an h minimax approach to the design of robust control systems abstract a realistic feedback design problem is posed based on the minimization of a weighted combination of the sensitivity and complementary sensitivity matrices a solution is obtained which makes use of the recently proposed methods for minimizing the sensitivity function alone

new kinds of continuities a sequence x n of points in a topological group is slowly oscillating if for any given neighborhood u of 0 there exist d d u 0 and n n u such that x m x n u if n n u and n m 1 d n it is well known that in a first countable hausdorff topological space a function f is continuous if and only if f x n is convergent whenever x n is applying this idea to slowly oscillating sequences one gets slowly oscillating continuity i e a function f defined on a subset of a topological group is slowly oscillating continuous if f x n is slowly oscillating whenever x n is slowly oscillating we study the concept of slowly oscillating continuity and investigate relations with statistical continuity lacunary statistical continuity and some other kinds of continuities in metrizable topological groups

pointwise i convergence and i convergence in measure of sequences of functions let i p n be an ideal we say that a sequence yn n n of real numbers is i convergent to y r if for every neighborhood u of y the set of n s satisfying yn u is in i basing upon this notion we define pointwise i convergence and i convergence in measure of sequences of measurable functions defined on a measure space with finite measure we discuss the relationship between these two convergences in particular we show that for a wide class of ideals including erd s ulam ideals and summable ideals the pointwise i convergence implies the i convergence in measure we also present examples of very regular ideals such that this implication does not hold

the matrix sign function a survey of the matrix sign function is presented including some historical background definitions and properties approximation theory and computational methods and condition theory and estimation procedures applications to areas such as control theory eigendecompositions and roots of matrices are outlined and some new theoretical results are also given

explicit polar decomposition and a near characteristic polynomial the 2 2 case abstract explicit algebraic formulas for the polar decomposition of a nonsingular real 2 2 matrix a are given as well as a classification of all integer 2 2 matrices that admit a rational polar decomposition these formulas lead to a functional identity which is satisfied by all nonsingular real 2 2 matrices a as well as by exactly one type of exceptional matrix a n for each n 2

lacunary statistical summability abstract the sequence x is said to be s convergent to l if for each 0 lim r 1 h r the number of k i r x k l 0 where k r is an increasing sequnece of integers such that k 0 0 h r k r k r 1 as r and i r k r 1 k r in this paper we define the s analog of the cauchy criterion for convergence and show that tt is equivalent to s convergence also s convergence is compared to other summability methods and it is shown that the s method can not be included by any matrix method in addition a tauberian theorem for s convergence is given

some results related with statistical convergence and berezin symbols abstract the concept of discrete statistical abel convergence is introduced in terms of berezin symbols we present necessary and sufficient condition under which a series n 0 a n with bounded sequence a n n 0 of complex numbers is discrete statistically abel convergent by using concept of statistical convergence we also give slight strengthening of a result of gokhberg and krein on compact operators

procrustes problems in finite dimensional indefinite scalar product spaces abstract some applications from multidimensional scaling in an environment of indefinite scalar products are investigated in particular the construction of points from given distances is considered and two variants of procrustes problems are discussed it is found that both procrustes problems can be solved with the help of h polar decompositions or g h polar decompositions of specified matrices furthermore a method for the numerical computation of h or g h polar decompositions of nonsingular matrices is described

roots of real matrices abstract an algorithm for computing the roots of a matrix with real elements and the real part of the eigenvalues not zero will be described the algorithm is based on the sign matrix given in an earlier paper and the projectors of the matrix

parallel algorithms for algebraic riccati equations the matrix sign function is the basis of a parallel algorithm for solving the generalized algebraic riccati equation three forms of the algorithm were implemented and tested on a distributed memory hypercube multiprocessor performance results indicate that the method is an excellent means of solving large scale problems on a parallel computer

ideal convergence characterization of the completion of linear n normed spaces in this paper we give the completion of a linear n normed space in terms of ideal convergence by introducing the concept of a uniformly continuous n norm

