Related papers: Shelling Coxeter-like Complexes and Sorting on Tre…
This paper is concerned with the approximation of high-dimensional functions in a statistical learning setting, by empirical risk minimization over model classes of functions in tree-based tensor format. These are particular classes of…
We introduce and study strongly vertex dismissible, vertex dismissible, and scalable simplicial complexes as non-pure extensions of vertex decomposability and shellability. Strong vertex dismissibility is defined recursively by relaxing the…
In 1989 Erd\H{o}s and Sz\'ekely showed that there is a bijection between (i) the set of rooted trees with $n+1$ vertices whose leaves are bijectively labeled with the elements of $[\ell]=\{1,2,\dots,\ell\}$ for some $\ell \leq n$, and (ii)…
In this paper we study natural reconfiguration spaces associated to the problem of distributing a fixed number of resources to labeled nodes of a tree network, so that no node is left empty. These spaces turn out to be cubical complexes,…
Connected acyclic graphs (trees) are data objects that hierarchically organize categories. Collections of trees arise in a diverse variety of fields, including evolutionary biology, public health, machine learning, social sciences and…
The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees, near-linear time algorithms…
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…
We introduce the notion of tensor t-structures on the bounded derived categories of schemes. For a Noetherian scheme $X$ admitting a dualizing complex, Bezrukavnikov-Deligne, and then independently Gabber and Kashiwara have shown that given…
In this paper, we study the lattice properties of posets of torsion pairs in the module category of a family of representation-finite gentle algebras called tiling algebras, introduced by Coelho Simoes and Parsons. We present a…
Let p be a prime number. We give a conjecture of a sheaf-theoretic nature which is equivalent to the strong form of the Tate conjecture for smooth, projective varieties X over F_p: for all n>0, the order of pole of the Hasse-Weil zeta…
In this paper, we give a new and efficient algebraic criterion for the pure as well as non-pure shellability of simplicial complex $\Delta$ over [n]. We also give an algebraic characterization of a leaf in a simplicial complex (defined in…
We introduce polytopal cell complexes associated with partial acyclic orientations of a simple graph, which generalize acyclic orientations. Using the theory of cellular resolutions, two of these polytopal cell complexes are observed to…
Daily internet communication relies heavily on tree-structured graphs, embodied by popular data formats such as XML and JSON. However, many recent generative (probabilistic) models utilize neural networks to learn a probability distribution…
We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give…
For the ordered set $[n]$ of $n$ elements, we consider the class $\Bscr_n$ of bases $B$ of tropical Pl\"ucker functions on $2^{[n]}$ such that $B$ can be obtained by a series of mutations (flips) from the basis formed by the intervals in…
Trees with labelled leaves and with all other vertices of degree three play an important role in systematic biology and other areas of classification. A classical combinatorial result ensures that such trees can be uniquely reconstructed…
Fix a field $k$. When $\Delta$ is a simplicial complex on $n$ vertices with Stanley-Reisner ideal $I_\Delta$, we define and study an invariant called the $\textit{type defect}$ of $\Delta$. Except when $\Delta$ is of a single simplex, the…
We describe an algorithm to identify a minimal set of "braid relations" which span and preserve all sets of involution words for twisted Coxeter systems of finite or affine type. We classify the cases in which adding the smallest possible…
Broadcasting on trees is a fundamental model from statistical physics that plays an important role in information theory, noisy computation and phylogenetic reconstruction within computational biology and linguistics. While this model…
Quot schemes of quotients of a trivial bundle of arbitrary rank on a nonsingular projective surface X carry perfect obstruction theories and virtual fundamental classes whenever the quotient sheaf has at most 1-dimensional support. The…