Abstract:
We present a unifying framework which reduces the construction of probabilistic component analysis techniques to a mere selection of the latent neighbourhood, thus providing an elegant and principled framework for creating novel component analysis models as well as constructing probabilistic equivalents of deterministic component analysis methods. Under our framework, we unify many very popular and well-studied component analysis algorithms, such as Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), Locality Preserving Projections (LPP) and Slow Feature Analysis (SFA), some of which have no probabilistic equivalents in literature thus far. We firstly define the Markov Random Fields (MRFs) which encapsulate the latent connectivity of the aforementioned component analysis techniques; subsequently, we show that the projection directions produced by all PCA, LDA, LPP and SFA are also produced by the Maximum Likelihood (ML) solution of a single joint probability density function, composed by selecting one of the defined MRF priors while utilising a simple observation model. Furthermore, we propose novel Expectation Maximization (EM) algorithms, exploiting the proposed joint PDF, while we generalize the proposed methodologies to arbitrary connectivities via parameterizable MRF products. Theoretical analysis and experiments on both simulated and real world data show the usefulness of the proposed framework, by deriving methods which well outperform state-of-the-art equivalents.

Abstract:
The aim of this study was to examine the effect of a classic Spartathlon race, which means continuous running for 246 km, on the concentration of 2,3 dinor 6-keto-prostaglandin (2, 3 dinor 6-keto PGF1a), the urinary metabolite of the prostacyclin (PGI2), 2, 3-dinor-thromboxane B2 (2, 3 dinor TXB2), the urinary metabolite of thromboxane A2 (TXB2), the 2, 3 dinor 6-keto-PGF1a:2, 3 dinor-TXB2 ratio, Total Cholesterol (TC), Triglycerides (TG), High-Density Lipoprotein Cholesterol (HDL-C), Low-Density Lipoprotein-L (LDL-L) and the TC: HDL-C ratio. It was hypothesized that these parameters would be changed after the completion of the Sparthathlon. For this study proposes blood and urine samples were obtained from19 male athletes, all of which finished the Spartathlon race in <35 h, before, at the end of and 24 h after the race. After result analysis, the levels of all the substances measured were different at the end of the race compared with before the race and these altered levels remained 24 h after the race. Importantly, it was observed that metabolism of 2, 3 dinor-6 keto-PGF1a at the end of the race was fourfold than before the race (p<0.001) and the concentration of 2, 3 dinor TXB2 after the race was tenfold than before the race (p<0.001).

Abstract:
It is evident in contemporary urban studies that the interest in city marketing both as a practice within urban centre management and as an academic sub-discipline has accelerated. There remain, however, several issues that need clarification before an agreement can be reached as to the exact effects and potential of city marketing as a tool of economic and socio-cultural development. A particular gap can be noticed between theoretical suggestions on the ways in which marketing should be understood and used within cities and the practical implementation as this can be observed in contemporary cities. A common view on this issue highlights the need for practitioners to follow theoretical ideas but the practice can also be a source of useful lessons that might enrich the theory. This paper investigates marketing and branding practices of two European cities in order to extract from the practice lessons that will support the theoretical development of city marketing and city branding and might contribute towards bridging this gap. The cities investigated are Amsterdam and Budapest, both of which provide valuable insights into the challenges of an effective city marketing implementation.

Abstract:
This paper investigates the global properties of a class of spherically symmetric spacetimes. The class contains the maximal development of asymptotically flat spherically symmetric initial data for a wide variety of coupled Einstein-matter systems. For this class, it is proven here that the existence of a single trapped surface or marginally trapped surface implies the completeness of future null infinity and the formation of an event horizon whose area radius is bounded by twice the final Bondi mass.

