on weighted statistical convergence based on p q integers and related approximation theorems for functions of two variables abstract in this work we first define a difference operator p q m of natural order m with respect to p q integers we then introduce the concepts of p q m statistical convergence statistical p q m summability and strong p q m summability of order by the weighted method furthermore based on the definition of statistical p q m summability we prove a korovkin type approximation theorem for functions of two variables by using p q analogue of bernstein operator of two variables we give an example which shows that proposed method successfully works finally some voronovskaja type approximation results are obtained

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

boundary minimum principles for the unilateral contact problems abstract the aim of the present paper is the derivation of boundary variational principles for the unilateral contact problem using the inequality constrained principles of minimum potential and complementary energy and the equivalent variational inequality formulations we derive first saddle point formulations for the problems using appropriate langrangian functions an elimination technique allows the formulation of two minimum principles on the boundary with respect to the unknown normal displacements and reactions of the contact region respectively it is also shown that these two minimum problems are equivalent to multivalued boundary integral equations involving symmetric operators the theory is illustrated by numerical examples solved both by the fem and the bem

on the generalized bi skew symmetric solutions of a linear matrix equation and its procrust problems in this paper the solvability conditions and the explicit expressions of the generalized bisymmetric and bi skew symmetric solutions of the matrix equation ax b are respectively established by applying two methods then the maximal and minimal ranks of the solutions are derived if the solvability conditions are not satisfied the generalized bisymmetric and bi skew symmetric least squares solutions of the matrix equation are considered and the generalized bisymmetric and bi skew symmetric least squares solutions with the minimum norm are also obtained in addition two algorithms are provided to compute the generalized bi skew symmetric least squares solution and some examples are given to illustrate that the algorithms are feasible

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

consistency for bi skew symmetric solutions to systems of generalized sylvester equations over a finite central algebra let be a finite dimensional central algebra with an involutorial antiautomorphism and char 2 n n be the set of all n n matrices over a aij n n is called bisymmetric if aij an i 1 n j 1 aji and biskewsymmetric if aij an i 1 n j 1 aji the following systems of generalized sylvester equations over a1x yb1 c1 i asx ybs cs a1xb1 c1xd1 e1 ii asxbs csxds es are considered necessary and sufficient conditions are given for the existence of constant solutions with bi skew symmetric constrains to i and ii as a particular case auxiliary results dealing with the system of sylvester equations are also presented

the computation of the principal square roots of centrosymmetric h matrices in this paper two algorithms for computing the principal square root of a centrosymmetric h matrix with positive diagonal entries are proposed it is showed that our algorithms ensure significant savings in computational costs as compared to the case of an arbitrary h matrix a with positive diagonal entries

symmetric centrosymmetric matrix vector multiplication abstract we present a method for the multiplication of an arbitrary vector by a symmetric centrosymmetric matrix requiring 5 4 n 2 o n floating point operations rather than the 2n 2 operations needed in the case of an arbitrary matrix combining this method with trench s algorithm for toeplitz matrix inversion yields a method for solving toeplitz systems with the same complexity as levinson s algorithm

a note on computing matrix vector products with generalized centrosymmetric centrohermitian matrices to reduce the costs of computing matrix vector product ax related to a centrosymmetric matrix a as compared to the case of an arbitrary matrix a two algorithms were proposed recently one was designed by melman a melman symmetric centrosymmetric matrix vector multiplication linear algebra appl 320 2000 193 198 for symmetric centrosymmetric matrices another was presented by fassbender and ikramov h fassbender k d ikramov computing matrix vector products with centrosymmetric and centrohermitian matrices linear algebra appl 364 2003 235 241 for general centrosymmetric matrices in this note we further discuss this topic of computing ax where a is a generalized centrosymmetric matrix we firstly investigate the reducibility of a generalized centrosymmetric matrix and then provide an algorithm which can be viewed as a generalization of one h fassbender k d ikramov computing matrix vector products with centrosymmetric and centrohermitian matrices linear algebra appl 364 2003 235 241 we show that our algorithm is suitable for computing many matrix vector products with the same matrix furthermore we show that similar results can be obtained for certain subclasses of generalized centrosymmetric matrices as compared to the case of a centrosymmetric matrix and analogue gains are available for generalized skew centrosymmetric or generalized centrohermitian matrices

