On the Pythagoras numbers of real analytic set germs

José F. Fernando; Jesús M. Ruiz

Displaying similar documents to “On the Pythagoras numbers of real analytic set germs”

The trivial locus of an analytic map germ

H. Hauser, G. Muller (1989)

Annales de l'institut Fourier

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We prove: For a local analytic family ${X_{s}}_{s \in S}$ of analytic space germs there is a largest subspace $T$ in $S$ such that the family is trivial over $T$ . Moreover the reduction of $T$ equals the germ of those points $s$ in $S$ for which $X_{s}$ is isomorphic to the special fibre $X_{0}$ .

The analytic $α$ -capacity and the Cauchy integral

K. Adžievski (1986)

Matematički Vesnik

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Analytic and $C^{k}$ approximations of norms in separable Banach spaces

Robert Deville, Vladimir Fonf, Petr Hájek (1996)

Studia Mathematica

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Approximation of $C^{\infty}$ -functions without changing their zero-set

F. Broglia, A. Tognoli (1989)

Annales de l'institut Fourier

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For a $C^{\infty}$ function $ϕ : M \to ℝ$ (where $M$ is a real algebraic manifold) the following problem is studied. If $ϕ^{- 1} (0)$ is an algebraic subvariety of $M$ , can $ϕ$ be approximated by rational regular functions $f$ such that $f^{- 1} (0) = ϕ^{- 1} (0) ?$ We find that this is possible if and only if there exists a rational regular function $g : M \to ℝ$ such that $g^{- 1} (0) = ϕ^{- 1} (0)$ and g(x) $\cdot ϕ (x) \geq 0$ for any $x$ in $ℝ^{n}$ . Similar results are obtained also in the analytic and in the Nash cases. For non approximable functions the minimal flatness locus...

Overstability and resonance

Augustin Fruchard, Reinhard Schäfke (2003)

Annales de l’institut Fourier

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We consider a singularity perturbed nonlinear differential equation $ε u^{'} = f (x) u + + ε P (x, u, ε)$ which we suppose real analytic for $x$ near some interval $[a, b]$ and small $| u |$ , $| ε |$ . We furthermore suppose that 0 is a turning point, namely that $x f (x)$ is positive if $x \neq 0$ . We prove that the existence of nicely behaved (as $ϵ \to 0$ ) local (at $x = 0$ ) or global, real analytic or $C^{\infty}$ solutions is equivalent to the existence of a formal series solution $\sum u_{n} (x) ε^{n}$ with $u_{n}$ analytic at $x = 0$ . The main tool of a proof is a new “principle of analytic continuation” for...

Spaces of D-paraanalytic elements

D. Przeworska-Rolewicz

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Let X be a linear space. Consider a linear equation(*) P(D)x = y, where y ∈ E ⊂ X,with a right invertible operator D ∈ L(X) and, in general, operator coefficients. The main purpose of this paper is to characterize those subspaces E ⊂ X for which all solutions of (*) belong to E (provided that they exist). This leads, even in the classical case of ordinary differential equations with scalar coefficients, to a new class of $C^{\infty}$ -functions, which properly contains the classes of analytic functions...

On certain regularity properties of Haar-null sets

Pandelis Dodos (2004)

Fundamenta Mathematicae

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Let X be an abelian Polish group. For every analytic Haar-null set A ⊆ X let T(A) be the set of test measures of A. We show that T(A) is always dense and co-analytic in P(X). We prove that if A is compact then T(A) is $G_{δ}$ dense, while if A is non-meager then T(A) is meager. We also strengthen a result of Solecki and we show that for every analytic Haar-null set A, there exists a Borel Haar-null set B ⊇ A such that T(A)∖ T(B) is meager. Finally, under Martin’s Axiom and the negation of...

On the analyticity of generalized eigenfunctions (case of real variables)

Eberhard Gerlach (1968)

Annales de l'institut Fourier

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On démontre que, dans les espaces fonctionnels propres de Hilbert (avec un noyau reproduisant), formés de fonctions analytiques de $n$ variables dans un domaine $G$ , pour tout opérateur auto-adjoint, les fonctions propres généralisées sont des fonctions réelles-analytiques dans $G$ .

Analytic regularity for the Bergman kernel

Gabor Françis, Nicholas Hanges (1998)

Journées équations aux dérivées partielles

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Let $Ω \subset ℝ^{2}$ be a bounded, convex and open set with real analytic boundary. Let $T_{Ω} \subset ℂ^{2}$ be the tube with base $Ω,$ and let $ℬ$ be the Bergman kernel of $T_{Ω}$ . If $Ω$ is strongly convex, then $ℬ$ is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation, we relate the off diagonal points where analyticity fails to the Trèves curves. These curves are symplectic invariants which are determined by the CR structure of the boundary of $T_{Ω}$ . Note that Trèves curves...