hp discontinuous galerkin method based on local higher order reconstruction we present a new adaptive higher order finite element method hp fem for the solution of boundary value problems formulated in terms of partial differential equations pdes the method does not use any information about the problem to be solved which makes it robust and equation independent it employs a higher order reconstruction scheme over local element patches which makes it faster and easier to parallelize compared to hp adaptive methods that are based on the solution of a reference problem on a globally hp refined mesh the method can be used for the solution of linear as well as nonlinear problems discretized by conforming or non conforming finite element methods and it can be combined with arbitrary a posteriori error estimators the performance of the method is demonstrated by several examples carried out by the discontinuous galerkin method

toward a universal adaptive finite element strategy part 3 design of meshes abstract in this third paper in our series the issue of designing h p meshes which are optimal in some sense is addressed criteria for h p meshes are derived which are based on the idea of minimizing the estimated error over a mesh with a fixed number of degrees of freedom an optimization algorithm is developed based on these criteria and is applied to several model one dimensional and two dimensional elliptic boundary value problems numerical results indicate that the approach can lead to exponential rates of convergence

on astala s theorem for martingales and fourier multipliers abstract we exhibit a large class of symbols m on r d d 2 for which the corresponding fourier multipliers t m satisfy the following inequality if d e are measurable subsets of r d with e d and d then d e t m e x d x e e ln d 2 e if e d 2 d e 1 2 d e ln e d e if e d 2 here denotes the lebesgue measure on i r d when d 2 these multipliers include the real and imaginary parts of the beurling ahlfors operator b and hence the inequality is also valid for b with the right hand side multiplied by 2 the inequality is sharp for the real and imaginary parts of b this work is motivated by k astala s celebrated results on the gehring reich conjecture concerning the distortion of area by quasiconformal maps the proof rests on probabilistic methods and exploits a family of appropriate novel sharp inequalities for differentially subordinate martingales these martingale bounds are of interest on their own right

linear bounds for calderon zygmund operators with even kernel on umd spaces it is well known that several classical results about calderon zygmund singular integral operators can be extended to x valued functions if and only if the banach space x has the umd property the dependence of the norm of an x valued calderon zygmund operator on the umd constant of the space x is conjectured to be linear we prove that this is indeed the case for sufficiently smooth calderon zygmund operators with cancellation associated to an even kernel our method uses the bellman function technique to obtain the right estimates for the norm of dyadic haar shift operators we then apply the representation theorem of t hytonen to extend the result to general calderon zygmund operators

majorization and d optimality for complete block designs under correlations summary d optimal designs are derived in the complete block design set up for comparing treatments using ordinary least squares estimators when the experimental errors are correlated it is shown that the d optimal design is different from the weakly universally optimal design of morgan and chakravarti and kiefer and wynn

a geometrical characterization of banach spaces in which martingale difference sequences are unconditional we study banach space valued martingale transforms and in particular characterize those banach spaces for which the classical theorems of the real valued case carry over for example if b is a banach space and 1 0 and zeta x y leq x y text if x leq 1 leq y

sharp lp estimates for discrete second order riesz transforms we show that multipliers of second order riesz transforms on products of discrete abelian groups enjoy the lp estimate p 1 where p max p q and p and q are conjugate exponents this estimate is sharp if one considers all multipliers of the form i iriri with i 1 and infinite groups in the real valued case we obtain better sharp estimates for some specific multipliers such as i iriri with 0 i 1 these are the first known precise lp estimates for discrete calderon zygmund operators

feedback and adaptive finite element solution of one dimensional boundary value problems this paper examines the concepts of feedback and adaptivity for the finite element method the model problem concernsc 0 elements of arbitrary fixed degree for a one dimensional two point boundary value problem three different feedback methods are introduced and a detailed analysis of their adaptivity is given

burkholder integrals morrey s problem and quasiconformal mappings inspired by morrey s problem on rank one convex functionals and the burkholder integrals of his martingale theory we find that the burkholder functionals bp p 2 are quasiconcave when tested on deformations of identity f id c with bp df x 0 pointwise or equivalently deformations such that df 6 p p 2jf in particular this holds in explicit neighbourhoods of the identity map among the many immediate consequences this gives the strongest possible l estimates for the gradient of a principal solution to the beltrami equation fz z fz for any p in the critical interval 2 6 p 6 1 1 f examples of local maxima lacking symmetry manifest the intricate nature of the problem

the h p and h p versions of the finite element methods in 1 dimension part iii the adaptive h p version abstract the paper is the third and final part in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimension the first part analyzed the p version and the second part concentrated on the h and h p version this paper analyzes a theoretical frame of the adaptive h p version and based on it the authors provide concrete algorithm for the one dimensional problem it is proven that in the case that the solution has x sub alpha type singularity the adaptive algorithm give an exponential rate of convergence very close to the optimal one analyzed in the second part of the paper additional keyword error analysis

burkholder inequalities for submartingales bessel processes and conformal martingales the motivation for this paper comes from the following question on comparison of norms of conformal martingales x y in bbb r d d geq 2 suppose that y is differentially subordinate to x for 0 p infty what is the optimal value of the constant c p d in the inequality y p leq c p d x p we answer this question by considering a more general related problem for nonnegative submartingales this enables us to study extension of the above inequality to the case when d1 is not an integer which has further interesting applications to stopped bessel processes and to the behavior of smooth functions on euclidean domains the inequality for conformal martingales which has its roots on the study of the l p norms of the beurling ahlfors singular integral operator extends a recent result of borichev janakiraman and volberg

hp optimization of finite element approximations analysis of the optimal mesh sequences in one dimension this paper addresses the problem of the construction of the optimal hp meshes for the singular solution x 12 in the one dimensional setting first a review of the asymptotic analysis of gui and babuska 1 is given and is used to characterize the optimal hp meshes which are geometrically graded toward the singular point the asymptotically optimal hp meshes are compared with the hp meshes which are constructed by employing the hp strategy of rachowicz et al 2 which employs the principle of maximum change of error per change in the number of degrees of freedom and the true optimal hp meshes which minimize the error for fixed number of degrees of freedom and which are determined by searching through the entire sequence of the admissible hp meshes it is shown that for the singular solution the meshes constructed by the hp strategy of rachowicz coincide with the true optimal meshes nevertheless there exist various solutions for which the algorithm which employs the principle of maximum change of error may not converge for this reason the hp strategy of rachowicz is combined with the classical feedback algorithm to obtain a robust hp strategy which always converges further it is shown that the asymptotically optimal hp meshes are very close to the true optimal hp meshes in the entire range of practical computation the effect of the accuracy of the error estimator on the optimality of the constructed hp meshes is also analyzed it is shown that unless the estimator is sufficiently accurate in the element adjacent to the singular point the meshes constructed by the adaptive hp strategy can differ substantially from the true optimal meshes

on singular integral and martingale transforms linear equivalences of norms of vector valued singular integral operators and vector valued martingale transforms are studied in particular it is shown that the umd constant of a banach space x equals the norm of the real or the imaginary part of the beurling ahlfors singular integral operator acting on l p x r 2 with p 1 moreover replacing equality by a linear equivalence this is found to be a typical property of even multipliers a corresponding result for odd multipliers and the hilbert transform is given as a corollary we obtain that the norm of the real part of the beurling ahlfors operator equals p 1 with p max p p p 1 where the novelty is the lower bound

a link between the e value and the robustness of block designs abstractthis article investigates the robustness of binary incomplete block designs against giving rise to a disconnected design in the event of observation loss a link is established between the e value of a planned design and the extent of observation loss that can be experienced while still guaranteeing an eventual design from which all treatment contrasts can be estimated patterns of missing observations covered include loss of entire blocks and loss of individual observations simple bounds are provided enabling practitioners to easily assess the robustness of a planned design

optimality and robustness to the unavailability of blocks in block designs summary this paper investigates the optimality of designs derived from binary variance balanced incomplete block designs bibds by removing t equal sized blocks which are not necessarily disjoint it is shown that the optimal design is derived by removing blocks which have disjoint sets of treatments and the worst design is derived by removing identical blocks for bibds and t 2 all resulting designs are ordered the implication for competing bibds which have the same parameters v b and k but different support sizes numbers of distinct blocks is addressed optimality and robustness are measured through the non zero eigenvalues of the c matrix of the design missing observations using the universal optimality criteria of bondar

