Proposal for the Formalization of Dialectical Logic*Mathematics* **2016**, *4*(4), 69; doi:10.3390/math4040069 (registering DOI) - 11 December 2016**Abstract **

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 (registering DOI) - 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 (registering DOI) - 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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Positive Solutions for Nonlinear Caputo Type Fractional *q*-Difference Equations with Integral Boundary Conditions*Mathematics* **2016**, *4*(4), 63; doi:10.3390/math4040063 - 2 November 2016**Abstract **
In this paper, by applying some well-known fixed point theorems, we investigate the existence of positive solutions for a class of nonlinear Caputo type fractional *q*-difference equations with integral boundary conditions. Finally, some interesting examples are presented to illustrate the main results.
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A Study of Controllability of Impulsive Neutral Evolution Integro-Differential Equations with State-Dependent Delay in Banach Spaces*Mathematics* **2016**, *4*(4), 60; doi:10.3390/math4040060 - 19 October 2016**Abstract **

In this paper, we study the problem of controllability of impulsive neutral evolution integro-differential equations with state-dependent delay in Banach spaces. The main results are completely new and are obtained by using Sadovskii’s fixed point theorem, theory of resolvent operators, and an abstract

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In this paper, we study the problem of controllability of impulsive neutral evolution integro-differential equations with state-dependent delay in Banach spaces. The main results are completely new and are obtained by using Sadovskii’s fixed point theorem, theory of resolvent operators, and an abstract phase space. An example is given to illustrate the theory.
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Nuclear Space Facts, Strange and Plain*Mathematics* **2016**, *4*(4), 61; doi:10.3390/math4040061 - 9 October 2016**Abstract **
We present a scenic but practical guide through nuclear spaces and their dual spaces, examining useful, unexpected, and often unfamiliar results both for nuclear spaces and their strong and weak duals.
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Interval Type 2 Fuzzy Set in Fuzzy Shortest Path Problem*Mathematics* **2016**, *4*(4), 62; doi:10.3390/math4040062 - 9 October 2016**Abstract **

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The shortest path problem (SPP) is one of the most important combinatorial optimization problems in graph theory due to its various applications. The uncertainty existing in the real world problems makes it difficult to determine the arc lengths exactly. The fuzzy set is

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The shortest path problem (SPP) is one of the most important combinatorial optimization problems in graph theory due to its various applications. The uncertainty existing in the real world problems makes it difficult to determine the arc lengths exactly. The fuzzy set is one of the popular tools to represent and handle uncertainty in information due to incompleteness or inexactness. In most cases, the SPP in fuzzy graph, called the fuzzy shortest path problem (FSPP) uses type-1 fuzzy set (T1FS) as arc length. Uncertainty in the evaluation of membership degrees due to inexactness of human perception is not considered in T1FS. An interval type-2 fuzzy set (IT2FS) is able to tackle this uncertainty. In this paper, we use IT2FSs to represent the arc lengths of a fuzzy graph for FSPP. We call this problem an interval type-2 fuzzy shortest path problem (IT2FSPP). We describe the utility of IT2FSs as arc lengths and its application in different real world shortest path problems. Here, we propose an algorithm for IT2FSPP. In the proposed algorithm, we incorporate the uncertainty in Dijkstra’s algorithm for SPP using IT2FS as arc length. The path algebra corresponding to the proposed algorithm and the generalized algorithm based on the path algebra are also presented here. Numerical examples are used to illustrate the effectiveness of the proposed approach.
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Effective Potential from the Generalized Time-Dependent Schrödinger Equation*Mathematics* **2016**, *4*(4), 59; doi:10.3390/math4040059 - 28 September 2016**Abstract **

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We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to

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We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode.
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Finite-Time Stabilization of Homogeneous Non-Lipschitz Systems*Mathematics* **2016**, *4*(4), 58; doi:10.3390/math4040058 - 24 September 2016**Abstract **

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This paper focuses on the problem of finite-time stabilization of homogeneous, non-Lipschitz systems with dilations. A key contribution of this paper is the design of a virtual recursive *Hölder*, non-Lipschitz state feedback, which renders the non-Lipschitz systems in the special case dominated

