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Recently translated father's 1968 Ph.D. Thesis, appears to be related to Hilbert's Problem #21, in the context of Theory of Elasticity using Complex Variables. The Hilbert kernel is stressed on various points and a complete solution for its elastic profile is given up to compromise. The crack is described using analytic functions.

## Contexts in source publication

**Context 1**

... the linear elements are parallel to the axes x and y respectively, we say that the stress tensor T on the given point has components the linear magnitudes σ x , σ y , τ xy and τ yx . (Fig. ...

**Context 2**

... we "project" the above tensor against the linear element ∆s (passing through the initial point), which is signed with respect to direction as ⃗ n, we find by using the Cauchy relations the two components X n and Y n of T according to ⃗ n as follows 5 : (Fig. 2) ...

**Context 3**

... to be a unit vector, then the component of n in the x direction is precisely the cosine of the angle between them. So this also matches the interpretation which has συν(n, * ) = cos(n, * ) in the case that n is a unit, in which case the more austere notation cos ⟨ˆn⟨ˆn, ˆ * ⟩ could have been used. 6 Stresses are linear magnitudes and hence Fig. 2 depicts vectors having components equal to the stresses. 7 In the above as well as in what follows, we accept that there are no forces due to Mass. ...

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## Citations

... Any such compression is (ultimately) characterized by the "crack" the compressive agent produces on the various elastic media it is applied upon. This, has been analyzed completely in [5] for the "generally" isotropic or anisotropic disk under tension 20 . The implosion type nuclear weapon on the other hand is another process which causes deformation and a "crack" (on a much larger scale). ...

... (6.10) 20 With the compressive agent being − → IA in the direction n, giving component stresses Xn and Yn on a linear element ∆s as in page 10, therein. This being the most general case, a model of any other case can be constructed with a subset of [5]. 21 As continuous however, it may not necessarily be linear. ...

... For example, under an appropriate coordinate transformation which takes the arguments ρ and φ onto a standard polar coordinate system ρ and θ, the curve can be seen to satisfy the polar form of the Cauchy-Riemann equations.5 The folia of Theorem 2.1 also bare similar p-polygonal symmetry to the symmetry of the orbit of the iterated exponential z z ··· on the Complex plane ([3]), with the exception of that when either m/ GCD(m, n) or n/ GCD(m, n) is divisible by 2, the period p is doubled. ...

The Spherical Harmonic Oscillator as a basis Model for the solution of various Classical or Relativistic IVP problems in Astrophysics and Mechanics