Quantitative Tverberg, Helly, & Carath\'eodory theorems

This paper presents sixteen quantitative versions of the classic Tverberg, Helly, & Caratheodory theorems in combinatorial convexity. Our results include measurable or enumerable information in the hypothesis and the conclusion. Typical measurements include the volume, the diameter, or the number of points in a lattice.Comment: 33 page

Quantitative combinatorial geometry for continuous parameters

We prove variations of Carath\'eodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lov\'asz's colorful Helly theorem, B\'ar\'any's colorful Carath\'eodory's theorem, and the colorful Tverberg theorem.Comment: 22 pages. arXiv admin note: substantial text overlap with arXiv:1503.0611

Quantitative Tverberg theorems over lattices and other discrete sets

This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$, we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. The proofs of the main results require new quantitative versions of Helly's and Carath\'eodory's theorems.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1503.0611

Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals

We revisit a well-known family of polynomial ideals encoding the problem of graph-$k$-colorability. Our paper describes how the inherent combinatorial structure of the ideals implies several interesting algebraic properties. Specifically, we provide lower bounds on the difficulty of computing Gr\"obner bases and Nullstellensatz certificates for the coloring ideals of general graphs. For chordal graphs, however, we explicitly describe a Gr\"obner basis for the coloring ideal, and provide a polynomial-time algorithm.Comment: 16 page

How to be causal: time, spacetime, and spectra

I explain a simple definition of causality in widespread use, and indicate how it links to the Kramers Kronig relations. The specification of causality in terms of temporal differential eqations then shows us the way to write down dynamical models so that their causal nature /in the sense used here/ should be obvious to all. To extend existing treatments of causality that work only in the frequency domain, I derive a reformulation of the long-standing Kramers Kronig relations applicable not only to just temporal causality, but also to spacetime "light-cone" causality based on signals carried by waves. I also apply this causal reasoning to Maxwell's equations, which is an instructive example since their casual properties are sometimes debated.Comment: v4 - add Appdx A, "discrete" picture (not in EJP); v5 - add Appdx B, cause classification/frames (not in EJP); v7 - unusual model case; v8 add reference

Design, Construction, Operation and Performance of a Hadron Blind Detector for the PHENIX Experiment

A Hadron Blind Detector (HBD) has been developed, constructed and successfully operated within the PHENIX detector at RHIC. The HBD is a Cherenkov detector operated with pure CF4. It has a 50 cm long radiator directly coupled in a window- less configuration to a readout element consisting of a triple GEM stack, with a CsI photocathode evaporated on the top surface of the top GEM and pad readout at the bottom of the stack. This paper gives a comprehensive account of the construction, operation and in-beam performance of the detector.Comment: 51 pages, 39 Figures, submitted to Nuclear Instruments and Method

Neutral Particles in Light of the Majorana-Ahluwalia Ideas

The first part of this article (Sections I and II) presents oneself an overview of theory and phenomenology of truly neutral particles based on the papers of Majorana, Racah, Furry, McLennan and Case. The recent development of the construct, undertaken by Ahluwalia [{\it Mod. Phys. Lett. A}{\bf 9} (1994) 439; {\it Acta Phys. Polon. B}{\bf 25} (1994) 1267; Preprints LANL LA-UR-94-1252, LA-UR-94-3118], could be relevant for explanation of the present experimental situation in neutrino physics and astrophysics. In Section III the new fundamental wave equations for self/anti-self conjugate type-II spinors, proposed by Ahluwalia, are re-casted to covariant form. The connection with the Foldy-Nigam-Bargmann-Wightman- Wigner (FNBWW) type quantum field theory is found. The possible applications to the problem of neutrino oscillations are discussed.Comment: REVTEX file. 21pp. No figure

Inclusive cross section and double helicity asymmetry for pi^0 production in p+p collisions at sqrt(s) = 62.4 GeV

The PHENIX experiment presents results from the RHIC 2006 run with polarized proton collisions at sqrt(s) = 62.4 GeV for inclusive pi^0 production at mid-rapidity. Unpolarized cross section results are measured for transverse momenta p_T = 0.5 to 7 GeV/c. Next-to-leading order perturbative quantum chromodynamics calculations are compared with the data, and while the calculations are consistent with the measurements, next-to-leading logarithmic corrections improve the agreement. Double helicity asymmetries A_LL are presented for p_T = 1 to 4 GeV/c and probe the higher range of Bjorken_x of the gluon (x_g) with better statistical precision than our previous measurements at sqrt(s)=200 GeV. These measurements are sensitive to the gluon polarization in the proton for 0.06 < x_g < 0.4.Comment: 387 authors from 63 institutions, 10 pages, 6 figures, 1 table. Submitted to Physical Review D. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm