On Some Extended Block Krylov Based Methods for Large Scale Nonsymmetric Stein Matrix Equations*Mathematics* **2017**, *5*(2), 21; doi:10.3390/math5020021 - 27 March 2017**Abstract **

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In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term $AXB-X+E{F}^{T}=0$ . These matrix equations appear in many applications in discrete-time control problems, filtering and image

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In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term $AXB-X+E{F}^{T}=0$ . These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the norm of the residual. We give some results on the residual and error norms and report some numerical experiments.
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In this paper, some properties of $\mathcal{F}$ -harmonic and conformal $\mathcal{F}$ -harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic $\mathcal{F}$ -harmonic maps are constructed.

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In this paper, some properties of $\mathcal{F}$ -harmonic and conformal $\mathcal{F}$ -harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic $\mathcal{F}$ -harmonic maps are constructed.
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A Generalization of *b*-Metric Space and Some Fixed Point Theorems*Mathematics* **2017**, *5*(2), 19; doi:10.3390/math5020019 - 23 March 2017**Abstract **
In this paper, inspired by the concept of *b*-metric space, we introduce the concept of extended *b*-metric space. We also establish some fixed point theorems for self-mappings defined on such spaces. Our results extend/generalize many pre-existing results in literature.
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Characterization of the Minimizing Graph of the Connected Graphs Whose Complements Are Bicyclic*Mathematics* **2017**, *5*(1), 18; doi:10.3390/math5010018 - 11 March 2017**Abstract **

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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

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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 ${\mathcal{G}}_{1,n}^{c}$ and ${\mathcal{G}}_{2,n}^{c}$ 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 ${\mathcal{G}}_{n}^{c}={\mathcal{G}}_{1,n}^{c}\cup {\mathcal{G}}_{2,n}^{c}$ , a class of the connected graphs of order *n* whose complements are bicyclic.
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On the Additively Weighted Harary Index of Some Composite Graphs*Mathematics* **2017**, *5*(1), 16; doi:10.3390/math5010016 - 7 March 2017**Abstract **

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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 ${H}_{A}\left(G\right)$ is a modification of the Harary index in which the contributions of

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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 ${H}_{A}\left(G\right)$ 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 *${H}_{A}\left(G\right)$ 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.
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Certain Concepts of Bipolar Fuzzy Directed Hypergraphs*Mathematics* **2017**, *5*(1), 17; doi:10.3390/math5010017 - 4 March 2017**Abstract **

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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

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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.
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Dialectical Multivalued Logic and Probabilistic Theory*Mathematics* **2017**, *5*(1), 15; doi:10.3390/math5010015 - 23 February 2017**Abstract **

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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,

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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.
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A Novel Iterative Algorithm Applied to Totally Asymptotically Nonexpansive Mappings in CAT(0) Spaces*Mathematics* **2017**, *5*(1), 14; doi:10.3390/math5010014 - 22 February 2017**Abstract **

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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

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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.
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A Few Finite Trigonometric Sums*Mathematics* **2017**, *5*(1), 13; doi:10.3390/math5010013 - 18 February 2017**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

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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.
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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; doi:10.3390/math5010011 - 11 February 2017**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

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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.
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Fractional Fokker-Planck Equation*Mathematics* **2017**, *5*(1), 12; doi:10.3390/math5010012 - 11 February 2017**Abstract **

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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

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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.
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Existence of Mild Solutions for Impulsive Fractional Integro-Differential Inclusions with State-Dependent Delay*Mathematics* **2017**, *5*(1), 9; doi:10.3390/math5010009 - 25 January 2017**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.
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Approximation in Müntz Spaces M_{Λ,p} of L_{p} Functions for 1 < *p* < ∞ and Bases*Mathematics* **2017**, *5*(1), 10; doi:10.3390/math5010010 - 25 January 2017**Abstract **

Müntz spaces satisfying the Müntz and gap conditions are considered. A Fourier approximation of functions in the Müntz spaces M_{Λ,p} of L_{p} functions is studied, where 1 < *p* < ∞. It is proven that up to an isomorphism and

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Müntz spaces satisfying the Müntz and gap conditions are considered. A Fourier approximation of functions in the Müntz spaces M_{Λ,p} of L_{p} 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 M_{Λ,p} is investigated.
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An Analysis of the Influence of Graph Theory When Preparing for Programming Contests*Mathematics* **2017**, *5*(1), 8; doi:10.3390/math5010008 - 20 January 2017**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

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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.
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Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity*Mathematics* **2017**, *5*(1), 7; doi:10.3390/math5010007 - 17 January 2017**Abstract **

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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

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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 *R*_{0} for two cases of no epidemic (*R*_{0} ≤ 1) and epidemic (*R*_{0} > 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 ${\overline{X}}_{DFE}$ ) and epidemic (EE ${\overline{X}}_{EE}$ ).
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Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation*Mathematics* **2017**, *5*(1), 6; doi:10.3390/math5010006 - 14 January 2017**Abstract **

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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 $\mathcal{P}$ . It is unusual, but significant to recognize that

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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 $\mathcal{P}$ . It is unusual, but significant to recognize that a $\mathcal{P}$ is a Grothendieck’s “dessin d’enfant” $\mathcal{D}$ and that a wealth of standard graphs and finite geometries $\mathcal{G}$ —such as near polygons and their generalizations—are stabilized by a $\mathcal{D}$ . In our paper, tripods $\mathcal{P}-\mathcal{D}-\mathcal{G}$ 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 ${\mathcal{G}}^{\prime}s$ have a contextuality parameter close to its maximal value of one.
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Logical Entropy of Dynamical Systems—A General Model*Mathematics* **2017**, *5*(1), 4; doi:10.3390/math5010004 - 6 January 2017**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

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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.
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On Autonomy Imposition in Zero Interval Limit Perturbation Expansion for the Spectral Entities of Hilbert–Schmidt Integral Operators*Mathematics* **2017**, *5*(1), 2; doi:10.3390/math5010002 - 6 January 2017**Abstract **

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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

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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.
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From the Underdamped Generalized Elastic Model to the Single Particle Langevin Description*Mathematics* **2017**, *5*(1), 3; doi:10.3390/math5010003 - 6 January 2017**Abstract **

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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

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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.
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Data Clustering with Quantum Mechanics*Mathematics* **2017**, *5*(1), 5; doi:10.3390/math5010005 - 6 January 2017**Abstract **

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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

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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.
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