Related papers: Hands-on Tropical Geometry
We present the Julia interface Polymake.jl to polymake, a software for research in polyhedral geometry. We describe the technical design and how the integration into Julia makes it possible to combine polymake with state-of-the-art…
In this paper we further develop the theory of geometric tropicalization due to Hacking, Keel and Tevelev and we describe tropical methods for implicitization of surfaces. More precisely, we enrich this theory with a combinatorial formula…
The tropical variety of a $d$-dimensional prime ideal in a polynomial ring with complex coefficients is a pure $d$-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing…
Most ocean models in current use are built upon structured meshes. It follows that most existing tools for extracting diagnostic quantities (volume and surface integrals, for example) from ocean model output are constructed using techniques…
This work aims to describe a mathematical model and a numerical method to simulate a thin anisotropic composite membrane moving and deforming in 3D space under a dynamic load of an arbitrary time and space profile. The model and the method…
Tropical polytopes are images of polytopes in an affine space over the Puiseux series field under the degree map. This viewpoint gives rise to a family of cellular resolutions of monomial ideals which generalize the hull complex of Bayer…
Underwater gliders offer effective means in oceanic surveys with a major task in reconstructing the three-dimensional hydrographic field of a mesoscale eddy. This paper considers three key issues in the hydrographic reconstruction of…
We consider optimization problems that are formulated and solved in the framework of tropical mathematics. The problems consist in minimizing or maximizing functionals defined on vectors of finite-dimensional semimodules over idempotent…
The advent of transformation optics has lead to the initiation of designing devices and applications associated to electromagnetic wave propagation in anisotropic media. Here, a method is suggested using a coordinate transformation with…
Methods were developed in Ref. [1] for constructing reference metrics (and from them differentiable structures) on three-dimensional manifolds with topologies specified by suitable triangulations. This note generalizes those methods by…
In this note I will explain how relative/log Gromov-Witten invariants of pairs $(X,D)$ with very ample smooth anticanonical divisor $D$ can be computed using algebro-combinatorial objects called scattering diagrams. The underlying principle…
This work introduces a novel technique for fabricating functional robots using 3D printers. Simultaneously depositing photopolymers and a non-curing liquid allows complex, pre-filled fluidic channels to be fabricated. This new printing…
This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are…
A method for creating 3D texture coordinates for a sequence of polygon meshes with changing topology and vertex motion vectors.
Visualization of implicit surfaces is an actively researched topic. While raytracing can produce high quality images, it is not well suited for creating a quick preview of the surface. Indirect algorithms (e.g. Marching Cubes) create an…
We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically…
Additive manufacturing, also known as 3D printing, is an advanced manufacturing technique that allows the fabrication of arbitrary macroscopic and microscopic objects. All 3D printing systems require large optical elements or nozzles in…
Modeling the geometry and the appearance of knitted fabrics has been challenging due to their complex geometries and interactions with light. Previous surface-based models have difficulties capturing fine-grained knit geometries;…
We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the…
With the emergence of data-driven approaches in science, there is growing interest in their application to manufacturing, particularly in surface precision engineering. However, generating large datasets required for model training is often…