# Results in Mathematics

## List of Papers (Total 61)

#### The Distant Graph of the Projective Line Over a Finite Ring with Unity

We discuss the projective line $\mathbb {P}(R)$ over a finite associative ring with unity. $\mathbb {P}(R)$ is naturally endowed with the symmetric and anti-reflexive relation “distant”. We study the graph of this relation on $\mathbb {P}(R)$ and classify up to isomorphism all distant graphs $G(R, \Delta )$ for rings R up to order $p^5$, p prime.

#### Remarks on Farah’s Theorems

The aim of this paper is to prove two Farah’s Theorems concerning approximate group homomorphisms, without some assumptions present in Farah’s Theorems. Farah’s approach to the Ulam’s type stability of homomorphisms between finite groups seems appropriate and interesting.

#### Non-optimality of the Greedy Algorithm for Subspace Orderings in the Method of Alternating Projections

The method of alternating projections involves projecting an element of a Hilbert space cyclically onto a collection of closed subspaces. It is known that the resulting sequence always converges in norm and that one can obtain estimates for the rate of convergence in terms of quantities describing the geometric relationship between the subspaces in question, namely their pairwise ...

#### On Positive-Characteristic Semi-parametric Local Uniform Reductions of Varieties over Finitely Generated $\mathbb {Q}$ -Algebras

We present a non-standard proof of the fact that the existence of a local (i.e. restricted to a point) characteristic-zero, semi-parametric lifting for a variety defined by the zero locus of polynomial equations over the integers is equivalent to the existence of a collection of local semi-parametric (positive-characteristic) reductions of such variety for almost all primes (i.e. ...

#### New Fixed Point Tools in Non-metrizable Spaces

The aim of this paper is to provide some sufficient conditions under which a self-mapping T defined on a non-empty set X endowed with some convergence property is a Picard operator. A relevant example showing that such a mapping T on a non-metrizable space is a Picard operator is given. Our results can be used to obtain some known fixed point theorems on generalized metric spaces.

#### Integral Representation of Continuous Operators with Respect to Strict Topologies

Let $\,X\,$ be a completely regular Hausdorff space and $\,\mathcal {B}o\,$ be the $\sigma$-algebra of Borel sets in X. Let $C_b(X)$ (resp. $B(\mathcal {B}o)$) be the space of all bounded continuous (resp. bounded $\mathcal {B}o$-measurable) scalar functions on X, equipped with the natural strict topology $\beta$. We develop a general integral representation theory ...

#### On Para-Complex Affine Hyperspheres

In this paper we introduce a notion of a para-complex affine hypersphere. We give a complete local classification of such hypersurfaces and give several examples. It turns out that every para-complex affine hypersphere can be constructed from (real) affine hyperspheres. As an application, we classify all 2-dimensional para-complex affine hyperspheres.

#### Nonassociative Differential Extensions of Characteristic $\varvec{p}$

Let F be a field of characteristic p. We define and investigate nonassociative differential extensions of F and of a finite-dimensional central division algebra over F and give a criterium for these algebras to be division. As special cases, we obtain classical results for associative algebras by Amitsur and Jacobson. We construct families of nonassociative division algebras which ...

In the present paper, inspired by methods contained in Gajda and Kominek (Stud Math 100:25–38, 1991) we generalize the well known sandwich theorem for subadditive and superadditive functionals to the case of delta-subadditive and delta-superadditive mappings. As a consequence we obtain the classical Hyers–Ulam stability result for the Cauchy functional equation. We also consider ...

#### Approximation of Integrable Functions by Wavelet Expansions

Walter (J Approx Theory 80:108–118, 1995), Xiehua (Approx Theory Appl 14(1):81–90, 1998) and Lal and Kumar (Lobachevskii J Math 34(2):163–172, 2013) established results on pointwise and uniform convergence of wavelet expansions. Working in this direction new more general theorems on degree of pointwise approximation by such expansions have been proved.

