Finitely-Generated Projective Modules over the θ-Deformed 4-Sphere

We investigate the “θ-deformed spheres” ${C(S^{3}_{\theta})}$ and ${C(S^{4}_{\theta})}$ , where θ is any real number. We show that all finitely-generated projective modules over ${C(S^{3}_{\theta})}$ are free, and that ${C(S^{4}_{\theta})}$ has the cancellation property. We classify and construct all finitely-generated projective modules over ${C(S^{4}_{\theta})}$ up to isomorphism. An interesting feature is that if θ is irrational then there are nontrivial “rank-1” modules over ${C(S^{4}_{\theta})}$ . In that case, every finitely-generated projective module over ${C(S^{4}_{\theta})}$ is a sum of a rank-1 module and a free module. If θ is rational, the situation mirrors that for the commutative case θ = 0.     

Super W_(1+∞) n-Algebra in the Supersymmetric Landau Problem

We analyze the super n-bracket built from associative operator products.Since the super n-bracket with n even satisfies the so-called generalized super Jacobi identity,we deal with the n odd case and give the generalized super Bremner identity.For the infinite conserved operators in the supersymmetric Landau problem,we derive the super W_(1+∞) n-algebra which satisfies the generalized super Jacobi and Bremner identities for the n even and odd cases,respectively.Moreover the super W_(1+∞) sub-2n-algebra is also given.     

Geometric Phases for Wave Packets of the Landau Problem

The Landau problem of a charged particle in a plane with a uniform perpendicular magnetic field is analysed in two oscillator modes. The coherent states for the problem have been found out using a general definition of displaced states. The time evolution and the associated nonadiabatic geometric phase for both initially displaced and non-displaced wave packets have been studied. The path integral is derived in a simple way through the calculation of Gaussian integrals via the concept of coherent state wavefunctions.     

The impact of solar and atmospheric parameter uncertainties * on the measurement of θ13 and δ

We present in this paper the analysis of the measurement of the unknown PMNS parameters θ₁₃ and δ at future LBL facilities performing complete three parameters fits, each time fully including in the fit one of the atmospheric and solar oscillation parameters within its present (future) error. We show that, due to the presence of degeneracies, present uncertainties on θ₂₃ and worsen significantly the precision on (θ₁₃,δ) at future LBL experiments. Only if a precision on the atmospheric parameters at least similar to what expected at T2K-I is reached, then the sensitivities to θ₁₃ and δ that have been presented in the literature for many facilities (where θ₂₃ and are generally considered as fixed external inputs) can indeed be almost recovered. On the other hand, the impact on this measurement of the uncertainties on the solar parameters, θ₁₂ and is already negligible. Our analysis has been performed using three reference setups: the SPL Super-Beam and the standard low-γ β-Beam, both aiming toward a Mton Water Čerenkov detector located at L = 130 km; the 50 GeV Neutrino Factory with a 40 kton Magnetized Iron Detector to look for the "golden channel" ν_e → ν_μ with baseline L = 3000 km and a 4 kton Emulsion Cloud Chamber to look for the "silver channel" ν_e → ν_τ with baseline L = 732 km. Received: 19 July 2005, Revised: 30 September 2005, Published online: 11 November 2005 PACS: 14.60.Pq, 14.60.Lm     

Inverse Spectral Problem with Partial Information¶on the Potential: The Case of the Whole Real Line

The Schrödinger operator -d²/dx²+q(x)-d^2/dx^2+q(x) is considered on the real axis. We discuss the inverse spectral problem where discrete spectrum and the potential on the positive half-axis determine the potential completely. We do not impose any restrictions on the growth of the potential but only assume that the operator is bounded from below, has discrete spectrum, and the potential obeys q(-|x|) 3 q(|x|)q(-|x|)\geq q(|x|). Under these assertions we prove that the potential for xS 0 and the spectrum of the problem uniquely determine the potential on the whole real axis. Also, we study the uniqueness under slightly different conditions on the potential. The method employed uses Weyl m-function techniques and asymptotic behavior of the Herglotz functions.     

The quark electric dipole moment and the induced θ-term in the Kobayashi-Maskawa model

Closed expressions are obtained for the quark electric dipole moment and the induced θ-term in the Kobayashi-Maskawa model in the approximation G²σ_s.     

