Related papers: Divisor package for Macaulay2
We study a projective Calabi-Yau threefold which has been constructed in an earlier paper. It is rigid and has Picard number two. We construct a pair of divisors which give a basis of the Picard group and determine all intersection numbers…
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful…
This tutorial illustrates how to use Grayson and Stillman's computer algebra system, Macaulay2, to study schemes. The examples are taken from the homework for an algebraic geometry class given at the University of California, Berkeley in…
This article summarizes new features and enhancements of the first major update of Package-X. Package-X 2.0 can now generate analytic expressions for arbitrarily high rank dimensionally regulated tensor integrals with up to four distinct…
We consider a class of linear differential operators acting on vector-valued function spaces with general coupled boundary conditions. Unlike in the more usual case of so-called quantum graphs, the boundary conditions can be nonlinear.…
In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at…
The woylier package implements tour interpolation paths between frames using Givens rotations. This provides an alternative to the geodesic interpolation between planes currently available in the tourr package. Tours are used to visualise…
In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with four invariant lines, including the line at infinity…
Consider an ample and globally generated line bundle $L$ on a smooth projective variety $X$ of dimension $N\geq 2$ over $\mathbb{C}$. Let $D$ be a smooth divisor in the complete linear system of $L$. We construct reflexive sheaves on $X$ by…
We obtain a global resolution for the sheaf of differential operators on smooth geometric quotients of free linear actions of algebraic groups. The terms of our resolution involve symmetric and alternating powers of vector bundles easily…
Diffractive optical elements that divide an input beam into a set of replicas are used in many optical applications ranging from image processing to communications. Their design requires time-consuming optimization processes, which, for a…
The goal of this paper is to discover a set of discriminative patches which can serve as a fully unsupervised mid-level visual representation. The desired patches need to satisfy two requirements: 1) to be representative, they need to occur…
Images are often obstructed by various obstacles due to capture limitations, hindering the observation of objects of interest. Most existing methods address occlusions from specific elements like fences or raindrops, but are constrained by…
The divisor graph is the non oriented graph whose vertices are the positive integers, and edges are the {a,b} such that a divides b or b divides a. Let F(x,y) be the maximum number of integers<= x belonging in one of y pairwise disjoint…
We give a computationally efficient method for constructing the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic…
Macaulay Duality, between quotients of a polynomial ring over a field, annihilated by powers of the variables, and finitely generated submodules of the ring's graded dual, is generalized over any Noetherian ring, and used to provide…
The main goal of this paper is to show that the notions of Weil and Cartier $\mathbb{Q}$-divisors coincide for $V$-manifolds and give a procedure to express a rational Weil divisor as a rational Cartier divisor. The theory is illustrated on…
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…
The crossover among two or more types of diffusive processes represents a vibrant theme in nonequilibrium statistical physics. In this work we propose two models to generate crossovers among different L\'evy processes: in the first model we…
We describe the qFunctions Mathematica package for $q$-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for $q$-shift equations and…