Rational index of Vector Addition Systems languages

Laurent Pierre; SylvianeR. Schwer

doi:10.1007/bf00263577

Inglehart-Welzel's Traditional vs. Rational Index Revisited: A Comparison between China and the West

SSRN Electronic Journal ◽

10.2139/ssrn.2179576 ◽

2012 ◽

Author(s):

Man-Li Gu

Keyword(s):

The West ◽

Rational Index

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Dipole Moments of para-Substituted N-Phenylsulfonyl-N'-Allylthioureas

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19950856 ◽

1995 ◽

Vol 60 (5) ◽

pp. 856-862

Author(s):

Gejza Suchár ◽

Ivan Danihel

Keyword(s):

Dipole Moments ◽

Vector Addition ◽

Stretching Vibrations

Dipole moments of a series of para-substituted N-phenylsulfonyl-N'-allylthioureas were determined. Comparison of the experimentally found dipole moments with those calculated by vector addition of bond and group moments has shown that (E) conformation at the N-C bonds is preferred. The same result was obtained from the N-H stretching vibrations. The results are compatible with a synperiplanar arrangement at the C-C bond of the allyl moiety.

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Vector Arithmetic in the Triangular Grid

Entropy ◽

10.3390/e23030373 ◽

2021 ◽

Vol 23 (3) ◽

pp. 373

Author(s):

Khaled Abuhmaidan ◽

Monther Aldwairi ◽

Benedek Nagy

Keyword(s):

Coordinate System ◽

Triangular Grid ◽

Vector Addition ◽

Plane Space ◽

Physical Simulations ◽

Rectangular Grids ◽

Cartesian Coordinate ◽

Zero Sum ◽

Coordinate Geometry ◽

Point Lattice

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.

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