The aim of this paper is to study in characteristic $2$ and degree at most $4$ the isotropy of involutions of the first kind and quadratic paires over the function field of a projective quadric. We also give a complete answer to the hyperbolicity over an inseparable quadratic extension in arbitrary degree. The case of a separable quadratic extension has been studied in [4].

We study modules of solutions of the equation $DF=F$, where $F$ is a function in the plane with values in the quaternions and $D$ is the Dirac operator. The functions $F$ will belong to the Sobolev-type space of all functions in $L^{p}(\Omega,|x|^{-3}dx)$ jointly with their angular and radial derivatives, and where $\Omega$ is the complement of the unit disk in $\mathbb{R}^{2}$. The resulting spaces are right Banach modules over the quaternions. When $p=2$ we calculate the reproducing kernel of this space and explain its reproducing properties when $p\neq2$.

We consider periodic problems driven by the ordinary scalar $p$-Laplacian with a Caratheodory nonlinearity. Using variational techniques, coupled with the method of upper and lower solutions, we obtain two nontrivial solutions, with one positive and the other negative.

Let $\Pi=(P,L)$ be a partial linear space in which any line contains three points and let $K$ be a field. Then by ${\cal L}_K(\Pi)$ we denote the free $K$-algebra generated by the elements of $P$ and subject to the relations $xy=0$ if $x$ and $y$ are noncollinear elements from $P$ and $xy=z$ for any triple $\{x,y,z\}\in L$. We prove that the algebra ${\cal L}_K(\Pi)$ is a Lie algebra if and only if $K$ is of even characteristic and $\Pi$ is a cotriangular space satisfying Pasch's axiom. Moreover, if $\Pi$ is a cotriangular space satisfying Pasch's axiom, then a connection between derivations of the Lie algebra ${\cal L}_K(\Pi)$ and geometric hyperplanes of $\Pi$ is used to determine the structure of the algebra of derivations of ${\cal L}_K(\Pi)$.

We define some subclasses of Orlicz spaces of functions and establish here a direct theorem of the approximation theory by rational functions.

We obtain a generalized continuous selection theorem and a coincidence theorem for generalized convex spaces. Some new Himmelberg type theorems and Eilenberg-Montgomery and Gorniéwicz type fixed point theorems for mappings with KKM property are established in noncompact LG-spaces. Moreover, applications to these fixed point theorems for existence of equilibria are given.

We characterize some eigenfunctions of Landau Hamiltonian on the hyperbolic disc which are Poisson integrals of square integrable functions at the disc boundary.

Denote by $\cal A'$, the class of functions $f$, analytic in $E$ which satisfy $f(0)=1$. Let $\alpha >0, \beta \in (0,1]$ be real numbers and let $\gamma, {\rm Re} \gamma >0$, be a complex number. For $p, q \in \cal A'$, the authors study the differential subordination of the form $$(p(z))^\alpha \left[1+\frac {\gamma zp'(z)}{p(z)}\right]^\beta \prec(q(z))^\alpha \left[1+\frac {\gamma zq'(z)}{q(z)}\right]^\beta, z\in E,$$ and obtain its best dominant. Its applications to univalent functions are also given.

To a finite Hopf-Galois extension $A | B$ we associate dual bialgebroids $S := End\,_BA_B$ and $T := (A \o_B A)^B$ over the centralizer $R$ using the depth two theory in [18,Kadison-Szlachányi]. First we extend results on the equivalence of certain properties of Hopf-Galois extensions with corresponding properties of the coacting Hopf algebra [21,8] to depth two extensions using coring theory [3]. Next we show that $T^{\rm op}$ is a Hopf algebroid over the centralizer $R$ via Lu's theorem \cite[23,5.1] for smash products with special modules over the Drinfel'd double, the Miyashita-Ulbrich action, the fact that $R$ is a commutative algebra in the pre-braided category of Yetter-Drinfel'd modules [28] and the equivalence of Yetter-Drinfel'd modules with modules over Drinfel'd double [24]. In our last section, an exposition of results of Sugano [29,30] leads us to a Galois correspondence between sub-Hopf algebroids of $S$ over simple subalgebras of the centralizer with finite projective intermediate simple subrings of a finite projective H-separable extension of simple rings $A \supseteq B$.

We show a general method to translate Tauberian theorems for summability methods in $\mathbb R$ into Tauberian theorems for the corresponding forms of statistical convergence in metric spaces. The main tools (distance functions and the Hausdorff metric) come from set-valued analysis.

Let $X$ be a compact plane set. $A(X)$ denotes the uniform algebra of all continuous complex-valued functions on $X$ which are analytic on int$X$. For $0<\alpha\leq 1$, Lipschitz algebra of order $\alpha$, $Lip(X,\alpha)$ is the algebra of all complex-valued functions $f$ on $X$ for which $p_\alpha(f)=\sup\{\frac{|f(x)-f(y)|}{|x-y|^{\alpha}}:x,y\in X, x\neq y\}<\infty.$ Let $Lip_A(X,\alpha)=A(X)\bigcap Lip(X,\alpha)$, and $Lip^{n}(X,\alpha)$ be the algebra of complex-valued functions on $X$ whose derivatives up to order $n$ are in $\Lip(X,\alpha)$. $Lip_A(X,\alpha)$ under the norm $\|f\|=\|f\|_X+p_\alpha(f)$, and $Lip^n(X,\alpha)$ for a certain plane set $X$ under the norm $\|f\|=\sum_{k=0}^{n}\frac{\|f^{(k)}\|_X+p_{\alpha}(f^{(k)})}{k!}$ are natural Banach function algebras, where $\|f\|_X = \sup_{x\in X } |f(x)|$. In this note we study endomorphisms of algebras $Lip_A(X,\alpha)$ and $Lip^n(X,\alpha)$ and investigate necessary and sufficient conditions for which these endomorphisms to be compact. Finally, we determine the spectra of compact endomorphisms of these algebras.