text:
Theory Design Lab/Theory of thinking

Thinking about thinking
To form a theory means to modify your preliminary understanding of a topic to meet the requirements, which should be the requirements to describe reality with sufficient accuracy. Unfortunately the wrong requirements are often applied, which means the understanding may be modified to confirm prejudices or to meet the requirements of other elements of small talk. One could probably describe that as a frequent cognitive bias.

Metacognition
...

See: Metacognition

Philosophizing
...

See: Philosophy

Learning
...

See: Learning

Your own theory of thinking
Describe your own theory of thinking and learning. The theory should include memory effects, learning methods and how they affect the learning results.

 * Does your own theory describe any relevant cognitive bias?
 * Does your theory allow to evaluate learning strategies?

title: Theory Design Lab/Theory of thinking


text:
Cisco Networking/CCNP

CCNP Cisco Certified Network Professional

There is currently one path to attain the CCNP R&S certification.

Path 1 - 3 exams.
 * 1) 300-101 ROUTE.
 * 2) 300-115 SWITCH.
 * 3) 300-135 TSHOOT

A valid CCNA certification and the successful passing of all three exams are required to become CCNP certified. While the CCNA is a prerequisite for the CCNP, there are no formal prerequisites for any CCIE certification.

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

title: Cisco Networking/CCNP


text:
User talk:Warat

My name is Warat Poonyasiri, and my nickname is Mo. I was born on Dec 21,1989. Now I'm 20 years old. I'm the oldest children in my family; i have one brother and one sister. I don't like mathematics; this made me repeated Intensive Math 3 times. I only have 23 credits so far, and this is my 4th terms already. So, I'm trying my best to get good grades this trimester.

Cloud Computing

1.How is cloud computing different than using a normal computer?

Cloud computing is a technology that uses the internet as central share network. It also allows other users to access the network. Users can save as many works as they want on the network. These help them to have more space for other important applications in storage. You can see if your friends are working. Moreover, using cloud computing helps users get new friends. In addition, it is convenient to do work. A good example of cloud computing is yahoo, g-mail, and Amazon. Cloud computing only needs the internet connection. Normal computing does not have internet access. Presently you can access to internet almost everywhere in the central business district, which makes life become more convenient.

2.How could you turn a classroom into a cloud computing classroom?

First of all, all the students in the classroom need to register network like, Amazon, Yahoo, and G-mail etc. These networks are connecting together. Moreover, students will have to plan their schedule, which uses to set for work as a team. Furthermore, students will use their ID number to access the network which is safe and personal. For example, Wikiversity offers us for more comfortable work. Instructor can post anything on Wikiversity and students can read them. They can share an assignment with others.

Section1 Warat Poonyasiri 5180833

1. What is the site you selected? Digg.com 2. What is the site's purpose and why is it considered a web 2.0 site? The purpose of the site is that people can share or can read whatever on the sites. There is various kinds of content in there like stories, videos, music,..etc. 3. What is your favorite website and why Digg.com because there is so many things in different kinds in there. You get to see different things.

Welcome
 Hello Warat, 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.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

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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. --Ottava Rima (talk) 16:08, 1 May 2010 (UTC)

title: User talk:Warat


text:
User talk:Jbpayton

 Hello Jbpayton, 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.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

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 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

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. --Ottava Rima (talk) 13:54, 22 April 2010 (UTC)

title: User talk:Jbpayton


text:
User talk:Origboman

 Hello Origboman, 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.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

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. --Ottava Rima (talk) 13:56, 22 April 2010 (UTC)

Warning
Please do not make. They are inappropriate. Please read through the links provided above. Ottava Rima (talk) 13:57, 22 April 2010 (UTC)

title: User talk:Origboman


text:
User talk:Risa5280166

My full name is Risa Treeweranuwat. Everyone can call me Yelly, which means jelly but Thai people named it yelly. I am the first year and TIM student. I am friendly and easy going with other people. I feel exciting to study in this class and looking forward to study new information of computer and technology.

Welcome
 Hello Risa5280166, 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.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

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. --Ottava Rima (talk) 14:02, 22 April 2010 (UTC)

title: User talk:Risa5280166


text:
User:U5280103

Best Smartphone OS?
In my opinion, I think the iPhone OS is the best operating system for mobile devices so far. As I'm using an iPod touch, which runs the same OS, the iPhone OS really is designed for the best user experience. It has very nice interface designed for touch screen, it is very user-friendly, it runs smoothly, and the multi-touch capabilities are making the phone easier and more fun to use. Unlike the old-fashioned Windows Mobile where users need a 'stylus pen' and 'luck' using it.

Additionally, with vast choices of more than 150,000 applications on the "AppStore," the phone became smarter and packed with more productivity. Users can get a simple application, like a unit converter, to a complex money tracking application. With iPhone OS, the phone became more than just a phone. Everyone can develop an application and submit it to the AppStore!

Last but not least, With the iPhone OS being distributed to the users via iTunes application, the developers can easily send updates or fixes to the device, or add more software features through an easy two-step without having to install any extra softwares that come bundled with the phone package.

