text:
Browse Questions

# Find the shortest distance between the lines : $\overrightarrow r = 6\hat i + 2 \hat j +2\hat k + \lambda(\hat i-2\hat j+2\hat k)\: and \: \overrightarrow r = -4\hat i -\hat k + \mu(3\hat i-2\hat j-2\hat k)$

Toolbox:
• The shortest distance between two lines is
• $d=\begin{vmatrix}\large\frac{(\overrightarrow b_1\times \overrightarrow b_2).(\overrightarrow a_2-\overrightarrow a_1)}{\overrightarrow b_1\times \overrightarrow b_2}\end{vmatrix}$
Step 1:
Let the given lines be
$\overrightarrow r=6\hat i+2\hat j+2\hat k+\lambda(\hat i-2\hat j+2\hat k)$----(1)
$\overrightarrow r=-4\hat i+\hat k+\mu(3\hat i-2\hat j-2\hat k)$----(2)
We know that the shortest distance between two lines,$\overrightarrow r=\overrightarrow a_1+\lambda \overrightarrow b_1$ and $\overrightarrow r=\overrightarrow a_2+\lambda \overrightarrow b_2$ is given by
$d=\begin{vmatrix}\large\frac{(\overrightarrow b_1\times \overrightarrow b_2).(\overrightarrow a_2-\overrightarrow a_1)}{\overrightarrow b_1\times \overrightarrow b_2}\end{vmatrix}$
Step 2:
From equ(1) and equ(2) we can determine,
$\overrightarrow a_1=6\hat i+2\hat j+2\hat k$
$\overrightarrow b_1=\hat i-2\hat j+2\hat k$
$\overrightarrow a_2=-4\hat i-\hat k$
$\overrightarrow b_2=3\hat i-2\hat j-2\hat k$
Let us determine $(\overrightarrow a_2-\overrightarrow a_1)$
$(\overrightarrow a_2-\overrightarrow a_1)=(-4\hat i-\hat k)-(6\hat i+2\hat j+2\hat k)$
$\qquad\qquad\;\;=-10\hat i-2\hat j-3\hat k$
Step 3:
Next let us determine $\overrightarrow b_1\times \overrightarrow b_2=\begin{vmatrix}\hat i &\hat j&\hat k\\1 & -2 & 2\\3 &-2 & -2\end{vmatrix}$
On expanding we get,
$\hat i(4+4)-\hat j(-2-6)+\hat k(-2+6)$
$\Rightarrow 8\hat i+8\hat j+4\hat k$
$\overrightarrow b_1\times \overrightarrow b_2=8\hat i+8\hat j+4\hat k$
Step 4:
$\mid \overrightarrow b_1\times \overrightarrow b_2\mid=\sqrt{8^2+8^2+4^2}$
$\qquad\qquad\;\;=\sqrt{64+64+16}$
$\qquad\qquad\;\;=\sqrt{144}$
$\qquad\qquad\;\;=12$
Step 5:
Now substituting the respective value we get
$d=\begin{vmatrix}\large\frac{(8\hat i+8\hat j+4\hat k).(-10\hat i-2\hat j-3\hat k)}{12}\end{vmatrix}$
$d=\begin{vmatrix}\large\frac{-80-16-12}{12}\end{vmatrix}$
$\;\;\;=\large\frac{108}{12}$
$\;\;\;=9$ units.
Hence the shortest distance between the given two lines is $9$ units.


text:
An event has $X\%$ chance of happening. What are the odds of the event happening more/less over time?

Say an event has $30%$ chances of happening $30/100$. If we run 100 experiments, what are the odds that the event will happen exactly $30$ times? What are the odds it will happen $50$ times or $10$ times? I am interested in learning the formula behind the calculations so I can run it with different numbers. Thanks.

-
Do you want the odds or the probability? The odds against the event occurring are $7$ to $3$; the _probability is $0.3$. –  Dilip Sarwate Nov 18 '11 at 16:46

The number of events in the $n$ trials is $Y_n = X_1 + X_2 + \ldots + X_n$, where $X_i$ is the indicator variable, equal to 1 if the event took place and zero otherwise. Each $X_i$ is independent and has Bernoulli distribution with parameter $p=\frac{3}{10}$.

The sum of $n$ independent Bernoulli random variable is equal in distribution to the Binomial random varaiable. Hence $Y_n \sim \mathrm{Binom}(n,p)$.

Then $$\mathbb{P}(Y_{n} = k) = \binom{n}{k} p^k (1-p)^{n-k}$$ With some computations we find $\mathbb{P}(Y_{100} = 30) = 0.0867839$, and $\mathbb{P}(Y_{100} = 50 \lor Y_{100} = 10) = \mathbb{P}(Y_{100} = 50) + \mathbb{P}(Y_{100} = 10) = 0.0000141966$.

-
Exactly what I was looking for. Thank you. –  Cristian Nov 18 '11 at 16:57
A mistake often made in the real world is assuming that the trials are independent. –  Ben Hocking Nov 18 '11 at 18:38

Say an event has probability $p$ of happening. If we run $n$ experiments, the most likely number of times for the event to happen is $np$ (assuming $np$ is an integer.) what are the odds that the event happens $np$ times? In particular, can we come up with an approximate formula for this that doesn't include binomial coefficients?

As Sasha points out, this is $$P(Y_n = np) = {n \choose np} p^{np} (1-p)^{n-np}.$$ Let $q = 1-p$ for ease of notation; then $$P(Y_n = np) = {n! \over (np)! (nq)!} (p^p q^q)^n.$$ Now by Stirling's approximation, we have $$n! \sim \sqrt{2\pi n} \left( {n \over e} \right)^n.$$ Therefore $$P(Y_n = np) \sim {\sqrt{2\pi n} (n/e)^n \over \sqrt{2\pi pn} (pn/e)^{pn} \sqrt{2\pi qn} (qn/e)^{qn}} (p^p q^q)^n$$ and after a lot of things cancel out you get $$P(Y_n = np) \sim {1 \over \sqrt{2\pi npq}}.$$ So the probability of getting the most likely number of successes goes down like $\sqrt{n}$. For example consider $p = 1/2$; then we get $$P(Y_n = n/2) \sim {1 \over \sqrt{n \pi/2}} \approx {0.8 \over \sqrt{n}}.$$ So if I flip 100 coins the probability of getting 50 heads is about $0.8/\sqrt{100} = 0.08$; if I flip 400 coins (four times as many) the probability of getting 200 heads is about $0.8/\sqrt{400} = 0.04$, half what it was.

-
I can only chose one answer and I chose Sasha's already. If I could I would chose you both. You're a very helpful bunch. Thanks. –  Cristian Nov 18 '11 at 17:07

format: plain


text:
# Electron Dot Structures, etc.

In this electron dot structures handout, new chemists read about electron arrangement and how to draw electron dot diagrams. They complete a chart with the mass number, atomic number, isotope notation, number of neutrons, and Bohr notation for 16 elements of their choice. A space is also provided for the electron dot drawing. This makes a terrific practice assignment.


text:
# Thread: [help] a generic autoclicker

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## [help] a generic autoclicker

Hello!

I'm new here and was looking for just a basic autoclicker. It'd have to click fast and no need for any fancy anti bans so just rapid clicking. I tried searching for one but couldn't find at least a working. Yes I am lazy can't really be bothered to learn scripting just for one simple script. Help would be much appreciated.

P.S Garys wont do cause i'd prefer being able to autoclick on multiple accs.

2. Code:
repeat
mouse(x, y, randomX, randomY, mouse_left); //srl6
wait(randomrange(time1, time2));
until false;

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Any chance for a bit further details? What else do I put on the simba? I'm completely new. Also how do I get the coordinates where I'd like to click on.

4. Does it have to be written in Simba? If not, Gary's Hood has worked for many years.

Edit: I can't read. Ignore me.

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This is more complicated I thought it'd be sigh. Is it even possible to have it autoclick in the background while I still use my own mouse for other stuff on pc.
Last edited by aww yiss; 01-20-2017 at 06:17 PM.

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program Autoclicker;
{$define SMART} {$i srl-6/srl.simba}

procedure autoType;
begin
SetupSRL;
repeat
mouse(1201,415, mouse_left); //srl6
wait(randomrange(100, 100));
until false;
end

This is what ive worked out so far, isnt working however.

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No more help needed thank you all.


text:
# “Least Squares” of Dirac Delta?

It is well known that the first $N$ terms of a Fourier series of an even function $f$ corresponds to the least squares approximation of $f$ on $[-\pi,\pi]$ using the functions $S = \{1,\cos(x), \cos(2x),\dots,\cos(Nx)\}$.

The least squares method doesn't make sense for approximating the delta function, since

$$\int_{-\pi}^\pi(f(x)-\delta(x))^2\,dx$$

diverges. However, the Fourier series technique can approximate this distribution without issue. Moreover, one can populate the least-squares matrix with values when approximating the delta function. In what sense, then, is the Fourier series approximating the delta function? Is there a generalized "least-squares" that will quantify how close a function is to the delta function?

Suppose $$S_N(x) = \sum_{n=0}^N a_n \cos(nx)$$ and $g$ is an even function with period $2\pi.$

Suppose the coefficients $a_n,\, n=0,\ldots , N$ are so chosen that $$\int_{-\pi}^\pi g(x)\cos(nx) \,dx = \frac 1 \pi \int_{-\pi}^\pi S_N(x) \cos(nx)\, dx \tag 1$$ for $n=0,\ldots,N.$ If $(1)$ holds, then $S_N$ is the $N$th-degree Fourier series approximation to $g.$

If $g$ is the delta function, then the integral on the left in $(1)$ is $1.$ In the integral on the right every term integrates to $0$ except the $n$th term, and that integrates to $a_n.$ Therefore $a_n=1$ if $S_N$ is the $N$th-degree Fourier series approximation of the delta function.

There is a certain sense in which this minimizes the integral of the square of the discrepancy between the finite trigonometric series and $\delta.$ Suppose we let \begin{align} \delta(x) & = \sum_{n=0}^\infty \cos(nx) \\ S_N(x) & = \sum_{n=0}^N \cos(nx) \\ g(x) & = \sum_{n=0}^N b_n \cos(nx) \end{align} Then $$g(x) - \delta(x) = \Big(g(x) - S_N(x)\Big) + \Big(S_N(x) - \delta(x)\Big)$$ and $$\underbrace{(g(x) - \delta(x))^2} = \Big(g(x) - S_N(x)\Big)^2 + {} \underbrace{\Big(S_N(x) - \delta(x)\Big)^2} {} + \Big(\text{something whose integral is 0}\Big)$$ The integrals of the two terms over the $\underbrace{\text{underbraces}}$ are infinite. But the first term on the right is finite and is made as small as possible, in fact is made equal to $0,$ by making $b_n=1$ for $n=0,\ldots,N.$ In other words, that portion of the integral of the square of the discrepancy that can be altered by altering the first $N$ coefficients is made as small as possible by making those coefficients equal to $1.$

(How, if possible, to make this logically rigorous is more than I will attempt to say at this point.)

