Related papers: Tropical coefficient dynamics for higher-dimension…
Using the APM cluster data we investigate whether the dynamical status of clusters is related to the large-scale structure of the Universe. We find that cluster substructure is strongly correlated with the tendency of clusters to be aligned…
Two dimensional magnetic films with perpendicular magnetization spontaneously form magnetic domain patterns that evolve or undergo symmetry transformations as a function of temperature. When the system is driven from equilibrium by a rapid…
For a fusion category, we prove some new integral properties concerning the dimension of a simple object that generates a Isaacs fusion subcategory. A stronger divisibility result is proven for any modular fusion category. This divisibility…
We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup…
Cluster media are dynamical, not static; observational evidence suggests they are turbulent. High-resolution simulations of the intracluster media (ICMs) and of idealized, similar media help us understand the complex physics and…
In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras.…
By virtue of their high galaxy space densities and their large spatial separations, clusters are efficient and accurate tracers of the large-scale density and velocity fields. Substantial progress has been made over the past decade in the…
We show that the algebraic invariants multiplicity and depth of a graded ideal in the polynomial ring are closely connected to the fan structure of its generic tropical variety in the constant coefficient case. Generically the multiplicity…
We study the behavior of the clustering coefficient in tagged networks. The rich variety of tags associated with the nodes in the studied systems provide additional information about the entities represented by the nodes which can be…
Molecular dynamics simulations are used to study the local dynamics of counterion-charged polymer association at charge densities above and below the counterion condensation threshold. Surprisingly, the counterions form weakly-interacting…
A polynomial complexity algorithm is designed which tests whether a point belongs to a given tropical linear variety.
We translate notions and results of decomposition and dimension theories for module categories, into the lattice environment. In particular we translate dimension theory in module categories to complete modular upper-continuous lattices.
Flexible spatial models that allow transitions between tail dependence classes have recently appeared in the literature. However, inference for these models is computationally prohibitive, even in moderate dimensions, due to the necessity…
We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate $$TC(X)\le TC(\pi_1(X))+\dim X$$ of the topological complexity of a space in terms of its dimension and the complexity of its…
We present structural properties of two-dimensional polymers as far as they can be described by percolation theory. The percolation threshold, critical exponents and fractal dimensions of clusters are determined by computer simulation and…
We define scrollar invariants of tropical curves with a fixed divisor of rank 1. We examine the behavior of scrollar invariants under specialization, and provide an algorithm for computing these invariants for a much-studied family of…
This is a survey on recent developments in Cohen-Macaulay representations via tilting and cluster tilting theory. We explain triangle equivalences between the singularity categories of Gorenstein rings and the derived (or cluster)…
Rapid intensification (RI) of tropical cyclones (TCs) poses a great challenge due to their highly nonlinear dynamics and inherent uncertainties. Conventional statistical dynamics and artificial intelligence prediction models typically rely…
We develop the basic properties of the higher commutator for congruence modular varieties.
Transformers play a central role in the inner workings of large language models. We develop a mathematical framework for analyzing Transformers based on their interpretation as interacting particle systems, which reveals that clusters…