The dimension of bivariate spline spaces of smoothnessr for degreed≥4r+1

We consider spaces of piecewise polynomials of degreed defined over a triangulation of a polygonal domain and possessingr continuous derivatives globally. Morgan and Scott constructed a basis in the case wherer=1 andd≥5. The purpose of this paper is to extend the dimension part of their result tor≥0 andd≥4r+l. We use Bézier nets as a crucial tool in deriving the dimension of such spaces.     

Optimal cubature formulas for calculation of multidimensional integrals in weighted Sobolev spaces

Optimal cubature formulas are constructed for calculations of multidimensional integrals in weighted Sobolev spaces. We consider some classes of functions defined in the cube Ω = [-1, 1]^l, l = 1, 2,..., and having bounded partial derivatives up to the order r in Ω and the derivatives of jth order (r < j ≤ s) whose modulus tends to infinity as power functions of the form (d(x, Г))^-(j-r), where x ∈ Ω Г, x = (x₁,..., x_l), Г = ?Ω, and d(x, Г) is the distance from x to Г.     

Bracketing Metric Entropy Rates and Empirical Central Limit Theorems for Function Classes of Besov- and Sobolev-Type

We derive ? ^r (μ)-bracketing metric and sup-norm metric entropy rates of bounded subsets of general function spaces defined over ? ^d or, more generally, over Borel subsets thereof, by adapting results of Haroske and Triebel (Math. Nachr. 167, 131–156, 1994; 278, 108–132, 2005). The function spaces covered are of (weighted) Besov, Sobolev, Hölder, and Triebel type. Applications to the theory of empirical processes are discussed. In particular, we show that (norm-)bounded subsets of the above mentioned spaces are Donsker classes uniformly in various sets of probability measures.     

Θ-type Calderón-Zygmund operators with non-doubling measures

Let µ be a Radon measure on ? ^d which may be non-doubling. The only condition that µ must satisfy is µ(B(x, r)) ≤ Cr ⁿ for all x∈? ^d , r > 0 and for some fixed 0 < n ≤ d. In this paper, under this assumption, we prove that θ-type Calderón-Zygmund operator which is bounded on L ²(µ) is also bounded from L ^∞(µ) into RBMO(µ) and from H _atb ^1,∞ (µ) into L ¹(µ). According to the interpolation theorem introduced by Tolsa, the L ^p (µ)-boundedness (1 < p < ∞) is established for θ-type Calderón-Zygmund operators. Via a sharp maximal operator, it is shown that commutators and multilinear commutators of θ-type Calderón-Zygmund operator with RBMO(µ) function are bounded on L ^p (µ) (1 < p < ∞).     

On some new sharp embedding theorems in minimal and pseudoconvex domains

We present new sharp embedding theorems for mixed-norm analytic spaces in pseudoconvex domains with smooth boundary. New related sharp results in minimal bounded homogeneous domains in higher dimension are also provided. Last domains we consider are domains which are direct generalizations of the well-studied so-called bounded symmetric domains in Cⁿ. Our results were known before only in the very particular case of domains of such type in the unit ball. As in the unit ball case, all our proofs are heavily based on nice properties of the r-lattice. Some results of this paper can be also obtained in some unbounded domains, namely tubular domains over symmetric cones.     

The packing problem for projective geometries over GF(3) with dimension greater than five

An (n, d) set in the projective geometry PG(r, q) is a set of n points, no d of which are dependent. The packing problem is that of finding n(r, q, d), the largest size of an (n, d) set in PG(r, q). The packing problem for PG(r, 3) is considered. All of the values of n(r, 3, d) for r ? 5 are known. New results for r = 6 are n(6, 3, 5) = 14 and 20 ? n(6, 3, 4) ? 31. In general, upper bounds on n(r, q, d) are determined using a slightly improved sphere-packing bound, the linear programming approach of coding theory, and an orthogonal (n, d) set with the known extremal values of n(r, q, d)—values when r and d are close to each other. The BCH constructions and computer searches are used to give lower bounds. The current situation for the packing problem for PG(r, 3) with r ? 15 is summarized in a final table.     

L 1 Spline Methods for Scattered Data Interpolation and Approximation

We generalize the L ₁ spline methods proposed in [4, 5] for scattered data interpolation and fitting using bivariate spline spaces of any degree d and any smoothness r (of course, r<d) over any triangulation. Some numerical experiments are presented to illustrate the better performance of the L ₁ spline methods as compared to the minimal energy method. We include some extensions for dealing with other surface design problems.     

On associated and co-recursive d-orthogonal polynomials

The associated sequence of order r for a given d-OPS (i.e. a sequence of orthogonal polynomials satisfying a (d + 1)-order recurrence relation), is again a d-OPS. In this paper we are interested in the determination of the corresponding dual sequence. The explicit form of the dual sequence of the first associated sequence and the corresponding formal Stieltjes function are given. Indeed, we construct by recurrence the dual sequence of the r-associated sequence and we give some properties of the corresponding Stieltjes function. Second, we give the definition of co-recursive polynomials of dimension d and some relations in the particular cases d = 3 and d = 4. Some properties of the dual sequence as well as of the corresponding Stieltjes functions are given.     

TT-cross approximation for multidimensional arrays

As is well known, a rank-r matrix can be recovered from a cross of r linearly independent columns and rows, and an arbitrary matrix can be interpolated on the cross entries. Other entries by this cross or pseudo-skeleton approximation are given with errors depending on the closeness of the matrix to a rank-r matrix and as well on the choice of cross. In this paper we extend this construction to d-dimensional arrays (tensors) and suggest a new interpolation formula in which a d-dimensional array is interpolated on the entries of some TT-cross (tensor train-cross). The total number of entries and the complexity of our interpolation algorithm depend on d linearly, so the approach does not suffer from the curse of dimensionality.We also propose a TT-cross method for computation of d-dimensional integrals and apply it to some examples with dimensionality in the range from d=100 up to d=4000 and the relative accuracy of order 10^-10. In all constructions we capitalize on the new tensor decomposition in the form of tensor trains (TT-decomposition).     

Group actions and Helly's theorem

We describe a connection between the combinatorics of generators for certain groups and the combinatorics of Helly's 1913 theorem on convex sets. We use this connection to prove fixed point theorems for actions of these groups on nonpositively curved metric spaces. These results are encoded in a property that we introduce called “property FAr”, which reduces to Serre's property FA when r=1. The method applies to S-arithmetic groups in higher Q-rank, to simplex reflection groups (including some nonarithmetic ones), and to higher rank Chevalley groups over polynomial and other rings (for example SLn(Z[x₁,…,xd]), n>2).