Wr,p(R)-splines

In [3] Golomb describes, for 1 < p < ∞, the H^r,p(R)-extremal extension $\text{F}{}_1$ of a function $\text{?:E → R}$ (i.e., the H^r,p-spline with knots in E) and studies the cone $\text{H}{}_{1E}{}^{\mathrm{r,p}}$ of all such splines. We study the problem of determining when $\text{F}{}_1$ is in W^r,p ≡ H^r,p ∩ L^p. If $\text{F}{}_1\text{? W}{}^{\mathrm{r,p}}$, then $\text{F}{}_1$ is called a W^r,p-spline, and we denote by $\text{W}{}_{1E}{}^{\mathrm{r,p}}$ the cone of all such splines. If E is quasiuniform, then $\text{F}{}_1\text{? W}{}^{\mathrm{r,p}}$ if and only if . The cone $\text{W}{}_{1E}{}^{\mathrm{r,p}}$ with E quasiuniform is shown to be homeomorphic to l^p. Similarly, $\text{H}{}_{1E}{}^{\mathrm{r,p}}$ is homeomorphic to h^r,p. Approximation properties of the W^r,p-splines are studied and error bounds in terms of the mesh size $\text{¦ E ¦}$ are calculated. Restricting ourselves to the case p = 2 and to quasiuniform partitions E, the second integral relation is proved and better error bounds in terms of $\text{¦ E ¦}$ are derived.     

R1^m+1中拟迷向类空超曲面

设f:M^m→R1^m+1是无脐点类空超曲面,则在Mm上可以定义四个基本的共形不变量:共形度量g,共形1-形式C,共形第二基本形式B,共形Blaschke张量A.如果存在光滑函数λ和常数μ,使得A+μB=Ag,则称M^m是拟迷向类空超曲面.本文不仅构造了拟迷向类空超曲面的例子,同时在相差R1^m+1的一个共形变换下,本文还完全分类了拟迷向类空超曲面.     

关于L2(En+1+,(dxdy)/(yn+1)中的柱面函数空间的正交直和分解

本文利用小波变换给出了L²(E_n+1⁺,(dxdy)/(yⁿ⁺¹)中的柱面函数空间的一种正交直和分解.在这种分解下定义了Toeplitz-Hankel型算子,得到了类似的Schatten-Von Neumann性质.     

The simultaneous diophantine equations 5Y2 − 20 = X2 and 2Y2 + 1 = Z2

The three numbers 1, 5, 10 have the property that the product of any two numbers decreased by 1 is a perfect square. In this paper it is proved that there is no other positive integer N which shares this property with 1, 5, and 10.     

Hardy空间H~1(R)的紧小波框架系数刻画

利用标准正交小波基下函数的展开系数来刻画Hardy空间H~1(R)已经得到了很好的证明.该文利用紧小波框架与标准正交小波基的关系及其性质,给出了Hardy空间H~1(R)在紧小波框架下函数展开系数的一个刻画.     

A note on the geometry of M 3 and symplectic structures on S 1 × M 3

We investigate the relationship between the geometry of a closed, oriented 3-manifold M and the symplectic structures on S ¹ × M. In most cases the existence of a symplectic structure on S ¹ × M and Thurstonșs geometrization conjecture imply the existence of a geometric structure on M. This observation together with the existence of geometric structures on most 3-manifolds which fiber over the circle suggests a different approach to the problem of finding a fibration of a 3-manifold over the circle in case its product with the circle admits a symplectic structure. This work was supported in part by a GEBIP grant from the Turkish Academy of Sciences and a CAREER grant from the Scientific and Technological Research Council of Turkey.     

Pseudo-Chebyshev subspaces in L 1

We give various characterizations of pseudo-Chebyshev subspaces in the spacesL ¹(S, μ) andC(T).     

INTERPOLATION SPACES BETWEEN H^1 AND L^∞ ON SPACES OF HOMOGENEOUS TYPE

Using the maximal function characterization of Hardy spaces,we study the interpolation spaces between H^1 and L^∞on spaces of homogeneous type.     

The diophantine equation x3 − 3xy2 − y3 = 1 and related equations

The diophantine equation of the title has been solved by Ljunggren, by indirect use of the p-adic method (use is made of intermediate algebraic extensions). It is generally accepted that an immediate application of the p-adic method for the aforementioned equation is impossible. In this paper, however, this view was overthrown by first solving x² + 3 = 4y³ and then x³ ? 3xy² ? y³ = 1 with direct application of the p-adic method, avoiding the use of intermediate algebraic extensions, fulfilling thus a desire of Professor Mordell. The method used in this paper has a general character, as it is shown in Appendix B, where three more examples are given.     

C 1 Lohner Algorithm

Abstract. We present a modification of the Lohner algorithm for the computation of rigorous bounds for solutions of ordinary differential equations together with partial derivatives with respect to initial conditions. The modified algorithm requires essentially the same computational effort as the original one. We applied the algorithm to show the existence of several periodic orbits for R?ssler equations and the 14-dimensional Galerkin projection of the Kuramoto—Sivashinsky partial differential equation.     

Exposed points and extremal problems in H1

If φ ∈ L_∞, we denote by T_φ the functional defined on the Hardy space H¹ by

$\text{T}{}_{\mathrm{\phi }}\text{(?) =}\text{∫}\text{?π}\text{π}\text{?(e}{}^{\mathrm{i\theta }}\text{) φ(e}{}^{\mathrm{i\theta }}\text{)}\text{dθ}\text{2π}$

. Let S_φ be the set of functions in H¹ which satisfy $\text{T}{}_{\mathrm{\phi }}\text{(?) = ∥T}{}_{\mathrm{\phi }}\text{∥}\text{and}\text{∥? ∥}{}_1\text{? 1}$. It is known that if φ is continuous, then S_φ is weak-¹ compact and not empty. For many noncontinuous φ each S_φ is weak-¹ compact and not empty. A complete descr ption of S_φ if S_φ is weak-¹ compact and not empty is obtained. S_φ is not empty if and only if $\text{S}{}_{\mathrm{\phi }}\text{= S}{}_{\mathrm{\psi }}\text{and}\text{ψ = ¦ ?¦}\text{?}$ for some nonzero ? in H¹. It is shown that if $\text{φ = ¦? ¦}\text{?}$ and $\text{? = pg}$, where p is an analytic polynomial and g is a strong outer function, then S_φ is weak-¹ compact. As the consequence, if $\text{? = p}$, then S_φ is weak-¹ compact.     

丢番图方程aX~4-bY~2=1

应用Thue-Siegel方法,我们证明:对任意正整数a,b,不定方程aX~4-bY~2=1至多只有两组正整数解(X,Y),这证实了Walsh提的一个猜测.