text:
User talk:Yaroslav Blanter

 Hello Yaroslav Blanter, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Yaroslav Blanter


text:
User:Yaroslav Blanter

meta:User:Yaroslav Blanter

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Yaroslav Blanter


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User:J.delanoy/monobook.js

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title: User:J.delanoy/monobook.js


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User:X!/monobook.js

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title: User:X!/monobook.js


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User:X!

en:User:Soxred93 fr:User:Soxred93 simple:User:X!

title: User:X!


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Template:Human Legacy

This user is the instructor for the Human Legacy Course.

title: Template:Human Legacy


text:
New journalism

New Journalism - An overview
New Journalism is a broad term for how journalism is changing as the internet and technology enhance how journalists report and deliver the news. New media involves blogging, podcasting, webcasting, and Twittering. It improves upon what journalism is a about: news. By breaking a story at almost the instant it happens, and spreading it around the net, the news travels fast.

The biggest factor in new journalism is that the professional journalist now must compete with the general populace when it comes to reporting. With the aid of blogs and smart phones, any amateur can become a field reporter. Many news agencies, such as CNN, have specialized iReport sites that allow the people to input their field work to gain better coverage of the news.

Common types of New Journalism

 * 1) Blogging
 * 2) Crowdsourcing
 * 3) Citizen journalism

title: New journalism


text:
User:Kuvaly/vector.js

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User talk:Tomlowenhaupt

 Hello Tomlowenhaupt, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Tomlowenhaupt


text:
User talk:Andrewdavis117

 Hello Andrewdavis117, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Andrewdavis117


text:
User talk:Robert Pappenheimer

 Hello Robert Pappenheimer, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
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title: User talk:Robert Pappenheimer


text:
Human Legacy Course/The Beginnings of Civilization (Prehistory - 1000 BC)

Early humans slowly spread from Africa to other parts of the world. These early people struggled to survive by using basic stone tools and weapons to hunt animals, catch fish, and gather plants and nuts. For hundreds of thousands of years, early people lived in this way. Then, as the last Ice Age ended, some people learned to farm. This breakthrough gave rise to villages and cities—and in time, to the first civilizations.

Scientific evidence suggests that the earliest humans lived in Africa. This week, you will learn how humans spread to other parts of the world and established the first civilizations.

Let's Begin!


text:
Human Legacy Course/Quiz 1

{What are the preserved remains or imprints of living things? -Hominids -Artifacts -Absolute Dating +Fossil
 * type=""}

{What was the name of the first part of the Stone Age? -Neanderthal +Paleolithic Era -Homo Sapiens -Society
 * type=""}

{Who were the people who hunted animals and gathered wild plants, seeds, fruits, and nuts to survive? -Anthropologists -Archaeologists +Hunter-Gatherers -Homo Erectus
 * type=""}

{What was the name for the early humans known for their upright posture? +Homo Erectus -Homo Sapiens -Neanderthal -Hunter-Gatherers
 * type=""}

{What was the time before the invention of writing? -Society -Hominids +Prehistory -Absolute Dating
 * type=""}

{What is the term to describe a scientist who digs into ancient settlements to find objects used by early people? -Anthropologist -Hunter-Gatherer -Hominid +Archaeologist
 * type=""}

{Who is the "Wise Man?" -Homo Erectus +Homo Sapiens -Neanderthal -Anthropologists
 * type=""}

{What is a community of people who share a common culture?
 * type=""}

-Absolute Dating -Artifacts +Society -Hominids

{What is the term used for early humanlike beings and humans?
 * type=""}

+Hominids -Homo Erectus -Homo Sapiens -Hunter-Gatherers

{What is the term for a scientist who studies fossils to learn more about human origins? -Archaeologist -Hominid +Anthropologist -Neanderthal
 * type=""}

title: Human Legacy Course/Quiz 1


text:
Human Legacy Course/Quiz 2

{The shift from hunting and gathering to farming is called the...? -Bronze Age -Iron Age -Ice Age +Neolithic Revolution
 * type=""}

{Which factor most influenced the types of crops grown? -Animals -Language +Geography -Tools
 * type=""}

{Domestication meant that early humans could...? -live in covered dwellings. +produce their own food. -predict climate changes. -build more efficient tools.
 * type=""}

{Which group was more likely to contain people of different social status? -Hunter-Gatherers -Neanderthals +Settled People -Paleolithic Hominids
 * type=""}

{Technological advances, social ranking, and trade developed once people...? +were not worried about basic survival. -discovered Çatal Hüyük. -were able to navigate mountains and deserts. -learned to polish stones.
 * type=""}

{What was one such animal-based disease at this time? -Bronze Age +Measles -Dog -Çatal Hüyük
 * type=""}

{What is one of the oldest farming sites discovered? -Bronze Age -Pottery -Horse +Çatal Hüyük
 * type=""}

{What was used for cooking, storing water, grains, and oils at this time? -Dog -Horse +Pottery -Measles
 * type=""}

{What was the period of history when people learned to use metal? -Çatal Hüyük +Bronze Age -Neolithic Revolution -Ice Age
 * type=""}

{What was the first animal to ever be domesticated? +Dog -Horse -Cat -Bird
 * type=""}

title: Human Legacy Course/Quiz 2


text:
Talk:Primary mathematics/Probability

Why is this article teaching about teaching and not focused on teaching probability?

