Mine surveying technique and steel storage tank base deformation removal

Introduction. Vertical steel tanks are extensively used in oil fields to hold crude oil and other liquids. Their construction requires continuous surveying control. However, measurement procedures, data processing, and deviations and documentation correction are not universal and cannot be applied for tanks with a capacity of less than 3000 m3. Moreover, the process of correcting the detected bottom irregularities is poorly substantiated and intuitive. Research objective is to improve the as-built survey accuracy and reliability of the low-capacity steel tank bottom, substantiate the minimum height of its irregularities, increase the objectivity and productivity of measurement processing, develop an unambiguous method for time-predictable correction of bottom irregularities, and substantiate the optimal contents of its relief’s as-built scheme. Methods of research. Geometric leveling for the bottom profile survey failed to provide adequate accuracy and was therefore replaced by the method of trigonometric leveling. A method has been developed for determining the deviations of the existent tank bottom profile from the design position by means of algebraic actions with surfaces of a topographic order. The accuracy of determining the smallest height of bottom irregularities has been estimated under the tacheometric survey. A method has been proposed for controlling the correction of tank bottom irregularities based on evaluation calculations of bottom deformations by the finite element method. Research results. All the development results are exemplified by a specific production example. It was found that for a full completion of work, two iterations of tank bottom irregularities correction are enough. Methods of optimal design for the facility’s as-built schemes are presented. Conclusions. A simple, accurate, low-cost, productive, and time-predictable method of mine surveying, mathematical processing and correction of deviations in the tank bottom profile has been developed. This technique reduces the construction period and increases the operating time of the facility.

Harmonic models and Bernoullicity

We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left-orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left-orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition.

New examples of Bernoulli algebraic actions

We give an example of a principal algebraic action of the non-commutative free group ${\mathbb {F}}$ of rank two by automorphisms of a connected compact abelian group for which there is an explicit measurable isomorphism with the full Bernoulli 3-shift action of ${\mathbb {F}}$ . The isomorphism is defined using homoclinic points, a method that has been used to construct symbolic covers of algebraic actions. To our knowledge, this is the first example of a Bernoulli algebraic action of ${\mathbb {F}}$ without an obvious independent generator. Our methods can be generalized to a large class of acting groups.

Group actions on quiver varieties and applications

We study algebraic actions of finite groups of quiver automorphisms on moduli spaces of quiver representations. We decompose the fixed loci using group cohomology and we give a modular interpretation of each component. As an application, we construct branes in hyperkähler quiver varieties, as fixed loci of such actions.

Mean dimension, mean rank, and von Neumann–Lück rank

We introduce an invariant, called mean rank, for any module{\mathcal{M}}of the integral group ring of a discrete amenable group Γ, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced Γ-action on the Pontryagin dual of{\mathcal{M}}, the mean rank of{\mathcal{M}}, and the von Neumann–Lück rank of{\mathcal{M}}all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin–Schnirelmann theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.

Garden of Eden and specification

We establish a Garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property, i.e. for any continuous equivariant map $T$ from the underlying space to itself, $T$ is pre-injective if and only if it is surjective. In particular, this applies to all expansive principal algebraic actions of amenable groups and expansive algebraic actions of $\mathbb{Z}^{d}$ with completely positive entropy.

Pseudo-orbit tracing and algebraic actions of countable amenable groups

Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved that they do. We provide examples showing that Chung and Li’s result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li’s algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.