Abstract:
Understanding the behaviour of linear waves on black hole backgrounds is a central problem in general relativity, intimately connected with the nonlinear stability of the black hole spacetimes themselves as solutions to the Einstein equations--a major open question in the subject. Nonetheless, it is only very recently that even the most basic boundedness and quantitative decay properties of linear waves have been proven in a suitably general class of black hole exterior spacetimes. This talk will review our current mathematical understanding of waves on black hole backgrounds, beginning with the classical boundedness theorem of Kay and Wald on exactly Schwarzschild exteriors and ending with very recent boundedness and decay theorems (proven in collaboration with Igor Rodnianski) on a wider class of spacetimes. This class of spacetimes includes in particular slowly rotating Kerr spacetimes, but in the case of the boundedness theorem is in fact much larger, encompassing general axisymmetric stationary spacetimes whose geometry is sufficiently close to Schwarzschild and whose Killing fields span the null generator of the horizon.

Abstract:
Two aspects of the widely accepted heuristic picture of the final state of gravitational collapse are the so-called Price law tails, describing the asymptotics of the exterior region of the black hole that forms, and Israel-Poisson's mass inflation scenario, describing the internal structure of the black hole. (The latter scenario, if valid, would indicate that the maximal development of initial data is extendible as a C^0 metric, putting into question the validity of Penrose's strong cosmic censorship conjecture.) In this talk, I shall discuss a series of rigorous results proving both Price's law and the mass inflation scenario in an appropriate spherically symmetric setting. The proof of Price's law is joint work with I. Rodnianski.

Abstract:
This paper explores ``black hole'' solutions of various Einstein-wave matter systems admitting an isometry of their domain of outer communications taking every point to its future. In the first two parts, it is shown that such solutions, assuming in addition that they are spherically symmetric and the matter has a certain structure, must be Schwarzschild or Reissner-Nordstrom. Non-trivial examples of matter for which the result applies are a wave map and a massive charged scalar field interacting with an electromagnetic field. The results thus generalize work of Bekenstein [1] and Heusler [12] from the static to the periodic case. In the third part, which is independent of the first two, it is shown that Dirac fields preserved by an isometry of a spherically symmetric domain of outer communications of the type described above must vanish. It can be applied in particular to the Einstein-Dirac-Maxwell equations or the Einstein-Dirac-Yang/Mills equations, generalizing work of Finster, Smoller, and Yau [9], [7], [8], and also [6].

Abstract:
We consider a spherically symmetric characteristic initial value problem for the Einstein-Maxwell-scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal domain of development has a future boundary, over which the spacetime is extendible as a continuous metric, but along which the Hawking mass blows up identically; thus, the spacetime is inextendible as a differentiable metric. In view of recent results of the author in collaboration with I. Rodnianski (gr-qc/0309115), which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the continuous extendibility result applies to the collapse of complete asymptotically flat spacelike data where the scalar field is compactly supported on the initial hypersurface. This shows that under Christodoulou's C^0 formulation, the strong cosmic censorship conjecture is false for this system.

Abstract:
This talk describes some recent results [16] regarding the problem of uniqueness in the large (also known as strong cosmic censorship) for the initial value problem in general relativity. In order to isolate the essential analytic features of the problem from the complicated setting of gravitational collapse in which it arises, some familiarity with conformal properties of certain celebrated special solutions of the theory of relativity will have to be developed. This talk is an attempt to present precisely these features to an audience of non-specialists, in a way which will hopefully fully motivate a certain characteristic initial value problem for the spherically-symmetric Einstein-Maxwell-Scalar Field system. The considerations outlined here leading to this particular initial value problem are well known in the physics relativity community, where the problem of uniqueness has been studied heuristically [1, 22] and numerically [2, 3]. In [16], the global behavior of solutions to this IVP, in particular, the issue of uniqueness, is mathematically completely understood. A statement of the relevant Theorems is included in Section 9. Only a sketch of the ideas of the proof is provided here, but the readers may refer to [16] for details.

Abstract:
An open problem in general relativity has been to construct an asymptotically flat solution to a reasonable Einstein-matter system containing a black hole in the future and yet past-causally geodesically complete, in particular, containing no white holes. We give such an example in this paper--in fact, a family of such examples, stable in a suitable sense--for the case of a self-gravitating scalar field.