Abstract:
The final version of the New Capital Accord, which includes operational risk, was released by the Basel Committee on Banking Supervision in June 2004. The article “Basel II approaches for the calculation of the regulatory capital for operational risk” is devoted to the issue of operational risk of credit financial institutions. The paper talks about methods of operational risk calculation, advantages and disadvantages of particular methods.

Abstract:
We prove that the functor $\Hat{P}$ of Radon probability measures transforms any open map between completely metrizable spaces into a soft map. This result is applied to establish some properties of Milyutin maps between completely metrizable spaces.

Abstract:
Let $M$ be a complete metric $ANR$-space such that for any metric compactum $K$ the function space $C(K,M)$ contains a dense set of Bing (resp., Krasinkiewicz) maps. It is shown that $M$ has the following property: If $f\colon X\to Y$ is a perfect surjection between metric spaces, then $C(X,M)$ with the source limitation topology contains a dense $G_\delta$-subset of maps $g$ such that all restrictions $g|f^{-1}(y)$, $y\in Y$, are Bing (resp., Krasinkiewicz) maps. We apply the above result to establish some mapping theorems for extensional dimension.

Abstract:
A characterization of $\varkappa$-metrizable compacta in terms of extension of functions and usco retractions into superextensions is established.

Abstract:
We prove that if $f\colon X\to Y$ is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class of space $\mathrm S$, then there exists an $F_\sigma$-set $A\subset X$ such that $A\in\mathrm S$ and $\dim f^{-1}(y)\backslash A=0$ for all $y\in Y$. Here, $\mathrm S$ can be one of the following classes: (i) $\{M:\mathrm{e-dim}M\leq K\}$ for some $CW$-complex $K$; (ii) $C$-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if $\mathrm S=\{M:\dim M\leq n\}$, then $\dim f\triangle g\leq 0$ for almost all $g\in C(X,\mathbb I^{n+1})$.

Abstract:
The notion of a regular operator with compact supports between function spaces is introduced. On that base we obtain a characterization of absolute extensors for zero-dimensional spaces in terms of regular extension operators having compact supports. Milyutin maps are also considered and it is established that some topological properties, like paracompactness, metrizability and k-metrizability, are preserved under Milyutin maps.

Abstract:
Using a factorization theorem due to Pasynkov we provide a short proof of the existence and density of parametric Bing and Krasinkiewicz maps. In particular, the following corollary is established: Let $f\colon X\to Y$ be a surjective map between paracompact spaces such that all fibers $f^{-1}(y)$, $y\in Y$, are compact and there exists a map $g\colon X\to\mathbb I^{\aleph_0}$ embedding each $f^{-1}(y)$ into $\mathbb I^{\aleph_0}$. Then for every $n\geq 1$ the space $C^*(X,\mathbb R^n)$ of all bounded continuous functions with the uniform convergence topology contains a dense set of maps $g$ such that any restriction $g|f^{-1}(y)$, $y\in Y$, is a Bing and Krasinkiewicz map.

Abstract:
Arvanitakis established recently a theorem which is a common generalization of Michael's convex selection theorem and Dugundji's extension theorem. In this note we provide a short proof of a more general version of Arvanitakis' result.

Abstract:
The following selection theorem is established:\\ Let $X$ be a compactum possessing a binary normal subbase $\mathcal S$ for its closed subsets. Then every set-valued $\mathcal S$-continuous map $\Phi\colon Z\to X$ with closed $\mathcal S$-convex values, where $Z$ is an arbitrary space, has a continuous single-valued selection. More generally, if $A\subset Z$ is closed and any map from $A$ to $X$ is continuously extendable to a map from $Z$ to $X$, then every selection for $\Phi|A$ can be extended to a selection for $\Phi$. This theorem implies that if $X$ is a $\kappa$-metrizable (resp., $\kappa$-metrizable and connected) compactum with a normal binary closed subbase $\mathcal S$, then every open $\mathcal S$-convex surjection $f\colon X\to Y$ is a zero-soft (resp., soft) map. Our results provide some generalizations and specifications of Ivanov's results (see \cite{i1}, \cite{i2}, \cite{i3}) concerning superextensions of $\kappa$-metrizable compacta.