On (0, 1)-matrices with prescribed row and column sum vectors

Minimal matrices in the Bruhat order for symmetric (0,1)-matrices

Transportation matrices are $m\times n$ nonnegative matrices with given row sum vector $R$ and column sum vector $S$. All such matrices form the convex polytope $\mathcal{U}(R,S)$ which is called a transportation polytope and its extreme points have been classified. In this article, we consider a new class of convex polytopes $\Delta(\bar{R},\bar{S},\sigma)$ consisting of certain transportation polytopes satisfying that the sum of all elements is $\sigma$, and the row and column sum vectors are dominated componentwise by the given positive vectors $\bar{R}$ and $\bar{S}$, respectively. We characterize the extreme points of $\Delta(\bar{R},\bar{S},\sigma)$. Moreover, we give the minimal term rank and maximal permanent of $\Delta(\bar{R},\bar{S},\sigma)$.

On asymmetric (0, 1)-matrices with given row and column sum vectors

On the [formula omitted]-term rank of a matrix

We introduce a generalization of alternating sign matrices (ASMs) called multiASMs and develop some of their properties. Classes of multiASMs with specified row and column sum vectors $R$ and $S$ extend the classes of $(0,1)$-matrices with specified $R$ and $S$. The special case when $R=S$ is a constant vector, in particular all 2's, is treated in more detail. We also investigate the polytope spanned by a class of multiASMs. Finally, we discuss the possibility of defining a Bruhat order on a class of multiASMs.

We consider a class of matrices whose row and column sum vectors are majorized by given vectors b and c, and whose entries lie in the interval [0, 1]. This class generalizes the class of doubly stochastic matrices. We investigate the corresponding polytope Ω(b|c) of such matrices. Main results include a generalization of the Birkhoff–von Neumann theorem and a characterization of the faces, including edges, of Ω(b|c).

We consider the problem of counting the number of contingency tables with given row and column sums. This problem is known to be #P-complete, even when there are only two rows [7]. In this paper we present the first fully-polynomial randomized approximation scheme for counting contingency tables when the number of rows is constant. A novel feature of our algorithm is that it is a hybrid of an exact counting technique with an approximation algorithm, giving two distinct phases. In the first, the columns are partitioned into "small" and "large". We show that the number of contingency tables can be expressed as the weighted sum of a polynomial number of new instances of the problem, where each instance consists of some new row sums and the original large column sums. In the second phase, we show how to approximately count contingency tables when all the column sums are large. In this case, we show that the solution lies in approximating the volume of a single convex body, a problem which is known to be solvable in polynomial time [5].

The maximum absolute value of the determinant of n x n nonsinguiar (0,1) matrices that have constant line sums (i.e., row sums and column sums) k is investigated. For n≠4k=2, this maximum determinant is determined to be 2t if n=3t or 3t+2, and 2t−1 if n=3t+1. Restriction to a subset of these matrices, namely those that are symmetric and have zero trace (their graphs are regular of degree k), leaves the maximum unchanged for k=2. For this restricted class, when n≥7k=n−3, the maximum absolute value of the determinant is (n−3)3[n/4]−-1, This maximum gives a lower bound for the maximum absolute value of the determinant for the larger class, but in general this bound is not tight. Other determinantal values and bounds for specific n and k are given.

Asymptotic enumeration of sparse 0–1 matrices with irregular row and column sums

Inn then last two decades the Content Based Image Retrieval (CBIR) considered as one of the topic of interest for theresearchers. It depending one analysis of the image’s visual content which can be done by extracting the color, texture and shapefeatures. Therefore, feature extraction is one of the important steps in CBIR system for representing the image completely. Color featureis the most widely used and more reliable feature among the image visual features. This paper reviews different methods, namely LocalColor Histogram, Color Correlogram, Row sum and Column sum and Colors Coherences Vectors were used to extract colors featurestaking in consideration the spatial information of the image.

Let m n and k (with k n m) be positive integers and let R = (r 1,…r m ) and S = (s 1,…s n ) be two partitions of k into m and n parts, respectively. In this work we present a simple algorithm for the direct calculation of the number of m × n (0, 1)-matrices having r i 1's in row i and s j 1's in column j. We also extend the above procedure to the general case of (0, l, 2,…l)-matrices with prefixed row sum and column sum vectors.

where the biological activity is expressed as the reciprocal of the concentration that elicit a given biological, pharmacological or toxicological endpoint such as ED 50 (effective dose 50%), IC 50 (inhibitory concentration 50%) or LD 50 (lethal dose 50%), and where x i is the i predictor (from a set of n predictors) raised to the powers from 1 to m and weighted by its coefficients a ij obtained from a multivariate statistical fitting [4,5]. Usually, most of the coefficients a ij can be zero, leading to much simpler looking QSAR models than the general form in Equation 1, as in the example below. Often, other functions of x i are used in the QSAR model such as the logarithmic or exponential function instead of a simple power expansion as in Equation 1. As a simple illustration of the operation of Equation 1 is the following example of a QSAR model to predict the inhibitory effect of 3-nitroflavones (3NF) based on a single predictor, x (the energy of the highest occupied molecular orbital, E HOMO ) raised to the power 1 (see also [6]). Invasive tumor growth and metastasis are sustained by angiogenesis (the formation of new blood vessels [7,8]), which led Folkman to suggest anticancer therapies by the use of antiangiogenic agents [9,10]. Lichtenberg et al. have shown that the antiangiogenicity of six 3NF is correlated with the energies of the highest occupied molecular orbital, E HOMO, the lower the oxidation potential the greater the inhibitory effects on several biomarkers of angiogenesis [11,12]. These authors report the following correlation between the E HOMO and the IC 50 of six 3NF for inhibition of proliferation of microvascular endothelial HMEC-1 cells after 72 h of incubation:

On a conjecture concerning the Bruhat order

Let A ( R , S ) \mathfrak {A}(R,S) denote the class of all m × n m \times n matrices of 0’s and 1’s with row sum vector R and column sum vector S. A set I × J ( I ⊆ { 1 , … , m } , J ⊆ { 1 , … , n } ) I \times J(I \subseteq \{ 1, \ldots ,m\} ,J \subseteq \{ 1, \ldots ,n\} ) is said to be invariant if each matrix in A ( R , S ) \mathfrak {A}(R,S) contains the same number of 1’s in the positions I × J I \times J . We prove that if there are no invariant singletons, then an invariant set I × J I \times J satisfies I = ∅ , I = { 1 , … , m } , J = ∅ I = \emptyset ,I = \{ 1, \ldots ,m\} ,J = \emptyset or J = { 1 , … , n } J = \{ 1, \ldots ,n\} .

We prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorthmic and combinatorial applications. Our main application is the first poly-time deterministic algorithm which approximates the partition functions associated with boolean matrices with prescribed row and column sums within simply exponential multiplicative factor. This new algorithm is a particular instance of new polynomial time deterministic algorithms related to the multiple partial differentiation of polynomials given by evaluation oracles.