• Title/Summary/Keyword: two-isometry

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GENERALIZED ISOMETRY IN NORMED SPACES

  • Zivari-Kazempour, Abbas
    • Communications of the Korean Mathematical Society
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    • v.37 no.1
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    • pp.105-112
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    • 2022
  • Let g : X ⟶ Y and f : Y ⟶ Z be two maps between real normed linear spaces. Then f is called generalized isometry or g-isometry if for each x, y ∈ X, ║f(g(x)) - f(g(y))║ = ║g(x) - g(y)║. In this paper, under special hypotheses, we prove that each generalized isometry is affine. Some examples of generalized isometry are given as well.

GEOMETRIC CLASSIFICATION OF ISOMETRIES ACTING ON HYPERBOLIC 4-SPACE

  • Kim, Youngju
    • Journal of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.303-317
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    • 2017
  • An isometry of hyperbolic space can be written as a composition of the reflection in the isometric sphere and two Euclidean isometries on the boundary at infinity. The isometric sphere is also used to construct the Ford fundamental domains for the action of discrete groups of isometries. In this paper, we study the isometric spheres of isometries acting on hyperbolic 4-space. This is a new phenomenon which occurs in hyperbolic 4-space that the two isometric spheres of a parabolic isometry can intersect transversally. We provide one geometric way to classify isometries of hyperbolic 4-space using the isometric spheres.

MAXIMUM SUBSPACES RELATED TO A-CONTRACTIONS AND QUASINORMAL OPERATORS

  • Suciu, Laurian
    • Journal of the Korean Mathematical Society
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    • v.45 no.1
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    • pp.205-219
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    • 2008
  • It is shown that if $A{\geq}0$ and T are two bounded linear operators on a complex Hilbert space H satisfying the inequality $T^*\;AT{\leq}A$ and the condition $AT=A^{1/2}TA^{1/2}$, then there exists the maximum reducing subspace for A and $A^{1/2}T$ on which the equality $T^*\;AT=A$ is satisfied. We concretely express this subspace in two ways, and as applications, we derive certain decompositions for quasinormal contractions. Also, some facts concerning the quasi-isometries are obtained.

GENERALIZATIONS OF ALESANDROV PROBLEM AND MAZUR-ULAM THEOREM FOR TWO-ISOMETRIES AND TWO-EXPANSIVE MAPPINGS

  • Khodaei, Hamid;Mohammadi, Abdulqader
    • Communications of the Korean Mathematical Society
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    • v.34 no.3
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    • pp.771-782
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    • 2019
  • We show that mappings preserving unit distance are close to two-isometries. We also prove that a mapping f is a linear isometry up to translation when f is a two-expansive surjective mapping preserving unit distance. Then we apply these results to consider two-isometries between normed spaces, strictly convex normed spaces and unital $C^*$-algebras. Finally, we propose some remarks and problems about generalized two-isometries on Banach spaces.

SUPERCYCLICITY OF TWO-ISOMETRIES

  • Ahmadi, M. Faghih;Hedayatian, K.
    • Honam Mathematical Journal
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    • v.30 no.1
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    • pp.115-118
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    • 2008
  • A bounded linear operator T on a complex separable Hilbert space H is called a two-isometry, if $T^{*2}T^2-2T^*T+1=0$. In this paper it is shown that every two-isometry is not supercyclic. This generalizes a result due to Ansari and Bourdon.

An Isometric Shape Interpolation Method on Mesh Models (메쉬 모델에 대한 아이소메트릭 형상 보간 방법)

  • Baek, Seung-Yeob;Lee, Kunwoo
    • Korean Journal of Computational Design and Engineering
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    • v.19 no.2
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    • pp.119-128
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    • 2014
  • Computing the natural-looking interpolation of different shapes is a fundamental problem of computer graphics. It is proved by some researchers that such an interpolation can be achieved by pursuing the isometry. In this paper, a novel coordinate system that is invariant under isometries is defined. The coordinate system can easily be converted from the global vertex coordinates. Furthermore, the global coordinates can be efficiently recovered from the new coordinates by simply solving two sparse least-squares problems. Since the proposed coordinate system is invariant under isometries, then transformations such as global rigid trans-formations, articulated posture deformations, or any other isometric deformations, do not change the coordinate values. Therefore, shape interpolation can be done in this framework without being affected by the distortions caused by the isometry.

Isometric Motion Recognition in Computer Animation

  • Lee, Myeong Won
    • Journal of the Korea Computer Graphics Society
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    • v.3 no.2
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    • pp.55-63
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    • 1997
  • This paper presents a method of detecting motion isometry from the motions of two objects in a three-dimensional space. We define the motion isometry based on the group theory and a newly defined coordinate system. Motion isometry can be detected using the coordinate system which we call Motion Specific Coordinate System(MSCS). In addition, we present an algorithm if two motions are isometric using the coordinate system. The algorithm can detect the difference in the motions of objects irrespective of their positions or the directions of their motions in a space. The algorithm can also detect the motion difference in the case of segmented objects which have several joints. The motion quantity is represented by translation values or rotation angles about some axes.

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CONTROLLABILITY OF ROLLING BODIES WITH REGULAR SURFACES

  • Moghadasi, S. Reza
    • Journal of the Korean Mathematical Society
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    • v.53 no.4
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    • pp.725-735
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    • 2016
  • A pair of bodies rolling on each other is an interesting example of nonholonomic systems in control theory. There is a geometric condition equivalent to the rolling constraint which enables us to generalize the rolling motions for any two-dimensional Riemannian manifolds. This system has a five-dimensional phase space. In order to study the controllability of the rolling surfaces, we lift the system to a six-dimensional space and show that the lifted system is controllable unless the two surfaces have isometric universal covering spaces. In the non-controllable case there are some three-dimensional orbits each of which corresponds to an isometry of the universal covering spaces.

SOME HYPERBOLIC SPACE FORMS WITH FEW GENERATED FUNDAMENTAL GROUPS

  • Cavicchioli, Alberto;Molnar, Emil;Telloni, Agnese I.
    • Journal of the Korean Mathematical Society
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    • v.50 no.2
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    • pp.425-444
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    • 2013
  • We construct some hyperbolic hyperelliptic space forms whose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane re ections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic 3-manifold with one cusp.