type cotype and twisted sums induced by complex interpolation abstract this paper deals with extensions or twisted sums of banach spaces that come induced by complex interpolation and the relation between the type and cotype of the spaces in the interpolation scale and the nontriviality and singularity of the induced extension the results are presented in the context of interpolation of families of banach spaces and are applied to the study of submodules of schatten classes we also obtain nontrivial extensions of spaces without the cap which also fail the cap

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

carl s inequality for quasi banach spaces we prove that for any two quasi banach spaces x and y and any 0 there exists a constant 0 such that sup1 k n k ek t sup1 k n k ck t holds for all linear and bounded operators t x y here ek t is the k th entropy number of t and ck t is the k th gelfand number of t for banach spaces x and y this inequality is widely used and well known as carl s inequality for general quasi banach spaces it is a new result

smallest singular value of random matrices and geometry of random polytopes abstract we study the behaviour of the smallest singular value of a rectangular random matrix i e matrix whose entries are independent random variables satisfying some additional conditions we prove a deviation inequality and show that such a matrix is a good isomorphism on its image then we obtain asymptotically sharp estimates for volumes and other geometric parameters of random polytopes absolutely convex hulls of rows of random matrices all our results hold with high probability that is with probability exponentially in dimension close to 1

a probabilistic approach to the geometry of the lᵨⁿ ball this article investigates by probabilistic methods various geometric questions on b n p the unit ball of ln p we propose realizations in terms of independent random variables of several distributions on b n p including the normalized volume measure these representations allow us to unify and extend the known results of the sub independence of coordinate slabs in b n p as another application we compute moments of linear functionals on b n p which gives sharp constants in khinchine s inequalities on b n p and determines the 2 constant of all directions on b n p we also study the extremal values of several gaussian averages on sections of b n p including mean width and l norm and derive several monotonicity results as p varies applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition

universal teichmuller space and diff s 1 s 1 we point out that the coset space diffs1 s1 is a dense complex submanifold of the universal teichmuller spaces of compact riemann spaces of genus g 1 a holomorphic map ofs into the inifinite dimensional segal diskd1 is constructed this is the universal analogue of the map of teichmuller spaces into the siegel disk provided by the period matrix the kahler potential for the general homogenous metric on diffs1 s1 is computed explicitly using the map intod1 some applications to string theory are discussed

r bounded approximating sequences and applications to semigroups abstract it is shown that on certain banach spaces including c 0 1 and l1 0 1 there is no strongly continuous semigroup tt 0

entropy numbers s numbers and eigenvalue problems abstract we establish inequalities between entropy numbers and approximation numbers for operators acting between banach spaces furthermore we derive inequalities between eigenvalues and entropy numbers for operators acting on a banach space the results are compared with the classical inequalities of bernstein and jackson

maximum norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter

schauder bases in quotients of subspaces of l 2 x introduction questions and problems related to schauder basis are among the most classical ones in the banach space theory and have been studied from the earliest days the fact that every infinite dimensional banach space has an infinite dimensional subspace with a basis has been observed already in banach s book ba at about the same time the existence of bases in classical banach spaces such as c 0 1 and lp 0 1 was established an importance of the concept of a basis lies in the fact that it provides a natural method of approximation of vectors and operators in the space a question related to the approximation property in banach spaces was asked by mazur in the scottish book scb this question and the problem of the existence of a basis in every banach space stayed open for about 40 years although not because of the lack of interest cf e g sin in 1972 p enflo constructed a banach space which fails the approximation property and hence does not admit a schauder basis his discovery triggered a flurry of activities resulting in constructions of subspaces failing the approximation property in all classical banach spaces not isomorphic to a hilbert space such subspaces have been also found in all banach spaces sufficiently far from the hilbert space cf e g l t l l t 2 and references therein techniques developed until that moment did not provide quantitative finitedimensional estimates for basis constants it took another 10 years before a new probabilistic argument was introduced by e d gluskin g l and consequently the existence of n dimensional banach spaces with basis constant tending to infinity was proved independently by gluskin g 2 and s j szarek sz l at last a circle got closed a method of constructing infinite dimensional examples out of finite dimensional ones was proposed by j bourgain b l and developed by szarek sz 3 to obtain an 2 sum of finite dimensional banach spaces which has no basis but obviously admits a so called finite dimensional decomposition in this paper we prove a striking isomorphic characterization of a hilbert space in terms of a schauder basis first observe that the class of banach spaces isomorphic to a hilbert space is closed under operations of passing to closed

