text:
# Table and caption formatting

I'm having some problems getting my table and caption formatting to look right. Below is a sample of the code from my thesis. I've included all the packages that I'm using in my thesis, so that you can get an idea of what else might be affecting the look of the table. I've also attached a picture of my output.

The main issues that I have with the table are:

1. The caption text sits too close to the top of the table. Is there a way to get this to sit a little higher (0.5-1 line spaces) above the table?
2. The rows are too closely spaced, making it a little hard to read the nuclide and atomic mass numbers for the elements. Does line spacing not affect tables? How can I can increase the spacing between rows?
3. The text underneath the table needs to sit directly beneath the table and be no wider than the width of the table. How do I go about this?

Thanks for the help!

.

\documentclass[12pt,a4paper]{report}
\usepackage{setspace}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{layout}
\usepackage{lscape}
\usepackage[round]{natbib}
\usepackage{array}
\usepackage{booktabs}
\usepackage{siunitx}
\usepackage{tensor}

\begin{document}
\doublespacing
\begin{table}
\centering
\caption{Naturally occurring long-lived radionuclides and their stable radiogenic daughters frequently used in geochronology.}\label{tab:001}
\addtolength{\tabcolsep}{-2pt} % slight reduction of intercolumn space
\begin{tabular}{*{4}{c} >{\raggedright\arraybackslash}p{3.5cm}}
\toprule
Parent & Decay Modes$^a$ & Half-life (years) & Daughters \\
\midrule
$^{40}_{19}$K & $\varepsilon$: $10.72\%$, $\beta^{+}$: $1.0\times 10^{-3}\%$ & $1.248 \times 10^{9}$ & $^{40}_{18}$Ar \\
$^{40}_{19}$K & $\beta^{-}$: $89.28\%$ & $1.248 \times 10^{9}$ & $^{40}_{20}$Ca \\
$^{87}_{37}$Rb & $\beta^{-}$: $100\%$ & $4.81\times 10^{10}$ & $^{87}_{38}$Sr \\
$^{147}_{62}$Sm & $\alpha$: $100\%$ & $1.06\times 10^{11}$ & $^{143}_{60}$Nd \\
$^{176}_{71}$Lu & $\beta^{-}$: $100\%$ & $3.76\times 10^{10}$ & $^{176}_{72}$Hf \\
$^{187}_{75}$Re & $\alpha$: $1.0\times 10^{-4}\%$, $\beta^{-}: 100\%$ & $4.33\times 10^{10}$ & $^{187}_{76}$Os \\
$^{232}_{90}$Th & $\alpha$: $100\%$, SF: $1.1\times 10^{-9}\%$ & $1.40\times 10^{10}$ & $^{208}_{82}$Pb \\
$^{235}_{92}$U & $\alpha$: $100\%$, SF: $7.0\times 10^{-9}\%$ & $7.04\times 10^{8}$ & $^{207}_{82}$Pb \\
$^{238}_{92}$U & $\alpha$: $100\%$, SF: $5.5\times 10^{-5}\%$ & $4.468\times 10^{9}$ & $^{206}_{82}$Pb \\
\bottomrule
\end{tabular}
\end{table}
$^{a}$ $\varepsilon$ = electron capture decay; $\beta^{+}$ = positron decay; $\beta^{-}$ = negatron decay; $\alpha$ = alpha decay.
\end{document}

• i'll try to tackle the rest of this later, but for a first suggestion, you'll probably need to put in some explicit vertical space between the caption and the tabular; try \medskip. also, the way the notes are now outside the table environment, they might well end up on a different page -- table is a floating body, while the notes will stay where they are. for the lines too close together, since you have loaded array, you can \renewcommand{\arraystretch}{1.2} or some other factor to expand the distance between lines (may require experimentation). Jan 24, 2014 at 19:32

## 3 Answers

1. It's enough to load the caption package (you could even set a bigger skip, if needed).

2. Redefine \arraystretch locally.

3. Build your table and its footnote using the ctable package.

The code:

\documentclass[12pt,a4paper]{report}
\usepackage{booktabs}
\usepackage{siunitx}
\usepackage{chemmacros}
\usepackage{caption}
\usepackage{ctable}

\begin{document}

{
\renewcommand\arraystretch{1.3}
\addtolength{\tabcolsep}{-2pt} % slight reduction of intercolumn space
\ctable[
caption = {Naturally occurring long-lived radionuclides and their stable radiogenic daughters frequently used in geochronology.},
label= {tab:001},
mincapwidth = \textwidth,
footerwidth
]
{ccS[table-figures-exponent=2,table-figures-integer=2,table-figures-decimal=3,table-number-alignment=center]c
}
{%
\tnote{$\varepsilon$ = electron capture decay; $\beta^{+}$ = positron decay; $\beta^{-}$ = negatron decay; \\ $\alpha$ = alpha decay.}%
}
{%
\toprule
Parent & Decay Modes & {Half-life (years)} & Daughters \\
\midrule
\ch{^{40}_{19}K}   & $\varepsilon$: \SI{10.72}{\percent}, $\beta^{+}$: \SI{1.0e-3}{\percent} & 1.248e9 & \ch{^{40}_{18}Ar} \\
\ch{^{40}_{19}K}   & $\beta^{-}$: \SI{89.28}{\percent} & 1.248e9 & \ch{^{40}_{20}Ca} \\
\ch{^{87}_{37}Rb}  & $\beta^{-}$: $100\%$ & 4.81e10 & \ch{^{87}_{38}Sr} \\
\ch{^{147}_{62}Sm} & $\alpha$: \SI{100}{\percent} & 1.06e11 & \ch{^{143}_{60}Nd} \\
\ch{^{176}_{71}Lu} & $\beta^{-}$: \SI{100}{\percent} & 3.76e10 & \ch{^{176}_{72}Hf} \\
\ch{^{187}_{75}Re} & $\alpha$: \SI{1.0e-4}{\percent}, $\beta^{-}$: \SI{100}{\percent} & 4.33e10 & \ch{^{187}_{76}Os} \\
\ch{^{232}_{90}Th} & $\alpha$: \SI{100}{\percent}, SF: \SI{1.1e-9}{\percent} & 1.40e10 & \ch{^{208}_{82}Pb} \\
\ch{^{235}_{92}U}  & $\alpha$: \SI{100}{\percent}, SF: \SI{7.0e-9}{\percent} & 7.04e8 & \ch{^{207}_{82}Pb} \\
\ch{^{238}_{92}U}  & $\alpha$: \SI{100}{\percent}, SF: \SI{5.5e-5}{\percent} & 4.468e9 & \ch{^{206}_{82}Pb} \\
\bottomrule
}
}

\end{document}

### Additional improvements:

1. The siunitx package was used to format the entries in the third column.

2. The chemmacros package was used to properly format the sub/superscripts in the first and third columns.

• I’d also suggest to add a small space, e.g. \,, before percent signs, especially when preceded by powers of ten. Is it standard practice to have nuclide and atomic mass numbers left-aligned? You already use siunitx, have a look at what tabular column styles it adds, e.g. regarding the half-life column. Finally, is it really plural “Daughters”? Jan 24, 2014 at 19:58
• @Crissov I incorporated your suggestions. Thanks. Jan 24, 2014 at 20:58
• @GonzaloMedina Thanks for the code - it's done the job beautifully! Jan 25, 2014 at 13:57
• @cjms85 You're welcome! Glad I could help. Jan 25, 2014 at 15:32

Another approach for the third problem is to use the »threeparttable« package. For formatting and alignment of the numbers it is suggestive to let »siunitx« do that job. The isotopes can be formatted easier by chemformula (from the »chemmacros« bundle).

\documentclass[12pt,a4paper]{report}
\usepackage[T1]{fontenc}
\usepackage{caption}
\usepackage{array,booktabs,threeparttable}
\usepackage{siunitx}
\usepackage{chemformula}

\begin{document}
\begin{table}[!ht]
\caption{Naturally occurring long-lived radionuclides and their stable radiogenic daughters frequently used in geochronology.}
\label{tab:nuclides}
\centering
\begin{threeparttable}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{
c
c
S[table-figures-exponent=2,table-figures-integer=2,table-figures-decimal=3,table-number-alignment=center]
c
} \toprule
Parent & Decay Modes\tnote{a} & {Half-life (years)} & Daughters \\ \midrule
\ch{^{40}_{19}K}   & $\varepsilon$: \SI{10.72}{\percent}, $\beta^{+}$: \SI{1.0e-3}{\percent} & 1.248e9 & \ch{^{40}_{18}Ar} \\
\ch{^{40}_{19}K}   & $\beta^{-}$: \SI{89.28}{\percent} & 1.248e9 & \ch{^{40}_{20}Ca} \\
\ch{^{87}_{37}Rb}  & $\beta^{-}$: $100\%$ & 4.81e10 & \ch{^{87}_{38}Sr} \\
\ch{^{147}_{62}Sm} & $\alpha$: \SI{100}{\percent} & 1.06e11 & \ch{^{143}_{60}Nd} \\
\ch{^{176}_{71}Lu} & $\beta^{-}$: \SI{100}{\percent} & 3.76e10 & \ch{^{176}_{72}Hf} \\
\ch{^{187}_{75}Re} & $\alpha$: \SI{1.0e-4}{\percent}, $\beta^{-}$: \SI{100}{\percent} & 4.33e10 & \ch{^{187}_{76}Os} \\
\ch{^{232}_{90}Th} & $\alpha$: \SI{100}{\percent}, SF: \SI{1.1e-9}{\percent} & 1.40e10 & \ch{^{208}_{82}Pb} \\
\ch{^{235}_{92}U}  & $\alpha$: \SI{100}{\percent}, SF: \SI{7.0e-9}{\percent} & 7.04e8 & \ch{^{207}_{82}Pb} \\
\ch{^{238}_{92}U}  & $\alpha$: \SI{100}{\percent}, SF: \SI{5.5e-5}{\percent} & 4.468e9 & \ch{^{206}_{82}Pb} \\ \bottomrule
\end{tabular}
\begin{tablenotes}
\footnotesize
\item[a]$\varepsilon$ = electron capture decay; $\beta^{+}$ = positron decay; $\beta^{-}$ = negatron decay; $\alpha$ = alpha decay
\end{tablenotes}
\end{threeparttable}
\end{table}
\end{document}

• Re »The isotopes can be formatted easier by chemformula« -- this also aligns the subscripts right. I think (although I'm not sure) that the alpha- and beta-particle also should be upright... Jan 24, 2014 at 20:18
• @cgnieder: That's why I used chemformula here. And regarding the α- and β-particles you are probably right. But I wanted to leave some minor tasks to the interested reader. Packages like »textgreek« or »upgreek« should do the trick in this regard. XeLaTeX or LuaLaTeX would make things easier with Unicode characters. Jan 24, 2014 at 20:54
• I just remembered that we have a question about them :) Jan 24, 2014 at 20:57

A solution using the caption and floatrow packages, so that the caption width is equal to the table width:

