a new family of extended gauss quadratures with an interior interval constraint starting from two sequences ga c n and gd b n of ordinary gauss quadrature formulae with an orthogonality measure d on the open intervals a c and d b respectively we construct a new sequence ga b e n of extended gaussian quadrature formulae for d on a b which is based on some preassigned points the nodes of ga c n gd b n and the e n zeros contained in c d of a nonclassical orthogonal polynomial on a b with respect to a linear functional the principal result gives explicit formulae relating these polynomials and shows how their recurrence coefficients in the three term recurrence formulae are related thus a new class of gaussian quadratures having some nodes contained in a given interior interval can be computed directly by standard software for ordinary gauss quadrature formulae

a stable gauss kronrod algorithm for cauchy principal value integrals abstract it is shown that the ratio of the precision of the stable kronrod extension to the precision of the stable version of the gauss rule for cauchy principal value integrals is approximately the same as the ratio of the precision of the kronrod extension to that of the gauss rule for ordinary integrals

covering classes strongly flat modules and completions we study some closely interrelated notions of homological algebra 1 we define a topology on modules over a not necessarily commutative ring r that coincides with the r topology defined by matlis when r is commutative 2 we consider the class mathcal sf of strongly flat modules when r is a right ore domain with classical right quotient ring q strongly flat modules are flat the completion of r in its r topology is a strongly flat r module 3 we prove some results related to the question whether mathcal sf a covering class implies mathcal sf closed under direct limits this is a particular case of the so called enochs conjecture whether covering classes are closed under direct limits some of our results concern right chain domains for instance we show that if the class of strongly flat modules over a right chain domain r is covering then r is right invariant in this case flat r modules are strongly flat

on the numerical resolution of the generalized airfoil equation with possio kernel in this paper we consider the so called generalized airfoil singular integral equation with a smooth input function and present ad hoc quadrature rules to compute efficiently the elements of collocation and galerkin matrices

on orthogonal polynomials for certain nondefinite linear functionals abstract we consider the non definite linear functionals l n f r w x f n x d x and prove the nonexistence of orthogonal polynomials with all zeros real in several cases the proofs are based on the connection with moment preserving spline approximation

h spaces semiperfect rings and self homotopy equivalences introductionthe group of self homotopy equivalences aut x of a space xconsists of homotopyclasses of maps x x that are homotopy equivalences see 22 for a detailedsurvey of the subject in 15 the second author proved a decomposition theoremfor groups of self homotopy equivalences of product spaces under the assumptionthat the self equivalences in question are in certain sense reducible and then in 17 he derived a series of computable homological criteria that imply reducibility these criteria are based on the detection of nilpotent self maps and so are bestsuited for the study of atomic spaces i e spaces for which every self map is eithernilpotent or an equivalence cf 1 it turns out that indecomposable p completeh spaces are always atomic while in general indecomposable p local h spaces arenot consequently the classical theorem of c wilkerson 23 that nite p localh spaces and simply connected nite p local coh spaces admit essentially uniquedecompositions into indecomposable factors did not provide su cient informationto imply reducibility for p local spaces motivated by the results obtained in 18 we use the theory of semiperfect rings to show that the endomorphism sets ofindecomposable factors of p local h spaces are in a certain sense local furthermore semiperfect rings turned out to be a powerful organizational principle that allowa uni ed and simpli ed treatment of decomposition theorems for both p completeand p local h and coh spaces the improved versions of the wilkerson s theoremare strong enough to imply reducibility of self equivalences and hence a canonical

common fixed point theorems in fuzzy metric spaces in this paper we prove common fixed point theorems for six mappings in fuzzy metric space our main results extend generalize and fuzzify some known results in fuzzy metric spaces probabilistic metric spaces and uniform spaces

suboptimal kronrod extension formulae for numerical quadrature we consider cases where the stieltjes polynomial associated with a gaussian quadrature formula has complex zeros in such cases a kronrod extension of the gaussian rule does not exist a method is described for modifying the stieltjes polynomial so that the resulting polynomial has no complex zeros the modification is performed in such a way that the kronrod type extension rule resulting from the addition of then 1 zeros of the modified stieltjes polynomial to the original knots of the gaussian rule has only slightly lower degree of precision than normally achieved when the kronrod extension rule exists as examples of the use of the method we present some new formulae extending the classical gauss hermite quadrature rules we comment on the limited success of the method in extending gauss laguerre rules and show that several modified extensions of the gauss gegenbauer formulae exist in cases where the standard kronrod extension does not