donsker classes of sets we study the central limit theorem clt and the law of large numbers lln for empirical processes indexed by a countable class of sets c the main result of purely measure theoretical nature relates different ways to measure the size of c it relies on a new rearrangement inequality that has been inspired by techniques used in the local theory of banach spaces as an application we give sharp necessary conditions for the clt that are in some sense the best possible we also obtain a way to compute the rate of convergence in the lln

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

geometric quantization of n 2 superstring a nonperturbative geometric formulation of the n 2 neveu schwarz superstring theory which has been recently interpreted by h ooguri and c vafa mod phys lett a5 1389 1990 as a consistent quantum theory of self dual gravity in four dimensions is constructed it is shown that the natural complex structure over the loop superspace md d associated to the n 2 neveu schwarz fermionic string is invariant under symmetry group osp 2 2 superdiffs1 2 moreover it is proved that there is a unique lorentz and osp 2 2 invariant complex structure on md d this result implies that the superspace of all admissible complex structures over md d is isomorphic to the homogeneous kahler supermanifold superdiffs1 2 osp 2 2 the ricci curvature of superdiffs1 2 osp 2 2 is calculated applying the method of geometric quantization to the n 2 neveu schwarz superstring theory along the lines suggested by m j bowick and s g rajeev nucl phys b361 469 1991 a representation is constructed of nonperturbativ

polynomial inequalities on the hamming cube let x vert cdot vert x be a banach space the purpose of this article is to systematically investigate dimension independent properties of vector valued functions f 1 1 n rightarrow x on the hamming cube whose spectrum is bounded above or below our proofs exploit contractivity properties of the heat flow induced by the geometry of the target space x vert cdot vert x combined with duality arguments and suitable tools from approximation theory and complex analysis we obtain a series of improvements of various well studied estimates for functions with bounded spectrum including moment comparison results for low degree walsh polynomials and bernstein markov type inequalities which constitute discrete vector valued analogues of freud s inequality in gauss space 1971 many of these inequalities are new even for scalar valued functions furthermore we provide a short proof of mendel and naor s heat smoothing theorem 2014 for functions in tail spaces with values in spaces of nontrivial type and we also prove a dual lower bound on the decay of the heat semigroup acting on functions with spectrum bounded from above finally we improve the reverse bernstein markov inequalities of meyer in seminar on probability xviii lecture notes in mathematics springer berlin 1984 https doi org 10 1007 bfb0100043 and mendel and naor publ math inst hautes etudes sci 119 1 95 2014 https doi org 10 1007 s10240 013 0053 2 for functions with narrow enough spectrum and improve the bounds of filmus et al isr j math 214 1 167 192 2016 https doi org 10 1007 s11856 016 1355 0 on the ell p sums of influences of bounded functions for p in big 1 frac 4 3 big

on the nontrivial projection problem abstract the nontrivial projection problem asks whether every finite dimensional normed space admits a well bounded projection of nontrivial rank and corank or equivalently whether every centrally symmetric convex body of arbitrary dimension is approximately affinely equivalent to a direct product of two bodies of nontrivial dimensions we show that this is true up to a logarithmic factor

