Related papers: An algebraic formula for the intersection number o…
On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear integrable systems is obtained, and the integration scheme for such equations is proposed.
The self intersection of an immersion i : S^2 \to R^3 dissects S^2 into pieces which are planar surfaces (unless i is an embedding). In this work we determine what collections of planar surfaces may be obtained in this way. In particular,…
We present the Macaulay2 package TropicalToric.m2 for toric intersection theory computations using tropical geometry.
An effective method of computing division polynomials in terms of Mumford coordinates is presented. As an example, division polynomials for $3$- and $4$-torsion divisors on a genus two curve are obtained explicitly in terms of Mumford…
We present integral representations of solutions to division problems involving matrices of polynomials in several complex variables. We also find estimates of the polynomial degree of the solutions by means of careful degree estimates of…
Cartesian-grid methods in combination with immersed-body and volume-of-fluid methods are ideally suited for simulating breaking waves around ships. A surface panelization of the ship hull is used as input to impose body-boundary conditions…
In this paper, we give a quantum algorithm which solves collision problem in an expected polynomial time. Especially, when the function is two-to-one, we present a quantum algorithm which can find a collision with certainty in a worst-case…
This article presents numerical methods in order to solve problems of tolerance analysis. A geometric specification, a contact specification and a functional requirement can be respectively characterized by a finite set of geometric…
Polynomial solving algorithms are essential to applied mathematics and the sciences. As such, reduction of their complexity has become an incredibly important field of topological research. We present a topological approach to constructing…
We investigate arcs on a pair of pants and present an algorithm to compute the self-intersection number of an arc. Additionally, we establish bounds for the self-intersection number in terms of the word length. We also prove that the…
We calculate the intersection ring of three-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this…
In this note, we use a natural desingularization of the conormal variety of the variety of n x n symmetric matrices of rank at most r to find a general formula for the algebraic degree in semidefinite programming.
A method is developed to compute analytically fully symmetric cubature rules on the triangle by using symmetric polynomials to express the two kinds of invariance inherent in these rules. Rules of degree up to 15, some of them new and of…
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
Intersection algorithms are very important in computation of geometrical problems. Algorithms for a line intersection with linear or quadratic surfaces are quite efficient. However, algorithms for a line intersection with other surfaces are…
We provided two explicit formulas for the intersection cohomology (as a graded vector space with pairing) of the symplectic quotient by a circle in terms of the $S^1$ equivariant cohomology of the original symplectic manifold and the fixed…
The polytopic definition introduced recently describing the topology of manifolds is used to formulate a generating function pertinent to its topological properties. In particular, a polynomial in terms of one variable and a tori underlying…
Let $\mathcal{T}$ be a set of $n$ flat (planar) semi-algebraic regions in $\mathbb{R}^3$ of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess $\mathcal{T}$ into a data structure so that for a query…
Let $\mathcal C$ be a real plane algebraic curve defined by the resultant of two polynomials (resp. by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces $P(x,y,z)=Q(x,y,z)=0$…
We propose a constructive proof for the Ambrosetti-Rabinowitz Mountain Pass Theorem providing an algorithm, based on a bisection method, for its implementation. The efficiency of our algorithm, particularly suitable for problems in high…