Mathematics

335 KiB

Open AccessArticle

The Schwartz Space: Tools for Quantum Mechanics and Infinite Dimensional Analysis

by Jeremy Becnel and Ambar Sengupta

Mathematics 2015, 3(2), 527-562; https://doi.org/10.3390/math3020527 - 16 Jun 2015

Cited by 13 | Viewed by 6895

An account of the Schwartz space of rapidly decreasing functions as a topological vector space with additional special structures is presented in a manner that provides all the essential background ideas for some areas of quantum mechanics along with infinite-dimensional distribution theory. Full article

(This article belongs to the Special Issue Mathematical physics)

209 KiB

Open AccessArticle

Effective Summation and Interpolation of Series by Self-Similar Root Approximants

by Simon Gluzman and Vyacheslav I. Yukalov

Mathematics 2015, 3(2), 510-526; https://doi.org/10.3390/math3020510 - 15 Jun 2015

Cited by 8 | Viewed by 4836

Abstract

We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general and applicable to different problems, as is illustrated by [...] Read more.

We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general and applicable to different problems, as is illustrated by a number of examples. The accuracy of the method is not worse, and in many cases better, than that of Padé approximants, when the latter can be defined. Full article

(This article belongs to the Special Issue Mathematical physics)

302 KiB

Open AccessArticle

The Fractional Orthogonal Difference with Applications

by Enno Diekema

Mathematics 2015, 3(2), 487-509; https://doi.org/10.3390/math3020487 - 12 Jun 2015

Cited by 3 | Viewed by 4703

Abstract

This paper is a follow-up of a previous paper of the author published in Mathematics journal in 2015, which treats the so-called continuous fractional orthogonal derivative. In this paper, we treat the discrete case using the fractional orthogonal difference. The theory is illustrated [...] Read more.

This paper is a follow-up of a previous paper of the author published in Mathematics journal in 2015, which treats the so-called continuous fractional orthogonal derivative. In this paper, we treat the discrete case using the fractional orthogonal difference. The theory is illustrated with an application of a fractional differentiating filter. In particular, graphs are presented of the absolutel value of the modulus of the frequency response. These make clear that for a good insight into the behavior of a fractional differentiating filter, one has to look for the modulus of its frequency response in a log-log plot, rather than for plots in the time domain. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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172 KiB

Open AccessLetter

The Complement of Binary Klein Quadric as a Combinatorial Grassmannian

by Metod Saniga

Mathematics 2015, 3(2), 481-486; https://doi.org/10.3390/math3020481 - 8 Jun 2015

Cited by 3 | Viewed by 5095

Abstract

Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286; 563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It [...] Read more.

Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286; 563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (286; 563)-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888). Full article

(This article belongs to the Special Issue Mathematical physics)

705 KiB

Open AccessArticle

Sinc-Approximations of Fractional Operators: A Computing Approach

by Gerd Baumann and Frank Stenger

Mathematics 2015, 3(2), 444-480; https://doi.org/10.3390/math3020444 - 5 Jun 2015

Cited by 12 | Viewed by 5904

Abstract

We discuss a new approach to represent fractional operators by Sinc approximation using convolution integrals. A spin off of the convolution representation is an effective inverse Laplace transform. Several examples demonstrate the application of the method to different practical problems. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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262 KiB

Open AccessArticle

The 1st Law of Thermodynamics for the Mean Energy of a Closed Quantum System in the Aharonov-Vaidman Gauge

by Allen D. Parks

Mathematics 2015, 3(2), 428-443; https://doi.org/10.3390/math3020428 - 1 Jun 2015

Cited by 1 | Viewed by 4566

Abstract

The Aharonov-Vaidman gauge additively transforms the mean energy of a quantum mechanical system into a weak valued system energy. In this paper, the equation of motion of this weak valued energy is used to provide a mathematical statement of an extended 1st Law [...] Read more.

