Related papers: Real solutions to equations from geometry

Modeling of physical systems includes extensive use of software packages that implement the accurate finite element method for solving differential equations considered along with the appropriate initial and boundary conditions. When the…

Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…

The notion of an equational shell is studied to involve the objects and their environment. Appropriate methods are studied as valid embeddings of refined objects. The refinement process determines the linkages between the variety of…

There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions…

Geometrical inequalities show how certain parameters of a physical system set restrictions on other parameters. For instance, a black hole of given mass can not rotate too fast, or an ordinary object of given size can not have too much…

Answering connectivity queries in real algebraic sets is a fundamental problem in effective real algebraic geometry that finds many applications in e.g. robotics where motion planning issues are topical. This computational problem is…

Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…

Continuum equations are ubiquitous in physical modelling of elastic, viscous, and viscoelastic systems. The equations of continuum mechanics take nontrivial forms on curved surfaces. Although the curved surface formulation of the continuum…

We study geometry and topology as complementary and dual aspects of the mathematical space. The same is used to get a better understanding of the Cosmological Constant. Having failed so far to include gravity in a proper unified framework…

We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.

Geometrization of physical theories have always played an important role in their analysis and development. In this contribution we discuss various aspects concerning the geometrization of physical theories: from classical mechanics to…

In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts…

Geometric problems are usually formulated by means of (exterior) differential systems. In this theory, one enriches the system by adding algebraic and differential constraints, and then looks for regular solutions. Here we adopt a dual…

This note is a survey on the topology of hyperplane arrangements. We mainly focus on the relationship between topology and the real structure, such as adjacent relations of chambers and stratifications related to real structures.

This article demonstrates how an understanding of the geometry of a family of cost functions can be used to develop efficient numerical algorithms for real-time optimisation. Crucially, it is not the geometry of the individual functions…

The reasons of the crisis in the contemporary (Riemannian) geometry are discussed. The conventional method of the generalized geometries construction, based on a use of the topology, leads to a overdetermination of the Riemannian geometry.…

Several problems which could be thought of as belonging to recreational mathematics are described. They are all such that solutions to the problem depend on finding rational points on elliptic curves. Many of the problems considered lead to…

It is shown how the dimension of any arbitrary over-determined system of differential equations can be reduced, which makes the system suitable for numerical solution modeling. Specifically, over-determined equations of hydrodynamics are…

We discuss and compare several geometric structures which imply an upper bound to the acceleration of a particle measured in its rest system. While all of them have the same implications on the motion of a point particle, they differ in…

The introduction of automated deduction systems in secondary schools face several bottlenecks, the absence of the subject of rigorous mathematical demonstrations in the curricula, the lack of knowledge by the teachers about the subject and…