Norm (mathematics)

Summary

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude of the vector. This norm can be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equal

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Dans la plupart des pays la hauteur autorisée des bâtiments en bois est encore limitée par la loi. Malgré les grands progrès de la recherche en matière de protection contre les incendies, la hauteur maximale reste de trois à six étages, et seuls quelques pays, comme la Norvège, autorisent la construction de bâtiments de plus de huit étages - hauteur à partir de laquelle un bâtiment est considéré comme "haut" - en bois, matériau de construction combustible. Ces dernières années, de plus en plus de normes et labels ont vu le jour afin de promouvoir la durabilité dans le domaine de la construction, montrant une volonté des autorités de diminuer son impact écologique. Cependant, le bois, bien qu’étant un matériau dont la durabilité n’est plus à prouver face à l’acier et au béton, continue d’être freiné par ces limites législatives. C’est pourquoi je me suis décidé à effectuer mon projet de Master sur ce sujet. Les bâtiments en bois de grande hauteur ont un énorme potentiel dans le futur de la construction et leur domaine de recherche reste encore très peu développé.

2022

Structure preservation: a challenge in computational control

Daniel Kressner, Volker Mehrmann

Current and future directions in the development of numerical methods and numerical software for control problems are discussed. Major challenges include the demand for higher accuracy, robustness of the method with respect to uncertainties in the data or the model, and the need for methods to solve large scale problems. To address these demands it is essential to preserve any underlying physical structure of the problem. At the same time, to obtain the required accuracy it is necessary to avoid all inversions or unnecessary matrix products. We will demonstrate how these demands can be met to a great extent for some important tasks in control, the linear-quadratic optimal control problem for first and second order systems as well as stability radius and H-infinity norm computations. (C) 2003 Elsevier Science B.V All rights reserved.

Elsevier2003

Sup-norms of Eigenfunctions on Arithmetic Ellipsoids

Philippe Michel

Let B be a positive quaternion algebra, and let O subset of B be an Eichler order. There is associated, in a natural way, a variety X = X(O) the connected components of which are indexed by the ideal classes of O and are isomorphic to spheres. This variety is naturally equipped with a Laplace operator and a large family of Hecke operators. For a joint eigenfunction phi of the Hecke algebra and of the Laplace operator with eigenvalue lambda, the hybrid sup norm bound parallel to phi parallel to(infinity) < (tV)(-delta)t(1/2)parallel to phi parallel to(2) for any delta < 1/60 is shown, where t = (1 + lambda)(1/2) and V = vol(X(O)).

Oxford University Press2011