Formulae for null curves deriving from elliptic curves

Any elliptic curve can be realised in the tangent bundle of the complex projective line as a double cover branched at four distinct points on the zero section. Such a curve generates, via classical osculation duality, a null curve in C³${\mathbb{C}}^3$ and thus an algebraic minimal surface in R³${\mathbb{R}}^3$. We derive simple formulae for the coordinate functions of such a null curve.     

Holomorphic curves from matrices

Membranes holomorphically embedded in flat non-compact space are constructed in terms of the degrees of freedom of an infinite collection of 0-branes. To each holomorphic curve we associate infinite-dimensional matrices which are static solutions to the matrix theory equations of motion, and which can be interpreted as the matrix theory representation of the holomorphically embedded membrane. The problem of finding such matrix representations can be phrased as a problem in geometric quantization, where ϵ ∝ l_P³/R plays the role of the Planck constant and parametrizes families of solutions. The concept of Bergman projection is used as a basic tool, and a local expansion for the action of the projection in inverse powers of curvature is derived. This expansion is then used to compute the required matrices perturbatively in ϵ. The first two terms in the expansion correspond to the standard geometric quantization result and to the result obtained using the metaplectic correction to geometric quantization.     

Integrable generalizations of Schrödinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable SO(3)$SO\left(3\right)$-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame. This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each model in the hierarchy. A generalization of these results to geometric surface flows is presented, where the surfaces are non-stretching in one direction while stretching in all transverse directions. Through the Frenet equations of a moving frame, such surface flows are shown to encode a hierarchy of 2+1 dimensional integrable SO(3)$SO\left(3\right)$-invariant vector models, along with their bi-Hamiltonian structure, recursion operator, and constants of motion, describing a geometric realization of 2+1 dimensional bi-Hamiltonian NLS and mKdV soliton equations. Based on the well-known equivalence between the Heisenberg model and the Schrödinger map equation in 1+1 dimensions, a geometrical formulation of these hierarchies of 1+1 and 2+1 vector models is given in terms of dynamical maps into the 2-sphere. In particular, this formulation yields a new integrable generalization of the Schrödinger map equation in 2+1 dimensions as well as a mKdV analog of this map equation corresponding to the mKdV spin model in 1+1 and 2+1 dimensions.     

Extraction of hysteresis loops from main magnetization curves

In firm catalogues,the basic properties of magnetic materials are often described by their main magnetization curves. Such curves may be used for the first analysis of circuits containing ferromagnetic cores. The analysis will be more accurate, if the curves are transformed into the families of hysteresis loops. To enable the reconstruction of such loops, we formulate a simple model of hysteresis, making main magnetization curve directly dependent on coercivity. In this way we can approximate hysteresis loops of most typical materials in a pretty wide range of magnetization. Application of variable coercivity enables extension of the model to stronger fields.     

The determination of magnetic anisotropy constants from torque curves

The procedures by which magnetic anisotropy constants are evaluated from torque curves are examined with reference to some calculated torque curves for an ideal uniaxial crystal. The results are compared with some torque measurements on a crystal of cobalt.     

Yield curves for photo-proton emission from carbon and beryllium

The yields of photo-protons at laboratory angles of 40° and 70° from carbon irradiated with bremsstrahlung of end-point energies between 125 and 1000 MeV are given. From these data and a more limited set for beryllium, the shapes of the cross-sections for photo-proton emission by monochromatic photons are deduced. From a separate series of measurements on carbon at 70° lab with bremsstrahlung of end-point energies 380 and 420 MeV, the energy spectrum of photo-protons emitted by photons of energies 400±20 MeV has been derived.     

Invariant curves for area-preserving twist maps far from integrable

One-parameter families of area-preserving twist maps of the formF (x, y)=(x +y +f(x),y +f(x)) are considered. Various invariant curves, for the maps corresponding tof(x)=sin andf(x)=sinx+(1/50) sin(5x), are rigorously constructed forlarge values of the nonlinearity parameter . For larger values of , close to critical, some numerical experiments are briefly discussed.     

Temperature dependence of the superconducting energy gap from conductance curves

Summary An analytic relationship between the values of the applied voltages which produce a maximum in the differential conductance and the energy gap is derived for BCS isotropic superconductors. The different methods of deduction of the temperature dependence of the gap in high-T _c superconductors are analysed.     

The determination of anisotropy constants from torque curves — II

This is an extension of a previous paper which examined the methods available for the determination of the anisotropy constant from torque curves when the anisotropy is governed by a single constant i.e. E = K₁ sin²φ. The two methods singled out in that paper as being of special merit are briefly re-examined when the anisotropy energy takes the form E = K₁ sin²φ + K₂ sin⁴φ.     

The dissociation energy of BiI derived from potential energy curves

The potential energy curve for the ground state of BiI has been constructed by the methods of Lakshman and Rao and of Jarmain. The dissociation energy is estimated to be 1.949±0.009 eV from fitting the three-parameter Lippincott potential function. The value recommended by Gaydon is 2.5±1.0 eV.     

Ionic potential analysis of RHEED rocking curves from fluoride structures

Rocking curves of reflection high energy electron diffraction from MnF₂ and CaF₂(1 1 1) surfaces have been calculated with ionic scattering potentials. The potentials were derived from tabulated X-ray scattering factors and expressed in the Doyle-Turner representation. The Coulomb potential of the ions was introduced into the calculations of diffracted intensity using dynamical diffraction theory. Comparison of the calculated and measured rocking curves for CaF₂(1 1 1) surface confirmed that it is bulk terminated; the correction to volume average potential was found to be 1.1 eV (much less than when using atomic potentials). Analysis of the rocking curves from ultra-thin MnF₂ layers on CaF₂(1 1 1) indicated that MnF₂ inherits the cubic lattice of fluorides up to a thickness of three molecular layers.     

