Network and Seiberg Duality

We define and study a new class of 4d N=1 superconformal quiver gauge theories associated with a planar bipartite network. While UV description is not unique due to Seiberg duality, we can classify the IR fixed points of the theory by a permutation, or equivalently a cell of the totally non-negative Grassmannian. The story is similar to a bipartite network on the torus classified by a Newton polygon. We then generalize the network to a general bordered Riemann surface and define IR SCFT from the geometric data of a Riemann surface. We also comment on IR R-charges and superconformal indices of our theories.Comment: 28 pages, 28 figures; v2: minor correction

On the structure of the adjacency matrix of the line digraph of a regular digraph

We show that the adjacency matrix M of the line digraph of a d-regular digraph D on n vertices can be written as M=AB, where the matrix A is the Kronecker product of the all-ones matrix of dimension d with the identity matrix of dimension n and the matrix B is the direct sum of the adjacency matrices of the factors in a dicycle factorization of D.Comment: 5 page

Sharp Total Variation Bounds for Finitely Exchangeable Arrays

In this article we demonstrate the relationship between finitely exchangeable arrays and finitely exchangeable sequences. We then derive sharp bounds on the total variation distance between distributions of finitely and infinitely exchangeable arrays

Inferring Biologically Relevant Models: Nested Canalyzing Functions

Inferring dynamic biochemical networks is one of the main challenges in systems biology. Given experimental data, the objective is to identify the rules of interaction among the different entities of the network. However, the number of possible models fitting the available data is huge and identifying a biologically relevant model is of great interest. Nested canalyzing functions, where variables in a given order dominate the function, have recently been proposed as a framework for modeling gene regulatory networks. Previously we described this class of functions as an algebraic toric variety. In this paper, we present an algorithm that identifies all nested canalyzing models that fit the given data. We demonstrate our methods using a well-known Boolean model of the cell cycle in budding yeast

On the least exponential growth admitting uncountably many closed permutation classes

We show that the least exponential growth of counting functions which admits uncountably many closed permutation classes lies between 2^n and (2.33529...)^n.Comment: 13 page