.俄罗斯世界杯厉害球员_『wn4.com_』世界杯亚洲预选赛1_w6n2c9o_2022年11月29日15时10分23秒_xscpuhpjp

(0.002 seconds)

21—30 of 790 matching pages

21: 26.16 Multiset Permutations

… ►Let

S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}

be the multiset that has

a_{j}

copies of

j

1\leq j\leq n

\mathfrak{S}_{S}

denotes the set of permutations of

S

for all distinct orderings of the

a_{1}+a_{2}+\cdots+a_{n}

integers. The number of elements in

\mathfrak{S}_{S}

is the multinomial coefficient (§26.4)

\genfrac{(}{)}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}

. … ►The $q$ -multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by …and again with

S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}

we have …

22: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►

\genfrac{(}{)}{0.0pt}{}{n}{n_{1},n_{2},\ldots,n_{k}}

is the number of ways of placing

n=n_{1}+n_{2}+\cdots+n_{k}

distinct objects into

k

labeled boxes so that there are

n_{j}

objects in the

j

th box. … ►These are given by the following equations in which

a_{1},a_{2},\ldots,a_{n}

are nonnegative integers such that …

M_{1}

is the multinominal coefficient (26.4.2): …For each

n

all possible values of

a_{1},a_{2},\ldots,a_{n}

are covered. … ►where the summation is over all nonnegative integers

n_{1},n_{2},\ldots,n_{k}

such that

n_{1}+n_{2}+\cdots+n_{k}=n

. …

23: 10.75 Tables

… ►

•

Achenbach (1986) tabulates $J_{0}\left(x\right)$ , $J_{1}\left(x\right)$ , $Y_{0}\left(x\right)$ , $Y_{1}\left(x\right)$ , $x=0(.1)8$ , 20D or 18–20S.

… ►

•

Makinouchi (1966) tabulates all values of $j_{\nu,m}$ and $y_{\nu,m}$ in the interval $(0,100)$ , with at least 29S. These are for $\nu=0(1)5$ , 10, 20; $\nu=\tfrac{3}{2}$ , $\tfrac{5}{2}$ ; $\nu=m/n$ with $m=1(1)n-1$ and $n=3(1)8$ , except for $\nu=\tfrac{1}{2}$ .

… ►

•

Abramowitz and Stegun (1964, Chapter 11) tabulates $\int_{0}^{x}J_{0}\left(t\right)\,\mathrm{d}t$ , $\int_{0}^{x}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 10D; $\int_{0}^{x}t^{-1}(1-J_{0}\left(t\right))\,\mathrm{d}t$ , $\int_{x}^{\infty}t^{-1}Y_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 8D.

… ►

•

Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of $K_{n}\left(z\right)$ , for $n=2(1)10$ , 29S.

… ►

•

Abramowitz and Stegun (1964, Chapter 11) tabulates $e^{-x}\int_{0}^{x}I_{0}\left(t\right)\,\mathrm{d}t$ , $e^{x}\int_{x}^{\infty}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)10$ , 7D; $e^{-x}\int_{0}^{x}t^{-1}(I_{0}\left(t\right)-1)\,\mathrm{d}t$ , $xe^{x}\int_{x}^{\infty}t^{-1}K_{0}\left(t\right)\,\mathrm{d}t$ , $x=0(.1)5$ , 6D.

…

24: 10 Bessel Functions

…

25: Bibliography F

… ►

V. N. Faddeyeva and N. M. Terent’ev (1961) Tables of Values of the Function

w(z)=e^{-z^{2}}(1+2i\pi^{-1/2}\int_{0}^{z}e^{t^{2}}dt)

for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: V. N. Faddeeva and N. M. Terent’ev (1954).
Encodings:: BibTeX

… ►

N. Fleury and A. Turbiner (1994) Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Shao-Ming Fei), ZentralBlatt
Referenced by:: §13.3(ii), §15.5(i), §16.3(i).
Encodings:: BibTeX

… ►

A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.13(i), §32.7(i), §32.7(iii).
Encodings:: BibTeX

… ►

P. J. Forrester and N. S. Witte (2004) Application of the

\tau

-function theory of Painlevé equations to random matrices:

\rm P_{\rm VI}

, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

ⓘ

External Links:: ISSN 0027-7630, MathReview, ZentralBlatt
Referenced by:: §32.10(vi), §32.6(v).
Encodings:: BibTeX

