Canonical Vector Research Articles

Influence in canonical correlation analysis

The perturbation theory of the generalized eigenproblem is used to derive influence functions of each squared canonical correlation coefficient and the corresponding canonical vector pair. Three sample versions of these functions are described and some properties are noted. As particular applications, the influence function of the squared multiple correlation coefficient and influence functions of eigenvalues and eigenvectors in correspondence analysis are obtained. Three numerical examples are briefly discussed.

Allozyme markers in breeding zone designation

Early studies of allozyme variation in plant populations suggested that allelic frequencies in some loci vary by geography. Since then, the expectation that allozymes might be useful in describing geographic patterns has generally not been borne out by single locus analyses, except on the broadest scale. Multi-locus analyses reveal the converse: canonical correlation analysis of individual, uniformly-spaced genotypes describe statistically-significant, complex patterns with geography. Multi-locus scores in four major species, Abies concolor, Pinus lambertiana, P. ponderosa, and Pseudotsuga menziesii, of the mixed conifer forest in the Sierra Nevada correlate 0.40 or greater with the first canonical vector of a geographical trend surface equation. The different species follow similar patterns by latitude and elevation. In contrast with patterns in the Sierra Nevada, large-scale differentiation is weak (R2 < 0.20) among populations of Pseudotsuga menziesii in the Coast Ranges and Siskyou Mountains of northern California and southern Oregon, where differentiation may be local. For the purpose of forming zones, we subdivided scores of the first two to four canonical vectors into groups and plotted them as multidimensional contour intervals. Reclassification by discriminant analysis serves as an approximate guide to transfer risks within and among these groups.

3-dimensional affine hypersurfaces in ℝ4 with parallel cubic form

In this paper, we study 3-dimensional locally strongly convex affine hypersurfaces in ℝ4. Since the publication of Blaschke’s book [B] in the early twenties, it is well-known that on a nondegenerate affine hyper-surface M there exists a canonical transversal vector field called the affine normal. The second fundamental form associated to the affine normal is called the affine metric. In the special case that M is locally strongly convex, this affine metric is a Riemannian metric. Also, using the affine normal, by the Gauss formula one can introduce an affine connection on M, called the induced connection ∇. So on M, we can consider two connections, namely the induced affine connection ∇ and the Levi Civita connection of the affine metric h.

On the Banach-Steinhaus Theorem and the Continuity of Bilinear Mappings

In this paper, the BANACH-STEINHAUS theorem is extended from its usual locally convex topological vector space setting to the much broader framework of convergence vector spaces. It is used to derive theorems yielding the joint continuity of separately continuous bilinear mappings. These results are used, in turn, to show that the convolution mapping is a jointly continuous bilinear mapping when the distribution spaces ℰ and carry the canonical convergence vector space structures.

Detection of homologous recombination between yeast artificial chromosomes with overlapping inserts

We have developed a system which facilitates the detection of recombination between Yeast Artificial Chromosomes (YAC's) carrying homologous inserts. The system consists of a classical YAC vector, a new YAC vector and two appropriately labelled yeast strains of opposite mating type. The new YAC vector differs in markers from the canonical YAC vector. To test whether homologous recombination takes place, phage lambda DNA was cloned in the two vectors to provide a region of homology. The two constructs were then introduced into yeast strains of opposite mating type in which the endogenous genes for the selective markers present in the vectors are not expressed. Artificial chromosomes obtained by meiotic recombination are detected in the spores resulting from the mating.

$G$-Central States of Almost Periodic Type on $C^*$-Algebras

Let (A, G, a) be a C*-dynamical system, namely a triple consisting of a C*algebra A, a locally compact group G and a homomorphism a from G into the automorphism group of A such that G3t-+ oct(x)EA is continuous for each x in A. Now assume that the C*-algebra A is unital. Then the set of a-invariant states of A, denoted by S, forms a weak* compact convex subset in the state space of A, and each extremal point in S is called an ergodic state (occasionally, G-ergodic state or oc-ergodic state). When we attempt to decompose an ainvariant state into ergodic states, it is most important to investigate the existence and uniqueness of the decomposition, and in order to carry out such investigation we need to require some abelianness of C* -dynamical systems, in particular, of invariant states. Now denote by (nv, u , H^, <^) the GNS covariant representation associated with an a-invariant state <p, that is, n^ is a representation of A on the Hilbert space Hv with the canonical cyclic vector <^ and u is a strongly continuous unitary representation of G on Hv defined by «?KW)^ = ^(«tM)^ for xeA and teG. Note that

Generalized Luneburg canonical varieties and vector fields on quasicaustics

Some aspects of a particular class of bifurcation varieties which are provided by simple and unimodal boundary singularities are studied. Their correspondence to diffraction theory is established. The generic caustics by diffraction on apertures are derived and their generating families for the corresponding Lagrangian varieties are calculated. It is proved that the quasicaustics associated to simple singularities are smooth hypersurfaces or Whitney’s cross-caps. The procedure for calculating the modules of logarithmic vector fields is given, and the minimal sets of the corresponding generators are explicitly calculated. The general boundary singularities are constructed and the structure of quasicaustics defined by parabolic singularities is investigated.