Modeling of superconductors based on the timedependent Ginsburg–Landau equations

Results of modeling of superconductor magnetization process based on a numerical solution of the timedependent Ginsburg-Landau equations are presented. Methods of grid approximation of the equations and method of finite elements are used. Two-dimensional patterns of changes in the order parameter and supercurrent distribution in superconductors are calculated and visualized. The main results are in agreement with the well-known representations for type I and II superconductors.     

The Weiss model and the Landau–Khalatnikov model for the switching of ferroelectrics

It is shown that the molecular field theory by P. Weiss formally leads to the switching kinetics of ferroelectrics, which is described by the well-known Landau–Khalatnikov equation. The switching has a critical character, taking place only at E_a>E_c (E_a: external field, E_c: coercive field). The results are checked by computer simulations.     

Cure to the Landau–Pomeranchuk and associated long-wavelength Fermi-surface instabilities on the lattice

The cure to the ??=?1 Landau–Pomeranchuk instabilities in translationally invariant fermions is shown to be a state with an anisotropic gap at the Fermi surface. For higher ? and for fermions on a lattice, the general criteria for long-wavelength instabilities and their cure are found in terms of the derivatives of the single particle self-energy with respect to momentum for spin-symmetric instabilities and with respect to magnetic field for spin-antisymmetric instabilities. The results may be relevant to identifying hidden order parameters found in many metals.     

The non-commutative oscillator, symmetry and the Landau problem

The isotropic oscillator on a plane is discussed where the coordinate and momentum space are both considered to be non-commutative. We also discuss the symmetry properties of the oscillator for three separate cases when the non-commutative parameters Θ and for x and p-space, respectively, satisfy specific relations. We compare the Landau problem with the isotropic oscillator on non-commutative space and obtain a relation between the two non-commutative parameters and the magnetic field of the Landau problem.     

The Ginzburg–Landau theory in application

A numerical approach to Ginzburg–Landau (GL) theory is demonstrated and we review its applications to several examples of current interest in the research on superconductivity. This analysis also shows the applicability of the two-dimensional approach to thin superconductors and the re-defined effective GL parameter κ$\kappa$. For two-gap superconductors, the conveniently written GL equations directly show that the magnetic behavior of the sample depends not just on the GL parameter of two bands, but also on the ratio of respective coherence lengths.     

The θ-structure in string theories: Superstrings

This paper is the continuation of a previous paper in which the θ-structure of bosonic strings was discussed. In this paper it is pointed out that the SUSYs are broken, both in open strings and closed strings by introducing non-zero parameters θ. The conclusion is that the compact subspace of space-time must not be multiply connected unless the positive energy of the ground state is regularized to be zero.     

The Hamilton equations of ferrohydrodynamics with the Landau–Lifshitz equation for magnetization

A system of Hamilton equations of motions for the ideal nonconducting magnetic fluid with the Landau–Lifshitz equation for the magnetization vector is constructed based on the functional of total energy derived in this work and the Poisson bracket method.     

The Ginzburg–Landau theory and the surface energy of a colour superconductor

We apply the Ginzburg–Landau theory to the colour superconducting phase of a lump of dense quark matter. We calculate the surface energy of a domain wall separating the normal phase from the super phase with the bulk equilibrium maintained by a critical external magnetic field. Because of the symmetry of the problem, we are able to simplify the Ginzburg–Landau equations and express them in terms of two components of the di-quark condensate and one component of the gauge potential. The equations also contain two dimensionless parameters: the Ginzburg–Landau parameter κ and ρ. The main result of this paper is a set of inequalities obeyed by the critical value of the Ginzburg–Landau parameter—the value of κ for which the surface energy changes sign—and its derivative with respect to ρ. In addition we prove a number of inequalities of the functional dependence of the surface energy on the parameters of the problem and obtain a numerical solution of the Ginzburg–Landau equations. Finally a criterion for the types of colour superconductivity (type I or type II) is established in the weak coupling approximation.     

The Duotron,Can It Solve the Solar Neutrino Problem?