\documentclass[12pt,a4paper]{report}
\usepackage{amsmath}
\usepackage{array}
\usepackage{booktabs}
\usepackage[justification = centerlast]{caption}
\usepackage{floatrow}
\floatsetup[table]{footnoterule = none}
\renewcommand{\arraystretch}{1.5}

\begin{document}

\begin{table}
\addtolength{\tabcolsep}{-2pt} % slight reduction of intercolumn space
\centering
\ttabbox%
{\caption{Naturally occurring long-lived radionuclides and their stable radiogenic  daughters frequently used in geochronology.}\label{tab:001}}
{\begin{tabular}{*{4}{c} >{\raggedright\arraybackslash}p{3.5cm}}
\toprule
Parent & Decay Modes\mpfootnotemark & Half-life (years) & Daughters \\
\midrule
$^{40}_{19}$K & $\varepsilon$: $10.72\%$, $\beta^{+}$: $1.0\times 10^{-3}\%$ &  $1.248 \times 10^{9}$ & $^{40}_{18}$Ar \\
$^{40}_{19}$K & $\beta^{-}$: $89.28\%$ & $1.248 \times 10^{9}$ & $^{40}_{20}$Ca \\
$^{87}_{37}$Rb & $\beta^{-}$: $100\%$ & $4.81\times 10^{10}$ & $^{87}_{38}$Sr \\
$^{147}_{62}$Sm & $\alpha$: $100\%$ & $1.06\times 10^{11}$ & $^{143}_{60}$Nd \\
$^{176}_{71}$Lu & $\beta^{-}$: $100\%$ & $3.76\times 10^{10}$ & $^{176}_{72}$Hf \\
$^{187}_{75}$Re & $\alpha$: $1.0\times 10^{-4}\%$, $\beta^{-}: 100\%$ &  $4.33\times 10^{10}$ & $^{187}_{76}$Os \\
$^{232}_{90}$Th & $\alpha$: $100\%$, SF: $1.1\times 10^{-9}\%$ & $1.40\times 10^{10}$ & $^{208}_{82}$Pb \\
$^{235}_{92}$U & $\alpha$: $100\%$, SF: $7.0\times 10^{-9}\%$ & $7.04\times 10^{8}$ & $^{207}_{82}$Pb \\
$^{238}_{92}$U & $\alpha$: $100\%$, SF: $5.5\times 10^{-5}\%$ & $4.468\times 10^{9}$ & $^{206}_{82}$Pb \\
\bottomrule
\end{tabular}
\floatfoot{\quad\textsuperscript{\thempfootnote}\:$\varepsilon$ = electron capture  decay; $\beta^{+}$ = positron decay; $\beta^{-}$ = negatron decay;  \\ $\alpha$ = alpha  decay.}}
\end{table}

\end{document}


text:
## FG7 Linear graphs

A linear graph is the simplest way of representing data or a functional relationship. Consequently an understanding of linear graphs is an essential requirement for many courses in science, engineering and mathematics. This module deals with linear graphs.

### The Cartesian Plane

The Cartesian plane is defined by a pair of mutually perpendicular coordinate axes. The horizontal axis is the x- axis and the vertical axis the y – axis. Their point of intersection is called the origin O.

Points on the plane are referred to by their horizontal distance ($$x$$ – coordinate) and vertical distance ($$y$$ – coordinate) from the origin. Distances to the right of, and up from the origin are positive. Distances to the left of, and down from the origin are negative. Consider the figure below.

The $$x$$ – coordinate is always given first.

The point A ($$2,3$$) has coordinates $$x=2$$ and $$y=3$$.

The point B ($$1,3$$) has coordinates $$x=1$$ and $$y=3$$.

The point C ($$3,3$$) has coordinates $$x=3$$ and $$y=3$$.

The point D ($$3,4$$) has coordinates $$x=3$$ and $$y=4$$.

### Sketch Graphs

When sketching a graph it is not necessary to plot large numbers of points. Only the basic shape of the graph is required, although some important points need to be clearly labeled.

#### Intercepts

These are the points where the graph crosses the $$x$$ and $$y$$ axes. The $$x$$-intercept is where the graph cuts the $$x$$-axis $$\left(y=0\right)$$ and the $$y$$-intercept is where the graph cuts the $$y$$-axis $$\left(x=0\right)$$.

### Linear Graphs

The graph of a linear relationship is a straight line. Only two points are needed to sketch the graph of a straight line.

#### Example

Sketch the function $$y=3x+6$$.
Solution:
Determine two points on the line.
The $$x$$ and $$y$$-intercepts are often the easiest to find.
For the $$y$$- intercept $$\left(x=0\right)$$, we have \begin{align*} y & =3x+6\\ & =6. \end{align*} Hence the $$y$$-intercept is at $$\left(0,6\right)$$.
Now we want the $$x$$- intercept $$\left(y=0\right).$$ So \begin{align*} y & =3x+6\\ 0 & =3x+6\\ 3x & =-6\\ x & =-2. \end{align*}

Hence the $$x$$-intercept is at $$\left(-2,0\right).$$
The y-intercept is the point $$(0,6$$).

\begin{align*} x\textrm{-intercept} & (y=0)\\ \textrm{When }y & =0\\ 0 & =3x+6\\ 3x & =-6\\ x & =-2 \end{align*}

The x-intercept is the point ( 2, 0).

Connect the two points with a straight line.

The $$x$$- and $$y$$-intercepts should be marked on the graph.

See Exercise 1

The slope of the straight line is called the gradient. This is the ratio of the vertical change (rise) to the horizontal change (run) between two points ($$x_{1},y_{1}$$) and ($$x_{2},y_{2}$$) on the line.

\begin{align*} Gradient & =\frac{rise}{run}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \end{align*}

#### Examples

1. Find the gradient of the line $$y=3x+6$$ (see previous example).
Solution:
The points ($$0,6$$) and ($$-2,0$$) are on the line $$y=3x+6$$. Let1 Note that it doesn’t matter which points you choose for $$\left(x_{1},y_{1}\right)$$ and $$\left(x_{2},y_{2}\right)$$. The only requirement is that both points must be on the line. For example we could let \begin{align*} \left(x_{1},y_{1}\right) & =\left(-2,0\right) \end{align*} \begin{align*} \left(x_{2},y_{2}\right) & =\left(0,6\right) \end{align*} then \begin{align*} Gradient & =\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\ & =\frac{6-0}{0-\left(-2\right)}\\ & =3 \end{align*} the same answer as at left. \begin{align*} \left(x_{1},y_{1}\right) & =\left(0,6\right) \end{align*} and \begin{align*} \left(x_{2},y_{2}\right) & =\left(-2,0\right) \end{align*} Then \begin{align*} Gradient & =\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{0-6}{-2-0}=\frac{-6}{-2}=3. \end{align*} The gradient is 3.

2. Find the gradient of the line joining the two points $$\left(1,3\right)$$ and $$\left(4,5\right).$$
Solution:
Let \begin{align*} \left(x_{1},y_{1}\right) & =\left(1,3\right) \end{align*} and \begin{align*} \left(x_{2},y_{2}\right) & =\left(4,5\right) \end{align*} Then \begin{align*} Gradient & =\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{5-3}{4-1}=\frac{2}{3}. \end{align*} The gradient is $$2/3.$$

See Exercise 2

### Equation of a Straight Line

The general equation of a straight line is \begin{align*} y & =mx+c \end{align*} where $$m$$ is the gradient, and $$c$$ is the value of the intercept on the $$y$$-axis. The figure below illustrates this.

The equation of a straight line can also be written as2 To see this, assume you know a point $$\left(x_{1},y_{1}\right)$$ on the line and the gradient $$m$$ of the line. Since $$\left(x_{1},y_{1}\right)$$ is on the line, it must satisfy the equation \begin{align*} y & =mx+c. & \left(1\right) \end{align*} and so \begin{align*} y_{1} & =mx_{1}+c. & \left(2\right) \end{align*} Now subtract $$\left(2\right)$$ from $$\left(1\right)$$ to get \begin{align*} y-y_{1} & =mx+c-\left(mx_{1}+c\right)\\ & =mx-mx_{1}+c-c\\ & =m\left(x-x_{1}\right). \end{align*} \begin{align*} y-y_{1} & =m(x-x_{1}) \end{align*} where $$m$$ is the gradient and ($$x_{1},y_{1}$$) is a point on the line. This is called point-slope form because it assumes you know

1. A point on the line $$\left(x_{1},y_{1}\right)$$ and

2. The gradient $$m$$ of the line.

### Some Properties of Linear Graphs

• A line that slopes up and to the right has a positive gradient.

• A line that slopes up and to the left has a negative gradient.

• Parallel lines have the same gradient.

• The gradients of two mutually perpendicular lines multiply to $$-1.$$

• A line parallel to the x – axis has a gradient of $$0$$.

• A line parallel to the y – axis; the gradient is undefined.

#### Example 1: Find the gradient and the equation of the straight line below:

Solution:

First note that the graph has a positive gradient3 Because it slopes up (increases) as you move to the right. You don’t have to write this but it is a good check to have in mind. If you calculate a negative gradient, you know something has gone wrong.. Let \begin{align*} \left(x_{1},y_{1}\right) & =\left(1,3\right) \end{align*} and \begin{align*} \left(x_{2},y_{2}\right) & =\left(3,7\right). \end{align*} Then the gradient $$m$$ is given by \begin{align*} m & =\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\ & =\frac{7-3}{3-1}\\ & =\frac{4}{2}\\ & =2. \end{align*} So the gradient $$m=2.$$ From the graph we see the $$y$$-intercept is $$1.$$ That is, $$c=1.$$ Now we can substitute in the equation for a line4 The equation for a line is \begin{align*} y & =mx+c. \end{align*} to get \begin{align*} y & =2x+1. \end{align*} The equation for the line is $$y=2x+1.$$

#### Example 2: Find the gradient and the equation of the straight line below:

Solution:

First note that the graph has a negative gradient.5 Because it slopes down (decreases) as you move to the right. You don’t have to write this but it is a good check to have in mind. If you calculate a positive gradient, you know something has gone wrong.

Let \begin{align*} \left(x_{1},y_{1}\right) & =\left(0,4\right) \end{align*} and \begin{align*} \left(x_{2},y_{2}\right) & =\left(6,0\right) \end{align*} Then the gradient $$m$$ is given by \begin{align*} m & =\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\ & =\frac{0-4}{6-0}\\ & =-\frac{2}{3}. \end{align*} The $$y$$-intercept is at $$\left(0,4\right)$$ and so $$c=4.$$ Hence the equation of the line is $$y=-\frac{2}{3}x+4.$$ This is a correct answer but the fraction makes it look a bit ugly. We can multiply both sides by $$3$$ to get rid of the fraction. The equation then becomes: $$3y=-2x+12$$ or $$3y+2x=12.$$

#### Example 3: Sketch the function $$2y=x-6$$ and state the gradient of the straight line.

Solution:

First find the $$x$$ and $$y$$ intercepts. For the $$x$$-intercept, set $$y=0$$ then \begin{align*} 0 & =x-6\\ \text{and }x & =6. \end{align*} Hence the $$x$$-intercept is at $$\left(6,0\right)$$.

For the $$y$$-intercept, set $$x=0$$ then \begin{align*} 2y & =-6\\ \text{and }y & =-3. \end{align*} Hence the $$y$$-intercept is at $$\left(0,-3\right).$$

Now draw a line between these two points as shown below:

To find the gradient, let \begin{align*} \left(x_{1},y_{1}\right) & =\left(0,-3\right) \end{align*} and \begin{align*} \left(x_{2},y_{2}\right) & =\left(6,0\right). \end{align*} Then the gradient $$m$$ is given by \begin{align*} m & =\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\ & \frac{0-\left(-3\right)}{6-0}\\ & =\frac{3}{6}\\ & =\frac{1}{2}. \end{align*} Alternatively, to find the gradient divide both sides of the equation by $$2$$ to get \begin{align*} 2y & =x-6\\ y & =\frac{1}{2}x-3. \end{align*} This is in the standard form $$y=mx+c$$ and we see that the gradient $$m=1/2$$ as before.

