Bivariate spline interpolation with optimal approximation order

Let Δ be a triangulation of some polygonal domain Ω ⊂ R² and let S_q^r(Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S _q ^r (Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh ^q +1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of S _q ^r (Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].     

An additive divisor problem with a growing number of factors

Let τk(n) be the number of representations ofn as the product ofk positive factors, τ(n)=τ(n). The asymptotics of Σ _n≤x τ _k (n)τ(n+1) for 80k ¹⁰ (lnlnx)³≤lnx is shown to be uniform with respect tok. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 391–406, March, 1997. Translated by N. K. Kulman     

On banach spaces which containl 1(τ) and types of measures on compact spaces

Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic properties of compact spaces, the other being a non-separable analogue of a result of Peŀczyński's about Banach spaces containingL ¹. Let τ be a regular cardinal satisfying the hypothesis that κ^ω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto [0, 1]^τ. 2) A Banach spaceX has a subspace isomorphic tol ¹(τ) if and only ifX ^∗ has a subspace isomorphic toL ¹({0, 1}^τ). An example is given to show that a more recent result of Rosenthal's about Banach spaces containingl ¹ does not have an obvious transfinite analogue. A second example (answering a question of Rosenthal's) shows that there is a Banach spaceX which contains no copy ofl ¹ (ω₁), while the unit ball ofX ^∗ is not weakly^∗ sequentially compact.     

On Two Questions About Topological (and Nonstandard) Extensions

We construct, under MA, a non-Hausdorff (T₁-)topological extension ^*ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ^*ω to ^*ω. We also show (in ZFC) that for every nontrivial topological extension ^*X of a countable set X there exists a topology τ_f on ^*X, strictly finer than the Star topology, and such that (^*X, τ_f) is still a topological extension of X with the same function extensions ^*f. This solves two questions raised by M. Di Nasso and M. Forti.     

Note on hit-and-miss topologies

This is a continuation of [19]. We characterize first and second countability of the general hit-and-miss hyperspace topologyτ ₊ ^Δ for weakly-R ₀ base spaces. Further, metrizability ofτ ₊ ^Δ is characterized with no preliminary conditions on the base space and the generating family of closed sets and a new proof on uniformizability (i.e. complete regularity) ofτ ₊ ^Δ is given in this general setting, thus generalizing results of [3], [5] and [6].     

Triangular finite elements of HCT type and classC ρ

Let τ be some triangulation of a planar polygonal domain Ω. Given a smooth functionu, we construct piecewise polynomial functionsv∈C ^ρ(Ω) of degreen=3 ρ for ρ odd, andn=3ρ+1 for ρ even on a subtriangulation τ₃ of τ. The latter is obtained by subdividing eachT∈ρ into three triangles, andv/T is a composite triangular finite element, generalizing the classicalC ¹ cubic Hsieh-Clough-Tocher (HCT) triangular scheme. The functionv interpolates the derivatives ofu up to order ρ at the vertices of τ. Polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements of this type.     

The size-Ramsey number of trees

IfG andH are graphs, let us writeG→(H)₂ ifG contains a monochromatic copy ofH in any 2-colouring of the edges ofG. Thesize-Ramsey number r _e(H) of a graphH is the smallest possible number of edges a graphG may have ifG→(H)₂. SupposeT is a tree of order |T|≥2, and lett ₀,t ₁ be the cardinalities of the vertex classes ofT as a bipartite graph, and let Δ(T) be the maximal degree ofT. Moreover, let Δ₀, Δ₁ be the maxima of the degrees of the vertices in the respective vertex classes, and letβ(T)=T ₀Δ₀+t ₁Δ₁. Beck [7] proved thatβ(T)/4≤r _e(T)=O{β(T)(log|T|)¹²}, improving on a previous result of his [6] stating thatr _e(T)≤Δ(T)|T|(log|T|)¹². In [6], Beck conjectures thatr _e(T)=O{Δ(T)|T|}, and in [7] he puts forward the stronger conjecture thatr _e(T)=O{β(T)}. Here, we prove the first of these conjectures, and come quite close to proving the second by showing thatr _e(T)=O{β(T)logΔ(T)}.     