on inverse eigenvalue problems for block toeplitz matrices with toeplitz blocks we propose an algorithm for solving the inverse eigenvalue problem for real symmetric block toeplitz matrices with symmetric toeplitz blocks it is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for toeplitz matrices first we expose the structure of the eigenvectors of the so called generalized centrosymmetric matrices then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum we have implemented our ideas in a matlab code numerical results produced with this code are included

inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i r symmetric matrices ii hermitian r symmetric matrices iii r skew symmetric matrices iv hermitian r skew symmetric matrices let z cn m with rank z m and diag 1 m the inverse eigenproblem consists of finding z such that the set s z a s az z is nonempty and to find the general form of a s z in all cases we use the special spectral properties of s to essentially characterize the set of admissible pairs z and the special structure of the members of s to obtain the general solution of the inverse eigenproblem given an arbitrary b s the approximation problem consists of finding the unique matrix ab s z that best approximates b in the frobenius norm it is not necessary to assume that r r in connection with the inverse eigenproblem for r symmetric or r skew symmetric matrices however we impose this additional assumption in connection with the inverse eigenproblem for hermitian r symmetric or r skew symmetric matrices and in connection with the approximation problem for i iv

bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations in this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra necessary and sufficient conditions are obtained for the matrix equation ax c and the following systemsa 1x c 1 a 1x c 1 xb 3 c 3 a 2x c 2 to have bisymmetric solutions and the systema 1x c 1 a 3xb 3 c 3 to have centrosymmetric solutions the expressions of such solutions of the matrix and the systems mentioned above are also given moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over are also mentioned

bcr method for solving generalized coupled sylvester equations over centrosymmetric or anti centrosymmetric matrix abstract this paper introduces another version of biconjugate residual method bcr for solving the generalized coupled sylvester matrix equations over centrosymmetric or anti centrosymmetric matrix we prove this version of bcr algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round off errors furthermore a method is provided for choosing the initial matrices to obtain the least norm solution of the problem at last some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed

the general solution to a system of real quaternion matrix equations in this paper we consider the system of matrix equations a 1x c 1 a 2x c 2 a 3xb 3 c 3 and a 4xb 4 c 4 over the real quaternion algebra a necessary and sufficient condition for the existence and the expression of the general solution to the system are given as particular cases the corresponding results on other systems over are also obtained

characterization and properties of r s commutative matrices let r pdiag 0im0 1im1 k 1imk 1 p 1 cm m and s qdiag 0 in0 1 in1 k 1 ink 1 q 1 cn n where m0 m1 mk 1 m n0 n1 nk 1 n 0 1 k 1 are distinct complex numbers and zktozk 0 1 k 1 we say that a cm n is r s commutative if ra as we characterize the class of r s commutative matrices and extend results obtained previously for the case where l e2 il k and l l modk 0 l k 1 with zk our results are independent of 0 1 k 1 so long as they are distinct i e if ra as for some choice of 0 1 k 1 all distinct then ra as for arbitrary of 0 1 k 1

frictional contact problems of kinked cracks modelled by a boundary integral method a numerical method is presented for the solution of two dimensional crack problems including the effects of crack kinks and frictional contact between crack faces the method is based on an integral equation for the resultant forces along a crack coulomb friction between contacting crack surfaces is taken into account the numerical implementation is demonstrated by considerations of surface and sub surface piece wise straight line cracks in a half plane numerical results are presented to illustrate the efficiency and the reliability of the presented method

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

matrix inverse problem and its optimal approximation problem for r symmetric matrices abstract let r cn n be a nontrivial involution i e r2 i and r i a matrix a cn n is called r skew symmetric if rar a the least squares solutions of the matrix inverse problem for r skew symmetric matrices with r r are firstly derived then the solvability conditions and the solutions of the matrix inverse problem for r skew symmetric matrices with r r are given the solutions of the corresponding optimal approximation problem with r r for r skew symmetric matrices are also derived at last an algorithm for the optimal approximation problem is given it can be seen that we extend our previous results g x huang f yin matrix inverse problem and its optimal approximation problem for r symmetric matrices appl math comput 189 2007 482 489 and the results proposed by zhou et al f z zhou l zhang x y hu least square solutions for inverse problem of centrosymmetric matrices comput math appl 45 2003 1581 1589

