Anti-PXK monoclonal antibody, clone 5E22

We show that a finite polyhedron P of nontrivial homotopy type cannot be homotopy equivalent to a proper Cartesian factor of itself. In other words, P similar or equal to PxK implies that K similar or equal to 1. Consequently, if P is a finite polyhedron such that P-n similar or equal to P, for some integer n >= 2, then the homotopy type of P is trivial. This answers positively the problem [1, Problem (34.10)] stated by K. Borsuk in 1971 (in an equivalent, shape-theoretical formulation) and a question from Borsuk's monograph "Theory of Shape" [2, Problem (7.13), p. 142]. We also prove that if (X, x) is an element of FANR, then Sh(n)(X, x) = Sh(X, x), for some integer n >= 2, implies that Sh(X) = 1. This resolves positively Problem (5.6) from [7] (1981). Similarly as for polyhedra, first we show that if X is an element of FANR and Sh(X) not equal 1, then X cannot have the shape of a proper Cartesian factor of itself. (C) 2017 Elsevier B.V. All rights reserved.

In this paper we consider the Procrustes problem on the manifold of orthogonal Stiefel matrices. Given matrices A is an element of R-mxk; B is an element of R-mxp, m >= p >= k; we seek the minimum of parallel to A - BQ parallel to(2) for all matrices Q is an element of R-pxk; Q(T)Q = I-kxk. We introduce a class of relaxation methods for generating sequences of approximations to a minimizer and offer a geometric interpretation of these methods. Results of numerical experiments illustrating the convergence of the methods are given.