virtually uniserial modules and rings abstract we study the class of virtually uniserial modules and rings as a nontrivial generalization of uniserial modules and rings an r module m is virtually uniserial if for every finitely generated submodule 0 k m k rad k is virtually simple also an r module m is called virtually serial if it is a direct sum of virtually uniserial modules and a left virtually uniserial resp left virtually serial ring is a ring which is virtually uniserial resp serial as a left r module we give some useful properties of virtually uni serial modules and rings in particular it is shown that every left virtually uniserial module is uniform and bezout also we show that if r is a left virtually serial ring then r j r i 1 t m n i d i where t n 1 n t n and each d i is a principal left ideal domain as a consequence we obtain that a ring r is left virtually serial with j r 0 if and only if r i 1 t m n i d i where t n 1 n t n and each d i is a principal left ideal domain with j d i 0 also several classes of rings for which every virtually uniserial module resp ring is uniserial are given noetherian left virtually uniserial rings are characterized finally we obtain some structure theorems for commutative rings over which every finitely generated module is virtually serial

on the convergence of an interpolatory product rule for evaluating cauchy principal value integrals etablissement des theoremes de convergence des regles de produits interpolatoires pour l evaluation d integrales singulieres de cauchy avec estimation asymptotique du reste amelioration et generalisation de resultats anterieurs

classes of modules with the exchange property in searching for isomorphic refinements of direct sum decompositions of modules9 crawley and jonsson introduced an exchange property for modules that is strongly reminiscent of steinitz exchange lemma for vector spaces an r module m where r is an associative ring with identity is said to have the finite exchange property if m can be fitted into any finite sum grid as follows whenever a4 occurs as a direct summand of a finite direct sum a bielai then a m ci for suitable submodules ci of the ai 2 this concept led to very general extensions of the classical refinement theorems see e g 2 3 171 many of the problems suggested by crawley and jonsson in their pioneering paper are still open the most salient one does the finite exchange property imply the unrestricted exchange property we show that the answer is yes for modules which possess decompositions into indecomposable summands in the background of this observation stands the following

stieltjes polynomials and functions of the second kind abstract in this paper we discuss known and new facts on the asymptotic behaviour of the stieltjes polynomials and the close relation to the asymptotic behaviour of the boundary values of the functions of the second kind on 1 1 most of the results are derived by new methods of proof where mainly the theory of real orthogonal polynomials is used

kronrod extensions with multiple nodes of quadrature formulas for fourier coefficients abstract we continue with analyzing quadrature formulas of high degree of precision for computing the fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials started recently by bojanov and petrova quadrature formulae for fourier coefficients j comput appl math 231 2009 378 391 and we extend their results construction of new gaussian quadrature formulas for the fourier coefficients of a function based on the values of the function and its derivatives is considered we prove the existence and uniqueness of kronrod extensions with multiple nodes of standard gaussian quadrature formulas with multiple nodes for several weight functions in order to construct some new generalizations of quadrature formulas for the fourier coefficients for the quadrature formulas for the fourier coefficients based on the zeros of the corresponding orthogonal polynomials we construct kronrod extensions with multiple nodes and highest algebraic degree of precision for this very desirable kind of extension there do not exist any results in the theory of standard quadrature formulas

k0 of a semilocal ring let r be a semilocal ring that is r modulo its jacobson radical j r is artinian then k0 r j r is a partially ordered abelian group with order unit isomorphic to zn u where denotes the componentwise order on zn and u is an order unit in zn moreover the canonical projection r r j r induces an embedding of partially ordered abelian groups with order unit k0 k0 r k0 r j r in this paper we prove that every embedding of partially ordered abelian groups with order unit g zn can be realized as the mapping k0 k0 r k0 r j r for a suitable hereditary semilocal ring r

associated generalized jacobi functions and polynomials abstract pointwise estimates are obtained for associated functions and polynomials corresponding to generalized jacobi weights these estimates hold uniformly in the interval of orthogonality

direct products of modules whose endomorphism rings have at most two maximal ideals abstract monogeny classes and epigeny classes have proved to be useful in the study of direct sums of uniserial modules and other classes of modules in this paper we show that they also turn out to be useful in the study of direct products

on uniserial modules that are not quasi small abstract we prove 0 th root property for uniserial modules and we show that the krull schmidt theorem holds for direct sums of uniserial modules that are not quasi small

on the remainder term of gauss radau quadratures for analytic functions for analytic functions the remainder term of gauss radau quadrature formulae can be represented as a contour integral with a complex kernel we study the kernel on elliptic contours with foci at the points 1 and a sum of semi axes r 1 for the chebyshev weight function of the second kind starting from explicit expressions of the corresponding kernels the location of their maximum modulus on ellipses is determined the corresponding gautschi s conjecture from on the remainder term for analytic functions of gauss lobatto and gauss radau quadratures rocky mountain j math 21 1991 209 226 is proved

stopping functionals for gaussian quadrature formulas abstract gaussian formulas are among the most often used quadrature formulas in practice in this survey an overview is given on stopping functionals for gaussian formulas which are of the same type as quadrature formulas i e linear combinations of function evaluations in particular methods based on extended formulas like the important gauss kronrod and patterson schemes and methods which are based on gaussian nodes are presented and compared