two singular value inequalities and their implications in h approach to control system design in this note we prove that if a and b are both nonnegative definite hermitian matrices and a b is also nonnegative definite then the singular values of a and b satisfy the inequalities sigma i a geq sigma i b where bar sigma cdot sigma 1 cdot geq sigma 2 cdot geq geq sigma m cdot underbar sigma denote the singular values of a matrix a consequence of this property is that in a nonsquare h infty optimization problem if sup omega bar sigma z j sigma underline underline delta sup omega bar sigma x j omega t y j omega t t lambda then the singular values of x and y satisfy the inequality lambda 2 geq max i sup omega sigma i 2 x sigma m i 1 2 y where m is the number of columns of the matrix z

factorizations into normal matrices in indefinite inner product spaces abstract we show that any nonsingular real or complex square matrix can be factorized into a product of at most three normal matrices one of which is unitary another is selfadjoint with eigenvalues in the open right half plane and the third one is normal involutory with a neutral negative eigenspace we call the last matrix normal neutral involutory here the words normal unitary selfadjoint and neutral are understood with respect to an indefinite inner product

numerically stable improved chebyshev halley type schemes for matrix sign function a general family of iterative methods including a free parameter is derived and proved to be convergent for computing matrix sign function under some restrictions on the parameter several special cases including global convergence behavior are dealt with it is analytically shown that they are asymptotically stable a variety of numerical experiments for matrices with different sizes is considered to show the effectiveness of the proposed members of the family

on relation between the ideal core and ideal cluster points abstract this article is motivated by the recent paper characterization of the ideal core by paolo leonetti jmaa 2019 we show that in the main result of that paper the first countability condition can be omitted and the compactness condition may be substituted by a strictly weaker one

strengthened h sup spl infin control via state feedback a majorization approach using algebraic riccati inequalities we extend the algebraic riccati equation approach to h sup spl infin h sup 2 state feedback design to incorporate additional constraints on the other nonmaximum singular values it is shown that a state feedback control law that satisfies a singular values singular vectors constraint exists if and only if a corresponding algebraic riccati inequality admits a stabilizing and positive definite solution the results are applied to devise a majorization approach for constructing state feedback controllers that strive to optimize some obvious design objectives

properties of simple density ideals abstract let g consist of all functions g 0 with g n and n g n 0 then for each g g the family z g a lim n card a n g n 0 is an ideal associated to the notion of so called upper density of weight g although those ideals have recently been extensively studied they do not have their own name in this paper for reader s convenience we propose to call them simple density ideals we partially answer 16 problem 5 8 by showing that every simple density ideal satisfies the property from 16 problem 5 8 earlier the only known example was the ideal z of sets of asymptotic density zero we show that there are c many non isomorphic in fact even incomparable with respect to kat tov order simple density ideals moreover we prove that for a given a g with card a b one can construct a family of cardinality c of pairwise incomparable with respect to inclusion simple density ideals which additionally are incomparable with all z g for g a we show that this cannot be generalized to kat tov order as z is maximal in the sense of kat tov order among all simple density ideals we examine how many substantially different functions g can generate the same ideal z g it turns out that the answer is either 1 or c depending on g

statistically monotonic and statistically bounded sequences of fuzzy numbers we introduce the statistical monotonicity and boundedness of a sequence of fuzzy numbers we also derive the analogue of monotone convergence theorem and prove the decomposition theorems for this type of sequences

on g continuity a function f on a topological space is sequentially continuous at a point u if given a sequence x n limx n u implies that limf x n f u this definition was modified by connor and grosse erdmann for real functions by replacing lim with an arbitrary linear functional g defined on a linear subspace of the vector space of all real sequences in this paper we extend this definition to a topological group x by replacing g a linear functional with an arbitrary additive function defined on a subgroup of the group of all x valued sequences and not only give new theorems in this generalized setting but also present theorems that have not been obtained for real functions so far