#### Example 4: Sketch the graph of $$3y+2x=0$$ and state the gradient of the straight line.

Solution:

First find the intercepts. For the $$x$$-intercept $$y=0$$. Substituting in the equation we get \begin{align*} 2x & =0\\ x & =0 \end{align*} and the $$x$$-intercept is at $$\left(0,0\right).$$ For the $$y$$-intercept, $$x=0.$$ Substituting in the equation we get \begin{align*} 3y & =0\\ y & =0 \end{align*} and the $$y$$-intercept is at $$\left(0,0\right).$$ But this means the $$x$$-intercept is equal to the $$y$$-intercept. This is because the line passes through the origin $$\left(0,0\right)$$6 We could have seen this by considering the equation of the line: $$3y+2x=0.$$ Rearranging the equation and dividing by $$3$$ we get \begin{align*} 3y+2x & =0\\ 3y & =-2x\\ y & =-\frac{2}{3}x. \end{align*} Comparing this to the standard form $$y=mx+c$$ we see that $$m=-2/3$$ and $$c=0.$$ That is the $$y$$-intercept is at $$\left(0,0\right).$$.

So far we know the graph goes through $$\left(0,0\right).$$ To sketch its graph we need another point. To find another point on the line, pick any value of $$x$$ (other than $$0$$). Let’s take $$x=3$$ then substituting into the equation we have \begin{align*} 3y+2\left(3\right) & =0\\ 3y & =-6\\ y & =-2 \end{align*} so the point $$\left(3,-2\right)$$ is on the graph.

To sketch the graph, we plot the points $$\left(0,0\right)$$ and $$\left(3,-2\right)$$ and draw a line through them as shown below.

To find the gradient, rearrange the equation to the form of $$y=mx+c$$:

\begin{align*} 3y+2x & =0\\ y & =-\frac{2}{3}x \end{align*}

The gradient of the line is $$-\frac{2}{3}$$.

#### Example 5: Find the equation of the line that passes through the points $$\left(-1,3\right)$$ and $$\left(2,0\right)$$.

Solution:

First find the gradient $$m.$$ Let \begin{align*} \left(x_{1},y_{1}\right) & =\left(-1,3\right) \end{align*} and \begin{align*} \left(x_{2},y_{2}\right) & =\left(2,0\right). \end{align*} Then the gradient $$m$$ is given by \begin{align*} m & =\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\ & \frac{0-3}{2-\left(-1\right)}\\ & =\frac{-3}{3}\\ & =-1. \end{align*}

The equation to our line is $$y=-x+c$$. We need to find $$c.$$ To find $$c$$ substitute either of the points on the line into the equation for the line. Using the point $$\left(-1,3\right)$$ we obtain \begin{align*} y & =-x+c\\ 3 & =-\left(-1\right)+c\\ c & =3-1\\ & =2. \end{align*}

The equation to the line is $$y=-x+2$$.

#### Example 6: Find the equation of the line that passes through the point $$\left(3,4\right)$$ and parallel to the line $$y=-2x+7.$$ Give your answer in point slope form.

Solution:

The point-slope form of the equation to a straight line is \begin{align*} y-y_{1} & =m\left(x-x_{1}\right). \end{align*}

The gradient of the line $$y=-2x+7$$ is $$-2$$, so $$m=-2$$.

The point on the line is ($$3,4$$) so $$x_{1}=3$$ and $$y_{1}=4$$.

Substituting in the point slope equation gives: \begin{align*} y-y_{1} & =m\left(x-x_{1}\right)\\ y-4 & =-2\left(x-3\right). \end{align*}

The equation to the line is $$y-4=-2(x-3)$$.

See Exercise 3

### Exercise 1

Sketch the graphs of the following equations.

$\begin{array}{lllllllclc} \textrm{(a)} & y=4x+3 & & & \textrm{(b)} & 2y=-x+6 & & & \textrm{(c)} & 3y+2x=9\\ \textrm{(d)} & y=2x & & & \textrm{(e)} & y=7 & & & \textrm{(f)} & 2y+x=0 \end{array}$

1. $$\textrm{a. }y=4x+3$$

1. $$2y=-x+6$$

1. $$3y+2x=9$$

1. $$y=2x$$

1. $$y=7$$

1. $$2y+x=0$$

### Exercise 2

Find the gradient of the following straight lines.

$\begin{array}{lllllllclc} \textrm{(a)} & y=-5x & & & \textrm{(b)} & 2y=6x+9 & & & \textrm{(c)} & 2x-3y=0\\ \textrm{(d)} & 3y+2x-2=0 & & & \textrm{(e)} & 5y=8 & & & \textrm{(f)} & x=-2 \end{array}$

\begin{aligned}\begin{array}{llll} 2. & \left(\text{a}\right)\,-5 & \left(\text{b}\right)\,3 & \left(\text{c}\right)\,2/3\\ \end{array}\end{aligned}

### Exercise 3

Find the equation of the straight lines below

1. Find the equation, in slope-point form, of the line with a gradient of $$3$$, passing through the point ($$0,-5$$).

\begin{aligned}\begin{array}{llll} 3. & \left(\text{a}\right)\,2y=-x+4 & \left(\text{b}\right)\,y=x+5 & \left(\text{c}\right)\,y+5=3x \end{array}\end{aligned}

format: plain


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# Thread: Finding the length of a triangle

1. ## Finding the length of a triangle

Find RS.

Find TU.

Any help is appreciated! (:

2. Originally Posted by 4AM
Find RS.
Let x denote PR, then RQ = 50-x. Using the tangens function you can set up 2 equations:

$|\overline{RS}| = \dfrac{x}{\tan(32^\circ)}$ ..... and ..... $|\overline{RS}| = \dfrac{50-x}{\tan(26^\circ)}$

Set equal the RHS of the two equations and solve for x, afterwards you can use one of the two equations to get the length of RS. I've got $|\overline{RS}| \approx 44.94$

Find TU.
I've modified your sketch a little bit. (see attachment)

According to the drawing you can set up 2 equations:

$\tan(53.8^\circ)=\dfrac xy$ ..... and ..... $\tan(38.8^\circ)=\dfrac{x}{y+12.5}$

Solve the second equation for y, plug in this term into the first equation and solve for x.

I've got $|\overline{TU}| = x \approx 24.42$


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# Opening ArcGIS Pro application from ArcGIS Pro Project (.aprx) file using ArcPy?

I need to open an ArcGIS Pro project (.aprx) file in the ArcGIS Pro application inside of an arcpy script.

Does anyone know of a way to accomplish this?

I am using ArcGIS pro 2.4.1.

By open, I mean having the application actually open up on my screen. Similar to opening it from the start menu but utilizing the script instead.

I have done something similar to open PDF files in Adobe Acrobat Reader in the past so I just looked up this answer to Opening PDF generated by Python AddIn using Report (*.rlf) file automatcally? and tested:

import os
myfile = r"C:\Temp\Projects\TestProject\TestProject.aprx"
os.system("start " + myfile)

and it worked.

• Woah! That works!! Thank you. Now is it possible to close that same file? – ketar Sep 30 '19 at 18:25
• @ketar I’ve never had a requirement to close an application opened this way. I always want the user to interact with it and then decide when they want to close it. – PolyGeo Sep 30 '19 at 22:31

You can use:

aprx = arcpy.mp.ArcGISProject(r"C:\path\to\project.aprx")

For further things you can do with aprx refer to the documentation.

• I meant open as in have it pop up, as if the icon was double clicked. Let me update the question. – ketar Sep 26 '19 at 19:50


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# Monitoring a data stream

You are monitoring a data stream which is delivering very many $32$-bit quantities at a rate of $10$ Megabytes per second. You know that either:

$A$: All values occur equally often, $or$

$B$: Half of the values occur $2^{10}$ times more often than others (but you dont know anything about which would be the more common values).

You are allowed to read the values off the data stream, but you only have $2^{20}$ bytes of memory.

Describe a method for determining which of the two situations, $A$ or $B$, occurs.

Roughly how many data values do you need to read to be confident of your result with a probability of $0.999$? [This is about the $3$ sigma level – $3$ standard deviations of a normal distribution.]

Please could I have some thoughts on the solution, it is a difficult problem.

• Not sure this helps, my 5c of top of my head. First, you probably assume samples are independent, right? Let $s_i$ be the sample $i$. In situation A $s_i \in [0, 2^{32}-1]$ with probability $1/2^{32}$ in situation $B$ you have $s_i \in {Set}_1~$ with probability $1/(2^{16} +2^6 )$ and $s_i \in {Set}_2~$ w.p $1/ (2^{26} +2^{16})$ (if I have done the arithmetics right). If you observe any given sequence of length $2^{20}/2^5$ you can write the odds of seeing this sequence under Model A and under the Model B given that you have uniform probability on selecting the $Set_1$ in model B.
– them
Jan 6 '17 at 19:50
• Yes we would assume samples are independent. I'm not sure how you calculated the situation $B$ probabilities.
– Hogg
Jan 6 '17 at 19:59
• One more comment, to get the rough estimates you might consider apriori fixing half of the numbers say $[0,2^{16}]$ and then count the number of values you expect to see out of this set, under situation A and under situation B (where the half of the numbers with lower probability are chosen uniformly from $[0, 2^{32}]$ ) if you observe one value out of the stream, then two values, and so on...
– them
Jan 6 '17 at 20:10
• @them It is not really an efficient approach. If you don't know which are the "high probability" numbers, for any given set you fix, about half of it will be high probability and half low probability ... and the cardinality of the intersection of this set with any given slice of the datastream will be the same you'd have seen if all the numbers had been equiprobable, until you start seeing lots of repetitions. That's what this problem is really after: repetitions in the datastream, which are more common if there is an uneven distribution of probabilities. Jan 10 '17 at 8:04

An sms/tweet answer: you need less than $4$ million samples, and less than $2$ seconds on most modern computing platforms (including some smartphones), without sophisticated algorithms.

Below, there's a more complete answer, in six parts. The first part provides a rough intuition of what one may want to do. The second part details an implementation that attempts to strike a balance between simplicity and efficiency. The third part provides a rigorous proof of correctness (essentially applying Chernoff bounds to the probability of an incorrect answer). The fourth part estimates the time and computational resources required. The fifth part describes some optimizations that are not really worth the effort in this case, but may be for different problem parameters. The sixth part looks at how things change if you change the parameters of the problem, e.g. increasing the keyspace, changing the probability distributions, or reducing the available memory.

1 A ROUGH INTUITION

The basic idea is to choose a data structure that

a) can be “filled” with values just once and then is both b) compact and c) efficient to search for the presence or absence of a given value.

The simplest implementation would probably be a sorted array: in $2^{20}$ bytes one can store $2^{18}$ values, and check the presence or absence in the array of any given value with just $18$ comparisons through binary search.

After filling the data structure, one then simply checks if each further value produced by the data stream is present or absent. Informally, if the data structure holds $k$ values, and every $32$-bit value has a probability $2^{-32}$ of appearing in a given sample, then the probability of finding that value in the data structure is $k 2^{-32}$. If, on the other hand, half the values have a probability $\approx 2^{-31}$ of appearing in a given sample, and half the values have a probability $\approx 2^{-41}$, then almost all the values in the data structure are “high probability” ones and the probability that each new sample will be present in the data structure is $\approx k 2^{-31}$, i.e. about twice that for the case of equiprobable values. Then, determining whether values are equiprobable or not is simply a matter of taking a “sufficiently large” number of samples $s$, checking for each if a collision occurs, i.e. if the item is present in the data structure, and determining if the number of collisions is closer to $s k 2^{-32}$ (what we’d expect with equiprobable values) or to $s k 2^{-31}$ (what we’d expect otherwise).

What is a "sufficiently large" number of samples? A simple Chernoff bound shows that if the probability gap between the "high probability" and "low probability" values is sufficiently large (as in this case), you need of the order of $30 \frac{values in keyspace}{values in data structure} \log_e\frac{1}{error probability}$ samples (in addition to the samples needed to fill your data structure in the first place). The simple algorithm below keeps about $0.25$M samples in the data structure, and needs to look at about $3.4$M additional samples.