on lacunary statistical convergence with respect to the intuitionistic fuzzy normed space the concept of statistical convergence was introduced by fast h fast sur la convergence statistique colloq math 2 1951 241 244 which was later on studied by many authors in j a fridy c orhan lacunary statistical convergence pacific j math 160 1993 43 51 fridy and orhan introduced the idea of lacunary statistical convergence quite recently the concept of statistical convergence of double sequences has been studied in intuitionistic fuzzy normed space by mursaleen and mohiuddine m mursaleen s a mohiuddine statistical convergence of double sequences in intuitionistic fuzzy normed spaces chaos solitons fractals 2008 doi 10 1016 j chaos 2008 09 018 in this paper we study lacunary statistical convergence in intuitionistic fuzzy normed space we also introduce here a new concept that is statistical completeness and show that ifns is statistically complete but not complete

characterization and properties of matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a there are positive integers r and s with r s n and matrices p cn r and q cn s such that p p ir q q is rp p and rq q we give an explicit representation of an arbitrary r symmetric matrix a in terms of p and q and show that solving az w and the eigenvalue problem for a reduce to the corresponding problems for matrices app cr r and aqq cs s we also express a 1 in terms of app 1 and aqq 1 under the additional assumption that r r we show that moore penrose inversion and singular value decomposition of a reduce to the corresponding problems for app and aqq we give similar results for r skew symmetric matrices these results are known for the case where r j i n j 1 i j 1n however our proofs are simpler even in this case we say that a cn n is r conjugate if rar a where r rn n and r r 1 i in this case r a is r symmetric and i a is r skew symmetric so our results provide explicit representations for r conjugate matrices in terms of p and q which are now in rn r and rn s respectively we show that solving az w inverting a and the eigenvalue problem for a reduce to the corresponding problems for a related matrix s rn n if rt r this is also true for moore penrose inversion and singular value decomposition of a

remark on lacunary statistical convergence of fuzzy numbers fatih nuray fuzzy sets and systems 99 3 1998 269 proved the inclusion relations between the sets of statistically convergent and lacunary statistically convergent sequences of fuzzy numbers in this paper we show that nuray s condition is sufficient as well as necessary

bounded multipliers of bounded a statistically convergent sequences abstract in this paper we provide an alternate proof of a result of kline it claims that the set of all a statistically convergent sequences cannot be given a locally convex fk topology where a is a nonnegative regular matrix whose rows spread we also study the bounded multiplier space of all bounded a statistically convergent sequences and using the n program we give an analogue of a result of fridy and miller for bounded multipliers

characterization and properties of matrices with k involutory symmetries we say that a matrix r cn n is k involutory if its minimal polynomial is xk 1 for some k 2 so rk 1 r 1 and the eigenvalues of r are 1 2 k 1 where e2 i k let 0 1 k 1 if r cm m a cm n s cn n and r and s are k involutory we say that a is r s symmetric if ras a we show that an r s symmetric matrix a can be usefully represented in terms of matrices fl cc l dl 0 l k 1 where cl and dl are respectively the dimensions of the l eigenspaces of r and s this continues a theme initiated in an earlier paper with the same title in which we assumed that 1 we say that a k involution is equidimensional with width d if all of its eigenspaces have dimension d we show that if r and s are equidimensional k involutions with widths d1 and d2 respectively then r s symmetric matrices are closely related to generalized circulants ras r r s 0k 1 where a0 a1 ak 1 cd1 d2 for this case our results are new even if 1 we also give an explicit formula for the moore penrose inverse of a unilevel block circulant as r r s 0k 1 for any 0 1 k 1 generalizing a result previously obtained for the case where gcd k 1

backward errors for eigenproblem of two kinds of structured matrices in this paper we consider backward errors in the eigenproblem of symmetric centrosymmetric and symmetric skew centrosymmetric matrices by making use of the properties of symmetric centrosymmetric and symmetric skew centrosymmetric matrices we derive explicit formulae for the backward errors of approximate eigenpairs

constrained inverse eigenproblem and associated approximation problem for anti hermitian r symmetric matrices abstract let r c n n be a nontrivial involution i e r 2 i and r i a c n n is called anti hermitian r symmetric if a a and rar a the presentation and some properties for an arbitrary anti hermitian r symmetric matrix with r r and the relations between the eigenproblem for a and the corresponding eigenproblems for anti hermitian matrices are given then the solutions of constrained inverse eigenproblem and approximation problem are essentially decomposed into the same kind subproblems for anti hermitian matrices in complex field with smaller dimensions the explicit solutions for the later subproblems are arrived the corresponding problems which are the formulations of constrained inverse eigenproblem and approximation problem in complex field was first given then the solutions of constrained inverse eigenproblem and approximation problem with r r are derived