strong and a statistical comparisons for sequences abstract let t and a be two nonnegative regular summability matrices and w t p l and ca b denote the spaces of all bounded strongly t summable sequences with index p 0 and bounded summability domain of a respectively in this paper we show among other things that n is a multiplier from w t p l into ca b if and only if any subset k of positive integers that has t density zero implies that k has a density zero these results are used to characterize the a statistical comparisons for both bounded as well as arbitrary sequences using the concept of a statistical tauberian rate we also show that n is not a multiplier from w t p l into ca b that leads to a steinhaus type result

ideal version of egorov s theorem for analytic p ideals we introduce the notion of equi ideal convergence and use it to prove an ideal variant of egorov s theorem we also show that this variant usually cannot be strengthen to a direct analogue of egorov s theorem

on statistical measure theory abstract the purpose of this paper is to unify various kinds of statistical convergence by statistical measure convergence and to present jordan decomposition of finitely additive measures it is done through dealing with the most generalized statistical convergence ideal convergence by applying geometric functional analysis and banach space theory we first show that for each type of ideal i 2 n convergence there exists a set s of statistical measures such that the measure s convergence is equivalent to the statistical convergence to search for jordan decomposition of measures of statistical type we show that the subspace x i span a a i is an ideal of the space l in the sense of banach lattice hence the quotient space l x i is isometric to a c k space we then prove that a statistical measure has a jordan decomposition if and only if its corresponding functional is norm attaining on l and which in turn induces an approximate null ideal preserved jordan decomposition theorem of finitely additive measures finally we show this characterization and the approximate decomposition theorem are true for finitely additive measures defined on a general measurable space

weak statistical convergence and weak filter convergence for unbounded sequences abstract for every weakly statistically convergent sequence x n with increasing norms in a hilbert space we prove that sup n x n n this estimate is sharp we study analogous problem for some other types of weak filter convergence in particular for the erdos ulam filters analytical p filters and f filters we present also a refinement of the recent aron garcia maestre result on weakly dense sequences that tend to infinity in norm

on iterative algorithms for the polar decomposition of a matrix and the matrix sign function in this paper we consider relations between the principal iterations from the pade family of kenney and laub for computing the matrix sign function and the principal iterations from the reciprocal pade family of greco iannazzo and poloni and the dual pade family of zi tak we show global convergence of the principal reciprocal pade iterations and the principal dual pade iterations we adopt the dual pade family of iterations to which the newton method belongs for computing the unitary polar factor of a nonsingular matrix we present numerical experiments with the scaled newton method of higham for computing the unitary polar factor which show how the quality of the computed inverses of matrices in the scaled newton method affects the accuracy of the computed polar factorization of a nonsingular matrix it indicates that assumptions under which the backward stability of the scaled newton method has been proved in the literature cannot be weaker

computing a nearest symmetric positive semidefinite matrix the nearest symmetric positive semidefinite matrix in the frobenius norm to an arbitrary real matrix a is shown to be b h 2 where h is the symmetric polar factor of b a at 2 in the 2 norm a nearest symmetric positive semidefinite matrix and its distance 2 a from a are given by a computationally challenging formula due to halmos we show how the bisection method can be applied to this formula to compute upper and lower bounds for 2 a differing by no more than a given amount a key ingredient is a stable and efficient test for positive definiteness based on an attempted choleski decomposition for accurate computation of 2 a we formulate the problem as one of zero finding and apply a hybrid newton bisection algorithm some numerical difficulties are discussed and illustrated by example

backward errors for eigenvalue and singular value decompositions we present bounds on the backward errors for the symmetric eigenvalue decomposition and the singular value decomposition in the two norm and in the frobenius norm through different orthogonal decompositions of the computed eigenvectors we can define different symmetric backward errors for the eigenvalue decomposition when the computed eigenvectors have a small residual and are close to orthonormal then all backward errors tend to be small consequently it does not matter how exactly a backward error is defined and how exactly residual and deviation from orthogonality are measured analogous results hold for the singular vectors we indicate the effect of our error bounds on implementations for eigenvector and singular vector computation in a more general context we prove that the distance of an appropriately scaled matrix to its orthogonal qr factor is not much larger than its distance to the closest orthogonal matrix

