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The property of symmetry in 3D objects is helpful in various applications such as object alignment, compression, symmetrical editing or reconstruction of incomplete objects. However, its robust and efficient detection is a challen...
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The property of symmetry in 3D objects is helpful in various applications such as object alignment, compression, symmetrical editing or reconstruction of incomplete objects. However, its robust and efficient detection is a challenging task. The two most commonly occurring types of symmetry are probably reflectional and rotational symmetry. While reflectional symmetry detection methods are quite plentiful, this does not seem to be the case with rotational symmetry detection. In this paper a use of approximate reflectional symmetries to derive plausible approximate rotational symmetries is proposed that can be integrated with multiple different approaches for reflectional symmetry detection. One such specific approach, based on maximizing a given symmetry measure, is chosen and combined with this idea. A modification of the maximization step for rotations is further proposed using a simple, yet efficient, quaternion-based parameterization of the rotation transformation which seems novel in the field of symmetry detection. The results confirm that this combination provides a robust and efficient solution for finding rotational symmetry in a 3D point set and can handle approximate symmetry, noisy input or even partial data.
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The Gauss map of non-degenerate surfaces in the three-dimensional Minkowski space are viewed as dynamical fields of the two-dimensional O(2, 1) Nonlinear Sigma Model. In this setting, the moduli space of solutions with rotational ...
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The Gauss map of non-degenerate surfaces in the three-dimensional Minkowski space are viewed as dynamical fields of the two-dimensional O(2, 1) Nonlinear Sigma Model. In this setting, the moduli space of solutions with rotational symmetry is completely determined. Essentially, the solutions are warped products of orbits of the 1-dimensional groups of isometries and elastic curves in either a de Sitter plane, a hyperbolic plane or an anti de Sitter plane. The main tools are the equivalence of the two-dimensional O(2, 1) Nonlinear Sigma Model and the Willmore problem, and the description of the surfaces with rotational symmetry. A complete classification of such surfaces is obtained in this paper. Indeed, a huge new family of Lorentzian rotational surfaces with a space-like axis is presented. The description of this new class of surfaces is based on a technique of surgery and a gluing process, which is illustrated by an algorithm.
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The authors of [1] describe a method for producing a set of k-bead, n-color necklaces for the group C_k 2 C_n and define two such necklaces to be equivalent if one can be mapped to the other by means of a rotation. They prove that...
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The authors of [1] describe a method for producing a set of k-bead, n-color necklaces for the group C_k 2 C_n and define two such necklaces to be equivalent if one can be mapped to the other by means of a rotation. They prove that each nonequivalent necklace in the set of k-bead, n-color necklaces can be used to form a degree-k representation of C_k 2 C_n and that the degree-k representations derived from necklaces having s-fold rotational symmetry will reduce into s irreducible representations of degree m = k/s. In what we call here the "necklace conjecture," the authors of [1] conjecture that the irreducible representations of C_k 2 C_n obtained from nonequivalent necklaces are mutually nonequivalent. Finally, they prove that if the necklace conjecture is true, then the set of irreducible representations of C_k 2 C_n obtained in this way is a minimal complete set of irreducible representations for C_k 2 C_n. In this paper, we prove the necklace conjecture, thereby establishing the result that the set of nonequivalent necklaces produces exactly one copy of each irreducible representation of C_k 2 C_n. Note that we employ the same terminology and notation as in [1], and throughout, we let ε_n = e~(2πi/n).
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We predict level degeneracy of the rotational type in diatomic molecules described by means of a cotangent-hindered rigid rotator. The problem is shown to be exactly solvable in terms of non-classical Romanovski polynomials. The e...
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We predict level degeneracy of the rotational type in diatomic molecules described by means of a cotangent-hindered rigid rotator. The problem is shown to be exactly solvable in terms of non-classical Romanovski polynomials. The energies of such a system are linear combinations of t(t + 1) and 1/[t (t + 1) + 1/4] terms with the non-negative integer principal quantum number t=n+|m?| being the sum of order n of the polynomials and the absolute value, |m?|, of the square root of the separation constant between the polar and azimuthal angular motions. The latter obeys, with respect to t, the same branching rule, |m?|=0,1t, as does the magnetic quantum number with respect to the angular momentum, l, and, in this fashion, the t quantum number presents itself formally indistinguishable from l. In effect, the spectrum of the hindered rotator has the same (2t + 1)-fold level multiplicity as the unperturbed one. For small t values, the wave functions and excitation energies of the perturbed rotator differ from the ordinary spherical harmonics, and the l(l + 1) law, respectively, while approaching them asymptotically with increasing t. In this fashion the breaking of the rotational symmetry at the level of the representation functions is opaqued by the level degeneracies. The model provides a tool for the description of rotational bands with anomalously large gaps between the ground state and its first excitation.
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The internal symmetry of composite relativistic systems is discussed. It is demonstrated that the Lorentz–Poincar′e symmetry implies the existence of internal moments associated with the Lorentz boost, which are Laplace–Runge–...
