scholarly journals Symmetries of Monocoronal Tilings

2015 ◽  
Vol Vol. 17 no.2 (Combinatorics) ◽  
Author(s):  
Dirk Frettlöh ◽  
Alexey Garber
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Dimensional Euclidean Space ◽  
Symmetry Groups ◽  
Translation Group ◽  
One Dimensional ◽  
Higher Dimensional ◽  
International Audience

International audience The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

scholarly journals The mathematical foundations of anelasticity: existence of smooth global intermediate configurations

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200462
Author(s):  
Christian Goodbrake ◽  
Alain Goriely ◽  
Arash Yavari
Keyword(s):  
Euclidean Space ◽  
Deformation Gradient ◽  
Deformation Tensor ◽  
Dimensional Euclidean Space ◽  
Multiplicative Decomposition ◽  
Mathematical Basis ◽  
Isometric Embeddings ◽  
Higher Dimensional ◽  
Mathematical Foundations ◽  
Central Tool

A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground. Here, we derive a sufficient condition for the existence of global intermediate configurations, starting from a multiplicative decomposition of the deformation gradient. We show that these global configurations are unique up to isometry. We examine the result of isometrically embedding these configurations in higher-dimensional Euclidean space, and construct multiplicative decompositions of the deformation gradient reflecting these embeddings. As an example, for a family of radially symmetric deformations, we construct isometric embeddings of the resulting intermediate configurations, and compute the residual stress fields explicitly.

The Development of Topological 4-manifold Theory

2021 ◽  
pp. 295-330
Author(s):  
Mark Powell ◽  
Arunima Ray
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Smooth Structures ◽  
Normal Bundles ◽  
Exotic Smooth Structures ◽  
Poincare Conjecture ◽  
Manifold Theory ◽  
Simply Connected

The development of topological 4-manifold theory is described in the form of a flowchart showing the interdependence among many key statements in the theory. In particular, the flowchart demonstrates how the theory crucially relies on the constructions in this book, what goes into the work of Quinn on smoothing, normal bundles, and transversality, and what is needed to deduce the famous consequences, such as the classification of closed, simply connected, topological 4-manifolds, the category preserving Poincaré conjecture, and the existence of exotic smooth structures on 4-dimensional Euclidean space. Precise statements of the results, brief indications of some proofs, and extensive references are provided.

Moments of coverage and coverage spaces

1966 ◽  
Vol 3 (02) ◽  
pp. 550-555 ◽  
Author(s):  
Gedalia Ailam
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Euclidean Plane ◽  
Random Sets ◽  
Dimensional Euclidean Space ◽  
Homogeneous Distribution ◽  
Transformation Formulas

Moments of the measure of the intersection of a given set with the union of random sets may be called moments of coverage. These moments specified for the case of independent random sets in the Euclidean space, were treated by several authors. Kolmogorov (1933) and Robbins (1944), (1945), (1947) derived transformation formulas for the moments of coverage under some specified conditions. Robbins' formula was applied by Robbins (1945), Santaló (1947) and Garwood (1947) to computations of expectations and variances of coverage in the cases of spheres and intervals in 2-dimensional and in n-dimensional Euclidean space. The computations were carried out under the assumption of a simple distribution function for the centers of the covering figures (a homogeneous distribution in most of the cases and a normal one in some of them). Neyman and Bronowski (1945) computed by a different method the variance of coverage for some specific cases in the Euclidean plane.

Inequalities for random flats meeting a convex body

1985 ◽  
Vol 22 (03) ◽  
pp. 710-716 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Finite Number ◽  
Random Point ◽  
Dimensional Euclidean Space ◽  
Random Direction ◽  
Higher Dimensional

We choose a uniform random point in a given convex bodyKinn-dimensional Euclidean space and through that point the secant ofKwith random direction chosen independently and isotropically. Given the volume ofK, the expectation of the length of the resulting random secant ofKwas conjectured by Enns and Ehlers [5] to be maximal ifKis a ball. We prove this, and we also treat higher-dimensional sections defined in an analogous way. Next, we consider a finite number of independent isotropic uniform random flats meetingK, and we prove that certain geometric probabilities connected with these again become maximal whenKis a ball.

scholarly journals NEW MEAN VALUES FOR HOMOGENEOUS SPATIAL TESSELLATIONS THAT ARE STABLE UNDER ITERATION

2010 ◽  
Vol 29 (3) ◽  
pp. 143 ◽  
Author(s):  
Christoph Thäle ◽  
Viola Weiss
Keyword(s):  
Euclidean Space ◽  
Convex Geometry ◽  
Dimensional Euclidean Space ◽  
Mean Values ◽  
3 Dimensional ◽  
Extremum Problems ◽  
Random Tessellations ◽  
Typical Cell

Homogeneous random tessellations in the 3-dimensional Euclidean space are considered that are stable under iteration – STIT tessellations. A classification of vertices, segments and flats is introduced and a couple of new metric and topological mean values for them and for the typical cell are calculated. They are illustrated by two examples, the isotropic and the cuboid case. Several extremum problems for these mean values are solved with the help of techniques from convex geometry by introducing an associated zonoid for STIT tessellations.

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