 * In the history of the math courses, we had two different groups, one emphasizing teaching students directly and another emphasizing teaching teachers.  We tried to divide those two different purposes between here (Wikiversity) and Wikibooks, but not always successfully. StuRat 12:14, 23 June 2010 (UTC)

title: Talk:Primary mathematics/Probability


text:
User:Darshan.Gole

south africa world cup is being now ,which coutry dou you support ?

title: User:Darshan.Gole


text:
User talk:Gardenworms

 Hello Gardenworms, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 07:18, 21 June 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:18, 21 June 2010 (UTC)

title: User talk:Gardenworms


text:
User talk:Yumihasegawa

 Hello Yumihasegawa, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:18, 21 June 2010 (UTC)

title: User talk:Yumihasegawa


text:
Talk:Micromechanics of composites/Average stress power in a RVE

Where is the Appendix?


text:
Talk:Regional Anatomy/Lesson 4

Still to Do

 * Still requires a few diagrams

title: Talk:Regional Anatomy/Lesson 4


text:
Regional Anatomy/Lesson 5

Introduction to Regional Anatomy - Lesson 5 The Facial Compartments and Potential Spaces of the Human Body

Reading:
Fascia are the wrapping, packaging and insulating tissues found within deeper structures of the body. The superficial fascia (subcutaneous fascia) is found just below the skin (cutaneous layer).

title: Regional Anatomy/Lesson 5


text:
Regional Anatomy/Lesson 6

Introduction to Regional Anatomy - Lesson 6

The Regions of the Human Body

Reading:
Anatomists have divided the body into several regions. These regions help localize disease names, surgeries, and have other medical implications.

The following are the general regions of the body. They will be covered in detail later:

Quiz:
{The _______ artery supplies blood to the armpit region. - Femoral + Axillary - Brachial - Antebrachial
 * type=""}

{The heart is contained in the ________ region. - Inguinal - Sural + Thoracic - Cervical
 * type=""}

{Antebrachial pertains to the anterior portion of the leg. -TRUE +FALSE
 * type=""}

title: Regional Anatomy/Lesson 6


text:
Regional Anatomy/Lesson 7

Introduction to Regional Anatomy - Lesson 7 The Systems of the Human Body (Relating to Regional Anatomy)

title: Regional Anatomy/Lesson 7


text:
Markets, Games, and Strategic Behavior/2010

his page is an experiment in student generated answers to homework questions. In Markets, Games, and Strategic Behavior, we covered a diverse number of topics. The lecturer will pose a question and the students will provide the answer(s). Feel free to improve on other students' answers, put alternate answers and pose new questions. Feel free to clarify the questions as well.

Question 1
Take from class, the Diamond Dybvig model with $$L=.5 $$ and $$R=2 $$, two impatient depositors and two patient depositors. Each depositor has $1000 to deposit in the bank. Let us say that deposits are insured up to fraction $$f $$. For what values of $$ f $$ is there only one equilibrium and what values are there two equilibria? (For each dollar put in the bank yesterday, early withdrawers are guaranteed to get $$1*f$$ and late get $$2*f$$.)

Answer 1
The bank expects the two impatient depositors to withdraw today and the two patient depositors to withdraw tomorrow. Hence, yesterday, the bank sets aside $2000 for today and invests $2000 for tomorrow. Now today, the depositors must decide whether to withdraw today or tomorrow. We assume that the impatient depositors withdraw today. Now We can examine this as a game between the two patient depositors. Each has to decide whether or not to withdraw today. When $$ f=0 $$ the payoffs as discussed in class.

For a general $$f $$, we must calculate again each of the payoffs. If both withdraw today, the bank can pay the first 3 depositors the $1000. The last depositor will receive $$f $$. Thus, the expected payoff is $$ 750+(f/4)*1000 $$. If one withdraws today and the other withdraws tomorrow, the bank will be able to pay all three today, and the depositor withdrawing tomorrow receives $$ 2000*f$$. Rewriting the game yields.

We see that if one patient depositor withdraws today, the second patient depositor only has incentive to withdraw today if $$ 750+1000*(f/4)>=2000*f $$. Hence, if and only if $$f<=3/7 $$, there is a possibility of two equilibria.

Question 2
Part A.

Examine the second treatment of the Beer-Quiche game where there is a 2/3 chance of the proposer being strong.

Payoffs: Proposer, Responder

Can there be a pooling equilibrium where both proposers choose Quiche and the responder flees? Does this seem reasonable to you? Part B.

