In this paper we characterize the family of secant lines of an ovoid of PG$(3,q)$ and the family of external lines to an ovoid of PG$(3,q)$.

We present several results about the extension of vector-valued holomorphic or meromorphic functions from an open domain in $\mathbb C$ to a larger domain on which the function has a weakly holomorphic or meromorphic extension.

In this paper, the fundamental solution of the Dirac equation on hyperbolic space will be calculated by means of the fundamental solution for the wave-operator in the $(m+1)$-dimensional Minkowski space-time of signature $(1,m)$. This leads to addition formulas for the fundamental solution in terms of the solution in a lower-dimensional Minkowski space-time. Certain identities between hypergeometric functions can then be used to obtain a closed form for the fundamental solution of the Dirac equation.

We say a positive integer $n$ is abundant if $\sigma(n)>2n$, where $\sigma(n)$ denotes the sum of the positive divisors of $n$. Number the primes in ascending order: $p_1=2$, $p_2=3$, and so forth. Let $A(k)$ denote the smallest abundant number not divisible by $p_1$, $p_2$, \dots, $p_k$. In this paper we find $A(k)$ for $1\leq k\leq 7$, and we show that for all $\epsilon>0$, $(1-\epsilon)(k\ln{k})^{2-\epsilon}<\ln{A(k)}<(1+\epsilon)(k\ln{k})^{2 +\epsilon}$ for all sufficiently large $k$.

In this paper we study existence of solutions to a class of semilinear degenerate elliptic equations in Weighted Sobolev spaces.

We are concerned with the following problem. Let $L$ and $K$ be fixed real numbers. When does the Koebe function $k(z)=z (1-z)^{-2}$ maximize the $N$th Taylor coefficient of $(1/f'(z))^L(z/f(z))^K$ for $f$ in the class $S$ of normalized schlicht functions? A sufficient condition for $L \geq -1$ is $1 \leq N \leq 2L+K+1$. A necessary condition is that a certain trigonometric sum involving hypergeometric functions is non-negative. These results generalize a recent theorem of Bertilsson and suggest a link between Brennan's conjecture in conformal mapping and Baernstein's theorem about integral means of functions in $S$.

We show that the sectional category of a non trivial fibration $p$ with fibre $ K(\mathbb{Q}, 2k) $ has sectional category $1$ although all $n$-fold fibre joins $ p * \dotsb * p $ are not trivial.

We study the nonexistence of global weak solutions for equations of the following type: \begin{equation} u_{tt}-\Delta\ u +g(t)\ u_t=|u|^p \label{*} \end{equation} where $g(t)$ behaves like $t^{\beta},\ 0\leq\beta< 1$. Then the situation is extended to systems of equations of the same type, and more general equation than $(\ref{*})$.

In this paper we describe various methods of constructing new paratopological vector spaces from given ones. First, we introduce the notion of a right dual of a paratopological vector space with the aim to define a right weak topology. We prove that, in a certain class of normed spaces, the classical weak topology is determined by a right weak topology. Next, the quotient topology in the context of paratopological vector spaces is discussed. Finally, we consider the projective limit of paratopological vector spaces and prove that every pseudoconvex space is a projective limit of quasi-normed spaces.

For $\lambda>0$ and $0<\mu<n$, let ${\mathcal U}_n(\lambda,\mu)$ denote the class of all normalized analytic functions $f$ in the unit disc $\Delta$ of the form $f(z)=z+\sum_{k=n+1}^{\infty}a_kz^k$ such that $$\left|f'(z)\left(\frac{z}{f(z)}\right)^{\mu+1}-1\right|<\lambda, ~ z\in \Delta, $$ where $n\in \mathbb N$ is fixed. In addition to the discussion of the basic properties of the class ${\mathcal U}_n(\lambda,\mu)$, we find conditions so that ${\mathcal U}_n(\lambda,\mu)$ is included in ${\mathcal S}_{\gamma}$, the class of all strongly starlike functions of order $\gamma$ $(0<\gamma\leq 1)$. We also find necessary conditions so that $f\in {\mathcal U}_n(\lambda,\mu)$ implies that $$\left|\frac{zf'(z)}{f(z)}-\frac{1}{2\beta}\right|<\frac{1}{2\beta},\quad \mbox{for all $z\in \Delta,$} $$ or $$\left|1+\frac{zf''(z)}{f'(z)}-\frac{1}{2\beta}\right| <\frac{1}{2\beta},\quad \mbox{for $|z|<r<1$},$$ where $r=r(\lambda,\mu,n)$ will be specified.

In this paper we consider various types of relative groups which naturally arise in surgery theory, and describe algebraic properties of them. Then we apply the obtained results to investigate the splitting obstruction groups $LS_*$ and the surgery obstruction groups $LP_*$ for a manifold pair. Finally, we introduce the lower $LS_*$- and $LP_*$-groups, and describe connections between them and the corresponding lower $L_*$-groups and surgery exact sequence.

A generalized quadrangle is classical if it has a grid of axes of symmetry.