erdős ulam ideals vs simple density ideals abstract the main aim of this paper is to bridge two directions of research generalizing asymptotic density zero sets this enables to transfer results concerning one direction to the other one consider a function g 0 such that lim n g n and n g n does not converge to 0 then the family z g a lim n card a n g n 0 is an ideal called simple density ideal or ideal associated to upper density of weight g we compare this class of ideals with erd s ulam ideals in particular we show that there are antichains of size c among erd s ulam ideals which are and are not simple density ideals in 12 it is shown that there is also such an antichain among simple density ideals which are not erd s ulam ideals we characterize simple density ideals which are erd s ulam as those containing the classical ideal of sets of asymptotic density zero we also characterize erd s ulam ideals which are simple density ideals in the latter case we need to introduce two new notions one of them called increasing invariance of an ideal i asserts that given b i and c with card c n card b n for all n we have c i this notion is inspired by 3 and is later applied in 12 for a partial solution of 15 problem 5 8 finally we pose some open problems

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

the matrix sign decomposition and its relation to the polar decomposition the sign function of a square matrix was introduced by roberts in 1971 we show that it is useful to regard s sign a as being part of a matrix sign decomposition a sn where n a2 1 2 this decomposition leads to the new representation sign a a a2 1 2 most results for the matrix sign decomposition have a counterpart for the polar decomposition a uh and vice versa to illustrate this we derive best approximation properties of the factors u h and s determine bounds for a s and a u and describe integral formulas for s and u we also derive explicit expressions for the condition numbers of the factors s and n an important equation expresses the sign of a block 2 2 matrix involving a in terms of the polar factor u of a we apply this equation to a family of iterations for computing s by pandey kenney and laub to obtain a new family of iterations for computing u the iterations have some attractive properties including suitability for parallel computation

an inequality for hilbert schmidt norm for the absolute value c c c 1 2 and the hilbert schmidt norm c hs trc c 1 2 of an operatorc the following inequality is proved for any bounded linear operatorsa andb on a hilbert space a b hs leqq 2 1 2 a b hs the corresponding inequality for two normal state and of a von neumann algebram is also proved in the following form d varphi psi leqq xi varphi xi psi leqq 2 1 2 d varphi psi here denotes the unique vector representative of a state in a natural positive conep form andd denotes the bures distance defined as the infimum which is also the minimum of the distance of vector representatives of and in particular xi varphi 1 xi varphi 2 leqq 2 1 2 xi 1 xi 2 for any vector representatives j of j j 1 2

the iterative methods for computing the polar decomposition of rank deficient matrix some iterative methods for computing the polar decomposition are introduced they can be used to compute the polar decomposition of rank deficient matrix which includes both square and rectangular matrices a very important application of the polar decomposition is about matrix sign function through the methods of this paper to compute the matrix sign function is convenient some examples are provided in this paper

computing fundamental matrix decompositions accurately via the matrix sign function in two iterations the power of zolotarev s functions the symmetric eigenvalue decomposition and the singular value decomposition svd are fundamental matrix decompositions with many applications conventional algorithms for computing these decomposi

remarks on ideal boundedness convergence and variation of sequences abstract we answer the questions asked by faisant et al 2005 2 the first main result states that for every admissible ideal i p n the quotient space l i c 0 i is complete the second main result states that consistently there is an admissible ideal i p n such that the sets w i of all real sequences with finite i variation and c i of all restrictively i convergent sequences are equal

