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## WeBWorK Problems

### randomizing word pairs in question text.

by tim Payer -
Number of replies: 11
Hello All,

I would like to use some randomized word pairs in my question text.
These word pairs do not have to be tied to the numerical answers of the problem that I am using.

For example
"Given that 23% of a flock of birds has the bird flu what is the probability of drawing two birds from the flock that have bird flu"?

Where in the example above the word variable of flock would be replaced by:
flock = gaggle, murder, murmuring, etc...
Paired with these flock names would be the associated bird name
birds = geese, crows, starlings, etc...

Much appreciated!

Tim

by John Jones -

$a = list_random(['gaggle', 'geese'],['murder','crows'],['murmuring', 'starlings']); Then if$a->[0] is murder, then $a->[1] is crows. John In reply to John Jones ### Re: randomizing word pairs in question text. by Paul Pearson - I like John's solution, but here's one that uses more basic data structures. @birds = ('geese','crows','starlings'); # create parallel arrays @flock = ('gaggle','murder','murmuring');$i = random(0, 2, 1); # choose an array index at random
# $i = random(0,$#birds, 1); # choose an array index at random, more fancy

A $flock[$i] of $birds[$i] ...

This approach could also be generalized using a hash (associative array).

Paul Pearson

### Re: randomizing word pairs in question text.

by tim Payer -
Thanks Paul!

I will give that a try!

What would happen if a had triples instead or pairs to account for the singular and plural spellings?

Could this also be used for that approach?

Tim

### Re: randomizing word pairs in question text.

by Paul Pearson -
# create three arrays of the same length "in parallel"
@bird = ('goose','crow','starling');
@birds = ('geese','crows','starlings');
@flock = ('gaggle','murder','murmuring');
$i = random(0, 2, 1); # choose an array index at random BEGIN_TEXT A$bird[$i] flew with a$flock[$i] of$birds[$i] ... END_TEXT BEGIN_PGML A [$bird[$i] ] flew with a [$flock[$i] ] of [$birds[$i] ] ... END_PGML In reply to Paul Pearson ### Re: randomizing word pairs in question text. by tim Payer - Hi Paul, I liked your solution to my word triple problem, but unfortunately the arrays are non-responsive as I have entered them. Can you see any problems? I think I entered your arrays as given... Thanks, for any help. Tim # DESCRIPTION Function translations # Translate descriptions of critical points and function # into correct notation. # WeBWorK problem written by TimPayer <tsp1@humboldt.edu> # ENDDESCRIPTION ## DBsubject(Probability) ## DBchapter(Random variables) ## DBsection(Expectation) ## Institution(Humboldt State University) ## Author(Tim Payer) ## KEYWORDS(probability, translate, notation) DOCUMENT(); loadMacros( "PGstandard.pl", "PGunion.pl", "MathObjects.pl", "parserPopUp.pl", "PGML.pl", "PGcourse.pl", ); #Text(beginproblem()); #uncomment install_problem_grader(~~&std_problem_grader);$showPartialCorrectAnswers = 1;

Context("Numeric");

# create three arrays of the same length "in parallel"
@bird = ('goose','crow','starling');
@birds = ('geese','crows','starlings');
@flock = ('gaggle','murder','murmuring');
$i = random(0, 2, 1); # choose an array index at random$popup1 = PopUp(
["probability notation", "P(V)", "P(N)", "P(VV)", "P(NN)", "P(VN)", "P(NV)",
"P(VN  U  NN)", "P(NV  U  NN)", "P(VN   $\cap$   NN)", "P(NV   $\cap$   NN)",
"P(VN   $\cap$   NV  U  NN)", "P(VN  U  NV  $\cap$  NN)", "P(VN  U  NV  U  VV)",
"P(VN   $\cap$   NV  $\cap$  NN)", "P(NN   $\cap$   NV  U  VN)", "P(NN  U  NV  $\cap$  VV)",
"P(NN   $\cap$   NV   $\cap$  VV)","P(VN   $\cap$  NV) + P(NN)", "P(VN  $\times$  NV) + P(NN)",
"P(VN   $\cap$   NN) + P(NV   $\cap$   NN)",], "P(VV)");

