The purpose of this paper is to study the famous Pappus configuration of 9 lines and its dual arrangement. We show among others that by applying the Pappus Theorem to the dual arrangement we obtain the configuration corresponding to the initial data of beginning configuration. We consider also the Pappus arrangements with some additional incidences and we establish algebraic conditions paralleling with these incidences.

Assuming initial data have small weighted $H^4\times H^3$ norm, we prove global existence of solutions to the Cauchy problem for systems of quasi-linear wave equations in three space dimensions satisfying the null condition of Klainerman. Compared with the work of Christodoulou, our result assumes smallness of data with respect to $H^4\times H^3$ norm having a lower weight. Our proof uses the space-time $L^2$ estimate due to Alinhac for some special derivatives of solutions to variable-coefficient wave equations. It also uses the conformal energy estimate for inhomogeneous wave equation $\Box u=F$. A new observation made in this paper is that, in comparison with the proofs of Klainerman and Hörmander, we can limit the number of occurrences of the generators of hyperbolic rotations or dilations in the bootstrap argument. This limitation allows us to obtain global solutions for radially symmetric data, when a certain norm with considerably lower weight is small enough.

In the present paper, we classify all surjective homomorphisms from the étale fundamental group of the configuration space of a hyperbolic curve (over an algebraically closed field of characteristic zero) to the étale fundamental group of a hyperbolic curve. We can show that such a surjective homomorphism is necessarily “geometric” in some sense, that is, it factors through one of the homomorphisms which arise from specific morphisms of schemes.

We explore the relationship in metric spaces between different properties related to the Besicovitch covering theorem, and also consider weak versions of doubling, in connection to the non-uniqueness of centers and radii in arbitrary metric spaces.

We consider the bilinear pseudo-differential operators with symbols in the bilinear Hörmander classes $BS_{\rho, \rho}^m$, $0 \lt \rho \lt 1$. In this paper, we show that the condition $1/p = 1/p_1 + 1/p_2$ is necessary to assure the boundedness from $H^{p_1} \times H^{p_2}$ to $L^p$ of those operators with the critical order $m$.

The main purpose of the present paper is to study the numerical properties of supersolvable resolutions of line arrangements. We provide upper-bounds on the so-called extension to supersolvability numbers for certain extreme line arrangements in $\mathbb{P}^{2}_{\mathbb{C}}$ and we show that these numbers are not determined by the intersection lattice of the given arrangement.

In the present paper we investigate the sufficiency conditions for which a normalized Hurwitz-Lerch Zeta function is convex in the open unit disk. A similar property as well as properties of close-to-convexity and starlikeness of a normalized Mittag-Leffler are also studied. The concluding section gives some examples which are deduced from the main results and for such special cases we also employ certain numerically set values for the parameters in order to find their image domains under the unit disk.

Our aim in this note is to establish Hardy-Sobolev inequalities in $\mathbb{R}^n$ for double phase functionals $\Phi(x,t)= t^p + (b(x) t)^q$, where $1 \le p \lt q$, $b(\cdot)$ is non-negative and Hölder continuous of order $\theta \in (0,1]$. The Sobolev functional of $\Phi$ is given by $\Phi^*(x,t)= t^{p^*} + (b(x) t)^{q^*}$, where $p^*$ and $q^*$ denote the Sobolev exponent of $p$ and $q$, respectively, that is, $1/p^* = 1/p - 1/n$ and $1/q^* = 1/q - 1/n$.