Remarks on Grassmannian supermanifolds

This paper studies some aspects of a particular class of examples of supermanifolds; the supergrassmannians, introduced in [Manin]. Their definition, in terms of local data and glueing isomorphisms, is reviewed. Explicit formulas in local coordinates are given for the Lie group action they come equipped with. It is proved that, for those supergrassmannians whose underlying manifold is an ordinary grassmannian, their structural sheaf can be realized as the sheaf of sections of the exterior algebra bundle of some canonical vector bundle. This realization holds true equivariantly for the Lie group action in question, thus making natural in these cases the identification given in [Batchelor]. The proof depends on the computation of the transition functions of the supercotangent bundle as defined in a previous work [OASV 2]. Finally, it is shown that there is a natural supergroup action involved (in the sense of [OASV 3]) and hence, the supergrassmannians may be regarded as examples of superhomogeneous spaces—a notion first introduced in [Kostant]. The corresponding Lie superalgebra action can be realized as superderivations of the structural sheaf; explicit formulas are included for those supergrassmannians identifiable with exterior algebra vector bundles.

Genetic divergence among some maintainer and restorer lines in relation to hybrid breeding in rice (Oryza sativa L.)

A representative group of 100 elite lines, 67 of which are restorers and 33 maintainers, from 68 crosses made at IRRI and 18 improved varieties from five countries were studied, using Mahalonobis, D(2)-statistic and canonical analysis, to understand the nature and magnitude of divergence and to assess the importance of a set of quantitative characters related to yield in genetic differentiation. The 100 genotypes were grouped into 13 clusters. There were three single variety clusters and the number of lines in the remaining clusters ranged from 2-36. Canonical vector values indicated the importance of yield followed by 1,000-grain weight, days to maturity and plant height in primary as well as in secondary differentiation. Results showed that yield, number of tillers per plant, days to maturity and 1,000-grain weight contributed largely to the divergence. There were no indications of a relationship between geographical diversity and genetic diversity in the present study. Disposition of IRRI developed maintainers and restorers into various clusters indicated the presence of large amounts of diversity within the IRRI elite lines which suggested that these materials could be used in crossing programs to produce heterotic F1 hybrids. Crossing of maintainers and restorers among the highly diverse groups was suggested as it may produce F1S that will give higher magnitudes of heterosis.

A Bose representation for the massless Dirac field in four dimensions

We present an attempt to extend to four dimensions the powerful, two-dimensional Fermi-Bose equivalence. Our approach rests on the formalism of the tomographic representation of quantized fields. Starting with canonical vector fields satisfying a duality relation it is shown how to construct a four-dimensional field satisfying the massless Dirac equation and canonical anitcommutation relations.

Divergent gene pools in rice improvement

The objective of the present study was to investigate the genetic architecture of yield in diverse populations of rice. Multivariate analysis by Mahalanobis's D(2) statistic and canonical (vector) analysis revealed that panicle weight, days to maturity, plant height and seed size were the important forces contributing towards divergence. One hundred rice strains were grouped into nine clusters with the help of D(2) and canonical analysis. The grouping pattern of the varieties were quite at random indicating that the geographical and genetic diversity were not related.

Ecological association of birds and vegetation structure in wet tropical forests of Australia

AbstractBirds and environments of tropical rainforests were studied in 18 selected areas from sea level to 1520 m within the humid tropical region of northeastern Australia. A total of 122 sites (20 m × 20 m each) were established within the 18 areas, and bird data (observation and mist‐netting) and environmental data (pro forma survey and foliage measurements) were collected at each site. The bird data were analysed numerically in two ways: abundance of 146 species in 18 areas and presence and absence of 108 species at 122 sites. Both classification and ordination analyses produced a major discontinuity between the tableland sites and the lowland sites, each divisible further into various types of vine forest and non‐vine forest vegetation. The few rare species did not contribute to the general pattern. The environmental data were reduced to a set of those attributes that showed variation among the sites. They included topographic and edaphic features, physiognomic and structural features of vegetation, fruiting and defoliating habits of plants, foliage height diversity and evenness, and canopy cover. The primary pattern in the data set of 135 attributes by 122 sites was vine forest versus nonvine forest, and altitudinal divisions appeared as a secondary pattern, in both classification and ordination.Canonical correlation coefficients between bird vectors and environmental vectors were reasonably high for the first four canonical vectors, which correlated well with major Gower vectors of birds and environments. The back‐correlation to the raw data produced bird species and matching environmental attributes with sites sharing them for both positive and negative ends of each canonical vector. In general, the analysis extracted elements of both fauna and environment that characterized the vine forest as opposed to the non‐vine forest and the tableland sites as opposed to the lowland sites. Canonical correlation analysis is limited by the unverifiable nature of derived correlations. Such correlations may suggest certain associations to be tested by independent means (hypothesis‐generating function), but may also formalize self‐evident or fortuitous associations.

