## English Mathspeak Steve Noble's samples. Locale: en, Style: Verbose.

 0 $-5\frac{1}{5}-6\frac{2}{3}=$ negative 5 and one fifth minus 6 and two thirds equals 1 $-7\frac{3}{4}-\left(-4\frac{7}{8}\right)=$ negative 7 and three fourths minus left parenthesis negative 4 and seven eighths right parenthesis equals 2 $-24.15-\left(13.7\right)=$ negative 24.15 minus left parenthesis 13.7 right parenthesis equals 3 $\left(-4\right)×3=-12$ left parenthesis negative 4 right parenthesis times 3 equals negative 12 4 $-12÷3=-4$ negative 12 divided by 3 equals negative 4 5 $-12÷\left(-4\right)=3$ negative 12 divided by left parenthesis negative 4 right parenthesis equals 3 6 $6×5$ 6 times 5 7 $6×\left(-5\right)$ 6 times left parenthesis negative 5 right parenthesis 8 $-6×5$ negative 6 times 5 9 $-6×\left(-5\right)$ negative 6 times left parenthesis negative 5 right parenthesis 10 $-8×7$ negative 8 times 7 11 $-8×\left(-7\right)$ negative 8 times left parenthesis negative 7 right parenthesis 12 $8×\left(-7\right)$ 8 times left parenthesis negative 7 right parenthesis 13 $8×7$ 8 times 7 14 $m\angle 1=\mathrm{30°}$ m angle 1 equals 30 degree 15 $m\angle 2=\mathrm{60°}$ m angle 2 equals 60 degree 16 $m\angle 1+m\angle 2=\mathrm{90°}$ m angle 1 plus m angle 2 equals 90 degree 17 $m\angle M+m\angle N=\mathrm{180°}$ m angle upper M plus m angle upper N equals 180 degree 18 $A=\frac{1}{2}bh$ upper A equals one half b h 19 StartFraction area of triangle Over area of square EndFraction equals StartFraction 1 unit squared Over 16 units squared EndFraction 20 ${0.6}^{2}$ 0.6 squared 21 ${1.5}^{2}$ 1.5 squared 22 $4\left(2x+3x\right)$ 4 left parenthesis 2 x plus 3 x right parenthesis 23 $36+4y-1{y}^{2}+5{y}^{2}-2$ 36 plus 4 y minus 1 y squared plus 5 y squared minus 2 24 $\left(5+9\right)-4+3=$ left parenthesis 5 plus 9 right parenthesis minus 4 plus 3 equals 25 $\stackrel{↔}{BC}$ ModifyingAbove upper B upper C With left right arrow 26 $\stackrel{\to }{PQ}$ ModifyingAbove upper P upper Q With right arrow 27 $\overline{GH}$ ModifyingAbove upper G upper H With bar 28 $\overline{WX}\cong \overline{YZ}$ ModifyingAbove upper W upper X With bar approximately equals ModifyingAbove upper Y upper Z With bar 29 $\angle BEF$ angle upper B upper E upper F 30 $\angle BED$ angle upper B upper E upper D 31 $\angle DEF$ angle upper D upper E upper F 32 $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$ x equals StartFraction negative b plus or minus StartRoot b squared minus 4 a c EndRoot Over 2 a EndFraction 33 $y={x}^{2}+8x+16$ y equals x squared plus 8 x plus 16 34 $y=\frac{1}{3}\left({3}^{x}\right)$ y equals one third left parenthesis 3 Superscript x Baseline right parenthesis 35 $y=10-2x$ y equals 10 minus 2 x 36 $y=2{x}^{3}+5$ y equals 2 x cubed plus 5 37 $y=\left({x}^{2}+1\right)\left({x}^{2}+3\right)$ y equals left parenthesis x squared plus 1 right parenthesis left parenthesis x squared plus 3 right parenthesis 38 $y={0.5}^{x}$ y equals 0.5 Superscript x 39 $y=22-2x$ y equals 22 minus 2 x 40 $y=\frac{3}{x}$ y equals StartFraction 3 Over x EndFraction 41 $y=\left(x+4\right)\left(x+4\right)$ y equals left parenthesis x plus 4 right parenthesis left parenthesis x plus 4 right parenthesis 42 $y=\left(4x-3\right)\left(x+1\right)$ y equals left parenthesis 4 x minus 3 right parenthesis left parenthesis x plus 1 right parenthesis 43 $y=20x-4{x}^{2}$ y equals 20 x minus 4 x squared 44 $y={x}^{2}$ y equals x squared 45 $y={3}^{x-1}$ y equals 3 Superscript x minus 1 46 $y=16-2\left(x+3\right)$ y equals 16 minus 2 left parenthesis x plus 3 right parenthesis 47 $y=4{x}^{2}-x-3$ y equals 4 x squared minus x minus 3 48 $y=x+\frac{1}{x}$ y equals x plus StartFraction 1 Over x EndFraction 49 $y=4x\left(5-x\right)$ y equals 4 x left parenthesis 5 minus x right parenthesis 50 $y=2\left(x-3\right)+6\left(1-x\right)$ y equals 2 left parenthesis x minus 3 right parenthesis plus 6 left parenthesis 1 minus x right parenthesis 51 $0.25>\frac{5}{16}$ 0.25 greater than five sixteenths 52 $32\cdot \left(5\cdot 7\right)$ 32 dot left parenthesis 5 dot 7 right parenthesis 53 $\left(\frac{1}{2}×\frac{1}{2}×\pi ×2\right)+\left(2×\frac{1}{2}×\pi ×5\right)$ left parenthesis one half times one half times pi times 2 right parenthesis plus left parenthesis 2 times one half times pi times 5 right parenthesis 54 $\underset{n\to \infty }{\text{liminf}}{E}_{n}=\bigcup _{n\ge 1}\bigcap _{k\ge n}{E}_{k},\phantom{\rule{0.2em}{0ex}}\underset{n\to \infty }{\text{limsup}}{E}_{n}=\bigcap _{n\ge 1}\bigcup _{k\ge n}{E}_{k}.$ liminf Underscript n right arrow infinity Endscripts upper E Subscript n Baseline equals union Underscript n greater than or equals 1 Endscripts intersection Underscript k greater than or equals n Endscripts upper E Subscript k Baseline comma limsup Underscript n right arrow infinity Endscripts upper E Subscript n Baseline equals intersection Underscript n greater than or equals 1 Endscripts union Underscript k greater than or equals n Endscripts upper E Subscript k Baseline period 55 $\begin{array}{ll}\text{(i)}\hfill & \phantom{\rule{0.2em}{0ex}}𝒮\in 𝒜;\hfill \\ \multicolumn{1}{c}{\text{(ii)}}& \phantom{\rule{0.2em}{0ex}}\text{if}E\in 𝒜\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}\stackrel{‾}{E}\in 𝒜;\hfill \\ \multicolumn{1}{c}{\text{(iii)}}& \phantom{\rule{0.2em}{0ex}}\text{if}{E}_{1},{E}_{2}\in 𝒜\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}{E}_{1}\cup {E}_{2}\in 𝒜.\hfill \end{array}$ StartLayout 1st Row 1st Column left parenthesis i right parenthesis 2nd Column script upper S element of script upper A semicolon 2nd Row 1st Column left parenthesis ii right parenthesis 2nd Column if upper E element of script upper A then upper E overbar element of script upper A semicolon 3rd Row 1st Column left parenthesis iii right parenthesis 2nd Column if upper E 1 comma upper E 2 element of script upper A then upper E 1 union upper E 2 element of script upper A period EndLayout 56 $\begin{array}{llll}\hfill & \hfill & \left(\mathrm{A}\mathrm{.}1\right)\phantom{\rule{0.2em}{0ex}}\mathrm{If}\phantom{\rule{0.2em}{0ex}}A\in ℱ\phantom{\rule{0.2em}{0ex}}\mathrm{then}\phantom{\rule{0.2em}{0ex}}0\le P\left\{A\right\}\le 1.\hfill & \left(1\right)\hfill \\ \multicolumn{1}{c}{}& \hfill & \left(\mathrm{A}\mathrm{.}2\right)\phantom{\rule{0.2em}{0ex}}P\left\{𝒮\right\}=1.\hfill & \left(2\right)\hfill \\ \multicolumn{1}{c}{}& \hfill & \left(\mathrm{A}\mathrm{.}3\right)\phantom{\rule{0.2em}{0ex}}\mathrm{If}\phantom{\rule{0.2em}{0ex}}\left\{{E}_{n},n\ge 1\right\}\in ℱ\phantom{\rule{0.2em}{0ex}}\text{is a sequence of}\phantom{\rule{0.2em}{0ex}}\text{disjoint}\hfill & \left(3\right)\hfill \end{array}$ StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column left parenthesis normal upper A period 1 right parenthesis upper I f upper A element of script upper F t h e n 0 less than or equals upper P left brace upper A right brace less than or equals 1 period 4th Column left parenthesis 1 right parenthesis 2nd Row 1st Column Blank 2nd Column Blank 3rd Column left parenthesis normal upper A period 2 right parenthesis upper P left brace script upper S right brace equals 1 period 4th Column left parenthesis 2 right parenthesis 3rd Row 1st Column Blank 2nd Column Blank 3rd Column left parenthesis normal upper A period 3 right parenthesis upper I f left brace upper E Subscript n Baseline comma n greater than or equals 1 right brace element of script upper F is a sequence of disjoint 4th Column left parenthesis 3 right parenthesis EndLayout 57 $P\left\{{B}_{j}|A\right\}=\frac{P\left\{{B}_{j}\right\}P\left\{A|{B}_{j}\right\}}{\sum _{j\prime \in J}P\left\{{B}_{j\prime }\right\}P\left\{A|{B}_{j\prime }\right\}}.$ upper P left brace upper B Subscript j Baseline vertical bar upper A right brace equals StartFraction upper P left brace upper B Subscript j Baseline right brace upper P left brace upper A vertical bar upper B Subscript j Baseline right brace Over sigma summation Underscript j prime element of upper J Endscripts upper P left brace upper B Subscript j prime Baseline right brace upper P left brace upper A vertical bar upper B Subscript j prime Baseline right brace EndFraction period 58 ${\mu }_{1}\left(B\right)={\int }_{B}f\left(x\right)d{\mu }_{2}\left(x\right)$ mu 1 left parenthesis upper B right parenthesis equals integral Underscript upper B Endscripts f left parenthesis x right parenthesis d mu 2 left parenthesis x right parenthesis 59 $\underset{n\to \infty }{\text{lim}}E\left\{|{X}_{n}-X|\right\}=E\left\{\underset{n\to \infty }{\text{lim}}|{X}_{n}-X|\right\}=0.