Related papers: Numerical Schubert Calculus in Macaulay2
We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary…
This paper presents some algorithmic techniques to compute explicitly the noetherian operators associated to a class of ideals and modules over a polynomial ring. The procedures we include in this work can be easily encoded in computer…
A general primal-dual splitting algorithm for solving systems of structured coupled monotone inclusions in Hilbert spaces is introduced and its asymptotic behavior is analyzed. Each inclusion in the primal system features compositions with…
In this paper, we investigate the relationship between the Hilbert functions and the associated properties of the graded modules. To attain this, we construct the graded modules from the sets of points in projective space, $\mathbb{P}_k^n$…
We consider algebras underlying Hilbert spaces used by quantum information algorithms. We show how one can arrive at equations on such algebras which define n-dimensional Hilbert space subspaces which in turn can simulate quantum systems on…
We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of…
The work is devoted to the development of numerical methods for computing "formal solutions" of interval systems of linear algebraic equations. These solutions are found in Kaucher interval arithmetic, which extends and completes the…
We are developing a Maple package of functions related to Rota's Umbral Calculus. A Mathematica version of this package is being developed in parallel.
In this paper, a quantum computational framework for algebraic topology based on simplicial set theory is presented. This extends previous work, which was limited to simplicial complexes and aimed mostly to topological data analysis. The…
We study the back stable Schubert calculus of the infinite flag variety. Our main results are: 1) a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part; 2) a novel…
Homotopy methods to solve polynomial systems are well suited for parallel computing because the solution paths defined by the homotopy can be tracked independently. Both the static and dynamic load balancing models are implemented in C with…
Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional…
A new robust algorithm for the numerical computation of biarcs, i.e. $G^1$ curves composed of two arcs of circle, is presented. Many algorithms exist but are based on geometric constructions, which must consider many geometrical…
The numerical Hilbert series combinatorics and the comodule Hilbert series combinatorics are introduced, and some applications are presented, including the MacMahon Master Theorem.
The Macaulay2 package DecomposableSparseSystems implements methods for studying and numerically solving decomposable sparse polynomial systems. We describe the structure of decomposable sparse systems and explain how the methods in this…
We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods and show that its h-polynomial coincides with the k-Narayana polynomial. We give a simplified formula for the h-polynomial of Schubert varieties.…
We present a package 'MixedMultiplicity' for computing mixed multiplicities of ideals in a Noetherian ring which is either local or a standard graded algebra over a field. This enables us to find mixed volumes of convex lattice polytopes…
Let X be a projective scheme; let M and N be two coherent O_X-modules. Given an integer m, we present an algorithm for computing the global extension module Ext^m(X;M,N). In particular, this allows one to compute the sheaf cohomology…
This introduces Rees algebras and some of their uses with illustrations via version 2.0 of the Macaulay2 package ReesAlgebra.m2.
In this paper, we propose methods for computing the Hilbert series of multigraded right modules over the free associative algebra. In particular, we compute such series for noncommutative multigraded algebras. Using results from the theory…