L-norms and M-norms on Banach lattices, unit-norms and base norms on ordered vector spaces are well known. In this paper m- and $m_{\le }$-norms are introduced on ordered normed spaces. They generalize the notions of the M-norm and the order-unit norm, possess also some interesting properties and can be characterized by means of the constants of reproducibility of cones. In particular, the dual norm of an ordered Banach space with a closed cone turns out to be additive on the dual cone if and only if the...

Let $\Omega \subset {ℝ}^n$, $n\ge 2$, be a bounded connected domain of the class $C^{1,\theta }$ for some $\theta \in (0,1]$. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem $$u\in W^1{L}^{\Phi }\left(\Omega \right),-div\left({\Phi }^{\text{'}}\left(\right|\nabla u\left|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi }^{\text{'}}\left(\right|u\left|\right)\frac{u}{\left|u\right|}=f(x,u)+\mu h\left(x\right)\text{in}\Omega ,\frac{\partial u}{\partial 𝐧}=0\text{on}\partial \Omega ,$$ where $\Phi$ is a Young function such that the space $W^1{L}^{\Phi }\left(\Omega \right)$ is embedded into exponential or multiple exponential Orlicz space, the nonlinearity $f(x,t)$ has the corresponding critical growth, $V\left(x\right)$ is a continuous potential,...

Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem $$u\in W_0^1{L}^{\Phi }\left(\Omega \right)and-div\left({\Phi }^{\text{'}}\left(\left|\nabla u\right|\right)\frac{\nabla u}{\left|\nabla u\right|}\right)+V\left(x\right){\Phi }^{\text{'}}\left(\left|u\right|\right)\frac{u}{\left|u\right|}=f\left(x,u\right)+\mu h\left(x\right)in\Omega ,$$ where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω =...

The generalized non-commutative torus $T_{\varrho }^k$ of rank n is defined by the crossed product $A_{m/k}{\times }_{\alpha ₃}\mathbb{Z}{\times }_{\alpha ₄}...{\times }_{\alpha ₙ}\mathbb{Z}$, where the actions ${\alpha }_i$ of ℤ on the fibre $M_k\left(ℂ\right)$ of a rational rotation algebra $A_{m/k}$ are trivial, and $C*\left(k\mathbb{Z}\times k\mathbb{Z}\right){\times }_{\alpha ₃}\mathbb{Z}{\times }_{\alpha ₄}...{\times }_{\alpha ₙ}\mathbb{Z}$ is a non-commutative torus $A_{\varrho }$. It is shown that $T_{\varrho }^k$ is strongly Morita equivalent to $A_{\varrho }$, and that $T_{\varrho }^k\otimes M_{p^{\infty }}$ is isomorphic to $A_{\varrho }\otimes M_k\left(ℂ\right)\otimes M_{p^{\infty }}$ if and only if the set of prime factors of k is a subset of the set of prime factors of p.

We investigate the amenability and its related homological notions for a class of $I\times I$-upper triangular matrix algebra, say $\mathrm{UP}(I,A)$, where $A$ is a Banach algebra equipped with a nonzero character. We show that $\mathrm{UP}(I,A)$ is pseudo-contractible (amenable) if and only if $I$ is singleton and $A$ is pseudo-contractible (amenable), respectively. We also study pseudo-amenability and approximate biprojectivity of $\mathrm{UP}(I,A)$.

The equicontinuous sets of locally convex generalized inducted limit (or mixed) topologies are characterized as generalized precompact sets. Uniformly pre-Lebesgue and Lebesgue topologies in normed Riesz spaces are investigated and it is shown that order precompactness and mixed topologies can be used to great advantage in the study of these topologies.

We produce generalized q-Gaussian random variables which have two parameters of deformation. One of them is, of course, q as for the usual q-deformation. We also investigate the corresponding Wick formulas, which will be described by some joint statistics on pair partitions.

2000 Mathematics Subject Classification: 35E45In this paper we study generalized Sobolev spaces H^sG of exponential type associated with the Dunkl operators based on the space G of test functions for generalized hyperfunctions and investigate their properties. Moreover, we introduce a class of symbols of exponential type and their associated pseudodifferential operators related to the Dunkl operators, which act naturally on H^sG.

In this paper first order systems of linear of ODEs are considered. It is shown that these systems admit unique solutions in the Colombeau algebra $ℒ\left({𝐑}^1\right)$.