a period mapping in universal teichmuller space in previous work it had been shown that the remarkable homogeneous space m diff s 1 psl 2 r sits as a complex analytic and kahler submanifold of the universal teichmuller space there is a natural immersion of m into the infinite dimensional version due to segal of the siegel space of period matrices that map is proved to be injective equivariant holomorphic and kahler isometric with respect to the canonical metrics regarding a period mapping as a map describing the variation of complex structure we explain why is an infinite dimensional period mapping

holomorphic semi groups and the geometry of banach spaces according to a fundamental theorem of dvoretzky cf 24 25 any infinite dimensional banach space x contains for each integer n and each e 0 a subspace xn which is 1 e isomorphic to 12n in the classical banach spaces lp and 1p with 1 p x it is well known that this result can be improved these spaces contain uniformly complemented ln 7 s this means that we can find subspaces xn as above with the additional property that there exist projections p x xn such that supn 11 pn 11 cx moreover it is also well known that this property no longer holds for 11 or lo this motivated the question cf 13 to determine which infinite dimensional banach spaces contain uniformly complemented 1 n s we will prove below that all uniformly convex spaces have this property moreover we will show that a banach space x verifies this property locally see definition 2 10 for more precision if and only if x does not contain l n s uniformly progressively the work of previous authors cf 14 12 and particularly 5 has reduced such so called geometric questions to a purely analytic problem consisting of verifying in x a certain inequality referred to as k convexity in this paper we connect this inequality with the holomorphy of a certain semi group of operators on lp x and we show how this leads to the results already mentioned

sections of the unit ball of ipn let bnp xi rn i li n xi p 1 1 p and let ek be a k dimensional subspace of rn it is proved that if p 2 resp then vol bpn ek vol bpk resp we give some applications to linear forms

type and cotype in vector valued nakano sequence spaces abstract given a sequence of banach spaces xn n and a sequence of real numbers pn n in 1 the vector valued nakano sequence spaces l pn xn consist of elements xn n in nxn for which there is a constant 0 such that n xn pn

isomorphic properties of intersection bodies we study isomorphic properties of two generalizations of intersection bodies the class ikn of k intersection bodies in rn and the class bpkn of generalized k intersection bodies in rn in particular we show that all convex bodies can be in a certain sense approximated by intersection bodies namely if k is any symmetric convex body in rn and 1 k n 1 then the outer volume ratio distance from k to the class bpkn can be estimated by o v r k bpkn inf c k 1n c bpkn k c cnklogenk where c 0 is an absolute constant next we prove that if k is a symmetric convex body in rn 1 k n 1 and its k intersection body ik k exists and is convex then dbm ik k b2n c k where c k is a constant depending only on k dbm is the banach mazur distance and b2n is the unit euclidean ball in rn this generalizes a well known result of hensley and borell we conclude the paper with volumetric estimates for k intersection bodies

functional calculus under the tadmor ritt condition and free interpolation by polynomials of a given degree for banach space operators t satisfying the tadmor ritt condition zi t 1 c z 1 1 z 1 we prove that the best possible constant ct n bounding the polynomial calculus for t p t ct n p deg p n behaves in the worst case as logn as n this result is based on a new free carleson type interpolation theorem for polynomials of a given degree

reliability and efficiency of an anisotropic zienkiewicz zhu error estimator abstract in this paper we study the efficiency and the reliability of an anisotropic a posteriori error estimator in the case of the poisson problem supplied with mixed boundary conditions the error estimator may be classified as a residual based one but its novelty is twofold firstly it employs anisotropic estimates of the interpolation error for linear triangular finite elements and secondly it makes use of the zienkiewicz zhu recovery procedure to approximate the gradient of the exact solution finally we describe the adaptive procedure used to obtain a numerical solution satisfying a given accuracy and we include some numerical test cases to assess the robustness of the proposed numerical algorithm

on 0 1 polytopes with many facets there exist n dimensional 0 1 polytopes with as many as cn logn n 4 facets this is our main result it answers a question of komei fukuda and gunter m ziegler c 2001 academic press