gauss kronrod integration rules for cauchy principal value integrals kronrod extensions to two classes of gauss and lobatto integration rules for the evaluation of cauchy principal value integrals are derived since in one frequently occurring case the kronrod extension involves evaluating the derivative of the integrand a new extension is introduced using n 2 points which requires only values of the integrand however this new rule does not exist for all n and when it does several significant figures are lost in its use 1 introduction in this paper we shall consider kronrod extensions ke to integration rules based on gauss and lobatto points for the evaluation of cauchy principal value cpv integrals of the form fix vyx 1 since the existence theory of ke s for regular integrals insures such extensions for only certain classes of weight functions w x we shall restrict our attention here to the most important class namely w x 1 x2y x 2 where 0 p 2 in the gauss case and p 1 in the lobatto case 9 the only other relevant weight functions are the jacobi weight functions w x 1 x a l x with a i s or s y a f and w x x2 i rx2 oo r 1 5 for our case w x 1 x2 m 1 2 the corresponding orthogonal polynomials are the gegenbauer polynomials c x usually written c x which have the following normalization 11 p 174 2 w x c x cmil x dx 8 mhnil where 3 hn i ft n 2p r p n u n r p r 2p

marcinkiewicz inequalities based on stieltjes zeros the authors find necessary and sufficient conditions for gdt weighted marcinkiewicz inequalities based at stieltjes zeros

stieltjes polynomials and related quadrature rules in this paper we present a survey of theoretical results some of them new and numerical evidence of others concerning stieltjes polynomials kronrod schemes and their generalizations applied to integrals with a classical nonnegative weight function the use of these schemes in automatic integration routines is briefly outlined

stieltjes polynomials and lagrange interpolation bounds are proved for the stieltjes polynomial e n 1 and lower bounds are proved for the distances of consecutive zeros of the stieltjes polynomials and the legendre polynomials p n this sharpens a known interlacing result of szego as a byproduct bounds are obtained for the geronimus polynomials g n applying these results convergence theorems are proved for the lagrange interpolation process with respect to the zeros of e n 1 and for the extended lagrange interpolation process with respect to the zeros of p n e n 1 in the uniform and weighted l p norms the corresponding lebesgue constants are of optimal order

gauss kronrod quadrature formulae for weight functions of bernstein szego type ii we study the kronrod extensions of gaussian quadrature rules whose weight functions on 1 l consist of any one of the four chebyshev weights divided by an arbitrary quadratic polynomial that remains positive on 1 11 we show that in almost all cases these extended gauss kronrod quadrature rules have all the desirable properties kronrod nodes interlacing with gauss nodes all nodes contained in 1 11 and all weights positive and representable by semiexplicit formulas exceptions to these properties occur only for small values of n the number of gauss nodes namely n 3 and are carefully identified the precise degree of exactness of each of these gauss kronrod formulae is determined and shown to grow like 4n rather than 3n as is normally the case our findings are the result of a detailed analysis of the underlying orthogonal polynomials and stieltjes polynomials the paper concludes with a study of the limit case of a linear divisor polynomial in the weight function

modified optimal quadrature extensions optimal extensions of quadrature rules are of importance in the construction of automatic integrators but many sequences fail to exist in usable form the paper considers some techniques for overcoming the problem of inextensibility with a minimal effect on integrating efficiency a modification to the extension procedure proposed recently by begumisa and robinson is shown to be just a special case of the standard theory for quadrature extension some illustrative examples are included

on convex and concave fuzzy mappings abstract in this paper we introduce the concepts of convex subsets of f 0 the set of all fuzzy numbers and monotonic fuzzy mappings on r2 and show that a monotonic fuzzy mapping f on a convex set c r1 or r2 is quasi convex and quasi concave we establish two characterizations for convex fuzzy mappings and two characterizations for concave fuzzy mappings we also prove the least upper bound property

on polynomials orthogonal on a circular arc abstract polynomials rk orthogonal on a circular arc with respect to the complex inner product f g f1 g1 w1 d where 0 1 2 and for f z the function f1 is defined by f1 f ir ei r2 1 1 2 r tan have been introduced by de bruin 1990 in this paper the functions of the second kind as well as the corresponding associated polynomials are introduced some recurrence relations and identities of christoffel darboux type are proved also the corresponding stieltjes polynomials which are orthogonal to all lower degree polynomials with respect to a complex measure on r z c z ir ei r2 1 1 2 tan r are investigated a class of polynomials orthogonal on a symmetrical circular arc in the down half plane is also introduced finally in the jacobi case w z 1 z 1 z 1 a linear second order differential equation for rn z is obtained

a note on compactness in a fuzzy metric space we study the compactness in a fuzzy metric space we also give a generalization of niemytzki tychonoff theorem for a fuzzy metric space