r s conjugate solution to a pair of linear matrix equations abstract let r and s be m m and n n nontrivial real symmetric involutions an m n complex matrix a is termed r s conjugate if a ras where a denotes the conjugate of a in this paper necessary and sufficient conditions are established for the existence of the r s conjugate solution to the system of matrix equations ax c and xb d the expression is also presented for such solution to this system in addition the explicit expression of this solution to the corresponding optimal approximation problem is obtained furthermore the least squares r s conjugate solution with least norm to this system mentioned above is considered the representation of such solution is also derived finally an algorithm and numerical examples are given

p skew symmetric common solutions to a pair of quaternion matrix equations abstract an n n quaternion matrix a is termed p symmetric or p skewsymmetric if a pap or a pap where p is an n n nontrivial quaternion involution in this paper we first establish necessary and sufficient conditions for the existence and the expression of the general solution to the system of quaternion matrix equations a 1 x 1 c 1 a 2 x 2 c 2 a 3 x 1 b 1 a 4 x 2 b 2 c b then use the results on the system mentioned above to give necessary and sufficient conditions for the existence and the representations of p symmetric and p skewsymmetric solutions to the system of quaternion matrix equations a a x c a and a b xb b c b furthermore we establish representations of p symmetric and p skewsymmetric quaternion matrices a numerical example is presented to illustrate the results of this paper

relations between k s 1 potent matrices and different classes of complex matrices in this paper k s 1 potent matrices are considered a matrix a cn n is called k s 1 potent when kas 1k a where k is an involutory matrix and s 1 2 3 specifically k s 1 potent matrices are analyzed considering their relations to different classes of complex matrices these classes of matrices are s 1 generalized projectors k hermitian matrices normal matrices and matrices b cn n anti commuting with k or such that kb is involutory hermitian or normal in addition some new relations for k generalized centrosymmetric matrices have been derived

lacunary ideal convergence of multiple sequences in probabilistic normed spaces an ideal i is a family of subsets of positive integers n n which is closed under finite unions and subsets of its elements the aim of this paper is to study the notion of lacunary i convergence of double sequences in probabilistic normed spaces as a variant of the notion of ideal convergence also lacunary i limit points and lacunary i cluster points have been defined and the relation between them has been established furthermore lacunary cauchy and lacunary i cauchy lacunary i cauchy lacunary i convergent double sequences are introduced and studied in probabilistic normed spaces finally we provided example which shows that our method of convergence in probabilistic normed space is more general

structured least squares problems and inverse eigenvalue problems for p q reflexive matrices abstract a new meaningful structured matrix p q reflexive matrix is defined without the common assumption that p or q is unitary a general solution is derived for its structured least squares problem as a necessary and sufficient condition being presented for the solvability of its structured inverse eigenvalue problem structured constrains are firstly given to guarantee the existence of the solution the optimal approximation problem is also considered under spectral constrains

an inverse eigenvalue problem and an associated approximation problem for generalized k centrohermitian matrices a partially described inverse eigenvalue problem and an associated optimal approximation problem for generalized k centrohermitian matrices are considered it is shown under which conditions the inverse eigenproblem has a solution an expression of its general solution is given in case a solution of the inverse eigenproblem exists the optimal approximation problem can be solved the formula of its unique solution is given

lacunary statistical convergence of sequences of fuzzy numbers the sequence x xk of fuzzy numbers is statistically convergent to the fuzzy number x0 provided that for each 0 limln the number ofk n d xk x0 0 in this paper we study a related concept of convergence in which the set k k n is replaced by k kr 1 k kr for some lacunary sequence kr also we introduce the concept of lacunary statistically cauchy sequence and show that it is equivalent to the lacunary statistical convergence in addition the inclusion relations between the sets of statistically convergent and lacunary statistically convergent sequences of fuzzy numbers are given

generalized statistical convergence recently kolk acta et comm univ tartu de math 2 1998 39 introduced the concept of b statistical convergence by considering a sequence of infinite matrices b bnk i in this paper we use this idea to define and study b statistical limit point b statistical cluster point stb limit inferior and stb limit superior

