non generic cusps we find the order of contact of the boundaries of the cusp for two parameter families of vector fields on the real line or diffeomorphisms of the real line for cusp bifurcations of codimensions 1 and 2 moreover we create a machinery that can be used for the same problem in higher codimensions and perhaps for other similar problems

a feti dp preconditioner with a special scaling for mortar discretization of elliptic problems with discontinuous coefficients we consider two dimensional elliptic problems with discontinuous coefficients discretized by the finite element method on geometrically conforming nonmatching triangulations across the interface using the mortar technique the resulting discrete problem is solved by a dual primal feti method in this paper we introduce and analyze a preconditioner with a special scaling of coefficients and step parameters and establish convergence bounds we show that the preconditioner is almost optimal with constants independent of the jumps of coefficients and step parameters extensive computational evidence is presented that illustrates an almost optimal convergence for a variety of situations distribution of subregions grid assignment grid ratios number of subregions for both continuous and discontinuous problems

locally connected models for julia sets let p be a polynomial with a connected julia set j we use continuum theory to show that it admits a finest monotone map onto a locally connected continuum j p i e a monotone map j j p such that for any other monotone map j j there exists a monotone map h with h then we extend onto the complex plane c keeping the same notation and show that monotonically semiconjugates p c to a topological polynomial g c c if p does not have siegel or cremer periodic points this gives an alternative proof of kiwi s fundamental results on locally connected models of dynamics on the julia sets but the results hold for all polynomials with connected julia sets we also give a characterization and a useful sufficient condition for the map not to collapse all of j into a point

a two level nonoverlapping schwarz algorithm for the stokes problem without primal pressure unknowns a two level nonoverlapping schwarz algorithm is developed for the stokes problem the main feature of the algorithm is that a mixed problem with both velocity and pressure unknowns is solved with a balancing domain decomposition by constraints bddc type preconditioner which consists of solving local stokes problems and one global coarse problem related to only primal velocity unknowns our preconditioner allows to use a smaller set of primal velocity unknowns than other bddc preconditioners without much concern on certain flux conditions on the subdomain boundaries and the inf sup stability of the coarse problem in the two dimensional case velocity unknowns at subdomain corners are selected as the primal unknowns in addition to them averages of each velocity component across common faces are employed as the primal unknowns for the three dimensional case by using its close connection to the dual primal finite element tearing and interconnecting feti dp algorithm siam j sci comput 2010 32 3301 3322 siam j numer anal 2010 47 4142 4162 it is shown that the resulting matrix of our algorithm has the same eigenvalues as the feti dp algorithm except zero and one the maximum eigenvalue is determined by h h the number of elements across each subdomains and the minimum eigenvalue is bounded below by a constant which does not depend on any mesh parameters convergence of the method is analyzed and numerical results are included copyright 2011 john wiley sons ltd

a feti dp formulation of three dimensional elasticity problems with mortar discretization in this paper a feti dp formulation for three dimensional elasticity on nonmatching grids over geometrically nonconforming subdomain partitions is considered to resolve the nonconformity of the finite elements a mortar matching condition is imposed on the subdomain interfaces faces a feti dp algorithm is then built by enforcing the mortar matching condition in dual and primal ways in order to make the feti dp algorithm scalable a set of primal constraints which include average and momentum constraints over interfaces are selected from the mortar matching condition a condition number bound c 1 log h h 2 is then proved for the feti dp formulation for the elasticity problems with discontinuous material parameters only some faces need to be chosen as primal faces on which the average and momentum constraints are imposed numerical results are included

inexact feti dp methods inexact feti dp domain decomposition methods are considered preconditioners based on formulations of feti dp as a saddle point problem are used which allow for an inexact solution of the coarse problem a positive definite reformulation of the preconditioned saddle point problem which also allows for approximate solvers is considered as well in the formulation that iterates on the original feti dp saddle point system it is also possible to solve the local neumann subdomain problems inexactly given good approximate solvers for the local and coarse problems convergence bounds of the same quality as for the standard feti dp methods are obtained numerical experiments which compare the convergence of the inexact methods with that of standard feti dp are shown for 2d and 3d elasticity using gmres and cg as krylov space methods based on parallel computations a comparison of one variant of the inexact feti dp algorithms and the standard feti dp method is carried out and similar parallel performance is achieved parallel scalability of the inexact variant is also demonstrated it is shown that for a very large number of subdomains and a very large coarse problem the inexact method can be superior copyright 2006 john wiley sons ltd

