(1.1) 0 −→ N f −−→M g −−→ P −→ 0 which is exact at N , M , and P . That means f is injective, g is surjective, and im f = ker g. Example 1.1. For an R-module M and submodule N , there is a short exact sequence 0 // N // M // M/N // 0, where the map N →M is the inclusion and the map M →M/N is reduction modulo N . Example 1.2. For R-modules N and P , the direct sum N ⊕ P fits into the short exact...

We consider the problem of multiple sequence alignment under a xed evolu tionary tree given a tree whose leaves are labeled by sequences nd ancestral sequences to label its internal nodes so as to minimize the total length of the tree where the length of an edge is the edit distance between the sequences labeling its endpoints We present a new polynomial time approximation algorithm for this pr...

In (2003), we proved the injective homotopy exact sequence of modules by a method that does not refer to any elements of the sets in the argument, so that the duality applies automatically in the projective homotopy theory (of modules) without further derivation. We inherit this fashion in this paper during our process of expanding the homotopy exact sequence. We name the resulting doubly infin...

Let φ be a w-continuous homomorphism from a dual Banach algebra to C. The notion of φ-Connes amenability is studied and some characterizations is given. A type of diagonal for dual Banach algebras is dened. It is proved that the existence of such a diagonal is equivalent to φ-Connes amenability. It is also shown that φ-Connes amenability is equivalent to so-called φ-splitting of a certain short...

1. Exact sequences A sequence of R-module maps A φ → B π → C is exact if the kernel of π and the image of φ are the exact same submodule of B. A longer sequence is exact by definition if every subsequence of two consecutive maps is exact. a). Show that B π → C → 0 is exact if and only if π is surjective. The kernel of C → 0 is all of C, so it is exact if the image of π is all of C—that is, if π...