Narrow Results

We study topological properties of Ind-groups $\mathrm{\text{Aut}}(K[x_1,\dots ,x_n])$ and $\mathrm{\text{Aut}}(K〈x_1,\dots ,x_n〉)$ of automorphisms of polynomial and free associative algebras via Ind-schemes, toric varieties, approximations, and singularities. We obtain a number of properties of $\mathrm{\text{Aut}}(\mathrm{\text{Aut}}(A))$, where $A$ is the polynomial or free associative algebra over the base field $K$. We prove that all Ind-scheme automorphisms of $\mathrm{\text{Aut}}(K[x_1,\dots ,x_n])$ are inner for $n\ge 3$, and all Ind-scheme automorphisms of $\mathrm{\text{Aut}}(K〈x_1,\dots ,x_n〉)$ are semi-inner. As an application, we prove that $\mathrm{\text{Aut}}(K[x_1,\dots ,x_n])$ cannot be embedded into $\mathrm{\text{Aut}}(K〈x_1,\dots ,x_n〉)$ by the natural abelianization. In other words, the Automorphism Group Lifting Problem has a negative solution. We explore close connection between the above results and the Jacobian conjecture, as well as the Kanel-Belov–Kontsevich conjecture, and formulate the Jacobian conjecture for fields of any characteristic. We make use of results of Bodnarchuk and Rips, and we also consider automorphisms of tame groups preserving the origin and obtain a modification of said results in the tame setting.

Roger Penrose’s 2020 Nobel Prize in Physics recognizes that his identification of the concepts of “gravitational singularity” and an “incomplete, inextendible, null geodesic” is physically very important. The existence of an incomplete, inextendible, null geodesic does not say much, however, if anything, about curvature divergence, nor is it a helpful definition for performing actual calculations. Physicists have long sought for a coordinate independent method of defining where a singularity is located, given an incomplete, inextendible, null geodesic, that also allows for standard analytic techniques to be implemented. In this essay, we present a solution to this issue. It is now possible to give a concrete relationship between an incomplete, inextendible, null geodesic and a gravitational singularity, and to study any possible curvature divergence using standard techniques.

This chapter reveals the depth of the transformational nature inherent in the Mereon Matrix showing how clusters and singles of polyhedra, 33 unified forms, scale to infinity, internally and externally. This chapter suggests significant physical, psychological and sociological implications inherent in the structure and dynamics of the Matrix’s outermost polyhedron. Referred to as the Context, this geosphere iterates between 120 and 180 faces as it opens and closes, expands and contracts. Image intense, it is revealed how localised environmental energy is required, received and processed to eventually produce new systems and/or product that expands the global environment.

The asymptotic expansion of the solution of a singularly perturbed parabolic problem in a general, bounded, and smooth domain is considered when the diffusivity parameter ∊ is small by using techniques of curvilinear coordinates and Laplace transformation. The boundedness of asymptotic expansion is estimated. Convergence results with respect to ∊ are carefully proved. Actual errors of all orders are performed, especially the optimal error order is obtained.