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 $\gamma$ and the integral differential density $\gamma^\ast$. 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 $dN/dz$ are extracted from the LF and used to calculate both $\gamma$ and $\gamma^\ast$ 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 $\gamma$ and $\gamma^\ast$ are calculated for two cosmological models: Einstein-de Sitter and an $\Omega_{m_0}=0.3$, $\Omega_{\Lambda_0}=0.7$ 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 $\gamma$ and $\gamma^\ast$ 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