Related papers: Escher squares and lattice links
It is shown that, given any $k$-dimensional lattice $\Lambda$, there is a lattice sequence $\Lambda_w$, $w\in \mathbb Z$, with sub-orthogonal lattice $\Lambda_o \subset \Lambda$, converging to $\Lambda$ (unless equivalence), also we discuss…
The main results of the paper points out the connection between the weak ordered relations and factor lattices defined by tolerances. It is proved that for any tolerance $T$ of a lattice $L$ the Dedekind-Mac Neille completion of $L/T$ is…
The projective space $\mathbb{P}_q(n)$, i.e. the set of all subspaces of the vector space $\mathbb{F}_q^n$, is a metric space endowed with the subspace distance metric. Braun, Etzion and Vardy argued that codes in a projective space are…
Let $L$ be a link in $S^3$. We consider the class of meridional presentations for $\pi_1(S^3\backslash L)$ in which the relations are witnessed by embedded two-spheres which can be represented simultaneously in a fixed diagram of $L$,…
Necessary and sufficient conditions for a finite connected graph with a strict partial order on vertices to be a combinatorial invariant of pseudoharmonic function are obtained.
We study lattice-theoretical extensions of the celebrated Sauer-Shelah-Perles Lemma. We conjecture that a general Sauer-Shelah-Perlem Lemma holds for a lattice $L$ if and only if $L$ is relatively complemented, and prove partial results…
In a recent paper, G. Cz\'edli and E.\,T. Schmidt present a structure theorem for planar semimodular lattices. In this note, we present an alternative proof.
The purpose of this article is to present my new proof of the the construction and the convergence theorem of spectral sequences of filtered complexes, which is much shorter and cleaner than the "standard" proof.
A well-known conjecture states that the Whitney numbers of the second kind of a geometric lattice (simple matroid) are logarithmically concave. We show this conjecture to be equivalent to proving an upper bound on the number of new copoints…
Some congruence relations satisfied by the theta series associated with the Leech lattice are given.
Fakhruddin has proved that for two lattice polygons P and Q any lattice point in their Minkowski sum can be written as a sum of a lattice point in P and one in Q, provided P is smooth and the normal fan of P is a subdivision of the normal…
We study linkedness of Cartesian product of graphs and prove that the product of an $a$-linked and a $b$-linked graphs is $(a+b-1)$-linked if the graphs are sufficiently large. Further bounds in terms of connectivity are shown. We determine…
Two mesh patterns are coincident if they are avoided by the same set of permutations. In this paper, we provide necessary conditions for this coincidence, which include having the same set of enclosed diagonals. This condition is sufficient…
We state the formula for the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given sublattice and give a partial proof of the formula. We show that the proof can be…
In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [3] for the equivalence between connections and parallel transport on vector bundles over manifolds. Out of the proof will emerge a new…
A detailed combinatorial analysis of planar convex lattice polygonal lines is presented. This makes it possible to answer an open question of Vershik regarding the existence of a limit shape when the number of vertices is constrained.
This paper gives a complete classification of linear repetitivity (LR) for a natural class of aperiodic Euclidean cut and project schemes with convex polytopal windows. Our results cover those cut and project schemes for which the lattice…
We study the equational theory of the Weihrauch lattice with multiplication, meaning the collection of equations between terms built from variables, the lattice operations $\sqcup$, $\sqcap$, the product $\times$, and the finite…
In this paper, we obtain a sufficient condition for the existence of parity factors in a regular graph in terms of edge-connectivity. Moreover, we also show that our condition is sharp.
We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. This proof uses only classic results in topology and combinatorial group theory.