Assume the odds of a strong hand is 80%. Find any equilibrium. Is it signalling or pooling? Extra hard: what happens if it is 60%?

Answer 2
(shaked zin)

Part A.

Pooling equilibrium 1 - always choose beer. Pooling equilibrium 2 - always choose quiche - can't be a real equilibrium since it doesn't make sense. TECHNICALLY, IT IS AN EQUILIBRIUM IF THE PROPOSER BELIEVES THE RESPONDER WILL THINK HE IS WEAK IF HE DRINKS BEER. IT DOESN'T MAKE SENSE (NOT INTUITIVE) ACCORDING TO CHO-KREPS SINCE A WEAK PROPOSER WOULD HAVE NO POTENTIAL GAIN FOR DRINKING BEER THUS A PROPOSER DEVIATING MUST BE STRONG.

Part B. Pooling equilibrium, when the odds are 80% to be strong any player would pretend to be strong and choose to raise the pot. because: 0.8*(-2)+0.2*2= -1.2 < -1

extra hard: 0.6*(-2)+0.4*2= -0.4 > -1 In this case a weak A knows B would call anyone if all weak A's lie - therefore a weak A will fold and B will fold to anyone raising. its worth mentioning that its good for weak A's to lie once in a while and pretend to be strong and raise - as long as not many of them does so - B will always fold unless more than 0.375 weak A's will lie.

what is it 0.375???

Question 4
There is a Beersheva to Haifa train line. Travellers either go between Haifa and Tel Aviv with demand $$12-p $$,	Tel Aviv and Beersheva 12-p,	Haifa and Beersheva. $$18-p$$, Say it is all owned by one profit maximizing monopolist with marginal cost of zero. For simplicity assume that the monopolist must set the price of the Haifa-Beersheva route equal to the sum of the other two. a) What would he charge for all three routes?

b) Now say the government thinks it needs to add competition to the rail industry. It divides things into two companies. One takes care of the Haifa-Tel Aviv route and the other the Tel Aviv-Beersheva route. The price of the combined trip is the sum of the other two.	What are the new prices?	Who wins and who loses?

ANSWERS למי שיש דעות אחרות בנושא?

אבל יש ביקוש עבור כל קו בנפרד, ולא מדובר על קו שלם. לכן לדעתי סכום של רווחים עבור כל קטע מהווה רווח של מונופול עבור קו שלם לכן תשובות שלי, כמו שרשום ב א & ב a) p1=3 p2=3 for each small trip   P=6 for all trip

b)

p1=p2=6 p=12

Answer (start):
a. The monopolist would want to choose a p1 and p2 to maximize (12-p1)p1+(12-p2)p2+(18-(p1+p2))(p1+p2)

b. With separate firms, firm 1 would choose p1 to maximize (12-p1)p1+(18-(p1+p2))p1 and firm 2 would choose p2 to maximize (12-p2)p2+(18-(p1+p2))p2.

ANSWERS
(according to Tomer Zalmanson, Ariel Sawicki & Shaked Zin)

a) P1=P2=5 P3=10

b) P1=P2=6 P3=12

Question 5
Students like to go to the Haifa Ball depending upon how many other students go there. Tickets cost 32 NIS each. There are 1000 students indexed by i from 1 to 1000. Student i has value vi=i. Student i has utility (in shekels) for going to the Ball of $$v_i{n \over 5000}$$, where n is the total number of students going to the Ball. (i) If everyone believes $$n=500 $$, which students will be willing to go to the ball? (ii) What is the threshold number of tickets sold above which it will be a success and below which it will be a failure? (iii) What is the equilibrium of tickets sold if the ball is a success? (iv) What is the equilibrium of tickets sold if the ball is a failure?

(i)n=680, students that their value vi>= 320

(ii) 200

(iii) 800

(iv) 0

Question 6
A monopoly has marginal cost of 5 and faces a demand of q=20-p. What price should he charge to maximize profits? Let us say it is a vertical market of two firms: supplier and retailer. What would the price would the supplier charge the retailer? What would be the price charged to the end consumer? If the supplier charged a franchise fee in addition to wholesale price, what would they be? Extra: Solve the above problem for the general case of marginal cost of c facing demand of q=A-p where (A>c).

Answers

1. a monopoly case: p=12.5

2. Vertical markets

Ps=12.5 Max ProfitR= (20-q-Ps)*q FOC: 20-2q-Ps=0 -> q= 10-Ps/2 = 3.75 -> Pc=20-3.75=16.25 3. Franchise: Ps=MC=5  -> q=10-Ps/2=7.5  Pc=12.5 Profit 0f retailer Pc*q-Ps*q-F= 56.25-F  -> F=56.25-E 4. General case

Ps=(A+c)/2 q=A/4-c/4 Pc=3A/4+c/4

Question 7
El Al and British Air are competing for passengers on the Tel Aviv- Heathrow route. Assume marginal cost is 4 and demand is Q = 18 ? P. If they choose prices simultaneously, what will be the Bertrand equilibrium? If they can collude together and fix prices, what would they charge. In practice with such competition under what conditions would you expect collusion to be strong and under what conditions would you expect it to be weak. Under what conditions should the introduction of BMI affect prices?

