Class numbers of cyclic 2-extensions and Gross conjecture over ℚ

The Gross conjecture over ? was first claimed by Aoki in 1991. However, the original proof contains too many mistakes and false claims to be considered as a serious proof. This paper is an attempt to find a sound proof of the Gross conjecture under the outline of Aoki. We reduce the conjecture to an elementary conjecture concerning the class numbers of cyclic 2-extensions of ?.     

Class numbers of cyclic 2-extensions and Gross conjecture over Q

The Gross conjecture over Q was first claimed by Aoki in 1991.However,the original proof contains too many mistakes and false claims to be considered as a serious proof.This paper is an attempt to find a sound proof of the Gross conjecture under the outline of Aoki.We reduce the conjecture to an elementary conjecture concerning the class numbers of cyclic 2-extensions of Q.     

Artin's primitive root conjecture for function fields revisited

Artin's primitive root conjecture for function fields was proved by Bilharz in his thesis in 1937, conditionally on the proof of the Riemann hypothesis for function fields over finite fields, which was proved later by Weil in 1948. In this paper, we provide a simple proof of Artin's primitive root conjecture for function fields which does not use the Riemann hypothesis for function fields but rather modifies the classical argument of Hadamard and de la Vallée Poussin in their 1896 proof of the prime number theorem.     

A simplified proof of the André-Oort conjecture for products of modular curves

In this paper we give a short proof of the André-Oort conjecture for products of modular curves under the Generalised Riemann Hypothesis using only simple Galois-theoretic and geometric arguments. We believe this method represents a strategy for proving the conjecture for a general Shimura variety under GRH without using ergodic theory. We also demonstrate a short proof of the Manin–Mumford conjecture for Abelian varieties using similar arguments.     

Generalized Kneser coloring theorems with combinatorial proofs

The Kneser conjecture (1955) was proved by Lovász (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matoušek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol’nikov, Alon-Frankl-Lovász, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver’s theorem. Oblatum 17-IV-2001 & 12-IX-2001?Published online: 19 November 2001 An erratum to this article is available at .     

Nakai's conjecture for varieties smoothed by normalization

The notion of D-simplicity is used to give a short proof that varieties whose normalization is smooth satisfy Ishibashi's extension of Nakai's conjecture to arbitrary characteristic. This gives a new proof of Nakai's conjecture for curves and Stanley-Reisner rings.

     

Counter-examples for refinement of conjectures and proofs in primary school mathematics

The purpose of this study is to explore how primary school students reexamine their conjectures and proofs when they confront counter-examples to the conjectures they have proved. In the case study, a pair of Japanese fifth graders thought that they had proved their primitive conjecture with manipulative objects (that is, they constructed an action proof), and then the author presented a counter-example to them. Confronting the counter-example functioned as a driving force for them to refine their conjectures and proofs. They understood the reason why their conjecture was false through their analysis of its proof and therefore could modify their primitive conjecture. They also identified the part of the proof which was applicable to the counter-example. This identification and their action proof were essential for their invention of a more comprehensive conjecture.     

Proof of semialgebraic covering mapping cylinder conjecture with semialgebraic covering homotopy theorem

In this paper we prove the semialgebraic version of Palais' covering homotopy theorem, and use this to prove Bredon's covering mapping cylinder conjecture positively in the semialgebraic category. Bredon's conjecture was originally stated in the topological category, and a topological version of our semialgebraic proof of the conjecture answers the original topological conjecture for topological G-spaces over “simplicial” mapping cylinders.     

A reformulation of Le's conjecture

We give a new proof of Le's conjecture on surface germs in ?³ having as link a topological sphere for the case of surface singularities containing a smooth curve. Our proof leads to a reformulation of the general case of the conjecture into a problem of plane curve singularities and their relative polar curves.     

Guest Editors' Foreword

This issue of Discrete & Computational Geometry contains the detailed proof by T. Hales and S.P. Ferguson of the Kepler conjecture that the densest packing of three-dimensional Euclidean space by equal spheres is attained by "cannonball" packing. This is a landmark result. This conjecture, formulated by Kepler in 1611, was stated in Hilbert's formulation of his 18th problem [8]. The proof consists of mathematical arguments and a massive computer verification of many inequalities.     

A Revision of the Proof of the Kepler Conjecture

The Kepler conjecture asserts that no packing of congruent balls in three-dimensional Euclidean space has density greater than that of the face-centered cubic packing. The original proof, announced in 1998 and published in 2006, is long and complex. The process of revision and review did not end with the publication of the proof. This article summarizes the current status of a long-term initiative to reorganize the original proof into a more transparent form and to provide a greater level of certification of the correctness of the computer code and other details of the proof. A final part of this article lists errata in the original proof of the Kepler conjecture.     