on the use of inexact subdomain solvers for bddc algorithms the standard bddc balancing domain decomposition by constraints preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model this results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model the cost is then often dominated by that of the problems on the subdomains an important role is also played both in theory and practice by an averaging operator and in addition exact dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains the use of inexact solvers for these problems and even the replacement of the dirichlet solvers by a trivial extension are considered it is established that one of the resulting algorithms has the same eigenvalues as the standard bddc algorithm and the connection of another with the feti dp algorithm with a lumped preconditioner is also considered multigrid methods are used in the experimental work and under certain assumptions it is established that the iteration count essentially remains the same as when exact solvers are used while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear

universal channel coding for general output alphabet we propose two classes of universal codes that are suited to two asymptotic regimes when the output alphabet is possibly continuous the first class has the property that the error probability decays exponentially fast and we identify an explicit lower bound on the error exponent the other class attains the epsilon capacity of the channel and we also identify the second order term in the asymptotic expansion the proposed encoder is essentially based on the packing lemma of the method of types for the decoder we first derive a renyi relative entropy version of clarke and barron s formula the distance between the true distribution and the bayesian mixture which is of independent interest the universal decoder is stated in terms of this formula and quantities used in the information spectrum method the methods contained herein allow us to analyze universal codes for channels with continuous and discrete output alphabets in a unified manner and to analyze their performances in terms of the exponential decay of the error probability and the second order coding rate

dual primal feti methods for three dimensional elliptic problems with heterogeneous coefficients in this paper certain iterative substructuring methods with lagrange multipliers are considered for elliptic problems in three dimensions the algorithms belong to the family of dual primal finite element tearing and interconnecting feti methods which recently have been introduced and analyzed successfully for elliptic problems in the plane the family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dual primal preconditioner it is shown that the condition numbers of several of the dual primal feti methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains the mesh size and jumps in the coefficients these results closely parallel those of other successful iterative substructuring methods of primal as well as dual type

three level bddc in three dimensions balancing domain decomposition by constraints bddc methods are nonoverlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from the discretization of elliptic boundary value problems their coarse problems are given in terms of a small number of continuity constraints for each subdomain which are enforced across the interface the coarse problem matrix is generated and factored by a direct solver at the beginning of the computation and it can ultimately become a bottleneck if the number of subdomains is very large in this paper two three level bddc methods are introduced for solving the coarse problem approximately for problems in three dimensions this is an extension of previous work for the two dimensional case edge constraints are considered in this work since vertex constraints alone which work well in two dimensions result in a noncompetitive algorithm in three dimensions some new technical tools are then needed in the analysis and this makes the three dimensional case more complicated estimates of the condition numbers are provided for two three level bddc methods and numerical experiments are also discussed

multiplier spaces for the mortar finite element method in three dimensions we consider the construction of multiplier spaces for use with the mortar finite element method in three spatial dimensions on globally or locally quasi uniform meshes a set of abstract conditions is given for the multiplier spaces which are sufficient to guarantee a stable and convergent mortar approximation three examples of multipliers satisfying these conditions are presented the first one is a dual basis example while the remaining two are based on finite volumes finally the results of computational examples illustrating the theory are reported

generalized autoregressive moving average models a class of generalized autoregressive moving average garma models is developed that extends the univariate gaussian arma time series model to a flexible observation driven model for non gaussian time series data the dependent variable is assumed to have a conditional exponential family distribution given the past history of the process the model estimation is carried out using an iteratively reweighted least squares algorithm properties of the model including stationarity and marginal moments are either derived explicitly or investigated using monte carlo simulation the relationship of the garma model to other models is shown including the autoregressive models of zeger and qaqish the moving average models of li and the reparameterized generalized autoregressive conditional heteroscedastic garch model providing the formula for its fourth marginal moment not previously derived the model is demonstrated by the application of the garma model with a negative binomial conditional distribution to

