Let $A$ be an $\mathcal{O}$-algebra with positive squares and $F\left( X_{1},...,X_{n}\right) \in\linebreak\mathbb{R}^{+}\left[ X_{1},...,X_{n}\right] $ be a homogeneous polynomial of degree $p$ $\left( p\in\mathbb{N}^{\ast },\text{ }p\neq2\right) $. It is shown that for all $0\leq a_{1},...,a_{n}\in A$ there exists $0\leq a$ $\in A$ such that $F\left( a_{1},...,a_{n}\right) =a^{p}$ . As an application we show that every algebra homomorphism $T$ from an $\mathcal{O}$-algebra $A$\textit{ }with positive squares into an Archimedean semiprime \textit{f-}algebra $B$ is positive. This improves a result of Render [14, Theorem 4.1], who proved it for the case of order bounded multiplicative functional $T$ from an $\mathcal{O}$-algebra $A$ with positive squares into $\mathbb{R}$.

We derive necessary and sufficient conditions for the existence of a least and a greatest fixed point of an operator which satisfies the hypothesis of Schauder's theorem. The so obtained results are applied to prove existence of extremal solutions for some initial and boundary value problems.

We consider function algebras whose elements are functions taking values from topological algebra bundles and study section algebras and their spectra of the topological algebra bundles involved. This naturally extends the so far standard theory of vector-valued function algebras to an appropriate {\it varying set-up.}

The goal of this note is to characterize Haagerup property for finite von Neumann algebras in terms of $||.||_{2}$-continuous semi-group of completely positive $L^{2}$-compact maps. The generator of this semi-group could be interpreted as an analogue for von Neumann algebras having the Haagerup property of a proper conditionally negative definite map on a group having Haagerup property.

Let $G$ be a simple group of finite Morley rank with a definable irreducible BN-pair of (Tits) rank $2$ where $B$ is solvable and let $\fP$ be the associated generalized $n$-gon. If $n$ is odd and $B$ connected, then $n=3$ and $G$ is definably isomorphic to $PSL_3(K)$ for some algebraically closed field $K$. Furthermore, $n\leq 14$ if $T=B\cap N\neq 1$. We also give sufficient conditions for $G$ to be a simple algebraic group.

Let $\perp$ be the polarity of $PG(5,q)$ defined by the elliptic quadric $Q^-(5,q)$. A locally Hermitian spread ${\cal S}$ of $Q^-(5,q)$, with respect to a line $L$, is associated in a canonical way with a spread ${\cal S}_{\Lambda}$ of the $3$-dimensional projective space $L^{\perp} = \Lambda$, and conversely. In this paper we give a geometric characterization of the regular spreads of $\Lambda$ which induce Hermitian spreads of $Q^-(5,q)$.

It is known that every 1-system of $W_{5}(q)$ is an SPG regulus and thus defines a semipartial geometry. In this paper, the semipartial geometries arising from locally hermitian 1-systems of $W_{5}(q)$, $q$ even, will be investigated. It will be shown that non-isomorphic locally hermitian 1-systems of $W_{5}(q)$ yield non-isomorphic semipartial geometries, which implies the existence of new semipartial geometries.

Ordinary differential equations are considered consisting of two equations with nonlinear coupling where the linear parts of the two equations have equilibria which are, respectively, a saddle and a center. Perturbation terms are added which correspond to damping and forcing. A reduced equation is obtained from the hyperbolic equation by setting to zero the variable from the center equation. Melnikov theory is used to obtain a transverse homoclinic solution, and hence chaos, in the reduced equation. Conditions are then established such that the chaos for the reduced equation is shadowed by chaos for the full equation. The resonant case is also studied when the chaos of the full system is not detected from the reduced equation. The techniques make use of exponential dichotomies.

In this paper we study the problem of decomposing a Hölder continuous $k$-grade multivector field $F_{k}$ on the boundary $\Gamma$ of an open bounded subset $\Omega$ in Euclidean space $\R^{n}$ into a sum $F_{k}=F_{k}^{+}+F_{k}^{-}$ of harmonic $k$-grade multivector fields $F_{k}^{\pm}$ in $\Omega_{+}=\Omega$ and $\Omega_{-}=\R^{n}\setminus (\Omega\cup\Gamma)$ respectively. The necessary and sufficient conditions upon $F_{k}$ we thus obtain complement those proved by Dyn'kin in [20,21] in the case where $F_{k}$ is a continuous $k$-form on $\Gamma$. Being obtained within the framework of Clifford analysis and hence being of a pure function theoretic nature, they once more illustrate the importance of the interplay between Clifford analysis and classical real harmonic analysis.

Uniform spaces can be Cauchy-completed; and if the base space was a first-order structure, this structure can be naturally extended to the completion. While common in algebra, this construction has been more recently used to produce new models of special set theories. We investigate here a natural way to ``twist'' the semantics of any structure according to a uniformity on its universe. We use it to relate the (classical first-order) theories of structures and dense substructures and apply it to the case of Cauchy-completions.

Let $C_{0}\left( \Omega ,X\right)$ be the linear space of all continuous functions from a locally compact normal space $\Omega $ into a normed space $X$ vanishing at infinity, equipped with the supremum-norm topology. The main result of the paper says that if $X$ is barrelled, then the space $C_{0}\left( \Omega ,X\right) $ is always barrelled. This answers a question posed by J. Horváth.

We study some partial differential equations with infinite delay which appear, for example, in models of population dynamics. Using the semigroup theory, we prove the existence of classical solutions of such equations.

In this short note we prove that the locally convex inductive limit of commutative locally m-convex algebras is already locally m-convex whenever multiplication is continuous.