$popup2 = PopUp( ["probability notation", "P(V)", "P(N)", "P(VV)", "P(NN)", "P(VN)", "P(NV)", "P(VN U NN)", "P(NV U NN)", "P(VN $\cap$ NN)", "P(NV $\cap$ NN)", "P(VN $\cap$ NV U NN)", "P(VN U NV $\cap$ NN)", "P(VN U NV U VV)", "P(VN $\cap$ NV $\cap$ NN)", "P(NN $\cap$ NV U VN)", "P(NN U NV $\cap$ VV)", "P(NN $\cap$ NV $\cap$ VV)","P(VN $\cap$ NV) + P(NN)", "P(VN $\times$ NV) + P(NN)", "P(VN $\cap$ NN) + P(NV $\cap$ NN)",], "P(VN U NV U VV)"); BEGIN_PGML 1. A flock of [$birds[$i] ] is described as a [@flock[$i] ] of [$birds[$i] ].
Consider the case where (percent)% of a [@flock[$i] ] of [$birds[$i] ] carry a bird flu virus, and that two [$birds[$i]] are drawn at random from the [@flock[$i] ], find the following probabilities given the following variable declarations:

Let V = Event that a [$bird[$i] ] has the bird flu virus.

Let N = Event that a [$bird[$i] ] does not have the bird flu virus.

1a.) What is the probability of randomly drawing two [$birds[$i] ] that both have the bird flu virus? Choose the correct probability notation below and then calculate its associated probability with fourth decimal accuracy.

[___]{$popup1} = 1b.) What is the probability of randomly drawing two [$birds[$i] ] such that at least one [$bird[$i]] has the bird flu virus? Choose the correct probability notation below and then calculate its associated probability with fourth decimal accuracy. [___]{$popup2} =

END_PGML

BEGIN_PGML_SOLUTION

END_PGML_SOLUTION

ENDDOCUMENT();

### Re: randomizing word pairs in question text.

by Paul Pearson -

https://raw.githubusercontent.com/openwebwork/pg/master/macros/PGML.pl

since PGML.pl versions before June 2015 did not handle nested square brackets properly.  If you don't have command line access on the server to install PGML.pl server wide, install it for a particular course by clicking "File Manager" and saving PGML.pl in the location

macros/PGML.pl

Also, you want to extract scalars (prefix $) from arrays (prefix @), so use BEGIN_PGML [$flock[$i] ] END_PGML but do not use BEGIN_PGML [ @flock[$i] ]
END_PGML

Or, just use the solution that recommends hashes.

### Re: randomizing word pairs in question text.

by Alex Jordan -
Another option is to use hashes, which are like functions with input and output (keys and values). Depending on taste, the code may be more to your liking.

$bird = list_random('goose', 'crow', 'starling'); %plural = (goose=>'geese', crow=>'crows', starling=>'starlings'); %flock= (goose=>'gaggle', crow=>'murder', starling=>'murmuring'); BEGIN_TEXT The plural of$bird is $plural{$bird}. A group of $plural{$bird} is called a $flock{$bird}.
END_TEXT

by Alex Jordan -

BEGIN_PGML
The plural of [$bird] is [$plural{$bird}]. A group of [$plural{$bird}] is called a [$flock{$bird}]. END_PGML In reply to Alex Jordan ### Re: randomizing word pairs in question text. by tim Payer - Thank you all but unfortunately I am not having any success with my flock of bird word pairs. Could you please help? Only the single instance of [$bird] is rendering. Everything else is not rendering.

I tried "hashes" as an alternative form and I tried indexing.
And no, I do not see where I can load the latest version of PGML through my file Manager.

I am using the "Free Trial" version 10 that was distributed last July.
As the hashes do NOT seem to work, must I get a newer version?

Thanks so much for your help.

The code block is below if you can see anything that I have missed.

# DESCRIPTION   Function translations
# Translate descriptions of critical points and function
# into correct notation.
# WeBWorK problem written by TimPayer <tsp1@humboldt.edu>
# ENDDESCRIPTION

## DBsubject(Probability)
## DBchapter(Random variables)
## DBsection(Expectation)
## Institution(Humboldt State University)
## Author(Tim Payer)
## KEYWORDS(probability, translate, notation)
DOCUMENT();

"PGstandard.pl",
"PGunion.pl",
"MathObjects.pl",
"parserPopUp.pl",
"PGML.pl",
"PGcourse.pl",
);