a bddc method for mortar discretizations using a transformation of basis a bddc balancing domain decomposition by constraints method is developed for elliptic equations with discontinuous coefficients discretized by mortar finite element methods for geometrically nonconforming partitions in both two and three space dimensions the coarse component of the preconditioner is defined in terms of one mortar constraint for each edge face which is the intersection of the boundaries of a pair of subdomains a condition number bound of the form c max i 1 log h i h i 2 is established under certain assumptions on the geometrically nonconforming subdomain partition in the three dimensional case here h i and h i are the subdomain diameters and the mesh sizes respectively in the geometrically conforming case and the geometrically nonconforming cases in two dimensions no assumptions on the subdomain partition are required this bddc preconditioner is also shown to be closely related to the neumann dirichlet version of the feti dp algorithm the results are illustrated by numerical experiments which confirm the theoretical results

an adaptive bddc algorithm in variational form for mortar discretizations abstract a balancing domain decomposition by constraints bddc algorithm with adaptive primal constraints in variational form is introduced and analyzed for high order mortar discretization of two dimensional elliptic problems with high varying and random coefficients some vector valued auxiliary spaces and operators with essential properties are defined to describe the variational algorithm and the coarse space is formed by using a transformation operator on each interface compared with the adaptive bddc algorithms for conforming galerkin approximations our algorithm is simpler because there have not any continuity constraints at subdomain vertices in the mortar method involved in this paper the condition number of the preconditioned system is proved to be bounded above by a user defined tolerance and a constant which is dependent on the maximum number of interfaces per subdomain and independent of the mesh size and the contrast of the given coefficients numerical results show the robustness and efficiency of the algorithm for various model problems

information geometry approach to parameter estimation in markov chains we consider the parameter estimation of markov chain when the unknown transition matrix belongs to an exponential family of transition matrices then we show that the sample mean of the generator of the exponential family is an asymptotically efficient estimator further we also define a curved exponential family of transition matrices using a transition matrix version of the pythagorean theorem we give an asymptotically efficient estimator for a curved exponential family primary 62m05

bddc algorithms for incompressible stokes equations the purpose of this paper is to extend the balancing domain decomposition by constraints bddc algorithm to saddle point problems that arise when mixed finite element methods are used to approximate the system of incompressible stokes equations the bddc algorithms are iterative substructuring methods which form a class of domain decomposition methods based on the decomposition of the domain of the differential equations into nonoverlapping subdomains they are defined in terms of a set of primal continuity constraints which are enforced across the interface between the subdomains and which provide a coarse space component of the preconditioner sets of such constraints are identified for which bounds on the rate of convergence can be established that are just as strong as previously known bounds for the elliptic case in fact the preconditioned operator is effectively positive definite which makes the use of a conjugate gradient method possible a close connection is also established between the bddc

a parallel implementation of dual primal feti methods for three dimensional linear elasticity using a transformation of basis dual primal feti methods for linear elasticity problems in three dimensions are considered these are nonoverlapping domain decomposition methods where some primal continuity constraints across subdomain boundaries are required to hold throughout the iterations whereas most of the constraints are enforced by lagrange multipliers an algorithmic framework for dual primal feti methods is described together with a transformation of basis to implement the primal constraints numerical results obtained from a parallel implementation of these algorithms applied to a model benchmark problem with structured meshes and to problems with more complicated geometries from industrial and biological applications using unstructured meshes are provided these results show that the presented dual primal feti algorithms are numerical and parallel scalable

two level schwarz algorithms with overlapping subregions for mortar finite elements preconditioned conjugate gradient methods based on two level overlapping schwarz methods often perform quite well such a preconditioner combines a coarse space solver with local components which are defined in terms of subregions that form an overlapping covering of the region on which the elliptic problem is defined precise bounds on the rate of convergence of such iterative methods have previously been obtained in the case of conforming lower order and spectral finite elements as well as in a number of other cases in this paper this domain decomposition algorithm and analysis are extended to mortar finite elements it is established that the condition number of the relevant iteration operator is independent of the number of subregions and varies with the relative overlap between neighboring subregions linearly as in the conforming cases previously considered

families of rational maps and iterative root finding algorithms in this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root finding algorithms we answer a question of smale s by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more we settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics and we give a classification of all such algorithms in the context of conformal dynamics our main result is the following a stable algebraic family of rational maps is either trivial all its members are conjugate by mobius transformations or affine its members are obtained as quotients of iterated addition on a family of complex tori our classification of generally convergent algorithms follows easily from this result as another consequence of rigidity we observe that the eigenvalues of a nonaffine rational map at its periodic points determine the map up to finitely many choices this implies that bounded analytic functions nearly separate points on the moduli space of a rational map

