New Examples of Willmore Surfaces in S n

A surface x: M S ⁿ is called a Willmore surface if it is a criticalsurface of the Willmore functional _M (S – 2H ²)dv, where H isthe mean curvature and S is the square of the length of the secondfundamental form. It is well known that any minimal surface is aWillmore surface. The first nonminimal example of a flat Willmoresurface in higher codimension was obtained by Ejiri. This example whichcan be viewed as a tensor product immersion of S ¹(1) and a particularsmall circle in S ²(1), and therefore is contained in S ⁵(1) gives anegative answer to a question by Weiner. In this paper we generalize theabove mentioned example by investigating Willmore surfaces in S ⁿ (1)which can be obtained as a tensor product immersion of two curves. We inparticular show that in this case too, one of the curves has to beS ¹(1), whereas the other one is contained either in S ²(1) or in S ³(1). In the first case, we explicitly determine the immersion interms of elliptic functions, thus constructing infinetely many newnonminimal flat Willmore surfaces in S ⁵. Also in the latter casewe explicitly include examples.     

Using self-similar properties of turbulent premixed flames to downsize chemical tables in high-performance numerical simulations

Detailed chemical mechanisms have to be incorporated in turbulent combustion modelling to predict flame propagation, ignition, extinction or pollutant formation. Unfortunately, hundreds of species and thousands of elementary reactions are involved in hydrocarbon chemical schemes and cannot be handled in practical simulations, because of the related computational costs and the need to model the complexity of their interaction with turbulent motions. Detailed chemistry may be handled using look-up tables, where chemical parameters such as reaction rates and/or species mass fractions are determined from a reduced set of coordinates, progress variables or mixture fractions, as proposed in ILDM, FPI or FGM methods. Nevertheless, these tables may require large computer memory spaces and non-negligible access times. This issue becomes of crucial importance when running on massively parallel computers: to implement these databases in shared memories would induce a large number of data exchanges, reducing the overall code performance; on the other hand duplicating databases in every local processor memory may become impossible either for large databases or small local memories. This work proposes to take advantage of the self-similar behaviour of turbulent premixed flames to reduce the size of these chemical databases, specifically when running on massively parallel machines, under the FPI (Flame Prolongation of ILDM) framework. Several approaches to reduce the database are investigated and discussed both in terms of memory requirements and access times. A very good compromise is obtained for methane–air turbulent premixed flames, where the size of the database is decreased by a factor of 1000, while the access time is reduced by about 60%.     

Analysing the sintering of heterogeneous powder structures by in situ microtomography

This work analyses the microstructure changes of various copper-based powder systems during sintering from 3D images provided by in situ synchrotron microtomography. The investigated systems include a copper powder with a wide particle size distribution of 0–63 µm poured into a quartz capillary, a pre-sintered copper compact with artificially created large pores and a mixture of copper and alumina particles. The experiments were carried out at the European Synchrotron in Grenoble, France. Powders were sintered up to 1060°C under reducing atmosphere in a furnace located between the X-ray source and the detector. During each experiment, 3D images were taken at various times of the thermal cycle. We have obtained images with a resolution of 1.5 µm and the time of acquisition of every image was ～1 min. Quantitative analysis of these images allowed the changes of various important parameters to be followed. Such parameters characterise the sintering process at the particle length scale: interparticle coordination, pore size distribution and particle centre-to-centre distance. Moreover, by tracking the displacement of each particle centre and comparing it to the displacement predicted by classical mean field assumption, we have been able to assess the magnitude of particle rearrangement occurring during sintering. From these data, the sintering behaviour of heterogeneous powder systems is discussed with particular emphasis of collective particle phenomena.     

Rotational analysis of the 7390- and 7937-Å bands of NO2 by means of Fourier transform spectroscopy

We have extended to higher N and to K_a = 3 and 4 the rotational analysis of the 7390-Å band of NO₂ performed by K. E. Hallin and A. J. Merer (Canad. J. Phys.55, 2101–2112 (1977)). The lines belong to a perturbed parallel band for which Hallin and others have proposed the vibrational assignment (2 13 1)-(0 0 0) within the electronic ground state. These authors presumed that this band borrows its intensity through a vibronic coupling (spin-orbit and/or Coriolis coupling) from the stronger (0 2 0)-(0 0 0) band of the $\text{A}\text{?}\text{-}\text{X}\text{?}$ electronic system at 7460 Å. We have observed about 900 transitions belonging to the K_a = 0, 1, 2, 3, 4 subbands of the (2 13 1)-(0 0 0) band for N values going up to about 23, and 300 lines of the “hot” band (2 13 1)-(0 1 0). We have also looked for spin-orbit-induced transitions and we have detected about 400 transitions with ΔN ≠ ΔJ. Among them ΔN = ±2 transitions with ΔK_a = 0 or ± 2 have been observed, indicating that N and K_a are no longer good quantum numbers, and demonstrating clearly the existence of rovibronic interactions perturbing the upper levels of the transitions.     

