On the Conformal Geometry of Transverse Riemann-Lorentz Manifolds

Physical reasons suggested in \cite{Ha-Ha} for the \emph{Quantum Gravity Problem} lead us to study \emph{type-changing metrics} on a manifold. The most interesting cases are \emph{Transverse Riemann-Lorentz Manifolds}. Here we study the conformal geometry of such manifolds

Black holes, cosmological singularities and change of signature

There exists a widespread belief that signature type change could be used to avoid spacetime singularities. We show that signature change cannot be utilised to this end unless the Einstein equation is abandoned at the suface of signature type change. We also discuss how to solve the initial value problem and show to which extent smooth and discontinuous signature changing solutions are equivalent.Comment: 14pages, Latex, no figur

First order systems of odes with nonlinear nonlocal boundary conditions

In this article, we prove an existence of solutions for a non-local boundary value problem with nonlinearity in a nonlocal condition. Our method is based upon the Mawhin's coincidence theory

Teoretyczne aspekty modelowania przestrzennego w badaniach regionalnych

Spatial modeling is currently one of the primary research tools used in regional analysis. Spatial models are an extension of traditional econometric models, which are included in the so called spatial effects: spatial dependence and spatial heterogeneity. The article presents the theoretical basis of spatial modelling, together with definitions of basic concepts and an analysis of their properties. Methods for estimating spatial models and diagnostics are presented. The study also indicates the complexity of spatial modeling, and the usefulness of this kind research approach. In this paper an outline the development trends of spatial modeling is delivered.Modelowanie przestrzenne jest obecnie jednym z podstawowych narzędzi badawczych wykorzystywanych w analizie regionalnej. Modele przestrzenne są rozszerzeniem klasycznych modeli ekonometrycznych, do których włączane są tak zwane efekty przestrzenne: przestrzenna zależność i przestrzenna heterogeniczność. Artykuł prezentuje podstawy teoretyczne modelowania przestrzennego wraz z definicjami podstawowych pojęć oraz analizą ich własności. Przedstawione są również metody estymacji i diagnostyki modeli przestrzennych. W pracy wskazuje się też z jednej strony na złożoność modelowania przestrzennego, a z drugiej strony na użyteczność takiego podejścia badawczego. Zarysowane zostały także trendy rozwojowe modelowania przestrzennego

The Application Of Local Indicators For Categorical Data (LICD) In The Spatial Analysis Of Economic Development

Firstly, we identify classes of regions presenting different economic development levels using taxonomic methods of multivariate data analysis. Secondly, we apply a join-count test to examine spatial dependencies between regions. It examines the tendency to form the spatial clusters. The global test indicates general spatial interactions between regions, while local tests give detailed results separately for each region. The global test detects spatial clustering of economically poor regions but is statistically insignificant as regards well-developed regions. Thus, the local tests are also applied. They indicate the occurrence of five spatial clusters and three outliers in Poland. There are three clusters of wealth. Their development is based on a diffusion impact of regional economic centres. The areas of eastern and north western Poland include clusters of poverty. The first one is impeded by the presense of three indiviual growth centres, while the second one is out of range of diffusion influence of bigger agglomerations

Actions for signature change

This is a contribution on the controversy about junction conditions for classical signature change. The central issue in this debate is whether the extrinsic curvature on slices near the hypersurface of signature change has to be continuous ({\it weak} signature change) or to vanish ({\it strong} signature change). Led by a Lagrangian point of view, we write down eight candidate action functionals $S_1$,\dots $S_8$ as possible generalizations of general relativity and investigate to what extent each of these defines a sensible variational problem, and which junction condition is implied. Four of the actions involve an integration over the total manifold. A particular subtlety arises from the precise definition of the Einstein-Hilbert Lagrangian density $|g|^{1/2} R[g]$. The other four actions are constructed as sums of integrals over singe-signature domains. The result is that {\it both} types of junction conditions occur in different models, i.e. are based on different first principles, none of which can be claimed to represent the ''correct'' one, unless physical predictions are taken into account. From a point of view of naturality dictated by the variational formalism, {\it weak} signature change is slightly favoured over {\it strong} one, because it requires less {\it \`a priori} restrictions for the class of off-shell metrics. In addition, a proposal for the use of the Lagrangian framework in cosmology is made.Comment: 36 pages, LaTeX, no figures; some corrections have been made, several Comments and further references are included and a note has been added

On geometry of hypersurfaces of a pseudoconformal space of Lorentzian signature

There are three types of hypersurfaces in a pseudoconformal space C^n_1 of Lorentzian signature: spacelike, timelike, and lightlike. These three types of hypersurfaces are considered in parallel. Spacelike hypersurfaces are endowed with a proper conformal structure, and timelike hypersurfaces are endowed with a conformal structure of Lorentzian type. Geometry of these two types of hypersurfaces can be studied in a manner that is similar to that for hypersurfaces of a proper conformal space. Lightlike hypersurfaces are endowed with a degenerate conformal structure. This is the reason that their investigation has special features. It is proved that under the Darboux mapping such hypersurfaces are transferred into tangentially degenerate (n-1)-dimensional submanifolds of rank n-2 located on the Darboux hyperquadric. The isotropic congruences of the space C^n_1 that are closely connected with lightlike hypersurfaces and their Darboux mapping are also considered.Comment: LaTeX, 21 page

Initial Value Problems and Signature Change

We make a rigorous study of classical field equations on a 2-dimensional signature changing spacetime using the techniques of operator theory. Boundary conditions at the surface of signature change are determined by forming self-adjoint extensions of the Schr\"odinger Hamiltonian. We show that the initial value problem for the Klein--Gordon equation on this spacetime is ill-posed in the sense that its solutions are unstable. Furthermore, if the initial data is smooth and compactly supported away from the surface of signature change, the solution has divergent $L^2$-norm after finite time.Comment: 33 pages, LaTeX The introduction has been altered, and new work (relating our previous results to continuous signature change) has been include

Signature change from Schutz's canonical quantum cosmology and its classical analogue

We study the signature change in a perfect fluid Friedmann-Robertson-Walker quantum cosmological model. In this work the Schutz's variational formalism is applied to recover the notion of time. This gives rise to a Schrodinger-Wheeler-DeWitt equation with arbitrary ordering for the scale factor. We use the eigenfunctions in order to construct wave packets and evaluate the time-dependent expectation value of the scale factor which coincides with the ontological interpretation. We show that these solutions exhibit signature transitions from a finite Euclidean to a Lorentzian domain. Moreover, such models are equivalent to a classical system where, besides the perfect fluid, a repulsive fluid is present.Comment: 15 pages, 4 figures, to appear in PR

Comment on "Failure of standard conservation laws at a classical change of signature"

Hellaby & Dray (gr-qc/9404001) have recently claimed that matter conservation fails under a change of signature, compounding earlier claims that the standard junction conditions for signature change are unnecessary. In fact, if the field equations are satisfied, then the junction conditions and the conservation equations are satisfied. The failure is rather that the authors did not make sense of the field equations and conservation equations, which are singular at a change of signature.Comment: 3 pages, Te