We proved the existence of a 2-dimensional nonaspherical simply connected cell-like Peano continuum (such a space was constructed in one of our earlier papers). We also indicated some relations between this space and the celebrated classical Griffiths' space from the 1950s.
COBISS.SI-ID: 15045977
One of the important concepts in computational topology is discrete Morse theory, which has many applications but it is not suitable for data analysis because of the discrete nature of data. Discrete Morse theory has similar properties and is better suited for the discrete domain. In the paper, an algorithm for the construction of ascending and descending disks of critical cells of discrete Morse functions is developed, which compared to similar algorithms, is not limited to 2 or 3 dimensional data. The algorithm was applied to qualitative analysis of natural and artificial data.
COBISS.SI-ID: 14994265
We found a surprising generalization and unification of two key problems by showing that the space of continuous functions from a countable polyhedron to an absolute neighbourhood retract has the homotopy type of a polyhedron if and only if it is actually an absolute neighbourhood retract, and, given Sakai's weak restrictions on dimension of the domain and isolated points of the codomain, that is if and only if the space of functions actually admits the structure of a Hilbert manifold.
COBISS.SI-ID: 14965849
We investigated cases when the Repovš-Semenov splitting problem for selections has an affirmative solution for continuous set-valued mappings. We considered the situation in infinite-dimensional uniformly convex Banach spaces and studied general geometric properties of uniformly convex sets. We also obtained an affirmative solution of the splitting problem for selections of certain set-valued mappings with uniformly convex images. The journal is very high on the Science Citation Index list, in the year 2008 it placed the 32th among 214 journals.
COBISS.SI-ID: 15236697
Let X be a finite spectrum. We prove that R(X_{(p)}), the endomorphism ring of the p-localization of X, is a semi-perfect ring. This implies, among other things, a strong form of unique factorization for finite p-local spectra. The main step in the proof is that the Jacobson radical of R(X_{(p)}) is idempotent-lifting, which is proved by a combination of geometric properties of finite spectra and algebraic properties of the p-localization.
COBISS.SI-ID: 15179097