a new polar decomposition in a scalar product space abstract there are various definitions of right and left polar decompositions of an m n matrix f k m n where k c or r with respect to bilinear or sesquilinear products defined by nonsingular matrices m k m m and n k n n the existence and uniqueness of such decompositions under various assumptions on f m and n have been studied here we introduce a new form of right and left polar decompositions f w s and f s w respectively where the matrix w has orthonormal columns w has orthonormal rows with respect to suitably defined scalar products which are functions of m n and f and the matrix s is selfadjoint with respect to the same suitably defined scalar products and has eigenvalues only in the open right half plane we show that our right and left decompositions exist and are unique for any nonsingular matrices m and n when the matrix f satisfies f m n n m f and f m n f f f m n respectively is nonsingular where f m n n 1 f m with f f t for real or complex bilinear forms and f f t for sesquilinear forms when m n our results apply to nonsingular square matrices f our assumptions on f m and n are in some respects weaker and in some respects stronger than those of previous work on polar decompositions

generic frechet differentiability of convex functions on non asplund spaces abstract let f be a continuous convex function on a banach space e this paper shows that every proper convex function g on e with g f is generically frechet differentiable if and only if the image of the subdifferential map f of f has the radon nikod m property and in this case it is equivalent to showing that the image of f is separable on each separable subspace of e

perturbation analysis of the matrix sign function perturbation expansions and new perturbation bounds for the matrix sign function are derived by using a simple technique the perturbation expansions are then used to derive frechet derivatives th

tauberian theorems via statistical convergence the concept of statistical convergence which is related to the usual concept of convergence in probability provides a regular summability method for abstract metric spaces by using probabilistic tools we provide some tauberian theorems which have best possible order tauberian conditions furthermore these methods can be used to unify and improve the classical pointwise tauberian theorems of summability theory for the random walk type methods as proved by bingham and hausdorff methods as proved by lorentz

statistical convergence in fuzzy normed linear spaces in this paper we introduce the notions of statistically convergent sequence and statistically cauchy sequence in a fuzzy normed linear space fns and establish some basic facts we also define statistical limit points and statistical cluster points of a sequence in an fns and investigate the relations between these concepts

the influence of orthogonality on the arnoldi method abstract many algorithms for solving eigenproblems need to compute an orthonormal basis the computation is commonly performed using a qr factorization computed using the classical or the modified gram schmidt algorithm the householder algorithm the givens algorithm or the gram schmidt algorithm with iterative reorthogonalization for the eigenproblem although textbooks warn users about the possible instability of eigensolvers due to loss of orthonormality few theoretical results exist in this paper we prove that the loss of orthonormality of the computed basis can affect the reliability of the computed eigenpair when we use the arnoldi method we also show that the stopping criterion based on the backward error and the value computed using the arnoldi method can differ because of the loss of orthonormality of the computed basis of the krylov subspace we also give a bound which quantifies this difference in terms of the loss of orthonormality

matrix factorizations and their perturbations there are several ways in which a matrix can be factorized as a product of two special matrices these factorizations are often used in numerical analysis and many perturbation bounds useful in such analysis have been proved by various authors in this paper a simple method which leads to these results and several new ones is discussed in the last section related results on the lipschitz continuity of the matrix absolute value are surveyed

palindromic matrix polynomials matrix functions and integral representations abstract we study the properties of palindromic quadratic matrix polynomials z p qz pz 2 i e quadratic polynomials where the coefficients p and q are square matrices and where the constant and the leading coefficients are equal we show that for suitable choices of the matrix coefficients p and q it is possible to characterize by means of z well known matrix functions namely the matrix square root the matrix polar factor the matrix sign and the geometric mean of two matrices finally we provide some integral representations of these matrix functions

ideal invariant injections abstract for an ideal i on we introduce the notions of i invariant and bi i invariant injections from to we study injections that are invariant with respect to selected classes of ideals we show some applications to ideal convergence

structured matrices and newton s iteration unified approach abstract recent progress in the study of structured matrices shows advantages of unifying the treatment of various classes of such matrices we recall some fundamental techniques for such a unification and then specify it in full details for newton s iteration which rapidly improves an initial approximation to the inverse matrix by performing two matrix multiplications per recursive step the iteration is particularly suitable for n n structured matrices represented with o n entries of their short generators rather than with their own n 2 entries based on such a representation matrix operations are performed much more rapidly and use much less memory space a major problem is to control the length of the generators which tends to grow quite rapidly in the iterative process two known methods solve this problem for toeplitz like and cauchy like matrices we extend both methods to a more general class of structured matrices and estimate the convergence rate as well as the computational complexity some novel techniques are introduced in this study in particular for the estimation of the norms of the inverse displacement operators