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This paper focuses on the problem of finite-time stabilization of homogeneous, non-Lipschitz systems with dilations. A key contribution of this paper is the design of a virtual recursive *Hölder*, non-Lipschitz state feedback, which renders the non-Lipschitz systems in the special case dominated by a lower-triangular nonlinear system finite-time stable. The proof is based on a recursive design algorithm developed recently to construct the virtual *Hölder* continuous, finite-time stabilizer as well as a *C*^{1} positive definite and proper Lyapunov function that guarantees finite-time stability of the non-Lipschitz nonlinear systems.
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Analysis of Dynamics in Multiphysics Modelling of Active Faults*Mathematics* **2016**, *4*(4), 57; doi:10.3390/math4040057 - 22 September 2016**Abstract **

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Instabilities in Geomechanics appear on multiple scales involving multiple physical processes. They appear often as planar features of localised deformation (faults), which can be relatively stable creep or display rich dynamics, sometimes culminating in earthquakes. To study those features, we propose a fundamental

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Instabilities in Geomechanics appear on multiple scales involving multiple physical processes. They appear often as planar features of localised deformation (faults), which can be relatively stable creep or display rich dynamics, sometimes culminating in earthquakes. To study those features, we propose a fundamental physics-based approach that overcomes the current limitations of statistical rule-based methods and allows a physical understanding of the nucleation and temporal evolution of such faults. In particular, we formulate the coupling between temperature and pressure evolution in the faults through their multiphysics energetic process(es). We analyse their multiple steady states using numerical continuation methods and characterise their transient dynamics by studying the time-dependent problem near the critical Hopf points. We find that the global system can be characterised by a homoclinic bifurcation that depends on the two main dimensionless groups of the underlying physical system. The Gruntfest number determines the onset of the localisation phenomenon, while the dynamics are mainly controlled by the Lewis number, which is the ratio of energy diffusion over mass diffusion. Here, we show that the Lewis number is the critical parameter for dynamics of the system as it controls the time evolution of the system for a given energy supply (Gruntfest number).
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Quantum Measurements, Stochastic Networks, the Uncertainty Principle, and the Not So Strange “Weak Values”*Mathematics* **2016**, *4*(3), 56; doi:10.3390/math4030056 - 15 September 2016**Abstract **

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Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the

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Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become “real”, and the mean values of a quantity (a functional) are directly related to the frequencies with which the paths are traveled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities’ amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one from distinguishing between interfering alternatives, while leaving the interference between them intact.
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Amenability Modulo an Ideal of Second Duals of Semigroup Algebras*Mathematics* **2016**, *4*(3), 55; doi:10.3390/math4030055 - 13 September 2016**Abstract **

The aim of this paper is to investigate the amenability modulo, an ideal of Banach algebras with emphasis on applications to homological algebras. In doing so, we show that amenability modulo, an ideal of ${A}^{**}$ implies amenability modulo, an ideal of

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The aim of this paper is to investigate the amenability modulo, an ideal of Banach algebras with emphasis on applications to homological algebras. In doing so, we show that amenability modulo, an ideal of ${A}^{**}$ implies amenability modulo, an ideal of *A*. Finally, for a large class of semigroups, we prove that ${l}^{1}{(S)}^{**}$ is amenable modulo ${I}_{\sigma}^{**}$ if and only if an appropriate group homomorphic image of *S* is finite, where ${I}_{\sigma}$ is the closed ideal induced by the least group congruence $\sigma $ .
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Quantum Incompatibility in Collective Measurements*Mathematics* **2016**, *4*(3), 54; doi:10.3390/math4030054 - 10 September 2016**Abstract **

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We study the compatibility (or joint measurability) of quantum observables in a setting where the experimenter has access to multiple copies of a given quantum system, rather than performing the experiments on each individual copy separately. We introduce the index of incompatibility as

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We study the compatibility (or joint measurability) of quantum observables in a setting where the experimenter has access to multiple copies of a given quantum system, rather than performing the experiments on each individual copy separately. We introduce the index of incompatibility as a quantifier of incompatibility in this multi-copy setting, as well as the notion of the compatibility stack representing various compatibility relations present in a given set of observables. We then prove a general structure theorem for multi-copy joint observables and use it to prove that all abstract compatibility stacks with three vertices have realizations in terms of quantum observables.
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Solution for Rational Systems of Difference Equations of Order Three*Mathematics* **2016**, *4*(3), 53; doi:10.3390/math4030053 - 3 September 2016**Abstract **

