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.     

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.     

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.

     

Primes between consecutive squares

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. The aim of this paper is to provide a bound for the exceptional set for this conjecture, unconditionally and under the assumption of some classical hypothesis. We also provide a conditional proof of the conjecture assuming an hypothesis about the behavior of Selberg's integral in short intervals.     

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.     

An easier proof of the maximal arcs conjecture

It was a long-standing conjecture in finite geometry that a Desarguesian plane of odd order contains no maximal arcs. A rather inaccessible and long proof was given recently by the authors in collaboration with Mazzocca. In this paper a new observation leads to a greatly simplified proof of the conjecture.

     

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.     

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.     

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.     

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.     

Remarks on Serre’s modularity conjecture

In this article we give a proof of Serre’s conjecture for the case of odd level and arbitrary weight. Our proof does not use any modularity lifting theorem in characteristic 2 (moreover, we will not consider at all characteristic 2 representations at any step of our proof). The key tool in the proof is a very general modularity lifting result of Kisin, which is combined with the methods and results of previous articles on Serre’s conjecture by Khare, Wintenberger, and the author, and modularity results of Schoof for abelian varieties of small conductor. Assuming GRH, infinitely many cases of even level will also be proved.     

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.     

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.     

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 .     

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).     

On the Union of Closed Balls Property

We provide a new analytical proof for a strengthened version of the variable radius form of the union of closed balls conjecture. We also introduce a strong version of this conjecture and discuss its validity.     

An answer to a question of Cao, Reilly and Xiong

We present a simple proof of a Banach-Stone type Theorem. The method used in the proof also provides an answer to a conjecture of Cao, Reilly and Xiong.     

Miklós–Manickam–Singhi conjectures on partial geometries

In this paper we give a proof of the Miklós–Manickam–Singhi (MMS) conjecture for some partial geometries. Specifically, we give a condition on partial geometries which implies that the MMS conjecture holds. Further, several specific partial geometries that are counterexamples to the conjecture are described.     

A New Derivation of the Inner Product Formula for the Macdonald Symmetric Polynomials

We give a short proof of the inner product conjecture for the symmetric Macdonald polynomials of type A_n-1. As a special case, the corresponding constant term conjecture is also proved.