关于非线性Lipschitz算子的Sderlind猜想

设ｆ是以Ｌ（ｆ）为最小上界Ｌｉｐｓｃｈｉｔｚ常数、以ρ（ｆ）为谱域半径、以ｒ（ｆ）为Ｇｅｒｓｃｈｇｏｒｉｍ域半径的有限维非线性Ｌｉｐｓｃｈｉｔｚ算子．本文证明了“存在等价范数‖·‖使Ｌ（ｆ）＝ｒ（ｆ）”的Ｓｄｅｒｌｉｎｄ猜想；给出反例否定了Ｓｄｅｒｌｉｎｄ的另一个猜想：“存在等价范数‖·‖使Ｌ（ｆ）ｒ（ｆ）”（注意ｒ（ｆ）与ｒ（ｆ）的区别），同时也否定了“ε＞０，存在等价范数‖·‖ε使Ｌε（ｆ）ρ（ｆ）＋ε”的猜想．作为以上所获结论的应用，本文将有关Ｄａｕｇａｖｅｔ方程的相应结果推广到了非线性算子情形．     

The Saturation Class of Shepard Operators

The saturation problem of the Shepard operators for 1 2 is completely settled.     

三个测度猜想的否定

给出三个非常容易让人误以为真的测度猜想,通过定理与λ-Cantor集及其余集的构造给出三个猜想的否定答案.     

Equistarable Graphs and Counterexamples to Three Conjectures on Equistable Graphs

Equistable graphs are graphs admitting positive weights on vertices such that a subset of vertices is a maximal stable set if and only if it is of total weight 1. Strongly equistable graphs are graphs such that for every and every nonempty subset T of vertices that is not a maximal stable set, there exist positive vertex weights assigning weight 1 to every maximal stable set such that the total weight of T does not equal c . General partition graphs are the intersection graphs of set systems over a finite ground set U such that every maximal stable set of the graph corresponds to a partition of U . General partition graphs are exactly the graphs every edge of which is contained in a strong clique. In 1994, Mahadev, Peled, and Sun proved that every strongly equistable graph is equistable, and conjectured that the converse holds as well. In 2009, Orlin proved that every general partition graph is equistable, and conjectured that the converse holds as well. Orlin's conjecture, if true, would imply the conjecture due to Mahadev, Peled, and Sun. An “intermediate” conjecture, posed by Miklavi? and Milani? in 2011, states that every equistable graph has a strong clique. The above conjectures have been verified for several graph classes. We introduce the notion of equistarable graphs and based on it construct counterexamples to all three conjectures within the class of complements of line graphs of triangle‐free graphs. We also show that not all strongly equistable graphs are general partition.     

拉格朗日插值算子的L_p收敛速度

本文得到了以第二类Ｔｃｈｅｂｙｃｈｅｆ多项式的零点为插值结点组的拉格朗日插值于Ｌｐ（１＜ｐ＜２）下的收敛速度     

Lp空间Shepard算子逼近的正逆定理

本文引入了一种修正的积分型Shepard算子,建立了相应的Jackson型定理,并通过建立Bernstein型不等式,给出了算子在L[0,1]p空间中一种新的逼近阶刻画的等价形式,得到了逼近的逆定理．     

Conjectures on GCn sets

The paper gives a generalization of the counterexample of C. de Boor on ? ^d and provides an example of GC ₂ set in ?⁴ with three maximal hyperplanes. Also, a conjecture on the number of maximal lines of GC _n sets in ? ^d is discussed and proved that it is true in the case when n = 2, d = 3.     

Conjectures on the dilogarithm

The paper expresses Bloch's dilogarithm D(z) in terms of Clausen's function Cl₂(t). Using Lichtenbaum's conjecture on #K ₂ O _F for quadratic imaginary fields F, we get more than 50 conjectural linear relations between the values of Cl₂(t). We cannot prove most of them.     

On the Approximation of Bounded Functions with Discontinuities of the First Kind by Generalized Shepard Operators

The authors investigate the approximation of bounded functions with discontinuities of the first kind by generalized Shepard operators. Estimate for the rate of convergence at a continuity point is obtained, while at the points of discontinuity it is shown that the sequence of generalized Shepard operators is almost always divergent. It is also shown that the sequence of Cesàro means of generalized Shepard operators is convergent everywhere for bounded functions which have only discontinuities of the first kind in [0,1].     

