On bijections for pattern-avoiding permutations

By considering bijections from the set of Dyck paths of length 2n onto each of Sn(321) and Sn(132), Elizalde and Pak in [S. Elizalde, I. Pak, Bijections for refined restricted permutations, J. Combin. Theory Ser. A 105 (2004) 207-219] gave a bijection that preserves the number of fixed points and the number of excedances in each σ∈Sn(321). We show that a direct bijection Γ:Sn(321)→Sn(132) introduced by Robertson in [A. Robertson, Restricted permutations from Catalan to Fine and back, Sém. Lothar. Combin. 50 (2004) B50g] also preserves the number of fixed points and the number of excedances in each σ. We also show that a bijection ?^∗:Sn(213)→Sn(321) studied in [J. Backelin, J. West, G. Xin, Wilf-equivalence for singleton classes, Adv. in Appl. Math. 38 (2007) 133-148] and [M. Bousquet-Melou, E. Steingrimsson, Decreasing subsequences in permutations and Wilf equivalence for involutions, J. Algebraic Combin. 22 (2005) 383-409] preserves these same statistics, and we show that an analogous bijection from Sn(132) onto Sn(213) does the same.     

Linear bijections preserving the Hölder seminorm

Let be a compact metric space and let be a real number with The aim of this paper is to solve a linear preserver problem on the Banach algebra of Hölder functions of order from into We show that each linear bijection having the property that for every where

is of the form for every where with is a surjective isometry and is a linear functional.

     

多项式不动点的进一步研究

对文[1]中关于多项式不动点的主要定理进行了修正和发展,进而研究了多项式的广义(高阶)不动点,证明了对任意给定的n个点t_1≤t_2≤…≤t_n,存在唯一的首项系数为α∈R(α≠0)的n次多项式P(x)以它们为广义不动点.     

超凸度量空间中非扩张可交换映象族的几乎不动点

获得了关于超凸度量空间中非扩张可交换映象族不动点存在性的更一般结果.证明了可数个非扩张可交换映象的公共ε-几乎不动点集的超凸性,并在一定条件下证明了任意个非扩张可交换映象的公共ε-几乎不动点集的超凸性.还讨论了Baillon的两个重要结果间的等价关系.     

多项式微分代数系统的奇点性质

本讨论多项式微分代数方程的奇点性质，证明了经用吴方法整序后的系统的奇点与原系统的相应奇点有相同的鞍点性质。     

Fixed points of commuting holomorphic mappings other than the Wolff point

Let be the unit disc of and let be such that . For 1$">, let . We study the behavior of on . In particular, we prove that . As a consequence, besides conditions for , we prove a conjecture of C. Cowen in case and are univalent mappings.

     

Identity principles for commuting holomorphic self-maps of the unit disc

Let f,g be two commuting holomorphic self-maps of the unit disc. If f and g agree at the common Wolff point up to a certain order of derivatives (no more than 3 if the Wolff point is on the unit circle), then f≡g.     

Rings with commuting projections

Si dimostra che in un anello con involuzione, sotto opportune ipotesi, ogni proiezione che commuta con un certo tipo di suoi coniugati è centrale.     

Partition bijections, a survey

We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises. 2000 Mathematics Subject Classification Primary—05A17; Secondary—05A30; 11P83 The author was partially supported by NSA and NSF grants.     

Partition identities and geometric bijections

We present a geometric framework for a class of partition identities. We show that there exists a unique bijection proving these identities, which satisfies certain linearity conditions. In particular, we show that Corteel's bijection enumerating partitions with nonnegative -th differences can be obtained by our approach. Other examples and generalizations are presented.

     

Memorandum on multiplicative bijections and order

Let denote the semigroup of continuous functions from the topological space X to , equipped with the pointwise multiplication. The paper studies semigroup homomorphisms , with emphasis on isomorphisms. The crucial observation is that, in this setting, homomorphisms preserve order, so isomorphisms preserve order in both directions and they are automatically lattice isomorphisms. Applications to uniformly continuous and Lipschitz functions on metric spaces are given. Sample result: if Y and X are complete metric spaces of finite diameter without isolated points, every multiplicative bijection has the form Tf=f○τ, where τ:X→Y is a Lipschitz homeomorphism. F. Cabello Sánchez and J. Cabello Sánchez are supported in part by DGICYT projects MTM2004-02635 and MTM2007-6994-C02-02. J. Cabello Sánchez is supported in part by a grant of the UEx (Programa Propio–Acción 2).     

Manifolds with commuting Jacobi operators

We characterize Riemannian manifolds of constant sectional curvature in terms of commutation properties of their Jacobi operators.     

Submanifolds with commuting normal vector field

This is a survey of studies on submanifolds of spaces of constant curvature with a parallel normal vector field, parallel second or third fundamental form, or with a parallel (or satisfying other geometric conditions) field of normal p-directions, and on conformally flat submanifolds.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 14, pp. 73–100, 1983.     

Moufang loops with commuting inner mappings

We investigate the relation between the structure of a Moufang loop and its inner mapping group. Moufang loops of odd order with commuting inner mappings have nilpotency class at most 2. The 6-divisible Moufang loops with commuting inner mappings have nilpotency class at most 2. There is a Moufang loop of order 2¹⁴ with commuting inner mappings and of nilpotency class 3.     

Toeplitz matrices commuting with tridiagonal matrices

we prove that if R is a nonscalar Toeplitz matrix R_{i, j}=r_?i?j? which commutes with a tridiagonal matrix with simple spectrum, then

$\text{r}{}_{\mathrm{k}}\text{r}{}_1\text{=}\text{u}{}_{\mathrm{k-1}}\text{r}{}_2\text{r}{}_1\text{cos p}\text{u}{}_{\mathrm{k-1}}\text{(cos p)}$

, k=4, 5,…, with U_k the Chebychev polynomial of the second kind, where p is determined from

$\text{cos p=}\text{1}\text{2}\text{r}{}^2{}_1\text{?r}{}_1\text{r}{}_3\text{r}{}^2{}_2\text{?r}{}_1\text{r}{}_3$

.