A variety of models for the membrane-mediated interaction of particles in lipid membranes, mostly well-established in theoretical physics, is reviewed from a mathematical perspective. We provide mathematically consistent formulations in a variational framework, relate apparently different modelling approaches in terms of successive approximation, and investigate existence and uniqueness. Numerical computations illustrate that the new variational formulations are directly accessible to effective numerical methods

Abstract. Multi-Level Monte-Carlo Finite Element (MLMC-FE) methods for the solution of stochastic elliptic variational inequalities are introduced, analyzed, and numerically investigated. Under suitable assumptions on the random diffusion coefficient, the random forcing function, and the deterministic obstacle, we prove existence and uniqueness of solutions of "mean-square" and "pathwise" formulations. Suitable regularity results for deterministic, elliptic obstacle problems lead to uniform pathwise error bounds, providing optimal-order error estimates of the statistical error and upper bounds for the corresponding computational cost for classical Monte-Carlo and novel MLMC-FE methods. Utilizing suitable multigrid solvers for the occurring sample problems, in two space dimensions MLMC-FE methods then provide numerical approximations of the expectation of the random solution with the same order of efficiency as for a corresponding deterministic problem, up to logarithmic terms. Our theoretical findings are illustrated by numerical experiments.

It is the purpose of this paper to develop the theory of the similarity transformation for matrices whose elements belong to a division algebra. In order to get a basis for generalization, the theory of the similarity transformation for matrices whose elements belong to a field is sketched in what seems to the authors a more suggestive method than those used heretofore.! L. A. Wolf's paper entitled Similarity of matrices in which the elements are real quaternions^ treats the case of the quaternion algebra over any subfield of the real field, by passing to the equivalent square matrices of order 2n with elements in the subfield.Some of the results of the present paper are given in an abstract and a subsequent paper by N. Jacobson.|| Jacobson's results are to a certain extent more general. In the present paper a usable rational process is given for determining the equivalence or non-equivalence of matrices whose elements belong to a division algebra, and a theorem is developed concerning the rank of a polynomial in a matrix, which is not indicated by Jacobson. Some of the results contained herein were given by Ingraham at the summer meeting of the Society in 1935.If I. The similarity transformation for the commutative case 1. Review of certain known theory. If M is an mXn matrix (i.e., a matrix with m rows and n columns) with elements in a field F, the rank of M is *

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