Certain System Of Linear Equations Research Articles

An Algorithm to Compute the Fundamental Classes of Spin Components of Strata of Differentials

Abstract We construct an algorithm for computing the cycle classes of the spin components of a stratum of differentials in the moduli space of stable curves $\overline {\mathcal {M}}_{g,n}$. In addition, we implement it within the Sage package admcycles. Our main strategy is to reconstruct these cycles by their restrictions to the boundary of $\overline {\mathcal {M}}_{g,n}$ via clutching maps. These restrictions can be computed recursively by smaller dimensional spin classes and determine the original class via a certain system of linear equations. To study the spin parities on the boundary of a stratum of differentials of even type, we make use of the moduli space of multi-scale differentials introduced in [4]. As an application of our algorithm, one can verify a conjecture on spin double ramification cycles stated in [18] in many examples, by using the results computed by our algorithm.

A Rescaling Bootstrap Approach For Imputed Survey Data

Abstract Imputation is usually used to deal with item nonresponse in surveys. Treating the imputed values as true observations may obviously lead to serious underestimation of the variance of point estimators. In this article, we propose a new bootstrap method under the rescaling bootstrap approach for estimating the variance of an imputed estimator obtained after applying deterministic regression or random hot-deck imputation. A novel technique is used to rescale the original data set through solving certain systems of linear equations. The proposed procedure can handle unequal response probabilities and large sampling fractions. Some simulation studies are conducted to show the great performance of the proposed method in terms of relative bias, relative efficiency, and coverage probability, for both population mean and median.

A Feasible Algorithm for Solubility of Transitive Graphs

Transitive, weighted graphs, with the weights coming from an Abelian group and satisfying a transitivity property, were studied in [1] and were referred to as the transitive systems. Of central importance is the question of completing a given system to a transitive system where every ordered pair of the vertices can be assigned a weight. It was proven in [1] that, in the finite case, the question can be reduced to the situation when the Abelian group is a finite cyclic group. Nevertheless, in practical applications, the most common assignments of the weights are coming from the groups of numbers. For this reason, we consider the case when the underlying Abelian group G is torsion-free. We show that within the class of all torsion-free Abelian groups, the underlying transitive and unweighted graph predetermines the possibility of completion of a given transitive system. Such a graph is G-soluble. The question of G-solubility of a graph is equivalent to its ℝ-solubility. This, in turn, is equivalent to solvability of a certain system of linear equations in ℝ. As a result, G-solubility of a transitive graph can be determined by means of a feasible algorithm.

Weighted composite outer inverses

In order to extend and unify the definitions of W-weighted DMP, W-weighted MPD, W-weighted CMP and composite outer inverses, we present the weighted composite outer inverses. Precisely, the notions of MNOMP, MPMNO and MPMNOMP inverses are introduced as appropriate expressions involving the (M,N)-weighted (B,C)-inverse and Moore–Penrose inverse. Basic properties and a number of characterizations for the MNOMP, MPMNO or MPMNOMP inverse are discovered. Various representations and characterizations of weighted composite outer inverses are studied. General solutions for certain systems of linear equations are given in terms of weighted composite outer inverses. Numerical examples are presented on randomly generated matrices of various orders.

On circulant like matrices properties involving Horadam, Fibonacci, Jacobsthal and Pell numbers

In this work a new type of matrix called circulant-like matrix is introduced. This type of matrix includes the classical k-circulant matrix, introduced in [4], in a natural sense. Its eigenvalues and its inverse and some other properties are studied, namely, it is shown that the inverse of a matrix of this type is also a matrix of this type and that its first row is the unique solution of a certain system of linear equations. Additionally, for some of these matrices whose entries are written as function of Horadam, Fibonacci, Jacobsthal and Pell numbers we study its eigenvalues and write it as function of those numbers. Moreover, the same study is done if the entries are arithmetic sequences.

Composite factors of binomials and linear systems in roots of unity

In this paper we completely classify binomials in one variable which have a nontrivial factor which is composite, i.e., of the shape g(h(x)) for polynomials g, h both of degree > 1. In particular, we prove that, if a binomial has such a composite factor, then deg g ≤ 2 (under natural necessary conditions). This is best-possible and improves on a previous bound deg g ≤ 24. This result provides evidence toward a conjecture predicting a similar bound when binomials are replaced by polynomials with any given number of terms. As an auxiliary result, which could have other applications, we completely classify the solutions in roots of unity of certain systems of linear equations.

