We study algebraic algorithms for expressing the number of non-negative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd classes of toric varieties via Gröbner bases, and rational generating functions as in Barvinok’s algorithm. We report polyhedral and computational results for two special cases: counting contingency ta...

We give sufficient conditions for the tautness of a transversely homogenous foliation defined on compact manifold, by computing its base-like cohomology. As an application, we prove that if is non-unimodular then either ambient closure leaves or total space associated principal bundle fiber over S 1 .

In this paper, we mainly study the embedding problem of unimodular row vectors, focusing on avoiding identification polynomial zeros. We investigate existence minimal syzygy module ZLP matrix and demonstrate that has structural properties are similar to fundamental solution system homogeneous linear equations found in algebra. Finally, provide several methods for vectors certain cases.

We show that on an n = 24m+ 8k-dimensional even unimodular lattice, if the shortest vector length is ≥ 2m, then as the number of vectors of length 2m decreases, the secrecy gain increases. We will also prove a similar result on general unimodular lattices. Furthermore, assuming the conjecture by Belfiore and Solé, we will calculate the difference between inverses of secrecy gains as the number ...

We prove that for the Activated Random Walks model on transitive unimodular graphs, if there is fixation, then every particle eventually fixates, almost surely. We deduce that the critical density is at most 1. Our methods apply for much more general processes on unimodular graphs. Roughly put, our result apply whenever the path of each particle has an automorphism invariant distribution and is...