#Text(beginproblem());   #uncomment

$showPartialCorrectAnswers = 1; Context("Numeric"); # create three arrays of the same length "in parallel" # Using "hashes":$bird = list_random('goose', 'crow', 'starling');
%plural = (goose=>'geese', crow=>'crows', starling=>'starlings');
%flock= (goose=>'gaggle', crow=>'murder', starling=>'murmuring');

#@bird = ('goose','crow','starling');  ##Apparently I do not have the latest PGML?
#@birds = ('geese','crows','starlings');
#@flock = ('gaggle','murder','murmuring');
#$i = random(0, 2, 1); # choose an array index at random$popup1 = PopUp(
["probability notation", "P(V = 0)", "P(V = 1)", "P(V = 2)", "P(V  > 0)", "P(V  > 1)","P(V < 2)",], "P(V = 2)");

$popup2 = PopUp( ["probability notation", "P(V = 0)", "P(V = 1)", "P(V = 2)", "P(V > 0)", "P(V > 1)","P(V < 2)",], "P(V > 0)");$popup3 = PopUp(
["probability notation", "P(V = 0)", "P(V = 1)", "P(V = 2)", "P(V = 3)","P(V  > 0)", "P(V > 1)","P(V < 2)","P(V < 3)",], "P(V  > 0)");

# Need if then for 1c depending on whether "at least one or at most 2

BEGIN_PGML

Trials to test:

The plural of [$bird] is [$plural{$bird}]. A group of [$plural{$bird}] is called a [$flock{$bird}]. 1. A flock of [$plural{$bird}] are described as a [$flock{$bird} ] of [$plural].
Consider the case where (percent)% of a [$flock[$i] ] of [$birds[$i] ] carry a bird flu virus, and that two [$birds[$i]] are drawn at random from the [$flock[$i] ], find the following probabilities but first complete the following event variable declaration:

Let V = The number of [$birds[$i] ] that have the bird flu virus from (two $k) random draws of the [$flock[$i] ]. 1a.) What is the probability of randomly drawing two [$birds[$i] ] that both have the bird flu virus? Choose the correct probability notation below and then calculate its associated probability with fourth decimal accuracy. [___]{$popup1}  =

1b.) What is the probability of randomly drawing two [$birds[$i] ] such that at least one [$bird[$i]] has the bird flu virus? Choose the correct probability notation below and then calculate its associated probability with fourth decimal accuracy.

[___]{$popup2} = 1c.) Consider the case where three [$birds[$i]] are drawn at random from the [$flock[$i] ]. What is the probability of drawing (draw$j at least one [$bird[$i] ] that has , at most two [$birds[$i] ]  that have ) the bird flu virus? First complete the declaration of the event variable below:

Let V = The number of [$birds[$i] ] that have the bird flu virus from (three $k) random draws of the [$flock[$i] ]. Choose the correct probability notation below and then calculate its associated probability with fourth decimal accuracy. [___]{$popup3} =

END_PGML

BEGIN_PGML_SOLUTION

END_PGML_SOLUTION

ENDDOCUMENT();

### Re: randomizing word pairs in question text.

by Alex Jordan -
The following (cut from your sample) works for me. But perhaps you have an older version of pgml.pl that doesn't parse things like [$flock{$bird}] correctly. I'm using 2.11. I think I have a few development branch pulls as well, and I can't say at the moment if pgml.pl is affected. Be wary of just acquiring the latest version of pgml.pl without a full upgrade, since it's possible that concurrent upgrades to deeper parts of the engine will be required for the newer pgml.pl.

Maybe if you have to stick with your current version, [@$flock{$bird}@] will work. I'm not sure though.

$bird = list_random('goose', 'crow', 'starling'); %plural = (goose=>'geese', crow=>'crows', starling=>'starlings'); %flock= (goose=>'gaggle', crow=>'murder', starling=>'murmuring'); BEGIN_PGML The plural of [$bird] is [$plural{$bird}]. A group of [$plural{$bird}] is called a [$flock{$bird}].

