相关论文: Shattering-extremal set systems of small VC-dimens…
We show that certain families of sets in $\mathbb{R}^2$ (or $\mathbb{R}^n$) which are neither definable nor have bounded VC-dimension are nonetheless uniformly approximately definable in the real field, an o-minimal structure.
For finite-dimensional linear semigroups which leave a proper cone invariant it is shown that irreducibility with respect to the cone implies the existence of an extremal norm. In case the cone is simplicial a similar statement applies to…
Following \cite{Visintin}, we exploit the fractional perimeter of a set to give a definition of fractal dimension for its measure theoretic boundary. We calculate the fractal dimension of sets which can be defined in a recursive way and we…
We provide the first known upper bounds for the packing dimension of weighted singular and weighted $\omega$-singular matrices. We also prove upper bounds for these sets when intersected with fractal subsets. The latter results, even in the…
The erosion of a set in Euclidean space by a radius r>0 is the subset of X consisting of points at distance >/-r from the complement of X. A set is resilient to erosion if it is similar to its erosion by some positive radius. We give a…
Consequences of explicit symmetry breaking in a physically motivated model of SU(N) antiferromagnet in spatial dimensions one and two are studied. It is shown that the case N=3, which can be realized in spin-1 cold atom systems, displays…
A graph is called set-sequential if its vertices can be labeled with distinct nonzero vectors in $\mathbb{F}_2^n$ such that when each edge is labeled with the sum$\pmod{2}$ of its vertices, every nonzero vector in $\mathbb{F}_2^n$ is the…
A method is proposed for generating compact fractal disordered media, by generalizing the random midpoint displacement algorithm. The obtained structures are invasive stochastic fractals, with the Hurst exponent varying as a continuous…
Given an infinite iterated function system (IFS) $\mathcal{F}$, we define its dimension spectrum $D(\mathcal{F})$ to be the set of real numbers which can be realised as the dimension of some subsystem of $\mathcal{F}$. In the case where…
Diestel, et al. (see Order 35 (2017), JCT-A 167 (2019), arXiv:1805.01439) introduced the notion of abstract separation systems that satisfy a submodularity property, and they call this structural submodularity. Williamson, Goemans, Mihail,…
The conventional definition of extremality of a finite collection of sets is extended by replacing a fixed point (extremal point) in the intersection of the sets by a collection of sequences of points in the individual sets with the…
The Aubry-Andre model is a one-dimensional lattice model for quasicrystals with localized and delocalized phases. At the localization transition point, the system displays fractal spectrum, which relates to the Hofstadter butterfly. In this…
We call a family $\mathcal{F}$ of subsets of $[n]$ $s$-saturated if it contains no $s$ pairwise disjoint sets, and moreover no set can be added to $\mathcal{F}$ while preserving this property (here $[n] = \{1,\ldots,n\}$). More than 40…
If F is a type-definable family of commensurable subsets, subgroups or sub-vector spaces in a metric structure, then there is an invariant subset, subgroup or sub-vector space commensurable with F. This in particular applies to…
We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases this…
We compare the dimension of a non-invertible self-affine set to the dimension of the respective invertible self-affine set. In particular, for generic planar self-affine sets, we show that the dimensions coincide when they are large and…
Using ultraproduct techniques we define a nonstandard Minkowski dimension which exists for all bounded sets and which has the property that $\dim(A\times B)=\dim(A)+\dim(B).$ That is, our new dimension is product-summable. To illustrate our…
We characterize the downsets of integer partitions (ordered by containment of Ferrers diagrams) and compositions (ordered by the generalized subword order) which have finite dimension in the sense of Dushnik and Miller. In the case of…
A set of quantum states is said to be absolutely entangled, when at least one state in the set remains entangled for any definition of subsystems, i.e. for any choice of the global reference frame. In this work we investigate the properties…
Let $\mathcal{K}$ be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space $\mathbb{R}^n$, there is a well-studied notion of "ultrametric orthogonality" in $\mathcal{K}^n$. In this paper,…