Towards the Eaton-Moretó conjecture on the minimal height of characters

Abstract

In this note we prove that the Eaton-Moretó conjecture holds for all blocks of finite general linear and unitary groups for all primes. Also, we show that no block of a finite quasi-simple group of classical Lie type provides a minimal counterexample to the conjecture, and so for ? > 5 no ?-block of any quasi-simple group can be a minimal counterexample.     

A Mordell-Lang plus Bogomolov type result for curves in $${\mathbb{G}_{m}^{2}}$$

We prove a sharper so-called Mordell-Lang plus Bogomolov type result for curves lying in the two-dimensional linear torus. We mainly follow the approach of Rémond in (Comp Math 134:337–366, 2002), using Vojta and Mumford type inequalities. In the special case we consider, we improve Rémond’s main result using a better Bogomolov property and an elementary arithmetic Bézout theorem.      

Heights and Determinants over Quaternion Algebras #

We provide here a very preliminary—or somehow experimental—result in the direction of the stability of free products of stable groups, restricting ourselves to quantifer-free definable sets and to bounded balls of free products.     

煤层倾角与覆岩变形破裂分带

地下采煤中三带(冒落带、裂隙带、弯曲带)高度的判断是预测采空引起地表变形破坏程度的依据。在诸多的影响因素中, 煤层倾角是控制“三带”高度的最主要因素。建立在实测统计基础上的经验范围值用于“三带”高度的判断带有一定的人为性。急倾斜煤层中冒落带和裂隙带高度随煤层倾角变化的规律已被褐示[7].通过弹塑性岩石材料的非线性有限元模拟, 本文提出了利用应力重分布图判断中、缓倾角煤层采空区覆岩“三带”高度的方法, 并应用于实际工程。     

Isomorphic Commutative Group Algebras of p-Mixed Warfield Groups

Let G be a p-mixed Warfield Abelian group and F a field of char F = p ≠ 0. It is proved that if for any group H the group algebras FH and FG are F-isomorphic, then H is isomorphic to G. This presentation enlarges a result of W. May argued when G is p-local Warfield Abelian and published in Proc. Amer. Math. Soc. （1988）.     

Linear equations and heights over division algebras

We make a start on the investigation of “small” solutions to systems of homogeneous linear equations over non-commutative division algebras. In this paper we prove some upper and lower bounds for the sizes of solutions to such systems. To measure solutions and coefficient matrices we define heights which satisfy natural invariance and finiteness properties.     

Vapour phase chemistry of oxovanadium iv β-diketonates

A series of fluorinated and one non-fluorinated β-diketonate complexes of oxovanadium IV have been prepared, their gas phase IR spectra examined, vapour pressures determined and their vapour phase thermal decomposition investigated. Decomposition of the fluorinated complexes occurred according to the overall reaction scheme VO(β-diket)₂»VOF₂ + Furanone. The furanones C₅HF₅O₂, C₅H₄F₂O₂ and C₈H₁₀F₂O₂ were isolated and identified. The non-fluorinated complex gave the parent diketone and a vanadium oxide as its principal products.     

The Lehmer inequality and the Mordell-Weil theorem for Drinfeld modules

In this paper we prove several Lehmer type inequalities for Drinfeld modules which will enable us to prove certain Mordell-Weil type structure theorems for Drinfeld modules.     

Arakelov theory of noncommutative arithmetic curves

The purpose of this article is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with Arakelov theory of noncommutative arithmetic curves. A noncommutative arithmetic curve is the spectrum of a Z-order O in a finite-dimensional semisimple Q-algebra. Our first main result is an arithmetic Riemann-Roch formula in this setup. We proceed with introducing the Grothendieck group of arithmetic vector bundles on a noncommutative arithmetic curve and show that there is a uniquely determined degree map , which we then use to define a height function HO. We prove a duality theorem for the height HO.