International tourism in Małopolskie Województwo: the present situation and prospects for development

Małopolskie Województwo is one of the most popular tourist regions in Poland. Due to many attractions, long traditions of hosting tourists and well-developed accommodation facilities, Małopolskie Województwo has become the destination for a growing number of international tourists in recent years. Significant growth in incoming international tourism is found both in Kraków and in mountain areas. The objective of this paper is to present the state of development of incoming international tourism in Małopolskie Województwo

The natural brackets on couples of vector fields and 1-forms

All natural bilinear operators transforming pairs of couples of vector fields and 1-forms into couples of vector fields and 1-forms are found. All natural bilinear operators as above satisfying the Leibniz rule are extracted. All natural Lie algebra brackets on couples of vector fields and 1-forms are collected

Stan i perspektywy rozwoju turystyki międzynarodowej w województwie małopolskim

Województwo małopolskie od lat zalicza się do najpopularniejszych regionów turystycznych w Polsce. Ze względu na liczne walory turystyczne, bogate tradycje w goszczeniu turystów, a także stosunkowo dobrze rozwiniętą bazę turystyczną, województwo małopolskie staje się w ostatnich latach celem coraz liczniejszych przyjazdów turystów zagranicznych. Znaczącą dynamikę wzrostu zagranicznej turystyki przyjazdowej obserwuje się zarówno w samym Krakowie, jak i na obszarach górskich. Celem artykułu jest próba przedstawienia stanu rozwoju zagranicznej turystyki przyjazdowej do województwa małopolskiego

On lifting of 2-vector fields to rr-jet prolongation of the tangent bundle

If $m \geq 3$ and $r \geq 1$, we prove that any natural linear operator $A$ lifting 2-vector fields $\Lambda \in \Gamma (\bigwedge^2 TM)$ (i.e., skew-symmetric tensor fields of type (2,0)) on $m$-dimensional manifolds $M$ into 2-vector fields $A(\Lambda)$ on $r$-jet prolongation $J^rTM$ of the tangent bundle $TM$ of $M$ is the zero one

The twisted gauge-natural bilinear brackets on couples of linear vector fields and linear p-forms

We completely describe all gauge-natural operators $C$ which send linear $(p+2)$-forms $H$ on vector bundles $E$ (with sufficiently large dimensional bases) into $\mathbf{R}$-bilinear operators $C_H$ transforming pairs $(X_1\oplus\omega_1,X_2\oplus\omega_2)$ of couples of linear vector fields and linear $p$-forms on $E$ into couples $C_H(X_1\oplus\omega_1, X_2\oplus\omega_2)$ of linear vector fields and linear $p$-forms on $E$. Further, we extract all $C$ (as above) such that $C_0$ is the restriction of the well-known Courant bracket and $C_H$ satisfies the Jacobi identity in Leibniz form for all closed linear $(p+2)$-forms $H$

On the existence of connections with a prescribed skew-symmetic Ricci tensor

We study the so-called inverse problem. Namely, given a prescribed skew-symmetric Ricci tensor we find (locally) a respective linear connection

The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

If $(M,g)$ is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism $TM\mathrel{\tilde=}T^*M$ given by $v\to g(v,-)$ between the tangent $TM$ and the cotangent $T^*M$ bundles of $M$. In the present note, we generalize this isomorphism to the one $T^{(r)}M\mathrel{\tilde=} T^{r*}M$ between the $r$-th order vector tangent $T^{(r)}M=(J^r(M,R)_0)^*$ and the $r$-th order cotangent $T^{r*}M=J^r(M,R)_0$ bundles of $M$. Next, we describe all base preserving  vector bundle maps $C_M(g):T^{(r)}M\to T^{r*}M$ depending on a Riemannian metric $g$ in terms of natural (in $g$) tensor fields on $M$

Lifting vector fields from manifolds to the rr-jet prolongation of the tangent bundle

If m≥3 and r≥0, we deduce that any natural linear operator lifting vector fields from an m-manifold M to the r-jet prolongation JrTM of the tangent bundle TM is the composition of the flow lifting Jr corresponding to the r-jet prolongation functor Jr with a natural linear operator lifting vector fields from M to TM. If 0≤s≤r and m≥3, we find all natural linear operators transforming vector fields on M into base-preserving fibred maps JrTM→JsTM