$ limit Underscript n right arrow infinity Endscripts upper E left brace StartAbsoluteValue upper X Subscript n Baseline minus upper X EndAbsoluteValue right brace equals upper E left brace limit Underscript n right arrow infinity Endscripts StartAbsoluteValue upper X Subscript n Baseline minus upper X EndAbsoluteValue right brace equals 0 period 60 $\begin{array}{l}{P}_{\mu ,\sigma }\left\{Y\ge {l}_{\beta }\left({\stackrel{‾}{Y}}_{n},{S}_{n}\right)\right\}={P}_{\mu ,\sigma }\left\{\left(Y-{\stackrel{‾}{Y}}_{n}\right)/\left(S·{\left(1+\frac{1}{n}\right)}^{1/2}\right)\ge -{t}_{\beta }\left[n-1\right]\right\}=\beta ,\hfill \\ \multicolumn{1}{c}{}& \left(1\right)\hfill \end{array}$ StartLayout 1st Row 1st Column upper P Subscript mu comma sigma Baseline left brace upper Y greater than or equals l Subscript beta Baseline left parenthesis upper Y overbar Subscript n Baseline comma upper S Subscript n Baseline right parenthesis right brace equals upper P Subscript mu comma sigma Baseline left brace left parenthesis upper Y minus upper Y overbar Subscript n Baseline right parenthesis divided by left parenthesis upper S dot left parenthesis 1 plus StartFraction 1 Over n EndFraction right parenthesis Superscript 1 divided by 2 Baseline right parenthesis greater than or equals minus t Subscript beta Baseline left bracket n minus 1 right bracket right brace equals beta comma 2nd Row 1st Column Blank 2nd Column left parenthesis 1 right parenthesis EndLayout 61 $L=\left(\begin{array}{cccccc}\hfill 1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}-1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}-1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}0\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}0\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}-1\hfill \end{array}\right).$ upper L equals Start 5 By 6 Matrix 1st Row 1st Column 1 2nd Column negative 1 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 2nd Row 1st Column Blank 2nd Column 1 3rd Column negative 1 4th Column Blank 5th Column 0 6th Column Blank 3rd Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 4th Row 1st Column Blank 2nd Column 0 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 5th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column 1 6th Column negative 1 EndMatrix period 62 $\sqrt{n}\left[{\stackrel{‾}{Y}}_{n}-\left(\mu +{z}_{\beta }\sigma \right)\right]/{S}_{n}~\frac{U+\sqrt{n}\phantom{\rule{0.2em}{0ex}}{z}_{1-\beta }}{\left({\chi }^{2}\left[n-1\right]/\left(n-1\right)\right){}^{1/2}}~t\left[n-1;\sqrt{n}\phantom{\rule{0.2em}{0ex}}{z}_{1-\beta }\right],$ StartRoot n EndRoot left bracket upper Y overbar Subscript n Baseline minus left parenthesis mu plus z Subscript beta Baseline sigma right parenthesis right bracket divided by upper S Subscript n Baseline tilde StartFraction upper U plus StartRoot n EndRoot z Subscript 1 minus beta Baseline Over left parenthesis chi squared left bracket n minus 1 right bracket divided by left parenthesis n minus 1 right parenthesis right parenthesis Superscript 1 divided by 2 Baseline EndFraction tilde t left bracket n minus 1 semicolon StartRoot n EndRoot z Subscript 1 minus beta Baseline right bracket comma 63 $\begin{array}{ll}\gamma \hfill & =P\left\{{E}_{p,q}\subset \left({X}_{\left(r\right)},{X}_{\left(s\right)}\right\}\hfill \\ \multicolumn{1}{c}{}& =\frac{n!}{\left(r-1\right)!}\sum _{j=0}^{s-r-1}\left(-1\right){}^{j}\frac{{p}^{r+j}}{\left(n-r-j\right)!j!}{I}_{1-q}\left(n-s+1,s-r-j\right).\hfill \end{array}$ StartLayout 1st Row 1st Column gamma 2nd Column equals upper P left brace upper E Subscript p comma q Baseline subset of left parenthesis upper X Subscript left parenthesis r right parenthesis Baseline comma upper X Subscript left parenthesis s right parenthesis Baseline right brace 2nd Row 1st Column Blank 2nd Column equals StartFraction n factorial Over left parenthesis r minus 1 right parenthesis factorial EndFraction sigma summation Underscript j equals 0 Overscript s minus r minus 1 Endscripts left parenthesis negative 1 right parenthesis Superscript j Baseline StartFraction p Superscript r plus j Baseline Over left parenthesis n minus r minus j right parenthesis factorial j factorial EndFraction upper I Subscript 1 minus q Baseline left parenthesis n minus s plus 1 comma s minus r minus j right parenthesis period EndLayout 64 ${S}_{i}\left[\begin{array}{c}t\\ x\end{array}\right]=\left[\begin{array}{cc}1/m& 0\\ {a}_{i}& {r}_{i}\end{array}\right]\left[\begin{array}{c}t\\ x\end{array}\right]+\left[\begin{array}{c}\left(i-1\right)/m\\ {b}_{i}\end{array}\right],$ upper S Subscript i Baseline StartBinomialOrMatrix t Choose x EndBinomialOrMatrix equals Start 2 By 2 Matrix 1st Row 1st Column 1 divided by m 2nd Column 0 2nd Row 1st Column a Subscript i Baseline 2nd Column r Subscript i Baseline EndMatrix StartBinomialOrMatrix t Choose x EndBinomialOrMatrix plus StartBinomialOrMatrix left parenthesis i minus 1 right parenthesis divided by m Choose b Subscript i Baseline EndBinomialOrMatrix comma 65 ${c}_{1}{h}^{4-2s}\le \frac{1}{2T}{\int }_{-T}^{T}{\left(f\left(t+h\right)-f\left(t\right)\right)}^{2}\mathrm{d}t\le {c}_{2}{h}^{4-2s}$ c 1 h Superscript 4 minus 2 s Baseline less than or equals StartFraction 1 Over 2 upper T EndFraction integral Subscript negative upper T Superscript upper T Baseline left parenthesis f left parenthesis t plus h right parenthesis minus f left parenthesis t right parenthesis right parenthesis squared normal d t less than or equals c 2 h Superscript 4 minus 2 s 66 $C\left(0\right)-C\left(h\right)\simeq c{h}^{4-2s}$ upper C left parenthesis 0 right parenthesis minus upper C left parenthesis h right parenthesis asymptotically equals c h Superscript 4 minus 2 s 67 $S\left(\omega \right)=\underset{T\to \infty }{\mathrm{lim}}\frac{1}{2T}{\left|{\int }_{-T}^{T},f,\left(,t,\right),{\mathrm{e}}^{\mathit{it}\omega },\mathrm{d},t\right|}^{2}.$ upper S left parenthesis omega right parenthesis equals limit Underscript upper T right arrow infinity Endscripts StartFraction 1 Over 2 upper T EndFraction StartAbsoluteValue integral Subscript negative upper T Superscript upper T Baseline comma f comma left parenthesis comma t comma right parenthesis comma normal e Superscript italic i t omega Baseline comma normal d comma t EndAbsoluteValue squared period 68 ${\int }_{0}^{1}\phantom{\rule{-0.2em}{0ex}}{\int }_{0}^{1}{\left[|f\left(t\right)-f\left(u\right){|}^{2}+|t-u{|}^{2}\right]}^{-s/2}\mathrm{d}t\mathrm{d}u<\infty$ integral Subscript 0 Superscript 1 Baseline integral Subscript 0 Superscript 1 Baseline left bracket StartAbsoluteValue f left parenthesis t right parenthesis minus f left parenthesis u right parenthesis EndAbsoluteValue squared plus StartAbsoluteValue t minus u EndAbsoluteValue squared right bracket Superscript negative s divided by 2 Baseline normal d t normal d u less than infinity 69 $\mathsf{E}\left(\sum _{I\in {E}_{k+1}}|I{|}^{s}\right)=\mathsf{E}\left(\sum _{I\in {E}_{k}}|I{|}^{s}\right)\mathsf{E}\left({R}_{1}^{s}+{R}_{2}^{s}\right).$ sans serif upper E left parenthesis sigma summation Underscript upper I element of upper E Subscript k plus 1 Baseline Endscripts StartAbsoluteValue upper I EndAbsoluteValue Superscript s Baseline right parenthesis equals sans serif upper E left parenthesis sigma summation Underscript upper I element of upper E Subscript k Baseline Endscripts StartAbsoluteValue upper I EndAbsoluteValue Superscript s Baseline right parenthesis sans serif upper E left parenthesis upper R 1 Superscript s Baseline plus upper R 2 Superscript s Baseline right parenthesis period 70 $\left({x}_{1},{y}_{1}\right)$ left parenthesis x 1 comma y 1 right parenthesis 71 $\left({x}_{2},{y}_{2}\right)$ left parenthesis x 2 comma y 2 right parenthesis 72 $d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}$ d equals StartRoot left parenthesis x 2 minus x 1 right parenthesis squared plus left parenthesis y 2 minus y 1 right parenthesis squared EndRoot 73 $ℝ$ double struck upper R 74 $ℝ=\left(-\infty ,\infty \right)$ double struck upper R equals left parenthesis negative infinity comma infinity right parenthesis 75 $\left\{1,2,3\right\}$ StartSet 1 comma 2 comma 3 EndSet 76 $1\in S$ 1 element of upper S 77 $3\in S$ 3 element of upper S 78 $4\notin S$ 4 not an element of upper S 79 $a=\sqrt{3x-1}+{\left(1+x\right)}^{2}$ a equals StartRoot 3 x minus 1 EndRoot plus left parenthesis 1 plus x right parenthesis squared 80 $a=\frac{{\left(b+c\right)}^{2}}{d}+\frac{{\left(e+f\right)}^{2}}{g}$ a equals StartFraction left parenthesis b plus c right parenthesis squared Over d EndFraction plus StartFraction left parenthesis e plus f right parenthesis squared Over g EndFraction 81 $x=\left[{\left(a+b\right)}^{2}{\left(c-b\right)}^{2}\right]+\left[{\left(d+e\right)}^{2}{\left(f-e\right)}^{2}\right]$ x equals left bracket left parenthesis a plus b right parenthesis squared left parenthesis c minus b right parenthesis squared right bracket plus left bracket left parenthesis d plus e right parenthesis squared left parenthesis f minus e right parenthesis squared right bracket 82 $x=\left[{\left(a+b\right)}^{2}\right]+\left[{\left(f-e\right)}^{2}\right]$ x equals left bracket left parenthesis a plus b right parenthesis squared right bracket plus left bracket left parenthesis f minus e right parenthesis squared right bracket 83 $x=\left[{\left(a+b\right)}^{2}\right]$ x equals left bracket left parenthesis a plus b right parenthesis squared right bracket 84 $x={\left(a+b\right)}^{2}$ x equals left parenthesis a plus b right parenthesis squared 85 $x=a+{b}^{2}$ x equals a plus b squared 86 $\frac{\frac{1}{2}}{\frac{3}{4}}=\frac{2}{3}$ StartFraction one half Over three fourths EndFraction equals two thirds 87 $2\left(\left(x+1\right)\left(x+3\right)-4\left(\left(x-1\right)\left(x+2\right)-3\right)\right)=y$ 2 left parenthesis left parenthesis x plus 1 right parenthesis left parenthesis x plus 3 right parenthesis minus 4 left parenthesis left parenthesis x minus 1 right parenthesis left parenthesis x plus 2 right parenthesis minus 3 right parenthesis right parenthesis equals y 88 $\mathrm{cos}x=1-\frac{{x}^{2}}{2!