the ricci curvature of diff s1 sl 2 r previous calculations of the ricci curvature for the manifold diff diff s1 s1 are extended to diff s1 sl 2r these manifolds are distinguished by being coadjoint orbits of diff s1 which admit compatible symphectic and complex structures making them kahler manifolds

hyperreflexivity and operator ideals abstract suppose b is an operator ideal and a is a linear space of operators between banach spaces x and y modifying the classical notion of hyperreflexivity we say that a is called b hyperreflexive if there exists a constant c such that for any t b x y with sup q t i the supremum runs over all isometric embeddings i into x and all quotient maps of y satisfying q a i 0 there exists a a for which t a c in this paper we give examples of b hyperreflexive spaces as well as of spaces failing this property in the last section we apply s e hyperreflexivity of operator algebras s e is a regular symmetrically normed operator ideal to constructing operator spaces with prescribed families of completely bounded maps

gaussian processes and mixed volumes on demontre la conjecture de dudley si l exposant du volume d un sous ensemble symetrique et convexe compact d un espace de hilbert est 1 alors k est un ensemble g c

poincare inequality in mean value for gaussian polytopes let k n g 1 g n be the absolute convex hull of n independent standard gaussian random points in mathbb r n with n n we prove that for any 1 lipschitz function f mathbb r n rightarrow mathbb r the polytope k n satisfies the following poincare inequality in mean value mathbb e omega int limits k n omega left f x frac 1 textup vol n left k n omega right int limits k n omega f y dy right 2 dx leq frac c n mathbb e omega int limits k n omega x 2dx where c 0 is an absolute constant this poincare inequality is the one suggested by a conjecture of kannan lovasz and simonovits for general convex bodies moreover we prove in mean value that the volume of the polytope k n is concentrated in a subexponential way within a thin euclidean shell with the optimal dependence of the dimension n an important tool of the proofs is a representation of the law of g 1 g n conditioned by the event that the convex hull of g 1 g n is a n 1 face of k n as an application we also get an estimate of the number of n 1 faces of the polytope k n valid for every n n

rademacher averages on noncommutative symmetric spaces let e be a separable or the dual of a separable symmetric function space let m be a semifinite von neumann algebra and let e m be the associated noncommutative function space let ek k 1 be a rademacher sequence on some probability space for finite sequences xk k 1 of e m we consider the rademacher averages kek xk as elements of the noncommutative function space e l m and study estimates for their norms kek xk e calculated in that space we establish general khintchine type inequalities in this context then we show that if e is 2 concave kek xk e is equivalent to the infimum of yk yk 12 zkzk 12 over all yk zk in e m such that xk yk zk for any k 1 dual estimates are given when e is 2 convex and has a nontrivial upper boyd index in this case kek xk e is equivalent to xk xk 12 xkxk 12 we also study rademacher averages i jei ej xij for doubly indexed families xij i j of e m

an a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes a new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes all considerations carry over to anisotropic triangular meshes with minor changes only the lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities in order to prove the upper error bound it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration to measure this correspondence a so called matching function is defined and its discussion shows it to be a useful tool with its help anisotropic interpolation estimates and subsequently the upper error bound are proven additionally it is pointed out how to treat robin boundary conditions in a posteriori error analysis on isotropic and anisotropic meshes a numerical example supports the anisotropic error analysis

projections of gibbs measures may be non gibbsian consider the phase of the two dimensional nearest neighbor ferromagnetic ising model at a temperature belowt c let be the restriction of this measure to a coordinate axis we prove that there is no one dimensional translation invariant summable interaction for which is a gibbs measure this is proven by showing that if such an interaction existed would have large deviation properties different from those it actually has percolation methods are used in the proof

