In the stable homotopy groups of spheres, we have Greek letter elements due to J. F. Adams [2], L. Smith [12] and H. Toda [13]. Here we study the nontriviality of certain products of the first alpha element, the first and the second beta elements and a gamma element in the homotopy groups.

In the present paper, we study the anabelian geometry of mixedcharacteristic local fields by an algorithmic approach. We begin by discussing some generalities on log-shells of mixed-characteristic local fields. One main topic of this discussion is the difference between the log-shell and the ring of integers. This discussion concerning log-shells allows one to establish mono-anabelian reconstruction algorithms for constructing some objects related to the $p$-adic valuations. Next, we consider open homomorphisms between profinite groups of MLF-type. This consideration leads us to a bi-anabelian result for absolutely unramified mixed-characteristic local fields. Next, we establish some mono-anabelian reconstruction algorithms related to each of absolutely abelian mixed-characteristic local fields, mixed-characteristic local fields of degree one, and Galois-specifiable mixed-characteristic local fields. For instance, we give a mono-anabelian reconstruction algorithm for constructing the Norm map with respect to the finite extension determined by the uniquely determined minimal mixed-characteristic local subfield. Finally, we apply various results of the present paper to prove some facts concerning outer automorphisms of the absolute Galois groups of mixedcharacteristic local fields that arise from field automorphisms of the mixed-characteristic local fields.

For a finite collection $\mathbf{A}=(A_i)_{i \in I}$ of locally closed sets in $\mathbb R ^n$, $n \geqslant 3$, with the sign $s_i=\pm 1$ prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the $\alpha$-Riesz kernel $|x − y|^{\alpha-n}, \alpha \in (0,2]$, over positive vector Radon measures $\mathbf{μ}=(\mu^i)_{i \in I}$ such that each $\mu^i, i \in I$, is carried by $A_i$ and normalized by $\mu^1(A_i)=a_i \in (0, \infty)$, while the interaction between $\mu^i, i \in I$ , is determined by the matrix $(s_is_j)_{i,j\in I^\bullet}$. We show that, though the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity, this problem has a solution $\mathbf{λ}_\mathbf{A}^\xi=(\lambda_\mathbf{A}^i)_{i \in I}$ (also in the presence of an external field) if we restrict ourselves to $\mathbf{μ}$ with $\mu^i\leqslant\xi^i, i \in I$, where the constraint $\mathbf{ξ}=(\xi^i)_{i \in I}$ is properly chosen. We establish the sharpness of the suffcient conditions on the solvability thus obtained, provide descriptions of the weighted vector $\alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the $\lambda_\mathbf{A}^i, i \in I$. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the $\alpha$-Riesz energy on a set of vector measures associated with $\mathbf{A}$, as well as on the establishment of an intimate relationship between the constrained minimum $\alpha$-Riesz energy problem and a constrained minimum $\alpha$-Green energy problem, suitably formulated. The results are illustrated by examples.