Ith Row Research Articles

Inequalities for totally nonnegative matrices: Gantmacher–Krein, Karlin, and Laplace

A real linear combination of products of minors which is nonnegative over all totally nonnegative (TN) matrices is called a determinantal inequality for these matrices. It is referred to as multiplicative when it compares two collections of products of minors and additive otherwise. Set theoretic operations preserving the class of TN matrices naturally translate into operations preserving determinantal inequalities in this class. We introduce index-row (and index-column) operations that act directly on all determinantal inequalities for TN matrices, and yield further inequalities for these matrices. These operations assist in revealing novel additive inequalities for TN matrices embedded in the classical identities due to Laplace [Mem. Acad. Sciences Paris 1772] and Karlin (1968). In particular, for any square TN matrix A, these derived inequalities generalize – to every ith row of A and jth column of adjA – the classical Gantmacher–Krein fluctuating inequalities (1941) for i=j=1. Furthermore, our index-row/column operations reveal additional undiscovered fluctuating inequalities for TN matrices.The introduced index-row/column operations naturally birth an algorithm that can detect certain determinantal expressions that do not form an inequality for TN matrices. However, the algorithm completely characterizes the multiplicative inequalities comparing products of pairs of minors. Moreover, the underlying index-row/column operations add that these inequalities are offshoots of certain “complementary/higher” ones. These novel results seem very natural, and in addition thoroughly describe and enrich the classification of these multiplicative inequalities due to Fallat–Gekhtman–Johnson [Adv. Appl. Math. 2003] and later Skandera [J. Algebraic Comb. 2004] (and revisited by Rhoades–Skandera [Ann. Comb. 2005, J. Algebra 2006]).

Interstage flow matrices: Population statistic derived from matrix population models

Abstract Many population statistics describe the characteristics of populations within and among species. These are useful for describing population dynamics, understanding how environmental factors alter demographic patterns, testing hypotheses related to the evolution of life history characteristics and informing the effective management of populations. In this study, we propose a population statistic: the interstage flow. The interstage flow is defined as the product of the element in the ith row, the jth column of the population projection matrix and the jth element of the normalized stable stage distribution. The sum of the interstage flow matrix elements is equal to the population growth rate (PGR), which is the dominant eigenvalue of the population projection matrix. The interstage flow matrix elements allow decomposition of PGR into component contributions made by transitions between developmental stages. We demonstrate the utility of interstage flow matrices using matrix population models from the COMPADRE plant matrix database. We compared interstage flows among four life history/functional groups (FGs) (semelparous herbs, iteroparous herbs, shrubs and trees) and described how PGR reflected individual transitions related to stasis, fecundity and growth. We found that the individual flows are different among FGs. Synthesis. The proposed population statistic, the interstage flow matrix, describes the contribution of individual developmental stage transitions to the PGR. The flow of individuals between developmental stages differs in distinctive ways among different life histories and FGs. The interstage flow matrix is a valuable statistic for describing these differences.

The complexity of regex crosswords

In a typical regex crossword puzzle, one is given two non-empty lists 〈R1,…,Rm〉 and 〈C1,…,Cn〉 of regular expressions (regexes) over some alphabet, and the challenge is to fill in an m×n grid of characters such that the string formed by the ith row is in L(Ri) and the string formed by the jth column is in L(Cj), for all 1≤i≤m and 1≤j≤n. We consider a restriction of this puzzle where all the Ri are equal to one another and similarly the Cj. We consider a 2-player version of this puzzle, showing it to be PSPACE-complete. Using a reduction from 3SAT, we also give a new, simple proof of the known result that the existence problem of a solution for the restricted (1-player) puzzle is NP-complete. We also prove NP-completeness where all the row and column regular expressions are equal to each other.

Listing the Positive Rationals

We start with a review of the usual method for listing the rationals. Consider an infinite array with the rows and columns indexed by the positive integers. The entry in the ith row and jth column ...

English

Let m and n be positive integers, and let R = (r1, . . . , rm) and S = (s1, . . . , sn) be nonnegative integral vectors. Let A(R,S) be the set of all m × n (0, 1)-matrices with row sum vector R and column vector S. Let R and S be nonincreasing, and let F(R) be the m × n (0, 1)-matrix, where for each i, the ith row of F(R,S) consists of ri 1’s followed by (n−ri) 0’s. Let A ∈ A(R,S). The discrepancy of A, disc(A), is the number of positions in which F(R) has a 1 and A has a 0. In this paper we investigate linear operators mapping m × n matrices over the binary Boolean semiring to itself that preserve sets related to the discrepancy. In particular, we show that bijective linear preservers of Ferrers matrices are either the identity mapping or, when m = n, the transpose mapping.

