A geometric formula for Haefliger knots

Haefliger has shown that a smooth embedding of the (4k−1)-sphere in the 6k-sphere can be knotted in the smooth sense. In this paper, we give a formula with which we can detect the isotopy class of such a Haefliger knot. The formula is expressed in terms of the geometric characteristics of an extension, analogous to a Seifert surface, of the given embedding. In particular, the Hopf invariant associated to the extension plays a crucial role. This leads us to a new characterisation of Haefliger knots.     

由Schr?dinger算子定义的Riesz变换的Lp估计

本文主要讨论了当非负位势 V(x)属于某逆Holder类时,由一致椭圆算子L=-div(A(x))+V(x)所定义的 Riesz变换在 L^p空间的有界性。     

A note on strongly π-regular rings

Let R be an associative ring with unit and let N(R) denote the set of nilpotent elements of R. R is said to be stronglyπ-regular if for each x∈R, there exist a positive integer n and an element y∈R such that x ⁿ=x ^{n +1} y and xy=yx. R is said to be periodic if for each x∈R there are integers m,n≥ 1 such that m≠n and x ^m=x ⁿ. Assume that the idempotents in R are central. It is shown in this paper that R is a strongly π-regular ring if and only if N(R) coincides with the Jacobson radical of R and R/N(R) is regular. Some similar conditions for periodic rings are also obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

INJECTIVE PRECOVERS AND MODULES OF GENERALIZED INVERSE POLYNOMIALS

Let (S,≤) be an ordered set. Recall that (S,≤) is artinian if every strictly decreasingsequence of elements of S is ?nite, and that (S,≤) is narrow if every subset of pairwiseorder-incomparable elements of S is ?nite. Let S be a commutative monoid. Unl…     

Recent Developments in Synchrotron Mössbauer Reflectometry

Hyperfine Interactions - Synchrotron Mössbauer Reflectometry (SMR), the grazing incidence nuclear resonant scattering of synchrotron radiation, can be applied to perform depth-selective phase...     

Magnetic behaviour of TbMnFe

Neutron diffraction and M?ssbauer measurements have been carried out on the cubic Laves phase intermetallic TbMnFe. The magnetic moment on the transition metal atom is found to be low, 0.2μ _B, at room temperature. This moment is temperature independent down to 10 K. Magnetic moment on the rare earth atom varies from 2.5μ _B at 296 K to 7.27μ _B at 10 K. M?ssbauer spectra recorded at 298 K and 78 K have magnetic character but there is a large distribution of hyperfine field values. Both these features arise due to magnetic frustration created in the sample due to the competing ferro and antiferromagnetic interactions between the transition metal atoms.     

Poly(phenylene vinylene)‐based copolymers containing 3,7‐phenothiazylene and 2,6‐pyridylene chromophores: Fluorescence sensors for acids,metal ions,and oxidation

A novel copolymer, poly(N‐hexyl‐3,7‐phenothiazylene‐1,2‐ethenylene‐2,6‐pyridylene‐1,2‐ethenylene) ( P3 ), containing N‐hexyl‐3,7‐phenothiazylene and 2,6‐pyridylene chromophores was synthesized to investigate the effect of protonation, metal complexation, and chemical oxidation on its absorption and photoluminescence (PL). Poly(N‐hexyl‐3,8‐iminodibenzyl‐1,2‐ethenylene‐1,3‐phenylene‐1,2‐ethenylene) and poly(N‐hexyl‐3,7‐phenothiazylene‐1,2‐ethenylene‐1,3‐phenylene‐1,2‐ethenylene) ( P2 ), consisting of 1,3‐divinylbenzene alternated with N‐hexyl‐3,8‐iminodibenzyl and N‐hexyl‐3,7‐phenothiazylene, respectively, were also prepared for comparison. Electrochemical investigations revealed that P3 exhibited lower band gaps (2.34 eV) due to alternating donor and acceptor conjugated units (push–pull structure). The absorption and PL spectral variations of P3 were easily manipulated by protonation, metal chelation, and chemical oxidation. P3 displayed significant bathochromic shifts when protonated with trifluoroacetic acid in chloroform. The complexation of P3 with Fe³⁺ led to a significant absorption change and fluorescence quenching, and this implied the coordination of ferric ions with the 2,6‐pyridylene groups in the backbone. Moreover, both phenothiazylene‐containing P2 and P3 showed conspicuous PL quenching with a slight redshift when oxidized with NOBF₄. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 1272–1284, 2004     

On the approximation of convex functions by Bernstein-type operators

As a consequence of Jensen's inequality, centered operators of probabilistic type (also called Bernstein-type operators) approximate convex functions from above. Starting from this fact, we consider several pairs of classical operators and determine, in each case, which one is better to approximate convex functions. In almost all the discussed examples, the conclusion follows from a simple argument concerning composition of operators. However, when comparing Szász-Mirakyan operators with Bernstein operators over the positive semi-axis, the result is derived from the convex ordering of the involved probability distributions. Analogous results for non-centered operators are also considered.