Relative separation theorems for $\scr L_{\kappa+\kappa}$.

Stochastic separation theorems

The problem of non-iterative one-shot and non-destructive correction of unavoidable mistakes arises in all Artificial Intelligence applications in the real world. Its solution requires robust separation of samples with errors from samples where the system works properly. We demonstrate that in (moderately) high dimension this separation could be achieved with probability close to one by linear ...

Separation and Interlacing Theorems

Separation Theorems for Simplicity 26-Surfaces

The main goal of this paper is to prove a Digital Jordan– Brouwer Theorem and an Index Theorem for simplicity 26-surfaces. For this, we follow the approach to Digital Topology introduced in [2], and find a digital space such that the continuous analogue of each simplicity 26-surface is a combinatorial 2-manifold. Thus, the separation theorems quoted above turn out to be an immediate consequence...

Colimit Theorems for Relative Homotopy Groups

The first paper [10] (whose results were announced in [8]) developed the necessary ‘algebra of cubes’. Categories G of ω-groupoids and C of crossed complexes were defined, and the principal result of [10] was an equivalence of categories γ : G → C. Also established were a version of the homotopy addition lemma, and properties of ‘thin’ elements, in an ω-groupoid. In particular it was proved tha...

Brooks-type theorems for choosability with separation

We consider the following type of problems. Given a graph G = (V; E) and lists L(v) of allowed colors for its vertices v 2 V such that jL(v)j = p for all v 2 V and jL(u) \ L(v)j c for all uv 2 E, is it possible to nd a \list coloring", i.e., a color f (v) 2 L(v) for each v 2 V , so that f (u) 6 = f (v) for all uv 2 E ? We prove that every graph of maximum degree admits a list coloring for every...