Mathematics

221 KiB

Open AccessArticle

Multiple q-Zeta Brackets

by Wadim Zudilin

Mathematics 2015, 3(1), 119-130; https://doi.org/10.3390/math3010119 - 20 Mar 2015

Cited by 12 | Viewed by 5164

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Mathematical physics)

373 KiB

Open AccessArticle

Quantum Measurements of Scattered Particles

by Marco Merkli and Mark Penney

Mathematics 2015, 3(1), 92-118; https://doi.org/10.3390/math3010092 - 19 Mar 2015

Cited by 2 | Viewed by 4906

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. [...] Read more.

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. Full article

(This article belongs to the Special Issue Mathematical physics)

1133 KiB

Open AccessArticle

Basic Results for Sequential Caputo Fractional Differential Equations

by Bhuvaneswari Sambandham and Aghalaya S. Vatsala

Mathematics 2015, 3(1), 76-91; https://doi.org/10.3390/math3010076 - 19 Mar 2015

Cited by 26 | Viewed by 5678

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 [...] Read more.

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. Full article

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

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330 KiB

Open AccessReview

Twistor Interpretation of Harmonic Spheres and Yang–Mills Fields

by Armen Sergeev

Mathematics 2015, 3(1), 47-75; https://doi.org/10.3390/math3010047 - 16 Mar 2015

Cited by 4 | Viewed by 5094

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 [...] Read more.

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 \(\mathbb R^4\). Full article

(This article belongs to the Special Issue Mathematical physics)

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645 KiB

Open AccessReview

Analyticity and the Global Information Field

by Evgeni A. Solov'ev

Mathematics 2015, 3(1), 40-46; https://doi.org/10.3390/math3010040 - 13 Mar 2015

Viewed by 4766

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 [...] Read more.

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. Full article

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203 KiB

Open AccessArticle

A Study on the Nourishing Number of Graphs and Graph Powers

by Sudev Naduvath and Germina Augustine

Mathematics 2015, 3(1), 29-39; https://doi.org/10.3390/math3010029 - 6 Mar 2015

Cited by 6 | Viewed by 4281

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 [...] Read more.

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. Full article

222 KiB

Open AccessCommunication

Existence Results for Fractional Neutral Functional Differential Equations with Random Impulses

by Annamalai Anguraj, Mullarithodi C. Ranjini, Margarita Rivero and Juan J. Trujillo

Mathematics 2015, 3(1), 16-28; https://doi.org/10.3390/math3010016 - 21 Jan 2015

Cited by 10 | Viewed by 5335

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. Full article

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

232 KiB

Open AccessArticle

On θ-Congruent Numbers, Rational Squares in Arithmetic Progressions, Concordant Forms and Elliptic Curves

by Erich Selder and Karlheinz Spindler

Mathematics 2015, 3(1), 2-15; https://doi.org/10.3390/math3010002 - 19 Jan 2015

Cited by 1 | Viewed by 4684

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 [...] Read more.

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. Full article

108 KiB

Open AccessEditorial

Acknowledgement to Reviewers of Mathematics in 2014

by Mathematics Editorial Office

Mathematics 2015, 3(1), 1; https://doi.org/10.3390/math3010001 - 9 Jan 2015

Viewed by 5703

Abstract

The editors of Mathematics would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2014:[...] Full article

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Mathematics, Volume 3, Issue 1 (March 2015) – 9 articles , Pages 1-130

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