Note that to a first approximation, the number of samples one has to take to achieve a given confidence is inversely proportional to the number of values in the data structure. That’s why requisite b) (a “compact” data structure) is important: a data structure with twice the capacity requires half the samples. Of course, one has to process samples sufficiently quickly to keep up with the data stream. That’s why requisite c) (efficiency in determining whether a given sample produced a collision) is also important.

2 A SIMPLE IMPLEMENTATION

This is the pseudocode of a simple implementation of the scheme above. Formatting is ugly, but it should be legible and very easy to recode in your favourite programming language.

1. $CAPACITY=2^{20}$ bytes/($4$ bytes/value)=$2^{18}$ values
2. $CONFIDENCE = 0.999$
• N.B. does not work for arbitrarily high CONFIDENCE, see analysis below
3. allocate new array, capable of holding CAPACITY values
4. for $i=0…CAPACITY-1$ array[i]=new_sample from datastream
5. sort array
6. $duplicates = 0$
7. for $i=1…CAPACITY$ if (array[i]==array[i-1]) then $duplicates=duplicates+1$
8. if $duplicates > CAPACITY/2$ restart from line 4!

9. $SAMPLES=\frac{30 \cdot 2^{32} \cdot \log_e(\frac{1}{1-CONFIDENCE})}{CAPACITY-duplicates}$

10. $THRESHOLD=40 \cdot \log_e(\frac{1}{1-CONFIDENCE})$

11. $collisions = 0$

12. for $i=0…SAMPLES$
12a. sample = new_sample from datastream
12b. if (array contains sample) then collisions=collisions+1
13. if ($collisions \leq THRESHOLD$) then return EQUIPROBABLE else return NOT_EQUIPROBABLE

Note: To check if array contains sample just use binary search:

1. $left=0$, $right=CAPACITY-1$
2. while ($left<right$)

2a. $middle =$ round_down$((left+right)/2)$

2b. if $(sample > array[middle])$ then $left = middle+1$ else $right = middle$

3. if array[left]==value return TRUE else return FALSE

3 TERMINATION AND CORRECTNESS

Note that the algorithm eventually terminates with an answer: the only possible loop is the "restart" on line 8, if more than half the values in memory are duplicates of some other value: this occurs with probability less than $1/2$ if the values stored in memory are less than $2^{30}$ (and with a really really tiny probability if they are $2^{18}$).

All we have to prove is that the probability of an incorrect answer is at most $1-CONFIDENCE$. In other words, we want to prove that the probability that $collisions > THRESHOLD$ if the values are equiprobable, and the probability that $collisions \leq THRESHOLD$ if the values are not equiprobable, are both at most $1-CONFIDENCE$.

To prove it, we use the simple Chernoff bound stating that

if the sum of an arbitrary number of independent random variables with values in $[0,1]$ has expectation $\mu$ then for any $\delta\in(0,1]$ the probabilities that the actual sum

a. is at least $(1+\delta)\mu$

b. is at most $(1-\delta)\mu$

are both no more than $e^{-\frac{\delta^2}{3}\mu}$.

Informally, since in one case we expect an average number of collisions that is about twice the other, we've chosen a threshold somewhere between the two, and a number of samples that ensures a number of collisions $\mu$ proportional to the logarithm of the inverse of the "tolerable" probability of error.

If the values are equiprobable, then it’s easy to see that the number of expected collisions is $expect_{equiprobable}=SAMPLES \cdot (CAPACITY -duplicates) \cdot 2^{-32}=30 \log_e(\frac{1}{1-CONFIDENCE})$. The probability that it's at least $(1+\frac{1}{3})$ times this value, i.e. at least THRESHOLD, is at most $e^{-\frac{(1/3)^2}{3}expect_{equiprobable}}<e^{- \log_e(\frac{1}{1-CONFIDENCE})}=1-CONFIDENCE$, and we are fine.

If the values are not equiprobable, let’s consider only collisions between “high probability” values and show that even just those alone are very likely to exceed THRESHOLD. We expect a fraction more than $1-2^{-10}$ of all values in the array, and of all subsequent samples from the data stream, to be “high probability” values. The probability that the fraction of “low probability” values exceeds $(1+\delta) 2^{-10}$ is in each case is at most, respectively, $e^{-\frac{\delta^2}{3}(2^{-10})(CAPACITY-duplicates)}$ and $e^{-\frac{\delta^2}{3}(2^{-10})(SAMPLES)}$. Taking the first of the two values (which is higher) and doubling it gives for e.g. $\delta=1$ a bound on the probability of having more than twice the expected number of “low probability” values. This probability is at most $p_{lowval}=2\cdot e^{-\frac{2^7}{12}}<0.00005$. If this is not the case, we are going to have at least $(1-2^{-9})(CAPACITY-duplicates)$ “high probability” values in our array, and at least $(1-2^{-9})SAMPLES$ “high probability” values amoung our samples. The expected number of collisions between “high probability” values is then at least $expect_{notequiprobable}=(1-2^{-9})^2 SAMPLES (CAPACITY-duplicates) 2^{-31}=60 (1-2^{-9})^2 \log_e(\frac{1}{1-CONFIDENCE})>\frac{3}{2} (1-2^{-8}) THRESHOLD$. Applying again the Chernoff bound with $1-\delta=\frac{1}{(3/2)(1-2^{-8})}$, and remembering to add the probability of more than twice the expected number of "low probability values" $p_{lowval}$ computed above, after some tedious number crunching we find again that the probability that not equiprobable samples yield more than $THRESHOLD$ collisions is at most $1-CONFIDENCE$; so again we are fine.

4 PERFORMANCE ANALYSIS

On most modern computing platforms (including laptops, and possibly some high performance smartphones) a relatively straightforward coding of the pseudocode above will need less than $4$ million samples and less than $2$ seconds to determine an answer with the desired confidence. To see this, note that the algorithm above and in fact any algorithm for similar tasks requires:

a) a preprocessing phase in which you “fill” the data structure

b) a certain number of additional samples from the datastream and

c) a certain amount of processing for each of these samples. If the processing time/sample is less than the time between samples, your algorithm will be waiting between one sample and the next; if it is greater, all that will happen is that you’ll be forced to discard a fraction of samples from the datastream, and take proportionally longer.

Let’s look at the terms individually.

a) In terms of preprocessing, the algorithm simply fills the memory with values, sorts them, and counts duplicates. Modern sorting routines can easily sort integers at well over $10MB/s$; one more pass to count duplicates takes almost negligible time, filling your $2^{20}$ bytes of data structure will take less than $0.2s$ (i.e. $0.1s$ to read, less than $0.1$s to process).

b) In terms of subsequent samples, for the desired accuracy you need $30 \cdot 2^{14} \cdot log_e (1/0.001)<3.4M$ samples. Your datastream produces them at $2.5M$/second, so if you can keep up with the datastream, you’ll need less than $1.5$ seconds.

c) Let’s look at the processing time for each sample (after initializing the data structure)! For each sample, the algorithm requires only $18$ lookups and comparisons/sample, plus possibly increasing two counters (collisions and samples seen) by $1$ each. On most modern computers (in fact, even on some smartphone processors) the entire $2^20$ bytes of the data structure fits in the processor cache, and performing those operations probably takes, in total, no more than $200$ cycles - so on a $1Ghz$ processor you can process $5$ million values per second, which is $20$ Mbytes/second or twice what you need.

5 FURTHER OPTIMIZATIONS

To a first approximation, if you can double the number of samples in your data structure, you can halve the number of samples you need afterwards.

One way to increase the number of values stored is to use some compression scheme; a simple one that works really well is to note that you can group your data into $2^{16}$ "bins" each with the same $2^{16}-bit$ prefix, which you then do not need to write down. This "reorganization" is almost free with most efficient sorting schemes, does not significantly slow down search (in fact, it can speed it up if for each "bin" you annotate the start and end position, and you quickly "zoom" to it) and saves you almost a factor $2$ in space (and thus in samples).

A more sophisticated scheme involves using data structures such as Bloom filters, or the even better [i]Cuckoo filters[/i], that only return a probabilistic answer to the query "have I seen this value before"? After all, since you are looking at two distributions that yield collisions differing by almost a factor $2$, a small fraction of errors only minimally increases the number of samples needed to separate them. If you are aiming for a $5\%$ error in your collision count, for example, you can do it with no more than $1$ byte/value. The problem with these solutions is that they are more complex to code, more complex to analyze, and ultimately more computationally onerous (for the current parameters of the system) than the simple "compressed-store-and search" above, so depending on your computing platform you may not manage to keep up with the data stream.

Note that with the current implementation and in the current scenario you have about $0.25$M values in the data structure, and about $3.4M$ subsequent samples (which is a bound that's loose, but not too loose), so you are already relatively close to the optimum, which would be a little less than $1M$ samples in the data structure and a little less than $1M$ subsequent samples - saving you about a factor $2$ in terms of time and samples. It's probably not worth the effort; but it may be if you change the parameters of the problem (see below).

6 CHANGING THE PARAMETERS OF THE PROBLEM

What if you want more confidence, or you have less space, or a larger key-space? Or if you have less memory?

In terms of accuracy, note that the amount of samples needed is roughly proportional to the logarithm of the inverse of the "tolerable" error rate. So if you want $0.999 999$ confidence you'll need twice the samples needed for $0.999$ confidence. In other terms, you can grow your confidence very quickly by adding few samples. This is not a property of the scheme above, but intrinsic to the fact that you are counting sums of independent variables.

What if one changes the probability of "low probability" values, or their fraction? Ultimately, what really counts is the expected number of collisions in the two cases. Remember the Chernoff bound above, which is not tight but reasonably so: your probability of erring on this count by a factor $1+\delta$ or $1-\delta$ is $e^{-\frac{\delta^2}{3}\mu}$, where $\mu$ is the expected number of collisions. If your two distributions produce a number of collisions within a factor $1+\epsilon$ of each other, you'll need to make $\delta\approx \epsilon/2$, and the number of samples you need will grow quadratically with $1/\epsilon$ (i.e. halve $\epsilon$, quadruple the number of samples needed). If the distribution with the highest collision count sees its collision count fall, $\mu$ will fall, and no matter how few collisions the other distribution has, your number of samples will have to increase, inversely proportional with $\mu$.

This also tells you what happens if your keyspace grows (e.g. from $2^{32}$ possible values to $2^{64}$ possible values): the number of collisions proportionally decreases, and the number of samples needed proportionally increases. E.g. moving up to $2^{64}$ possible values increases the number of needed samples by a huge factor $2^{32}$.

It also tells you what happens if you reduce the available memory - again, your proportionally decrease the number of collisions, and proportionally increase the number of samples: half the memory, half the collisions, twice the samples.

In situation $A$, we're drawing from $2^{32}$ possible values. To a first approximation, in situation $B$, we're drawing from $2^{31}$ possible values.

The only reasonable thing to do is to count collisions.

# Option A: Fill memory with samples, then count

Read the first $2^{18}$ values and store them in memory. To a first approximation, they are all different.

Now read $N$ more values, and increment a counter whenever the value collides with one of the values we've stored. The final counter $c$ is approximately a Poisson distribution. In situation $A$, its mean and variance are equal to $2^{-14}N$, and in situation $B$, its mean and variance are equal to $2^{-13}N$. Let us guess $A$ if $c<x$ and $B$ if $c>x$ for some decision boundary $x$.

