strictly%20increasing
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11: 26.2 Basic Definitions
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►A k-dimensional lattice path is a directed path composed of segments that connect vertices in so that each segment increases one coordinate by exactly one unit.
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12: 1.11 Zeros of Polynomials
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►Resolvent cubic is with roots , , , and , , .
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►with real coefficients, is called stable if the real parts of all the zeros are strictly negative.
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13: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►An equivalent form states that the th prime (when the primes are listed in increasing order) is asymptotic to as :
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14: 21.1 Special Notation
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positive integers. | |
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complex, symmetric matrix with strictly positive definite, i.e., a Riemann matrix. | |
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15: 23.20 Mathematical Applications
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►The boundary of the rectangle , with vertices , , , , is mapped strictly monotonically by onto the real line with , , , , .
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►The two pairs of edges and of are each mapped strictly monotonically by onto the real line, with , , ; similarly for the other pair of edges.
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16: 27.17 Other Applications
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►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series.
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17: 33.23 Methods of Computation
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►Cancellation errors increase with increases in and , and may be estimated by comparing the final sum of the series with the largest partial sum.
Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21.
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►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii.
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►This implies decreasing for the regular solutions and increasing
for the irregular solutions of §§33.2(iii) and 33.14(iii).
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18: 8 Incomplete Gamma and Related
Functions
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19: 28 Mathieu Functions and Hill’s Equation
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