a counterexample concerning the relation between decoupling constants and umd constants for banach spaces x and y and a bounded linear operator t x y we let t inf c such that av l 1

inequalities for hilbert operator and its extensions the probabilistic approach we present a probabilistic study of the hilbert operator tf x 1 0f y dyx y x 0 tf x 1 0 f y dyx y x 0 defined on integrable functions ff on the positive halfline using appropriate novel estimates for orthogonal martingales satisfying the differential subordination we establish sharp moment weak type and inequalities for tt we also show similar estimates for higher dimensional analogues of the hilbert operator and by the further careful modification of martingale methods we obtain related sharp localized inequalities for hilbert and riesz transforms

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

a fast hp adaptive finite element mesh design an optimum hp adaptive mesh design is accomplished by minimizing the number of equations for a specified error limit this new approach leads to a problem in which the h and p mesh parameters appear explicitly in the formulation the optimal conditions yields a non linear equation for each element which simultaneously supplies the optimum values for h and p parameters the methodology here developed is applied to the numerical solution of several unidimensional elliptic boundary value problems

optimization of the shape of regions supporting boundary conditions this article deals with the optimization of the shape of the regions assigned to different types of boundary conditions in the definition of a physical partial differential equation at first we analyze a model situation involving the solution u varomega to a laplace equation in a domain varomega the boundary partial varomega is divided into three parts vargamma d vargamma and vargamma n supporting respectively homogeneous dirichlet homogeneous neumann and inhomogeneous neumann boundary conditions the shape derivative j prime varomega theta of a general objective function j varomega of the domain is calculated in the framework of hadamard s method when the considered deformations theta are allowed to modify the geometry of vargamma d vargamma and vargamma n i e theta does not vanish on the boundary of these regions the structure of this shape derivative turns out to depend very much on the regularity of u varomega near the boundaries of the regions vargamma d vargamma and vargamma n for this reason in particular j prime varomega is difficult to calculate and to evaluate numerically when the transition overline vargamma d cap overline vargamma between homogeneous dirichlet and homogeneous neumann boundary conditions is subject to optimization to overcome this difficulty an approximation method is proposed in which the considered exact laplace equation with mixed boundary conditions is replaced with a smoothed version featuring robin boundary conditions on the whole boundary partial varomega with coefficients depending on a small parameter varepsilon we prove the consistency of this approach in our model context the approximate objective function j varepsilon varomega and its shape derivative converge to their exact counterparts as varepsilon vanishes although it is rigorously justified only in a model problem this approximation methodology may be adapted to many more complex situations for example in three space dimensions or in the context of the linearized elasticity system various numerical examples are eventually presented in order to appraise the efficiency of the proposed approximation process

a goal oriented anisotropic hp mesh adaptation method for linear convection diffusion reaction problems abstract we deal with the numerical solution of linear convection diffusion reaction equations using the h p variant of the discontinuous galerkin method on triangular grids we develop a mesh adaptive algorithm which modifies the size and shape of mesh elements and the corresponding polynomial approximation degrees in order to decrease the error in a target functional under the given tolerance we recall some theoretical results and describe in detail several technical aspects of this approach numerical experiments demonstrating the accuracy efficiency and robustness of the algorithm are presented

multilinear operator valued calderon zygmund theory abstract we develop a general theory of multilinear singular integrals with operator valued kernels acting on tuples of umd banach spaces this in particular involves investigating multilinear variants of the r boundedness condition naturally arising in operator valued theory we proceed by establishing a suitable representation of multilinear operator valued singular integrals in terms of operator valued dyadic shifts and paraproducts and studying the boundedness of these model operators via dyadic probabilistic banach space valued analysis in the bilinear case we obtain a t 1 type theorem without any additional assumptions on the banach spaces other than the necessary umd higher degrees of multilinearity are tackled via a new formulation of the rademacher maximal function rmf condition in addition to the natural umd lattice cases our rmf condition covers suitable tuples of non commutative l p spaces we employ our operator valued theory to obtain new multilinear multi parameter operator valued theorems in the natural setting of umd spaces with property