Related papers: Obtainable Sizes of Topologies on Finite Sets
The classical $k$-means algorithm for partitioning $n$ points in $\mathbb{R}^d$ into $k$ clusters is one of the most popular and widely spread clustering methods. The need to respect prescribed lower bounds on the cluster sizes has been…
Given a dominating set, how much smaller a dominating set can we find through elementary operations? Here, we proceed by iterative vertex addition and removal while maintaining the property that the set forms a dominating set of bounded…
Consider the projections of a finite set $A\subset R^n$ onto the coordinate hyperplanes. How small can the sum of the sizes of these projections be, given the size of $A$? In a different form, this problem has been studied earlier in the…
For random graphs, the containment problem considers the probability that a binomial random graph $G(n,p)$ contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the…
Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As…
New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable space…
The Graph Minor Theorem of Robertson and Seymour implies a finite set of obstructions for any minor closed graph property. We show that there are only three obstructions to knotless embedding of size 23, which is far fewer than the 92 of…
Finite topological spaces became much more essential in topology, with the development of computer science. The task of this paper is to study and investigate some properties of such spaces with the existence of an ordered relation between…
We present a method for computing the topological entropy of one-dimensional maps. As an approximation scheme, the algorithm converges rapidly and provides both upper and lower bounds.
What sets A \subset Z^n can be written in the form (K-K) \cap Z^n, where K is a compact subset of R^n such that K+Z^n=R^n? Such sets A are called achievable, and it is known that if A is achievable, then < A >=Z^n. This condition completely…
We introduce an algorithm which can be directly used to feasible and optimum search in linear programming. Starting from an initial point the algorithm iteratively moves a point in a direction to resolve the violated constraints. At the…
We introduce a new class of possibly noncompact n-dimensional manifolds without boundary associated to finite data which we call topological automata. This class is large enough to contain many interesting examples of open 2-dimensional and…
In this paper we present a novel two-scale framework to optimize the structure and the material distribution of an object given its functional specifications. Our approach utilizes multi-material microstructures as low-level building blocks…
Some new classes of compacta $K$ are considered for which $C(K)$ endowed with the pointwise topology has a countable cover by sets of small local norm--diameter.
We present a level-set based topology optimization algorithm for design optimization problems involving an arbitrary number of different materials, where the evolution of a design is solely guided by topological derivatives. Our method can…
We investigate the partitioning of partial orders into a minimal number of heapable subsets. We prove a characterization result reminiscent of the proof of Dilworth's theorem, which yields as a byproduct a flow-based algorithm for computing…
We study the problem of finding the smallest set of nodes in a network whose removal results in an empty $k$-core; where the $k$-core is the sub-network obtained after the iterative removal of all nodes of degree smaller than $k$. This…
A design is a finite set of points in a space on which every "simple" functions averages to its global mean. Illustrative examples of simple functions are low-degree polynomials on the Euclidean sphere or on the Hamming cube. We prove lower…
Starting from filters over the set of indices, we introduce structures in a product of sets where the coordinate sets have the given structures.