On stable Baire classes

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps \({f : X\to Y}\) of the class \({\alpha}\) for wide classes of topological spaces. In particular, we prove that for a topological space X and a contractible space Y a map \({f : X \to Y}\) belongs to the nth stable Baire class if and only if there exist a sequence \({(f_k)_{k=1}^\infty}\) of continuous maps \({f_k : {X \to Y}}\) and a sequence \({(F_k)_{k=1}^\infty}\) of functionally ambiguous sets of the nth class in X such that \({f|_{F_k}=f_k|_{F_k}}\) for every k. Moreover, we show that every monotone function \({f : \mathbb{R} \to \mathbb{R}}\) is of the \({\alpha}\) th stable Baire class if and only if it belongs to the first stable Baire class.     

Structure changes in Al80Ni15Y5 amorphous alloy

The structure of Al₈₀Ni₁₅Y₅ amorphous alloy at various temperatures have been studied with X-ray diffraction methods. The obtained scattered intensities, structure factors, pair correlation functions and main structure parameters have been analyzed. Temperature dependences of the parameters suggest formation of Al, Al₃Ni and Al₂₃Ni₆Y₄ phases upon crystallization of an amorphous alloy.     

The structure of Al–Cu and Al–Si eutectic melts

The structure of Al₈₃Cu₁₇ and Al₈₈Si₁₂ liquid eutectic alloys has been studied using the X-ray diffraction (XRD) and reverse Monte-Carlo (RMC) methods. The total and partial structure factors and pair correlation functions as well as structure parameters obtained therefrom have been analyzed. Chemically ordered Al–Cu groups significantly effect the structure in Al₈₃Cu₁₇, while a tendency to chemical ordering between atoms of an unlike kind is negligible in Al₈₈Si₁₂.     

On Gibson functions with connected graphs

A function f: X → Y between topological spaces is said to be a weakly Gibson function if $f(\bar G) \subseteq \overline {f(G)} $ for any open connected set G ? X. We call a function f: X → Y segmentary connected if X is topological vector space and f([a, b]) is connected for every segment [a, b] ? X. We show that if X is a hereditarily Baire space, Y is a metric space, f: X → Y is a Baire-one function and one of the following conditions holds:

1. X is a connected and locally connected space and f is a weakly Gibson function
2. X is an arcwise connected space and f is a Darboux function
3. X is a topological vector space and f is a segmentary connected function, then f has a connected graph.

     

A lattice approach to narrow operators

It is known that if a rearrangement invariant function space E on [0,1] has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L₁ is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain space is a vector lattice. Our main result asserts that the set N_r(E, F) of all narrow regular operators is a band in the vector lattice L_r(E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserving operators is the orthogonal complement to N_r(E, F) in L_r(E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : E → F from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = T_D + T_N where T_D is a sum of an order absolutely summable family of disjointness preserving operators and T_N is narrow. Supported by Ukr. Derzh. Tema N 0103Y001103.