Compressed sensing reconstruction of synthetic transmit aperture dataset for volumetric diverging wave imaging

A high volume rate and high performance ultrasound imaging method based on a matrix array is proposed by using compressed sensing (CS) to reconstruct the complete dataset of synthetic transmit aperture (STA) from three-dimensional (3D) diverging wave transmissions (i.e. 3D CS-STA). Hereto, a series of apodized 3D diverging waves are transmitted from a fixed virtual source, with the ith row of a Hadamard matrix taken as the apodization coefficients in the ith transmit event. Then CS is used to reconstruct the complete dataset, based on the linear relationship between the backscattered echoes and the complete dataset of 3D STA. Finally, standard STA beamforming is applied on the reconstructed complete dataset to obtain the volumetric image. Four layouts of element numbering for apodizations and transmit numbers of 16, 32 and 64 are investigated through computer simulations and phantom experiments. Furthermore, the proposed 3D CS-STA setups are compared with 3D single-line-transmit (SLT) and 3D diverging wave compounding (DWC). The results show that, (i) 3D CS-STA has competitive lateral resolutions to 3D STA, and their contrast ratios (CRs) and contrast-to-noise ratios (CNRs) approach to those of 3D STA as the number of transmit events increases in noise-free condition. (ii) the tested 3D CS-STA setups show good robustness in complete dataset reconstruction in the presence of different levels of noise. (iii) 3D CS-STA outperforms 3D SLT and 3D DWC. More specifically, the 3D CS-STA setup with 64 transmit events and the Random layout achieves ~31% improvement in lateral resolution, ~14% improvement in ratio of the estimated-to-true cystic areas, a higher volume rate, and competitive CR/CNR when compared with 3D DWC. The results demonstrate that 3D CS-STA has great potential of providing high quality volumetric image with a higher volume rate.

Vertex types in some lexicographic products of graphs

ABSTRACTLet be a symmetric matrix, or equivalently, a weighted graph whose edge ij has the weight . The eigenvalues of are the eigenvalues of M. We denote by the principal submatrix of M obtained by deleting from M both the ith row and the ith column. If μ is an eigenvalue of M, and thus of , of multiplicity , then vertex i of is a downer, or a neutral, or a Parter vertex, depending whether the multiplicity of μ in or, equivalently, in , is k−1, k, or k+1, respectively. In this paper, for a fixed μ, we consider vertex types according to the above classification in graphs which are generalized lexicographic products of an arbitrary graph over cliques and co-cliques, or connected regular graphs. In addition, we add some comments on constructions of large families of cospectral and integral graphs.

Heini Halberstam, 1926-2014

Heini Halberstam, 1926-2014

1D-Bernoulli Chaos Sequences Based Collaborative-CDMA: A Novel Secure High Capacity CDMA Scheme

Collaborative multi-user transmission and detection is a contemporary technique used for the up-link of code-division multiple access (CDMA). CDMA has contributed much to the field of wireless communication and still addresses many of the capacity and security requirements. The collaborative multi-user technique achieves an increase in capacity by sharing the spreading sequences among the users. In such grouped CDMA schemes, the sum of large numbers of statistically uncorrelated pseudo-random (PN) sequences results in multiple access interference. This restricts the number of PN sequences available for spreading, thereby limiting the total number of full rate users. Furthermore, PN sequences can be detected by their bit timings and consequently the security of the users' data is a matter of concern. To solve this problem, a novel high-capacity and secure system, called the 1D-Bernoulli-chaotic-collaborative-CDMA (BCC-CDMA), is proposed. In the BCC-CDMA, a set of specific `initial-conditions' are used, and identified for the 1D-Bernoulli Map, that generates row elements of an N × N matrix of mutually orthogonal chaotic sequences,. The ith row of this matrix acts as the spreading sequence for the ith group of collaborating users in BCC-CDMA. The proposed scheme eliminates the constraints in security and user capacity outperforming the existing multi-user detection schemes. Simulations show the bit-error-rate performance of BCC-CDMA supporting 120 full-rate users compared to the almost 90 and 30 users offered by conventional schemes.

