60 research outputs found
Improving Hölder's inequality
International audienceWe show that the remainder in Hölder's inequality (Rogers, 1888; Hölder, 1889) may be computed exactly. It satisfies functional equations, and possesses monotonicity and scaling properties. We obtain as a consequence improvements of recent sharpenings (Aldaz, 2008) of the classical inequality
Boundary blow-up and degenerate equations
AbstractLet Ω⊂R2 be a bounded domain of class C2+α, 0<α<1. We show that if u is the solution of Δu=4exp(2u) which tends to +∞ as (x,y)→∂Ω, then the hyperbolic radius v=exp(−u) is also of class C2+α up to the boundary. The proof relies on new Schauder estimates for degenerate elliptic equations of Fuchsian type
Brahmagupta's derivation of the area of a cyclic quadrilateral
International audienceThis paper shows that Propositions XII.21–27 of Brahmagupta's Br¯ ahma-sphut. asiddh¯ anta (628 a.d.) constitute a coherent mathematical discourse leading to the expression of the area of a cyclic quadrilateral in terms of its sides. The radius of the circumcircle is determined by considering two auxiliary quadrilaterals. Observing that a cyclic quadrilateral is split by a diagonal into two triangles with the same circumcenter and the same circumradius, the result follows, using the tools available to Brahmagupta. The expression for the diagonals (XII.28) is a consequence. The shortcomings of earlier attempts at reconstructing Brahmagupta's method are overcome by restoring the mathematical consistency of the text. This leads to a new interpretation of Brahmagupta's terminology for quadrilaterals of different types. Résumé. On montre que les propositions XII.21–27 du Br¯ ahmasphut. asiddh¯ anta (628 ap. J.-C.) forment un discours cohérent conduisantà conduisantà l'expression de l'aire d'un quadrilatère cyclique en termes de ses côtés. Le rayon du cercle circonscrit est déterminé en considérant deux quadrilatères auxiliaires. Exprimant que le quadrilatère cyclique est partagé par une diagonale en deux triangles ayant en commun le centre et le rayon de leur cercle circonscrit, on obtient l'aire du quadrilatère, ` a l'aide des outils connus de Brahmagupta. L'expression des diagonales (XII.28) en découle. Les difficultés des tentatives antérieures en vue de retrouver la démarche de Brahmagupta sont résolues en restituant la cohérence mathématique du texte. On est ainsi conduità conduità une nouvelle interprétation des termes qu'utilise Brahmagupta pour désigner des quadrilatères de différentes classes
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