למישהו יש רעינות נוספים?

ANSWER: If they choose prices simultaneously, what will be the Bertrand equilibrium? p1=p2=4 - the player cut off the price till it become MC

If they can collude together and fix prices, what would they charge. p=11

Question 8
Solve a three stage ultimatum game where in the first stage player A offers player B an offer for a $10 pie. If this offer is rejected, then the pie shrinks to $8 and player B makes the offer. If this offer is rejected, then the pie shrinks to $6 and player A makes the offer. If this final offer is rejected, then the payoffs are 0 to both players. (Assume the possibility of continuous offers.)

(8,2)

Question 9
You get in a taxi. Should you bargain over the price at the beginning or end of the trip? Why?

Question 10
Home Box Office is a pay-TV service that is based in the US. After showing only movies they decided to increase subscribers by introducing shows such as Sex and the City and the Sopranos. A person enjoys a show for its quality and whether they can talk about it next to the water cooler the next day at work. Given that someone has seen the show, the probability that one can talk about it f is just the number of people who have seen the show divided by the total number of people. We index the possible viewers by i (from 1 to 1000). Viewer i has parameter vi where vi = i. Viewer i values subscribing to HBO 9+ (vi/10)· f. Note that 9 is the value of HBO from the sheer quality. The price charged for HBO is 30. (i) If everyone believes f = .4, which people will subscribe to HBO? (ii) What is the threshold number of subscribers above which HBO will be a success and below which HBO will be a failure? (iii) What is the equilibrium number of subscribers if HBO is a success? (iv) What is the equilibrium number of subscribers if HBO is a failure? i.475 ii.300 iii.700 iv.0

Lab report 2 questions: Bertrand Complements
Which price should be higher: monopoly or Bertrand price competition?

Which to consumers should be higher: monopoly or Bertrand Complements?

When demand is 15-p and mc=6, what should a monopolist charge?

When demand is 15-(p1+p2) and mc=3 for both firms, what is the equilibrium price (it is Bertrand Complements)?

Answers to Lab Report 2
Monopoly price is higher than Bertrand competition.

Bertrand complements price is higher than monopoly price

The monopoly will charge 10.5$

p1=p2=6, P=12

Lab report 3 questions: Bank Runs
If they value R tomorrow less than 1 today, Impatient depositors will always want to withdraw today: True or False?

For no matter what values of L and R (L<1, R>1) and any number of depositors, there will always be two equilibria in the Diamond Dybvig model: True or False

For 4 depositors (2 impatient and 2 patient), L=.6 and R=1.5 what are the payoffs for (today, today)?

For 4 depositors (2 impatient and 2 patient), L=.6 and R=1.5 what are the payoffs for (today, tomorrow)?

Answers

1. True

2. False

3. 800/800

4. 1000/500

Lab report 4 questions: Network Externalities
If in the Network Experiment values were drawn from 0,10 and p=2.1, then what is the expected proportion of consumers buying in the success equilibrium. .2 .3 .4 .5 .6 .7 *** .8

If in the Network Experiment values were drawn from 0,10 and p=2.1, then what is the threshold/tipping point? .2 .3 *** .4 .5 .6 .7 .8

If in the Network Experiment values were drawn from 0,10, then for what prices is failure the only equilibrium? p=2.5

Give an example of Network Externalities that wasn't mentioned in class.

Lab report 5 questions: Vertical Market
The profit of a monopolist is higher than the upstream firm in a vertical monopoly. True *** False

If the upstream firm can charge a franchise fee, how much would it charge per item in addition to this fee if its marginal cost is 3 0 between 0 and 3. 3 *** twice the marginal cost (i.e., 6). = this other

Say demand is D=16-p and mc=0. How much would a monopolist charge? p=8

Say demand is D=16-p and mc=0. How much would the consumer pay if there is a vertical market (upstream and downstream)? Pc=12

Lab report 6 questions: Holdup
A buyer's investment cost is C. The gains in profits is G. The supplier can raise price by R. If the buyer switches the loss is B. By what amount R (in a one shot game) will it be rational for the buyer to stay with that firm? any R no R. R>G G>R G-C-R>B (tomer)

When will the buyer choose not to invest? G-C-R<0 *** G-C>0 G-B<0 G-C-B<0

Assume C=2000. Give a case of R, B, G where the subgame equilibrium is inefficient?

Assume C=2000. Give a case of R, B, G where the subgame equilibrium is efficient?

