Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDuality of antidiagonals and pipe dreams
156
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [A. Knutson and E. Miller, Grobner geometry of Schubert polynomials, Ann. Math. 161, 1245-1318]. The original proof of the duality was roundabout, relying on the algebra of certain monomial ideals and a recursive characterization of reduced pipe dreams. This paper provides a direct combinatorial proof.
rate research
Read More
Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it, i.e., the rotor-routing action is actually independent of the choice of root vertex. It is well-known that the spanning trees of a planar graph G are in canonical bijection with those of its planar dual G*, and furthermore that the sandpile groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing actions, of the sandpile group of G on its spanning trees, and of the sandpile group of G* on its spanning trees, compatible under plane duality? In this paper, we give an affirmative answer to this question, which had been conjectured by Baker.
Drilling and Extraction Automated System (DREAMS) is a fully automated prototype-drilling rig that can drill, extract water and assess subsurface density profiles from simulated lunar and Martian subsurface ice. DREAMS system is developed by the Texas A&M drilling automation team and composed of four main components: 1- tensegrity rig structure, 2- drilling system, 3- water extracting and heating system, and 4- electronic hardware, controls, and machine algorithm. The vertical and rotational movements are controlled by using an Acme rod, stepper, and rotary motor. DREAMS is a unique system and different from other systems presented before in the NASA Rascal-Al competition because 1- It uses the tensegrity structure concept to decrease the system weight, improve mobility, and easier installation in space. 2- It cuts rock layers by using a short bit length connected to drill pipes. This drilling methodology is expected to drill hundreds and thousands of meters below the moon and Martian surfaces without any anticipated problems (not only 1 m.). 3- Drilling, heating, and extraction systems are integrated into one system that can work simultaneously or individually to save time and cost.
Property A is a form of weak amenability for groups and metric spaces introduced as an approach to the famous Novikov higher signature conjecture, one of the most important unsolved problems in topology. We show that property A can be reduced to a sequence of linear programming optimization problems on finite graphs. We explore the dual problems, which turn out to have interesting interpretations as combinatorial problems concerning the maximum total supply of flows on a network. Using isoperimetric inequalities, we relate the dual problems to the Cheeger constant of the graph and explore the role played by symmetry of a graph to obtain a striking characterization of the difference between an expander and a graph without property A. Property A turns out to be a new measure of connectivity of a graph that is relevant to graph theory. The dual linear problems can be solved using a variety of methods, which we demonstrate on several enlightening examples. As a demonstration of the power of this linear programming approach we give elegant proofs of theorems of Nowak and Willett about graphs without property A.
Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer partitions, allowing us to connect number of standard Young tableaux and numbers of permutations. Here, we use operads, that algebraic devices abstracting the notion of composition of combinatorial objects, to build pairs of graded graphs. For this, we first construct a pair of graded graphs where vertices are syntax trees, the elements of free nonsymmetric operads. This pair of graphs is dual for a new notion of duality called $phi$-diagonal duality, similar to the ones introduced by Fomin. We also provide a general way to build pairs of graded graphs from operads, wherein underlying posets are analogous to the Young lattice. Some examples of operads leading to new pairs of graded graphs involving integer compositions, Motzkin paths, and $m$-trees are considered.
A {it sign pattern matrix} is a matrix whose entries are from the set ${+,-, 0}$. The minimum rank of a sign pattern matrix $A$ is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of $A$. It is shown in this paper that for any $m times n$ sign pattern $A$ with minimum rank $n-2$, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is shown that for each integer $ngeq 9$, there exists a nonnegative integer $m$ such that there exists an $ntimes m$ sign pattern matrix with minimum rank $n-3$ for which rational realization is not possible. A characterization of $mtimes n$ sign patterns $A$ with minimum rank $n-1$ is given (which solves an open problem in Brualdi et al. cite{Bru10}), along with a more general description of sign patterns with minimum rank $r$, in terms of sign vectors of certain subspaces. A number of results on the maximum and minimum numbers of sign vectors of $k$-dimensional subspaces of $mathbb R^n$ are obtained. In particular, it is shown that the maximum number of sign vectors of $2$-dimensional subspaces of $mathbb R^n$ is $4n+1$. Several related open problems are stated along the way.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
