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Precise 3D Tracking of Highly Non-Planar Eukaryotic Flagellar Beating Patterns Using Digital Holographic Microscopy. [PDF]

open access: yesSmall Methods
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Planar Separators

SIAM Journal on Discrete Mathematics, 1994
A separator in a graph \(G\) is a partition \((A,B,C)\) of \(V(G)\) such that \(| A|\), \(| B|\leq {2\over 3}| V(G)|\) and no vertex in \(A\) is adjacent to any vertex in \(B\); its order is \(| C|\). The authors give a short proof of a theorem of Lipton and Tarjan that for any planar graph with \(n\) vertices there is a separator of order \(\leq 2 ...
Noga Alon   +2 more
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Planar and non-planar borepins

Molecular Physics, 1996
Theoretical studies of borepin and its B-substituted derivatives show planar structures with weak out-of-plane bending modes, often leading to boat-shaped borepins. The lowest vibrational frequency of borepin is 167 cm-1 at both HF/6-31G* and MP2/6-31G* levels, compared to 248 cm-1 calculated for the isoelectronic tropylium ion.
JEROME SCHULMAN, RAYMOND DISCH
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Planar Crossovers

IEEE Transactions on Computers, 1981
Summary: Those bases which permit the realization of a planar crossover are characterized.
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PLANAR LACES

Journal of Knot Theory and Its Ramifications, 2002
Let S be a connected open subset of 2-sphere S2 which is identified with the extended plane R2 ∪ {∞}. We assume that S contains the n segments {1, 2, …, n} × [-1, 1]. An n-lace ℓ (in S) is the union ℓ1 ∪ … ∪ ℓn of disjoint simple arcs in S such that ∂ ℓi = {(i, 1), (π (i),-1)}, i = 1, …, n, for some permutation π of {1, 2, …, n}.
Jin, GT Jin, Gyo Taek, Kim, H
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ACHIRALITY AND PLANARITY

Communications in Contemporary Mathematics, 2000
An embedding of a space Y into the 3-sphere S 3 is said to be strictly achiral if its image is pointwise fixed by an orientation reversing homeomorphism of S 3 . A space Y is said to be abstractly planar if it can be embedded into the 2-sphere S
Jiang, Boju, Wang, Shicheng
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Planar Rings

Results in Mathematics, 1996
All rings that are planar near-rings are characterized and constructed. For each vector space over a nontrivial division ring there is precisely one isomorphism class of such rings.
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