}+\frac{{x}^{4}}{4!}-\dots$ cosine x equals 1 minus StartFraction x squared Over 2 factorial EndFraction plus StartFraction x Superscript 4 Baseline Over 4 factorial EndFraction minus ellipsis 89 $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$ x equals StartFraction negative b plus or minus StartRoot b squared minus 4 a c EndRoot Over 2 a EndFraction 90 $x+{y}^{\frac{2}{k+1}}$ x plus y Superscript StartFraction 2 Over k plus 1 EndFraction 91 $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}x}{x}=1$ limit Underscript x right arrow 0 Endscripts StartFraction sine x Over x EndFraction equals 1 92 $d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}$ d equals StartRoot left parenthesis x 2 minus x 1 right parenthesis squared plus left parenthesis y 2 minus y 1 right parenthesis squared EndRoot 93 ${F}_{n}={F}_{n-1}+{F}_{n-2}$ upper F Subscript n Baseline equals upper F Subscript n minus 1 Baseline plus upper F Subscript n minus 2 94 $\mathbf{\Pi }=\left(\begin{array}{cccccc}\hfill {\pi }_{11}& \hfill {\pi }_{12}\hfill & {\pi }_{12}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\pi }_{12}& \hfill {\pi }_{11}\hfill & {\pi }_{12}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\pi }_{12}& \hfill {\pi }_{12}\hfill & {\pi }_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0& \hfill 0\hfill & 0\hfill & \hfill {\pi }_{44}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0& \hfill 0\hfill & 0\hfill & \hfill 0\hfill & \hfill {\pi }_{44}\hfill & \hfill 0\hfill \\ \hfill 0& \hfill 0\hfill & 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\pi }_{44}\hfill \end{array}\right)$ bold upper Pi equals Start 6 By 6 Matrix 1st Row 1st Column pi 11 2nd Column pi 12 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 2nd Row 1st Column pi 12 2nd Column pi 11 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 3rd Row 1st Column pi 12 2nd Column pi 12 3rd Column pi 11 4th Column 0 5th Column 0 6th Column 0 4th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column pi 44 5th Column 0 6th Column 0 5th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column pi 44 6th Column 0 6th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 6th Column pi 44 EndMatrix 95 ${s}_{11}=\frac{{c}_{11}+{c}_{12}}{\left({c}_{11}-{c}_{12}\right)\left({c}_{11}+2{c}_{12}\right)}$ s 11 equals StartFraction c 11 plus c 12 Over left parenthesis c 11 minus c 12 right parenthesis left parenthesis c 11 plus 2 c 12 right parenthesis EndFraction 96 $\mathrm{Si}{\mathrm{O}}_{2}+6\mathrm{H}\mathrm{F}\to {\mathrm{H}}_{2}\mathrm{Si}{\mathrm{F}}_{6}+2{\mathrm{H}}_{2}\mathrm{O}$ upper S i normal upper O 2 plus 6 normal upper H normal upper F right arrow normal upper H 2 upper S i normal upper F 6 plus 2 normal upper H 2 normal upper O 97 $\frac{\text{d}}{\text{d}x}\left(E\left(x\right)A\left(x\right)\frac{\text{d}w\left(x\right)}{\text{d}x}\right)+p\left(x\right)=0$ StartFraction d Over d x EndFraction left parenthesis upper E left parenthesis x right parenthesis upper A left parenthesis x right parenthesis StartFraction d w left parenthesis x right parenthesis Over d x EndFraction right parenthesis plus p left parenthesis x right parenthesis equals 0 98 ${\text{TCS}}_{\text{gas}}=-\frac{1}{2}\left(\frac{{P}_{\text{seal}}}{{P}_{\text{max}}}\right)\left(\frac{1}{{T}_{\text{seal}}}\right)$ TCS Subscript gas Baseline equals minus one half left parenthesis StartFraction upper P Subscript seal Baseline Over upper P Subscript max Baseline EndFraction right parenthesis left parenthesis StartFraction 1 Over upper T Subscript seal Baseline EndFraction right parenthesis 99 ${B}_{p}=\frac{\frac{7-{v}^{2}}{3}\left(1+\frac{{c}^{2}}{{a}^{2}}+\frac{{c}^{4}}{{a}^{4}}\right)+\frac{{\left(3-v\right)}^{2}{c}^{2}}{\left(1+v\right){a}^{2}}}{\left(1-v\right)\left(1-\frac{{c}^{4}}{{a}^{4}}\right)\left(1-\frac{{c}^{2}}{{a}^{2}}\right)}$ upper B Subscript p Baseline equals StartStartFraction StartFraction 7 minus v squared Over 3 EndFraction left parenthesis 1 plus StartFraction c squared Over a squared EndFraction plus StartFraction c Superscript 4 Baseline Over a Superscript 4 Baseline EndFraction right parenthesis plus StartFraction left parenthesis 3 minus v right parenthesis squared c squared Over left parenthesis 1 plus v right parenthesis a squared EndFraction OverOver left parenthesis 1 minus v right parenthesis left parenthesis 1 minus StartFraction c Superscript 4 Baseline Over a Superscript 4 Baseline EndFraction right parenthesis left parenthesis 1 minus StartFraction c squared Over a squared EndFraction right parenthesis EndEndFraction 100 ${Q}_{\text{tank}}^{\text{series}}=\frac{1}{{R}_{\text{s}}}\sqrt{\frac{{L}_{\text{s}}}{{C}_{\text{s}}}}$ upper Q Subscript tank Superscript series Baseline equals StartFraction 1 Over upper R Subscript s Baseline EndFraction StartRoot StartFraction upper L Subscript s Baseline Over upper C Subscript s Baseline EndFraction EndRoot 101 $\text{Δ}{\varphi }_{\text{peak}}={tan}^{-1}\left({k}^{2}{Q}_{\text{tank}}^{\text{series}}\right)$ upper Delta phi Subscript peak Baseline equals tangent Superscript negative 1 Baseline left parenthesis k squared upper Q Subscript tank Superscript series Baseline right parenthesis 102 $f=1.013\frac{W}{{L}^{2}}\sqrt{\frac{E}{\rho }}\sqrt{\left(1+0.293\frac{{L}^{2}}{{\text{EW}}^{2}}\sigma \right)}$ f equals 1.013 StartFraction upper W Over upper L squared EndFraction StartRoot StartFraction upper E Over rho EndFraction EndRoot StartRoot left parenthesis 1 plus 0.293 StartFraction upper L squared Over EW squared EndFraction sigma right parenthesis EndRoot 103 ${u}_{n}\left(x\right)={\gamma }_{n}\left(\mathrm{cosh}{k}_{n}x-\mathrm{cos}{k}_{n}x\right)+\left(\mathrm{sinh}{k}_{n}x-\mathrm{sin}{k}_{n}x\right)$ u Subscript n Baseline left parenthesis x right parenthesis equals gamma Subscript n Baseline left parenthesis hyperbolic cosine k Subscript n Baseline x minus cosine k Subscript n Baseline x right parenthesis plus left parenthesis hyperbolic sine k Subscript n Baseline x minus sine k Subscript n Baseline x right parenthesis 104 $\begin{array}{cc}B\hfill & =\frac{\frac{{F}_{0}}{m}}{\sqrt{\left({\omega }_{0}^{2}-{\omega }^{2}{\right)}^{2}+4{n}^{2}{\omega }^{2}}}\hfill \\ & =\frac{\frac{{F}_{0}}{k}}{\sqrt{\left(1-\left(\omega /{\omega }_{0}^{2}{\right)}^{2}{\right)}^{2}+4\left(n/{\omega }_{0}{\right)}^{2}\left(\omega /{\omega }_{0}{\right)}^{2}}}\hfill \end{array}$ StartLayout 1st Row 1st Column upper B 2nd Column equals StartStartFraction StartFraction upper F 0 Over m EndFraction OverOver StartRoot left parenthesis omega 0 squared minus omega squared right parenthesis squared plus 4 n squared omega squared EndRoot EndEndFraction 2nd Row 1st Column Blank 2nd Column equals StartStartFraction StartFraction upper F 0 Over k EndFraction OverOver StartRoot left parenthesis 1 minus left parenthesis omega divided by omega 0 squared right parenthesis squared right parenthesis squared plus 4 left parenthesis n divided by omega 0 right parenthesis squared left parenthesis omega divided by omega 0 right parenthesis squared EndRoot EndEndFraction EndLayout 105 $\mathrm{p}\left(A\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}B\right)=\mathrm{p}\left(A\right)\phantom{\rule{3.33333pt}{0ex}}\mathrm{p}\left(B|A\right)$ normal p left parenthesis upper A a n d upper B right parenthesis equals normal p left parenthesis upper A right parenthesis normal p left parenthesis upper B vertical bar upper A right parenthesis 106 $\mathrm{PMF}\left(x\right)\propto {\left(\frac{1}{x}\right)}^{\alpha }$ upper P upper M upper F left parenthesis x right parenthesis proportional to left parenthesis StartFraction 1 Over x EndFraction right parenthesis Superscript alpha 107 $f\left(x\right)=\frac{1}{\sqrt{2\pi }}exp\left(-{x}^{2}/2\right)$ f left parenthesis x right parenthesis equals StartFraction 1 Over StartRoot 2 pi EndRoot EndFraction exp left parenthesis minus x squared slash 2 right parenthesis 108 $\frac{dx}{d\theta }=\frac{\beta }{{cos}^{2}\theta }$ StartFraction d x Over d theta EndFraction equals StartFraction beta Over cosine squared theta EndFraction 109 $s/\sqrt{2\left(n-1\right)}$ s divided by StartRoot 2 left parenthesis n minus 1 right parenthesis EndRoot

