6.3 HL Triangle Congruence
... so that AB = 6 cm. He used the software’s compass tool to construct a circle centered at point A with radius 3 cm. Based on this construction, is there a unique △ABC with m∠ABC = 20°, AB = 6 cm, and AC = 3 cm? Explain. ...
... so that AB = 6 cm. He used the software’s compass tool to construct a circle centered at point A with radius 3 cm. Based on this construction, is there a unique △ABC with m∠ABC = 20°, AB = 6 cm, and AC = 3 cm? Explain. ...
Convex Sets and Convex Functions on Complete Manifolds
... The aim of this paper is to show how convex sets and functions give strong restrictions to the topology of a certain class of complete Riemannian manifolds without boundary. The idea of convexity plays an essential role for the proofs of "finiteness theorems", which give a priori estimates for the n ...
... The aim of this paper is to show how convex sets and functions give strong restrictions to the topology of a certain class of complete Riemannian manifolds without boundary. The idea of convexity plays an essential role for the proofs of "finiteness theorems", which give a priori estimates for the n ...
THE EULER CLASS OF A SUBSET COMPLEX 1. Introduction Let G
... In the rest of the paper, we consider abelian p-groups. First we show that ζG is zero when G is isomorphic to Z/8 or (Z/4)2 ×Z/2, and hence conclude that any group which has a subquotient isomorphic to one of these groups has zero Euler class. This shows that if G is a 2-group with ζG 6= 0, then G i ...
... In the rest of the paper, we consider abelian p-groups. First we show that ζG is zero when G is isomorphic to Z/8 or (Z/4)2 ×Z/2, and hence conclude that any group which has a subquotient isomorphic to one of these groups has zero Euler class. This shows that if G is a 2-group with ζG 6= 0, then G i ...
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings.Initially proved as Riemann's inequality by Riemann (1857), the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student Gustav Roch (1865). It was later generalized to algebraic curves, to higher-dimensional varieties and beyond.