Variations on the theme of solvability by radicals

We discuss the problem of representability and nonrepresentability of algebraic functions by radicals. We show that the Riemann surfaces of functions that are the inverses of Chebyshev polynomials are determined by their local behavior near branch points. We find lower bounds on the degrees of equations to which sufficiently general algebraic functions can be reduced by radicals. We also begin to classify rational functions of prime degree whose inverses are representable by radicals. Original Russian Text ? A.G. Khovanskii, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 86–105. To Vladmir Igorevich Arnold, mathematical idol of my generation     

The Algebraic Perturbation Method for Generalized Inverses

Algebraic perturbation methods were first proposed for the solution of nonsingular linear systems by R. E. Lynch and T. J. Aird [2]. Since then, the algebraic perturbation methods for generalized inverses have been discussed by many scholars [3]-[6]. In [4], a singular square matrix was perturbed algebraically to obtain a nonsingular matrix, resulting in the algebraic perturbation method for the Moore-Penrose generalized inverse. In [5], some results on the relations between nonsingular perturbations and generalized inverses of $m\times n$ matrices were obtained, which generalized the results in [4]. For the Drazin generalized inverse, the author has derived an algebraic perturbation method in [6]. In this paper, we will discuss the algebraic perturbation method for generalized inverses with prescribed range and null space, which generalizes the results in [5] and [6]. We remark that the algebraic perturbation methods for generalized inverses are quite useful. The applications can be found in [5] and [8]. In this paper, we use the same terms and notations as in [1].     

Inner Generalized Inverses with Prescribed Idempotents

We define and characterize inner generalized inverses with prescribed idempotents. These classes of generalized inverses are natural algebraic extension of generalized inverses of linear operators with prescribed range and kernel. We consider the reverse order rule for inner generalized inverses of elements of a ring, some perturbation bounds, and we construct an iterative method for computing a (p, q)-inner inverse in Banach algebras.     

On inverses and algebraic loops of co-H-spaces

In this paper we study the properties of homotopy inverses of comultiplications and algebraic loops of co-H-spaces based on a wedge of spheres. We also investigate a method to construct new comultiplications out of old ones by using a group action. We are primarily interested in the algebraic loops which have inversive, power-associative and Moufang properties for some comultiplications.     

ON INVERSES AND ALGEBRAIC LOOPS OF CO-H-SPACES

In this paper we study the properties of homotopy inverses of comultiplications and Mgebraic loops of co-H-spaces based on a wedge of spheres. We also investigate a method to construct new comultiplications out of old ones by using a group action. We are primarily interested in the algebraic loops which have inversive, power-associative and Moufang properties for some comultiplications.     

Estimations of parametric functions under a system of linear regression equations with correlated errors

Estimations of parametric functions under a system of linear regression equations with correlated errors across equations involve many complicated operations of matrices and their generalized inverses. In the past several years, a useful tool -- the matrix rank method was utilized to simplify various complicated operations of matrices and their generalized inverses. In this paper, we use the matrix rank method to derive a variety of new algebraic and statistical properties for the best linear unbiased estimators （BLUEs） of parametric functions under the system. In particular, we give the necessary and sufficient conditions for some equalities, additive and block decompositions of BLUEs of parametric functions under the system to hold.     

Improved outer inverses for the numerical solution of Euler–Lagrange equations

In this paper, improved outer inverses are used for solving the overdetermined differential‐algebraic equations that arise in the simulation of mechanical systems. The method is based on Newton‐type iterations using outer inverses. Copyright © 2004 John Wiley & Sons, Ltd.     

Resolvent expansions of matrices and applications

Various explicit expansions of the resolvent of a square complex matrix in a neighborhood of the origin, including the well-known Laurent expansion, are obtained. Simple proofs using algebraic arguments rather than the theory of complex functions are given. Cesaro-average expansions of the resolvent on the spectral circle are also developed. Several applications of these expansions are given. In particular, these include some new inverse nonnegativity properties of the Drazin inverses of M-matrices.     

On general frame decompositions

We provide a characterization and construction of general frame decompositions. We show that generating all duals for a given frame amounts to finding left inverses of an one-to-one mapping. A general parametric and algebraic formula for all duals is derived. An application of the theory to Weyl-Heisenberg (WH) frames is discussed. Besides the (usual) dual frame that preserves the translation and modulation structure, we construct a class of duals that attain such a structure. We also show constructively that there are duals to WH frames which are not the translation and modulation of a single function.     

Remark on convergence of algebraic multigrid in the form of matrix decomposition

We introduce the convergence of algebraic multigrid in the form of matrix decomposition. The convergence is proved in block versions of the multi-elimination incomplete LU (BILUM) factorization technique and the approximation of their inverses to preserve sparsity. The convergence theorem can be applied to general interpolation operator. Furthermore, we discuss the error caused by the error matrix.     

