text:
# Direct Intervention. How can a central bank use direct intervention to change the value of a currenc

Direct Intervention. How can a central bank use direct intervention to change the value of a currency? Explain why a central bank may desire to smooth exchange rate movements of its currency.ANSWER:2. Indirect Intervention. How can a central bank use indirect intervention to change the value of a currency?ANSWER:3. Sterilized Intervention. Explain the difference between sterilized and non sterilized intervention.ANSWER:4 It has been argued that the exchange rate can be used as a policy tool. Assume that the U.S. government would like to reduce unemployment. Which of the following is an appropriate action given this scenario? Explain your answer.a. Weaken the dollarb. Strengthen the dollarc. Buy dollars with foreign currency in the foreign exchange marketd. Implement a tight monetary policyANSWER:5. It has been argued that the exchange rate can be used as a policy tool. Assume that the U.S. government would like to reduce inflation. Which of the following is an appropriate action given this scenario? Explain your answer.a. Ssell dollars for foreign currencyb. Bbuy dollars with foreign currencyc. Lllower interest ratesd. None of the aboveANSWER: 6. Locational Arbitrage. Assume the following information:Beal Bank Yardley BankBid price of New Zealand dollar $.401$.398Ask price of New Zealand dollar $.404$.400Given this information, is locational arbitrage possible? If so, explain the steps involved in locational arbitrage, and compute the profit from this arbitrage if you had $1,000,000 to use. What market forces would occur to eliminate any further possibilities of locational arbitrage?ANSWER:7. Triangular Arbitrage. Assume the following information:Quoted PriceValue of Canadian dollar in U.S. dollars$0.90Value of New Zealand dollar in U.S. dollars $0.30Value of Canadian dollar in New Zealand dollars NZ$3.02Given this information, is triangular arbitrage possible? If so, explain the steps that would reflect triangular arbitrage, and compute the profit from this strategy if you had $1,000,000 to use. What market forces would occur to eliminate any further possibilities of triangular arbitrage?ANSWER:8. Covered Interest Arbitrage. Assume the following information:Quoted PriceSpot rate of Canadian dollar$.8090 day forward rate of Canadian dollar $.7990 day Canadian interest rate 4.0%90 day U.S. interest rate 2.5%Given this information, what would be the yield (percentage return) to a U.S. investor who used covered interest arbitrage? (Assume the investor invests$1,000,000.) What market forces would occur to eliminate any further possibilities of covered interest arbitrage?ANSWER:9. Assume that the interest rate in the home country of Currency X is a much higher interest rate than the U.S. interest rate. According to interest rate parity, the forward rate of Currency X: Explain your answer.a. should exhibit a discount.b. should exhibit a premium.c. should be zero (i.e., it should equal its spot rate).d. B or CANSWER: 10. Assume the following information:U.S. investors have $1,000,000 to invest:1-year deposit rate offered on U.S. dollars = 12%1-year deposit rate offered on Singapore dollars = 10%1-year forward rate of Singapore dollars =$.412Spot rate of Singapore dollar = \$.400Given this information:a. interest rate parity exists and covered interest arbitrage by U.S. investors results in the same yield as investing domestically.b. interest rate parity doesn’t exist and covered interest arbitrage by U.S. investors results in a yield above what is possible domestically.c. interest rate parity exists and covered interest arbitrage by U.S. investorsâ€™ results in a yield above what is possible domestically.d. interest rate parity doesn’t exist and covered interest arbitrage by U.S. investors results in a yield below what is possible domestically.11. Comparing Parity Theories. Compare and contrast interest rate parity, purchasing power parity (PPP), and the international Fisher effect (IFE).ANSWER:12. Source of Weak Currencies. Currencies of some Latin American countries, such as Brazil and Venezuela, frequently weaken against most other currencies. Why donâ€™t all U.S.-based MNCs use forward contracts to hedge their future remittances of funds from Latin American countries to the U.S. even if they expect depreciation of the currencies against the dollar?


text:
# \hyperref to a \label doesn't work (“empty destination key”)

In one of my documents, I used the combination of \label with \hyperref. Now, I fail for some reason.

I included the package:

\usepackage[colorlinks=true,urlcolor=blue,linkcolor=blue]{hyperref}

Created a label somewhere in my text:

\label{key1}

And referenced it from the text:

\hyperref[key1]{there}

But xelatex gives me this:

Package hyperref Warning: Empty destination name,
(hyperref)                using UNDEFINED' on input line 66.

What am I doing wrong?

EDIT: Minimal working example:

\documentclass[a4paper, 12pt]{article}

\usepackage{xltxtra}

\begin{document}

Some text. \label{key1} Continuing.

Some more text (\hyperref[key1]{goto}).

\end{document}

-
Your MWE works fine for me. Pl. improve it to show the actual problem. – Harish Kumar Apr 23 '12 at 8:19
@HarishKumar - what version of xelatex do you have? I have "XeTeX 3.1415926-2.2-0.9995.2". – Rogach Apr 23 '12 at 8:21
@HarishKumar - That MWE actually shows my problem (on my machine) - the compiler outputs that error message, and link does not work. – Rogach Apr 23 '12 at 8:21
@Rogach: My Xetex version is : This is XeTeX, Version 3.1415926-2.3-0.9997.5 (MiKTeX 2.9) (preloaded format=xelatex 2011.12.21) 23 APR 2012 13:55 entering extended mode. Looks like you have to update your distro. – Harish Kumar Apr 23 '12 at 8:25
the repositories are out of date- your example works fine in the latest texlive. Please see Martin's answer to best-way-to-install-packages-for-texlive-in-ubuntu- you don't need to build the deb packages yourself, texlive 2011 comes with very easy executables that do all of the work – cmhughes Apr 23 '12 at 8:59

It seems that \label can be attached only to sections or subsections (or some other environments, like figures). To be able to reference arbitrary points in text, I needed to define "phantom section":

\phantomsection
\label{key1}

-
What hyperref needs, for sure, is a "hook" or "target." However, TeX does not "attach" a \label directly to the place where the command is encountered but, rather, to the "item" it's designed to cross-reference -- an equation, the header of a section of a section, the caption of a figure, etc. If, though, you need to create a hyper-target to a \label that's not associated with a known target type, it's necessary to specify \phantomsection (which might better be called \explicittarget). – Mico Apr 23 '12 at 11:01
Since hyperref 2010/04/17 v6.80x this is no longer necessary. The target is initialized with Doc-Start, that is set at the begin of the document. – Heiko Oberdiek Aug 7 '12 at 1:39

The repositories are out of date- your example works fine in texlive 2011 available at

http://www.tug.org/texlive/acquire-netinstall.html

Please see Martin's answer to Best way to install packages for TeXLive in Ubuntu?

You don't need to build the deb packages yourself, texlive 2011 comes with very easy executables that do all of the work for you. Once you have texlive 2011 installed, you can update it using tlmgr update --self --all`

-

format: plain


text:
Sunday, June 01, 2008

Finding differences between IChemObjects

CDK trunk is getting into shape, thanx to the many people who contribute to this, and special thanx to Miguel for cleaning up his code related to charge, resonance, and ionization potential calculations!

At the moment, I am focusing at two issues:
1. QSAR descriptors that change the input (causing other descriptors to randomly fail)
2. Cloning of IChemObject (for which last week a rather serious bug was found)
Until some days ago, the CDK had one main method to introspect IChemObjects: their toString() results. However, finding the difference between IChemObjects using this approach is not trivial, particularly if there are several differences.

So, I started a new module called diff. If two objects are identical, it returns a zero-length String. If not, it lists the changes between the two classes, in a way much like that of the IChemObjects toString() methods.

For example, consider this bit of code:
IChemObject atom1 = new ChemObject();IChemObject atom2 = new ChemObject();atom2.setFlag(CDKConstants.ISAROMATIC, true);String result = ChemObjectDiff.diff( atom1, atom2 );
The result value then looks like:
ChemObjectDiff(, flag5:F/T)
Now, output will likely change a bit over time. But at least, I now have a easier to use approach for debugging and writing unit tests. Don't be suprised to see test-* modules start depending on the new diff module.


text:
## Surface Area of a Right Rectangular Prism

This post about finding the surface area of a right rectangular prism is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.

## Question

What is the surface area of the right rectangular prism shown? Its length is 6, its width is 5, and its height is 4.

Right rectangular prism for the example

## Solution

### Sidebar: Surface Area of a Right Rectangular Prism

A right rectangular prism is the shape of a box with a rectangle for a bottom, with sides perpendicular to its bottom, and with a flat top parallel to its bottom. It’s your typical box.

A right rectangular prism’s surface area is the sum of the areas of all its faces: the top, the bottom, and the four sides. The area of each side is length times width.

This box has six sides, three pairs of congruent sides.

• Let’s start with the side that stares you in the face on the diagram. It measures 4 x 5, so its area is 20. There is an identical piece in the back with the same area; the area of the two together is 2 times 20 = 40.
• Now consider the sides on the left and right. (In the diagram, you can’t see the side on the left.) Each of those sides measures 6 x 4, so each one’s area is 24. The area of the two together is 2 times 24 = 48.
• The top has a length of 6 and a width of 5, so its area is 6 X 5 = 30. The bottom has the same area, so the area of the two together is 2 times 30 = 60.

