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Topological Space‐Time Photonic Transitions in Angular‐Momentum‐Biased Metasurfaces
Advanced Optical Materials, 2020In this paper, topological space‐time photonic transition of light in angular‐momentum‐biased metasurfaces is established, which yields a superposition of orbital‐angular‐momentum (OAM)‐carrying beams at distinct frequency harmonics upon scattering whose
H. Barati Sedeh +2 more
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On β*-supra topological spaces
Journal of Discrete Mathematical Sciences & Cryptography, 2023In this paper, a new type of supra closed sets is introduced which we called supra β*-closed sets in a supra topological space. A new set of separation axioms is defined, and its many properties are examined. The relationships between supra β*-Ti –spaces
Ehab A. Ghaloob, A. R. Sadek
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Triangulating topological spaces
Proceedings of the tenth annual symposium on Computational geometry - SCG '94, 1994Given a subspace [Formula: see text] and a finite set S⊆ℝd, we introduce the Delaunay complex, [Formula: see text], restricted by [Formula: see text]. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets [Formula: see text] in a non-empty set. By the nerve theorem, [Formula: see text] and [Formula: see text]
Herbert Edelsbrunner, Nimish R. Shah
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2006
Publisher Summary This chapter provides an overview of the topological spaces. A topological space (X, τ) is a set X with a topology τ, that is, a collection of subsets of X with the following properties: (1) X ∈ τ, o ∈ τ; (2) if A, B ∈ τ then A ∩ B ∈ τ ; and (3) for any collection {Aα}α, if all Aα ∈ τ , then UαAα ∈ τ .
Elena Deza, Michel-Marie Deza
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Publisher Summary This chapter provides an overview of the topological spaces. A topological space (X, τ) is a set X with a topology τ, that is, a collection of subsets of X with the following properties: (1) X ∈ τ, o ∈ τ; (2) if A, B ∈ τ then A ∩ B ∈ τ ; and (3) for any collection {Aα}α, if all Aα ∈ τ , then UαAα ∈ τ .
Elena Deza, Michel-Marie Deza
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Topology and Topological Spaces
1997We have now seen four different proofs of the Fundamental Theorem of Algebra. The first two were purely analysis, while the second pair involved a wide collection of algebraic ideas. However, we should realize that even in these algebraic proofs we did not totally leave analysis.
Gerhard Rosenberger, Benjamin Fine
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Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions
Nature Physics, 2021Topological phases of matter connect mathematical principles to real materials, and may shape future electronic and quantum technologies. So far, this discipline has mostly focused on single-gap topology described by topological invariants such as Chern ...
B. Jiang +7 more
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Nature Photonics, 2014
Applying the mathematical concept of topology to the wave-vector space of photonics yields exciting opportunities for creating new states of light with useful properties such as unidirectional propagation and the ability to flow around imperfections. The
Ling Lu, J. Joannopoulos, M. Soljačić
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Applying the mathematical concept of topology to the wave-vector space of photonics yields exciting opportunities for creating new states of light with useful properties such as unidirectional propagation and the ability to flow around imperfections. The
Ling Lu, J. Joannopoulos, M. Soljačić
semanticscholar +1 more source