A generalization of the Bott–Duffin inverse and its applications

The ‘constrained inverse’ was introduced by Bott and Duffin in 1953. Since then various generalizations have arisen. In this paper, we propose a generalization of the Bott–Duffin inverse. Applications of the projection methods for solving sparse linear systems, the truncated methods for solving least squares problems, and solvers for generalized saddle point problems are also presented. The relationships with other generalized Bott–Duffin inverses are explored. The results indicate that the generalized Bott–Duffin inverse may be a useful tool in matrix computations. Copyright © 2008 John Wiley & Sons, Ltd.     

On Presentations of Bruck–Reilly Extensions

We first consider the class of monoids in which every left invertible element is also right invertible, and prove that if a monoid belonging to this class admits a finitely presented Bruck–Reilly extension then it is finitely generated. This allow us to obtain necessary and sufficient conditions for the Bruck–Reilly extensions of this class of monoids to be finitely presented. We then prove that thes 𝒟-classes of a Bruck–Reilly extension of a Clifford monoid are Bruck–Reilly extensions of groups. This yields another necessary and sufficient condition for these Bruck–Reilly extensions to be finitely generated and presented. Finally, we show that a Bruck–Reilly extension of a Clifford monoid is finitely presented as an inverse monoid if and only if it is finitely presented as a monoid, and that this property cannot be generalized to Bruck–Reilly extensions of arbitrary inverse monoids.     

Towards an Operator-Algebraic Construction of Integrable Global Gauge Theories

The recent construction of integrable quantum field theories on two-dimensional Minkowski space by operator-algebraic methods is extended to models with a richer particle spectrum, including finitely many massive particle species transforming under a global gauge group. Starting from a two-particle S-matrix satisfying the usual requirements (unitarity, Yang–Baxter equation, Poincaré and gauge invariance, crossing symmetry, . . .), a pair of relatively wedge-local quantum fields is constructed which determines the field net of the model. Although the verification of the modular nuclearity condition as a criterion for the existence of local fields is not carried out in this paper, arguments are presented that suggest it holds in typical examples such as non-linear O(N)   σ-models. It is also shown that for all models complying with this condition, the presented construction solves the inverse scattering problem by recovering the S-matrix from the model via Haag–Ruelle scattering theory, and a proof of asymptotic completeness is given.     

Kinematic and dynamic analysis of the Gantry-Tau,a 3-DoF translational parallel manipulator

This paper deals with the mathematical modeling of kinematics and dynamics of the 3-degrees-of-freedom Gantry-Tau manipulator. Compared to many other parallel robots, Gantry-Tau offers a large accessible workspace and high stiffness. The kinematics of Gantry-Tau is presented which includes inverse kinematics formulation for the position, velocity and acceleration of the mechanism. Also, based on the obtained Jacobin matrices, singular configurations of the robot are studied. Afterwards, the equations of the inverse dynamic model of the Gantry-Tau are obtained through two different methods, i.e., virtual work and Newton–Euler. Finally, a case study is performed to verify the correctness of the derived models and investigate their computational efficiency.     

Nonnegative Generalized Inverses and Interval Linear Programs

Connections between nonnegative generalized inverses and interval linear programs form the subject matter of this article. Characterizations of nonnegativity of the Moore–Penrose inverse and the group inverse are presented and their role in the study of a special class of optimization spaces, namely, Ben-Israel–Charnes spaces is brought out. The interplay between Ben-Israel–Charnes spaces and other notions of generalized inverse positivity are also studied.     

Characterizations of m-EP elements in rings

Let R be a ring with involution. In this paper, we extend the notions of m-EP matrices and m-EP operators to an arbitrary ring case. A number of new characterizations of m-EP elements in rings are presented. In particular, the existence criteria for 1-EP (i.e. EP) elements are obtained by means of the group inverse, Moore–Penrose inverse, and core inverse. Some properties of 2-EP are also given.     

Direct and inverse boundary value problems for models of stationary reaction–convection–diffusion

Direct and inverse boundary value problems for models of stationary reaction–convection–diffusion are investigated. The direct problem consists in finding a solution of the corresponding boundary value problem for given data on the boundary of the domain of the independent variable. The peculiarity of the direct problem consists in the inhomogeneity and irregularity of mixed boundary data. Solvability and stability conditions are specified for the direct problem. The inverse boundary value problem consists in finding some traces of the solution of the corresponding boundary value problem for given standard and additional data on a certain part of the boundary of the domain of the independent variable. The peculiarity of the inverse problem consists in its ill-posedness. Regularizing methods and solution algorithms are developed for the inverse problem.     

A numerical method for determining the localized initial condition in the FitzHugh-Nagumo and Aliev-Panfilov models

The inverse problem for the FitzHugh-Nagumo and Aliev-Panfilov models describing wave propagation in excitable media is considered. The problem lies in determining a localized initial condition from measurements on the external boundary of a plane region. A numerical method for solving the inverse problem is proposed, and the results from a numerical solution of the inverse problem for regions similar to different sections of a heart are presented.     

Two New Mixture Models Related to the Inverse Gaussian Distribution

This article presents a new family of logarithmic distributions to be called the sinh mixture inverse Gaussian model and its associated life distribution referred as the extended mixture inverse Gaussian model. Specifically, the density, distribution function, and moments are developed for the sinh mixture inverse Gaussian distribution. Next, the extended mixture inverse Gaussian distribution is characterized. A graphical analysis of the densities of the new models is also provided. In addition, a lifetime analysis is presented for the extended mixture inverse Gaussian distribution. Finally, an example with a real data set is given to illustrate the methodology, which indicates that the new models result in a better fit to the data than some other well-known distributions.     

Estimates for the inverse elements of tridiagonal matrices

In this work, new upper and lower bounds for the inverse entries of the tridiagonal matrices are presented. The bounds improve the bounds in D. Kershaw [Inequalities on the elements of the inverse of a certain tridiagonal matrix, Math. Comput. 24 (1970) 155–158], P.N. Shivakumar, C.X. Ji [Upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices, Linear Algebr. Appl. 247 (1996) 297–316], R. Nabben [Two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear Algebr. Appl. 287 (1999) 289–305] and R. Peluso, T. Politi [Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear. Algebr. Appl. 330 (2001) 1–14].     

指数族广义非线性随机系数模型的变离差检验

指数族广义非线性随机系数模型是Smith &; Heitjan［１０］和 Wei et al［１１］所研究模型的推广。该文分别在模型离差 (dispersion) 的权不变和变异时，讨论了指数族 广义非线性随机系数模型的变离差的检验问题，得到了score检验统计量。并利用欧洲野兔数据，分别对正态分布模型、Γ 分布模型和 逆高斯分布模型说明检验方法的有效性。     

An Algorithm for Solving Inverse Geoelectrics Problems Based on the Neural Network Approximation

A neural network approximation algorithm for solving inverse geoelectrics problems in the class of grid (block) models of media is presented. The algorithm is based on using neural networks for constructing an approximate inverse operator and enables formalized construction of solutions of inverse geoelectrics problem with a total number of sought-for medium parameters of ~ n · 10³. The correctness of the problem of constructing neural network inverse operators is considered. A posteriori estimates of the degree of ambiguity of solutions of the resulting inverse problem are calculated. The operation of the algorithm is illustrated by examples of 2D and 3D inversions of synthetic and field geoelectric data obtained by the MTS method.     

Application of Tikhonov’s regularization method to solve inverse problems for two population models

Tikhonov’s regularization method is applied to numerical solution of inverse problems for two population models. For the first model we solve the inverse problem that involves simultaneous determination of the mortality rate and the initial distribution of individuals given supplementary information on population density. For the second model we determine the growth rate of the individuals given additional information about their density. Examples of numerical solution are presented for both inverse problems. __________ Translated from Prikladnaya Matematika i Informatika, No. 23, pp. 5–14, 2006.     

The free idempotent generated locally inverse semigroup

In this paper we construct a model for the free idempotent generated locally inverse semigroup on a set X. The elements of this model are special vertex-labeled bipartite trees with a pair of distinguished vertices. To describe this model, we need first to introduce a variation of a model for the free pseudosemilattice on a set X presented in Auinger and Oliveira (On the variety of strict pseudosemilattices. Stud Sci Math Hungarica 50:207–241, 2013). A construction of a graph associated with a regular semigroup was presented in Brittenham et al. (Subgroups of free idempotent generated semigroups need not be free. J Algebra 321:3026–3042, 2009) in order to give a first example of a free regular idempotent generated semigroup on a biordered set E with non-free maximal subgroups. If G is the graph associated with the free pseudosemilattice on X, we shall see that the models we present for the free pseudosemilattice on X and for the free idempotent generated locally inverse semigroup on X are closely related with the graph G.     

Testing for Varying Dispersion in Exponential Family Nonlinear Models

A diagnostic model and several new diagnostic statistics are proposed for testing for varying dispersion in exponential family nonlinear models. A score statistic and an adjusted score statistic based on Cox and Reid (1987, J. Roy. Statist. Soc. Ser. B, 55, 467-471) are derived in normal, inverse Gaussian, and gamma nonlinear models. An adjusted likelihood ratio statistic is also given for normal and inverse Gaussian nonlinear models. The results of simulation studies are presented, which show that the adjusted tests keep their sizes better and are more powerful than the ordinary tests.     

Further results on generalized inverses of tensors via the Einstein product

The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.     

A problem in elasticity theory

The problem of determining the dimensions of the transverse cross-sections of a beam from the given frequencies of its natural vibrations is examined. Frequency spectra are indicated that determine the dimenions of the transverse cross-sections of the beam uniquely, an effective procedure is presented for solving the inverse problem, and a uniqueness theorem is proved. The method of standard models /1/ is used to solve the inverse problem.     

An efficient method for computing the inverse of arrowhead matrices

In this paper we propose a simple and effective method to find the inverse of arrowhead matrices which often appear in wide areas of applied science and engineering such as wireless communications systems, molecular physics, oscillators vibrationally coupled with Fermi liquid, and eigenvalue problems. A modified Sherman–Morrison inverse matrix method is proposed for computing the inverse of an arrowhead matrix. The effectiveness of the proposed method is illustrated and numerical results are presented along with comparative results.     

Variational methods of data assimilation and inverse problems for studying the atmosphere, ocean, and environment

Methods for the combined use of mathematical models and observational data for studying and forecasting the evolution of natural processes in the atmosphere, ocean, and environment are presented. Variational principles for estimation of functionals defined on a set of functions of state, parameters and sources of models of processes are a theoretical background. Mathematical models with allowance for uncertainties are considered as constraints to the class of functions. Attention is focused on methods of successive data assimilation and on inverse problems.     

The Inverse Problem for Ellis–Gohberg Orthogonal Matrix Functions

This paper deals with the inverse problem for the class of orthogonal functions that for the scalar case was introduced by Ellis and Gohberg (J Funct Anal 109:155–198, 1992). The problem is reduced to a linear equation with a special right hand side. This reduction allows one to solve the inverse problem for square matrix functions under conditions that are natural generalizations of those appearing in the scalar case. These conditions lead to a unique solution. Special attention is paid to the polynomial case. A number of partial results are obtained for the non-square case. Various examples are given to illustrate the main results and some open problems are presented.