Constant Row Research Articles

Use of a multivariate model using allele frequency distributions to analyse patterns of genetic differentiation among populations

Very few studies have attempted to relate the properties of some ordination techniques to classical tools of population genetics as F-statistics. A multivariate model to analyse population genetics data based on the properties of ‘joint scaling’ of populations and loci is developed. The design of population genetics data means that this model deals with a modified version of the classical Multiple Correspondence Analysis which is called Constant Row Total-Multiple Correspondence Analysis (CRT-MCA) and is an original tool in population genetics. Such a model allows estimates of the degree of population differentiation by studying the variability of the distribution of allele frequencies in different samples. Some clear relationships exist between some model parameters and the classical Fst statistics. The CRT-MCA also allows all the studied loci to be considered simultaneously and the role of each locus in patterns of population differentiation to be expressed. Such a multivariate approach prevents the use of any pooling strategy as is classically used in studies of hierarchical F-statistics. The relevance of the CRT-MCA model is illustrated by the analysis of population structure of 15 dogwhelk (Nucella lapillus) populations in south-west England. The advantages and limitations of CRT-MCA are presented.

Circum-Euclidean distance matrices and faces

We study the structure of circum-Euclidean distance matrices, those Euclidean distance matrices generated by points lying on a hypersphere. We show, for example, that such Euclidean distance matrices are characterized as having constant row sums and they constitute the interior of the cone of all Euclidean distance matrices. Also, we provide a formula for computing the radius of a representing configuration in the smallest embedding dimension r and show that rk D = r + 1. Finally we obtain a geometric characterization of the faces of this cone. Given a configuration of points and its Euclidean distance matrix D, any matrix in the minimal face containing D comes from a configuration that is a linear perturbation of the points that generate D.

Minimum positive determinant of integer matrices with constant row and column sums

The least possible positive determinant of zero-one matrices that have constant row and column sums is determined, thus proving a conjecture of Newman. The result is extended to n×n integer matrices.

Directed Moore hypergraphs

For graphs with maximum degree Δ and diameter D, an upper bound on the number of vertices is 1 + Δ∑ d − 1 i = 0 ( Δ − 1) i . This bound is called the Moore bound for graphs and the graphs that attain it are called Moore graphs. Similar bounds for directed graphs and for hypergraphs have been defined and the existence of directed Moore graphs and of Moore hypergraphs has been studied. In this article, we define a Moore bound for directed hypergraphs and show that directed Moore hypergraphs either are directed cycles, or have diameter one. Any directed hypergraph may be regarded as a factorization of a square nonnegative integer matrix into a pair of (0, 1)-matrices. In particular, the directed Moore hypergraphs of diameter 1 that have n vertices and m hyperarcs can be identified with factorizations J − I = XY where J is the n × n all-ones matrix, I is the n × n identity matrix, X is an n × m (0,1)-matrix, Y is an m × n (0, 1)-matrix, and X and Y have constant row sums. We conclude with a survey of results on factorizations of J − I.

Odd Magic Powers

(which is obtained from the famous example in Diirer's Melancholia by interchanging the first two rows and the first two columns) has the additional property that all of its 8 diagonals (e.g. 5,2, 12, 15; 11, 13,6,4; 10, 13,7,4) also have the same sum. A magic square with this extra property is called a pandiagonal magic square. Conventionally, additional restrictions are placed on the entries: (a) that they be (non-negative) integers; and often, (b) that they be the consecutive integers from 1 to n2 (when the array is n x n). We shall not require either of these restrictions and the numbers may be rational, real or complex. The purpose of this note is to consider the multiplicative properties of sets of magic squares (using the normal matrix multiplication). As van den Essen [31 has shown, some of these turn out to be quite surprising. We extend his results by showing that the product of any odd number of 3 x 3 magic squares is magic, that if such a square is invertible, then the inverse is magic and that these results extend to 4 x 4 and 5 x 5 pandiagonal magic squares. Because our method of proof is somewhat different from that of [3] and for completeness, we present the whole argument. As C. Small [2] points out, results of this type provide good examples for a Linear Algebra class. The first question of a multiplicative type is to ask whether a magic square is necessarily invertible. Clearly the answer is no-take the trivial case with all entries equal (even 0!) but a less trivial one is the example above (the vector (3, 1, 3, 1)T is in the null space). Our notation and terminology is standard except that we shall use first and second diagonal to mean the main diagonal and the 'other' one respectively. Also, we shall call a matrix with constant row sums an affine matrix and an affine matrix with constant column sums will be called doubly affine. Since we often need various permutation matrices, these will be defined by listing the columns as elements of the usual basis {el, e2,... , ej} of R'. For example: O 0 1 p = [e3 ,e2, el] = 1 0. -1 0 0-

A New Lower Bound Via Projection for the Quadratic Assignment Problem

New lower bounds for the quadratic assignment problem QAP are presented. These bounds are based on the orthogonal relaxation of QAP. The additional improvement is obtained by making efficient use of a tractable representation of orthogonal matrices having constant row and column sums. The new bound is easy to implement and often provides high quality bounds under an acceptable computational effort.

