equality conditions for matrix kantorovich type inequalities abstract following a recent paper of 7 we present sufficient and necessary conditions snecs under which equalities occur in those corresponding matrix kantorovich type inequalities we also present several relevant inequalities

fully automatic hp adaptivity for maxwell s equations i report on the development of a fully automatic hp adaptive strategy for the solution of time harmonic maxwell equations the strategy produces a sequence of grids that deliver exponential convergence for both regular and singular solutions given a coarse mesh we refine it first globally in both h and p and solve the problem on the resulting fine mesh we consider then the projection based interpolants of the fine mesh solution with respect to both current and next to be determined coarse grid and introduce the interpolation error decrease rate equal to the difference of the old and new coarse mesh interpolation errors vs number of degrees of freedom added the optimal hp refinements leading to the next coarse grid are then determined by maximizing the interpolation error decrease rate

a matrix version of the wielandt inequality and its applications to statistics abstract suppose that a is an n n positive definite hermitian matrix let x and y be n p and n q matrices respectively such that x y 0 the present article proves the following inequality x ay y ay y ax 1 n 1 n 2 x ax where 1 and n are respectively the largest and smallest eigenvalues of a and m stands for a generalized inverse of m this inequality is an extension of the well known wielandt inequality in which both x and y are vectors the inequality is utilized to obtain some interesting inequalities about covariance matrix and various correlation coefficients including the canonical correlation multiple and simple correlations some applications in parameter estimation are also given

more operator versions of the schwarz inequality some new operator versions of the schwarz inequality are obtained one of them is a counterpart of the variance covariance inequality in the context of noncommutative probability

some comments on six inequalities associated with the inefficiency of ordinary least squares with one regressor abstract the study of the inefficiency of the ordinary least squares estimator olse with one regressor by watson 1951 required a lower bound for the efficiency defined as the ratio of the variance of the best linear unbiased estimator blue to the variance of the olse such a lower bound was provided by the cassels inequality 1951 which we note is closely related to five other inequalities including the well known inequality usually attributed to kantorovich 1948 but which was established already by frucht 1943 the main purpose in this paper is to show how these six inequalities are related with a historical perspective we present some proofs and conclude that all six inequalities are essentially equivalent in the sense that any one inequality implies the other five we identify conditions for equality in each inequality and present the six continuous integral analogues we end the paper with english translations of the seminal papers by frucht 1943 and schweitzer 1914 respectively from the spanish and hungarian and a fairly extensive bibliography

a direct proof and a generalization for a kantorovich type inequality a direct proof for a kantorovich type inequality due to bauer and householder is presented a generalization of the inequality is also established by the theory of compound matrices

some further results on hermitian matrix inequalities abstract some new results on the lowner partial ordering between certain sums products and direct products of matrices are derived proofs of these results are based on a necessary and sufficient condition for the nonnegative definiteness of the difference of two hermitian matrices which follows from the criterion for the nonnegative definiteness of a partitioned hermitian matrix due to albert 1969

some further matrix extensions of the cauchy schwarz and kantorovich inequalities with some statistical applications abstract the well known cauchy schwarz and kantorovich inequalities may be expressed in terms of vectors and a positive definite matrix we consider what happens to these inequalities when the vectors are replaced by matrices the positive definite matrix is allowed to be positive semidefinite singular and the usual inequalities are replaced by lowner partial orderings some examples in the context of linear statistical models are presented

goal oriented error estimation and adaptivity for the finite element method abstract in this paper we study a new approach in a posteriori error estimation in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm these so called quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs we present here the theory with respect to a class of elliptic boundary value problems and in particular show how to obtain accurate estimates as well as upper and lower bounds on the error we also study the new concept of goal oriented adaptivity which embodies mesh adaptation procedures designed to control error in specific quantities numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates

the inefficiency of least squares extensions of the kantorovich inequality abstract four different measures of inefficiency of the simple least squares estimator in the general gauss markoff model are considered previous work on the bounds to some of these measures is briefly reviewed and new bounds are obtained for a particular measure

on the bundle of connections and the gauge orbit manifold in yang mills theory in an appropriate mathematical framework we supply a simple proof that the quotienting of the space of connections by the group of gauge transformations in yang mills theory is ac principal fibration the underlying quotient space the gauge orbit space is seen explicitly to be ac manifold modelled on a hilbert space

the eigenvalues range of a class of matrices and some applications in cauchy schwarz inequality and iterative methods abstract this paper discusses the range of the eigenvalues of a class of matrices by using the eigenvalues range of a class of matrices an extension of the inner product type cauchy schwarz inequality is obtained the convergence proof of the least squares based iterative algorithm for solving the coupled sylvester matrix equations is given and the best convergence factor is determined moreover by using the eigenvalues range of this class of matrices an iterative algorithm for solving linear matrix equation is established three numerical examples are offered to illustrate the effectiveness of the results suggested in this paper

characterizations of the best linear unbiased estimator in the general gauss markov model with the use of matrix partial orderings abstract under the general gauss markov model y x 2 v two new characterizations of blue x are derived involving the lowner and rank subtractivity partial orderings between the dispersion matrix of blue x and the dispersion matrix of y as particular cases of these characterizations three new criteria for the equality between olse x and blue x are given

a converse to the jensen inequality its matrix extensions and inequalities for minors and eigenvalues we obtain a scalar inequality converse to the jensen inequality we also derive an operator converse to the jensen inequality as special cases we obtain inequalities similar to the kantorovich one as well as some operator generalizations of them using some exterior algebra we prove a generalization of the sylvester determinant theorem we also deduce some determinant analogs of the additive and multiplicative kantorovich inequalities

