Measurability implies continuity for solutions of functional equations - even with few variables

Summary. We prove that - under certain conditions - measurable solutions $f$ of the functional equation $f(x)=h(x,y,f(g_{1}(x,y)),\ldots,f(g_{n}(x,y))),\quad(x,y)\in D \subset \mathbb{R}^{s} \times \mathbb{R}^{l}$ are continuous, even if $1\le l\le s$. As a tool we introduce new classes of functions which - roughly speaking - interpolate between continuous and Lebesgue measurable functions. Connection between these classes are also investigated.     

Characterization of Cds clusters in zeolite-A grown in alkaline solution

CdS clusters were synthesized in A type zeolite by reaction in alkaline aqueous solution at temperatures from 30 to 70 °C. The optical properties of the samples were studied by diffuse reflectance and photoluminescence spectroscopy. Their crystalline structure and morphology were studied by X-ray diffraction and by transmission and scanning electron microscopy. We found that at lower temperatures the CdS clusters are encapsulated in the zeolite cages. We compared the properties of these clusters with those encapsulated in the cages of zeolites X and Y, prepared by similar methods. CdS clusters smaller than the CdS exciton diameter are also formed outside the cages in the zeolite matrix. The size of these clusters increases with temperature producing a red-shift of the absorption edge in the optical absorption spectra.     

Pressure-induced polymorphism in enstatite (MgSiO3) at room temperature: clinoenstatite and orthoenstatite

The phase transition of a synthetic clinoenstatite in a diamond-anvil cell has been studied by using Raman spectroscopy at various pressures and room temperature. The phenomena observed in clinoenstatite have been compared with that observed in orthoenstatite. It is found that the pressure-induced phase transitions in the two enstatites are reversible, but with different transition pressures and transition behavior. An analysis of Raman spectra has revealed that the two enstatites have different high-pressure polymorphs. This result suggests that the space group of the high-pressure polymorph of orthoenstatite is not of C2/c, and that orthoenstatite and orthoferrosilite have different transition routes at room temperature and high pressure. The compressional behavior of the high-PC2/c enstatite is also discussed according to the pressure dependences of Raman frequencies.     

Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

For a commutative ring R with set of zero-divisors Z(R), the zero-divisor graph of R is Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we show that Γ(T(R)) and Γ(R) are isomorphic as graphs, where T(R) is the total quotient ring of R, and that Γ(R) is uniquely complemented if and only if either T(R) is von Neumann regular or Γ(R) is a star graph. We also investigate which cardinal numbers can arise as orders of equivalence classes (related to annihilator conditions) in a von Neumann regular ring.     

The convergence rate of continued fractions representing solutions of a Riccati equation

The aim of this note is to generalize and apply results on matrix continued fractions representing the solution of discrete matrix Riccati equations. Assuming uniform bounds for the norm of the matrix coefficients of the continued fraction, the minimal and maximal solutions of the corresponding algebraic Riccati equation can be accurately enclosed.     

Degree bounds for syzygies of invariants

Suppose that G is a linearly reductive group. Good degree bounds for generators of invariant rings were given in (Proc. Amer. Math. Soc. 129 (4) (2001) 955). Here we study minimal free resolutions of invariant rings. For finite linearly reductive groups G it was recently shown in (Adv. Math. 156 (1) (2000) 23, Electron Res. Announc. Amer. Math. Soc. 7 (2001) 5, Adv. Math. 172 (2002) 151) that rings of invariants are generated in degree at most the group order |G|. In characteristic 0 this degree bound is a classical result by Emmy Noether (see Math. Ann. 77 (1916) 89). Given an invariant ring of a finite linearly reductive group G, we prove that the ideal of relations of a minimal set of generators is generated in degree at most ?2|G|.     

The solution of the neutron transport equation in a slab with anisotropic scattering

The neutron transport equation for a slab geometry with the extremely anisotropic scattering kernel is considered. The albedo and transmission factors are calculated using the variation method. The effect of the extremely anisotropic parameter on the variation of the slab albedo and transmission factor is calculated. The obtained results are compared with the published data.     

Development of measuring magnetic Compton profiles by grazing incidence geometry

A novel technique of measuring a magnetic Compton profile using the grazing angle geometry against a sample surface (Grazing Incidence Magnetic Compton Profile) has been successively developed. Measurements of a magnetic moment and a magnetic Compton profile are possible for a Fe 200 nm film on a thick glass substrate. The estimated thinnest limit for measurements is 100 nm for a Fe film.     

Another approach to the heat flow for Plateau problem

For the minimal surfaces in Rⁿ with Plateau boundary condition and establish the global existence and uniqueness of the flow as well as the continuous dependence of the initial datum.     

Surface bodies and p-affine surface area

The surface body is a generalization of the floating body. Its relation to p-affine surface area is studied.