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Topological and differentiable rigidity of submanifolds in space forms
Let be an -dimensional simply connected space form with
nonnegative constant curvature . We prove that if is a compact
submanifold in , and if where is the mean
curvature of , then is homeomorphic to a sphere. We also show that the
pinching condition above is sharp. Moreover, we obtain a new differentiable
sphere theorem for submanifolds with positive Ricci curvature.Comment: 12 page
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Highly flexible, foldable, and rollable microsupercapacitors on an ultrathin polyimide substrate with high power density.
The design and functionality of extremely flexible, foldable, and rollable microsupercapacitors (MSCs) with in-plane interdigital electrodes that consist of single-walled carbon nanotube (SWCNT) networks on an ultrathin polyimide substrate are demonstrated through experiments and finite element simulations. The all-solid-state MSCs can be reversibly bent, folded, and rolled purely elastically without degradation of their electrical performance. The simulation results confirm that the deformation in bent, folded, and rolled MSCs is purely elastic. The high power density (1125 W cm-3) and small time constant (1 ms) of the present MSCs are comparable to those of aluminum electrolytic capacitors. The MSCs operate at scan rates of up to 1000 V s-1, are characterized by a volumetric capacitance of 18 F cm-3 and an energy density of 1.6 mWh cm-3, and exhibit superior electrochemical stability with 96% capacity retention even after 100,000 charge/discharge cycles. The developed MSCs demonstrate high potential for integration in flexible and wearable electronic systems
"Hard-scattering" approach to very hindered magnetic-dipole transitions in quarkonium
For a class of hindered magnetic dipole () transition processes, such as
(the discovery channel of the meson),
the emitted photon is rather energetic so that the traditional approaches based
on multipole expansion may be invalidated. We propose that a "hard-scattering"
picture, somewhat analogous to the pion electromagnetic form factor at large
momentum transfer, may be more plausible to describe such types of transition
processes. We work out a simple factorization formula at lowest order in the
strong coupling constant, which involves convolution of the Schr\"odinger wave
functions of quarkonia with a perturbatively calculable part induced by
exchange of one semihard gluon between quark and antiquark. This formula,
without any freely adjustable parameters, is found to agree with the measured
rate of rather well, and can also reasonably
account for other recently measured hindered transition rates. The
branching fractions of are also
predicted.Comment: v3; 5 pages, 1 figure and 1 table; title changed, presentation
improve
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