Splittings of Banach spaces induced by Clifford algebras

Let be an infinite-dimensional Hilbert space of density character . By representing as a module over an appropriate Clifford algebra, it is proved that possesses a family of proper closed nonzero subspaces such that

Analogous results are proved for spaces and for and () when is an arbitrary nonzero Banach space.

     

Splittings

We investigate some notions of splitting families and estimate sizes of the corresponding cardinal coefficients. In particular we solve a problem of P. Simon. Received July 12, 1995     

Splittings of Surfaces

The main result characterizes small actions of surface groups on -trees.

     

Heegaard分解的重新分块

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔＭｂｅａｃｏｍｐａｃｔ 3 ｍａｎｉｆｏｌｄ .ＩｆｔｈｅｒｅｉｓａｐｒｏｐｅｒｌｙｅｍｂｅｄｄｅｄｃｌｏｓｅｄｓｕｒｆａｃｅＳｉｎＭｗｈｉｃｈｓｅｐａｒａｔｅｓＭｉｎｔｏｔｗｏｃｏｍｐｒｅｓｓｉｏｎｂｏｄｉｅｓＨ1ａｎｄＨ2 ,ｔｈｅｎＭｃａｎｂｅｗｒｉｔｔｅｎａｓＭ =Ｈ1∪ＳＨ2 .ＴｈｉｓｓｔｒｕｃｔｕｒｅｏｎＭｉｓｃａｌｌｅｄａＨｅｅｇａａｒｄｓｐｌｉｔｔｉｎｇｏｆＭａｎｄＳｉｓａｓｐｌｉｔｔｉｎｇｓｕｒｆａｃｅ .Ｈ1∪ＳＨ2 ｉｓｓａｉｄｔｏｂｅｒｅｄｕｃｉｂｌｅ…     

Obstructions to Compatible Splittings

Suppose one has a map of split short exact sequences in a category of modules, or more generally, in any abelian category. Do the short exact sequences split compatibly, i.e., does there exist a splitting of each short exact sequence which commutes with the map of short exact sequences? The answer is sometimes “yes” and sometimes “no”. We define and prove basic properties of a group of obstructions to the existence of compatible splittings.     

Cohomology Splittings of Stiefel Manifolds

The complex Stiefel manifolds admit a stable decomposition asThom spaces of certain bundles over Grassmannians. The purposeof the paper is to identify the splitting in any complex orientedcohomology theory.     

Heegaard splittings 的曲面连通和

Suppose Mi = Vi ∪ Wi （i = 1,2） are Heegaard splittings. A homeomorphism f ： F1 → F2 produces an attached manifold M = M1 ∪F1=F2 M2, where Fi ∪→ δ_Wi. In this paper we define a surface sum of Heegaard splittings induced from the Heegaard splittings of M1 and M2, and give a sufficient condition when the surface sum of Heegaard splitting is stabilized. We also give examples showing that the surface sum of Heegaard splittings can be unstabilized. This indicates that the surface sum of Heegaard splittings and the amalgamation of Heegaard splittings can give different Heegaard structures.     

Splittings of certain Fuchsian groups

We provide an explicit splitting of Fuchsian groups of type as free products with amalgamation.

     

边界可约的Heegaard分解

本文证明了任意边界可约流形的Heegaard分解都是n个不可约的、边界不可约的三维流形的Heegaard分解通过连通和、边界连通和及边界自连通和运算而得到.     

Thin Position of Heegaard Splittings

§１．ＢａｃｋｇｒｏｕｎｄａｎｄＭａｉｎＲｅｓｕｌｔｓＳｉｎｃｅｔｈｅｃｏｎｃｅｐｔｏｆｔｈｉｎｐｏｓｉｔｉｏｎｆｏｒｋｎｏｔｓｉｎｔｈｅ３－ｓｐｈｅｒｅｗａｓｄｅｖｅｌｏｐｅｄｂｙＧａｂａｉ［１］，ｍａｎｙｉｔｓａｐｐｌｉｃａｔｉｏｎｓｔｏｄｅａｌ...     

关于Z－矩阵的弱正则分裂

本文主要考虑一类Ｚ－矩阵弱正则分裂的迭代矩阵的谱半径和特征值１的指数问题，推广了Ｒｏｓｅ在１９８４年得到的一个定理．     

On graph compatible Splittings of M-matrices

Let A ϵ Rⁿⁿ. Then A = M − N is called a regular splitting of A if M^-1 ⩾ 0 and N ⩾ 0, and A = M − N is called a graph compatible splitting of A if ⊼[Ggr](A)⊼ ⊇ [Ggr](M), where [Ggr](A) is the graph of A and ⊼[Ggr](A)⊼ its reflexive transitive closure. We answer in the negative a question raised by H. Schneider whether a regular splitting A = M − N of an M-matrix is graph compatible if all diagonal elements of A (or M) are positive.     

Two Splittings of a Square Matrix

In this note it is shown that any square matrix AC ^n×n can be represented as the sum A= , where is complex symmetric and rank . The corresponding persymmetric result can be used in finding the terms of a small rank perturbed Toeplitz matrix via an O(n ²) computation. This allows one to perform fast matrix–vector products in case n is large.     

On Separation Index of Heegaard Splittings

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｃｏｎｃｅｐｔｏｆｒｅｄｕｃｉｂｌｅＨｅｅｇａａｒｄｓｐｌｉｔｉｎｇｓｗａｓｆｉｒｓｔｄｅｖｅｌｏｐｅｄｂｙＨａｋｅｎ［１］．Ｉｔｓｒｅｌａ－ｔｉｏｎｔｏｔｈｅｃｏｒｒｅｓｐｏｎｄｉｎｇ３－ｍａｎｉｆｏｌｄｓｃｏｎ...