Information om | Engelska ordet LIPSCHITZ

Exempel på hur man kan använda LIPSCHITZ i en mening

• Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1).
• However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map).
• One consequence of the Banach fixed-point theorem is that small Lipschitz perturbations of the identity are bi-lipschitz homeomorphisms.
• Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity).
• He first became known in the 1980s for his results in geometric topology, notably the solution of the classical recognition problem for 3-manifolds, the proof of the 4-dimensional Cellularity Criterion, and the proof of the Lipschitz case of the classical Hilbert–Smith conjecture.
• They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps.
• Lipschitz distance between metric spaces is the infimum of numbers r such that there is a bijective bi-Lipschitz map between these spaces with constants exp(-r), exp(r).
• Both the Hurwitz and Lipschitz quaternions are examples of noncommutative domains which are not division rings.
• Calderón's work ranged over a wide variety of topics: from singular integral operators to partial differential equations, from interpolation theory to Cauchy integrals on Lipschitz curves, from ergodic theory to inverse problems in electrical prospection.
• If the gradient of the cost function is globally Lipschitz continuous, with Lipschitz constant L, and learning rate is chosen of the order 1/L, then the standard version of SGD is a special case of backtracking line search.
• For the case of a function with at most countably many critical points (such as a Morse function) and compact sublevels, as well as with Lipschitz continuous gradient where one uses standard GD with learning rate <1/L (see the section "Stochastic gradient descent"), then convergence is guaranteed, see for example Chapter 12 in.
• It turns out that the standard notion of spin structure is too restrictive for applications to supergravity and string theory, and that the correct notion of spinorial structure for the mathematical formulation of these theories is a "Lipschitz structure".
• Rådström was also the (initial) advisor of Martin Ribe, who wrote a thesis on metric linear spaces that need not be locally convex; Ribe also used a few of Enflo's ideas on metric geometry, especially "roundness", in obtaining independent results on uniform and Lipschitz embeddings (Benyamini and Lindenstrauss).
• Kahan discovered that polynomials with a particular set of multiplicities form what he called a pejorative manifold and proved that a multiple root is Lipschitz continuous if the perturbation maintains its multiplicity.
• After attending school in Burscheid and Barmen, Germany, he proceeded to Bonn University in 1884, where he studied science and mathematics under such eminent instructors as Rudolf Lipschitz, Rudolf Clausius, Eduard Schönfeld, and August Kekulé.
• Asplund spaces were introduced in 1968 by the mathematician Edgar Asplund, who was interested in the Fréchet differentiability properties of Lipschitz functions on Banach spaces.
• While many criticisms have been made of Norton's thought experiment, such as it being a violation of the principle of Lipschitz continuity (the force that appears in Newton's second law is not a Lipschitz continuous function of the particle's trajectory -- this allows evasion of the local uniqueness theorem for solutions of ordinary differential equations), or in violation of the principles of physical symmetry, or that it is somehow in some other way "unphysical", there is no consensus among its critics as to why they regard it as invalid.
• Singers with the Hudson-DeLange Orchestra included Ruth Gaylor, in 1936; Mitchell Ayres (né Meyer Agress; 1910–1969), in 1937; Georgia Gibbs (formerly Fredda Gibson; née Frieda Lipschitz; 1919–2006); and Nan Wynn.
• Training the generator in Wasserstein GAN is just gradient descent, the same as in GAN (or most deep learning methods), but training the discriminator is different, as the discriminator is now restricted to have bounded Lipschitz norm.
• After an introductory chapter on the geometry of hypersurfaces, It includes results of Ecker and Huisken concerning "locally Lipschitz entire graphs", and Huisken's theorem that uniformly convex surfaces remain smooth and convex, converging to a sphere, before they collapse to a point.

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