The geometry of manifolds and the perception of space

This essay discusses the development of key geometric ideas in the 19th century which led to the formulation of the concept of an abstract manifold (which was not necessarily tied to an ambient Euclidean space) by Hermann Weyl in 1913. This notion of manifold and the geometric ideas which could be formulated and utilized in such a setting (measuring a distance between points, curvature and other geometric concepts) was an essential ingredient in Einstein's gravitational theory of space-time from 1916 and has played important roles in numerous other theories of nature ever since.Comment: arXiv admin note: substantial text overlap with arXiv:1301.064

Conformal Invariance in Einstein-Cartan-Weyl space

We consider conformally invariant form of the actions in Einstein, Weyl, Einstein-Cartan and Einstein-Cartan-Weyl space in general dimensions($>2$) and investigate the relations among them. In Weyl space, the observational consistency condition for the vector field determining non-metricity of the connection can be obtained from the equation of motion. In Einstein-Cartan space a similar role is played by the vector part of the torsion tensor. We consider the case where the trace part of the torsion is the Kalb-Ramond type of field. In this case, we express conformally invariant action in terms of two scalar fields of conformal weight -1, which can be cast into some interesting form. We discuss some applications of the result.Comment: 10 pages, version to appear MPL

Possible way out of the Hawking paradox: Erasing the information at the horizon

We show that small deviations from spherical symmetry, described by means of exact solutions to Einstein equations, provide a mechanism to "bleach" the information about the collapsing body as it falls through the aparent horizon, thereby resolving the information loss paradox. The resulting picture and its implication related to the Landauer's principle in the presence of a gravitational field, is discussed.Comment: 11 pages, Latex. Some comments added to answer to some raised questions. Typos corected. Final version, to appear in Int. J. Modern. Phys.

The biology of Tilapia rendalli and Oreochromis mossambicus (Pisces: Cichlidae) in a subtropical lake in Mozambique

This study of age and growth, reproduction and juvenile recruitment of Tilapia rendalli and Oreochromis mossambicus was undertaken in Lake Chicamba, a large, clear subtropical lake in central Mozambique. Sectioned otoliths were used to estimate age. Marginal zone analysis showed that annulus formation occurred during winter, in July and August. Maximum age estimates for T. rendalli and O. mossambicus were 16 and 10 years, respectively. Growth was described by the 3 parameter von Bertalanffy model as Lt = 238.74 (1-e-0,636(1 + 00905)) mm TL for T. rendalli and Lt = 266.06 (1-e-0 70904(1 + 0269)) mm TL for O. mossambicus. The length-at-50% maturity (Lm50) of male T. rendalli was 218 mm total lenglh (TL) and 205 mm TL in females. In O. mossambicus the Lm50 for males was 251 mm TL and for females 223 mm TL. Both species spawned throughout summer. Lake level was found to have no effect on spawning periodicity although recruitment was flood dependent

New results for the missing quantum numbers labeling the quadrupole and octupole boson basis

The many $2^k$-pole boson states, $|N_kv_k\alpha_k I_kM_k>$ with $k=2,3$, realize the irreducible representation (IR) for the group reduction chains $SU(2k+1)\supset R_{2k+1}\supset R_3\supset R_2$. They have been analytically studied and widely used for the description of nuclear systems. However, no analytical expression for the degeneracy $d_v(I)$ of the $R_{2k+1}$'s IR, determined by the reduction $R_{2k+1}\supset R_3$, is available. Thus, the number of distinct values taken by $\alpha_k$ has been so far obtained by solving some complex equations. Here we derive analytical expressions for the degeneracy $d_v(I)$ characterizing the octupole and quadrupole boson states, respectively. The merit of this work consists of the fact that it completes the analytical expressions for the $2^k$-pole boson basis.Comment: 10page

Variational formulation of Eisenhart's unified theory

Eisenhart's classical unified field theory is based on a non-Riemannian affine connection related to the covariant derivative of the electromagnetic field tensor. The sourceless field equations of this theory arise from vanishing of the torsion trace and the symmetrized Ricci tensor. We formulate Eisenhart's theory from the metric-affine variational principle. In this formulation, a Lagrange multiplier constraining the torsion becomes the source for the Maxwell equations.Comment: 7 pages; published versio

The inception of Symplectic Geometry: the works of Lagrange and Poisson during the years 1808-1810

The concept of a symplectic structure first appeared in the works of Lagrange on the so-called "method of variation of the constants". These works are presented, together with those of Poisson, who first defined the composition law called today the "Poisson bracket". The method of variation of the constants is presented using today's mathematical concepts and notations.Comment: Presented at the meeting "Poisson 2008" in Lausanne, July 2008. Published in Letters in Mathematical Physics. 22 page

Accelerated black holes in an anti-de Sitter universe

The C-metric is one of few known exact solutions of Einstein's field equations which describes the gravitational field of moving sources. For a vanishing or positive cosmological constant, the C-metric represents two accelerated black holes in asymptotically flat or de Sitter spacetime. For a negative cosmological constant the structure of the spacetime is more complicated. Depending on the value of the acceleration, it can represent one black hole or a sequence of pairs of accelerated black holes in the spacetime with an anti-de Sitter-like infinity. The global structure of this spacetime is analyzed and compared with an empty anti-de Sitter universe. It is illustrated by 3D conformal-like diagrams.Comment: 14 pages, 17 figures [see http://utf.mff.cuni.cz/~krtous/physics/CADS/ for the version with the high quality figures and for related animations and interactive 3D diagrams

Monomial integrals on the classical groups

This paper presents a powerfull method to integrate general monomials on the classical groups with respect to their invariant (Haar) measure. The method has first been applied to the orthogonal group in [J. Math. Phys. 43, 3342 (2002)], and is here used to obtain similar integration formulas for the unitary and the unitary symplectic group. The integration formulas turn out to be of similar form. They are all recursive, where the recursion parameter is the number of column (row) vectors from which the elements in the monomial are taken. This is an important difference to other integration methods. The integration formulas are easily implemented in a computer algebra environment, which allows to obtain analytical expressions very efficiently. Those expressions contain the matrix dimension as a free parameter.Comment: 16 page

Research in interactive scene analysis

An interactive scene interpretation system (ISIS) was developed as a tool for constructing and experimenting with man-machine and automatic scene analysis methods tailored for particular image domains. A recently developed region analysis subsystem based on the paradigm of Brice and Fennema is described. Using this subsystem a series of experiments was conducted to determine good criteria for initially partitioning a scene into atomic regions and for merging these regions into a final partition of the scene along object boundaries. Semantic (problem-dependent) knowledge is essential for complete, correct partitions of complex real-world scenes. An interactive approach to semantic scene segmentation was developed and demonstrated on both landscape and indoor scenes. This approach provides a reasonable methodology for segmenting scenes that cannot be processed completely automatically, and is a promising basis for a future automatic system. A program is described that can automatically generate strategies for finding specific objects in a scene based on manually designated pictorial examples