Face tracking using a hyperbolic catadioptric omnidirectional system

In the first part of this paper, we present a brief review on catadioptric omnidirectional systems. The special case of the hyperbolic omnidirectional system is analysed in depth. The literature shows that a hyperboloidal mirror has two clear advantages over alternative geometries. Firstly, a hyperboloidal mirror has a single projection centre [1]. Secondly, the image resolution is uniformly distributed along the mirror’s radius [2]. In the second part of this paper we show empirical results for the detection and tracking of faces from the omnidirectional images using Viola-Jones method. Both panoramic and perspective projections, extracted from the omnidirectional image, were used for that purpose. The omnidirectional image size was 480x480 pixels, in greyscale. The tracking method used regions of interest (ROIs) set as the result of the detections of faces from a panoramic projection of the image. In order to avoid losing or duplicating detections, the panoramic projection was extended horizontally. Duplications were eliminated based on the ROIs established by previous detections. After a confirmed detection, faces were tracked from perspective projections (which are called virtual cameras), each one associated with a particular face. The zoom, pan and tilt of each virtual camera was determined by the ROIs previously computed on the panoramic image. The results show that, when using a careful combination of the two projections, good frame rates can be achieved in the task of tracking faces reliably

Equations of the reaction-diffusion type with a loop algebra structure

A system of equations of the reaction-diffusion type is studied in the framework of both the direct and the inverse prolongation structure. We find that this system allows an incomplete prolongation Lie algebra, which is used to find the spectral problem and a whole class of nonlinear field equations containing the original ones as a special case.Comment: 16 pages, LaTex. submitted to Inverse Problem

Large Firm Dynamics and the Business Cycle

Do large firm dynamics drive the business cycle? We answer this question by developing a quantitative theory of aggregate fluctuations caused by firm-level disturbances alone. We show that a standard heterogeneous firm dynamics setup already contains in it a theory of the business cycle, without appealing to aggregate shocks. We offer a complete analytical characterization of the law of motion of the aggregate state in this class of models – the firm size distribution – and show that the resulting closed form solutions for aggregate output and productivity dynamics display: (i) persistence, (ii) volatility and (iii) time-varying second moments. We explore the key role of moments of the firm size distribution – and, in particular, the role of large firm dynamics – in shaping aggregate fluctuations, theoretically, quantitatively and in the data

Non-equilibrated post freeze out distributions

We discuss freeze out on the hypersurface with time-like normal vector, trying to answer how realistic is to assume thermal post freeze out distributions for measured hadrons. Using simple kinetic models for gradual freeze out we are able to generate thermal post FO distribution, but only in highly simplified situation. In a more advanced model, taking into account rescattering and re-thermalization, the post FO distribution gets more complicated. The resulting particle distributions are in qualitative agreement with the experimentally measured pion spectra. Our study also shows that the obtained post FO distribution functions, although analytically very different from the Juttner distribution, do look pretty much like thermal distributions in some range of parameters.Comment: 14 pages, 2 figures, EPJ style, submitted to EPJ

Continuous approximation of binomial lattices

A systematic analysis of a continuous version of a binomial lattice, containing a real parameter $\gamma$ and covering the Toda field equation as $\gamma\to\infty$, is carried out in the framework of group theory. The symmetry algebra of the equation is derived. Reductions by one-dimensional and two-dimensional subalgebras of the symmetry algebra and their corresponding subgroups, yield notable field equations in lower dimensions whose solutions allow to find exact solutions to the original equation. Some reduced equations turn out to be related to potentials of physical interest, such as the Fermi-Pasta-Ulam and the Killingbeck potentials, and others. An instanton-like approximate solution is also obtained which reproduces the Eguchi-Hanson instanton configuration for $\gamma\to\infty$. Furthermore, the equation under consideration is extended to $(n+1)$--dimensions. A spherically symmetric form of this equation, studied by means of the symmetry approach, provides conformally invariant classes of field equations comprising remarkable special cases. One of these $(n=4)$ enables us to establish a connection with the Euclidean Yang-Mills equations, another appears in the context of Differential Geometry in relation to the socalled Yamabe problem. All the properties of the reduced equations are shared by the spherically symmetric generalized field equation.Comment: 30 pages, LaTeX, no figures. Submitted to Annals of Physic

Freeze out of the expanding system

The freeze out (FO) of the expanding systems, created in relativistic heavy ion collisions, is discussed. We start with kinetic FO model, which realizes complete physical FO in a layer of given thickness, and then combine our gradual FO equations with Bjorken type system expansion into a unified model. We shall see that the basic FO features, pointed out in the earlier works, are not smeared out by the expansion.Comment: 3 pages, 2 figure

A bound on 6D N=1 supergravities

We prove that there are only finitely many distinct semi-simple gauge groups and matter representations possible in consistent 6D chiral (1,0) supergravity theories with one tensor multiplet. The proof relies only on features of the low-energy theory; the consistency conditions we impose are that anomalies should be cancelled by the Green-Schwarz mechanism, and that the kinetic terms for all fields should be positive in some region of moduli space. This result does not apply to the case of the non-chiral (1,1) supergravities, which are not constrained by anomaly cancellation.Comment: 23 pages, no figures; two paragraphs added to the proof in Appendix A covering the SU(2) and SU(3) case, other minor correction

Picard group of hypersurfaces in toric 3-folds

We show that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties. This sufficient condition is always satisfied by generic K3 surfaces embedded in Fano toric 3-folds.Comment: 14 pages. v2: some typos corrected. v3: Slightly changed title. Final version to appear in Int. J. Math., incorporates many (mainly expository) changes suggested by the refere