224 research outputs found
Researches in non-associative algebra
I have frequently been asked by biologists for mathematical
help in connection with their problems. I was working on one
such problem when an algebraist, observing my work without
knowing what it was about, remarked that I was apparently using
hypercomplex numbers. I was considering a certain type of
inheritance specified by formulae which could be regarded as
forming the multiplication table of a non -associative linear
algebra; and my calculations could be regarded as manipulations
of hypercomplex numbers in this algebra, or in another algebra
derived from it by a process which I later called "duplication;I then realised that there are many such "genetic algebras ",
representing different types of inheritance. They are in all
cases non-associative as regards multiplication, though they can
always be taken to be commutative. I found that a large class of
genetic algebras (viz. those for "symmetrical inheritance" as
defined in Paper VI, p. 2) possessed certain distinctive
properties which seemed worthy of investigation for their own
sake, and also for the sake of possible exploitation in genetics.Part Three, the main part of this thesis, consists of four
papers in which this investigation is given - or rather is begun,
for there are a good many problems left untackled.Part One consists of four papers (one written in collaboration
with Dr A. Erdélyi) on some purely combinatory problems of non - associative algebra, suggested by the notations which I employed
for products and powers in the genetic algebras. The combinatory per t 0.4-01.5 rt. theory is continued in theAconcluding postscipt which follows
Paper X.Part Two shows how genetic algebras arise and are manipulate
The multiplication table of a genetic algebra, the multiplication
of hypercomplex numbers, and the above mentioned process of
duplication, are simply a translation into symbols of the relevant
essentials in the processes; of inheritance; and the symbolism as
a whole is a convenient shorthand for reckoning with combinations
and statistical distributions of genetic types, enabling one to
dispense with some of the verbal arguments and the chessboard
diagrams commonly used for the same purpose. In paper VI the
treatment is made as general as possible with the object of showirg
the relationship between different genetic algebras and something
of their structure; and the concepts to be discussed in Part Three
are here defined. In Paper V, which was published later but mostly
written earlier than VI, the explanation is given in very much
simpler mathematical language (for it was intended to be read by
geneticists), and with more attention to practical applications.
It can be explained very simply why multiplication in the
genetic algebras is non- associative, that is to say(AB) C ≠A (BC)This statement is interpreted:- "If the offspring of A and B
mates with C, the probability distribution of genetic types in
the progeny will not be the same as if A mates with the offspring
of B and C."My symbolism was not essentially new: the novelty lay in is
interpretationlin terms of hypercomplex numbers. In fact it
could be said that genetic algebras had been used by geneticists
in a primitive way for quite a long time without having been
recognised. explicitly. Their explicit recognition is I believe
more than a mere change of notation. Apart from greater brevity
achieved in some applications, general theorems on linear algebras
can be applied; transformations can be used which are quite
meaningless genetically but which lead to genetically significant
conclusions; and the use of an index notation and summation
convention reduces the symbolism to manageable proportions when,
with inheritance involving many genes, it threatens to become too
heavy to handle.Biological considerations were thus the root of these
researches, and I intend to return to the genetical applications
later; for I believe that genetic algebras may throw light on some
deeper problems of genetics. I cannot at present give solid
justification for this belief in the sense of having successfully
tackled problems otherwise unsolved, and I therefore wish that this
thesis may not be judged as a finished achievement in biological
investigation; but may be judged primarily as a contribution to
algebra, suggested by biological problems, and perhaps having
possibilities of application beyond the simple ones so far
demonstrated
A new Tolman test of a cosmic distance duality relation at 21 cm
Under certain general conditions in an expanding universe, the luminosity
