5 it is evident that the maximum values of the dynamic radial stresses in a foam medium with a tunnel elliptic cross-section cavity are 2.7 (for the ratio of semiaxis
of the ellipse 2) and 4.97 (for the ratio of semiaxis
of the ellipse 5) times less than the corresponding values for the case of a circular cross-section cavity.
 __, Embedding theorems with an exponential weight on the real semiaxis
As is known (see, e.g., [10, 11]) a canonical differential system of an order n on the semiaxis
[R.sub.+] = [0, [infinity]) is of the form
[l.sub.n] are the lengths of the n semiaxis
of the ellipsoid.
For more control over the shape and size of a cross section, the frontal and lateral sides were modeled according to (1), the sides shared the same exponent (p) and had identical lengths of the common semiaxis
, and they may however have different lengths assigned to the opposite semiaxes as depicted in Figure 2.
magnitudes (a and b) presented in Table 1 are computed experimentally with an estimated 95% confidence level.
Therefore, the largest portion generated in L division will represent the measure of the largest c semiaxis
, and the other portion will represent [absolute value of [sigma]] measure.
The intersection of the first truncated cone (adjacent to the tunnel face) with the circular tunnel face is an ellipse [[summation].sub.1], with semiaxis
lengths of [a.sub.1] and [b.sub.1] and with area of [A.sub.1] that are calculated as follows (cf.
A Bernstein type operator approximating continuous functions on semiaxis
. Indagationes Mathematicae, v.
This [[lambda].sub.1] is also the FTLE for the longer semiaxis
of a material ellipse.
Let [??](y) = f(x), where x = (2/[pi]) [cos.sup.-1] y, be analytic in the region bounded by an ellipse with major semiaxis
length R and with foci 0 and 1, The corresponding domain of analyticity of f is denoted by D(R), If f is analytic in the domain D(R), with R > 1/2, then the solution [g.sub.n] to the Problem 1 satisfies
This ellipse is defined by its major semiaxis
a = [bar.OQ] and its eccentricity e = c/a, 0 [less than or equal to] e < 1, where c is the focal semidistance c = [bar.FF' /2], and the minor semiaxis
b is defined as b = a[square root of 1 - [e.sup.2]].