uniserial modules over valuation rings this is the first in a series of papers deahng with the theory of modules over valuation rings it has been observed that a variety of concepts and basic results in abelian group theory can be extended mutatis mutandis to modules over valuation rings and several other results which require drastic modification for valuation rings can also be dealt with by suitably generalizing ideas of abelian group theory our purpose is to develop techniques for modules over valuation rings actually these are the simplest kind of commutative non noetherian rings our point of departure is abelian group theory in the local case when the groups are merely modules over z the integers localized at a prime p which is a discrete rank one valuation domain in the process of generalization the most attractive and frequently used properties are sacrilied pleasant properties we are so accustomed to in abelian groups are gone in return we learn new features and discover new phenomena in the behavior of modules several results are known on modules over valuation rings r see the references 3 lo or the survey article 121 this paper is devoted to the study of uniseriul r modules i e those r modules in which the submodules form a chain as far as the simplicity of the structure is concerned these are second only to cyclic modules they have been investigated by shores and lewis 9 we make use of their results especially their description of endomorphism rings we study various aspects of these modules with special emphasis on their quasiand pure injectivity as well as on the existence of pure uniserial submodules in torsion r modules in some cases as expected more explicit results can be established only under the additional hypothesis that r is almost maximal

some new formulae for the stieltjes polynomials relative to classical weight functions using known limit formulae which connect the gegenbauer and jacobi polynomials with the hermite and laguerre polynomials respectively similar formulae are derived for the corresponding stieltjes polynomials the usefulness of these formulae is demonstrated by employing them to show the nonexistence of gauss kronrod quadrature formulae for the gegenbauer and jacobi weight functions with real nodes and positive weights

on stratified extensions of gauss laguerre and gauss hermite quadrature formulas for the practical estimation of the error of gauss laguerre and gauss hermite quadrature formulas it is well known that real positive kronrod formulas do not exist among the alternatives which are available in the literature the stratified rules introduced by laurie are of particular interest in this note we determine the degree optimal stratified extensions for the gauss laguerre and gauss hermite formulas and we investigate their properties

direct sum decompositions of modules semilocal endomorphism rings and krull monoids abstract commutative monoids yield an analogy between the theory of factorization in commutative integral domains and the theory of direct sum decompositions of modules we show that the monoid v c of isomorphism classes of a class c of modules with semilocal endomorphism rings is a krull monoid theorem 3 4 krull monoids often appear in the study of factorizations of elements in integral domains and are defined as the monoids v for which there is a divisor homomorphism of v into a free commutative monoid in particular we consider the case in which c is the class of biuniform modules for this class the validity of a weak form of the krull schmidt theorem is explained via a representation of v c as a subdirect product of free commutative monoids

stieltjes and geronimus type polynomials abstract this work is an extension to the formal case of a previous work by monegato 5 a stieltjes type polynomial is a polynomial of degree n 1 e n 1 orthogonal with respect to the functional s c i c p n x x i i 0 1 2 s where c is a functional defined by the series f t i 0 c i t i e n 1 is of important use in the estimation of the error in pade approximation an iteration of the construction of e n 1 is attempted sections 1 2 3 4 in the last section we study the properties of the polynomial s n associated with e n 1 with respect to another functional c s n is called a geronimus type polynomial it is shown that s n verifies the system c p n s n g k 0 for k 1 n where the g k s are orthogonal with respect to c

on fuzzy metric spaces abstract in this paper we introduce the concept of a fuzzy metric space the distance between two points in a fuzzy metric space is a non negative upper semicontinuous normal and convex fuzzy number properties of fuzzy metric spaces are studied and some fixed point theorems are proved

noninterpolatory integration rules for cauchy principal value integrals etude de la convergence des regles d integration interpolat pour l approximation des integrales de cauchy

stieltjes polynomials and related quadrature formulae for a class of weight functions consider a nonnegative measure d with support in the interval a b such that the respective orthogonal polynomials above a specific index l satisfy a three term recurrence relation with constant coefficients we show that the corresponding stieltjes polynomials above the index 2l 1 have a very simple and useful representation in terms of the orthogonal polynomials as a result of this the gauss kronrod quadrature formulae for d have all the desirable properties namely the interlacing of nodes their inclusion in the closed interval a b under an additional assumption on d and the positivity of all weights furthermore the interpolatory quadrature formulae based on the zeros of the stieltjes polynomials have positive weights and both of these quadrature formulae have elevated degrees of exactness

