Formulas For Mesavage's Cubic-Foot Volume Table

Table of contents Volume 7 Issue 2

Recursive formulas for embedding distributions of cubic outerplanar graphs

Recently, the first author and his coauthor proved a k-order homogeneous linear recursion for the genus polynomials of any H-linear family of graphs (called path-like graph families by Mohar). Cubic outerplanar graphs are tree-like graph families. In this paper, we derive a recursive formula for the total embedding distribution of any cubic outerplanar graph. We also obtain explicit formulas fo...

Monotone Volume Formulas for Geometric Flows

We consider a closed manifold M with a Riemannian metric gij(t) evolving by ∂t gij = −2Sij where Sij(t) is a symmetric two-tensor on (M, g(t)). We prove that if Sij satisfies the tensor inequality D(Sij , X) ≥ 0 for all vector fields X on M , where D(Sij , X) is defined in (1.6), then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latt...

Solution formulas for cubic equations without or with constraints

We present a convention (for square/cubic root) which provides correct interpretations of Lagrange’s formula for all cubic polynomial equations with real coefficients. Using this convention, we also present a real solution formulas for the general cubic equation with real coefficients under equality and inequality constraints.

A Cubic Algorithm for Computing Gaussian Volume

We present randomized algorithms for sampling the standard Gaussian distribution restricted to a convex set and for estimating the Gaussian measure of a convex set, in the general membership oracle model. The complexity of the integration algorithm is O(n) while the complexity of the sampling algorithm is O(n) for the first sample and O(n) for every subsequent sample. These bounds improve on th...