Related papers: Real solutions to equations from geometry

We discuss fun problems, vaguely related to notions and theorems of a course in differential geometry. This paper can be regarded as a weekend "treasure chest" supplementing the course weekday lecture notes. The problems and solutions are…

We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.

The concept of number and its generalization has played a central role in the development of mathematics over many centuries and many civilizations. Noteworthy milestones in this long and arduous process were the developments of the real…

We apply symmetry and invariance methods to analyse systems of difference equations. Non trivial symmetries are derived and their exact solutions obtained.

In black hole physics, inflationary cosmology, and quantum field theories, it is conjectured that the physical laws are subject to radical changes below the Planck length. Such changes are due to effects of quantum gravity believed to…

Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is…

A class of two-dimensional systems of second-order ordinary differential equations is identified in which a system requires fewer Lie point symmetries than required to solve it. The procedure distinguishes among those which are…

Given a single (differential-algebraic) input-output equation, we present a method for finding different representations of the associated system in the form of rational realizations; these are dynamical systems with rational right-hand…

This is a survey on formality results relying on weight structures. A weight structure is a naturally occurring grading on certain differential graded algebras. If this weight satisfies a purity property, one can deduce formality. Algebraic…

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous…

Given a functional for a one-dimensional physical system, a classical problem is to minimize it by finding stationary solutions and then checking the positive definiteness of the second variation. Establishing the positive definiteness is,…

This is a survey of the theory of real trees and their applications.

This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…

This is not a new result. Purpose of this work is to describe a method to search the analytical expression of the general real solution of the two-dimensional Laplace differential equation. This thing is not easy to find in scientific…

A powerful method for solving non-linear first-order ordinary differential equations, which is based on geometrical understanding of the corresponding dynamics of the so called Lie systems, is developed. This method allows us not only to…

This is a study of a problem in geodesy with methods from complex algebraic geometry: for a fixed number of measure points and target points at unknown position in the Euclidean plane, we study the problem of determining their relative…

The author has been interested in regions surrounded by cylinders of real algebraic hypersurfaces and their shapes and polynomials associated to them. Here, we formulate and investigate natural decompositions into such cylinders of real…

We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger…

We study orbit-finite systems of linear equations, in the setting of sets with atoms. Our principal contribution is a decision procedure for solvability of such systems. The procedure works for every field (and even commutative ring) under…

The unprecedented prowess of measurement techniques provides a detailed, multi-scale look into the depths of living systems. Understanding these avalanches of high-dimensional data -- by distilling underlying principles and mechanisms --…