Lab report 7 questions: Draft
If a team goes up in order of the draft (can select earlier), then the team will always get better players the team will never get better players the team will sometimes get worse players *** the team will always get the same players

Teams selecting players by a draft will Always be Pareto Efficient Only sometimes be Pareto Efficient *** Never be Pareto Efficient Can always be improved by allowing trading afterwards

If Team 1 prefers A to B to C to D (A>B>C>D) and Team 2 prefers B to C to D to A, then what are the sincere choices. 1 - AC 2 - BD

If Team 1 prefers A to B to C to D (A>B>C>D) and Team 2 prefers B to C to D to A, then what are the sophisticated choices. 1 - BA 2 - CD

Lab report 8 questions: Subgame Perfection
In the centipede game, the subgame perfect equilibrium will be: The first player will stop. *** The game will continue to the end. The game will stop just be for the end. None of the above.

In the centipeded game, the outcome with the highest joint payoffs will be the subgame perfect equilibrium. True False ***

Explain why in experiments the results tend not to coincide with the subgame perfect equilibrium.

Is it benefically to be smart (and this trait known) in the centipede game?

Lab report 9 questions: Kiyotaki Wright Model
In the model, one would never gain from trading for an object one didn't need. True False ***

What was the commodity money? 1*** 2 3 there was none.

Explain how (at least for some parameters) there may be a speculative equilibrium.

Explain how under the parameters of the experiment, assuming that a player always would accept his desired good, a type 3 player should never accept a (2) good from a type 1 player.

Lab report 10 questions: Signalling
A peacock's tail is an example of signalling. True False

In treatment 2, what is the minimum fraction of strong types needed for a pooling equilibrium? 3/4 2/3 1/2 1/3

What was the reason that Cho Kreps gave for why pooling could not occur having both types each quiche?

Give an example of signalling in the real world.

Lab report 11 questions: Information Cascades
If you see a red ball you should always choose the cup that is color of your ball. True False***

If there are 10 people choosing before you and all 10 choose blue then at least 9/10 times blue would be correct choice. True False***

If everyone chooses rationally and there are 5 blue cups chosen out of 5 choices what are the odds that the cup is blue?

80%

מישהו יכול להסביר למה זה יוצא 80%... בבקשה

מחשבים הסתברות רק לפי שניים הראשונים כי כל השאר מפל מידע, נקבע על סמך 2 ראשונים

What is the connection between this experiment and a cascade?

Lab report 12 questions: Rent Seeking
In a rent-seeking game, in equilibrium players make on average zero profits. True False***

In a rent-seeking game, in equilibrium a player's profits goes down with his marginal cost. True False***

What is the equilibrium effort when there are 5 players competing for $10 with marginal cost of $2? - 0.8

When the number of players goes to infinity, what is the limit of the total effort? - v/mc

Lab report 13 questions: Taxes
A seller would prefer which of the following? A $2 per unit tax on sellers A $2 per unit tax on buyers depends upon the shape of the demand and supply curves indifferent between the two options.

A buyer would prefer which of the following? A $2 per unit tax on sellers A $2 per unit tax on buyers depends upon the shape of the demand and supply curves indifferent between the two options.

Buyers have valuations 10, 9, 8, 7, 5 Sellers have costs 1 4 5 6 7 what is the equilibrium price/quantity?

Buyers have valuations 10, 9, 8, 6, 5 Sellers have costs 1 4 5 6 7 what is the equilibrium price/quantity?


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Home | Lesson 1 | Lesson 2 | Lesson 3 | Lesson 4 | Lesson 5 | Lesson 6 | Lesson 7

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User:Egm6341.s10.team3.heejun/Meeting 8 transcript

 '''MTG 8: Thu. 14 Jan 10'''

[[media: Egm6341.s10.mtg8.djvu | Page 8-1]]

'''Trap. rule (simple)'''    fig.   [[media:Egm6341.s10.mtg7.djvu|p.7-1]] * as a particular case of Newton Cotes * approximated by function of straight line $$ \displaystyle \Rightarrow $$ Two points and One interval

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle [a,b],\,\,\,\, x_{0}=a\,\, and\,\, x_{1}=b\,\,\,\, when\,\, n=1 $$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

&= \sum_{i=0}^{1}l_{i}(x)f(x_{i}) \\ &= l_{0}(x)\underbrace{f(x_{0})}_{\color{blue}{x_{0}=a}} + l_{1}(x)\underbrace{f(x_{1})}_{\color{blue}{x_{1}=b}} \\\end{align}\ $$
 * $$ \begin{align} \displaystyle f_{1}(x) &= p_{1}(x) \\
 * $$ \begin{align} \displaystyle f_{1}(x) &= p_{1}(x) \\
 * }
 * }

If recall the formula of Lagrange Interpolation, then $$ l_{0}\,\, and\,\, l_{1}$$ will be
 * {| style="width:100%" border="0" align="left"

&= 1\,\,\,\, for\,\, \color{blue}{x=x_{0}} \\ & \color{blue}{or}\\ &= 0\,\,\,\, for\,\, \color{blue}{x=x_{1}} \\\end{align}\ $$
 * $$ \begin{align} \displaystyle l_{0} &= \frac{x-x_{1}}{x_{0}-x_{1}} \\
 * $$ \begin{align} \displaystyle l_{0} &= \frac{x-x_{1}}{x_{0}-x_{1}} \\
 * }
 * }

