The Fa,b,c conjecture is true, II

In 1977 a five-part conjecture was made about a family of groups related to trivalent graphs and subsequently two parts of the conjecture were proved. The conjecture completely determines all finite members of the family. Here we complete the proof of the conjecture by giving proofs for the remaining three parts. (c) 2006 Elsevier Inc. All rights reserved.

On some Results Related to Köthe's Conjecture

The Köthe conjecture states that if a ring R has no nonzero nil ideals then R has no nonzero nil one-sided ideals. Although for more than 70 years significant progress has been made, it is still open in general. In this paper we survey some results related to the Köthe conjecture as well as some equivalent problems.

Generalization of a Conjecture in the Geometry of Polynomials

In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by the zeros of the polynomials.

Toward Clemens' Conjecture in Degrees between 10 and 24

1 Supported in part by the Norwegian Research Council for Science and the Humanities. It is a pleasure for this author to thank the Department of Mathematics of the University of Sofia for organizing the remarkable conference in Zlatograd during the period August 28-September 2, 1995. It is also a pleasure to thank the M.I.T. Department of Mathematics for its hospitality from January 1 to July 31, 1993, when this work was started. 2Supported in part by NSF grant 9400918-DMS.

Hilbert-Smith Conjecture for K - Quasiconformal Groups

MSC 2010: 30C60

Smale's Conjecture on Mean Values of Polynomials and Electrostatics

2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.

A Note About the Nowicki Conjecture on Weitzenböck Derivations

2000 Mathematics Subject Classification: 13N15, 13A50, 16W25.

Finite dimensional ordered vector spaces with Riesz interpolation and Effros-Shen's unimodularity conjecture

Peer reviewed

The proof of Kontsevich's periodicity conjecture on noncommutative birational transformations

For an arbitrary associative unital ring RR, let J1J1 and J2J2 be the following noncommutative, birational, partly defined involutions on the set M3(R)M3(R) of 3×33×3 matrices over RR: J1(M)=M−1J1(M)=M−1 (the usual matrix inverse) and J2(M)jk=(Mkj)−1J2(M)jk=(Mkj)−1 (the transpose of the Hadamard inverse).

We prove the surprising conjecture by Kontsevich that (J2∘J1)3(J2∘J1)3 is the identity map modulo the DiagL×DiagRDiagL×DiagR action (D1,D2)(M)=D−11MD2(D1,D2)(M)=D1−1MD2 of pairs of invertible diagonal matrices. That is, we show that, for each MM in the domain where (J2∘J1)3(J2∘J1)3 is defined, there are invertible diagonal 3×33×3 matrices D1=D1(M)D1=D1(M) and D2=D2(M)D2=D2(M) such that (J2∘J1)3(M)=D−11MD2(J2∘J1)3(M)=D1−1MD2.

Tutte's 5-flow conjecture

The aim of this thesis is to show and put together the results, obtained so far, useful to tackle a conjecture of graph theory proposed in 1954 by William Thomas Tutte. The conjecture in question is Tutte's 5-flow conjecture, which states that every bridgeless graph admits a nowhere-zero 5-flow, namely a flow with non-zero integer values between -4 and 4. We will start by giving some basics on graph theory, useful for the followings, and proving some results about flows on oriented graphs and in particular about the flow polynomial. Next we will treat two cases: graphs embeddable in the plane $\mathbb{R}^2$ and graphs embeddable in the projective plane $\mathbb{P}^2$. In the first case we will see the correlation between flows and colorings and prove a theorem even stronger than Tutte's conjecture, using the 4-color theorem. In the second case we will see how in 1984 Richard Steinberg used Fleischner's Splitting Lemma to show that there can be no minimal counterexample of the conjecture in the case of graphs in the projective plane. In the fourth chapter we will look at the theorems of François Jaeger (1976) and Paul D. Seymour (1981). The former proved that every bridgeless graph admits a nowhere-zero 8-flow, the latter managed to go even further showing that every bridgeless graph admits a nowhere-zero 6-flow. In the fifth and final chapter there will be a short introduction to the Tutte polynomial and it will be shown how it is related to the flow polynomial via the Recipe Theorem. Finally we will see some applications of flows through the study of networks and their properties.

