Remark on the Generalized Graph Extension 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).

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 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...

Generalized Caratheodory Extension Theorem on Fuzzy Measure Space

and Applied Analysis 3 3. Main Results Throughout this paper, we will consider lattices as complete lattices,X will denote space, and μ is a membership function of any fuzzy set X. Definition 3.1. If m : σ L → R ∪ {∞} satisfies the following properties, then m is called a lattice measure on the lattice σ-algebra σ L . i m ∅ L0. ii For all f, g ∈ σ L such that m f , m g ≥ L0 : f ≤ g ⇒ m f ≤ m g ...