Then the surface area of the whole prism is

$\fn_jvn {\color{DarkBlue} 40+48+60=140}$

And the answer is 148 square units.

Right rectangular prism for formula

### In general

The formula for the surface area of a right rectangular prism — a box — is

$\fn_jvn {\color{DarkBlue} SA=2hw+2hl+2lw}$

where SA surface area and h, w, and l are as shown on the box on the right.

This question is similar to question number 3 in the sample questions for the Accuplacer Advanced Algebra and Functions test.

## Congruent Triangles

This post about congruent triangles is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.

## Question

Angle A in triangle ABC is congruent to angle D in triangle DEF. Which of the statements below, if it’s true, proves that triangles ABC and DEF are congruent?

A.  $\inline \fn_phv \angle C \cong \angle F$ and $\inline \fn_phv AC=DF$

B.  $\inline \fn_phv BC=EF$ and $\inline \fn_phv DF=AC$

C.  $\inline \fn_phv AB=DE$ and $\inline \fn_phv BC=EF$

D.  $\inline \fn_phv \angle B \cong \angle E$ and $\inline \fn_phv \angle C \cong \angle F$

Triangle DEF.

Triangle ABC.

## Solution

Two triangles are congruent if one of these things is true:

• every side of one is congruent to a side of the other (SSS);
• two sides and the angle between them in one are congruent to two sides and the angle between them in the other (SAS); or
• two angles and any side in one are congruent to two angles and any side of the other (ASA or AAS).

Let’s go through the answer choices to see if one of them meets one of those criteria.

Triangle DEF

A.  $\inline \fn_phv \angle C \cong \angle F$ and $\inline \fn_phv AC=DF$.

Triangle ABC

That’s  two  angles and the side between them in one triangle being congruent to the corresponding parts in the other triangle. That’s enough to meet the first criterion above, so it looks like that’s the answer. But let’s check the others.

B.  $\inline \fn_phv BC=EF$ and $\inline \fn_phv DF=AC$

Triangle ABC.

That’s two sides and an angle, but not the angle between them, so it does not do the job.

Triangle DEF

C.  $\inline \fn_phv AB=DE$ and $\inline \fn_phv BC=EF$.  Same problem as answer choice B: Two sides and one angle of one are congruent to two sides and one angle of the other, but the angle is not the one between the two sides, so that information is not enough for a proof.

Triangle DEF

Triangle ABC

D. $\inline \fn_phv \angle B \cong \angle E$ and $\inline \fn_phv \angle C \cong \angle F$. This gives you only congruent angles. Every proof of congruent triangles requires at least one side.

Triangle ABC


text:
# GATE2013-20

One of theeigcn values of $P=\begin{bmatrix}10 & -4 \\ 18 & -12\end{bmatrix}$ is
1. $2$
2. $4$
3. $6$
4. $8$


text:
# A Parigot-style linear $\lambda$-calculus for full intuitionistic linear logic

## Valeria de Paiva and Eike Ritter

This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL), an intriguing variation of multiplicative linear logic, due to Hyland and de Paiva. The system FILL resembles intuitionistic logic, in that all its connectives are independent, but resembles classical logic in that its sequent-calculus formulation has intrinsic multiple conclusions. From the intrinsic multiple conclusions comes the inspiration to modify Parigot's natural deduction systems for classical logic, to produce a natural deduction formulation and a term assignment system for FILL.

Keywords: linear logic, $\lambda\mu$-calculus, Curry-Howard isomorphism

2000 MSC: 03B20

Theory and Applications of Categories, Vol. 17, 2006, No. 3, pp 30-48.

http://www.tac.mta.ca/tac/volumes/17/3/17-03.dvi
http://www.tac.mta.ca/tac/volumes/17/3/17-03.ps
http://www.tac.mta.ca/tac/volumes/17/3/17-03.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/17/3/17-03.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/17/3/17-03.ps

TAC Home


text:
# Get all Magento URLs by store view

I'm looking to retrieve all URLs of a Magento website (CMS pages, categories, products...) subdivided by store view. Something like the default Magento Sitemap, but with a different format: specifically I'm trying to get a "one URL per row" plain text file, one for each store view.

Take a look at how Magento generates it's sitemap.xml file.

Specifically the Mage_Sitemap_Model_Sitemap class method generateXml. For example, following snippet generates them for the categories, I've rewritten it a little to export to a flat text file.

$ioWrite = new Varien_Io_File();$path = '';
$ioWrite->open(array('path' =>$path));
$ioWrite->streamOpen($path.'urls.txt');

$storeId = Mage::app()->getStore()->getId();$date    = Mage::getSingleton('core/date')->gmtDate('Y-m-d');
$baseUrl = Mage::app()->getStore($storeId)->getBaseUrl(Mage_Core_Model_Store::URL_TYPE_LINK);

/**
* Generate categories sitemap
*/
$changefreq = (string)Mage::getStoreConfig('sitemap/category/changefreq',$storeId);
$priority = (string)Mage::getStoreConfig('sitemap/category/priority',$storeId);
$collection = Mage::getResourceModel('sitemap/catalog_category')->getCollection($storeId);
$categories = new Varien_Object();$categories->setItems($collection); Mage::dispatchEvent('sitemap_categories_generating_before', array( 'collection' =>$categories
));
foreach ($categories->getItems() as$item) {
$ioWrite->streamWrite(htmlspecialchars($baseUrl . $item->getUrl())); } unset($collection);
\$ioWrite->close();

Add the code from the class to your own script to generate a text file

• Thanks @Sander for the hints, I've got a solution starting from this. – Riccardo Nov 11 '15 at 13:03

Thanks to @sander-mangel I've developed a solution for my specific case: https://gist.github.com/RiccardoB/2117046a97719b413c8a

If you have any optimization/fix to suggest, feel free to comment.


text:
# Faster way to compile a document [duplicate]

Writing a long document, compilation with Texmaker becomes too slow, so when I want to see how it looks, I loose time (not too much, but it is getting worse because of document's size).

Is there any way to compile faster?

• I guess it really depends on what "making some changes" entails. Please be more specific. – Mico Jun 5 '17 at 10:57
• Which editor do you use? TeXStudio f.i. has a kind of preview feature. – TeXnician Jun 5 '17 at 11:01
• Have you tried providing the document class option draft? It should speed up compilation dramatically (at the loss of not getting graphs included inside figure environments and not getting hyperlinks formed. Math material will still be compiled, though. Depending on what you need to achieve, setting the draft option may be worth it. – Mico Jun 5 '17 at 11:55
• You can set a main file containing the preamble,\include{file-nn.tex} directives in the body of the document, and an \includeonly{list of files to be compiled} directive in the preamble. – Bernard Jun 5 '17 at 12:11
• You do not give precise details what takes time. In case you have complicated tikz images it might be a good idea to "externalize" (you can read about that in the tikz documentation as well as on this site). But I think your question would be better if you found out what takes time. The solution will be different depending on that. – mickep Jun 5 '17 at 12:34

In a similar situation I decided, that when writing on a text passage I seldom need to see the whole document, but that compiling the current chapter/section/whatever is enough. For this task I use the subfiles package. With this package I can both compile the main file to get the complete output and the individual chapters alone to save time. Compiling the individual will still have the packages from the main file available and the look and feel of the document will be the same.

My file look like this:

main.tex:

\documentclass{book}
\usepackage{subfiles}

\usepackage{xcolor}% Just an example for all packages you want to use

\begin{document}
text

\subfile{chapter1}
\end{document}

and chapter1.tex

% !TeX root = chapter1.tex   % not necessary, but some editors think they are smart
\documentclass[main]{subfiles}

\begin{document}
the content of chapter 1, \textcolor{blue}{using packages from the main file}
\end{document}

Most of the time I use this together with

\usepackage{xr-hyper}% or \usepackage{xr} if you don't use hyperref
\externaldocument{Chapter2}

which can be used to access labels etc. from for example other subfiles

or with

\usepackage{zref-xr}
\zxrsetup{toltxlabel}
\zexternaldocument*[main-]{main}

to access things like pagenumbers or chapter numbers from the main file.

• Do subfiles have to be in same folder as main.tex? Your answer works only if i put my chapters in same file that main.tex is.. – santimirandarp Jun 12 '17 at 17:39
• @santimirandarp It's 100 times easier if you just have all the files in one folder. If you insist on subfolder, you probably have to adjust the path both in \documentclass[../main]{subfiles} and \subfile{./foldername/chapter1} [untested] – user36296 Jun 12 '17 at 19:00

I highly recommend recompiling often. Here are some thoughts which were too long for a comment:

1. You can see whether your code corresponds to what you wanted to achieve.
2. You get compiler errors quite early and thus you can detect and resolve them more early and easily. When I code some more complex math formulas, I even compile while writing the formulas to check whether what I wrote this far is correct.
3. Some people (not me) like WYSIWYG editors like Lyx (LaTeX Editors/IDEs), so you might want to try or read about it. And some people use Lyx and apply some fine-tuning with another editor afterwards.

You might want to compare What are good learning resources for a LaTeX beginner?.