Sob o signo de Darwin? Sobre o mau uso de uma quimera

No ano de 2003 Francisco de Oliveria publicou um artigo intitulado "O Ornitorrinco" no qual fez considerações críticas sobre a conjectura politico-social daquele momento histórico. Tal artigo é permeado por um paralelo entre o evolucionismo darwinista e a visão do autor sobre a sociedade brasileira contemporânea. Entretanto, ao fazer tal analogia ele incorre numa série de equívocos teóricos sobre a teoria evolucionista. Tais equívocos consistem, em grande parte, numa substuição indevida entre aquilo que ficou conhecido como Darwinismo Social e a teoria neodarwinista como entendida pelos seus atuais proponentes. O presente trabalho identifica estes equívocos e os contextualiza dentro da teoria neodarwiniana. Além disso, fazemos um recorte histórico do processo de formação do pensamento evolucionista para enfatizar que a associação entre biologia e darwinismo social é mais complexa do que geralmente se assume.

Atividade óptica de um meio dielétrico diluído: Pasteur e as simetrias moleculares

Em 1848 Pasteur conjeturou que a rotação do plano de polarização da luz em um meio diluído é gerada pelas propriedades de simetria das moléculas do meio no qual a luz se propaga. O objetivo do nosso artigo é de mostrar que Pasteur estava correto usando conhecimentos de eletromagnetismo e mecânica quântica de um curso de graduação em física. Faremos um breve retrospecto das ideias básicas da teoria eletromagnética necessárias para o estudo da atividade óptica. A seguir, usando a teoria de perturbações em mecânica quântica e levando em conta as simetrias das moléculas calcularemos a atividade óptica do meio. Mostraremos que as previsões teóricas, que estão plenamente de acordo com os resultados experimentais, comprovam a hipótese de Pasteur.

First stars XIV. Sulfur abundances in extremely metal-poor stars

Context. Precise S abundances are important in the study of the early chemical evolution of the Galaxy. In particular the site of the formation remains uncertain because, at low metallicity, the trend of this alpha-element versus [Fe/H] remains unclear. Moreover, although sulfur is not bound significantly in dust grains in the ISM, it seems to behave differently in DLAs and old metal-poor stars. Aims. We attempt a precise measurement of the S abundance in a sample of extremely metal-poor stars observed with the ESO VLT equipped with UVES, taking into account NLTE and 3D effects. Methods. The NLTE profiles of the lines of multiplet 1 of S I were computed with a version of the program MULTI, including opacity sources from ATLAS9 and based on a new model atom for S. These profiles were fitted to the observed spectra. Results. We find that sulfur in EMP stars behaves like the other alpha-elements, with [S/Fe] remaining approximately constant below [Fe/H] = -3. However, [S/Mg] seems to decrease slightly with increasing [Mg/H]. The overall abundance patterns of O, Na, Mg, Al, S, and K are most closely matched by the SN model yields by Heger & Woosley. The [S/Zn] ratio in EMP stars is solar, as also found in DLAs. We derive an upper limit to the sulfur abundance [S/Fe] < +0.5 for the ultra metal-poor star CS 22949-037. This, along with a previously reported measurement of zinc, argues against the conjecture that the light-element abundance pattern of this star (and by analogy, the hyper iron-poor stars HE 0107-5240 and HE 1327-2326) would be due to dust depletion.

Perturbations of black p-branes

We consider black p-brane solutions of the low-energy string action, computing scalar perturbations. Using standard methods, we derive the wave equations obeyed by the perturbations and treat them analytically and numerically. We have found that tensorial perturbations obtained via a gauge-invariant formalism leads to the same results as scalar perturbations. No instability has been found. Asymptotically, these solutions typically reduce to a AdSd((p+2)) x Sd((8-p)) space which, in the framework of Maldacena's conjecture, can be regarded as a gravitational dual to a conformal field theory defined in a (p+1)-dimensional flat space-time. The results presented open the possibility of a better understanding the AdS/CFT correspondence, as originally formulated in terms of the relation among brane structures and gauge theories.