We would like a wrong guess to be equally unlikely in both cases, so we want $c$ to be a bit closer to the expectation of $A$, but since everything is "rough", let's put it halfway between the means, at a distance of $2^{-15}N$. Now if we want that to be at least three standard deviations away, with a worst-case variance of $2^{-13}N$, then $$2^{-15}N\geq3\sqrt{2^{-13}N}$$ $$2^{-30}N^2\geq9\times2^{-13}N$$ $$N \geq 9\times2^{17}= 1179648$$ so we need about 1.2 million values. Adding that to the original $2^{18}$, we need about 1.5 million values in all. You can get away with less, if you calculate a better decision boundary, but since we're getting 10 MB/sec, it's not really worth the added effort.

Actually, I doubt that this algorithm can keep up with the data stream; it's probably limited by the time it takes to search the storage for the incoming values.

# Option B: Bloom filter

A Bloom filter would be both faster and more space-efficient, allowing for a better balance between the storage phase and the test phase. It would also allow us to combine the two phases, testing for collisions with every addition. You could probably get under half a million values if you're clever about it. You'd want to tune the filter error rate carefully.


text:
Complete graph edge colouring in two colours: lower bound for number of monochromatic triangles

Say $K_n$ is a complete graph. Show that any coloring of edges of $K_n$ with $n \ge 6$ in two colors contains at least $$\frac1{20}\binom{n}3$$ monochromatic triangles. Any ideas on how to use Ramsey theory to solve this?

Any help is much appreciated!

-

Let $K_n$ be 2-colored. A bi-angle is a configuration of two edges with different colors meeting at a vertex. If $n=2m+1$ then the greatest possible number of bi-angles involving any particular vertex is $m^2$, so the total number of bi-angles in the coloring is at most $(2m+1)m^2$. The number of triangles in $K_n$ is $n\choose3$. Each nonmonochromatic triangle uses up 2 bi-angles, so the number of nonmonochromatic triangles is at most $(2m+1)m^2/2$. This gives you a lower bound on the number of monochromatic triangles, in the $n$ odd case. A simple modification handles the $n$ even case.
thanks for the lead! however, i don't understand why $$\frac1{20}\binom{n}3$$ = $binom{n}3$ - (2m+1)m^2/2 –  Gogol Nov 4 '12 at 23:15
so, clearly there are $n\choose3$ triangles in $K_n$. I'm still not sure how we get the bi-angles number. is there no way to use Ramsey's theorem to prove this? Please help me –  Gogol Nov 5 '12 at 1:02
I don't know whether there is a way to use Ramsey theory here. For the bi-angles, let's look at $n=7$. Each vertex has 6 edges. Some are red, some are blue. How many ways are there to choose 2 of these 6 edges, the two being different colors (that is, how many bi-angles are there at that vertex)? Well, that depends on how many of the 6 edges are red, and how many are blue. But which way of coloring those 6 edges would result in the greatest possible number of bi-angles at that vertex? And how many bi-angles would there then be at that vertex? And if there were the same number of biangles... –  Gerry Myerson Nov 5 '12 at 1:49
(continued)...at each vertex, how many would there be in all? And what if instead of 7 vertices, there were $2m+1$? So, that's how you get the number of bi-angles. –  Gerry Myerson Nov 5 '12 at 1:50
Then you want to prove $${n\choose3}-{(2m+1)m^2\over2}\ge{1\over20}{n\choose3}$$ and that's just a simple inequality where you multiply everything out and see what you get. –  Gerry Myerson Nov 5 '12 at 1:52


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# Regular Polyhedrons

In $\mathbb{R}^3$, there are five regular polyhedrons (up to similarity), and can be parametrized by number of vertices, edges and faces. What is the number of regular polyhedrons in $\mathbb{R}^n$, and their parametrization? Please suggest the reference(s) also. (Thanks in advance.)

-
There is a pretty decent article here which gives the basic facts and pointers to references. –  Old John Aug 6 '12 at 8:31

In short, what happens is the following. The $n$-dimensional analogue of a Platonic solid is called a regular polytope. In any dimension you are guaranteed three "boring" regular polytopes: the $n$-dimensional version of the tetrahedron (the $n$-simplex), the $n$-dimensional hypercube, and its dual, the $n$-dimensional version of the octahedron. In three dimensions, as you know, there are two others. In four dimensions there are others as well, called the 24-cell, 120-cell and 600-cell. In dimensions five and above the boring regular polytopes are the only ones that exist.

The wiki pages http://en.wikipedia.org/wiki/Regular_polytopes and http://en.wikipedia.org/wiki/Convex_regular_4-polytope are good places to start. Coxeter's book Regular Polytopes is very comprehensive. Another approach is to look at these things through their reflection symmetry groups: Coxeter's book is a good source for this too, see also http://en.wikipedia.org/wiki/Coxeter_group

-

Just a comment, but "comments" don't support gif's: The Tesseract

$\hskip2.1in$

-
... don't get hypnotized! –  draks ... Aug 6 '12 at 8:47


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# Radial wave function matrix elements

For hydrogen atom radial wave function is the analytic form of the matrix elements, $$\langle n'\ell'|r^k|n\ell\rangle,$$ known? I am especially interested in $$k=-2$$ and $$k=-3$$.

Notation: $$|n\ell\rangle=2^{\ell+1} e^{-\frac{r}{a n}} \sqrt{\frac{(-\ell+n-1)!}{a^3 n^4 (\ell+n)!}} \left(\frac{r}{a n}\right)^\ell L_{-\ell+n-1}^{2 \ell+1}\left(\frac{2 r}{a n}\right)$$ and, $$\langle n'\ell'|r^k|n\ell\rangle=\int_0^\infty 2^{\ell^\prime+1} e^{-\frac{r}{a n^\prime}} \sqrt{\frac{(-\ell^\prime+n^\prime-1)!}{a^3 n^{\prime4} (\ell^\prime+n^\prime)!}} \left(\frac{r}{a n^\prime}\right)^{\ell^\prime} L_{-\ell^\prime+n^\prime-1}^{2 \ell^\prime+1}\left(\frac{2 r}{a n^\prime}\right)2^{\ell+1} e^{-\frac{r}{a n}} \sqrt{\frac{(-\ell+n-1)!}{a^3 n^4 (\ell+n)!}} \left(\frac{r}{a n}\right)^\ell L_{-\ell+n-1}^{2 \ell+1}\left(\frac{2 r}{a n}\right)r^kr^2dr$$

• This paper has a formula for the case when $n=n’$. – G. Smith May 10 '20 at 16:35


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# Instability of a 2nd order differential equations

I have a system of $$N$$ coupled differential equations that are numerically integrated in time. Each equation $$i$$ is coupled to the other equations $$(j,k,l,etc...)$$ by

$$\ddot{x}_i = -\sum_iK_ix_i - \sum_{jk}K_{ijk}x_jx_k-\sum_{jkl}K_{ijkl}x_jx_kx_l - \cdots$$

The only constraint is that $$K_i > 0$$. The constants $$K_{ijk}$$, $$K_{ijkl}$$, etc. can be anything, but I'm looking for values such that the system is numerically "stable" in the sense that values of $$x_i$$ oscillate about a small value given some small perturbation as an initial condition.

Given random intitial conditions of $$x_i$$, I numerically integrate this equation in time. I notice some peculiarities that lead to questions:

1. The system is always "stable" in the sense that values of $$x_i$$ oscillate about $$0$$ if only the first term on the right hand side, $$-\sum_iK_ix_i$$, is retained. Why do higher order terms cause instability?
2. Are there nonzero constants $$K_{ijk}$$, $$K_{ijkl}$$ that ensure stability?
3. If so, how do you find these constants?
4. Is there any field of math that studies the dynamical "stability" of systems of coupled equations like these?


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# Molecular and Ionic Equations

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# Molecular and Ionic Equations

“Just a little rain...”

One of the unfortunate by-products of our industrialized society is acid rain. Sulfur dioxide from burning coal and nitric oxides from vehicle emissions both form acids. When these acids react with limestone (calcium carbonate), reactions occur that dissolve the limestone and release water and carbon dioxide. Over a period of time, serious damage is caused to the structures.

### Molecular and Ionic Equations

When ionic compounds are dissolved into water, the polar water molecules break apart the solid crystal lattice, resulting in the hydrated ions being evenly distributed through the water. This process is called dissociation and is the reason that all ionic compounds are strong electrolytes. When two different ionic compounds that have been dissolved in water are mixed, a chemical reaction may occur between certain pairs of the hydrated ions.

Consider the double-replacement reaction that occurs when a solution of sodium chloride is mixed with a solution of silver nitrate.

$\text{NaCl}(aq)+\text{AgNO}_3 (aq) \rightarrow \text{NaNO}_3 (aq)+\text{AgCl}(s)$

The driving force behind this reaction is the formation of the silver chloride precipitate (see below)

The white silver chloride precipitate instantly forms when a solution of silver nitrate is added to a solution of sodium chloride.

This is called a molecular equation. A molecular equation is an equation in which the formulas of the compounds are written as though all substances exist as molecules. However, there is a better way to show what is happening in this reaction. All of the aqueous compounds should be written as ions because they are present in the water as separated ions because of their dissociation.

$\text{Na}^+(aq)+\text{Cl}^-(aq)+\text{Ag}^+(aq)+\text{NO}^-_3 (aq) \rightarrow \text{Na}^+ (aq)+\text{NO}^-_3 (aq)+\text{AgCl}(s)$

This equation is called an ionic equation , an equation in which dissolved ionic compounds are shown as free ions.

Some other double-replacement reactions do not produce a precipitate as one of the products. The production of a gas and/or a molecular compound such as water may also drive the reaction. For example, consider the reaction of a solution of sodium carbonate with a solution of hydrochloric acid (HCl). The products of the reaction are aqueous sodium chloride, carbon dioxide, and water. The balanced molecular equation is:

$\text{Na}_2 \text{CO}_3 (aq)+2\text{HCl}(aq) \rightarrow 2\text{NaCl}(aq)+\text{CO}_2(g)+\text{H}_2\text{O}(l)$

The ionic equation is:

$2\text{Na}^+(aq)+\text{CO}^{2-}_3 (aq)+2\text{H}^+ (aq)+2\text{Cl}^-(aq) \rightarrow 2\text{Na}^+(aq)+2\text{Cl}^-(aq)+\text{CO}_2 (g)+\text{H}_2\text{O}(l)$

A single-replacement reaction is one in which an element replaces another element in a compound. An element is in either the solid, liquid, or gas state and is not an ion. The example below shows the reaction of solid magnesium metal with aqueous silver nitrate to form aqueous magnesium nitrate and silver metal.

Balanced molecular equation:

$\text{Mg}(s)+2\text{AgNO}_3 (aq) \rightarrow \text{Mg(NO}_3)_2 (aq)+2\text{Ag}(s)$

Ionic equation:

$\text{Mg}(s)+2\text{Ag}^+ (aq)+2\text{NO}^-_3 (aq) \rightarrow \text{Mg}^{2+}(aq)+2\text{NO}^-_3 (aq)+2\text{Ag}(s)$

This type of single-replacement reaction is called a metal replacement. Other common categories of single-replacement reactions are hydrogen replacement and halogen replacement.

#### Summary

• Examples of molecular and ionic equations are shown.

