General relativity histories theory II: Invariance groups

We show in detail how the histories description of general relativity carries representations of both the spacetime diffeomorphisms group and the Dirac algebra of constraints. We show that the introduction of metric-dependent equivariant foliations leads to the crucial result that the canonical constraints are invariant under the action of spacetime diffeomorphisms. Furthermore, there exists a representation of the group of generalised spacetime mappings that are functionals of the four-metric: this is a spacetime analogue of the group originally defined by Bergmann and Komar in the context of the canonical formulation of general relativity. Finally, we discuss the possible directions for the quantization of gravity in histories theory.Comment: 24 pages, submitted to Class. Quant. Gra

The local hh-vector of the cluster subdivision of a simplex

The cluster complex $\Delta (\Phi)$ is an abstract simplicial complex, introduced by Fomin and Zelevinsky for a finite root system $\Phi$. The positive part of $\Delta (\Phi)$ naturally defines a simplicial subdivision of the simplex on the vertex set of simple roots of $\Phi$. The local $h$-vector of this subdivision, in the sense of Stanley, is computed and the corresponding $\gamma$-vector is shown to be nonnegative. Combinatorial interpretations to the entries of the local $h$-vector and the corresponding $\gamma$-vector are provided for the classical root systems, in terms of noncrossing partitions of types $A$ and $B$. An analogous result is given for the barycentric subdivision of a simplex.Comment: 21 pages, 4 figure

A symmetric unimodal decomposition of the derangement polynomial of type BB

The derangement polynomial $d_n (x)$ for the symmetric group enumerates derangements by the number of excedances. The derangement polynomial $d^B_n(x)$ for the hyperoctahedral group is a natural type $B$ analogue. A new combinatorial formula for this polynomial is given in this paper. This formula implies that $d^B_n (x)$ decomposes as a sum of two nonnegative, symmetric and unimodal polynomials whose centers of symmetry differ by a half and thus provides a new transparent proof of its unimodality. A geometric interpretation, analogous to Stanley's interpretation of $d_n (x)$ as the local $h$-polynomial of the barycentric subdivision of the simplex, is given to one of the summands of this decomposition. This interpretation leads to a unimodal decomposition and a new formula for the Eulerian polynomial of type $B$. The various decomposing polynomials introduced here are also studied in terms of recurrences, generating functions, combinatorial interpretations, expansions and real-rootedness.Comment: Changes in Remark 7.3 and the bibliograph

General relativity histories theory I: The spacetime character of the canonical description

The problem of time in canonical quantum gravity is related to the fact that the canonical description is based on the prior choice of a spacelike foliation, hence making a reference to a spacetime metric. However, the metric is expected to be a dynamical, fluctuating quantity in quantum gravity. We show how this problem can be solved in the histories formulation of general relativity. We implement the 3+1 decomposition using metric-dependent foliations which remain spacelike with respect to all possible Lorentzian metrics. This allows us to find an explicit relation of covariant and canonical quantities which preserves the spacetime character of the canonical description. In this new construction, we also have a coexistence of the spacetime diffeomorphisms group, and the Dirac algebra of constraints.Comment: 23 pages, submitted to Class. Quant. Gra

The cold frontal depression that affected the area of Cyprus between 28 and 29 January 2008

The baroclinic depression that affected the area of Cyprus during the cold period, between 28 and 29 January 2008 was thoroughly studied and is presented in the present paper. A small perturbation on a northwesterly flow to the north of Cyprus has initiated the generation of the depression and in 24 h this developed into a deep baroclinic system. This depression was associated with intense weather phenomena, such as heavy thunderstorms with hail and near gale force winds. Strong cold advection resulted in a significant temperature decrease; precipitation even in lower altitudes was in the form of snow, while the accumulated rainfall corresponded to the 25% of the monthly normal. January 2008 is considered as a dry month, despite the fact that, on the average, January is considered as the wettest month of the year. In this study, the evolution and development of the depression was investigated from synoptic, dynamic, energetic and thermodynamic perspectives, in order to enhance our knowledge on the life cycle and behaviour of similar depressions over the area with extreme characteristics