Multi-dimensional backward stochastic differential equations of diagonally quadratic generators

In this paper, we study a multi-dimensional BSDE with a “diagonally” quadratic generator, the quadratic part of whose ith component depends only on the ith row of the second unknown variable. Local and global solutions are given, which seem to be the first systematic (positive) results on the general solvability of multi-dimensional quadratic BSDEs. In our proofs, it is natural and crucial to apply both John–Nirenberg and reverse Hölder inequalities for BMO martingales. Our results are finally illustrated to solve the system of “diagonally” quadratic BSDEs arising from a nonzero-sum risk-sensitive stochastic differential game, which answers the open problem posed in El Karoui and Hamadène [Stochastic Process. Appl. 107 (2003), page 164].

Non-Singularity Conditions for Two Z-Matrix Types

A real square matrix whose non-diagonal elements are non-positive is called a Z-matrix. This paper shows a necessary and sufficient condition for non-singularity of two types of Z-matrices. The first is for the Z-matrix whose row sums are all non-negative. The non-singularity condition for this matrix is that at least one positive row sum exists in any principal submatrix of the matrix. The second is for the Z-matrix which satisfies where . Let be the ith row and the jth column element of , and be the jth element of . Let be a subset of which is not empty, and be the complement of if is a proper subset. The non-singularity condition for this matrix is such that or such that for . Robert Beauwens and Michael Neumann previously presented conditions similar to these conditions. In this paper, we present a different proof and show that these conditions can be also derived from theirs.

A matrix sequence [formula omitted] might converge even if the matrix A is not primitive

It is well-known that, for an irreducible Boolean (0,1)-matrix A, the matrix sequence {Am}m=1∞ converges if and only if A is primitive. In this paper, we introduce an operation Γ on the set of Boolean (0,1)-matrices such that a matrix sequence {Γ(Am)}m=1∞ might converge even if the matrix A is not primitive. Given a Boolean (0,1)-matrix A, we define a matrix Γ(A) so that the (i,j)-entry of Γ(A) equals 0 if for i≠j, the inner product of the ith row and jth row of A is 0 and equals 1 otherwise.The aim of this paper is to study the convergence of {Γ(Am)}m=1∞ for a Boolean (0,1)-matrix A whose digraph has at most two strong components. We show that {Γ(Am)}m=1∞ converges to a very special type of matrix as m increases if A is an irreducible Boolean matrix. Furthermore, we completely characterize a Boolean (0,1)-matrix A whose digraph has exactly two strongly connected components and for which {Γ(Am)}m=1∞ converges, and find the limit of {Γ(Am)}m=1∞ in terms of its digraph when it converges. We derive these results in terms of the competition graph of the digraph of A.

The Properties of Case-Weight Model when the Row and Column of Design Matrix Simultaneous Mitted

This paper studies that the properties of case-weighted model when simultaneous omission of the ith row and the jth column of design matrix X. In other words, basing on case-weighted model, this article obtains the relation of total-data model and select-mean-shift outlier model about regression coefficient, leverage and potency.

On matrices with different tropical and Kapranov ranks

The main object of tropical mathematics is the tropical semiring, i.e., the set R of real numbers with the operations of tropical addition, ⊕, and tropical multiplication, ⊗, which are defined by the rules a⊕ b = min{a, b} and a⊗ b = a+ b for any a, b ∈ R. Note that the element 0 ∈ R is neutral with respect to the tropical multiplication. Tropical mathematics is obtained from traditional mathematics as a result of Maslov dequantization (see [1]–[3]). Methods of tropical mathematics turn out to be useful for numerous applications (see [4]–[6]) and are employed in algebraic geometry (see [7], [8]). The notion of rank of tropical matrices is of great interest (see [4], [9], [10]). In contrast to the case of matrices over fields, there is a lot of different rank functions for tropical matrices, and many of them are described in [4], [9], and [10]. The present note is devoted to the notions of tropical rank and Kapranov rank. We use the symbol F to denote a field and F∗ for the set of nonzero elements of F. Denote by aij the entries of a matrix A, by A(j) the jth column and by A(i) the ith row of the matrix, and by A the matrix transposed to A. A submatrix of a matrix A formed by the rows with indices r1, . . . , rk is denoted by A[r1, . . . , rk] and a submatrix of the matrix A[r1, . . . , rk] formed by the columns with indices c1, . . . , cl is denoted by A[r1, . . . , rk|c1, . . . , cl]. Denote byHF the field (see [11]) consisting of formal sums of the form

