equivalent

SearchAdvancedHelp

(0.000 seconds)

11—20 of 107 matching pages

11: 34.6 Definition: $\mathit{9}j$ Symbol

… ►The $\mathit{9}j$ symbol may be defined either in terms of $\mathit{3}j$ symbols or equivalently in terms of $\mathit{6}j$ symbols: … ►The $\mathit{9}j$ symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

12: 27.12 Asymptotic Formulas: Primes

… ► $\pi \left(x\right)-\mathrm{li}\left(x\right)$ changes sign infinitely often as $x\to \mathrm{\infty }$; see Littlewood (1914), Bays and Hudson (2000). ►The Riemann hypothesis (§25.10(i)) is equivalent to the statement that for every $x\ge 2657$, … ►For example, if $2^n\cancel{\equiv }2(modn)$, then $n$ is composite. … ►A Carmichael number is a composite number $n$ for which $b^n\equiv b(modn)$ for all $b\in \mathbb{N}$. …

13: Mathematical Introduction

… ► ►►►

$ℂ$	complex plane (excluding infinity).
…
$⟺$	is equivalent to.
…

► ►►►

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$mod$ or modulo	$m\equiv n(modp)$ means $p$ divides $m-n$, where $m$, $n$, and $p$ are positive integers with $m>n$.
…

…

14: 25.16 Mathematical Applications

… ►The prime number theorem (27.2.3) is equivalent to the statement ►

25.16.3 $$\psi \left(x\right)=x+o\left(x\right),$$ $x\to \mathrm{\infty }$.

ⓘ

Symbols:

$\psi \left(x\right)$: Chebyshev $\psi$-function, $\sim$: asymptotic equality, $o\left(x\right)$: order less than and $x$: real variable

Keywords:

asymptotic approximation

Source:

Apostol (1976, (10), p. 79); since $\psi \left(x\right)\sim x$ is equivalent to $\psi \left(x\right)=x+o\left(x\right)$

Referenced by:

§25.10(i)

Permalink:

http://dlmf.nist.gov/25.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(i), §25.16 and Ch.25

►The Riemann hypothesis is equivalent to the statement …

15: 31.3 Basic Solutions

… ►

§31.3(iii) Equivalent Expressions

… ►There are 192 automorphisms in all, so there are $192/8=24$ equivalent expressions for each of the 8. … ►The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).

16: 20.12 Mathematical Applications

… ►The space of complex tori $ℂ/(\mathbb{Z}+\tau \mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers $z_1$ and $z_2$ are regarded as equivalent if there exist integers $m,n$ such that $z_1-z_2=m+\tau n$) is mapped into the projective space $P^3$ via the identification $z\to ({\theta }_1\left(2z\right|\tau ),{\theta }_2\left(2z\right|\tau ),{\theta }_3\left(2z\right|\tau ),{\theta }_4\left(2z\right|\tau ))$. …

17: 27.14 Unrestricted Partitions

… ►implies $p\left(5n+4\right)\equiv 0(mod5)$. Ramanujan also found that $p\left(7n+5\right)\equiv 0(mod7)$ and $p\left(11n+6\right)\equiv 0(mod11)$ for all $n$. …Ono proved that for every prime $q>3$ there are integers $a$ and $b$ such that $p\left(an+b\right)\equiv 0(modq)$ for all $n$. For example, $p\left(\mathrm{1575\hspace{0.33em}25693}n+\mathrm{1\hspace{0.33em}11247}\right)\equiv 0(mod13)$. … ►

27.14.20 $$\tau \left(n\right)\equiv {\sigma }_{11}\left(n\right)(mod691).$$

ⓘ

Symbols:

$\tau \left(n\right)$: Ramanujan’s tau function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\equiv$: modular equivalence and $n$: positive integer

Referenced by:

§27.20, §27.20

Permalink:

http://dlmf.nist.gov/27.14.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(vi), §27.14 and Ch.27

…

18: 21.6 Products

… ►

21.6.1 $$𝒦={\mathbb{Z}}^{g\times h}𝐓/({\mathbb{Z}}^{g\times h}𝐓\cap {\mathbb{Z}}^{g\times h}),$$

ⓘ

Defines:

$𝒦$: set of matrices (locally)

Symbols:

$\mathbb{Z}$: set of all integers, $\cap$: intersection, $g$: positive integer and $h$: positive integer

Permalink:

http://dlmf.nist.gov/21.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

►that is, $𝒦$ is the set of all $g\times h$ matrices that are obtained by premultiplying $𝐓$ by any $g\times h$ matrix with integer elements; two such matrices in $𝒦$ are considered equivalent if their difference is a matrix with integer elements. … ►

21.6.2 $$𝒟=|{𝐓}^{\mathrm{T}}{\mathbb{Z}}^h/({𝐓}^{\mathrm{T}}{\mathbb{Z}}^h\cap {\mathbb{Z}}^h)|,$$

ⓘ

Defines:

$𝒟$: number of elements (locally)

Symbols:

$\mathbb{Z}$: set of all integers, $\cap$: intersection, ${𝐀}^{\mathrm{T}}$: transpose of matrix and $h$: positive integer

Permalink:

http://dlmf.nist.gov/21.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

… ►Two such vectors are considered equivalent if their difference is a vector with integer elements. …

19: 1.12 Continued Fractions

… ►

Equivalence

►Two continued fractions are equivalent if they have the same convergents. ► $b_0+{\displaystyle \frac{a_1}{b_1+}\frac{a_2}{b_2+}\mathrm{\cdots }}$ is equivalent to $b_0^{\prime }+{\displaystyle \frac{a_1^{\prime }}{b_1^{\prime }+}\frac{a_2^{\prime }}{b_2^{\prime }+}\mathrm{\cdots }}$ if there is a sequence ${\{d_n\}}_{n=0}^{\mathrm{\infty }}$, $d_0=1$,
$d_n\ne 0$, such that … ►The even part of $C$ exists iff $b_{2k}\ne 0$, $k=1,2,\mathrm{\dots }$, and up to equivalence is given by …The odd part of $C$ exists iff $b_{2k+1}\ne 0$, $k=0,1,2,\mathrm{\dots }$, and up to equivalence is given by …

20: 22.15 Inverse Functions

… ►Equations (22.15.1) and (22.15.4), for $\mathrm{arcsn}(x,k)$, are equivalent to (22.15.12) and also to … ►

§22.15(ii) Representations as Elliptic Integrals

… ►