Related papers: The combinatorics of tau-covers
The main theme of this paper is the introduction of a new type of polarizations, suited for some open symplectic manifolds, and their applications. These applications include symplectic embedding results that answer a question by…
Three combinatorial matrices are considered and their LU-decompositions were found. This is typically done by (creative) guessing, and necessary proofs are more or less routine calculations.
Cylindrical-like coordinates for constant-curvature 3-spaces are introduced and discussed. This helps to clarify the geometrical properties, the coordinate ranges and the meaning of free parameters in the static vacuum solution of Linet and…
Regular $A_n$-, $B_n$- and $C_n$-crystals are edge-colored directed graphs, with ordered colors $1,2,...,n$, which are related to representations of quantized algebras $U_q(\mathfrak{sl}_{n+1})$, $U_q(\mathfrak{sp}_{2n})$ and…
Packing problems, even of objects with regular geometries, are in general non-trivial. For few special shapes, the features of crystalline as well as random, irregular two-dimensional (2D) packings are known. The packing of 2D crosses does…
The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some numerical characteristics showing the cardinality of equivalence classes are introduced. A combinatorial identity that is in relationship to these…
In this paper it is shown that the compact linearization approach, that has been previously proposed only for binary quadratic problems with assignment constraints, can be generalized to arbitrary linear equations with positive coefficients…
The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.
Let T be a tile in the Cartesian plane made up of finitely many rectangles whose corners have rational coordinates and whose sides are parallel to the coordinate axes. This paper gives necessary and sufficient conditions for a square to be…
Integer compositions, integer partitions, Fibonacci numbers, and generalizations of these have recently been shown to be interconnected via two-toned tilings of horizontal grids. In this article, we present refinements of two-toned tilings,…
Using GCH, we force the following: There are continuum many simple cardinal characteristics with pairwise different values.
There are many fundamental algorithmic problems on triangulated 3-manifolds whose complexities are unknown. Here we study the problem of finding a taut angle structure on a 3-manifold triangulation, whose existence has implications for both…
We study symplectic structures on four-dimensional small covers. Our main result shows that every symplectic four-dimensional small cover is aspherical. We then classify symplectic small covers over products of two polygons, proving that…
We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a…
Let $\tau\colon\tilde{\mathcal{E}}\to\mathcal{E}$ be a differential covering of a PDE $\tilde{\mathcal{E}}$ over $\mathcal{E}$. We prove that if $\mathcal{E}$ possesses infinite number of symmetries and/or conservation laws then…
We investigate the combinatorial analogues, in the context of normal surfaces, of taut and transversely measured (codimension 1) foliations of 3-manifolds. We establish that the existence of certain combinatorial structures, a priori weaker…
Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the…
We prove the existence of $N$ distinct pairs of nontrivial solutions for critical $p$-Laplacian problems in ${\mathbb R}^N$, as well as in bounded domains. To overcome the difficulties arising from the lack of compactness, we use a recent…
We obtain homological properties of the second symmetric product of P^4 and the double cover of the symmetric determinantal quintic hypersurface in P^{14} (the double quintic symmetroids), which indicate the homological projective duality…
We begin the study of the consequences of the existence of certain infinite matrices. Our present application is to compactness of products of topological spaces.