Automata in Groups and Dynamics and Induced Systems of PDE in Tropical Geometry

In this paper we develop a dynamical scaling limit from rational dynamics to automata in tropical geometry. We compare these dynamics and induce uniform estimates of their orbits. We apply these estimates to introduce a comparison analysis of theory of automata groups in geometric group theory with analysis of rational dynamics and some hyperbolic PDE systems. Frameworks of characteristic properties of automata groups are inherited to the corresponding rational or PDE dynamics. As an application we study the Burnside problem in group theory and translate the property as the infinite quasi-recursiveness in rational dynamics.     

Deformations of Functions Without Real Critical Points

Abstract

Given a real analytic function f(x, y) with one critical point P ₀, we study deformations f _t of fsuch that, for any t ≠ 0, the analytic function f _t has no critical points in a neighborhood of P ₀. We give explicitly a deformation without real critical points for any function which has only one real branch with characteristic exponents (4, 2q, r).     

Lower Bounds in Real Algebraic Geometry and Orientability of Real Toric Varieties

The real solutions to a system of sparse polynomial equations may be realized as a fiber of a projection map from a toric variety. When the toric variety is orientable, the degree of this map is a lower bound for the number of real solutions to the system of equations. We strengthen previous work by characterizing when the toric variety is orientable. This is based on work of Nakayama and Nishimura, who characterized the orientability of smooth real toric varieties.     

Geometry of Truncated Symmetric Products and Real Roots of Real Polynomials

We study the conditions for truncated symmetric products ofmanifolds to be manifolds. In particular, we show that suitablydefined spaces of systems of real roots of real polynomialsare homeomorphic to real projective spaces. 1991 MathematicsSubject Classification 57N99, 26C10.     

Geometry of Real Grassmann Manifolds. IV

Classical geodesic embeddings in the Grassmann manifold are described algebraically by means of the exterior algebra (V). The group I ₀(G ⁺) of isometries of the Grassmannian is described without the theory of Lie groups and algebras. Interior and exterior properties of Grassmann manifolds are proved with the help of the equivariant Plücker embedding. The isomorphism type of the group of twists of the Grassmann manifold around an arbitrary geodesic is described. Bibliography: 16 titles.     

On Affine Geometry of Purley Real Submanifolds

We propose to study n-dimensional purely real submanifolds of the standard affine complex space Cⁿ from an affine point of view. Such submanifolds are naturally endowed with a unique transversal bundle. Fundamental theorems are given as well as a theorem of Cartan–Norden type. Examples illustrating particular affine properties of such submanifolds are provided.     

Rigidity and moduli space in Real Algebraic Geometry

Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Z over R, the set of nonconstant rational maps from Z to X is finite. A bijective map from an affine algebraic variety over R to X is called weak change of the algebraic structure of X if it is regular and φ⁻¹ is a Nash map, which preserves nonsingular points. We prove the following rigidity theorem: every affine algebraic variety over R is UMR up to a weak change of its algebraic structure. Let us introduce another notion. Let Y be an affine algebraic variety over R. We say that X and Y are algebraically unfriendly if all the rational maps from X to Y and from Y to X are trivial, i.e., Zariski locally constant. From the preceding theorem, we infer that, if dim (X)≥1, then there exists a set of weak changes of the algebraic structure of X such that, for each t,s ∈ R with t≠s, and are algebraically unfriendly. This result implies the following expected fact: For each (nonsingular) affine algebraic variety X over R of positive dimension, the natural Nash structure of X does not determine the algebraic structure of X. In fact, the moduli space of birationally nonisomorphic (nonsingular) affine algebraic varieties over R, which are Nash isomorphic to X, has the same cardinality of R. This result was already known under the special assumption that R is the field of real numbers and X is compact and nonsingular. The author is a member of GNSAGA of CNR, partially supported by MURST and European Research Training Network RAAG 2002–2006 (HPRN–CT–00271).     

Topics on the Geometry of Rational Homogeneous Spaces

This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent role. This selection is largely arbitrary and mainly reflects the interests of the author.     

Spherical Tropical Geometry: a Survey of Recent Developments

This is a survey of some recent results on spherical tropical geometry.     

有理数、无理数与实数

本文用通俗的语言向大学一、二年级学生介绍有理数,无理数与实数的基本性质,旨在把抽象的概念与艰涩的证明转化成有趣易懂的概念与方法.     

Piecewise Rational Approximations of Real Algebraic Curves

1.IntroductionAnaJgebraicplanecurveCofdegreedinn2isimplicitlydefinedbyasinglepolynomialequationf(x,y)=Oofdegreedwithcoefficientsinn.Arationalalgebraiccurveofdegreedinn2canadditionaJlybedefinedbyrationalparametricequationswhicharegivenas(x=G1(u),y=G2(u)),whereG1andG2arerationalfunctionsinuofdegreed,i.e-,eachisaquotientofpolynomiaJ8inuofmtalmumdegreedwithcoefficientsinn.ffetionalcurvesaxeonlyasubsetofimplicitalgebraiccurvesofdegreed+1.Whi1eaJldegreetwocurves(conics)arerational,oIilyasubsetof…     

Dynamics of a Rational Difference Equation

The authors investigate the global behavior of the solutions of the difference equation xn＋1=axn-1xn-k/bxn-p＋cxn-q,n=0,1,…where the initial conditions x-r, x-r＋1, x-r＋2,… , x0 are arbitrary positive real numbers, r = max{l, k,p, q） is a nonnegative integer and a, b, c are positive constants. Some special cases of this equation are also studied in this paper.     

Integral Geometry of Real Surfaces in the Complex Projective Plane

We give a Poincaré formula for any real surfaces in the complex projective plane which states that the mean value of the intersection numbers of two real surfaces is equal to the integral of some terms of their Kähler angles.     

Nonlinear Dynamics on the Plane and Integrable Hierarchies of Infinitesimal Deformations

A class of nonlinear problems on the plane, described by nonlinear inhomogeneous     -equations, is considered. It is shown that the corresponding dynamics, generated by deformations of inhomogeneous terms (sources), is described by Hamilton–Jacobi-type equations associated with hierarchies of dispersionless integrable systems. These hierarchies are constructed by applying the quasiclassical     -dressing method.     

Rational and Homological Equivalence of Real Algebraic Cycles

In this paper we study algebraic cycles on smooth quasiprojective real algebraic varieties. We are interested under what conditions an algebraic cycle homologous to zero is rationally equivalent to a cycle supporting no real points.