distance (d_L) and angular diameter distance (d_A) are connected by the
Etherington relation as d_L = d_A (1 + z)^2. The Tolman test suggests the use
of objects of known surface brightness, to test this relation. In this letter,
we propose the use of redshifted 21 cm signal from disk galaxies, where neutral
hydrogen (HI) masses are seen to be almost linearly correlated with surface
area, to conduct a new Tolman test. We construct simulated catalogs of
galaxies, with the observed size-luminosity relation and realistic redshift
evolution of HI mass functions, likely to be detected with the planned Square
Kilometer Array (SKA). We demonstrate that these observations may soon provide
the best implementation of the Tolman test to detect any violation of the
Etherington relation.Comment: 4 pages, 2 figures, 1 table, v2: published versio
Supernova Brightening from Chameleon-Photon Mixing
Measurements of standard candles and measurements of standard rulers give an
inconsistent picture of the history of the universe. This discrepancy can be
explained if photon number is not conserved as computations of the luminosity
distance must be modified. I show that photon number is not conserved when
photons mix with chameleons in the presence of a magnetic field. The strong
magnetic fields in a supernova mean that the probability of a photon converting
into a chameleon in the interior of the supernova is high, this results in a
large flux of chameleons at the surface of the supernova. Chameleons and
photons also mix as a result of the intergalactic magnetic field. These two
effects combined cause the image of the supernova to be brightened resulting in
a model which fits both observations of standard candles and observations of
standard rulers.Comment: 17 pages, 3 figure
Cosmology With A Dark Refraction Index
We review Gordon's optical metric and the transport equations for the
amplitude and polarization of a geometrical optics wave traveling in a gravity
field. We apply the theory to the FLRW cosmologies by associating a refraction
index with the cosmic fluid. We then derive an expression for the accumulated
effect of a refraction index on the distance redshift relations and fit the
Hubble curve of current supernova observations with a non-accelerating
cosmological model. We also show that some observational effects caused by
inhomogeneities, e.g. the Sachs-Wolfe effect, can be interpreted as being
caused by an effective index of refraction, and hence this theory could extend
to other speed of light communications such as gravitational radiation and
neutrino fluxes.Comment: 21 pages, 3 figure
An Improved Treatment of Optics in the Lindquist-Wheeler Models
We consider the optical properties of Lindquist-Wheeler (LW) models of the
Universe. These models consist of lattices constructed from regularly arranged
discrete masses. They are akin to the Wigner-Seitz construction of solid state
physics, and result in a dynamical description of the large-scale Universe in
which the global expansion is given by a Friedmann-like equation. We show that
if these models are constructed in a particular way then the redshifts of
distant objects, as well as the dynamics of the global space-time, can be made
to be in good agreement with the homogeneous and isotropic
Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions of Einstein's equations,
at the level of <3% out to z~2. Angular diameter and luminosity distances, on
the other hand, differ from those found in the corresponding FLRW models, while
being consistent with the 'empty beam' approximation, together with the
shearing effects due to the nearest masses. This can be compared with the large
deviations found from the corresponding FLRW values obtained in a previous
study that considered LW models constructed in a different way. We therefore
advocate the improved LW models we consider here as useful constructions that
appear to faithfully reproduce both the dynamical and observational properties
of space-times containing discrete masses.Comment: 7 pages, 5 figure
Local and non-local measures of acceleration in cosmology
Current cosmological observations, when interpreted within the framework of a
homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) model,
strongly suggest that the Universe is entering a period of accelerating
expansion. This is often taken to mean that the expansion of space itself is
accelerating. In a general spacetime, however, this is not necessarily true. We
attempt to clarify this point by considering a handful of local and non-local
measures of acceleration in a variety of inhomogeneous cosmological models.
