the fixed point set of a nonexpansive mapping and the zero point set of an accretive operator in Banach space. Under certain conditions, this paper obtains the strong convergence results of the proposed algorithm and improves the relevant results of researchers in this field. In the end, this paper gives numerical examples to support the main results. Full article

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

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7 pages, 237 KiB

Open AccessArticle

Interpolative Reich–Rus–Ćirić Type Contractions on Partial Metric Spaces

by Erdal Karapinar, Ravi Agarwal and Hassen Aydi

Mathematics 2018, 6(11), 256; https://doi.org/10.3390/math6110256 - 16 Nov 2018

Cited by 134 | Viewed by 4556

Abstract

By giving a counter-example, we point out a gap in the paper by Karapinar (Adv. Theory Nonlinear Anal. Its Appl. 2018, 2, 85–87) where the given fixed point may be not unique and we present the corrected version. We also state the celebrated [...] Read more.

By giving a counter-example, we point out a gap in the paper by Karapinar (Adv. Theory Nonlinear Anal. Its Appl. 2018, 2, 85–87) where the given fixed point may be not unique and we present the corrected version. We also state the celebrated fixed point theorem of Reich–Rus–Ćirić in the framework of complete partial metric spaces, by taking the interpolation theory into account. Some examples are provided where the main result in papers by Reich (Can. Math. Bull. 1971, 14, 121–124; Boll. Unione Mat. Ital. 1972, 4, 26–42 and Boll. Unione Mat. Ital. 1971, 4, 1–11.) is not applicable. Full article

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

26 pages, 313 KiB

Open AccessArticle

Interval-Valued Cores and Interval-Valued Dominance Cores of Cooperative Games Endowed with Interval-Valued Payoffs

by Hsien-Chung Wu

Mathematics 2018, 6(11), 255; https://doi.org/10.3390/math6110255 - 16 Nov 2018

Cited by 4 | Viewed by 2019

Abstract

Cooperative games endowed with interval-valued payoffs are studied in this paper. Based on the interval-valued payoff and the different types of orderings, we can propose many types of so-called interval-valued cores and interval-valued dominance cores. The main issue of this paper is to [...] Read more.

Cooperative games endowed with interval-valued payoffs are studied in this paper. Based on the interval-valued payoff and the different types of orderings, we can propose many types of so-called interval-valued cores and interval-valued dominance cores. The main issue of this paper is to establish the equalities of different types of interval-valued cores and interval-valued dominance cores under a mild assumption. Without considering the individual rationality, we also establish the equalities of different types of interval-valued pre-cores and interval-valued dominance pre-cores without any extra assumptions. Full article

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

19 pages, 2299 KiB

Open AccessArticle

Rule-Based Production of Mathematical Expressions

by Mir Mohammad Reza Alavi Milani, Sahereh Hosseinpour and Huseyin Pehlivan

Mathematics 2018, 6(11), 254; https://doi.org/10.3390/math6110254 - 15 Nov 2018

Cited by 2 | Viewed by 4647

Abstract

There are situations in which one needs to write various kinds of mathematical expressions, such as practicing tests and school exams. There is a variety of methods to produce such expressions, but they are usually based on a database. This paper addresses the [...] Read more.

There are situations in which one needs to write various kinds of mathematical expressions, such as practicing tests and school exams. There is a variety of methods to produce such expressions, but they are usually based on a database. This paper addresses the production of new expressions using the template ones that can be derived from the evaluation process or entered by users. With special limitations on the values of parameters, some templates can be dynamically constructed for the automatic generation of mathematical expressions and represented in the form of classes. For this purpose, a new type of grammar is proposed. This grammar is similar to Context-Free Grammar, but it empowers the producer to gain control over the generation of rules for different expressions. Our work mainly focuses on generating mathematical expressions in a user-oriented way, using a predefined set of templates of production rules. The production of expressions is not completely random, and is based on the defined subject. Full article

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9 pages, 1067 KiB

Open AccessArticle

Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation

by Aditya Kamath and Sergei Manzhos

Mathematics 2018, 6(11), 253; https://doi.org/10.3390/math6110253 - 15 Nov 2018

Cited by 10 | Viewed by 3054

Abstract

We explore the use of inverse multiquadratic (IMQ) functions as basis functions when solving the vibrational Schrödinger equation with the rectangular collocation method. The quality of the vibrational spectrum of formaldehyde (in six dimensions) is compared to that obtained using Gaussian basis functions [...] Read more.

