Multiplier theorem for Hankel transform on Hardy spaces

The aim of this paper is to prove a multiplier theorem for the Hankel transform on the atomic Hardy space H ¹(X), where X = ((0, ∞), x ^α dx) is the space of homogeneous type in the sense of Coifman–Weiss. The main tool is a maximal function characterization of H ¹(X).     

Multiplier theorem for Hankel transform on Hardy spaces

The aim of this paper is to prove a multiplier theorem for the Hankel transform on the atomic Hardy space H ¹(X), where X = ((0, ∞), x ^α dx) is the space of homogeneous type in the sense of Coifman–Weiss. The main tool is a maximal function characterization of H ¹(X).     

Stochastic evolution of 2D crystals

We study a crystalline version of the modified Stefan problem in the plane. The feature of our approach is that, we consider a model with stochastic perturbations and assume the interfacial curve to be a polygon. The existence of solution to our stochastic system is established. Galerkin’s method is one of the main tools used in the proof of our assertion.     

Generalizations of Arnold’s version of Euler’s theorem for matrices

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has ${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k}). We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A,  B are congruent modulo p ^k then the characteristic polynomials of A ^p and B ^p are congruent modulo p ^k+1, and then we show that Arnold’s conjecture follows from it easily. Using this result, we prove the following generalization of Euler’s theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A ^Φ(n) and A ^Φ(n)-ϕ(n) are congruent modulo n. Here ϕ is the Euler function, ?_i=1^l p_i^a_i\prod_{i=1}^{l} p_i^{\alpha_i} is a prime factorization of n and $\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2.     

On some fixed-point theorems for generalized contractions and their perturbations

The aim of this paper is to prove a collection of fixed-point theorems for mappings which can be roughly called generalized contractions or their perturbations. In particular, we are going to consider operators (single-valued or multi-valued) in Banach spaces with a quasimodulus, in hyperconvex subsets of normed spaces, or finally in non-Archimedean spaces. A particular attention will be paid to Krasnoselskii-type fixed-point theorems as well as to a Schaefer-type fixed-point theorem. Some applications to nonlinear functional-integral equations will be given. Our results extend and complement some commonly known theorems.     

Spectroscopic and Theoretical Studies of Fluorescence Effects in 2-Methylamino-5-(2,4-dihydroxyphenyl)-1,3,4-thiadiazole Induced by Molecular Aggregation

The article presents the results of fluorescence analyses of 2-methylamino-5-(2,4-dihydroxyphenyl)-1,3,4-thiadiazole (MDFT) in an aqueous environment. MDFT dissolved in aqueous solutions with a pH value in the range from 1 to 4.5 yielded an interesting effect of two clearly separated fluorescence emissions. In turn, a single fluorescence was observed in MDFT dissolved in water solutions with a pH value from 4.5 to 12. As it was suggested in the previous investigations of other 1,3,4-thiadiazole compounds, these effects may be associated with conformational changes in the structure of the analysed molecule accompanied by aggregation effects. Crystallographic data showed that the effect of the two separated fluorescence emissions occurred in a conformation with the –OH group in the resorcyl ring bound on the side of the sulphur atom from the 1,3,4-thiadiazole ring. The hypothesis of aggregation as the mechanism involved in the change in the spectral properties at low pH is supported by the results of (Time-Dependent) Density Functional Theory calculations. The possibility of rapid analysis of conformational changes with the fluorescence spectroscopy technique may be rather important outcome obtained from the spectroscopic studies presented in this article. Additionally, the presented results seem to be highly important as they can be easily observed in solutions and biologically important samples.