Practical interpretation of X-ray rocking curves from semiconductor heteroepitaxial layers

X-ray rocking curves are widely used to study semiconductor heteroepitaxial structures. Examples are given of the use of rocking curves to investigate crystal growth problems, determine layer growth rates, confirm layer thicknesses and to determine the state of relaxation in layers which are above the critical thickness. The application of dynamic simulation and Fourier transformation to the raw data as tools to interpret data from multilayer structures is discussed.     

Properties of SN Ia progenitors from light curves and spectra

With recent advances in theory and observations, direct connections emerge between the progenitors of Type Ia Supernovae (SNe Ia) and the observed light curves and spectra. A direct link is important for our understanding of the supernovae physics, the diversity of SNe Ia and the use of SNe Ia for high-precision cosmology because the details of the explosion depends sensitively on the initial conditions and the explosion scenario(s) realized in nature. Do SNe Ia originate from SD- or DD systems, and do they lead to M_Ch mass explosions or dynamical mergers? Does the statistical distribtion of SNe Ia depend on their environment which can be expected to change with redshift? In this contribution, we will exam from the theoretical point of view the tell-tails for this connection, their consistency with the observations, and future directions. In a first section, we present the physics of the explosion, light curves and spectral formation in a nutshell to help understanding the connection. For details of the progenitor evolution and explosion physics, we refer to reviews and the other contributions in this issue. Each of the topical sections starts with a brief general review followed by a more detailed discussion of specific results. Because the youth of the field, some bias is unavoidable towards results obtained within our collaborations (and FSU). The imprint of the metallicity, progenitor stars and properties such as the central density of the exploding WD are presented. IR spectroscopy, polarimetry and imaging of SNR remnants are discussed as a tool to test for the WD properties, magnetic fields and asymmetries. We discuss different classes of Type Ia supernovae, and their environment. Possible correlations between the spectroscopic and light curve properties of SN Ia are discussed. Finally, the overall emerging picture and future developments are discussed.     

Comparing dark matter models,modified Newtonian dynamics and modified gravity in accounting for galaxy rotation curves

We compare six models(including the baryonic model,two dark matter models,two modified Newtonian dynamics models and one modified gravity model) in accounting for galaxy rotation curves.For the dark matter models,we assume NFW profile and core-modified profile for the dark halo,respectively.For the modified Newtonian dynamics models,we discuss Milgrom's MOND theory with two different interpolation functions,the standard and the simple interpolation functions.For the modified gravity,we focus on Moffat's MSTG theory.We fit these models to the observed rotation curves of 9 high-surface brightness and 9 low-surface brightness galaxies.We apply the Bayesian Information Criterion and the Akaike Information Criterion to test the goodness-of-fit of each model.It is found that none of the six models can fit all the galaxy rotation curves well.Two galaxies can be best fitted by the baryonic model without involving nonluminous dark matter.MOND can fit the largest number of galaxies,and only one galaxy can be best fitted by the MSTG model.Core-modified model fits about half the LSB galaxies well,but no HSB galaxies,while the NFW model fits only a small fraction of HSB galaxies but no LSB galaxies.This may imply that the oversimplified NFW and core-modified profiles cannot model the postulated dark matter haloes well.     

Perturbed nonlinear models from noncommutativity

Some discrete two-dimensional many particle systems are investigated inside the context of a deformed Newtonian mechanics obtained as a classical limit of a noncommutative quantum mechanics where the noncommutativity includes all degrees of freedom of the theory. Also, applying the continuous limit some alternative noncommutative generalizations of two-dimensional field theories have been constructed.     

Fermionic composite models from complementarity

Composite models for (in principle massless) quarks and leptons without fundamental scalars are constructed with the aim of providing for fermionic realizations of models which include elementary bosons (by Abbott and Farhi, Casalbuoni and Gatto and Barbieri, Mohapatra and Masiero). The models use one confining unitary (subcolor) group (with left-handed fermions in the fundamental, in its conjugate, and either in the adjoint, or in the symmetric, or in the antisymmetric representation of subcolor) or two confining groups. Families may arise from discrete symmetries.     

Fluid models from kinetic models using a geometric averaging procedure

We interpret the Lorentz force equation as a geodesic equation associated with a non-linear connection. Using a geometric averaging procedure, we prove that for narrow and smooth one-particle distribution functions whose supports are invariant under the flow of the Lorentz equation, a bunch of charged point particles can be described by a charged cold fluid model in the ultra-relativistic regime. The method used to prove this result does not require additional hypotheses on the higher moments of the distribution. This is accomplished by estimating the expressions that include the differential operators appearing in the charged cold fluid model equation. Under the specified conditions of narrowness and ultra-relativistic dynamics, it turns out that these differential expressions are close to zero, justifying the use of the charged cold fluid model. The method presented in this work can also be applied to justify the use of warm plasmas and other models. Finally, a possible relation with chromohydrodynamics is discussed.     

Multi-matrix models from jet coefficients

We present a very natural framework in which to discuss multi-matrix models of two-dimensional quantum gravity. Multi-matrix model actions, string equations, and other quantities can be compactly expressed in terms of the jets of the resolvents of the relevant differential operators. This allows one to write down equations describing minimal matter coupled to two-dimensional quantum gravity directly in terms of known functionals.