## English Mathspeak Steve Noble's samples. Locale: en, Style: Brief.

 0 $-5\frac{1}{5}-6\frac{2}{3}=$ negative 5 and one fifth minus 6 and two thirds equals 1 $-7\frac{3}{4}-\left(-4\frac{7}{8}\right)=$ negative 7 and three fourths minus left p'ren negative 4 and seven eighths right p'ren equals 2 $-24.15-\left(13.7\right)=$ negative 24.15 minus left p'ren 13.7 right p'ren equals 3 $\left(-4\right)×3=-12$ left p'ren negative 4 right p'ren times 3 equals negative 12 4 $-12÷3=-4$ negative 12 divided by 3 equals negative 4 5 $-12÷\left(-4\right)=3$ negative 12 divided by left p'ren negative 4 right p'ren equals 3 6 $6×5$ 6 times 5 7 $6×\left(-5\right)$ 6 times left p'ren negative 5 right p'ren 8 $-6×5$ negative 6 times 5 9 $-6×\left(-5\right)$ negative 6 times left p'ren negative 5 right p'ren 10 $-8×7$ negative 8 times 7 11 $-8×\left(-7\right)$ negative 8 times left p'ren negative 7 right p'ren 12 $8×\left(-7\right)$ 8 times left p'ren negative 7 right p'ren 13 $8×7$ 8 times 7 14 $m\angle 1=\mathrm{30°}$ m angle 1 equals 30 degree 15 $m\angle 2=\mathrm{60°}$ m angle 2 equals 60 degree 16 $m\angle 1+m\angle 2=\mathrm{90°}$ m angle 1 plus m angle 2 equals 90 degree 17 $m\angle M+m\angle N=\mathrm{180°}$ m angle upper M plus m angle upper N equals 180 degree 18 $A=\frac{1}{2}bh$ upper A equals one half b h 19 StartFrac area of triangle Over area of square EndFrac equals StartFrac 1 unit squared Over 16 units squared EndFrac 20 ${0.6}^{2}$ 0.6 squared 21 ${1.5}^{2}$ 1.5 squared 22 $4\left(2x+3x\right)$ 4 left p'ren 2 x plus 3 x right p'ren 23 $36+4y-1{y}^{2}+5{y}^{2}-2$ 36 plus 4 y minus 1 y squared plus 5 y squared minus 2 24 $\left(5+9\right)-4+3=$ left p'ren 5 plus 9 right p'ren minus 4 plus 3 equals 25 $\stackrel{↔}{BC}$ ModAbove upper B upper C With left right arrow 26 $\stackrel{\to }{PQ}$ ModAbove upper P upper Q With right arrow 27 $\overline{GH}$ ModAbove upper G upper H With bar 28 $\overline{WX}\cong \overline{YZ}$ ModAbove upper W upper X With bar approximately equals ModAbove upper Y upper Z With bar 29 $\angle BEF$ angle upper B upper E upper F 30 $\angle BED$ angle upper B upper E upper D 31 $\angle DEF$ angle upper D upper E upper F 32 $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$ x equals StartFrac negative b plus or minus StartRoot b squared minus 4 a c EndRoot Over 2 a EndFrac 33 $y={x}^{2}+8x+16$ y equals x squared plus 8 x plus 16 34 $y=\frac{1}{3}\left({3}^{x}\right)$ y equals one third left p'ren 3 Sup x Base right p'ren 35 $y=10-2x$ y equals 10 minus 2 x 36 $y=2{x}^{3}+5$ y equals 2 x cubed plus 5 37 $y=\left({x}^{2}+1\right)\left({x}^{2}+3\right)$ y equals left p'ren x squared plus 1 right p'ren left p'ren x squared plus 3 right p'ren 38 $y={0.5}^{x}$ y equals 0.5 Sup x 39 $y=22-2x$ y equals 22 minus 2 x 40 $y=\frac{3}{x}$ y equals StartFrac 3 Over x EndFrac 41 $y=\left(x+4\right)\left(x+4\right)$ y equals left p'ren x plus 4 right p'ren left p'ren x plus 4 right p'ren 42 $y=\left(4x-3\right)\left(x+1\right)$ y equals left p'ren 4 x minus 3 right p'ren left p'ren x plus 1 right p'ren 43 $y=20x-4{x}^{2}$ y equals 20 x minus 4 x squared 44 $y={x}^{2}$ y equals x squared 45 $y={3}^{x-1}$ y equals 3 Sup x minus 1 46 $y=16-2\left(x+3\right)$ y equals 16 minus 2 left p'ren x plus 3 right p'ren 47 $y=4{x}^{2}-x-3$ y equals 4 x squared minus x minus 3 48 $y=x+\frac{1}{x}$ y equals x plus StartFrac 1 Over x EndFrac 49 $y=4x\left(5-x\right)$ y equals 4 x left p'ren 5 minus x right p'ren 50 $y=2\left(x-3\right)+6\left(1-x\right)$ y equals 2 left p'ren x minus 3 right p'ren plus 6 left p'ren 1 minus x right p'ren 51 $0.25>\frac{5}{16}$ 0.25 greater than five sixteenths 52 $32\cdot \left(5\cdot 7\right)$ 32 dot left p'ren 5 dot 7 right p'ren 53 $\left(\frac{1}{2}×\frac{1}{2}×\pi ×2\right)+\left(2×\frac{1}{2}×\pi ×5\right)$ left p'ren one half times one half times pi times 2 right p'ren plus left p'ren 2 times one half times pi times 5 right p'ren 54 $\underset{n\to \infty }{\text{liminf}}{E}_{n}=\bigcup _{n\ge 1}\bigcap _{k\ge n}{E}_{k},\phantom{\rule{0.2em}{0ex}}\underset{n\to \infty }{\text{limsup}}{E}_{n}=\bigcap _{n\ge 1}\bigcup _{k\ge n}{E}_{k}.$ liminf Underscript n right arrow infinity Endscripts upper E Sub n Base equals union Underscript n greater than or equals 1 Endscripts intersection Underscript k greater than or equals n Endscripts upper E Sub k Base comma limsup Underscript n right arrow infinity Endscripts upper E Sub n Base equals intersection Underscript n greater than or equals 1 Endscripts union Underscript k greater than or equals n Endscripts upper E Sub k Base period 55 $\begin{array}{ll}\text{(i)}\hfill & \phantom{\rule{0.2em}{0ex}}𝒮\in 𝒜;\hfill \\ \multicolumn{1}{c}{\text{(ii)}}& \phantom{\rule{0.2em}{0ex}}\text{if}E\in 𝒜\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}\stackrel{‾}{E}\in 𝒜;\hfill \\ \multicolumn{1}{c}{\text{(iii)}}& \phantom{\rule{0.2em}{0ex}}\text{if}{E}_{1},{E}_{2}\in 𝒜\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}{E}_{1}\cup {E}_{2}\in 𝒜.\hfill \end{array}$ StartLayout 1st Row 1st Column left p'ren i right p'ren 2nd Column script upper S element of script upper A semicolon 2nd Row 1st Column left p'ren ii right p'ren 2nd Column if upper E element of script upper A then upper E overbar element of script upper A semicolon 3rd Row 1st Column left p'ren iii right p'ren 2nd Column if upper E 1 comma upper E 2 element of script upper A then upper E 1 union upper E 2 element of script upper A period EndLayout 56 $\begin{array}{llll}\hfill & \hfill & \left(\mathrm{A}\mathrm{.}1\right)\phantom{\rule{0.2em}{0ex}}\mathrm{If}\phantom{\rule{0.2em}{0ex}}A\in ℱ\phantom{\rule{0.2em}{0ex}}\mathrm{then}\phantom{\rule{0.2em}{0ex}}0\le P\left\{A\right\}\le 1.\hfill & \left(1\right)\hfill \\ \multicolumn{1}{c}{}& \hfill & \left(\mathrm{A}\mathrm{.}2\right)\phantom{\rule{0.2em}{0ex}}P\left\{𝒮\right\}=1.\hfill & \left(2\right)\hfill \\ \multicolumn{1}{c}{}& \hfill & \left(\mathrm{A}\mathrm{.}3\right)\phantom{\rule{0.2em}{0ex}}\mathrm{If}\phantom{\rule{0.2em}{0ex}}\left\{{E}_{n},n\ge 1\right\}\in ℱ\phantom{\rule{0.2em}{0ex}}\text{is a sequence of}\phantom{\rule{0.2em}{0ex}}\text{disjoint}\hfill & \left(3\right)\hfill \end{array}$ StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column left p'ren normal upper A period 1 right p'ren upper I f upper A element of script upper F t h e n 0 less than or equals upper P left brace upper A right brace less than or equals 1 period 4th Column left p'ren 1 right p'ren 2nd Row 1st Column Blank 2nd Column Blank 3rd Column left p'ren normal upper A period 2 right p'ren upper P left brace script upper S right brace equals 1 period 4th Column left p'ren 2 right p'ren 3rd Row 1st Column Blank 2nd Column Blank 3rd Column left p'ren normal upper A period 3 right p'ren upper I f left brace upper E Sub n Base comma n greater than or equals 1 right brace element of script upper F is a sequence of disjoint 4th Column left p'ren 3 right p'ren EndLayout 57 $P\left\{{B}_{j}|A\right\}=\frac{P\left\{{B}_{j}\right\}P\left\{A|{B}_{j}\right\}}{\sum _{j\prime \in J}P\left\{{B}_{j\prime }\right\}P\left\{A|{B}_{j\prime }\right\}}.$ upper P left brace upper B Sub j Base vertical bar upper A right brace equals StartFrac upper P left brace upper B Sub j Base right brace upper P left brace upper A vertical bar upper B Sub j Base right brace Over sigma summation Underscript j prime element of upper J Endscripts upper P left brace upper B Sub j prime Base right brace upper P left brace upper A vertical bar upper B Sub j prime Base right brace EndFrac period 58 ${\mu }_{1}\left(B\right)={\int }_{B}f\left(x\right)d{\mu }_{2}\left(x\right)$ mu 1 left p'ren upper B right p'ren equals integral Underscript upper B Endscripts f left p'ren x right p'ren d mu 2 left p'ren x right p'ren 59 $\underset{n\to \infty }{\text{lim}}E\left\{|{X}_{n}-X|\right\}=E\left\{\underset{n\to \infty }{\text{lim}}|{X}_{n}-X|\right\}=0.$ limit Underscript n right arrow infinity Endscripts upper E left brace StartAbsoluteValue upper X Sub n Base minus upper X EndAbsoluteValue right brace equals upper E left brace limit Underscript n right arrow infinity Endscripts StartAbsoluteValue upper X Sub n Base minus upper X EndAbsoluteValue right brace equals 0 period 60 $\begin{array}{l}{P}_{\mu ,\sigma }\left\{Y\ge {l}_{\beta }\left({\stackrel{‾}{Y}}_{n},{S}_{n}\right)\right\}={P}_{\mu ,\sigma }\left\{\left(Y-{\stackrel{‾}{Y}}_{n}\right)/\left(S·{\left(1+\frac{1}{n}\right)}^{1/2}\right)\ge -{t}_{\beta }\left[n-1\right]\right\}=\beta ,\hfill \\ \multicolumn{1}{c}{}& \left(1\right)\hfill \end{array}$ StartLayout 1st Row 1st Column upper P Sub mu comma sigma Base left brace upper Y greater than or equals l Sub beta Base left p'ren upper Y overbar Sub n Base comma upper S Sub n Base right p'ren right brace equals upper P Sub mu comma sigma Base left brace left p'ren upper Y minus upper Y overbar Sub n Base right p'ren divided by left p'ren upper S dot left p'ren 1 plus StartFrac 1 Over n EndFrac right p'ren Sup 1 divided by 2 Base right p'ren greater than or equals minus t Sub beta Base left brack n minus 1 right brack right brace equals beta comma 2nd Row 1st Column Blank 2nd Column left p'ren 1 right p'ren EndLayout 61 $L=\left(\begin{array}{cccccc}\hfill 1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}-1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}-1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}0\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}0\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}-1\hfill \end{array}\right).