.6月20足球赛世界杯『wn4.com』国足进过世界杯吗.w6n2c9o.2022年11月29日23时12分3秒.eu6me84ke.cc

SearchAdvancedHelp

(0.004 seconds)

21—30 of 605 matching pages

21: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

22: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

23: Publications

… ►

Using Numerical Grid Generation to Facilitate 3D Visualization of Complicated Mathematical Functions, Technical Report NISTIR 6413 (November 1999), National Institute of Standards and Technology.

… ►

Boundary/Contour Fitted Grid Generation for Effective Visualizations in a Digital Library of Mathematical Functions, Proceedings of the 9th International Conference on Numerical Grid Generation in Computational Field Simulations, San Jose, June 11–18, 2005. pp. 61–71.

►

Web-Based 3D Visualization in a Digital Library of Mathematical Functions, Proceedings of the Web3D Symposium, Bangor, UK, March 29–April 1, 2005.

►

From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243.

… ►

NIST’s Digital Library of Mathematial Functions, Physics Today 71, 2, 48 (2018), pp. 48–53.

24: 26.2 Basic Definitions

… ►Thus $231$ is the permutation $\sigma (1)=2$, $\sigma (2)=3$, $\sigma (3)=1$. … ►If, for example, a permutation of the integers 1 through 6 is denoted by $256413$, then the cycles are $\left(1,2,5\right)$, $\left(3,6\right)$, and $\left(4\right)$. …The function $\sigma$ also interchanges 3 and 6, and sends 4 to itself. … ►As an example, $\{1,3,4\}$, $\{2,6\}$, $\{5\}$ is a partition of $\{1,2,3,4,5,6\}$. … ►

Table 26.2.1: Partitions $p\left(n\right)$.

► ►►►

$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
3	3	20	627	37	21637
…

►

25: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►Table 26.4.1 gives numerical values of multinomials and partitions $\lambda ,M_1,M_2,M_3$ for $1\le m\le n\le 5$. …$M_3$ is the number of set partitions of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ subsets of size 1, $a_2$ subsets of size 2, $\mathrm{\dots }$, and $a_n$ subsets of size $n$: ►

26.4.8 $$M_3=\frac{n!}{{(1!)}^{a_1}(a_1!){(2!)}^{a_2}(a_2!)\mathrm{\cdots }{(n!)}^{a_n}(a_n!)}.$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $n$: nonnegative integer, $a_n$: nonnegative integers and $M_3$: number of partitions

Permalink:

http://dlmf.nist.gov/26.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.4(i), §26.4 and Ch.26

… ►

Table 26.4.1: Multinomials and partitions.

► ►►►►

$n$	$m$	$\lambda$	$M_1$	$M_2$	$M_3$
…
$5$	$2$	$2^1,3^1$	$10$	$20$	$10$
$5$	$3$	$1^2,3^1$	$20$	$20$	$10$
…

►

…

26: 22.3 Graphics

… ►

►

►

Figure 22.3.13: $\mathrm{sn}(x,k)$ for $k=1-{\mathrm{e}}^{-n}$, $n=0$ to 20, $-5\pi \le x\le 5\pi$. Magnify 3D Help

►

►

►

Figure 22.3.14: $\mathrm{cn}(x,k)$ for $k=1-{\mathrm{e}}^{-n}$, $n=0$ to 20, $-5\pi \le x\le 5\pi$. Magnify 3D Help

►

►

►

Figure 22.3.15: $\mathrm{dn}(x,k)$ for $k=1-{\mathrm{e}}^{-n}$, $n=0$ to 20, $-5\pi \le x\le 5\pi$. Magnify 3D Help

… ►

►►

Figure 22.3.28: Density plot of $|\mathrm{sn}(20,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. Grayscale, running from 0 (black) to 10 (white), with $|\mathrm{sn}(20,k)|>10$ truncated to 10. … Magnify

…

27: 26.3 Lattice Paths: Binomial Coefficients

… ►

Table 26.3.1: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$.

► ►►►►

$m$	$n$
	…
3	1	3	3	1
…
6	1	6	15	20	15	6	1
…

►

►

Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

► ►►►►

$m$	$n$
	0	1	2	3	4	5	6	7	8
…
3	1	4	10	20	35	56	84	120	165
…

►

… ►

26.3.11 $$\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)=\frac{2^n(2n-1)(2n-3)\mathrm{\cdots }3\cdot 1}{n!}.$$

ⓘ

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$) and $n$: nonnegative integer

A&S Ref:

24.1.1

Permalink:

http://dlmf.nist.gov/26.3.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(iv), §26.3 and Ch.26

…

28: 3.4 Differentiation

… ►

$$B_{-3}^6=-\frac{1}{720}(12-8t-45t^2+20t^3+15t^4-6t^5),$$

►

$$B_{-2}^6=\frac{1}{60}(9-9t-30t^2+20t^3+5t^4-3t^5),$$

… ►

$$B_2^6=-\frac{1}{60}(9+9t-30t^2-20t^3+5t^4+3t^5),$$

►

$$B_3^6=\frac{1}{720}(12+8t-45t^2-20t^3+15t^4+6t^5).$$

… ►

$$B_3^7=\frac{1}{720}(48+8t-192t^2-20t^3+85t^4+6t^5-7t^6),$$

…

29: 5.22 Tables

… ►Abramowitz and Stegun (1964, Chapter 6) tabulates $\mathrm{\Gamma }\left(x\right)$, $\ln \mathrm{\Gamma }\left(x\right)$, $\psi \left(x\right)$, and ${\psi }^{\prime }\left(x\right)$ for $x=1(.005)2$ to 10D; ${\psi }^{\prime \prime }\left(x\right)$ and ${\psi }^{(3)}\left(x\right)$ for $x=1(.01)2$ to 10D; $\mathrm{\Gamma }\left(n\right)$, $1/\mathrm{\Gamma }\left(n\right)$, $\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, $\psi \left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{3}\right)$, ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{1}{2}\right)$, and ${\log }_{10}\mathrm{\Gamma }\left(n+\frac{2}{3}\right)$ for $n=1(1)101$ to 8–11S; $\mathrm{\Gamma }\left(n+1\right)$ for $n=100(100)1000$ to 20S. …

30: 6.16 Mathematical Applications

… ►

6.16.1

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/6.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.16(i), §6.16 and Ch.6

… ►

►►

Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify