Published in Forum Mathematicum
We study a parametric Robin problem driven by a nonlinear nonhomogeneous differential operator and with a superlinear Carathéodory reaction term. We prove a bifurcation-type theorem for small values of the parameter. Also, we show that as the parameter λ > 0 {\lambda>0} approaches zero, we can find positive solutions with arbitrarily big and arbitr...
Published in Forum Mathematicum
It is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover to be homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on s...
Published in Forum Mathematicum
Answering a question of Pálfy and Pyber, we first prove the following extension of the k ( G V ) {k(GV)} -problem: Let G be a finite group and let A be a coprime automorphism group of G. Then the number of conjugacy classes of the semidirect product G ⋊ A {G\rtimes A} is at most | G | {\lvert G\rvert} . As a consequence, we verify Brauer’s k ...
Published in Forum Mathematicum
In the present paper, right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras will be studied. For this purpose, first the structure of the set of all right 2-Engel elements of a Lie algebra will be examined and then, by taking advantage of it, a number of interesting results about central and commuting automorphis...
Published in Forum Mathematicum
The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum twisted by the action of the orthogonal group. We prove that simple arithmetic formulas hold for some basic class...
Published in Forum Mathematicum
Given measurable functions ϕ, ψ on ℝ + {\mathbb{R}^{+}} and a kernel function k ( x , y ) ≥ 0 {k(x,y)\geq 0} satisfying the Oinarov condition, we study the Hardy operator K f ( x ) = ψ ( x ) ∫ 0 x k ( x , y ) ϕ ( y ) f ( y ) 𝑑 y , x > 0 , Kf(x)=\psi(x)\int_{0}^{x}k(x,y)\phi(y)f(y)\,dy,\quad x>0, between Orlicz–Lorentz spaces Λ...
Published in Forum Mathematicum
In this paper, we estimate the shifted convolution sum ∑ n ⩾ 1 λ 1 ( 1 , n ) λ 2 ( n + h ) V ( n X ) , \sum_{n\geqslant 1}\lambda_{1}(1,n)\lambda_{2}(n+h)V\Big{(}\frac{n}{X}\Big{)}, where V is a smooth function with support in [ 1 , 2 ] {[1,2]} , 1 ⩽ | h | ⩽ X {1\leqslant|h|\leqslant X} , and λ 1 ( 1 , n ) {\lambda_{1}(1,n)} and λ 2 (...
Published in Forum Mathematicum
It is shown that the projection lattice of a von Neumann algebra, or more generally every orthomodular lattice X, admits a natural embedding into a group G ( X ) {G(X)} with a lattice ordering so that G ( X ) {G(X)} determines X up to isomorphism. The embedding X ↪ G ( X ) {X\hookrightarrow G(X)} appears to be a universal (non-commutative) gr...
Published in Forum Mathematicum
In 1959, Philip Hall introduced the locally finite group 𝒰 {\mathcal{U}} , today known as Hall’s universal group. This group is countable, universal, simple, and any two finite isomorphic subgroups are conjugate in 𝒰 {\mathcal{U}} . It can explicitly be described as a direct limit of finite symmetric groups. It is homogeneous in the model-theoretic...
Published in Forum Mathematicum
In the paper, our main aim is to generalize the dual affine quermassintegrals to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new affine geometric quantity by calculating the first-order variation of the dual affine quermassintegrals, and call it the Orlicz dual affine quermassintegral. The fundamental...
