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11: 10.62 Graphs

§10.62 Graphs

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12: 27.19 Methods of Computation: Factorization

… ►Type II probabilistic algorithms for factoring $n$ rely on finding a pseudo-random pair of integers $(x,y)$ that satisfy $x^2\equiv y^2(modn)$. …

13: Philip J. Davis

… ►He also had a big influence on the development of the NBS Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (A&S), which became one of the most widely distributed and highly cited publications in NIST’s history. … ►After receiving an overview of the project and watching a short demo that included a few preliminary colorful, but static, 3D graphs constructed for the first Chapter, “Airy and Related Functions”, written by Olver, Davis expressed the hope that designing a web-based resource would allow the team to incorporate interesting computer graphics, such as function surfaces that could be rotated and examined. … ►Davis’s comments about our uninspired graphs sparked the research and design of techniques for creating interactive 3D visualizations of function surfaces, which grew in sophistication as our knowledge and the technology for developing 3D graphics on the web advanced over the years. Today the DLMF contains close to 600 2D and 3D graphs and more than 200 interactive 3D visualizations. …

14: 10.48 Graphs

§10.48 Graphs

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Figure 10.48.7: ${𝗂}_5^{(1)}\left(x\right)$, ${𝗂}_5^{(2)}\left(x\right)$, ${𝗄}_5\left(x\right)$, $0\le x\le 8$. Magnify

15: 10.26 Graphics

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Figure 10.26.7: ${\tilde{I}}_{1/2}\left(x\right)$, ${\tilde{K}}_{1/2}\left(x\right)$, $0.01\le x\le 3$. Magnify

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Figure 10.26.8: ${\tilde{I}}_1\left(x\right)$, ${\tilde{K}}_1\left(x\right)$, $0.01\le x\le 3$. Magnify

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Figure 10.26.9: ${\tilde{I}}_5\left(x\right)$, ${\tilde{K}}_5\left(x\right)$, $0.01\le x\le 3$. Magnify

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Figure 10.26.10: ${\tilde{K}}_5\left(x\right)$, $0.01\le x\le 3$. Magnify

16: 18.38 Mathematical Applications

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Random Matrix Theory

►Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. …

17: 6.3 Graphics

… ►For a graph of $\mathrm{li}\left(x\right)$ see Figure 6.16.2. …

18: 9.3 Graphics

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§9.3(i) Real Variable

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19: Bibliography F

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Application of the $\tau$-function theory of Painlevé equations to random matrices: PIV, PII and the GUE. Comm. Math. Phys. 219 (2), pp. 357–398.

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External Links:

ISSN 0010-3616, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§32.10(iv), §32.14, §32.6(v).

Encodings:

BibTeX

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Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\mathrm{P}}_{\mathrm{V}}$, ${\mathrm{P}}_{\mathrm{III}}$, the LUE, JUE, and CUE. Comm. Pure Appl. Math. 55 (6), pp. 679–727.

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External Links:

ISSN 0010-3640, Document, MathReview (Mark Adler), ZentralBlatt

Referenced by:

§32.10(iii), §32.10(v), §32.14, §32.6(iv), §32.6(v).

Encodings:

BibTeX

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Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\mathrm{P}}_{\mathrm{VI}}$, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

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External Links:

ISSN 0027-7630, MathReview, ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v).

Encodings:

BibTeX

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Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond. In Recent Perspectives in Random Matrix Theory and Number Theory, London Math. Soc. Lecture Note Ser., Vol. 322, pp. 31–78.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

BibTeX

20: 26.9 Integer Partitions: Restricted Number and Part Size

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Table 26.9.1: Partitions $p_k\left(n\right)$.

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$n$	$k$
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►A useful representation for a partition is the Ferrers graph in which the integers in the partition are each represented by a row of dots. … ►

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Figure 26.9.1: Ferrers graph of the partition $7+4+3+3+2+1$.

►The conjugate partition is obtained by reflecting the Ferrers graph across the main diagonal or, equivalently, by representing each integer by a column of dots. …