Non-imprisonment conditions on spacetime

The non-imprisonment conditions on spacetimes are studied. It is proved that the non-partial imprisonment property implies the distinction property. Moreover, it is proved that feeble distinction, a property which stays between weak distinction and causality, implies non-total imprisonment. As a result the non-imprisonment conditions can be included in the causal ladder of spacetimes. Finally, totally imprisoned causal curves are studied in detail, and results concerning the existence and properties of minimal invariant sets are obtained.Comment: 12 pages, 2 figures. v2: improved results on totally imprisoned curves, a figure changed, some misprints fixe

Isometry groups among topological groups

It is shown that a topological group G is topologically isomorphic to the isometry group of a (complete) metric space iff G coincides with its G-delta-closure in the Rajkov completion of G (resp. if G is Rajkov-complete). It is also shown that for every Polish (resp. compact Polish; locally compact Polish) group G there is a complete (resp. proper) metric d on X inducing the topology of X such that G is isomorphic to Iso(X,d) where X = l_2 (resp. X = Q; X = Q\{point} where Q is the Hilbert cube). It is demonstrated that there are a separable Banach space E and a nonzero vector e in E such that G is isomorphic to the group of all (linear) isometries of E which leave the point e fixed. Similar results are proved for an arbitrary complete topological group.Comment: 30 page

Dimension of graphoids of rational vector-functions

Let $F$ be a countable family of rational functions of two variables with real coefficients. Each rational function $f\in F$ can be thought as a continuous function $f:dom(f)\to\bar R$ taking values in the projective line $\bar R=R\cup\{\infty\}$ and defined on a cofinite subset $dom(f)$ of the torus $\bar R^2$. Then the family \F determines a continuous vector-function $F:dom(F)\to\bar R^F$ defined on the dense $G_\delta$-set $dom(F)=\bigcap_{f\in F}dom(F)$ of $\bar R^2$. The closure $\bar\Gamma(F)$ of its graph $\Gamma(F)=\{(x,f(x)):x\in dom(F)\}$ in $\bar R^2\times\bar R^F$ is called the {\em graphoid} of the family $F$. We prove the graphoid $\bar\Gamma(F)$ has topological dimension $dim(\bar\Gamma(F))=2$. If the family $F$ contains all linear fractional transformations $f(x,y)=\frac{x-a}{y-b}$ for $(a,b)\in Q^2$, then the graphoid $\bar\Gamma(F)$ has cohomological dimension $dim_G(\bar\Gamma(F))=1$ for any non-trivial 2-divisible abelian group $G$. Hence the space $\bar\Gamma(F)$ is a natural example of a compact space that is not dimensionally full-valued and by this property resembles the famous Pontryagin surface.Comment: 20 page

Wokół pierwszej polskiej systematyki zamówień. O zapomnianej propozycji Józefa Obrębskiego

The author discusses the first Polish systematics of magic spells against illness, which was proposed by Józef Obrębski in a brochure Index for „Treating the Polish people” by Henryk Biegeleisen in 1931. Obrębski introduced a division of spells into 9 groups, separated on the basis  of their  content,  form and  function.  It  was  not  an  autonomous  development.  It  have been created for the needs of someone else’s work and on the margin of a broader classification of material in the field of folk medicine as a whole. However, the proposal of Obrębski  can  be  regarded  as  a  serious  outline covering  all  material  concerning  the  systematics  of spells against illness. In the following parts of the article the author sets the discussed work in  a  due  context,  locating  it  in  the  history  of Polish research  on  the  genre  of  spells.  She recalls the first nineteenth-century works and analyses a possible impact of the Obrębski’s systematics on the subsequent approaches to the subject.The author discusses the first Polish systematics of magic spells against illness, which was proposed by Józef Obrębski in a brochure Index for „Treating the Polish people” by Henryk Biegeleisen in 1931. Obrębski introduced a division of spells into 9 groups, separated on the basis  of their  content,  form and  function.  It  was  not  an  autonomous  development.  It  have been created for the needs of someone else’s work and on the margin of a broader classification of material in the field of folk medicine as a whole. However, the proposal of Obrębski  can  be  regarded  as  a  serious  outline covering  all  material  concerning  the  systematics  of spells against illness. In the following parts of the article the author sets the discussed work in  a  due  context,  locating  it  in  the  history  of Polish research  on  the  genre  of  spells.  She recalls the first nineteenth-century works and analyses a possible impact of the Obrębski’s systematics on the subsequent approaches to the subject

