The extremal convex bodies of constant width for the Minkowski measure of asymmetry are discussed. A result, similar to that of H. Groemer＇s and of H. Lu＇s, is obtained, which states that, for the Minkowski measure ...The extremal convex bodies of constant width for the Minkowski measure of asymmetry are discussed. A result, similar to that of H. Groemer＇s and of H. Lu＇s, is obtained, which states that, for the Minkowski measure of asymmetry, the most asymmetric convex domains of constant width in R2 are Reuleaux triangles.展开更多
In this paper,we introduce the Minkowski measure of asymmetry for the spherical bodies of constant width.Then we prove that the spherical balls are the most symmetric bodies among all spherical bodies of constant widt...In this paper,we introduce the Minkowski measure of asymmetry for the spherical bodies of constant width.Then we prove that the spherical balls are the most symmetric bodies among all spherical bodies of constant width,and the completion of the spherical regular simplexes are the most asymmetric bodies.展开更多
In this paper, we show that when Minkowski measure of asymmetry of convex body K of constant width is bigger than a（n-1）, K has at least n＋1 critical chords, where a（n）=n＋（2n（n＋1））√1/2/n＋2.
Slow evaporation method was used to grow pure and KCl (10 mol%) doped KAP single crystal. The solubility and metastable zone width of aqueous solutions of pure and KCl (10 mol%) doped KAP crystal were evaluated to ana...Slow evaporation method was used to grow pure and KCl (10 mol%) doped KAP single crystal. The solubility and metastable zone width of aqueous solutions of pure and KCl (10 mol%) doped KAP crystal were evaluated to analyze the crystallization process. Measuring the induction period τ, the critical nucleation parameters like interfacial energy (σ), energy of formation of the critical nucleus (ΔG*) were determined using the classical theory of nucleation. The structural properties and optical constants of the grown crystals have been put to test and observed that the addition of KCl results in an enhancement of properties of the crystal. Grown crystals were characterized by powder X-ray diffraction. FTIR spectra confirmed the presence of KCl in pure KAP crystal. UV- Visible spectroscopic studies revealed that addition of KCl in pure KAP crystal increased transparency from 75% to 80%. The analysis of the optical absorption data revealed the presence of both indirect and direct transitions and both of these band gaps increased with the addition of KCl. The transmittance data was analyzed to calculate the refractive index, oscillator energy, dispersion energy, electric susceptibility, zero-frequency dielectric constant and both the real and imaginary parts of the dielectric permittivity as a function of photon energy. The moments of ε(E) were also determined. The dispersion i.e. spectral dependence of the refractive index was discussed according to the single-effective oscillator model proposed by Wemple and DiDomenico.展开更多
基金The NSF (08KJD110016) of Jiangsu Hight Education
文摘The extremal convex bodies of constant width for the Minkowski measure of asymmetry are discussed. A result, similar to that of H. Groemer＇s and of H. Lu＇s, is obtained, which states that, for the Minkowski measure of asymmetry, the most asymmetric convex domains of constant width in R2 are Reuleaux triangles.
基金Supported by the National Natural Science Foundation of China(12071334,12071277)。
文摘In this paper,we introduce the Minkowski measure of asymmetry for the spherical bodies of constant width.Then we prove that the spherical balls are the most symmetric bodies among all spherical bodies of constant width,and the completion of the spherical regular simplexes are the most asymmetric bodies.
基金Supported by the Innovative Project of College Students of Jiangsu Province(201710332019Z)the Natural Science Foundation of Jiangsu Province(BK20171218)the National Natural Science Foundation of China(11671293)
文摘In this paper, we show that when Minkowski measure of asymmetry of convex body K of constant width is bigger than a（n-1）, K has at least n＋1 critical chords, where a（n）=n＋（2n（n＋1））√1/2/n＋2.
文摘Slow evaporation method was used to grow pure and KCl (10 mol%) doped KAP single crystal. The solubility and metastable zone width of aqueous solutions of pure and KCl (10 mol%) doped KAP crystal were evaluated to analyze the crystallization process. Measuring the induction period τ, the critical nucleation parameters like interfacial energy (σ), energy of formation of the critical nucleus (ΔG*) were determined using the classical theory of nucleation. The structural properties and optical constants of the grown crystals have been put to test and observed that the addition of KCl results in an enhancement of properties of the crystal. Grown crystals were characterized by powder X-ray diffraction. FTIR spectra confirmed the presence of KCl in pure KAP crystal. UV- Visible spectroscopic studies revealed that addition of KCl in pure KAP crystal increased transparency from 75% to 80%. The analysis of the optical absorption data revealed the presence of both indirect and direct transitions and both of these band gaps increased with the addition of KCl. The transmittance data was analyzed to calculate the refractive index, oscillator energy, dispersion energy, electric susceptibility, zero-frequency dielectric constant and both the real and imaginary parts of the dielectric permittivity as a function of photon energy. The moments of ε(E) were also determined. The dispersion i.e. spectral dependence of the refractive index was discussed according to the single-effective oscillator model proposed by Wemple and DiDomenico.