a characterization of banach spaces with separable duals via weak statistical convergence abstract let b be a banach space a b valued sequence xk is weakly statistically null provided lim n 1 n k n f x k e 0 for all e 0 and every continuous linear functional f on b a banach space is finite dimensional if and only if every weakly statistically null b valued sequence has a bounded subsequence if b is separable b is separable if and only if every bounded weakly statistically null b valued sequence contains a large weakly null sequence a characterization of spaces containing an isomorphic copy of l1 is given and it is also shown that l2 has a statistical m basis which is not a schauder basis

on δ quasi slowly oscillating sequences a sequence x n of points in a topological group is called d quasi slowly oscillating if dx n is quasi slowly oscillating and is called quasi slowly oscillating if dx n is slowly oscillating a function f defined on a subset of a topological group is quasi slowly respectively d quasi slowly oscillating continuous if it preserves quasi slowly respectively d quasi slowly oscillating sequences i e f x n is quasi slowly respectively d quasi slowly oscillating whenever x n is we study these kinds of continuities and investigate relations with statistical continuity lacunary statistical continuity and some other types of continuities in metrizable topological groups

a sixth order iterative method for approximating the polar decomposition of an arbitrary matrix a new iterative method for computing the polar decomposition of any rectangular complex matrix is presented and analyzed the study of the convergence shows that this method has order of convergence six some numerical tests confirm the theoretical results and allow us to compare the proposed iterative scheme with other known ones

fuzzifying ideal convergence in fuzzifying topological linear spaces in this study we propose a framework for fuzzifying ideal convergence in fuzzifying topological linear spaces after defining fuzzifying ideal convergence we discuss many properties of fuzzifying ideal convergence in fuzzifying topological linear spaces we also introduce a stronger convergence than fuzzifying ideal convergence and a sufficient condition called ap fuzzifying condition under which fuzzifying ideal convergence implies the stronger convergence above is proved

statistical convergence and ideal convergence for sequences of functions abstract let i p n stand for an ideal containing finite sets we discuss various kinds of statistical convergence and i convergence for sequences of functions with values in r or in a metric space for real valued measurable functions defined on a measure space x m we obtain a statistical version of the egorov theorem when x we show that in its assertion equi statistical convergence on a big set cannot be replaced by uniform statistical convergence also we consider statistical convergence in measure and i convergence in measure with some consequences of the riesz theorem we prove that outer and inner statistical convergences in measure for sequences of measurable functions are equivalent if the measure is finite

three kinds of convergence and the associated i baire classes abstract we consider ideal versions of pointwise discrete and equal convergence of sequences of functions defining in a natural way ideal pointwise discrete equal baire classes of functions we show that these classes are equal to their classical counterparts for ideals for which there is a winning strategy in a game introduced by laflamme 1996 10 in the proofs we make extensive use of a characterization in terms of filters f which are diagonalizable by f universal sets of a winning strategy this article extends results of laczkovich and reclaw 2009 9 and debs and saint raymond 2009 5

convex invertible sets and matrix sign function it is proved that in two dimensional unital real algebras a step by step construction of minimal convex invertible sets terminates in two steps an analogous construction of minimal convex conditionally invertible sets of the complex field also terminates in two steps and the complex field is the only finite dimensional complex algebra with this property connections are made with the matrix sign function the matrix sign function of a matrix belongs to the convex invertible set generated by the matrix an example is given to show that the matrix sign function cannot always be obtained in two steps by following the step by step construction