END_PGML


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# Highlights

Awards and Honors

### Hertha-Sponer-Preis 2011 der DPG

Martina Hentschel vom Dresdner Max-Planck-Institut für Physik komplexer Systeme erhält den mit 3.000 Euro dotierten „Hertha-Sponer-Preis“ der Deutschen Physikalischen Gesellschaft. Forschungsschwerpunkt der 39-jährigen Wissenschaftlerin ist die theoretische Beschreibung mesoskopischer elektronischer und optischer Systeme. Ihre Arbeiten sind für die Entwicklung miniaturisierter Laser von praktischer Bedeutung.
Publication Highlights

### A Homonuclear Molecule with a Permanent Electric Dipole Moment

Permanent electric dipole moments in molecules require a breaking of parity symmetry. Conventionally, this symmetry breaking relies on the presence of heteronuclear constituents. We report the observation of a permanent electric dipole moment in a homonuclear molecule in which the binding is based on asymmetric electronic excitation between the atoms. These exotic molecules consist of a ground-state rubidium (Rb) atom bound inside a second Rb atom electronically excited to a high-lying Rydberg state. Detailed calculations predict appreciable dipole moments on the order of 1 Debye, in excellent agreement with the observations. Weibin Li, Thomas Pohl, Jan-Michael Rost, Seth T. Rittenhouse, Hossein R. Sadeghpour, Johannes Nipper, Bjoern Butscher, Jonathan Balweski, Vera Bendowsky, Robert Löw, Tilman Pfau Science 334, 1110 (2011)
Awards and Honors

### Paul Ehrlich- und Ludwig Darmstädter-Nachwuchspreis 2011 der Paul Ehrlich-Stiftung

Wie differenzieren sich Zellen in die verschiedenen Zelltypen, die einen lebenden Organismus ausmachen? Neben molekularen Mechanismen spielen dabei mechanische Prozesse eine wesentliche Rolle. Wie diese Parameter miteinander interagieren, steht im Zentrum der Forschungsaktivitäten von Dr. Stephan Grill. Der Biophysiker hat dazu eine Methode entwickelt, mit der die mechanischen Kräfte in lebenden Zellen gemessen werden können. Mit Hilfe eines Lasers kann er bestimmte Zellstrukturen minimal-invasiv zerstören. Deren Fragmente bewegen sich danach voneinander weg, falls die Struktur unter mechanischer Spannung stand. Auf diese Weise erhält der Wissenschaftler einen Überblick darüber, wo in der Zelle mechanische Kräfte walten.
Publication Highlights

### Polarization of PAR Proteins by Advective Triggering of a Pattern-Forming System

In the Caenorhabditis elegans zygote, a conserved network of partitioning defective 4 (PAR) polarity proteins segregate into an anterior and a posterior domain, facilitated by flows of the cortical actomyosin meshwork. The physical mechanisms by which stable asymmetric PAR distributions arise from transient cortical flows remain unclear. We present evidence that PAR polarity arises from coupling of advective transport by the flowing cell cortex to a multistable PAR reaction-diffusion system. By inducing transient PAR segregation, advection serves as a mechanical trigger for the formation of a PAR pattern within an otherwise stably unpolarized system. We suggest that passive advective transport in an active and flowing material may be a general mechanism for mechanochemical pattern formation in developmental systems. W. Goehring, Philipp Khuc Trong, Justin S. Bois, Debanjan Chowdhury, Ernesto M. Nicola, Anthony A. Hyman, Stephan W. Grill Science 334, 1137 (2011)
Awards and Honors

### Marian-Smoluchowski-Emil-Warburg-Preis 2011 der DPG

Der Dresdner Physiker Peter Fulde, emeritierte Direktor des Max-Planck-Instituts für Physik komplexer Systeme, erhält den deutsch-polnischen „Marian-Smoluchowski-Emil-Warburg-Preis“, der mit 3.000 Euro dotiert ist. Der 74-Jährige wird für seine Beiträge zur Theorie der Festkörperphysik ausgezeichnet. Peter Fulde hat insbesondere zur Erforschung der Supraleitung, des Magnetismus und zum Verständnis „elektronischer Korrelationen“ maßgeblich beigetragen.
Institute's News

### New research group 'Computational Biology and Evolutionary Genomics'

The joint junior research group between our institute and the MPI of Molecular Cellbiology and Genetics is headed by Dr. Michael Hiller and uses computational approaches to link phenotypic differences between species to differences in their genomes, which is key to understand how nature’s phenotypic diversity evolved.
Institute's News

### New research group 'Collective Dynamics of Cells'

The research group headed by Dr. Vasily Zaburdaev develops and applies methods of statistical physics helping to understand complex biological phenomena.
Publication Highlights