$ upper L equals Start 5 By 6 Matrix 1st Row 1st Column 1 2nd Column negative 1 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 2nd Row 1st Column Blank 2nd Column 1 3rd Column negative 1 4th Column Blank 5th Column 0 6th Column Blank 3rd Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 4th Row 1st Column Blank 2nd Column 0 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 5th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column 1 6th Column negative 1 EndMatrix period 62 $\sqrt{n}\left[{\stackrel{‾}{Y}}_{n}-\left(\mu +{z}_{\beta }\sigma \right)\right]/{S}_{n}~\frac{U+\sqrt{n}\phantom{\rule{0.2em}{0ex}}{z}_{1-\beta }}{\left({\chi }^{2}\left[n-1\right]/\left(n-1\right)\right){}^{1/2}}~t\left[n-1;\sqrt{n}\phantom{\rule{0.2em}{0ex}}{z}_{1-\beta }\right],$ StartRoot n EndRoot left brack upper Y overbar Sub n Base minus left p'ren mu plus z Sub beta Base sigma right p'ren right brack divided by upper S Sub n Base tilde StartFrac upper U plus StartRoot n EndRoot z Sub 1 minus beta Base Over left p'ren chi squared left brack n minus 1 right brack divided by left p'ren n minus 1 right p'ren right p'ren Sup 1 divided by 2 Base EndFrac tilde t left brack n minus 1 semicolon StartRoot n EndRoot z Sub 1 minus beta Base right brack comma 63 $\begin{array}{ll}\gamma \hfill & =P\left\{{E}_{p,q}\subset \left({X}_{\left(r\right)},{X}_{\left(s\right)}\right\}\hfill \\ \multicolumn{1}{c}{}& =\frac{n!}{\left(r-1\right)!}\sum _{j=0}^{s-r-1}\left(-1\right){}^{j}\frac{{p}^{r+j}}{\left(n-r-j\right)!j!}{I}_{1-q}\left(n-s+1,s-r-j\right).\hfill \end{array}$ StartLayout 1st Row 1st Column gamma 2nd Column equals upper P left brace upper E Sub p comma q Base subset of left p'ren upper X Sub left p'ren r right p'ren Base comma upper X Sub left p'ren s right p'ren Base right brace 2nd Row 1st Column Blank 2nd Column equals StartFrac n factorial Over left p'ren r minus 1 right p'ren factorial EndFrac sigma summation Underscript j equals 0 Overscript s minus r minus 1 Endscripts left p'ren negative 1 right p'ren Sup j Base StartFrac p Sup r plus j Base Over left p'ren n minus r minus j right p'ren factorial j factorial EndFrac upper I Sub 1 minus q Base left p'ren n minus s plus 1 comma s minus r minus j right p'ren period EndLayout 64 ${S}_{i}\left[\begin{array}{c}t\\ x\end{array}\right]=\left[\begin{array}{cc}1/m& 0\\ {a}_{i}& {r}_{i}\end{array}\right]\left[\begin{array}{c}t\\ x\end{array}\right]+\left[\begin{array}{c}\left(i-1\right)/m\\ {b}_{i}\end{array}\right],$ upper S Sub i Base StartBinomialOrMatrix t Choose x EndBinomialOrMatrix equals Start 2 By 2 Matrix 1st Row 1st Column 1 divided by m 2nd Column 0 2nd Row 1st Column a Sub i Base 2nd Column r Sub i Base EndMatrix StartBinomialOrMatrix t Choose x EndBinomialOrMatrix plus StartBinomialOrMatrix left p'ren i minus 1 right p'ren divided by m Choose b Sub i Base EndBinomialOrMatrix comma 65 ${c}_{1}{h}^{4-2s}\le \frac{1}{2T}{\int }_{-T}^{T}{\left(f\left(t+h\right)-f\left(t\right)\right)}^{2}\mathrm{d}t\le {c}_{2}{h}^{4-2s}$ c 1 h Sup 4 minus 2 s Base less than or equals StartFrac 1 Over 2 upper T EndFrac integral Sub negative upper T Sup upper T Base left p'ren f left p'ren t plus h right p'ren minus f left p'ren t right p'ren right p'ren squared normal d t less than or equals c 2 h Sup 4 minus 2 s 66 $C\left(0\right)-C\left(h\right)\simeq c{h}^{4-2s}$ upper C left p'ren 0 right p'ren minus upper C left p'ren h right p'ren asymptotically equals c h Sup 4 minus 2 s 67 $S\left(\omega \right)=\underset{T\to \infty }{\mathrm{lim}}\frac{1}{2T}{\left|{\int }_{-T}^{T},f,\left(,t,\right),{\mathrm{e}}^{\mathit{it}\omega },\mathrm{d},t\right|}^{2}.$ upper S left p'ren omega right p'ren equals limit Underscript upper T right arrow infinity Endscripts StartFrac 1 Over 2 upper T EndFrac StartAbsoluteValue integral Sub negative upper T Sup upper T Base comma f comma left p'ren comma t comma right p'ren comma normal e Sup italic i t omega Base comma normal d comma t EndAbsoluteValue squared period 68 ${\int }_{0}^{1}\phantom{\rule{-0.2em}{0ex}}{\int }_{0}^{1}{\left[|f\left(t\right)-f\left(u\right){|}^{2}+|t-u{|}^{2}\right]}^{-s/2}\mathrm{d}t\mathrm{d}u<\infty$ integral Sub 0 Sup 1 Base integral Sub 0 Sup 1 Base left brack StartAbsoluteValue f left p'ren t right p'ren minus f left p'ren u right p'ren EndAbsoluteValue squared plus StartAbsoluteValue t minus u EndAbsoluteValue squared right brack Sup negative s divided by 2 Base normal d t normal d u less than infinity 69 $\mathsf{E}\left(\sum _{I\in {E}_{k+1}}|I{|}^{s}\right)=\mathsf{E}\left(\sum _{I\in {E}_{k}}|I{|}^{s}\right)\mathsf{E}\left({R}_{1}^{s}+{R}_{2}^{s}\right).$ sans serif upper E left p'ren sigma summation Underscript upper I element of upper E Sub k plus 1 Base Endscripts StartAbsoluteValue upper I EndAbsoluteValue Sup s Base right p'ren equals sans serif upper E left p'ren sigma summation Underscript upper I element of upper E Sub k Base Endscripts StartAbsoluteValue upper I EndAbsoluteValue Sup s Base right p'ren sans serif upper E left p'ren upper R 1 Sup s Base plus upper R 2 Sup s Base right p'ren period 70 $\left({x}_{1},{y}_{1}\right)$ left p'ren x 1 comma y 1 right p'ren 71 $\left({x}_{2},{y}_{2}\right)$ left p'ren x 2 comma y 2 right p'ren 72 $d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}$ d equals StartRoot left p'ren x 2 minus x 1 right p'ren squared plus left p'ren y 2 minus y 1 right p'ren squared EndRoot 73 $ℝ$ double struck upper R 74 $ℝ=\left(-\infty ,\infty \right)$ double struck upper R equals left p'ren negative infinity comma infinity right p'ren 75 $\left\{1,2,3\right\}$ StartSet 1 comma 2 comma 3 EndSet 76 $1\in S$ 1 element of upper S 77 $3\in S$ 3 element of upper S 78 $4\notin S$ 4 not an element of upper S 79 $a=\sqrt{3x-1}+{\left(1+x\right)}^{2}$ a equals StartRoot 3 x minus 1 EndRoot plus left p'ren 1 plus x right p'ren squared 80 $a=\frac{{\left(b+c\right)}^{2}}{d}+\frac{{\left(e+f\right)}^{2}}{g}$ a equals StartFrac left p'ren b plus c right p'ren squared Over d EndFrac plus StartFrac left p'ren e plus f right p'ren squared Over g EndFrac 81 $x=\left[{\left(a+b\right)}^{2}{\left(c-b\right)}^{2}\right]+\left[{\left(d+e\right)}^{2}{\left(f-e\right)}^{2}\right]$ x equals left brack left p'ren a plus b right p'ren squared left p'ren c minus b right p'ren squared right brack plus left brack left p'ren d plus e right p'ren squared left p'ren f minus e right p'ren squared right brack 82 $x=\left[{\left(a+b\right)}^{2}\right]+\left[{\left(f-e\right)}^{2}\right]$ x equals left brack left p'ren a plus b right p'ren squared right brack plus left brack left p'ren f minus e right p'ren squared right brack 83 $x=\left[{\left(a+b\right)}^{2}\right]$ x equals left brack left p'ren a plus b right p'ren squared right brack 84 $x={\left(a+b\right)}^{2}$ x equals left p'ren a plus b right p'ren squared 85 $x=a+{b}^{2}$ x equals a plus b squared 86 $\frac{\frac{1}{2}}{\frac{3}{4}}=\frac{2}{3}$ StartFrac one half Over three fourths EndFrac equals two thirds 87 $2\left(\left(x+1\right)\left(x+3\right)-4\left(\left(x-1\right)\left(x+2\right)-3\right)\right)=y$ 2 left p'ren left p'ren x plus 1 right p'ren left p'ren x plus 3 right p'ren minus 4 left p'ren left p'ren x minus 1 right p'ren left p'ren x plus 2 right p'ren minus 3 right p'ren right p'ren equals y 88 $\mathrm{cos}x=1-\frac{{x}^{2}}{2!}+\frac{{x}^{4}}{4!}-\dots$ cosine x equals 1 minus StartFrac x squared Over 2 factorial EndFrac plus StartFrac x Sup 4 Base Over 4 factorial EndFrac minus ellipsis 89 $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$ x equals StartFrac negative b plus or minus StartRoot b squared minus 4 a c EndRoot Over 2 a EndFrac 90 $x+{y}^{\frac{2}{k+1}}$ x plus y Sup StartFrac 2 Over k plus 1 EndFrac 91 $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}x}{x}=1$ limit Underscript x right arrow 0 Endscripts StartFrac sine x Over x EndFrac equals 1 92 $d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}$ d equals StartRoot left p'ren x 2 minus x 1 right p'ren squared plus left p'ren y 2 minus y 1 right p'ren squared EndRoot 93 ${F}_{n}={F}_{n-1}+{F}_{n-2}$ upper F Sub n Base equals upper F Sub n minus 1 Base plus upper F Sub n minus 2 94 $\mathbf{\Pi }=\left(\begin{array}{cccccc}\hfill {\pi }_{11}& \hfill {\pi }_{12}\hfill & {\pi }_{12}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\pi }_{12}& \hfill {\pi }_{11}\hfill & {\pi }_{12}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\pi }_{12}& \hfill {\pi }_{12}\hfill & {\pi }_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0& \hfill 0\hfill & 0\hfill & \hfill {\pi }_{44}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0& \hfill 0\hfill & 0\hfill & \hfill 0\hfill & \hfill {\pi }_{44}\hfill & \hfill 0\hfill \\ \hfill 0& \hfill 0\hfill & 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\pi }_{44}\hfill \end{array}\right)$ bold upper Pi equals Start 6 By 6 Matrix 1st Row 1st Column pi 11 2nd Column pi 12 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 2nd Row 1st Column pi 12 2nd Column pi 11 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 3rd Row 1st Column pi 12 2nd Column pi 12 3rd Column pi 11 4th Column 0 5th Column 0 6th Column 0 4th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column pi 44 5th Column 