The Aharonov-Vaidman gauge additively transforms the mean energy of a quantum mechanical system into a weak valued system energy. In this paper, the equation of motion of this weak valued energy is used to provide a mathematical statement of an extended 1st Law of Thermodynamics that is applicable to the mean energy of a closed quantum system when the mean energy is expressed in the Aharonov-Vaidman gauge, i.e., when the system’s energy is weak valued. This is achieved by identifying the generalized heat and work exchange terms that appear in the equation of motion for weak valued energy. The complex valued contributions of the additive gauge term to these generalized exchange terms are discussed and this extended 1st Law is shown to subsume the usual 1st Law that is applicable for the mean energy of a closed quantum system. It is found that the gauge transformation introduces an additional energy uncertainty exchange term that—while it is neither a heat nor a work exchange term—is necessary for the conservation of weak valued energy. A spin-1/2 particle in a uniform magnetic field is used to illustrate aspects of the theory. It is demonstrated for this case that the extended 1st Law implies the existence of a gauge potential ω and that it generates a non-vanishing gauge field F. It is also shown for this case that the energy uncertainty exchange accumulated during the evolution of the system along a closed evolutionary cycle C in an associated parameter space is a geometric phase. This phase is equal to both the path integral of ω along C and the integral of the flux of F through the area enclosed by C. Full article

(This article belongs to the Special Issue Mathematical physics)

216 KiB

Open AccessArticle

Subordination Principle for a Class of Fractional Order Differential Equations

by Emilia Bazhlekova

Mathematics 2015, 3(2), 412-427; https://doi.org/10.3390/math3020412 - 26 May 2015

Cited by 25 | Viewed by 5701

Abstract

The fractional order differential equation \(u'(t)=Au(t)+\gamma D_t^{\alpha} Au(t)+f(t), \ t>0\), \(u(0)=a\in X\) is studied, where \(A\) is an operator generating a strongly continuous one-parameter semigroup on a Banach space \(X\), \(D_t^{\alpha}\) is the Riemann–Liouville fractional derivative of order \(\alpha \in (0,1)\), \(\gamma>0\) and [...] Read more.

The fractional order differential equation \(u'(t)=Au(t)+\gamma D_t^{\alpha} Au(t)+f(t), \ t>0\), \(u(0)=a\in X\) is studied, where \(A\) is an operator generating a strongly continuous one-parameter semigroup on a Banach space \(X\), \(D_t^{\alpha}\) is the Riemann–Liouville fractional derivative of order \(\alpha \in (0,1)\), \(\gamma>0\) and \(f\) is an \(X\)-valued function. Equations of this type appear in the modeling of unidirectional viscoelastic flows. Well-posedness is proven, and a subordination identity is obtained relating the solution operator of the considered problem and the \(C_{0}\)-semigroup, generated by the operator \(A\). As an example, the Rayleigh–Stokes problem for a generalized second-grade fluid is considered. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

227 KiB

Open AccessArticle

Implicit Fractional Differential Equations via the Liouville–Caputo Derivative

by Juan J. Nieto, Abelghani Ouahab and Venktesh Venktesh

Mathematics 2015, 3(2), 398-411; https://doi.org/10.3390/math3020398 - 25 May 2015

Cited by 48 | Viewed by 6354

Abstract

We study an initial value problem for an implicit fractional differential equation with the Liouville–Caputo fractional derivative. By using fixed point theory and an approximation method, we obtain some existence and uniqueness results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

216 KiB

Open AccessArticle

The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials

by Tom Bella and Jenna Reis

Mathematics 2015, 3(2), 382-397; https://doi.org/10.3390/math3020382 - 14 May 2015

Cited by 4 | Viewed by 4229

Abstract

The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many [...] Read more.

The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change–of–basis matrix, this approach is computationally expensive. A recent approach to solving the connection problem involves the use of the spectral connection matrix, which is a matrix whose eigenvector matrix is the desired change–of–basis matrix. In Bella and Reis (2014), it is shown that for the connection problem between any two different classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer families, the related spectral connection matrix has quasiseparable structure. This result is limited to the case where both the source and target families are one of the Hermite, Laguerre, or Gegenbauer families, which are each defined by at most a single parameter. In particular, this excludes the large and common class of Jacobi polynomials, defined by two parameters, both as a source and as a target family. In this paper, we continue the study of the spectral connection matrix for connections between real orthogonal polynomial families. In particular, for the connection problem between any two families of the Hermite, Laguerre, or Jacobi type (including Chebyshev, Legendre, and Gegenbauer), we prove that the spectral connection matrix has quasiseparable structure. In addition, our results also show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral connection matrix are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014). Full article

(This article belongs to the Special Issue New Trends in Applications of Orthogonal Polynomials and Special Functions)

265 KiB

Open AccessArticle

The Role of the Mittag-Leffler Function in Fractional Modeling

by Sergei Rogosin

Mathematics 2015, 3(2), 368-381; https://doi.org/10.3390/math3020368 - 13 May 2015

Cited by 58 | Viewed by 6505

Abstract

This is a survey paper illuminating the distinguished role of the Mittag-Leffler function and its generalizations in fractional analysis and fractional modeling. The content of the paper is connected to the recently published monograph by Rudolf Gorenflo, Anatoly Kilbas, Francesco Mainardi and Sergei [...] Read more.

This is a survey paper illuminating the distinguished role of the Mittag-Leffler function and its generalizations in fractional analysis and fractional modeling. The content of the paper is connected to the recently published monograph by Rudolf Gorenflo, Anatoly Kilbas, Francesco Mainardi and Sergei Rogosin. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

904 KiB

Open AccessArticle

High-Precision Arithmetic in Mathematical Physics

by David H. Bailey and Jonathan M. Borwein

Mathematics 2015, 3(2), 337-367; https://doi.org/10.3390/math3020337 - 12 May 2015

Cited by 45 | Viewed by 9429

Abstract

For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This [...] Read more.

For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This article discusses the challenge of high-precision computation, in the context of mathematical physics, and highlights what facilities are required to support future computation, in light of emerging developments in computer architecture. Full article

(This article belongs to the Special Issue Mathematical physics)

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180 KiB

Open AccessArticle

Action at a Distance in Quantum Theory

by Jerome Blackman

Mathematics 2015, 3(2), 329-336; https://doi.org/10.3390/math3020329 - 6 May 2015

Viewed by 3781

Abstract

The purpose of this paper is to present a consistent mathematical framework that shows how the EPR (Einstein. Podolsky, Rosen) phenomenon fits into our view of space time. To resolve the differences between the Hilbert space structure of quantum theory and the manifold [...] Read more.

The purpose of this paper is to present a consistent mathematical framework that shows how the EPR (Einstein. Podolsky, Rosen) phenomenon fits into our view of space time. To resolve the differences between the Hilbert space structure of quantum theory and the manifold structure of classical physics, the manifold is taken as a partial representation of the Hilbert space. It is the partial nature of the representation that allows for action at a distance and the failure of the manifold picture. Full article

(This article belongs to the Special Issue Mathematical physics)

247 KiB

Open AccessArticle

There Are Quantum Jumps

by Erkki J. Brändas

Mathematics 2015, 3(2), 319-328; https://doi.org/10.3390/math3020319 - 5 May 2015

Viewed by 4848

Abstract

In this communication we take up the age-old problem of the possibility to incorporate quantum jumps. Unusually, we investigate quantum jumps in an extended quantum setting, but one of rigorous mathematical significance. The general background for this formulation originates in the Balslev-Combes theorem [...] Read more.

In this communication we take up the age-old problem of the possibility to incorporate quantum jumps. Unusually, we investigate quantum jumps in an extended quantum setting, but one of rigorous mathematical significance. The general background for this formulation originates in the Balslev-Combes theorem for dilatation analytic Hamiltonians and associated complex symmetric representations. The actual jump is mapped into a Jordan block of order two and a detailed derivation is discussed for the case of the emission of a photon by an atom. The result can be easily reassigned to analogous cases as well as generalized to Segrè characteristics of arbitrary order. Full article

(This article belongs to the Special Issue Mathematical physics)

145 KiB

Open AccessArticle

On the Duality of Discrete and Periodic Functions

by Jens V. Fischer

Mathematics 2015, 3(2), 299-318; https://doi.org/10.3390/math3020299 - 30 Apr 2015

Cited by 11 | Viewed by 7540

Abstract

Although versions of Poisson’s Summation Formula (PSF) have already been studied extensively, there seems to be no theorem that relates discretization to periodization and periodization to discretization in a simple manner. In this study, we show that two complementary formulas, both closely related [...] Read more.

