The Theory of Isometries

Millman, Richard S.; Parker, George D.

doi:10.1007/978-1-4684-0130-1_11

Cited by 3 publications

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“…It means a mapping that preserves distances. It is a bijective mapping, characterized as one-to-one mapping of a group onto itself or onto another in various transformational ways such as reflections, translation, or rotations (Miillman, R and Parker, G 1981).…”

Section: Introductionmentioning

confidence: 99%

Optinalysis: Isometric Isomorphism and Automorphism Through A Looking-Glass

Abdullahi¹

2020

Preprint

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Measures of graph symmetry, similarity, and identity have been extensively studied in graph automorphism and isomorphism detection problems. Nevertheless, graph isomorphism detection remains an open (unsolved) problem for many decades. In this paper, a new and efficient methodological paradigm, called optinalysis, is proposed for symmetry detections, similarity, and identity measures between isometric isomorphs or automorphs. Optinalysis is explained and expressed in clearly stated definitions and prove theorems, which conform to the definitions and theorems of isometry, isomorphism, and automorphism. Analogous to the polynomiality formalization for an efficient algorithm for graph isomorphism detection, optinalysis is however deterministic on polynomial and non-polynomial graph models.

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Section: Introductionmentioning

confidence: 99%

Optinalysis: Isometric Isomorphism and Automorphism Through A Looking-Glass

Abdullahi¹

2020

Preprint

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Section: Introductionmentioning

confidence: 99%

A Moment-Based Method of Optinalysis

Abdullahi¹

2020

Preprint

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Graph symmetry detection, similarity, and identity measures have been extensively studied in graph automorphism and isomorphism problems. Nevertheless, graph isomorphism and automorphism detection remain an open (unsolved) problem for many decades. In this paper, a new optinalytic coefficient termed as an optical moment coefficient was introduced for optinalysis. Its characteristic efficiency was tested for bijective property, invariance, deterministic polynomiality and non-polynomiality. The test results show that the new optical moment coefficient is very efficient for symmetry detections, similarity and identity measures between two isometric isomorphs and automorphs; and deterministic on polynomial and non-polynomial graph models.

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Kabirian-based optinalysis: A conceptually grounded framework for symmetry/asymmetry, similarity/dissimilarity and identity/unidentity estimations in mathematical structures and biological sequences

Abdullahi

2023

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The Theory of Isometries

Cited by 3 publications

References 0 publications

Optinalysis: Isometric Isomorphism and Automorphism Through A Looking-Glass

Optinalysis: Isometric Isomorphism and Automorphism Through A Looking-Glass

A Moment-Based Method of Optinalysis

Kabirian-based optinalysis: A conceptually grounded framework for symmetry/asymmetry, similarity/dissimilarity and identity/unidentity estimations in mathematical structures and biological sequences

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