Related papers: On dually-CPT and strong-CPT posets
A graph $G=(V,E)$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree $T$ and two non-negative real numbers $d_{min}$ and $d_{max}$, $d_{min} \leq d_{max}$, such that each node $u \in V$ is uniquely associated to a…
We prove the 3-fold DT/PT correspondence for K-theoretic vertices via wall-crossing techniques. We provide two different setups, following Mochizuki and following Joyce; both reduce the problem to q-combinatorial identities on word…
Let $P$ be a finite poset. We will show that for any reasonable $P$-persistent object $X$ in the category of finite topological spaces, there is a $P-$ weighted graph, whose clique complex has the same $P$-persistent homology as $X$.
We study the counting problem known as #PPM, whose input is a pair of permutations $\pi$ and $\tau$ (called pattern and text, respectively), and the task is to find the number of subsequences of $\tau$ that have the same relative order as…
Given a symmetric monoidal category $C$ with product $\sqcup$, where the neutral element for the product is an initial object, we consider the poset of $\sqcup$-complemented subobjects of a given object $X$. When this poset has finite…
The dimension of a poset $P$, denoted $\dim(P)$, is the least positive integer $d$ for which $P$ is the intersection of $d$ linear extensions of $P$. The maximum dimension of a poset $P$ with $|P|\le 2n+1$ is $n$, provided $n\ge2$, and this…
A plane poset is a finite set with two partial orders, satisfying a certain incompatibility condition. The set PP of isoclasses of plane posets owns two products, and an infinitesimal Hopf algebra structure is defined on the vector space…
The choice of data representation is a key factor in the success of deep learning in geometric tasks. For instance, DUSt3R recently introduced the concept of viewpoint-invariant point maps, generalizing depth prediction and showing that all…
Geometric complexity theory (GCT) is an approach to the $P$ vs. $NP$ and related problems through algebraic geometry and representation theory. This article gives a high-level exposition of the basic plan of GCT based on the principle,…
We introduce so-called consistent posets which are bounded posets with an antitone involution ' where the lower cones of x,x' and of y,y' coincide provided x,y are different form 0,1 and, moreover, if x,y are different form 0 then their…
In \cite{TY}, we investigate the pair $(P, \Supp(P))$ of minimal path $P$ and its supporting sub-digraph $\Supp(P)$ in the path complex of a digraph $G$ under the strongly regular condition. In this paper, first, we consider the special…
The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees, near-linear time algorithms…
For a partially ordered set P, we denote by Co(P) the lattice of order-convex subsets of P. We find three new lattice identities, (S), (U), and (B), such that the following result holds. Theorem. Let L be a lattice. Then L embeds into some…
In 1980, Edelman defined a poset on objects called the noncrossing 2-partitions. They are closely related with noncrossing partitions and parking functions. To some extent, his definition is a precursor of the parking space theory, in the…
A geometric graph G(bar) is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call G(bar) a geometric realization of the underlying abstract graph G. A geometric homomorphism is a vertex map that…
We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all…
The poset of maximal tubings of a graph generalizes several well-known and remarkable partial orders. Notable examples include the weak Bruhat order and the Tamari lattice, posets of maximal tubings for the complete graph and the path…
For a discrete poset $\mathcal X$, McCord proved that the natural map $|{\mathcal X}|\to {\mathcal X}$, from the order complex to the poset with the Up topology, is a weak homotopy equivalence. Much later, \u{Z}ivaljevi\'{c} defined the…
A strongly separating path system in a graph $G$ is a collection $\mathcal{P}$ of paths in $G$ such that, for every two edges $e$ and $f$ of $G$, there is a paths in $\mathcal{P}$ with $e$ and not $f$, and vice-versa. The minimum number of…
This thesis deals with General Probabilistic Theories (GPTs) and Entanglement Structures (ESs). An ES is a possible structure of a quantum composite system in GPTs, which is not uniquely determined as the Standard Entanglement Structure…