Geometry from Information Geometry

We use the method of maximum entropy to model physical space as a curved statistical manifold. It is then natural to use information geometry to explain the geometry of space. We find that the resultant information metric does not describe the full geometry of space but only its conformal geometry -- the geometry up to local changes of scale. Remarkably, this is precisely what is needed to model "physical" space in general relativity.Comment: Presented at MaxEnt 2015, the 35th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (July 19-24, 2015, Potsdam NY, USA

Inclusive V0V^0 Production Cross Sections from 920 GeV Fixed Target Proton-Nucleus Collisions

Inclusive differential cross sections $d\sigma_{pA}/dx_F$ and $d\sigma_{pA}/dp_t^2$ for the production of \kzeros, \lambdazero, and \antilambda particles are measured at HERA in proton-induced reactions on C, Al, Ti, and W targets. The incident beam energy is 920 GeV, corresponding to $\sqrt {s} = 41.6$ GeV in the proton-nucleon system. The ratios of differential cross sections \rklpa and \rllpa are measured to be $6.2\pm 0.5$ and $0.66\pm 0.07$, respectively, for \xf $\approx-0.06$. No significant dependence upon the target material is observed. Within errors, the slopes of the transverse momentum distributions $d\sigma_{pA}/dp_t^2$ also show no significant dependence upon the target material. The dependence of the extrapolated total cross sections $\sigma_{pA}$ on the atomic mass $A$ of the target material is discussed, and the deduced cross sections per nucleon $\sigma_{pN}$ are compared with results obtained at other energies.Comment: 17 pages, 7 figures, 5 table

Single-Player and Two-Player Buttons & Scissors Games

We study the computational complexity of the Buttons \& Scissors game and obtain sharp thresholds with respect to several parameters. Specifically we show that the game is NP-complete for $C = 2$ colors but polytime solvable for $C = 1$. Similarly the game is NP-complete if every color is used by at most $F = 4$ buttons but polytime solvable for $F \leq 3$. We also consider restrictions on the board size, cut directions, and cut sizes. Finally, we introduce several natural two-player versions of the game and show that they are PSPACE-complete.Comment: 21 pages, 15 figures. Presented at JCDCG2 2015, Kyoto University, Kyoto, Japan, September 14 - 16, 201

Geometry in the Transition from Primary to Post-Primary

This article is intended as a kind of precursor to the document Geometry for Post-primary School Mathematics, part of the Mathematics Syllabus for Junior Certicate issued by the Irish National Council for Curriculum and Assessment in the context of Project Maths. Our purpose is to place that document in the context of an overview of plane geometry, touching on several important pedagogical and historical aspects, in the hope that this will prove useful for teachers.Comment: 19 page

Octonionic Geometry

We extend vector formalism by including it in the algebra of split octonions, which we treat as the universal algebra to describe physical signals. The new geometrical interpretation of the products of octonionic basis units is presented. Eight real parameters of octonions are interpreted as the space-time coordinates, momentum and energy. In our approach the two fundamental constants, $c$ and $\hbar$, have the geometrical meaning and appear from the condition of positive definiteness of the octonion norm. We connect the property of non-associativity with the time irreversibility and fundamental probabilities in physics.Comment: 11 pages, no figure

Evidence for the Rare Decay B -> K*ll and Measurement of the B -> Kll Branching Fraction

We present evidence for the flavor-changing neutral current decay $B\to K^*\ell^+\ell^-$ and a measurement of the branching fraction for the related process $B\to K\ell^+\ell^-$, where $\ell^+\ell^-$ is either an $e^+e^-$ or $\mu^+\mu^-$ pair. These decays are highly suppressed in the Standard Model, and they are sensitive to contributions from new particles in the intermediate state. The data sample comprises $123\times 10^6$ $\Upsilon(4S)\to B\bar{B}$ decays collected with the Babar detector at the PEP-II $e^+e^-$ storage ring. Averaging over $K^{(*)}$ isospin and lepton flavor, we obtain the branching fractions ${\mathcal B}(B\to K\ell^+\ell^-)=(0.65^{+0.14}_{-0.13}\pm 0.04)\times 10^{-6}$ and ${\mathcal B}(B\to K^*\ell^+\ell^-)=(0.88^{+0.33}_{-0.29}\pm 0.10)\times 10^{-6}$, where the uncertainties are statistical and systematic, respectively. The significance of the $B\to K\ell^+\ell^-$ signal is over $8\sigma$, while for $B\to K^*\ell^+\ell^-$ it is $3.3\sigma$.Comment: 7 pages, 2 postscript figues, submitted to Phys. Rev. Let

Etale cohomology, cofinite generation, and p-adic L-functions

For a prime number p and a number field k, we first study certain etale cohomology groups with coefficients associated to a p-adic Artin representation of its Galois group, where we twist the coefficients using a modified Tate twist with a p-adic index. We show that those groups are cofinitely generated and explicitly compute an additive Euler characteristic. When k is totally real and the representation is even, we relate the order of vanishing of the p-adic L-function at a point of its domain and the corank of such a cohomology group with a suitable p-adic twist. If the groups are finite, then the value of the p-adic L-function is non-zero and its p-adic absolute value is related to a multiplicative Euler characteristic. For a negative integer n (and for 0 in certain cases), this gives a proof of a conjecture by Coates and Lichtenbaum, and a short proof of the equivariant Tamagawa number conjecture for classical L-functions that do not vanish at n. For p=2 our results involving p-adic L-functions depend on a conjecture in Iwasawa theory.Comment: Updated version. The final version will appear in the Annales de l'Institut Fourie

Vacuum Geometry

We analyse symmetry breaking in general gauge theories paying particular attention to the underlying geometry of the theory. In this context we find two natural metrics upon the vacuum manifold: a Euclidean metric associated with the scalar sector, and another generally inequivalent metric associated with the gauge sector. Physically, the interplay between these metrics gives rise to many of the non-perturbative features of symmetry breaking.Comment: 20 pages. no figure