#### Practice

Read the material at the link below and solve the practice equations:

#### Review

1. What do the small letters after the compounds indicate?
2. Why is it important to show the state of each compound?
3. What do ionic equations tell us?


text:
Stokers

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It aint exactly arctic like in the engineering spaces while in the Persian Gulf
by LABBATTS50 on Dec 30, 2005

Stokers!

by LABBATTS50 on Dec 30, 2005

Where is the EZ Glide jar?

by IkeaMan on Dec 30, 2005

Beats working in the ship's office!

by Ice_man on Dec 30, 2005

Hairy bags @ home...miss the good old days.

by ShepherdsDog on Dec 30, 2005

1 in 2 watch in that heat and noise, no thanks.

by RoyalHighlander on Dec 30, 2005

Was an oiler for 2 years myself...At least in Dec you were warm and cozy...lol

by rcnsailor on Dec 31, 2005

life as a hairy bag ...

by Hester on Dec 31, 2005

I haven't seen the engineering dept stand 1 in 2 since 96. For two months. Straight back to 1 in 3 without turning to.

by ShepherdsDog on Jan 1, 2006

1 in 4 was ideal. I think that's what the hull techs did when I was on the Yukon(2 squadron did this as well for all trades, lucky harbour patrol pricks). The stokers had a 1in 2 and the rest of us had a 1 in 3, unless you were an OD or an Officer Cadidiot. I loved ADRs though.

by Wullu on Jan 1, 2006

They got it easy now. Steaming 1 in 4 and not turning to. Course ya need a good Chief ERA to get that to stick.

by Nuggie77 on Feb 18, 2010

There is no 1 in 4 now, not enough people and too many taskings.

by Heavy_Metal on Feb 19, 2010

bunch-o-seamen XD

by stokes on Feb 23, 2010

Looks like the Aft Engine room by #2 LP Air compressor and #3 chiller, I miss being at sea especially now that I am in recruiting.

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text:
## The binary Be star $δ$ Scorpii at high spectral and spatial resolution : II The circumstellar disk evolution after the periastron    [PDF]

Anthony Meilland, Philippe Stee, Alain Spang, Fabien Malbet, Fabrizio Massi, Dieter Schertl
Classical Be stars are hot non-supergiant stars surrounded by a gaseous circumstellar disk that is responsible for the observed infrared (IR) excess and emission lines. The influence of binarity on these phenomena remains controversial. We followed the evolution of the environment surrounding the binary Be star $\delta$ Scorpii one year before and one year after the 2011 periastron to check for any evidence of a strong interaction between its companion and the primary circumstellar disk. We used the VLTI/AMBER spectro-interferometric instrument operating in the K band in high (12000) spectral resolution to obtain information on both the disk geometry and kinematics. Observations were carried out in two emission lines: Br$\gamma$ (2.172\,$\mu$m) and $\ion{He}{i}$ (2.056\,$\mu$m). We detected some important changes in $\delta$ Scorpii's circumstellar disk geometry between the first observation made in April 2010 and the new observation made in June 2012. During the last two years the disk has grown at a mean velocity of 0.2\,km\,s$^{-1}$. This is compatible with the expansion velocity previously found during the 2001-2007 period. The disk was also found to be asymmetric at both epochs, but with a different morphology in 2010 and 2012. Considering the available spectroscopic data showing that the main changes in the emission-line profiles occurred quickly during the periastron, it is probable that the differences between the 2010 and 2012 disk geometry seen in our interferometric data stem from a disk perturbation caused by the companion tidal effects. However, taking into account that no significant changes have occurred in the disk since the end of the 2011 observing season, it is difficult to understand how this induced inhomogeneity has been "frozen" in the disk for such a long period.
View original: http://arxiv.org/abs/1301.1842


text:
# 5.16 Digital signal processing problems

 Page 1 / 6
(Blank Abstract)

## Sampling and filtering

The signal $s(t)$ is bandlimited to 4 kHz. We want to sample it, but it has been subjected to various signal processingmanipulations.

1. What sampling frequency (if any works) can be used to sample the result of passing $s(t)$ through an RC highpass filter with $R=10\mathrm{k\Omega }$ and $C=8\mathrm{nF}$ ?
2. What sampling frequency (if any works) can be used to sample the derivative of $s(t)$ ?
3. The signal $s(t)$ has been modulated by an 8 kHz sinusoid having an unknown phase: the resultingsignal is $s(t)\sin (2\pi {f}_{0}t+\phi )$ , with ${f}_{0}=8\mathrm{kHz}$ and $\phi =?$ Can the modulated signal be sampled so that the original signal can be recovered from the modulated signal regardless of the phase value $\phi$ ? If so, show how and find the smallest sampling rate that can be used; if not,show why not.

## Non-standard sampling

Using the properties of the Fourier series can ease finding a signal's spectrum.

1. Suppose a signal $s(t)$ is periodic with period $T$ . If ${c}_{k}()$ represents the signal's Fourier series coefficients, what are the Fourier seriescoefficients of $s(t-\frac{T}{2})$ ?
2. Find the Fourier series of the signal $p(t)$ shown in [link] .
3. Suppose this signal is used to sample a signal bandlimited to $\frac{1}{T}\mathrm{Hz}$ . Find an expression for and sketch the spectrum of the sampled signal.
4. Does aliasing occur? If so, can a change in sampling rate prevent aliasing;if not, show how the signal can be recovered from these samples.

## A different sampling scheme

A signal processing engineer from Texas A&M claims to have developed an improved sampling scheme. He multiplies the bandlimited signal by the depicted periodic pulse signal to perform sampling ( [link] ).

1. Find the Fourier spectrum of this signal.
2. Will this scheme work? If so, how should ${T}_{S}$ be related to the signal's bandwidth? If not, why not?

## Bandpass sampling

The signal $s(t)$ has the indicated spectrum.

1. What is the minimum sampling rate for this signal suggested by the Sampling Theorem?
2. Because of the particular structure of this spectrum, one wonders whether a lower sampling ratecould be used. Show that this is indeed the case, and find the system that reconstructs $s(t)$ from its samples.

## Sampling signals

If a signal is bandlimited to $W$ Hz, we can sample it at any rate $\frac{1}{{T}_{s}}> 2W$ and recover the waveform exactly. This statement of the Sampling Theorem can be taken to mean that allinformation about the original signal can be extracted from the samples. While true in principle, you do haveto be careful how you do so. In addition to the rms value of a signal, an important aspect of a signal isits peak value, which equals $\max\{\left|s(t)\right|\}$ .

1. Let $s(t)$ be a sinusoid having frequency $W$  Hz. If we sample it at precisely the Nyquist rate, how accurately do thesamples convey the sinusoid's amplitude? In other words, find the worst case example.
2. How fast would you need to sample for the amplitude estimate to be within 5% of the truevalue?
3. Another issue in sampling is the inherent amplitude quantization produced by A/D converters. Assume themaximum voltage allowed by the converter is ${V}_{\mathrm{max}}()$ volts and that it quantizes amplitudes to $b$ bits. We can express the quantized sample $Q(s(n{T}_{s}))$ as $s(n{T}_{s})+\epsilon (t)$ , where $\epsilon (t)$ represents the quantization error at the ${n}^{\mathrm{th}}()$ sample. Assuming the converter rounds, how large is maximum quantization error?
4. We can describe the quantization error as noise, with apower proportional to the square of the maximum error. What is the signal-to-noise ratio of thequantization error for a full-range sinusoid? Express your result in decibels.

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
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text:
## Putnam, Hilary Whitehall

Compute Distance To:
 Author ID: putnam.hilary-w Published as: Putnam, Hilary; Putnam, H.; Putnam, Hilary W. External Links: MGP · Wikidata · Math-Net.Ru · dblp · GND · IdRef
 Documents Indexed: 41 Publications since 1957, including 3 Books 3 Contributions as Editor · 10 Further Contributions Biographic References: 2 Publications Co-Authors: 19 Co-Authors with 24 Joint Publications 382 Co-Co-Authors
all top 5

### Co-Authors

 23 single-authored 7 Davis, Martin David 3 Hensel, G. 3 Quine, Willard Van Orman 2 Baaz, Matthias 2 Boolos, George S. 2 Finkelstein, David Ritz 2 Gudder, Stanley P. 2 Harper, Charles L. jun. 2 Hooker, Clifford A. 2 Jauch, Josef Maria 2 Kreisel, Georg 2 Leeds, Stephen 2 Omodeo, Eugenio Giovanni 2 Papadimitriou, Christos Harilaos 2 Policriti, Alberto 2 Scott, Dana Stewart 2 von Neumann, John 2 Wang, Hao 1 Abernathy, Robert 1 Agassi, Joseph 1 Ayer, A. J. 1 Benacerraf, Paul 1 Bernays, Paul 1 Beth, Evert Willem 1 Birkhoff, Garrett 1 Boyd, Richard K. 1 Brouwer, Luitzen Egbertus Jan 1 Bub, Jeffrey 1 Capek, Milic 1 Carnap, Rudolf 1 Carson, Daniel F. 1 Catlin, Donald E. 1 Chao, Ren 1 Chomsky, Noam Avram 1 Cohen, I. Bernard 1 Cohen, Robert S. 1 Curry, Haskell Brooks 1 Davydov, G. V. 1 Dennett, Daniel C. 1 Dishkant, Herman 1 Dreben, Burton 1 Dummett, Michael Anthony Eardley 1 Dunham, Bradford 1 Enderton, Herbert B. 1 Feyerabend, Paul K. 1 Finch, Peter D. 1 Foulis, David James 1 Frege, Gottlob 1 Fridshal, R. 1 Friedman, Michael Lee 1 Fürth, Reinhold 1 Gelernter, Herbert L. 1 Gilmore, Paul C. 1 Gleason, Andrew Mattei 1 Gleason, H. A. jun. 1 Gödel, Kurt 1 Goodfield, June 1 Goodman, Nelson 1 Greechie, Richard J. 1 Grünbaum, Adolf 1 Grunstra, Bernard R. 1 Halle, Morris 1 Harary, Frank 1 Hardegree, Gary M. 1 Havas, Peter 1 Heelan, Patrick Aidan 1 Hempel, Carl Gustav 1 Herzberger, Hans G. 1 Heyting, Arend 1 Hilbert, David 1 Hiz, Henry 1 Hockett, Charles F. 1 Holland, Samuel Shaheen jun. 1 Jakobson, Roman 1 Kamber, A. 1 Kanger, Stig 1 Kochen, Simon 1 Kripke, Saul A. 1 Kronfli, N. S. 1 Lambek, Joachim 1 Logemann, George 1 Łoś, Jerzy 1 Loveland, Donald W. 1 Luckham, David C. 1 Lukas, Joan D. 1 Mandelbrot, Benoit B. 1 Markovic, Mihailo I. 1 Maslov, S. Yu. 1 Mayr, Ernst 1 Meltzer, Bernard 1 Mints, Grigoriĭ Efroimovich 1 Newell, Allen 1 North, J. H. 1 Oettinger, Anthony G. 1 Orevkov, V. P. 1 Petersen, Aage 1 Peterson, Gordon E. 1 Piron, Constantin 1 Poincaré, Henri 1 Pools, J. C. T. ...and 37 more Co-Authors
all top 5

### Serials

 7 The Journal of Symbolic Logic 3 Archiv für Mathematische Logik und Grundlagenforschung 3 Notre Dame Journal of Formal Logic 2 Fundamenta Mathematicae 2 Proceedings of the American Mathematical Society 2 Synthese 2 Transactions of the American Mathematical Society 2 Historia Mathematica 2 The University of Western Ontario Series in Philosophy of Science 1 American Mathematical Monthly 1 The British Journal for the Philosophy of Science 1 Dialectica 1 Illinois Journal of Mathematics 1 Journal of the Association for Computing Machinery 1 Logic Journal of the IGPL 1 Annals of Mathematics. Second Series 1 Proceedings of Symposia in Applied Mathematics 1 Synthese Library
all top 5

### Fields

 34 Mathematical logic and foundations (03-XX) 11 General and overarching topics; collections (00-XX) 11 History and biography (01-XX) 5 Quantum theory (81-XX) 3 Computer science (68-XX) 2 Number theory (11-XX) 1 Mechanics of particles and systems (70-XX) 1 Relativity and gravitational theory (83-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)

