Some results in graph theory and their application to abstract relational biology

It has been shown in earlier work that one approach to what Rashevsky has called “abstract biology” is through the study of the class of ( )-systems that can be formed in an arbitrary subcategory of the category of sets. The concept of the ( )-system, however, depends on the availability of mappings that contain other mappings in their range. It is shown that, by introducing an appropriate measure for this property, the problem of characterizing those categories suitable for a rich theory of ( )-systems reduces to a problem familiar from the general theory of graphs. Some new results in these directions are obtained, and it is then shown that any category with mappings that possess properties we might expect to hold in the physical world will also admit a rich theory of ( )-systems. In particular, it is shown that a sufficiently large family of mappings drawn at random from such a category will with overwhelming probability contain an ( )-system. This research was supported by the United States Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command, under Grant No. AF-AFOSR-9-63.     

A note on Wallemia sebi

A review of the available information on the monotypic genus Wallemia supports the interpretation that W. sebi is probably a basidiomycete and may be a teleomorph. It has dolipore/parenthesome septa similar to those found in the suborder Tremellineae, except that the associated parenthesome vesicles are very small and composed of a single membrane. A conjectural interpretation is made that the spore chains, which are composed of repeating sets of four, are meiospores generated from a permanently diploid mother cell that divides repeatedly to generate meiocyte daughter cells.     

A note on imitative behavior

In a preceding paper (Bull. Math. Biophysics,27, 175–185) the distribution function ofφ=ɛ ₁-ɛ ₂,—the difference of excitations in the two mutually inhibiting centers, has been derived in terms of the distribution functionsf ₁(ɛ ₁) andf ₂(ɛ ₂) of the two excitations. In the present note some properties of the distribution functionf(ϕ) in terms of the propertiesf ₁(ɛ ₁) andf ₂(ɛ ₂) are derived.     

A note on neural nets

A neural mechanism is described which provides for a perception which is invariant with respect to movement of the stimulus.     

A note on Gauss--Hermite quadrature

For Gauss—Hermite quadrature, we consider a systematicmethod for transforming the variable of integration so thatthe integrand is sampled in an appropriate region. The effectivenessof the quadrature then depends on the ratio of the integrandto some Gaussian density being a smooth function, well approximatedby a low-order polynomial. It is pointed out that, in this approach,order one Gauss-Hermite quadrature becomes the Laplace approximationmxHermitequadrature becomes the Laplace approximationmxHermite quadraturebecomes the Laplace approximationmxHermite quadrature becomesthe Laplace approximation. Thus the quadrature as implementedhere can be thought of as a higher-order Laplace approximation.     

A note on the horopter

By assuming the fixity (but not the symmetry) of corresponding points on the two retinae, it is possible to derive the equation of any horopter when one is known. In particular when, as experiment shows, one horopter is linear, then all horopters must be conics. These have the form given by Ogle, but whereas Ogle leaves one parameter undetermined at each fixation, on our assumption the only arbitrary parameter is determined by the position of the linear horopter.