In this paper we study the contractive mappings in vector space with vector norm. We suppose that values of the norm belong to a $\sigma$-complete vector lattice. The results extend the Banach fixed point theorem for differential equations, under the condition to retain the convergence of successive approximations to the unique solution.

We establish some conditions under which the differential subordination of the type $p(z) + zp'(z)/p(z) \prec Q(z)$ yields $p \prec q$ in $\mathcal U$. Functions $Q$ and $q$ are chosen so that they map the unit disk onto domains enclosed by conic sections. Some applications of obtained results are given.

Pour $0<s<\frac{N}{2}$, $\frac{1}{p}=\frac{1}{2}-\frac{s}{N}\; (N=2d+2)$, on démontre que toute suite bornée de ${\dot H}^s(\mathbb H^d)$ s'écrit, à une sous-suite près, comme la somme presque orthogonale d'un terme tendant vers $0$ dans $L^{p}({\mathbb H}^{d})$ et d'une superposition de termes du type $h_{n}^{-\frac{N}{p}}$ $\psi (\delta _{h_{n}^{-1}}(v_{n}^{-1}.v))$, où $\psi \in {\dot{H}}^s({\mathbb H }^d)$, $(h_n)_n$ est une suite de réels positifs, et $(v_n)_n$ est une suite de points de ${\mathbb H}^d$.

Most of the algebras appearing in the theory of rings of differential operators, quantized algebras of different kinds (including many quantum groups), regular algebras in projective non-commutative geometry, etc... come equipped with a natural gradation or filtration controlled by some finite dimensional vector space(s), e.g. the degree one part of filtration or gradation. In this note we relate the valuations of the algebras considered to unramified sub-lattices in some vector space(s).

In this paper, we obtain new characterizations of weakly unconditionally Cauchy series and unconditionally convergent series through the summability obtained by the Banach-Lorentz convergence. We study new spaces associated to a series in a Banach space and obtain a new version of the Orlicz-Pettis theorem by means of the almost summability.

Let $\Omega_1$ and $\Omega_2$ be non empty open subsets of $\mathbb R^r$ and $\mathbb R^s$ respectively and let $\omega_1$ and $\omega_2$ be weights. We introduce the spaces of ultradifferentiable functions $\mathcal{E}_{(\omega_1,\omega_2)}(\Omega_1 \times \Omega_2)$, $\mathcal{D}_{(\omega_1,\omega_2)}(\Omega_1 \times \Omega_2)$, $\mathcal{E}_{\{\omega_1,\omega_2\}}(\Omega_1 \times \Omega_2)$ and $\mathcal{D}_{\{\omega_1,\omega_2\}}(\Omega_1 \times \Omega_2)$, study their locally convex properties, examine the structure of their elements and also consider their links with the tensor products $\mathcal{E}_{*}(\Omega_1) \otimes \mathcal{E}_{*}(\Omega_2)$ and $\mathcal{D}_{*}(\Omega_1) \otimes \mathcal{D}_{*}(\Omega_2)$ endowed with the $\varepsilon$-, $\pi$- or $i$-topologies. This leads to kernel theorems.

As an application of the Gottlieb sequence of fibration, we give an upper bound of the rank of Gottlieb group $G(E) =\oplus_{i>0}G_i(E)$ of the total space $E$ of a fibration $\xi :X\to E\to B$ and define the {\it Gottlieb type $(a,b,c;s,t,u)$}, which describes a rational homotopical condition of fibration with $\rank G(E)=s+t+u$. We also note various examples showing the different situations that can occur. Finally we comment about an interaction with a Halperin's conjecture on fibration.

Let $g:B\to \mathbb C^1$ be a holomorphic map of the unit ball $B$. We give a complete picture regarding the boundedness and compactness of the following two integral operators $$ T_gf(z)=\int_0^1f(tz)\Re g(tz)\frac{dt}{t}\text{ and } L_gf(z)= \int_0^1 \Re f(tz) g(tz)\frac{dt}{t},\quad z\in B, $$ between different weighted Bergman spaces.

We study span and stable span of Dold manifolds $P(m,n)$. We present several lower and upper bounds for these invariants, and for some pairs $(m,n)$, we determine the exact values of span or stable span. In the case $P(m,1)$, $m\not\equiv15\pmod{16}$, we show how the maximum number of everywhere linearly independent vector fields can be constructed.

In this article we introduce and study two classes of finite groups for which the orders of their subgroups satisfy a certain inequality. These are closely connected to some well-known arithmetic classes of natural numbers.

Suppose that a group $G$ has socle $L$ a simple large-rank classical group. Suppose furthermore that $G$ acts transitively on the set of lines of a linear space $\mathcal{S}$. We prove that, provided $L$ has dimension at least $25$, then $G$ acts transitively on the set of flags of $\mathcal{S}$ and hence the action is known. For particular families of classical groups our results hold for dimension smaller than $25$. The group theoretic methods used to prove the result (described in Section 3) are robust and general and are likely to have wider application in the study of almost simple groups acting on finite linear spaces.

We show that the symmetric track groups $\operatorname{Sym}_\Box(n)$, which are extensions of the symmetric groups $\operatorname{Sym}(n)$ associated to the second Stiefel-Whitney class, act as crossed modules on the secondary homotopy groups of a pointed space.