A Note on Cheeger’s Isoperimetric Constant

On Asymptotic Isoperimetric Constant of Tori

In this note we continue the study of asymptotic invariants of Riemannian tori (e.g. see [BuI] and references there). By asymptotic invariants we mean invariants which do not change under passing to finite covers. In [BuI] we show that the asymptotic volume growth of a Riemannian torus is at least as fast as that of a flat one. One may ask what are the possible values of “asymptotic isoperimetr...

On Asymptotic Isoperimetric Constant of Torid

Isoperimetric-type inequalities on constant curvature manifolds

By exploiting optimal transport theory on Riemannian manifolds and adapting Gromov’s proof of the isoperimetric inequality in the Euclidean space, we prove an isoperimetric-type inequality on simply connected constant curvature manifolds.

A Note on Artin’s Constant

P p(pi−1) , where the summation is over first N prime pi, k ∈ Z+. The classical summation formula is as follows: A = limN→∞Σ(N), where Σ(N) = 1 N ∑ φ(pi−1) pi−1 . The changes needed for arbitrary g are addressed in Theorem 1, a good exercise in basic analytic and algebraic number theory. The same procedure can be applied to other number-theoretic constants like A (see, e.g., [Ni]). In Theorem 2...

A Note on Edge Isoperimetric Numbers and Regular Graphs

This note resolves an open problem asked by Bezrukov in the open problem session of IWOCA 2014. It shows an equivalence between regular graphs and graphs for which a sequence of invariants presents some symmetric property. We extend this result to a few other sequences.