#### Generalized Pexider Equation on an Open Domain

Inspired by the papers by Abbas, Aczél and by Chudziak and Tabor, we consider the problem of existence and uniqueness of extensions for the generalized Pexider equation $$k(x+y)=l(x)+m(x)n(y) \;\;\; {\rm for} \;\;\; (x,y)\in D,$$where D is a nonempty open subset of a normed space. We show that the connectedness of D, assumed in the mentioned above papers, can be weakened.

#### The Hilbert–Schmidt Analyticity Associated with Infinite-Dimensional Unitary Groups

The article is devoted to the problem of Hilbert–Schmidt type analytic extensions in Hardy spaces over the infinite-dimensional unitary group endowed with an invariant probability measure. Reproducing kernels of Hardy spaces, integral formulas of analytic extensions and their boundary values are considered.

#### Explicit Solutions of the Invariance Equation for Means

Extending the notion of projective means we first generalize an invariance identity related to the Carlson log given in Kahlig and Matkowski (Math Inequal Appl 18(3):1143–1150, 2015), and then, more generally, given a bivariate symmetric, homogeneous and monotone mean M, we give explicit formula for a rich family of pairs of M-complementary means. We prove that this method cannot ...

#### Free Cyclic Submodules in the Context of the Projective Line

We discuss the free cyclic submodules over an associative ring R with unity. Special attention is paid to those which are generated by outliers. This paper describes all orbits of such submodules in the ring of lower triangular 3 × 3 matrices over a field F under the action of the general linear group. Besides rings with outliers generating free cyclic submodules, there are also ...

#### A Singular Behaviour of a Set-Valued Approximate Orthogonal Additivity

We show that unlikely to the single-valued case, the set-valued orthogonally additive equation is unstable. After presenting an example showing this phenomenon, we provide some special cases where a set-valued approximately orthogonally additive function can be approximated by the one which satisfies the equation of orthogonal additivity exactly.

#### Inequalities of the Hermite–Hadamard Type Involving Numerical Differentiation Formulas

We observe that the Hermite–Hadamard inequality written in the form $$f\left(\frac{x+y}{2}\right)\leq\frac{F(y)-F(x)}{y-x}\leq\frac{f(x)+f(y)}{2}$$may be viewed as an inequality between two quadrature operators ${f\left(\frac{x+y}{2}\right)}$ ${\frac{f(x)+f(y)}{2}}$ and a differentiation formula ${\frac{F(y)-F(x)}{y-x}}$. We extend this inequality, replacing the middle term ...

#### Popoviciu Type Equations on Cylinders

Using a correspondence between the Popoviciu type functional equations and the Fréchet equation we investigate the solutions of the Popoviciu type functional equations on cylinders.

#### Inclusions Characterizing Polynomial-Type Multifunctions

This work is motivated by some earlier papers concerning a pair of functional inequalities characterizing polynomials. This system is also related to the notion of microperiodic function. We study multifunctions satisfying two simultaneous conditional functional inclusions. An explicit formula for the solution to this system of inclusions is given. Applying this result we obtain ...

#### Stability of the Drygas Functional Equation on Restricted Domain

We study the stability of the Drygas functional equation on a restricted domain. The main tool used in the proofs is the fixed point theorem for functional spaces.

#### J-Tangent Affine Hyperspheres

In this paper we study J-tangent affine hyperspheres. Under some additional conditions we give a local characterization of 3-dimensional J-tangent affine hyperspheres.

#### Continuous Solutions of Conditional Composite Type Functional Equations

It is known that some problems in meteorology and fluid dynamics lead to Goła̧b–Schinzel type equations on a restricted domain. Inspired by a question of Professor L. Reich we determine the solutions of conditional composite type functional equations related to the Goła̧b–Schinzel equation.

#### Symmetric 2-Structures, a Classification

We classify symmetric 2-structures ${(P, \mathfrak{G}_1, \mathfrak{G}_2, \mathfrak{K})}$, i.e. chain structures which correspond to sharply 2-transitive permutation sets (E, Σ) satisfying the condition: “ ${(*) \, \, \forall \sigma, \tau \in \Sigma : \sigma \circ \tau^{-1} \circ \sigma \in \Sigma}$ ”. To every chain ${K \in \mathfrak{K}}$ one can associate a reflection ...