Multivariate analysis of genetic divergence in blackgram (Vigna mungo) (L.) Hepper.)

Forty five types of blackgram were grouped into ten clusters with the aid of Mahalanobis's D statistic bringing out the presence of wide genetic diversity. The clus- tering pattern indicated that the geographic diversity was not the only factor for determining the genetic diversity. The intercluster distance between the type No.55 and the types in the cluster VII were found maximum. The clustering pattern of the more or less confirmed with the clustering pattern of the canonical (Vector) analysis carried out with the same data. Among the characters yield plant contributed to the maximum towards the genetic divergence.

A water quality index for lake beaches

Discriminant analysis of ten years of data from seven beaches located on three lakes in Gatineau Park (Québec) leads to a canonical vector accounting for most of the variability of the three microbial count variables recorded. This discriminant axis orders the beaches along a cline of increasing pollution. It is then transformed into a microbial water quality index calibrated for the given set of beaches, which is in turn used to study the medium-term variability and the long-term evolution of water quality of the beaches.

Construction of orthonormal vector sets for atoms and molecules by means of recursive variation

It is shown using a recursive variation method that the Schmidt orthonormalized vectors are optimal in the sense that the least squared distances are minimized, and that the use of maximal squared overlaps yields a Schmidt canonical orthonormalized vector set which corresponds to the Löwdin canonical vector set. By introducing the Krylov sequence, the method is applied to the derivation of a variational expression for the Lanczos vectors which tridiagonalize any self-adjoint operator. The repeated use of two-step variations is shown to derive a pseudopotential which is useful for the calculations of correlated pair functions.

Order of the canonical vector bundle on $C_{n}(K)/\Sigma_{k}$

Order of the canonical vector bundle on $C_{n}(K)/\Sigma_{k}$

Symmetry and separation of variables for the Hamilton–Jacobi equation W2t−W2x −W2y =0

We present a detailed group theoretical study of the problem of separation of variables for the characteristic equation of the wave equation in one time and two space dimensions. Using the well-known Lie algebra isomorphism between canonical vector fields under the Lie bracket operation and functions (modulo constants) under Poisson brackets, we associate, with each R-separable coordinate system of the equation, an orbit of commuting constants of the motion which are quadratic members of the universal enveloping algebra of the symmetry algebra o (3,2). In this, the first of two papers, we essentially restrict ourselves to those orbits where one of the constants of the motion can be split off, giving rise to a reduced equation with a nontrivial symmetry algebra. Our analysis includes several of the better known two-body problems, including the harmonic oscillator and Kepler problems, as special cases.

On separable Lindenstrauss spaces

Isometric embeddings from l ∞ n in l ∞ n + 1 can be described by a i,n, i ⩽ n, with ∑ i = 1 n ¦ a i,n ¦ ⩽ 1 , such that e i, n = e i, n + 1 + a i, n e n + 1, n + 1 ; i = 1,…, n; holds, where e i, n and e i, n + 1 are the elements of the canonical unit vector bases of l ∞ n and l ∞ n + 1 , respectively (negative signs may occur). We study the connections between a triangular substochastic matrix A, whose nth column consists of the elements a i, n , i = 1,…, n, and the Banach space a i,n, E n ⊂ E n + 1 , E n ≅ l ∞ n , where A determines the embeddings of the E n . The class of these Banach spaces is the class of all separable Lindenstrauss spaces. Sufficient and necessary conditions are stated for a matrix A to represent c 0and c. Furthermore, we characterize the class of all extreme triangular substochastic matrices which represents C( K), where K is the Cantor set. We investigate how the special biface structure of the dual unit ball of X is reflected in the elements of a matrix A representing the separable Lindenstrauss space X. This is applicable to Gurarij spaces; we give a new proof for the maximality property of Gurarij spaces and show that they are isomorphic to A( S) where S is a Choquet simplex with dense extreme points.

Theory of Currents and the Nonstrong Interactions

General perturbations around Sugawara's theory of currents are considered. The usual forms for the semistrong, electromagnetic, and weak interactions are easily incorporated, and, using a modified interaction picture, a covariant Feynman-graphical approach to the perturbation is achieved. There is some degree of uniqueness in the allowed perturbation: If infinite polynomials are to be avoided in the theory, the perturbations must be of the form ${V}_{\ensuremath{\mu}}{V}^{\ensuremath{\mu}}$, or ${V}_{\ensuremath{\mu}}{C}^{\ensuremath{\mu}}$ (${C}^{\ensuremath{\mu}}$ is a canonical vector field), or $\ensuremath{\sigma}$ (the scalar density due to Bardakci, Frishman, and Halpern).

Canonical vector fields on spheres

Canonical vector fields on spheres