—U5280103 03:27, 18 June 2010 (UTC)

title: User:U5280103


text:
User talk:U5280103

My name is Sirijit Supavarodom. Everyone call me New. I'm the freshmen, and i'm majoring in TIM. I like to go to shopping or cinema when i have free time. I like to eat all kinds of dessert.--U5280103 13:59, 22 April 2010 (UTC)

Cloud computing
Q1. Cloud computing is different from a normal computer in the terms that cloud computing can be accessed by any user, anytime, and any computer, while a normal computer cannot. Users need to use only one specific computer in order to get the same information. Besides, Cloud computing does not have limited memory spaces, but a normal computer does. Q2. We can turn a normal classroom into a cloud computing classroom by use the website like ' Wikiversity', as an important material, so every student can easily learn and find out something that they do not understand in the website immediately.--U5280103 16:01, 27 April 2010 (UTC)

My photo
--U5280103 14:08, 30 April 2010 (UTC)

title: User talk:U5280103


text:
User talk:Palitaoyl

--Palitaoyl 14:07, 22 April 2010 (UTC)My name is Palita Poonpetchkul. People always call me "OYL". Now I'm twenty years old. I'm MUIC students, and my major is TIM. I'm a talkative person. In my leisure time, my friends and I usually hang out together to find some tasty food to eat. My favourite activities are shopping, sleeping, eating, and watching a movie.

title: User talk:Palitaoyl


text:
User:Peaky

Welcome ! And HELLO! My name is Siwat Sakuldeelert. But just call me Peak. My MUIC ID is 5180785. I'm just an ordinary guy who loves to play video game, read books, watch TV and sleep. And of course, what I hate is homework ! Honestly, I'm not good at remembering electronics or computer components and I don't have any knowledge about computer. The thing I'm expecting is that I would like to be more familiar with computer equipments and know something more about computer. I'm looking forward to study and I'll do my best.

Q 1 : How is cloud computing different than using computers?

A: The file storage is different. For cloud computing, users keep the file on the internet where they can share with others,edit the file and can be accessed from anywhere. While normal file and informations in normal computers cannot be accessed publicly.

Q 2 : How could you turn a classroom into a cloud computer classroom?

A: For cloud computer classroom to be created, first, we need to choose a website that provides places for files to be saved on the internet. Secondly, inform students to use the chosen site and register so that they can share, view and edit files in order to help each other in improving the work.

Answers for social networking

Q1: What site you selected?

A: I choose Digg

Q2: What is the site's purpose and why is it considered a web 2.0 site?

A: Digg is the site that people can share or post anything. For instance, we can post any news, picture, video, links and informations. It's also the place that every people can view the posted topic and they can share their thoughts and opinion. In conclusion, this site helps people in communicating with each other in a particular topic.

Q3: What is your favorite website and why?

A: Personally, I like the web that I've done which is digg.com. What I like about this site is that any people can share things that are interesting or things that they are interested in. In addition, people can view and discuss about that topic and we are able to know other people by discussing the topic.

The Best PC

Details :

CPU

 * Product name: Core I7 980X 3.33 GHz.
 * Intel Core i7 Processors.
 * Number processor i7-980X.
 * Cache memory, 12 MB L3.
 * Maximum memory 24GB.
 * Number Core 6 Core.
 * Number of Threads 12 Threads.
 * Signals Clock Speed 3.33 GHz.
 * Max Turbo Frequency 3.60 GHz.
 * Intel QPI Speed 6.4 GT / s.
 * Yes Hyper-Threading Technology.

Motherboard
 * Product: GIGABYTE X58A-UD9.
 * Chipset Intel X58 / ICH10R.
 * CPU Socket 1366 Core i7.
 * Audio 8-channel HD Audio.
 * Support DDR3 800 to 2200 (OC).
 * RAM socket supports up to 24GB.
 * Network, LAN (10/100/1000).
 * Expansion / Connectivity.
 * Devices 6 x SATA 3Gb / s.
 * Devices 2 x eSATA 3Gb / s.
 * Devices 1 x IDE connector.
 * Connect the device 2 x Realtek RTL8111E.
 * Devices 6 x USB back panel.
 * Devices 4 x USB brackets.
 * Devices 6 x HD Audio Jack.
 * Connect the device 4 x PCI E 2.0 x16 at x16.
 * Connect the device 3 x PCI E 2.0 x1 at x4.
 * Devices 6 x RAM slots.

RAM
 * Product: KINGSTON HYPER-X (CL9).
 * Product Type RAM PC.
 * Manufacturer Product KINGSTON.
 * memory 6 GB
 * Use System Triple-Channel 2GBx3.
 * Type of RAM DDR3.
 * Size of the RAM bus 1600.
 * Control signals CL 9-9-9-27.

Hard disk VGA Card
 * Product: SEAGATE (B / F 64MB).
 * The connection type of SATA II.
 * Data capacity 2 TB.
 * Sheets magnetic speed 7200 rpm.
 * Buffer memory 64 MB.
 * Guaranteed life of three years.
 * Products Asus GTX285.
 * Type of chip card Nvidia.
 * Type of card rooms DDR3.
 * Size of memory 4GB.
 * Signal Engine Clock 648Mhz.
 * Signal Memory Clock 2400MHz.
 * Forms of memory 1024-bit.
 * Connect Slot Type PCI-E.
 * The Connection ALL IN ONE.

