Strongly Invariant Subgroups

Călugăreanu, Grigore

doi:10.1017/s0017089514000391

Cited by 22 publications

(37 citation statements)

References 8 publications

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“…In topological analysis, three parameters, linking number ( L k ), twist ( T w ), and writhe ( W r ), are applied to define the geometry of a closed twisted ribbon. The three parameters are connected by the Călugăreanu theorem as: L k = W r + T w 7. The linking number L k is an integer, with even numbers of L k describing orientable and two‐sided ribbons, whereas odd numbers of L k denote non‐orientable and one‐sided ribbons, such as Möbius systems.…”

Section: Methodsmentioning

confidence: 99%

Aggregation‐Induced‐Emission‐Active Macrocycle Exhibiting Analogous Triply and Singly Twisted Möbius Topologies

Wang

Zhao

et al. 2015

Chemistry A European J

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Molecules with Möbius topology have drawn increasing attention from scientists in a variety of fields, such as organic chemistry, inorganic chemistry, and material science. However, synthetic difficulties and the lack of functionality impede their fundamental understanding and practical applications. Here, we report the facile synthesis of an aggregation-induced-emission (AIE)-active macrocycle (TPE-ET) and investigate its analogous triply and singly twisted Möbius topologies. Because of the twisted and flexible nature of the tetraphenylethene units, the macrocycle adjusts its conformations so as to accommodate different guest molecules in its crystals. Moreover, theoretical studies including topological and electronic calculations reveal the energetically favorable interconversion process between triply and singly twisted topologies.

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Section: Methodsmentioning

confidence: 99%

Aggregation‐Induced‐Emission‐Active Macrocycle Exhibiting Analogous Triply and Singly Twisted Möbius Topologies

Wang

Zhao

et al. 2015

Chemistry A European J

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“…In general the self-linking number SL, can be decomposed into global geometric quantities, and one can show [19,20] …”

Section: Introductionmentioning

confidence: 99%

Helicity conservation under quantum reconnection of vortex rings

Zuccher

Ricca

2015

Phys. Rev. E

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Here we show that under quantum reconnection, simulated by using the three-dimensional GrossPitaevskii equation, self-helicity of a system of two interacting vortex rings remains conserved. By resolving the fine structure of the vortex cores, we demonstrate that total length of the vortex system reaches a maximum at the reconnection time, while both writhe helicity and twist helicity remain separately unchanged throughout the process. Self-helicity is computed by two independent methods, and topological information is based on the extraction and analysis of geometric quantities such as writhe, total torsion and intrinsic twist of the reconnecting vortex rings.

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“…Experimentally, the force and linking number of a topologically closed DNA double helix can be specified in a single-molecule manipulation experiment with torsional control [2]. Following a theorem that states that the imposed torsion or excess linking number (Lk) partitions into interconverting components of twist (Tw) and writhe (Wr) [3][4][5], the DNA molecule can either twist about its backbone or writhe into more complex shapes that depend on force and solution conditions. These DNA shapes produce mechanical signatures observable by experiment.At a critical torque, twisted DNA buckles: It loops upon itself and wraps into a superhelical writhed structure called a plectoneme (Fig.…”

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Evidence for a Solenoid Phase of Supercoiled DNA

Dittmore

Neuman

2018

Preprint

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In mechanical manipulation experiments, a single DNA molecule overwound at constant force undergoes a discontinuous drop in extension as it buckles and forms a superhelical loop (a plectoneme). Further overwinding the DNA, we observe an unanticipated cascade of highly regular discontinuous extension changes associated with stepwise plectoneme lengthening. This phenomenon is consistent with a model in which the force-extended DNA forms barriers to plectoneme lengthening caused by topological writhe. Furthermore, accounting for writhe in a fluctuating solenoid gives an improved description of the measured force-dependent effective torsional modulus of DNA, providing a reliable formula to estimate DNA torque. Our data and model thus provide context for further measurements and theories that capture the structures and mechanics of supercoiled biopolymers.Supercoiling of DNA permits its compaction in a cell and is crucial to the biological activities of DNA interacting proteins. Understanding the structures and mechanics of DNA supercoiling is therefore a topic of fundamental interest [1]. Experimentally, the force and linking number of a topologically closed DNA double helix can be specified in a single-molecule manipulation experiment with torsional control [2]. Following a theorem that states that the imposed torsion or excess linking number (Lk) partitions into interconverting components of twist (Tw) and writhe (Wr) [3][4][5], the DNA molecule can either twist about its backbone or writhe into more complex shapes that depend on force and solution conditions. These DNA shapes produce mechanical signatures observable by experiment.At a critical torque, twisted DNA buckles: It loops upon itself and wraps into a superhelical writhed structure called a plectoneme (Fig. 1). The plectoneme lengthens with additional imposed turns. Although this is widely assumed to be a smooth and continuous process, here we present experimental evidence that a DNA plectoneme lengthens in discrete steps that occur with a periodicity that exceeds the characteristic plectoneme repeat length (Fig. 1a). This behavior indicates supercoiling involves unexpected periodic free-energy barriers of unknown origin.The observed plectoneme lengthening steps can be interpreted as discontinuous exchanges of twist for writhe. At fixed force and torsion, the conjugate variables, extension and torque, are free to fluctuate. Since torque increases with increasing twist, the discrete plectoneme lengthening phenomenon corresponds to a progressive buildup, then release, of DNA torque (Fig. 1b).To explain this phenomenon, we propose a model in which barriers to plectoneme lengthening arise from the force-extended DNA adopting a solenoid structure (Fig. 2). If the force-extended region were straight without writhe, it would have continuous translational symmetry and adding length to the plectoneme would increase energy smoothly. In contrast, a solenoid shape is periodic and has discrete translational symmetry, resulting in a periodic energy associated with addin...

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Strongly Invariant Subgroups

Abstract: As a special case of fully invariant subgroups, strongly invariant subgroups are introduced and studied for Abelian groups.

Cited by 22 publications

References 8 publications

Aggregation‐Induced‐Emission‐Active Macrocycle Exhibiting Analogous Triply and Singly Twisted Möbius Topologies

Aggregation‐Induced‐Emission‐Active Macrocycle Exhibiting Analogous Triply and Singly Twisted Möbius Topologies

Helicity conservation under quantum reconnection of vortex rings

Evidence for a Solenoid Phase of Supercoiled DNA

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