the set of maps f_ ɑ b x x ɑ b 2π sin 2πx with any given rotation interval is contractible consider the two parameter family of real analytic maps f b x x b 2 sin 2 x which are lifts of degree one endomorphisms of the circle the purpose of this paper is to provide a proof that for any closed interval i the set of maps f b whose rotation interval is i form a contractible set

isogeometric bddc preconditioners with deluxe scaling a balancing domain decomposition by constraints bddc preconditioner with a novel scaling introduced by dohrmann for problems with more than one variable coefficient and here denoted as deluxe scaling is extended to isogeometric analysis of scalar elliptic problems this new scaling turns out to be more powerful than the standard rho and stiffness scalings considered in a previous isogeometric bddc study our h analysis shows that the condition number of the resulting deluxe bddc preconditioner is scalable with a quasi optimal polylogarithmic bound which is also independent of coefficient discontinuities across subdomain interfaces extensive numerical experiments support the theory and show that the deluxe scaling yields a remarkable improvement over the older scalings in particular for large isogeometric polynomial degree and high regularity

julia fatou sullivan theory for real one dimensional dynamics our aim is to show that the julia fatou sullivan structure theory for the dynamics of rational maps is also valid for smooth endomorphisms of the circle and of the interval under extremely mild smoothness and non flatness conditions in order to stress the similarity between real and complex one dimensional dynamics let us recall the main results from the fatou julia sullivan theory if f is a rational map then there is a dynamical decomposition of the riemann sphere into the disjoint union of two totally invariant i e both forward and backward invariant sets j f f f here f f is the domain of normality of the family of iterates of f and is called the fatou set its complement which is called the julia set off is a compact set which contains all the complications of the dynamics off the connected components of the open set f f are mapped onto each other by f hence the orbit of a component of f f is the union of some components of f f julia proved in the beginning of the century that if a component of f f is periodic and contains an attracting periodic point then the orbit of this component must contain a critical point sullivan in the remarkable paper su showed via quasi conformal deformations that the components are eventually periodic and fall into finitely many orbits let n be either the circle s j or a compact interval of the real line and f n n be a smooth endomorphism a critical point of f is a point where the derivative vanishes a critical point is non flat if some higher derivative is non zero a critical point is an inflection point if it has a neighbourhood where f is monotone otherwise it is called a turning point assume f is not a homeomorphism i f f is a homeomorphism then these maps correspond to degree 1 rational maps of the riemann sphere and the situation is

a preconditioner for the feti dp formulation with mortar methods in two dimensions in this paper we consider a dual primal feti feti dp method for elliptic problems on nonmatching grids the feti dp method is a domain decomposition method that uses lagrange multipliers to match solutions continuously across subdomain boundaries in the sense of dual primal variables we use the mortar matching condition as the continuity constraints for the feti dp formulation we construct a preconditioner for the feti dp operator and show that the condition number of the preconditioned feti dp operator is bounded by c max i 1 ldots n left 1 log left h i h i right right 2 where h i and h i are sizes of domain and mesh for each subdomain respectively and c is a constant independent of h i s and h i s we allow jumps of coefficients of elliptic problems across subdomain boundaries numerical results are included

the parameter planes of λz m exp z for m 2 we consider the families of entire transcendental maps given by f m z zm exp z where m 2 all functions f m have a superattracting fixed point at z 0 and a critical point at z m in the parameter planes we focus on the capture zones i e values for which the critical point belongs to the basin of attraction of z 0 denoted by a 0 in particular we study the main capture zone parameter values for which the critical point lies in the immediate basin a 0 and prove that is bounded connected and simply connected all other capture zones are unbounded and simply connected for each parameter in the main capture zone a 0 consists of a single connected component with non locally connected boundary for all remaining values of a 0 is a quasidisk on a different approach we introduce some families of holomorphic maps of mathbb c which serve as a model for f m in the sense that they are related by means of quasiconformal surgery to f m

double standard maps we investigate the family of double standard maps of the circle onto itself given by f a b x 2x a b pi sin 2 pi x mod 1 where the parameters a b are real and 0 b 1 similarly to the well known family of arnold standard maps of the circle a a b x x a b 2 pi sin 2 pi x mod 1 any such map has at most one attracting periodic orbit and the set of parameters a b for which such orbit exists is divided into tongues however unlike the classical arnold tongues that begin at the level b 0 for double standard maps the tongues begin at higher levels depending on the tongue moreover the order of the tongues is different for the standard maps it is governed by the continued fraction expansions of rational numbers for the double standard maps it is governed by their binary expansions we investigate closer two families of tongues with different behavior

on conditioning for model based inference in survey sampling summary a discontinuity of sampling distribution inference between a prediction problem and an estimation problem in model based survey sampling is discussed the problem is resolved by conditioning on an appropriate ancillary within a curved exponential family framework the ignorability of the sampling design for inference is discussed

bifurcations of circle maps arnol d tongues bistability and rotation intervals we study the bifurcations of two parameter families of circle maps that are similar tofb w x x w b 2 sin 2 x mod1 the bifurcation diagram is constructed in terms of setstr wheretr is the set of parameter values b w for whichfb w has an orbit with rotation numberr we show that the known structure whenb 1 wherefb w is no longer injective and can have an interval of rotation numbers specifically the tongues overlap in a uniform monotonic manner and forr irrational tr opens into a tongue our other theorems give information about the dynamics offb w e g bistability or aperiodicity for b w in various subsets of parameter space

dual primal feti methods for linear elasticity dual primal feti methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations as in primal iterative substructuring methods while most of the constraints are enforced by lagrange multipliers as in one level feti methods these methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems the purpose of this article is to develop strategies for selecting these constraints which are enforced throughout the iterations such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them the algorithms are described in terms of a change of basis that has proven to be quite robust in practice a theoretical analysis is provided for the case of linear elasticity and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the young s modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain the strategies have already proven quite successful in large scale implementations of these iterative methods 2006 wiley periodicals inc

the scaling of arnol d tongues for differentiable homeomorphisms of the circle we show both for diffeomorphisms of the circle and for differentiable homeomorphisms that are not diffeomorphisms that the widths of the arnol d tongues in a one parameter family scale asq 3 whenq is the denominator of the rotation number

three level bddc in two dimensions balancing domain decomposition by constraints bddc methods are non overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems they are similar to the balancing neumann neumann algorithm however in bddc methods a small number of continuity constraints are enforced across the interface and these constraints form a new coarse global component an important advantage of using such constraints is that the schur complements that arise in the computation will all be strictly positive definite the matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation however this problem can ultimately become a bottleneck if the number of subdomains is very large in this paper two three level bddc methods are introduced for solving the coarse problem approximately in two dimensional space while still maintaining a good convergence rate estimates of the condition numbers are provided for the two three level bddc methods and numerical experiments are also discussed copyright 2006 john wiley sons ltd

on the selection of primal unknowns for a feti dp formulation of the stokes problem in two dimensions selection of primal unknowns is important in convergence of feti dp dual primal finite element tearing and interconnecting methods which are known to be the most scalable dual iterative substructuring methods a feti dp algorithm for the stokes problem without primal pressure unknowns was developed and analyzed by kim et al 2010 1 only the velocity unknowns at the subdomain vertices are selected to be the primal unknowns and convergence of the algorithm with a lumped preconditioner is determined by the condition number bound c h h 1 log h h where h h is the number of elements across subdomains in this work primal unknowns corresponding to the averages on edges are introduced and a better condition number bound c h h is proved for such a selection of primal unknowns numerical results are included

transition to topological chaos for circle maps abstract many biperiodic flows can be modelled by maps of a circle to itself for such maps the transition from zero to positive topological entropy can be achieved in several ways we describe all the possible routes for smooth circle maps and discuss the relevance of our results to the transition to chaos for two frequency systems

universality of critical circle covers for a class of critical circle covers we show that properly re scaled first return maps to a neighborhood of the critical point converge to universal limits for that to occur the critical point has to be sufficiently flat

computing the invariant measure and the lyapunov exponent for one dimensional maps using a measure preserving polynomial basis we consider a generalisation of ulam s method for approximating invariant densities of one dimensional maps rather than use piecewise constant polynomials to approximate the density we use polynomials of degree n which are defined by the requirement that they preserve the measure on n 1 neighbouring subintervals over the whole interval this results in a discontinuous piecewise polynomial approximation to the density we prove error results where this approach is used to approximate smooth densities we also consider the computation of the lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself together with using cubic polynomials in the density approximation this yields a very efficient method for computing highly accurate estimates of the lyapunov exponent we illustrate the theoretical findings with some examples

