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11—19 of 19 matching pages
11: 28.33 Physical Applications
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►We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass per unit area, and radial tension per unit arc length.
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►If the parameters of a physical system vary periodically with time, then the question of stability arises, for example, a mathematical pendulum whose length varies as .
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12: 21.9 Integrable Equations
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►Particularly important for the use of Riemann theta functions is the Kadomtsev–Petviashvili (KP) equation, which describes the propagation of two-dimensional, long-wave length surface waves in shallow water (Ablowitz and Segur (1981, Chapter 4)):
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13: 19.24 Inequalities
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►Approximations and one-sided inequalities for follow from those given in §19.9(i) for the length
of an ellipse with semiaxes and , since
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19.24.7
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14: 1.2 Elementary Algebra
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►A column vector of length
is an matrix
…and the corresponding transposed row vector of length
is
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►Column vectors and of the same length
have a scalar product
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►Special cases are the Euclidean length or
norm
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►Unless otherwise indicated, matrices are assumed square, of order ; and, when vectors are combined with them, these are of length
.
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15: 34.4 Definition: Symbol
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►Except in degenerate cases the combination of the triangle inequalities for the four symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths
; see Figure 34.4.1.
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16: 1.6 Vectors and Vector-Valued Functions
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►then the length of a path for is
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17: Bibliography B
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On the distribution of the length of the longest increasing subsequence of random permutations.
J. Amer. Math. Soc. 12 (4), pp. 1119–1178.
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A monotonicity property involving and comparisons of the classical approximations of elliptical arc length.
SIAM J. Math. Anal. 32 (2), pp. 403–419.
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18: 1.4 Calculus of One Variable
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►For historical reasons, is also sometimes referred to as a density, as, for example, the mass per unit length at point , see Shohat and Tamarkin (1970, p vii).
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