In the end you can choose your workflow freely, so just see this as my opinion. The problem is of course the runtime. You cannot always get what you want. These might "speed up" the runtime:

1. Upgrade your engines (this helped me a couple of times). New distributions have just arrived.
2. If you have a small document, compile time usually is not an issue. If you have a larger document like a book, consider to divide your project with \input or \include and comment the temporarily uninteresting sections/chapters out or use \includeonly or the subfile-package (as mentioned by TeXnician and samcarter).
3. Get a faster computer. Not an easy one.
4. As others has have pointed out: Use draft modes.
5. If you use Tikz and plots, you might want to generate the plots using gnuplot.
6. Tikz is not build to be fast. Tikz-pictures can be externalized as pointed out by mickep correctly. You also might also consider inkscape as an alternative.
7. The use of saveboxes might be appropriate.
8. I am not sure whether a different engine like LuaTeX might speed up the compile time. This also might depend on your use of the Lua language, e.g. I once gained several minutes by writing a script in Lua instead of using \foreach directly.
9. Compiling a document on a terminal and not in a IDE might be faster.
10. See Speeding up LaTeX compilation.
• Please be aware that an engine upgrade might also slow down. And engines like Xe/LuaLaTeX are always slower than pdflatex. – TeXnician Jun 5 '17 at 11:15


text:
Search by Topic

Resources tagged with Mathematical reasoning & proof similar to Some Cubes:

Filter by: Content type:
Stage:
Challenge level:

There are 182 results

Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof

A Biggy

Stage: 4 Challenge Level:

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

Sums of Squares and Sums of Cubes

Stage: 5

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

Number Rules - OK

Stage: 4 Challenge Level:

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

How Many Solutions?

Stage: 5 Challenge Level:

Find all the solutions to the this equation.

N000ughty Thoughts

Stage: 4 Challenge Level:

How many noughts are at the end of these giant numbers?

Perfectly Square

Stage: 4 Challenge Level:

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Postage

Stage: 4 Challenge Level:

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

Magic W Wrap Up

Stage: 5 Challenge Level:

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

Polite Numbers

Stage: 5 Challenge Level:

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

Big, Bigger, Biggest

Stage: 5 Challenge Level:

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

Sixational

Stage: 4 and 5 Challenge Level:

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

Common Divisor

Stage: 4 Challenge Level:

Find the largest integer which divides every member of the following sequence: 1^5-1, 2^5-2, 3^5-3, ... n^5-n.

Ordered Sums

Stage: 4 Challenge Level:

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

For What?

Stage: 4 Challenge Level:

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

The Root Cause

Stage: 5 Challenge Level:

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Mod 3

Stage: 4 Challenge Level:

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Take Three from Five

Stage: 4 Challenge Level:

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Telescoping Functions

Stage: 5

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

Pent

Stage: 4 and 5 Challenge Level:

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

Rhombus in Rectangle

Stage: 4 Challenge Level:

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Cube Net

Stage: 5 Challenge Level:

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

Plus or Minus

Stage: 5 Challenge Level:

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

Symmetric Tangles

Stage: 4

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

Square Pair Circles

Stage: 5 Challenge Level:

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

Modulus Arithmetic and a Solution to Dirisibly Yours

Stage: 5

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Archimedes and Numerical Roots

Stage: 4 Challenge Level:

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Transitivity

Stage: 5

Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.

More Sums of Squares

Stage: 5

Tom writes about expressing numbers as the sums of three squares.

Why 24?

Stage: 4 Challenge Level:

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

Stage: 5 Challenge Level:

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

Doodles

Stage: 4 Challenge Level:

Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?

Tetra Inequalities

Stage: 5 Challenge Level:

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

Knight Defeated

Stage: 4 Challenge Level:

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

Rational Roots

Stage: 5 Challenge Level:

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Proximity

Stage: 4 Challenge Level:

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Russian Cubes

Stage: 4 Challenge Level:

I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?

Shape and Territory

Stage: 5 Challenge Level:

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Similarly So

Stage: 4 Challenge Level:

ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.

Stage: 5 Short Challenge Level:

Can you work out where the blue-and-red brick roads end?

Without Calculus

Stage: 5 Challenge Level:

Given that u>0 and v>0 find the smallest possible value of 1/u + 1/v given that u + v = 5 by different methods.

Mouhefanggai

Stage: 4

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Mediant

Stage: 4 Challenge Level:

If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.

Generally Geometric

Stage: 5 Challenge Level:

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

Unit Interval

Stage: 4 and 5 Challenge Level:

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Matter of Scale

Stage: 4 Challenge Level:

Prove Pythagoras' Theorem using enlargements and scale factors.

Round and Round

Stage: 4 Challenge Level:

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

A Computer Program to Find Magic Squares

Stage: 5

This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

Circle Box

Stage: 4 Challenge Level:

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?

Composite Notions

Stage: 4 Challenge Level:

A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.


text:
apache_beam.coders.typecoders module¶

Type coders registration.

This module contains functionality to define and use coders for custom classes. Let’s say we have a class Xyz and we are processing a PCollection with elements of type Xyz. If we do not register a coder for Xyz, a default pickle-based fallback coder will be used. This can be undesirable for two reasons. First, we may want a faster coder or a more space efficient one. Second, the pickle-based coder is not deterministic in the sense that objects like dictionaries or sets are not guaranteed to be encoded in the same way every time (elements are not really ordered).

Two (sometimes three) steps are needed to define and use a custom coder:
• define the coder class
• associate the code with the class (a.k.a. coder registration)
• typehint DoFns or transforms with the new class or composite types using the class.

A coder class is defined by subclassing from CoderBase and defining the encode_to_bytes and decode_from_bytes methods. The framework uses duck-typing for coders so it is not strictly required to subclass from CoderBase as long as the encode/decode methods are defined.

Registering a coder class is made with a register_coder() call:

from apache_beam import coders
...
coders.registry.register_coder(Xyz, XyzCoder)

Additionally, DoFns and PTransforms may need type hints. This is not always necessary since there is functionality to infer the return types of DoFns by analyzing the code. For instance, for the function below the return type of ‘Xyz’ will be inferred:

def MakeXyzs(v):
return Xyz(v)

If Xyz is inferred then its coder will be used whenever the framework needs to serialize data (e.g., writing to the shuffler subsystem responsible for group by key operations). If a typehint is needed it can be specified by decorating the DoFns or using with_input_types/with_output_types methods on PTransforms. For example, the above function can be decorated:

@with_output_types(Xyz)
def MakeXyzs(v):
return complex_operation_returning_Xyz(v)

See apache_beam.typehints.decorators module for more details.


text:
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distance between 2 parallel lines in 3d calculator

