A Remark on the Gromov Convergence Theorem

A Remark on the Gromov Convergence Theorem

In [3] M. Gromov introduced the concept of convergence of Riemannian manifolds and he proved the convergence theorem. Since that time the theorem has been developed in detail (see [5], [7], [2]), and we know that it contains some interesting applications. Nevertheless there seems to be an inadequate way of applying the convergence theorem. The purpose of this paper is to present an example whic...

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem

A Remark on Zoloterav’s Theorem

Let n ≥ 3 be an odd integer. For any integer a prime to n, define the permutation γ a,n of {1,. .. , (n − 1)/2} by γ a,n (x) = n − {ax} n if {ax} n ≥ (n + 1)/2, {ax} n if {ax} n ≤ (n − 1)/2, where {x} n denotes the least nonnegative residue of x modulo n. In this note, we show that the sign of γ a,n coincides with the Jacobi symbol a n if n ≡ 1 (mod 4), and 1 if n ≡ 3 (mod 4).

A remark on the Erdös-Szekeres theorem

The Erdős–Szekeres theorem states that for any , any sufficiently large set of points in general position in the plane contains points in convex position. We show that for any and any finite sequence ! ! #" , with #$% & ( ')(+*, ! ! ), any sufficiently large set of points in general position in the plane contains either an empty convex ) gon or convex polygons . ! ! /. " , where 0 . $ 0,(& $ an...

On a small cancellation theorem of Gromov

We give a combinatorial proof of a theorem of Gromov, which extends the scope of small cancellation theory to group presentations arising from