Quasi‐Relativistic Heat Equation via Lévy Stable Distributions: Exact Solutions

We introduce and study an extension of the heat equation relevant to relativistic energy formula involving square root of differential operators. We furnish exact solutions of corresponding Cauchy (initial) problem using the operator formalism invoking one‐sided Lévy stable distributions. We note a natural appearance of Bessel polynomials which allow one to obtain closed form solutions for a number of initial conditions. The resulting diffusion is slower than the non‐relativistic one, although it still can be termed a normal one. Its detailed statistical characterization is presented in terms of exact evaluation of arbitrary moments and kurtosis and is compared with the non‐relativistic case.     

A Raman spectroscopic study of the different vanadate groups in solid‐state compounds—model case: mineral phases vésigniéite [BaCu3(VO4)2(OH)2] and volborthite [Cu3V2O7(OH)2·2H2O]

Raman spectroscopy has been used to study vanadates in the solid state. The molecular structure of the vanadate minerals vésigniéite [BaCu₃(VO₄)₂(OH)₂] and volborthite [Cu₃V₂O₇(OH)₂·2H₂O] have been studied by Raman spectroscopy and infrared spectroscopy. The spectra are related to the structure of the two minerals. The Raman spectrum of vésigniéite is characterized by two intense bands at 821 and 856 cm⁻¹ assigned to ν₁ (VO₄)³⁻ symmetric stretching modes. A series of infrared bands at 755, 787 and 899 cm⁻¹ are assigned to the ν₃ (VO₄)³⁻ antisymmetric stretching vibrational mode. Raman bands at 307 and 332 cm⁻¹ and at 466 and 511 cm⁻¹ are assigned to the ν₂ and ν₄ (VO₄)³⁻ bending modes. The Raman spectrum of volborthite is characterized by the strong band at 888 cm⁻¹, assigned to the ν₁ (VO₃) symmetric stretching vibrations. Raman bands at 858 and 749 cm⁻¹ are assigned to the ν₃ (VO₃) antisymmetric stretching vibrations; those at 814 cm⁻¹ to the ν₃ (VOV) antisymmetric vibrations; that at 508 cm⁻¹ to the ν₁ (VOV) symmetric stretching vibration and those at 442 and 476 cm⁻¹ and 347 and 308 cm⁻¹ to the ν₄ (VO₃) and ν₂ (VO₃) bending vibrations, respectively. The spectra of vésigniéite and volborthite are similar, especially in the region of skeletal vibrations, even though their crystal structures differ. Copyright © 2011 John Wiley & Sons, Ltd.     

Raman spectroscopy of selected borate minerals of the pinakiolité group

The Raman spectra of a series of related minerals of the pinakiolité group have been collected and the spectra related to the mineral structure. These minerals are based upon an isolated BO₃³⁻ ion. The site symmetry is reduced from D_3h to C₁. Intense Raman bands are observed for the minerals takeuchiité, pinakiolité, fredrikssonité and azoproité at 1084, 1086, 1086 and 1086 cm⁻¹. These bands are assigned to the ν₁ BO₃³⁻ symmetric stretching mode. Low‐intensity Raman bands are observed for the minerals at 1345, 1748; 1435, 1748; 1435, 1750; and 1436, 1749 cm⁻¹, respectively. One probable assignment is to ν₃ BO₃³⁻ antisymmetric stretching mode. Intense Raman bands of the studied minerals at 712 cm⁻¹ are attributed to the ν₂ out‐of‐plane bending mode. Importantly, through the comparison of the Raman spectra, the molecular structure of borate minerals with ill‐defined structures can be obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Geometric micron-moiré

A high spatial resolution and high-sensitivity geometric micron-moiré method is presented. The method is based upon classical geometric moiré method and microscopic principle, the frequency of the specimen and reference gratings used in this method can be higher than 1000 lines/mm. By a microscopic moiré system, the high-frequency gratings could be magnified to a low-frequency level, so geometric moiré patterns can be formed. Experiments show that both the spatial resolution and the sensitivity can reach or exceed micron-level.     

Moiré topography: alternative technique in health care

The instrumentation and application of moiré topography in the diagnosis of corporal asymmetries associated to different illnesses of the osseous-muscular system in early ages is presented in this paper. A sample comprising 203 adolescents ranging between 12 and 15 years were studied. Some moiré topography analysis of human legs and the sole of the foot are also given as examples of moiré applications. The results show the advantages of using this technique in the health care and treatment of different illnesses in adolescents. Some of these advantages are provided by the easy diagnosis and the low cost of the moiré technique.