A proof of Sethares'''' conjecture

Let (?)(z) be holomorphic in the unit disk △ and meromorphic on △. Suppose / is a Teichmuller mapping with complex dilatation In 1968, Sethares conjectured that f is extremal if and only if either (i)(?) has a double pole or (ii)(?) has no pole of order exceeding two on (?)△. The "if" part of the conjecture had been solved by himself. We will give the affirmative answer to the "only if" part of the conjecture. In addition, a more general criterion for extremality of quasiconformal mappings is constructed in this paper, which generalizes the "if" part of Sethares' conjecture and improves the result by Reich and Shapiro in 1990.     

A proof of snevily’s conjecture

We prove Snevily’s conjecture, which states that for any positive integer k and any two k-element subsets {a ₁, …, a _k } and {b ₁, …, b _k } of a finite abelian group of odd order there exists a permutation π ∈ S _k such that all sums a _i + b _π(i) (i ∈ [1, k]) are pairwise distinct.     

Students’ ways of understanding a proof

We present a model for describing the growth of students’ understandings when reading a proof. The model is composed of two main paths. One is focused on becoming aware of the deductive structure of the proof, in other words, understanding the proof at a semantic level. Generalization, abstraction, and formalization are the most important transitions in this path. The other path focuses on the surface-level form of the proof, and the use of symbolic representations. At the end of this path, students understand how and why symbolic computations formally establish a claim, at a syntactic level. We make distinctions between states in the model and illustrate them with examples from early secondary students’ mathematical activity. We then apply the model to one student’s developing understanding in order to show how the model works in practice. We close with some suggestions for further research.     

A contraction-mapping proof of Koenigs’ theorem

We give a simple, functional analytic proof of Koenigs’ theorem on the linearisation of a complex analytic function in a neighbourhood of a hyperbolic fixed point. The proof uses the contraction mapping principle in the nonlinearity norm.     

A proof of the Lyusternik–Graves theorem

In this paper, we give a new proof of the Lyusternik–Graves theorem, based on an intermediate result regarding linear openness inspired by works of Frankowska and Ursescu.     

An alternative proof of Lickorish–Wallace theorem

We introduce the concept of s-distance of an unstabilized Heegaard splitting. We prove if a 3-manifold admits an unstabilized genus g Heegaard splitting with s-distance m  , then surgery on some (m−1)$(m-1)$ components link may produce a 3-manifold which admits a stabilized genus g Heegaard splitting. We also give an alternative proof of the fundamental theorem of surgery theory, which states that every closed orientable 3-manifold is obtained by surgery on some link in 3-sphere.     

New proof of a Calabi’s theorem

A Calabi’s theorem says that on a compact Riemann surface, an extremal metric is a constant scalar curvature metric. In this paper, we use a new method to prove this theorem. Then we give an interesting corollary.     

A short proof of Levinson’s theorem

We give a short proof of Levinson’s result that over 1/3 of the zeros of the Riemann zeta function are on the critical line.     

A simple proof of Olatinwo’s theorems

In this paper we provide a simple proof of the existence coupled fixed point theorem in complete cone metric spaces due to Sabetghadam et al. (Fixed Point Theory Appl 2009:8, 2009) and due to Olatinwo (Annali Dell’Universita’Di Ferrara 57:173–180, 2011). In particular we prove that these results are spacial cases of Rezapour and Hamlbarani’s theorems (J Math Anal Appl 345(2):719–724, 2008).     

An elementary proof of Tverberg’s theorem

A new proof is suggested for Tverberg’s familiar theorem saying that an arbitrary set of q: (d + 1)(p − 1) + 1 points in ℝ ^d can be split into p parts such that their convex hulls have a nonempty intersection. Bibliography: 9 titles.     

A direct proof of Gromov’s theorem

A new proof of a theorem by Gromov is given: for any positive C and any integer n greater than 1, there exists a function Δ _C,n (δ) such that if the Gromov–Hausdorff distance between two complete Riemannian n-manifolds V and W is at most δ, their sectional curvaturcs |K _σ | do not exceed C, and their injectivity radii are at least 1/C, than the Lipschitz distance between V and W is less than Δ _C,n (δ), and Δ _C,n (δ) → 0 as δ → 0. Bibliography: 6 titles.     

On Stöhr-Voloch's proof of Weil's theorem

LetX be a smooth, projective curve of genusg defined over a finite field. Then there exists a finite extension fieldk=F _q such that, for all sufficiently largen, (1)X imbeds in P ⁿ in such a way that the set of pointsx ofX mapped via Frobenius into the osculating hyperplane ofX atx is the support of a positive divisor and (2) the numberN of points inX(k) satisfies

Yet another proof of Marstrand’s theorem

In a paper from 1954 Marstrand proved that if K ⊂ ℝ² is a Borel set with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we give a combinatorial proof of this theorem, extending the techniques developed in our previous paper [9].     

An alternative proof of Painlevé's theorem

In this article we show some aspects of analytical and numerical solution of the n-body problem, which arises from the classical Newtonian model for gravitation attraction. We prove the non-existence of stationary solutions and give an alternative proof for Painlevé's theorem.     

A motivated proof of Gordon’s identities

We generalize the ??motivated proof?? of the Rogers?CRamanujan identities given by Andrews and Baxter to provide an analogous ??motivated proof?? of Gordon??s generalization of the Rogers?CRamanujan identities. Our main purpose is to provide insight into certain vertex-algebraic structure being developed.     

An elementary proof of the Loomis–Whitney theorem

Loomis and Whitney proved an inequality between volume and areas of projections of an open set in n-dimensional space related to the isoperimetric inequality. They reduced the problem to a combinatorial theorem proved by a repeated use of Hölder inequality. In this paper we prove a general inequality between real numbers which easily implies the combinatorial theorem of Loomis and Whitney.

     

A short proof of the Doob–Meyer theorem

Every submartingale S$S$ of class D$D$ has a unique Doob–Meyer decomposition S=M+A$S=M+A$, where M$M$ is a martingale and A$A$ is a predictable increasing process starting at 0.