strictly%20decreasing

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1: 7.8 Inequalities

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7.8.5 $x\ge 0$.

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Symbols:

$𝖬\left(x\right)$: Mills’ ratio and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

… ►The function $F\left(x\right)/\sqrt{1-{\mathrm{e}}^{-2x^2}}$ is strictly decreasing for $x>0$. …

2: 1.4 Calculus of One Variable

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§1.4(i) Monotonicity

… ►If the $\le$ sign is replaced by , then $f(x)$ is increasing (also called strictly increasing) on $I$. Similarly for nonincreasing and decreasing (strictly decreasing) functions. Each of the preceding four cases is classified as monotonic; sometimes strictly monotonic is used for the strictly increasing or strictly decreasing cases. …

3: 26.12 Plane Partitions

… ►A plane partition, $\pi$, of a positive integer $n$, is a partition of $n$ in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. … ►

Table 26.12.1: Plane partitions.

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$n$	$\mathrm{pp}\left(n\right)$	$n$	$\mathrm{pp}\left(n\right)$	$n$	$\mathrm{pp}\left(n\right)$
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3	6	20	75278	37	903 79784
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… ►A strict shifted plane partition is an arrangement of the parts in a partition so that each row is indented one space from the previous row and there is weak decrease across rows and strict decrease down columns. … ►A descending plane partition is a strict shifted plane partition in which the number of parts in each row is strictly less than the largest part in that row and is greater than or equal to the largest part in the next row. …

4: 18.16 Zeros

… ►Then ${\theta }_{n,m}$ is strictly increasing in $\alpha$ and strictly decreasing in $\beta$; furthermore, if $\alpha =\beta$, then ${\theta }_{n,m}$ is strictly increasing in $\alpha$. … ►Arrange them in decreasing order: …

5: 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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8	0	1	5	10	15	18	20	21	22	22	22
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… ►equivalently, partitions into at most $k$ parts either have exactly $k$ parts, in which case we can subtract one from each part, or they have strictly fewer than $k$ parts. …

6: 4.12 Generalized Logarithms and Exponentials

… ►and is strictly increasing when $0\le x\le 1$. …It, too, is strictly increasing when $0\le x\le 1$, and …

7: Notices

… ►Reproduction, copying, or distribution for any commercial purpose is strictly prohibited. …

8: 20 Theta Functions

Chapter 20 Theta Functions

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9: 1.11 Zeros of Polynomials

… ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. … ►with real coefficients, is called stable if the real parts of all the zeros are strictly negative. …

10: 2.2 Transcendental Equations

… ►Let $f(x)$ be continuous and strictly increasing when and …