a new method for estimating subgroup means under misclassification summary this paper considers the estimation of the subgroup means of a response variable when the individuals in a random sample are allocated to the subgroups or classes by a fallible device a double sampling scheme under which we determine also the true classes of the individuals in a subsample is utilized a simple distributional free estimator is proposed and the associated large sample properties are also derived the proposed estimator is also demonstrated by simulation studies to be more efficient than selen s estimator 1986

a colimit of classifying spaces abstract we recall a description of the first non vanishing homotopy group of a certain n 1 ad of spaces and show how it yields group theoretic formulae for homotopy and homology groups of several specific spaces

solution of weakly singular fractional integro differential equations by using a new operational approach abstract in this study an operational approach based on the shifted jacobi polynomials is presented to solve a class of weakly singular fractional integro differential equations the fractional derivative operators are considered as the caputo sense in addition to finding the operational matrices of integration and product a new operational matrix is derived to be approximated the singular kernels of this class of the functional equations these operational matrices together with the collocation method are applied to convert the main equation into a linear or non linear system of algebraic equations by presenting some theorems conditions of the existence and uniqueness of the solution of the main equation and its corresponding algebraic system are investigated also the stability and convergence of the proposed method are studied and an error bound is obtained in a jacobi weighted sobolev space finally some illustrative examples are provided to demonstrate the efficiency and simplicity of the suggested approach

a new approach to the numerical solution of weakly singular volterra integral equations we consider linear weakly singular volterra integral equations of the second kind with kernels of the form k x v x v k x v 0 1 or k x v log x v k x v k x v being a smooth function the solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration by a transformation of the unknown function we obtain an equation which is still weakly singular but whose solution is as smooth as we like this resulting equation is then solved by standard product integration methods

numerical methods for a volterra integral equation with non smooth solutions we consider the numerical treatment of a singular volterra integral equation with an infinite set of solutions one of which is smooth and all others have infinite gradient at the origin this equation has been the subject of previous works where we have dealt with the approximation of the smooth solution here we present numerical methods which enable us to obtain approximations to any of the infinite class of solutions some numerical examples are given which illustrate the performance of the methods employed

artin braid groups and homotopy groups we study the brunnian subgroups and the boundary brunnian subgroups of the artin braid groups the general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the artin braid groups modulo the boundary brunnian braids as well as given as a summand of the center of the quotient groups of artin pure braid groups modulo boundary brunnian braids these results give some deep and fundamental connections between the braid groups and the general higher homotopy groups of spheres

brunnian braids and lie algebras abstract brunnian braids have interesting relations with homotopy groups of spheres in this work we study the graded lie algebra of the descending central series related to brunnian subgroup of the pure braid group a presentation of this lie algebra is obtained as well as the rank in each grading is calculated

variable transformations in the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels extensions to fredholm integral equations abstract a number of techniques that use variable transformations in numerical integration have been developed recently cf sidi numerical integration iv h brass g hammerlin eds birkhauser basel 1993 pp 359 373 laurie j comput appl math 66 1996 337 344 the use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule in this paper the application of variable transformation techniques of sidi and laurie type to the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels is considered since the transformations are such that the end points of integration need not be used as mesh points the methods introduced can be used for vie with both continuous and weakly singular kernel in a uniform way the methods have also the advantages of simplicity of application and of achieving high order of convergence the application of the idea to fredholm integral equations with continuous and weakly singular equations is also considered numerical results are included and they verify the expected increased order of convergence they were obtained by using the trapezoidal formula for the evaluation of the transformed integrals

a hybrid collocation method for volterra integral equations with weakly singular kernels the commonly used graded piecewise polynomial collocation method for weakly singular volterra integral equations may cause serious round off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time dependent equations to round off errors the singularity preserving nonpolynomial collocation method is known to have only local convergence to overcome the shortcoming of these well known methods we introduce a hybrid collocation method for solving volterra integral equations of the second kind with weakly singular kernels in this hybrid method we combine a singularity preserving nonpolynomial collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain we prove the optimal order of global convergence for this method the convergence analysis of this method is based on a singularity expansion of the exact solution of the equations we prove that the solutions of such equations can be decomposed into two parts with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function a numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method

