Divergent trajectories and ℚ-rank

The author proves a conjecture of the author: IfG is a semisimple real algebraic defined over ℚ, Γ is an arithmetic subgroup (with respect to the given ℚ-structure) andA is a diagonalizable subgroup admitting a divergent trajectory inG/Γ, then dimA≤ rank_ℚ G.     

On quadratic fields with large 3-rank

Davenport and Heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields, which has been used to tabulate the latter. We give a simpler proof of their theorem then analyze and improve the table-building algorithm. It computes the multiplicities of the general cubic discriminants (real or imaginary) up to in time and space , or more generally in time and space for a freely chosen positive . A variant computes the -ranks of all quadratic fields of discriminant up to with the same time complexity, but using only units of storage. As an application we obtain the first real quadratic fields with , and prove that is the smallest imaginary quadratic field with -rank equal to .

     

Tame kernels and further 4-rank densities

There has been recent progress on computing the 4-rank of the tame kernel for F a quadratic number field. For certain quadratic number fields, this progress has led to “density results” concerning the 4-rank of tame kernels. These results were first mentioned in Conner and Hurrelbrink (J. Number Theory 88 (2001) 263) and proven in Osburn (Acta Arith. 102 (2002) 45). In this paper, we consider some additional quadratic number fields and obtain further density results of 4-ranks of tame kernels. Additionally, we give tables which might indicate densities in some generality.     

3-rank of ambiguous class groups of cubic Kummer extensions

Periodica Mathematica Hungarica - Let $$k=k_0(\root 3 \of {d})$$ be a cubic Kummer extension of $$k_0=\mathbb {Q}(\zeta _3)$$ with $$d>1$$ a cube-free integer and $$\zeta _3$$ a primitive...     

A note on the 3-rank of quadratic fields

The difference between the 3-rank of the ideal class group of an imaginary quadratic field and that of the associated real quadratic field is equal to 0 or 1. In this note, we give an infinite family of examples in each case.Received: 9 September 2002     

8-rank of the class group and isotropy index

Suppose F = Q(√-p1 pt) is an imaginary quadratic number field with distinct primes p1,..., pt,where pi≡ 1(mod 4)(i = 1,..., t- 1) and pt ≡ 3(mod 4). We express the possible values of the 8-rank r8 of the class group of F in terms of a quadratic form Q over F2 which is defined by quartic symbols. In particular,we show that r8 is bounded by the isotropy index of Q.     

1/N expansion for scalar fields

Models of scalar field theories with a large number N of isotopic degrees of freedom are considered. A theory of perturbations with respect to a small parameter is developed in the formalism of path integration for a space-time dimensionD =2, 3, 4.The particle spectrum obtained in basic order with respect to N^?1 is compared with the spectrum in the path approach. It is shown that whenD =4 the chiral field model is turned, as a result of renormalization, into a model with four interactions. The limitations of the applicability of the 1/N expansion are discussed.     

MIMOl 1-optimization with a scalar control

In Ref. 1, the author showed that some scalar mixed sensitivity minimization problems have rationall ¹-optimal solutions and computed these solutions. Here, a similar approach is applied to a multi-input multi-output (MIMO) plant. The general flavor of the solution is the same as in Ref. 1, but there is one distinct new feature. We prove that some of the outputs may be ignored, in the sense that they have no influence on the optimal solution and that all the operators corresponding to the remaining outputs have the same optimal norm. This fact suggests alternative methods for solving the problem and it is employed in the construction of an exact rationall ¹-optimal solution of a particular problem.This work was partly done while the author was a visiting professor at the Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia. Supported in part by a joint grant from NSF and the Academy of Finland.     

Proportionality principle for the simplicial volume of families of \mathbb Q -rank 1 locally symmetric spaces

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for $\mathbb Q $ -rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the proportionality principle for the locally finite simplicial volume and the relative simplicial volume of $\mathbb Q $ -rank $1$ locally symmetric spaces with amenable cusp groups established by Löh and Sauer [26].     

Divergent torus orbits in homogeneous spaces of ℚ-rank two

LetG be a semisimple algebraic ℚ-group, let Γ be an arithmetic subgroup ofG, and letT be an ℝ-split torus inG. We prove that if there is a divergentT _ℝ-orbit in Γ\G _ℝ, and ℚ-rankG≤2, then dimT≤ℚ-rankG. This provides a partial answer to a question of G. Tomanov and B. Weiss.     

On the 4-rank of class groups of quadratic number fields

We prove that the 4-rank of class groups of quadratic number fields behaves as predicted in an extension due to Gerth of the Cohen–Lenstra heuristics. Mathematics Subject Classification (2000)  11R29, 11R11, 11R45     

Class groups of quadratic fields of 3-rank at least 2: Effective bounds

In this paper, we give parametric families of both real and complex quadratic number fields whose class group has 3-rank at least 2. As a consequence, we obtain that for all large positive real numbers x, the number of both real and complex quadratic fields whose class group has 3-rank at least 2 and absolute value of the discriminant ?x is >cx1/3, where c is some positive constant.     

Über den 4-rank der Klassengruppe quadratischer Zahlkörper

Relations between the 4-ranks of various quadratic number fields are investigated using the classical criterion of Rédei and Reichardt.     

On the 3-rank of tame kernels of certain pure cubic number fields

In this paper, we present some explicit formulas for the 3-rank of the tame kernels of certain pure cubic number fields, and give the density results concerning the 3-rank of the tame kernels. Numerical examples are given in Tables 1 and 2.