179,265 research outputs found
A Remark on Soliton Equation of Mean Curvature Flow
In this short note, we consider self-similar immersions of the Graphic Mean Curvature Flow of higher co-dimension. We
show that the following is true: Let be
a graph solution to the soliton equation
Assume . Then there exists a
unique smooth function such that
and for any real number , where Comment: 6 page
Modular transformation and twist between trigonometric limits of elliptic R-matrix
We study the modular transformation of -symmetric elliptic
R-matrix and construct the twist between the trigonometric degeneracy of the
elliptic R-matrix.Comment: 8 pages, latex, reference revise
Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems
Nanomechanical resonators can now be realized that achieve fundamental resonance frequencies exceeding 1 GHz, with quality factors (Q) in the range 10^3<=Q<=10^5. The minuscule active masses of these devices, in conjunction with their high Qs, translate into unprecedented inertial mass sensitivities. This makes them natural candidates for a variety of mass sensing applications. Here we evaluate the ultimate mass sensitivity limits for nanomechanical resonators operating in vacuo that are imposed by a number of fundamental physical noise processes. Our analyses indicate that nanomechanical resonators offer immense potential for mass sensing—ultimately with resolution at the level of individual molecules
The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension
We identify a region \Bbb{W}_{\f{1}{3}} in a Grassmann manifold
\grs{n}{m}, not covered by a usual matrix coordinate chart, with the
following important property. For a complete submanifold in \ir{n+m} \,
(n\ge 3, m\ge2) with parallel mean curvature whose image under the Gauss map
is contained in a compact subset K\subset\Bbb{W}_{\f{1}{3}}\subset\grs{n}{m},
we can construct strongly subharmonic functions and derive a priori estimates
for the harmonic Gauss map. While we do not know yet how close our region is to
being optimal in this respect, it is substantially larger than what could be
achieved previously with other methods. Consequently, this enables us to obtain
substantially stronger Bernstein type theorems in higher codimension than
previously known.Comment: 36 page
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