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The Dawning of the Age of Stochasticity

David Mumford — 2000

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

For over two millennia, Aristotle's logic has ruled over the thinking of western intellectuals. All precise theories, all scientific models, even models of the process of thinking itself, have in principle conformed to the straight-jacket of logic. But from its shady beginnings devising gambling strategies and counting corpses in medieval London, probability theory and statistical inference now emerge as better foundations for scientific models, especially those of the process of thinking and as...

Riemannian geometries on spaces of plane curves

Peter W. MichorDavid Mumford — 2006

Journal of the European Mathematical Society

We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from S 1 to the plane modulo the group of diffeomorphisms of S 1 , acting as reparametrizations. In particular we investigate the metric, for a constant A > 0 , G c A ( h , k ) : = S 1 ( 1 + A κ c ( θ ) 2 ) h ( θ ) , k ( θ ) | c ' ( θ ) | d θ where κ c is the curvature of the curve c and h , k are normal vector fields to c . The term A κ 2 is a sort of geometric Tikhonov regularization because, for A = 0 , the geodesic distance between any two distinct curves is 0, while for A > 0 the...

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