Optical Drive

Monitor
 * LG BEO06LU ( LightScribe )
 * Blu-Ray, HD-DVD
 * Display Setup External.
 * Ability to read sheet BD-R/ROM (SL) 6x, BD-R/ROM (DL) 4x, BD-RE (SL / DL) 2x, HD DVD-ROM (SL / DL) 3x, DVD-ROM (. SL) 16x, CD-ROM 40x.
 * Ability to burn BD-R (SL / DL) 6x/2x, BD-RE (SL / DL) 2x, DVD + R / R DL 16x/4x, DVD + RW /-RW 8x/6x, CD-R. / RW 40x/24x.
 * Operating system support Windows Vista, XP, 2000.
 * Warranty 1 year

 * Product SAMSUNG B2430H.
 * 24 '' screen size
 * Type screen LCD Wide screen.
 * Resolution 1920x1080.
 * Response time 5ms.
 * Brightness 300 cd / m².
 * Contrast ratio 1000:1.
 * Dynamic Contrast Ratio: 70000:1.

title: User:Peaky


text:
User talk:Peaky

 Hello Peaky, 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.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

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. --Ottava Rima (talk) 16:17, 22 April 2010 (UTC)

title: User talk:Peaky


text:
User talk:Kamolpat 5180367

Homework 1 Hello!!!

I'm Kamolpat, nickname is Pat.

studying at MUIC, TIM major.

really love Comp Ess class and mr. Brian!

Homework 2 How is cloud computing different than using a normal computer? The cloud computer is very convenient for all pepople because when we do our projects and and assignmentsm, we can maintain it in the servers. Moreover, files and datas are not lost when our computers are broken. We can get out datas anywhere that provides internet to us. On the other hand, normal computers are unconvenient for all people because we have to save a data on a computer, and we need to use a thumb drive or a floppy drive to save them. Furethermore, when our computers are error and broken, we can not bring our work back, and noone reposible for it. How could you turn a classroom into a cloud computing classroom? We need to register the some websites such as google, yahoo, or wiki. Therefore, when teachers assign homework, or when we have to send homework, we can put it on that servers. We will not lost datas

Copyright problem with File:25717 431022666112 563966112 5765019 8259897 n.jpg
Homework 3 What is the best smart phone operating system and why?

I think Palm Pre is the best smart phone in my point of view because the Palm Pre signaled the resurrection of Palm. It is well-designed, and has a very promising OS, which if tweaked and improved slightly, can challenge any OS on the market.

Operating System: Palm webOS

title: User talk:Kamolpat 5180367


text:
BCP/Potentilla argentea

Potentilla argentea

title: BCP/Potentilla argentea


text:
BCP/Potentilla argentea/Logs

Recent Logs

 * Trinity507 16:42, 22 April 2010 (UTC)

title: BCP/Potentilla argentea/Logs


text:
Biblical Studies (NT)/In Conclusion...

 NEW TESTAMENT

''' In Conclusion... '''

Congratulations!
If you have worked your way through all of the lessons in this course, together with their multiple choice tests and assigned readings... Congratulations! You should be commended for your persistence and self-motivation. Hopefully, it has been a rewarding study which has enriched your knowledge of the New Testament. If you haven't already done so, follow the instructions in Lesson 1 to grade yourself. (The instructions are also on the score sheet.) You may also wish to leave a sentence or two expressing your thoughts on the course (click on the "discussion" tab, above.)

What to Do Next?
Bear in mind that this course is just an introduction to the New Testament. Perhaps it has fulfilled your needs as far as New Testament studies are concerned, or perhaps it has whetted your appetite to know more. If the latter is the case, you can increase your knowledge and understanding by taking courses which provide a more in-depth focus on specific Biblical books and topics. A few courses are currently available in Wikiversity, and no doubt more will be added as time goes by. Colleges, universities, and churches also offer courses and study groups.

If you prefer to study the Bible on your own, you can use the study aids listed under Other Resources on the Contents page. If you do not currently own any of these, it is recommended that you begin by obtaining the handbook, the commentary, and the Bible dictionary or encyclopedia (the encyclopedia is just a larger, more comprehensive version of the dictionary).

What You Should Know about Published Works
Be aware that not all commentaries, dictionaries, etc. are alike. They vary in quality, but more importantly, in their approach to the Bible. Some take a high view of the integrity of the Biblical text. They are written by scholars who believe that the Bible has been copied and handed down faithfully through the centuries (the Dead Sea Scrolls seem to bear this out). Such scholars accept the Bible's claims for authorship, dating, historical accuracy, etc.

Other scholars take the position that the Biblical text has been edited extensively over the years, and believe that its claims for authorship, dating, and historicity are unreliable. Scholars in this group often dismiss the notion of divine inspiration, and take the position that Bible predictions which seem to have been fulfilled must have been written after the fact, a belief which has serious implications for authorship, dating, and the Bible's integrity. Most published works on the Bible, including commentaries and dictionaries, consistently adhere to one or the other of these paradigms (one can often tell which by looking at the publisher). As we get to know more about the subject, our sympathies may also lean toward one or the other of these positions. However, there are respected scholars in both camps, so it is good to keep an open mind and to be familiar with the rationale behind both schools of thought.

It is also worth noting that, while it is convenient to divide scholarship into these broad categories, the reality of the world of Christian theology is far more complex. Within these two broad schools, there are a diversity of views on every conceivable Biblical topic, and we are just as entitled as anyone else to draw our own conclusions, once we become acquainted with the facts. Published works often speak as though their position on a given theological topic is the only position, but that is rarely the case, and the student should not be taken in by such dogmatism.