almost uniserial rings and modules abstract we study the class of almost uniserial rings as a straightforward common generalization of left uniserial rings and left principal ideal domains a ring r is called almost left uniserial if any two non isomorphic left ideals of r are linearly ordered by inclusion i e for every pair i j of left ideals of r either i j or j i or i j also an r module m is called almost uniserial if any two non isomorphic submodules are linearly ordered by inclusion we give some interesting and useful properties of almost uniserial rings and modules it is shown that a left almost uniserial ring is either a local ring or its maximal left ideals are cyclic a noetherian left almost uniserial ring is a local ring or a principal left ideal ring also a left artinian principal left ideal ring r is almost left uniserial if and only if r is left uniserial or r m 2 d where d is a division ring finally we consider artinian commutative rings which are almost uniserial and we obtain a structure theorem for these rings

uncertainty quantification in discrete fracture network models we consider flows in fractured media described by discrete fracture network dfn models we perform an uncertainty quantification analysis assuming the fractures transmissivity coefficients to be random variables two probability distributions log uniform and log normal are used within different laws that express the coefficients in terms of a family of independent stochastic variables truncated karhunenlove expansions provide instances of such laws the approximate computation of quantities of interest such as mean value and variance for outgoing fluxes is based on a stochastic collocation approach that uses suitable sparse grids in the range of the stochastic variables whose number defines the stochastic dimension of the problem this produces a non intrusive computational method in which the dfn flow solver is applied as a black box a very fast error decay related to the analytical dependence of the observed quantities upon the stochastic variables is obtained in the low dimensional cases using isotropic sparse grids comparisons with monte carlo results show a clear gain in efficiency for the proposed method for increasing dimensions attained via successive truncations of karhunenlove expansions results are still good although the rates of convergence are progressively reduced resorting to suitably tuned anisotropic grids is an effective way to contrast such curse of dimensionality in the explored range of dimensions the resulting convergence histories are nearly independent of the dimension

asymptotic behaviour of stieltjes polynomials for ultraspherical weight functions abstract for the ultraspherical weight functions w x 1 x 2 1 2 an asymptotic representation of the stieltjes polynomials is proved for 1 n n the main result is applied to prove positivity results for kronrod extensions of gauss and lobatto quadrature formulae

generalized noninterpolatory rules for cauchy principal value integrals consider the cauchy principal value integral i kf 1 1 k x f x x dx 1 1 if we approximate f x by n 0 a j p j x w where p j is a sequence of orthonormal polynomials with respect to an admissible weight function w and a j f p j then an approximation to i kf is given by n 0 2ji kp j if in turn we approximate a j by a j m i 1im f x im p j x im then we get a double sequence of approximations q m n f to i kf we study the convergence of this sequence by relating it to the sequence of approximations associated with i wf which has been investigated previously

rings over which every rd projective module is a direct sums of cyclically presented modules abstract we study direct sum decompositions of rd projective modules in particular we investigate the rings over which every rd projective right module is a direct sum of cyclically presented right modules or a direct sum of finitely presented cyclic right modules or a direct sum of right modules with local endomorphism rings ssp rings ssp rings are necessarily semiperfect for instance the superlocal rings introduced by puninski prest and rothmaler in 28 and the semilocal strongly regular rings introduced by kaplansky in 24 are ssp rings in the case of a noetherian ring r with further additional hypotheses an rd projective r module m turns out to be either a direct sum of finitely presented cyclic modules or of the form m t m p where t m is the torsion part of m elements of m annihilated by a regular element of r and p is a projective module

modified stieltjes polynomials and gauss kronrod quadrature rules modified stieltjes polynomials are defined and used to construct suboptimal extensions of gaussian rules with one or two degrees less of polynomial exactness than the corresponding kronrod extension we prove that for wide classes of weight functions and a sufficiently large number of nodes the extended quadratures have positive weights and simple nodes on the interval 1 1 the classes of weight functions considered complement those for which the gauss kronrod rule is known to exist also strong asymptotic representations on the whole interval 1 1 are given for the modified stieltjes polynomials which prove that they behave asymptotically as orthogonal polynomials finally we provide some numerical examples

the theory of global fuzzy neighborhood structures iii fuzzy uniform structures in this paper a new notion of fuzzy uniform structure is presented and investigated fuzzy uniform structures are given as special fuzzy filters of special interest are those fuzzy uniform structures which are sup fuzzy filters they can be characterized by a related notion of fuzzy syntopogenous structure as is shown fuzzy metrics gahler and gahler 1994 generate fuzzy uniform structures in our sense each fuzzy uniform structure has an associated global homogeneous fuzzy neighborhood structure and hence also an associated fuzzy topology in case of a 0 1 fuzzy uniform structure and even in a more general case a fuzzy uniform structure of the first type can be associated there are given relation to huttons s and katsaras notions of fuzzy uniform structure hutton 1977 katsaras 1979 and to the notion of fuzzy syntopogenous structure in sense of katsaras and petalas 1984

