Homology of spaces of non-resultant polynomial systems in R^2 and C^2

The resultant veriety in the space of systems of homogeneous polynomials of given degrees consists of such systems having non-trivial solutions. We calculate the integer cohomology groups of all spaces of non-resultant systems of polynomials $R^2 \to R$, and also the rational cohomology groups of similar systems in $C^2$

Extreme TeV Blazars and Lower Limits on Intergalactic Magnetic Fields

The intergalactic magnetic field (IGMF) in cosmic voids can be indirectly probed through its effect on electromagnetic cascades initiated by a source of TeV gamma rays, such as blazars, a subclass of active galactic nuclei. Blazars that are sufficiently luminous at TeV energies, "extreme TeV blazars", can produce detectable levels of secondary radiation from inverse Compton scattering of the electrons in the cascade, provided that the IGMF is not too large. We reveiw recent work in the literature which utilizes this idea to derive constraints on the IGMF for three TeV-detected blazars-1ES 0229+200, 1ES 1218+304, and RGB J0710+591, and we also investigate four other hard-spectrum TeV blazars in the same framework. Through a recently developed detailed 3D particle tracking Monte Carlo simulation code, incorporating all major effects of QED and cosmological expansion, we research effects of major uncertainties such as the spectral properties of the source, uncertainty in the intensity of the UV - far IR extragalactic background light (EBL), under-sampled Very High Energy (VHE; energy > 100 GeV) coverage, past history of gamma-ray emission, source vs. observer geometry, and jet AGN Doppler factor. The implications of these effects on the recently reported lower limits of the IGMF are thoroughly examined to conclude that presently available data are compatible with a zero IGMF hypothesis.Comment: 2012 Fermi Symposium proceedings - eConf C12102

Simplified numerical form of universal finite type invariant of Gauss words

In the present paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degree 4, 5, and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.Comment: 12 pages, 3 table