Vulnerabilities related to weak passwords are a pressing global economic and security issue. We report a novel, simple, and effective approach to address the weak password problem. Building upon chaotic dynamics, criticality at phase transitions, CAPTCHA recognition, and computational round-off errors we design an algorithm that strengthens security of passwords. The core idea of our method is to split a long and secure password into two components. The first component is memorized by the user. The second component is transformed into a CAPTCHA image and then protected using evolution of a two-dimensional dynamical system close to a phase transition, in such a way that standard brute-force attacks become ineffective. We expect our approach to have wide applications for authentication and encryption technologies. T.V.Laptyeva, S. Flach, K. Kladko arXiv:1103.6219v1 (2011)
### Emerging local Kondo screening and spatial coherence in the heavy-fermion metal YbRh$_2$Si$_2$
The entanglement of quantum states is both a central concept in fundamental physics and a potential tool for realizing advanced materials and applications. The quantum superpositions underlying entanglement are at the heart of the intricate interplay of localized spin states and itinerant electronic states that gives rise to the Kondo effect in certain dilute magnetic alloys. In systems where the density of localized spin states is sufficiently high, they can no longer be treated as non-interacting; if they form a dense periodic array, a Kondo lattice may be established1. Such a Kondo lattice gives rise to the emergence of charge carriers with enhanced effective masses, but the precise nature of the coherent Kondo state responsible for the generation of these heavy fermions remains highly debated. Here we use atomic-resolution tunnelling spectroscopy to investigate the low-energy excitations of a generic Kondo lattice system, YbRh$_2$Si$_2$. We find that the hybridization of the conduction electrons with the localized 4f electrons results in a decrease in the tunnelling conductance at the Fermi energy. In addition, we observe unambiguously the crystal-field excitations of the Yb$^{3+}$ ions. A strongly temperature-dependent peak in the tunnelling conductance is attributed to the Fano resonance resulting from tunnelling into the coherent heavy-fermion states that emerge at low temperature. Taken together, these features reveal how quantum coherence develops in heavy 4f-electron Kondo lattices. Our results demonstrate the efficiency of real-space electronic structure imaging for the investigation of strong electronic correlations, specifically with respect to coherence phenomena, phase coexistence and quantum criticality. S. Ernst, S. Kirchner, C. Krellner, C. Geibel, G. Zwicknagel, F. Steglich & S. Wirth Nature 474, 362 (2011)

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# Tag Info

8

I have calculated pressure in container one and that is p1 = 342817.92 Pa, and in container two p2 = 326979.2275 Pa. Then I have substracted this two values which gives me 15838.6925 Pa. After that I have divided this value with two, to figure out how much do I have to decrease pressure in container 1, and that gives me 7919.34625 Pa. You cannot generally ...

5

Unfortunately this is not feasible Current catalytic converters are designed to remove noxious pollutants from exhausts. Things like carbon monoxide (poisonous) and nitrogen oxides (noxious and irritating). The trouble with doing this for carbon dioxide is fundamental. Carbon dioxide and water are the ultimate products of combustion of hydrocarbons because ...

5

Chalk cannot at given conditions explode chemically, spontaneously turning into other compounds. It could locally "explode" mechanically as consequence of mechanical stress caused by thermal dilation due changes to temperature. But calling it explosion would be very much exaggerated. It would be rather just cracking. There is a way such cracking ...

3

Nothing would happen. The reaction would not shift to produce more product. Let's suppose you allow this reaction to equilibrate in a sealed container. Equilibrate means you've reached the equilibrium value for $p_{\text{O}_2}$ at that temperature. Now further suppose you were able to remove all the product without changing the free volume (say you ...

3

I am using a saturated sodium chloride solution as an example here because it is more common and easier to reproduce at home. This system is at equilibrium, with salt dissolving and crystallizing at the same rate. If you take away the solid NaCl from a saturated solution, both the forward and the reverse reactions stop. Obviously, salt crystals are no longer ...

2

Let's compare the two compounds. Solubility: Both compounds are miscible with water, so there is no difference. With the straight chain, 1-propanol may be slightly more lipophilic, and thus better able to act as a surfactant. However, 1-propanol is a really poor surfactant. Volatility: Isopropanol has a lower boiling point (82.5 degrees C) than 1-propanol (...

1

It might help to think of the system as containing three phases: $\ce{P4(s)}$, $\ce{O2(g)}$ and $\ce{P4O10(s)}$. As long as the pressure, temperature and compositions of each phase remain constant (and P,T and chemical potentials across phases are in equilibrium i.e. equal), we can alter their extents arbitrarily without altering the condition for ...