13:14 09-Th12-2020

Distance Between Two Parallel Planes. {{a_1}}&{{b_1}}&{{c_1}} Free parallel line calculator - find the equation of a parallel line step-by-step This website uses cookies to ensure you get the best experience. Online space geometric calculator to find the shortest distance between given two lines in space, each passing through a point and parallel to a vector. In three-dimensional geometry, one of the most crucial elements is a straight line.Any two straight lines can be differently related to each other in the Cartesian plane in the sense that they may be intersecting each other, skewed lines or parallel lines. $$. Mark leftmost line as line 1. (x_0, y_0, z_0) and is parallel {a_0}&{b_0}&{c_0}\\ I want to calculate the closest distance between the two lines and I also want to know where it happens (z2+delta). In the case of non-parallel coplanar intersecting lines, the distance between them is zero.For non-parallel and non-coplanar lines (), a shortest distance between nearest points can be calculated. So far my approach has been as follow: I have chosen some reference z values and recalculated all the deltas for both lines to these reference z values. Alternatively we can find the distance between two parallel lines as follows: Considers two parallel lines $\begin{gathered} ax + by + c = 0 \\ ax + by + {c_1} = 0 \\ \end{gathered}$ Now the distance between two parallel lines can be found with the following formula: The shortest distance between two parallel lines is equal to determining how far apart lines are. It does not matter which perpendicular line you are choosing, as long as two points are on the line. Parallel Lines in 3D Geometry. {c_0}&{c_1}&{c_2} (x_1, y_1, z_1),$$ \end{array}} \right|}^2} + {{\left| {\begin{array}{*{20}{c}} A line parallel to Vector (p,q,r) through Point (a,b,c) is expressed with $$\hspace{20px}\frac{x-a}{p}=\frac{y-b}{q}=\frac{z-c}{r}$$ Math Calculators, Lessons and Formulas. Skew lines are the lines which are neither intersecting nor parallel. We can solve for this parallel distance of separation by fixing the value of one parameter and using either equation to solve for the other. First of all, you don't need to equate the lines. \mathbb R^3 R3 is equal to the distance between parallel planes that contain these lines. Non-parallel planes have distance 0. shortest distance between two lines in 3d pdf,shortest distance between two parallel lines,perpendicular distance between two parallel lines,shortest distance between two skew lines cartesian form,shortest distance between two points,shortest distance formula in 3d,distance between two non parallel lines,distance between two lines calculator,shortest distance between two parallel lines In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. - \infty &< t < + \infty A command _DIST exists, using object snap _ENDPOINT and _PERPENDICULAR (for the second point) can you show the distance between 2 lines.. A general command to find the minimum distance between objects (e.g. $(a_0, b_0, c_0)$, $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are parallel Also, we need to rewrite the equations of the lines a bit because the line parameters k are not the same thing in both lines. If two lines are parallel, then the shortest distance between will be given by the length of the perpendicular drawn from a point on one line form another line. a&b $(x_1, y_1, z_1)$, in direction 3D lines: Distance between two points. The distance between two parallel lines in the plane is the minimum distance between any two points lying on the lines. Given the equations of two non-vertical, non-horizontal parallel lines, = + = +, the distance between the two lines can be found by locating two points (one on each line) that lie on a common perpendicular to the parallel lines and calculating the distance between them. Elevations are not considered in the calculations. We know that slopes of two parallel lines are equal. L2(t): x = t. y = -1. z = -t When ac–b 2 = 0, the two equations are dependant, the two lines are parallel, and the distance between the lines is constant. and : A point P(x, y, z) is on the line L if and only if the direction numbers determined by P 0 and P 1 are proportional to those determined by P 1 and P 2. This is either given to you, or you can choose it in the form (x₀,y₀). This lesson lets you understand the meaning of skew lines and how the shortest distance between them can be calculated. y &= y_0 + (y_1 - y_0) t \\ Calculator Academy© - All Rights Reserved 2020, find the equation of the line that is parallel to this line and passes through the point calculator, find the slope of a line parallel to each given line, how to find the distance between two parallel lines, how to find a line parallel to another line, find the distance between two parallel lines, how to find distance between parallel lines, parallel line calculator given two points, distance between parallel lines calculator, distance between two parallel lines calculator, formula of distance between two parallel lines, find the distance between the parallel lines, how to tell if two equations are parallel, how to find the slope of a line parallel to each given line, parallel lines and transversals calculator, how to find the parallel line of an equation, distance between 2 parallel lines formula, find equations of the tangent lines to the curve that are parallel to the line, find the distance of the point from the plane measured parallel to the line, parallel lines cut by a transversal calculator, formula to find distance between two parallel lines, find the distance between two parallel lines calculator, how to find the distance between two parallel lines with given equations, find the distance between the parallel lines l and m using a set square, formula for distance between parallel lines, how to determine if two lines are parallel without graphing, how to find a line parallel to another line through a point, how to calculate distance between two parallel lines, parallel line calculator slope intercept form, find the slope of a line parallel to each given line calculator, how to find the distance between 2 parallel lines, find the distance between the pair of parallel lines, finding an equation of a line parallel to a given line, distance formula between two parallel lines, equation of plane through point and parallel to plane, find the equation of a line parallel calculator, find the distance between each pair of parallel lines, how to find equation of a line parallel to another line, find the slope of a parallel line calculator, how to find the shortest distance between two parallel lines, find the slope of a line parallel to the graph of each equation, how do you find the distance between two parallel lines, write the equation of a line parallel calculator, find equation of plane through point and parallel to plane, find parallel line with equation and given point calculator, find a line parallel to another line calculator, find the equation of the tangent line to the curve that is parallel to the line, distance between 2 parallel lines calculator, how to find the slope intercept form of a parallel line, how to determine if two equations are parallel, shortest distance between two parallel lines formula, how to find an equation of a line parallel to another, find the equation of the line parallel calculator, write an equation of the line containing the given point and parallel to the given line calculator, how to calculate angles in parallel lines, how to find an equation of a line parallel, find the line parallel to given equation calculator, how to find a parallel line that goes through a point. x &= x_0 + (x_1 - x_0) t \\ I do believe 7.59 is correct, could you please explain why I got 2.53? mathhelp@mathportal.org, Analytic geometry of three dimensions: (lesson 1 of 2), More help with analytic geometry in three dimensions at mathportal.org. D = \frac{{\left| {\begin{array}{*{20}{c}} I can handle non-parallel lines and the minimum distance between them (by using the projection of the line and the normal vector to both direction vectors in the line), however, in parallel lines, I'm not sure on how to start. To find the distance, choose any point on one of the lines. Finally you need to calculate the slope of the new line, but guess what, it’s the same as the first line! It is time to solve your math problem. because <4, -2 , -4> = (1/2) <2, -1 , -2> so M1 (-2 , 3 , -3) is on line L1. \end{array}} \right|}^2}}}{{{a^2} + {b^2} + {c^2}}}} Next, you need to determine a point that the parallel line passes through. We verify that the plane and the straight line are parallel using the scalar product between the governing vector of the straight line, $$\vec{v}$$, and the normal vector of the plane $$\vec{n}$$. and : Line passing through two points. Shortest distance between two lines(d) We are considering the two line in space as line1 and line2. , In three-dimensional space, the line passing through the points So the y-axis is also an asymptote. No other pixels (noise) apart from the lines; My proposed solution: Almost same as given above. Enter the slope intercept form of the first equation, and a coordinate the second line passes through to calculate the equation of the second line. \end{array}} \right|}^2} + {{\left| {\begin{array}{*{20}{c}} If you want to contact me, probably have some question write me using the contact form or email me on \end{array}} \right| = 0 {y_0 - y_1}&{z_o - z_1}\\ To find that distance first find the normal vector of those planes - it is the cross product of directional vectors of the given lines. The shortest distance between the lines is the distance which is perpendicular to both the lines given as compared to any other lines that joins these two skew lines. We can find the distance between this point and the other line by putting the second line into the form : subtract the whole right side from both sides . (a_1, b_1, c_1), b) Find a point on the line that is located at a distance of 2 units from the point (3, 1, 1). The Distance Between Parallel Planes. In 2-D lines are either parallel or intersecting. {z_0 - z_1}&{x_o - x_1}\\ z &= z_0 + (z_1 - z_0) t \\ That’s the definition of parallel lines. Free parallel line calculator - find the equation of a parallel line step-by-step. We will first define what it means for two lines to be parallel, and then learn how to compute the distance between such planes. Example 1: Find a) the parametric equations of the line passing through the points P 1 (3, 1, 1) and P 2 (3, 0, 2). \begin{aligned} (x_0, y_0, z_0) and \left| {\begin{array}{*{20}{c}} Black holes result from God dividing the universe by zero. There are no skew lines in 2-D. to (a, b, c) has parametric equations, Vector Form We shall consider two skew lines L 1 and L 2 and we are to calculate the distance between them. Show Instructions. Plugging in 2 into the first equation can generate our first point: this gives us the point . The shortest distance between the two parallel lines can be determined using the length of the perpendicular segment between the lines. let's take a point of L2 for t : M2(t) ( -1+2t , 2-t , -2-2t ) the right point of L2 giving the distance is the one. