Some computations on the spectra of Pisot and Salem numbers

Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdos, Joó and Komornik in 1990, is the determination of for Pisot numbers , where

Although the quantity is known for some Pisot numbers , there has been no general method for computing . This paper gives such an algorithm. With this algorithm, some properties of and its generalizations are investigated.

A related question concerns the analogy of , denoted , where the coefficients are restricted to ; in particular, for which non-Pisot numbers is nonzero? This paper finds an infinite class of Salem numbers where .

     

Quincunx fundamental refinable functions and quincunx biorthogonal wavelets

We analyze the approximation and smoothness properties of quincunx fundamental refinable functions. In particular, we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in . Their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order. In addition, these examples are minimally supported with symmetry. For two special families of such quincunx interpolatory masks, we prove that their symbols are nonnegative. Finally, a general way of constructing quincunx biorthogonal wavelets is presented. Several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed.

     

Sets of uniqueness for spherically convergent multiple trigonometric series

A subset of the -dimensional torus is called a set of uniqueness, or -set, if every multiple trigonometric series spherically converging to outside vanishes identically. We show that all countable sets are -sets and also that sets are -sets for every . In particular, , where is the Cantor set, is an set and hence a -set. We will say that is a -set if every multiple trigonometric series spherically Abel summable to outside and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for sets. In addition, every -set has measure , and a countable union of closed -sets is a -set.

     

Absorption Spectra of Excited Ruby Crystals

We have investigated the absorption spectra of excited laser single crystals of ruby. The longlived increase in absorption depending on the degree of excitation is observed in optically excited crystals of ruby in a wide spectral range. Moreover, in the absorption spectrum of a postexcited ruby we discovered the excitationinduced longlived kinetic instability, namely, the noiselike quasiline component variable in time and over the spectrum and imposed on the smooth spectral curve.     

Initiation of Light Pulses by a Coaxial H‐Compressed Discharge

An optimum configuration of the discharge channel of a coaxial Hcompressed discharge is determined to exclude the screening influence of an envelope on discharge radiation in the UV region of a spectrum. It is shown that the use of an Hcompressed discharge of coaxial shape makes it possible to increase the radiation yield in the UV spectrum as compared to a plane shape at the same parameters of electric power supply.     

Beatings in the Spin Echo Signal in Media Having a Doublet Structure of the Spectrum

We have obtained an analytical expression for the twopulse echo signal arising in nonresonant excitation of a condensed medium with a nonuniformly broadened spectrum of doublet structure by two pulses of an electromagnetic field of different durations and of the same amplitude. It is shown that in the case where the differences between the carrying frequency of exciting pulses and the central frequencies of the spectral lines differ considerably, beatings are formed in the signal of the twopulse echo. The rate of decay of the amplitude of these beatings is found to depend only on the time of reversible and irreversible relaxation, and their circular frequency is determined by the shift of the spectral lines relative to each other. A technique for separate determination of the inhomogeneous width of the lines and time of the irreversible transverse relaxation is suggested which is based on the analysis of the twopulse echo signal beatings.     

Analysis of the Structure of the Bands in the IR Spectrum of β-D Glucose by the Regularized Method of Deconvolution

Deconvolution of the IR absorption spectrum of -D glucose in the spectral range 1500–450 cm^–1 has been carried out. The results of the deconvolution were compared with the IR and Raman spectra recorded at room and low temperatures and with the data obtained by theoretical calculations for the frequencies of the normal vibrations of the -D glucose molecule in the crystalline state. It is shown that deconvolution of the IR spectra recorded at room temperature makes it possible to separate the bands observed experimentally only at a very low temperature of the sample and a number of components that were not resolved earlier. The number of bands separated on deconvolution of the IR spectra of -D glucose in the spectral range 1500–450 cm^–1 is more than twice the number of visible absorption maxima in the usual spectrum. The results of deconvolution of the IR spectrum of -D glucose are in good agreement with the data of theoretical calculations for the frequencies of the normal vibrations of the -D glucose molecule in the crystalline state. The existence of the factor-group (Davydov) splitting of a number of frequencies of the nondegenerate fundamental vibrations of molecules in a crystal cell has been revealed in the IR spectrum of -D glucose. It was concluded that the model of an isolated molecule is insufficient for detailed theoretical interpretation of the vibrational spectra of carbohydrates.     

On the Continuity of the Generalized Nemytskii Operator on Spaces of Differentiable Functions

We obtain sufficient conditions for the continuity of the general nonlinear superposition operator (generalized Nemytskii operator) acting from the space of differentiable functions on a bounded domain to the Lebesgue space . The values of operators on a function are locally determined by the values of both the function itself and all of its partial derivatives up to order inclusive. In certain particular cases, the sufficient conditions obtained are proved to be necessary as well. The results are illustrated by several examples, and an application to the theory of Sobolev spaces is also given.     

Performance of laser-ultrasonic F-SAFT imaging

The resolution and signal-to-noise ratio of laser-ultrasonics to detect small and buried defects can be greatly enhanced by using the synthetic aperture focusing technique (SAFT). Originally developed in the time domain, SAFT can also be implemented in the frequency domain (F-SAFT) using the angular spectrum approach for a significant reduction in processing time. In this paper, an F-SAFT based data processing method especially adapted to laser-ultrasonic data is presented. This method allows for further significant improvements towards laser-ultrasonic imaging of small defects. It includes temporal deconvolution of the waveform data, control for an optimal aperture and frequency bandwidth as well as spatial interpolation of the subsurface images. All the above operations are well adapted to the frequency domain calculations and embedded in the F-SAFT data processing. Also, the aperture control and spatial interpolation allow a reduction of sampling requirements to further decrease both inspection and processing times. The above improvements are illustrated using laser-ultrasonic data taken from an aluminum sample with flat-bottom holes.     

长余辉发光材料的紫外-可见反射光谱测定

用“湿法”制备了长余辉发光材料,原料通过水溶液液相分子水平上的均匀混合,利用金属硝酸盐和有机还原剂在较低的温度下发生氧化还原燃烧反应,一步快速生成产品。加热起燃温度低至500℃,反应时间短,所制得的产品成份均匀,晶粒小,外观呈蓬松状态,易研磨粉碎,粉体表观密度小。以紫外－可见分光光度计测定分析了所制备样品在蓄光前后的反射光谱特征并作了探讨。结果表明,除表观密度外,“显法”与“干法”制备的长余辉发光材料的主要性质相同,紫外－可见反射光谱可以准确描述长余辉发光材料的紫外－可见光谱性能特征。