### Citations contained in zbMATH Open

41 Publications have been cited 832 times in 762 Documents Cited by Year
A computing procedure for quantification theory. Zbl 0212.34203
Davis, M.; Putnam, H.
1960
The decision problem for exponential diophantine equations. Zbl 0111.01003
Davis, Martin; Putnam, Hilary; Robinson, Julia
1961
Trial and error predicates and the solution to a problem of Mostowski. Zbl 0193.30102
Putnam, H.
1965
Philosophy of mathematics. Selected readings. 2nd ed. Zbl 0548.03002
1983
Eine Unableitbarkeitsbeweismethode für den intuitionistischen Aussagenkalkül. Zbl 0079.00702
Kreisel, Georg; Putnam, H.
1957
Models and reality. Zbl 0443.03003
Putnam, Hilary
1980
The logico-algebraic approach to quantum mechanics. Vol. I: Historical evolution. Zbl 0429.03045
1975
Degrees of unsolvability of constructible sets of integers. Zbl 0188.32701
Boolos, G.; Putnam, H.
1968
Mathematics, matter and method. Philosophical papers. Vol. 1. Zbl 0311.00035
Putnam, Hilary
1975
What is mathematical truth? Zbl 0325.02004
Putnam, Hilary
1975
Decidability and essential undecidability. Zbl 0078.24501
Putnam, Hilary
1957
Mathematics, matter and method. Philosophical papers, Vol. 1. 2nd ed. Zbl 0426.00022
Putnam, Hilary
1979
Peirce the logician. Zbl 0487.01012
Putnam, Hilary
1982
A note on the hyperarithmetical hierarchy. Zbl 0205.30803
Enderton, H. B.; Putnam, H.
1971
Recursively enumerable classes and their application to recursive sequences of formal theories. Zbl 0242.02046
Pour-El, Marian Boykan; Putnam, Hilary
1965
A philosopher looks at quantum mechanics (again). Zbl 1098.81011
Putnam, Hilary
2005
Diophantine sets over polynomial rings. Zbl 0113.00604
Davis, M.; Putnam, H.
1963
Automation of reasoning. 1: Classical papers on computational logic 1957–1966. Zbl 0567.03001
1983
Quantum logic, conditional probability, and interference. Zbl 0402.03016
Friedman, Michael; Putnam, Hilary
1978
Nonstandard models and Kripke’s proof of the Gödel theorem. Zbl 1005.03054
Putnam, Hilary
2000
What theories are not. Zbl 0147.24702
Putnam, H.
1962
The logico-algebraic approach to quantum mechanics. Vol. II: Contemporary consolidation. Zbl 0429.03046
1979
A recursion-theoretic characterization of the ramified analytical hierarchy. Zbl 0207.01203
Boyd, Richard; Hensel, G.; Putnam, H.
1969
Meaning as functional classification. (A perspective on the relation of syntax to semantics.) With comments by Daniel Dennett, Hilary Putnam, Saul Kripke and W. V. Quine. Zbl 0308.02012
Sellars, Wilfrid
1974
Philosophical papers. Vol. 2: Mind, language and reality. Zbl 0485.01026
Putnam, Hilary
1979
Structure of language and its mathematical aspects. Zbl 0111.16102
1961
Exact separation of recursively enumerable sets within theories. Zbl 0118.25201
Putnam, H.; Smullyan, R. M.
1960
A note on constructible sets of integers. Zbl 0192.04303
Putnam, H.
1963
How to think quantum-logically. Zbl 0338.02003
Putnam, Hilary
1974
After Gödel. Zbl 1111.03002
Putnam, Hilary
2006
On the notational independence of various hierarchies of degrees of unsolvability. Zbl 0137.00903
Hensel, G.; Putnam, H.
1965
Normal models and the field $$\sum_ 1^ *$$. Zbl 0193.30201
Hensel, G.; Putnam, H.
1969
An intrinsic characterization of the hierarchy of constructible sets of integers. Zbl 0234.02027
Leeds, Stephen; Putnam, Hilary
1971
On hierarchies and systems of notations. Zbl 0237.02011
Putnam, Hilary
1964
Kurt Gödel and the foundations of mathematics. Horizons of truth. Zbl 1253.00009
2011
On families of sets represented in theories. Zbl 0126.02101
Putnam, H.
1964
Paradox revisited. II: Sets – a case of all or none? Zbl 0986.03011
Putnam, Hilary
2000
Proceedings of the Boston colloquium for the philosophy of science 1966/1968. Zbl 0172.28503
1969
Recursive functions and hierarchies. Zbl 0268.02024
Putnam, Hilary
1973
Solution to a problem of Gandy’s. Zbl 0325.02028
Leeds, Stephen; Putnam, Hilary
1974
How to think quantum-logically. Zbl 0328.02003
Putnam, Hilary
1976
Kurt Gödel and the foundations of mathematics. Horizons of truth. Zbl 1253.00009
2011
After Gödel. Zbl 1111.03002
Putnam, Hilary
2006
A philosopher looks at quantum mechanics (again). Zbl 1098.81011
Putnam, Hilary
2005
Nonstandard models and Kripke’s proof of the Gödel theorem. Zbl 1005.03054
Putnam, Hilary
2000
Paradox revisited. II: Sets – a case of all or none? Zbl 0986.03011
Putnam, Hilary
2000
Philosophy of mathematics. Selected readings. 2nd ed. Zbl 0548.03002
1983
Automation of reasoning. 1: Classical papers on computational logic 1957–1966. Zbl 0567.03001
1983
Peirce the logician. Zbl 0487.01012
Putnam, Hilary
1982
Models and reality. Zbl 0443.03003
Putnam, Hilary
1980
Mathematics, matter and method. Philosophical papers, Vol. 1. 2nd ed. Zbl 0426.00022
Putnam, Hilary
1979
The logico-algebraic approach to quantum mechanics. Vol. II: Contemporary consolidation. Zbl 0429.03046
1979
Philosophical papers. Vol. 2: Mind, language and reality. Zbl 0485.01026
Putnam, Hilary
1979
Quantum logic, conditional probability, and interference. Zbl 0402.03016
Friedman, Michael; Putnam, Hilary
1978
How to think quantum-logically. Zbl 0328.02003
Putnam, Hilary
1976
The logico-algebraic approach to quantum mechanics. Vol. I: Historical evolution. Zbl 0429.03045
1975
Mathematics, matter and method. Philosophical papers. Vol. 1. Zbl 0311.00035
Putnam, Hilary
1975
What is mathematical truth? Zbl 0325.02004
Putnam, Hilary
1975
Meaning as functional classification. (A perspective on the relation of syntax to semantics.) With comments by Daniel Dennett, Hilary Putnam, Saul Kripke and W. V. Quine. Zbl 0308.02012
Sellars, Wilfrid
1974
How to think quantum-logically. Zbl 0338.02003
Putnam, Hilary
1974
Solution to a problem of Gandy’s. Zbl 0325.02028
Leeds, Stephen; Putnam, Hilary
1974
Recursive functions and hierarchies. Zbl 0268.02024
Putnam, Hilary
1973
A note on the hyperarithmetical hierarchy. Zbl 0205.30803
Enderton, H. B.; Putnam, H.
1971
An intrinsic characterization of the hierarchy of constructible sets of integers. Zbl 0234.02027
Leeds, Stephen; Putnam, Hilary
1971
A recursion-theoretic characterization of the ramified analytical hierarchy. Zbl 0207.01203
Boyd, Richard; Hensel, G.; Putnam, H.
1969
Normal models and the field $$\sum_ 1^ *$$. Zbl 0193.30201
Hensel, G.; Putnam, H.
1969
Proceedings of the Boston colloquium for the philosophy of science 1966/1968. Zbl 0172.28503
1969
Degrees of unsolvability of constructible sets of integers. Zbl 0188.32701
Boolos, G.; Putnam, H.
1968
Trial and error predicates and the solution to a problem of Mostowski. Zbl 0193.30102
Putnam, H.
1965
Recursively enumerable classes and their application to recursive sequences of formal theories. Zbl 0242.02046
Pour-El, Marian Boykan; Putnam, Hilary
1965
On the notational independence of various hierarchies of degrees of unsolvability. Zbl 0137.00903
Hensel, G.; Putnam, H.
1965
On hierarchies and systems of notations. Zbl 0237.02011
Putnam, Hilary
1964
On families of sets represented in theories. Zbl 0126.02101
Putnam, H.
1964
Diophantine sets over polynomial rings. Zbl 0113.00604
Davis, M.; Putnam, H.
1963
A note on constructible sets of integers. Zbl 0192.04303
Putnam, H.
1963
What theories are not. Zbl 0147.24702
Putnam, H.
1962
The decision problem for exponential diophantine equations. Zbl 0111.01003
Davis, Martin; Putnam, Hilary; Robinson, Julia
1961
Structure of language and its mathematical aspects. Zbl 0111.16102
1961
A computing procedure for quantification theory. Zbl 0212.34203
Davis, M.; Putnam, H.
1960
Exact separation of recursively enumerable sets within theories. Zbl 0118.25201
Putnam, H.; Smullyan, R. M.
1960
Eine Unableitbarkeitsbeweismethode für den intuitionistischen Aussagenkalkül. Zbl 0079.00702
Kreisel, Georg; Putnam, H.
1957
Decidability and essential undecidability. Zbl 0078.24501
Putnam, Hilary
1957
all top 5

### Cited by 1,015 Authors

 12 Matiyasevich, Yuriĭ Vladimirovich 8 Plaisted, David Alan 6 Bonacina, Maria Paola 6 Eisenträger, Kirsten 6 Friedman, Sy-David 6 Marques-Silva, João P. 6 Schaub, Torsten H. 6 Szeider, Stefan 6 Van Gelder, Allen 5 Arslanov, Marat M. 5 Becker, Bernd 5 Beyersdorff, Olaf 5 Chen, Jian-er 5 Kelly, Kevin T. 5 Miglioli, Pierangelo 5 Miller, Russell G. 5 Putnam, Hilary Whitehall 5 Sorbi, Andrea 5 Ternullo, Claudio 4 Ábrahám, Erika 4 Biere, Armin 4 Bueno, Otávio 4 de Moura, Leonardo 4 Dose, Titus 4 Eggers, Andreas 4 Giunchiglia, Enrico 4 Glaßer, Christian 4 Heule, Marijn J. H. 4 Hooker, John N. jun. 4 Itsykson, Dmitry M. 4 Jones, James P. 4 Kupferschmid, Stefan 4 Lauria, Massimo 4 Lê Văn Băng 4 Lempp, Steffen 4 Lynce, Inês 4 Myasnikov, Alexei G. 4 Nordström, Jakob 4 Pasten, Hector V. 4 Sabharwal, Ashish 4 Schulte, Oliver 4 Selman, Bart 4 Stephan, Frank 4 Teige, Tino 4 Yu, Liang 3 Anellis, Irving H. 3 Badaev, Serikzhan A. 3 Beigel, Richard 3 Bringsjord, Selmer 3 Bruni, Renato 3 Bryant, Randal E. 3 Chong, Chi Tat 3 Cook, Roy T. 3 Cooper, Stuart Barry 3 Ferrari, Mauro 3 Franco, John V. 3 Fränzle, Martin 3 Garreta, Albert 3 Gebser, Martin 3 Hirsch, Edward A. 3 Järvisalo, Matti 3 Jeroslow, Robert G. 3 Jonsson, Peter A. 3 Kanamori, Akihiro 3 Kaufmann, Benjamin 3 Kirousis, Lefteris Miltiades 3 Kleine Büning, Hans 3 Kratsch, Dieter 3 Kullmann, Oliver 3 Li, Chumin 3 Linnebo, Øystein 3 Nieuwenhuis, Robert 3 Omodeo, Eugenio Giovanni 3 Ovchinnikov, Denis 3 Pitassi, Toniann 3 Sanders, Sam 3 van Maaren, Hans 3 Vardi, Moshe Ya’akov 3 Visser, Albert 3 Wang, Jianxin 3 Wang, Jinchang 3 Welch, Philip D. 3 Xu, Chao 3 Yamaleev, Mars Mansurovich 3 Zakharyaschev, Michael Viktorovich 3 Zhao, Xishun 2 Achlioptas, Dimitris 2 Alekhnovich, Michael 2 Alexander, Samuel Allen 2 Antos, Carolin 2 Armando, Alessandro 2 Arrigoni, Tatiana 2 Badban, Bahareh 2 Barth, Dominik 2 Baxa, Christoph 2 Beame, Paul W. 2 Beck, Moritz 2 Benhamou, Belaid 2 Bernardi, Claudio 2 Bidoit, Nicole ...and 915 more Authors
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### Cited in 147 Serials