You can think of the delta function and the function you're comparing it to as both being linear functionals on some fixed normed space, so they come equipped with a norm from that dual space. For example, although you cannot think of $\delta$ as a linear functional on $L^2$, it is a linear functional on the Sobolev space $H^1=W^{1,2}$, because Sobolev embedding implies that $H^1$ functions in one dimension are continuous functions. Thus you can measure the distance between any bounded linear functional on $H^1$ and $\delta$ through the norm on the dual of $H^1$ like this:

$$\| f - \delta \| := \sup_{\| g \|_{H^1}=1} |f(g)-\delta(g)|.$$

• But note that there are several different (equivalent) norms used on $H^1$. It isn't clear to me that one obtains the same best approximation for $|f|_2+|f'|_2$ and $\sqrt{|f|_2^2+|f'|_2^2}$, for example. – David E Speyer Apr 13 '18 at 12:33
• @DavidESpeyer Sure. The point is just to have a space where $f$ and $\delta$ both live which has some metric, and that space can't be a space of functions, nor can it really be a full space of distributions (since those topologies aren't given by a metric). You could pick a different metric and space, such as the dual of the continuous functions, if you wanted. I picked this because it has some flavor of squaring built into it while still allowing $\delta$ to make sense (even though it doesn't involve squaring function values in any sense). – Ian Apr 13 '18 at 14:43

In crude physical terms it is the Energy Principle expressed by Parseval's Theorem that tells you "how near/far" two signals are from each other. And that is quite intuitive.

If you take the frequency spectrum of the two signals, then its difference is the spectrum of the difference between the signals (the "error"). The sum/integral of the square of that is proportional to the energy/power of the "error" signal.

The delta function has a flat spectrum with unitary amplitude, and the set $\{\cos {(kx)}\quad |0 \le k \le N \}$ has a unit spectrum clipped at $N$.
The conclusion will be obvious.

• I get the intuitive point, but this is sort of taking Parseval's theorem and running with it..."I can measure the $L^2$ distance on the Fourier side, so I can basically measure distance on the Fourier side even when the metric I actually wanted to use makes no sense". – Ian Apr 13 '18 at 14:46
• @Ian: I get (more or less) your perplexity. In any case Parseval's theorem allow us to work either in time and in frequency to measure the $L^2$ "error". Now the delta has an infinite energy, being evenly spread all over the spectrum. However, among all the signals with a given limited bandwidth, we can tell the one which better approximates the delta as being that with a "box" spectrum: can't we ? – G Cab Apr 13 '18 at 18:04
• Technically no, because the metric still gives you infinity. In that respect, although it's true that a widening box will eventually fill out the frequency spectrum of the Dirac delta, there's no particular bias in favor of a box centered at zero--it could be centered at some other frequency, or even have a sufficiently slowly moving center, and the situation would be the same. You need something a bit better, like a frequency cap in the OP, to reduce to the symmetric case. – Ian Apr 13 '18 at 18:18

Generally in Banach spaces $\|u\|=\sup_{\|l\|=1}|\langle l,u\rangle|$, which means that least squares solution realizes $\inf_u\sup_{\|l\|=1}|\langle l,u\rangle|$, where $l$ runs over the unit ball of the dual space. Just having $|\langle l,u_n\rangle|$ go to $0$ for any dual $l$ makes sense for Sobolev spaces to which $f(x)-\delta(x)$ belongs, and their duals, it even makes sense for more general locally convex spaces to which distributions belong. If one wants a single measure of deviation Fréchet metric can be used in many cases (when the space has a countable basis of seminorms).

$f(x)-\delta(x)$ belongs to $H^{-1}$, for example, from the standard Hilbert scale $H^k$. It is straightforward to see that there will be weak convergence of the Fourier series to it, i.e. if $l$-s are taken from $H^{1}$ (it may not converge by norm in $H^{-1}$ in all cases). Alternatively, one can use weak convergence of measures, their space is dual to $C^0$. Both convergences are metrizable (on bounded subsets), see e.g. Metrization of weak convergence of signed measures on MO.

In what sense, then, is the Fourier series approximating the delta function? Is there a generalized "least-squares" that will quantify how close a function is to the delta function?

1. The Dirac Delta is not a function and you can't square it, so using the $L^2$-norm does not make sense. I think that the most natural space for the Dirac Delta is the space of (Radon) measures: The Dirac Delta acts on continuous functions and thus, is an element in the dual space of these, and the dual space of the continuous functions (vanishing on the boundary) is, by Riesz-Markov, the space of Radon measures. This gives a natural norm to work with, namely $$\|f-\delta\| = \sup\{\int \phi d(f-\delta)\ :\ \|\phi\|_\infty\leq 1\}$$ where $\phi$ runs over continuous functions and $\int \phi d(f-\delta) = \int \phi f - \phi(0)$. This is the Radon-norm on the space of measures. Actually, solving minimization problems involving the Radon norm is (at least numerically) not that hard, since you end up with a convex-concave saddle point problem, in this case $$\min_f\max_{\|\phi\|_\infty\leq 1}\int \phi f - \phi(0).$$ A generalization to for fancy "norms" has been linked in Conifold's answer. The Wasserstein-1-norm (also called dual Lipschitz norm) mentioned in the linked answers simple amount to the problem $$\min_f\max_{\|\phi\|_\infty\leq 1, \|\phi'\|_\infty\leq 1}\int \phi f - \phi(0)$$ i.e., you maximize over Lipschitz continuous functions $\phi$ with Lipschitz constant $\leq 1$.
2. If you insist on least squares, note that formally expanding the square gives $$\|f-\delta\|_2^2 = \|f\|_2^2 - 2\langle f,\delta\rangle + \|\delta\|_2^2.$$ The latter term does not make sense, but we want to minimize over $f$ anyway, so it's value does not play a role. Thus, we can drop it. The term in the middle makes sense as soon as $f$ is continuous, and then it evaluates to $-2f(0)$ thus we can give a meaning to the least squares problem as $$\min_f \|f\|_2^2 - 2f(0)$$ where, in your case, you minimize over linear combinations of cosines.
• That last problem isn't bounded below. You need something like $H^1$ to force the value at zero to give some contribution to the norm, so that there is a meaningful competition. As it is, you can use extremely narrow spikes near zero to get continuous functions with arbitrarily small $\| f \|_2^2-2f(0)$. These kind of issues are why I brought up Sobolev spaces; $L^p$ spaces alone don't give any relationship between single point values and their norms. – Ian Apr 13 '18 at 14:48
• Sure you need something to get the objective bounded from below. The OP has the additional constraint that they minimize over trig polynomials up to a certain degree. You could also use different penalties or constraint (besides Sobolev norms one could use many different things, e.g. bounded Lipschitz constant (to be fair, this is $W^{1,\infty}$)). – Dirk Apr 13 '18 at 14:53


text:
The following are some common issues and questions that have arisen when building your textbook with Jekyll.

## How can I update my book?

Sometimes Jupyter Book will get updates that you want to incorporate into a book you’ve already built. The easiest way to do this is to use the Command-Line Interface to upgrade your book.

To upgrade a pre-existing Jupyter Book, run the following command:

jupyter-book upgrade path/to/mybook

This will do the following:

1. Generate a fresh Jupyter Book in mybook_UPGRADED using the content files in your current book.
2. If this succeeds, copy over the contents of mybook_UPGRADED into your current book folder.
3. If this succeeds, delete the mybook_UPGRADED folder.

Note that only the content that you can manually specify via the jupyter-book create command will be retained in your upgraded book. For a list of these options, see the help menu for this command:

jupyter-book create -h

You should check out the content in your upgraded book to make sure it looks correct, then commit the changes to your repository.

## Does the book behave differently depending on the browser?

Maybe - Jupyter Book does use some features that might have different behaviors in some browsers. For example, Safari tends to treat downloadable objects differently for some reason.

The two browsers on which Jupyter Book should always behave as expected are Firefox and Chrome.

## Why isn’t my math showing up properly?

This site uses MathJax to render all math, with $ denoting inline math, and $\$ denoting multi-line math blocks. Make sure that all of your math is wrapped in these characters.

Another tip is to make sure that your math isn’t being escaped improperly. Jekyll strips escape characters, so you should make sure to add two escape characters when needed. This is done automatically for many escape characters in generate_book.py, and if you notice something that should be included in this script, please open an issue at the textbook template issues page

## How can I include interactive Plotly figures?

To display interactive Plotly figures, they must first be generated in a Jupyter notebook using the offline mode. You must then plot the figure with plotly.offline.plot(), which generates an HTML file (plotly.offline.iplot() does not), and then load the HTML into the notebook with display(HTML('file.html')) prior to saving your *.ipynb file.

e.g.

from IPython.core.display import display, HTML
from plotly.offline import init_notebook_mode, plot
init_notebook_mode(connected=True)
.
.
.
plot(fig, filename = 'figure.html')
display(HTML('figure.html')

Note that, if viewing the file on a Jupyter Notebook session, the figure will not be displayed there (iplot is needed for this). However, if working on a JupyterLab session, the figure can be displayed there using the plot code above by having the JupyterLab plotly extension installed.

## What if I have an issue or question?

If you’ve got questions, concerns, or suggestions, please open an issue at at the jupyter book issues page


text:
# How Is Server-Timing used on the web?

I was curious to see where Server-Timing was implemented on the web, so I started searching the HTTP Archive for sites using it. Interestingly enough, there were no sites in the HTTP Archive that had Server-Timing response headers before 3/1/2017. Since then it’s usage has been gradually increasing each month. As of July 2017, there are 72 sites and 352 HTTP responses containing Server-Timing headers.

Here’s a query that I wrote to find a list of sites with Server-Timing headers. You can run it on BigQuery here :

    SELECT p.url, p.rank, count(*) as freq
FROM
( SELECT pageid, url, rank
FROM [httparchive:runs.latest_pages]
) p
JOIN
FROM [httparchive:runs.latest_requests]
) r
ON p.pageid = r.pageid
GROUP BY p.url, p.rank
ORDER BY freq DESC

Next I was curious to see the actual Server-Timing headers to understand how they are being used. I wrote this query to strip out the server-timing headers from the respOtherHeaders column. The query has a lot of string manipulations to extract the text. I’ve added some comments to the query to help make it easier to follow -

    SELECT HOST(url), status, mimeType,
IF (
-- In respOtherHeaders, remove everything to the left of the server-timing header.   Is there a ; in the remaining string?
-- Then strip out everything to the left of the server-timing header and everything to the right of the ";" character.
LEFT(
INSTR(
"server-timing")+1),
";")
),
-- If there are no ; chars, then the respOtherHeaders column ends with server-timing.  Remove everything to the left of server-timing
"server-timing")+1)
FROM [httparchive:runs.latest_requests]

You can see it on BigQuery here

Looking at the Server-Timing headers, I can see 3 different types of data being sent in these headers:

Timing Metrics (such as application time, gateway time, image resize time, etc)

Server-Timing = app=133.99219512939;
Server-Timing = gateway-time=25;
Server-Timing = resizer=0.015;
server-timing = view=2.188, db=0.000, total=61.743, x-runtime = 0.069661, x-ua-compatible = ie=edge,chrome=1, x-frame-options = SAMEORIGIN
Server-Timing = app=53.393840789795;

Caching Information

server-timing = hit, cf-cache-status = HIT
server-timing = cold, cf-cache-status = MISS
server-timing = hit, cf-cache-status = HIT
server-timing = cold, cf-cache-status = MISS

Informational

server-timing = ibs_5318e9209631=1
server-timing = ibs_1964fd84c9cea3=1
server-timing = ibs_1880bf4b0f6311=1

Looking at all of the data in the 7/1/2017 tables, I can see that the current examples in the wild are split between these 3 types of data, although the caching ones are mostly on images, the informational ones are set on JS and responses w/ no content (ie, 204s -likely beacon responses), and the actual timing metrics are spread across a wide range of content.

