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Axiomatic Fredholm theory in unital C*-algebras was established in [D. Kečkić, Z. Lazović, Acta Sci. Math., 83(3-4):629--655, 2017]. Following the pure algebraic approach by Kečkić and Lazović, in the author's paper [S. Ivković, Banach J. Math. Anal., 17:51, 2023] we extended further this theory to axiomatic semi-Fredholm and semi-Weyl theory in unital C*-algebras. However, recently, in [S. Ivković, arXiv:2306.01133] we developed another approach to axiomatic Fredholm theory in unital C*-algebras which is based on the theory of Hilbert modules and is equivalent to the algebraic approach by Kečkić and Lazović. In this paper, we extend further that new Hilbert-module approach from Fredholm theory to semi-Fredholm and semi-Weyl theory in unital C*-algebras. Hence, we provide new proofs to the results in [S. Ivković, Banach J. Math. Anal., 17:51, 2023].

The paper summarizes some results on algebraic geometry presence in the homotopy theory. For the homotopy group πm(Sn), denote by πalgm(Sn) its subset of homotopy classes represented by ℝ-entire rational maps Sm→Sn of spheres. The main result of this paper concerns to the study of πn+kalg(Sn) for k=0,...,7.

In this paper, we consider the extension of the analytic functions of two variables by special families of functions — continued fractions. In particular, we establish new symmetric domains of the analytical continuation of three ratios of Horn's confluent hypergeometric function $\mathrm{H}_7$ with certain conditions on real and complex parameters using their continued fraction representations. We use Worpitzky's theorem, the multiple parabola theorem, and a technique that extends the convergence, already known for a small domain, to a larger domain to obtain domains of convergence of continued fractions, and the PC method to prove that they are also domains of analytical continuation.

In this paper, we study the ultrametric 2-normed spaces and the ultrametric 2-Banach spaces. In particular, we establish some results on Cauchy sequences in ultrametric 2-normed spaces. Also, we introduce and study the notion of bounded linear 2-functionals on ultrametric 2-Banach spaces and we give some of its properties. On the other hand, the new norm on the ultrametric 2-normed space is constructed. The concepts of closed operators between ultrametric 2-normed spaces and $b$-linear functionals in ultrametric 2-normed spaces are introduced. Finally, a necessary and sufficient condition for a linear operator to be closed in terms of its graph is proved and some results on bounded $b$-linear functionals in ultrametric 2-normed spaces are given.

In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.

In this paper we classify the norming set of a bilinear form on the plane with a certain norm whose unit ball has only four extreme points. We obtain the results of [6, 8] as corollary.

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $G/\zeta(G)$ of a group $G$ is finite, then its derived subgroup $[G,G]$ is also finite. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, $n$-groups, associative algebras, Lie algebras, Lie $n$-algebras, Lie rings, Leibniz algebras. In 2021, L.A. Kurdachenko, O.O. Pypka and I.Ya. Subbotin proved an analogue of Schur theorem for Poisson algebras: if the center of the Poisson algebra $P$ has finite codimension, then $P$ includes an ideal $K$ of finite dimension such that $P/K$ is abelian. In this paper, we continue similar studies for another algebraic structure. An analogue of Schur theorem for Poisson (2-3)-algebras is proved.

In this paper, we give simple proofs for the bounds (some of them sharp) of the difference of the moduli of the second and the first logarithmic coefficient for the general class of univalent functions and for the class of convex univalent functions.

The presented paper investigates the exponential asymptotics at a point of the complex plane of the ring Q-homeomorphisms with respect to p-modulus for p>2. Examples showing sharpness of the obtained results have been constructed.

The rational homotopy type of a mapping space is a way to describe the structure of the space using the algebra of its homotopy groups and the differential graded algebra of its cochains. An L∞-model is a graded Lie algebra with a family of higher-order brackets satisfying the generalized Jacobi identity and antisymmetry. It can be used to study the rational homotopy type of a space. The nilpotency index of an L∞-model is useful in understanding a space's algebraic structure. In this paper, we compute the rational homotopy type of the component of some mapping spaces between projective spaces and determine the nilpotency index of corresponding L∞-models.

In the paper, we consider nonlinear control systems that are linear with respect to controls with output; vector fields defining the system and the output are supposed to be real analytic. Following the algebraic approach, we consider series $S$ of iterated integrals corresponding to such systems. Iterated integrals form a free associative algebra, and all our constructions use its properties. First, we consider the set of all (formal) functions of such series $f(S)$ and define the set $N_S$ of terms of minimal order for all such functions. We introduce the definition of the maximal graded Lie generated left ideal ${\mathcal J}_S^{\rm max}$ which is orthogonal to the set $N_S$. We describe the relation between this maximal left ideal and the left ideal ${\mathcal J}_S$ generated by the core Lie subalgebra of the system which realizes the series. Namely, we show that ${\mathcal J}_S\subset {\mathcal J}_S^{\rm max}$. In particular, this implies that the graded Lie subalgebra that generates the left ideal ${\mathcal J}_S^{\rm max}$ has a finite codimension. Also, we give the algorithm which reduces the series $S$ to the triangular form and propose the definition of the homogeneous approximation for the series $S$. Namely, homogeneous approximation is a homogeneous series with components that are terms of minimal order in each component of this triangular form. We prove that the set $N_S$ coincides with the set of all shuffle polynomials of components of a homogeneous approximation. Unlike the case when the output is identical, the homogeneous approximation is not completely defined by the ideal ${\mathcal J}_S^{\rm max}$. In order to describe this property, we introduce two different concepts of equivalence of series: algebraic equivalence (when two series have the same homogeneous approximation) and weak algebraic equivalence (when two series have the same maximal left ideal and therefore have the same minimal realizing system). We prove that if two series are algebraically equivalent, then they are weakly algebraically equivalent. The examples show that in general the converse is not true.

In this paper we study pseudo-Riemannian spaces with recurrent tensor of conformal curvature, which admit a Kähler structure. It is proved that Kähler conformally recurrent spaces other than recurrent spaces do not exist, if their dimension is four. Recurrent Kähler spaces are divided into two types. For each type, the internal necessary characteristic is given. Some properties of four-dimensional Kähler conformally recurrent Kähler spaces are studied.

The paper is devoted to the study of properties of pseudo-Riemannian spaces admitting nontrivial geodesic mappings. Necessary and sufficient conditions are found for A-threesymmetric spaces to admit nontrivial geodesic mappings. The research is carried out locally, in tensor form without restrictions to the sign of the metric tensor and the signature of the space.