The Importance of the Old Testament to the New
Finally, any serious student of the New Testament should consider becoming acquainted with the Old Testament (Hebrew scriptures). Some people focus their attention entirely on the New Testament to the exclusion of the Old. However, the Old Testament provides background which is crucial to a complete understanding of the New Testament and its authors.

For example, the chief evidence for Jesus' Messiahship for the New Testament authors was not that he performed miracles, but that he fulfilled Old Testament prophecies. To establish this, they filled their writings with quotes from, and allusions to, the Hebrew scriptures, and it is helpful to know the context of these quotes and allusions (obtaining a Bible with a good cross-referencing system is the first step).

Jesus, himself, made frequent references to the Hebrew scriptures. When he spoke his famous words on the cross, "My God, my God, why have you forsaken me?" (Mt 27:46), he was quoting the opening line of Psalm 22, a song of worship which would have been familiar to the Jews who were gathered there. Psalm 22 is classified as a "Messianic psalm" and it contains details which pertain to the crucifixion. Students who wish to take their knowledge to a deeper level, therefore, should acquire a basic knowledge of the Old Testament at some point.

Never Stop Learning!
Thank you for using this Wikiversity course, and may your appetite for learning continue to bring you personal growth and satisfaction.

Back to Home Page:

title: Biblical Studies (NT)/In Conclusion...


text:
User talk:Nongauu

--Nongauu 18:17, 22 April 2010 (UTC) My name is Aekachai Termkunanon. You can also call me "Auu". I'm 20 year olds and studying in TIM major in Mahidol university. I graduated from Saint Gabriel College and studied there for 12 years. Playing all kinds of sport is my hobby. I like many categories of music such as blues, jazz, pop, folk, etc. I enjoy watching films and learning many kinds of art. Drama is my favorite type of movie.

 Hello Nongauu, 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.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

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. --Ottava Rima (talk) 18:41, 22 April 2010 (UTC)

title: User talk:Nongauu


text:
User talk:Repair~enwikiversity

 Hello Repair, 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.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

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. --Geoff Plourde 20:47, 22 April 2010 (UTC) Geoff Plourde 20:47, 22 April 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Repair. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Repair~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:50, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:19, 19 April 2015 (UTC)

title: User talk:Repair~enwikiversity


text:
Portal:Agape World Fellowship/Deptnews/Archive

Back to News 20 March 2010:AWF School created!


text:
User talk:Roastedalmonds

 Hello Roastedalmonds, 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.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

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. --Ottava Rima (talk) 21:22, 22 April 2010 (UTC)

title: User talk:Roastedalmonds


text: File:HW 7-8.jpg

title: File:HW 7-8.jpg


text:
Portal:Agape World Fellowship/Links

 * Agape World Fellowship - Website
 * Agape World Fellowship School - Moodle Website
 * United Christian Faith Ministries - Website

title: Portal:Agape World Fellowship/Links


text: File:Pythagorean.JPG

title: File:Pythagorean.JPG


text:
User:Egm6341.s10.team3.Min/HW7

= (1) Linearization about $$ x_{max} $$ = Ref: Lecture Notes [[media:Egm6341.s10.mtg38.djvu|p.38-4]]

Problem Statement

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

$$
 * $$\displaystyle x=\hat{x}+ y
 * $$\displaystyle x=\hat{x}+ y
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle f(x)=rx \left(1- \frac{x}{x_{max}} \right)
 * $$\displaystyle f(x)=rx \left(1- \frac{x}{x_{max}} \right)
 * }
 * }

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

$$ Linearization about equil. pt $$\hat {x} $$
 * $$\displaystyle \hat{x}=x_{max}
 * $$\displaystyle \hat{x}=x_{max}
 * }
 * }

Solution

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

$$
 * $$\displaystyle At \hat{x}=x_{max}
 * $$\displaystyle At \hat{x}=x_{max}
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

\begin{align} f(x)&=f(x_{max}+y)\\ &=r(x_{max}+y)(1- \frac{x_{max}+y}{x_{max}})\\ &=-ry \left(1+ \frac{y}{x_{max}} \right) \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle where yy<<x_{max} thus 1+ \frac{y}{x_{max}} \approx 1
 * $$\displaystyle where yy<<x_{max} thus 1+ \frac{y}{x_{max}} \approx 1
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle f(x)\approx -ry
 * $$\displaystyle f(x)\approx -ry
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle \frac{dx}{dt}= \frac{d(x_{max}+y)}{dt}= \frac{dy}{dt}
 * $$\displaystyle \frac{dx}{dt}= \frac{d(x_{max}+y)}{dt}= \frac{dy}{dt}
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

$$ -- By Min Zhong 12:00, 22 Apr 2010 (UTC)
 * $$\displaystyle \Rightarrow \frac{dy}{dt}=-ry \Rightarrow y=y_0e^{-rt}
 * $$\displaystyle \Rightarrow \frac{dy}{dt}=-ry \Rightarrow y=y_0e^{-rt}
 * }
 * }

= (2) Verify and solve the differential equation = Ref: Lecture Notes [[media:Egm6341.s10.mtg39.djvu|p.39-1]]