Biclique coverings of regular bigraphs and minimum semiring ranks of regular matrices

We study the minimum number of complete bipartite subgraphs needed to cover and partition the edges of a k-regular bigraph on 2 n vertices. Bounds are determined on the minima of these numbers for fixed n and k. Exact values of the minima are found for all n and k ≤ 4. The same results hold for directed graphs. Equivalently, we have determined bounds on the minimum value of the Boolean and nonnegative integer ranks of binary n × n matrices with constant row and column sum k for fixed n and k, obtaining the exact values of the minimum for k ≤ 4.

Anatomical study of the arterial blood supply of the skin of the lower leg

AbstractThe arteries leaving the intermuscular fissures of the lower leg form six constant rows. Two further vascular groups emerge from the lateral and the medial heads of the gastrocnemius muscle. The vessels form a fine epifascial network, the radial and longitudinal meshes of which anastomose together. The epifascial network is complemented by a weaker, subfascial one. The arteries can be subdivided into cutaneous, musculocutaneous, and cutaneomuscular groups, corresponding to their supplying areas. A significant difference in vessel pattern could not be found between the left and the right lower leg of the same person. The different areas of supply are constant in the middle one‐third of the lower leg, but vary considerably in the upper one‐third and moderately in the lower one‐third. The borders of the different areas show interlocking patterns.

The oxidation resistance and constitution of compounds of groups IIIA and IVA with d-transition elements

The thermal oxidation resistance of compounds of types RCo, R3Ni and RCu, where R is a rare earth element, has been tested by an isochronal method. It is found that the ratioTp/Tm of the oxidation parameter to the melting point decreases from about 0.75 to 0.6 with increasing atomic number, indicating a relaxation of chemical binding, and displays ‘saw-tooth’ variability as we pass from a compound of one element to another. This variability may be due either to weaknesses from valency-induced changes with elements such as terbium, or to deep-seated changes between odd- and even-Z elements. In a further development of the work, contours forTp/Tm of 0.6 and 0.7 have been charted on synoptic diagrams characterized by a constant row of 4f/5d elements on one ordinate and a variable 3d element (V to aluminium) on the other. We note that the largest area of high ground is found for cobalt compounds, followed by nickel, then copper; and that with copper only elements from barium to gadolinium contribute, in contrast with cobalt and nickel (lanthanum to at least thulium). Evidently a strong transition metal contribution to bonding in cobalt compounds has weakened in the copper counterparts, and s-d resonance/attenuation across the composite divide from the 4f element into the copper d-shell is only possible for the lighter rare earths.

Singular (0,1) matrices with constant row and column sums

We address the problem of explicitly constructing an n×n (0, 1) matrix with constant line sum k and rank r for each r<n. The corresponding problem for r=n was solved by Newman and also by Houck and Paul. We solve the problem for all possible ranks r < n when (1) k⩽ 3 for all n, (2) n≡0 (mod k for all k, and (3) n ≡k2 (mod k) for all even k. We also solve the problem for all but at most the first 2k−4 possible ranks r<n, for all k when n>3k.

Peanut‐Corn Intercrop Performance as Affected by Within‐Row Corn Spacing at a Constant Row Spacing

