In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval $\left[ a,b\right] $ into a Banach space $X.$ It is shown that a Denjoy-Bochner integrable function on $ \left[ a,b\right] $ is Denjoy-Riemann integrable on $\left[ a,b\right] $, that a Denjoy-Riemann integrable function on $\left[ a,b\right] $ is Denjoy-McShane integrable on $\left[ a,b\right] $ and that a Denjoy-McShane integrable function on $\left[ a,b\right] $ is Denjoy-Pettis integrable on $\left[ a,b\right].$ In addition, it is shown that for spaces that do not contain a copy of $c_{0}$, a measurable Denjoy-McShane integrable function on $\left[ a,b\right] $ is McShane integrable on some subinterval of $\left[ a,b\right].$ Some examples of functions that are integrable in one sense but not another are included.
Let $A$ and $B$ be two Archimedean vector lattices and let $( A^{\prime }) _n'$ and $( B') _n'$ be their order continuous order biduals. If $\Psi \colon A\times A\rightarrow B$ is a positive orthosymmetric bimorphism, then the triadjoint $\Psi ^{\ast \ast \ast }\colon ( A') _n'\times ( A') _n'\rightarrow ( B') _n'$ of $\Psi $ is inevitably orthosymmetric. This leads to a new and short proof of the commutativity of almost $f$-algebras.
In this paper, we give necessary and sufficient conditions on $(p_n)$ for $| R,p_n| _k$, $k\ge 1$, to be translative. So we extend the known results of Al-Madi [1] and Cesco $\left[ 4\right] $ to the case $k>1$.
We show that each element in the semigroup $S_n$ of all $n \times n$ non-singular upper (or lower) triangular stochastic matrices is generated by the infinitesimal elements of $S_n$, which form a cone consisting of all $n \times n$ upper (or lower) triangular intensity matrices.
We give some explicit values of the constants $C_{1}$ and $C_{2}$ in the inequality $C_{1}/{\sin (\frac{\pi }{p})}\le \left| P\right| _{p}\le C_{2}/{\sin (\frac{\pi }{p})}$ where $\left| P\right| _{p}$ denotes the norm of the Bergman projection on the $L^{p}$ space.
There exists a rich literature on systems of connections and systems of vector fields, stimulated by the irimportance in geometry and physis. In the previous papers [T1], [T2] we examined a simple type of systems of vector fields, called parameter dependent vector fields, and established their varionational equation.
In this paper we generalize the above equation to the projectable system of vector fields. The material is organized as follows: in the first section the geometry of the product bundle is presented. In the second we introduce the notion of derivative along a direction and prove Theorem 1. The third section is devoted to Theorem
2, which represents the main result of the paper. Some examples are presented in the last section. In a further paper we will apply the results in order to investigate some special systems as strong systems, “nice” systems and systems of connections generated by systems of vector fields.