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Let $M$ be a noncompact Riemannian manifold of dimension $n$ . Then there exists a proper embedding of $M$ into $R^{2 n + 1}$ by harmonic functions on $M$ . It is easy to find $2 n + 1$ harmonic functions which give an embedding. However, it is more difficult to achieve properness. The proof depends on the theorems of Lax-Malgrange and Aronszajn-Cordes in the theory of elliptic equations.

Ends of varieties

H. Alexander (1992)

Bulletin de la Société Mathématique de France

Energy and the law of the interated logarithm.

J.R. Baxter, G.A. Brosamler (1976)

Mathematica Scandinavica

Enlarging a subspace of C(X) without changing the Choquet boundary.

Eggert Briem (1979)

Mathematica Scandinavica

Entire functions and logarithmic sums over nonsymmetric sets of the real line.

Pedersen, Henrik L. (2000)

Annales Academiae Scientiarum Fennicae. Mathematica

Envelope of Dirichlet problems on a domain in $C^{n}$

Eric Bedford (1985)

Annales Polonici Mathematici

Envelopes of Continuous, Plurisubharmonic Functions.

Eric Bedford (1980)

Mathematische Annalen

Equality cases for condenser capacity inequalities under symmetrization

Dimitrios Betsakos, Stamatis Pouliasis (2012)

Annales UMCS, Mathematica

It is well known that certain transformations decrease the capacity of a condenser. We prove equality statements for the condenser capacity inequalities under symmetrization and polarization without connectivity restrictions on the condenser and without regularity assumptions on the boundary of the condenser.

Equality cases in the symmetrization inequalities for Brownian transition functions and Dirichlet heat kernels.

Betsakos, Dimitrios (2008)

Annales Academiae Scientiarum Fennicae. Mathematica

Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Matthew H. Baker, Robert Rumely (2006)

Annales de l’institut Fourier

Given a rational function $ϕ (T)$ on $ℙ^{1}$ of degree at least 2 with coefficients in a number field $k$ , we show that for each place $v$ of $k$ , there is a unique probability measure $μ_{ϕ, v}$ on the Berkovich space $ℙ_{Berk, v}^{1} / ℂ_{v}$ such that if ${z_{n}}$ is a sequence of points in $ℙ^{1} (\bar{k})$ whose $ϕ$ -canonical heights tend to zero, then the $z_{n}$ ’s and their $Gal (\bar{k} / k)$ -conjugates are equidistributed with respect to $μ_{ϕ, v}$ .The proof uses a polynomial lift $F (x, y) = (F_{1} (x, y), F_{2} (x, y))$ of $ϕ$ to construct a two-variable Arakelov-Green’s function $g_{ϕ, v} (x, y)$ for each $v$ . The measure $μ_{ϕ, v}$ is obtained by taking the...

Erratum to : “Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel–Leader graphs”

Sara Brofferio, Wolfgang Woess (2006)

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

Erratum to "M-Harmonic Besov p-spaces and Hankel operators in the Bergman Space on the Ball in Cn".

E.H. Youssfi, Kyong T. Hahn (1991)

Manuscripta mathematica

Error estimates for mixed methods

R. S. Falk, J. E. Osborn (1980)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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Éléments extrémaux pour le balayage

Elements generating balayages.

Ellipses and harmonic Möbius transformations.

Elliptic points in one-dimensional harmonic spaces

Embedding of open riemannian manifolds by harmonic functions