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Concept# Three-dimensional space

Summary

In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, the Euclidean n-space of dimension n=3 that models physical space. More general three-dimensional spaces are called 3-manifolds.
Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space. The set of these n-tuples is commonly denoted \R^n, and can be identified to the pair formed by a n-dimensional Euclidean space and a Cartesian coordinate system.
When n = 3, this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). It serves as a model of the physical universe (when relativity theory is not considered), in which all known matter exists. While this space re

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Single-molecule imaging methods are of importance in structural biology, and specifically in the imaging of proteins, since they can elucidate conformational variability and structural changes that might be lost in imaging methods relying on averaging processes. Low-energy electron holography (LEEH) is a promising technique for imaging individual proteins with negligible radiation damage, which can be combined with a sample preparation process by native electrospray ion beam deposition (native ES-IBD) ensuring the creation of chemically pure samples suitable for holographic imaging.The central step in the analysis of data measured by LEEH is the numerical reconstruction of the object from the experimentally acquired holograms.Full information about the imaged object consists of both amplitude and phase information imprinted on the scattered wave during the interaction of the electron beam with the molecule and encoded in the hologram created by the interference of the scattered wave and the transmitted incident wave. A propagation-based algorithm with the goal of reconstructing the wave field in the plane of the object is presented and applied to holograms of individual proteins prepared by ES-IBD and measured with a low-energy electron holography microscope. The information retrieved from the reconstruction of the amplitude distribution in the object plane is discussed by analysing holograms of antibodies, demonstrating that the inherent conformational variability of these molecules can be mapped by LEEH. The influence of the sample preparation process on the surface conformations is tracked by tuning the landing energy of the proteins during deposition.To complement the amplitude data, an iterative phase retrieval algorithm is implemented to reconstruct the phase distribution in the object plane along with the amplitude distribution. The algorithm's performance and robustness is thoroughly evaluated and simulations regarding multiple scattering effects and element-dependent variations in scattering strength are carried out to provide a reference for the interpretation of phase data retrieved from experimentally acquired holograms. The iterative phase retrieval algorithm is then applied to protein data, indicating that both molecular density and charges can be related to features in the phase reconstructions, while the presence of metals does not correlate with specific phase signals at the current resolution obtainable from the experimental data.Since proteins are inherently three-dimensional, approaches towards three-dimensional reconstruction schemes are discussed, which will be the focus of future work.

Particle dampers have received considerable attention in recent years as a novel damping technique. However, due to its high degree of nonlinearity, no mature damping model has been constructed. This article proposes an equivalent raft model that takes into account both the three-dimensional geometry and the material property. The model is validated by comparing the benchmark results with experimental observations and to simulations using the discrete element method. The equivalent raft model predicts the behavior of the structure-damper system in the presence of ground motion. Additionally, it can compute the time evolution of particles' displacement and force in three different function modes. Moreover, the energy dissipation mechanism of particle damper is investigated using the suggested model to analyze the impact of load amplitude, load frequency, container size, particle mass, and particle material.

The so-called first selection lemma states the following: given any set P of n points in a"e (d) , there exists a point in a"e (d) contained in at least c (d) n (d+1)-O(n (d) ) simplices spanned by P, where the constant c (d) depends on d. We present improved bounds on the first selection lemma in a"e(3). In particular, we prove that c (3)a parts per thousand yen0.00227, improving the previous best result of c (3)a parts per thousand yen0.00162 by Wagner (On k-sets and applications. Ph.D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points and flats. Discrete Comput. Geom., 2010) (where it is proven that c (3)a parts per thousand currency sign1/4(4)a parts per thousand 0.00390) and Boros and Furedi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69-77, 1984) (where the two-dimensional case was settled).

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