Jordan curve theorem

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21: 27.22 Software

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Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below ${10}^{16}$. Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard $p-1$, and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

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ECMNET Project. Links to software for elliptic curve methods of factorization and primality testing.

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22: Bibliography

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Algorithm 39: Areas under the normal curve. The Computer Journal 12 (2), pp. 197–198.

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Notes:

Includes Algol program, maximum accuracy 12D.

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Document

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§7.25(ii).

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Characterization theorems for orthogonal polynomials. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 1–24.

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Link, MathReview (Walter Van Assche), ZentralBlatt

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§18.3.

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Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172 (2), pp. 289–312.

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ISSN 0377-0427, Document, MathReview (Živorad Tomovski), ZentralBlatt

Referenced by:

§19.9(i).

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Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.

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MathReview (N. G. de Bruijn), ZentralBlatt

Referenced by:

(25.11.41), (25.11.42), §25.11(xii).

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A Centennial History of the Prime Number Theorem. In Number Theory, Trends Math., pp. 1–14.

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MathReview (Giovanni Coppola), ZentralBlatt

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(25.16.2), (25.16.4), §25.16(i), §25.16.

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23: Bibliography F

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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt

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§18.39(i).

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Transverse cusp diffraction catastrophes produced by the reflection of ultrasonic tone bursts from a curved surface in water. J. Acoust. Soc. Amer. 92 (5), pp. 2869–2877.

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Document

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§36.14(iv).

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Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

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Document

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§36.14(iv).

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24: Bibliography M

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Sixth Painlevé Equation, Universal Elliptic Curve, and Mirror of ${𝐏}^2$ . In Geometry of Differential Equations, A. Khovanskii, A. Varchenko, and V. Vassiliev (Eds.), Amer. Math. Soc. Transl. Ser. 2, Vol. 186, pp. 131–151.

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External Links:

MathReview (Bruce Hunt), ZentralBlatt

Referenced by:

§32.2(iv).

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Elliptic Curves. Cambridge University Press, Cambridge.

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Notes:

Reprint with corrections of original edition, published in 1997 by Cambridge University Press

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ISBN 0-521-58228-8; 0-521-65817-9, MathReview (Amílcar Pacheco), ZentralBlatt

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§20.1, §20.11(i), §20.11(ii), §20.12(i), §20.12(i), §20.12(ii), §20.7(iii), §20.7(v), §20.9(i), §20.9(ii), Ch.20, §22.18(ii), §22.18(iv), Ch.22, §23.1, §23.15(i), §23.20(i), §23.20(ii), §23.20(ii), §23.21(ii), §23.22(ii), §23.6(iv), Ch.23, Notations T.

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Table of the ratio: Area to bounding ordinate, for any portion of normal curve. Biometrika 18, pp. 395–400.

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FullText, ZentralBlatt

Referenced by:

§7.8.

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A $q$-analog of the ${}_5{}^{}F_4^{}(1)$ summation theorem for hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ . Adv. in Math. 57 (1), pp. 14–33.

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ISSN 0001-8708, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

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§17.15.

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A $q$-analog of the Gauss summation theorem for hypergeometric series in $U(n)$ . Adv. in Math. 72 (1), pp. 59–131.

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ISSN 0001-8708, Document, MathReview (David M. Bressoud), ZentralBlatt

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§17.15.

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25: 24.6 Explicit Formulas

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26: 19.35 Other Applications

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§19.35(i) Mathematical

►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …

27: 28.17 Stability as $x\to \pm \mathrm{\infty }$

… ►The boundary of each region comprises the characteristic curves $a=a_n\left(q\right)$ and $a=b_n\left(q\right)$; compare Figure 28.2.1. …

28: Bibliography C

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The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.

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MathReview (K. Goldberg), ZentralBlatt

Referenced by:

§24.6.

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New proof of the addition theorem for Gegenbauer polynomials. SIAM J. Math. Anal. 2, pp. 347–351.

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ISSN 1095-7154, Document, MathReview (Mary L. Boas), ZentralBlatt

Referenced by:

§18.18(ii).

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Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9 (3), pp. 524–528.

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Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.26(i).

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The universal von Staudt theorems. Trans. Amer. Math. Soc. 315 (2), pp. 591–603.

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ISSN 0002-9947, FullText, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.16(iii).

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Algorithms for Modular Elliptic Curves. 2nd edition, Cambridge University Press, Cambridge.

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ISBN 0-521-59820-6, MathReview (Joseph H. Silverman), ZentralBlatt

Referenced by:

§23.20(ii).

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29: 18.39 Applications in the Physical Sciences

… ►An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with $L^2$ eigenfunctions vanishing at the end points, in this case $\pm \mathrm{\infty }$ see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. …Both satisfy Sturm’s theorem. … ►

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Figure 18.39.1: Graphs of the first and fourth excited state eigenfunctions of the harmonic oscillator, for $\mathrm{\hslash }=k=m=1$, of (18.39.13), in ${\psi }_1(x)$, ${\psi }_4(x)$ and those of the rational potential of (18.39.19), in ${\widehat{\psi }}_3(x)$, ${\widehat{\psi }}_6(x)$. Both sets satisfy the Sturm oscillation theorem. Magnify

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Figure 18.39.2: Coulomb–Pollaczek weight functions, $x\in [-1,1]$, (18.39.50) for $s=10$, $l=0$, and $Z=\pm 1$. For $Z=+1$ the weight function, red curve, has an essential singularity at $x=-1$, as all derivatives vanish as $x\to -1+$; the green curve is ${\int }_{-1}^x{w}^{\mathrm{CP}}(y)dy$, to be compared with its histogram approximation in §18.40(ii). For $Z=-1$ the weight function, blue curve, is non-zero at $x=-1$, but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as $k\to \mathrm{\infty }$, $x_k\to -1-$. Magnify

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30: Bibliography D

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Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.

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ISSN 0167-2789, Document, MathReview (John B. Little), ZentralBlatt

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3rd item.

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Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.

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MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

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Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

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ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§35.7(ii), §35.8(iv), §35.8(v).

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On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.

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ZentralBlatt

Referenced by:

§15.4(ii).

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