The main results:(1) Let L be a nilpotent CW complex and F the homotopy fiber of the inclusion i of L into its infinite symmetric product SP(L). If X is a metrizable space such that X\tau K(H_k(L),k) for all k\ge 1, then X\tau K(\pi_k(F),k) and X\tau K(\pi_k(L),k) for all k\ge 2. (2) Let X be a metrizable space such that \dim (X) < \infty or X\in ANR. Suppose L is a nilpotent CW complex. If X\tau SP(L), then X\tau L in the following cases: (a) H_1(L) is finitely generated, or (b) H_1(L) is a torsion group.

COBISS.SI-ID: 14551385

Two important questions are answered in the negative: (1) If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic? (2) Can adding arcs to a space cause an essential curve to become nulhomotopic? The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and \pi_1-shape injective.

COBISS.SI-ID: 14657625

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

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 establish the complete bifurcation diagram for a class of nonlinear problems on the whole space. Our model corresponds to a class of semilinear elliptic equations with logistic type nonlinearity and absorption. Since this problem arises in population dynamics or in fishery or hunting management, we are interested only in situations allowing the existence of positive solutions. The proofs combine elliptic estimates with the method of sub- and super-solutions. The journal is very high on the Science Citation Index list, in the year 2008 it placed the 15th among 214 journals.

COBISS.SI-ID: 15094617