a new method for computing the stable invariant subspace of a real hamiltonian matrix abstract a new backward stable structure preserving method of complexity o n3 is presented for computing the stable invariant subspace of a real hamiltonian matrix and the stabilizing solution of the continuous time algebraic riccati equation the new method is based on the relationship between the invariant subspaces of the hamiltonian matrix h and the extended matrix 0 h h 0 and makes use of the symplectic urv like decomposition that was recently introduced by the authors

the sign matrix and the separation of matrix eigenvalues abstract the sign matrices uniquely associated with the matrices m j i 2 where j are the corners of a rectangle oriented at 4 to the axes of a cartesian coordinate system may be used to compute the number of eigenvalues of the arbitrarily chosen matrix m which lie within the rectangle and to determine the left and right invariant subspaces of m associated with these eigenvalues this paper is concerned with the proof of this statement and with the details of the computation of the required sign matrices

a newton squaring algorithm for computing the negative invariant subspace of a matrix by combining newton s method for the matrix sign function with a squaring procedure a basis for the negative invariant subspace of a matrix can be computed efficiently the algorithm presented is a variant of multiplication rich schemes for computing the matrix sign function such as the well known inversion free schulz method which requires two matrix multiplications per step however by avoiding a complete computation of the matrix sign and instead concentrating only on the negative invariant subspace the final newton steps can be replaced by steps which require only one matrix squaring each this efficiency is attained without sacrificing the quadratic convergence of newton s method

a measure theoretical subsequence characterization of statistical convergence the concept of statistical convergence of a sequence was first introduced by h fast statistical convergence was generalized by r c buck and studied by other authors using a regular nonnegative summability matrix a in place of c1 the main result in this paper is a theorem that gives meaning to the statement s s converges to l statistically t if and only if most of the subsequences of s converge in the ordinary sense to l here t is a regular nonnegative and triangular matrix corresponding results for lacunary statistical convergence recently defined and studied by j a fridy and c orhan are also presented introduction the concept of the statistical convergence of a sequence of reals s sn was first introduced by h fast 9 the sequence s sn is said to converge statistically to l and we write lim s l stat if for every e n oo rim n k e1 0 where jai denotes the cardinality of the set a properties of statistically convergent sequences were studied in 5 6 12 and 16 in 13 fridy and miller gave a characterization of statistical convergence for bounded sequences using a family of matrix summability methods statistical convergence can be generalized by using a regular nonnegative summability matrix a in place of c this idea was first mentioned by r c buck 3 in 1953 and has been further studied by sember and freedman 10 and 11 and connor 5 and 7 regular nonnegative summability matrices turn out to be too general for our purposes here instead we use the concept of a mean a matrix t amn will be called a mean if amn 0 when n m e iamn 1 for all m and limm 0oam 0 for each n if t amn is a mean following buck a sequence s sn is said to be statistically t summable to l and we write sn l stat t if for every e 0 received by the editors august 18 1993 and in revised form february 14 1994 originally communicated to the proceedings of the ams by andrew bruckner 1991 mathematics subject classification primary 40d25 secondary 40g99 28a12 1995 american mathematical society 0002 9947 95 1 00 s 25 per page

on statistical limit points and the consistency of statistical convergence abstract this article extends the concept of a statistical limit cluster point of a sequencex as introduced by fridy to at statistical limit cluster point wheretis a nonnegative regular matrix summability method these definitions are reformulated in the setting of n n it is shown that for a bounded sequencex the set oft statistical cluster points ofxforms a compact subset of r it is also shown that iftandrare two nonnegative regular summability matrices thent statistical convergence andr statistical convergence are consistent if and only if the support sets oftandrhave nonempty intersection

solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

generalized statistical convergence and convergence free spaces abstract many results about statistical convergence in a general setting can be derived by consideration of the convergence free space of null sequences and its related cofilter and filter we derive algebraic and order properties preservation under uniform convergence cauchy properties and properties of cluster points we also show that for certain types of statistical convergence stronger convergence properties also hold finally we discuss the relationship between generalized statistical convergence and subsets of the stone cech compactification of the integers