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

peano kernels and bounds for the error constants of gaussian and related quadrature rules for cauchy principal value integrals we show that if f in c k 1 1 k ge 2 the error term of every modified positive interpolatory quadrature rule for cauchy principal value integrals of the type mathop int nolimits 1 1 w x f x over x lambda dx lambda in 1 1 fulfills r n f lambda o n k ln n uniformly for all lambda in 1 1 and hence it is of optimal order of magnitude in the classes c k 1 1 k 2 3 4 ldots here w is a weight function with the property 0 le w x sqrt 1 x 2 le c we give explicit upper bounds for the peano type error constants of such rules this improves and completes earlier results by criscuolo and mastroianni calcolo 22 1985 391 441 and numer math 54 1989 445 461 and ioakimidis math comp 44 1985 191 198 for the special case of the gaussian rule we show that the restriction k ge 2 can be dropped the results are based on a new representation of the peano kernels of these formulae via the peano kernels of the underlying classical quadrature formulae this representation may also be useful in connection with some different problems

rings of square stable range one abstract a ring s is said to have square stable range one written ssr s 1 if a s b s s implies that a 2 b x is a unit for some x s in the commutative case this extends the class of rings of stable range 1 and allows many new examples such as rings of real valued continuous functions and real holomorphy rings on the other hand ssr s 1 sometimes forces s to have stable range 1 for instance this is the case for exchange rings s for which ssr s 1 is characterized by s rad s being reduced or abelian or right quasi duo we also characterize rings s whose von neumann regular elements are strongly regular by using an element wise notion of square stable range one extending a result of estes and ohm we show that a possibly noncommutative infinite domain with stable range one or square stable range one must have a non artinian group of units

an algebraic study of gauss kronrod quadrature formulae for jacobi weight functions we study gauss kronrod quadrature formulae for the jacobi weight function w d t 1 t a 1 t and its special case a f a of the gegenbauer weight function we are interested in delineating regions in the a f plane resp intervals in a for which the quadrature rule has a the interlacing property i e the gauss nodes and the kronrod nodes interlace b all nodes contained in 1 1 c all weights positive d only real nodes not necessarily satisfying a and or b we determine the respective regions numerically for n 1 1 20 4 40 in the gegenbauer case and for n 1 1 10 in the jacobi case where n is the number of gauss nodes algebraic criteria in particular the vanishing of appropriate resultants and discriminants are used to determine the boundaries of the regions identifying properties a and d the regions for properties b and c are found more directly a number of conjectures are suggested by the numerical results finally the gauss kronrod formula for the weight w al 2 is obtained from the one for the weight w ca and similarly the gauss kronrod formula with an odd number of gauss nodes for the weight function w t itlv 1 t2 c is derived from the gauss kronrod formula for the weight w a l y 2

on fuzzy metric spaces in the present paper the authors define f open sets f closed sets f adherent points f limit points f isolated points f isolated sets f derived sets f closures f interior points f interior f exterior points f exterior f everywhere dense sets f nowhere dense sets and make some characterizations of fuzzy metric spaces

smolyak cubature of given polynomial degree with few nodes for increasing dimension some recent investigations see e g gerstner and griebel 5 novak and ritter 9 and 10 novak ritter and steinbauer 11 wasilkowski and wo niakowski 18 or petras 13 show that the so called smolyak algorithm applied to a cubature problem on the d dimensional cube seems to be particularly useful for smooth integrands the problem is still that the numbers of nodes grow polynomially but fast for increasing dimensions we therefore investigate how to obtain smolyak cubature formulae with a given degree of polynomial exactness and the asymptotically minimal number of nodes for increasing dimension d and obtain their characterization for a subset of smolyak formulae error bounds and numerical examples show their good behaviour for smooth integrands a modification can be applied successfully to problems of mathematical finance as indicated by a further numerical example

hermite and hermite fejer interpolation for stieltjes polynomials let x 1 x 2 1 2 and p n be the ultraspherical polynomials with respect to x then we denote by e n 1 the stieltjes polynomials with respect to x satisfying in this paper we show uniform convergence of the hermite fejer interpolation polynomials h n 1 and h 2n 1 based on the zeros of the stieltjes polynomials e n 1 and the product e n 1 p n for 0 1 and 0 a 1 2 respectively to prove these results we prove that the lebesgue constants of hermite fejer interpolation operators for the stieltjes polynomials e n 1 and the product e n 1 p n are optimal that is the lebesgue constants h n 1 0 a 1 and h 2n 1 0 a 1 2 have optimal order o 1 in the case of the hermite fejer interpolation polynomials h 2n 1 for 1 2 1 we prove weighted uniform convergence moreover we give some convergence theorems of hermite fejer and hermite interpolation polynomials for 0 1 in weighted lp norms

piecewise polynomial quadratures for cauchy singular integrals in this paper we propose piecewise polynomial methods for the approximation of cauchy principal value integrals and develop a simple efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise polynomial quadratures we present two examples to illustrate the advantages of these quadratures versus the gauss jacobi quadratures