We explore the use of inverse multiquadratic (IMQ) functions as basis functions when solving the vibrational Schrödinger equation with the rectangular collocation method. The quality of the vibrational spectrum of formaldehyde (in six dimensions) is compared to that obtained using Gaussian basis functions when using different numbers of width-optimized IMQ functions. The effects of the ratio of the number of collocation points to the number of basis functions and of the choice of the IMQ exponent are studied. We show that the IMQ basis can be used with parameters where the IMQ function is not integrable. We find that the quality of the spectrum with IMQ basis functions is somewhat lower that that with a Gaussian basis when the basis size is large, and for a range of IMQ exponents. The IMQ functions are; however, advantageous when a small number of functions is used or with a small number of collocation points (e.g., when using square collocation). Full article

(This article belongs to the Special Issue Computational Spectroscopy)

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1 pages, 160 KiB

Open AccessCorrection

Correction: Kuniya, T. Stability Analysis of an Age-Structured SIR Epidemic Model with a Reduction Method to ODEs. Mathematics 2018, 6, 147

by Toshikazu Kuniya

Mathematics 2018, 6(11), 252; https://doi.org/10.3390/math6110252 - 15 Nov 2018

Viewed by 2078

Abstract

I have found an error in Equation (17) in my paper [1] [...] Full article

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

18 pages, 1377 KiB

Open AccessArticle

Geometric Construction of Some Lehmer Means

by Ralph Høibakk, Dag Lukkassen, Annette Meidell and Lars-Erik Persson

Mathematics 2018, 6(11), 251; https://doi.org/10.3390/math6110251 - 14 Nov 2018

Cited by 2 | Viewed by 3044

Abstract

The main aim of this paper is to contribute to the recently initiated research concerning geometric constructions of means, where the variables are appearing as line segments. The present study shows that all Lehmer means of two variables for integer power k and [...] Read more.

The main aim of this paper is to contribute to the recently initiated research concerning geometric constructions of means, where the variables are appearing as line segments. The present study shows that all Lehmer means of two variables for integer power k and for

k = \frac{m}{2}

, where m is an integer, can be geometrically constructed, that Lehmer means for power

k = 0, 1

and 2 can be geometrically constructed for any number of variables and that Lehmer means for power

k = 1 / 2

and

- 1

can be geometrically constructed, where the number of variables is

n = 2^{m}

and m is a positive integer. Full article

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17 pages, 1415 KiB

Open AccessArticle

A Generalized Framework for Analyzing Taxonomic, Phylogenetic, and Functional Community Structure Based on Presence–Absence Data

by János Podani, Sandrine Pavoine and Carlo Ricotta

Mathematics 2018, 6(11), 250; https://doi.org/10.3390/math6110250 - 12 Nov 2018

Cited by 9 | Viewed by 4376

Abstract

Community structure as summarized by presence–absence data is often evaluated via diversity measures by incorporating taxonomic, phylogenetic and functional information on the constituting species. Most commonly, various dissimilarity coefficients are used to express these aspects simultaneously such that the results are not comparable [...] Read more.

