SEMIGROUP CONSTRUCTION ON POLYGONAL NUMBERS

Polygonal Numbers

In the article the formal characterization of triangular numbers (famous from [15] and words “EYPHKA! num = ∆ + ∆ + ∆”) [17] is given. Our primary aim was to formalize one of the items (#42) from Wiedijk’s Top 100 Mathematical Theorems list [33], namely that the sequence of sums of reciprocals of triangular numbers converges to 2. This Mizar representation was written in 2007. As the Mizar lang...

On Universal Sums of Polygonal Numbers

For m = 3, 4, . . . , the polygonal numbers of order m are given by pm(n) = (m−2) ` n 2 ́ +n (n = 0, 1, 2, . . . ). For positive integers a, b, c and i, j, k > 3 with max{i, j, k} > 5, we call the triple (api, bpj , cpk) universal if for any n = 0, 1, 2, . . . there are nonnegative integers x, y, z such that n = api(x)+bpj(y)+cpk(z). We show that there are only 95 candidates for universal triple...

On Wiener numbers of polygonal nets

Polygonal Designs: Existence and Construction

Polygonal designs form a special class of partially balanced incomplete block designs. We resolve the existence problem for polygonal designs with various parameter sets and present several construction methods with blocks of small sizes.

Bass Numbers of Semigroup-graded Local Cohomology

Given a module M over a ring R that has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology H I(M) at any graded ideal I in terms of Ext modules. We use this method to obtain finiteness results for the local cohomology of graded modules over semigroup rings. In particular we prove that for a semigroup Q whose saturation Q is simplicial, and a finitely g...