It’ll be interesting to see how this evolves as Server-Timing adoption increases over time.

Originally published at https://discuss.httparchive.org/t/how-is-server-timing-used-on-the-web/1034


text:
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Concept. Go to the latest version.

Series Circuits

A closed circuit in which the current follows through one path.

%
Progress
Progress
%
Lights in Series

Lights in Series

Credit: Laura Guerin
Source: CK-12 Foundation

When three light bulbs hooked in series are attached to a battery, they light up. If you continue to add more light bulbs in series, do the lights dim, get brighter or do some of the light bulbs go out?

News You Can Use

• A series circuit is a circuit in which the current in the system has only one path for the current to flow though. For a series circuit, any resistors that are added along the path changes the total resistance of the circuit according to the following rule:

\begin{align*}R_{eq}=\sum\limits_{i}R_i=R_1+R_2+R_3+ \cdots\end{align*}

• When the current flows through the 3 light bulbs in the image above, the current through each light bulb is the same but the voltage across each bulb is different. According to Kirchoff's rule which states that the sum of the voltage drops in a closed loop must be equal to 0, the above situation can be written as

\begin{align*}V_{battery}-(IR_1)_{bulb \ \# 1}-(IR_1)_{bulb \ \# 2}-(IR_1)_{bulb \ \# 3}=0\end{align*}

• (If you factor out \begin{align*}I\end{align*}, from each of the terms in parenthesis and divide by the current \begin{align*}I\end{align*}, you get the first equation above.)
• Watch a demonstration of light bulbs in series being lit up:

Explore More

Using the information provided above, answer the following questions.

1. As more light bulbs are added in series, what happens to the overall brightness of the light bulbs? Why?
2. What is Kirchoff's loop rule say for a system that only has a battery and 1 light bulb attached?
3. If one of the light bulbs in the series burns out, what happens to the other light bulbs? Why?

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes


text:
## Chemistry 12th Edition

We know that the solution of $Na_2SO_4$ is colorless which shows that $SO_4^-2$ is colorless and the blue color is because of $Cu^+2$.


text:
Division of Complex Numbers

Dividing Complex Numbers of the form a + bi

Tip: This isn't the place to ask a question because the teacher can't reply.

Key Questions

By definition if $z = a + b i$ is a complex number, then his conjugate is $\overline{z} = a - b i$

Explanation:

A conjugate of a complex number is the other complex number with the same real part and opposite imaginary part

• You can do it by multiplying the numerator and the denominator by the complex conjugate of the denominator.

$\frac{1}{a + b i} = \frac{1}{a + b i} \cdot \frac{a - b i}{a - b i} = \frac{a - b i}{{a}^{2} + {b}^{2}}$

I hope that this was helpful.

• If complex numbers ${z}_{1}$ and ${z}_{2}$ in polar form are

$\left\{\begin{matrix}{z}_{1} = {r}_{1} \left(\cos {\theta}_{1} + i \sin {\theta}_{1}\right) \\ {z}_{2} = {r}_{2} \left(\cos {\theta}_{2} + i \sin {\theta}_{2}\right)\end{matrix}\right.$,

then we can write in exponential form

$\left\{\begin{matrix}{z}_{1} = {r}_{1} {e}^{i {\theta}_{1}} \\ {z}_{2} = {r}_{2} {e}^{i {\theta}_{2}}\end{matrix}\right.$.

So, the quotient ${z}_{1} / {z}_{2}$ can be written as

${z}_{1} / {z}_{2} = \frac{{r}_{1} {e}^{i {\theta}_{1}}}{{r}_{2} {e}^{i {\theta}_{2}}} = {r}_{1} / {r}_{2} {e}^{i \left({\theta}_{1} - {\theta}_{2}\right)}$

$= {r}_{1} / {r}_{2} \left[\cos \left({\theta}_{1} - {\theta}_{2}\right) + i \sin \left({\theta}_{1} - {\theta}_{2}\right)\right]$

I hope that this was helpful.

• This key question hasn't been answered yet.

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text:
Busbar section

The com.powsybl.iidm.network.BusbarSection interface is used to model a busbar section, which is a non impedant element used in a node/breaker substation topology to connect equipments. In IIDM, it is a sub interface of Injection.

# Characteristics

Attribute Type Unit Required Default value Description
V double kV no - The voltage magnitude of the busbar section
Angle double ° no - The voltage angle of the busbar section

# Examples

This example shows how to create a new Busbar Section in a network:

BusbarSection bbs = voltageLevel.getNodeBreakerView().newBusbarSection()
.setId("BBS")
.setNode(1)


text:
# How do we define division?

Multiplication is defined as repeated addition.

3x5 = 5+5+5

How do we define 10/2?

## Answers and Replies

Related Linear and Abstract Algebra News on Phys.org
Originally posted by AndersHermansson
Multiplication is defined as repeated addition.

3x5 = 5+5+5

How do we define 10/2?
Repeated subtraction?
10-2-2-2-2-2=0 5 equal parts.

10/3
10-3-3-3=1 3.3333333333333 parts. 3 and 1/3

HallsofIvy
Science Advisor
Homework Helper
You also posted this under "general mathematics" where Hurkyl pointed out, correctly, that multiplication is NOT defined as "repeated addition".

Soroban
Reminds me of a truly awful joke . . .

Q: How many times can you subtract 5 from 21
and what do you have left?

A: I can subtract 5 from 21 as many times as I like,
and I always have 16 left.

If A/B = C, then I would define C as the the number of times that you have to subtract B from a quantity that starts out at A until you get to 0. Hence, 10/5 = 2, because you have to subtract 5 twice from a quantity that starts out at 10 in order to get 0, and 0/0 is undefined, because you always have 0, no matter how many times you subtract, and 10/0 is undefined because there is no answer(There is no amount of times that you can subtract in order to arrive at zero--"no amount of times" is NOT the same as "zero times", because zero is an amount of times; you have an empty set, as opposed to a set with an element 0).

hypnagogue
Staff Emeritus
Science Advisor
Gold Member
Originally posted by Dissident Dan
If A/B = C, then I would define C as the the number of times that you have to subtract B from a quantity that starts out at A until you get to 0. Hence, 10/5 = 2, because you have to subtract 5 twice from a quantity that starts out at 10 in order to get 0, and 0/0 is undefined, because you always have 0, no matter how many times you subtract, and 10/0 is undefined because there is no answer(There is no amount of times that you can subtract in order to arrive at zero--"no amount of times" is NOT the same as "zero times", because zero is an amount of times; you have an empty set, as opposed to a set with an element 0).
What about something like 1/3? How many times do you have to subtract 3 from 1 to get 0? The answer is "1/3 of 3" times, but the answer here using the above formulation doesn't get us any closer to a meaningful answer than the initial question. It's circular.

rolandmath
we define division of two integer numbers a and b as follows a/b is
a=b(q)+r
for some integers q and 0<= r< b. Here q and r are uniquely deteremine. 28/5 is the same thing as 28=5(5)+3 If you want to get fancy schmancy. Take the integers Z and since Z is an integral domain define the field of rationals Q as all the numbers(quotients) that satisfy the following equation for x

xm=n

all solution are n/m where m and n are integers, here you have the field of quotients or Q


text:
Share

# Oil-Production Costs Reach New Lows, Making Deepwater One of the Cheapest Sources of Novel Supply

## In an updated projection, Rystad said that a breakeven of only $50/bbl is needed to produce 100 million BOPD in 2025, reflecting the reduction in oil-production costs. A Rystad Energy analysis of oil-production costs revealed the average breakeven price for all unsanctioned projects has dropped to around$50/bbl, down around 10% over the past 2 years and 35% since 2014. This means that oil is much cheaper to produce now compared to 6 years ago, with the clear cost-savings winner being new offshore deepwater developments.

Our cost-of-supply curve for liquids now reveals that the required oil price for producing 100 million B/D in 2025 has been in continuous decline in recent years, with its updated projection showing that an oil price of only $50/bbl is needed to keep oil production at this level. Previously, in 2014, we had estimated that the required oil price for producing 100 million B/D in 2025 was close to$90/bbl, an estimate which we then revised in 2018 to around $55/bbl. The updated curve also shows another key trend. From 2014 to 2018, the cost-of-supply curve moved to the right. In 2014, we estimated that the total 2025 liquids potential was only 105 million B/D. In 2018, this number had increased considerably to around 115 million B/D. However, since 2018, we have revised down the potential to about 108 million B/D, moving the cost curve to the left. The implication of falling breakeven prices is that the upstream industry over the past 2 years has become more competitive than ever and is able to supply more volumes at a lower price. However, the average breakeven prices for most of the sources remain higher than the current oil price. This is a clear indication that for upstream investments to rebound, oil prices must recover from their current values. When it comes to breakeven prices and potential liquids supply in 2025 for the main sources of new production, Rystad Energy data show that from 2014 to 2018, tight oil and OPEC came out on top. Back in 2014, the average breakeven price for tight oil was estimated to be$82/bbl and potential supply in 2025 at 12 million B/D.

Since then, the breakeven price has come down and the potential supply has increased for tight oil. In 2018, an average breakeven price of $47/bbl and a potential supply of 22 million B/D was estimated. After 2018, the breakeven price for tight oil has continued to fall, reaching a current average of$44/bbl. However, the potential of tight-oil production has dropped from our 2018 estimate.

Now we estimate that tight oil can potentially supply around 18 million B/D of liquids in 2025. This drop is due to the sharp reduction in tight-oil production during the first half of this year. The lower activity this year, and a potentially slow recovery next year, will remove tight-oil supply from the market.

Between 2014 and 2018, shelf and deepwater projects experienced a cost reduction of around 30%. However, the lack of new sanctioning during the same period reduced the offshore potential liquids supply for 2025. Since 2018, breakeven prices have been falling for offshore, with deepwater down 16% and shallow water down 10%.

This cost reduction puts average breakeven prices for deepwater just below those of tight oil. At the same time, the potential 2025 supply from offshore developments has not changed too much. This makes offshore a winner out of all the supply sources over the last 2 years when it comes to cost improvements and supply potential.

Onshore Middle East is the least-expensive source of new production with an average breakeven price of around $30/bbl. This is also the segment with one of the largest resource potential estimates. Offshore deepwater is the second-cheapest source of new production, with an average breakeven price of$43/bbl, while onshore supply in Russia remains one of the more expensive resources due to the high gross taxes in the country. Shelf remains the segment with the largest resource potential with 131 billion bbl of unsanctioned volumes.

