Related papers: Some real and unreal enumerative geometry for flag…
We prove that there exist solutions for a non-parametric capillary problem in a wide class of Riemannian manifolds endowed with a Killing vector field. In other terms, we prove the existence of Killing graphs with prescribed mean curvature…
A standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. By adapting an approach for determining connectivity in complements of real hypersurfaces by Hong,…
We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This Littlewood-Richardson homotopy is based on Vakil's geometric proof of the Littlewood-Richardson rule. Its start…
We produce algorithms to detect whether a complex affine variety computed and presented numerically by the machinery of numerical algebraic geometry corresponds to an associated component of a polynomial ideal.
This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…
We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial…
In this paper, we present a unified approach using model category theory and an associative law to compare some classic variants of the geometric realization functor.
In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear…
We present a general technique, based on parametric search with some twist, for solving a variety of optimization problems on a set of semi-algebraic geometric objects of constant complexity. The common feature of these problems is that…
The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, yields insight into the hierarchy of complexity classes. In particular, the theory produces algebraic tests for a…
Algorithms for embedding certain types of nilpotent subalgebras in maximal subalgebras of the same type are developed, using methods of real algebraic groups. These algorithms are applied to determine non-conjugate subalgebras of the…
We study the properties of the constructive linear programing problems. The parameters of linear functions in such problems are constructive real numbers. To solve such a problem is to find the optimal plan with the constructive real number…
Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other…
This article is concerned with an example of complex planar geometry arising from flat origami challenges. The complexity of solution algorithms is illustrated, depending on the depth of the initial analysis of the problem, starting from…
In certain circumstances tools of Riemannian geometry are sufficient to address questions arising in the more general Finslerian context. We show that one such instance presents itself in the characterisation of geodesics in Randers spaces…
We give an overview of progress on homogeneous Einstein metrics on large classes of homogeneous manifolds, such as generalized flag manifolds and Stiefel manifolds. The main difference between these two classes of homogeneous spaces is that…
The edge isoperimetric problem for a graph $G$ is to determine, for each $n$, the minimum number of edges leaving any set of $n$ vertices. In general this problem is NP-hard, but exact solutions are known in some special cases, for example…
If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…
Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical…
We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients.…