Related papers: Maximum rank of a Legendrian web
Kendall's Similarity Shape Theory for constellations of N points in the carrier space $\mathbb{R}^d$ as quotiented by the similarity group was developed for use in Probability and Statistics. It was subsequently shown to reside within…
Spatially constrained planar networks are frequently encountered in real-life systems. In this paper, based on a space-filling disk packing we propose a minimal model for spatial maximal planar networks, which is similar to but different…
We present old and recent results on rank problems and linearizability of geodesic planar webs.
The entities in directed networks arising from real-world interactions are often naturally organized under some hierarchical structure. Given a directed, weighted, graph with edges and node labels, we introduce ranking problem where the…
A geometric approach is used to study a family of higher-order nonlinear Abel equations. The inverse problem of the Lagrangian dynamics is studied in the particular case of the second-order Abel equation and the existence of two alternative…
We determine all toric phases for the $2d$ $(0,2)$ theories on D1-branes probing the complex cones over the 18 smooth Fano 3-folds, whose toric diagrams correspond to the regular reflexive polytopes in 3 dimensions. These results…
Geometry of networks endowed with a causal structure is discussed using the conventional framework of equilibrium statistical mechanics. The popular growing network models appear as particular causal models. We focus on a class of tree…
We study the family of elliptic curves $y^2=x(x-a^2)(x-b^2)$ parametrized by Pythagorean triples $(a,b,c)$. We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over $\mathbb{Q}$ is 1, and for some…
Let $\Lambda$ be a Legendrian in the jet space of some manifold $X$. To a generating family presentation of $\Lambda$, we associate a constructible sheaf on $X \times \mathbb{R}$ whose singular support at infinity is $\Lambda$, and such…
Let $\bar{G} = (G, \omega)$ be a vertex-weighted graph, and $\delta$ a divisor class on $G$. Let $r_{\bar{G}}(\delta)$ denote the combinatorial rank of $\delta$. Caporaso has introduced the algebraic rank…
D3 branes stretching between webs of (p,q) 5branes provide an interesting class of 3d N=2 theories. For generic pq-webs however the low energy field theory is not known. We use 3d mirror symmetry and Type IIB S-duality to construct Abelian…
In the recent articles by Alper, Eastwood and Isaev, it was conjectured that all rational $GL_n({\mathbb C})$-invariant functions of forms of degree $d\ge 3$ on ${\mathbb C}^n$ can be extracted, in a canonical way, from those of forms of…
We study the $b$-matching problem in bipartite graphs $G=(S,R,E)$. Each vertex $s\in S$ is a server with individual capacity $b_s$. The vertices $r\in R$ are requests that arrive online and must be assigned instantly to an eligible server.…
A new modelling approach for the analysis of weighted networks with ordinal/polytomous dyadic values is introduced. Specifically, it is proposed to model the weighted network connectivity structure using a hierarchical multilayer…
We show that the conditions imposed on a second order linear differential equation with rational coefficients on the complex line by requiring it to have regular singularities with fixed exponents at the points of a finite set $P$ and…
Let K be a cyclic Galois extension of degree f over k and T a generator of the Galois group. For any v=(v_1,... , v_m)\in K^m such that v is linearly independent over k, and any 0< d < m the Gabidulin-like code C(v, T , d) is a maximum rank…
In this paper, we give a complete coarse classification of strongly exceptional Legendrian realizations of connected sum of two Hopf links in contact 3-spheres. This is the first classification result about exceptional Legendrian…
We extend the sutured framework to the case of Legendrians with boundary. Using ideas from Lagrangian Floer theory, we define the cylindrical and the wrapped sutured Legendrian homologies of a pair of sutured Legendrians. They fit together…
We prove the following statement, which has been conjectured since 1985: There exists a constant $K$ such that for all natural numbers $d,g$ with $g\le Kd^{3/2}$ there exists an irreducible component of the Hilbert scheme of $\mathbb{P}^3$…
Ryser's max term rank formula with graph theoretic terminology is equivalent to a characterization of degree sequences of simple bipartite graphs with matching number at least $\ell$. In a previous paper by the authors, a generalization was…