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On triangles determined by subsets of the Euclidean plane, the associated bilinear operators and applications to discrete geometry
triangles determined Euclidean plane associated bilinear operators discrete geometry
2010/12/6
We prove that if the Hausdorff dimension of the set E R2 is greater than 74 , then the three-dimensional Lebesgue measure of the set of triangles determined by E is positive. We
establish this fact...
Cyclic polygons in classical geometry
cyclic polygon Euclidean geometry hyperbolic geometry spherical geometry
2010/12/7
Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.
A result of [2] extends the Marsden-Ratiu reduction theorem [4] in Poisson geometry, and is proven by means of graded geometry. In this note we provide the background material about graded geometry ne...
Geometry of Darboux-Manakov-Zakharov systems and its application
Darboux-Manakov-Zakharov systems Geometry
2010/4/2
The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov systems of semilinear partial differential equations \label{GDMZabstract} \frac{\partial^2 u}{\partial x_i\partial x_j}=f_{ij...
An overview of some recent results on geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space...
Recent Development on the Geometry of the Teichuller and Moduli Spaces of Riemann Surfaces and Polarized Calabi-Yau Manifolds
Geometry of the Teichuller Moduli Spaces Riemann Surfaces Polarized Calabi-Yau Manifolds
2018/4/19
The moduli spaceMg,k of Riemann surfaces of genus g with k punctures plays an important role in many area of mathematics and theoretical physics.In this article we first survey some of our recent work...
Integrable GL(2) Geometry and Hydrodynamic Partial Differential Equations
Integrable GL(2) Geometry Hydrodynamic Partial Differential Equations
2010/4/7
This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). ...
Visually Dynamic Presentation of Proofs in Plane Geometry Part 1. Basic Features and the Manual Input Method
Geometry theorem proving Visually dynamic presentation of proof Dynamic geometry Unordered geometry Gelernter Java Geometry Expert Morley’s theorem Feuerbach’s theorem Pythagorean theorem Pedal triangle
2013/9/9
With dynamic mediums such as computer displays, we propose a new kind of visually dynamic presentation of proofs in plane geometry. In asingle diagram for the proof, when the proof text goes on step b...
Visually Dynamic Presentation of Proofs in Plane Geometry
Geometry theorem proving Proof with hierarchical structures Visually dynamic presentation of proof Dynamic geometry Full-angle Deductive database Fixpoint Unordered geometry Hilbert geometry Tarski geometry
2013/9/9
We present the method for automated generation of visually dynamic presentations of plane geometry proofs based on the full-angle method. The proof generated by the full-angle method is organized hier...
In this paper we briefly survey our recent results about the Mumford goodness of several canonical metrics on the moduli spaces of Riemann surfaces. We will also prove the dual Nakano negativity of th...
Annals of Global Analysis Geometry
Annals Geometry
2008/1/14
The Annals of Global Analysis and Geometry contribute to an enlargement of the international exchange of research results in the fields of global analysis and geometry. The journal treats, in particul...
THE GEOMETRY OF HIGHER PATH-SPACES
2007/12/13
<正> IntroductionThe geometry in a space R_n(x~1,…, x~n)in whit.h there is givena system of differential equations of the r-th order:(?)has recently been studied by various writers.It seems to me thatt...
<正> INTRODUGTION.1.In the proiective theory of a curve in ordinary space thereare introduced various systems of tetrahedrons associated to a pointof the curve.Most of them are,more or less,artificial ...
<正> In a recent paper the present author has established somerepresentations for a plane curve C in the neighbourhood of a singularpoint O characterized by the property that the tangent t(?) of C at O...
PROJECTIVE DIFFERENTIAL GEOMETRY
2007/12/12
<正> By hypothesis the curve C_m has with C a contact of order m+1;the constant term and the coefficient of the term s in the expansion off/s~m must vanish,namely……