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. What Is The Distance Formula. to a common plane if and only if, Distance between the point mathportal.org . (a_0, b_0, c_0), and the line The strategy behind determining the distance between 2 skew lines is to find two parallel planes passing through each line; this is because the distance between two planes is easy to calculate … The distance between two parallel planes is understood to be the shortest distance between their surfaces. This can be done by measuring the length of a line that is perpendicular to both of them. Skew Lines . {a_1}&{b_1}&{c_1} \end{aligned} {{b_0}}&{{c_o}}\\ {{b_1}}&{{c_1}} \end{array}} \right|}^2} + {{\left| {\begin{array}{*{20}{c}} … Distance between two lines is equal to the length of the perpendicular from point A to line (2). We will look at both, Vector and Cartesian equations in this topic. Hi, >> Is there a command to list out the minimum distance >> between two lines/object? The line1 is passing though point A (a 1 ,b 1 ,c 1 ) and parallel to vector V 1 and The line2 is passing though point B(a 2 ,b 2 ,c 2 ) and parallel to vector V 2 . This needs to be in the form of y = mx + b, where m is the slope, and b is the y intercept of that line. This is what I’m talking about.. Let the equations of the lines be ax+by+c 1 =0 and ax+by+c 2 =0. (x_1 - x_0, y_1 - y_0, z_1 - z_0), In three-dimensional space, the line passing through the point The direction vector of the plane orthogonal to the given lines is collinear or coincides with their direction vectors that is N = s = ai + b j … D = \sqrt {\frac{{{{\left| {\begin{array}{*{20}{c}} Doceri is free in the iTunes app store. Also, y → ∞ as t → 0 from the right, and the distance between the curve and the y-axis is t which approaches 0 as t → 0. It equals the perpendicular distance from any point on one line to the other line.. First, suppose we have two planes \Pi_1 and \Pi_2. The line1 is passing though point A (a 1 ,b 1 ,c 1 ) and parallel to vector V 1 and The line2 is passing though point B(a 2 ,b 2 ,c 2 ) and parallel to vector V 2 . To find the distance, choose any point on one of the lines. b&c … L2: < -1, 2, -2 > + <4, -2 , -4>t or: < -1, 2, -2 > + <2, -1 , -2>t . isnt it supposed to be -2 Yes then y u wrote 2 nm anyway thx ; Therefore, the x-axis is an asymptote of the curve. This web site owner is mathematician Miloš Petrović. For the normal vector of the form (A, B, C) equations representing the planes are: Ax + By + Cz + D_1 = 0 Ax +B y +C z +D1 If and determine the lines r and… \end{array}} \right|}^2}} }} This website uses cookies to ensure you get the best experience. This lesson lets you understand the meaning of skew lines and how the shortest distance between them can be calculated. {{a_0}}&{{b_0}}&{{c_0}}\\ Perpendicular and parallel lines in space are very similar to those in 2D and finding if lines are perpendicular or parallel in space requires an understanding of the equations of lines in 3D. You seem to to know more on this then my teacher does. Non-parallel planes have distance 0. eval(ez_write_tag([[970,90],'calculator_academy-medrectangle-4','ezslot_1',107,'0','0'])); The next step is to calculate the y intercept, b, of the parallel line. The distance between the two points (x 1,y 1) and (x 2,y 2) is . We may derive a formula using this approach and use this formula directly to find the shortest distance between two parallel lines. You have only two continuous lines without any break in between. \end{aligned} {a_0}&{a_1}&{a_2}\\ (x_1, y_1, z_0): \frac{x - x_0}{x_1 - x_0} = \frac{y - y_0}{y_1 - y_0} = \frac{z - z_0}{z_1 - z_0}, This line is parallel to the vector {x_0 - x_1}&{y_o - y_1}\\ 2 nurbs surfaces) does not exist. Distance between two 3D lines Parametric line equation: L 1: x = + t: y = + t: z = + t: L 2: x = + s: Line equation: L 1: x + = We will look at both, Vector and Cartesian equations in this topic. 2. By using this website, you agree to our Cookie Policy. Therefore, distance between the lines (1) and (2) is |(–m)(–c1/m) + (–c2)|/√(1 + m2) or d = |c1–c2|/√(1+m2). We may derive a formula using this approach and use this formula directly to find the shortest distance between two parallel lines. Skew lines are new, and are lines that are not parallel, yet never intersect. Think about that; if the planes are not parallel, they must intersect, eventually. The shortest distance between two parallel lines is equal to determining how far apart lines are. (x_0, y_0, z_0) in direction If the straight line and the plane are parallel the scalar product will be zero:\vec{v}\cdot\vec{n}=(1,1,1)\cdot(1,1,-2)=1+1-2… \begin{aligned} {b_0}&{b_1}&{b_2}\\ find the distance between two parallel lines calculator: how to find the distance between two parallel lines with given equations: parallel wire capacitance calculator: find the distance between the parallel lines l and m using a set square: how do you find parallel lines: write equation of x axis and y axis: parallel formula math They must of the same slope. Thus the distance d betw… Welcome to MathPortal. Mark rightmost line as line 2. This calculator will find the equation of a line (in the slope-intercept, point-slope and general forms) given two points or the slope and one point, with steps shown. {x_1 - x_0}&{y_1 - y_0}&{z_1 - z_0}\\ By using this website, you agree to our Cookie Policy. x &= x_0 + at \\ Distance between the line through(x_1, y_1, z_1)$in direction$(a, b, c): ~x= e are two parallel planes, then their distance is |e−d| |~n|. This command can help you design for a minimum distance between an alignment centerline and the right-of-way, for example. DISTANCE POINT-PLANE (3D). Analytical geometry line in 3D space. To do this you use this equation: b = y₀ – m * x₀, where m is the slope of the original line. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). through (x_0, y_0, z_0) and the line Line through (x_1, y_1, z_1) in direction intersect if: If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. {{a_1}}&{{b_1}} If we select an arbitrary point on either plane and then use the other plane's equation in the formula for the distance between a point and a plane, then we will have obtained the distance between both planes. Let's Begin! Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is (1) 5/2 (2) 7/2 (3) 9/2 (4) 3/2Hello student, Given planes are2x + y + 2z = 8 a ; Free parallel line calculator - find the equation of a parallel line step-by-step This website uses cookies to ensure you get the best experience. {{c_0}}&{{a_o}}\\(x_0, y_0, z_0)and Question to the reader: what is the distance between the point (−5,2,4) and the sphere (x+2)2 +(y−2)2 +z2 = 1? Math Tests; Math Lessons; Math Formulas; Online Calculators; Math Lessons > Analytic Geometry > Three Dimensions > Lines in Three Dimensions ; Planes » Analytic geometry of three dimensions: (lesson 1 of 2) Line in three dimensions. Proof: use the distance for-mula between point and plane. / Space geometry Calculates the shortest distance between two lines in space. Uses cookies to ensure you get the best experience steps below for the... Into the first line you are going to be parallel to y axis at a distance of 1.1666, 1... 1 distance between 2 parallel lines in 3d calculator be any point on one line to the length of the perpendicular between the lines! Believe 7.59 is correct, could you please explain why I got 2.53 parallel. From the lines be ax+by+c 1 =0 and ax+by+c 2 =0 about.. Let the of! Is what I ’ m talking about.. Let the equations of the origin is skew lines and the between! Equation can generate our first point: this gives us the point on iPad! And… Math calculators, lessons and formulas any two points lying on the line planes that contain lines., we can now easily calculate the distance for-mula between point and a line that perpendicular. Between any two points are on the lines which are neither intersecting nor parallel and got a of... The right-of-way, for example illustration show the minimum distance distance between 2 parallel lines in 3d calculator > two. The steps below for calculating the equation of the perpendicular distance from any point on one line the... To y axis at a distance of 2 units to the length of A̅C̅ 3... Understand the meaning of skew lines L 1 and L 2 and we are considering the two points are the. Straightforward – the distance between two lines is equal to distance between 2 parallel lines in 3d calculator left of the perpendicular between the two line space... As given above 1 =0 distance between 2 parallel lines in 3d calculator ax+by+c 2 =0 and wrote all the,. Must intersect, eventually given above lower distance of 2.53, however my went! Screencast was created with Doceri on an iPad distance from any point one! Between two parallel lines using this website uses cookies to ensure you get the best.! 3.166 ] from r to s line has lower distance of 2.53, however my teacher went through it and. To know where it happens ( z2+delta ) d ) we are considering the two lines and I also to! Illustration show the minimum distance between the two line in space as line1 and.... ( s ): x = -1 + s. y = -s. z = 1 never intersect of,... Ax+By+C 2 =0: 1 = -1 + s. y = -s. z = 1 both of them that. Distance found, formulas and calculators now easily calculate the closest distance between two lines and I also to... From point a to line ( 2 ) only two continuous lines without any break in.. Pixels ( noise ) apart from the lines ~x= e are two parallel lines in the form x₀! Continuous lines without any break in between to be the shortest distance between two parallel lines is equal to length... L 2 and we are considering the two lines is equal to the distance between two. To list out the minimum distance found help you design for a minimum distance between two planes... – PB the formula of the first line you are choosing, as long distance between 2 parallel lines in 3d calculator two points lesson you! Equation of the perpendicular distance from any point on one of the perpendicular distance from any point on of. Teacher went through it, and got a distance of 1.1666 lines r and… Math calculators, lessons and.!: use the Pythagorean Theorem to find the shortest distance between two parallel and! By using this website, you agree to our Cookie Policy between an alignment centerline and the,! Between skew lines L 1 and L 2 and we are considering the two line in as! Units to the distance between 2 parallel lines in 3d calculator of the first line you are choosing, as long two. X₀, y₀ ) a minimum distance between lines y=-3x+10 and y=-3+2 illustration the! The first line you are choosing, as long as two points are on the lines lines in the (... Video screencast was created with Doceri on an iPad = 3 – 1 = 2 lines and... Can skip the multiplication sign, so  5x  is equivalent to  *... Know where it happens ( z2+delta ) first point: this gives us the point x -1. > is there a command to list out the minimum distance > > between lines/object!, lessons and formulas is relatively easily point on one of the lines be ax+by+c 1 =0 ax+by+c. And how the shortest distance between any two points lying on the lines to y axis at a distance 2. Lets you understand the meaning of skew lines L 1 and L and! Multiplication sign, so  5x  is equivalent to  5 * x  holes result from God the... Illustration show the minimum distance between two lines and I also want to know more on this then my does. \Pi_1 $and$ \Pi_2 \$ for example directly to find the between. About that ; if the planes are not parallel, yet never intersect |e−d| |~n| any. Lines is equal to determining how far apart lines are new, and are lines do... Have no distance -- 0 distance -- 0 distance -- between them can be done measuring... 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And ax+by+c 2 =0 how far apart lines are the lines which are neither intersecting parallel! To calculate the distance between them can be calculated are the lines which are neither nor! From a point that the parallel line step-by-step why I got 2.53 can it! Will look at both, Vector and Cartesian equations in this topic is simple, it only... We can now easily calculate the distance between any two points are on the lines r and… calculators. Are not parallel, they must intersect, eventually for calculating the equation a... Relatively easily their surfaces selecting P and [ 2 3.166 ] from r to s line has distance. Help you design for a minimum distance between the two lines ( )! The other line explain why I got a distance of 7.59 I distance between 2 parallel lines in 3d calculator m talking about.. the! Where it happens ( z2+delta ) and L 2 and we are to calculate the between! Line step-by-step this website uses cookies to ensure you get the best experience you get best. Both, Vector and Cartesian equations in this topic blue lines in the coordinate plane, you use. You agree to our Cookie Policy site and wrote all the lessons, formulas and calculators 1! Line has lower distance of 1.1666 ( noise ) apart from the lines without break. Describing the straight line: 1 lines in the following illustration show minimum... You need the formula of the lines from a point to calculate distance... If the planes are not parallel, yet never intersect following illustration show the distance... Happens ( z2+delta ) will look at both, Vector and Cartesian equations this! In this topic parallel lines is the difference between the two points are on the lines – distance! This is what I ’ m talking about.. Let the equations the. Do not believe that mathematics is simple, it is only because they do not that! Two parallel lines easily calculate the distance between lines y=-3x+10 and y=-3+2 formulas... ; my proposed solution: Almost same as given above lines y=-3x+10 and y=-3+2 3! Do believe 7.59 is correct, could you please explain why I got 2.53 lets you understand the meaning skew... Are considering the two line in space as line1 and line2 ): x = -1 + s. y -s.... Parallel to y axis at a distance of 2.53, however my teacher through! Their surfaces only because they do not realize how complicated life is skew ( non-parallel lines that n't. You, or you can follow the steps below for calculating the equation of parallel... That contain these lines lying on the lines your formula to find the shortest distance between lines y=-3x+10 y=-3+2. And use this formula directly to find the distance between them can be calculated s. y -s.! And [ 2 3.166 ] from r to s line has lower distance of 2.53, my! Equation can generate our first point: this gives us the point point: this us. = 3 – 1 = 2 > is there a command to list the!, Vector and Cartesian equations in this topic are the lines be ax+by+c =0... Can help you design for a minimum distance > > between two parallel lines is equal to the left the! S line has lower distance of 1.1666 use the distance between two parallel lines is equal to distance! It happens ( z2+delta ) x  two continuous lines without any break in between equation!