0 6th Column 0 5th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column pi 44 6th Column 0 6th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 6th Column pi 44 EndMatrix 95 ${s}_{11}=\frac{{c}_{11}+{c}_{12}}{\left({c}_{11}-{c}_{12}\right)\left({c}_{11}+2{c}_{12}\right)}$ s 11 equals StartFrac c 11 plus c 12 Over left p'ren c 11 minus c 12 right p'ren left p'ren c 11 plus 2 c 12 right p'ren EndFrac 96 $\mathrm{Si}{\mathrm{O}}_{2}+6\mathrm{H}\mathrm{F}\to {\mathrm{H}}_{2}\mathrm{Si}{\mathrm{F}}_{6}+2{\mathrm{H}}_{2}\mathrm{O}$ upper S i normal upper O 2 plus 6 normal upper H normal upper F right arrow normal upper H 2 upper S i normal upper F 6 plus 2 normal upper H 2 normal upper O 97 $\frac{\text{d}}{\text{d}x}\left(E\left(x\right)A\left(x\right)\frac{\text{d}w\left(x\right)}{\text{d}x}\right)+p\left(x\right)=0$ StartFrac d Over d x EndFrac left p'ren upper E left p'ren x right p'ren upper A left p'ren x right p'ren StartFrac d w left p'ren x right p'ren Over d x EndFrac right p'ren plus p left p'ren x right p'ren equals 0 98 ${\text{TCS}}_{\text{gas}}=-\frac{1}{2}\left(\frac{{P}_{\text{seal}}}{{P}_{\text{max}}}\right)\left(\frac{1}{{T}_{\text{seal}}}\right)$ TCS Sub gas Base equals minus one half left p'ren StartFrac upper P Sub seal Base Over upper P Sub max Base EndFrac right p'ren left p'ren StartFrac 1 Over upper T Sub seal Base EndFrac right p'ren 99 ${B}_{p}=\frac{\frac{7-{v}^{2}}{3}\left(1+\frac{{c}^{2}}{{a}^{2}}+\frac{{c}^{4}}{{a}^{4}}\right)+\frac{{\left(3-v\right)}^{2}{c}^{2}}{\left(1+v\right){a}^{2}}}{\left(1-v\right)\left(1-\frac{{c}^{4}}{{a}^{4}}\right)\left(1-\frac{{c}^{2}}{{a}^{2}}\right)}$ upper B Sub p Base equals StartStartFrac StartFrac 7 minus v squared Over 3 EndFrac left p'ren 1 plus StartFrac c squared Over a squared EndFrac plus StartFrac c Sup 4 Base Over a Sup 4 Base EndFrac right p'ren plus StartFrac left p'ren 3 minus v right p'ren squared c squared Over left p'ren 1 plus v right p'ren a squared EndFrac OverOver left p'ren 1 minus v right p'ren left p'ren 1 minus StartFrac c Sup 4 Base Over a Sup 4 Base EndFrac right p'ren left p'ren 1 minus StartFrac c squared Over a squared EndFrac right p'ren EndEndFrac 100 ${Q}_{\text{tank}}^{\text{series}}=\frac{1}{{R}_{\text{s}}}\sqrt{\frac{{L}_{\text{s}}}{{C}_{\text{s}}}}$ upper Q Sub tank Sup series Base equals StartFrac 1 Over upper R Sub s Base EndFrac StartRoot StartFrac upper L Sub s Base Over upper C Sub s Base EndFrac EndRoot 101 $\text{Δ}{\varphi }_{\text{peak}}={tan}^{-1}\left({k}^{2}{Q}_{\text{tank}}^{\text{series}}\right)$ upper Delta phi Sub peak Base equals tangent Sup negative 1 Base left p'ren k squared upper Q Sub tank Sup series Base right p'ren 102 $f=1.013\frac{W}{{L}^{2}}\sqrt{\frac{E}{\rho }}\sqrt{\left(1+0.293\frac{{L}^{2}}{{\text{EW}}^{2}}\sigma \right)}$ f equals 1.013 StartFrac upper W Over upper L squared EndFrac StartRoot StartFrac upper E Over rho EndFrac EndRoot StartRoot left p'ren 1 plus 0.293 StartFrac upper L squared Over EW squared EndFrac sigma right p'ren EndRoot 103 ${u}_{n}\left(x\right)={\gamma }_{n}\left(\mathrm{cosh}{k}_{n}x-\mathrm{cos}{k}_{n}x\right)+\left(\mathrm{sinh}{k}_{n}x-\mathrm{sin}{k}_{n}x\right)$ u Sub n Base left p'ren x right p'ren equals gamma Sub n Base left p'ren hyperbolic cosine k Sub n Base x minus cosine k Sub n Base x right p'ren plus left p'ren hyperbolic sine k Sub n Base x minus sine k Sub n Base x right p'ren 104 $\begin{array}{cc}B\hfill & =\frac{\frac{{F}_{0}}{m}}{\sqrt{\left({\omega }_{0}^{2}-{\omega }^{2}{\right)}^{2}+4{n}^{2}{\omega }^{2}}}\hfill \\ & =\frac{\frac{{F}_{0}}{k}}{\sqrt{\left(1-\left(\omega /{\omega }_{0}^{2}{\right)}^{2}{\right)}^{2}+4\left(n/{\omega }_{0}{\right)}^{2}\left(\omega /{\omega }_{0}{\right)}^{2}}}\hfill \end{array}$ StartLayout 1st Row 1st Column upper B 2nd Column equals StartStartFrac StartFrac upper F 0 Over m EndFrac OverOver StartRoot left p'ren omega 0 squared minus omega squared right p'ren squared plus 4 n squared omega squared EndRoot EndEndFrac 2nd Row 1st Column Blank 2nd Column equals StartStartFrac StartFrac upper F 0 Over k EndFrac OverOver StartRoot left p'ren 1 minus left p'ren omega divided by omega 0 squared right p'ren squared right p'ren squared plus 4 left p'ren n divided by omega 0 right p'ren squared left p'ren omega divided by omega 0 right p'ren squared EndRoot EndEndFrac EndLayout 105 $\mathrm{p}\left(A\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}B\right)=\mathrm{p}\left(A\right)\phantom{\rule{3.33333pt}{0ex}}\mathrm{p}\left(B|A\right)$ normal p left p'ren upper A a n d upper B right p'ren equals normal p left p'ren upper A right p'ren normal p left p'ren upper B vertical bar upper A right p'ren 106 $\mathrm{PMF}\left(x\right)\propto {\left(\frac{1}{x}\right)}^{\alpha }$ upper P upper M upper F left p'ren x right p'ren proportional to left p'ren StartFrac 1 Over x EndFrac right p'ren Sup alpha 107 $f\left(x\right)=\frac{1}{\sqrt{2\pi }}exp\left(-{x}^{2}/2\right)$ f left p'ren x right p'ren equals StartFrac 1 Over StartRoot 2 pi EndRoot EndFrac exp left p'ren minus x squared slash 2 right p'ren 108 $\frac{dx}{d\theta }=\frac{\beta }{{cos}^{2}\theta }$ StartFrac d x Over d theta EndFrac equals StartFrac beta Over cosine squared theta EndFrac 109 $s/\sqrt{2\left(n-1\right)}$ s divided by StartRoot 2 left p'ren n minus 1 right p'ren EndRoot

## English Mathspeak Steve Noble's samples. Locale: en, Style: Superbrief.

 0 $-5\frac{1}{5}-6\frac{2}{3}=$ negative 5 and one fifth minus 6 and two thirds equals 1 $-7\frac{3}{4}-\left(-4\frac{7}{8}\right)=$ negative 7 and three fourths minus L p'ren negative 4 and seven eighths R p'ren equals 2 $-24.15-\left(13.7\right)=$ negative 24.15 minus L p'ren 13.7 R p'ren equals 3 $\left(-4\right)×3=-12$ L p'ren negative 4 R p'ren times 3 equals negative 12 4 $-12÷3=-4$ negative 12 divided by 3 equals negative 4 5 $-12÷\left(-4\right)=3$ negative 12 divided by L p'ren negative 4 R p'ren equals 3 6 $6×5$ 6 times 5 7 $6×\left(-5\right)$ 6 times L p'ren negative 5 R p'ren 8 $-6×5$ negative 6 times 5 9 $-6×\left(-5\right)$ negative 6 times L p'ren negative 5 R p'ren 10 $-8×7$ negative 8 times 7 11 $-8×\left(-7\right)$ negative 8 times L p'ren negative 7 R p'ren 12 $8×\left(-7\right)$ 8 times L p'ren negative 7 R p'ren 13 $8×7$ 8 times 7 14 $m\angle 1=\mathrm{30°}$ m angle 1 equals 30 degree 15 $m\angle 2=\mathrm{60°}$ m angle 2 equals 60 degree 16 $m\angle 1+m\angle 2=\mathrm{90°}$ m angle 1 plus m angle 2 equals 90 degree 17 $m\angle M+m\angle N=\mathrm{180°}$ m angle upper M plus m angle upper N equals 180 degree 18 $A=\frac{1}{2}bh$ upper A equals one half b h 19 Frac area of triangle Over area of square EndFrac equals Frac 1 unit squared Over 16 units squared EndFrac 20 ${0.6}^{2}$ 0.6 squared 21 ${1.5}^{2}$ 1.5 squared 22 $4\left(2x+3x\right)$ 4 L p'ren 2 x plus 3 x R p'ren 23 $36+4y-1{y}^{2}+5{y}^{2}-2$ 36 plus 4 y minus 1 y squared plus 5 y squared minus 2 24 $\left(5+9\right)-4+3=$ L p'ren 5 plus 9 R p'ren minus 4 plus 3 equals 25 $\stackrel{↔}{BC}$ ModAbove upper B upper C With L R arrow 26 $\stackrel{\to }{PQ}$ ModAbove upper P upper Q With R arrow 27 $\overline{GH}$ ModAbove upper G upper H With bar 28 $\overline{WX}\cong \overline{YZ}$ ModAbove upper W upper X With bar approximately equals ModAbove upper Y upper Z With bar 29 $\angle BEF$ angle upper B upper E upper F 30 $\angle BED$ angle upper B upper E upper D 31 $\angle DEF$ angle upper D upper E upper F 32 $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$ x equals Frac negative b plus or minus Root b squared minus 4 a c EndRoot Over 2 a EndFrac 33 $y={x}^{2}+8x+16$ y equals x squared plus 8 x plus 16 34 $y=\frac{1}{3}\left({3}^{x}\right)$ y equals one third L p'ren 3 Sup x Base R p'ren 35 $y=10-2x$ y equals 10 minus 2 x 36 $y=2{x}^{3}+5$ y equals 2 x cubed plus 5 37 $y=\left({x}^{2}+1\right)\left({x}^{2}+3\right)$ y equals L p'ren x squared plus 1 R p'ren L p'ren x squared plus 3 R p'ren 38 $y={0.5}^{x}$ y equals 0.5 Sup x 39 $y=22-2x$ y equals 22 minus 2 x 40 $y=\frac{3}{x}$ y equals Frac 3 Over x EndFrac 41 $y=\left(x+4\right)\left(x+4\right)$ y equals L p'ren x plus 4 R p'ren L p'ren x plus 4 R p'ren 42 $y=\left(4x-3\right)\left(x+1\right)$ y equals L p'ren 4 x minus 3 R p'ren L p'ren x plus 1 R p'ren 43 $y=20x-4{x}^{2}$ y equals 20 x minus 4 x squared 44 $y={x}^{2}$ y equals x squared 45 $y={3}^{x-1}$ y equals 3 Sup x minus 1 46 $y=16-2\left(x+3\right)$ y equals 16 minus 2 L p'ren x plus 3 R p'ren 47 $y=4{x}^{2}-x-3$ y equals 4 x squared minus x minus 3 48 $y=x+\frac{1}{x}$ y equals x plus Frac 1 Over x EndFrac 49 $y=4x\left(5-x\right)$ y equals 4 x L p'ren 5 minus x R p'ren 50 $y=2\left(x-3\right)+6\left(1-x\right)$ y equals 2 L p'ren x minus 3 R p'ren plus 6 L p'ren 1 minus x R p'ren 51 $0.25>\frac{5}{16}$ 0.25 greater than five sixteenths 52 $32\cdot \left(5\cdot 7\right)$ 32 dot L p'ren 5 dot 7 R p'ren 53 $\left(\frac{1}{2}×\frac{1}{2}×\pi ×2\right)+\left(2×\frac{1}{2}×\pi ×5\right)$ L p'ren one half times one half times pi times 2 R p'ren plus L p'ren 2 times one half times pi times 5 R p'ren 54 $\underset{n\to \infty }{\text{liminf}}{E}_{n}=\bigcup _{n\ge 1}\bigcap _{k\ge n}{E}_{k},\phantom{\rule{0.2em}{0ex}}\underset{n\to \infty }{\text{limsup}}{E}_{n}=\bigcap _{n\ge 1}\bigcup _{k\ge n}{E}_{k}.