A remark on sets having the Steinhaus property

A point set satisfies the Steinhaus property if no matter how it is placed on a plane, it covers exactly one integer lattice point. Whether or not such a set exists, is an open problem. Beck has proved [1] that any bounded set satisfying the Steinhaus property is not Lebesgue measurable. We show that any such set (bounded or not) must have empty interior. As a corollary, we deduce that closed sets do not have the Steinhaus property, fact noted by Sierpinski [3] under the additional assumption of boundedness.     

Classical and statistical analysis of40Ar(300 MeV)+197Au deep inelastic interaction

We examine the way in which local field effects in the neutron refractive index affect the values of coherent scattering lengths determined by various kinds of neutron optical measurements. We find that under typical experimental conditions these effects are negligible for interferometry measurements but that they are significant for gravity refractometry measurements, producing changes in the effective scattering length of as much as two or three standard deviations in some cases. Refined values of the scattering lengths are obtained for the thirteen elements for which data are presently available. The special role of local field effects in neutron transmission is also discussed.     

Irreducible representations of infinite-dimensional Lie algebras

Some results concerning the operatorially irreducible, topologically irreducible, operatorially and topologically irreducible representations of a class of infinite-dimensional Lie algebras by bounded operators in a complex Banach space are obtained.     

Stereochemie und1H-NMR-Spektren einiger vom Pentaerythrit abgeleiteten Spiro-1,3-dioxane

Summary Mono- and trispiro-1,3-dioxanes with mobile, fixed and semimobile structures were obtained by the condensation reaction of pentaerythritol with aldehydes and symmetrical or unsymmetrical ketones. The conformational analysis of the obtained compounds was undertaken by means of¹H-NMR spectroscopy. The spiro-1,3-dioxanes obtained with aldehydes or nonsymmetrical ketones represent fixed (anancomeric) structures. Ketalisation with 4-t-butyl-cyclohexanone leads to a semimobile structure of a trispiro-1,3-dioxane in which the two marginal cyclohexanic rings are fixed, while the two middle ones are mobile, continuously flipping at room temperature. The solvent effect on the NMR spectra is also discussed. Utilisation of benzene-d ₆ makes it possible to interpret some of the complex spectra with superposed signals on the basis of the aromatic solvent induced shift (ASIS).

     

Theoretical determination of the ablation rate of metals in multiple-nanosecond laser pulses irradiation regime

A detailed understanding of the physical determinants of the ablation rate in multiple nanosecond laser pulses regime is of key importance for technological applications such as patterning and pulsed-laser deposition. Here, theoretical modeling is employed to investigate the ablation of thick metallic plates by intense, multiple nanosecond laser pulses. A new photo-thermal model is proposed, in which the complex phenomena associated to the ablation process are accounted for as supplementary terms of the classical heat equation. The pulsed laser ablation in the nanosecond regime is considered as a competition between thermal vapourization and melt ejection under the action of the plasma recoil pressure. Computer simulations using the photo-thermal model presented here and the comparison of the theoretical results with experiment indicate two different mechanisms that contribute to the decrease of the ablation efficiency. First, during the ablation process the vapour/plasma plume expanding above the irradiated target attenuates the laser beam that reaches the sample, leading to a marked decrease of the ablation efficiency. Additional attenuation of the laser beam incident on the sample is produced due to the heating of the plasma by the absorption of the laser beam into the plasma plume. The second mechanism by which the ablation efficiency decreases consists of the reduction of the incident laser intensity with the lateral area, and of the melt ejection velocity with the depth of the hole.     

Some Pólya-Type Irreducibility Criteria for Multivariate Polynomials

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y ? f ₁(X))…(Y ? f _n (X)) + g(X), with f, f ₁, ?, f _n , g univariate polynomials over an arbitrary field.     

An Algebraic Perspective on Multivariate Tight Wavelet Frames

Recent advances in real algebraic geometry and in the theory of polynomial optimization are applied to answer some open questions in the theory of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) are given in terms of Hermitian sums of squares of certain nonnegative Laurent polynomials and in terms of semidefinite programming. These formulations merge recent advances in real algebraic geometry and wavelet frame theory and lead to an affirmative answer to the long-standing open question of the existence of tight wavelet frames in dimension d=2. They also provide, for every d, efficient numerical methods for checking the existence of tight wavelet frames and for their construction. A class of counterexamples in dimension d=3 show that, in general, the so-called sub-QMF condition is not sufficient for the existence of tight wavelet frames. Stronger sufficient conditions for determining the existence of tight wavelet frames in dimension d≥3 are derived. The results are illustrated on several examples.     

Complex symmetric operators and applications II

A bounded linear operator on a complex Hilbert space is called complex symmetric if , where is a conjugation (an isometric, antilinear involution of ). We prove that , where is an auxiliary conjugation commuting with . We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition also extends to the class of unbounded -selfadjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.