The Mechanization of the Comrade Matrix Approach in Determining the GCD of Orthogonal Polynomials

This paper revisits the comrade matrix approach in finding the greatest common divisor (GCD) of two orthogonal polynomials. The present work investigates on the applications of the QR decomposition with iterative refinement (QRIR) to solve certain systems of linear equations which is generated from the comrade matrix. Besides iterative refinement, an alternative approach of improving the conditioning behavior of the coefficient matrix by normalizing its columns is also considered. As expected the results reveal that QRIR is able to improve the solutions given by QR decomposition while the normalization of the matrix entries do improves the conditioning behavior of the coefficient matrix leading to a good approximate solutions of the GCD.

Reconstructing a phylogenetic level-1 network from quartets.

We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on [Formula: see text] taxa from the set of all quartets containing a certain fixed taxon, in [Formula: see text] time. We also present a more general method which can handle more diverse quartet data, but which takes [Formula: see text] time. Both methods proceed by solving a certain system of linear equations over the two-element field [Formula: see text]. For a general dense quartet set, i.e. a set containing at least one quartet on every four taxa, our [Formula: see text] algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an [Formula: see text]-sized certificate of inconsistency otherwise. This answers a question raised by Gambette, Berry and Paul regarding the complexity of reconstructing a level-1 network from a dense quartet set, and more particularly regarding the complexity of constructing a cyclic ordering of taxa consistent with a dense quartet set.

A Method for Constructing Invariants of a Lie Group

For the adjoint action of a Lie group G (or a subgroup of G) on the Lie algebra Lie (G), we suggest a method for constructing its invariants. The method is easy to implement and may shed light on the algebraical independence of invariants. The main idea is to extend automorphisms of the Cartan subalgebra to automorphisms of the whole Lie algebra Lie (G). The corresponding matrices in the linear space V ∼ = Lie (G) define the Reynolds operator, which “gathers” invariants of the torus T ⊂ G into special polynomials. The condition for a linear combination of polynomials to be G-invariant is equivalent to the existence of a solution for a certain system of linear equations in the coefficients of the combination. As an example, we consider the adjoint action of the Lie group SL(3) (and its subgroup SL(2)) on the Lie algebra sl(3).

The Schlesinger system and isomonodromic deformations of bundles with connections on Riemann surfaces

We introduce a way to represent pairs (E,∇), where E is a bundle on a Riemann surface and ∇ is a logarithmic connection in E, based on a representation of the surface as the quotient of the exterior of the unit disc. In this representation, we write the local isomonodromic deformation conditions for the pairs (E,∇). These conditions are written as a modified Schlesinger system on a Riemann sphere (reduced to the ordinary Schlesinger system in the typical case) supplemented by a certain system of linear equations.

A matrix-based ranking method with application to tennis

We introduce, and investigate, a ranking methodology which may be of interest in sports like tennis. The approach may also be of interest in decision-making situations based on pairwise comparisons. The method is based on linear algebra and one computes a score for each player by solving a certain linear system of equations – from these scores one finds the ranking. The input is a set of matches, and weights representing the importance of the matches; this is represented by a weighted directed graph. We prove a number of properties of the method, including uniqueness of scores, connection to M-matrices and combinatorial interpretations. A case study from ranking in professional tennis is discussed in detail.

Selectivity in probabilistic causality: Where psychology runs into quantum physics

Given a set of several inputs into a system (e.g., independent variables characterizing stimuli) and a set of several stochastically non-independent outputs (e.g., random variables describing different aspects of responses), how can one determine, for each of the outputs, which of the inputs it is influenced by? The problem has applications ranging from modeling pairwise comparisons to reconstructing mental processing architectures to conjoint testing. A necessary and sufficient condition for a given pattern of selective influences is provided by the Joint Distribution Criterion, according to which the problem of “what influences what” is equivalent to that of the existence of a joint distribution for a certain set of random variables. For inputs and outputs with finite sets of values this criterion translates into a test of consistency of a certain system of linear equations and inequalities (Linear Feasibility Test) which can be performed by means of linear programming. While new in the behavioral context, both this test and the Joint Distribution Criterion on which it is based have been previously proposed in quantum physics, in dealing with generalizations of Bell inequalities for the quantum entanglement problem. The parallels between this problem and that of selective influences in behavioral sciences are established by observing that noncommuting measurements in quantum physics are mutually exclusive and can therefore be treated as different levels of one and the same factor.

Equivalence of the Navier-Stokes equation and the infinite-dimensional Burgers equation

In this paper, we prove the equivalence of the Navier-Stokes equation and the infinite-dimensional Burgers-type equation with the Laplace-Lévy operator. An explicit formula for the solution of a certain system of linear equations arising in studying the circulation of the solution of the Navier-Stokes equation is presented.

On the computational complexity of partial covers of Theta graphs

By use of elementary geometric arguments we prove the existence of a special integral solution of a certain system of linear equations. The existence of such a solution then yields the NP-hardness of the decision problem on the existence of locally injective homomorphisms to Theta graphs with three distinct odd path lengths.

Numerically Stable Implementations of the Structured Covariance Expectation-Maximization Algorithm

Numerically stable methods for the computations required in the expectation-maximization (EM) algorithm for maximum-likelihood structured covariance estimation are presented. It is shown that the basic computational task at each iteration is the calculation of a pseudoinverse of a certain linear system of equations. In the no-noise case this is a hard-decision, or Moore–Penrose, pseudoinverse, whereas in the additive noise case it is a soft-decision pseudoinverse. An approach to computing the soft-decision pseudoinverse, which can handle all combinations of dimension and rank in this system of equations, based on the singular value decomposition (SVD), is proposed. An alternative method based on the LQ factorization, which is applicable in certain circumstances, is also proposed. The intermediate calculations required in the EM algorithm can be used to calculate the log-likelihood and the gradient of the log-likelihood.

SIGEST

How often is it the case that not only the applications but also the methods of analysis of an important mathematical property range all the way from the deeply mathematical to the realm of the social sciences (and beyond)? Or, for that matter, that one can learn about the property under consideration in some detail on Wikipedia? The {small world phenomenon} is an instance of all of these characteristics.For the mathematical community, the small world phenomenon came alive with the 1998 paper of Watts and Strogatz that looked at the phenomenon from the perspective of a lattice network model and showed that the addition of a relatively small number of randomly chosen long—range connections to an otherwise locally connected lattice leads to a very good model of the small world phenomenon. Their work and others showed further that this type of model describes the actual connectivity and behavior of applications in areas as diverse as the brain of a small organism, the electrical power grid, and connectivity on the World Wide Web.This issue's SIGEST paper, ’’A Matrix Perturbation View of the Small World Phenomenon’’ by D. J. Higham that originally was published in the { SIAM Journal on Matrix Analysis and Applications} in 2003, makes an inspired and important contribution to the mathematical understanding of the small world phenomenon by looking at it from a new perspective. As Higham discusses in the new preamble to the paper, he was motivated to formulate a model that could display the small world phenomenon and be amenable to rigorous mathematical analysis. He succeeded by considering a Markov chain that also can be thought of as a one—dimensional periodic random walk, where the standard connections to the two nearest neighbors are augmented by low probability uniform jumps to any state, also referred to as shortcuts. Higham uses matrix perturbation theory methods to study the correspondence between the probabilities of these uniform jumps and the mean hitting time (the average number of steps to get from some random state to a given fixed state) of the random walk. In particular, the mean hitting times of a Markov chain can be computed by solving a certain system of linear equations, and by use of matrix perturbation theory, Higham analyzes how these times vary with the probability of the uniform jumps. The paper shows that this model is very effective in explaining the small world phenomenon mathematically, and that in addition, its mathematical behavior corroborates behavior that was observed experimentally by Watts and Strogatz. Higham's model also has a close connection to techniques used in Internet search engines.We hope that many SIAM readers will enjoy this interesting, accessible, and very nicely written paper that adds great insight into an important property of the natural and social worlds. And those of you with Hollywood connections will have something novel to bring up the next time the conversation turns to the ’’Six Degrees of Kevin Bacon.’’

