相关论文: Uniform Growth, Actions on Trees and $GL_2$

We study the isoperimetric and spectral profiles of certain families of finitely generated groups defined via actions on labelled Schreier graphs and simple {\em gluing} of such. In one of our simplest constructions---the {\em…

We improve the previously best known lower and upper bounds on the number n_g of numerical semigroups of genus g. Starting from a known recursive description of the tree T of numerical semigroups, we analyze some of its properties and use…

In this note, we prove a conjecture of Puder on an extension of the co-growth formula to any non-negative function defined on a bi-regular tree. A key component of our proof is the establishment of a resolvent identity, which serves as an…

Let k be an algebraically closed field of characteristic p>0 and C a connected nonsingular projective curve over k with genus g>1. Let (C,G) be a "big action", i.e. a pair (C,G) where G is a p-subgroup of the k-automorphism group of C such…

We present Collaborative Trees, a novel tree model designed for regression prediction, along with its bagging version, which aims to analyze complex statistical associations between features and uncover potential patterns inherent in the…

In this article we present a Bernstein inequality for sums of random variables which are defined on a graphical network whose nodes grow at an exponential rate. The inequality can be used to derive concentration inequalities in…

We study the problem of learning a latent tree graphical model where samples are available only from a subset of variables. We propose two consistent and computationally efficient algorithms for learning minimal latent trees, that is, trees…

This paper uses combinatorics and group theory to answer questions about the assembly of icosahedral viral shells. Although the geometric structure of the capsid (shell) is fairly well understood in terms of its constituent subunits, the…

Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a power of a prime $p$. Recently, a particular action of the group $\mathrm{GL}_2(\mathbb F_q)$ on irreducible polynomials in $\mathbb F_q[x]$ has been introduced and…

Suppose we are given a profinite group $G$ acting on a formal moduli stack $\mathcal{M}$, and we want to understand the group action, and compute cohomology related to this group action. How can we do it? This prolegomenon surveys two…

In recent years, there has been an effort to extend the classical notion of phylogenetic balance, originally defined in the context of trees, to networks. One of the most natural ways to do this is with the so-called $B_2$ index. In this…

We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such…

The use of cumulative incidence functions for characterizing the risk of one type of event in the presence of others has become increasingly popular over the past decade. The problems of modeling, estimation and inference have been treated…

We generalize a class of groups defined by Rostislav Grigorchuk to a much larger class of groups, and provide upper and lower bounds for their word growth (they are all of intermediate growth) and period growth (under a small additional…

An evolutionary tree is a rooted tree where each internal vertex has at least two children and where the leaves are labeled with distinct symbols representing species. Evolutionary trees are useful for modeling the evolutionary history of…

Ensemble methods such as random forests have transformed the landscape of supervised learning, offering highly accurate prediction through the aggregation of multiple weak learners. However, despite their effectiveness, these methods often…

Given an $\mathbb{N}$-weighted tree automaton, we give a decision procedure for exponential vs polynomial growth (with respect to the input size) in quadratic time, and an algorithm that computes the exact polynomial degree of growth in…

We define a kind of 'operational calculus' for $GL_2(R)$. Namely, the group $GL_2(R)$ can be regarded as an open dense chart in the Grassmannian of 2-dimensional subspaces in $R^4$. Therefore the group $GL_4(R)$ acts in $L^2$ on $GL_2(R)$.…

Perturbation of the GUE are known in physics to be related to enumeration of graphs on surfaces. We investigate this idea and show that for a small convex perturbation, we can perform a genus expansion: the moments of the empirical measure…

We give a sufficient condition for a sequence of normal subgroups of a free group to have the property that both, their growths tend to the upper bound and their cogrowths tend to the lower bound. The condition is represented by planarity…