scholarly journals About the geographic grid of the three-dimensional hypersphere

2020 ◽  
Vol 223 ◽  
pp. 02005
Author(s):  
Pavel Kim
Keyword(s):  
Population Distribution ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The State ◽  
Dimensional Manifold ◽  
State Assignment ◽  
Distribution Process ◽  
Technological Means ◽  
Simply Connected ◽  
Three Dimensional Space

According to the Poincaré conjecture (1904) proved by Grigory Perelman (2002-2003) that any simply connected compact three- dimensional manifold without edges is homeomorphic to a three- dimensional hypersphere [1], to solve the problems of visualizing four- dimensional objects in three-dimensional space [2], it is proposed to choose a suitable manifold, in in this case, a ball, establishing a homeomorphism between objects located in different spaces by technological means of cartography. As a result of this work, it seems possible to build a dynamic video of the population distribution process on a map of the globe, which provides informational four-dimensional data flow, following the ideas embodied in 4D Anatomy [3]. The proposed technology opens up new ways of visualizing four-dimensional space This work was performed within the framework of the state assignment of the ICM MG SB RAS (project 0315-2019-0003).

scholarly journals Harmonic morphisms and conformal foliations by geodesics of three-dimensional space forms

1991 ◽  
Vol 51 (1) ◽  
pp. 118-153 ◽  
Author(s):  
Paul Baird ◽  
John C. Wood
Keyword(s):  
Space Form ◽  
Holomorphic Mapping ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Complete Classification ◽  
Harmonic Morphisms ◽  
Connected Space ◽  
Space Forms ◽  
Simply Connected ◽  
Three Dimensional Space

AbstractA complete classification is given of harmonic morphisms to a surface and conformal foliations by geodesics, with or without isolated singularities, of a simply-connected space form. The method is to associate to any such a holomorphic map from a Riemann surface into the space of geodesics of the space form. Properties such as nonintersecting fibres (or leaves) are translated into conditions on the holomorphic mapping which show it must have a simple form corresponding to a standard example.

scholarly journals Assessment of an isogeometric approach with Catmull–Clark subdivision surfaces using the Laplace–Beltrami problems

2020 ◽  
Vol 66 (4) ◽  
pp. 851-876
Author(s):  
Zhaowei Liu ◽  
Andrew McBride ◽  
Prashant Saxena ◽  
Paul Steinmann
Keyword(s):  
Convergence Rate ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Beltrami Equation ◽  
Subdivision Surfaces ◽  
Dimensional Manifold ◽  
Adaptive Quadrature ◽  
Quadrature Scheme ◽  
Fitting Method ◽  
Three Dimensional Space

Abstract An isogeometric approach for solving the Laplace–Beltrami equation on a two-dimensional manifold embedded in three-dimensional space using a Galerkin method based on Catmull–Clark subdivision surfaces is presented and assessed. The scalar-valued Laplace–Beltrami equation requires only $$C^0$$ C 0 continuity and is adopted to elucidate key features and properties of the isogeometric method using Catmull–Clark subdivision surfaces. Catmull–Clark subdivision bases are used to discretise both the geometry and the physical field. A fitting method generates control meshes to approximate any given geometry with Catmull–Clark subdivision surfaces. The performance of the Catmull–Clark subdivision method is compared to the conventional finite element method. Subdivision surfaces without extraordinary vertices show the optimal convergence rate. However, extraordinary vertices introduce error, which decreases the convergence rate. A comparative study shows the effect of the number and valences of the extraordinary vertices on accuracy and convergence. An adaptive quadrature scheme is shown to reduce the error.

Perceptual-cognitive universals as reflections of the world

2001 ◽  
Vol 24 (4) ◽  
pp. 581-601 ◽  
Author(s):  
Roger N. Shepard
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
Euclidean Group ◽  
Dimensional Vector Space ◽  
The World ◽  
Geodesic Paths ◽  
Abstract Spaces ◽  
The Universe ◽  
Three Dimensional Space

The universality, invariance, and elegance of principles governing the universe may be reflected in principles of the minds that have evolved in that universe – provided that the mental principles are formulated with respect to the abstract spaces appropriate for the representation of biologically significant objects and their properties. (1) Positions and motions of objects conserve their shapes in the geometrically fullest and simplest way when represented as points and connecting geodesic paths in the six-dimensional manifold jointly determined by the Euclidean group of three-dimensional space and the symmetry group of each object. (2) Colors of objects attain constancy when represented as points in a three-dimensional vector space in which each variation in natural illumination is canceled by application of its inverse from the three-dimensional linear group of terrestrial transformations of the invariant solar source. (3) Kinds of objects support optimal generalization and categorization when represented, in an evolutionarily-shaped space of possible objects, as connected regions with associated weights determined by Bayesian revision of maximum-entropy priors.

scholarly journals Electromechanical Catastrophe

2016 ◽  
Vol 08 (07) ◽  
pp. 1640005 ◽  
Author(s):  
Tongqing Lu ◽  
Sibo Cheng ◽  
Tiefeng Li ◽  
Tiejun Wang ◽  
Zhigang Suo
Keyword(s):  
Nonlinear System ◽  
Horizontal Plane ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Vertical Axis ◽  
Mechanical Force ◽  
The State ◽  
Constant Force ◽  
Constant Voltage ◽  
Three Dimensional Space

A transducer is a system that couples two loads. For example, an electromechanical transducer couples a mechanical force and an electrical voltage. A two-load, nonlinear system can exhibit rich behavior of bifurcation, which can be displayed in a three-dimensional space, with the horizontal plane representing the two loads, and the vertical axis representing the state of the system. In this three-dimensional space, a state of equilibrium at fixed loads corresponds to a point on a surface. The surface is smooth, but its projection to the load plane results in singularities of two types: fold and cusp. Here we identify the fold and cusp for a dielectric elastomer transducer by a combination of experiment and calculation. We conduct two kinds of experiment: electrical actuation under a constant force and mechanical pulling under a constant voltage. The theory and the experiment agree well. The fold and cusp are essential in the design of loading paths to avoid or harness the bifurcation.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

A Peptoid with Extended Shape in Water

2019 ◽  
Author(s):  
Jumpei Morimoto ◽  
Yasuhiro Fukuda ◽  
Takumu Watanabe ◽  
Daisuke Kuroda ◽  
Kouhei Tsumoto ◽  
...  
Keyword(s):  
Spectroscopic Analysis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Biomolecular Recognition ◽  
Biological Application ◽  
New Class ◽  
Proteolytic Resistance ◽  
Pharmacokinetic Properties ◽  
Optically Pure ◽  
Three Dimensional Space

<div> <div> <div> <p>“Peptoids” was proposed, over decades ago, as a term describing analogs of peptides that exhibit better physicochemical and pharmacokinetic properties than peptides. Oligo-(N-substituted glycines) (oligo-NSG) was previously proposed as a peptoid due to its high proteolytic resistance and membrane permeability. However, oligo-NSG is conformationally flexible and is difficult to achieve a defined shape in water. This conformational flexibility is severely limiting biological application of oligo-NSG. Here, we propose oligo-(N-substituted alanines) (oligo-NSA) as a new peptoid that forms a defined shape in water. A synthetic method established in this study enabled the first isolation and conformational study of optically pure oligo-NSA. Computational simulations, crystallographic studies and spectroscopic analysis demonstrated the well-defined extended shape of oligo-NSA realized by backbone steric effects. The new class of peptoid achieves the constrained conformation without any assistance of N-substituents and serves as an ideal scaffold for displaying functional groups in well-defined three-dimensional space, which leads to effective biomolecular recognition. </p> </div> </div> </div>

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