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603 KiB

Open AccessArticle

Two Dimensional Temperature Distributions in Plate Heat Exchangers: An Analytical Approach

by Amir Reza Ansari Dezfoli, Mozaffar Ali Mehrabian and Mohamad Hasan Saffaripour

Mathematics 2015, 3(4), 1255-1273; https://doi.org/10.3390/math3041255 - 16 Dec 2015

Cited by 4 | Viewed by 4496

Abstract

Analytical solutions are developed to work out the two-dimensional (2D) temperature changes of flow in the passages of a plate heat exchanger in parallel flow and counter flow arrangements. Two different flow regimes, namely, the plug flow and the turbulent flow are considered. [...] Read more.

Analytical solutions are developed to work out the two-dimensional (2D) temperature changes of flow in the passages of a plate heat exchanger in parallel flow and counter flow arrangements. Two different flow regimes, namely, the plug flow and the turbulent flow are considered. The mathematical formulation of problems coupled at boundary conditions are presented, the solution procedure is then obtained as a special case of the two region Sturm-Liouville problem. The results obtained for two different flow regimes are then compared with experimental results and with each other. The agreement between the analytical and experimental results is an indication of the accuracy of solution method. Full article

(This article belongs to the Special Issue Mathematical physics)

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363 KiB

Open AccessArticle

Free W^*-Dynamical Systems From p-Adic Number Fields and the Euler Totient Function

by Ilwoo Cho and Palle E. T. Jorgensen

Mathematics 2015, 3(4), 1095-1138; https://doi.org/10.3390/math3041095 - 2 Dec 2015

Viewed by 3673

Abstract

In this paper, we study relations between free probability on crossed product W * -algebras with a von Neumann algebra over p-adic number fields ℚ_p (for primes p), and free probability on the subalgebra Φ, generated by the Euler totient function ϕ, [...] Read more.

In this paper, we study relations between free probability on crossed product W * -algebras with a von Neumann algebra over p-adic number fields ℚ_p (for primes p), and free probability on the subalgebra Φ, generated by the Euler totient function ϕ, of the arithmetic algebra A , consisting of all arithmetic functions. In particular, we apply such free probability to consider operator-theoretic and operator-algebraic properties of W * -dynamical systems induced by ℚp under free-probabilistic (and hence, spectral-theoretic) techniques. Full article

(This article belongs to the Special Issue Mathematical physics)

234 KiB

Open AccessArticle

Gauge Invariance and Symmetry Breaking by Topology and Energy Gap

by Franco Strocchi and Carlo Heissenberg

Mathematics 2015, 3(4), 984-1000; https://doi.org/10.3390/math3040984 - 22 Oct 2015

Cited by 2 | Viewed by 3854

Abstract

For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra A, with a non-trivial center Z, describes observables, the other Weyl operators playing the role of intertwiners [...] Read more.

For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra A, with a non-trivial center Z, describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations of A. In particular, this gives rise to a gauge symmetry described by the action of Z. A distinguished case is when the center of the observables arises from the fundamental group of the manifold of the positions of the quantum system. Symmetries that do not commute with the topological invariants represented by elements of Z are then spontaneously broken in each irreducible representation of the observable algebra, compatibly with an energy gap; such a breaking exhibits a mechanism radically different from Goldstone and Higgs mechanisms. This is clearly displayed by the quantum particle on a circle, the Bloch electron and the two body problem. Full article

(This article belongs to the Special Issue Mathematical physics)

259 KiB

Open AccessArticle

Photon Localization Revisited

by Izumi Ojima and Hayato Saigo

Mathematics 2015, 3(3), 897-912; https://doi.org/10.3390/math3030897 - 23 Sep 2015

Cited by 4 | Viewed by 3921

Abstract

In the light of the Newton–Wigner–Wightman theorem of localizability question, we have proposed before a typical generation mechanism of effective mass for photons to be localized in the form of polaritons owing to photon-media interactions. In this paper, the general essence of this [...] Read more.