Sur les espaces de Banach contenantl 1(τ)

Letτ be a cardinal with cf(τ)>ℵ₀. Then a Banach spaceE contains a subspace isomorphic tol ^l(τ) if and only if [0,1] ^r is a continuous image of the unit ballE′₁ ofE′, provided with the w*-topology. It follows that, for each cardinalκ, ifE′₁ contains a copy ofβκ, thenE has a quotient isomorphic tol ^∞(κ). In this situation we show thatE has even a quotientisometric tol ^∞(κ).      

On voronoi diagrams in theL p-metric in ℝD

We prove upper bounds on the number ofL _p-spheres passing throughD+1 points in general position in ℝ”, and on the sum of the Betti numbers of the intersection of bisectors in theL _p-metric, wherep is an even positive integer. The bounds found do not depend onp. Our result implies that the complexity of Voronoi diagrams (for point sites in general position) in theL _p-metric is bounded for increasingp. The proof for this upper bound involves the techniques of Milnor [12] and Thom [16] for finding a bound on the sum of the Betti numbers of algebraic varieties, but instead of the usual degree of polynomials we use their additive complexity, and apply results of Benedetti and Risler [2], [13]. Furthermore, we prove that inD dimensions and for evenp the number ofL _p-spheres passing throughD+1 points in general position is odd. In particular, combined with results of [8], [9], our results clarify the structure of Voronoi diagrams based on theL _p-metric (with evenp) in three dimensions. For the proof we use the theory of degree of continuous mappings in ℝ^D, which is a tool widely applied in nonlinear analysis [14]. This work was partially supported by Deutsche Forschungsgemeinschaft, Grant K1 655/2-1. A preliminary version of this paper was presented at the 11th Annual Symposium on Theoretical Aspects of Computer Science, France, 1994.     

Solution of the Transmission Problem

The solution of the following transmission problem for the Laplace equation is constructed: Δu ₊=0 in G ⁺, Δu ₋=0 in G ⁻, u ₊−u ₋=f in ∂ G ⁺, n⋅(∇ u ₊−a ∇ u ₋)+b τ⋅(∇ u ₊−∇ u ₋)+h ₊ u ₊+h ₋ u ₋=g in ∂ G ⁺.     

The spectral projections and the resolvent for scattering metrics

In this paper, we consider a compact manifold with boundaryX equipped with a scattering metricg as defined by Melrose [9]. That is,g is a Riemannian metric in the interior ofX that can be brought to the formg=x ⁻⁴ dx²+x^{−2 h’} near the boundary, wherex is a boundary defining function andh’ is a smooth symmetric 2-cotensor which restricts to a metrich on ϖX. LetH=Δ+V, whereV∈x ²C^∞ (X) is real, soV is a ‘short-range’ perturbation of Δ. Melrose and Zworski started a detailed analysis of various operators associated toH in [11] and showed that the scattering matrix ofH is a Fourier integral operator associated to the geodesic flow ofh on ϖX at distance π and that the kernel of the Poisson operator is a Legendre distribution onX×ϖX associated to an intersecting pair with conic points. In this paper, we describe the kernel of the spectral projections and the resolvent,R(σ±i0), on the positive real axis. We define a class of Legendre distributions on certain types of manifolds with corners and show that the kernel of the spectral projection is a Legendre distribution associated to a conic pair on the b-stretched productX _b ² (the blowup ofX ² about the corner, (ϖX)²). The structure of the resolvent is only slightly more complicated. As applications of our results, we show that there are ‘distorted Fourier transforms’ forH, i.e., unitary operators which intertwineH with a multiplication operator and determine the scattering matrix; we also give a scattering wavefront set estimate for the resolventR(σ±i0) applied to a distributionf.     

Homeomorphisms of universal dendrites

Homeomorphisms of some special universal dendrites are considered in the paper. In particular, dendrites are characterized for which the action onX of the group of autohomeomorphisms ofX has exactlyn≥3 orbits, and for each orbitB ofX and for each areA ⊂X the intersectionA∪B is a dense subset ofA. Some of the results generalize earlier ones from the author’s paper [2].     

On the topological types of symmetries of elliptic-hyperelliptic Riemann surfaces

LetX be a Riemann surface of genusg. The surfaceX is called elliptic-hyperelliptic if it admits a conformal involutionh such that the orbit spaceX/〈h〉 has genus one. The involutionh is then called an elliptic-hyperelliptic involution. Ifg>5 then the involutionh is unique, see [A]. We call symmetry to any anticonformal involution ofX. LetAut ^±(X) be the group of conformal and anticonformal automorphisms ofX and letσ, τ be two symmetries ofX with fixed points and such that {σ, hσ} and {τ, hτ} are not conjugate inAut ^±(X). We describe all the possible topological conjugacy classes of {σ, σh, τ, τh}. As consequence of our study we obtain that, in the moduli space of complex algebraic curves of genusg (g even >5), the subspace whose elements are the elliptic-hyperelliptic real algebraic curves is not connected. This fact contrasts with the result in [Se]: the subspace whose elements are the hyperelliptic real algebraic curves is connected. The authors are supported by BFM2002-04801.     

