相关论文: Simple proofing of Jordan's theorem
We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of elliptic ruled surfaces. This gives a positive answer to a question of Vladimir L. Popov.
We introduce the notion of a free associative $\mathcal{Z}_2$-algebra on the union of two disjoint sets and prove a generalization of Cohn's Theorem on Jordan algebras.
A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.
Proofs that a smooth morphism is flat available in the literature are long and difficult. We give a short proof of this fact.
This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…
In this note we give two proofs of Brooks' Theorem. The first is obtained by modifying an earlier proof and the second by combining two earlier proofs. We believe these proofs are easier to teach in Computer Science courses.
In this paper we provide a proof of the Carleson $\varepsilon^2$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated…
In the early 1960s, Brown and Mazur proved the general Jordan-Schoenflies theorem. This fundamental theorem states: If we embed an $(n-1)$ sphere $S^{(n-1)}$ locally flatly in an $n$ sphere $S^{n}$, then it decomposes $S^{n}$ into two…
We give a remarkably elementary proof of the Brouwer fixed point theorem. The proof is verifiable for most of the mathematicians.
In this paper we prove that if $\gamma$ is a Jordan curve on $\mathbb{S}^2$ then there is a smooth curve shortening flow defined on $(0,T)$ which converges to $\gamma$ in $\mathcal{C}^0$ as $t\to 0^+ $. Another perspective is that the…
An elementary proof of Bertrand's theorem is given by examining the radial orbit equation, without needing to solve complicated equations or integrals.
In this paper, we mainly study the derivation algebras of semi-simple Jordan algebras over a field of characteristic $0$ and give sufficient and necessary conditions that the derivation algebras of them are simple. As an application, we…
We present a short proof, relaying on the divergence theorem, verifying that minimal sets in the plane are trivial.
Most of the assertions in the theory of well ordered sets are quite simple. However, one of its central statements, Zermelo's theorem, stands out of this rule, for its well-known proofs are rather complicated. The aim of the current paper…
An explicit upper bound is derived for the modulus of divided difference for a smooth(not necessarily analytic) function defined on a smooth Jordan arc (or a smooth Jordan curve) in the complex plane. As an immediate application, an error…
We expose here a short proof of Cramer's theorem in R based on convex duality.
This paper has two main goals. First, we give a complete, explicit, and computable solution to the problem of when two simple closed curves on a surface are equivalent under the Johnson kernel. Second, we show that the Johnson filtration…
We prove that the closure of every Jordan class J in a semisimple simply connected complex group G at a point x with Jordan decomposition x = rv is smoothly equivalent to the union of closures of those Jordan classes in the centraliser of r…
A proof of Petri's general conjecture on the unobstructedness of linear systems on a general curve is proposed, using only the local properties of the deformation space of the pair (curve, line bundle).
We examine a class of geometric theorems on cyclic 2n-gons. We prove that if we take n disjoint pairs of sides, each pair separated by an even number of polygon sides, then there is a linear combination of the angles between those sides…