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**Mathematics, Vol. 3, Pages 563-603: Singular Bilinear Integrals in Quantum**

Physics**Authors:***Brian Jefferies***Pages:**563 - 603**Abstract:**Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how ``decoupled'' bilinear integration may be used to find solutions \(X\) of operator equations \(AX-XB=Y\) with respect to the spectral measure of \(A\) and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space \(L^1((P\otimes Q)_{\mathcal L(\mathcal H)})\) of double operator integrable functions for spectral measures \(P\), \(Q\) acting in a Hilbert space \(\mathcal H\) and applied to the representation of the trace of \(\int_{\Lambda\times\Lambda}\varphi\,d(PTP)\) for a trace class operator \(T\). The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function.**PubDate:**2015-06-29**DOI:**10.3390/math3030563**Issue No:***Vol. 3, No. 3 (2015)*

**Mathematics, Vol. 3, Pages 604-614: Topological Integer Additive**

Set-Sequential Graphs**Authors:***Sudev Naduvath, Germina Augustine, Chithra Sudev***Pages:**604 - 614**Abstract:**Let \(\mathbb{N}_0\) denote the set of all non-negative integers and \(X\) be any non-empty subset of \(\mathbb{N}_0\). Denote the power set of \(X\) by \(\mathcal{P}(X)\). An integer additive set-labeling (IASL) of a graph \(G\) is an injective function \(f : V (G) \to P(X)\) such that the image of the induced function \(f^+: E(G) \to \mathcal{P}(\mathbb{N}_0)\), defined by \(f^+(uv)=f(u)+f(v)\), is contained in \(\mathcal{P}(X)\), where \(f(u) + f(v)\) is the sumset of \(f(u)\) and \(f(v)\). If the associated set-valued edge function \(f^+\) is also injective, then such an IASL is called an integer additive set-indexer (IASI). An IASL \(f\) is said to be a topological IASL (TIASL) if \(f(V(G))\cup \{\emptyset\}\) is a topology of the ground set \(X\). An IASL is said to be an integer additive set-sequential labeling (IASSL) if \(f(V(G))\cup f^+(E(G))= \mathcal{P}(X)-\{\emptyset\}\). An IASL of a given graph \(G\) is said to be a topological integer additive set-sequential labeling of \(G\), if it is a topological integer additive set-labeling as well as an integer additive set-sequential labeling of \(G\). In this paper, we study the conditions required for a graph \(G\) to admit this type of IASL and propose some important characteristics of the graphs which admit this type of IASLs.**PubDate:**2015-07-03**DOI:**10.3390/math3030604**Issue No:***Vol. 3, No. 3 (2015)*

**Mathematics, Vol. 3, Pages 615-625: Reproducing Kernel Hilbert Space vs.**

Frame Estimates**Authors:***Palle Jorgensen, Myung-Sin Song***Pages:**615 - 625**Abstract:**We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn H into a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω . We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.**PubDate:**2015-07-08**DOI:**10.3390/math3030615**Issue No:***Vol. 3, No. 3 (2015)*

**Mathematics, Vol. 3, Pages 626-643: Time Automorphisms on C-Algebras****Authors:***R. Hilfer***Pages:**626 - 643**Abstract:**Applications of fractional time derivatives in physics and engineering require the existence of nontranslational time automorphisms on the appropriate algebra of observables. The existence of time automorphisms on commutative and noncommutative C*-algebras for interacting many-body systems is investigated in this article. A mathematical framework is given to discuss local stationarity in time and the global existence of fractional and nonfractional time automorphisms. The results challenge the concept of time flow as a translation along the orbits and support a more general concept of time flow as a convolution along orbits. Implications for the distinction of reversible and irreversible dynamics are discussed. The generalized concept of time as a convolution reduces to the traditional concept of time translation in a special limit.**PubDate:**2015-07-16**DOI:**10.3390/math3030626**Issue No:***Vol. 3, No. 3 (2015)*

**Mathematics, Vol. 3, Pages 644-652: On the Nature of the**

Tsallis–Fourier Transform**Authors:***A. Plastino, Mario Rocca***Pages:**644 - 652**Abstract:**By recourse to tempered ultradistributions, we show here that the effect of a q-Fourier transform (qFT) is to map equivalence classes of functions into other classes in a one-to-one fashion. This suggests that Tsallis’ q-statistics may revolve around equivalence classes of distributions and not individual ones, as orthodox statistics does. We solve here the qFT’s non-invertibility issue, but discover a problem that remains open.**PubDate:**2015-07-21**DOI:**10.3390/math3030644**Issue No:***Vol. 3, No. 3 (2015)*