One of the key drivers of the improved costs and breakeven prices for upstream developments are the lower unit prices within the industry. After the 2015 oil-price collapse, oilfield service companies needed to reduce the prices they charged exploration and production companies in order to remain competitive in the challenging market conditions.

For all sources, unit prices dropped close to 25% between 2014 and 2016 but it was tight oil that faced the largest reduction. Since then, the supply segments have recovered differently. While offshore and conventional onshore supply only saw marginal price increases, tight oil unit prices increased considerably during 2018.

However, given that tight-oil activity fell last year and with the COVID-19 inflicted market crisis this year, tight-oil unit prices have started to drop again since 2018. For all the main upstream segments, current unit prices have come down around 30% compared to 2014 levels.

Espen Erlingsen is the head of upstream research at Rystad Energy. He can be contacted at espen.erlingsen@rystadenergy.com.


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# User:Damonwang

This is my proposal for Summer of Code 2010. I would welcome any comments on the associated discussion page, and especially answers to the questions in my notes. Thank you for taking the time to read this!

## Identity

Name: Damon Wang

Email: damonwang uchicago edu

Project title: Towards a Python port of Texvc

## Contact/working info

Timezone: Chicago or New York (UTC-5 or -4)

Typical working hours: noon to four, eight to midnight (UTC 17:00-21:00, 1:00-5:00)

IRC or IM networks/handle(s): damonwang on freenode

## Project summary

Mediawiki pretty-prints math formulae by taking a subset of AMS-TeX and displaying either a PNG rendered via dvi2png or, if the formula is simple enough, an HTML approximation. However, AMS-TeX has numerous features allowing arbitrary code execution, unbounded render time, and similarly undesirable behavior. The current solution is texvc[1], an OCaml parser which accepts only a safe subset of AMS-TeX, and produces the appropriate HTML or PNG output. Unfortunately, the obscurity of the source language seems to have discouraged maintenance: texvc has some fifty open bugs[2] going back six years and including a request to make it easier to install[3]; of the texvc bugs that have been resolved, very few involved edits to the OCaml code. It was therefore suggested that texvc be ported to a more popular language[4].

A PHP port would have been the ideal solution. Mediawiki already has a large pool of PHP developers, calling the external binary is currently the source of a major outstanding bug[5], and a PHP dependency for a project written in PHP is as good as eliminating a dependency entirely. In fact, a PHP port has been attempted[6], but omitted all validation of the input, probably because PHP has no existing parsing packages. Judging from the LaTeX2MathML code, which does implement a LaTeX parser in PHP, backward to write the parser manually. Much of the clarity, concision, and robustness of the OCaml texvc comes from the fact we need only maintain a BNF-like input to a parser-generator, rather than the parser itself. As it would be well beyond the scope of GSoC to write a general PHP parsing package, I propose instead to port texvc into Python, which offers the following benefits:

• popularity: easier to find maintainers, easier for sysadmins to install, easier for new developers to "read themselves into" the codebase for quick patches
• ubiquity: although it does need an interpreter at run-time, this is not a difficult dependency to satisfy; probably even easier than an OCaml compiler at install-time or a binary distributions
• several mature parsing packages: potentially no regression from the OCaml version in terms of concision, clarity, and "elegance"

We don't just care about your project -- you are a person, and that matters to us! What drives you? What makes you want to make this the most awesomest wiki enhancement ever?

You don't need to write out your life story (we can read your blog if we want that), but we want to know a little about what makes you tick. Are you a Wikipedia addict wanting to make your own experience better? Did a wiki with usability problems run over your dog, and you're seeking revenge? What does making this project happen mean to you?

I am a third-year undergraduate who was first started programming in a high school data structures course taught with Scheme and C. Perhaps because I started out with homework exercises, my interest was in small toy problems such as those offered by Project Euler[7], the USA Computing Olympiad[8], the Python Challenge [9], and the system scripts on my laptop. As a relative newcomer to open-source software, especially on the scale of Mediawiki, it's a relief to find a rather isolated corner of the project where I can make a concrete contribution without needing to grok the entire codebase.

Although Scheme was my first language in high school and Haskell my first language at university, my work as a sysadmin at the Mac Lab on campus[10] these last two years has taught me that big libraries and side effects are often the easiest way to get a job done. For practical programming, Python is my first choice, with shell a close second for scripting and Perl tying with Common Lisp as very distant thirds[11]. The texvc port offers almost the best of both worlds: exposure to a new functional language while improving my Python by coding to the standards of a larger, mature project.

Working at the MacLab has also generated a healthy respect for written records. Since our policy of hiring exclusively students enforces high turnover, inadequate documentation would quickly turn sysadmins into detectives or archaeologists. At one point, I wasted two hours configuring the wrong web server before someone pointed out that this machine had a second Apache installation in a nonstandard location. I've since set up a lab bug tracker to help people work together without overlaps or gaps, and a searchable mail archive to record discussions on our mailing list. Although I don't like writing documentation any more than the next guy, I do love using it, and so one of the core deliverables for my Python texvc will be a complete set of documentation.

## Deliverables

It should be possible to break down your project into some bullet points describing particular features or milestones which can be reached individually. Consider that we may wish to roll out the system for testing when at an intermediate stage of completion, and that time estimates might vary, leaving you with more time than you expected or (more likely) a lot less -- some features can be pushed back if you end up short.

Since this is a port, and I'll essentially be regression-testing against the OCaml texvc, "correct" means "like the OCaml texvc". Whether I reproduce erroneous behavior will be decided on a case-by-case basis: as a rule of thumb, I won't give special effort to either recreating bugs or fixing them, unless specifically noted.

### Required deliverables

#### texvctest.sh

A testing script which determines if two version of texvc produce identical output on stdout (accept/reject, HTML or MathML as necessary). This script will not attempt to check the PNGs, since we trust TeX to render the correct output once texvc proper decides that the input is safe enough and complicated enough to pass on. In order to account for intended deviations, it will either check that two texvc binaries produce the same output when run, or that a single texvc binary produces output identical to what is cached in a file.

#### texvcstats

Write a script which, given a dump of the [itex] elements from a Mediawiki instance, counts the possible rendering modes. This will primarily be used to decide how much HTML support to incorporate into the Python texvc as a required deliverable. If, for example, we find that half of enwiki's math can be rendered as conservative HTML and almost all of the rest needs PNGs, then moderate and liberal HTML support would probably become optional deliverables.

is there an easier way to get these data?

#### test cases

Many TeX files mimicking valid and invalid Math input, grouped into a few sets approximately in order of descending importance:

1. manually constructed invalid examples involving various security holes (calling other executables, catcodes, etc.)
2. manually constructed valid examples involving each of the supported commands, both in isolation and in simple combinations
3. manually constructed invalid examples involving common mistakes like unsupported commands, mismatched delimiters (braces or \begin...\end blocks), etc.
4. a dump of all math on enwiki and dewiki[12], to guarantee no regression against anything currently accepted
Is there any way to log Math input on one of the production servers, in order to capture real examples of invalid input? If I've learned one thing as a sysadmin it's how effectively a thousand users will fuzz-test any application.

#### correct accept/reject decision

OCaml and Python versions should agree on what is valid and invalid input. In particular, Python texvc should accept all examples from enwiki and dewiki, since Bugzilla has no bug reports for the OCaml texvc in this area. It should also reject all of the contrived examples involving various security holes (this takes precedence over all other aspects of the Python texvc's behavior)

#### correct HTML decision and output

Python and OCaml versions should agree what can be rendered as HTML and how that should be done. I will definitely support the conservative HTML output, and leave the less stringent options as optional deliverables depending on how often they are actually used according to the stats script.

#### documentation

In three formats:

1. Python docstrings in compliance with Google's Python style guide[13] as might be expected by future Python programmers
I'd appreciate if someone with Python experience would weigh in on how necessary he considers docstrings. I use them frequently because I use iPython and I like how docstrings let me switch fluidly between experimenting with different calls and reading about them. I don't know if this is generally true of Python programmers.
2. inline documentation in compliance with the Mediawiki project's Doxygen style[14] as might be expected by future maintainers with other Mediawiki development experience
I imagine this one is considered non-negotiable?
3. Manually written full-paragraphs-of-English documentation explaining
• the overall architecture, including not only the code but also the test suite and my development workflow
I don't expect that future maintainers will follow my workflow, but I thought it might help people unfamiliar with this corner of the codebase get up to speed for a quick patch. I'd love to hear anyone else's opinions on this point.
• reasoning behind more nontrivial design decisions
• pointers for common expected modifications (e.g., "to add support for a new command, use these modules and edit these files and run these tests...")

As a concession to logistics, I'll probably only maintain the docstrings during development, and port it over to Doxygen at various milestones. Since the two systems contain the same information, I don't expect this to generate any oversights.

### Optional Deliverables

#### higher-resolution statistics

Extend the stats script to count occurrences of each supported math command. At the most basic, this would give coverage information for the wikidump tests to show how well we were really testing.

These data would not be used to justify dropping any currently supported commands, since that would be a very user-visible regression.

If I have time, there could be some interesting work to be done with the data, which essentially thousands of points in a space of as many dimensions as supported math commands. We could, for example, try some principal component analysis or clustering. I've been itching for a chance to really learn these techniques hands-on, so it'd be an exciting epilogue after I finished all my required deliverables.

#### texvc no-PNG flag

If automated testing (which might require running texvc on dozens of examples during development and thousands or more during overnight tests) turns out to be really slow, add a flag to prevent texvc from actually calling tex or dvipng in the case that it decides to produce PNG output. Since the image is not examined by texvctest.sh, this is unnecessary work and just slows down automated testing.

For every test, check texvc output with and without no-PNG before accepting the no-PNG output for testing purposes. While we're testing the no-PNG flag, log how great an improvement it provides. This isn't worth merging back into the trunk unless a significant increase is observed.

#### Windows support

Fix bug 13518[5] preventing Math support on Windows servers. This depends on getting a Windows development environment, so I've scheduled it quite late.

#### texvc-fuzz

As the very last finishing touch, write a generator which can produce valid and almost-valid TeX randomly for "fuzz-testing". This is so pie-in-the-sky I won't even leave room for it in the schedule.

The best way I can imagine to generate a mix of valid and invalid TeX is a Markov chain, but that doesn't account for the matched \begin...\end delimiters. Is there a more appropriate technique? The hard part might actually be generating invalid TeX, since otherwise I could just reverse the parser.

#### MathML support

Supposedly, texvc also supports MathML to some extent[1]. Although dropping it in the Python port would technically be a regression against the OCaml texvc, it doesn't seem like a very important feature. IE requires a (probably never installed) plugin for MathML support and Opera has rendering problems[15], so correct MathML export would be of limited use to most users anyway. I'd prefer to focus my time on more commonly used features like PNG and HTML rendering.

is this acceptable? I really don't like to drop anything that was supported by the OCaml texvc, but at the same time it seems so unimportant compared to getting PNG and HTML rendering right.