Outer inverses and multi body system simulation

In this paper, a numerical method for solving overdetermined differential algebraic equations that arise in multi body dynamics is proposed. The method is based on Newton type iterations using outer inverses. We prove that the ssf method and tangent space parametrization can be regarded as particular cases of our method     

Classroom note: Sawtooth functions

Using MAPLE enables students to consider many examples which would be very tedious to work out by hand. This applies to graph plotting as well as to algebraic manipulation. The challenge is to use these observations to develop the students’ understanding of mathematical concepts. In this note an interesting relationship arising from inverse trigonometric functions is analysed. To understand what is going on students have to develop an understanding of how to deal with inverses where a function is not 1–1, by restricting the domain. The piece of work developed here also provides some interesting exercises in proof by induction.     

函子的广义逆

本文建立了函子的广义逆．得到：（１）函子的广义逆从不同于视矩阵为态射的另一角度，包含了“矩阵的广义逆”．（２）函子的广义道与态射的广义逆互不蕴含，但却有异曲同工之处．     

Estimates for coefficients of univalent functions from integral means and Grunsky inequalities

Sharp bounds are obtained for the coefficients of inverses of univalent functions in the class Σ(p) by using results on integral means and generalized Grunsky inequalities. A new and elementary proof is given for a result due to Löwner about sharp bounds for coefficients of inverses of functions in the classS.     

A note on generalized inverses

Motivated by too restrictive or even incorrect statements about generalized inverses in the literature, properties about these functions are investigated and proven. Examples and counterexamples show the importance of generalized inverses in mathematical theory and its applications.     

Algorithms for finding the minimal polynomials and inverses of resultant matrices

In this paper, algorithms for computing the minimal polynomial and the common minimal polynomial of resultant matrices over any field are presented by means of the approach for the Gröbner basis of the ideal in the polynomial ring, respectively, and two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for the inverse of partitioned matrix with resultant blocks over any field is given, which can be realized by CoCoA 4.0, an algebraic system over the field of rational numbers or the field of residue classes of modulo prime number. We get examples showing the effectiveness of the algorithms.     

Numerical integration of positive linear differential-algebraic systems

In the simulation of dynamical processes in economy, social sciences, biology or chemistry, the analyzed values often represent non-negative quantities like the amount of goods or individuals or the density of a chemical or biological species. Such systems are typically described by positive ordinary differential equations (ODEs) that have a non-negative solution for every non-negative initial value. Besides positivity, these processes often are subject to algebraic constraints that result from conservation laws, limitation of resources, or balance conditions and thus the models are differential-algebraic equations (DAEs). In this work, we present conditions under which both these properties, the positivity as well as the algebraic constraints, are preserved in the numerical simulation by Runge–Kutta or multistep discretization methods. Using a decomposition approach, we separate the dynamic and the algebraic equations of a given linear, positive DAE to give positivity preserving conditions for each part separately. For the dynamic part, we generalize the results for positive ODEs to DAEs using the solution representation via Drazin inverses. For the algebraic part, we use the consistency conditions of the discretization method to derive conditions under which this part of the approximation overestimates the exact solution and thus is non-negative. We analyze these conditions for some common Runge–Kutta and multistep methods and observe that for index-1 systems and stiffly accurate Runge–Kutta methods, positivity is conditionally preserved under similar conditions as for ODEs. For higher index problems, however, none of the common methods is suitable.     

Leveraging Structure: Logical Necessity in the Context of Integer Arithmetic

Looking for, recognizing, and using underlying mathematical structure is an important aspect of mathematical reasoning. We explore the use of mathematical structure in children’s integer strategies by developing and exemplifying the construct of logical necessity. Students in our study used logical necessity to approach and use numbers in a formal, algebraic way, leveraging key mathematical ideas about inverses, the structure of our number system, and fundamental properties. We identified the use of carefully chosen comparisons as a key feature of logical necessity and documented three types of comparisons students made when solving integer tasks. We believe that logical necessity can be applied in various mathematical domains to support students to successfully engage with mathematical structure across the K–12 curriculum.     

Reconstructing birational maps from their face functions

We first prove that any birational map, from an affine space of dimension ≥ 2 to itself, is not determined by its face functions. On the other hand, we prove that a birational map with irreducibly polynomial inverse is completely determined, within the class of all birational maps with irreducibly polynomial inverses, by its face functions. We show also how to effectively reconstruct such a map from its face functions. Supported partly by the Centre Interuniversitaireen Calcul Mathématique Algébrique.