Hurwitz' theorem implies Rouché's theorem

It is well known that Hurwitz's theorem is easily proved from Rouché's theorem. We show that conversely, Rouché's theorem is readily proved from Hurwitz' theorem. Since Hurwitz' theorem is easily proved from the formula giving the number of roots of an analytic function, our result thus gives also a simple proof of Rouché's theorem.     

Position dependent and stochastic thinning of point processes

A short probabilistic proof of Kallenberg's theorem [2] on thinning of point processes is given. It is extended to the case where the probability of deletion of a point depends on the position of the point and is itself random. The proof also leads easily to a statement about the rate of convergence in Renyi's theorem on thinning a renewal process.     

Hypergraphs and Whitney's theorem on edge-isomorphisms of graphs

This paper presents a new proof of Whitney's theorem on edge-isomorphisms of graphs and extends Whitney's theorem to hypergraphs. Whitney's theorem asserts that any two edge-isomorphic graphs of order at least 5 have their edge-isomorphism induced by a node-isomorphism isomorphism. Previous results of Gardner and of Berge and Rado are used.     

Note on a theorem of J. Folkman on transversals of infinite families with finitely many infinite members

In this note we show by a simple direct proof that Folkman's necessary and sufficient condition for an infinite family of sets with finitely many infinite members to have a transversal implies Woodall's condition. A short proof of Folkman's theorem results by combining with Woodall's proof.     

Planarity and duality of finite and infinite graphs

We present a short proof of the following theorems simultaneously: Kuratowski's theorem, Fary's theorem, and the theorem of Tutte that every 3-connected planar graph has a convex representation. We stress the importance of Kuratowski's theorem by showing how it implies a result of Tutte on planar representations with prescribed vertices on the same facial cycle as well as the planarity criteria of Whitney, MacLane, Tutte, and Fournier (in the case of Whitney's theorem and MacLane's theorem this has already been done by Tutte). In connection with Tutte's planarity criterion in terms of non-separating cycles we give a short proof of the result of Tutte that the induced non-separating cycles in a 3-connected graph generate the cycle space. We consider each of the above-mentioned planarity criteria for infinite graphs. Specifically, we prove that Tutte's condition in terms of overlap graphs is equivalent to Kuratowski's condition, we characterize completely the infinite graphs satisfying MacLane's condition and we prove that the 3-connected locally finite ones have convex representations. We investigate when an infinite graph has a dual graph and we settle this problem completely in the locally finite case. We show by examples that Tutte's criterion involving non-separating cycles has no immediate extension to infinite graphs, but we present some analogues of that criterion for special classes of infinite graphs.     

On pairs of multidimensional matrices

Recently Bapat applied a topological theorem of Kronecker and generalized a theorem of Sinkhorn on positive matrices. Here we give an alternative proof of a slightly stronger version of his generalization. This proof combines Kakutani's fixed point theorem and the duality theorem of linear programming and gives yet another proof of a theorem of Bacharach and Menon on pairs of nonnegative matrices.     

A simple proof that every matroid is an intersection of fundamental transversal matroids

We give a new proof of a theorem of Bondy and Welsh. Our proof is simpler than previous ones in that it makes no use of Hall's theorem on the existence of a transversal of a family of sets.     

A generalization of theorems of Krasnoselskii and Juberg

In this paper, we use the author's diagram proof of Riesz's theorem together with special seminorms to show that two theorems proved by the author generalize a compact interpolation theorem of Krasnoselskii, and a compact interpolation theorem of Juberg.     

An elementary proof of the Petersson-Knopp theorem on Dedekind sums

A brief and elementary proof of Petersson and Knopp's recent theorem on Dedekind sums is given. The proof is based on a result of Subrahmanyam.     

Analytic proofs of a network feasibility theorem and a theorem of Fulkerson

We provide an elementary proff of Fulkerson's theorem which gives the permutation matrices as extreme points of a certain unbounded convex polyhedron. An adaptation of the proof also establishes an analogous feasibility theorem for network flows which has Fulkerson's theorem as a corollary.     

Permutations and riemann surfaces

Another combinatorial proof of a theorem of Ree's on permutations is offered. This proof makes use of the notion of the genus of pair of permutations.     

A new proof of Kiselman's minimum principle for plurisubharmonic functions

We give a new proof of Kiselman's minimum principle for plurisubharmonic functions, inspired by Demailly's regularization of plurisubharmonic functions by using Ohsawa–Takegoshi's extension theorem.     

On constructivity and the Rosser property: a closer look at some Gödelean proofs

The proofs of Kleene, Chaitin and Boolos for Gödel's First Incompleteness Theorem are studied from the perspectives of constructivity and the Rosser property. A proof of the incompleteness theorem has the Rosser property when the independence of the true but unprovable sentence can be shown by assuming only the (simple) consistency of the theory. It is known that Gödel's own proof for his incompleteness theorem does not have the Rosser property, and we show that neither do Kleene's or Boolos' proofs. However, we show that a variant of Chaitin's proof can have the Rosser property. The proofs of Gödel, Rosser and Kleene are constructive in the sense that they explicitly construct, by algorithmic ways, the independent sentence(s) from the theory. We show that the proofs of Chaitin and Boolos are not constructive, and they prove only the mere existence of the independent sentences.     

Hamiltonian paths in oriented graphs

A short proof is given of Meyniel's theorem on Hamiltonian cycles in oriented graphs. Analogous conditions are obtained for a graph to be Hamiltonianconnected.     

An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space

J. B. Baillon [C. R. Acad. Sci. Paris Ser. A.280 (1975), 1511–1514] proved an ergodic theorem for a single nonexpansive mapping in a Hilbert space, which is a nonlinear version of von Neumann's mean ergodic theorem. In this paper, we study the ergodic behavior of a semigroup of nonexpansive mappings. We try to find a sequence of means on the semigroup, generalizing the Cesàro means on $\text{N}$, such that the corresponding sequence of nonexpansive mappings converges to a projection onto the set of common fixed-points. Our method of proof is an appropriate modification of A. Pazy's proof [Israel J. Math.26 (1977), 197–204] of Baillon's theorem.     

Remark on Wielandt's visibility theorem

A short proof is given of Wielandt's visibility theorem, using a special case of a theorem of Rédei, which was proved in an elementary way by Lóvasz and Schriver.     

A remark on Burnside's theorem on matrix algebras

We present a simple proof of Burnside's theorem using techniques from elementary linear algebra.     

A subspace theorem approach to integral points on curvesPoints entiers sur les courbes et théorème des sous-espaces

We present a proof of Siegel's theorem on integral points on affine curves, through the Schmidt subspace theorem, rather than Roth's theorem. This approach allows one to work only on curves, avoiding the embedding into Jacobians and the subsequent use of tools from the arithmetic of Abelian varieties. To cite this article: P. Corvaja, U. Zannier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 267–271.     

Matrix Tree Theorems

A simple proof of a directed graph generalization of the Matrix Tree Theorem, sometimes called Maxwell's rule or Kirchhoff's rule, is given. It is based on the idea A. Rényi used to prove Cayley's tree counting formula. The theorem counts rooted arborescences (analogs of forests) in a directed graph with the determinant of a submatrix in a special adjacency matrix. In the proof we show two n-k degree homogeneous polynomials in n variables are equal by applying induction to those terms lacking one variable. An application to a well-known identity and related theorems are given.     

Some further remarks on genus field

Gurak's characterization of the relative (narrow) genus field of a normal extension is extended to any finite extension. Using this, a purely algebraic proof for a theorem of the author on the (narrow) genus field is given.