## Project schedule

Try to break your deliverables into "milestone" points which can be reached in sequence. Block out your estimated schedule of when you'll reach each functional milestone. Don't forgot that real time may change -- leave enough wiggle room for your required features to be completed!

This schedule is highly provisional in two senses. As mentioned in the participation section, I hope to discuss it weekly with my mentor in order to adjust it for the last week's progress. Also, if I start falling behind, I will cut features before blowing a deadline, with the intention of having source that will compile and run at some level of functionality. This is especially true in weeks 10 and 11, where the number of existing bugtracker tickets I close will be dictated purely by how much time I have left.

The official start of coding come two weeks before end of spring term at my university, but, conveniently, official end of coding comes five weeks before start of autumn term. Would it be all right if we were to shift Google Summer of Code down two weeks? It makes evaluations a little odd, since I'd have only three weeks before midterms, but I think we could make it work.

Weeks are numbered with zero at my finals week, so that I would start coding in week 1 assuming my schedule change is accepted. Ranges are inclusive.

### weeks -7 through -2

21 April (week -7)
accepted students announced
26 May (week -2)
official start of coding
• find out if it would be possible to log invalid input on a production Mediawiki server
• try to find someone who would be willing to answer specific OCaml questions. Supposedly my department has two world-class OCaml researchers as faculty, and I know at least one grad student whose research involves ML.
• ask on wikitech-l whether people want Math split off as an extension, and how best to handle the upgrade path
• research security holes offered by TeX. A few ways I might start:
• look through the TeXBook, TeX: The Program and TeX by Topic[16]
• "read myself into" the project:
• the Mediawiki style guide[17], supplemented by the Google style guide[13] for the Python-specific bits
• development policy[18]
• quick-start guide[19], although I shouldn't need touch the database or the rest of the codebase except for testing installation
• outstanding texvc bugs[2]
• set up a development environment: an OCaml compiler, a local subversion repository, MantisBT, backups onto another machine, etc.
I'd also like a Windows environment for testing, but I have no IIS or general Windows administration experience, and no way to get either Server 2003 or Server 2008 licenses except by paying out of pocket.
I'd also like to find someone with a Gentoo Hardened system who could check that my port doesn't recreate the problem with PaX[20]
• update
• try to fix bug #13518[5] which breaks Math on Windows, just to get some experience with relevant parts of the Mediawiki codebase and contribution guidelines. I'd have to test on a desktop Windows environment, so the patch would have to be reviewed by someone with a server.

### weeks -1 through 0

3-4 June
7-11 June
university finals

### week 1: set-up

13 June
actual start of coding
• write and document texvctest.sh
• write and document the statistics script
• produce some visualisations showing how many of the math blocks on enwiki require each of the supported commands, so that through the rest of the summer we know how bad immediate deployment would be.
• decide at what level to support HTML output

At the end, I should know what counts as a feature-complete port.

### week 2: test suites

• write test cases covering all the dangerous bits of TeX discovered during weeks -6 through -2 (very simple usage, not any kind of elaborate exploit)
• write test cases covering all the supported commands (in very simple usage, just to make sure the parser recognizes them as valid tokens)
• sort the enwiki dump according to the commands in each snippet, to assemble test suites for each of the downstream milestones
• check that OCaml texvc passes all the valid tests and fails all the invalid tests
• depending on how long it took to run the full suit the first time, optionally implement a no-PNG flag to complete in reasonable time and make sure its output matches OCaml texvc without no-PNG so future testing goes faster
• decide whether dewiki significantly adds to the diversity of the test suites, and, if so, sort its dump the same way
• discuss with mentor the possibility of reordering the milestones so that the parser supports as much of enwiki's math as early as possible
• document the layout of the test cases
I plan to do the sorting manually, though I could be convinced otherwise. Certainly an automated way to assemble test suites from dumps off the production servers would be a great thing to leave for the next maintainer. On the other hand, some things are best done by putting an undergrad in a room for a week...

At the end, I should have test suites for all the supported commands and all of the forbidden ones.

### weeks 3 through 6: parsing

7 July (week 4)
start midterms
14 July (week 5)
end midterms
1. Translate all the BNF-like parsing rules over from OCaml, with stubs for all the parser actions a single hardcoded example for each token
2. Fill in the tokenizer patterns and the parser actions

The order in which I implement these may change if it turns out some commands are notably more popular than others, but at the moment, based on the Math help page[21], I'd divide the parser into four categories:

1. grouping with braces
2. commands that just substitute a symbol, like the Greek alphabet, operators, etc., in the order listed
3. commands that modify display, like typefaces and color
4. commands which modify layout, like in "Larger Expressions"

At the end, Python texvc should call TeX on all valid cases and reject all invalid cases. I will also provide documentation for that parser, and a document in the style of a quick-start guide which explains how to add support for a new command.

### weeks 7 through 8: HTML

At the end, Python texvc should match OCaml texvc for the supported HTML output modes.

1. extend texvctest.sh to allow each of the three HTML rendering policies: always always PNG, HTML if simple, HTML if possible
2. implement conservative HTML output
3. possibly also implement moderate and liberal HTML output, depending on statistics and scheduling

Depending on how far we've fallen behind at this point, whether due to schedule slip or feature creep, I'd discuss with my mentor at this point how important moderate and liberal HTML support is.

### week 9: Windows compatibility or refactoring as extension

11 August (week 9)
"pencils down"

I've left this until last because it's not certain whether I'll be able to get a Windows testing environment. I'll have been working throughout the previous weeks to try to find one, or, failing that, find someone with one who could work with me. However, if at all possible I would like to test the Python texvc on Windows.

If I haven't solved bug 13518[5] yet, this would be the time to pick it up again.

At the end, the Python texvc should work when called from Cygwin, the native prompt ("PowerShell"?), and the web server.

If I haven't been able to get a Windows testing environment, then I'll refactor the Math code as an extension without pulling the original PHP out of core. Apparently there's some disagreement over whether Math should now be moved to an extension, and that hasn't been settled yet. In particular, upgrading older installation past the core-to-extension transition might be confusing.

### weeks 10 through 11: clean-up

18 August (week 10)
official end of coding, start of finals
• check that Python texvc doesn't need write access to its own executable like OCaml texvc[20]
• go through my bugtracker and close as many tickets as possible
• check the Python texvc for each bug reported against the OCaml texvc in the official Bugzilla; solve them, if possible
• submit remaining tickets from my bugtracker to the official Bugzilla

### week 12: documentation

3 September (week 12)
start code submissions, actual end of coding
• revise and copy-edit inline documentation
• port the docstrings over to Doxygen-style inline documentation as required by Mediawiki style guide
• Now that I have the entire summer's experience behind me, look over my notes and write up a document in the style of How to become a MediaWiki hacker, with solutions to any unobvious problems a new contributor might run into setting up to write a quick patch for Python texvc
• Add a Math blurb to How to become a MediaWiki hacker
• copy any remaining open tickets from my bugtracker to the official Bugzilla
• summarize my notes into an English document in the style of Mediawiki's Manual:Code[22], explaining at a high level the architecture of Python texvc
I realize the documentation sounds a bit grandiose considering that I've only ported one tiny corner of the Mediawiki codebase, but considering this project started off because the OCaml version wasn't getting enough maintenance, I want to make it as easy as I can for people to start contributing.

## Participation

We don't just want to know what you plan to accomplish; we want to know how. Briefly describe your work style: how you plan to communicate progress, where you plan to publish your source code while you're working, how and where you plan to ask for help. (We will tend to favor applicants that demonstrate a clear vision for what it means to be an active participant in our development community)

From a previous project at work, I've discovered a tendency to get lost in the nitpicky details unless I broke the project into smaller tasks with their own deadlines, and reminded myself frequently of the big picture. A few things that helped:

• a bug tracker which I used as a glorified to-do list and scratch pad
• writing myself an email each night summarizing my progress for the day and laying out my goals for the next
• explicitly scheduling time for things like eating, sleeping, and miscellaneous not-programming fun, in order to avoid burn-out
• taking copious notes as I went, like documentation for myself but less formally structured, usually entered into the bug tracker or a permanent scratch file
• putting my code in source control and actually exercising some SCM discipline, like writing useful commit logs and committing frequently
• an automated way to run some fairly comprehensive test suites with a few commands

I'd like to propose three different kinds of communication:

• specific questions about the code, like stubborn bugs, done very quickly and informally over email or IRC
• a weekly email to my mentor specifically to report my progress and plan the next week, and a meeting on IRC to discuss the schedule and vaguer, higher-level questions that need more back-and-forth than would be convenient over email
• code review after each milestone

Generally, my programming could be called some kind of infantile Agile or test-driven methodology:

1. set up a development environment
2. write a testing framework
3. write a functional specification more detailed than "like the OCaml texvc does it"[23]
4. repeat until everything has been implemented:
1. write some tests that the OCaml texvc never fails, which define some minimally functional slice of texvc
2. code until those tests pass
3. write more stringent tests, fixing the code as necessary so they too pass
4. document this
5. Refactor (using the test suites to avoid regression) until I run out of time

## Past open source experience

Do you have any past experience working in open source projects (MediaWiki or otherwise). If so, tell us about it!

I haven't got any projects whose code would be available for people to read, but my work as a sysadmin has occasionally required minor bits of software development. For example, I have maintained an in-house print system for two years, which involved Python and MySQL in the server process and and PHP web interface. Since a core feature was the ability to filter out print jobs which violated lab policy, my modifications included improved copy-counting for MSOffice documents that rendered to raw PostScript, a temporary ban feature so users could not flood the system with decoy jobs, a new keep-alive mechanism using Mac OS X's LaunchDaemon system, and various other modifications as users developed new strategies for obfuscating their documents or deceiving the desk staff.

Although this project is not open-source, it was similar to distributed open-source development in one critical way: I picked up a mature project which was actively developed by multiple people at the same time, brought myself up to speed by reviewing the code and what documentation there was, and communicated with my fellow contributors and the original developers primarily by email. I hope the same skills will serve me well as I port texvc.

## Any other info

This project has been discussed on wikitech-l in a very long thread titled, "GSoC project advice: port texvc to Python?"[24].

### Second mentor

Prof. John Reppy[25] has agreed to answer any OCaml-specific questions that I couldn't find in the manual or online. I hope that, with his expertise available, the issue of not actually knowing any OCaml will be a serious problem in this project.

### PHP or Python?

One of the most difficult decisions in drawing up this proposal was choosing between Python and PHP. Both are similarly popular, well-supported, and mature. Neither would constitute a significant burden upon sysadmins to set up or maintain, and either would enormously expand texvc's pool of potential contributors and maintainers.

Since PHP is the language of choice for the rest of Mediawiki, though, a PHP port would have a few advantages which Python could not match, namely,

• a PHP dependency is essentially not a dependency at all, if the user wants to install the rest of Mediawiki
• while it wouldn't be difficult to find Python experience, Mediawiki already has plenty of experienced PHP developers
• a native PHP port would immediately eliminate the quoting problems that currently break TeX support on Windows[5]

The reason I chose Python anyway was the availability of mature parser-generator packages, which PHP cannot match. To summarize my argument on the wikitech-l mailing list[26], if the port is motivated by maintainability concerns due to the obscurity of the OCaml language, then it would be a step backwards to port to PHP because at this point, manually written LALR parsers would be even more obscure than OCaml.

