Effective Potential from the Generalized Time-Dependent Schrödinger Equation*Mathematics* **2016**, *4*(4), 59; doi:10.3390/math4040059 (registering DOI) - 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
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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
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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
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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
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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
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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
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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: ${x}_{n+1}={\displaystyle \frac{{y}_{n-2}}{-1+{y}_{n-2}{x}_{n-1}{y}_{n}}}$ , ${y}_{n+}$ [...] Read more.

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
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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
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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
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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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Preparational Uncertainty Relations for N Continuous Variables*Mathematics* **2016**, *4*(3), 49; doi:10.3390/math4030049 - 19 July 2016**Abstract **

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A smooth function of the second moments of N continuous variables gives rise to an uncertainty relation if it is bounded from below. We present a method to systematically derive such bounds by generalizing an approach applied previously to a single continuous
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A smooth function of the second moments of N continuous variables gives rise to an uncertainty relation if it is bounded from below. We present a method to systematically derive such bounds by generalizing an approach applied previously to a single continuous variable. New uncertainty relations are obtained for multi-partite systems that allow one to distinguish entangled from separable states. We also investigate the geometry of the “uncertainty region” in the $N(2N+1)$ -dimensional space of moments. It is shown to be a convex set, and the points on its boundary are found to be in one-to-one correspondence with pure Gaussian states of minimal uncertainty. For a single degree of freedom, the boundary can be visualized as one sheet of a “Lorentz-invariant” hyperboloid in the three-dimensional space of second moments.
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Sharing of Nonlocality of a Single Member of an Entangled Pair of Qubits Is Not Possible by More than Two Unbiased Observers on the Other Wing*Mathematics* **2016**, *4*(3), 48; doi:10.3390/math4030048 - 16 July 2016**Abstract **

We address the recently posed question as to whether the nonlocality of a single member of an entangled pair of spin $1/2$ particles can be shared among multiple observers on the other wing who act sequentially and independently of each
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We address the recently posed question as to whether the nonlocality of a single member of an entangled pair of spin $1/2$ particles can be shared among multiple observers on the other wing who act sequentially and independently of each other. We first show that the optimality condition for the trade-off between information gain and disturbance in the context of weak or non-ideal measurements emerges naturally when one employs a one-parameter class of positive operator valued measures (POVMs). Using this formalism we then prove analytically that it is impossible to obtain violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality by more than two Bobs in one of the two wings using unbiased input settings with an Alice in the other wing.
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Uncertainty Relations for Quantum Coherence*Mathematics* **2016**, *4*(3), 47; doi:10.3390/math4030047 - 16 July 2016**Abstract **

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Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can there be some trade-off relation between the coherence measures in
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Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can there be some trade-off relation between the coherence measures in different reference bases? We show that the quantum coherence of a state as quantified by the relative entropy of coherence in two or more noncommuting reference bases respects uncertainty like relations for a given state of single and bipartite quantum systems. In the case of bipartite systems, we find that the presence of entanglement may tighten the above relation. Further, we find an upper bound on the sum of the relative entropies of coherence of bipartite quantum states in two noncommuting reference bases. Moreover, we provide an upper bound on the absolute value of the difference of the relative entropies of coherence calculated with respect to two incompatible bases.
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Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices*Mathematics* **2016**, *4*(3), 46; doi:10.3390/math4030046 - 9 July 2016**Abstract **

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We focus on inverse preconditioners based on minimizing $F(X)=1-cos(XA,I)$ , where $XA$ is the preconditioned matrix and A is symmetric and positive definite. We present and analyze gradient-type
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We focus on inverse preconditioners based on minimizing $F(X)=1-cos(XA,I)$ , where $XA$ is the preconditioned matrix and A is symmetric and positive definite. We present and analyze gradient-type methods to minimize $F(X)$ on a suitable compact set. For this, we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of $F(X)$ on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included.
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Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations*Mathematics* **2016**, *4*(3), 45; doi:10.3390/math4030045 - 1 July 2016**Abstract **