banach spaces in various positions we formulate a general theory of positions for subspaces of a banach space we define equivalent and isomorphic positions study the automorphy index a y x that measures how many non equivalent positions y admits in x and obtain estimates of a y x for x a classical banach space such as lp lp l1 c or c 0 1 then we study different aspects of the automorphic space problem posed by lindenstrauss and rosenthal namely does there exist a separable automorphic space different from c0 or l2 recall that a banach space x is said to be automorphic if every subspace y admits only one position in x i e a y x 1 for every subspace y of x we study the notion of extensible space and uniformly finitely extensible space ufo which are relevant since every automorphic space is extensible and every extensible space is ufo we obtain a dichotomy theorem every ufo must be either an l space or a weak type 2 near hilbert space with the maurey projection property we show that a banach space all of whose subspaces are ufo called hereditarily ufo spaces must be asymptotically hilbertian while a banach space for which both x and x are ufo must be weak hilbert we then refine the dichotomy theorem for banach spaces with some additional structure in particular we show that an ufo with unconditional basis must be either c0 or a superreflexive weak type 2 space that a hereditarily ufo kothe function space must be hilbert and that a rearrangement invariant space ufo must be either l or a superreflexive type 2 banach lattice

affine quermassintegrals of random polytopes abstract a question related to some conjectures of lutwak about the affine quermassintegrals of a convex body k in r n asks whether for every convex body k in r n and all 1 k n k k vol n k 1 n g n k vol k p f k n d n k f 1 k n c n k where c 0 is an absolute constant we provide an affirmative answer for some broad classes of random polytopes we also discuss upper bounds for k k when k b 1 n the unit ball of l 1 n and explain how this special instance has implications for the case of a general unconditional convex body k

the ritt property of subordinated operators in the group case abstract let g be a locally compact abelian group let be a regular probability measure on g let x be a banach space let g b x be a bounded strongly continuous representation consider the average or subordinated operator s g t d t x x we show that if x is a umd banach lattice and has bounded angular ratio then s is a ritt operator with a bounded h functional calculus next we show that if is the square of a symmetric probability measure and x is k convex then s is a ritt operator we further show that this assertion is false on any non k convex space x

coadjoint orbits of the virasoro group the coadjoint orbits of the virasoro group which have been investigated by lazutkin and pankratova and by segal should according to the kirillov kostant theory be related to the unitary representations of the virasoro group in this paper the classification of orbits is reconsidered with an explicit description of the possible centralizers of coadjoint orbits the possible orbits are diff s1 itself diff s1 s1 and diff s1 sl n 2 r withsl n 2 r a certain discrete series of embeddings ofsl 2 r in diff s1 and diffs1 t wheret may be any of certain rather special one parameter subgroups of diffs1 an attempt is made to clarify the relation between orbits and representations it appears that quantization of diffs1 s1 is related to unitary representations with nondegenerate kac determinant unitary verma modules while quantization of diffs1 sl n 2 r is seemingly related to unitary representations with null vectors in leveln a better understanding of how to quantize the relevant orbits might lead to a better geometrical understanding of virasoro representation theory in the process of investigating virasoro coadjoint orbits we observe the existence of left invariant symplectic structures on the virasoro group manifold as is described in an appendix these give rise to lie bialgebra structures with the virasoro algebra as the underlying lie algebra

consistency of support vector machines and other regularized kernel classifiers it is shown that various classifiers that are based on minimization of a regularized risk are universally consistent i e they can asymptotically learn in every classification task the role of the loss functions used in these algorithms is considered in detail as an application of our general framework several types of support vector machines svms as well as regularization networks are treated our methods combine techniques from stochastics approximation theory and functional analysis

s numbers of integral operators with holder continuous kernels over metric compacta abstract we determine the asymptotic behaviour of kolmogorov and approximation numbers of integral operators which act in c x where x is a compact metric space and which are defined by means of holder continuous kernels the results show the connection of the decay of these s numbers with the entropy and in this way with the dimension of x

the norm of sums of independent noncommutative random variables in lp l1 we investigate the norm of sums of independent vector valued random variables in noncommutative lp spaces this allows us to obtain a uniform family of complete embeddings of the schatten class s n q in sp m q with optimal order m n 2 using these embeddings we show the surprising fact that the sharp type cotype index in the sense of operator spaces for lp 0 1 is min p p 0 max p p 0 similar techniques are used to show that the operator space notions of b convexity and k convexity are equivalent

