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Register a new userTwo-dimensional self-interlocking structures in three-space
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It is well known that if there exists a finite set of convex bodies on the plane with non-overlapping interiors, then there is at least one extremal one among them, i.e., some one which can be continuously taken away to the infinity (outside a large ball containing all other bodies). In 3-space a phenomenon of self-interlocking structures takes place. A self-interlocking structure is such a set of three-dimensional convex bodies with non-overlapping interiors that any infinitesimal move of any of them is possible only as a part of the move of all bodies as a solid body. Previously known self-interlocking structures are based on configurations of cut cubes, tetrahedra, and octahedra. In the present paper we discover a principally new phenomenon of 2-dimensional self-interlocking structures: a family of 2-dimensional polygons in 3-space where no infinitesimal move of any piece is possible. (Infinitely thin) tiles are used to create {em decahedra}, which, in turn, used to create columns, which turn out to be stable when we fix some two extreme tiles. Seemingly, our work is the first appearance of the structure which is stable if we fix just two tiles (and not all but one). Two-dimensional self-interlocking structures naturally lead to three-dimensional structures possessing the same properties.
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This paper proves the following statement: {it If a convex body can form a twofold translative tiling in $mathbb{E}^3$, it must be a parallelohedron.} In other words, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron.
This paper proves the following statement: If a convex body can form a three or fourfold translative tiling in three-dimensional space, it must be a parallelohedron. In other words, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron.
A subset $X$ in the $d$-dimensional Euclidean space is called a $k$-distance set if there are exactly $k$ distances between two distinct points in $X$. Einhorn and Schoenberg conjectured that the vertices of the regular icosahedron is the only 12-point three-distance set in $mathbb{R}^3$ up to isomorphism. In this paper, we prove the uniqueness of 12-point three-distance sets in $mathbb{R}^3$.
The phenomenon of transparency in two-dimensional and three-dimensional superlattices is analyzed on the basis of the Boltzmann equation with a collision term encompassing three distinct scattering mechanisms (elastic, inelastic and electron-electron) in terms of three corresponding distinct relaxation times. On this basis, we show that electron heating in the plane perpendicular to the current direction drastically changes the conditions for the occurrence of self-induced transparency in the superlattice. In particular, it leads to an additional modulation of the current amplitudes excited by an applied biharmonic electric field with harmonic components polarized in orthogonal directions. Furthermore, we show that self-induced transparency and dynamic localization are different phenomena with different physical origins, displaced in time from each other, and, in general, they arise at different electric fields.
We discuss a new pseudometric on the space of all norms on a finite-dimensional vector space (or free module) $mathbb{F}^k$, with $mathbb{F}$ the real, complex, or quaternion numbers. This metric arises from the Lipschitz-equivalence of all norms on $mathbb{F}^k$, and seems to be unexplored in the literature. We initiate the study of the associated quotient metric space, and show that it is complete, connected, and non-compact. In particular, the new topology is strictly coarser than that of the Banach-Mazur compactum. For example, for each $k geqslant 2$ the metric subspace ${ | cdot |_p : p in [1,infty] }$ maps isometrically and monotonically to $[0, log k]$ (or $[0,1]$ by scaling the norm), again unlike in the Banach-Mazur compactum. Our analysis goes through embedding the above quotient space into a normed space, and reveals an implicit functorial construction of function spaces with diameter norms (as well as a variant of the distortion). In particular, we realize the above quotient space of norms as a normed space. We next study the parallel setting of the - also hitherto unexplored - metric space $mathcal{S}([n])$ of all metrics on a finite set of $n$ elements, revealing the connection between log-distortion and diameter norms. In particular, we show that $mathcal{S}([n])$ is also a normed space. We demonstrate embeddings of equivalence classes of finite metric spaces (parallel to the Gromov-Hausdorff setting), as well as of $mathcal{S}([n-1])$, into $mathcal{S}([n])$. We conclude by discussing extensions to norms on an arbitrary Banach space and to discrete metrics on any set, as well as some questions in both settings above.
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