1

Depends on how accurately performed is each measurement. One would have to take into account precision of ruler and asymmetry of cube, for example, in the first case, and adhesion of water to cube and cylinder, causing meniscus and other errors. However, in some cases, ruler measurement would be far more accurate than water displacement: Archimedes would ...

1

Gypsum is the major constituent of plaster. It is non-toxic. Our lungs are capable of filtering out particles greater than 50 micrometers in diameter. Any dust finer than that is potentially dangerous.

Only top voted, non community-wiki answers of a minimum length are eligible


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LNG

# BP Receives Key Permit for Tangguh LNG Expansion

## Environmental permit will allow work on the $12-billion project to begin on site. BP received approval from the Government of Indonesia's Ministry of Environment for the Tangguh Expansion Project Integrated AMDAL environmental and social impact assessment. The Indonesian government issued the project an environmental permit as a result. The AMDAL includes Tangguh’s environmental and social commitments. It also outlines the role of the local administration and central government. Approval is required for project activities to begin at the site. The$12-billion Tangguh Expansion Project will deliver significant value to the Indonesian government and will help meet the growing energy demand in Indonesia. As part of the expansion project, BP and its partners will supply 40% of the LNG output from Train 3 (1.5 mtpa) to Indonesia’s state electricity company PT PLN (Persero) for the Indonesian domestic market.

“This is a significant milestone for the Tangguh Expansion Project, and I would like to give my highest appreciation to the Ministry of Environment, along with the Papua Barat, Teluk Bintuni, and Fakfak Administrations for their efforts and cooperation in reaching this achievement,” said Christina Verchere, BP regional president Asia Pacific. “We look forward to receiving the remaining approvals from the government to realize the project, which will bring significant benefit to Indonesia.”

Tangguh LNG, located in Teluk Bintuni, Papua Barat province, is the third-largest LNG supply facility in Indonesia and the first fully combined upstream and downstream LNG operation in the country. The current operations comprise two identical LNG trains (Trains 1 and 2) with production capacity of 7.6 mtpa.

Other key government approvals that are still in process are required to continue with the planning, design, and procurement of the expansion project.

“We are engaging closely with SKK Migas, the Ministry of Energy and Mineral Resources, and other related government agencies to receive these critical approvals," added Verchere. "We remain hopeful that they will be received soon.”


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# Three geometry questions

In figure AB$\parallel PQ \parallel CD$ Prove that $\frac 1 x + \frac 1 y = \frac 1 z$

Equilateral triangles APB, BQC and ASC are described on each side of a right-angled triangle ABC, right angled at B. Then prove that ar($\triangle$APB)+ar($\triangle$BQC)=ar($\triangle$ASC).

-
what's the meaning of ar($\triangle$APB)+ar($\triangle$BQC)+ar($\triangle$ASC)? – lab bhattacharjee Sep 23 '12 at 14:18
Im guessing that its three external angles based on the respective sides, but i could be wrong... These people are paid to make confusing questions D: – Aayush Agrawal Sep 23 '12 at 14:19
Could you tell me what's to be proven? – lab bhattacharjee Sep 23 '12 at 14:21
Prove that ar(△APB)+ar(△BQC)=ar(△ASC) – Aayush Agrawal Sep 23 '12 at 14:26

As $\triangle ABD,\triangle PQD$ are similar as $\angle ABD=\angle PQD$ ,so $$\frac z x= \frac{QD}{BD}$$

Similarly, as $\triangle BCD,\triangle BPQ$ are similar ,so $$\frac z y = \frac{BQ}{BD}$$

So, $$\frac z x+\frac z y=\frac{QD}{BD}+ \frac{BQ}{BD}=1$$

The area of $\triangle APB=\frac{\sqrt3}2|AB|^2$

So, the area$(\triangle APB)$+ area $(\triangle BQC)$ $=\frac{\sqrt3}4(|AB|^2+|BC|^2)=\frac{\sqrt3}4 |CA|^2=$area$(\triangle ASC)$ as $|AB|^2+|BC|^2=|CA|^2$

-
Thx, i just added a second question too, just realised i forgot to add that earlier. Can you take a loot at that too? Thx alot! – Aayush Agrawal Sep 23 '12 at 14:16
@AndréNicolas, thanks for your observation, I've rectified now. – lab bhattacharjee Sep 23 '12 at 15:08