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In this paper, we consider the solution and periodicity of the following systems of difference equations: ${x}_{n+1}={\displaystyle \frac{{y}_{n-2}}{-1+{y}_{n-2}{x}_{n-1}{y}_{n}}}$ , ${y}_{n+1}$

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In this paper, we consider the solution and periodicity of the following systems of difference equations: ${x}_{n+1}={\displaystyle \frac{{y}_{n-2}}{-1+{y}_{n-2}{x}_{n-1}{y}_{n}}}$ , ${y}_{n+1}={\displaystyle \frac{{x}_{n-2}}{\pm 1\pm {x}_{n-2}{y}_{n-1}{x}_{n}}}$ , with initial conditions ${x}_{-2},\phantom{\rule{0.277778em}{0ex}}{x}_{-1},\phantom{\rule{4pt}{0ex}}{x}_{0},\phantom{\rule{3.33333pt}{0ex}}{y}_{-2},\phantom{\rule{0.277778em}{0ex}}{y}_{-1},\phantom{\rule{0.277778em}{0ex}}$ and$\phantom{\rule{0.277778em}{0ex}}{y}_{0}\phantom{\rule{3.33333pt}{0ex}}$ are nonzero real numbers.
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Role of Measurement Incompatibility and Uncertainty in Determining Nonlocality*Mathematics* **2016**, *4*(3), 52; doi:10.3390/math4030052 - 15 August 2016**Abstract **

It has been recently shown that measurement incompatibility and fine grained uncertainty—a particular form of preparation uncertainty relation—are deeply related to the nonlocal feature of quantum mechanics. In particular, the degree of measurement incompatibility in a no-signaling theory determines the bound on the

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It has been recently shown that measurement incompatibility and fine grained uncertainty—a particular form of preparation uncertainty relation—are deeply related to the nonlocal feature of quantum mechanics. In particular, the degree of measurement incompatibility in a no-signaling theory determines the bound on the violation of Bell-CHSH inequality, and a similar role is also played by (fine-grained) uncertainty along with steering, a subtle non-local phenomenon. We review these connections, along with comments on the difference in the roles played by measurement incompatibility and uncertainty. We also discuss why the toy model of Spekkens (Phys. Rev. A 75, 032110 (2007)) shows no nonlocal feature even though steering is present in this theory.
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A New Approach to Study Fixed Point of Multivalued Mappings in Modular Metric Spaces and Applications*Mathematics* **2016**, *4*(3), 51; doi:10.3390/math4030051 - 8 August 2016**Abstract **

The purpose of this paper is to present a new approach to study the existence of fixed points for multivalued *F*-contraction in the setting of modular metric spaces. In establishing this connection, we introduce the notion of multivalued *F*-contraction and prove

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The purpose of this paper is to present a new approach to study the existence of fixed points for multivalued *F*-contraction in the setting of modular metric spaces. In establishing this connection, we introduce the notion of multivalued *F*-contraction and prove corresponding fixed point theorems in complete modular metric space with some specific assumption on the modular. Then we apply our results to establish the existence of solutions for a certain type of non-linear integral equations.
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Complete Classification of Cylindrically Symmetric Static Spacetimes and the Corresponding Conservation Laws*Mathematics* **2016**, *4*(3), 50; doi:10.3390/math4030050 - 8 August 2016**Abstract **

In this paper we find the Noether symmetries of the Lagrangian of cylindrically symmetric static spacetimes. Using this approach we recover all cylindrically symmetric static spacetimes appeared in the classification by isometries and homotheties. We give different classes of cylindrically symmetric static spacetimes

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In this paper we find the Noether symmetries of the Lagrangian of cylindrically symmetric static spacetimes. Using this approach we recover all cylindrically symmetric static spacetimes appeared in the classification by isometries and homotheties. We give different classes of cylindrically symmetric static spacetimes along with the Noether symmetries of the corresponding Lagrangians and conservation laws.
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