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# Mathematics, Volume 5, Issue 1 (March 2017) – 18 articles

Cover Story (view full-size image): We provide the derivation of the tracer equation in a many-body underdamped interacting system with generalized non-local interactions. This system could be a polymer, a surface, a single file model, a fluctuating interface according to the type and degree of the interaction. The proposed equation describe initial ballisitic, diffusional and asymptotical subdiffusive stages attained by the probe. Moreover this equation takes the form of a generalized Langevin equation with the damping kernel and the thermal noise satisfying the FD theorem at any given time. View this paper
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Article
Characterization of the Minimizing Graph of the Connected Graphs Whose Complements Are Bicyclic
Mathematics 2017, 5(1), 18; https://doi.org/10.3390/math5010018 - 11 Mar 2017
Cited by 6 | Viewed by 2829
Abstract
In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to [...] Read more.
In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. Let $G 1 , n c$ and $G 2 , n c$ be the classes of the connected graphs of order n whose complements are bicyclic with exactly two and three cycles, respectively. In this paper, we characterize the unique minimizing graph among all the graphs which belong to $G n c = G 1 , n c ∪ G 2 , n c$ , a class of the connected graphs of order n whose complements are bicyclic. Full article
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Article
On the Additively Weighted Harary Index of Some Composite Graphs
Mathematics 2017, 5(1), 16; https://doi.org/10.3390/math5010016 - 07 Mar 2017
Cited by 5 | Viewed by 2690
Abstract
The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The additively weighted Harary index $H A ( G )$ is a modification of the Harary index in which the contributions of [...] Read more.
The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The additively weighted Harary index $H A ( G )$ is a modification of the Harary index in which the contributions of vertex pairs are weighted by the sum of their degrees. This new invariant was introduced in (Alizadeh, Iranmanesh and Došlić. Additively weighted Harary index of some composite graphs, Discrete Math, 2013) and they posed the following question: What is the behavior of $H A ( G )$ when G is a composite graph resulting for example by: splice, link, corona and rooted product? We investigate the additively weighted Harary index for these standard graph products. Then we obtain lower and upper bounds for some of them. Full article
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Article
Certain Concepts of Bipolar Fuzzy Directed Hypergraphs
Mathematics 2017, 5(1), 17; https://doi.org/10.3390/math5010017 - 04 Mar 2017
Cited by 12 | Viewed by 4074
Abstract
A hypergraph is the most developed tool for modeling various practical problems in different fields, including computer sciences, biological sciences, social networks and psychology. Sometimes, given data in a network model are based on bipolar information rather than one sided. To deal with [...] Read more.
A hypergraph is the most developed tool for modeling various practical problems in different fields, including computer sciences, biological sciences, social networks and psychology. Sometimes, given data in a network model are based on bipolar information rather than one sided. To deal with such types of problems, we use mathematical models that are based on bipolar fuzzy (BF) sets. In this research paper, we introduce the concept of BF directed hypergraphs. We describe certain operations on BF directed hypergraphs, including addition, multiplication, vertex-wise multiplication and structural subtraction. We introduce the concept of $B = ( m + , m − )$ -tempered BF directed hypergraphs and investigate some of their properties. We also present an algorithm to compute the minimum arc length of a BF directed hyperpath. Full article
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Article
Dialectical Multivalued Logic and Probabilistic Theory
Mathematics 2017, 5(1), 15; https://doi.org/10.3390/math5010015 - 23 Feb 2017
Cited by 4 | Viewed by 3010
Abstract
There are two probabilistic algebras: one for classical probability and the other for quantum mechanics. Naturally, it is the relation to the object that decides, as in the case of logic, which algebra is to be used. From a paraconsistent multivalued logic therefore, [...] Read more.
There are two probabilistic algebras: one for classical probability and the other for quantum mechanics. Naturally, it is the relation to the object that decides, as in the case of logic, which algebra is to be used. From a paraconsistent multivalued logic therefore, one can derive a probability theory, adding the correspondence between truth value and fortuity. Full article
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Article
A Novel Iterative Algorithm Applied to Totally Asymptotically Nonexpansive Mappings in CAT(0) Spaces
Mathematics 2017, 5(1), 14; https://doi.org/10.3390/math5010014 - 22 Feb 2017
Cited by 8 | Viewed by 3340
Abstract
In this paper we introduce a new iterative algorithm for approximating fixed points of totally asymptotically quasi-nonexpansive mappings on CAT(0) spaces. We prove a strong convergence theorem under suitable conditions. The result we obtain improves and extends several recent results stated by many [...] Read more.
