116 research outputs found
A decomposition theorem for binary matroids with no prism minor
The prism graph is the dual of the complete graph on five vertices with an
edge deleted, . In this paper we determine the class of binary
matroids with no prism minor. The motivation for this problem is the 1963
result by Dirac where he identified the simple 3-connected graphs with no minor
isomorphic to the prism graph. We prove that besides Dirac's infinite families
of graphs and four infinite families of non-regular matroids determined by
Oxley, there are only three possibilities for a matroid in this class: it is
isomorphic to the dual of the generalized parallel connection of with
itself across a triangle with an element of the triangle deleted; it's rank is
bounded by 5; or it admits a non-minimal exact 3-separation induced by the
3-separation in . Since the prism graph has rank 5, the class has to
contain the binary projective geometries of rank 3 and 4, and ,
respectively. We show that there is just one rank 5 extremal matroid in the
class. It has 17 elements and is an extension of , the unique splitter
for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's
result identifying the binary internally 4-connected matroids with no prism
minor [5]
Light Front Theory Of Nuclear Matter
A relativistic light front formulation of nuclear dynamics is applied to
infinite nuclear matter. A hadronic meson-baryon Lagrangian, consistent with
chiral symmetry, leads to a nuclear eigenvalue problem which is solved,
including nucleon-nucleon (NN) correlations, in the one-boson-exchange
approximation for the NN potential. The nuclear matter saturation properties
are reasonably well reproduced, with a compression modulus of 180 MeV. We find
that there are about 0.05 excess pions per nucleon.Comment: 6 pages, Revtex, one figure; version resubmitted to Phys. Lett.
Cubic Augmentation of Planar Graphs
In this paper we study the problem of augmenting a planar graph such that it
becomes 3-regular and remains planar. We show that it is NP-hard to decide
whether such an augmentation exists. On the other hand, we give an efficient
algorithm for the variant of the problem where the input graph has a fixed
planar (topological) embedding that has to be preserved by the augmentation. We
further generalize this algorithm to test efficiently whether a 3-regular
planar augmentation exists that additionally makes the input graph connected or
biconnected. If the input graph should become even triconnected, we show that
the existence of a 3-regular planar augmentation is again NP-hard to decide.Comment: accepted at ISAAC 201
Time and Dirac Observables in Friedmann Cosmologies
A cosmological time variable is emerged from the Hamiltonian formulation of
Friedmann model to measure the evolution of dynamical observables in the
theory. A set of observables has been identified for the theory on the null
hypersurfaces that its evolution is with respect to the volume clock introduced
by the cosmological time variable.Comment: 11 page
Koszul binomial edge ideals
It is shown that if the binomial edge ideal of a graph defines a Koszul
algebra, then must be chordal and claw free. A converse of this statement
is proved for a class of chordal and claw free graphs
Time and Observables in Unimodular General Relativity
A cosmological time variable is emerged from the hamiltonian formulation of
unimodular theory of gravity to measure the evolution of dynamical observables
in the theory. A set of constants of motion has been identified for the theory
on the null hypersurfaces that its evolution is with respect to the volume
clock introduced by the cosmological time variable.Comment: 16 page
A Bichromatic Incidence Bound and an Application
We prove a new, tight upper bound on the number of incidences between points
and hyperplanes in Euclidean d-space. Given n points, of which k are colored
red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between
the k red points and m hyperplanes spanned by all n points provided that m =
\Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal
and Aronov.
We use this incidence bound to prove that a set of n points, no more than n-k
of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also
provide an infinite family of counterexamples to a conjecture of Purdy's on the
number of hyperplanes spanned by a set of points in dimensions higher than 3,
and present new conjectures not subject to the counterexample.Comment: 12 page
Higher Dimensional Dark Energy Investigation with Variable and
Time variable and are studied here under a phenomenological
model of through an () dimensional analysis. The relation of
Zeldovich (1968) between and is
employed here, where is the proton mass and is Planck's constant. In
the present investigation some key issues of modern cosmology, viz. the age
problem, the amount of variation of and the nature of expansion of the
Universe have been addressed.Comment: 7 Latex pages with few change
Lines, Circles, Planes and Spheres
Let be a set of points in , no three collinear and not
all coplanar. If at most are coplanar and is sufficiently large, the
total number of planes determined is at least . For similar conditions and
sufficiently large , (inspired by the work of P. D. T. A. Elliott in
\cite{Ell67}) we also show that the number of spheres determined by points
is at least , and this bound is best
possible under its hypothesis. (By , we are denoting the
maximum number of three-point lines attainable by a configuration of
points, no four collinear, in the plane, i.e., the classic Orchard Problem.)
New lower bounds are also given for both lines and circles.Comment: 37 page
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