Infinitely many positive harmonic functions with an oscillating boundary condition on exterior regions

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we writewhere λ > 1, then the equationdefines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such thatThe problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

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Growth of positive harmonic functions and Kleinian group limit sets of zero planar measure and hausdorf dimension two In celebration of Nico Kuiper's sixtieth birthday

Lecture Notes in Mathematics - Geometry Symposium Utrecht 1980 ◽

10.1007/bfb0096228 ◽

1981 ◽

pp. 127-144 ◽

Cited By ~ 12

Author(s):

Dennis Sullivan

Keyword(s):

Harmonic Functions ◽

Kleinian Group ◽

Sixtieth Birthday ◽

Limit Sets ◽

Group Limit ◽

Positive Harmonic Functions

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Damped Vibration Response of an Axially Moving Wire Subject to an Oscillating Boundary Condition and the Application to Slurry Wiresaws

Journal of Vibration and Acoustics ◽

10.1115/1.4049269 ◽

2020 ◽

pp. 1-37

Author(s):

Liming Li ◽

Imin Kao

Keyword(s):

Boundary Condition ◽

Analytical Solution ◽

Traveling Waves ◽

Solution Space ◽

Vibration Response ◽

Temporal Variables ◽

Rocking Motion ◽

Vibration Responses ◽

Oscillating Boundary ◽

Axially Moving

Abstract This paper presents the analysis of a new class of differential continuum system with a solution of traveling waves containing coupled spatial and temporal variables. Herein, we derive the analytical solution of the damped vibration response of a longitudinally moving wire with damping, subject to an oscillating boundary condition. The vibration response is the outcome of combining four traveling waves, induced by a wave initiating from the oscillating boundary, and traveling between the two boundaries. The four different traveling waves are the independent bases of the vibration responses that span the solution space of vibration of such continuum system. The combination, or the interference, of these traveling waves in the undamped condition produces nodal points in the vibration response, which can be formulated through the analytical solution. The impacts of wire speed, oscillating frequency at the boundary and damping factors on the vibration response are investigated. Furthermore, the vibration induced by the oscillating motion of the boundary has a profound impact on the effectiveness of slicing ingots with rocking motion of oscillating wire guides in wiresaw manufacturing processes.

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