179,265 research outputs found

    A Remark on Soliton Equation of Mean Curvature Flow

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    In this short note, we consider self-similar immersions F:RnRn+kF: \mathbb{R}^n \to \mathbb{R}^{n+k} of the Graphic Mean Curvature Flow of higher co-dimension. We show that the following is true: Let F(x)=(x,f(x)),xRnF(x) = (x,f(x)), x \in \mathbb{R}^{n} be a graph solution to the soliton equation Hˉ(x)+F(x)=0. \bar{H}(x) + F^{\bot}(x) = 0. Assume supRnDf(x)C0<+\sup_{\mathbb{R}^{n}}|Df(x)| \le C_{0} < + \infty. Then there exists a unique smooth function f:RnRkf_{\infty}: \mathbb{R}^{n}\to \mathbb{R}^k such that f(x)=limλfλ(x) f_{\infty}(x) = \lim_{\lambda \to \infty}f_{\lambda}(x) and f(rx)=rf(x) f_{\infty}(r x)=r f_{\infty}(x) for any real number r0r\not= 0, where fλ(x)=λ1f(λx). f_{\lambda}(x) = \lambda^{-1}f(\lambda x). Comment: 6 page

    Modular transformation and twist between trigonometric limits of sl(n)sl(n) elliptic R-matrix

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    We study the modular transformation of Zn{\bf Z}_n-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

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    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

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    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 nn-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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