On the fields generated by the lengths of closed geodesics in locally symmetric spaces

This paper is the next installment of our analysis of length-commensurable locally symmetric spaces begun in Publ. math. IHES 109(2009), 113-184. For a Riemannian manifold $M$, we let $L(M)$ be the weak length spectrum of $M$, i.e. the set of lengths of all closed geodesics in $M$, and let $\mathcal{F}(M)$ denote the subfield of $\mathbb{R}$ generated by $L(M)$. Let now $M_i$ be an arithmetically defined locally symmetric space associated with a simple algebraic $\mathbb{R}$-group $G_i$ for $i = 1, 2$. Assuming Schanuel's conjecture from transcendental number theory, we prove (under some minor technical restrictions) the following dichotomy: either $M_1$ and $M_2$ are length-commensurable, i.e. $\mathbb{Q} \cdot L(M_1) = \mathbb{Q} \cdot L(M_2)$, or the compositum $\mathcal{F}(M_1)\mathcal{F}(M_2)$ has infinite transcendence degree over $\mathcal{F}(M_i)$ for at least one $i = 1$ or $2$ (which means that the sets $L(M_1)$ and $L(M_2)$ are very different)

Local-global principles for embedding of fields with involution into simple algebras with involution

In this paper we prove local-global principles for embedding of fields with involution into central simple algebras with involution over a global field. These should be of interest in study of classical groups over global fields. We deduce from our results that in a group of type D_n, n>4 even, two weakly commensurable Zariski-dense S-arithmetic subgroups are actually commensurable. A consequence of this result is that given an absolutely simple algebraic K-group G of type D_n, n>4 even, K a number field, any K-form G' of G having the same set of isomorphism classes of maximal K-tori as G, is necessarily K-isomorphic to G. These results lead to results about isolength and isospectral compact hyperbolic spaces of dimension 2n-1 with n even

Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces

The article contains a survey of results on length-commensurable and isospectral locally symmetric spaces and related problems in the theory of semi-simple algebraic groups.Comment: New material has been added in section

Weakly commensurable groups, with applications to differential geometry

The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic groups. We have included a discussion of very recent results and conjectures on absolutely almost simple algebraic groups having the same maximal tori and finite-dimensional division algebras having the same maximal subfields.Comment: Improved exposition, updated bibliography. arXiv admin note: substantial text overlap with arXiv:1212.121

Generation of Magnetic Field in the Pre-recombination Era

We study the possibility of generating magnetic fields during the evolution of electron, proton, and photon plasma in the pre-recombination era. We show that a small magnetic field can be generated in the second order of perturbation theory for scalar modes with adiabatic initial conditions. The amplitude of the field is \la 10^{-30} \rm G at the present epoch for scales from sub-kpc to \ga 100 \rm Mpc.Comment: 8 page

Computational aspects of helicopter trim analysis and damping levels from Floquet theory

Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated