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We propose a new field description of limb regeneration in developmental biology. Central to our approach is the recognition of the importance of vector fields as descriptors in problems involving polarities or a sense of direction. In limb regeneration these vector fields represent the organisms sense of the distal direction, and its sense of the circumferential ordering of structures. Essentially all experimental results obtained to date are consistent with or explained by the model.

The extensive use of weighting patterns in the analysis and synthesis of linear, finite-dimensional systems relies heavily on a realizability result involving a decomposition of the weighting pattern $W( t,\tau )$ into an inner product of two vector terms $\psi ( t )$ and $\beta ( \tau )$. This result, as stated for the continuous-time case, depends on the invertibility of the transition matrix $\Phi ( t,\tau )$ for the general case. Hence, the realizability condition must be modified for the discrete-time case, where the transition matrix is not invertible in general.This paper presents and proves a discrete-time realizability condition which closely parallels the continuous-time results. At the same time, some insight is conveyed regarding the differences between the continuous- and discrete-time cases. Finally, auxiliary results are proved along established lines.

The negative result of theChan, De Celles andPatonG2 vector meson bootstrap, based on their generation in two pseudoscalar mesons scattering, and a similar result in two vector meson scattering, are shown to derive from the fact that this model cannot assign the existing mesons to its adjoint representation.

A quantized non-conserved four-component field theory is constructed in view to identify the axial currents with fields of this kind. The identification of non-strange vector and axial vector currents is done with the usual ϱ and A1 fields, while the strangeness carrying currents are assummed to be dominated by two vector or axial vector particles. A new sum rule relating the K decay amplitude to the decay amplitudes (assumed to be the same) and the masses of the two strange axial vector particles is derived. The ratios FK/Fπ and FK∗/Fϱ are deduced and their values are found to be in good agreement with experiment. The second Weinberg sum rule for SU(2) ⊗ SU(2) is seen to be valid for non-strange isomultiplets and violated by an amount of about 15% for the strange ones.

The so-called ‘Gauss-Jordan’ procedure for inverting matrices has been described in many papers and is well known. The present paper gives an unusual approach to this method insofar as the inverse is obtained by repeated application of a certain non-similarity transformation to the original matrix. Of course this transformation is related to the Gauss elimination process, but formally has rather the character of an exchange of components in two vector spaces.

An inequality between the norm of the difference of two vector states of a W* -algebra and the innmum distance between vectors representing the states in a fixed representation is derived.

Introduction.! As this paper is the expression of my work as a student under Prof. R. Brauer, I wish at the outset to acknowledge my deep indebtedness to my former teacher. For an associative algebra a, having a unit element and lying over a given field K, two regular representations T? and (E are defined. As we shall be concerned with the absolutely irreducible constituents of the regular representations we shall assume that K is algebraically closed. The decomposition of St and t is our first goal. Let 23 be a representation of an algebra a by linear transformations of a vector space V. If V is the direct sum of two vector spaces V, and V2, both invariant under A, then Z is decomposable. By adapting the co-ordinate system in V to the decomposition, we obtain Q3 in the form

A locally asymptotically most powerful test for composite hypotheses with t independent linear constraints on the parameters has been developed for the case where the observed random variables {X nk , k=1,2,...,n} are independently but not necessarily identically distributed. However, their distributions depend on two vector parameters, one ξ = (ξ 1, ξ 2, ..., ξ t) being under test, and the other θ = (θ 1, θ 2, ..., θ s) being the nuisance parameter.

The interaction between nucleon and antinucleon has been treated from a bosonic viewpoint where also the short-range part is described by the exchange of vector bosons (introduction of a suitable form factor). The energy levels of the bound- and resonance states are explicitly calculated whereby the various bosonic decay processes (up to two vector bosons) are included. In these calculations the coupling constants are taken from the nucleon-nucleon problem whereas slight variations of the form factors were considered. The results (in particular the number of the bound states exhibit some similarities with experimental facts.

A theory of polarization-holographic recording is considered in the general case of a reaction of a photoanisotropic medium involving linear and nonlinear effects when the medium subjected to elliptically polarized light assumes the properties of a gyrotropic crystal. The matrix describing conversion of the state of polarization of a reconstructed image is obtained in the linear approximation. This matrix depends on a scalar reaction and two vector reactions of a photoanisotropic medium and also on the state of polarization of the reference and reconstructing waves. The approximation of an infinitesimally thin hologram is used to derive the equation for the characteristic ellipse of a photoanisotropic medium, which describes an elliptically polarized reference wave matched to the characteristics of the medium so as to ensure that the reconstructed image is undistorted in respect of the polarization.

Prim and Truesdell have recently given a simple vector proof of Zorawski's condition for the permanence of vector-lines in a moving fluid.' Their argument is necessarily by its form confined to threedimensional space, and it seems of interest to investigate the problem in a Euclidean space of N dimensions, using a slightly different approach. We consider a Euclidean space EN with rectangular cartesians xi. Two vector fields are given, vi(x, t), ci(x, t), where t is a parameter (the time); we shall call v* the primary and ci the secondary vector fields; vi plays the part of velocity. Any vector field defines for given t a congruence of curves in an obvious way, the direction of the curve at each point coinciding with that of the vector field. We are not particularly interested here in the congruence defined by the primary vector field (stream-lines), and we shall use the expression vector-lines to refer exclusively to the congruence defined by the secondary field, that is, those curves which satisfy

We try to explain the succession of periodic and chaotic behaviours observed in B.Z.Z. reaction (See many contributions in this volume). We define an abstract mathematical object: “Mixed vector fields”which is very similar to ROSSLER models with, however, some new features inspired by the above mentioned experiments. Our model is not purely speculative and is closely related to a modified version of the Oregonator taken from TYSON.

As pointed out in Section 5 of Chapter 1, the standard regression problem is related to the problem of finding the maximum correlation between a scalar and a vector random variable. Indeed, the formulation of the problem is in terms of finding a linear combination of the elements of the vector random variable exhibiting maximum correlation with the given scalar variable. In this section we deal with a natural generalization in which we seek to define the correlation (or set of correlations) between two vector random variables. Specifically, the problem dealt with is as follows: Let (x 1, x 2, …, x m1 and x m1+1 , …, x m1+m2 be two sets of variables, (m 1≤m 2).

In this note we give an easy proof of the existence of generically smooth components of the expected dimension of certain Brill--Noether loci of stable rank 2 vector bundles on a curve with general moduli, with related applications to Hilbert scheme of scrolls., 9 pages, submitted preprint

It has long been known that no continuous field of unit tangent vectors can exist on the 2n-dimensional sphere S2". On the other hand, if n = 2k 1, 4k 1, or 8k 1, there exist 1, 3, 7 everywhere independent continuous vector fields over S'. It was recently proved independently by B. Eckmann2 and the author3 that if n 1 (mod 4) every two vector fields over St are somewhere dependent. In the present paper an analogous result is obtained for the case n 3 (mod 8): every four vector fields over Sn are dependent at some point. The principal tools are the notion of fibre space developed by Huxewicz and Steenrod4 andthe results of Freudenthal5 on the homotopy groups of spheres. As a corollary to the above results and a result of C. Ehresmann,6 it is shown that no sphere of dimension > k can be a k-sphere bundle over any complex B if k = 2m, 4m + 1, or 8m + 3 with m > O.6"