Introduction to the Mathematical Background of Classical Potential Theory

AbstractIt is shown that ifϕdenotes a harmonic morphism of type Bl between suitable Brelot harmonic spacesXandY, then a functionf, defined on an open setV ⊂ Y, is superharmonic if and only iff ∘ ϕis superharmonic onϕ–1(V) ⊂ X. The “only if” part is due to Constantinescu and Cornea, withϕdenoting any harmonic morphism between two Brelot spaces. A similar result is obtained for finely superharmonic functions defined on finely open sets. These results apply, for example, to the case whereϕis the projection from ℝNto ℝn(N > n ≥1) or whereϕis the radial projection from ℝN\ {0} to the unit sphere in ℝN(N≥ 2).

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On the Dynamic Stability of Fluid-Conveying Pipes

Journal of Applied Mechanics ◽

10.1115/1.3422971 ◽

1973 ◽

Vol 40 (1) ◽

pp. 48-52 ◽

Cited By ~ 83

Author(s):

D. S. Weaver ◽

T. E. Unny

Keyword(s):

Potential Theory ◽

Dynamic Stability ◽

Thin Shell ◽

Critical Flow ◽

Thin Shells ◽

Classical Potential ◽

Classical Potential Theory ◽

Limiting Case ◽

Beam Mode ◽

Fluid Conveying Pipes

This paper presents a general analysis of the dynamic stability of a finite-length, fluid-conveying pipe. The Flu¨gge-Kempner equation is used in conjunction with classical potential theory so that circumferential modes as well as the usual beam modes may be considered. The cylinders are found to become unstable statically at first but flutter is predicted for higher velocities. The critical flow velocities for short, thin shells are associated with a number of circumferential waves. This number reduces for thicker and longer shells until the instability is in a beam mode. When the limiting case of a long thin shell is taken, it is found to agree with previous results obtained using a simpler beam approach.

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INTEGER OPERATORS IN FINITE VON NEUMANN ALGEBRAS

Journal of Topology and Analysis ◽

10.1142/s1793525311000635 ◽

2011 ◽

Vol 03 (04) ◽

pp. 433-450

Author(s):

ANDREAS THOM

Keyword(s):

Potential Theory ◽

Spectral Properties ◽

Von Neumann Algebras ◽

Logarithmic Capacity ◽

Integral Group ◽

Von Neumann ◽

Classical Potential ◽

Classical Potential Theory ◽

Adjoint Operators ◽

Finite Von Neumann Algebras

Motivated by the study of spectral properties of self-adjoint operators in the integral group ring of a sofic group, we define and study integer operators. We establish a relation with classical potential theory and in particular the circle of results obtained by Fekete and Szegö, see [3, 4, 13]. More concretely, we use results by Rumely, see [12], on equidistribution of algebraic integers to obtain a description of those integer operator which have spectrum of logarithmic capacity less than or equal to one. Finally, we relate the study of integer operators to a recent construction by Petracovici and Zaharescu, see [10].

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