 * {| style="width:100%" border="0" align="left"

&= 0\,\,\,\, for\,\, \color{blue}{x=x_{0}} \\ & \color{blue}{or}\\ &= 1\,\,\,\, for\,\, \color{blue}{x=x_{1}} \\\end{align}\ $$
 * $$ \begin{align} \displaystyle l_{1} &= \frac{x-x_{0}}{x_{1}-x_{0}} \\
 * $$ \begin{align} \displaystyle l_{1} &= \frac{x-x_{0}}{x_{1}-x_{0}} \\
 * }
 * }

* Ref. for Lagrange Interpolation:    (2)   [[media:Egm6341.s10.mtg7.djvu|p.7-3]]

[[media: Egm6341.s10.mtg8.djvu | Page 8-2]]

 * {| style="width:100%" border="0" align="left"

&= 1\,\,\,\, for\,\, \color{blue}{i=j} \\ & \color{blue}{or}\\ &= 0\,\,\,\, for\,\, \color{blue}{i \neq j} \\\end{align}\ $$
 * $$ \begin{align} \displaystyle l_{i}(x_{j}) &= \underbrace{\delta_{ij}}_{\color{blue}{\delta = Kronecker\,\, delta}} \\
 * $$ \begin{align} \displaystyle l_{i}(x_{j}) &= \underbrace{\delta_{ij}}_{\color{blue}{\delta = Kronecker\,\, delta}} \\
 *   (1)
 * }.
 * }.

In this case, $$ \displaystyle l_{0}\,\, and\,\, l_{1}\,\, $$ are linear functions.

Therefore, $$ \displaystyle p_{1}(x) $$ is a linear combination of $$ \displaystyle l_{0}\,\, and\,\, l_{1}\,\, $$, and must be linear.

Integration of $$ \displaystyle f(x): $$

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle I= \int_{a}^{b}f(x)dx \cong I_{1} = \int_{a}^{b}\color{blue}{p_{1}} \color{black}{(x)dx} = (\int_{a}^{b}l_{0}(x)dx)f(x_{0}) + (\int_{a}^{b}l_{1}(x)dx)f(x_{1}) $$
 *   (2)
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * * p = polynomial and 1 = 1st order (linear)
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

HW:
 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style="width:100%; padding:10px; border:2px solid #8888aa" |

Use   (2)  $$\displaystyle \int\limits_{a}^{b} \sum_{i=0}^{n=1} l_i(x)f(x_i)\, dx $$ to obtain    (1)  $$\displaystyle I=\dfrac{b-a}{2}(f(a)+f(b)) $$     [[media:Egm6341.s10.mtg7.djvu|p.7-1]]
 * style = |
 * }.
 * }.

Simpson's rule (simple) * approximated by function of parabola (2nd order polynomial)

$$ \displaystyle \Rightarrow $$ Three points and Two intervals

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle [a,b],\,\,\,\, x_{0}=a,\,\, x_{1}= \frac{a+b}{2}\,\, and\,\, x_{2}=b $$
 * }
 * }
 * }

[[media: Egm6341.s10.mtg8.djvu | Page 8-3]]

How many methods do we have to get $$ \displaystyle f_{2}(x) $$ ?

Basically, 2 methods.

Method 1:

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle f_{2}(x) = p_{2} (x) = c_{2}x^{2} + c_{1}x^{1} + c_{0} $$
 *   (1)
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle c_{0},\,\, c_{1}\,\, and\,\, c_{2} = 3\,\, unknowns $$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle p_{2}(x_{i}) = f_{x_{i}}\,\,\,\, where\,\, i=0,1\,\, and\,\, 2 $$
 *   (2)
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle 3\,\, equations\,\, to\,\, slove\,\, for\,\, 3\,\, unknowns,\,\, c_{0},\,\, c_{1}\,\, and\,\, c_{2} $$
 * }.
 * }.
 * }.

Method 2:

Use the Lagrange polynomial,   (1)  and    (2)     [[media:Egm6341.s10.mtg7.djvu|p.7-3]], to get $$ \displaystyle p_{2}(x) $$

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle p_{2}(x) = \sum_{i=0}^{n=2} l_{i}(x)f(x_{i}) $$
 *   (3)
 * }
 * }
 * }

Equavalent of Method 1 and 2:

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle p_{2}(x_{j}) = \sum_{i=0}^{n=2} \underbrace{l_{i}(x_{j})}_{\color{blue}{\delta _{ij}}} f(x_{i})=f(x_{j})\,\,\,\, where\,\, j=0,1 \,\, and\,\, 2 $$
 *   (4)
 * }.
 * }.
 * }.

