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Over a non-archimedean local field the absolute value, raised to any positive power $α > 0$ , is a negative definite function and generates (the analogue of) the symmetric stable process. For $α \in (0, 1)$ , this process is transient with potential operator given by M. Riesz’ kernel. We develop this potential theory purely analytically and in an explicit manner, obtaining special features afforded by the non-archimedean setting ; e.g. Harnack’s inequality becomes an equality.

Analytic roots of invertible matrix functions.

Rodman, Leiba, Spitkovsky, Ilya M. (2005)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Analytic Self-Mapping Reducing to the Identity Mapping

Takao Kato (1974)

Commentarii mathematici Helvetici

Applications of Quaternionic Holomorphic Geometry to minimal surfaces

K. Leschke, K. Moriya (2016)

Complex Manifolds

In this paper we give a survey of methods of Quaternionic Holomorphic Geometry and of applications of the theory to minimal surfaces. We discuss recent developments in minimal surface theory using integrable systems. In particular, we give the Lopez–Ros deformation and the simple factor dressing in terms of the Gauss map and the Hopf differential of the minimal surface. We illustrate the results for well–known examples of minimal surfaces, namely the Riemann minimal surfaces and the Costa surface....

Applications of the $p$ -adic Nevanlinna theory to functional equations

Abdelbaki Boutabaa, Alain Escassut (2000)

Annales de l'institut Fourier

Let $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We apply the $p$ -adic Nevanlinna theory to functional equations of the form $g = R \circ f$ , where $R \in K (x)$ , $f, g$ are meromorphic functions in $K$ , or in an “open disk”, $g$ satisfying conditions on the order of its zeros and poles. In various cases we show that $f$ and $g$ must be constant when they are meromorphic in all $K$ , or they must be quotients of bounded functions when they are meromorphic in an “open disk”. In particular,...