Problem Statement
The exact expression of x(t) is :
 * {| style="width:100%" border="0" align="left"

$$ i) Verify that:
 * $$\displaystyle x(t)= \frac{x_0x_{max}e^{rt}}{x_{max}+x_0(e^{rt}-1)}
 * $$\displaystyle x(t)= \frac{x_0x_{max}e^{rt}}{x_{max}+x_0(e^{rt}-1)}
 * }
 * }
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle \frac{dx}{dt}=rx \left( 1- \frac{x}{x_{max}}\right)
 * $$\displaystyle \frac{dx}{dt}=rx \left( 1- \frac{x}{x_{max}}\right)
 * }
 * }

ii) solve the above differential equation with hint on Lecture Notes [[media:Egm6341.s10.mtg40.djvu|p.40-1]]

Solution
i) Verify the differential equation
 * {| style="width:100%" border="0" align="left"

\begin{align} \frac{dx(t)}{dt}&= \frac {x_0x_{max}e^{rt}r}{x_{max}+x_0(e^{rt}-1)}+x_0x_{max}e^{rt}(-1) \left( \frac{1}{x_{max}+x_0(e^{rt}-1)} \right) ^2x_0e^{rt}r \\ &= \frac{x_ox_{max}e^{rt}}{x_{max}+x_0 \left( e^{rt}-1\right)}r+ \frac{r}{x_{max}}  \left( \frac{x_0x_{max}e^{rt}}{x_{max}+x_0 \left( e^{rt}-1\right)} \right)^2\\ &=xr-\frac{r}{x_{max}}x^2\\ &=rx \left(1- \frac{x}{x_{max}} \right) \end{align} $$ End the verification.
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

ii)Solve the differential equation Partial fraction:
 * {| style="width:100%" border="0" align="left"

\begin{align} \frac{1}{x \left(1- \frac{x}{x_{max}} \right)}&= \frac{a}{x}+ \frac{b}{1- \frac{x}{x_{max}}}\\ &= \frac{a \left(1- \frac{x}{x_{max}}\right)+bx}{x \left(1- \frac{x}{x_{max}} \right)}\\ \end{align} $$ Let
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle a\left(1- \frac{x}{x_{max}}\right)+bx=1 \Rightarrow a-x \left(b- \frac{a}{x_{max}} \right)=1
 * $$\displaystyle a\left(1- \frac{x}{x_{max}}\right)+bx=1 \Rightarrow a-x \left(b- \frac{a}{x_{max}} \right)=1
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

\Rightarrow \begin{cases} & a= 1 \\ & b=  \frac{1}{x_{max}} \end{cases} $$ Separate the variables of the differential equation:
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle \int_{x_0}^{x}  \frac{dx}{x \left(1- \frac{x}{x_{max}} \right)}= \int_{0}^{t}rdt
 * $$\displaystyle \int_{x_0}^{x}  \frac{dx}{x \left(1- \frac{x}{x_{max}} \right)}= \int_{0}^{t}rdt
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle  \Rightarrow \int_{x_0}^{x}  \frac{1}{x} +  \frac{ \frac{1}{x_{max}}}{1- \frac{x}{x_{max}}}= \int_{0}^{t}rdt
 * $$\displaystyle  \Rightarrow \int_{x_0}^{x}  \frac{1}{x} +  \frac{ \frac{1}{x_{max}}}{1- \frac{x}{x_{max}}}= \int_{0}^{t}rdt
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

$$ :{| style="width:100%" border="0" align="left" $$ :{| style="width:100%" border="0" align="left" $$
 * $$\displaystyle  \Rightarrow  \left[ lnx-ln \left(1- \frac{x}{x_{max}} \right) \right]_{x_0}^x =  \left[rt \right]_{0}^t
 * $$\displaystyle  \Rightarrow  \left[ lnx-ln \left(1- \frac{x}{x_{max}} \right) \right]_{x_0}^x =  \left[rt \right]_{0}^t
 * }
 * }
 * $$\displaystyle  \Rightarrow  \left[ lnx-ln \left(1- \frac{x}{x_{max}} \right) \right]_{x_0}^x =  rt
 * $$\displaystyle  \Rightarrow  \left[ lnx-ln \left(1- \frac{x}{x_{max}} \right) \right]_{x_0}^x =  rt
 * }
 * }
 * $$\displaystyle  \Rightarrow ln \left( \frac{x}{x_0} \right) \left( \frac{1- \frac{x_0}{x_{max}}}{1- \frac{x}{x_{max}}} \right) =  rt
 * $$\displaystyle  \Rightarrow ln \left( \frac{x}{x_0} \right) \left( \frac{1- \frac{x_0}{x_{max}}}{1- \frac{x}{x_{max}}} \right) =  rt
 * }
 * }

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

$$
 * $$\displaystyle  \Rightarrow x= \frac{x_0x_{max}e^{rt}}{x_{max}+x_0(e^{rt}-1)}
 * $$\displaystyle  \Rightarrow x= \frac{x_0x_{max}e^{rt}}{x_{max}+x_0(e^{rt}-1)}
 * }
 * }

-- By Min Zhong 12:00, 22 Apr 2010 (UTC)

= (13) Change of variable  = Ref: Lecture Notes [[media:Egm6341.s10.mtg42.djvu|p.42-2]]

