135 research outputs found
A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line
It is demonstrated that the magic three-qubit Veldkamp line occurs naturally
within the Veldkamp space of combinatorial Grassmannian of type ,
. The lines of the ambient symplectic polar space are
those lines of whose cores feature an odd number of
points of . After introducing basic properties of three different types
of points and six distinct types of lines of , we
explicitly show the combinatorial Grassmannian composition of the magic
Veldkamp line; we first give representatives of points and lines of its core
generalized quadrangle GQ, and then additional points and lines of a
specific elliptic quadric (5,2), a hyperbolic quadric
(5,2) and a quadratic cone (4,2) that
are centered on the GQ. In particular, each point of
(5,2) is represented by a Pasch configuration and its
complementary line, the (Schl\"afli) double-six of points in
(5,2) comprise six Cayley-Salmon configurations and six
Desargues configurations with their complementary points, and the remaining
Cayley-Salmon configuration stands for the vertex of
(4,2).Comment: 6 pages, 2 figure
On an Observer-Related Unequivalence Between Spatial Dimensions of a Generic Cremonian Universe
Given a generic Cremonian space-time, its three spatial dimensions are shown
to exhibit an intriguing, "two-plus-one" partition with respect to standard
observers. Such observers are found to form three distinct, disjoint groups
based on which one out of the three dimensions stands away from the other two.
These two subject-related properties have, to our knowledge, no analogue in any
of the existing physical theories of space-time.Comment: 4 pages, no figures, submitted to CS
Cremonian Space-Time(s) as an Emergent Phenomenon
It is shown that the notion of fundamental elements can be extended to_any_,
i.e. not necessarily homaloidal, web of rational surfaces in a
three-dimensional projective space. A Cremonian space-time can then be viewed
as an_emergent_ phenomenon when the condition of "homaloidity" of the
corresponding web is satisfied. The point is illustrated by a couple of
particular types of "almost-homaloidal" webs of quadratic surfaces. In the
first case, the quadrics have a line and two distinct points in common and the
corresponding pseudo-Cremonian manifold is endowed with just two spatial
dimensions. In the second case, the quadrics share six distinct points, no
three of them collinear, that lie in quadruples in three different planes, and
the corresponding pseudo-Cremonian configuration features three time
dimensions. In both the cases, the limiting process of the emergence of generic
Cremonian space-times is explicitly demonstrated.Comment: 5 pages, no figures, submitted to CS
Combinatorial Intricacies of Labeled Fano Planes
Given a seven-element set , there are 30 ways to
define a Fano plane on it. Let us call a line of such Fano plane, that is to
say an unordered triple from , ordinary or defective according as the sum of
two smaller integers from the triple is or is not equal to the remaining one,
respectively. A point of the labeled Fano plane is said to be of order , , if there are {\it defective} lines passing through it. With
such structural refinement in mind, the 30 Fano planes are shown to fall into
eight distinct types. Out of the total of 35 lines, nine ordinary lines are of
five different kinds, whereas the remaining 26 defective lines yield as many as
ten distinct types. It is shown, in particular, that no labeled Fano plane can
have all points of zeroth order, or feature just one point of order two. A
connection with prominent configurations in Steiner triple systems is also
pointed out.Comment: 5 pages, 2 figure
On an Intriguing Signature-Reversal Exhibited by Cremonian Spacetimes
It is shown that a generic quadro-quartic Cremonian spacetime, which is
endowed with one spatial and three time dimensions, can continuously evolve
into a signature-reversed configuration, i.e. into the classical spacetime
featuring one temporal and three space dimensions. An interesting cosmological
implication of this finding is mentioned.Comment: 3 pages, 1 table, no figures; submitted to Chaos, Solitons & Fractal
The Complement of Binary Klein Quadric as a Combinatorial Grassmannian
Given a hyperbolic quadric of PG(5,2), there are 28 points off this quadric
and 56 lines skew to it. It is shown that the -configuration
formed by these points and lines is isomorphic to the combinatorial
Grassmannian of type . It is also pointed out that a set of seven
points of whose labels share a mark corresponds to a Conwell heptad of
PG(5,2). Gradual removal of Conwell heptads from the -configuration yields a nested sequence of binomial configurations
identical with part of that found to be associated with Cayley-Dickson algebras
(arXiv:1405.6888).Comment: 4 pages, 4 table
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