configurations braids and homotopy groups the main results of this article are certain connections between braid groups and the homotopy groups of the 2 sphere the connections are given in terms of brunnian braids over the disk and over the 2 sphere the techniques arise from the natural structure of simplicial and structures on fundamental groups of configuration spaces

solution of a class of volterra integral equations with singular and weakly singular kernels a class of volterra integral equations are solved in this paper where the involved kernel may be weakly singular singular and hypersingular for various cases existence and uniqueness of such volterra integral equations are established furthermore by reducing such equations to ordinary differential equations analytic solutions have been determined explicitly when restricting the solutions to certain special function classes the number of suitable solutions is given

brunnian subgroups of mapping class groups and braid groups this is the second of a pair of papers on the delta group structure on the braid and mapping class groups of a surface we obtain a description of the homotopy groups of these delta groups and generalize to an arbitrary surface the berrick cohen wong wu exact sequence relating the brunnian braid groups of the 2 sphere to its homotopy groups we prove a similar result for brunnian mapping class groups 1 introduction and statement of results recent work on braid groups has shown that varying the number of strands can give rise to a combinatorial structure that of a group that contains remarkable information an example is given by the group of brunnian braids brun pk 1 m of a riemann surface m comprising those pure braids on k 1 strands that become trivial when any strand is deleted the paper 2 obtains striking results about these brunnian braid groups when the surfacem is either the disc d or sphere s in general the brunnian groups are free groups of in nite rank and would therefore seem a rather crude instrument for calculating the fundamental building blocks of homotopy theory the nite abelian groups that are the homotopy groups of spheres the following theorem reveals how this counterintuitive phenomenon occurs theorem 2 whenever k 4 there is an exact sequence 1 brun pk 2 s brun pk 1 d c brun pk 1 s k s 1 the above result has been described as one of those mysterious alluring connections which makes mathematics worth doing 27 part of its allure is the possibility it raises of relating seemingly purely geometric data to topological maps of spheres for example 2 subsection 7 2 demonstrates how the hopf bration that provides the generator of the stable 1 stem in homotopy groups of spheres corresponds to the brunnian 3 strand braid with which young women have for centuries plaited their hair and whose closure forms the classical borromean rings in general a group g comprises an in nite sequence of groups g0 g1 related by face maps these face maps di gn gn 1 i 0 n are 2010 mathematics subject classi cation primary 20f36 secondary 55q40 55r80 55u10 57m07 57s05 key words and phrases braid groups mapping class groups con guration spaces simplicial structures homotopy groups of spheres brunnian groups

numerical analysis of a singular integral equation in this paper we give details of the numerical analysis of an integral equation of the form 1 y t f t email protected 0ts m 1t my s email protected 0 the distinctive feature of the equation is the presence of a singularity at t 0 for all values of m 0 and at s 0 for all values of t 0 for 0 m 1 this means that conventional analytical and numerical theory does not apply in fact for 0 m 1 the equation has an infinite family of solutions as background we give details of the analytical results on existence and uniqueness of solutions to 1 and we give new results on the use of numerical schemes that will yield approximations to any specific solution we conclude with some numerical results that show that our methods enable us to find approximations of any chosen order to any of the infinite class of exact solutions to 1

application of the collocation method for solving nonlinear fractional integro differential equations in this paper using the collocation method we solve the nonlinear fractional integro differential equations nfide of the form f t y t a cd t a 0y t a cd t a ry t email protected t y t a tk t s f s y s ds y k a d k k 0 1 m 0 1 we study the convergence and the stability analysis of this method for f t y t a cd t a 0y t a cd t a ry t y t email protected j 0 rb j a cd t a jy t g t some numerical examples are given to show the efficiency of the presented method