twisted sums with c k spaces if x is a separable banach space we consider the existence of non trivial twisted sums 0 c k y x 0 where k 0 1 or for the case k 0 1 we show that there exists a twisted sum whose quotient map is strictly singular if and only if x contains no copy of 1 if k we prove an analogue of a theorem of johnson and zippin for k 0 1 by showing that all such twisted sums are trivial if x is the dual of a space with summable szlenk index e g x could be tsirelson s space a converse is established under the assumption that x has an unconditional finite dimensional decomposition we also give conditions for the existence of a twisted sum with c with strictly singular quotient map

approximation of zonoids by zonotopes on etudie une propriete d approximation des corps convexes de r n qui sont des limites de sommes de segments

improved bounds in the metric cotype inequality for banach spaces abstract it is shown that if x x is a banach space with rademacher cotype q then for every integer n there exists an even integer m n 1 1 q such that for every f z m n x we have 1 j 1 n e x f x m 2 e j f x x q m q e e x f x e f x x q where the expectations are with respect to uniformly chosen x z m n and e 1 0 1 n and all the implied constants may depend only on q and the rademacher cotype q constant of x this improves the bound of m n 2 1 q from mendel and naor 2008 13 the proof of 1 is based on a smoothing and approximation procedure which simplifies the proof of the metric characterization of rademacher cotype of mendel and naor 2008 13 we also show that any such smoothing and approximation approach to metric cotype inequalities must require m n 1 2 1 q

markov chains in smooth banach spaces and gromov hyperbolic metric spaces a metric space x has markov type 2 if for any reversible flnite state markov chain fztg with z0 chosen according to the stationary distribution and any map f from the state space to x the distance dt from f z0 to f zt satisfles e d 2 k 2 te d 2 for some k k x 2 has markov type 2 this proves a conjecture of ball we also show that trees hyperbolic groups and simply connected riemannian manifolds of pinched negative curvature have markov type 2 our results are applied to settle several conjectures on lipschitz extensions and embeddings in particular we answer a question posed by johnson and lindenstrauss in 1982 by showing that for 1 q 2 p 1 any lipschitz mapping from a subset of lp to lq has a lipschitz extension deflned on all of lp

a solution to the problem of l p maximal regularity we give a negative solution to the problem of the l p maximal regularity on various classes of banach spaces including l q spaces with 1 q neq 2 infty

interpolative construction and the generalized cotype of abstract lorentz spaces the banach operator ideals generated by an interpolative construction depending on concave functions are studied these ideals are described in terms of factorization through abstract interpolation lorentz spaces the abstract notion of rademacher type and cotype for operators between banach spaces is introduced it is shown that abstract interpolation lorentz spaces that appeared in the factorization theorem are of the generalized rademacher cotype determined by orlicz sequence spaces

on the weak amenability of a x and its relation with the approximation property abstract we investigate the weak amenability of the algebra a x of approximable operators on a banach space x and its relation with the bounded approximation property in particular it will be shown that the bounded approximation property is neither necessary nor sufficient for the weak amenability of a x

duality of metric entropy abstractfor two convex bodies kand t in r n the covering number of kbyt denoted n k t is de ned as the minimal number of translates oft needed to cover k let us denote by k the polar body of k andby d the euclidean unit ball in r n we prove that the two functionsof t n k td and n d tk are equivalent in the appropriate sense uniformly over symmetric convex bodies k r n and over n n inparticular this veri es the duality conjecture for entropy numbers of linearoperators posed by pietsch in 1972 in the central case when either thedomain or the range of the operator is a hilbert space 1 introduction for two convex bodies kand t in r n the covering number of kby t denoted n k t is de ned as the minimal number of translates of tneeded to cover kn k t min n x 1 x n r n k i n x i t we denote by dthe euclidean unit ball in r n in this paper we prove thefollowing duality result for covering numbers theorem 1 main theorem there exist two universal constants

