An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions

We present a new, easy, and elementary proof of Jensen’s Theorem on the uniqueness of infinity harmonic functions. The idea is to pass to a finite difference equation by taking maximums and minimums over small balls.     

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.     

A note on the uniqueness theorem of meromorphic mappings

In this article, some uniqueness theorems of meromorphic mappings are proved.     

A counting proof of the Graham–Pollak Theorem

We give a counting based proof of the Graham–Pollak Theorem.     

A tropical proof of the Brill–Noether Theorem

We produce Brill–Noether general graphs in every genus, confirming a conjecture of Baker and giving a new proof of the Brill–Noether Theorem, due to Griffiths and Harris, over any algebraically closed field.     

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.     

A simple proof of the Rogers–Selberg identities

A short and simple proof of the Rogers–Selberg identities is obtained from the work of Andrew Sills on identities of the Rogers–Ramanujan type.     

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 simple proof of the generalized Craig–Sakamoto theorem

The Craig–Sakamoto theorem establishes the independence of two quadratic forms in normal variates. In this article, we provide a simple proof of a generalized Craig–Sakamoto theorem.     

A new topological proof of the Abhyankar–Moh theorem

We present a new simple proof of the famous theorem of Abhyankar, Moh and Suzuki about rational curves in a plane. This proof relies on the Poincaré–Hopf theorem. This work was supported by Polish KBN Grant No 2 P03A 041 15Mathematics Subject Classification (2002):14E25.     

A proof of the basic measure extension theorem of Carathéodory

In this note a direct elementary proof of Carathéodory's measure extension theorem is presented. It is based on an approximation argument for outer measures where elements of the -algebra are approached by elements of the underlying algebra of sets with respect to the symmetric difference. Received: 3 April 2000 / Accepted: 20 September 2000     

A uniqueness property for the quantization of Teichmüller Spaces

Chekhov, Fock and Kashaev introduced a quantization of the Teichmüller space of a punctured surface S, and an exponential version of this construction was developed by Bonahon and Liu. The construction of crucially depends on certain coordinate change isomorphisms between the Chekhov–Fock algebras associated to different ideal triangulations of S. We show that these coordinate change isomorphisms are essentially unique, once we require them to satisfy a certain number of natural conditions.      

The geometry of manipulation — A quantitative proof of the Gibbard-Satterthwaite theorem

We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show that a uniformly chosen voter profile for a neutral social choice function f of q ?? 4 alternatives and n voters will be manipulable with probability at least 10^?4??² n ^?3 q ^?30, where ?? is the minimal statistical distance between f and the family of dictator functions. Our results extend those of [11], which were obtained for the case of 3 alternatives, and imply that the approach of masking manipulations behind computational hardness (as considered in [4,6,9,15,7]) cannot hide manipulations completely. Our proof is geometric. More specifically it extends the method of canonical paths to show that the measure of the profiles that lie on the interface of 3 or more outcomes is large. To the best of our knowledge our result is the first isoperimetric result to establish interface of more than two bodies.     

A proof of Terras’ conjecture on the radius of convergence of the Ihara zeta function

We prove the conjecture by Audrey Terras (2011) on the inequalities among the spectral radius ${\rho }_X$ of a finite graph $X$, the radius of convergence $R$ of its Ihara zeta function $Z_X\left(u\right)$, and the average degree $\overline{d_X}$ of $X$. Relating to Terras’ conjecture, we propose a new conjecture between $R$ and certain Rayleigh quotients.     

A short proof of the generalized Faà di bruno's formula

A short proof of the generalized Faa di Bruno formula is given and an explicit parametrization of the set of indices involved in the coefficient of a specific term of the formula is provided. An application is also included.