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.

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.

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 ...

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. ...

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.

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 ...

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.

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 ...

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.

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 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.

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 ...

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 ...

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.

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 ...

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.

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 ...

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.

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.

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.

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 ...

In a symmetric 2-structure \({\Sigma =(P,\mathfrak{G}_1,\mathfrak{G}_2,\mathfrak{K})}\) we fix a chain \({E \in \mathfrak{K}}\) and define on E two binary operations “+” and “·”. Then (E,+) is a K-loop and for \({E^* := E {\setminus}\{o\}}\), (E *,·) is a Bol loop. If \({\Sigma}\) is even point symmetric then (E,+ ,·) is a quasidomain and one has the set \({Aff(E,+,\cdot) := ...