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Mathematics, Volume 4, Issue 3 (September 2016) – 14 articles

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Open AccessArticle
Quantum Measurements, Stochastic Networks, the Uncertainty Principle, and the Not So Strange “Weak Values”
Mathematics 2016, 4(3), 56; https://doi.org/10.3390/math4030056 - 15 Sep 2016
Cited by 7 | Viewed by 1948
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Amenability Modulo an Ideal of Second Duals of Semigroup Algebras
Mathematics 2016, 4(3), 55; https://doi.org/10.3390/math4030055 - 13 Sep 2016
Cited by 2 | Viewed by 1317
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 [...] Read more.
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 σ * * if and only if an appropriate group homomorphic image of S is finite, where I σ is the closed ideal induced by the least group congruence σ . Full article
Open AccessArticle
Quantum Incompatibility in Collective Measurements
Mathematics 2016, 4(3), 54; https://doi.org/10.3390/math4030054 - 10 Sep 2016
Cited by 2 | Viewed by 1712
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Open AccessArticle
Solution for Rational Systems of Difference Equations of Order Three
Mathematics 2016, 4(3), 53; https://doi.org/10.3390/math4030053 - 03 Sep 2016
Cited by 4 | Viewed by 1551
Abstract
In this paper, we consider the solution and periodicity of the following systems of difference equations: x n + 1 = y n 2 1 + y n 2 x n 1 y n , y n + 1 [...] Read more.
In this paper, we consider the solution and periodicity of the following systems of difference equations: x n + 1 = y n 2 1 + y n 2 x n 1 y n , y n + 1 = x n 2 ± 1 ± x n 2 y n 1 x n , with initial conditions x 2 , x 1 , x 0 , y 2 , y 1 , and y 0 are nonzero real numbers. Full article
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Open AccessArticle
Role of Measurement Incompatibility and Uncertainty in Determining Nonlocality
Mathematics 2016, 4(3), 52; https://doi.org/10.3390/math4030052 - 15 Aug 2016
Cited by 6 | Viewed by 1492
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
Open AccessArticle
A New Approach to Study Fixed Point of Multivalued Mappings in Modular Metric Spaces and Applications
Mathematics 2016, 4(3), 51; https://doi.org/10.3390/math4030051 - 08 Aug 2016
Cited by 3 | Viewed by 1862
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fixed Point Theorems and Applications)
Open AccessArticle
Complete Classification of Cylindrically Symmetric Static Spacetimes and the Corresponding Conservation Laws
Mathematics 2016, 4(3), 50; https://doi.org/10.3390/math4030050 - 08 Aug 2016
Cited by 10 | Viewed by 1334
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 [...] Read more.
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. Full article
Open AccessFeature PaperArticle
Preparational Uncertainty Relations for N Continuous Variables
Mathematics 2016, 4(3), 49; https://doi.org/10.3390/math4030049 - 19 Jul 2016
Cited by 6 | Viewed by 1769
Abstract
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. [...] Read more.
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 ( 2 N + 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. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Open AccessArticle
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; https://doi.org/10.3390/math4030048 - 16 Jul 2016
Cited by 14 | Viewed by 1630
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 other. [...] Read more.
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. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
Open AccessArticle
Uncertainty Relations for Quantum Coherence
Mathematics 2016, 4(3), 47; https://doi.org/10.3390/math4030047 - 16 Jul 2016
Cited by 19 | Viewed by 2267
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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Open AccessArticle
Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices
Mathematics 2016, 4(3), 46; https://doi.org/10.3390/math4030046 - 09 Jul 2016
Cited by 1 | Viewed by 1835
Abstract
We focus on inverse preconditioners based on minimizing F ( X ) = 1 cos ( X A , I ) , where X A is the preconditioned matrix and A is symmetric and positive definite. We present and analyze gradient-type methods [...] Read more.
We focus on inverse preconditioners based on minimizing F ( X ) = 1 cos ( X A , I ) , where X A 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. Full article
(This article belongs to the Special Issue Numerical Linear Algebra with Applications)
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Open AccessArticle
Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations
Mathematics 2016, 4(3), 45; https://doi.org/10.3390/math4030045 - 01 Jul 2016
Cited by 2 | Viewed by 2950
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Differential and Difference Equations)
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Open AccessArticle
Cohen Macaulayness and Arithmetical Rank of Generalized Theta Graphs
Mathematics 2016, 4(3), 43; https://doi.org/10.3390/math4030043 - 29 Jun 2016
Viewed by 1567
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Homological and Homotopical Algebra and Category Theory)
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Open AccessArticle
Exact Discrete Analogs of Canonical Commutation and Uncertainty Relations
Mathematics 2016, 4(3), 44; https://doi.org/10.3390/math4030044 - 28 Jun 2016
Cited by 2 | Viewed by 1579
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 the [...] Read more.
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. Full article
(This article belongs to the Special Issue Mathematics of Quantum Uncertainty)
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