*    (4)  $$ \displaystyle \equiv $$    (3)

 * {| style="width:100%" border="0" align="left"

HW: Expand   (4)      p.8-3  to obtain $$ P_2 (x_j)= \sum_{i=0}^2  l_i(x_j) f(x_i) = f(x_j)$$
 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style = |
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle \underbrace{l_{0}(x)}_{\color{blue}{i=0}} = \prod_{j=0, j \neq i=0}^{n=2} \frac{x-x_{j}}{x_{0}-x_{j}} = \frac{(x-x_{1})(x-x_{2})}{(x_{0}-x_{1})(x_{0}-x_{2})} $$
 *   (4)
 * }.
 * }.
 * }.

[[media: Egm6341.s10.mtg8.djvu | Page 8-4]]

It can be verified that:

$$ \displaystyle l_{0}(x_{0})=1, \,\, l_{0}(x_{1})=l_{0}(x_{2})=0, \,\, l_{i}(x_{j})= \,\, \delta_{ij} \,\, \,\, where\,\, i,j = 0,1 \,\, and \,\, 2 $$


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Boundary Value Problems/Lesson 1

Return to Boundary Value Problems

Objectives

 * 1) Develop the student's ability to create qualitative elemental model of mass flow process that will allow extrapolation to heat flow process.

Methods

 * 1) Reading material on development of flux model based on conservatio of mass.
 * 2) Consider modeling the movement of a small quantity of material  $$ \delta M $$ through a incremental surface,$$ \delta  \Omega $$ within a unit of time.
 * 3) Have student(s) post elemental models using online simulation package.
 * 4) Provide examples of modeling heat flow.
 * 1) Provide examples of modeling heat flow.

Assessment
Our goal is to understand Partial Derivatives and equations that are composed of terms that include PD's. Such equations are called Partial Differential Equations or PDEs. More specifically, our interest is to solve PDEs, that is find a function $$ f(x,y) $$ such that $$ a_1(x,y) \frac {\partial^2 f}{\partial x^2} + a_2(x,y) \frac {\partial^2 f}{\partial x \partial y} + a_3(x,y) \frac {\partial^2 f}{\partial y^2} + a_4(x,y) \frac {\partial f}{\partial x}+ a_5(x,y) \frac {\partial f}{\partial y}+ a_6(x,y) f +a_7(x,y) =0    $$
 * 1) Provide feedback on a student's mode: critique by "experts", comments by other students, providing student with a document that shows them how to determine the validity of their model, a self grading checklist.
 * 2) Multipe choice or short answer Quizzes

title: Boundary Value Problems/Lesson 1


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Wikiversity talk:Main Page/Welcome

Proposed new layout for "Welcome" Template
I noticed the welcome template looked a bit wonky, this occurred in both safari and google chrome, so here is a fix. --=Benjamin= (t)·(c)·(e) 15:53, 21 June 2010 (UTC)

Welcome to Wikiversity Set learning free  with Special:Statistics learning resources and growing.

 * I just found this page... why was this template not used!?

WikiJournal
Dave, may WikiJournal be inserted for indefinite amount of time until WikiJournal is spun off from Wikiversity? WikiJournal still needs more attention, especially when proposed presentations about it were not accepted into Wikimania 2017. Thanks. --George Ho (discuss • contribs) 21:30, 14 August 2017 (UTC)

 * WikiJournal is currently nothing more than a soft redirect to meta:. There's nothing there to feature. If you're referring to the WikiJournal of Medicine, I think it should be added as one of the featured learning projects, and perhaps some of the older projects removed from rotation. -- Dave Braunschweig (discuss • contribs) 22:07, 14 August 2017 (UTC)
 * ...Okay, Dave, you can add WikiJournal of Medicine alternatively. Meanwhile, WikiJournal of Science is not yet initiated officially, so I would leave that one out for now. I don't know which older projects, but feel free to reinsert them if you wish. --George Ho (discuss • contribs) 22:12, 14 August 2017 (UTC)

Change "Schools" Link
Would it maybe make more sense to send the 'Schools' link to Category:Wikiversity_schools rather than Category:Schools? --Jaxter184 (discuss • contribs) 16:34, 3 February 2020 (UTC)

 * I concur. Please replace the line:
 * Schools •
 * with
 * Schools •
 * - Daask (discuss • contribs) 18:35, 7 July 2021 (UTC)

title: Wikiversity talk:Main Page/Welcome


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User:Clarknj~enwikiversity

This is my userpage.

title: User:Clarknj~enwikiversity


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User:Egm6341.s10.team3.heejun/Meeting 10 transcript

 MTG 10: Tue. 19 Jan 10

[[media: Egm6341.s10.mtg10.djvu | Page 10-1]]

Theorem of interpolation error (Atkinson. p.134)

This theorem is very important to know how big the error is committed after interpolating function (Lagrange Interpolation in previous lectures)

$$ \displaystyle f: \underbrace{\mathbb{R}}_{\color{blue}{domain}} \rightarrow \underbrace{\mathbb{R}}_{\color{blue}{range}}  $$  (set of real numbers)