stieltjes polynomials and gauss kronrod quadrature for jacobi weight functions let w beta t 1 t 1 t 1 denote the jacobi weight function for 0 5 2 the stieltjes polynomial has only few real zeros and that gauss kronrod quadrature is not possible so far results of the above type have been known only for ultraspherical weight functions w lambda t big 1 t 2 big lambda 1 2 for 0 2 for 3 the impossibility of gauss kronrod quadrature has been proved by the authors recently

equivalence of diagonal matrices over local rings it is proved that two diagonal matrices diag a1 an and diag b1 bn over a local ring r are equivalent if and only if there are two permutations of 1 2 n such that r air l r b i r l and r air e r b i r e for every i 1 2 n here r ar e denotes the epigeny class of r ar and r ar l denotes the lower part of r ar in some particular cases like for instance in the case of r local commutative diag a1 an is equivalent to diag b1 bn if and only if there is a permutation of 1 2 n with air b i r for every i 1 n these results are obtained studying the direct sum decompositions of finite direct sums of cyclically presented modules over local rings the theory of these decompositions turns out to be incredibly similar to the theory of direct sum decompositions of finite direct sums of uniserial modules over arbitrary rings

pointwise simultaneous convergence of extended lagrange interpolation with additional knots in numerical analysis it is important to construct interpolating polynomials approximating a given function and its derivatives simultaneously the authors define some new good interpolating matrices with many nodes close to the endpoints of the interval and also give error estimates

computation of gauss kronrod of quadrature rules recently laurie presented a new algorithm for the computation of 2n 1 point gauss kronrod quadrature rules with real nodes and positive weights this algorithm first determines a symmetric tridiagonal matrix of order 2n 1 from certain mixed moments and then computes a partial spectral factorization we describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined and thereby avoids computations that can be sensitive to perturbations our algorithm uses the consolidation phase of a divide and conquer algorithm for the symmetric tridiagonal eigenproblem we also discuss how the algorithm can be applied to compute kronrod extensions of gauss radau and gauss lobatto quadrature rules throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible numerical examples illustrate the performance of the algorithm

on the computing gauss kronrod quadrature formulae on discute l application de la methode de newton au calcul des formules de quadrature de gauss kronrod a partir de moments modifies et on analyse les applications non lineaires sous jacentes on determine le polynome dont les zeros sont les n uds de kronrod on traite des exemples de formules pour les integrales a singularite logarithmique ou algebrique

uniform convergence of derivatives of extended lagrange interpolation the authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm they prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation

stieltjes polynomials and related quadrature formulae for a class of weight functions ii consider a nonnegative measure d sigma with support in the interval a b such that the respective orthogonal polynomials satisfy a three term recurrence relation with coefficients alpha n left begin array ll alpha e n hbox even alpha o n hbox odd end array right beta n beta hbox for n ge ell where alpha e alpha o beta and ell are specific constants we show that the corresponding stieltjes polynomials above the index 2 ell 1 have a very simple and useful representation in terms of the orthogonal polynomials as a result of this the gauss kronrod quadrature formula for d sigma has all the desirable properties namely the interlacing of nodes their inclusion in the closed interval a b under an additional assumption on d sigma and the positivity of all weights while the formula enjoys an elevated degree of exactness furthermore the interpolatory quadrature formula based on the zeros of the stieltjes polynomials has positive weights and also elevated degree of exactness it turns out that this formula is the anti gaussian formula for d sigma while the resulting averaged gaussian formula coincides with the gauss kronrod formula for this measure moreover we show that the only positive and even measure d sigma on a a for which the gauss kronrod formula is almost of chebyshev type i e it has almost all of its weights equal is the measure d sigma t a 2 t 2 1 2 dt

a krull schmidt theorem for infinite products of modules abstract we prove a unique decomposition theorem for direct products of finitely generated modules over certain classes of rings which is analogous to the classical krull schmidt remak azumaya theorem for direct sum decompositions of modules

a comment on the completion of fuzzy metric spaces we argue that in some recent papers discussing the completion of fuzzy metric and fuzzy normed spaces proper credit is not given to earlier works on the subject

a note on peano constants of gauss kronrod quadrature schemes abstract for the error r 2 n 1 gk of the gauss kronrod quadrature formula q 2 n 1 gk we prove the precise asymptotic value of the low order peano constants c s r 2 n 1 gk where s is fixed for increasing n the result is applied for comparisons of the gauss kronrod formula q 2 n 1 gk with the gauss formula q n g and with the gauss formula q 2 n 1 g with respect to this often used quality measure

gauss quadratures and jacobi matrices for weight functions not of one sign construction of gauss quadratures with prescribed knots via jacobi matrices is extended to the case where not all orthogonal polynomials exist due to the weight function changing sign an algorithm is described and is demonstrated by calculating the knots of kronrod schemes and other gauss quadratures with prescribed knots

