On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

Hausdorff operators originated from some classical summation methods. Now this is an active research field. In the present article, a spectral representation for multidimensional normal Hausdorff operator is given. We show that normal Hausdorff operator in {L^{2}(\mathbb{R}^{n})} is unitary equivalent to the operator of multiplication by some matrix-valued function (its matrix symbol) in the space {L^{2}(\mathbb{R}^{n};\mathbb{C}^{2^{n}})}. Several corollaries that show that properties of a Hausdorff operator are closely related to the properties of its symbol are considered. In particular, the norm and the spectrum of such operators are described in terms of the symbol.

Parabolic by Shilov systems with variable coefficients

Because of the parabolic instability of the Shilov systems to change their coefficients, the definition parabolicity of Shilov for systems with time-dependent $t$ coefficients, unlike the definition parabolicity of Petrovsky, is formulated by imposing conditions on the matricant of corresponding dual by Fourier system. For parabolic systems by Petrovsky with time-dependent coefficients, these conditions are the property of a matricant, which follows directly from the definition of parabolicity. In connection with this, the question of the wealth of the class Shilov systems with time-dependent coefficients is important.A new class of linear parabolic systems with partial derivatives to the first order by the time $t$ with time-dependent coefficients is considered in this work. It covers the class by Petrovsky systems with time-dependent younger coefficients. A main part of differential expression of each such system is parabolic (by Shilov) expression with constant coefficients. The fundamental solution of the Cauchy problem for systems of this class is constructed by the Fourier transform method. Also proved their parabolicity by Shilov. Only the structure of the system and the conditions on the eigenvalues of the matrix symbol were used. First of all, this class characterizes the wealth by Shilov class of systems with time-dependents coefficients.Also it is given a general method for investigating a fundamental solution of the Cauchy problem for Shilov parabolic systems with positive genus, which is the development of the well-known method of Y.I. Zhitomirskii.

Study of Effect of Laser Marking Data Matrix Symbols on Titanium Alloy Sheets Using Nd:YAG Laser

Laser direct-part marking is finding increasing use in the tracking of products as they pass through a manufacturing process. But too high heat input of the laser may cause deformation of the product beyond the limit of the drawing. In order to investigate the effect of laser marked Data Matrix symbol on titanium alloy, a Q-switched lamp pumped Nd:YAG laser was used in this research. Two Data Matrix symbols using 16A and 28A of electric current were marked respectively in the center of two titanium alloy sheets with 50mm length and 50mm width and 2mm thickness. To analyze the quality of the laser marking, environmental scanning electron microscope (ESEM) and X-ray diffractometer (XRD) were used. Experimental results showed the HAN’S Q-switched lamp pumped Nd:YAG laser marking system did not cause damage to the titanium alloy.

Laser Direct-Part Marking of Data Matrix Symbols on Carbon Steel Substrates

Certain applications have recently appeared in industry where a traditional bar code printed on a label will not survive because the item to be tracked has to be exposed to harsh environments. Laser direct-part marking is a manufacturing process used to create permanent marks on a substrate that could help to alleviate this problem. In this research, a 532 nm Nd:YAG laser was utilized to produce Data Matrix symbols onto carbon steel substrates. The quality of the laser marked Data Matrix symbol was then evaluated according to the ISO/IEC 16022 bar code technology specification for Data Matrix. Several experiments were conducted to explore the effects that different parameters have on the quality of the laser direct-part marked symbols. Parameters such as type of carbon steel, percent of laser tool path overlap, profile speed, average power, and frequency were found to have significant effects on the quality of the Data Matrix symbols produced with the laser. The analysis of the results indicated that contrast and print growth were the critical ISO/IEC 16022 standard performance measures that limited the laser marked Data Matrix symbols from achieving a higher final grade.