text:
# Estimating treatment effects with optimal inverse probability weighting

### Abstract

The average treatment effect is defined as the difference in the expectation of an outcome of interest between treatment groups. It is a popular summary measure, but when the treatment assignment is not random, it may not be directly interpretable. This is because potentially relevant factors may be unequally distributed across the treatment groups, and any observed differences in the outcome may not be accounted for by the treatment alone. Inverse probability weighting is often used to adjust for this potential unbalance. This method, however, is known to yield erratic and inefficient inference when outlying weights are present. Different approaches have been proposed to alleviate this limitation. These frequently entail introducing simplifying assumptions in the model for estimating the probability of being treated. While these approaches generally reduce the variability in the weights, and the consequent sampling variability of the weighted estimators, they can also introduce substantial bias. We present optimal inverse probability weighting, which minimizes the bias of the weighted estimator of the average treatment effect for any specified level of its standard error. The optimal weights are defined as the solution to a nonlinear constrained optimization problem. The method is evaluated in a simulation study and applied in the assessment of the timing of treatment initiation in individuals infected by the human immunodeficiency virus. The simulation study suggests that optimal inverse probability weighting has some desirable properties, such as: (1) it provides an estimated ATE with minimum bias while controlling for precision; (2) it allows the researcher to search for a suitable balance between bias and variance directly; (3) it can be implemented with available R packages, such as ipoptr'' andnloptr”; and (4) it maintains all its properties regardless of the chosen estimator for the ATE.

Date
Location
Glasgow, UK


text:
# Kopp's law

Hermann Franz Moritz Kopp

Kopp's law can refer to either of two relationships discovered by the German chemist Hermann Franz Moritz Kopp (1817–1892).

1. Kopp found "that the molecular heat capacity of a solid compound is the sum of the atomic heat capacities of the elements composing it; the elements having atomic heat capacities lower than those required by the Dulong–Petit law retain these lower values in their compounds."[1]
2. In studying organic compounds, Kopp found a regular relationship between boiling points and the number of CH2 groups present.[2]

## Kopp–Neumann law

The Kopp–Neumann law, named for Kopp and Franz Ernst Neumann, states that the specific heat C per unit mass (in J·kg−1·K−1) for alloys can be calculated from the following equation:

${\displaystyle C=\sum _{i=1}^{N}(C_{i}\cdot f_{i})}$ ,

where N is the total number of alloy constituents, and Ci and fi denote the specific heat and mass fraction of the i-th constituent.

## References

• Frederick Seitz, The Modern Theory of Solids, McGraw-Hill, New York, USA, 1940, ASIN: B000OLCK08
1. ^ See page 96 of Millard, Earl Bowman (1921). Physical Chemistry for Colleges. New York: McGraw-Hill.
2. ^ See page 942 of Miller, William Allen (1869). Elements of Chemistry. London: Longmans, Green, Reader, and Dyer.


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# How to?What's the maximum size of the thread text ?

#### Sergey Vontsov

What's the maximum size of the thread text ?
thanks

#### Sergey Vontsov

I mean the text you/I type here. I've created another thread but was unable to insert all the text I wrote

#### vefatica

I mean the text you/I type here. I've created another thread but was unable to insert all the text I wrote
If there's a limit, I don't know what it is. Since you can attach files and images (even inline the images) I don't think the space used by plain text will bother anyone. Often (and this has not been explained) the forum will not let you post certain words ("Oops we ran into some problems"). One of those words is (I can't write it properly) CMD<dot>EXE.

Last edited:

#### Sergey Vontsov

You are right - this is the error message I see !!!

#### AnrDaemon

That's cloudfucking blindly blocking some words in passing traffic. Not forum software per se.

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I was surprised to find that we did not have a question on this Stack Exchange on this particular problem in decision theory. So I have chosen to add a question on Newcomb's Paradox, though I would like to limit the scope of it to its economic considerations more than its philosophical considerations.

For those unfamiliar with the problem, here is the usual formulation of the problem.

You are presented with two boxes.

• Box A is clear and has \$1,000 in it. • Box B is opaque and either has \$0 or \$1,000,000 in it, which is decided by another player, let's say a machine here. You can choose to take just Box B, or both Box A and B, but before you choose, a machine makes a decision about what is in Box B, based on what it thinks you will do. The machine has played this game many times before and empirically is almost always correct in predicting what the player will do. • If it thinks you are going to choose both boxes, the machine will place \$0 in Box B.
• If it thinks you are going to choose just Box B, the machine will place \$1,000,000 in Box B. Empirically, people are split on whether to play the one box or two box strategy. The intuition for both cases goes likes this. For those who pick one box, the intuition is that if the machine is almost always right, then the expected value/utility of picking just Box B will be much much higher than picking both boxes. So obviously one should pick just Box B and not be greedy. For those who pick two boxes, the intuition is that picking both boxes seems to be a dominant strategy. If the machine has already picked putting nothing in Box B, it doesn't hurt, and can only help, if you pick both boxes. If the machine has already picked putting \$1,000,000 in Box B, then...well you should still want both boxes!

So what's the rub? Philosophers like getting into the weeds with this question in discussing causality or free will. But perhaps one criticism of this thought experiment is that the scenario described here is not rigorously defined within the realm of game theory. The machine's only goal is to predict the player's move correctly, but it does not actually have a defined payoff for making the correct prediction. One can map the set of choices and player payoffs as a tree, where the player cannot observe the choice of the machine, but the information set needs more elaboration than this. How does the machine make its prediction exactly so that it is "almost always right"? Why does the machine get to completely preclude mixed strategies and just...know what the player will do most of the time? Essentially, the whole game, strategies and payoffs and information, seems all very hazy. Should it even be considered a game? Or is this just a weird decision theory question?

With paradoxes, trying to add in extra assumptions, or showing why an existing assumption is ill-defined, this can go a long way to providing a possible solution to a paradox. So my question(s) are:

• How could one alter the formulation of Newcomb's Paradox to make it decisively analyzable with game theory or decision theory, and what strategy/decision should the player make in that case?
• If you would choose not to alter the formulation of the paradox, what is the correct analysis of the problem?

Two ending points.

$$^1$$ I hope I have formulated my question so that there could be multiple correct answers that are objectively correct, and not just opinion based.

$$^2$$ If you think Newcomb's Paradox is just so incoherent that one can't really alter it without really altering what makes the paradox interesting, you can talk about that too. I am not asking that as a question because I think that would be too opinion based.

For the second option, the machine's decision-making process needs to be clarified; what does it mean for the machine to be "almost always correct"? I'd implement this as follows: the machine has access to an oracle that will tell it the player's choice, but which, with probability $$p > 0$$, will give an incorrect answer. The machine always follows the oracle and puts \\$1,000,000 into box B iff the oracle says the player did not choose both boxes. This scenario could then be analyzed using standard tools, and the player's optimal strategy determined for both one-off and iterated games, depending on the value of $$p$$.


text:
5

# 1-a) In how many permutations of the 26 letters in the alphabet. do the patterns ads dog and sol not appear? b) In how many ofthe permutations of {1,2._ 10} exactly...

## Question

###### 1-a) In how many permutations of the 26 letters in the alphabet. do the patterns ads dog and sol not appear? b) In how many ofthe permutations of {1,2._ 10} exactly 3 of the even numbers and one of the odd numbers are in the correct position?

1-a) In how many permutations of the 26 letters in the alphabet. do the patterns ads dog and sol not appear? b) In how many ofthe permutations of {1,2._ 10} exactly 3 of the even numbers and one of the odd numbers are in the correct position?

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# Office Hours

Please see piazza for the office hours schedule, including any changes for the current week.


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# Zipf–Mandelbrot law

Parameters ${\displaystyle N\in \{1,2,3\ldots \}}$ (integer) ${\displaystyle q\in [0;\infty )}$ (real) ${\displaystyle s>0\,}$ (real) ${\displaystyle k\in \{1,2,\ldots ,N\}}$ ${\displaystyle {\frac {1/(k+q)^{s}}{H_{N,q,s}}}}$ ${\displaystyle {\frac {H_{k,q,s}}{H_{N,q,s}}}}$ ${\displaystyle {\frac {H_{N,q,s-1}}{H_{N,q,s}}}-q}$ ${\displaystyle 1\,}$ ${\displaystyle {\frac {s}{H_{N,q,s}}}\sum _{k=1}^{N}{\frac {\ln(k+q)}{(k+q)^{s}}}+\ln(H_{N,q,s})}$

In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution. Also known as the Pareto-Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf who suggested a simpler distribution called Zipf's law, and the mathematician Benoit Mandelbrot, who subsequently generalized it.

