Displaying similar documents to “Fermat’s method of quadrature”

Functional equations stemming from numerical analysis

Tomasz Szostok

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Always when a numerical method gives exact results an interesting functional equation arises. And, since no regularity is assumed, some unexpected solutions may appear. Here we deal with equations constructed in this spirit. The vast majority of this paper is devoted to the equation i = 0 l ( y - x ) i [ f 1 , i ( α 1 , i x + β 1 , i y ) + + f k i , i ( α k i , i x + β k i , i y ) ] = 0 ( 1 ) and its particular cases. We use Sablik’s lemma to prove that all solutions of (1) are polynomial functions. Since a continuous polynomial function is an ordinary polynomial, the crucial problem throughout...

Multi-Harnack smoothings of real plane branches

Pedro Daniel González Pérez, Jean-Jacques Risler (2010)

Annales scientifiques de l'École Normale Supérieure

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Let Δ 𝐑 2 be an integral convex polygon. G. Mikhalkin introduced the notion of, a class of real algebraic curves, defined by polynomials supported on Δ and contained in the corresponding toric surface. He proved their existence, viamethod, and that the topological type of their real parts is unique (and determined by Δ ). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch ( C , 0 ) . We introduce the class ofsmoothings of ( C , 0 ) by...

A new kind of augmentation of filtrations

Joseph Najnudel, Ashkan Nikeghbali (2011)

ESAIM: Probability and Statistics

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Let (Ω, , ( t ), ) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to (the-algebra generated by ( t )) a coherent family of probability measures ( t ) indexed by , each of them being defined on t . It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative...

The finite element solution of parabolic equations

Josef Nedoma (1978)

Aplikace matematiky

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In contradistinction to former results, the error bounds introduced in this paper are given for fully discretized approximate soltuions of parabolic equations and for arbitrary curved domains. Simplicial isoparametric elements in n -dimensional space are applied. Degrees of accuracy of quadrature formulas are determined so that numerical integration does not worsen the optimal order of convergence in L 2 -norm of the method.

Optional splitting formula in a progressively enlarged filtration

Shiqi Song (2014)

ESAIM: Probability and Statistics

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Let 𝔽 F be a filtration andbe a random time. Let 𝔾 G be the progressive enlargement of 𝔽 F with. We study the following formula, called the optional splitting formula: For any 𝔾 G-optional process, there exists an 𝔽 F-optional process and a function defined on [0∞] × (ℝ × ) being [ 0 , ] 𝒪 ( 𝔽 ) ℬ[0,∞]⊗x1d4aa;(F) measurable, such that Y = Y ' 1 [ 0 , τ ) + Y ' ' ( τ ) 1 [ τ , ) . Y=Y′1[0,τ)+Y′′(τ)1[τ,∞). (This formula can also be formulated for multiple random times ...

Constructive quantization: approximation by empirical measures

Steffen Dereich, Michael Scheutzow, Reik Schottstedt (2013)

Annales de l'I.H.P. Probabilités et statistiques

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In this article, we study the approximation of a probability measure μ on d by its empirical measure μ ^ N interpreted as a random quantization. As error criterion we consider an averaged p th moment Wasserstein metric. In the case where 2 p l t ; d , we establish fine upper and lower bounds for the error, a. Moreover, we provide a universal estimate based on moments, a . In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

The jacobian map, the jacobian group and the group of automorphisms of the Grassmann algebra

Vladimir V. Bavula (2010)

Bulletin de la Société Mathématique de France

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There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras (e.g., the Grassmann algebras are dual of polynomial algebras as quadratic algebras). This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the Jacobian set– the set of all algebra endomorphisms of a polynomial algebra with the Jacobian...

Curvature measures, normal cycles and asymptotic cones

Xiang Sun, Jean-Marie Morvan (2013)

Actes des rencontres du CIRM

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The purpose of this article is to give an overview of the theory of the and to show how to use it to define a on singular surfaces embedded in an (oriented) Euclidean space 𝔼 3 . In particular, we will introduce the notion of associated to a Borel subset of 𝔼 3 , generalizing the defined at each point of a smooth surface. For simplicity, we restrict our singular subsets to polyhedra of the 3 -dimensional Euclidean space 𝔼 3 . The coherence of the theory lies in a convergence theorem: If a...

Necessary and sufficient condition for the existence of a Fréchet mean on the circle

Benjamin Charlier (2013)

ESAIM: Probability and Statistics

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Let ( 𝕊 1 , d 𝕊 1 S 1 , d S 1 ) be the unit circle in ℝ endowed with the arclength distance. We give a sufficient and necessary condition for a general probability measure to admit a well defined Fréchet mean on ( 𝕊 1 , d 𝕊 1 S 1 , d S 1 ). We derive a new sufficient condition of existence() with no restriction on the support of the measure. Then, we study the convergence of the empirical Fréchet mean to the Fréchet mean and we give an algorithm to compute it.

On the estimate of the error of quadrature formulae

Miloš Zlámal (1966)

Aplikace matematiky

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An explicit expression for the exact estimate of the error E ( f ) = 0 1 f ( x ) d x - k = 1 m c k f ( x k ) of an arbitrary quadrature formula is given in terms of the coefficients c k and the nodes x k .

A priori bounds for some infinitely renormalizable quadratics: II. Decorations

Jeremy Kahn, Mikhail Lyubich (2008)

Annales scientifiques de l'École Normale Supérieure

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A decoration of the Mandelbrot set M is a part of M cut off by two external rays landing at some tip of a satellite copy of M attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. For this class of maps we prove bounds. They imply local connectivity of the corresponding Julia...

On the optimality of the empirical risk minimization procedure for the convex aggregation problem

Guillaume Lecué, Shahar Mendelson (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We study the performance of (ERM), with respect to the quadratic risk, in the context of , in which one wants to construct a procedure whose risk is as close as possible to the best function in the convex hull of an arbitrary finite class F . We show that ERM performed in the convex hull of F is an optimal aggregation procedure for the convex aggregation problem. We also show that if this procedure is used for the problem of model selection aggregation, in which one wants to mimic the...

A proof of Reidemeister-Singer’s theorem by Cerf’s methods

François Laudenbach (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Heegaard splittings and Heegaard diagrams of a closed 3-manifold M are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on M . We make use in a very simple setting of techniques which Jean Cerf developed for solving a famous problem. In passing, we show how to cancel the supernumerary local extrema in a generic path of functions when dim M > 2 . The main tool that we introduce is an which could be useful elsewhere.

Estimate of the pressure when its gradient is the divergence of a measure. Applications

Marc Briane, Juan Casado-Díaz (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper, a W - 1 , N ' estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on N , or on a regular bounded open set of  N . The proof is based partially on the Strauss inequality [Strauss, 23 (1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [ 9 (2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions which read as the divergence of a measure, and...

Waring’s problem for Beatty sequences and a local to global principle

William D. Banks, Ahmet M. Güloğlu, Robert C. Vaughan (2014)

Journal de Théorie des Nombres de Bordeaux

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We investigate in various ways the representation of a large natural number N as a sum of s positive k -th powers of numbers from a fixed Beatty sequence. , a very general form of the local to global principle is established in additive number theory. Although the proof is very short, it depends on a deep theorem of M. Kneser.