In the light of the Newton–Wigner–Wightman theorem of localizability question, we have proposed before a typical generation mechanism of effective mass for photons to be localized in the form of polaritons owing to photon-media interactions. In this paper, the general essence of this example model is extracted in such a form as quantum field ontology associated with the eventualization principle, which enables us to explain the mutual relations, back and forth, between quantum fields and various forms of particles in the localized form of the former. Full article

(This article belongs to the Special Issue Mathematical physics)

401 KiB

Open AccessArticle

Chern-Simons Path Integrals in S² × S¹

by Adrian P. C. Lim

Mathematics 2015, 3(3), 843-879; https://doi.org/10.3390/math3030843 - 21 Aug 2015

Cited by 3 | Viewed by 4014

Abstract

Using torus gauge fixing, Hahn in 2008 wrote down an expression for a Chern-Simons path integral to compute the Wilson Loop observable, using the Chern-Simons action \(S_{CS}^\kappa\), \(\kappa\) is some parameter. Instead of making sense of the path integral over the space of [...] Read more.

Using torus gauge fixing, Hahn in 2008 wrote down an expression for a Chern-Simons path integral to compute the Wilson Loop observable, using the Chern-Simons action \(S_{CS}^\kappa\), \(\kappa\) is some parameter. Instead of making sense of the path integral over the space of \(\mathfrak{g}\)-valued smooth 1-forms on \(S^2 \times S^1\), we use the Segal Bargmann transform to define the path integral over \(B_i\), the space of \(\mathfrak{g}\)-valued holomorphic functions over \(\mathbb{C}^2 \times \mathbb{C}^{i-1}\). This approach was first used by us in 2011. The main tool used is Abstract Wiener measure and applying analytic continuation to the Wiener integral. Using the above approach, we will show that the Chern-Simons path integral can be written as a linear functional defined on \(C(B_1^{\times^4} \times B_2^{\times^2}, \mathbb{C})\) and this linear functional is similar to the Chern-Simons linear functional defined by us in 2011, for the Chern-Simons path integral in the case of \(\mathbb{R}^3\). We will define the Wilson Loop observable using this linear functional and explicitly compute it, and the expression is dependent on the parameter \(\kappa\). The second half of the article concentrates on taking \(\kappa\) goes to infinity for the Wilson Loop observable, to obtain link invariants. As an application, we will compute the Wilson Loop observable in the case of \(SU(N)\) and \(SO(N)\). In these cases, the Wilson Loop observable reduces to a state model. We will show that the state models satisfy a Jones type skein relation in the case of \(SU(N)\) and a Conway type skein relation in the case of \(SO(N)\). By imposing quantization condition on the charge of the link \(L\), we will show that the state models are invariant under the Reidemeister Moves and hence the Wilson Loop observables indeed define a framed link invariant. This approach follows that used in an article written by us in 2012, for the case of \(\mathbb{R}^3\). Full article

(This article belongs to the Special Issue Mathematical physics)

523 KiB

Open AccessArticle

Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics

by Kundeti Muralidhar

Mathematics 2015, 3(3), 781-842; https://doi.org/10.3390/math3030781 - 20 Aug 2015

Cited by 7 | Viewed by 8979

Abstract

A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in [...] Read more.

A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows the generation of higher n-dimensional geometric algebra from (n — 1)-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics. Full article

(This article belongs to the Special Issue Mathematical physics)

297 KiB

Open AccessArticle

The Segal–Bargmann Transform for Odd-Dimensional Hyperbolic Spaces

by Brian C. Hall and Jeffrey J. Mitchell

Mathematics 2015, 3(3), 758-780; https://doi.org/10.3390/math3030758 - 18 Aug 2015

Cited by 1 | Viewed by 4155

Abstract

We develop isometry and inversion formulas for the Segal–Bargmann transform on odd-dimensional hyperbolic spaces that are as parallel as possible to the dual case of odd-dimensional spheres. Full article

(This article belongs to the Special Issue Mathematical physics)

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375 KiB

Open AccessArticle

From Classical to Discrete Gravity through Exponential Non-Standard Lagrangians in General Relativity

by Rami Ahmad El-Nabulsi

Mathematics 2015, 3(3), 727-745; https://doi.org/10.3390/math3030727 - 14 Aug 2015

Cited by 7 | Viewed by 4606

Abstract

Recently, non-standard Lagrangians have gained a growing importance in theoretical physics and in the theory of non-linear differential equations. However, their formulations and implications in general relativity are still in their infancies despite some advances in contemporary cosmology. The main aim of this [...] Read more.