… ►

L. W. Fullerton (1972) Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.

ⓘ

Notes:: Includes Fortran code. For a Remark see ACM Trans. Math. Software 4 (1978), no. 3, pp. 296–304.
External Links:: ISSN 0001-0782, Document, Remark
Referenced by:: §8.28(ii).
Encodings:: BibTeX

…

26: 19.29 Reduction of General Elliptic Integrals

… ►Let …where … ►Next, for

j=1,2

, define

Q_{j}(t)=f_{j}+g_{j}t+h_{j}t^{2}

, and assume both

Q

’s are positive for

y<t<x

. …where …If

Q_{1}(t)=(a_{1}+b_{1}t)(a_{2}+b_{2}t)

, where both linear factors are positive for

y<t<x

, and

Q_{2}(t)=f_{2}+g_{2}t+h_{2}t^{2}

, then (19.29.25) is modified so that …

27: 3.9 Acceleration of Convergence

… ►A transformation of a convergent sequence

\{s_{n}\}

with limit

\sigma

into a sequence

\{t_{n}\}

is called limit-preserving if

\{t_{n}\}

converges to the same limit

\sigma

. … ►This transformation is accelerating if

\{s_{n}\}

is a linearly convergent sequence, i. … ►Then the transformation of the sequence

\{s_{n}\}

into a sequence

\{t_{n,2k}\}

is given by … ►Then

t_{n,2k}=\varepsilon_{2k}^{(n)}

. … ►We give a special form of Levin’s transformation in which the sequence

s=\{s_{n}\}

of partial sums

s_{n}=\sum_{j=0}^{n}a_{j}

is transformed into: …

28: Bibliography B

… ►

R. Barakat (1961) Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

ⓘ

External Links:: FullText, MathReview (J. C. P. Miller), ZentralBlatt
Referenced by:: §8.7.
Encodings:: BibTeX

… ►

B. C. Berndt, S. Bhargava, and F. G. Garvan (1995) Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt
Referenced by:: §15.8(v), §20.11(iii).
Encodings:: BibTeX

… ►

F. Bethuel (1998) Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

ⓘ

External Links:: ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §17.17.
Encodings:: BibTeX

… ►

A. Bhattacharyya and L. Shafai (1988) Theoretical and experimental investigation of the elliptical annual ring antenna. IEEE Trans. Antennas and Propagation 36 (11), pp. 1526–1530.

ⓘ

External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

… ►

R. L. Bishop (1981) Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

ⓘ

External Links:: ISSN 0035-9149, Document, MathReview
Referenced by:: Sidebar 9.SB1.
Encodings:: BibTeX

…

29: 26.2 Basic Definitions

… ►

Table 26.2.1: Partitions

p\left(n\right)

► ►►►►►

$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
…
6	11	23	1255	40	37338
…
11	56	28	3718	45	89134
12	77	29	4565	46	1 05558
…

►

30: Bibliography W

… ►

C. A. Wills, J. M. Blair, and P. L. Ragde (1982) Rational Chebyshev approximations for the Bessel functions

J_{0}(x)

J_{1}(x)

Y_{0}(x)

Y_{1}(x)

. Math. Comp. 39 (160), pp. 617–623.

ⓘ

Notes:: Tables on microfiche
External Links:: ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt
Referenced by:: 9th item.
Encodings:: BibTeX

… ►

J. A. Wilson (1980) Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11 (4), pp. 690–701.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (R. G. Buschman), ZentralBlatt
Referenced by:: §18.25(i), §18.25(i), §18.25(ii), §18.25(iii), §18.26(i), §18.26(ii), §18.38(iii).
Encodings:: BibTeX

►

J. A. Wilson (1991) Asymptotics for the

{}_{4}F_{3}

polynomials. J. Approx. Theory 66 (1), pp. 58–71.

ⓘ

External Links:: ISSN 0021-9045, Document, MathReview (R. G. Buschman), ZentralBlatt
Referenced by:: §18.26(v).
Encodings:: BibTeX

… ►

G. Wolf (2008) On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.

ⓘ

External Links:: ISSN 1044-677X, FullText
Referenced by:: §28.4(vii).
Encodings:: BibTeX

… ►

E. M. Wright (1940b) The generalized Bessel function of order greater than one. Quart. J. Math., Oxford Ser. 11, pp. 36–48.

ⓘ

External Links:: Document, MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

…