In this paper we introduce a new iterative algorithm for approximating fixed points of totally asymptotically quasi-nonexpansive mappings on CAT(0) spaces. We prove a strong convergence theorem under suitable conditions. The result we obtain improves and extends several recent results stated by many others; they also complement many known recent results in the literature. We then provide some numerical examples to illustrate our main result and to display the efficiency of the proposed algorithm. Full article
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Article
A Few Finite Trigonometric Sums
Mathematics 2017, 5(1), 13; https://doi.org/10.3390/math5010013 - 18 Feb 2017
Cited by 1 | Viewed by 4050
Abstract
Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known; however, sums with products of trigonometric functions can become complicated, and [...] Read more.
Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known; however, sums with products of trigonometric functions can become complicated, and may not have a simple expression in a number of cases. Some of these sums have interesting properties, and can have amazingly simple values. However, only some of them are available in the literature. We obtain a number of such sums using the method of residues. Full article
Article
Fractional Fokker-Planck Equation
Mathematics 2017, 5(1), 12; https://doi.org/10.3390/math5010012 - 11 Feb 2017
Cited by 10 | Viewed by 5534
Abstract
We shall discuss the numerical solution of the Cauchy problem for the fully fractional Fokker-Planck (fFP) equation in connection with Sinc convolution methods. The numerical approximation is based on Caputo and Riesz-Feller fractional derivatives. The use of the transfer function in Laplace and [...] Read more.
We shall discuss the numerical solution of the Cauchy problem for the fully fractional Fokker-Planck (fFP) equation in connection with Sinc convolution methods. The numerical approximation is based on Caputo and Riesz-Feller fractional derivatives. The use of the transfer function in Laplace and Fourier spaces in connection with Sinc convolutions allow to find exponentially converging computing schemes. Examples using different initial conditions demonstrate the effective computations with a small number of grid points on an infinite spatial domain. Full article
(This article belongs to the Special Issue Fractional Differential and Difference Equations)
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Article
The Split Common Fixed Point Problem for a Family of Multivalued Quasinonexpansive Mappings and Totally Asymptotically Strictly Pseudocontractive Mappings in Banach Spaces
Mathematics 2017, 5(1), 11; https://doi.org/10.3390/math5010011 - 11 Feb 2017
Cited by 2 | Viewed by 3346
Abstract
In this paper, we introduce an iterative algorithm for solving the split common fixed point problem for a family of multi-valued quasinonexpansive mappings and totally asymptotically strictly pseudocontractive mappings, as well as for a family of totally quasi-ϕ-asymptotically nonexpansive mappings and [...] Read more.
In this paper, we introduce an iterative algorithm for solving the split common fixed point problem for a family of multi-valued quasinonexpansive mappings and totally asymptotically strictly pseudocontractive mappings, as well as for a family of totally quasi-ϕ-asymptotically nonexpansive mappings and k-quasi-strictly pseudocontractive mappings in the setting of Banach spaces. Our results improve and extend the results of Tang et al., Takahashi, Moudafi, Censor et al., and Byrne et al. Full article
Article
Existence of Mild Solutions for Impulsive Fractional Integro-Differential Inclusions with State-Dependent Delay
Mathematics 2017, 5(1), 9; https://doi.org/10.3390/math5010009 - 25 Jan 2017
Cited by 9 | Viewed by 3410
Abstract
In this manuscript, we implement Bohnenblust–Karlin’s fixed point theorem to demonstrate the existence of mild solutions for a class of impulsive fractional integro-differential inclusions (IFIDI) with state-dependent delay (SDD) in Banach spaces. An example is provided to illustrate the obtained abstract results. Full article
(This article belongs to the Special Issue Fractional Differential and Difference Equations)
Article
Approximation in Müntz Spaces MΛ,p of Lp Functions for 1 < p < ∞ and Bases
Mathematics 2017, 5(1), 10; https://doi.org/10.3390/math5010010 - 25 Jan 2017
Cited by 2 | Viewed by 2478
Abstract
Müntz spaces satisfying the Müntz and gap conditions are considered. A Fourier approximation of functions in the Müntz spaces MΛ,p of Lp functions is studied, where 1 < p < ∞. It is proven that up to an isomorphism and [...] Read more.
Müntz spaces satisfying the Müntz and gap conditions are considered. A Fourier approximation of functions in the Müntz spaces MΛ,p of Lp functions is studied, where 1 < p < ∞. It is proven that up to an isomorphism and a change of variables, these spaces are contained in Weil–Nagy’s class. Moreover, the existence of Schauder bases in the Müntz spaces MΛ,p is investigated. Full article
Article
An Analysis of the Influence of Graph Theory When Preparing for Programming Contests
Mathematics 2017, 5(1), 8; https://doi.org/10.3390/math5010008 - 20 Jan 2017
Cited by 1 | Viewed by 3448
Abstract
The subject known as Programming Contests in the Bachelor’s Degree in Computer Engineering course focuses on solving programming problems frequently met within contests such as the Southwest Europe Regional Contest (SWERC). In order to solve these problems one first needs to model the [...] Read more.
The subject known as Programming Contests in the Bachelor’s Degree in Computer Engineering course focuses on solving programming problems frequently met within contests such as the Southwest Europe Regional Contest (SWERC). In order to solve these problems one first needs to model the problem correctly, find the ideal solution, and then be able to program it without making any mistakes in a very short period of time. Leading multinationals such as Google, Apple, IBM, Facebook and Microsoft place a very high value on these abilities when selecting candidates for posts in their companies. In this communication we present some preliminary results of an analysis of the interaction between two optional subjects in the Computer Science Degree course: Programming Contests (PC) and Graphs, Models and Applications (GMA). The results of this analysis enabled us to make changes to some of the contents in GMA in order to better prepare the students to deal with the challenges they have to face in programming contests. Full article
Article
Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity
Mathematics 2017, 5(1), 7; https://doi.org/10.3390/math5010007 - 17 Jan 2017
Cited by 36 | Viewed by 10687
Abstract
In this paper, the author proposes a new SEIRS model that generalizes several classical deterministic epidemic models (e.g., SIR and SIS and SEIR and SEIRS) involving the relationships between the susceptible S, exposed E, infected I, and recovered R individuals [...] Read more.
In this paper, the author proposes a new SEIRS model that generalizes several classical deterministic epidemic models (e.g., SIR and SIS and SEIR and SEIRS) involving the relationships between the susceptible S, exposed E, infected I, and recovered R individuals for understanding the proliferation of infectious diseases. As a way to incorporate the most important features of the previous models under the assumption of homogeneous mixing (mass-action principle) of the individuals in the population N, the SEIRS model utilizes vital dynamics with unequal birth and death rates, vaccinations for newborns and non-newborns, and temporary immunity. In order to determine the equilibrium points, namely the disease-free and endemic equilibrium points, and study their local stability behaviors, the SEIRS model is rescaled with the total time-varying population and analyzed according to its epidemic condition R0 for two cases of no epidemic (R0 ≤ 1) and epidemic (R0 > 1) using the time-series and phase portraits of the susceptible s, exposed e, infected i, and recovered r individuals. Based on the experimental results using a set of arbitrarily-defined parameters for horizontal transmission of the infectious diseases, the proportional population of the SEIRS model consisted primarily of the recovered r (0.7–0.9) individuals and susceptible s (0.0–0.1) individuals (epidemic) and recovered r (0.9) individuals with only a small proportional population for the susceptible s (0.1) individuals (no epidemic). Overall, the initial conditions for the susceptible s, exposed e, infected i, and recovered r individuals reached the corresponding equilibrium point for local stability: no epidemic (DFE $X ¯ D F E$ ) and epidemic (EE $X ¯ E E$ ). Full article
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Article
Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation
Mathematics 2017, 5(1), 6; https://doi.org/10.3390/math5010006 - 14 Jan 2017
Cited by 9 | Viewed by 5007
Abstract
Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not necessarily minimal) permutation representations $P$ . It is unusual, but significant to recognize that [...] Read more.
Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not necessarily minimal) permutation representations $P$ . It is unusual, but significant to recognize that a $P$ is a Grothendieck’s “dessin d’enfant” $D$ and that a wealth of standard graphs and finite geometries $G$ —such as near polygons and their generalizations—are stabilized by a $D$ . In our paper, tripods $P − D − G$ of rank larger than two, corresponding to simple groups, are organized into classes, e.g., symplectic, unitary, sporadic, etc. (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All of the defined geometries $G ′ s$ have a contextuality parameter close to its maximal value of one. Full article
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Article
Data Clustering with Quantum Mechanics
Mathematics 2017, 5(1), 5; https://doi.org/10.3390/math5010005 - 06 Jan 2017
Cited by 7 | Viewed by 7079
Abstract
Data clustering is a vital tool for data analysis. This work shows that some existing useful methods in data clustering are actually based on quantum mechanics and can be assembled into a powerful and accurate data clustering method where the efficiency of computational [...] Read more.
Data clustering is a vital tool for data analysis. This work shows that some existing useful methods in data clustering are actually based on quantum mechanics and can be assembled into a powerful and accurate data clustering method where the efficiency of computational quantum chemistry eigenvalue methods is therefore applicable. These methods can be applied to scientific data, engineering data and even text. Full article
(This article belongs to the Special Issue Numerical Linear Algebra with Applications)
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Article
Logical Entropy of Dynamical Systems—A General Model
Mathematics 2017, 5(1), 4; https://doi.org/10.3390/math5010004 - 06 Jan 2017
Cited by 15 | Viewed by 3921
Abstract
In the paper by Riečan and Markechová (Fuzzy Sets Syst. 96, 1998), some fuzzy modifications of Shannon’s and Kolmogorov-Sinai’s entropy were studied and the general scheme involving the presented models was introduced. Our aim in this contribution is to provide analogies of these [...] Read more.
In the paper by Riečan and Markechová (Fuzzy Sets Syst. 96, 1998), some fuzzy modifications of Shannon’s and Kolmogorov-Sinai’s entropy were studied and the general scheme involving the presented models was introduced. Our aim in this contribution is to provide analogies of these results for the case of the logical entropy. We define the logical entropy and logical mutual information of finite partitions on the appropriate algebraic structure and prove basic properties of these measures. It is shown that, as a special case, we obtain the logical entropy of fuzzy partitions studied by Markechová and Riečan (Entropy 18, 2016). Finally, using the suggested concept of entropy of partitions we define the logical entropy of a dynamical system and prove that it is the same for two dynamical systems that are isomorphic. Full article
Article
From the Underdamped Generalized Elastic Model to the Single Particle Langevin Description
Mathematics 2017, 5(1), 3; https://doi.org/10.3390/math5010003 - 06 Jan 2017
Cited by 1 | Viewed by 3545
Abstract
The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. While usually defined in the overdamped case, in this paper we formally include the inertial term to account for the initial diffusive stages [...] Read more.
The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. While usually defined in the overdamped case, in this paper we formally include the inertial term to account for the initial diffusive stages of the stochastic dynamics. We derive the generalized Langevin equation for a probe particle and we show that this equation reduces to the usual Langevin equation for Brownian motion, and to the fractional Langevin equation on the long-time limit. Full article
(This article belongs to the Special Issue Fractional Differential and Difference Equations)
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Article
On Autonomy Imposition in Zero Interval Limit Perturbation Expansion for the Spectral Entities of Hilbert–Schmidt Integral Operators
Mathematics 2017, 5(1), 2; https://doi.org/10.3390/math5010002 - 06 Jan 2017
Cited by 2 | Viewed by 2904
Abstract
In this work, we deal with the autonomy issue in the perturbation expansion for the eigenfunctions of a compact Hilbert–Schmidt integral operator. Here, the autonomy points to the perturbation expansion coefficients of the relevant eigenfunction not depending on the perturbation parameter explicitly, but [...] Read more.
In this work, we deal with the autonomy issue in the perturbation expansion for the eigenfunctions of a compact Hilbert–Schmidt integral operator. Here, the autonomy points to the perturbation expansion coefficients of the relevant eigenfunction not depending on the perturbation parameter explicitly, but the dependence on this parameter arises from the coordinate change at the zero interval limit. Moreover, the related half interval length is utilized as the perturbation parameter in the perturbative analyses. Thus, the zero interval limit perturbation for solving the eigenproblem under consideration is developed. The aim of this work is to show that the autonomy imposition brings an important restriction on the kernel of the corresponding integral operator, and the constructed perturbation series is not capable of expressing the exact solution approximately unless a specific type of kernel is considered. The general structure for the encountered constraints is revealed, and the specific class of kernels is identified to this end. Error analysis of the developed method is given. These analyses are also supported by certain illustrative implementations involving the kernels, which are and are not in the specific class addressed above. Thus, the efficiency of the developed method is shown, and the relevant analyses are confirmed. Full article
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Article
Solution of the Master Equation for Quantum Brownian Motion Given by the Schrödinger Equation
Mathematics 2017, 5(1), 1; https://doi.org/10.3390/math5010001 - 22 Dec 2016
Viewed by 3086
Abstract
We consider the master equation of quantum Brownian motion, and with the application of the group invariant transformation, we show that there exists a surface on which the solution of the master equation is given by an autonomous one-dimensional Schrödinger Equation. Full article
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