$ liminf Underscript n R arrow infinity Endscripts upper E Sub n Base equals union Underscript n greater than or equals 1 Endscripts intersection Underscript k greater than or equals n Endscripts upper E Sub k Base comma limsup Underscript n R arrow infinity Endscripts upper E Sub n Base equals intersection Underscript n greater than or equals 1 Endscripts union Underscript k greater than or equals n Endscripts upper E Sub k Base period 55 $\begin{array}{ll}\text{(i)}\hfill & \phantom{\rule{0.2em}{0ex}}𝒮\in 𝒜;\hfill \\ \multicolumn{1}{c}{\text{(ii)}}& \phantom{\rule{0.2em}{0ex}}\text{if}E\in 𝒜\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}\stackrel{‾}{E}\in 𝒜;\hfill \\ \multicolumn{1}{c}{\text{(iii)}}& \phantom{\rule{0.2em}{0ex}}\text{if}{E}_{1},{E}_{2}\in 𝒜\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}{E}_{1}\cup {E}_{2}\in 𝒜.\hfill \end{array}$ Layout 1st Row 1st Column L p'ren i R p'ren 2nd Column script upper S element of script upper A semicolon 2nd Row 1st Column L p'ren ii R p'ren 2nd Column if upper E element of script upper A then upper E overbar element of script upper A semicolon 3rd Row 1st Column L p'ren iii R p'ren 2nd Column if upper E 1 comma upper E 2 element of script upper A then upper E 1 union upper E 2 element of script upper A period EndLayout 56 $\begin{array}{llll}\hfill & \hfill & \left(\mathrm{A}\mathrm{.}1\right)\phantom{\rule{0.2em}{0ex}}\mathrm{If}\phantom{\rule{0.2em}{0ex}}A\in ℱ\phantom{\rule{0.2em}{0ex}}\mathrm{then}\phantom{\rule{0.2em}{0ex}}0\le P\left\{A\right\}\le 1.\hfill & \left(1\right)\hfill \\ \multicolumn{1}{c}{}& \hfill & \left(\mathrm{A}\mathrm{.}2\right)\phantom{\rule{0.2em}{0ex}}P\left\{𝒮\right\}=1.\hfill & \left(2\right)\hfill \\ \multicolumn{1}{c}{}& \hfill & \left(\mathrm{A}\mathrm{.}3\right)\phantom{\rule{0.2em}{0ex}}\mathrm{If}\phantom{\rule{0.2em}{0ex}}\left\{{E}_{n},n\ge 1\right\}\in ℱ\phantom{\rule{0.2em}{0ex}}\text{is a sequence of}\phantom{\rule{0.2em}{0ex}}\text{disjoint}\hfill & \left(3\right)\hfill \end{array}$ Layout 1st Row 1st Column Blank 2nd Column Blank 3rd Column L p'ren normal upper A period 1 R p'ren upper I f upper A element of script upper F t h e n 0 less than or equals upper P L brace upper A R brace less than or equals 1 period 4th Column L p'ren 1 R p'ren 2nd Row 1st Column Blank 2nd Column Blank 3rd Column L p'ren normal upper A period 2 R p'ren upper P L brace script upper S R brace equals 1 period 4th Column L p'ren 2 R p'ren 3rd Row 1st Column Blank 2nd Column Blank 3rd Column L p'ren normal upper A period 3 R p'ren upper I f L brace upper E Sub n Base comma n greater than or equals 1 R brace element of script upper F is a sequence of disjoint 4th Column L p'ren 3 R p'ren EndLayout 57 $P\left\{{B}_{j}|A\right\}=\frac{P\left\{{B}_{j}\right\}P\left\{A|{B}_{j}\right\}}{\sum _{j\prime \in J}P\left\{{B}_{j\prime }\right\}P\left\{A|{B}_{j\prime }\right\}}.$ upper P L brace upper B Sub j Base vertical bar upper A R brace equals Frac upper P L brace upper B Sub j Base R brace upper P L brace upper A vertical bar upper B Sub j Base R brace Over sigma summation Underscript j prime element of upper J Endscripts upper P L brace upper B Sub j prime Base R brace upper P L brace upper A vertical bar upper B Sub j prime Base R brace EndFrac period 58 ${\mu }_{1}\left(B\right)={\int }_{B}f\left(x\right)d{\mu }_{2}\left(x\right)$ mu 1 L p'ren upper B R p'ren equals integral Underscript upper B Endscripts f L p'ren x R p'ren d mu 2 L p'ren x R p'ren 59 $\underset{n\to \infty }{\text{lim}}E\left\{|{X}_{n}-X|\right\}=E\left\{\underset{n\to \infty }{\text{lim}}|{X}_{n}-X|\right\}=0.$ limit Underscript n R arrow infinity Endscripts upper E L brace AbsoluteValue upper X Sub n Base minus upper X EndAbsoluteValue R brace equals upper E L brace limit Underscript n R arrow infinity Endscripts AbsoluteValue upper X Sub n Base minus upper X EndAbsoluteValue R brace equals 0 period 60 $\begin{array}{l}{P}_{\mu ,\sigma }\left\{Y\ge {l}_{\beta }\left({\stackrel{‾}{Y}}_{n},{S}_{n}\right)\right\}={P}_{\mu ,\sigma }\left\{\left(Y-{\stackrel{‾}{Y}}_{n}\right)/\left(S·{\left(1+\frac{1}{n}\right)}^{1/2}\right)\ge -{t}_{\beta }\left[n-1\right]\right\}=\beta ,\hfill \\ \multicolumn{1}{c}{}& \left(1\right)\hfill \end{array}$ Layout 1st Row 1st Column upper P Sub mu comma sigma Base L brace upper Y greater than or equals l Sub beta Base L p'ren upper Y overbar Sub n Base comma upper S Sub n Base R p'ren R brace equals upper P Sub mu comma sigma Base L brace L p'ren upper Y minus upper Y overbar Sub n Base R p'ren divided by L p'ren upper S dot L p'ren 1 plus Frac 1 Over n EndFrac R p'ren Sup 1 divided by 2 Base R p'ren greater than or equals minus t Sub beta Base L brack n minus 1 R brack R brace equals beta comma 2nd Row 1st Column Blank 2nd Column L p'ren 1 R p'ren EndLayout 61 $L=\left(\begin{array}{cccccc}\hfill 1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}-1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}-1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}0\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}0\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}\hfill & \hfill \phantom{\rule{0.2em}{0ex}}1\hfill & \hfill \phantom{\rule{0.2em}{0ex}}-1\hfill \end{array}\right).$ upper L equals 5 By 6 Matrix 1st Row 1st Column 1 2nd Column negative 1 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 2nd Row 1st Column Blank 2nd Column 1 3rd Column negative 1 4th Column Blank 5th Column 0 6th Column Blank 3rd Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 4th Row 1st Column Blank 2nd Column 0 3rd Column Blank 4th Column Blank 5th Column Blank 6th Column Blank 5th Row 1st Column Blank 2nd Column Blank 3rd Column Blank 4th Column Blank 5th Column 1 6th Column negative 1 EndMatrix period 62 $\sqrt{n}\left[{\stackrel{‾}{Y}}_{n}-\left(\mu +{z}_{\beta }\sigma \right)\right]/{S}_{n}~\frac{U+\sqrt{n}\phantom{\rule{0.2em}{0ex}}{z}_{1-\beta }}{\left({\chi }^{2}\left[n-1\right]/\left(n-1\right)\right){}^{1/2}}~t\left[n-1;\sqrt{n}\phantom{\rule{0.2em}{0ex}}{z}_{1-\beta }\right],$ Root n EndRoot L brack upper Y overbar Sub n Base minus L p'ren mu plus z Sub beta Base sigma R p'ren R brack divided by upper S Sub n Base tilde Frac upper U plus Root n EndRoot z Sub 1 minus beta Base Over L p'ren chi squared L brack n minus 1 R brack divided by L p'ren n minus 1 R p'ren R p'ren Sup 1 divided by 2 Base EndFrac tilde t L brack n minus 1 semicolon Root n EndRoot z Sub 1 minus beta Base R brack comma 63 $\begin{array}{ll}\gamma \hfill & =P\left\{{E}_{p,q}\subset \left({X}_{\left(r\right)},{X}_{\left(s\right)}\right\}\hfill \\ \multicolumn{1}{c}{}& =\frac{n!}{\left(r-1\right)!}\sum _{j=0}^{s-r-1}\left(-1\right){}^{j}\frac{{p}^{r+j}}{\left(n-r-j\right)!j!}{I}_{1-q}\left(n-s+1,s-r-j\right).\hfill \end{array}$ Layout 1st Row 1st Column gamma 2nd Column equals upper P L brace upper E Sub p comma q Base subset of L p'ren upper X Sub L p'ren r R p'ren Base comma upper X Sub L p'ren s R p'ren Base R brace 2nd Row 1st Column Blank 2nd Column equals Frac n factorial Over L p'ren r minus 1 R p'ren factorial EndFrac sigma summation Underscript j equals 0 Overscript s minus r minus 1 Endscripts L p'ren negative 1 R p'ren Sup j Base Frac p Sup r plus j Base Over L p'ren n minus r minus j R p'ren factorial j factorial EndFrac upper I Sub 1 minus q Base L p'ren n minus s plus 1 comma s minus r minus j R p'ren period EndLayout 64 ${S}_{i}\left[\begin{array}{c}t\\ x\end{array}\right]=\left[\begin{array}{cc}1/m& 0\\ {a}_{i}& {r}_{i}\end{array}\right]\left[\begin{array}{c}t\\ x\end{array}\right]+\left[\begin{array}{c}\left(i-1\right)/m\\ {b}_{i}\end{array}\right],$ upper S Sub i Base BinomialOrMatrix t Choose x EndBinomialOrMatrix equals 2 By 2 Matrix 1st Row 1st Column 1 divided by m 2nd Column 0 2nd Row 1st Column a Sub i Base 2nd Column r Sub i Base EndMatrix BinomialOrMatrix t Choose x EndBinomialOrMatrix plus BinomialOrMatrix L p'ren i minus 1 R p'ren divided by m Choose b Sub i Base EndBinomialOrMatrix comma 65 ${c}_{1}{h}^{4-2s}\le \frac{1}{2T}{\int }_{-T}^{T}{\left(f\left(t+h\right)-f\left(t\right)\right)}^{2}\mathrm{d}t\le {c}_{2}{h}^{4-2s}$ c 1 h Sup 4 minus 2 s Base less than or equals Frac 1 Over 2 upper T EndFrac integral Sub negative upper T Sup upper T Base L p'ren f L p'ren t plus h R p'ren minus f L p'ren t R p'ren R p'ren squared normal d t less than or equals c 2 h Sup 4 minus 2 s 66 $C\left(0\right)-C\left(h\right)\simeq c{h}^{4-2s}$ upper C L p'ren 0 R p'ren minus upper C L p'ren h R p'ren asymptotically equals c h Sup 4 minus 2 s 67 $S\left(\omega \right)=\underset{T\to \infty }{\mathrm{lim}}\frac{1}{2T}{\left|{\int }_{-T}^{T},f,\left(,t,\right),{\mathrm{e}}^{\mathit{it}\omega },\mathrm{d},t\right|}^{2}.$ upper S L p'ren omega R p'ren equals limit Underscript upper T R arrow infinity Endscripts Frac 1 Over 2 upper T EndFrac AbsoluteValue integral Sub negative upper T Sup upper T Base comma f comma L p'ren comma t comma R p'ren comma normal e Sup italic i t omega Base comma normal d comma t EndAbsoluteValue squared period 68 ${\int }_{0}^{1}\phantom{\rule{-0.2em}{0ex}}{\int }_{0}^{1}{\left[|f\left(t\right)-f\left(u\right){|}^{2}+|t-u{|}^{2}\right]}^{-s/2}\mathrm{d}t\mathrm{d}u<\infty$ integral Sub 0 Sup 1 Base integral Sub 0 Sup 1 Base L brack AbsoluteValue f L p'ren t R p'ren minus f L p'ren u R p'ren EndAbsoluteValue squared plus AbsoluteValue t minus u EndAbsoluteValue squared R brack Sup negative s divided by 2 Base normal d t normal d u less than infinity 69 $\mathsf{E}\left(\sum _{I\in {E}_{k+1}}|I{|}^{s}\right)=\mathsf{E}\left(\sum _{I\in {E}_{k}}|I{|}^{s}\right)\mathsf{E}\left({R}_{1}^{s}+{R}_{2}^{s}\right).