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Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in the one-dimensional case are introduced and analysed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of the Laplacian subject to some
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Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in the one-dimensional case are introduced and analysed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of the Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space-fractional partial differential equations. The approximated stochastic space-fractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in the ${L}^{2}$ -norm are obtained, and numerical examples are given.
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Cohen Macaulayness and Arithmetical Rank of Generalized Theta Graphs*Mathematics* **2016**, *4*(3), 43; doi:10.3390/math4030043 - 29 June 2016**Abstract **

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In this paper, we study some algebraic invariants of the edge ideal of generalized theta graphs, such as arithmetical rank, big height and height. We give an upper bound for the difference between the arithmetical rank and big height. Moreover, all Cohen-Macaulay
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In this paper, we study some algebraic invariants of the edge ideal of generalized theta graphs, such as arithmetical rank, big height and height. We give an upper bound for the difference between the arithmetical rank and big height. Moreover, all Cohen-Macaulay (and unmixed) graphs of this type will be characterized.
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Exact Discrete Analogs of Canonical Commutation and Uncertainty Relations*Mathematics* **2016**, *4*(3), 44; doi:10.3390/math4030044 - 28 June 2016**Abstract **

An exact discretization of the canonical commutation and corresponding uncertainty relations are suggested. We prove that the canonical commutation relations of discrete quantum mechanics, which is based on standard finite difference, holds for constant wave functions only. In this paper, we use
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An exact discretization of the canonical commutation and corresponding uncertainty relations are suggested. We prove that the canonical commutation relations of discrete quantum mechanics, which is based on standard finite difference, holds for constant wave functions only. In this paper, we use the recently proposed exact discretization of derivatives, which is based on differences that are represented by infinite series. This new mathematical tool allows us to build sensible discrete quantum mechanics based on the suggested differences and includes the correct canonical commutation and uncertainty relations.
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Exponential Energy Decay of Solutions for a Transmission Problem With Viscoelastic Term and Delay*Mathematics* **2016**, *4*(2), 42; doi:10.3390/math4020042 - 9 June 2016**Abstract **

In our previous work (Journal of Nonlinear Science and Applications 9: 1202–1215, 2016), we studied the well-posedness and general decay rate for a transmission problem in a bounded domain with a viscoelastic term and a delay term. In this paper, we continue
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In our previous work (Journal of Nonlinear Science and Applications 9: 1202–1215, 2016), we studied the well-posedness and general decay rate for a transmission problem in a bounded domain with a viscoelastic term and a delay term. In this paper, we continue to study the similar problem but without the frictional damping term. The main difficulty arises since we have no frictional damping term to control the delay term in the estimate of the energy decay. By introducing suitable energy and Lyapunov functionals, we establish an exponential decay result for the energy.
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Entropic Uncertainty Relations for Successive Generalized Measurements*Mathematics* **2016**, *4*(2), 41; doi:10.3390/math4020041 - 7 June 2016**Abstract **

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We derive entropic uncertainty relations for successive generalized measurements by using general descriptions of quantum measurement within two distinctive operational scenarios. In the first scenario, by merging two successive measurements into one we consider successive measurement scheme as a method to perform
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We derive entropic uncertainty relations for successive generalized measurements by using general descriptions of quantum measurement within two distinctive operational scenarios. In the first scenario, by merging two successive measurements into one we consider successive measurement scheme as a method to perform an overall composite measurement. In the second scenario, on the other hand, we consider it as a method to measure a pair of jointly measurable observables by marginalizing over the distribution obtained in this scheme. In the course of this work, we identify that limits on one’s ability to measure with low uncertainty via this scheme come from intrinsic unsharpness of observables obtained in each scenario. In particular, for the Lüders instrument, disturbance caused by the first measurement to the second one gives rise to the unsharpness at least as much as incompatibility of the observables composing successive measurement.
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Morphisms and Order Ideals of Toric Posets*Mathematics* **2016**, *4*(2), 39; doi:10.3390/math4020039 - 4 June 2016**Abstract **

Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes
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Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper, we define toric intervals and toric order-preserving maps, which lead to toric analogues of poset morphisms and order ideals. We develop this theory, discuss some fundamental differences between the toric and ordinary cases, and outline some areas for future research. Additionally, we provide a connection to cyclic reducibility and conjugacy in Coxeter groups.
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