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We enumerate the edges in the Hasse diagram of several lattices arising in the combinatorial context of lattice paths. Specifically, we will consider the case of Dyck, Grand Dyck, Motzkin, Grand Motzkin, Schr\"oder and Grand Schr\"oder…
We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a $(2,n)$-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov…
I show in this letter that it is possible to construct a Hamiltonian description for Lorentzian General Relativity in terms of two real $SO(3)$ connections. The constraints are simple polynomials in the basic variables. The present…
We generalize the classical notion of adjoint of a linear operator and the Aron-Schottenloher notion of adjoint of a homogeneous polynomial. The general notion is shown to enjoy several properties enjoyed by the classical ones, nevertheless…
Topological string theory with twistor space as the target makes visible some otherwise difficult to see properties of perturbative Yang-Mills theory. But left-right symmetry, which is obvious in the standard formalism, is highly unclear…
We consider linear port-Hamiltonian differential-algebraic equations (pH-DAEs). Inspired by the geometric approach of Maschke and van der Schaft and the linear algebraic approach of Mehl, Mehrmann and Wojtylak, we present another view by…
The purpose of this paper is to study the famous Pappus configuration of $9$ lines and its dual arrangement. We show among others that by applying the Pappus Theorem to the dual arrangement we obtain the configuration corresponding to the…
In this short note we give an elementary proof of the fact that connections and their geometric parallel-transport counterpart are equivalent notions.
We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study…
We use Picard-Lefschetz theory to introduce a new local model for the planar projective twists $\tau_{\mathbb{A}\mathbb{P}^2} \in \mathrm{Symp}_{ct}(T^*\mathbb{A}\mathbb{P}^2), \ \mathbb{A} \in \{ \mathbb{R}, \mathbb{C} \}$. In each case,…
Wei's celebrated Duality Theorem is generalized in several ways, expressed as duality theorems for linear codes over division rings and, more generally, duality theorems for matroids. These results are further generalized, resulting in two…
Linear Geometry describes geometric properties that depend on the fundamental notion of a line. In this paper we survey basic notions and results of Linear Geomery that depend on the flat hulls: flats, exchange, rank, regularity,…
The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable…
We consider the category of linear relations over an arbitrary commutative ring, and identify it as a subcategory of the category of Kronecker representations. We observe that this subcategory forms a definable, faithful and hereditary…
We formalize the general principle of significance with respect to binary relations which is a universal tool for description and analysis of various situations in and apart from mathematics. We derive the basic properties and focus on a…
We introduce and study strongly invertible Legendrian links in the standard contact three-dimensional space. We establish the equivariant analogs of basic results separately well-known for strongly invertible and Legendrian links, i.e. the…
A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric.
The relationship between linear relations and matrix pencils is investigated. Given a linear relation, we introduce its Weyr characteristic. If the linear relation is the range (or the kernel) representation of a given matrix pencil, we…
A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran…
Networks are a useful representation for data on connections between units of interests, but the observed connections are often noisy and/or include missing values. One common approach to network analysis is to treat the network as a…