A Classification of Mappings of the Three-Dimensional Complex into the Two-Dimensional Sphere

2019 ◽  
pp. 223-260
Author(s):  
R. V. Gamkrelidze
Keyword(s):  
Three Dimensional ◽  
Dimensional Sphere ◽  
Two Dimensional

Einführung in eine anorganische Strukturchemie

1986 ◽  
Vol 175 (3-4) ◽  
Author(s):  
E. Hellner
Keyword(s):  
Three Dimensional ◽  
Standard Setting ◽  
Transformation Matrix ◽  
Two Dimensional ◽  
Coordination Polyhedra ◽  
Systematic Description ◽  
Point Configurations ◽  
Inorganic Structure

AbstractA systematic description and classification of inorganic structure types is proposed on the basis of homogeneous or heterogeneous point configurations (Bauverbände) described by invariant lattice complexes and coordination polyhedra; subscripts or matrices explain the transformation of the complexes in respect (M) to their standard setting; the value of the determinant of the transformation matrix defines the order of the complex. The Bauverbände (frameworks) may be described by three-dimensional networks or two-dimensional nets explicitely shown with structures types of the

scholarly journals GEOMETRIC MOMENTUM IN THE MONGE PARAMETRIZATION OF TWO-DIMENSIONAL SPHERE

2013 ◽  
Vol 10 (03) ◽  
pp. 1220031 ◽  
Author(s):  
D. M. XUN ◽  
Q. H. LIU
Keyword(s):  
Quantum Mechanics ◽  
Laplace Operator ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Constrained Motion ◽  
Geometric Invariant ◽  
Gradient Operator ◽  
The Laplace Operator

A two-dimensional (2D) surface can be considered as three-dimensional (3D) shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of 2D sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [xi, pj] = iℏ(δij - xixj/r2) rather than [xi, pj] = iℏδij that does not hold true anymore. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

Comparison of three-dimensional and two-dimensional computed tomographies in the classification of acetabular fractures

2019 ◽  
Vol 27 (2) ◽  
pp. 157-164 ◽  
Author(s):  
Thanat Kanthawang ◽  
Tanawat Vaseenon ◽  
Patumrat Sripan ◽  
Nuttaya Pattamapaspong
Keyword(s):  
Three Dimensional ◽  
Acetabular Fractures ◽  
Two Dimensional

scholarly journals Automated classification of blood cell neutrophils.

1977 ◽  
Vol 25 (7) ◽  
pp. 633-640 ◽  
Author(s):  
J K Mui ◽  
K S Fu ◽  
J W Bacus
Keyword(s):  
Blood Cell ◽  
White Blood Cell ◽  
Classification Accuracy ◽  
Three Dimensional ◽  
Classification Problem ◽  
Differential Count ◽  
Two Dimensional ◽  
Automated Classification ◽  
Exact Shape

The classification of white blood cell neutrophils into band neutrophils (bands) and segmented neutrophils (segs) is a subproblem of the white blood cell differential count. This classification problem is not well defined for at least two reasons: (a) there are no unique quantitative definitions for bands and segs and (b) existing definitions use the shape of the nucleus as the only discriminating criterion. When cells are classified on a slide, decisions are made from the two-dimensional views of these three-dimensional cells. A problem arises because the exact shape of the nucleus becomes indeterminate when the nucleus overlaps so that the filament is hidden. To assess the importance of this problem, this paper quantitates the classification errors due to overlapped nuclei (ON). The results indicate that, using only neutrophils without ON, the classification accuracy is 89%. For neutrophils with ON, the classification accuracy is 65%. This suggests a classification strategy of first classifying neutrophils into three categories: (a) bands without ON, (b) segs without ON and (c) neutrophils with ON. Category III can then be further classified into segs and bands by other stretegies.

scholarly journals ON RELATION BETWEEN GEOMETRIC MOMENTUM AND ANNIHILATION OPERATORS ON A TWO-DIMENSIONAL SPHERE

2013 ◽  
Vol 10 (06) ◽  
pp. 1320007 ◽  
Author(s):  
Q. H. LIU ◽  
Y. SHEN ◽  
D. M. XUN ◽  
X. WANG
Keyword(s):  
Angular Momentum ◽  
Coherent States ◽  
Three Dimensional ◽  
Extrinsic Curvature ◽  
Flat Space ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Intrinsic Geometry

With a recently introduced geometric momentum that depends on the extrinsic curvature and offers a proper description of momentum on two-dimensional sphere, we show that the annihilation operators whose eigenstates are coherent states on the sphere take the expected form αx + iβp, where α and β are two operators that depend on the angular momentum and x and p are the position and the geometric momentum, respectively. Since the geometric momentum is manifestly a consequence of embedding the two-dimensional sphere in the three-dimensional flat space, the coherent states reflects some aspects beyond the intrinsic geometry of the surfaces.

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