Recently, non-standard Lagrangians have gained a growing importance in theoretical physics and in the theory of non-linear differential equations. However, their formulations and implications in general relativity are still in their infancies despite some advances in contemporary cosmology. The main aim of this paper is to fill the gap. Though non-standard Lagrangians may be defined by a multitude form, in this paper, we considered the exponential type. One basic feature of exponential non-standard Lagrangians concerns the modified Euler-Lagrange equation obtained from the standard variational analysis. Accordingly, when applied to spacetime geometries, one unsurprisingly expects modified geodesic equations. However, when taking into account the time-like paths parameterization constraint, remarkably, it was observed that mutually discrete gravity and discrete spacetime emerge in the theory. Two different independent cases were obtained: A geometrical manifold with new spacetime coordinates augmented by a metric signature change and a geometrical manifold characterized by a discretized spacetime metric. Both cases give raise to Einstein’s field equations yet the gravity is discretized and originated from “spacetime discreteness”. A number of mathematical and physical implications of these results were discussed though this paper and perspectives are given accordingly. Full article

(This article belongs to the Special Issue Mathematical physics)

291 KiB

Open AccessArticle

Zeta Function Expression of Spin Partition Functions on Thermal AdS₃

by Floyd L.Williams

Mathematics 2015, 3(3), 653-665; https://doi.org/10.3390/math3030653 - 28 Jul 2015

Cited by 3 | Viewed by 3835

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Mathematical physics)

201 KiB

Open AccessArticle

On the Nature of the Tsallis–Fourier Transform

by A. Plastino and Mario C. Rocca

Mathematics 2015, 3(3), 644-652; https://doi.org/10.3390/math3030644 - 21 Jul 2015

Cited by 2 | Viewed by 3270

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Mathematical physics)

475 KiB

Open AccessArticle

Singular Bilinear Integrals in Quantum Physics

by Brian Jefferies

Mathematics 2015, 3(3), 563-603; https://doi.org/10.3390/math3030563 - 29 Jun 2015

Cited by 1 | Viewed by 3550

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Mathematical physics)

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 6711

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. 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 4682

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)

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 4490

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)

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 9242

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 3714

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 4780

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)

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 7333

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)

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 5090

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 [...] 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 4848

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)

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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 5029

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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905 KiB

Open AccessEssay

A Moonshine Dialogue in Mathematical Physics

by Michel Planat

Mathematics 2015, 3(3), 746-757; https://doi.org/10.3390/math3030746 - 14 Aug 2015

Cited by 2 | Viewed by 6044

Abstract

Phys and Math are two colleagues at the University of Saçenbon (Crefan Kingdom), dialoguing about the remarkable efficiency of mathematics for physics. They talk about the notches on the Ishango bone and the various uses of psi in maths and physics; they arrive [...] Read more.

Phys and Math are two colleagues at the University of Saçenbon (Crefan Kingdom), dialoguing about the remarkable efficiency of mathematics for physics. They talk about the notches on the Ishango bone and the various uses of psi in maths and physics; they arrive at dessins d’enfants, moonshine concepts, Rademacher sums and their significance in the quantum world. You should not miss their eccentric proposal of relating Bell’s theorem to the Baby Monster group. Their hyperbolic polygons show a considerable singularity/cusp structure that our modern age of computers is able to capture. Henri Poincaré would have been happy to see it. Full article

(This article belongs to the Special Issue Mathematical physics)

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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 5036

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)

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