A unified theory of<Emphasis Type="Italic">θ</Emphasis>-continuity for functions

We introduce a new notion ofθ-M-continuous functions as functions from a set satisfying some minimal conditions into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. The functions enable us to formulate a unified theory ofθ-continuity [15],θ-semi-continuity [4], quasi-irresoluteness [9],θ-preirresoluteness [29], and weakβ-irresoluteness [30].     

On weakly (τ, m)-continuous functions

In this paper we introduce a new notion of weakly (τ, m)-continuous functions as functions from a topological space into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. This function leads to the formulation of a unified theory of weak continuity [20], almosts-continuity [33],p(θ)-continuity [10] andp-continuity [41].     

An asymptotic property of gap series. II

An upper bound estimate in the law of the iterated logarithm for Σf(n _k ω) where n_k+1∫n_k≧ 1 + ck ^-α (α≧0) is investigated. In the case α<1/2, an upper bound had been given by Takahashi [15], and the sharpness of the bound was proved in our previous paper [8]. In this paper it is proved that the upper bound is still valid in case α≧1/2 if some additional condition on {n _k} is assumed. As an application, the law of the iterated logarithm is proved when {n _k} is the arrangement in increasing order of the set B(τ)={₁ ⁱ ₁...q_τ ⁱ _τ|i₁,...,i_τ∈N ₀}, where τ≧ 2, N ₀=NU{0}, and q ₁,...,q _τ are integers greater than 1 and relatively prime to each others. This revised version was published online in June 2006 with corrections to the Cover Date.     

Solution of the twisting problem for skew group algebras

LetK be any field of characteristicp>0 and letG be a finite group acting onK via a map τ. The skew group algebraK _τG may be nonsemisimple (precisely whenP|(H), H=Kert). In [1] necessary conditions were given for the existence of a class α∈H ²(G,K^*) which “twists” the skew group algebraK _τG into a semisimple crossed productK _τ ^αG . The “twisting problem” asks whether these conditions are sufficient. In [1] we showed that this is indeed so in many cases. In this paper we prove it in general. During the period of this research the second author was an Associate at the Center for Advanced Study, Urbana, Illinois.     

Pointwise estimates of polynomials orthogonal on a circle with respect to a weight not belonging to the spaces Lr (r > 1)

Two-sided pointwise estimates are established for polynomials that are orthogonal on the circle |z| = 1 with respect to the weight ϕ(τ): = h(τ)|sin(τ/2)|⁻¹ g(|sin(τ/2)|) (τ ∈ ℝ), where g(t) is a concave modulus of continuity slowly changing at zero such that t ⁻¹ g(t) ∈ L ¹[0, 1] and h(τ) is a positive function from the class C _2π with a modulus of continuity satisfying the integral Dini condition. The obtained estimates are applied to find the order of the distance from the point t = 1 to the greatest zero of a polynomial orthogonal on the segment [−1, 1].     

Sum of sequence spaces and matrix transformations mapping in s α 0 ((Δ ? λ I ) h ) + s β ( c ) ((Δ ? μ I ) l )

We deal with the sum of sequence spaces. Then we apply these results to characterize matrix transformations mapping between s ^h,l (λ, μ) = s _α ⁰((Δ − λI) ^h ) + s _β ^(c)((Δ − μI) ^l ) and s _γ . Among other things the aim of this paper is to reduce the set (s ^h,l (λ, μ), s _γ to a set of the form S _τ,γ .      

Gruppo delle terne quasi pitagoriche primitive

We study the groupG _m of primitive solution of the diophantine equationx ²+my²=z² (m>1, squarefree). Form∈3 this group is torsion free, form=3 it has a torsion element of order 3; moreover for a finite number of values ofm we prove thatG _m is a direct sum of infinite cyclic groups and we give the generators ofG _m in terms of the primes represented by the quadratic forms of discriminant Δ=−4m.