Two-way model with random cell sizes

We consider inference for row effects in the presence of possible interactions in a two-way fixed effects model when the numbers of observations are themselves random variables. Let Nij be the number of observations in the (i,j) cell, πij be the probability that a particular observation is in that cell and μij be the expected value of an observation in that cell. We assume that the {Nij} have a joint multinomial distribution with parameters n and {πij}. Then μ¯i.=∑jπijμij/∑jπij is the expected value of a randomly chosen observation in the ith row. Hence, we consider testing that the μ¯i. are equal. With the {πij} unknown, there is no obvious sum of squares and F-ratio computed by the widely available statistical packages for testing this hypothesis. Let Y¯i‥ be the sample mean of the observations in the ith row. We show that Y¯i‥ is an MLE of μ¯i., is consistent and is conditionally unbiased. We then find the asymptotic joint distribution of the Y¯i‥ and use it to construct a sensible asymptotic size α test of the equality of the μ¯i. and asymptotic simultaneous (1−α) confidence intervals for contrasts in the μ¯i..

Min-rank conjecture for log-depth circuits

A completion of an m-by- n matrix A with entries in { 0 , 1 , ⁎ } is obtained by setting all ⁎-entries to constants 0 and 1. A system of semi-linear equations over G F 2 has the form M x = f ( x ) , where M is a completion of A and f : { 0 , 1 } n → { 0 , 1 } m is an operator, the ith coordinate of which can only depend on variables corresponding to ⁎-entries in the ith row of A. We conjecture that no such system can have more than 2 n − ϵ ⋅ mr ( A ) solutions, where ϵ > 0 is an absolute constant and mr ( A ) is the smallest rank over G F 2 of a completion of A. The conjecture is related to an old problem of proving super-linear lower bounds on the size of log-depth boolean circuits computing linear operators x ↦ M x . The conjecture is also a generalization of a classical question about how much larger can non-linear codes be than linear ones. We prove some special cases of the conjecture and establish some structural properties of solution sets.

A hybrid heuristic for the set covering problem

In this paper, we present a hybrid approach combining an artificial bee colony (ABC) algorithm with a local search to solve the non-unicost Set Covering Problem (SCP). Given a 0–1 matrix where each column is associated with a non-negative cost. A 1 at the jth column of ith row of this matrix indicates that row i is covered by column j, whereas, a 0 indicates that it is not. The objective of the SCP is to find a subset of columns with minimum cost such that each row of the matrix is covered by at least one column. The ABC algorithm is a recent metaheuristic technique based on the intelligent foraging behavior of honey bee swarm. Computational results show that our ABC algorithm is competitive in terms of solution quality with other metaheuristic approaches for the SCP problem.

Uniqueness of solutions for multidimensional BSDEs with uniformly continuous generators

Hamadène (2003) [2] obtained an existence result of solutions for multidimensional backward stochastic differential equations (BSDEs) with uniformly continuous generators, provided that the ith component g i ( t , y , z ) of the generator g depends only on the ith row of the matrix z. The uniqueness of solutions for this kind of BSDEs is proved in this Note.

Silver Cubes

An n × n matrix A is said to be silver if, for i = 1,2,...,n, each symbol in {1,2,...,2n − 1} appears either in the ith row or the ith column of A. The 38th International Mathematical Olympiad asked whether a silver matrix exists with n = 1997. More generally, a silver cube is a triple (K n d , I, c) where I is a maximum independent set in a Cartesian power of the complete graph K n , and $$c:V(K_n^d)\rightarrow \{1,2,\dots,d(n-1)+1\}$$is a vertex colouring where, for v ∈ I, the closed neighbourhood N[v] sees every colour. Silver cubes are related to codes, dominating sets, and those with n a prime power are also related to finite geometry. We present here algebraic constructions, small examples, and a product construction. The nonexistence of silver cubes for d = 2 and some values of n, is proved using bounds from coding theory.

Unipotent Elements, Standard Involutions, and the Divisor Matrix

We give an explicit description of a matrix T(n) in GL n (ℤ) with the property that T(n) = T(n)−1 and , where D n = (d ij ) is the divisor matrix whose (i,j)-entry is For all n ∈ ℕ, the matrices D n and T(n) are obtained as the truncations of semi-infinite matrices D and T also satisfying the relations T = T −1 and T D T −1 = D −1. By encoding the entries of the ith row of the semi-infinite matrix T in a Dirichlet series we give a description of the coefficients of T.