Problem Statement
By Changing the variable of integration,
 * {| style="width:100%" border="0" align="left"

$$ show that:
 * $$\displaystyle  x=cos \theta
 * $$\displaystyle  x=cos \theta
 * }
 * }
 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle  I= \int_{-1}^{+1}f(x)dx= \int_{0}^{\pi}f(cos  \theta)sin \theta d\theta
 * $$\displaystyle  I= \int_{-1}^{+1}f(x)dx= \int_{0}^{\pi}f(cos  \theta)sin \theta d\theta

$$
 * }
 * }

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

$$ and
 * $$\displaystyle  x=cos \theta \Rightarrow dx=-sin \theta d \theta
 * $$\displaystyle  x=cos \theta \Rightarrow dx=-sin \theta d \theta
 * }
 * }
 * {| style="width:100%" border="0" align="left"

\begin{cases} & x=-1;  \theta =\pi \\ & x=+1;\theta=0 \end{cases} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle  I= \int_{-1}^{+1}f(x)dx= \int_{\pi}^{0}f(cos\theta)(-1)sin \theta d\theta=\int_{0}^{\pi}f(cos\theta)sin \theta d\theta
 * $$\displaystyle  I= \int_{-1}^{+1}f(x)dx= \int_{\pi}^{0}f(cos\theta)(-1)sin \theta d\theta=\int_{0}^{\pi}f(cos\theta)sin \theta d\theta

$$ End the proof -- By Min Zhong 12:00, 22 Apr 2010 (UTC)
 * }
 * }

= (14)Find the item $$a_k$$ in cosine series of $$f(cos\theta)$$  = Ref: Lecture Notes [[media:Egm6341.s10.mtg42.djvu|p.42-3]]

Problem Statement
The known cosine series of f(x) is as follows:
 * {| style="width:100%" border="0" align="left"

$$ Find out the expression for $$a_k$$
 * $$\displaystyle  f(cos \theta)= \frac{a_0}{2}+ \sum_{k=1}^{\infty}a_kcos \left( k\theta\right)
 * $$\displaystyle  f(cos \theta)= \frac{a_0}{2}+ \sum_{k=1}^{\infty}a_kcos \left( k\theta\right)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle a_k= \frac{2}{\pi} \int_{0}^{\pi}f(cos\theta)cos(k\theta)d\theta
 * $$\displaystyle a_k= \frac{2}{\pi} \int_{0}^{\pi}f(cos\theta)cos(k\theta)d\theta
 * }
 * }

Solution
The proof starts by multiplying $$cos(k\theta)$$ on both sides of cosine series of $$ f(cos\theta)$$
 * {| style="width:100%" border="0" align="left"

f(cos \theta) = \frac{a_0}{2}+ \sum_{k=1}^{\infty}a_kcos \left( k\theta\right) $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

f(cos \theta)cos(k\theta) = \frac{a_0}{2}cos(k\theta)+ \sum_{k=1}^{\infty}a_kcos \left( k\theta\right)^2 $$ Do integration from $$ [0,\pi] $$ on both sides of the above equation:
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }
 * {| style="width:100%" border="0" align="left"

\int_{0}^{\pi}f(cos \theta)cos(k\theta) = \int_{0}^{\pi}\frac{a_0}{2}cos(k\theta)+  \int_{0}^{\pi}\sum_{k=1}^{\infty}a_kcos \left( k\theta\right)^2 $$ where
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }
 * {| style="width:100%" border="0" align="left"

\begin{align} \int_{0}^{\pi}\frac{a_0}{2}cos(k\theta)&= \frac{a_0}{2k} \left[sin(k\theta) \right]_0^\pi\\ &= \frac{a_0}{2k} \left( sin(k\pi)-sin(0)\right)\\ &=0\\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

\begin{align} \int_{0}^{\pi}\sum_{k=1}^{\infty}a_kcos \left( k\theta\right)^2&= \sum_{k=1}^{\infty}a_k \int_{0}^{\pi}cos^2(k\theta)d\theta\\ &=\sum_{k=1}^{\infty}a_k\int_{0}^{\pi} \frac{cos(2\theta+1)}{2}\\ &= \frac{1}{2} \sum_{k=1}^{\infty}a_k \left[ \frac{1}{2}sin(2\theta)+\theta\right]_0^\pi\\ &= \frac{1}{2} \sum_{k=1}^{\infty}a_k \left(  \frac{1}{2} sin(2\pi)+\pi- \frac{1}{2}sin0-0\right)\\ &= \frac{\pi}{2} \sum_{k=1}^{\infty}a_k\\
 * $$\displaystyle
 * $$\displaystyle

\end{align} $$
 * }
 * }

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

\int_{0}^{\pi}f(cos \theta)cos(k\theta) = \frac{\pi}{2} \sum_{k=1}^{\infty}a_k $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

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

\Rightarrow a_k= \frac{2}{\pi} \int_{0}^{\pi}f(cos\theta)cos(k\theta)d\theta $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

-- By Min Zhong 12:00, 23 Apr 2010 (UTC)

= (15)Find the expression of integration of fourier transfer  = Ref: Lecture Notes [[media:Egm6341.s10.mtg42.djvu|p.42-3]]

Problem Statement
The following is known:
 * {| style="width:100%" border="0" align="left"

I= \int_{-1}^{+1}f(x)dx= \int_{0}^{\pi}f(cos\theta)sin\theta d\theta $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

