Interaction in the TlSe-Pr2Se3 system

The interaction of components in the TlSe-Pr₂Se₃ system was studied by differential thermal, X-ray powder diffraction, and microstructural analyses and also by microhardness and density measurements. The state diagram of this system was constructed. It was found that the section TlSe-Pr₂Se₃ is a quasi-binary section of the Tl-Pr-Se ternary system. In the TlSe-Pr₂Se₃ system at the component ratio 1: 1, a new chemical compound, TlPr₂Se₄, forms, which melts congruently at 1295°C. In the system based on TlSe, solid solutions form until 3.5 mol % Pr₂Se₃, and in the system based on Pr₂Se₃, solid solutions occur until 2.5 mol % TlSe.     

Liquidus surface projection for the Er-Sn-Se ternary system

The Er-Sn-Se system was studied by differential thermal analysis, X-ray powder diffraction, and microstructure examination and also by microhardness and density measurements. A congruent triangulation scheme and a liquidus surface projection were constructed. Fourteen primary crystallization fields, including those of the ternary compounds ErSnSe₂, Er₂SnSe₄, Er₂SnSe₅, and Er₂Sn₃Se₉, were found, and the types and coordinates of nonvariant and monovariant equilibria were determined.     

A short proof of the Doob–Meyer theorem

Every submartingale S$S$ of class D$D$ has a unique Doob–Meyer decomposition S=M+A$S=M+A$, where M$M$ is a martingale and A$A$ is a predictable increasing process starting at 0.     

Edgeworth expansion for the pre-averaging estimator

In this paper, we study the Edgeworth expansion for a pre-averaging estimator of quadratic variation in the framework of continuous diffusion models observed with noise. More specifically, we obtain a second order expansion for the joint density of the estimators of quadratic variation and its asymptotic variance. Our approach is based on martingale embedding, Malliavin calculus and stable central limit theorems for continuous diffusions. Moreover, we derive the density expansion for the studentized statistic, which might be applied to construct asymptotic confidence regions.     

Phase equilibria in the InS-Sb2Te3 system

The interaction character of the InS-Sb₂Te₃ system was studied by differential thermal analysis, X-ray powder diffraction, microstructure examinations, microhardness measurements, and density determinations, and its phase diagram was constructed. The InS-Sb₂Te₃ phase diagram is a partially non-quasi-binary section of the In,Sb∥S,Te ternary reciprocal system. Two compounds are formed in the InS-Sb₂Te₃ system, namely: In₃Sb₂S₃Te₃ and InSb₂Te₃S. Sb₂Te₃-based solid solutions at room temperature have an extent of up to 6 mol % InS, while InS-based solid solutions are virtually nonexistent.