numerical simulation of borehole acoustic logging in the frequency and time domains with hp adaptive finite elements accurate numerical simulation of borehole acoustic measurements is of great relevance to improving the efficacy of acoustic logging techniques and to computationally estimating elastic formation properties such simulations require sound physical modeling combined with accurate and efficient numerical discretization and solution techniques the objective of this paper is to concomitantly model acoustic wave propagation in a fluid filled borehole coupled with elastic wave propagation both in the probed rock formation and in the elastic logging tool to ensure the accuracy and efficiency of our simulations we use a self adaptive finite element discretization method enhanced with perfectly matched layer spatial domain truncation this work constitutes the first application of automatic hp adaptivity to a coupled multi physics problem which requires the non trivial capability of propagating refinements between acoustics and elasticity subdomains through their common interface computations are carried out in the frequency domain subsequently using an inverse fourier transform frequency domain solutions are transformed into the time domain to obtain waveforms at the receiver positions numerical results are presented for monopole and dipole sources with and without the presence of the logging tool and for a layered formation to validate our method we compare our results to published reference data and to results obtained using an in house finite difference code convergence to a user specified tolerance for the discretization error confirms the accuracy delivered by our method in the presence of complex geometrical and physical conditions and indicates its potential for the simulation of borehole acoustic measurements

back propagating modes in elastic logging while drilling collars and their effect on pml stability the perfectly matched layer pml approach is widely used to implement the absorbing boundary conditions for coupled multi physics wave propagation problems however it has been recognized that the solution in the pml absorbing layer can become unstable in the presence of back propagating modes this paper analyzes the spectral location of those modes excited by monopole acoustic sources in logging while drilling cylindrical tools to recover the stability of the solution for such a class of problems involving the modeling of elastic cylindrical waveguides we propose the implementation of artificial attenuation in the waveguide to dampen undesirable modes thereby making their amplitude negligible in the pml absorbing layer due to the limited numerical accuracy and round off errors

new inequalities of the kantorovich type for bounded linear operators in hilbert spaces some new inequalities of the kantorovich type are established they hold for larger classes of operators and subsets of complex numbers than considered before in the literature and provide refinements of the classical results in the case when the involved operator satisfies the usual conditions several new reverse inequalities for the numerical radius of a bounded linear operator are obtained as well

adaptive neural output feedback control of output constrained nonlinear systems with unknown output nonlinearity this paper addresses the problem of adaptive neural output feedback control for a class of special nonlinear systems with the hysteretic output mechanism and the unmeasured states a modified bouc wen model is first employed to capture the output hysteresis phenomenon in the design procedure for its fusion with the neural networks and the nussbaum type function two key lemmas are established using some extended properties of this model to avoid the bad system performance caused by the output nonlinearity a barrier lyapunov function technique is introduced to guarantee the prescribed constraint of the tracking error in addition a robust filtering method is designed to cancel the restriction that all the system states require to be measured based on the lyapunov synthesis a new neural adaptive controller is constructed to guarantee the prescribed convergence of the tracking error and the semiglobal uniform ultimate boundedness of all the signals in the closed loop system simulations are implemented to evaluate the performance of the proposed neural control algorithm in this paper

a unified version of cauchy schwarz and wielandt inequalities abstract let a be an n n positive definite symmetric real matrix with n eigenvalues 1 2 n and let x and y be two n 1 vectors with the angle this paper proves the following inequality x t ay 2 max i j i cos 2 2 j sin 2 2 i cos 2 2 j sin 2 2 2 x t ax y t ay it is a unified version of the cauchy schwarz inequality and the wielandt one

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

on goal oriented error estimation for elliptic problems application to the control of pointwise errors a method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal oriented error estimation in the latter the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm goal oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates the performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one dimensional problem

kantorovich and cauchy schwarz inequalities involving positive semidefinite matrices and efficiency comparisons for a singular linear model abstract matrix kantorovich inequalities involving two positive semidefinite matrices are presented corresponding cauchy schwarz inequalities are discussed some of these are used to compare several efficient and inefficient estimators for a singular linear model

non representability of cohomology classes by bi invariant forms gauge and kac moody groups we give a necessary topological condition on a cohomology class of any lie group modelled on a frechet space to be representable by a bi invariant form on as a corollary we show that if for somed 0 then there exists a cohomology class in which cannot be represented by any bi invariant form in particular we conclude that there are many cohomology generators in general in the case of gauge groups and also kac moody groups which cannot be represented by bi invariant forms although very often they are representable by left invariant forms

nonnegative and positive definiteness of matrices modified by two matrices of rank one abstract necessary and sufficient conditions are given for the nonnegative and positive definiteness of matrices of the form a a 1 a 1 a 2 a 2 and a a 1 a 1 a 2 a 2 where a is a hermitian matrix and a 1 a 2 are complex vectors they are derived using some auxiliary results which seem to be of independent interest as well some particular cases of these conditions are also discussed especially in the context of related results known in the literature

operator trigonometry of statistics and econometrics a new and useful geometric point of view for the understanding and analysis of certain matrix methods as they are used in statistics and econometrics is presented applications to statistical efficiency parameter estimation and correlation theory are given in particular we show that worst case relative least squares efficiency although achieved by maximally inefficient regressors is also achieved by maximal covariance matrix turning vectors also we elaborate geometrically a commutator trace efficiency result of p bloomfield and g s watson biometrika 62 1975 121 well established lagrange multiplier methods for constrained optimizations are compared to use of an euler equation from the new geometric theory