相关论文: Real solutions to equations from geometry

The relationship between micro-structure and macro-structure of complex systems using information geometry has been dealt by several authors. From this perspective, we are going to apply it as a geometrical structure connecting both…

Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous…

How to understand deep learning systems remains an open problem. In this paper we propose that the answer may lie in the geometrization of deep networks. Geometrization is a bridge to connect physics, geometry, deep network and quantum…

Given an orientable ideally triangulated $3$--manifold $M$, we define a system of real valued equations and inequalities whose solutions can be used to construct projective structures on $M$. These equations represent a unifying framework…

This paper wants to show how practical geometry, created to give a concrete help to people involved in trade, in land-surveying and even in astronomy, underwent a transformation that underlined its didactical value and turned it first into…

Purely real space versions of the differential equations describing the kinematics of a dislocated crystalline medium are considered. The differential geometric structures associated with them are revealed.

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…

Real algebraic geometry is the study of semi-algebraic sets, subsets of $\R^k$ defined by Boolean combinations of polynomial equalities and inequalities. The focus of this thesis is to study quantitative results in real algebraic geometry,…

We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…

Existing structural analysis methods may fail to find all hidden constraints for a system of differential-algebraic equations with parameters if the system is structurally unamenable for certain values of the parameters. In this paper, for…

The unsatisfactory status of the search for a consistent and predictive quantization of gravity is taken as motivation to study the question whether geometrical laws could be more fundamental than quantization procedures. In such an…

Regular resolution is a refinement of the resolution proof system requiring that no variable be resolved on more than once along any path in the proof. It is known that there exist sequences of formulas that require exponential-size proofs…

A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real…

Chapters 1 to 4 are the lecture notes of my course "Real Algebraic Geometry I" from the winter term 2020/2021. Chapters 5 to 8 are the lecture notes of its continuation "Real Algebraic Geometry II" from the summer term 2021. Chapters 9 and…

This work gathers new results concerning the semi-geostrophic equations: existence and stability of measure valued solutions, existence and uniqueness of solutions under certain continuity conditions for the density, convergence to the…

We discuss the problem of whether a given problem in enumerative geometry can have all of its solutions be real. In particular, we describe an approach to problems of this type, and show how this can be used to show some enumerative…

It is shown the construction of a module structure [2] with universe over a set of a particular kind of mathematical proofs, the base ring of this module will be built on a maximal consistent extension of a set of propositions, this…

Optimisation algorithms are commonly compared on benchmarks to get insight into performance differences. However, it is not clear how closely benchmarks match the properties of real-world problems because these properties are largely…

We study nodal lines of solutions to the heat equations. We are interested in the global geometry of nodal sets, in the whole domain of definition of the solution. The local structure of nodal sets is a well understander subject, while the…

Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…