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

quasi invariance of brownian measures on the group of circle homeomorphisms and infinite dimensional riemannian geometry abstract a cameron martin type theorem is proved for the canonical brownian motion on the group of homeomorphisms of the circle

analysis of recovery type a posteriori error estimators for mildly structured grids some recovery type error estimators for linear finite elements are analyzed under o h 1 0 regular grids superconvergence of order o h 1 0 p is established for recovered gradients by three different methods as a consequence a posteriori error estimators based on those recovery methods are asymptotically exact

weyl numbers in sequence spaces and sections of unit balls using techniques of a geometrical nature the following result is proved denoting by iqp m lpm lqm the natural embedding and by xk the kth weyl number if 0 p q 2 there are positive constants c1 and c2 such that for all k m n with k m2 c1k1q 1p xk iqp m c2k1q 1p the upper estimate being even valid for all k 1 m as a consequence of the approach used some results about sections of unit balls are also derived namely volh h bpm volk bpk for 0 p 2 where bpm bpk are the closed unit balls centred at zero of the spaces lpm and lpk respectively h is a k dimensional subspace of lpm and volk volh denote lebesgue measures in rk and h respectively

lower a posteriori error estimates on anisotropic meshes lower a posteriori error bounds obtained using the standard bubble function approach are reviewed in the context of anisotropic meshes a numerical example is given that clearly demonstrates that the short edge jump residual terms in such bounds are not sharp hence for linear finite element approximations of the laplace equation in polygonal domains a new approach is employed to obtain essentially sharper lower a posteriori error bounds and thus to show that the upper error estimator in the recent paper kopteva in numer math 137 607 642 2017 is efficient on partially structured anisotropic meshes

malliavin calculus and decoupling inequalities in banach spaces abstract we develop a theory of malliavin calculus for banach space valued random variables using radonifying operators instead of symmetric tensor products we extend the wiener ito isometry to banach spaces in the white noise case we obtain two sided l p estimates for multiple stochastic integrals in arbitrary banach spaces it is shown that the malliavin derivative is bounded on vector valued wiener ito chaoses our main tools are decoupling inequalities for vector valued random variables in the opposite direction we use meyer s inequalities to give a new proof of a decoupling result for gaussian chaoses in umd banach spaces

energy norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter to deal with anisotropic triangulations a special quasi interpolation operator is employed that may be of independent interest

edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes summary both for the h 1 and l 2 norms we prove that up to higher order perturbation terms edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes we also show that with a correct scaling edge residuals yield a robust error estimator for a singularly perturbed reaction diffusion equation

volume ratio and other s numbers of operators related to local properties of banach spaces abstract volume ratio numbers are studied and inequalities relating them to other s numbers of operators are established characterizations of spaces of weak cotype 2 in terms of s numbers and of operator ideal norms are obtained

type and cotype of musielak orlicz sequence spaces let n be a musielak orlicz function and l the musielak orlicz sequence space generated by it is shown that satisfies condition qiff is equivalent to aq concave function n it is also proved thatl is of cotypeq typep if and only if satisfies condition q 2and pconditions respectively as applications we obtain characterization of the type and cotype of various sequence spaces nakano weighted orlicz and orlicz sequence spaces furthermore a relation betweenq concavity p convexity lowerq estimate upperp estimate and cotype type of l is given

on the volume of non central sections of a cube abstract let q n be the cube of side length one centered at the origin in r n and let f be an affine n d dimensional subspace of r n having distance to the origin less than or equal to 1 2 where 0 d n we show that the n d dimensional volume of the section q n f is bounded below by a value c d depending only on the codimension d but not on the ambient dimension n or a particular subspace f in the case of hyperplanes d 1 we show that c 1 1 17 is a possible choice we also consider a complex analogue of this problem for a hyperplane section of the polydisc