This paper deals with $\varphi \left(q\right)$ calculus which is an extension of finite operator calculus due to Rota, and leading results of Rota’s calculus are easily $\varphi$-extendable. The particular case ${\varphi }_n\left(q\right)={\left[n_{q^1}\right]}^{-1}$ is known to be relevant for quantum group investigations. It is shown here that such $\varphi \left(q\right)$ umbral calculus leads to infinitely many new $\varphi$-deformed quantum like oscillator algebra representations. The authors point to several references dealing with new applications of $q$ umbral and $\varphi \left(q\right)$ calculus in which new families of $\varphi \left(q\right)$ extensions...

A multisymplectic 3-structure on an $n$-dimensional manifold $M$ is given by a closed smooth 3-form $\omega$ of maximal rank on $M$ which is of the same algebraic type at each point of $M$, i.e. they belong to the same orbit under the action of the group $GL(n,ℝ)$. This means that for each point $x\in M$ the form ${\omega }_x$ is isomorphic to a chosen canonical 3-form on ${ℝ}^n$. R. Westwick [Linear Multilinear Algebra 10, 183–204 (1981; Zbl 0464.15001)] and D. Ž. Djoković [Linear Multilinear Algebra 13, 3–39 (1983; Zbl 0515.15011)] obtained...

We give a survey of the joint papers of Lawrence Somer and Michal Křížek and discuss the beginning of this collaboration.

The authors prove that all natural affinors (i.e. tensor fields of type (1,1) on the extended $r$-th order tangent bundle $E^{r}M$ over a manifold $M$) are linear combinations (the coefficients of which are smooth functions on $ℛ$) of four natural affinors defined in this work.

One studies the flow prolongation of projectable vector fields with respect to a bundle functor of order $(r,s,q)$ on the category of fibered manifolds. As a result, one constructs an operator transforming connections on a fibered manifold $Y$ into connections on an arbitrary vertical bundle over $Y$. It is deduced that this operator is the only natural one of finite order and one presents a condition on vertical bundles over $Y$ under which every natural operator in question has finite order.

Let $F$ be a $p$-dimensional foliation on an $n$-manifold $M$, and $T^{r}M$ the $r$-tangent bundle of $M$. The purpose of this paper is to present some reltionship between the foliation $F$ and a natural lifting of $F$ to the bundle $T^{r}M$. Let $L_q^r\left(F\right)$