In 1997 Pták defined generalized Hankel operators as follows: Given two contractions $T_1\in {\mathcal B}({\mathcal H}_1)$ and $T_2 \in {\mathcal B}({\mathcal H}_2)$, an operator $X \:{\mathcal H}_1 \rightarrow {\mathcal H}_2$ is said to be a generalized Hankel operator if $T_2X=XT_1^*$ and $X$ satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations of $T_1$ and $T_2$. This approach, call it (P), contrasts with a previous one developed by Pták and Vrbová in 1988, call it (PV), based on the existence of a previously defined generalized Toeplitz operator. There seemed to be a strong but somewhat hidden connection between the theories (P) and (PV) and we clarify that connection by proving that (P) is more general than (PV), even strictly more general for some $T_1$ and $T_2$, and by studying when they coincide. Then we characterize the existence of Hankel operators, Hankel symbols and analytic Hankel symbols, solving in this way some open problems proposed by Pták.
In the theory of accessible categories, pure subobjects, i.e. filtered colimits of split monomorphisms, play an important role. Here we investigate pure quotients, i.e., filtered colimits of split epimorphisms. For example, in abelian, finitely accessible categories, these are precisely the cokernels of pure subobjects, and pure subobjects are precisely the kernels of pure quotients.
Stochastic optimization problem is, as a rule, formulated in terms of expected cost function. However, the criterion based on averaging does not take in account possible variability of involved random variables. That is why the criterion considered in the present contribution uses selected quantiles. Moreover, it is assumed that the stochastic characteristics of optimized system are estimated from the data, in a non-parametric setting, and that the data may be randomly right-censored. Therefore, certain theoretical results concerning estimators of distribution function and quantiles under censoring are recalled and then utilized to prove consistency of solution based on estimates. Behavior of solutions for finite data sizes is studied with the aid of randomly generated example of a newsvendor problem.
Quasi-homogeneity of copulas is introduced and studied. Quasi-homogeneous copulas are characterized by the convexity and strict monotonicity of their diagonal sections. As a by-product, a new construction method for copulas when only their diagonal section is known is given.
A subgroup H of a finite group G is said to be conjugate-permutable if HHg = HgH for all g\in G. More generaly, if we limit the element g to a subgroup R of G, then we say that the subgroup H is R-conjugate-permutable. By means of the R-conjugatepermutable subgroups, we investigate the relationship between the nilpotence of G and the R-conjugate-permutability of the Sylow subgroups of A and B under the condition that G = AB, where A and B are subgroups of G. Some results known in the literature are improved and generalized in the paper., Xianhe Zhao, Ruifang Chen., and Obsahuje seznam literatury
We introduce the rainbowness of a polyhedron as the minimum number $k$ such that any colouring of vertices of the polyhedron using at least $k$ colours involves a face all vertices of which have different colours. We determine the rainbowness of Platonic solids, prisms, antiprisms and ten Archimedean solids. For the remaining three Archimedean solids this parameter is estimated.
Let T, T′ be weak contractions (in the sense of Sz.-Nagy and Foia¸s), m, m′ the minimal functions of their C0 parts and let d be the greatest common inner divisor of m, m′ . It is proved that the space I(T, T′ ) of all operators intertwining T, T′ is reflexive if and only if the model operator S(d) is reflexive. Here S(d) means the compression of the unilateral shift onto the space H 2 ⊖dH2 . In particular, in finite-dimensional spaces the space I(T, T′ ) is reflexive if and only if all roots of the greatest common divisor of minimal polynomials of T, T′ are simple. The paper is concluded by an example showing that quasisimilarity does not preserve hyperreflexivity of I(T, T′ ).
We extend a result of Rangaswamy about regularity of endomorphism rings of Abelian groups to arbitrary topological Abelian groups. Regularity of discrete quasi-injective modules over compact rings modulo radical is proved. A characterization of torsion LCA groups $A$ for which ${\rm End}_c(A)$ is regular is given.