Recently, James Salsman turned my argument on its head when he suggested LIME[27], which purports to be a

Complete LALR(1) parser generator and engine (like BISON or YACC) but it's all done in PHP, and the input grammar is easier and more maintainable.

If this is actually true, I'd be all in favor of a PHP port. However, I'm concerned about LIME's stability. It hasn't been updated in almost two years, and I can't find anybody talking about it or filing bugs against it, which suggests that it has lots of hidden flaws that haven't been discovered yet because it's not ready for general use and it's not being actively tested or developed any more. Aside from a recommendation on stackoverflow[28], I can't find anyone who will recommend it. I'd love to be proven wrong, but I don't think LIME is stable enough for production use.


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USF Home > College of Arts and Sciences > Department of Mathematics & Statistics

Mathematics & Statistics

# Discrete Mathematics (Leader: Prof. Greg McColm <mccolm (at) usf.edu>document.write('Prof. Greg McColm');)

## Monday, December 3, 2012

Title
Speaker
Time
Place

Sub(hypergraph) frequency algorithms
Brendan Nagle
3:05pm-3:55pm
CMC 120

Abstract

Let $$F$$ be a fixed $$r$$-graph ($$r$$-uniform hypergraph) on $$f$$ vertices. Let $$H=H_n$$ be a given $$r$$-graph on $$n$$ vertices. How quickly can you count the number of induced copies of $$F$$ in $$H$$? In particular, how much better than $$O(n^f)$$ can you do?

In the case of graphs ($$r=2$$), this problem is well understood, and the best algorithms come from Fast Matrix Multiplication. In the case of hypergraphs ($$r\ge 3$$), only a few partial results are known. We survey both the graph and hypergraph case. We conclude the talk with an elementary proof of the speaker that the induced $$F$$-count in $$H$$ can always be done in time $$o(n^f)$$, which had been an open question of R. Yuster.

We also survey what happens, in both the graph and hypergraph cases, if one replaces an exact count with a close estimate. When $$r=2,3$$ and $$|H|=\Omega(n^r)$$, one can give asymptotic estimates on the induced $$F$$-count in $$H$$ with surprising improvements to the running time. These details depend, however, on algorithmic versions of (strong) regularity lemmas, and so we omit such proofs.

## Monday, November 26, 2012

Title
Speaker
Time
Place

Iterated monodromy groups and $$L$$-presented groups
Dmytro Savchuk
3:05pm-3:55pm
CMC 120

Abstract

A relatively recent discovery (mainly due to Nekrashevych) of the beautiful connection between holomorphic dynamics and groups generated by automata is realized by iterated monodromy groups. Here is a rough idea: with every (nice) $$d$$-fold self-covering of a (nice) topological space $$X$$ one can associate the group acting on the tree of the preimages of any point in $$X$$ by automorphisms. It can be shown that under certain labeling of the tree this group can be generated by states of an automaton, which, in turn, may be used to understand the structure of the group.

In particular, it is known that many iterated monodromy groups admit finite $$L$$-presentations. $$L$$-presented groups are finitely generated groups that admit a presentation involving finitely many relators and their iterations by substitution. Such presentations are at the first level of complexity after the finite presentations and quite often provide the simplest way to describe a group that is not finitely presented. In this talk I will give an introduction to iterated monodromy groups, including a baby example. Then I will spend some time discussing $$\mathrm{IMG}(z^2+i)$$.

## Monday, November 19, 2012

Title
Speaker
Time
Place

A New Approach to Permutation Polynomials over Finite Fields
Neranga Fernando
3:05pm-3:55pm
CMC 120

Abstract

Let $$p$$ be a prime and $$q=p^k$$. A polynomial $$f(x)\in\mathbb{F}_q[{\tt x}]$$ is called a permutation polynomial (PP) over finite field $$\mathbb{F}_q$$ if it induces a bijection from $$\mathbb{F}_q$$ to itself. The polynomial $$g_{n,q}\in\mathbb{F}_p[x]$$ defined by the functional equation $$\sum_{a\in\mathbb{F}_q} (x+a)^n=g_{n,q}\left(x^q-x\right)$$ gives rise to many permutation polynomials over finite fields. We are interested in triples $$(n,e;q)$$ for which $$g_{n,q}$$ is a permutation polynomial of $$\mathbb{F}_{q^e}$$. We present recent discoveries of permutation polynomials in form of $$g_{n,q}$$.

## Monday, November 12, 2012

No seminar this week due to the holiday.

## Monday, November 5, 2012

Title
Speaker
Time
Place

A Duality between Pro-Filters and Varieties of the Form $$V(\Omega(A))$$
Joseph Van Name
3:05pm-3:55pm
CMC 120

Abstract

The category of pro-filters is the category of inverse systems of objects of the form $$(X,\mathcal{F})$$ such that $$\mathcal{F}$$ is a filter on the set $$X$$. If $$A$$ is an infinite set, then let $$\Omega(A)$$ to be the algebra with underlying set $$A$$ where each function $$f:A^{n}\rightarrow A$$ is a fundamental operation on $$A$$ and each $$a\in A$$ is a fundamental constant. I shall give an equivalence between a full subcategory of the category of pro-filters and the variety $$V(\Omega(A))$$ generated by $$\Omega(A)$$.

## Monday, October 29, 2012

Title

Speaker
Time
Place

Hashing and Indexing of Foreground vs. Background to Speed-up Tiling Puzzle: Solutions Using Distributed Processing Clusters
Apurva Bhatty
3:05pm-3:55pm
CMC 120

Abstract

Tiling puzzles may allow only translations (Wang tiles) or both rotations and translations (jigsaw puzzles, shredded documents). Tiling puzzles may have edges that are non-ordered (Wang Tiles) or that have a partial order. I will review some heuristics useful in solving these different classes of tiling puzzles. Using two examples from the DARPA Shredder Challenge Puzzle #5, different ways will be presented that these heuristics (and an algorithm for partially-ordered edges) can be carried out on distributed processing systems. A preliminary result about the likelihood of the existence of multiple solutions based on the number of colors leads to a conjecture about the “difficulty” of the puzzles.

## Monday, October 22, 2012

Title
Speaker

Time
Place

Lucas Sabalka
Saint Louis University
3:05pm-3:55pm
CMC 120

Abstract

The LYM inequality for the boolean lattice says that if you give the subset $$S$$ of $$\{1,\dotsc,n\}$$ a weight of $$1/{n \choose |S|}$$ then the sum of the weights in an antichain is at most $$1$$. This fact gives an easy proof of Sperner's Theorem, that the size of a largest antichain in the boolean lattice is $${n \choose n/2}$$. In this talk we will discuss what happens when you use other weights on other graded posets. In particular, we will characterize weightings that give an LYM inequality or a strict LYM inequality, and (in some sense) determine when these can be used to find large antichains. If time permits, we will prove a more general version of a sharpening of the LYM inequality due to Aydinian and Peter Erdős, which is itself a generalization of a theorem of Ahlswede and Zhang. For the most part, these results follow easily by adapting proof techniques from previously know special cases. This talk will be accessible to graduate students.

## Monday, October 15, 2012

Title
Speaker
Time
Place

Geometry of a separating curve complex analogue for $$\mathrm{Out}\,\left(F_n\right)$$
Dmytro Savchuk
3:05pm-3:55pm
CMC 120

Abstract

One of the most popular questions about $$\mathrm{Out}\,\left(F_n\right)$$ — a group of outer automorphisms of a free group of rank $$n$$ — is the construction of a Gromov hyperbolic space on which $$\mathrm{Out}\,\left(F_n\right)$$ naturally acts, extending the deep analogy between $$\mathrm{Out}\,\left(F_n\right)$$, mapping class groups, and linear groups. Each mapping class group naturally acts on a curve complex of the surface, which is hyperbolic due to a celebrated theorem by Masur and Minsky. Many spaces which could serve as an analogue for this space were proposed in recent years. We will define some of them and concentrate mostly on a space named the “Edge Splitting Graph” (also known as “Free Factorization Graph”). The main outcome of our work is that this space in not hyperbolic, so it should not be considered as a good analogue to curve complex. I will give some background and introduce the ideas and some techniques used in the proof. This is a joint work with Lucas Sabalka.

## Monday, October 8, 2012

Title
Speaker
Time
Place

Genus Range of $$4$$-regular Rigid Vertex Graphs
Nataša Jonoska
3:05pm-3:55pm
CMC 120

Abstract

Graphs with $$4$$-valent and $$1$$-valent rigid vertices, called assembly graphs, have been used to model homologous DNA recombination. We study the topological complexity of spatial embeddings of assembly graphs through studying the genus ranges of these graphs as a function of the number of vertices in the graphs. We show several observations about the genus ranges and state a couple of conjectures.

## Monday, October 1, 2012

Title
Speaker
Time
Place

Schreier graphs and Schreier Dynamical Systems of Thompson's group $$F$$
Dmytro Savchuk
3:05pm-3:55pm
CMC 120

Abstract

Schreier graphs are graphs constructed from the action of a group on a set that encode orbits. These graphs are direct generalizations of the Cayley graph of a group. For Thompson's group $$F$$, one of the motivations for considering Schreier graphs comes from the most famous question about this group: whether or not Thompson's group $$F$$ is amenable (which was very recently announced to have a positive answer). We discuss amenability and consider some examples. We then construct the Schreier graphs for the action of Thompson's group $$F$$ on the Cantor set with respect to the standard generating set, classify these graphs up to isomorphism, and study the corresponding Schreier dynamical systems.

## Monday, September 24, 2012

Title
Speaker
Time
Place

On some problems in interpolation related to combinatorial algebra
Boris Shektman
3:05pm-3:55pm
CMC 120

Abstract

In hope to solicit interest, advise and/or cooperation from the participants of the seminar, I will outline several problems that I came about through my work in multivariate interpolation. These problems turned out to be closely related to algebraic questions on hyperplane arrangements. I will talk about what little (and it is very little) results I have and what I would like to know and not afraid to ask.

## Monday, September 17, 2012

Title
Speaker
Time
Place

Periodic Graphs: Towards a Definition, Part II
Greg McColm
3:05pm-3:55pm
CMC 120

## Monday, September 10, 2012

Title
Speaker
Time
Place

Periodic Graphs: Towards a Definition
Greg McColm
3:05pm-3:55pm
CMC 120

Abstract

In textbooks, definitions are often taken for granted and the elbow grease is poured into theorems and occasionally examples. But a lot of effort goes into getting the right definitions. We look at one case study: the definition of “periodic graphs”. Intuitively, a periodic graph should be a graph embedded in a Euclidean space such that for some basis of that space, the corresponding translations induce automorphisms of that graph. But there is more to it than that.


