相关论文: Elementary classes of finite VC-dimension
The Euler Characteristic Transform (ECT) is a robust method for shape classification. It takes an embedded shape and, for each direction, computes a piecewise constant function representing the Euler Characteristic of the shape's sublevel…
Let $G$ be a finite, connected, undirected, and simple graph and $W$ be a set of vertices in $G$. A representation multiset of a vertex $u$ in $V(G)$ with respect to $W$ is defined as the multiset of distances between $u$ and the vertices…
A classical enumerative result states that, given a graph $G$ and a vertex $u$, the number of connected subgraphs of $G$ is equal to the number of orientations of $G$ such that every vertex can reach $u$ by a directed path. We show that…
We show that any countable model of a model complete theory has an elementary extension with a "pseudofinite-like" quasidimension that detects dividing.
To work more accurately with elements of the semigroup of the Stone Cech compactification of the discrete semigroup of natural numbers N under multiplication. We divided these elements into ultrafilters which are on finite levels and…
We introduce a simple lattice model in which percolation is constructed on top of critical percolation clusters, and show that it can be repeated recursively any number $n$ of generations. In two dimensions, we determine the percolation…
We consider and characterize classes of finite and countably categorical structures and their theories preserved under $E$-operators and $P$-operators. We describe $e$-spectra and families of finite cardinalities for structures belonging to…
Federer's characterization states that a set $E\subset \mathbb{R}^n$ is of finite perimeter if and only if $\mathcal H^{n-1}(\partial^*E)<\infty$. Here the measure-theoretic boundary $\partial^*E$ consists of those points where both $E$ and…
When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the…
We study the projective dimensions of the restriction of functors Hom(-,X) to a contravariantly finite rigid subcategory T of a triangulated category C. We show that the projective dimension of Hom(-,X)|T is at most one if and only if there…
In this article, we prove that if $R$ is an affine algebra of dimension $d\geq 4$ over $\overline{\mathbb{F}}_{p}$ and $1/(d-1)! \in R,$ then any unimodular row over $R$ of length $d$ can be mapped to a factorial row by elementary…
We show that a vertex in the reduced crystal is i-external for a residue i if the defect is less than the absolute value of the i-component of the hub. We demonstrate the existence of a bound on the degree after which all vertices of a…
We present several new model-theoretic applications of the fact that, under the assumption that there exists a proper class of almost strongly compact cardinals, the powerful image of any accessible functor is accessible. In particular, we…
We invent the notion of a {\it dimension of a variety} $V$ as the cardinality of all its proper {\it derived} subvarieties (of the same type). The dimensions of varieties of lattices, varieties of regular bands and other general algebraic…
We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already,…
We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that…
For a finite partially ordered set we calculate the dimension of the variety of its subspace representations having fixed dimension vector. The dimension is given in terms of the Euler quadratic form associated with a partially ordered set,…
This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple…
This paper classifies the complexity of various teaching models by their position in the arithmetical hierarchy. In particular, we determine the arithmetical complexity of the index sets of the following classes: (1) the class of uniformly…
We consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and…