Each of the chosen measures corresponds to a theoretical or observational
procedure that has previously been used to study acceleration in cosmology, and
all measures reduce to the same quantity in the limit of exact spatial
homogeneity and isotropy. In statistically homogeneous and isotropic
spacetimes, we find that the acceleration inferred from observations of the
distance-redshift relation is closely related to the acceleration of the
spatially averaged universe, but does not necessarily bear any resemblance to
the average of the local acceleration of spacetime itself. For inhomogeneous
spacetimes that do not display statistical homogeneity and isotropy, however,
we find little correlation between acceleration inferred from observations and
the acceleration of the averaged spacetime. This shows that observations made
in an inhomogeneous universe can imply acceleration without the existence of
dark energy.Comment: 19 pages, 10 figures. Several references added or amended, some minor
clarifications made in the tex
Modification to the Luminosity Distance Redshift Relation in Modified Gravity Theories
We derive an expression for the luminosity distance as a function of redshift
for a flat Robertson-Walker spacetime perturbed by arbitrary scalar
perturbations possibly produced by a modified gravity theory with two different
scalar perturbation potentials. Measurements of the luminosity distance as
function of redshift provide a constraint on a combination of the scalar
potentials and so they can complement weak lensing and other measurements in
trying to distinguish among the various alternative theories of gravity.Comment: 15 pages, we discuss in more detail how the luminosity distance
expression can be used to differentiate among various theories of gravit
Distance-redshift from an optical metric that includes absorption
We show that it is possible to equate the intensity reduction of a light wave
caused by weak absorption with a geometrical reduction in intensity caused by a
"transverse" conformal transformation of the spacetime metric in which the wave
travels. We are consequently able to modify Gordon's optical metric to account
for electromagnetic properties of ponderable material whose properties include
both refraction and absorption. Unlike refraction alone however, including
absorption requires a modification of the optical metric that depends on the
eikonal of the wave itself. We derive the distance-redshift relation from the
modified optical metric for Friedman-Lema\^itre-Robertson-Walker spacetimes
whose cosmic fluid has associated refraction and absorption coefficients. We
then fit the current supernovae data and provide an alternate explanation
(other than dark energy) of the apparent acceleration of the universe.Comment: 2 figure
On the Possibility of Anisotropic Curvature in Cosmology
In addition to shear and vorticity a homogeneous background may also exhibit
anisotropic curvature. Here a class of spacetimes is shown to exist where the
anisotropy is solely of the latter type, and the shear-free condition is
supported by a canonical, massless 2-form field. Such spacetimes possess a
preferred direction in the sky and at the same time a CMB which is isotropic at
the background level. A distortion of the luminosity distances is derived and
used to test the model against the CMB and supernovae (using the Union
catalog), and it is concluded that the latter exhibit a higher-than-expected
dependence on angular position. It is shown that future surveys could detect a
possible preferred direction by observing ~ 20 / (\Omega_{k0}^2) supernovae
over the whole sky.Comment: Extended SNe analysis and corrected some CMB results. Text also
extended and references added. 8 pages, 5 figure
Differential Density Statistics of Galaxy Distribution and the Luminosity Function
This paper uses data obtained from the galaxy luminosity function (LF) to
calculate two types of radial number densities statistics of the galaxy
distribution as discussed in Ribeiro (2005), namely the differential density
and the integral differential density . By applying the
theory advanced by Ribeiro and Stoeger (2003), which connects the relativistic
cosmology number counts with the astronomically derived LF, the differential
number counts are extracted from the LF and used to calculate both
and with various cosmological distance definitions,
namely the area distance, luminosity distance, galaxy area distance and
redshift distance. LF data are taken from the CNOC2 galaxy redshift survey and
and are calculated for two cosmological models:
Einstein-de Sitter and an , standard
cosmology. The results confirm the strong dependency of both statistics on the
distance definition, as predicted in Ribeiro (2005), as well as showing that
plots of and against the luminosity and redshift
distances indicate that the CNOC2 galaxy distribution follows a power law
pattern for redshifts higher than 0.1. These findings bring support to
Ribeiro's (2005) theoretical proposition that using different cosmological
distance measures in statistical analyses of galaxy surveys can lead to
significant ambiguity in drawing conclusions about the behavior of the observed
large scale distribution of galaxies.Comment: LaTeX, 37 pages, 6 tables, 10 figures. Accepted for publication in
"The Astrophysical Journal
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