**Mathematics, Vol. 3, Pages 653-665: Zeta Function Expression of Spin**

Partition Functions on Thermal AdS3**Authors:***Floyd L.Williams***Pages:**653 - 665**Abstract:**We find a Selberg zeta function expression of certain one-loop spin partition functions on three-dimensional thermal anti-de Sitter space. Of particular interest is the partition function of higher spin fermionic particles. We also set up, in the presence of spin, a Patterson-type formula involving the logarithmic derivative of zeta.**PubDate:**2015-07-28**DOI:**10.3390/math3030653**Issue No:***Vol. 3, No. 3 (2015)*

**Mathematics, Vol. 3, Pages 131-152: Fractional Diffusion in Gaussian Noisy**

Environment**Authors:***Guannan Hu, Yaozhong Hu***Pages:**131 - 152**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 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.**PubDate:**2015-03-31**DOI:**10.3390/math3020131**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 153-170: Analytical Solution of Generalized**

Space-Time Fractional Cable Equation**Authors:***Ram Saxena, Zivorad Tomovski, Trifce Sandev***Pages:**153 - 170**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 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.**PubDate:**2015-04-09**DOI:**10.3390/math3020153**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 171-189: Asymptotic Expansions of Fractional**

Derivatives andTheir Applications**Authors:***Tohru Morita, Ken-ichi Sato***Pages:**171 - 189**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 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.**PubDate:**2015-04-15**DOI:**10.3390/math3020171**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 190-257: Maxwell–Lorentz Electrodynamics**

Revisited via the Lagrangian Formalism and Feynman Proper Time Paradigm**Authors:***Nikolai Bogolubov, Anatolij Prykarpatski, Denis Blackmore***Pages:**190 - 257**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 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.**PubDate:**2015-04-17**DOI:**10.3390/math3020190**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 258-272: Fractional Euler-Lagrange Equations**

Applied to Oscillatory Systems**Authors:***Sergio David, Carlos Jr.***Pages:**258 - 272**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 (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.**PubDate:**2015-04-20**DOI:**10.3390/math3020258**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 273-298: The Fractional Orthogonal Derivative****Authors:***Enno Diekema***Pages:**273 - 298**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 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.**PubDate:**2015-04-22**DOI:**10.3390/math3020273**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 299-318: On the Duality of Discrete and**

Periodic Functions**Authors:***Jens Fischer***Pages:**299 - 318**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 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.**PubDate:**2015-04-30**DOI:**10.3390/math3020299**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 319-328: There Are Quantum Jumps****Authors:***Erkki Brändas***Pages:**319 - 328**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 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.**PubDate:**2015-05-05**DOI:**10.3390/math3020319**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 329-336: Action at a Distance in Quantum Theory****Authors:***Jerome Blackman***Pages:**329 - 336**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 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.**PubDate:**2015-05-06**DOI:**10.3390/math3020329**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 337-367: High-Precision Arithmetic in**

Mathematical Physics**Authors:***David Bailey, Jonathan Borwein***Pages:**337 - 367**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 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.**PubDate:**2015-05-12**DOI:**10.3390/math3020337**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 368-381: The Role of the Mittag-Leffler**

Function in Fractional Modeling**Authors:***Sergei Rogosin***Pages:**368 - 381**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 Rogosin.**PubDate:**2015-05-13**DOI:**10.3390/math3020368**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 382-397: The Spectral Connection Matrix for Any**

Change of Basis within the Classical Real Orthogonal Polynomials**Authors:***Tom Bella, Jenna Reis***Pages:**382 - 397**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 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).**PubDate:**2015-05-14**DOI:**10.3390/math3020382**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 398-411: Implicit Fractional Differential**

Equations via the Liouville–Caputo Derivative**Authors:***Juan Nieto, Abelghani Ouahab, Venktesh Venktesh***Pages:**398 - 411**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.**PubDate:**2015-05-25**DOI:**10.3390/math3020398**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 412-427: Subordination Principle for a Class of**

Fractional Order Differential Equations**Authors:***Emilia Bazhlekova***Pages:**412 - 427**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 \(f\) 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.**PubDate:**2015-05-26**DOI:**10.3390/math3020412**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 428-443: The 1st Law of Thermodynamics for the**

Mean Energy of a Closed Quantum System in the Aharonov-Vaidman Gauge**Authors:***Allen Parks***Pages:**428 - 443**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 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.**PubDate:**2015-06-01**DOI:**10.3390/math3020428**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 444-480: Sinc-Approximations of Fractional**

