a remark on a remark by macaulay or enhancing lazard structural theorem

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).

Remark on a Theorem of Lindström

In a recent paper LindstrSm [I] proves a theorem on finite sets and he also proves a transfinite extension. In this note we only concern ourselves with the transfinite case of Lindstriim’s theorem. He in fact proves the following theorem: Let / Y I = K be an infinite set, and let A,, l K, be subsets of Y. Then for every p -=z m there are p disj...

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...

A Remark on a Theorem by Kodama and Shimizu

We prove a characterization theorem for the unit polydisc ∆ ⊂ C in the spirit of a recent result due to Kodama and Shimizu. We show that if M is a connected n-dimensional complex manifold such that (i) the group Aut(M) of holomorphic automorphisms of M acts on M with compact isotropy subgroups, and (ii) Aut(M) and Aut(∆) are isomorphic as topological groups equipped with the compact-open topolo...

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...