On compatibility between isogenies and polarisations of abelian varieties

We discuss the notion of polarised isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarisations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show that certain questions about polarised isogenies can be reduced to questions about unpolarised isogenies or vice versa. Our main theorem concerns abelian varieties B which are isogenous to a fixed abelian variety A. It establishes the existence of a polarised isogeny A to B whose degree is polynomially bounded in n, if there exist both an unpolarised isogeny A to B of degree n and a polarised isogeny A to B of unknown degree. As a further result, we prove that given any two principally polarised abelian varieties related by an unpolarised isogeny, there exists a polarised isogeny between their fourth powers. The proofs of both theorems involve calculations in the endomorphism algebras of the abelian varieties, using the Albert classification of these endomorphism algebras and the classification of Hermitian forms over division algebras

Change in blood glucose level in rats after immobilization

Experiments were carried out on male white rats divided into four groups. In group one the blood glucose level was determined immediately after immobilization. In the other three groups, two hours following immobilization, the blood glucose level was determined every 20 minutes for 3 hours 40 minutes by the glucose oxidase method. Preliminary immobilization for 2 hours removed the increase in the blood glucose caused by the stress reaction. By the 2nd hour of immobilization in the presence of continuing stress, the blood glucose level stabilized and varied within 42 + or - 5.5 and 47 + or - 8.1 mg %. Within 2 hours after the immobilization, the differences in the blood glucose level of the rats from the control groups were statistically insignificant

Peripheral separability and cusps of arithmetic hyperbolic orbifolds

For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold. A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-32.abs.htm

Superrigid subgroups and syndetic hulls in solvable Lie groups

This is an expository paper. It is not difficult to see that every group homomorphism from the additive group Z of integers to the additive group R of real numbers extends to a homomorphism from R to R. We discuss other examples of discrete subgroups D of connected Lie groups G, such that the homomorphisms defined on D can ("virtually") be extended to homomorphisms defined on all of G. For the case where G is solvable, we give a simple proof that D has this property if it is Zariski dense. The key ingredient is a result on the existence of syndetic hulls.Comment: 17 pages. This is the final version that will appear in the volume "Rigidity in Dynamics and Geometry," edited by M. Burger and A. Iozzi (Springer, 2002

Arbitrarily large families of spaces of the same volume

In any connected non-compact semi-simple Lie group without factors locally isomorphic to SL_2(R), there can be only finitely many lattices (up to isomorphism) of a given covolume. We show that there exist arbitrarily large families of pairwise non-isomorphic arithmetic lattices of the same covolume. We construct these lattices with the help of Bruhat-Tits theory, using Prasad's volume formula to control their covolumes.Comment: 9 pages. Syntax corrected; one reference adde

Economic aspects of organizing high-technology resource saving production in the forest sector of Russia

Purpose: The article aims at assessing the possibility of organizing high-tech production in the Russian forest industry. Structure/Methodology/Approach: To switch to high-tech and resource-saving technologies in the production of sleepers, it is necessary to assess the influence of a few factors: the cost of production of innovative products, the demand for innovative products, the lifetime of innovative products. To assess the demand, we have used an expert survey. Results: The cost of producing railroad sleepers from modified wood has been established. The use of cheaper softwood wood as raw material (birch, aspen, alder, poplar, etc.) which makes it possible to reduce the cost of production of the sleepers is a competitive advantage of the innovative technology. Practical implications: It has been established that in the future we should expect an increase in demand for the sleepers made of modified wood, especially in the European part of Russia. Originality/Value. The main contribution of this study is the rationale for the transition to the concept of environmental savings in the Russian forest industry. The use of modified wood in the production of sleepers is economically viable and can be an alternative when diversifying production.peer-reviewe