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# A general formula for Hecke-type false theta functions

Dec 2022

In recent work where Matsusaka generalizes the relationship betweenHabiro-type series and false theta functions after Hikami, five families ofHecke-type double-sums of the form \begin{equation*} \left( \sum_{r,s\ge 0}-\sum_{r,s<0}\right)(-1)^{r+s}x^ry^sq^{a\binom{r}{2}+brs+c\binom{s}{2}},\end{equation*} where $b^2-ac<0$, are decomposed into sums of products of thetafunctions and false theta functions. Here we obtain a general formula for suchdouble-sums in terms of theta functions and false theta functions, whichsubsumes the decompositions of Matsusaka. Our general formula is similar instructure to the case $b^2-ac>0$, where Mortenson and Zwegers obtain adecomposition in terms of Appell functions and theta functions.

Q1论文试图解决什么问题？
Q2这是否是一个新的问题？
Q3这篇文章要验证一个什么科学假设？
0


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## Engage NY Eureka Math 3rd Grade Module 7 Lesson 1 Answer Key

### Eureka Math 3 Module 7 Lesson 1 Pattern Sheet Answer Key

Multiply

multiply by 3(1 – 5)

Explanation:
3 × 1 = 3
3 × 2 = 6
3  × 3 = 9
3 × 4 = 12
3  × 5 = 15.

### Eureka Math Grade 3 Module 7 Lesson 1 Problem Set Answer Key

Lena’s family visits Little Tree Apple Orchard. Use the RDW process to solve the problems about Lena’s visit to the orchard. Use a letter to represent the unknown in each problem.

Question 1.
The sign below shows information about hayrides at the orchard.

a. Lena’s family buys 2 adult tickets and 2 child tickets for the hayride. How much does it cost Lena’s family to go on the hayride?
Total cost of tickets for Lena’s family to go on the hayride = $22. Explanation: Cost of each adults tickets =$7
=> Cost of Two adults tickets = $7 × 2 =$14.
Cost of Each child ticket = $4 => Cost of Two Child tickets =$4 × 2 = $8 Total cost of tickets for Lena’s family to go on the hayride = Cost of Two adults tickets + Cost of Two Child tickets =$14 + $8 =$22.

b. Lena’s mom pays for the tickets with $5 bills. She receives$3 in change. How many $5 bills does Lena’s mom use to pay for the hayride? Answer: Number of$5 bills she uses to pay the bill = 5.

Explanation:
Total cost of tickets for Lena’s family to go on the hayride = $22 Amount of change she receives =$3
Total amount paid for tickets = Total cost of tickets for Lena’s family to go on the hayride + Amount of change she receives
= $22 +$3
= $25 Number of$5 bills she uses to pay the bill = Total amount paid for tickets  ÷ 5
= $25 ÷ 5 = 5. c. Lena’s family wants to go on the fourth hayride of the day. It’s 11:38 now. How many minutes do they have to wait for the fourth hayride? Answer: Time left for Lena’s family to wait for their fourth hayride = 7 minutes. Explanation: Time of their hayride of the day now = 11:38 Leaves every 15minutes hayride starting from 11:00. Number of hayride lens family going = Fourth of the day => Time of first hayride = 11:00 to 11:15 => Time of Second hayride = 11:15 to 11.30 => Time of Third hayride = 11:30 to 11:45 => Time of Fourth hayride = 11:45 to 12:00. Time left for Lena’s family to wait for their fourth hayride = Time of Fourth hayride – Time of their hayride of the day now = 11:45 – 11:38 = 7 minutes. Question 2. Lena picked 17 apples, and her brother picked 19. Lena’s mom has a pie recipe that requires 9 apples. How many pies can Mom make with the apples that Lena and her brother picked? Answer: Number of pies Mom can make with the apples that Lena and her brother picked = 4. Explanation: Number of Apples Lena picked = 17 Number of Apples Lena’s brother picked = 19 Number of Apples Lena’s mom needs to prepare a pie = 9 Number of Apples Lena and her brother picked = Number of Apples Lena picked + Number of Apples Lena’s brother picked = 19 + 17 = 36 Number of pies Mom can make with the apples that Lena and her brother picked = Number of Apples Lena and her brother picked ÷ Number of Apples Lena’s mom needs to prepare a pie = 36 ÷ 9 = 4. Question 3. Lena’s dad gives the cashier$30 to pay for 6 liters of apple cider. The cashier gives him $6 in change. How much does each liter of apple cider cost? Answer: Cost for the liter of apple cider =$4.

Explanation:
Amount of money Lena’s dad gives to the cashier to pay for 6 liters of apple cider = $30 Amount of money Lena’s dad gets in change =$6
Cost  for the 6 liters of apple cider = Amount of money Lena’s dad gives to the cashier to pay for 6 liters of apple cider  – Amount of money Lena’s dad gets in change
= $30 –$6
= $24. Number of liters of apple cider purchased = 6 Cost for the liter of apple cider = Cost for the 6 liters of apple cider ÷ Number of liters of apple cider purchased =$24 ÷ 6
= $4. Question 4. The apple orchard has 152 apple trees. There are 88 trees with red apples. The rest of the trees have green apples. How many more trees have red apples than green apples? Answer: Number of Green apple trees = 64. Number of Red apples more than Green apples = 24. Explanation: Number of Apple trees the Apple orchard has = 152 Number of Red apple trees = 88 Number of Green apple trees = Number of Apple trees the Apple orchard has – Number of Red apple trees = 152 – 88 = 64. Number of Red apples more than Green apples = Number of Red apple trees – Number of Green apple trees = 88 – 64 = 24. ### Eureka Math Grade 3 Module 7 Lesson 1 Exit Ticket Answer Key Use the RDW process to solve the problem below. Use a letter to represent the unknown. Sandra keeps her sticker collection in 7 albums. Each album has 40 stickers in it. She starts a new album that has 9 stickers in it. How many total stickers does she have in her collection? Answer: Total number of stickers Sandra has in each album = 640. Explanation: Number of stickers Sandra has in each album of 7albums = 40 × 7 = 280. Number of stickers Sandra has in each album of 9albums = 40 × 9 = 360 Total number of stickers Sandra has in each album = Number of stickers Sandra has in each album of 7albums + Number of stickers Sandra has in each album of 9albums = 280 + 360 = 640. ### Eureka Math Grade 3 Module 7 Lesson 1 Homework Answer Key Max’s family takes the train to visit the city zoo. Use the RDW process to solve the problems about Max’s trip to the zoo. Use a letter to represent the unknown in each problem. Question 1. The sign below shows information about the train schedule into the city. a. Max’s family buys 2 adult tickets and 3 child tickets. How much does it cost Max’s family to take the train into the city? Answer: Total amount for the Max’s family to take the train into the city =$34.

Explanation:
Amount of money for Adult ticket = $8 => Amount of money for 2 Adult ticket =$8 × 2 = $16. Amount of money for Child ticket =$6
=> Amount of money for 3 Child ticket =$6 × 3 =$18.
Total amount for the Max’s family to take the train into the city = Amount of money for 2 Adult ticket + Amount of money for 3 Child ticket
= $16 +$18
= $34. b. Max’s father pays for the tickets with$10 bills. He receives $6 in change. How many$10 bills does Max’s father use to pay for the train tickets?
Number of $10 bills Max’s father used to make the payment for the tickets = 4. Explanation: Amount of money paid by Max’s father for the tickets with$10 bills.
Amount of money received as change = $6 Total amount for the Max’s family to take the train into the city =$34
Number of $10 bills Max’s father used to make the payment for the tickets = Total amount for the Max’s family to take the train into the city + Amount of money received as change ÷ 6 =$34 + $6 ÷ 10 =$40 ÷ 10
=4.

c. Max’s family wants to take the fourth train of the day. It’s 6:38 a.m. now. How many minutes do they have to wait for the fourth train?
Time left for Max’s family to wait for their fourth train = 7 minutes.

Explanation:
Time of their Fourth train of the day now = 6:38 a.m.
Leaves every 15minutes train starting from 6:00 a.m.
Number of the train Max’s family to go = Fourth of the day
=> Time of first train  = 6:00 to 6:15
=> Time of Second train  = 6:15 to 6.30
=> Time of Third train  = 6:30 to 6:45
=> Time of Fourth train = 6:45 to 7:00.
Time left for Max’s family to wait for their fourth train = Time of Fourth train – Time of their train of the day now
= 6:45 – 6:38
= 7 minutes.

Question 2.
At the city zoo, they see 17 young bats and 19 adult bats. The bats are placed equally into 4 areas. How many bats are in each area?
Number of Bats in each area = 9.

Explanation:
Number of Young bats in the zoo = 17
Number of Adults bats in the zoo = 19
The bats are placed equally into 4 areas.
=> Number of areas bats placed equally = 4
Number of Bats in each area = (Number of Adults bats in the zoo + Number of Young bats in the zoo) ÷ Number of areas bats placed  equally
= (19 + 17) ÷ 4
= 36 ÷ 4
= 9.

Question 3.
Max’s father gives the cashier $20 to pay for 6 water bottles. The cashier gives him$8 in change. How much does each water bottle cost?
Cost of each water bottle = $2. Explanation: Amount of money given to cashier by Max’s father to pay for 6 water bottles =$20
Amount of money received as change = $8 Total number of water bottles purchased = 6 Cost of each water bottle = Amount of money given to cashier by Max’s father to pay for 6 water bottles – Amount of money received as change ÷ Total number of water bottles purchased =$20 – $8 ÷ 6 =$12 ÷ 6
= \$2.

Question 4.
The zoo has 112 types of reptiles and amphibians in their exhibits. There are 72 types of reptiles, and the rest are amphibians. How many more types of reptiles are there than amphibians in the exhibits?


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# Find the smallest $t$ such that the object reaches the height of $96\text{ feet}$ at time $t$.

An object is tossing upwards with an initial speed of $64 \text{ feet/sec}$.Suppose the gravitational acceleration is $32\text{ feet/sec}^2$. Find the smallest $t$ such that the object reaches the height of $96\text{ feet}$ at time $t$.

My problem: Which formula I have to use to find $t$?

Thanks.

• It seems like object will never reach $96$ feet. – Jihad Oct 24 '14 at 17:10
• @Jihad But what is the formula to solve such problem?$s=v_0+(1/2)gt^2$? – Flip Oct 24 '14 at 17:13
• $S = S_0 + v_0t + \frac{gt^2}{2}$ – Andrei Rykhalski Oct 24 '14 at 17:13
• @AndreiRykhalski what is $S_0$? – Flip Oct 24 '14 at 17:14
• @Flip Starting space coordinate, in your case an object is on a ground level, so $S_0 = 0$. – Andrei Rykhalski Oct 24 '14 at 17:15

We have $$s = ut + \frac{1}{2}at^2$$
which we can rearrange to get $$t^2 + 2 \dfrac{2u}{a} - 2 \dfrac{s}{a} = 0$$ and applying the quadratic equation to that (with correct values of s, u, and a) should get you your answer!


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## Book Store

Currently only available for.
CBSE Gujarat Board Haryana Board

## Previous Year Papers

Download the PDF Question Papers Free for off line practice and view the Solutions online.
Currently only available for.
Class 10 Class 12

In terms of band theory, what is the difference between a conductor and an insulator?

Solution:

Molecular orbirals of metals are formed by atomic orbitals. These orbitals are so close to each other as they form band or valence band.