Community structure as summarized by presence–absence data is often evaluated via diversity measures by incorporating taxonomic, phylogenetic and functional information on the constituting species. Most commonly, various dissimilarity coefficients are used to express these aspects simultaneously such that the results are not comparable due to the lack of common conceptual basis behind index definitions. A new framework is needed which allows such comparisons, thus facilitating evaluation of the importance of the three sources of extra information in relation to conventional species-based representations. We define taxonomic, phylogenetic and functional beta diversity of species assemblages based on the generalized Jaccard dissimilarity index. This coefficient does not give equal weight to species, because traditional site dissimilarities are lowered by taking into account the taxonomic, phylogenetic or functional similarity of differential species in one site to the species in the other. These, together with the traditional, taxon- (species-) based beta diversity are decomposed into two additive fractions, one due to taxonomic, phylogenetic or functional excess and the other to replacement. In addition to numerical results, taxonomic, phylogenetic and functional community structure is visualized by 2D simplex or ternary plots. Redundancy with respect to taxon-based structure is expressed in terms of centroid distances between point clouds in these diagrams. The approach is illustrated by examples coming from vegetation surveys representing different ecological conditions. We found that beta diversity decreases in the following order: taxon-based, taxonomic (Linnaean), phylogenetic and functional. Therefore, we put forward the beta-redundancy hypothesis suggesting that this ordering may be most often the case in ecological communities, and discuss potential reasons and possible exceptions to this supposed rule. Whereas the pattern of change in diversity may be indicative of fundamental features of the particular community being studied, the effect of the choice of functional traits—a more or less subjective element of the framework—remains to be investigated. Full article

(This article belongs to the Special Issue New Paradigms and Trends in Quantitative Ecology)

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13 pages, 337 KiB

Open AccessArticle

The Combination Projection Method for Solving Convex Feasibility Problems

by Songnian He and Qiao-Li Dong

Mathematics 2018, 6(11), 249; https://doi.org/10.3390/math6110249 - 12 Nov 2018

Cited by 2 | Viewed by 2616

Abstract

In this paper, we propose a new method, which is called the combination projection method (CPM), for solving the convex feasibility problem (CFP) of finding some [...] Read more.

In this paper, we propose a new method, which is called the combination projection method (CPM), for solving the convex feasibility problem (CFP) of finding some

x^{*} \in C : = \cap_{i = 1}^{m} {x \in H | c_{i} (x) \leq 0}

, where m is a positive integer,

H

is a real Hilbert space, and

{c_{i}}_{i = 1}^{m}

are convex functions defined as

H

. The key of the CPM is that, for the current iterate

x^{k}

, the CPM firstly constructs a new level set

H_{k}

through a convex combination of some of

{c_{i}}_{i = 1}^{m}

in an appropriate way, and then updates the new iterate

x^{k + 1}

only by using the projection

P_{H_{k}}

. We also introduce the combination relaxation projection methods (CRPM) to project onto half-spaces to make CPM easily implementable. The simplicity and easy implementation are two advantages of our methods since only one projection is used in each iteration and the projections are also easy to calculate. The weak convergence theorems are proved and the numerical results show the advantages of our methods. Full article

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

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10 pages, 225 KiB

Open AccessArticle

Bounds of Riemann-Liouville Fractional Integrals in General Form via Convex Functions and Their Applications

by Ghulam Farid, Waqas Nazeer, Muhammad Shoaib Saleem, Sajid Mehmood and Shin Min Kang

Mathematics 2018, 6(11), 248; https://doi.org/10.3390/math6110248 - 12 Nov 2018

Cited by 45 | Viewed by 2811

Abstract

In this article, we establish bounds of sum of the left and right sided Riemann Liouville (RL) fractional integrals and related inequalities in general form. A new and novel approach is followed to obtain these results for general Riemann Liouville (RL) fractional integrals. [...] Read more.

In this article, we establish bounds of sum of the left and right sided Riemann Liouville (RL) fractional integrals and related inequalities in general form. A new and novel approach is followed to obtain these results for general Riemann Liouville (RL) fractional integrals. Monotonicity and convexity of functions are used with some usual and straight forward inequalities. The presented results are also have connection with some known and already published results. Applications and motivations of presented results are briefly discussed. Full article

21 pages, 352 KiB

Open AccessArticle

Seeking for the Maximum Symmetric Rank

by Alessandro De Paris

Mathematics 2018, 6(11), 247; https://doi.org/10.3390/math6110247 - 12 Nov 2018

Cited by 1 | Viewed by 2159

Abstract

We present the state-of-the-art on maximum symmetric tensor rank, for each given dimension and order. After a general discussion on the interplay between symmetric tensors, polynomials and divided powers, we introduce the technical environment and the methods that have been set up in [...] Read more.

