Abstract. We give the complete stably rational classification of algebraic tori of dimensions 4 and 5 over a field k. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank 4 and 5 is given. We show that there exist exactly 487 (resp. 7, resp. 216) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 4, and there exist exactly 3051 (resp. 25, resp. 3003) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension 5. We make a procedure to compute a flabby resolution of a G-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a G-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby G-lattices of rank up to 6 and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for G-lattices holds when the rank ≤ 4, and fails when the rank is 5. Indeed, there exist exactly 11 (resp. 131) G-lattices of rank 5 (resp. 6) which are decomposable into two different ranks. Moreover, when the rank is 6, there exist exactly 18 G-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that H 1 (G, F ) = 0 for any Bravais group G of dimension n ≤ 6 where F is the flabby class of the corresponding G-lattice of rank n. In particular, H 1 (G, F ) = 0 for any maximal finite subgroup G ≤ GL(n, ) where n ≤ 6. As an application of the methods developed, some examples of not retract (stably) rational fields over k are given.

Oxidative stress is a persistent threat to the genome and is associated with major causes of human mortality, including cancer, atherosclerosis, and aging. Here we established a method to generate libraries of genomic DNA fragments containing oxidatively modified bases by using specific monoclonal antibodies to immunoprecipitate enzyme-digested genome DNA. We applied this technique to two different base modifications, 8-hydroxyguanine and 1,N6-propanoadenine (acrotein-Ade), in a ferric nitrilotriacetate-induced murine renal carcinogenesis model. Renal cortical genomic DNA derived from 10- to 12-week-old male C57BL/6 mice, of untreated control or 6 hours after intraperitoneal injection of 3 mg iron/kg ferric nitrilotriacetate, was enzyme digested, immunoprecipitated, cloned, and mapped to each chromosome. The results revealed that distribution of the two modified bases was not random but differed in terms of chromosomes, gene size, and expression, which could be partially explained by chromosomal territory. In the wild-type mice, low GC content areas were more likely to harbor the two modified bases. Knockout of OGG1, a repair enzyme for genomic 8-hydroxyguanine, increased the amounts of acrolein-Ade as determined by quantitative polymerase chain reaction analyses. This versatile technique would introduce a novel research area as a high-throughput screening method for critical genomic loci under oxidative stress.

Let K be a field of characteristic not two and K(x, y, z) the rational function field over K with three variables x, y, z. Let G be a finite group acting on K(x, y, z) by monomial K-automorphisms. We consider the rationality problem of the fixed field K(x, y, z) G under the action of G, namely whether K(x, y, z) G is rational (that is, purely transcendental) over K or not. We may assume that G is a subgroup of GL(3, Z) and the problem is determined up to conjugacy in GL(3, Z). There are 73 conjugacy classes of G in GL(3, Z). By results of Endo-Miyata, Voskresenskiȋ, Lenstra, Saltman, Hajja, Kang and Yamasaki, 8 conjugacy classes of 2-groups in GL(3, Z) have negative answers to the problem under certain monomial actions over some base field K, and the necessary and sufficient condition for the rationality of K(x, y, z) G over K is given. In this paper, we show that the fixed field K(x, y, z) G under monomial action of G is rational over K except for possibly negative 8 cases of 2-groups and unknown one case of the alternating group of degree four. Moreover we give explicit transcendental bases of the fixed fields over K. For the unknown case, we obtain an affirmative solution to the problem under some conditions. In particular, we show that if K is quadratically closed field then K(x, y, z) G is rational over K. We also give an application of the result to 4-dimensional linear Noether's problem.

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Three-dimensional monomial Noether problem can have negative solutions for 8 groups by the suitable choice of the coefficients. We find the necessary and sufficient condition for the coefficients to have a negative solution. The results are obtained by two criteria of irrationality using Galois cohomology.