3 research outputs found

    Minimal blocks of binary even-weight vectors

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    AbstractOdd circuits are minimal 1-blocks over GF(2) and the odd circuit of size 2t+1 can be represented by the vectors of Hamming weight 2t in a (2t+1)-dimensional vector space over GF(2). This is the tip of an iceberg. Let f(2t,k,2) be the maximum number of binary k-dimensional column vectors such that for all s, 1ā©½sā©½t, no 2s columns sum to the zero vector. If k=2, k=3, k=4, or kā©¾5 and 2t is sufficiently large (for example, 2tā©¾2kāˆ’k+1 suffices), then the set of vectors of weight 2t in a (f(2t,k,2)+2tāˆ’1)-dimensional vector space over GF(2) is a minimal k-block over GF(2)

    An Archaeology of the Fragment: The Transition from the Antique Fragment to the Historical Fragment in French Architecture Between 1750 and 1850

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    Although architects before the time of the French Enlightenment often made use of historical forms in their designs, this practice radically changed between the years 1750 and 1850. The fragment itself changed, as did the ways it was used. The transformation of the fragment followed three stages: it changed from the antique, to the elemental, to the historical fragment. Through the course of this transformation, design also changed, it came to be understood as composition. This dissertation describes the history of this transformation in consideration of writings by French author-architects, as well as their designs. It also shows how the new conception of the fragment gave birth to the next stage of architectural history: eclecticism. Mid eighteenth-century changes in European architecture were prompted by growing familiarity with recent archaeological work especially in Italy, the country of ancient ruins. In France, antique fragments were adopted initially as formal and spatial motifs that enriched architectural design by means of picturesque effects, inspired by paintings and Piranesian etchings. Later, these fragments gradually became regular elements of architectural composition. Charles Percier and Jean-Nicolas-Louis Durand, two disciples of BoullĆ©e, took over his imagery and technique of composing with antique fragments, but relied less than he did on the building\u27s picturesque and sensationalist aspects. Composition in elementary antique fragments underlay the neo-classical architectural education at both the Ecole des Beaux-Arts and the Ecole Polytechnique in the beginning of the nineteenth-century. In the 1830s, a group of pensionnaires argued for freer assembly of architectural elements that would allow diachronic reading of historical fragments as opposed to synchronic antique-looking motifs. Architects like Henri Labrouste, LĆ©on Vaudoyer, and FĆ©lix Duban preferred imitating the historical progress of architecture over Greco-Roman elements and compositions. Eclecticism taught them that mixture of antithetical things gave birth to something new after a transitory phase. While neo-classical architecture imitated the mature architectural representation of a distant past, eclectic architecture of the romantic-rationalists imitated the immature expressions of the architecture in transition. The buildings of the second group revealed a new problem of representation in architecture, a problem that had begun to emerge already in the architecture of the eighteenth-century: the problem of style, expressed most famously if pathetically in the early nineteenth-century as a question: ā€œin what style shall we build?ā€
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