We prove that the Reeb space of a proper definable map in an arbitrary o-minimal expansion of the reals is realizable as a proper definable quotient. We also show that the Betti numbers of the Reeb space of a map f can be arbitrarily large compared to those of X, unlike in the special case of Reeb graphs of manifolds. Nevertheless, in the special case when f : X → Y is a semi-algebraic map and ...

Lie Algebraic methodsl implemented with truncated power series algebra packages2 or libraries,3 have become important in nonlinear single-particle dynamics studies.4 Algorithms for nonlinear normal forms5 and one-turn map tracking, such as kick factorization and integrable polynomial factorization,7 have been becoming common numerical practices. Much progress has been made in the one-turn map t...

‎This paper extends the scope of algebraic computation based on a non standard $times$ to ‎the more basic case of a non standard $+$‎, ‎where standard means associative ‎and commutative‎. ‎Two physically meaningful examples of a non standard $+$ are ‎provided by the observation of motion in Special Relativity‎, ‎from either ‎outside (3D) or inside (2D or more)‎, ‎We revisit the ``gyro''-theory ...

We discuss recent progress in the study of the space of harmonic maps from the 2sphere to the unit n-sphere in Euclidean (n+1)-space. We consider the structure of this space as an algebraic variety, the existence of non-manifold points in this space, and the relation between this question and the integrability of Jacobi fields along harmonic maps. One of the main tools used is that of the twist...

In this paper, a vector version of the intermediate value theorem is established. The main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{On fixed point theorems for monotone increasing vector valued mappings via scalarizing}, Positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

The theorem of Ax says that any regular selfmapping of a complex algebraic variety is either surjective or non-injective; this property is called surjunctivity and investigated in the present paper in the category of proregular mappings of proalgebraic spaces. We show that such maps are surjunctive if they commute with sufficiently large automorphism groups. Of particular interest is the case o...

An algebraic characterization of monads which are abstract partial map classi$ers is provided, without the assumption that the categories of total maps possess products. By an abstract partial map classi$er we mean a monad whose Kleisli category is a full subcategory of a partial map category wherein the induced comonad classi$es partial maps in the usual sense. A construction of the correspond...

Abstract Let $X \subset \mathbb{C}^{n}$ be an algebraic variety, and let $\Lambda a discrete subgroup whose real complex spans agree. We describe the topological closure of image $X$ in $\mathbb{C}^{n} / \Lambda $, thereby extending result Peterzil–Starchenko case when $ is cocompact. also obtain similar extension $X\subset \mathbb{R}^{n}$ definable o-minimal structure with no restrictions on a...

We will investigate the stability behavior of quadratic maps in higher dimensions. To check stability, we will use infinitesimal V -stability of critical points of the map; since the infinitesimal V -stability of a map at all of its critical points is equivalent to the stability of the map. We will establish the connection between infinitesimal V-stability of solutions to the Algebraic Riccati ...

What is tropical geometry about? Back in 2005 an influential paper by Richter-Gebert, Sturmfels and Theobald [11] answered that question in the following way: “Tropical algebraic geometry is the geometry of the tropical semiring (R,min,+). Its objects are polyhedral cell complexes which behave like complex algebraic varieties.” Let us look at plane algebraic curves and their tropicalizations to...