TERMIUM Plus^®

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The Hausdorff dimension, also called the Hausdorff-Besicovitch dimension in one of the possible dimensions.  4, record 1, English, - Hausdorff%20dimension

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The Hausdorff dimension is defined for any subset of R [exponent] n that you can probably imagine. This includes all open sets, closed sets, countable unions of closed sets, countable intersections of open sets, etc. It will follow easily from the definition that Hausdorff dimension is not invariant under homeomorphisms. However, the topological dimension of E is the infimum of the Hausdorff dimensions of its homeomorphic images h(E). Later we will discover that Hausdorff dimension is invariant under something slightly stronger-quasi-isometries(sometimes called "bi-Lipschitz maps" or "Lipeomorphic maps"). Indeed, we will see that the equivalence class of quasi-isometric self-similar curves is completely determined by Hausdorff dimension!  5, record 1, English, - Hausdorff%20dimension

Record 1,  Textual support,  French

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La plus ancienne des dimensions fractales possibles, assignée en 1918 par Felix Hausdorff aux courbes de type flocon de neige et développée en 1935 par Besicovitch.  2, record 1, French, - dimension%20de%20Hausdorff%2DBesicovitch

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Le concept mathématique «dimension de Hausdorff» constitue une généralisation de la notion de dimension euclidienne.  2, record 1, French, - dimension%20de%20Hausdorff%2DBesicovitch

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Record 2,  Textual support,  English

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Some of the basic properties of objects with anomalous dimension were noticed and investigated at the beginning of this century mainly by Hausdorff(1919) and Besicovich(1935). They were considered "monstruous" and "pathological".  1, record 2, English, - anomalous%20dimension

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See fractal dimension, anomalous diffusion law, anomalous scaling.  1, record 2, English, - anomalous%20dimension

Record 2,  Textual support,  French

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Anomal : se dit d'une forme ou d'une construction qui présente un caractère aberrant par rapport à un type ou à une règle (sans être incorrecte ou anormale).  2, record 2, French, - dimension%20anomale

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Record 3,  Textual support,  English

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The idea of the capacity of a set was originally developed for treating electrostatic problems.... and extended to more general laws of attraction in... potential theory. It turns out that the Hausdorff dimension and the capacity of a set are related, and it is sometimes more convenient to use the latter concept when studying dimensional properties.  1, record 3, English, - capacity

Record 3,  Textual support,  French

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Vitesse de croissance du nombre de boules-unités de rayon donné, ou de cubes de côté uniforme, nécessaire à recouvrir un ensemble borné, à mesure que ce rayon décroît.  3, record 3, French, - indice%20de%20recouvrement

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La différence entre la dimension de Hausdorff et la densité logarithmique tient au fait que dans un cas on utilise des recouvrements par des boules égales, alors que dans l'autre, on autorise des recouvrements plus complexes. C. Tricot (1981) donne douze définitions possibles de la densité logarithmique.  3, record 3, French, - indice%20de%20recouvrement

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Record 4,  Textual support,  English

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For strictly self-similar mathematical fractals, such as the Sierpinski gasket and the Cantor dust, the mass dimension is the same as the Hausdorff dimension(and any other fractal dimension considered here). They are all given by the similarity dimension of the scaling law, defined by the initiator and the generator that generates the fractal.  2, record 4, English, - scaling%20law

Record 4,  Textual support,  French

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Loi de la géométrie, selon laquelle les propriétés physiques telles la masse ou la conductivité dépendent de la taille du système de façon algébrique.  2, record 4, French, - loi%20d%27%C3%A9chelle

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Par exemple, la masse d'un cube est proportionnelle à la longueur de côté élevée à une puissance D (dimension fractale) - un nombre positif et inférieur à la valeur 3 qui correspond à la dimension euclidienne à laquelle obéissent les objets homogènes.  2, record 4, French, - loi%20d%27%C3%A9chelle

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Record 5,  Textual support,  English

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For strictly self-similar mathematical fractals, such as the Sierpinski gasket and the Cantor dust, the mass dimension is the same as the Hausdorff dimension.... They are all given by the similarity dimension of the scaling law, defined by the initiator and the generator that generates the fractal. But for practical fractals, there are significant differences....  2, record 5, English, - mass%20dimension

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Record 6,  Textual support,  English

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A non-zero but finite s-dimensional Hausdorff measure.  2, record 6, English, - Besicovitch%20measure

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A fractal measure devised in 1935 by Besicovitch based on Felix Hausdorff's work(1919) on non-integer dimensions for geometrically very complex objects. See fractional dimension, fractal dimension, Hausdorff dimension.  3, record 6, English, - Besicovitch%20measure

Record 6,  Textual support,  French

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Dans le contexte triadique de Cantor, partons d'une masse distribuée sur (0,1) avec la densité uniforme égale à 1. Effacer le tiers central divise cette masse en parties égales .. réparties avec les densités 3/2, o, et 3/2 [...] Quand on aura répété la procédure à l'infini, la masse formera ce que nous allons appeler une mesure de Besicovitch [...] Il est clair, pour commencer, qu'aucun intervalle ouvert ne constitue une intermission absolue.  1, record 6, French, - mesure%20de%20Besicovitch

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Record 7,  Textual support,  English

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More generally, whenever S is of zero Lebesgue measure one can prove that Df is finite and at most equal to the Hausdorff Besicovitch dimension D of S.  2, record 7, English, - zero%20Lebesgue%20measure

Record 7,  Textual support,  French

Record 7,  Textual support,  Spanish