computing matrix vector products with centrosymmetric and centrohermitian matrices an algorithm proposed recently by melman reduces the costs of computing the product ax with a symmetric centrosymmetric matrix a as compared to the case of an arbitrary a we show that the same result can be achieved by a simpler algorithm which requires only that a be centrosymmetric however if a is hermitian or symmetric this can be exploited to some extent also we show that similar gains are possible when a is a skew centrosymmetric or a centrohermitian matrix

derivation of the variational inequalities and extremum principles of the frictionless elastic contact problem abstract for two contacting elastic bodies with arbitrary sufficiently smooth shapes the statical and kinematical conditions are established assuming small deformations and allowing the contact region to be large from these ones the principles of virtual displacements and virtual forces as well as the corresponding complementary variational principles are derived in two different versions in the first one only integrals over the contact surfaces in the second one also integrals over the volumes of both bodies are involved alternatively an equivalent derivation based on a mixed principle of virtual work and a saddle type functional is presented several representations known in the literature are contained as special cases

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

some properties of generalized k centrosymmetric h matrices every nxn generalized k centrosymmetric matrix a can be reduced into a 2x2 block diagonal matrix see z liu h cao h chen a note on computing matrix vector products with generalized centrosymmetric centrohermitian matrices appl math comput 169 2 2005 1332 1345 this block diagonal matrix is called the reduced form of the matrix a in this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices finally exploiting these properties the development of structure preserving algorithms for certain computations for generalized k centrosymmetric h matrices is discussed

a linear finite element approach to the solution of the variational inequalities arising in contact problems of structural dynamics the present paper deals with the theoretical and numerical treatment of dynamic unilateral problems the governing equations are formulated as an equivalent variational inequality expressing d alembert s principle in its inequality form the discretization with respect to time and space leads to a static nonlinear programming problem which is solved by an appropriate algorithm some properties of dynamic unilateral problems are outlined and the influence of several parameters on the solution is investigated by means of numerical examples

on variations via statistical convergence abstract the paper introduces some notions of variation using the concept of statistical convergence in particular the ideas of variation such as regular translationally regular o regular and rapid are presented in statistical context first some results are established connecting some statistical variations with their non statistical counterparts then certain conditions are explored under which various statistical variations can occur finally the paper examined some conditions to connect two different statistical variations

contact problems of kinked cracks modelled by a boundary integral method based on the integral equation for the resultant forces along a crack a numerical method is developed for the solution of two dimensional kinked crack problems taking crack contact into account the method is demonstrated by a consideration of an elastic half plane containing a piece wise straight line crack two numerical examples are presented and compared to finite element calculations the numerical results indicate that the present method can be an effective and reliable tool for investigations of kinked crack problems associated with crack contact

a perturbed lagrangian formulation for the finite element solution of contact problems making use of a perturbed lagrangian formulation a finite element procedure for contact problems is developed for the general case in which node to node contact no longer holds the proposed procedure leads naturally to a discretization of the contact interface into contact segments within the context of a bilinear interpolation for the displacement field a mixed finite element approximation is introduced by assuming discont naous contact pressure constant on the contact segment because of this piece wise constant approximation the gap function enters into the formulation in an average sense instead of through a point wise definition numerical examples are presented that illustrate the performance of the proposed procedure current finite element formulations for contact problems based on either the classical lagrange parameter procedure l 3 12 203 or the penalty function method 4 6 11 are characterized by a point wise enforcement of the contact constraint condition in the sense that penetration of the bodies is established on a nodal basis moreover in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure within the framework of classical lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables lagrange multipliers these extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms penalty methods on the other hand enable one to transform the constrained problem into an unconstrained one without introducing additional variables the constraint condition is now satisfied only approximately for finite values of the penalty parameter the main difhculty associated with these methods however lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition this is a well understood phenomenon particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics e g see l 22 25 for a review recently augmented lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and regularize the penalty formulation e g see the survey in 7 8 within the framework of zinearited e u c it is possible to restrict the finite element formulation of contact problems by assuming that node to node contact occurs this is in fact the case often considered in the literature l 2 4 6 12 20 211 in the general context of fully