Part 1,

In, $\Delta ABD, \frac{z}{x}=\frac{QD}{BD}$

In, $\Delta BCD, \frac{z}{y}=\frac{BQ}{BD}$

These equations $\implies \frac{z}{x}+\frac{z}{y}=\frac{BQ+QD}{BD}=1\implies \frac{1}{x}+\frac{1}{y}=\frac{1}{z}$

Part 2,

ar($\triangle$APB)+ar($\triangle$BQC)=$\frac{\sqrt 3}{4}(|AB|^2+|BC|^2)=\frac{\sqrt 3}{4}|AC|^2=$ar($\triangle$ASC)$- Thx man, i just realised i forgot to add a second question :( can you take a look at that one too – Aayush Agrawal Sep 23 '12 at 14:16 For the added question, note that the area of an equilateral triangle of side$a$is$\lambda a^2$for some constant$\lambda$. By using properties of special angles, or otherwise, it can be shown that actually$\lambda=\frac{\sqrt{3}}{4}$, but that is not really needed here. For details, draw the height from some vertex, and note that the height is$a\sin(60^\circ)$. From the Pythagorean Theorem, we have$(BA)^2+(BC)^2=(AC)^2$. Multiplying through by$\lambda$we get$\lambda(BA)^2+\lambda(BC)^2=\lambda(AC)^2\$, which gives the desired area result.

Remark: Adding to a question is not a good idea. This should have been an entirely new question.

-


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# Comparing effects y1 ON x vs. y2 ON x in different regressions

Let's assume we have predictors $$x_1-x_5$$ and dependent variables $$y_1-y_9$$ in one dataset. We have a certain hypothesis about $$x_1$$: it should have differently strong effects on our 9 dependent variables $$y_1-y_9$$.

We perform 9 regressions, and find that $$x_1$$ significantly predicts all y with $$p < .001$$. Now we want to find out whether these (highly significant) effects are different from each other (just because they are significant effects does not mean they are equally strong). Two questions:

1. What information that a regression provides us with would give us insight into this question? Unstandardized beta? Standardized beta? t?

2. Is there a statistical test we can perform to find out whether the strengths of predictions are different between (y1 ON x1) and (y2 ON x1) and (y3 ON x1)?

Software I can use is R and SPSS; field is psychology/medicine.

• Is that $y_1 \lt -x_1$ or $y_1 \leftarrow x_1$ or possibly the R code y1 <- x1? Nov 15 '12 at 7:15
• With y <- x I mean the regression effect of x on y. y ON x, as MPLUS puts it. Nov 15 '12 at 15:09
• Torvon, are you asking whether there is a way to compare which of the three outcomes, $y_1,y_2,y_3$ that $x_1$ has the greatest effect on? Also: do $y_1,y_2,$ and $y_3$ all have the same units? Nov 20 '21 at 17:44


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# Intuition/Motivation behind Algebras (R-Algebras, Q-Algebras, etc.)

I'm currently reading Ch. 10 in Dummit & Foote (3rd Edition) and towards the end of the first section, it defines an R-algebra. Dummit & Foote do a decent job of motivating certain definitions/ideas, but this one really seems important and just kinda comes out of nowhere. I was wondering if anyone could possibly give me some motivation or intuition to where this idea of an 'algebra' comes from? Thanks!

• What are you looking for in terms of motivation? The complex numbers and quaternions form algebras over the reals, and so do the octonions (though this last algebra is not associative and may not be encompassed by the definition in the textbook). – Matt Samuel Mar 8 '16 at 6:25
• Algebras lie at the intersection of module and ring street. As such, both theories (especially when we talk about $F$-modules, that is-vector spaces) can be used as avenues of attack. – David Wheeler Mar 9 '16 at 3:29
• To add on/clarify David's comment, basically by saying something is an $k$-algebra where $k$ is a field, you should always remember it is a ring AND a $k$-vector space. In particular, the natural place to do representation theory is over $k$ (and any extension of $k$), i.e. you want to consider $k$-algebra hom (i.e. ring hom and vector space hom) $\rho: A \to GL_n(k)$, or equivalently $A$-modules which are also $k$-vector spaces. – Aaron Mar 10 '16 at 11:47
• Thank you all! These were just the answers that I needed. – hijasonno Mar 12 '16 at 2:01