A systematic formula for the asymptotic expansion of singular integrals

Asymptotic theories like the lifting-line, the slender body or the slender ship lead to lineintegrals with singular kernels. Sometimes these integrals are improper, that is to say that they are defined only by their Finite Part. To find asymptotic expansions of these integrals, the Matched Asymptotic Expansion Method is widely used along with other more specific methods depending on the kernel type. The first method is laborious and not systematic, and the other methods are sometimes too much specific to treat general cases. Moreover, all of them are not well adapted to deal with Finite Part integrals.Here, a new method is proposed to avoid the previous difficulties. This method is systematic for homogeneous kernels and gives approximations up to any order, as long as the derivative of the weight function exists at this given order. Moreover the occurrence of logarithmic terms in the expansion is explained and easily predictable. An elliptic integral and the classical lifting-line theory are treated to illustrate the ease of this method.

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Résumé Les théories asymptotiques telles que la ligne portante, le corps élancé ou le navire de grand allongement conduisent à des intégrales curvilignes à noyaux singuliers. Parfois, ces intégrales sont impropres c'est à dire qu'elles sont définies en Parties Finies. Différentes méthodes ont été mises au point pour trouver les développements asymptotiques de ces intégrales. Généralement elles dépendent fortement de la nature du noyau, et c'est finalement la méthode des développements raccordés qui est utilisées quand le noyau est trop compliqué. Cependant, cette méthode est laborieuse et comme les précèdentes non adaptée aux intégrales défines par leur Partie Finie.Une nouvelle méthode est proposée pour surmonter ces difficultés. Cette méthode est systématique pour les noyaux homogènes et donne les approximations à tout ordre pourvu que les dérivées de la fonction poids existent jusqu'à cet ordre. De plus la présence de termes logarithmiques dans le développement est expliquée et aisément prédictible.Une intégrale elliptique, ainsi que la fameuse théorie de la ligne portante sont traités pour illustrer les possibilités de la méthode.

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Nomenclature D domain of integration - f(x) weight function - FP Finite Part - h(x) weight function - I ,I _o bounded intervals - j, J integers - K(x, ) singular kernel - L, L integers - M integer defining the approximation order - P _k (x) Legendre polynomial - R set of real numbers - R(ß) equals 1 if is an integer and 0 if not - R _{f, J} ,R _{K, L} remainders of Taylor developments - S () equals either 1 or the sign function:sgn() - t, u, v, x variable of integration - , real numbers - homogeneity order of the kernel - F () Euler's integral (gamma function) - small parameter - [.] integer part of     

A new mechanism for gravitational-wave emission in core-collapse supernovae

We present a new theory for the gravitational-wave signatures of core-collapse supernovae. Previous studies identified axisymmetric rotating core collapse, core bounce, postbounce convection, and anisotropic neutrino emission as the primary processes and phases for the radiation of gravitational waves. Our results, which are based on axisymmetric Newtonian supernova simulations, indicate that the dominant emission process of gravitational waves in core-collapse supernovae may be the oscillations of the protoneutron star core. The oscillations are predominantly of mode character, are excited hundreds of milliseconds after bounce, and typically last for several hundred milliseconds. Our results suggest that even nonrotating core-collapse supernovae should be visible to current LIGO-class detectors throughout the Galaxy, and depending on progenitor structure, possibly out to megaparsec distances.     

Symmetries of the heat equation on the lattice

Discrete versions of the heat equation on two-dimensional uniform lattices are shown to possess the same symmetry algebra as their continuum limits. Solutions with definite symmetry properties are presented.     

On a bosonic-parafermionic realization ofU_q (\widehat{sl(2)})

We realize the current algebra at an arbitrary level in terms of one deformed free bosonic field and a pair of deformed parafermionic fields. It is shown that the operator product expansions of these parafermionic fields involve an infinite number of simple poles and simple zeros, which then condensate to form a branch cut in the classical limitq1. Our realization coincides with those of Frenkel-Jing and Bernard when the levelk takes the values 1 and 2, respectively.     

Unreasonable effectiveness of symmetry in physics

We prove that in some reasonable sense, every possible physical law can be reformulated in terms of symmetries. This result explains the well-known success of the group-theoretic approach in physics.