 53 Theoretical Computer Science 37 Journal of Automated Reasoning 35 The Journal of Symbolic Logic 26 Artificial Intelligence 21 Discrete Applied Mathematics 21 Synthese 21 Annals of Pure and Applied Logic 20 Journal of Philosophical Logic 19 Studia Logica 17 Annals of Mathematics and Artificial Intelligence 15 Transactions of the American Mathematical Society 14 The Bulletin of Symbolic Logic 11 History and Philosophy of Logic 11 Information and Computation 10 The Mathematical Intelligencer 9 Information Processing Letters 9 Proceedings of the American Mathematical Society 9 Archive for Mathematical Logic 8 Journal of Soviet Mathematics 8 Journal of Mathematical Sciences (New York) 8 Studies in History and Philosophy of Science. Part B. Studies in History and Philosophy of Modern Physics 7 Information Sciences 7 Annals of Operations Research 7 Erkenntnis 6 Notre Dame Journal of Formal Logic 6 Journal of Symbolic Computation 6 Foundations of Science 5 Archiv für Mathematische Logik und Grundlagenforschung 5 Journal of Computer and System Sciences 5 JETAI. Journal of Experimental & Theoretical Artificial Intelligence 5 European Journal of Operational Research 5 Foundations of Physics 5 Logica Universalis 4 Algebra and Logic 4 Applied Mathematics and Computation 4 Journal of Algebra 4 Formal Methods in System Design 4 Constraints 4 Theory and Practice of Logic Programming 4 The Review of Symbolic Logic 3 Acta Informatica 3 Computers & Mathematics with Applications 3 International Journal of General Systems 3 International Journal of Theoretical Physics 3 Israel Journal of Mathematics 3 Advances in Mathematics 3 SIAM Journal on Computing 3 Operations Research Letters 3 Journal of Computer Science and Technology 3 Computers & Operations Research 3 Logic and Logical Philosophy 3 Logical Methods in Computer Science 2 Archive for History of Exact Sciences 2 Discrete Mathematics 2 BIT 2 Computing 2 Journal of Number Theory 2 Siberian Mathematical Journal 2 Advances in Applied Mathematics 2 Mathematical Social Sciences 2 International Journal of Parallel Programming 2 Journal of the American Mathematical Society 2 Mathematical and Computer Modelling 2 Random Structures & Algorithms 2 Bulletin of the American Mathematical Society. New Series 2 RAIRO. Informatique Théorique et Applications 2 Mathematical Programming. Series A. Series B 2 Computational Complexity 2 Journal of Applied Non-Classical Logics 2 Mathematical Logic Quarterly (MLQ) 2 New Journal of Physics 2 International Journal of Applied Mathematics and Computer Science 2 Lobachevskii Journal of Mathematics 2 International Studies in the Philosophy of Science 2 Journal of Discrete Algorithms 2 Journal of Applied Logic 2 Discrete Optimization 2 Proceedings of the Steklov Institute of Mathematics 2 Axiomathes 2 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 1 Educational Studies in Mathematics 1 Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV) 1 Journal of the Franklin Institute 1 Mathematical Notes 1 Mathematical Proceedings of the Cambridge Philosophical Society 1 Mathematische Semesterberichte 1 Reviews in Mathematical Physics 1 Science & Education 1 Annali di Matematica Pura ed Applicata. Serie Quarta 1 Bulletin de la Société Mathématique de France 1 Fuzzy Sets and Systems 1 International Journal of Computer & Information Sciences 1 International Journal of Game Theory 1 Journal of the Association for Computing Machinery 1 Le Matematiche 1 Mathematische Annalen 1 Mathematics and Computers in Simulation 1 Mathematische Nachrichten 1 Mathematica Slovaca 1 Publications of the Research Institute for Mathematical Sciences, Kyoto University ...and 47 more Serials
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### Cited in 34 Fields

 417 Mathematical logic and foundations (03-XX) 363 Computer science (68-XX) 51 General and overarching topics; collections (00-XX) 49 Number theory (11-XX) 44 Operations research, mathematical programming (90-XX) 34 History and biography (01-XX) 23 Quantum theory (81-XX) 21 Combinatorics (05-XX) 15 Information and communication theory, circuits (94-XX) 8 Group theory and generalizations (20-XX) 8 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 7 Order, lattices, ordered algebraic structures (06-XX) 7 Field theory and polynomials (12-XX) 7 Algebraic geometry (14-XX) 7 Probability theory and stochastic processes (60-XX) 7 Statistics (62-XX) 6 Numerical analysis (65-XX) 3 Commutative algebra (13-XX) 3 Biology and other natural sciences (92-XX) 2 General algebraic systems (08-XX) 2 Category theory; homological algebra (18-XX) 2 Real functions (26-XX) 2 General topology (54-XX) 1 Associative rings and algebras (16-XX) 1 Measure and integration (28-XX) 1 Ordinary differential equations (34-XX) 1 Partial differential equations (35-XX) 1 Functional analysis (46-XX) 1 Convex and discrete geometry (52-XX) 1 Algebraic topology (55-XX) 1 Mechanics of particles and systems (70-XX) 1 Fluid mechanics (76-XX) 1 Relativity and gravitational theory (83-XX) 1 Systems theory; control (93-XX)

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# American Institute of Mathematical Sciences

January  2011, 15(1): 75-92. doi: 10.3934/dcdsb.2011.15.75

## Numerical simulations of diffusion in cellular flows at high Péclet numbers

 1 Department of Mathematics, University of Houston, Houston, TX 77204, United States 2 Department of Mathematics, Ajou University, Suwon 443-749, South Korea 3 Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, United States

Received  November 2009 Revised  April 2010 Published  October 2010

We study numerically the solutions of the steady advection-diffu-sion problem in bounded domains with prescribed boundary conditions when the Péclet number Pe is large. We approximate the solution at high, but finite Péclet numbers by the solution to a certain asymptotic problem in the limit Pe $\to \infty$. The asymptotic problem is a system of coupled 1-dimensional heat equations on the graph of streamline-separatrices of the cellular flow, that was developed in [21]. This asymptotic model is implemented numerically using a finite volume scheme with exponential grids. We conclude that the asymptotic model provides for a good approximation of the solutions of the steady advection-diffusion problem at large Péclet numbers, and even when Pe is not too large.
Citation: Yuliya Gorb, Dukjin Nam, Alexei Novikov. Numerical simulations of diffusion in cellular flows at high Péclet numbers. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 75-92. doi: 10.3934/dcdsb.2011.15.75
##### References:

show all references

##### References:
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2018 Impact Factor: 1.008


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## PWM Expansion

The PWM Expansion allows you to generate up to 16 different Pulse Width Modulated (PWM) signals to control anything from Servo Motors (servos), DC Motor speed, LED brightness, etc.

This Expansion communicates with the Omega using the I2C protocol, it’s I2C device address is 0x5a. If you’re curious, check out the article on I2C.

### The Hardware

The PWM Expansion has 16 channels that can output individual PWM signals to drive motors and servos. The expansion can either be powered through the dock, or through a DC barrel jack to enable driving of motors up to 6V.

### Connecting to a Dock

To use the PWM Expansion, plug it into a Dock that has Expansion header pins (Expansion Dock, Power Dock, Arduino Dock R2).

You can safely stack other Expansions on top of it. However, be mindful of wires that are connected to the header pins underneath.

Note: You may only have one PWM Expansion stacked onto an Omega at a time.

### The PWM Channels

The PWM Expansion has 16 channels (outputs) that can be controlled simultaneously. Each channel has the following male header pins from top to bottom:

• GND - ground
• Vcc - 5V output
• SIGNAL - the wire that carries the PWM signal
• This channel is marked with a white header pin.

### Connecting Servos

We’ve placed the PWM pins in the same order that most* servo motors have them, so you can plug them right in!

One exception we’ve found is Airtronics servos, which may have the Vcc and SIGNAL pins on the outside.

Take a look at your servo’s connector. Typically the GND wire is black; line it up with the black GND pin on the servo channel’s header. The SIGNAL wire will then plug into the white header pin on the bottom. The photo below shows a Hitec servo properly plugged into channel 1 (SIGNAL is the yellow wire).

Check out the PWM Expansion hooked up to the servos on our Spiderbot!

### The DC Barrel Jack Adapter

The Omega supplies enough power to move only one or two servos under light load. In order to power projects with a bunch of servos, we’ve included an onboard DC barrel jack connector that can be used to supply power to any connected servos.

It is not recommended to attempt to power more than 2 servos without an external power supply.

The DC voltage supplied through the barrel jack will be provided on the Vcc (and GND) pins. The PWM voltage will also be stepped up to this voltage.

We’ve safely tested up to 6V, so try not to go beyond that or you risk damaging your Expansion. To get more power at this stage, look for power supplies with higher current ratings rather than higher voltages.

Please check the datasheets of your servomotors for guidance on the power ratings of the power supply.

#### When used with the Arduino Dock

Note that the Omega cannot be powered through this connector and still requires its own power supply since the DC Barrel Jack is isolated from the Dock’s 5V power supply. This was done to allow the servos to be powered by power supplies that provide more than 5V while preventing damage to the Omega and Dock.

Note: on the Arduino Dock, the barrel jack’s bottom tabs have been found to make contact with the metal case of the Omega2 and Omega2+ creating a short circuit. We recommend inserting a thin plastic layer between the top of the Omega and the bottom of the PWM Expansion to break this short. If the DC connector is shorted, it may cause damage to any components connected to the PWM Expansion.

### The Oscillator

The onboard chip has an oscillator that can generate PWM signals with a frequency in the range of 24 Hz to 1526 Hz. The default frequency is 50 Hz which is set to match most servomotors.

All signals generated by the PWM Expansion run at the same frequency set by the one oscillator, so you will not be able to control servos using different frequencies.

### Mechanical Drawings

We’ve made available a detailed diagram of the dimensions and geometry of the PWM Expansion.

### Using the PWM Expansion

Some of the things you can do with the PWM Expansion are as follows:

• Robotics
• LED light shows

Read our guide to using the PWM Expansion to learn how to control it using software.


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# Kinetic Energy of a Rolling Object

Written by Jerry Ratzlaff on . Posted in Kinematics

Rolling kinetic energy, abbreviated as $$KE_{rolling}$$, is the sum of translational kinetic energy and rotational kinetic energy.

## Kinetic Energy of a Rolling Object formula

 $$\large{ KE_{rolling} = \frac {1}{2}\; m \; v_{cm}^2 + \frac {1}{2}\; I_{cm} \; \omega^2 }$$

### Where:

$$\large{ KE_{rolling} }$$ = rolling kinetic energy

$$\large{ \omega }$$  (Greek symbol omega) = angular velocity

$$\large{ m }$$ = mass

$$\large{ I _{cm}}$$ = moment of inertia center of mass

$$\large{ v_{cm} }$$ = velocity center of mass without slipping