We present the state-of-the-art on maximum symmetric tensor rank, for each given dimension and order. After a general discussion on the interplay between symmetric tensors, polynomials and divided powers, we introduce the technical environment and the methods that have been set up in recent times to find new lower and upper bounds. Full article

(This article belongs to the Special Issue Decomposability of Tensors)

7 pages, 214 KiB

Open AccessArticle

Trans-Sasakian 3-Manifolds with Reeb Flow Invariant Ricci Operator

by Yan Zhao, Wenjie Wang and Ximin Liu

Mathematics 2018, 6(11), 246; https://doi.org/10.3390/math6110246 - 09 Nov 2018

Cited by 5 | Viewed by 2959

Abstract

Let M be a three-dimensional trans-Sasakian manifold of type

(α, β)

. In this paper, we obtain that the Ricci operator of M is invariant along Reeb flow if and only if M is an

α

-Sasakian manifold, cosymplectic manifold [...] Read more.

Let M be a three-dimensional trans-Sasakian manifold of type

(α, β)

. In this paper, we obtain that the Ricci operator of M is invariant along Reeb flow if and only if M is an

α

-Sasakian manifold, cosymplectic manifold or a space of constant sectional curvature. Applying this, we give a new characterization of proper trans-Sasakian 3-manifolds. Full article

(This article belongs to the Special Issue Differential Geometry)

9 pages, 241 KiB

Open AccessArticle

New Numerical Method for Solving Tenth Order Boundary Value Problems

by Ali Akgül, Esra Karatas Akgül, Dumitru Baleanu and Mustafa Inc

Mathematics 2018, 6(11), 245; https://doi.org/10.3390/math6110245 - 08 Nov 2018

Cited by 6 | Viewed by 2694

Abstract

In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate [...] Read more.

In this paper, we implement reproducing kernel Hilbert space method to tenth order boundary value problems. These problems are important for mathematicians. Different techniques were applied to get approximate solutions of such problems. We obtain some useful reproducing kernel functions to get approximate solutions. We obtain very efficient results by this method. We show our numerical results by tables. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2019)

8 pages, 724 KiB

Open AccessArticle

A New Identity Involving the Chebyshev Polynomials

by Yixue Zhang and Zhuoyu Chen

Mathematics 2018, 6(11), 244; https://doi.org/10.3390/math6110244 - 07 Nov 2018

Cited by 18 | Viewed by 2591

Abstract

In this paper, firstly, we introduced a second order non-linear recursive sequence, then we use this sequence and the combinatorial methods to perform a deep study on the computational problem concerning one kind sums, which includes the Chebyshev polynomials. This makes it possible [...] Read more.

In this paper, firstly, we introduced a second order non-linear recursive sequence, then we use this sequence and the combinatorial methods to perform a deep study on the computational problem concerning one kind sums, which includes the Chebyshev polynomials. This makes it possible to simplify a class of complex computations involving the second type Chebyshev polynomials into a very simple problem. Finally, we give a new and interesting identity for it. Full article

10 pages, 297 KiB

Open AccessArticle

Edge Version of Metric Dimension and Doubly Resolving Sets of the Necklace Graph

by Jia-Bao Liu, Zohaib Zahid, Ruby Nasir and Waqas Nazeer

Mathematics 2018, 6(11), 243; https://doi.org/10.3390/math6110243 - 07 Nov 2018

Cited by 35 | Viewed by 4007

Abstract

Consider an undirected and connected graph

G = (V_{G}, E_{G})

, where

V_{G}

and

E_{G}

represent the set of vertices and the set of edges respectively. The concept of edge version of metric dimension and doubly [...] Read more.

Consider an undirected and connected graph

G = (V_{G}, E_{G})

, where

V_{G}

and

E_{G}

represent the set of vertices and the set of edges respectively. The concept of edge version of metric dimension and doubly resolving sets is based on the distances of edges in a graph. In this paper, we find the edge version of metric dimension and doubly resolving sets for the necklace graph. Full article

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

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Mathematics, Volume 6, Issue 11 (November 2018) – 58 articles

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