A statistical analysis of sounding derived indices and parameters for extreme and non-extreme thunderstorm events over Cyprus

The main purpose of this study is to provide a simple statistical analysis of several stability indices and parameters for extreme and non-extreme thunderstorm events during the period 1997 to 2001 in Cyprus. For this study, radiosonde data from Athalassa station (35°1´ N, 33°4´ E) were analyzed during the aforementioned period. The stability indices and parameters set under study are the K index, the Total Totals (TT) index, the Convective Available Potential Energy related parameters such as Convective Available Potential Energy (CAPE), Downdraft CAPE (DCAPE) and the Convective Inhibition (CIN), the Vorticity Generator Parameter (VGP), the Bulk Richardson Number (BRN), the BRN Shear and the Storm Relative Helicity (SRH). An event is categorized as extreme, if primarily, CAPE was non zero and secondary, if values of both the K and the TotalTotals (TT) indices exceeded 26.9 and 50, respectively. The cases with positive CAPE but lower values of the other indices, were identified as non-extreme. By calculating the median, the lower and upper limits, as well as the lower and upper quartiles of the values of these indices, the main characteristics of their distribution were determined

Synoptic, thermodynamic and agroeconomic aspects of severe hail events in Cyprus

Hail is a hazardous weather element often accompanying a thunderstorm, as a result of either thermal instability or instability associated with baroclinic synoptic-scale systems (i.e. frontal depressions). Nevertheless, instability of any kind and thunderstorm activity does not always lead to the formation of hail of adequate size to reach the ground. The broader the knowledge concerning hail events the better the understanding of the underlying thermodynamic and dynamic mechanisms, as well as the physical processes associated with its formation. <br><br> In the present study, the severe hail events that were recorded in Cyprus during the ten-year period from 1996 until 2005 were examined, first by grouping them into two clusters, namely, the "thermal instability cluster" and the "frontal depression cluster". Subsequently, the spatial and temporal evolution of the synoptic, dynamic and thermodynamic characteristics of these hail events was studied in depth. Also, the impact of hailstorms on the local economy of the island is presented in terms of the compensations paid by the Agricultural Insurance Organization of the country

Canonical Lagrangian Dynamics and General Relativity

Building towards a more covariant approach to canonical classical and quantum gravity we outline an approach to constrained dynamics that de-emphasizes the role of the Hamiltonian phase space and highlights the role of the Lagrangian phase space. We identify a "Lagrangian one-form" to replace the standard symplectic one-form, which we use to construct the canonical constraints and an associated constraint algebra. The method is particularly useful for generally covariant systems and systems with a degenerate canonical symplectic form, such as Einstein Cartan gravity, to which we apply the method explicitly. We find that one can demonstrate the closure of the constraints without gauge fixing the Lorentz group or introducing primary constraints on the phase space variables. Finally, using geometric quantization techniques, we briefly discuss implications of the formalism for the quantum theory.Comment: Version published in Classical and Quantum Gravity. Significant content and references adde

Preliminary verification results of the DWD limited area model LME and evaluation of its storm forecasting skill over the area of Cyprus

A preliminary verification and evaluation is made of the forecast fields of the non-hydrostatic limited area model LME of the German Weather Service (DWD), for a recent three month period. For this purpose, observations from two synoptic stations in Cyprus are utilized. In addition, days with depressions over the area were selected in order to evaluate the model's forecast skill in storm forecasting

Centrifuge modelling of contaminant transport processes

Over the past decade, research workers have started to investigate problems of subsurface contaminant transport through physical modelling on a geotechnical centrifuge. A major advantage of this apparatus is its ability to model complex natural systems in a controlled laboratory environment In this paper, we discusses the principles and scaling laws related to the centrifugal modelling of contaminant transport, and presents four examples of recent work that has been carried out in this area. The first two of these examples illustrate the use of centrifugal techniques to investigate contaminant transport mechanisms in geologic formations, while the latter two illustrate the use of the centrifuge as a tool for investigating site remediation strategies. The scope of this work serves to demonstrate the contribution that centrifuge modelling techniques can make in the areas of environmental engineering and contaminant hydrology