Reaction of dibenzylphosphine oxide with α,β-unsaturated O-methyloximes

The reaction of dibenzylphosphine oxide with O-methyloximes of some 45t6w0557276558/xxlarge945.gif" alt="agr" align="BASELINE" BORDER="0">,45t6w0557276558/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">-unsaturated ketones results in the phosphorylation at the 45t6w0557276558/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">-carbon atom to form methoxyiminophosphine oxides, whereas the reaction of dibenzylphosphine oxide with O-methyloximes of 45t6w0557276558/xxlarge945.gif" alt="agr" align="BASELINE" BORDER="0">,45t6w0557276558/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">-unsaturated aldehydes affords aminodihydrophosphole oxides.     

On coverings of plane convex sets by translates of strips

In the euclidean planeE ² letS ₁,S ₂, ... be a sequence of strips of widthsw ₁,w ₂, .... It is shown thatE ² can be covered by translates of the stripsS _i if w ₁ ^3/2 = . Further results concern conditions in order that a compact convex domain inE ² can be covered by translates ofS ₁,S ₂, ....This research was supported by National Science Foundation Research Grant MCS 76-06111.     

Perturbation bounds for theLDL H andLU decompositions

LetA andA+A be Hermitian positive definite matrices. Suppose thatA=LDL ^H and (A+A)=(L+L)(D+D)(L+L)^H are theLDL ^H decompositons ofA andA+A, respectively. In this paper upper bounds on |D| _F and |L| _F are presented. Moreover, perturbation bounds are given for theLU decomposition of a complexn ×n matrix.     

Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations

Summary We discuss the application of a class of spline collocation methods to first-order Volterra integro-differential equations (VIDEs) which contain a weakly singular kernel (t–s)^{–45.gif" alt="agr" align="BASELINE" BORDER="0">} with 0<45.gif" alt="agr" align="BASELINE" BORDER="0"><1. It will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and graded meshes. The grading exponents of graded meshes used are not greater thanm (the polynomial degree) which is independent of 45.gif" alt="agr" align="BASELINE" BORDER="0">. This is in contrast to the theories of spline collocation methods for Volterra (or Fredholm) integral equation of the second kind. Numerical examples are given to illustrate the theoretical results.     

The bestL 2-approximation by finite sums of functions with separable variables

Summary We consider the problem of the best approximation of a given functionh 4578532551/xxlarge8712.gif" alt="isin" align="MIDDLE" BORDER="0"> L ² (X × Y) by sums4578532551/xxlarge8721.gif" alt="sum" align="MIDDLE" BORDER="0"> _k=1 ⁿ f _k f _k, with a prescribed numbern of products of arbitrary functionsf _k 4578532551/xxlarge8712.gif" alt="isin" align="MIDDLE" BORDER="0">L ² (X) andg _k 4578532551/xxlarge8712.gif" alt="isin" align="MIDDLE" BORDER="0">L ² (Y). As a co-product we develop a new proof of the Hilbert—Schmidt decomposition theorem for functions lying inL ² (X × Y).     

Toeplitz and Hankel type operators on the upper half-plane

An orthogonal decomposition of admissible wavelets is constructed via the Laguerre polynomials, it turns to give a complete decomposition of the space of square integrable functions on the upper half-plane with the measurey ^{45.gif" alt="agr" align="BASELINE" BORDER="0">} dxdy. The first subspace is just the weighted Bergman (or Dzhrbashyan) space. Three types of Ha-plitz operators are defined, they are the generalization of classical Toeplitz, small and big Hankel operators respectively. Their boundedness, compactness and Schatten-von Neumann properties are studied.Research was supported by the National Natural Science Foundation of China.     

Quasi-interpolation with translates of a function having noncompact support

We establish a result related to a theorem of de Boor and Jia [1]. Their theorem, in turn, corrected and extended a result of Fix and Strang [5] concerning controlled approximation. In our result, the approximating functions are not required to have compact support, but satisfy instead conditions on their behavior at . Our theorem includes some recent results of Jackson [6] and is closely related to the work of Buhmann [2].Communicated by Carl de Boor     

Dirichlet norms,capacities and generalized isoperimetric inequalities for Markov operators

We define the notion of p-capacity for a reversible Markov operator on a general measure space and prove that uniform estimates for the ratio of capacity and measure are equivalent to certain imbedding theorems for the Orlicz and Dirichlet norms. As a corollary we get results on connections between embedding theorems and isoperimetric properties for general Markov operators and, particularly, a generalization of the Kesten theorem on the spectral radius of random walks on amenable groups for the case of arbitrary graphs with non-finitely supported transition probabilities.     

Convex hull of powers of a complex number,trinomial equations and the Farey sequence

The color of a complex number is defined as the number of vertices of the convex hull of powers of that number. This induces a coloring of the unit disk. The structure of the set 454tv2k07551k11/xxlarge923.gif" alt="Lambda" align="BASELINE" BORDER="0"> of points where the color changes is investigated here. It is observed that there is a connection between this fractal set 454tv2k07551k11/xxlarge923.gif" alt="Lambda" align="BASELINE" BORDER="0"> and some family of trinomial equations. Three algorithms for coloring the unit disk are described, the last one (related to the Farey sequence) arising out of a conjecture. This conjecture is formulated and proved in this presentation.