Reducing the Impact of Measurement Errors When Reconstructing Spatial Dynamic Forces

Inferring external spatially distributed dynamic forces from measured structural responses is necessary when direct measurement of these forces is not possible. The finite difference method and the modal method have been previously developed for reconstructing these forces. However, the accuracy of these methods is often hindered due to the amplification of measurement errors in the computation process. In order to analyze these error amplification effects by using the singular value decomposition approach, the mathematic expressions for these two force reconstruction methods are first transformed into a certain linear system of equations. Then, a regularization method, the Tikhonov method, is applied to increase computational stability. In order to achieve a good regularized result, the L-curve method is used in conjunction with the Tikhonov method. The effectiveness in reducing the influence of the measurement errors when applying the regularization method to the finite difference method and the modal method is investigated analytically and numerically. It is found that when the regularization method is appropriately applied, reliable computational results for the reconstructed force can be achieved.

Breakdown of equivalence between the minimal [formula omitted]-norm solution and the sparsest solution

Finding the sparsest solution to a set of underdetermined linear equations is NP-hard in general. However, recent research has shown that for certain systems of linear equations, the sparsest solution (i.e. the solution with the smallest number of nonzeros), is also the solution with minimal ℓ1 norm, and so can be found by a computationally tractable method.For a given n by m matrix Φ defining a system y=Φα, with n<m making the system underdetermined, this phenomenon holds whenever there exists a ‘sufficiently sparse’ solution α0. We quantify the ‘sufficient sparsity’ condition, defining an equivalence breakdown point (EBP): the degree of sparsity of α required to guarantee equivalence to hold; this threshold depends on the matrix Φ.In this paper we study the size of the EBP for ‘typical’ matrices with unit norm columns (the uniform spherical ensemble (USE)); Donoho showed that for such matrices Φ, the EBP is at least proportional to n. We distinguish three notions of breakdown point—global, local, and individual—and describe a semi-empirical heuristic for predicting the local EBP at this ensemble. Our heuristic identifies a configuration which can cause breakdown, and predicts the level of sparsity required to avoid that situation. In experiments, our heuristic provides upper and lower bounds bracketing the EBP for ‘typical’ matrices in the USE. For instance, for an n×m matrix Φn,m with m=2n, our heuristic predicts breakdown of local equivalence when the coefficient vector α has about 30% nonzeros (relative to the reduced dimension n). This figure reliably describes the observed empirical behavior. A rough approximation to the observed breakdown point is provided by the simple formula 0.44·n/log(2m/n).There are many matrix ensembles of interest outside the USE; our heuristic may be useful in speeding up empirical studies of breakdown point at such ensembles. Rather than solving numerous linear programming problems pern,m combination, at least several for each degree of sparsity, the heuristic suggests to conduct a few experiments to measure the driving term of the heuristic and derive predictive bounds. We tested the applicability of this heuristic to three special ensembles of matrices, including the partial Hadamard ensemble and the partial Fourier ensemble, and found that it accurately predicts the sparsity level at which local equivalence breakdown occurs, which is at a lower level than for the USE. A rough approximation to the prediction is provided by the simple formula 0.65·n/log(1+10m/n).

On polynomials orthogonal to all powers of a given polynomial on a segment

In this paper we investigate the following “polynomial moment problem”: for a given complex polynomial P ( z ) and distinct a , b ∈ C to describe polynomials q ( z ) orthogonal to all powers of P ( z ) on [ a , b ] . We show that for given P ( z ) , q ( z ) the condition that q ( z ) is orthogonal to all powers of P ( z ) is equivalent to the condition that branches of the algebraic function Q ( P −1 ( z ) ) , where Q ( z ) = ∫ q ( z ) d z , satisfy a certain system of linear equations over Z . On this base we provide the solution of the polynomial moment problem for wide classes of polynomials. In particular, we give the complete solution for polynomials of degree less than 10.

On the computational complexity of partial covers of Theta graphs

By use of elementary geometric arguments we prove the existence of a special integral solution of a certain system of linear equations. The existence of such a solution then yields the NP-hardness of the decision problem on the existence of locally injective homomorphisms to Theta graphs with three distinct odd path lengths.