spline wavelets on triangulations

In this paper we investigate spline wavelets on general triangulations. In particular, we are interested in wavelets generated from piecewise quadratic polynomials. By using the Powell-Sabin elements, we set up a nested family of spaces of quadratic splines, which are suitable for multiresolution analysis of Besov spaces. Consequently, we construct wavelet bases on general triangulations and give explicit expressions for the wavelets on the three-direction mesh. A general theory is developed so as to verify the global stability of these wavelets in Besov spaces. The wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations.

     

A counterexample concerning the -projector onto linear spline spaces

For the -orthogonal projection onto spaces of linear splines over simplicial partitions in polyhedral domains in , , we show that in contrast to the one-dimensional case, where independently of the nature of the partition, in higher dimensions the -norm of cannot be bounded uniformly with respect to the partition. This fact is folklore among specialists in finite element methods and approximation theory but seemingly has never been formally proved.

     

Spaces between and

In this paper we consider the spaces that lie between and . We discuss their interpolation properties and the behavior of maximal functions and singular integrals acting on them.

     

An interpolation error estimate in based on the anisotropic measures of higher order derivatives

In this paper, we introduce the magnitude, orientation, and anisotropic ratio for the higher order derivative (with ) of a function to characterize its anisotropic behavior. The magnitude is equivalent to its usual Euclidean norm. The orientation is the direction along which the absolute value of the -th directional derivative is about the smallest, while along its perpendicular direction it is about the largest. The anisotropic ratio measures the strength of the anisotropic behavior of . These quantities are invariant under translation and rotation of the independent variables. They correspond to the area, orientation, and aspect ratio for triangular elements. Based on these measures, we derive an anisotropic error estimate for the piecewise polynomial interpolation over a family of triangulations that are quasi-uniform under a given Riemannian metric. Among the meshes of a fixed number of elements it is identified that the interpolation error is nearly the minimum on the one in which all the elements are aligned with the orientation of , their aspect ratios are about the anisotropic ratio of , and their areas make the error evenly distributed over every element.

     

On a fragment of the universal Baire property for sets

There is a well-known global equivalence between sets having the universal Baire property, two-step generic absoluteness, and the closure of the universe under the sharp operation. In this note, we determine the exact consistency strength of sets being -cc-universally Baire, which is below . In a model obtained, there is a set which is weakly -universally Baire but not -universally Baire.

     

A new construction of -manifolds

This paper provides a topological method to construct all simply-connected, spin, smooth -manifolds with torsion-free homology using simply-connected, smooth -manifolds as building blocks. We explicitly determine the invariants that classify these -manifolds from the intersection form and specific homology classes of the -manifold building blocks.

     

Principal groupoid -algebras with bounded trace

Suppose is a second countable, locally compact, Hausdorff, principal groupoid with a fixed left Haar system. We define a notion of integrability for groupoids and show is integrable if and only if the groupoid -algebra has bounded trace.

     

Unconditional stability and convergence of fully discrete schemes for viscous fluids models with mass diffusion

In this work we develop fully discrete (in time and space) numerical schemes for two-dimensional incompressible fluids with mass diffusion, also so-called Kazhikhov-Smagulov models. We propose at most -conformed finite elements (only globally continuous functions) to approximate all unknowns (velocity, pressure and density), although the limit density (solution of continuous problem) will have regularity. A backward Euler in time scheme is considered decoupling the computation of the density from the velocity and pressure.

Unconditional stability of the schemes and convergence towards the (unique) global in time weak solution of the models is proved. Since a discrete maximum principle cannot be ensured, we must use a different interpolation inequality to obtain the strong estimates for the discrete density, from the used one in the continuous case. This inequality is a discrete version of the Gagliardo-Nirenberg interpolation inequality in domains. Moreover, the discrete density is truncated in some adequate terms of the velocity-pressure problem.

     

A note on -estimates for stable integrals with drift

Let be of the form where is a symmetric stable process of index with . We obtain various -estimates for the process . In particular, for and any measurable, nonnegative function we derive the inequality

As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation for any initial value .

