553 research outputs found
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 Tverberg theorems over lattices and other discrete sets
This paper presents a new variation of Tverberg's theorem. Given a discrete
set of , we study the number of points of needed to guarantee the
existence of an -partition of the points such that the intersection of the
convex hulls of the parts contains at least points of . 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
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
Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals
We revisit a well-known family of polynomial ideals encoding the problem of
graph--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
Online Ramsey theory for a triangle on -free graphs
Given a class of graphs and a fixed graph , the online
Ramsey game for on is a game between two players Builder and
Painter as follows: an unbounded set of vertices is given as an initial state,
and on each turn Builder introduces a new edge with the constraint that the
resulting graph must be in , and Painter colors the new edge either
red or blue. Builder wins the game if Painter is forced to make a monochromatic
copy of at some point in the game. Otherwise, Painter can avoid creating a
monochromatic copy of forever, and we say Painter wins the game.
We initiate the study of characterizing the graphs such that for a given
graph , Painter wins the online Ramsey game for on -free graphs. We
characterize all graphs such that Painter wins the online Ramsey game for
on the class of -free graphs, except when is one particular graph.
We also show that Painter wins the online Ramsey game for on the class of
-minor-free graphs, extending a result by Grytczuk, Ha{\l}uszczak, and
Kierstead.Comment: 20 pages, 10 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
Single Top Quark Production as a Probe for Anomalous Moments at Hadron Colliders
Single production of top quarks at hadron colliders via fusion is
examined as a probe of possible anomalous chromomagnetic and/or chromoelectric
moment type couplings between the top and gluons. We find that this channel is
far less sensitive to the existence of anomalous couplings of this kind than is
the usual production of top pairs by or fusion. This result is
found to hold at both the Tevatron as well as the LHC although somewhat greater
sensitivity for anomalous couplings in this channel is found at the higher
energy machine.Comment: New discussion and 10 new figures added. uuencoded postscript fil
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