Garsia and Milne's bijective proof of the inclusion-exclusion principle

Although the following proof is implicit in Garsia and Milne's paper [1], it is so elegant that we felt that it should be presented by itself for the benefit of the general mathematical public. The idea behind the proof was further exploited by Remmel [2] and Wilf [3].     

A simplified proof of parikh's theorem

A strengthened form of the pumping lemma for context-free languages is used to give a simple proof of Parikh's Theorem.     

A short proof of Kneser's conjecture

In the paper a short proof is given for Kneser's conjecture. The proof is based on Borsuk's theorem and on a theorem of Gale.     

A short proof of Meyniel's theorem

Meyniel's theorem states that a strict diconnected digraph has a directed Hamilton cycle if d(u) + d(v) ? 2n ? 1 for every pair u, v of nonadjacent vertices. We give short proof of this theorem.     

A short proof of Minc's conjecture

A short proof is given of the following conjecture of Minc, proved in 1973 by Brègman. Let A be a n × n ? (0, 1)-matrix with r_i ones in row i. Then per .     

A minimum principle for superharmonic functions subject to interface conditions

Let D be a bounded domain in R² with smooth boundary. Let B₁, …, B_m be non-intersecting smooth Jordan curves contained in D, and let D′ denote the complement of ∪_{i ? 1}^mB_i respect to D. Suppose that $\text{u ? C}{}^2\text{(D′) ∩ C(}\text{D}\text{?}$) and Δu ? 0 in D′ (where Δ is the Laplacian), while across each “interface” B_i, i = 1,…, m, there is “continuity of flux” (as suggested by the theory of heat conduction). It is proved here that the presence of the interfaces does not alter the conclusions of the classical minimum principle (for Δu ? 0 in D). The result is extended in several regards. Also it is applied to an elliptic free boundary problem and to the proof of uniqueness for steady-state heat conduction in a composite medium. Finally this minimum principle (which assumes “continuity of flux”) is compared with one due to Collatz and Werner which employs an alternative interface condition.     

Local indicators for plurisubharmonic functions

The notion of index, classical in number theory and its calculation by P. Lelong (1997) for plurisubharmonic functions, allows to define an indicator which is applied to the study of the Monge–Ampère operator and a pluricomplex Green function.     

Another proof of dilworth's decomposition theorem

Dilworth's famous theorem [1] states that if the maximal sized antichains of a finite poset X have n elements, then X can be covered by n chains. The number n is called the width of X. Apart from proofs relating the theorem to other key theorems of combinatorics (see [1–4]), there have been a number of direct proofs (see [1, 2, 5, 6]). The shortest of these is the one by Perles [5], the outline of which is as follows.     

A class of maximal plurisubharmonic functions

In this note, we introduce a class of maximal plurisubharmonic functions and use that class to prove some properties of maximal plurisubharmonics functions.     

A short proof of Chvátal's Watchman Theorem

This note contains a short proof of Chvátal's Watchman Theorem using the existence of a three-coloring of a triangulated polygon.     

Approximation of plurisubharmonic functions

We extend a result by Fornaæss and Wiegerinck [Ark. Mat. 1989;27:257–272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.     

A new proof of Cheung’s characterization of comonotonicity

It is well known that if a random vector with given marginal distributions is comonotonic, it has the largest sum in the sense of the convex order. Cheung (2008) proved that the converse of this assertion is also true, provided that all marginal distribution functions are continuous and that the underlying probability space is atomless. This continuity assumption on the marginals was removed by Cheung (2010). In this short note, we give a new and simple proof of Cheung’s result without the assumption that the underlying probability space is atomless.     

An elementary proof for the majorization principle for Wright-convex functions

We give another elementary proof for the majorization principle for Wright-convex functions. This inequality is due to Ng.