188,495 research outputs found

    Categorification of the colored sl₃-invariant

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    Calibrated embeddings in the special Lagrangian and coassociative cases

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    Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of compact calibrated submanifolds with nontrivial deformation spaces.Comment: AMS-TeX v. 2.1, 26 pages, uses amsppt.sty (2.1h), minor typos correcte

    Comment on "On an identity for the volume integral of the square of a vector field"

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    Stewart has provided a position-space derivation of an identity for the volume integral of the square of a vector field that was quoted by Gubarev, Stodolsky and Zakharov. In this comment, I provide a momentum-space derivation of this result, generalized to the scalar product of two complex vector fields. This approach demonstrates the effective use of the Fourier transform technique in the context of vector analysis at a level suitable for undergraduate instruction.Comment: 5 page

    Technological Potentials in Home Construction

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    Para-Legal Authority and Its Perils

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    Four cases of System Identification courses are discussed and described: A short introduction to participants without any prior knowledge, a course for industry, an undergraduate course and a graduate course.The discussion is carried out in terms of how to convey six basic messages to the course participants
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