(i) Difference between conductor and insulator - In conductors there is no energy gap between the valence band, which facilitates the flow of electrons easily under an applied electric field and metals show conductivity.

While in insulators there is large energy gap between the valence band and electrons cannot jump to it i.e. large energy gap prevents the flow of electricity.

(ii)Difference between conductors and semiconductor - In conductors there is no energy gap between the valence band and conduction band, which facilitates the flow of electrons easily under an applied electric field and metals show conductivity.

While in semi conductors, there is small energy gap between valence bond and conduction band. The small gap between band facilitates some electrons to jump to the conduction band by acquiring extra energy.

231 Views

Why are solids rigid?

Solids are rigid because the internuclear distance between the  molecules of a solid are very less and closely packed and their positions are fixed due to the strong forces of attraction between them.
2488 Views

Classify the following as amorphous or crystalline solids: Polyurethane, naphthalene, benzoic acid, teflon, potassium nitrate, cellophane, polyvinyl chloride, fibre glass, copper.

Amorphous solids: Polyurethane, teflon, cellophane, polyvinyal chloride, fibre glass.
Crystalline solids: naphthlene, benzoic acid, potassium nitrate, copper.
1237 Views

Why do solids have a definite volume?

A solid have a definite volume because the inter molecular distance  between its molecules are fixed.
1439 Views

Refractive index of a solid is observed to have the same value along all directions. Comment on the nature of this solid. Would it show clevage property?

Refractive index of a solid which have the same value along all directions are isotropic in nature. It would not show cleavage property.
1055 Views

Why is glass considred a super-cooled liquid?

Glass is amorphous solids and  have a tendency to flow, though very slowly. Therefore, these are called pseudo solids or super cooled liquids .
1368 Views


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# Galois group of $x^p-x-a$

$F$ is a field of characteristic $p$ and $a\neq c^p-c$ for $c\in F$. Then determine the galois group of $x^p-x-a$.

First I showed that this is an irreducible polynomial and has no multiple roots. This will imply that the polynomial had $p$- distinct roots in its splitting field. Hence we can get a $p$-cycle. This means that the Galois group is $S_p$. Is my reasoning correct?

• Your $c^p - a$ should be $c^p - c.$ Concerning the question, just having a $p$-cycle is hardly sufficient. A separable irreducible cubic has Galois group $A_3$ or $S_3$. Both contain a $3$-cycle. You should think about how the roots are related to each other. If you have one root $\alpha$ then find a formula for the remaining roots in terms of $\alpha$. – KCd Apr 26 '15 at 0:46

Actually, if $\alpha$ is a root, then so is $\alpha+n$, for all $n\in\mathbb F_p$. Thus, adjoining one root gives the entire splitting field, and thus the splitting field is an extension of degree $p$. The Galois Group is thus isomorphic to $C_p$.


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## FANDOM

1,022 Pages

The product operator multiplies the terms of a sequence or partial sequence. It is denoted as

$\prod_{k=1}^n a_k=(a_1)(a_2)\cdots(a_{n-1})(a_n)$

Any infinite product of an will converge to a nonzero real number if and only if

$\sum_{n=1}^\infty\ln(a_n)=r$

converges. This can be determined by any convergence test.

A common application of the product operator is in factorials, where

$n!=\prod_{k=0}^{n-1}(n-k)=n(n-1)(n-2)\cdots(2)(1)$


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# Computing a derivative using logarithmic differentiation

Using the Logarithmic Differentiation find the derivative of $y=\sqrt{x(x-1)/(x-2)}$...so I tried,but the result is not correct..can you show me a hint?

so $\ln y= 0.5\ln[x(x-1)/(x-2)]$

$$\ln y=0.5[\ln x+\ln(x-1)-\ln(x-2)]$$

$$\ln y=0.5\ln x+0.5\ln(x-1)+0.5\ln(x-2)$$

$$y'=0.5\ln x+0.5\ln(x-1)-0.5\ln(x-2)[\sqrt{x(x-1)/(x-2)}]$$

-
I gave your question a more descriptive title, and added the calculus tag. –  Nate Eldredge Nov 20 '12 at 21:04
Check the plusses and minuses in the third row. Also, it'd be helpful if said what you got when you went on, instead of just saying it's not the same as what you want... –  Micah Nov 20 '12 at 21:21
Your last line should be $y'=\frac 12 \left(\frac 1x+\frac 1{x-1}-\frac 1{x-2}\right)\sqrt{x(x-1)/(x-2)}$ (you forgot to differentiate the $\ln$ terms and the parenthesis). –  Raymond Manzoni Nov 20 '12 at 22:05

## 2 Answers

Hint: Remember that the derivative of $\ln y$ is $\frac{y'}{y}$, by the chain rule. Also, you seem to have flipped a sign in your last step; the last term of the third line should be $-0.5 \ln(x-2)$.

Edit: After seeing what you did, you forgot to differentiate the terms with $x$! Suppose we have $\ln y = g(x)$. Then, differentiating both sides, $\frac{y'}{y} = g'(x)$, or $y'(x) = g'(x) y(x)$. You forgot to differentiate $g$, which in this case is $g(x) = \frac12[\ln x + \ln(x-1) - \ln(x-2)]$.

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wait,i will edit it.. –  AAS Nov 20 '12 at 21:21

Your error is that you didn't actually differentiate $\ln x$ and the other logarithms on the right side. $$\frac{d}{dx} (0.5\ln x) = 0.5\cdot\frac1x = \frac{0.5}x.$$

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# Are carboxylic acids strong acids?

May 14, 2016

No, the equilibrium lies to the left.

$R C \left(= O\right) O H + {H}_{2} O r i g h t \le f t h a r p \infty n s R C \left(= O\right) {O}^{-} + {H}_{3} {O}^{+}$

#### Explanation:

As acids go, carboxylic acids are fairly weak species, and acidity decreases as the length of the alkyl chain increases. See the table on this site for representative values.


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## Thursday, December 27, 2007 ... //

### Johannes Kepler: 436th birthday

Johannes Kepler was born prematurely near Stuttgart on 12/27/1571. His grandfather was a mayor of their town but once Johannes was born, the family's fortunes were already dropping. His father was a mercenary and left the family when Johannes was five. His mother was a healer and a witch which has also led to some legal problems.

Johannes was a brilliant child with early inclinations to astronomy. In Graz (1594-1600), he was defending the Copernican heliocentric system. At that time, there was no clear difference between astronomy and astrology. Therefore, Kepler also invented the ADE classification of planets orbiting the Sun. ;-) This attempt resembled, but was not identical to, Garrett Lisi's hopeless attempt to unify. Kepler also wrote that the Universe had to be stationary.

In 1600, Kepler finally met Tycho Brahe in Benátky nad Jizerou (see the picture), a central Bohemian town where Brahe built an observatory. Brahe quickly recognized Kepler's magic theoretical powers. Their negotiations about the new Kepler's job in Prague were accompanied by arguments and tension. Fortunately for the Czech capital, Kepler had more serious problems with Graz where they expected him to convert to Catholicism. Finally, Kepler moved to Prague, including his family.

He became the imperial mathematician of our monarchy, an advisor to Emperor Rudolph II, a predecessor of Václav Klaus and a great sponsor of arts and sciences pictured above, and 11 most productive years of Kepler's life were just getting started. Kepler was also giving his political recommendations to the empire although his common sense was more instrumental than the stars.

In Prague, Kepler established modern optics (he understood geometry of lunar shadows, the inverse-square law controlling the light intensity, and other things). In 1604, he started to observe SN 1604, a supernova also known as Kepler's star, in the constellation Serpentarius, the 13th sign of the zodiac in which your humble correspondent was born. Click at the link and see Kepler's beautiful drawing of Ophiuchus, as the constellation should now be called. It started to be clear to him that the heavens were not as constant as Aristotle used to think.

Kepler's laws

In 1602, Kepler discovered the law that we derive from the conservation of the angular momentum these days. He had a somewhat strange, non-Newtonian interpretation of it: the Sun provides the planets with motive power that decreases as they get further from the Sun which means that when the planets are far away from the Sun, they move more slowly. ;-)

Tycho's very accurate data about Mars were very important for Kepler. Incidentally, Mars may be hit by an asteroid in one month: the probability exceeds 1 percent.

After 40 years of failed attempts, Kepler finally got the right idea about the shape of the orbit in 1605. Why didn't he think about the ellipse - the shape dictated by his first law - earlier? Because it was too simple and Kepler thought that the astronomers would have figured out such a solution a long time ago if it were correct. This story hides at least two general lessons.

The first lesson is that it is sometimes easier to learn important insights from examples that are not the simplest ones because their patterns are sharper, more visible, and more characteristic and they cannot be confused with others. The simplest cases and solutions often look too singular and their modest internal structure is a bad starting point for generalizations. More complex examples typically "pinpoint" the right general law or algorithm uniquely or almost uniquely.

The second lesson is a sociological one. While it may be more likely that a simple solution should have already been found by others, they may have overlooked it, too. And if they did, such a simple solution might be much more valuable. I think it follows that scientists shouldn't ignore a topic just because it was found uninteresting or unrealistic by many others.

On the other hand, you are never guaranteed to succeed. If you attempt to do something simple that has been tried by many others, you are less likely to find something new, especially if you are less gifted than Kepler.

Kepler realized that the Mars data agreed with the ellipse beautifully and he has abruptly and correctly deduced that all planets had elliptical orbits even though he couldn't have done the numerical calculations for all the planets (he had no postdocs and grad students). In 1610, Kepler also had a healthy and friendly exchange with Galileo Galilei, supporting his discovery of the moons and helping him and others to improve the telescope.

The last law of planetary motion I didn't mention was the third one: the orbital times squared are proportional to the distances cubed. Kepler included this law as an example of the harmonies that the Creator used to decorate the heavens.

Religious tension in Prague

Unfortunately, politics slowed the progress down in 1611. Rudolph II became seriously ill - and died in 1612 - and his brother Matthias who was 5 years younger and who was already controlling Austria, Hungary, and Moravia was able to grab the kingdom of Bohemia, too. This, of course, meant a dramatic decrease of the influence of so-far-dominant Bohemia within the Holy Roman Empire. It also meant a dramatic weakening of the conditions to do research in Prague (and elsewhere, for that matter). While Matthias confirmed Kepler's job and salary, he allowed him to leave for Linz, Austria. In Linz, Kepler was teaching at the district school and he was calculating the year of Christ's birth.

That was a pretty bad development for science. At least, Kepler's second marriage was much happier than the first. However, his writing got much less quantitative than during the golden years in Prague. It became somewhat astrological and similar to his early years. A monument to Brahe and Kepler at the picture above is in Prague 6, Pohořelec, a former place of another observatory of Brahe near the Prague Castle.

Jacob Bernoulli

Another guy who was born on December 27th was Jacob Bernoulli, namely in 1654. His parents wanted him to do theology but he preferred mathematics and astronomy. The Bernoulli numbers that appear in Martin Schnabl's solution to string field theory and elsewhere belong among his discoveries. However, you should be careful: there were 8 good mathematicians and physicists in his family. For example, the laws of hydrodynamics and aerodynamics are due to Daniel Bernoulli.