We present an introduction to periodic and stochastic homogenization of elliptic partial differential equations. The first part is concerned with the qualitative theory, which we present for equations with periodic and random coefficients in a unified approach based on Tartar's method of oscillating test functions. In particular, we present a self-contained and elementary argument for the construction of the sublinear corrector of stochastic homogenization. (The argument also applies to elliptic systems and in particular to linear elasticity). In the second part we briefly discuss the representation of the homogenization error by means of a two-scale expansion. In the last part we discuss some results of quantitative stochastic homogenization in a discrete setting. In particular, we discuss the quantification of ergodicity via concentration inequalities, and we illustrate that the latter in combination with elliptic regularity theory leads to a quantification of the growth of the sublinear corrector and the homogenization error.

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This proceeding is based on the lecture given by the author at GSIS international winter school 2017 on ``Stochastic homogenization and its applications'' at Tohoku University. The main purpose of this proceeding is to present ideas of the works of [5, 6] on (periodic) homogenization for the Hamilton–Jacobi equation, and give a recent result of [16] on the selection problem for the cell problem as simply as possible.

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We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schrödinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors' recent work where similar conclusions have been obtained for bounded random potentials.

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In this note, we give some restrictions on the number of vectors of weight d/2+1 in the shadow of a singly even self-dual [n,n/2,d] code. This eliminates some of the possible weight enumerators of singly even self-dual [n,n/2,d] codes for (n,d)=(62,12), (72,14), (82,16), (90,16) and (100,18).

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Fixed points of the self-power map over a finite field have been studied in cryptology as a special case of modular exponentiation. In this note, we define an elliptic-curve version of the self-power map, enumerate the number of curves that contain at least one fixed point, and give its upper and lower bounds. Our result is a partial solution to the open question raised by Glebsky and Shparlinski in 2010.

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