Generalizing the notion of the almost free group we introduce almost Butler groups. An almost $B_2$-group $G$ of singular cardinality is a $B_2$-group. Since almost $B_2$-groups have preseparative chains, the same result in regular cardinality holds under the additional hypothesis that $G$ is a $B_1$-group. Some other results characterizing $B_2$-groups within the classes of almost $B_1$-groups and almost $B_2$-groups are obtained. A theorem of stating that a group $G$ of weakly compact cardinality $\lambda $ having a $\lambda $-filtration consisting of pure $B_2$-subgroup is a $B_2$-group appears as a corollary.
We consider almost hyper-Hermitian structures on principal fibre bundles with one-dimensional fiber over manifolds with almost contact 3-structure and study relations between the respective structures on the total space and the base. This construction suggests the definition of a new class of almost contact 3-structure, which we called trans-Sasakian, closely connected with locally conformal quaternionic Kähler manifolds. Finally we give a family of examples of hypercomplex manifolds which are not quaternionic Kähler.
We say that a binary operation $*$ is associated with a (finite undirected) graph $G$ (without loops and multiple edges) if $*$ is defined on $V(G)$ and $uv\in E(G)$ if and only if $u\ne v$, $u * v=v$ and $v*u=u$ for any $u$, $v\in V(G)$. In the paper it is proved that a connected graph $G$ is geodetic if and only if there exists a binary operation associated with $G$ which fulfils a certain set of four axioms. (This characterization is obtained as an immediate consequence of a stronger result proved in the paper).
The neutral differential equation (1.1) $$ \frac{{\mathrm{d}}^n}{{\mathrm{d}} t^n} [x(t)+x(t-\tau)] + \sigma F(t,x(g(t))) = 0, $$ is considered under the following conditions: $n\ge 2$, $\tau >0$, $\sigma = \pm 1$, $F(t,u)$ is nonnegative on $[t_0, \infty) \times (0,\infty)$ and is nondecreasing in $u\in (0,\infty)$, and $\lim g(t) = \infty$ as $t\rightarrow \infty$. It is shown that equation (1.1) has a solution $x(t)$ such that (1.2) $$ \lim_{t\rightarrow \infty} \frac{x(t)}{t^k}\ \text{exists and is a positive finite value if and only if} \int^{\infty}_{t_0} t^{n-k-1} F(t,c[g(t)]^k){\mathrm{d}} t < \infty\text{ for some }c > 0. $$ Here, $k$ is an integer with $0\le k \le n-1$. To prove the existence of a solution $x(t)$ satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.
My aim is to show that some properties, proved to be true for the square matrices, are true for some not necessarily linear operators on a linear space, in particular, for Hammerstein-type operators.
The basis number of a graph $G$ is defined by Schmeichel to be the least integer $h$ such that $G$ has an $h$-fold basis for its cycle space. MacLane showed that a graph is planar if and only if its basis number is $\le 2$. Schmeichel proved that the basis number of the complete graph $K_n$ is at most $3$. We generalize the result of Schmeichel by showing that the basis number of the $d$-th power of $K_n$ is at most $2d+1$.
It is not the purpose of this paper to construct approximations but to establish a class of almost periodic functions which can be approximated, with an arbitrarily prescribed accuracy, by continuous periodic functions uniformly on ${\mathbb R} = (-\infty ; +\infty )$.