Operators: A Computing Approach**Authors:***Gerd Baumann, Frank Stenger***Pages:**444 - 480**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.**PubDate:**2015-06-05**DOI:**10.3390/math3020444**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 481-486: The Complement of Binary Klein Quadric**

as a Combinatorial Grassmannian**Authors:***Metod Saniga***Pages:**481 - 486**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 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).**PubDate:**2015-06-08**DOI:**10.3390/math3020481**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 487-509: The Fractional Orthogonal Difference**

with Applications**Authors:***Enno Diekema***Pages:**487 - 509**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 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.**PubDate:**2015-06-12**DOI:**10.3390/math3020487**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 510-526: Effective Summation and Interpolation**

of Series by Self-Similar Root Approximants**Authors:***Simon Gluzman, Vyacheslav Yukalov***Pages:**510 - 526**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 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.**PubDate:**2015-06-15**DOI:**10.3390/math3020510**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 527-562: The Schwartz Space: Tools for Quantum**

Mechanics and Infinite Dimensional Analysis**Authors:***Jeremy Becnel, Ambar Sengupta***Pages:**527 - 562**Abstract:**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.**PubDate:**2015-06-16**DOI:**10.3390/math3020527**Issue No:***Vol. 3, No. 2 (2015)*

**Mathematics, Vol. 3, Pages 1: Acknowledgement to Reviewers of Mathematics**

in 2014**Authors:***Mathematics Office***Pages:**1 - 1**Abstract:**The editors of Mathematics would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2014:[...]**PubDate:**2015-01-09**DOI:**10.3390/math3010001**Issue No:***Vol. 3, No. 1 (2015)*

**Mathematics, Vol. 3, Pages 2-15: On θ-Congruent Numbers, Rational**

Squares in Arithmetic Progressions, Concordant Forms and Elliptic Curves**Authors:***Erich Selder, Karlheinz Spindler***Pages:**2 - 15**Abstract:**The correspondence between right triangles with rational sides, triplets of rational squares in arithmetic succession and integral solutions of certain quadratic forms is well-known. We show how this correspondence can be extended to the generalized notions of rational θ-triangles, rational squares occurring in arithmetic progressions and concordant forms. In our approach we establish one-to-one mappings to rational points on certain elliptic curves and examine in detail the role of solutions of the θ-congruent number problem and the concordant form problem associated with nontrivial torsion points on the corresponding elliptic curves. This approach allows us to combine and extend some disjoint results obtained by a number of authors, to clarify some statements in the literature and to answer some hitherto open questions.**PubDate:**2015-01-19**DOI:**10.3390/math3010002**Issue No:***Vol. 3, No. 1 (2015)*

**Mathematics, Vol. 3, Pages 16-28: Existence Results for Fractional Neutral**

Functional Differential Equations with Random Impulses**Authors:***Annamalai Anguraj, Mullarithodi Ranjini, Margarita Rivero, Juan Trujillo***Pages:**16 - 28**Abstract:**In this paper, we investigate the existence of solutions for the fractional neutral differential equations with random impulses. The results are obtained by using Krasnoselskii’s fixed point theorem. Examples are added to show applications of the main results.**PubDate:**2015-01-21**DOI:**10.3390/math3010016**Issue No:***Vol. 3, No. 1 (2015)*

**Mathematics, Vol. 3, Pages 29-39: A Study on the Nourishing Number of**

Graphs and Graph Powers**Authors:***Sudev Naduvath, Germina Augustine***Pages:**29 - 39**Abstract:**Let \(\mathbb{N}_{0}\) be the set of all non-negative integers and \(\mathcal{P}(\mathbb{N}_{0})\) be its power set. Then, an integer additive set-indexer (IASI) of a given graph \(G\) is defined as an injective function \(f:V(G)\to \mathcal{P}(\mathbb{N}_{0})\) such that the induced edge-function \(f^+:E(G) \to\mathcal{P}(\mathbb{N}_{0})\) defined by \(f^+ (uv) = f(u)+ f(v)\) is also injective, where \(f(u)+f(v)\) is the sumset of \(f(u)\) and \(f(v)\). An IASI \(f\) of \(G\) is said to be a strong IASI of \(G\) if \( f^+(uv) = f(u) \, f(v) \) for all \(uv\in E(G)\). The nourishing number of a graph \(G\) is the minimum order of the maximal complete subgraph of \(G\) so that \(G\) admits a strong IASI. In this paper, we study the characteristics of certain graph classes and graph powers that admit strong integer additive set-indexers and determine their corresponding nourishing numbers.**PubDate:**2015-03-06**DOI:**10.3390/math3010029**Issue No:***Vol. 3, No. 1 (2015)*