At a summary of the four present measurements for the solar neutrino problem one receives for the quotient of the number of the really observed events and the theoretically by the standard solar model exspected value the result This says that in the mean only 33.25% with a possible mistake of 19.25% of the exspected solar neutrinos have been observed. If one goes out from the assumption, that the attempts hitherto proposed for the solution of the problem on the solar neutrino deficits as to be regarded as very unsatisfactory on the one side, while on the other the neutrinos of the β-decay may not exist at all, this looks very troublesome. But there is another possibility of a new interpretation. One has to consider, that in connection with each theoretically until now assumed neutrino of β-decay at the solar p-p- and all joined following fusion -processes precisely one positron will be produced. Exactly this particle will with the probability 1/2 in a triplett state of fixing with an electron of the solar centre plasma at a binding energy of −2029.7 eV create an electric-neutral duotron. This particle assumes fully, inclusive the very welcome factor 1/2, the rǒle of the theoretically dispensable neutrino. This can be regarded as the solution of the solar neutrino-problem.     

Remarks on Undecidability,Incompleteness and the Integrability Problem

In their seminal paper Undecidability and incompleteness in classical mechanics (Int. J. Theor. Phys. 30:1041–1073, 1991), N.C.A. da Costa and F.A. Doria introduced a powerful method for studying the appearance of undecidability and incompleteness in mathematics and theoretical physics. In this work their results are applied to integrability theory. Specifically, it is pointed out that it is not possible to expect the existence of an algorithm able to decide whether a given partial differential equation is integrable or not.     

Detection of Supernova Neutrinos on the Earth for Large θ_(13)

Supernova （SN） neutrinos detected on the Earth are subject to the shock wave effects, the Mikheyev- Smirnov-Wolfenstein （MSW） effects, the neutrino collective effects and the Earth matter effects. Considering the recent experimental result about the large mixing angle 013 （-8.8°） provided by the Daya Bay Collaboration and applying the available knowledge for the neutrino conversion probability in the high resonance region of SN, PH , which is in the form of hypergeometric function in the case of large 813, we deduce the expression of PH taking into account the shock wave effects. It is found that PH is not zero in a certain range of time due to the shock wave effects. After considering all the four physical effects and scanning relevant parameters, we calculate the event numbers of SN neutrinos for the ＂Garehing＂ distribution of neutrino energy spectrum. From the numerical results, it is found that the behaviors of neutrino event numbers detected on the Earth depend on the neutrino mass hierarchy and neutrino spectrum parameters including the dimensionless pinching parameter βa （where a refers to neutrino flavor）, the average energy 〈Ea〉, and the SN neutrino luminosities La. Finally, we give the ranges of SN neutrino event numbers that will be detected at the Daya Bay experiment.     

The Effect of Finiteness in the Saffman–Taylor Viscous Fingering Problem

We derive a family of exact time-evolving solutions for the the evolution of a finite blob of fluid confined to a channel in a Hele–Shaw cell. We show rigorously that, for large fluid volume, there are solutions for which one of the interfaces approaches the steady Saffman–Taylor finger solution of arbitrary width λ∈(0, 1). On the basis of this, we argue that the far-field effects of a displaced second interface do not provide a selection mechanism for the formation of a width- $ \frac{1}{2} $ finger when surface tension, or any other regularization, is ignored.     

A Note on the Problem of Proper Time in Weyl Space–Time

We discuss the question of whether or not a general Weyl structure is a suitable mathematical model of space–time. This is an issue that has been in debate since Weyl formulated his unified field theory for the first time. We do not present the discussion from the point of view of a particular unification theory, but instead from a more general standpoint, in which the viability of such a structure as a model of space–time is investigated. Our starting point is the well known axiomatic approach to space–time given by Elhers, Pirani and Schild (EPS). In this framework, we carry out an exhaustive analysis of what is required for a consistent definition for proper time and show that such a definition leads to the prediction of the so-called “second clock effect”. We take the view that if, based on experience, we were to reject space–time models predicting this effect, this could be incorporated as the last axiom in the EPS approach. Finally, we provide a proof that, in this case, we are led to a Weyl integrable space–time as the most general structure that would be suitable to model space–time.     

An ?-Deformed Virasoro Algebra as Hidden Symmetry of the Restricted sine-Gordon Model

As the Yangian double with center, which is deformed from affine algebra by the additive loop parameter ?, we get the commutation relation and the bosonization of quantum ?-deformed Virasoro algebra. The corresponding Miura transformation, the associated screening operators and the BRST charge have been studied. Moreover, we also construct the bosonization for type Ⅰ and type Ⅱ intertwiner vertex operators. Finally, we show that the commutation relations of these vertex operators in the case of p^′ = γ, p = γ - 1 and ? = π actually give the exact scattering matrix of the restricted sine-Gordon model.