quadrature formulas for fourier coefficients we consider quadrature formulas of high degree of precision for the computation of the fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials in particular we show the uniqueness of a multiple node formula for the fourier tchebycheff coefficients given by micchelli and sharma and construct new gaussian formulas for the fourier coefficients of a function based on the values of the function and its derivatives

the equivalence of convergences of sequences of fuzzy numbers and its applications to the characterization of compact sets in this paper with the help of the convergence of sequences of fuzzy numbers with respect to the lebesgue measure we study the relationship between convergences of sequences of fuzzy numbers with respect to the endograph metric the sendograph metric and the l p metric we prove that these convergences are equivalent under proper conditions in addition by applying our result we give a new characterization of compact sets in fuzzy number space with the sendograph metric

nested quadrature rules for cauchy principal value integrals abstract we consider quadrature formulas for the numerical evaluation with error estimate of integrals of the form a b f x x d x and we discuss them pointing out the computational aspects then we present an automatic integrator and we compare our numerical results with the ones of another published integrator for cauchy principal value integrals

graded sets and points a stratified approach to fuzzy sets and points abstract in this paper we define the concepts of graded set and graded point as well as the usual theoretical set operations and relations and the extension of mappings between graded sets these definitions provide a method to extend concepts of classical set theory this method is related to fuzzy set theory and generalizes some properties of the family of cuts moreover the graded points verify the following two properties which cannot happen with fuzzy points

simplification of erceg s fuzzy metric function and its application abstract in this paper we greatly simplify the definition of erceg s p q p metric on a fuzzy lattice l on the basis of this a new pointwise p q p metric is introduced being defined on the set of all non zero join irreducible elements of a fuzzy lattice l meanwhile we prove that it can be directly extended to erceg s p q p metric defined on the fuzzy lattice l these results are exactly what erceg wanted to obtain furthermore we investigate the application of the pointwise metric to fuzzy p metric spaces

calculation of gauss kronrod quadrature rules the jacobi matrix of the 2n 1 point gauss kronrod quadrature rule for a given measure is calculated efficiently by a five term recurrence relation the algorithm uses only rational operations and is therefore also useful for obtaining the jacobi kronrod matrix analytically the nodes and weights can then be computed directly by standard software for gaussian quadrature formulas

stieltjes type polynomials on the unit circle stieltjes type polynomials corresponding to measures supported on the unit circle t are introduced and their asymptotic properties away from t are studied for general classes of measures as an application we prove the convergence of an associated sequence of interpolating rational functions to the corresponding caratheodory function in turn this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the gauss kronrod rule provided that the integrand is analytic in a neighborhood of t

ultraspherical stieltjes polynomials and gauss kronrod quadrature behave nicely for lambda 0 we show that the zeros of the stieltjes polynomials associated with the ultraspherical weight function with parameter lambda 0 are real and simple and except for two of them belong to 1 1 we also prove that they strictly interlace with the zeros of the corresponding ultraspherical polynomials consequently a gauss kronrod quadrature formula with two nodes outside the interval 1 1 is obtained we prove that the coefficients of such quadrature rules are positive finally an asymptotic representation of stieltjes polynomials which converges uniformly on the whole interval 1 1 is provided

weak krull schmidt for infinite direct sums of uniserial modules w x the well known krull schmidt azumaya theorem 2 says that if m s m s n are two indecomposable decompositions of a i j ig i jg j module m where m has local endomorphism ring for each i g i then i there is a bijection s i a j such that m n for every i g i in this i s i theorem the local endomorphism ring hypothesis is crucial to ensure the uniqueness of the decomposition of m into indecomposable summands and it is always tempting to ask whether under some weaker hypotheses the indecomposable decomposition of a module is unique this was done w x for example by anderson and fuller 1 theorem 12 4 who showed that the conclusion of the krull schmidt azumaya theorem still remains valid for modules with an indecomposable decomposition that complements maximal direct summands in general however there exist various examples in the literature which show that krull schmidt may fail even under rather good hypotheses on the rings or on the indecomposable sum w x w x w x mands see e g 5 7 and 8 in 1975 warfield proved that every finitely presented module over a serial ring is a finite direct sum of uniserial modules and asked whether

on gauss kronrod quadrature formulae of chebyshev type we prove that there is no positive measure da on a b such that the corresponding gauss kronrod quadrature formula is also a chebyshev quadrature formula the same is true if we consider measures of the form do t a t dt where w t is even on a symmetric interval a a and the gauss kronrod formula is required to have equal weights only for n even we also show that the only positive and even measure da t da t on 1 1 for which the gauss kronrod formula has all weights equal if n 1 or has the form i f t da t w j t to l w 2 fil wi l r f for all n 2 is the chebyshev measure of the first kind doc t t2 ll2dt

a family of gauss kronrod quadrature formulae etude d une famille de formules de quadrature de gauss kronrad formules explicites des poids