$ sans serif upper E L p'ren sigma summation Underscript upper I element of upper E Sub k plus 1 Base Endscripts AbsoluteValue upper I EndAbsoluteValue Sup s Base R p'ren equals sans serif upper E L p'ren sigma summation Underscript upper I element of upper E Sub k Base Endscripts AbsoluteValue upper I EndAbsoluteValue Sup s Base R p'ren sans serif upper E L p'ren upper R 1 Sup s Base plus upper R 2 Sup s Base R p'ren period 70 $\left({x}_{1},{y}_{1}\right)$ L p'ren x 1 comma y 1 R p'ren 71 $\left({x}_{2},{y}_{2}\right)$ L p'ren x 2 comma y 2 R p'ren 72 $d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}$ d equals Root L p'ren x 2 minus x 1 R p'ren squared plus L p'ren y 2 minus y 1 R p'ren squared EndRoot 73 $ℝ$ double struck upper R 74 $ℝ=\left(-\infty ,\infty \right)$ double struck upper R equals L p'ren negative infinity comma infinity R p'ren 75 $\left\{1,2,3\right\}$ Set 1 comma 2 comma 3 EndSet 76 $1\in S$ 1 element of upper S 77 $3\in S$ 3 element of upper S 78 $4\notin S$ 4 not an element of upper S 79 $a=\sqrt{3x-1}+{\left(1+x\right)}^{2}$ a equals Root 3 x minus 1 EndRoot plus L p'ren 1 plus x R p'ren squared 80 $a=\frac{{\left(b+c\right)}^{2}}{d}+\frac{{\left(e+f\right)}^{2}}{g}$ a equals Frac L p'ren b plus c R p'ren squared Over d EndFrac plus Frac L p'ren e plus f R p'ren squared Over g EndFrac 81 $x=\left[{\left(a+b\right)}^{2}{\left(c-b\right)}^{2}\right]+\left[{\left(d+e\right)}^{2}{\left(f-e\right)}^{2}\right]$ x equals L brack L p'ren a plus b R p'ren squared L p'ren c minus b R p'ren squared R brack plus L brack L p'ren d plus e R p'ren squared L p'ren f minus e R p'ren squared R brack 82 $x=\left[{\left(a+b\right)}^{2}\right]+\left[{\left(f-e\right)}^{2}\right]$ x equals L brack L p'ren a plus b R p'ren squared R brack plus L brack L p'ren f minus e R p'ren squared R brack 83 $x=\left[{\left(a+b\right)}^{2}\right]$ x equals L brack L p'ren a plus b R p'ren squared R brack 84 $x={\left(a+b\right)}^{2}$ x equals L p'ren a plus b R p'ren squared 85 $x=a+{b}^{2}$ x equals a plus b squared 86 $\frac{\frac{1}{2}}{\frac{3}{4}}=\frac{2}{3}$ Frac one half Over three fourths EndFrac equals two thirds 87 $2\left(\left(x+1\right)\left(x+3\right)-4\left(\left(x-1\right)\left(x+2\right)-3\right)\right)=y$ 2 L p'ren L p'ren x plus 1 R p'ren L p'ren x plus 3 R p'ren minus 4 L p'ren L p'ren x minus 1 R p'ren L p'ren x plus 2 R p'ren minus 3 R p'ren R p'ren equals y 88 $\mathrm{cos}x=1-\frac{{x}^{2}}{2!}+\frac{{x}^{4}}{4!}-\dots$ cosine x equals 1 minus Frac x squared Over 2 factorial EndFrac plus Frac x Sup 4 Base Over 4 factorial EndFrac minus ellipsis 89 $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$ x equals Frac negative b plus or minus Root b squared minus 4 a c EndRoot Over 2 a EndFrac 90 $x+{y}^{\frac{2}{k+1}}$ x plus y Sup Frac 2 Over k plus 1 EndFrac 91 $\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}x}{x}=1$ limit Underscript x R arrow 0 Endscripts Frac sine x Over x EndFrac equals 1 92 $d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}$ d equals Root L p'ren x 2 minus x 1 R p'ren squared plus L p'ren y 2 minus y 1 R p'ren squared EndRoot 93 ${F}_{n}={F}_{n-1}+{F}_{n-2}$ upper F Sub n Base equals upper F Sub n minus 1 Base plus upper F Sub n minus 2 94 $\mathbf{\Pi }=\left(\begin{array}{cccccc}\hfill {\pi }_{11}& \hfill {\pi }_{12}\hfill & {\pi }_{12}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\pi }_{12}& \hfill {\pi }_{11}\hfill & {\pi }_{12}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\pi }_{12}& \hfill {\pi }_{12}\hfill & {\pi }_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0& \hfill 0\hfill & 0\hfill & \hfill {\pi }_{44}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0& \hfill 0\hfill & 0\hfill & \hfill 0\hfill & \hfill {\pi }_{44}\hfill & \hfill 0\hfill \\ \hfill 0& \hfill 0\hfill & 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\pi }_{44}\hfill \end{array}\right)$ bold upper Pi equals 6 By 6 Matrix 1st Row 1st Column pi 11 2nd Column pi 12 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 2nd Row 1st Column pi 12 2nd Column pi 11 3rd Column pi 12 4th Column 0 5th Column 0 6th Column 0 3rd Row 1st Column pi 12 2nd Column pi 12 3rd Column pi 11 4th Column 0 5th Column 0 6th Column 0 4th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column pi 44 5th Column 0 6th Column 0 5th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column pi 44 6th Column 0 6th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column 0 5th Column 0 6th Column pi 44 EndMatrix 95 ${s}_{11}=\frac{{c}_{11}+{c}_{12}}{\left({c}_{11}-{c}_{12}\right)\left({c}_{11}+2{c}_{12}\right)}$ s 11 equals Frac c 11 plus c 12 Over L p'ren c 11 minus c 12 R p'ren L p'ren c 11 plus 2 c 12 R p'ren EndFrac 96 $\mathrm{Si}{\mathrm{O}}_{2}+6\mathrm{H}\mathrm{F}\to {\mathrm{H}}_{2}\mathrm{Si}{\mathrm{F}}_{6}+2{\mathrm{H}}_{2}\mathrm{O}$ upper S i normal upper O 2 plus 6 normal upper H normal upper F R arrow normal upper H 2 upper S i normal upper F 6 plus 2 normal upper H 2 normal upper O 97 $\frac{\text{d}}{\text{d}x}\left(E\left(x\right)A\left(x\right)\frac{\text{d}w\left(x\right)}{\text{d}x}\right)+p\left(x\right)=0$ Frac d Over d x EndFrac L p'ren upper E L p'ren x R p'ren upper A L p'ren x R p'ren Frac d w L p'ren x R p'ren Over d x EndFrac R p'ren plus p L p'ren x R p'ren equals 0 98 ${\text{TCS}}_{\text{gas}}=-\frac{1}{2}\left(\frac{{P}_{\text{seal}}}{{P}_{\text{max}}}\right)\left(\frac{1}{{T}_{\text{seal}}}\right)$ TCS Sub gas Base equals minus one half L p'ren Frac upper P Sub seal Base Over upper P Sub max Base EndFrac R p'ren L p'ren Frac 1 Over upper T Sub seal Base EndFrac R p'ren 99 ${B}_{p}=\frac{\frac{7-{v}^{2}}{3}\left(1+\frac{{c}^{2}}{{a}^{2}}+\frac{{c}^{4}}{{a}^{4}}\right)+\frac{{\left(3-v\right)}^{2}{c}^{2}}{\left(1+v\right){a}^{2}}}{\left(1-v\right)\left(1-\frac{{c}^{4}}{{a}^{4}}\right)\left(1-\frac{{c}^{2}}{{a}^{2}}\right)}$ upper B Sub p Base equals NestFrac Frac 7 minus v squared Over 3 EndFrac L p'ren 1 plus Frac c squared Over a squared EndFrac plus Frac c Sup 4 Base Over a Sup 4 Base EndFrac R p'ren plus Frac L p'ren 3 minus v R p'ren squared c squared Over L p'ren 1 plus v R p'ren a squared EndFrac NestOver L p'ren 1 minus v R p'ren L p'ren 1 minus Frac c Sup 4 Base Over a Sup 4 Base EndFrac R p'ren L p'ren 1 minus Frac c squared Over a squared EndFrac R p'ren NestEndFrac 100 ${Q}_{\text{tank}}^{\text{series}}=\frac{1}{{R}_{\text{s}}}\sqrt{\frac{{L}_{\text{s}}}{{C}_{\text{s}}}}$ upper Q Sub tank Sup series Base equals Frac 1 Over upper R Sub s Base EndFrac Root Frac upper L Sub s Base Over upper C Sub s Base EndFrac EndRoot 101 $\text{Δ}{\varphi }_{\text{peak}}={tan}^{-1}\left({k}^{2}{Q}_{\text{tank}}^{\text{series}}\right)$ upper Delta phi Sub peak Base equals tangent Sup negative 1 Base L p'ren k squared upper Q Sub tank Sup series Base R p'ren 102 $f=1.013\frac{W}{{L}^{2}}\sqrt{\frac{E}{\rho }}\sqrt{\left(1+0.293\frac{{L}^{2}}{{\text{EW}}^{2}}\sigma \right)}$ f equals 1.013 Frac upper W Over upper L squared EndFrac Root Frac upper E Over rho EndFrac EndRoot Root L p'ren 1 plus 0.293 Frac upper L squared Over EW squared EndFrac sigma R p'ren EndRoot 103 ${u}_{n}\left(x\right)={\gamma }_{n}\left(\mathrm{cosh}{k}_{n}x-\mathrm{cos}{k}_{n}x\right)+\left(\mathrm{sinh}{k}_{n}x-\mathrm{sin}{k}_{n}x\right)$ u Sub n Base L p'ren x R p'ren equals gamma Sub n Base L p'ren hyperbolic cosine k Sub n Base x minus cosine k Sub n Base x R p'ren plus L p'ren hyperbolic sine k Sub n Base x minus sine k Sub n Base x R p'ren 104 $\begin{array}{cc}B\hfill & =\frac{\frac{{F}_{0}}{m}}{\sqrt{\left({\omega }_{0}^{2}-{\omega }^{2}{\right)}^{2}+4{n}^{2}{\omega }^{2}}}\hfill \\ & =\frac{\frac{{F}_{0}}{k}}{\sqrt{\left(1-\left(\omega /{\omega }_{0}^{2}{\right)}^{2}{\right)}^{2}+4\left(n/{\omega }_{0}{\right)}^{2}\left(\omega /{\omega }_{0}{\right)}^{2}}}\hfill \end{array}$ Layout 1st Row 1st Column upper B 2nd Column equals NestFrac Frac upper F 0 Over m EndFrac NestOver Root L p'ren omega 0 squared minus omega squared R p'ren squared plus 4 n squared omega squared EndRoot NestEndFrac 2nd Row 1st Column Blank 2nd Column equals NestFrac Frac upper F 0 Over k EndFrac NestOver Root L p'ren 1 minus L p'ren omega divided by omega 0 squared R p'ren squared R p'ren squared plus 4 L p'ren n divided by omega 0 R p'ren squared L p'ren omega divided by omega 0 R p'ren squared EndRoot NestEndFrac EndLayout 105 $\mathrm{p}\left(A\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}B\right)=\mathrm{p}\left(A\right)\phantom{\rule{3.33333pt}{0ex}}\mathrm{p}\left(B|A\right)$ normal p L p'ren upper A a n d upper B R p'ren equals normal p L p'ren upper A R p'ren normal p L p'ren upper B vertical bar upper A R p'ren 106 $\mathrm{PMF}\left(x\right)\propto {\left(\frac{1}{x}\right)}^{\alpha }$ upper P upper M upper F L p'ren x R p'ren proportional to L p'ren Frac 1 Over x EndFrac R p'ren Sup alpha 107 $f\left(x\right)=\frac{1}{\sqrt{2\pi }}exp\left(-{x}^{2}/2\right)$ f L p'ren x R p'ren equals Frac 1 Over Root 2 pi EndRoot EndFrac exp L p'ren minus x squared slash 2 R p'ren 108 $\frac{dx}{d\theta }=\frac{\beta }{{cos}^{2}\theta }$ Frac d x Over d theta EndFrac equals Frac beta Over cosine squared theta EndFrac 109 $s/\sqrt{2\left(n-1\right)}$ s divided by Root 2 L p'ren n minus 1 R p'ren EndRoot