Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

We prove lower bounds for the error of optimal cubature formulae for d‐variate functions from Besov spaces of mixed smoothness in the case , and , where is either the d‐dimensional torus or the d‐dimensional unit cube . In addition, we prove upper bounds for QMC integration on the Fibonacci‐lattice for bivariate periodic functions from in the case , and . A non‐periodic modification of this classical formula yields upper bounds for if . In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from and indicate that a corresponding result is most likely also true in case . This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids can never achieve the optimal worst‐case error.     

Oscillation criteria for higher order nonlinear dynamic equations

In this paper, we will consider the higher‐order functional dynamic equations of the form on an above‐unbounded time scale , where and , . The function is a rd‐continuous function such that . The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.     

On an elementary operator with M‐hyponormal operator entries

A Hilbert space operator is M‐hyponormal if there exists a positive real number M such that for all . Let be M‐hyponormal and let denote either the generalized derivation or the elementary operator . We prove that if are M‐hyponormal, then satisfies the generalized Weyl's theorem and satisfies the generalized a‐Weyl's theorem for every f that is analytic on a neighborhood of .     

Dimension of images of subspaces under mappings in Triebel‐Lizorkin spaces

Let and let be a ‐quasicontinuous representative of a mapping in the Triebel‐Lizorkin space . We find an optimal value of such that for a.e. the Hausdorff dimension of is at most α. We construct examples to show that the value of β is optimal and we show that it does not increase once p goes below the critical value α.     

Pre‐Hilbert spaces with anomalous splitting and orthogonally‐closed subspace structures

Four classes of closed subspaces of an inner product space S that can naturally replace the lattice of projections in a Hilbert space are: the complete/cocomplete subspaces , the splitting subspaces , the quasi‐splitting subspaces and the orthogonally‐closed subspaces . It is well‐known that in general the algebraic structure of these families differ remarkably and they coalesce if and only if S is a Hilbert space. It is also known that when S is a hyperplane in its completion i.e. then and . On the other extreme, when i.e. then and . Motivated by this and in contrast to it, we show that in general the codimension of S in bears very little relation to the properties of these families. In particular, we show that the equalities and can hold for inner product spaces with arbitrary codimension in . At the end we also contribute to the study of the algebraic structure of by testing it for the Riesz interpolation property. We show that may fail to enjoy the Riesz interpolation property in both extreme situations when S is “very small” (i.e. and when S is ‘very big’ (i.e. .     

Fractional integration operators of variable order: continuity and compactness properties

Let be a Lebesgue‐almost everywhere positive function. We consider the Riemann‐Liouville operator of variable order defined by as an operator from to . Our first aim is to study its continuity properties. For example, we show that is always bounded (continuous) in provided that . Surprisingly, this becomes false for . In order to be bounded in L₁[0, 1], the function has to satisfy some additional assumptions. In the second, central part of this paper we investigate compactness properties of . We characterize functions for which is a compact operator and for certain classes of functions we provide order‐optimal bounds for the dyadic entropy numbers .     

A strict topology on Orlicz spaces

Let φ be a Young function, Ω be a locally compact space, and μ be a positive Radon measure on Ω. We consider a strict topology (in the sense of Sentilles‐Taylor) on the Orlicz function space and investigate various properties of this locally convex topology. We also study the Orlicz space of a locally compact group G with a left Haar measure under the strict topology and certain other natural locally convex topologies. Finally we present some results on various continuity properties of convolution operators on under the topology and other natural ones.     

Regularity theory for a class of quasilinear equations

This paper addresses the analysis of the weak solution of in a bounded domain Ω subject to the boundary condition on , when the data f belongs to and . We prove existence and uniqueness of solution for this problem in the Nikolskii space . Moreover, we obtain energy estimates regarding the Nikolskii norm of ω in terms of the norm of f.     

On massive sets for subordinated random walks

We study massive (reccurent) sets with respect to a certain random walk defined on the integer lattice , . Our random walk is obtained from the simple random walk S on by the procedure of discrete subordination. can be regarded as a discrete space and time counterpart of the symmetric α‐stable Lévy process in . In the case we show that some remarkable proper subsets of , e.g. the set of primes, are massive whereas some proper subsets of such as the Leitmann primes are massive/non‐massive depending on the function h. Our results can be regarded as an extension of the results of McKean (1961) about massiveness of the set of primes for the simple random walk in . In the case we study massiveness of thorns and their proper subsets. The case is presented in the recent paper Bendikov and Cygan 2 .     

