Broken Triangles Revisited

International audienceA broken triangle is a pattern of (in)compatibilities between assignments in a binary CSP (constraint satisfaction problem). In the absence of certain broken triangles, satisfiability-preserving domain reductions are possible via merging of domain values. We investigate the possibility of maximising the number of domain reduction operations by the choice of the order in which they are applied, as well as their interaction with arc consistency operations. It turns out that it is NP-hard to choose the best order

One-zone SSC model for the core emission of Centaurus A revisited

Aims: We investigate the role of the second synchrotron self-Compton (SSC) photon generation to the multiwavelength emission from the compact regions of sources that are characterized as misaligned blazars. For this, we focus on the nearest high-energy emitting radio galaxy Centaurus A and we revisit the one-zone SSC model for its core emission. Methods: We have calculated analytically the peak luminosities of the first and second SSC components by, first, deriving the steady-state electron distribution in the presence of synchrotron and SSC cooling and, then, by using appropriate expressions for the positions of the spectral peaks. We have also tested our analytical results against those derived from a numerical code where the full emissivities and cross-sections were used. Results: We show that the one-zone SSC model cannot account for the core emission of Centaurus A above a few GeV, where the peak of the second SSC component appears. We, thus, propose an alternative explanation for the origin of the high energy ($\gtrsim 0.4$ GeV) and TeV emission, where these are attributed to the radiation emitted by a relativistic proton component through photohadronic interactions with the photons produced by the primary leptonic component. We show that the required proton luminosities are not extremely high, e.g. $\sim 10^{43}$ erg/s, provided that the injection spectra are modelled by a power-law with a high value of the lower energy cutoff. Finally, we find that the contribution of the core emitting region of Cen A to the observed neutrino and ultra-high energy cosmic-ray fluxes is negligible.Comment: 12 pages, 6 figures, 3 tables, accepted for publication in A&

The 4d one component lattice ϕ4\phi^4 model in the broken phase revisited

Measurements of various physical quantities in the symmetry broken phase of the one component lattice $\phi^4_4$ with standard action, are shown to be consistent with the critical behavior obtained by renormalization group analyses. This is in contrast to recent conclusions by another group, who further claim that the unconventional scaling behavior they observe, when extended to the complete Higgs sector of the Standard Model, would alter the conventional triviality bound on the mass of the Higgs.Comment: 15 pages, 3 figure

Alexandrov geometry: preliminary version no. 1

This is a preliminary version of our book. It goes up to the definition of dimension, which is about 30% of the material we plan to include. If you use it as a reference, do not forget to include the version number since the numbering will be changed.Comment: 238 pages, 35 figure

Valence-bond crystal in the extended kagome spin-1/2 quantum Heisenberg antiferromagnet: A variational Monte Carlo approach

The highly-frustrated spin-1/2 quantum Heisenberg model with both nearest ($J_1$) and next-nearest ($J_2$) neighbor exchange interactions is revisited by using an extended variational space of projected wave functions that are optimized with state-of-the-art methods. Competition between modulated valence-bond crystals (VBCs) proposed in the literature and the Dirac spin liquid (DSL) is investigated. We find that the addition of a {\it small} ferromagnetic next-nearest-neighbor exchange coupling $|J_2|>0.09 J_1$ leads to stabilization of a 36-site unit cell VBC, although the DSL remains a local minimum of the variational parameter landscape. This implies that the VBC is not trivially connected to the DSL: instead it possesses a non-trivial flux pattern and large dimerization.Comment: 5 pages, 4 figure

Random Triangle Theory with Geometry and Applications

What is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis, random matrix theory, the Hopf fibration, and much much more. One of the best distributions of random triangles takes all six vertex coordinates as independent standard Gaussians. Six can be reduced to four by translation of the center to $(0,0)$ or reformulation as a 2x2 matrix problem. In this note, we develop shape theory in its historical context for a wide audience. We hope to encourage other to look again (and differently) at triangles. We provide a new constructive proof, using the geometry of parallelians, of a central result of shape theory: Triangle shapes naturally fall on a hemisphere. We give several proofs of the key random result: that triangles are uniformly distributed when the normal distribution is transferred to the hemisphere. A new proof connects to the distribution of random condition numbers. Generalizing to higher dimensions, we obtain the "square root ellipticity statistic" of random matrix theory. Another proof connects the Hopf map to the SVD of 2 by 2 matrices. A new theorem describes three similar triangles hidden in the hemisphere. Many triangle properties are reformulated as matrix theorems, providing insight to both. This paper argues for a shift of viewpoint to the modern approaches of random matrix theory. As one example, we propose that the smallest singular value is an effective test for uniformity. New software is developed and applications are proposed

Generalised Moore spectra in a triangulated category

In this paper we consider a construction in an arbitrary triangulated category T which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of T satisfying some finite tilting assumptions, we obtain a functor which "approximates" objects of the module category of the endomorphism algebra of C in T. This generalises and extends a construction of Jorgensen in connection with lifts of certain homological functors of derived categories. We show that this new functor is well-behaved with respect to short exact sequences and distinguished triangles, and as a consequence we obtain a new way of embedding the module category in a triangulated category. As an example of the theory, we recover Keller's canonical embedding of the module category of a path algebra of a quiver with no oriented cycles into its u-cluster category for u>1.Comment: 26 pages, improvement to exposition of the proof of Theorem 3.