The probability mass function is given by:

${\displaystyle f(k;N,q,s)={\frac {1/(k+q)^{s}}{H_{N,q,s}}}}$

where ${\displaystyle H_{N,q,s}}$ is given by:

${\displaystyle H_{N,q,s}=\sum _{i=1}^{N}{\frac {1}{(i+q)^{s}}}}$

which may be thought of as a generalization of a harmonic number. In the formula, ${\displaystyle k}$ is the rank of the data, and ${\displaystyle q}$ and ${\displaystyle s}$ are parameters of the distribution. In the limit as ${\displaystyle N}$ approaches infinity, this becomes the Hurwitz zeta function ${\displaystyle \zeta (s,q)}$. For finite ${\displaystyle N}$ and ${\displaystyle q=0}$ the Zipf–Mandelbrot law becomes Zipf's law. For infinite ${\displaystyle N}$ and ${\displaystyle q=0}$ it becomes a Zeta distribution.

## Applications

The distribution of words ranked by their frequency in a random text corpus is approximated by a power-law distribution, known as Zipf's law.

If one plots the frequency rank of words contained in a moderately sized corpus of text data versus the number of occurrences or actual frequencies, one obtains a power-law distribution, with exponent close to one (but see Powers, 1998 and Gelbukh & Sidorov, 2001). Zipf's law implicitly assumes a fixed vocabulary size, but the Harmonic series with s=1 does not converge, while the Zipf-Mandelbrot generalization with s>1 does. Furthermore, there is evidence that the closed class of functional words that define a language obeys a Zipf-Mandelbrot distribution with different parameters from the open classes of contentive words that vary by topic, field and register.[1]

In ecological field studies, the relative abundance distribution (i.e. the graph of the number of species observed as a function of their abundance) is often found to conform to a Zipf–Mandelbrot law.[2]

Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandelbrot distributions.[3]

## Notes

1. ^ Powers, David M W (1998). "Applications and explanations of Zipf's law". Association for Computational Linguistics: 151–160.
2. ^ Mouillot, D; Lepretre, A (2000). "Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity". Environmental Monitoring and Assessment. Springer. 63 (2): 279–295. doi:10.1023/A:1006297211561. Retrieved 24 Dec 2008.
3. ^ Manaris, B; Vaughan, D; Wagner, CS; Romero, J; Davis, RB. "Evolutionary Music and the Zipf-Mandelbrot Law: Developing Fitness Functions for Pleasant Music". Proceedings of 1st European Workshop on Evolutionary Music and Art (EvoMUSART2003). 611.

## References

• Mandelbrot, Benoît (1965). "Information Theory and Psycholinguistics". In B.B. Wolman and E. Nagel. Scientific psychology. Basic Books. Reprinted as
• Mandelbrot, Benoît (1968) [1965]. "Information Theory and Psycholinguistics". In R.C. Oldfield and J.C. Marchall. Language. Penguin Books.
• Powers, David M W (1998). "Applications and explanations of Zipf's law". Association for Computational Linguistics: 151–160.
• Zipf, George Kingsley (1932). Selected Studies of the Principle of Relative Frequency in Language. Cambridge, MA: Harvard University Press.


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# 24: Modeling Continuous Relationships

## Learning Objectives

• Describe the concept of the correlation coefficient and its interpretation
• Compute the correlation between two continuous variables
• Describe the potential causal influences that can give rise to a correlation.

Most people are familiar with the concept of correlation, and in this chapter we will provide a more formal understanding for this commonly used and misunderstood concept.

24: Modeling Continuous Relationships is shared under a CC BY-NC 2.0 license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


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Question

# Determine whether the ordered pair is a solution to the given system of linear equations.(5,3)x-y=2x+y=8

Forms of linear equations

Determine whether the ordered pair is a solution to the given system of linear equations.
(5,3)
$$x-y=2$$
$$x+y=8$$

2021-03-13

Given
$$x-y=2$$----(1)
$$x+y=8$$----(2)
The given ordered pair for this system is (5, 3).
The ordered pair is the solution of the system it must be satisfy both the equations.
Step 1
Substitute $$x=5$$ and $$y=3$$ in the left side of the equation (1).
$$LHS=x-y$$
$$=5-3$$
$$=2$$
$$=RHS$$
Substitute $$x=5$$ and $$y=3$$ in the left side of the equation(2)
$$LHS=x+y$$
$$=5+3$$
$$=8$$
$$=RHS$$
Since the ordered pair satisfies the system of linear equations.
Hence, the ordered pair (5, 3) is a solution to the given system of linear equations.


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The definition operator is typically used in DSL packages like ggvis and data.table. It is also used in the tidyverse as a way of unquoting names (see quasiquotation).

• is_definition() returns TRUE for calls to :=.

• is_formulaish() returns TRUE for both formulas and colon-equals operators.

is_definition(x)

new_definition(lhs, rhs, env = caller_env())

is_formulaish(x, scoped = NULL, lhs = NULL)

## Arguments

x An object to test. Expressions for the LHS and RHS of the definition. The evaluation environment bundled with the definition.

## Details

The recommended way to use it is to capture arguments as expressions or quosures. You can then give a special function definition for the := symbol in an overscope. Note that if you capture dots with exprs() or quos(), you need to disable interpretation of := by setting .unquote_names to FALSE.

From rlang and data.table perspectives, this operator is not meant to be evaluated directly at top-level which is why the exported definitions issue an error.

## Life cycle

These functions are experimental.

## Examples

# A predicate is provided to distinguish formulas from the
# colon-equals operator:
is_definition(quote(a := b))#> [1] TRUEis_definition(a ~ b)#> [1] FALSE

# is_formulaish() tests for both definitions and formulas:
is_formulaish(a ~ b)#> [1] TRUEis_formulaish(quote(a := b))#> [1] TRUE


text:
## anonymous 3 years ago A right triangle has a hypotenuse of length 10 and a leg of length 7. What is the area of the triangle?

1. anonymous

a=35?

2. ash2326

|dw:1358830749694:dw| Find the missing side

3. ash2326

@marcoduuuh Are you trying?

4. anonymous

yes!

5. ash2326

Find the side then use the area formula, $\frac 1 2 \times base\times \ height$

6. anonymous

hey just an idea here take half of seven and than half of 10 and times them together

7. anonymous

I got 35?

8. anonymous

1/2 of 7 is 3.5 and half of 10 is 5 so what is the answer?

9. ash2326

|dw:1358831111900:dw| You need to find the height first, formula doesn't involve hypotenuse's length

10. anonymous

17.5!

11. ash2326

@marcoduuuh Can you use Pythagoras to find height?

12. anonymous

better use a2+b2=c2 may be that would work here

13. anonymous

that's Pythagoras

14. anonymous

yes

15. anonymous

hey ash am I missing something here??

16. anonymous

10^2+7^2= 149. sqrt of 149 is: 12.2065556

17. anonymous

does this problem have answer choices or anything??

18. anonymous

Nope, it doesn't. ):

19. anonymous

one sec checking something here

20. anonymous

ok

21. anonymous

wait I know now

22. anonymous

opp/ adj what is the function?

23. anonymous

you still there?


text:
# Propositional formula in DNF can be decided in polynomial time?

For a given propositional formula f in DNF, one can decide in polynomial time, if the formula is satisfiable: Just walk through all subformulas (l_1 and ... and l_k) and check, wheter it has NO complementary pair of literals. Formula f is satisfiable iff such subformula exists.

Is my approach above correct ?

If yes, I'm wondering why all modern SAT solvers get a CNF as input format, and don't just use DNF.

The conversion from CNF to DNF can come at an exponential cost. For example $(a_1 \lor b_1) \land \cdots \land (a_n \lor b_n)$ expands to $2^n$ many terms. As you comment, for DNF satisfiability is easy - it is falsifiability which is hard. If the problem is trivial, you don't input it to a SAT solver, and that's why SAT solvers accept CNFs instead of DNFs.

If you believe that P is different from NP, then this implies that there is no polynomial time way to convert CNF satisfiability to DNF satisfiability, since the former is NP-complete while the latter is in P.

• Thank you. I was just wondering: When the SAT problem for DNF formulas is in P, why not just describe problems directly into DNF (instead of CNF) ? Because translating a formula into CNF might take exponential cost as well as translating it into DNF. – John Threepwood Dec 17 '12 at 22:26
• @JohnThreepwood most problems aren't easily expressed in DNF. And, the conversions to DNF usually take exponential time, and necessarily so, for example if there are exponential satisfiable solutions. – Realz Slaw Dec 18 '12 at 2:57
• @RealzSlaw Thank you. So most real-life problems are easier encoded in CNF. Is this because most problems are 'naturally' structured as constraints (in CNF each clause is a constraint) ? – John Threepwood Dec 18 '12 at 16:10
• @JohnThreepwood If a problem can be expressed as DNF, a satisfiable assignment is there naturally. There would be nothing to solve. And intuitively yes, I think the problems solved using SAT solvers are naturally described via constraints which CNF is useful for. – Realz Slaw Dec 18 '12 at 17:29
• @RealzSlaw Good point. That helped understanding. – John Threepwood Dec 18 '12 at 18:05


text:
# How do you calculate the number of sodium ions, Na^+, in 14.2 g of sodium sulfate? [Relative atomic masses Na = 23, S = 32, O = 16]?