adjusting for errors in classification and measurement in the analysis of partly and purely categorical data abstract this article examines a method of adjusting subgroup means when there are errors both in the allocation of elements to the subgroups and in the measurement of the response the advantages with the method are that it is simple it is also applicable when there are many subgroups and it requires no distributional assumptions for the response variable the method is compared to the maximum likelihood method for a dichotomous response and an example from the swedish level of living survey is given

on symmetric commutator subgroups braids links and homotopy groups in this paper we investigate some applications of commutator subgroups to homotopy groups and geometric groups in particular we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the higher homotopy groups this gives a connection between links and homotopy groups similar results hold for braid and surface groups

a fast numerical algorithm based on the second kind chebyshev polynomials for fractional integro differential equations with weakly singular kernels a spectral method based on operational matrices of the second kind chebyshev polynomials skcps is employed for solving fractional integro differential equations with weakly singular kernels firstly properties of shifted skcps operational matrix of fractional integration and product operational matrix are introduced and then utilized to reduce the given equation to the solution of a system of linear algebraic equations this new approach provides a significant computational advantage by converting the given original problem to an equivalent linear volterra integral equation of the second kind with the same initial conditions approximate solution is achieved by expanding the functions in terms of skcps and employing operational matrices unknown coefficients are determined by solving final system of linear equations an estimation of the error is given finally illustrative examples are included to demonstrate the high precision fast computation and good performance of the new scheme

estimating strata means in double sampling with corrections based on second phase sampling an estimator for strata means is presented for double sampling for stratification where the first phase strata estimates are corrected by second phase subsampling a variance estimator is derived for the estimated strata weights and for the adjusted strata means these are compared with two bootstrap variance estimators in a simulation study the stratum mean estimator has negligible bias and either bootstrap variance estimator is clearly better than the classical estimator in estimating the true variance the variability of the strata weights is most reliably estimated by the classical variance estimator

numerical solution of variable order fractional nonlinear quadratic integro differential equations based on the sixth kind chebyshev collocation method abstract in this paper a sixth kind chebyshev collocation method will be considered for solving a class of variable order fractional nonlinear quadratic integro differential equations v ofnqides the operational matrix of variable order fractional derivative for sixth kind chebyshev polynomials is derived and then a collocation approach is employed to reduce the v ofnqide to a system of nonlinear algebraic equations convergence analysis of the proposed method is evaluated and the rate of convergence is established finally some numerical test examples are investigated to validate the accuracy and robustness of the proposed approach

numerical solution of nonlinear fractional integro differential equations with weakly singular kernels via a modification of hat functions abstract in the present paper a modification of hat functions mhfs has been considered for solving a class of nonlinear fractional integro differential equations with weakly singular kernels numerically the fractional order operational matrix of integration is introduced we provide an error estimation for the approximation of a function by a series of mhfs to suggest a numerical method the main problem is converted to an equivalent volterra integral equation of the second kind and operational matrices of mhfs are used to reduce the problem to the solution of bivariate polynomial equations finally illustrative examples are provided to confirm the accuracy and validity of the proposed method

numerical solution of a nonuniquely solvable volterra integral equation using extrapolation methods in this work the numerical solution of a volterra integral equation with a certain weakly singular kernel depending on a real parameter is considered although for certain values of this equation possesses an infinite set of solutions we have been able to prove that euler s method converges to a particular solution it is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize so that general extrapolation algorithms like the e algorithm can be applied to improve the numerical results this is illustrated by means of some examples

a braided simplicial group by studying the braid group action on milnor s construction of the 1 sphere we show that the general higher homotopy group of the 3 sphere is the fixed set of the pure braid group action on certain combinatorially described groups this establishes a relation between the braid groups and the homotopy groups of the sphere 2000 mathematical subject classification 20f36 55p35 55q05 55q40 55u10

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence