In this paper, we show that, for any spatial embedding $\Gamma:G\to\mathbf{R}^3$ of a connected planar graph $G$, there exists a disk/band surface of $\Gamma(G)$ satisfying a certain linking condition. As an application of this result, it is proved that the homology class of $\Gamma(G)$ is determined only by the linking numbers of disjoint pairs in the set of boundary/outermost cycles with respect to a fixed planar embedding of $G$.

In this paper a class of contact, non Sasakian, Riemannian manifolds of constant $\varphi$-sectional curvature is found and studied.

We show that the topological symmetry group of a canonically embedded complete graph of $n\geq 7$ vertices in the 3-sphere is isomorphic to a dihedral group of order $2n$.

If $(X,\mathscr{A})$ is a measurable space, $(\mathscr{P}_n)_{n\in\mathbf{N}}$ is an increasing sequence of nonempty sets $\mathscr{P}_n$ of probability measures and $\mathscr{B}_n$ is a sub-$\sigma$-field of $\mathscr{A}$ which is sufficient for the statistical experiment $(X,\mathscr{A},\mathscr{P}_n)$, $n\in\mathbf{N}$, then the terminal $\sigma$-field of the sequence $(\mathscr{B}_n)_{n\in\mathbf{N}}$ contains a $\sigma$-field which is sufficient for $\bigcup_{n\in\mathbf{N}}\mathscr{P}_n$.

Let $(E_i)_{i\in\mathbf{N}}$ be a projective system of Banach vector bundles whose limit is a Fréchet bundle of fibre type $\mathbf{F}$. We construct a generalized bundle of frames $\mathbf{P}(E)$ of $E$ by revising entirely the classical notion and by substituting $GL(\mathbf{F})$ with an appropriate enlarged structure group. This is imposed by the pathology of $GL(\mathbf{F})$, which renders meaningless the ordinary frame bundle. As a result, we prove that $E$ is associated with $\mathbf{P}(E)$ and linear connections of $E$ correspond to (principal) connections of $\mathbf{P}(E)$. In particular, the former are necessarily projective limits of connections on the bundles $E_i$.

We prove Cauchy's integral formula for complex regular functions in complex Clifford analysis. We claim that our formula is valid in any dimension. Furthermore, we study some properties of manifolds where the integration in the formula is defined and relax the conditions imposed on the manifolds.

We define a property of strong-mixing in the theory of $C^*$-dynamical systems and show that the property follows from the existence of a base of Walsh type. Moreover as applications we analyze covariant representations of chaotic dynamical systems.

The aim of this paper is to study the stability of travelling wave solutions with shock profiles for one-dimensional viscoelastic materials with the non-degenerate and the degenerate shock conditions by means of an elementary weighted energy method. The stress function is not necessarily assumed to be convex or concave, and the third derivative of this stress function is also not necessarily assumed to be non-negative or non-positive. The travelling waves are proved to be stable for suitably small initial disturbance and shock strength, which improves recent stability results. The key points of our proofs are to choose the suitable weight function and weighted Sobolev spaces of the solutions.