The Graph Isomorphism Problem and approximate categories

It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many pairs of graphs, but the pairs of non-isomorphic graphs constructed by Cai, Furer and Immerman it cannot distinguish. If d is fixed, then the WL-algorithm runs in polynomial time. We will formulate the Graph Isomorphism Problem as an Orbit Problem: Given a representation V of an algebraic group G and two elements v_1,v_2 in V, decide whether v_1 and v_2 lie in the same G-orbit. Then we attack the Orbit Problem by constructing certain approximate categories C_d(V), d=1,2,3,... whose objects include the elements of V. We show that v_1 and v_2 are not in the same orbit by showing that they are not isomorphic in the category C_d(V) for some d. For every d this gives us an algorithm for isomorphism testing. We will show that the WL-algorithms reduce to our algorithms, but that our algorithms cannot be reduced to the WL-algorithms. Unlike the Weisfeiler-Lehman algorithm, our algorithm can distinguish the Cai-Furer-Immerman graphs in polynomial time.Comment: 29 page

Signal Constellations for Multilevel Coded Modulation with Sparse Graph Codes

A method to combine error-correction coding and spectral efficient modulation for transmission over channels with Gaussian noise is presented. The method of modulation leads to a signal constellation in which the constellation symbols have a nonuniform distribution. This gives a so-called shape gain which can be as high as e 6 (1:5 dB). A sparse graph code is constructed which is based on a LDPC code and includes the method of modulation. An efficient decoding algorithm can be derived for this sparse graph code. Simulation results show that the performance of the code is quite good compared\ud to other coded modulation schemes proposed in literature

The relationship between German banks and large German firms

German banks are often criticized, or praised, depending on a person's viewpoint, for owning German industry and for playing an active part in corporate control. The author argues that this misrepresents German banking. First, the number of German firms a bank can own or control, although significant, is limited. Second, although most of the largest 100 firms have a bank member on their supervisory board, this does not imply effective bank control. Third, the role of the banker in the supervisory board has to be viewed in the light of the rigorous standards of corporate governance imposed on German public firms. fourth, bank ownership of industry is not pervasive, but is in fact limited to a few special cases. Fifth and last, proxy voting is more important than stock ownership as a potential means of control. The author argues that the German system of corporate governance represents an efficient attempt to minimize socially wasteful behavior. The negotiated consensus achieved in the board room provides better incentives to management to maximize firm value and social welfare than the factionalized U.S. system.Banks&Banking Reform,Financial Intermediation,Financial Crisis Management&Restructuring,Municipal Financial Management,Microfinance

A Compact High Energy Camera (CHEC) for the Gamma-ray Cherenkov Telescope of the Cherenkov Telescope Array

The Gamma-ray Cherenkov Telescope (GCT) is one of the Small Size Telescopes (SSTs) proposed for the Cherenkov Telescope Array (CTA) aimed at the 1 TeV to 300 TeV energy range. GCT will be equipped with a Compact High-Energy Camera (CHEC) containing 2048 pixels of physical size about 6$\times$6~mm$^2$, leading to a field of view of over 8 degrees. Electronics based on custom TARGET ASICs and FPGAs sample incoming signals at a gigasample per second and provide a flexible triggering scheme. Waveforms for every pixel in every event are read out are on demand without loss at over 600 events per second. A GCT prototype in Meudon, Paris saw first Cherenkov light from air showers in late 2015, using the first CHEC prototype, CHEC-M. This contribution presents results from lab and field tests with CHEC-M and the progress made to a robust camera design for deployment within CTA.Comment: All CTA contributions at arXiv:1709.0348