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Mathematics, Volume 4, Issue 4 (December 2016)

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Research

Open AccessArticle Analysis of Dynamics in Multiphysics Modelling of Active Faults
Mathematics 2016, 4(4), 57; doi:10.3390/math4040057
Received: 30 April 2016 / Revised: 9 September 2016 / Accepted: 14 September 2016 / Published: 22 September 2016
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Abstract
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). Full article
(This article belongs to the Special Issue Applied Mathematics and Mechanics)
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Open AccessFeature PaperArticle Finite-Time Stabilization of Homogeneous Non-Lipschitz Systems
Mathematics 2016, 4(4), 58; doi:10.3390/math4040058
Received: 17 June 2016 / Revised: 22 August 2016 / Accepted: 13 September 2016 / Published: 24 September 2016
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Abstract
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 C1 positive definite and proper Lyapunov function that guarantees finite-time stability of the non-Lipschitz nonlinear systems. Full article
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Open AccessArticle Effective Potential from the Generalized Time-Dependent Schrödinger Equation
Mathematics 2016, 4(4), 59; doi:10.3390/math4040059
Received: 5 August 2016 / Revised: 20 September 2016 / Accepted: 21 September 2016 / Published: 28 September 2016
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Abstract
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. Full article
(This article belongs to the Special Issue Fractional Differential and Difference Equations)
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Open AccessArticle 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
Received: 20 July 2016 / Revised: 16 September 2016 / Accepted: 21 September 2016 / Published: 19 October 2016
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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. Full article
Open AccessArticle Nuclear Space Facts, Strange and Plain
Mathematics 2016, 4(4), 61; doi:10.3390/math4040061
Received: 31 May 2016 / Revised: 23 September 2016 / Accepted: 27 September 2016 / Published: 9 October 2016
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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. Full article
Open AccessArticle Interval Type 2 Fuzzy Set in Fuzzy Shortest Path Problem
Mathematics 2016, 4(4), 62; doi:10.3390/math4040062
Received: 10 March 2016 / Revised: 13 September 2016 / Accepted: 22 September 2016 / Published: 9 October 2016
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Abstract
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. Full article
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Open AccessArticle Positive Solutions for Nonlinear Caputo Type Fractional q-Difference Equations with Integral Boundary Conditions
Mathematics 2016, 4(4), 63; doi:10.3390/math4040063
Received: 9 September 2016 / Revised: 19 October 2016 / Accepted: 21 October 2016 / Published: 2 November 2016
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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. Full article
Open AccessArticle Viability for Semilinear Differential Equations with Infinite Delay
Mathematics 2016, 4(4), 64; doi:10.3390/math4040064
Received: 1 September 2016 / Revised: 25 October 2016 / Accepted: 2 November 2016 / Published: 8 November 2016
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Abstract
Let X be a Banach space, A:D(A)XX the generator of a compact C0-semigroup S(t):XX,t0, D(·):(
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Let X be a Banach space, A : D ( A ) X X the generator of a compact C 0 -semigroup S ( t ) : X X , t 0 , D ( · ) : ( a , b ) 2 X a tube in X, and f : ( a , b ) × B X a function of Carathéodory type. The main result of this paper is that a necessary and sufficient condition in order that D ( · ) be viable of the semilinear differential equation with infinite delay u ( t ) = A u ( t ) + f ( t , u t ) , t [ t 0 , t 0 + T ] , u t 0 = ϕ B is the tangency condition lim inf h 0 h 1 d ( S ( h ) v ( 0 ) + h f ( t , v ) ; D ( t + h ) ) = 0 for almost every t ( a , b ) and every v B with v ( 0 ) D ( t ) . Full article
Open AccessArticle Some Determinantal Expressions and Recurrence Relations of the Bernoulli Polynomials
Mathematics 2016, 4(4), 65; doi:10.3390/math4040065
Received: 27 April 2016 / Revised: 8 November 2016 / Accepted: 17 November 2016 / Published: 24 November 2016
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Abstract
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. Full article
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Open AccessArticle Best Proximity Point Theorems in Partially Ordered b-Quasi Metric Spaces
Mathematics 2016, 4(4), 66; doi:10.3390/math4040066
Received: 29 September 2016 / Revised: 15 November 2016 / Accepted: 21 November 2016 / Published: 26 November 2016
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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. Full article
Open AccessFeature PaperArticle Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus
Mathematics 2016, 4(4), 67; doi:10.3390/math4040067
Received: 1 November 2016 / Revised: 29 November 2016 / Accepted: 2 December 2016 / Published: 9 December 2016
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Abstract
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. Full article
(This article belongs to the Special Issue Fractional Differential and Difference Equations)
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Open AccessArticle 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
Received: 19 October 2016 / Revised: 22 November 2016 / Accepted: 22 November 2016 / Published: 10 December 2016
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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 ( ψ , φ ) g -generalized weakly contractive conditions. Full article
Open AccessFeature PaperArticle Proposal for the Formalization of Dialectical Logic
Mathematics 2016, 4(4), 69; doi:10.3390/math4040069
Received: 18 August 2016 / Revised: 27 November 2016 / Accepted: 6 December 2016 / Published: 11 December 2016
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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. Full article
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