Oct 1, 2016

Use the number of moles of sodium per mole of sodium sulfate.

#### Explanation:

Sodium sulfate is $N {a}_{2}$$S {O}_{4}$
First, calculate the moles of sodium sulfate you have in a 14.2g sample.

1. Find the molar mass of sodium sulfate
molar mass = atomic mass in grams
2(23) + 32 + 4(16) = 142 grams per mole
2. Divide the mass of the sample by the molar mass to obtain moles
$\frac{14.2 g}{142 g m o {l}^{-} 1}$ = 0.1 mol

Now, for every 1 mol of $N {a}_{2}$$S {O}_{4}$ you have 2 moles of $N {a}^{+}$ ions. Multiple the moles of $N {a}_{2}$$S {O}_{4}$ that you found just then by 2 to obtain the moles of sodium ions.

Use Avogadro's number (6.022 x ${10}^{23}$) to find the number of ions present. (Recall that Avogrado's number is the number of particles per mole of a substance).

6.022 x ${10}^{23}$ x 0.2 mol = 1.2044 x ${10}^{23}$ ions


text:
Angular Momentum Conundrum

1. Oct 3, 2009

e2m2a

I have searched high and low on the internet for the correct equation to determine the angular velocity of a rigid body under the following circumstances, and have not been able to find one. Could someone show me the correct equation?
Suppose a long thin rigid rod lies on the surface of a cart, which initially is at rest. One end of the rod can rotate freely around a vertical axis attached to the cart. A small object prevents the rod from rotating clockwise. The cart can move parallel to the y-axis of an x-y coordinate system. The rod initially lies 45 degrees with respect to the x-axis. At some point in time we push the cart, giving the cart and rod an initial velocity equal to v. At a later point in time the cart makes an inelastic collision with a front bumper. Immediately, the rod rotates in the counter-clock wise direction. I need to know what the angular velocity of the rod is after the collision. We denote the mass of the rod as m, the mass of the cart as M, the initial angle of the rod as theta, the length of the rod as l, and the moment of inertia of the rod as:

I = 1/3 m l sq

I know we CANNOT use the below equation to determine the final angular velocity rod:

L = m v r cos theta

because this equation applies to point particles, but the rod is a solid, rigid body. (Assume for the sake of symmetry, there is an equal small hole at the other end of the rod so that the center of mass of the rod is at its geometric center.)
I have no idea what the correct equation is. Could someone please show me and also a general equation that applies for any shape of a rigid body.

2. Oct 3, 2009

tiny-tim

Hi e2m2a!

(have a theta: θ and try using the X2 tag just above the Reply box )

Yes, use 1/3 ml2 for the angular momentum after the collision.

For the angular momentum before the collision (about the point where the pivot ends up), use dL = (m/l)dr vrcosθ, and integrate.

Last edited: Oct 3, 2009
3. Oct 3, 2009

e2m2a

Hello Tiny-Tim,

In your reply you stated: "Yes, use 1/3ml² for the angular momentum after the collision."
Did you mean the moment of inertia, not the angular momentum? Also, I assume the equation for the angular momentum before the collision applies to after the collsion, correct? Also, I am not clear how to use your equation. I integrate the right side of:
dL = (m/l)dr v r cosθ with respect to r, treating everything else as a constant. Correct? Also, is r the distance from the axis or pivot point to the center of mass of the rod? Thanks for your help.

4. Oct 3, 2009

tiny-tim

Hello e2m2a!

Yes, I meant use it for the moment of inertia to calculate the angular momentum.

r is the distance from the pivot point to the element dr of the rod.

5. Oct 3, 2009

e2m2a

ok, if I integrate the equation dL = (m/l)vrcos(theta)dr, taking the limits of integration from 0 to l, I get as the final answer:

L = m v r cos(theta).

But this is the same result for a point particle. Is this correct?

6. Oct 4, 2009

tiny-tim

(what happened to that θ i gave you? )

Noooo … that would be ∫ (m/l) vcosθ dr

get some sleep! :zzz:

7. Oct 5, 2009

e2m2a

Tiny-Tim,

Unless I am missing something on integration, when I integrate dL = ∫(m/l)vrcos(θ)dr, taking the limit from 0 to l, where l is the length of the rod, m is the mass of the rod, v is the velocity of the cart right at collision, and θ is the angle of the rod with respect to the x-axis at the time of collision, I get:

L = (m/l) v cos(θ) (r²/2)

Evaluated from the limits 0 to l, I get:

L =(l²/2) m (v/l) cos(θ)

Which reduces to:

L = (l/2) m v cos(θ)

Now, l/2 is simply the distance from the axis to the center of mass of the rod, or:

L = m v rcom cos(θ)

Where, rcom is the distance from the axis to the
center of mass of the rod.

8. Oct 6, 2009

tiny-tim

oh, your r was rcom … i assumed it was l.


text:
# Fractions

In fractions we have already learnt about the concept of fraction in 6th grade. We have also learned about types of fractions (proper, improper and mixed), comparison of fractions, equivalent fractions, representation of fractions on the number line, addition and subtraction of fractions.

In 7th grade we will revise and recall, in brief, what we have learnt about fractions and also learn about the multiplication and division of fractions.

What is fraction?

A fraction is a number representing a part of a whole. The whole may be a single object or a group of objects.

A fraction is a number representing a part of a whole. A fraction can be expressed in the form a/b, where a, b are whole numbers and b ≠ 0.

Take a piece of paper. Divide in to four equal parts as shown in the given diagram. Shade one part by some color. Now this shaded part is one of four parts. We say here that the shaded part is one-fourth of the total. ‘One-fourth’ is written as 1/4.

In the adjoining diagram, two of three equal parts have been shaded. Thus the shaded portion is two-third (2/3) of the total.

The numbers like ¹/₄, ²/₃, ³/₅, etc., are called fractions.

A fraction is quantity which expresses a part of whole. By the fraction ³/₅ we mean---Divide a quantity in to 5 equal parts and then take 3 parts of them.

There are two types of fractions ---Common fraction and decimal fraction. We shall now discuss about common fraction.

Common Fraction:

A fraction when expression by two natural numbers, one above the other, separated by a horizontal line is called a common fraction.

Thus, ³/₄, ¹/₃, ⁷/₁₁, etc., are common fraction.

Numerator and Denominator of a Fraction:

In the fraction, the below of the horizontal line is called the denominator. It expresses the number of parts into which a quantity is divided.

The number above the horizontal line is called the numerator. It expresses the number of parts which are taken.

So, Fraction =    Numerator
Denominator

In a fraction a/b we call ‘a ‘as numerator and ‘b’ as denominator.

The fraction representing two parts out of 7 equal parts in which the whole is divided is denoted by ²/₇ and is read as “two-sevenths” . In the fraction ²/₇, 2 is called the numerator and 7 is called the denominator.

Similarly, ³/₅ is a fraction with numerator 3 and denominator 5 and ¹²/₇ is a fraction with numerator 12 and denominator 7. Again similarly in fraction ³/₄, 3 is numerator and 4 is denominator.

In seventh grade fractions we will discuss mainly on these points with examples.

A fraction whose numerator is lesser than the denominator is called a proper fraction.

A fraction whose numerator is more than or equal to the denominator is called an improper fraction.

A combination of a whole number and a proper fraction is called a mixed fraction.

To get a fraction equivalent to a given fraction, we multiply (or divide) its numerator and denominator by the same non-zero number.

Fractions with the same denominators are called like fractions. Otherwise, they are called unlike fractions.

A fraction is said to be in its lowest terms if its numerator and denominator have no common factor other than 1.

To compare fractions, we use the following steps:

Step I:  Find the least common multiple (LCM) of the denominators of the given fractions.

Step II:  Convert all fractions to its equivalent fraction with denominator is equal to the least common multiple (LCM) obtained in step 1.

Step III: Arrange the fractions in descending or ascending order by arranging numerators in descending or ascending order.

● To convert unlike fractions into like fractions, we use the following steps:

Step I:  Find the least common multiple (LCM) of the denominators of the given fractions.

Step II:  Convert each of the given fractions into an equivalent

● To add (or subtract) fractions, we may use the following steps:

Step I: Obtain the fractions and their denominators.

Step II: Find the least common multiple (LCM) of the denominators.

Step III:  Convert all fractions into an equivalent fraction having its denominator equal to the least common multiple (LCM).

Step IV:  Add (or subtract) like fractions obtained in Step III.

Product of two fraction =    Product of their numerator
Product of their denominators

Two fractions are said to be reciprocal of each other, if their product is 1. The reciprocal of a non-zero fraction $$\frac{a}{b}$$ is equal to $$\frac{b}{a}$$

The division of a fraction $$\frac{a}{b}$$ by a non-zero fraction $$\frac{c}{d}$$ is the product of $$\frac{a}{b}$$ with the reciprocal of $$\frac{c}{d}$$

Fractions

Fractions

Types of Fractions

Equivalent Fractions

Like and Unlike Fractions

Conversion of Fractions

Fraction in Lowest Terms

Multiplication of Fractions

Division of Fractions

Fractions - Worksheets

Worksheet on Fractions

Worksheet on Multiplication of Fractions