We give a short proof of the Hass–Lagarias theorem on an upper bound on the minimal number of elementary moves to unknot in a triangulated 3-manifold. Our method uses a normal form for surfaces whose boundary is contained in the 1-skeleton of a triangulated 3-manifold. We also obtain a significantly better upper bound of 2, where t is the number of tetrahedra, and improve the Hass–Lagarias uppe...

Wilf’s eigenvalue upper bound on the chromatic number is extended to the setting of digraphs. The proof uses a generalization of Brooks’ Theorem to digraph colorings.

Clarkson Local Minima of Arrangements 6 Next to show that if v has outdegree i then v 0 is an i-minimum. Since v 0 j < 0 if and only if v 0 is below h j , we need to show that a coordinate v 0 j 6 = 0 corresponds to an oriented edge (v; q) where wv ? wq = w(v ? q) has the same sign as v 0 j. Suppose (v; q) is an edge of P. Then ^ Av = ^ b ^ Aq, with one strict inequality a j v = b j > a j q, an...

An upper bound for McMullen's problem on projective transformations in R is derived from Redei's classical theorem on Hamilton paths in tournaments.

We shall show that there is a effectively computable upper bound of the heights of solutions for an inequality in Roth-Ridout’s theorem.

We prove that a locally compact space with an upper curvature bound is topological manifold if and only all of its spaces directions are homotopy equivalent not contractible. discuss applications to homology manifolds, limits Riemannian manifolds deduce sphere theorem.

The first part of this paper is devoted to proving a comparison theorem for Kähler manifolds with holomorphic bisectional curvature bounded from below. The model spaces being compared to are CP, C, and CH. In particular, it follows that the bottom of the spectrum for the Laplacian is bounded from above by m for a complete, m-dimensional, Kähler manifold with holomorphic bisectional curvature bo...

This note complements an earlier paper of the author by providing a lower bound theorem for Minkowski sums of polytopes. In [AS16], we showed an analogue of McMullen’s Upper Bound theorem for Minkowski sums of polytopes, estimating the maximal complexity of such a sum of polytopes. A common question in reaction to that research was the question for an analogue for the Barnette’s Lower Bound The...

We present a natural restriction of Hindman’s Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In...

This paper introduces an isometric extension procedure for Riemannian manifolds with boundary, which preserves some lower curvature bound and produces a totally geodesic boundary. As immediate applications of this construction, one obtains in particular upper volume bounds, an upper intrinsic diameter bound for the boundary, precompactness, and a home-omorphism finiteness theorem for certain cl...