**Mathematics, Vol. 3, Pages 40-46: Analyticity and the Global Information**

Field**Authors:***Evgeni Solov'ev***Pages:**40 - 46**Abstract:**The relation between analyticity in mathematics and the concept of a global information field in physics is reviewed. Mathematics is complete in the complex plane only. In the complex plane, a very powerful tool appears—analyticity. According to this property, if an analytic function is known on the countable set of points having an accumulation point, then it is known everywhere. This mysterious property has profound consequences in quantum physics. Analyticity allows one to obtain asymptotic (approximate) results in terms of some singular points in the complex plane which accumulate all necessary data on a given process. As an example, slow atomic collisions are presented, where the cross-sections of inelastic transitions are determined by branch-points of the adiabatic energy surface at a complex internuclear distance. Common aspects of the non-local nature of analyticity and a recently introduced interpretation of classical electrodynamics and quantum physics as theories of a global information field are discussed.**PubDate:**2015-03-13**DOI:**10.3390/math3010040**Issue No:***Vol. 3, No. 1 (2015)*

**Mathematics, Vol. 3, Pages 47-75: Twistor Interpretation of Harmonic**

Spheres and Yang–Mills Fields**Authors:***Armen Sergeev***Pages:**47 - 75**Abstract:**We consider the twistor descriptions of harmonic maps of the Riemann sphere into Kähler manifolds and Yang–Mills fields on four-dimensional Euclidean space. The motivation to study twistor interpretations of these objects comes from the harmonic spheres conjecture stating the existence of the bijective correspondence between based harmonic spheres in the loop space \(\Omega G\) of a compact Lie group \(G\) and the moduli space of Yang–Mills \(G\)-fields on \(\mathbb R^4\).**PubDate:**2015-03-16**DOI:**10.3390/math3010047**Issue No:***Vol. 3, No. 1 (2015)*

**Mathematics, Vol. 3, Pages 76-91: Basic Results for Sequential Caputo**

Fractional Differential Equations**Authors:***Bhuvaneswari Sambandham, Aghalaya Vatsala***Pages:**76 - 91**Abstract:**We have developed a representation form for the linear fractional differential equation of order q when 0 < q < 1, with variable coefficients. We have also obtained a closed form of the solution for sequential Caputo fractional differential equation of order 2q, with initial and boundary conditions, for 0 < 2q < 1. The solutions are in terms of Mittag–Leffler functions of order q only. Our results yield the known results of integer order when q = 1. We have also presented some numerical results to bring the salient features of sequential fractional differential equations.**PubDate:**2015-03-19**DOI:**10.3390/math3010076**Issue No:***Vol. 3, No. 1 (2015)*

**Mathematics, Vol. 3, Pages 92-118: Quantum Measurements of Scattered**

Particles**Authors:***Marco Merkli, Mark Penney***Pages:**92 - 118**Abstract:**We investigate the process of quantum measurements on scattered probes. Before scattering, the probes are independent, but they become entangled afterwards, due to the interaction with the scatterer. The collection of measurement results (the history) is a stochastic process of dependent random variables. We link the asymptotic properties of this process to spectral characteristics of the dynamics. We show that the process has decaying time correlations and that a zero-one law holds. We deduce that if the incoming probes are not sharply localized with respect to the spectrum of the measurement operator, then the process does not converge. Nevertheless, the scattering modifies the measurement outcome frequencies, which are shown to be the average of the measurement projection operator, evolved for one interaction period, in an asymptotic state. We illustrate the results on a truncated Jaynes–Cummings model.**PubDate:**2015-03-19**DOI:**10.3390/math3010092**Issue No:***Vol. 3, No. 1 (2015)*

**Mathematics, Vol. 3, Pages 119-130: Multiple q-Zeta Brackets****Authors:***Wadim Zudilin***Pages:**119 - 130**Abstract:**The multiple zeta values (MZVs) possess a rich algebraic structure of algebraic relations, which is conjecturally determined by two different (shuffle and stuffle) products of a certain algebra of noncommutative words. In a recent work, Bachmann constructed a q-analogue of the MZVs—the so-called bi-brackets—for which the two products are dual to each other, in a very natural way. We overview Bachmann’s construction and discuss the radial asymptotics of the bi-brackets, its links to the MZVs, and related linear (in)dependence questions of the q-analogue.**PubDate:**2015-03-20**DOI:**10.3390/math3010119**Issue No:***Vol. 3, No. 1 (2015)*