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

multilevel matrices with involutory symmetries and skew symmetries let n n1 n2 nk where k 1 and n1 nk are integers 1 for 1 i k let pi j 1i 1nj and qi j i 1knj and suppose that ui cni ni is a nontrivial involution i e ui ui 1 ini let ri ipi ui iqi 1 i k and denote r r1 rk if 0 1 l 2k 1 let i 1kli 2i 1 be its binary expansion we say that a cn n is r symmetric if riari 1 li a 1 i k thus we are considering matrices with k levels of block structure and an involutory symmetry or skew symmetry at each level we characterize the class of all r symmetric matrices and study their properties the theory divides into two parts corresponding to 0 and 0 problems involving an r 0 symmetric matrix split into the corresponding problems for 2k 1 matrices with orders summing to n while problems involving an r symmetric matrix with 0 split into the corresponding problems for 2k 1 1 matrices with orders summing to n the latter is also true of a b c where b is r 0 symmetric and c is r symmetric with 0

on the consideration of unilateral contact and friction in cracks the boundary integral method the scope of the present paper is the study of cracks having a given geometry by taking into account unilateral contact and friction phenomena between the two sides of the crack the arising problem is highly non classical because of the inequality relations describing the unilateral contact and friction interface conditions the indirect biem is extended appropriately in order to treat this problem numerical examples concerning the calculation of stress intensity factors under the unilateral contact and friction interface conditions illustrate the developed method

a linear transformation and its properties with special applications in time series filtering abstract the main purpose of this paper is to introduce a linear transformation called t and to derive its algebraic properties by means of permutation matrices that represent it to demonstrate the importance of the t transformation for the estimation of latent variables in time series decomposition we obtain a general expression for smoothing matrices characterized by symmetric and asymmetric weighting systems we show that the submatrix of the symmetric weights to be applied to central observations is t invariant whereas the submatrices of the asymmetric weights to be applied to initial and final observations are the t transform of each other by virtue of this relation the properties of the t transformation provide useful information on the smoothing of time series data finally we illustrate the role of the t transformation on the weighting systems of several smoothers often applied for trend cycle estimation such as the locally weighted regression smoother loess the cubic smoothing spline the gaussian kernel and the 13 term trend cycle henderson filter

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

unilateral contact friction and related interactions in cracks the direct boundary integral method abstract the paper presents a theory for the study of cracks having a given geometry by taking into account all types of actions of monotone type like unilateral contact and friction phenomena between the two crack sides the arising problems are of a non classical nature due to the interface conditions expressed in terms of non differentiable convex superpotentials the direct b i e m is extended appropriately in order to treat this type of problem the developed method is illustrated by a numerical example concerning the calculation of stress intensity factors under the unilateral contact and friction interface conditions

minimization problems for r s symmetric and r s skew symmetric matrices let r cm m and s cn n be nontrivial involutions thus r r 1 i and s s 1 i we say that a cm n is r s symmetric r s skew symmetric if ras a ras a let s be the class of m n r s symmetric matrices or the class of m n r s skew symmetric matrices let z cn q and w cm q we study the following problems i give necessary and sufficient conditions for the existence of an a s such that az w and find all such matrices if the conditions are met ii find z w mina s az w and characterize the class s z w a s az w z w iii if b cm n is arbitrary find z w b mina s z w a b and find a s z w such that a b z w b we obtain explicit formulas for z w b z w and all the matrices in question

procrustes problems for p q η reflexive matrices this paper characterizes p q h reflexive matrices showing that a p q h reflexive matrix can be represented in terms of k matrices a a b c m a x n b where a b h modk m a and n b are dimensions of the t a and t b eigenspaces of p and q respectively a general solution of procrustes problems of p q h reflexive matrices is presented in terms of lower order matrices a 1 h 1 a h 1 1 a h k a h 1 k 1 a k h under the condition that p and q are unitary

on κ real and κ hermitian matrices abstract a body of theory for real and hermitian matrices is developed some basic results for these matrices their spectral properties and characterizations of linear transformations which preserve them are given

hermitian hermitian r symmetric and hermitian r skew symmetric procrustes problems let r cn n be a nontrivial unitary involution i e r r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i hermitian matrices ii hermitian r symmetric matrices iii hermitian r skew symmetric matrices given z w cn m we characterize the matrices a in s that minimize az w frobenius norm and given an arbitrary e cn n we find the unique matrix among the minimizers of az w in s that minimizes a e we also obtain necessary and sufficient conditions for existence of a s such that az w and assuming that the conditions are satisfied characterize the set of all such a