Volume estimates for sections of certain convex bodies

This paper deals with volume estimates for hyperplane sections of the simplex and for m‐codimensional sections of powers of m‐dimensional Euclidean balls. In the first part we consider sections through the centroid of the n‐dimensional regular simplex. We state a volume formula and give a lower bound for the volume of sections through the centroid. In the second part we study the extremal volumes of m‐codimensional sections “perpendicular” to of unit balls in the space for all . We give volume formulas and use them to show that the normal vector (1, 0, …, 0) yields the minimal volume. Furthermore we give an upper bound for the ‐dimensional volumes for natural numbers . This bound is asymptotically attained for the normal vector as .     

On the asymptotic behaviour of linear combinations of Mellin‐Picard type operators

Here we give a Voronovskaja formula for linear combination of Mellin‐Picard type convolution operators where is the Mellin‐Picard kernel. This approach provides a better order of pointwise approximation.     

On the frame bundle adapted to a submanifold

Let M be a submanifold of a Riemannian manifold . M induces a subbundle of adapted frames over M of the bundle of orthonormal frames . The Riemannian metric g induces a natural metric on . We study the geometry of a submanifold in . We characterize the horizontal distribution of and state its correspondence with the horizontal lift in induced by the Levi–Civita connection on N. In the case of extrinsic geometry, we show that minimality is equivalent to harmonicity of the Gauss map of the submanifold M with a deformed Riemannian metric. In the case of intrinsic geometry we compute the curvatures and compare this geometry with the geometry of M.     

Convolution powers of complex functions on

Repeated convolution of a probability measure on leads to the central limit theorem and other limit theorems. This paper investigates what kinds of results remain without positivity. It reviews theorems due to Schoenberg, Greville, and Thomée which are motivated by applications to data smoothing (Schoenberg and Greville) and finite difference schemes (Thomée). Using Fourier transform arguments, we prove detailed decay bounds for convolution powers of finitely supported complex functions on . If M is an hermitian contraction, an estimate for the off‐diagonal entries of the powers of is obtained. This generalizes the Carne–Varopoulos Markov chain estimate.     

Determinants of classical SG‐pseudodifferential operators

We introduce a generalized trace functional TR in the spirit of Kontsevich and Vishik's canonical trace for classical SG‐pseudodifferential operators on and suitable manifolds, using a finite‐part integral regularization technique. This allows us to define a zeta‐regularized determinant for parameter‐elliptic operators , , . For , the asymptotics of as and of as are derived. For suitable pairs we show that coincides with the so‐called relative determinant .     

Hyponormality of bounded‐type Toeplitz operators

In this paper we deal with the hyponormality of Toeplitz operators with matrix‐valued symbols. The aim of this paper is to provide a tractable criterion for the hyponormality of bounded‐type Toeplitz operators (i.e., the symbol is a matrix‐valued function such that Φ and are of bounded type). In particular, we get a much simpler criterion for the hyponormality of when the co‐analytic part of the symbol Φ is a left divisor of the analytic part.     

Push forward measures and concentration phenomena

In this note we study how a concentration phenomenon can be transferred from one measure μ to a push‐forward measure ν. In the first part, we push forward μ by , where , and obtain a concentration inequality in terms of the medians of the given norms (with respect to μ) and the Banach‐Mazur distance between them. This approach is finer than simply bounding the concentration of the push forward measure in terms of the Banach‐Mazur distance between K and L. As a consequence we show that any normed probability space with exponential type concentration is far (even in an average sense) from subspaces of . The sharpness of this result is shown by considering the spaces.     

Uniqueness results for semilinear elliptic systems on

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form As an application we consider the nonlinear Schrödinger system for and exponents q which satisfy in case and in case . Generalizing the results of Wei and Yao for we find new sufficient conditions and necessary conditions on such that precisely one positive solution exists. Our results dealing with the special case are optimal. Finally, an application to a multi‐component nonlinear Schrödinger system is given.     

Random attractors for stochastic reaction‐diffusion equations with multiplicative noise in

In this paper, we study the random dynamical system generated by a stochastic reaction‐diffusion equation with multiplicative noise and prove the existence of an ‐random attractor for such a random dynamical system. The nonlinearity f is supposed to satisfy some growth of arbitrary order .     

Kolmogorov equations for randomly perturbed generalized Newtonian fluids

We consider incompressible generalized Newtonian fluids in two space dimensions perturbed by an additive Gaussian noise. The velocity field of such a fluid obeys a stochastic partial differential equation with fully nonlinear drift due to the dependence of viscosity on the shear rate. In particular, we assume that the extra stress tensor is of power law type, i.e. a polynomial of degree , , i.e. the shear thinning case. We prove that the associated Kolmogorov operator K admits at least one infinitesimally invariant measure μ satisfying certain exponential moment estimates. Moreover, K is L²‐unique w.r.t. μ provided , where is the second root of , approximately .