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle f $$  differentiable with $$ \displaystyle (n+1) $$   continuous derivative on
 * $$ \displaystyle f $$  differentiable with $$ \displaystyle (n+1) $$   continuous derivative on

 $$ \displaystyle I_{t} := \mathbb{H} (t, x_{0}, x_{1}, ..., x_{n}) := $$ smallest interval containing points $$ \displaystyle (t, x_{0}, x_{1}, ... , x_{n})\,\,\,\, where\,\, x_{0} < x_{1} < ... < x_{n}\,\, and\,\, t\,\, is\,\, anywhere $$
 *  (1)
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

 * Case 1: $$ \displaystyle \color{blue}{ t < x_{0}} $$
 * Case 1: $$ \displaystyle \color{blue}{ t < x_{0}} $$

$$ \displaystyle I_{t}=[t,x_{n}] $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * Case 2: $$ \displaystyle \color{blue}{t \in [x_{0}, x_{n}]} $$
 * Case 2: $$ \displaystyle \color{blue}{t \in [x_{0}, x_{n}]} $$

$$ \displaystyle I_{t}=[x_{0},x_{n}] $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

$$ \displaystyle I_{t}=[x_{0},t] $$
 * Case 3: $$ \displaystyle \color{blue}{x_{n} < t} $$
 * Case 3: $$ \displaystyle \color{blue}{x_{n} < t} $$
 * }.
 * }.

Then, (Then means "imply", denoted by $$\displaystyle \Rightarrow $$ )

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle f_{n}(x)= \sum_{i=0}^{n}l_{i,n}(x)f(x_{i}) $$
 * $$ \displaystyle f_{n}(x)= \sum_{i=0}^{n}l_{i,n}(x)f(x_{i}) $$

 *  (2)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle f(t) - f_{n}(t)= \underbrace{\frac{q_{n+1}(t)}{(n+1)!}f^{(n+1)}(\xi)}_{\color{blue}{error\,\, term}} $$
 * $$ \displaystyle f(t) - f_{n}(t)= \underbrace{\frac{q_{n+1}(t)}{(n+1)!}f^{(n+1)}(\xi)}_{\color{blue}{error\,\, term}} $$

 *  (3)
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle \xi \in I_{t}, $$
 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

&= \prod_{j=0}^{n}(x-x_{j}) \in \color{blue}{\mathbb{P}_{n+1}} \\\end{align}\ $$
 * $$ \begin{align} \displaystyle q_{n+1}&:=(x-x_{0})(x-x_{1})...(x-x_{n}) \\
 * $$ \begin{align} \displaystyle q_{n+1}&:=(x-x_{0})(x-x_{1})...(x-x_{n}) \\
 *
 (4)
 * }.
 * }.

[[media: Egm6341.s10.mtg10.djvu | Page 10-2]]

Proof: (Similar proof technique used in other error analyses)

Note:

1) Montesquieu

Similarity and difference between  (3)   p.10-1 (i.e, approx error) and Taylor series remainder   (1)   [[media: Egm6341.s10.mtg2.djvu | P.2-3]]

2) Consider $$ \displaystyle t=x_{j},\,\,\,\, j=0,1,...,n $$

RHS  (3)   p.10-1 = 0 since $$ \displaystyle q_{n+1}(x_{j})=0\,\,\,\, j=0,1,...,n $$

LHS  (3)   p.10-1 = 0 since

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle f(x_{j})- \underbrace{f_{n}(x_{j})}_{\color{blue}{=\,\, f(x_{j})}} = 0 $$

 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle f_{n}(x_{j})= \sum_{i=0}^{n} \underbrace{l_{i,n}(x_{j})}_{\color{blue}{=\,\, \delta_{ij}}} f(x_{j}) = f(x_{j}) $$

 * }
 * }
 * }

 * Refer to fig. [[media: Egm6341.s10.mtg7.djvu | P.7-3]] for $$ \displaystyle f_{n}(x_{j})\,\, =\,\, f(x_{j}) $$ and  (1) & (4)  [[media: Egm6341.s10.mtg8.djvu | P.8-1 & 3]] for $$ \displaystyle l_{i,n}(x_{j})\,\, =\,\, \delta_{ij} $$

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle E(x) := f(x)-f_{n}(x)\,\,\,\, \color{blue}{interpolating\,\, error} $$

 * }.
 * }.
 * }.

[[media: Egm6341.s10.mtg10.djvu | Page 10-3]]

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle G(x) := E(x)- \frac{q_{n+1}(x)}{q_{n+1}(t)}E(t)\,\,\,\, x \in I_{t} $$

 * }
 * }
 * }

 * {| style="width:100%" border="0" align="left"

 * $$ \displaystyle G(\cdot)\,\, has\,\, (n+1)\,\, cont.\,\, deriv.\,\, $$

 * }
 * }
 * }


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User:Memo18

Hi! See my page in romanian.

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