The Arc‐Weighted Version of the Second Neighborhood Conjecture

Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. Seymour's conjecture has been verified in several special cases, most notably for tournaments by Fisher  6 . One extension of the conjecture that has been used by several researchers is to consider vertex‐weighted digraphs. In this article we introduce a version of the conjecture for arc‐weighted digraphs. We prove the conjecture in the special case of arc‐weighted tournaments, strengthening Fisher's theorem. Our proof does not rely on Fisher's result, and thus can be seen as an alternate proof of said theorem.     

Poncelet’s theorem and billiard knots

Let D be any elliptic right cylinder. We prove that every type of knot can be realized as the trajectory of a ball in D. This proves a conjecture of Lamm and gives a new proof of a conjecture of Jones and Przytycki. We use Jacobi??s proof of Poncelet??s theorem by means of elliptic functions.     

Webb’s conjecture for fusion systems

We give a short and conceptual proof of Webb’s conjecture. Our methods are general enough to prove an analogue of the conjecture for saturated fusion systems. The author was supported by an EPSRC grant EP/D506484/1.     

Corrigenda: Mackey Functors And Control of Fusion

In the introduction to [3] it is mistakenly claimed that Mislin'sproof uses Carlsson's proof of Segal's Burnside ring conjecture.In fact, Mislin uses instead the work of Carlsson, Miller andLannes on Sullivan's fixed-point conjecture, and work of Dwyerand Zabrodski [1]. There is, however, a proof by Snaith [2]that depends on a version of the Segal conjecture. 2000 MathematicsSubject Classification 20J06 (primary), 20C05 (secondary).     

A unified approach to known and unknown cases of Berge's conjecture

Berge's elegant dipath partition conjecture from 1982 states that in a dipath partition P of the vertex set of a digraph minimizing , there exists a collection C^k of k disjoint independent sets, where each dipath P∈P meets exactly min{|P|, k} of the independent sets in C. This conjecture extends Linial's conjecture, the Greene–Kleitman Theorem and Dilworth's Theorem for all digraphs. The conjecture is known to be true for acyclic digraphs. For general digraphs, it is known for k=1 by the Gallai–Milgram Theorem, for k?λ (where λis the number of vertices in the longest dipath in the graph), by the Gallai–Roy Theorem, and when the optimal path partition P contains only dipaths P with |P|?k. Recently, it was proved (Eur J Combin (2007)) for k=2. There was no proof that covers all the known cases of Berge's conjecture. In this article, we give an algorithmic proof of a stronger version of the conjecture for acyclic digraphs, using network flows, which covers all the known cases, except the case k=2, and the new, unknown case, of k=λ?1 for all digraphs. So far, there has been no proof that unified all these cases. This proof gives hope for finding a proof for all k.     

Bieberbach’s conjecture, the de Branges and Weinstein functions and the Askey-Gasper inequality

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [4]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [5] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [2] about certain hypergeometric functions played a crucial role in de Branges’ proof. In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [72] follows, and it is shown how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated. This article is dedicated to Dick Askey on occasion of his seventieth birthday. 2000 Mathematics Subject Classification Primary—30C50, 30C35, 30C45, 30C80, 33C20, 33C45, 33F10, 68W30     

Partial theta function identities from Wang and Ma's conjecture

Recently, Wang and Ma proposed a conjecture, embedding the Andrews–Warnaar partial theta function identity in an infinite family of such identities. In this paper we use q-series methods to give a proof of the Wang–Ma conjecture. We also present a result which may be regarded as the inverse of the Wang–Ma conjecture.     

Heegaard surfaces and measured laminations, I: The Waldhausen conjecture

We give a proof of the so-called generalized Waldhausen conjecture, which says that an orientable irreducible atoroidal 3-manifold has only finitely many Heegaard splittings in each genus, up to isotopy. Jaco and Rubinstein have announced a proof of this conjecture using different methods.     

Sphere Packing, IV. Detailed Bounds

This paper is the fourth in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. In a previous paper in this series, a continuous function f on a compact space was defined, certain points in the domain were conjectured to give the global maxima, and the relation between this conjecture and the Kepler conjecture was established. The function f can be expressed as a sum of terms, indexed by regions on a unit sphere. In this paper detailed estimates of the terms corresponding to general regions are developed. These results form the technical heart of the proof of the Kepler conjecture, by giving detailed bounds on the function f. The results rely on long computer calculations.