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The internal symmetry of composite relativistic systems is discussed. It is demonstrated that the Lorentz–Poincar′e symmetry implies the existence of internal moments associated with the Lorentz boost, which are Laplace–Runge–Lenz (LRL) vectors. The LRL symmetry is thus found to be the internal symmetry universally associated with the global Lorentz transformations, in much the same way as internal spatial rotations are associated with global spatial rotations. Two applications are included for an interacting two-body system and for an interaction-free many-body system of particles. The issue of localizability of the relativistic center-of-mass coordinate is also discussed.
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Symmetry, in the sense of repetitive spatial arrangements, takes many specific forms that we encounter routinely, usually recognize visually, and have some difficulty in quantifying. As there are many types of symmetry, some of th...
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Symmetry, in the sense of repetitive spatial arrangements, takes many specific forms that we encounter routinely, usually recognize visually, and have some difficulty in quantifying. As there are many types of symmetry, some of them partial or imperfect, so there are many measurement approaches. Some of these consider only the outline or boundary of an object, others include the interior structure, some apply to an entire image, while others operate on individual objects. Examples of the various classes of symmetry and several methods for analysis are presented.
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Exploring the symmetries underlying a previously proposed encryption scheme that relies on single-qubit rotations, we derive an improved upper bound on the maximum information that an eavesdropper might extract from all the availa...
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Exploring the symmetries underlying a previously proposed encryption scheme that relies on single-qubit rotations, we derive an improved upper bound on the maximum information that an eavesdropper might extract from all the available copies of the public key. Subsequently, the robustness of the scheme is investigated in the context of attacks that address each public-key qubit independently. The attacks under consideration make use of projective measurements on single qubits and their efficiency is compared to attacks that address many qubits collectively and require complicated quantum operations.
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Two conditions on symmetries are identified as necessary for a linear scattering system to be able to rotate the linear polarization of light: Lack of at least one mirror plane of symmetry and electromagnetic duality symmetry. Dua...
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Two conditions on symmetries are identified as necessary for a linear scattering system to be able to rotate the linear polarization of light: Lack of at least one mirror plane of symmetry and electromagnetic duality symmetry. Duality symmetry is equivalent to the conservation of the helicity of light in the same way that rotational symmetry is equivalent to the conservation of angular momentum. When the system is a solution of a single species of particles, the lack of at least one mirror plane of symmetry leads to the familiar requirement of chirality of the individual particle. With respect to helicity preservation, according to the analytical and numerical evidence presented in this paper, the solution preserves helicity if and only if the individual particle itself preserves helicity. However, only in the particular case of forward scattering the helicity preservation condition on the particle is relaxed: We show that the random orientation of the molecules endows the solution with an effective rotational symmetry; at its turn, this leads to helicity preservation in the forward scattering direction independently of any property of the particle. This is not the case for a general scattering direction. These results advance the current understanding of the phenomena of molecular optical activity and provide insight for the design of polarization control devices at the nanoscale.
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This paper concentrates on the fold number detection problem for the shapes with monotonic radii. The proposed method is extremely simple. Two monotonicity conditions are derived to ensure that the smallest positive integer l maki...
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This paper concentrates on the fold number detection problem for the shapes with monotonic radii. The proposed method is extremely simple. Two monotonicity conditions are derived to ensure that the smallest positive integer l making integral integral((r,0)is an element of S) r(2) e(il theta) dr d theta nonzero is exactly the fold number of the given shape S. The fold numbers of regular polygons, roses, bolt nuts, and other kinds of shapes discussed in the present paper, can therefore be detected quite easily. Note especially that the proposed method uses no matching procedure, a procedure essential in many reported methods. Theoretical properties, mathematical proofs, illustrative figures, and experimental results, are all included in this paper. (C) 1996 Pattern Recognition Society. Published by Elsevier Science Ltd. [References: 10]
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This paper considers the blind detection of spatial modulation multiple-input multiple-output (SM-MIMO) systems based on clustering algorithms. Firstly, we propose a novel framework (framework 1) that can guarantee the excellent d...
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This paper considers the blind detection of spatial modulation multiple-input multiple-output (SM-MIMO) systems based on clustering algorithms. Firstly, we propose a novel framework (framework 1) that can guarantee the excellent detection performance of clustering detectors to implement the signal detection even if in the case of short block length by considering the rotational symmetry of digital modulation constellations. Then, we propose a novel rule by defining the concept of pseudo rotational symmetry of cluster centroids, which can check the quality of the cluster centroids of clustering algorithms. Besides, based on the proposed rule, we investigate two scenarios and propose framework 2 and framework 3 for the problems in the scenarios. Specifically, the framework 2 can early terminate the clustering algorithms that are performed multiple times different initializations to avoid poor detection performance, and it significantly reduces computational complexity while maintaining excellent performance. The framework 3 can adjust the data length of clustering algorithms according to different environmental conditions that can improve the detection efficiency while improving performance. Moreover, the proposed framework 1 can be combined with proposed framework 2 or framework 3, which further enhances the detection performance. In addition, we conduct the performance analysis for the SM-MIMO systems when the perfect channel state information (CSI) is unavailable. Simulation results demonstrate the clustering detectors combined with proposed frameworks behave better than the clustering detectors without the proposed frameworks in terms of performance or complexity.
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