Finding an ordinary conic and an ordinary hyperplane

Given a finite set of non-collinear points in the plane, there exists a line that passes through exactly two points. Such a line is called an ordinary line. An efficient algorithm for computing such a line was proposed by Mukhopadhyay et al. In this note we extend this result in two directions. We first show how to use this algorithm to compute an ordinary conic, that is, a conic passing through exactly five points, assuming that all the points do not lie on the same conic. Both our proofs of existence and the consequent algorithms are simpler than previous ones. We next show how to compute an ordinary hyperplane in three and higher dimensions.Comment: 7 pages, 2 figure

Zoology of condensed matter: Framids, ordinary stuff, extra-ordinary stuff

We classify condensed matter systems in terms of the spacetime symmetries they spontaneously break. In particular, we characterize condensed matter itself as any state in a Poincar\'e-invariant theory that spontaneously breaks Lorentz boosts while preserving at large distances some form of spatial translations, time-translations, and possibly spatial rotations. Surprisingly, the simplest, most minimal system achieving this symmetry breaking pattern---the "framid"---does not seem to be realized in Nature. Instead, Nature usually adopts a more cumbersome strategy: that of introducing internal translational symmetries---and possibly rotational ones---and of spontaneously breaking them along with their space-time counterparts, while preserving unbroken diagonal subgroups. This symmetry breaking pattern describes the infrared dynamics of ordinary solids, fluids, superfluids, and---if they exist---supersolids. A third, "extra-ordinary", possibility involves replacing these internal symmetries with other symmetries that do not commute with the Poincar\'e group, for instance the galileon symmetry, supersymmetry or gauge symmetries. Among these options, we pick the systems based on the galileon symmetry, the "galileids", for a more detailed study. Despite some similarity, all different patterns produce truly distinct physical systems with different observable properties. For instance, the low-energy $2\to 2$ scattering amplitudes for the Goldstone excitations in the cases of framids, solids and galileids scale respectively as $E^2$, $E^4$, and $E^6$. Similarly the energy momentum tensor in the ground state is "trivial" for framids ($\rho +p=0$), normal for solids ($\rho+p>0$) and even inhomogenous for galileids.Comment: 58 pages, 1 table, 1 free cut-and-paste project for rainy days in Appendi

Ordinary Morality Implies Atheism

I present a "moral argument" for the nonexistence of God. Theism, I argue, can’t accommodate an ordinary and fundamental moral obligation acknowledged by many people, including many theists. My argument turns on a principle that a number of philosophers already accept as a constraint on God’s treatment of human beings. I defend the principle against objections from those inclined to reject i

(Extra)Ordinary Gauge Mediation

We study models of "(extra)ordinary gauge mediation," which consist of taking ordinary gauge mediation and extending the messenger superpotential to include all renormalizable couplings consistent with SM gauge invariance and an R-symmetry. We classify all such models and find that their phenomenology can differ significantly from that of ordinary gauge mediation. Some highlights include: arbitrary modifications of the squark/slepton mass relations, small mu and Higgsino NLSP's, and the possibility of having fewer than one effective messenger. We also show how these models lead naturally to extremely simple examples of direct gauge mediation, where SUSY and R-symmetry breaking occur not in a hidden sector, but due to the dynamics of the messenger sector itself.Comment: 50 pages, 11 figure

Ordinary and Extraordinary Hadrons

Resonances and enhancements in meson-meson scattering can be divided into two classes distinguished by their behavior as the number of colors N_c in QCD becomes large: The first are ordinary mesons that become stable as N_c goes to infinity. This class includes textbook q-bar q mesons as well as glueballs and hybrids. The second class, extraordinary mesons, are enhancements that disappear as N_c goes to infinity; they subside into the hadronic continuum. This class includes indistinct and controversial objects that have been classified as q-bar q-bar q q mesons or meson-meson molecules. Pelaez's study of the N_c dependence of unitarized chiral dynamics illustrates both classes: the p-wave pi-pi and K-pi resonances, the rho(770) and K*(892), behave as ordinary mesons; the s-wave pi-pi and K-pi enhancements, the sigma(600) and kappa(800), behave like extraordinary mesons. Ordinary mesons resemble Feshbach resonances while extraordinary mesons look more like effects due to potentials in meson-meson scattering channels. I build and explore toy models along these lines. Finally I discuss some related dynamical issues affecting the interpretation of extraordinary mesons.Comment: 18 pages, 10 figures, talk presented at the 2006 Yukawa International Seminar: New Frontiers in QCD, Kyoto University, November 2006. This talk is dedicated to the memory of R. H. Dalit

Zeno dynamics yields ordinary constraints

The dynamics of a quantum system undergoing frequent measurements (quantum Zeno effect) is investigated. Using asymptotic analysis, the system is found to evolve unitarily in a proper subspace of the total Hilbert space. For spatial projections, the generator of the "Zeno dynamics" is the Hamiltonian with Dirichlet boundary conditions.Comment: 6 page