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.

The authors study some geometrical constructions on the cotangent bundle $T^{*}M$ from the viewpoint of natural operations. First they deduce that all natural operators transforming functions on $T^{*}M$ into vector fields on $T^{*}M$ are linearly generated by the Hamiltonian vector field with respect to the canonical symplectic structure of $T^{*}M$ and by the Liouville vector field of $T^{*}M$. Then they determine all natural operators transforming pairs of functions on $T^{*}M$ into functions on $T^{*}M$. In this case, the main generator is...

Let $F_1$ be a natural bundle of order $r_1$; a basis of the $s$-th order differential operators of $F_1$ with values in $r_2$-th order bundles is an operator $D$ of that type such that any other one is obtained by composing $D$ with a suitable zero-order operator. In this article a basis is found in the following two cases: for $F_1=\text{semi}{F}^{r_1}$ (semi-holonomic $r_1$-th order frame bundle), $s=0$, $r_2<r_1$ and $F_1=F^1$ ($1$-st order frame bundle), $r_2\le s$. The author uses here the so-called method of orbit reduction which provides one with a criterion for checking...

Let $Y\to M$ be a fibered manifold over a manifold $M$ and $\mu :A\to B$ be a homomorphism between Weil algebras $A$ and $B$. Using the results of Mikulski and others, which classify product preserving bundle functors on the category of fibered manifolds, the author classifies all natural operators $T_{\text{proj}}Y\to T^{\mu }Y$, where $T_{\text{proj}}Y$ denotes the space of projective vector fields on $Y$ and $T^{\mu }$ the bundle functors associated with $\mu$.

A general definition of a quantum predicate and quantum labelled transition systems for finite quantum computation systems is presented. The notion of a quantum predicate as a positive operator-valued measure is developed. The main results of this paper are a theorem about the existence of generalised predicates for quantum programs defined as completely positive maps and a theorem about the existence of a GSOS format for quantum labelled transition systems. The first theorem is a slight generalisation...

In some other context, the question was raised how many nearly Kähler structures exist on the sphere ${𝕊}^6$ equipped with the standard Riemannian metric. In this short note, we prove that, up to isometry, there exists only one. This is a consequence of the description of the eigenspace to the eigenvalue $\lambda =12$ of the Laplacian acting on $2$-forms. A similar result concerning nearly parallel ${\mathrm{G}}_2$-structures on the round sphere ${𝕊}^7$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated.

A new method for computation of eigenvalues of the radial Schrödinger operator $-d^2/d{x}^2+v\left(x\right),x\ge 0$ is presented. The potential $v\left(x\right)$ is assumed to behave as $x^{-2+ϵ}$ if $x\to 0_{+}$ and as $x^{-2-ϵ}$ if $x\to +\infty ,ϵ\ge 0$. The Schrödinger equation is transformed to a non-linear differential equation of the first order for a function $z(x,\aleph )$. It is shown that the eigenvalues are the discontinuity points of the function $z(\infty ,\aleph )$. Moreover, it is shown how to obtain an arbitrarily accurate approximation of eigenvalues. The method seems to be much more economical in comparison...

A new method to solve stationary one-dimensional Schroedinger equation is investigated. Solutions are described by means of representation of circles with multiple winding number. The results are demonstrated using the well-known analytical solutions of the Schroedinger equation.

We consider partial abelian monoids, in particular generalized effect algebras. From the given structures, we construct new ones by introducing a new operation $\oplus$, which is given by restriction of the original partial operation + with respect to a special subset called preideal. We bring some derived properties and characterizations of these new built structures, supporting the results by illustrative examples.