Although versions of Poisson’s Summation Formula (PSF) have already been studied extensively, there seems to be no theorem that relates discretization to periodization and periodization to discretization in a simple manner. In this study, we show that two complementary formulas, both closely related to the classical Poisson Summation Formula, are needed to form a reciprocal Discretization-Periodization Theorem on generalized functions. We define discretization and periodization on generalized functions and show that the Fourier transform of periodic functions are discrete functions and, vice versa, the Fourier transform of discrete functions are periodic functions. Full article

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314 KiB

Open AccessArticle

The Fractional Orthogonal Derivative

by Enno Diekema

Mathematics 2015, 3(2), 273-298; https://doi.org/10.3390/math3020273 - 22 Apr 2015

Cited by 5 | Viewed by 4276

Abstract

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder [...] Read more.

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann–Liouville fractional derivative is considered by replacing the n-th derivative by its approximation in the formula for the fractional derivative. In the case of, for instance, Jacobi polynomials, an explicit formula for the kernel of this approximate fractional derivative can be given. Next, we consider the fractional derivative as a filter and compute the frequency response in the continuous case for the Jacobi polynomials and in the discrete case for the Hahn polynomials. The frequency response in this case is a confluent hypergeometric function. A different approach is discussed, which starts with this explicit frequency response and then obtains the approximate fractional derivative by taking the inverse Fourier transform. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

1856 KiB

Open AccessArticle

Fractional Euler-Lagrange Equations Applied to Oscillatory Systems

by Sergio Adriani David and Carlos Alberto Valentim, Jr.

Mathematics 2015, 3(2), 258-272; https://doi.org/10.3390/math3020258 - 20 Apr 2015

Cited by 17 | Viewed by 7241

Abstract

In this paper, we applied the Riemann-Liouville approach and the fractional Euler-Lagrange equations in order to obtain the fractional nonlinear dynamic equations involving two classical physical applications: “Simple Pendulum” and the “Spring-Mass-Damper System” to both integer order calculus (IOC) and fractional order calculus [...] Read more.

In this paper, we applied the Riemann-Liouville approach and the fractional Euler-Lagrange equations in order to obtain the fractional nonlinear dynamic equations involving two classical physical applications: “Simple Pendulum” and the “Spring-Mass-Damper System” to both integer order calculus (IOC) and fractional order calculus (FOC) approaches. The numerical simulations were conducted and the time histories and pseudo-phase portraits presented. Both systems, the one that already had a damping behavior (Spring-Mass-Damper) and the system that did not present any sort of damping behavior (Simple Pendulum), showed signs indicating a possible better capacity of attenuation of their respective oscillation amplitudes. This implication could mean that if the selection of the order of the derivative is conveniently made, systems that need greater intensities of damping or vibrating absorbers may benefit from using fractional order in dynamics and possibly in control of the aforementioned systems. Thereafter, we believe that the results described in this paper may offer greater insights into the complex behavior of these systems, and thus instigate more research efforts in this direction. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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498 KiB

Open AccessArticle

Maxwell–Lorentz Electrodynamics Revisited via the Lagrangian Formalism and Feynman Proper Time Paradigm

by Nikolai N. Bogolubov, Jr., Anatolij K. Prykarpatski and Denis Blackmore

Mathematics 2015, 3(2), 190-257; https://doi.org/10.3390/math3020190 - 17 Apr 2015

Cited by 5 | Viewed by 7456

Abstract

We review new electrodynamics models of interacting charged point particles and related fundamental physical aspects, motivated by the classical A.M. Ampère magnetic and H. Lorentz force laws electromagnetic field expressions. Based on the Feynman proper time paradigm and a recently devised vacuum field [...] Read more.

We review new electrodynamics models of interacting charged point particles and related fundamental physical aspects, motivated by the classical A.M. Ampère magnetic and H. Lorentz force laws electromagnetic field expressions. Based on the Feynman proper time paradigm and a recently devised vacuum field theory approach to the Lagrangian and Hamiltonian, the formulations of alternative classical electrodynamics models are analyzed in detail and their Dirac type quantization is suggested. Problems closely related to the radiation reaction force and electron mass inertia are analyzed. The validity of the Abraham-Lorentz electromagnetic electron mass origin hypothesis is argued. The related electromagnetic Dirac–Fock–Podolsky problem and symplectic properties of the Maxwell and Yang–Mills type dynamical systems are analyzed. The crucial importance of the remaining reference systems, with respect to which the dynamics of charged point particles is framed, is explained and emphasized. Full article

(This article belongs to the Special Issue Mathematical physics)

336 KiB

Open AccessArticle

Asymptotic Expansions of Fractional Derivatives andTheir Applications

by Tohru Morita and Ken-ichi Sato

Mathematics 2015, 3(2), 171-189; https://doi.org/10.3390/math3020171 - 15 Apr 2015

Cited by 5 | Viewed by 5062

Abstract

We compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two [...] Read more.

We compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two fIs. When this happens, this fact occurs also for the fractional derivative (fD).This method is applied to the derivation of the asymptotic expansion of the confluenthypergeometric function, which is a solution of Kummer’s differential equation. In thepresent paper, the solutions of the equation in the forms of the Riemann–Liouville fI orfD and the Liouville fI or fD are obtained by using the method, which Nishimoto used insolving the hypergeometric differential equation in terms of the Liouville fD. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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305 KiB

Open AccessArticle

Analytical Solution of Generalized Space-Time Fractional Cable Equation

by Ram K. Saxena, Zivorad Tomovski and Trifce Sandev

Mathematics 2015, 3(2), 153-170; https://doi.org/10.3390/math3020153 - 9 Apr 2015

Cited by 6 | Viewed by 5286

Abstract

In this paper, we consider generalized space-time fractional cable equation in presence of external source. By using the Fourier-Laplace transform we obtain the Green function in terms of infinite series in H-functions. The fractional moments of the fundamental solution are derived and their [...] Read more.

In this paper, we consider generalized space-time fractional cable equation in presence of external source. By using the Fourier-Laplace transform we obtain the Green function in terms of infinite series in H-functions. The fractional moments of the fundamental solution are derived and their asymptotic behavior in the short and long time limit is analyzed. Some previously obtained results are compared with those presented in this paper. By using the Bernstein characterization theorem we find the conditions under which the even moments are non-negative. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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339 KiB

Open AccessArticle

Fractional Diffusion in Gaussian Noisy Environment

by Guannan Hu and Yaozhong Hu

Mathematics 2015, 3(2), 131-152; https://doi.org/10.3390/math3020131 - 31 Mar 2015

Cited by 8 | Viewed by 4879

Abstract

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: \(D_t^{(\alpha)} u(t, x)=\textit{B}u+u\cdot \dot W^H\), where \(D_t^{(\alpha)}\) is the Caputo fractional derivative of order \(\alpha\in (0,1)\) with respect to the [...] Read more.

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: \(D_t^{(\alpha)} u(t, x)=\textit{B}u+u\cdot \dot W^H\), where \(D_t^{(\alpha)}\) is the Caputo fractional derivative of order \(\alpha\in (0,1)\) with respect to the time variable \(t\), \(\textit{B}\) is a second order elliptic operator with respect to the space variable \(x\in\mathbb{R}^d\) and \(\dot W^H\) a time homogeneous fractional Gaussian noise of Hurst parameter \(H=(H_1, \cdots, H_d)\). We obtain conditions satisfied by \(\alpha\) and \(H\), so that the square integrable solution \(u\) exists uniquely. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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Mathematics, Volume 3, Issue 2 (June 2015) – 20 articles , Pages 131-562

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