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Recent advances in micromachining technology have made possible the fabrication of a number of electrostatic devices, including variable capacitance micromotors. These variable capacitance micromotors employ one or more sets of conducting plates that act as capacitors. An accurate estimate of the capacitance between these sets of plates for various configurations is necessary to design and simulate such devices. Capacitance calculations can be performed using a variety of methods, including finite element techniques. An efficient method of calculating the capacitance is to reduce the problem to a set of integral equations that are then approximated by a set of algebraic equations. This method was previously used to calculate the capacitance of a linear micromotor with an infinite set of conducting lands on both the rotor and the stator, which introduces spatial periodicity that greatly simplifies the problem. Recent micromotor designs, however, employ only a few sets of conducting lands per phase, which makes it unclear if the previous results can be applied. In this paper, a new perturbation method for solving the integral equations and thereby estimating the capacitance is presented. This method does not make the assumption of infinite sets of conducting lands to introduce spatial periodicity. The perturbation method is computationally efficient and can be used for any design, including those that use very few repeated sets of conducting lands. The results obtained by this method are compared with the previous results which include experimental data. Finally, the method is extended to rotary micromotors and to three dimensional problems.

Let G be a connected semisimple algebraic group defined over the field R of real numbers. An element x of G(R) is called R-regular if the number of eigenvalues, counted with multiplicity, of modulus 1 of Ad x is minimum possible. (If G is R-anisotropic, i.e., the group G(R) is compact, every element of G(R) is Rregular.) The existence of R-regular elements in an arbitrary subsemigroup Γ of G(R) which is Zariski-dense in G was proved by Y. Benoist and F. Labourie [3] using Oseledet's multiplicative ergodic theorem, and then reproved by the firstnamed author [15] by a direct argument. Recently G.A. Margulis and G.A. Soifer asked us a question, which arose in their joint work with H. Abels on the Auslander problem, about the existence of R-regular elements with some special properties. The purpose of this note is to answer their question in the affirmative. Before formulating the result, we recall (cf. [16], Remark 1.6(1)) that an R-regular element x is necessarily semisimple, so if in addition it is regular, then T := Z G (x) • is a maximal torus; moreover, x belongs to T (see [4], Corollary 11.12). Theorem 1. Let G be a connected semisimple real algebraic group. Then any Zariski-dense subsemigroup Γ of G(R) contains a regular R-regular element x such that the cyclic subgroup generated by it is a Zariski-dense subgroup of the maximal torus T := Z G (x) •. Remark 1. Let Γ, x and T = Z G (x) • be as in Theorem 1. Let T s (resp., T a) be the maximal R-split (resp., R-anisotropic) subtorus of T. Then T = T s • T a (an almost direct product), T s is a maximal R-split torus of G since x is R-regular (see [16], Lemma 1.5), and T a (R) (R/Z) r , where r = dim T a. There is a positive integer d such that x d = y • z with y ∈ T s (R) and z ∈ T a (R). Then the cyclic group C generated by z is dense in T a in the Zariski-topology and since T a (R) is a compact Lie group, C is actually dense in T a (R) in the usual compact Hausdorff topology on the latter. Thus, in particular, if G(R) is compact, then any dense subsemigroup contains a Kronecker element, i. e. an element such that the closure of the subsemigroup generated by it is a maximal torus. Also, since the cyclic subgroup generated by x is dense in T in the Zariskitopology, Z G (x) = Z G (T) = T. Thus the centralizer of x is connected.

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Abstract. In this paper we prove local-global principles for the existence of an embedding .E; / ,! .A; / of a given global field E endowed with an involutive automorphism into a simple algebra A given with an involution in all situations except where A is a matrix algebra of even degree over a quaternion division algebra and is orthogonal (Theorem A of the introduction). Rather surprisingly, in the latter case we have a result which in some sense is opposite to the local-global principle, viz. algebras with involution locally isomorphic to .A; / are distinguished by their maximal subfields invariant under the involution (Theorem B of the introduction). These results can be used in the study of classical groups over global fields. In particular, we use Theorem B to complete the analysis of weakly commensurable Zariski-dense S -arithmetic groups in all absolutely simple algebraic groups of type different from D 4 which was initiated in our paper [23]. More precisely, we prove that in a group of type D n , n even > 4, two weakly commensurable Zariski-dense S-arithmetic subgroups are actually commensurable. As indicated in [23], this fact leads to results about length-commensurable and isospectral compact arithmetic hyperbolic manifolds of dimension 4n C 7, with n > 1. The appendix contains a Galois-cohomological interpretation of our embedding theorems.