339 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
The initial singularity of ultrastiff perfect fluid spacetimes without symmetries
We consider the Einstein equations coupled to an ultrastiff perfect fluid and
prove the existence of a family of solutions with an initial singularity whose
structure is that of explicit isotropic models. This family of solutions is
`generic' in the sense that it depends on as many free functions as a general
solution, i.e., without imposing any symmetry assumptions, of the
Einstein-Euler equations. The method we use is a that of a Fuchsian reduction.Comment: 16 pages, journal versio
Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity
Fuchsian equations provide a way of constructing large classes of spacetimes
whose singularities can be described in detail. In some of the applications of
this technique only the analytic case could be handled up to now. This paper
develops a method of removing the undesirable hypothesis of analyticity. This
is applied to the specific case of the Gowdy spacetimes in order to show that
analogues of the results known in the analytic case hold in the smooth case. As
far as possible the likely strengths and weaknesses of the method as applied to
more general problems are displayed.Comment: 14 page
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