Piecewise linear embeddings and isotopies

Piecewise Constant Controlled Linear Fuzzy Systems

Definable isotopies

Let X be a definable C manifold, Y1, Y2 definably compact definable C r submanifolds of X such that dimY1 + dimY2 < dimX and Y1 has a trivial normal bundle. We prove that there exists a definable isotopy {hp}p∈J such that h0 = idX and h1(Y1) ∩ Y2 = ∅. 2010 Mathematics Subject Classification. 14P10, 14P20, 58A05, 03C64.

Piecewise-linear and birational toggling

We define piecewise-linear and birational analogues of toggle-involutions, rowmotion, and promotion on order ideals of a poset P as studied by Striker and Williams. Piecewise-linear rowmotion relates to Stanley’s transfer map for order polytopes; piecewise-linear promotion relates to Schützenberger promotion for semistandard Young tableaux. When P = [a] × [b], a reciprocal symmetry property rec...

Constructing Piecewise Linear

Let P = fp1; : : : ; png and Q = fq1; : : : ; qng be two point sets lying in the interior of rectangles in the plane. We show how to construct a piecewise linear homeomorphism of size O(n2) between the rectangles which maps pi to qi for each i. This bound is optimal in the worst case; i.e., there exist point sets for which any piecewise linear homeomorphism has size (n2). Introduction A homeomo...

Piecewise-Linear Manifold Learning

The need to reduce the dimensionality of a dataset whilst retaining inherent manifold structure is key in many pattern recognition, machine learning and computer vision tasks. This process is often referred to as manifold learning since the structure is preserved during dimensionality reduction by learning the intrinsic low-dimensional manifold that the data lies on. Since the inception of mani...