Cohomology of the space of commuting n-tuples in a compact Lie group

Consider the space Hom(Z^n,G) of pairwise commuting n-tuples of elements in a compact Lie group G. This forms a real algebraic variety, which is generally singular. In this paper, we construct a desingularization of the generic component of Hom(Z^n,G), which allows us to derive formulas for its ordinary and equivariant cohomology in terms of the Lie algebra of a maximal torus in G and the action of the Weyl group. This is an application of a general theorem concerning G-spaces for which every element is fixed by a maximal torus.Comment: 11 pages Changes made: Implemented referee recommendations, in particular to use the Vietoris mapping theorem to generalize results and simplify argument

The structure of a the C*-algebra of a locally injective surjection

We obtain a description of the C*-algebras which can occur as a simple quotient of the C*-algebra of a locally injective surjection on a compact metric space of finite covering dimension

Characterization of some causality conditions through the continuity of the Lorentzian distance

A classical result in Lorentzian geometry states that a strongly causal spacetime is globally hyperbolic if and only if the Lorentzian distance is finite valued for every metric choice in the conformal class. It is proven here that a non-total imprisoning spacetime is globally hyperbolic if and only if for every metric choice in the conformal class the Lorentzian distance is continuous. Moreover, it is proven that a non-total imprisoning spacetime is causally simple if and only if for every metric choice in the conformal class the Lorentzian distance is continuous wherever it vanishes. Finally, a strongly causal spacetime is causally continuous if and only if there is at least one metric in the conformal class such that the Lorentzian distance is continuous wherever it vanishes.Comment: 14 pages, 2 figure. v2: Added material on global hyperbolicity. The title has changed. Previous title: Characterization of causal simplicity and causal continuity through the continuity of the Lorentzian distance. v3: Some misprints fixed. Final versio

Faithful compact quantum group actions on connected compact metrizable spaces

We construct faithful actions of quantum permutation groups on connected compact metrizable spaces. This disproves a conjecture of Goswami.Comment: 8 page

On the Chabauty space of locally compact abelian groups

This paper contains several results about the Chabauty space of a general locally compact abelian group. Notably, we determine its topological dimension, we characterize when it is totally disconnected or connected; we characterize isolated points.Comment: 24 pages, 0 figur

Statistical evaporation of rotating clusters. IV. Alignment effects in the dissociation of nonspherical clusters

Unimolecular evaporation in rotating, non-spherical atomic clusters is investigated using Phase Space Theory in its orbiting transition state version. The distributions of the total kinetic energy release epsilon_tr and the rotational angular momentum J_r are calculated for oblate top and prolate top main products with an arbitrary degree of deformation. The orientation of the angular momentum of the product cluster with respect to the cluster symmetry axis has also been obtained. This statistical approach is tested in the case of the small 8-atom Lennard-Jones cluster, for which comparison with extensive molecular dynamics simulations is presented. The role of the cluster shape has been systematically studied for larger, model clusters in the harmonic approximation for the vibrational densities of states. We find that the type of deformation (prolate vs. oblate) plays little role on the distributions and averages of epsilon_tr and J_r except at low initial angular momentum. However, alignment effects between the product angular momentum and the symmetry axis are found to be significant, and maximum at some degree of oblateness. The effects of deformation on the rotational cooling and heating effects are also illustrated.Comment: 15 pages, 9 figure