Interpolatory Wavelet Packets

In this note, we present a construction of interpolatory wavelet packets. Interpolatory wavelet packets provide a finer decomposition of the 2^jth dilate cardinal interpolation space and hence give a better localization for an adaptive interpolation. This can lead to a more efficient compression scheme which, in turn, provides an interpolation algorithm with a smaller set of data for use in applications.     

Two Conjectures on Graceful Digraphs

A digraph D with p vertices and q arcs is labeled by assigning a distinct integer value g(v) from {0,1, … , q} to each vertex v. The vertex values, in turn, induce a value g(u, v) on each arc (u, v) where g(u, v) = (g(v)? g(u))(mod q + 1). If the arc values are all distinct then the labeling is called a graceful labeling of a digraph. Bloom and Hsu (SIAM J Alg Discr Methods 6:519–536, 1985) conjectured that, all unicyclic wheels are graceful. Also, Zhao et al. (J Prime Res Math 4:118–126, 2008) conjectured that, for any positive even n and any integer m ≥ 14, the digraph ${n-\overrightarrow{C_m}}$ is graceful. In this paper, we prove both the conjectures.     

关于三个算子的算子不等式(英文)

As a generalization of grand Furuta inequality,recently Furuta obtain：If A≥ B≥0 with A0,then for t∈[0,1]and p1,p2,p3,p4≥1, A t 2[A- t 2{A t 2（A/ t 2 Bp 1A /t2 ）p 2A t 2}p 3A -t2 ]p 4A t 2 1 [{（p1/t）p2＋t}p3-t]p4＋t]≤A. In this paper,we generalize this result for three operators as follow：If A≥B≥C≥0 with B0,t∈[0,1]and p1,p2,···,p2n/1,p2n≥1 for a natural number n.Then the following inequalities hold for r≥t, A1/t＋r≥ [A r 2[B /t 2{B t 2······[B /t 2{B t 2（B /t 2 ←B /t 2 n times Bt 2 n/1 times by turns Cp 1B /t 2）p 2B t 2}p 3B /t 2]p 4···B t 2}p 2n/1B /t 2 B /t 2 n times Bt 2 n/1 times by turns→ ]p 2nA r 2] 1/t＋r q[2n]＋r/t, where q[2n]≡{···[{[（p1-t）p2＋t]p3/t}p4＋t]p5/···/t}p2n＋t /t and t alternately n times appear .     

Three Questions About Complex Symmetric Operators

We pose three questions about the structure and application of complex symmetric operators.     

Two Interpolatory Cubic Splines

The third order and the uniform cubic spline are defined andshown to have O(h³) and O(h⁴) convergence respectively whenused for interpolation.     

Interpolatory Hermite splines on rectangular domains

Based on Bittner and Urban’s construction of interpolatory multiwavelets [Kai Bittner, Karsten Urban, On interpolatory divergence-free wavelets, Math. Comput. 76 (258) (2007) 903-929], we use the truncation method to obtain interpolatory multiwavelets on rectangular domains in this paper. In addition, the characterization for the corresponding functional spaces is given.     

Interpolatory Band-Limited Wavelet Bases on the Sphere

In the present paper, we construct space-localized bases for the space $W_n^n:=\oplus_{k=n+1}^{2n} Harm_k({\Bbb S}^2)$ of band-limited functions on the sphere. Each of the basis functions is a zonal polynomial centered at a point $\eta_i\in{\Bbb S}^2$. The goal of this work is to describe explicit fundamental systems $\lbrace\eta_j\rbrace_{j=1,\dots,M_n}$ for the space $W_n^n$ which finally lead to space- and frequency-localized polynomial bases for $L^2({\Bbb S}^2)$.     

Conjectures and problems on Bochner-Riesz means

The aim of this paper is to state some conjectures and problems on Bochner-Riesz means in multiple Fourier series and integrals. The progress on these conjectures and problems are also mentioned.     

Estimates on Conjectures of Minkowski and woods

Let ℝ ⁿ be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in ℝ ⁿ of radius $ \sqrt {n/4} $ \sqrt {n/4} contains a point of ∧. This is known to be true for n ≤ 8. Here we give estimates on a more general conjecture of Woods for n ≥ 9. This leads to an improvement for 9 ≤ n ≤ 22 on estimates of Il’in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms.