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

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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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Solution of the Master Equation for Quantum Brownian Motion Given by the Schrödinger Equation*Mathematics* **2017**, *5*(1), 1; doi:10.3390/math5010001 - 22 December 2016**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.
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Proposal for the Formalization of Dialectical Logic*Mathematics* **2016**, *4*(4), 69; doi:10.3390/math4040069 - 11 December 2016**Abstract **

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Classical logic is typically concerned with abstract analysis. The problem for a synthetic logic is to transcend and unify available data to reconstruct the object as a totality. Three rules are proposed to pass from classic logic to synthetic logic. We present the

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Classical logic is typically concerned with abstract analysis. The problem for a synthetic logic is to transcend and unify available data to reconstruct the object as a totality. Three rules are proposed to pass from classic logic to synthetic logic. We present the category logic of qualitative opposition using examples from various sciences. This logic has been defined to include the neuter as part of qualitative opposition. The application of these rules to qualitative opposition, and, in particular, its neuter, demonstrated that a synthetic logic allows the truth of some contradictions. This synthetic logic is dialectical with a multi-valued logic, which gives every proposition a truth value in the interval [0 ,1 ] that is the square of the modulus of a complex number. In this dialectical logic, contradictions of the neuter of an opposition may be true.
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Results on Coincidence and Common Fixed Points for (ψ,φ)_{g}-Generalized Weakly Contractive Mappings in Ordered Metric Spaces*Mathematics* **2016**, *4*(4), 68; doi:10.3390/math4040068 - 10 December 2016**Abstract **

Inspired by a metrical-fixed point theorem from Choudhury et al. (*Nonlinear Anal*. **2011**, *74*, 2116–2126), we prove some order-theoretic results which generalize several core results of the existing literature, especially the two main results of Harjani and Sadarangani (

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Inspired by a metrical-fixed point theorem from Choudhury et al. (*Nonlinear Anal*. **2011**, *74*, 2116–2126), we prove some order-theoretic results which generalize several core results of the existing literature, especially the two main results of Harjani and Sadarangani (*Nonlinear Anal*. **2009**, *71*, 3403–3410 and **2010**, *72*, 1188–1197). We demonstrate the realized improvement obtained in our results by using a suitable example. As an application, we also prove a result for mappings satisfying integral type ${(\mathsf{\psi},\mathsf{\phi})}_{g}$ -generalized weakly contractive conditions.
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Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus*Mathematics* **2016**, *4*(4), 67; doi:10.3390/math4040067 - 9 December 2016**Abstract **

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The main objective of this paper is to generalize the Extended Irreversible Thermodynamics in order to include the anomalous transport in systems in non-equilibrium conditions. Considering the generalized entropy, the corresponding flux and entropy production, and using the time fractional derivative, we have

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The main objective of this paper is to generalize the Extended Irreversible Thermodynamics in order to include the anomalous transport in systems in non-equilibrium conditions. Considering the generalized entropy, the corresponding flux and entropy production, and using the time fractional derivative, we have derived a space-time generalized telegrapher’s equation with a fractional nested hierarchy which can be used in separate developments for the mass transport, for the heat conduction and for the flux of ions. We have obtained a new formalism which includes the contribution of fast of higher-order fluxes in the mesoscopic and inhomogeneous media. The results take the form of continued fraction expansions. The balance equations are used in a scheme of continued fractions, and they appear as a closure condition. In this way the transport equation and its corresponding wave number-frequency relation are obtained, both of them in the mathematical structure of the continued fraction scheme. Numerical examples are included to show the dispersive nature of the solutions, and the generalized fractional transport equation in the same mathematical form, which can be applied to the mass transport, the heat conduction and the flux of ions.
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Best Proximity Point Theorems in Partially Ordered *b*-Quasi Metric Spaces*Mathematics* **2016**, *4*(4), 66; doi:10.3390/math4040066 - 26 November 2016**Abstract **
In this paper, we introduce the notion of an ordered rational proximal contraction in partially ordered *b*-quasi metric spaces. We shall then prove some best proximity point theorems in partially ordered *b*-quasi metric spaces.
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Some Determinantal Expressions and Recurrence Relations of the Bernoulli Polynomials*Mathematics* **2016**, *4*(4), 65; doi:10.3390/math4040065 - 24 November 2016**Abstract **

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In the paper, the authors recall some known determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, and present several new recurrence

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In the paper, the authors recall some known determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, and present several new recurrence relations for the Bernoulli numbers and polynomials.
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Viability for Semilinear Differential Equations with Infinite Delay*Mathematics* **2016**, *4*(4), 64; doi:10.3390/math4040064 - 8 November 2016**Abstract **

Let *X* be a Banach space, $A:D(A)\subset X\to X$ the generator of a compact ${C}_{0}$ -semigroup $S(t):X\to X,t\ge 0$ , $D(\xb7):($

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Let *X* be a Banach space, $A:D(A)\subset X\to X$ the generator of a compact ${C}_{0}$ -semigroup $S(t):X\to X,t\ge 0$ , $D(\xb7):(a,b)\to {2}^{X}$ a tube in *X*, and $f:(a,b)\times \mathcal{B}\to X$ a function of Carathéodory type. The main result of this paper is that a necessary and sufficient condition in order that $D(\xb7)$ be viable of the semilinear differential equation with infinite delay ${u}^{\prime}(t)=Au(t)+f(t,{u}_{t}),t\in [{t}_{0},{t}_{0}+T],{u}_{{t}_{0}}=\varphi \in \mathcal{B}$ is the tangency condition ${lim\; inf}_{h\downarrow 0}{h}^{-1}d(S(h)v(0)+hf(t,v);D(t+h))=0$ for almost every $t\in (a,b)$ and every $v\in \mathcal{B}$ with $v(0)\in D(t)$ .
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