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571.
572.
Cubature over the sphere in Sobolev spaces of arbitrary order   总被引:2,自引:1,他引:1  
This paper studies numerical integration (or cubature) over the unit sphere for functions in arbitrary Sobolev spaces Hs(S2), s>1. We discuss sequences of cubature rules, where (i) the rule Qm(n) uses m(n) points and is assumed to integrate exactly all (spherical) polynomials of degree ≤n and (ii) the sequence (Qm(n)) satisfies a certain local regularity property. This local regularity property is automatically satisfied if each Qm(n) has positive weights. It is shown that for functions in the unit ball of the Sobolev space Hs(S2), s>1, the worst-case cubature error has the order of convergence O(n-s), a result previously known only for the particular case . The crucial step in the extension to general s>1 is a novel representation of , where P is the Legendre polynomial of degree ℓ, in which the dominant term is a polynomial of degree n, which is therefore integrated exactly by the rule Qm(n). The order of convergence O(n-s) is optimal for sequences (Qm(n)) of cubature rules with properties (i) and (ii) if Qm(n) uses m(n)=O(n2) points.  相似文献   
573.
This paper is concerned with numerical integration on the unit sphere Sr of dimension r≥2 in the Euclidean space ℝr+1. We consider the worst-case cubature error, denoted by E(Qm;Hs(Sr)), of an arbitrary m-point cubature rule Qm for functions in the unit ball of the Sobolev space Hs(Sr), where s>, and show that The positive constant cs,r in the estimate depends only on the sphere dimension r≥2 and the index s of the Sobolev space Hs(Sr). This result was previously only known for r=2, in which case the estimate is order optimal. The method of proof is constructive: we construct for each Qm a `bad' function fm, that is, a function which vanishes in all nodes of the cubature rule and for which Our proof uses a packing of the sphere Sr with spherical caps, as well as an interpolation result between Sobolev spaces of different indices.  相似文献   
574.
This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S 2. A spherical n-design is a point set on S 2 that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree ≤n. The s-energy E s (X) of a point set of m distinct points is the sum of the potential for all pairs of distinct points . A sequence Ξ = {X m } of point sets X m S 2, where X m has the cardinality card(X m )=m, is well separated if for each pair of distinct points , where the constant λ is independent of m and X m . For all s>0, we derive upper bounds in terms of orders of n and m(n) of the s-energy E s (X m(n)) for well separated sequences Ξ = {X m(n)} of spherical n-designs X m(n) with card(X m(n))=m(n).   相似文献   
575.
Bovine and human serum albumins and recombinant human albumin, all non-covalently complexed with 5- and 16-doxyl stearic acids, were investigated by ESR spectroscopy in solution over a range of pH values (5.5–8.0) and temperatures (25–50 °C), with respect to the allocation and mobility of fatty acid (FA) molecules bound to the proteins and conformation of the binding sites. In all proteins bound FA undergo a permanent intra-albumin migration between the binding sites and inter-domain residence. Nature identity of the recombinant human albumin to its serum-derived analog was observed. However, the binding sites of bovine albumin appeared shorter in length and wider in diameter than those of human albumin. Presumably, less tightly folded domains in bovine albumin allow better penetration of water molecules in the interior of the globule that resulted in higher activation energy of FA dissociation from the binding site. Thus, the sensitive technique based on ESR non-covalent spin labeling allowed quantitative analysis and reliable comparison of the fine features of binding proteins.  相似文献   
576.
We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely and . Proofs and numerical counterexamples are given in situations where the zeros of Rn, and Sn, respectively, interlace (or do not in general) with the zeros of , , k=n or n−1. The results we prove hold for continuous, as well as integral, shifts of the parameter α.  相似文献   
577.
We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases.  相似文献   
578.
This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \mathbbS2\mathbb{S}^2, we discuss tensor product rules with n 2/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n 3) nodes for numerical integration over spherical caps on \mathbbS2\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \mathbbSd\mathbb{S}^d that have O(n d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \mathbbSd\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n d ) nodes and possess a certain regularity property.  相似文献   
579.
580.
Heterogeneous light-driven catalysis is a cornerstone of sustainable energy conversion. Most catalytic studies focus on bulk analyses of the hydrogen and oxygen evolved, which impede the correlation of matrix heterogeneities, molecular features, and bulk reactivity. Here, we report studies of a heterogenized catalyst/photosensitizer system using a polyoxometalate water oxidation catalyst and a model, molecular photosensitizer that were co-immobilized within a nanoporous block copolymer membrane. Via operando scanning electrochemical microscopy (SECM), light-induced oxygen evolution was determined using sodium peroxodisulfate (Na2S2O8) as sacrificial electron acceptor. Ex situ element analyses provided spatially resolved information on the local concentration and distribution of the molecular components. Infrared attenuated total reflection (IR-ATR) studies of the modified membranes showed no degradation of the water oxidation catalyst under the reported light-driven conditions.  相似文献   
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