     

Quantum symmetric derivatives

For , a one-parameter family of symmetric quantum derivatives is defined for each order of differentiation as are two families of Riemann symmetric quantum derivatives. For , symmetrization holds, that is, whenever the th Peano derivative exists at a point, all of these derivatives of order also exist at that point. The main result, desymmetrization, is that conversely, for , each symmetric quantum derivative is a.e. equivalent to the Peano derivative of the same order. For and , each th symmetric quantum derivative coincides with both corresponding th Riemann symmetric quantum derivatives, so, in particular, for and , both th Riemann symmetric quantum derivatives are a.e. equivalent to the Peano derivative.

     

-actions with -dimensional fixed point set

We describe the equivariant cobordism classification of smooth actions of the group , considered as the group generated by two commuting involutions, on closed smooth -dimensional manifolds , for which the fixed point set of the action is a connected manifold of dimension and or . For , the classification is known.

     

Polyharmonic splines on grids and their limits

Radial Basis Functions (RBF) have found a wide area of applications. We consider the case of polyharmonic RBF (called sometimes polyharmonic splines) where the data are on special grids of the form having practical importance. The main purpose of the paper is to consider the behavior of the polyharmonic interpolation splines on such grids for the limiting process 0.$"> For a large class of data functions defined on it turns out that there exists a limit function This limit function is shown to be a polyspline of order on strips. By the theory of polysplines we know that the function is smooth up to order everywhere (in particular, they are smooth on the hyperplanes , which includes existence of the normal derivatives up to order while the RBF interpolants are smooth only up to the order The last fact has important consequences for the data smoothing practice.

     

Integral homology -spheres and the Johnson filtration

The mapping class group of an oriented surface of genus with one boundary component has a natural decreasing filtration , where is the kernel of the action of on the nilpotent quotient of . Using a tree Lie algebra approximating the graded Lie algebra we prove that any integral homology sphere of dimension has for some a Heegaard decomposition of the form , where and is such that . This proves a conjecture due to S. Morita and shows that the ``core' of the Casson invariant is indeed the Casson invariant.

     

Global existence for energy critical waves in -d domains

We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on for any smooth (compact) domain . The main ingredient in the proof is an spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.

     

Hyperbolic sets exhibiting -persistent homoclinic tangency for higher dimensions

For any manifold of dimension at least three, we give a simple construction of a hyperbolic invariant set that exhibits -persistent homoclinic tangency. It provides an open subset of the space of -diffeomorphisms in which generic diffeomorphisms have arbitrary given growth of the number of attracting periodic orbits and admit no symbolic extensions.

     

Interpolation between and

We show that if is a rearrangement invariant space on that is an interpolation space between and and for which we have only a one-sided estimate of the Boyd index 1/p, 1 < p < \infty$">, then is an interpolation space between and . This gives a positive answer for a question posed by Semenov. We also present the one-sided interpolation theorem about operators of strong type and weak type .

     

The -primary classifying space of the fiber of the double suspension

Gray showed that the homotopy fiber of the double suspension has an integral classifying space , which fits in a homotopy fibration . In addition, after localizing at an odd prime , is an -space and if , then is homotopy associative and homotopy commutative, and is an -map. We positively resolve a conjecture of Gray's that the same multiplicative properties hold for as well. We go on to give some exponent consequences.

     

On the essential commutant of QC

Let (QC) (resp. ) be the -algebra generated by the Toeplitz operators QC (resp. ) on the Hardy space of the unit circle. A well-known theorem of Davidson asserts that (QC) is the essential commutant of . We show that the essential commutant of (QC) is strictly larger than . Thus the image of in the Calkin algebra does not satisfy the double commutant relation. We also give a criterion for membership in the essential commutant of (QC).

     

-regularity, -fibrant Hochschild homology, and a conjecture of Vorst

In this paper we prove that for an affine scheme essentially of finite type over a field and of dimension , -regularity implies regularity, assuming that the characteristic of is zero. This verifies a conjecture of Vorst.

     

Optimal anisotropic meshes for minimizing interpolation errors in -norm

In this paper, we present a new optimal interpolation error estimate in norm ( ) for finite element simplicial meshes in any spatial dimension. A sufficient condition for a mesh to be nearly optimal is that it is quasi-uniform under a new metric defined by a modified Hessian matrix of the function to be interpolated. We also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the norm. Some numerical examples are also given to support the theoretical estimates.