AbstractThe advantages of intercropping a cereal with a grain legume are often limited by the dominance of the cereal, which severely inhibits the legume. A field experiment was conducted in 1983 and 1984 in southern Maryland to determine the effects of within‐row corn (Zea mays L.) spacing on a peanut (Arachis hypogaea L.)‐corn intercrop. ‘Southern States 35A’ corn was planted in rows 1 m apart, with a double row of ‘Florunner’ peanut planted midway between. Peanut spacing was uniform, but within‐row corn spacing varied from 0.125 to 0.335 m in a systematic design with spacing in adjacent pairs of corn rows differing by 0.03 m. Separate sole‐crop plots of either four corn rows with 0.155‐m spacing or four double rows of peanut were used as controls. Yield components of both crops, shading on the peanut canopy, and nitrogenase activity of the peanut were measured. Averaged across corn spacings, intercropping reduced peanut yields by 44 and 64% in 1983 and 1984, respectively. Corn yields were not significantly affected by intercropping, but decreased linearly from 4.2 to 3.5 Mg ha−1 with decreased population density of corn. Conversely, peanut pod yield per hectare, pods per plant, and nitrogenase activity per plant increased linearly with decreased corn population density. Because of these countervailing trends, the combined yield and the land equivalency ratio (LER) (1.49 in 1983, a dry year, and 1.28 in 1984, a wetter year) of the intercrops were little affected by corn spacing. The 0.125‐m within‐row corn spacing resulted in the highest corn and combined dry seed yields, while the 0.335‐111 within‐row corn spacing resulted in the greatest combined economic value because of greater peanut yields. Peanut in monoculture gave the greatest economic value. This study pointed out that very low corn densities («30 000 plants ha−1) should be considered in peanut‐corn intercrops and that an LER &gt;1.0 does not necessarily imply an economic advantage to intercropping.

On the minimum rank of regular classes of matrices of zeros and ones

We investigate the minimum rank over a class of n × n matrices of zeros and ones with constant row and column sum k. We derive general lower and upper bounds, and obtain the exact values for k = 3 and for other special values.

The use of correspondence analysis in geochemistry

When factor analysis is used in geochemistry, it may be useful for factors to be transformed by rotations in order to be identified either to the end members of a mixing model (Miesch, 1976a),or to known chemical equilibriums. It requires that the formula for recomputing data from the factors may be written in a factored manner, which is generally not the case in correspondence analysis. The present paper shows that this becomes possible with data having constant row sums. As an example, the method is tested on the lavas of Paricutin Volcano, already examined by using an extended Q-mode factor analysis (Miesch, 1979).Recomputation of the data after projection gives simular results for both methods. Otherwise, the fact that correspondence analysis provides centered factors makes it well suited to the study of chemical reactions leading to constant mass transformations.

On the ranks of some (0, 1)-matrices with constant row sums

AbstractLet g(n, m) denote the maximal number of distinct rows in any (0, 1 )-matrix with n columns, rank &lt; n, – 1, and all row sums equal to m. This paper determines g(n, m) in all cases:In addition, it is shown that if V is a k-dimensional vector subspace of any vector space, then V contains at most 2k vectors all of whose coordinates are 0 or 1.

A generalization of extendedQ-mode factor analysis to data matrices with variable row sums

The extended method of Q-mode factor analysis developed by Miesch for data matrices with constant row sums is generalized to data matrices with variable row sums. With the algorithm provided it is possible to compute factor scores in the metric of the original data and compute goodness-of-fit statistics and model geological systems unconstrained by constancy of row sums of data points.

Generalized doubly stochastic and permutation matrices over a ring

Let R be a ring with unity. A combinatorial argument is used to show that the R -module Δ n ( R ) of all n × n matrices over R with constant row and column sums has a basis consisting of permutation matrices. This is used to characterize orthogonal matrices which are linear combinations of permutation matrices. It is shown that all bases of Δ n ( R ) consisting of permutation matrices have the same cardinality, and other properties of bases of Δ n ( R ) are investigated.

Non-singular 0-1 matrices with constant row and column sums

This paper explicitly constructs non-singular 0-1 matrices of dimensions n with constant row and column sum k, subject only to the conditions 0< k< n and ( k,n)≠(2,4). An application of such matrices to finite geometries can be found in [1].

On the plane term rank of a three dimensional matrix

A lower bound is obtained for the plane term rank of a three dimensional matrix in which each of the horizontal two dimensional matrices has constant row and column sums.

On the Plane Term Rank of a Three Dimensional Matrix

A lower bound is obtained for the plane term rank of a three dimensional matrix in which each of the horizontal two dimensional matrices has constant row and column sums.

The 1-width of (0,1)-matrices having constant row sum 3

In this note we establish upper bounds for the 1-width of an m × n matrix of 0's and 1's having three 1's in every row and having a constant number, c, of 1's in every column. When c = 3, this upper bound is n 2 and when c = 4 this estimate is 5n 9 . In these cases the upper bound is best possible, in the sense that for every possible size there exist matrices with this maximal 1-width. The technique of proof is also used to improve the known bound for the 1-width of (0, 1)-matrices with constant line sum 4.