Multiple-image Network Graphics is a graphics file format, published in 2001, for animated images. Its specification is publicly documented and there are free software reference implementations available. MNG is related to the PNG image format; when PNG development started in early 1995, developers decided not to incorporate support for animation, because the majority of the PNG developers felt that overloading a single file type with both still and animation features is a bad design, both for users and for web servers. However, work soon started on MNG as an animation-supporting version of PNG. Version 1.0 of the MNG specification was released on 31 January 2001. Gwenview has native MNG support. GIMP can export images as MNG files. Imagemagick can create a MNG file from a series of PNG files. With the MNG plugin, Irfanview can read a MNG file. If MPlayer is linked against libmng, as it is, MPlayer and thus all graphical front-ends like Gnome MPlayer can display MNG files. Mozilla browsers and Netscape 6.0, 6.01 and 7.0 included native support for MNG until the code was removed in 2003 due to code size and little actual usage, causing complaints on the Mozilla development site.
Mozilla added support for APNG as a simpler alternative. Early versions of the Konqueror browser included MNG support but it was dropped. MNG support was never included in Internet Explorer, Opera, or Safari. Web servers don't come pre-configured to support MNG files; the MNG developers had hoped that MNG would replace GIF for animated images on the World Wide Web, just as PNG had done for still images. However, with the expiration of LZW patents and existence of alternative file formats such as Flash and SVG, combined with lack of MNG-supporting viewers and services, web usage was far less than expected; the structure of MNG files is the same as that of PNG files, differing only in the different signature and the use of a much greater variety of chunks to support all the animation features that it provides. Images to be used in the animation are stored in the MNG file as encapsulated JNG images. Two versions of MNG of reduced complexity are defined: MNG-LC and MNG-VLC; these allow applications to include some level of MNG support without having to implement the entire MNG specification, just as the SVG standard offers the "SVG Basic" and "SVG Tiny" subsets.
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In mathematics, a weak Lie algebra bundle ξ = is a vector bundle ξ over a base space X together with a morphism θ: ξ ⊗ ξ → ξ which induces a Lie algebra structure on each fibre ξ x. A Lie algebra bundle ξ = is a vector bundle in which each fibre is a Lie algebra and for every x in X, there is an open set U containing x, a Lie algebra L and a homeomorphism ϕ: U × L → p − 1 such that ϕ x: x × L → p − 1 is a Lie algebra isomorphism. Any Lie algebra bundle is a weak Lie algebra bundle; as an example of a weak Lie algebra bundle, not a strong Lie algebra bundle, consider the total space s o × R over the real line R. Let denote the Lie bracket of s o and deform it by the real parameter as: x = x ⋅ for X, Y ∈ s o and x ∈ R. Lie's third theorem states that every bundle of Lie algebras can locally be integrated to a bundle of Lie groups. In general globally the total space might fail to be Hausdorff.. But if all fibres of a real Lie algebra bundle over a topological space are mutually isomorphic as Lie algebras it is a locally trivial Lie algebra bundle.
This result was proved by proving that the real orbit of a real point under an algebraic group is open in the real part of its complex orbit. Suppose the base space is Hausdorff and fibers of total space are isomorphic as Lie algebras there exists a Hausdorff Lie group bundle over the same base space whose Lie algebra bundle is isomorphic to the given Lie algebra bundle.. Every semi simple Lie algebra bundle is locally trivial. Hence there exist a Hausdorff Lie group bundle over the same base space whose Lie algebra bundle is isomorphic to the given Lie algebra bundle. Douady, Adrien. "Espaces fibrés en algèbres de Lie et en groupes". Inventiones Mathematicae. 1: 133–151. Doi:10.1007/BF01389725. Kiranagi, B. S.. "On semisimple Lie algebra bundles". Journal of Algebra and Its Applications. 14: 1550009. Doi:10.1142/S0219498815500097. Algebra bundle Adjoint bundle