a two level nonoverlapping schwarz algorithm for the stokes problem numerical study abstract a general framework of a two level nonoverlapping schwarz algorithm for the stokes problem is developed by relaxing average zero condition on pressure unknowns this framework allows both discontinuous and continuous pressure finite element spaces the coarse problem is built by algebraic manipulation after selecting appropriate primal unknowns just like in bddc algorithms performance of the suggested algorithm is presented depending on the selection of finite elements and primal unknowns under the same set of primal unknowns the algorithm for the case with discontinuous pressure functions outperforms one with continuous pressure functions for the two dimensional stokes problem the algorithm with a set of primal unknowns consisting of velocity unknowns at corners averages of velocity components over common edges and pressure unknowns at corners presents good scalability when continuous pressure test functions are used in both two and three dimensional stokes problems an improvement can be made for the case with continuous pressure test functions by applying the suggested algorithm to the interface problem which is obtained by eliminating velocity unknowns and pressure unknowns interior to each subdomains

bddc preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions balancing domain decomposition by constraints bddc algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with gauss lobatto legendre spectral elements in three dimensions initially mixed spectral elements are employed to discretize the almost incompressible elasticity system but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped appropriate sets of primal constraints can be associated with the subdomain vertices edges and faces so that the resulting bddc methods have a fast convergence rate independent of the almost incompressibility of the material in particular the condition number of the bddc preconditioned operator is shown to depend only weakly on the polynomial degree n the ratio h h of subdomain and element diameters and the inverse of the inf sup constants of the subdomains and the underlying mixed formulation while being scalable i e independent of the number of subdomains and robust i e independent of the poisson ratio and young s modulus of the material considered these results also apply to the related dual primal finite element tearing and interconnect feti dp algorithms defined by the same set of primal constraints numerical experiments carried out on parallel computing systems confirm these results

iterative substructuring preconditioners for mortar element methods in two dimensions the mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains the resulting methods are nonconforming but still yield optimal approximations in this paper we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric second order elliptic equations in two dimensions both spectral and finite element methods for geometrically conforming as well as nonconforming domain decompositions are studied in each case we obtain a polylogarithmic bound on the condition number of the preconditioned matrix

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

asymptotic conditional inference for regular nonergodic models with an application to autoregressive processes a conditional limit theorem is derived for a certain class of stochastic processes whose distributions constitute a nonergodic family the limit theorem allows us to study the asymptotic behaviour under the conditional model of some standard statistical procedures by making use of results for ergodic families explosive gaussian autoregressive processes are studied in some detail here the conditional process is shown to be a nonexplosive gaussian autoregression bearing a simple relation to the original process some optimality results under the conditional model are given for estimators and tests based on the unconditional likelihood

a three level bddc algorithm for mortar discretizations in this paper a three level balancing domain decomposition by constraints bddc algorithm is developed for the solutions of large sparse algebraic linear systems arising from the mortar discretization of elliptic boundary value problems the mortar discretization is considered on geometrically nonconforming subdomain partitions in two level bddc algorithms the coarse problem needs to be solved exactly however its size will increase with the increase of the number of the subdomains to overcome this limitation the three level algorithm solves the coarse problem inexactly while a good rate of convergence is maintained this is an extension of previous work the three level bddc algorithms for standard finite element discretization estimates of the condition numbers are provided for the three level bddc method and numerical experiments are also discussed

a feti dp formulation for the stokes problem without primal pressure components a scalable feti dp dual primal finite element tearing and interconnecting algorithm for the stokes problem that employs a lumped preconditioner is developed and analyzed a pair of inf sup stable velocity and pressure finite element spaces is used to obtain a discrete problem differently from previous approaches no primal pressure unknowns are selected and only velocity primal unknowns at subdomain corners are selected this leads to a symmetric and positive definite coarse problem matrix in the feti dp operator while a larger and indefinite coarse problem appears in the previous approaches in addition its condition number bound is proved to be the same as the feti dp algorithm with a lumped preconditioner for elliptic problems numerical results are included