The known cosine series of f(x) is as follows:
 * {| style="width:100%" border="0" align="left"

$$ Find the expression of Integration as:
 * $$\displaystyle  f(cos \theta)= \frac{a_0}{2}+ \sum_{k=1}^{\infty}a_kcos \left( k\theta\right)
 * $$\displaystyle  f(cos \theta)= \frac{a_0}{2}+ \sum_{k=1}^{\infty}a_kcos \left( k\theta\right)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

$$
 * $$\displaystyle I=a_0+ \sum_{j=1}^{\infty} \frac{2a_{2j}}{1- \left( 2j\right)^2}
 * $$\displaystyle I=a_0+ \sum_{j=1}^{\infty} \frac{2a_{2j}}{1- \left( 2j\right)^2}
 * }
 * }

Solution

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

\begin{align} I&= \int_{0}^{\pi}f(cos\theta)sin\theta d\theta\\ &= \int_{0}^{\pi} \left[  \frac{a_0}{2}+ \sum_{j=1}^{\infty}a_{2j}cos(2j\theta)\right]sin\theta d\theta\\ &= \frac{a_0}{2}\int_{0}^{\pi}sin\theta d \theta +  \sum_{j=1}^{\infty} a_{2j}\int_{0}^{\pi}sin\theta cos(2j\theta) d \theta\\ &= - \frac{a_0}{2} \left[ cos\theta\right]_0^\pi+ \sum_{j=1}^{\infty}a_{2j} \int_{0}^{\pi} \frac{1}{2} \left [sin(1+2j) \theta +sin (1-2j)\theta \right] d\theta\\ &= a_0+ \sum_{j=1}^{\infty}a_{2j} \frac{1}{2} \left[\left( \frac{-1}{1+2j} cos(1+2j) \theta \right)_0^\pi+ \left(  \frac{-1}{1-2j}cos(1-2j) \theta\right) _0^\pi \right]\\ &=a_0+ \sum_{j=1}^{\infty}a_{2j} \frac{1}{2} \left[ \frac{2}{1+2j} + \frac{2}{1-2j}\right]\\ &=a_0+ \sum_{j=1}^{\infty}a_{2j}  \left( \frac{1}{2}\right) \frac{4}{1-(2j)^2}\\ &=a_0+ \sum_{j=1}^{\infty} \frac{2a_{2j}}{1-(2j)^2}\\ \end{align} $$ The end of proof
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

-- By Min Zhong 12:00, 23 Apr 2010 (UTC)

title: User:Egm6341.s10.team3.Min/HW7


title: File:2.jpg


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text:
User talk:Yuiyui

HI!, My name is Aticha Supanwattanachai, but you can call me 'yui'. I am 20 years old and has just graduate from Sarasas Ektra school, it is a bilingual school on sathupradit road. During my freetime, I like to play the piano and play lots of video games because I have nothing better to do. Nice to meet you.

title: User talk:Yuiyui


text:
Natural number

The natural numbers (or counting numbers) are the fundamental mathematical set on which all other arithmetic is based. They do not include negative numbers. (The set of integers provides negative numbers.) Whether the natural numbers include zero is a matter of taste—zero can be adjoined to the set in much the same way that the definition of the integers adjoins negative values. Sometimes the "natural numbers" are posited to include zero while the "counting numbers" are not. One can use whatever definition suits one's purpose at a given time.

Mathematicians denote the set of natural numbers with an ornate capital letter: $$\mathbb{N}$$. They are the 1st item in this hierarchy of types of numbers:

 * The "natural numbers"-5, -4, -3, -2, -1, —$$\mathbb{N}$$, 1, 2, 3, ...
 * The "integers"—$$\mathbb{Z}$$, positive, negative, NOT zero
 * The "rational numbers"—$$\mathbb{Q}$$, or fractions, like 355/113
 * The "real numbers"—$$\mathbb{R}$$, including irrational numbers
 * The "complex numbers"—$$\mathbb{C}$$, which give solutions to polynomial equations

Sometimes the symbol $$\mathbb{N}^+$$ is used to denote the set not including zero, while not $$\mathbb{N}$$ includes zero.

People generally understand the arithmetic operations on natural numbers, at an intuitive level, long before they appreciate the significance of careful mathematical definitions and proofs. Indeed, students often prove theorems about algebra and calculus, involving a good understanding of the rational, real, or complex numbers, before they think carefully about the axiomatic basis for the natural numbers. The seemingly intuitively fundamental existence of the natural numbers is perhaps best summed up by this saying from Leopold Kronecker:

God made the integers; all else is the work of man.

(In modern terminology, we would say that he was talking about the natural numbers.)

Can we put the natural numbers on a sound "axiomatic" basis?
Without a rigorous foundation for the natural numbers, the rest of mathematics could be considered flawed. In the 19th century much work was done on the foundations of set theory and arithmetic. Guiseppe Peano, along with Richard Dedekind, are generally credited with the solution to the problem of the natural numbers. In 1889, Peano provided 5 axioms, from which the natural numbers could be developed rigorously. The axioms are generally called the Peano axioms, and sometimes called the Dedekind-Peano axioms.

The Peano axioms
(Also called the Peano postulates.)

Before we give the axioms and define the natural numbers, there are a few preliminary things to say:
 * We will define the numbers starting with 1. Zero can be added later, when the integers are defined.
 * We assume elementary notions of set theory and logic. For example, we assume a notion of "things" being equal.  And that $$x=x$$ always, if $$x=y$$ then $$y=x$$, and if $$x=y$$ and  $$y=z$$, then $$x=z$$.

We will follow the presentation in the classic Foundations of Analysis by Edmund Landau.

Here are the axioms:
 * 1) There is a (natural) number, called "1".
 * 2) There is a successor function. We denote it with a prime.  So, for example, 1' is a number.  It's usually called "2".  1' ' is also a number, called "3".  And so on.
 * 3) If x' = y', then x = y. That is, we don't have two different numbers with the same successor.  (We already know that if x = y, then x' = y'; that follows from what we know about how functions work.)
 * 4) 1 is not the successor of any number.
 * 5) If a set S contains 1, and for every x in S, x' is also in S, then S is the entire set of (natural) numbers.

This last axiom is called the axiom of induction, and is the axiom underlying all proofs by induction.

Intuitively, the axioms can be thought of as making the following statements about the natural numbers:
 * Axiom 1 says that the set is not empty.
 * Axiom 2 says that the numbers go on forever. There is no "last" number.  Also, because functions are single-valued, the sequence does not "split" going upward.  It is one sequence.
 * Axiom 3 says that the sequence does not "split" going downward either. There are no branches that converge.
 * Axiom 4 says that they don't go on forever in the downward direction. There is a "first" number, and it is 1.  Of course, zero and the negative integers are formulated later, outside of the Peano axioms.
 * Axiom 5 says that there are no isolated "islands" of numbers. The number sequence starting from 1 is everything that there is.

Axiom 5 is generally used in a less formal way—rather than talking about a set S, we prove something for 1, and prove it for x' (that is, x+1) assuming it is true for x, and conclude that it is true for all x.

Addition
We can now define the fundamental notions about arithmetic, and prove as theorems the fundamental laws that people take for granted, such as associativity, commutativity, and distributivity.

Definition: There is an operation of addition, denoted x+y, as follows:
 * For all x, x+1 = x'                     (clause 1)
 * For all x and y, x+y' = (x+y)'   (clause 2)

This is well-defined for all x and y, by induction on y: Clause 1 defines it for y=1 and all x, and clause 2 defines it for y' and all x, assuming it is defined for y and all x.

Theorem 1: 1+x = x'    (or, by clause 1, 1+x = x+1)

Proof by induction on x: If x=1, this is 1+1 = 1', which is true by clause 1. Otherwise, assume it is true for x. Then:
 * 1+x' = (1+x)' (clause 2)
 * = (x+1)' (induction)
 * = x' ' (clause 1)

Theorem 2 (associative law of addition): (x+y)+z = x+(y+z)

Proof by induction on z, for all x and y:
 * If z=1, (x+y)+1 = (x+y)' (clause 1)
 * = x+y' (clause 2)
 * = x+(y+1) (clause 1)
 * = x+(y+z)

Now assume it is true for some z, and for all x and y:
 * (x+y)+z' = ((x+y)+z)' (clause 2)
 * = (x+(y+z))' (induction)
 * = x+(y+z)' (clause 2)
 * = x+(y+z') (clause 2)

Theorem 3 (commutative law of addition): x+y = y+x

Proof by induction on x, for all y:
 * If x=1, 1+y = y+1 (theorem 1)

Now assume it is true for some x, and for all y:
 * x'+y = (x+1)+y (clause 1)
 * = (1+x)+y (theorem 1)
 * = 1+(x+y) (associativity)
 * = 1+(y+x) (induction)
 * = (1+y)+x (associativity)
 * = (y+1)+x (theorem 1)
 * = y+(1+x) (associativity)
 * = y+x' (theorem 1)

Ordering
Lemma 1: We never have x+y = x

Proof by induction on x, for all y: If x=1, this would be y+1 = y' = 1, which is impossible by axiom 4. Otherwise, suppose x'+y = x'. Then:
 * (x+y)' = x' (clause 2)
 * x+y = x (axiom 3)
 * which is impossible by induction.

Lemma 2: If y ≠ z, then x+y ≠ x+z

Proof by induction on x, for all y and z: If x=1, this is y' ≠ z', which follows from axiom 3. Otherwise, assume x+y ≠ x+z. Then:
 * (x+y)' ≠ (x+z)' (axiom 3)
 * so x'+y = (x+y)' ≠ (x+z)' = x'+z (clause 2)

Lemma 3: If y ≠ 1, there is some v for which y = v'

In other words, every number other than 1 has a predecessor. We will call that predecessor "y-1". We will say more about subtraction shortly.

We will show that, for all y, either y=1 or y = v' for some v.

Proof by induction on y. For y = 1, the statement is trivial. Otherwise, we need to show that y' = 1 or y' = v' for some v. Letting v = y satisfies this.

The trichotomy law, and subtraction
Theorem 4 (trichotomy law): For a given x and y, exactly one of the following is true:
 * (A) x = y
 * (B) x = y+u for some u
 * (C) y = x+v for some v

Proof: We first show that at most one of these is true. (A) and (B) can't both be true by lemma 1. Similarly, (A) and (C) can't both be true. If (B) and (C) are both true, then
 * x = (x+v)+u = x+(v+u), which is impossible by lemma 1.

Now we show that one of them is true. The proof is by induction on x, for fixed y. For x=1, either y=1, so it's case (A), or y ≠ 1, so y = v' by lemma 3, so y = 1+v = x+v, so it's case (C).

Assume the theorem is true for x; we must prove it for x'.
 * If case (A) was true, x = y, so x' = y+1, so case (B) holds for x'.
 * If case (B) was true, x = y+u, so x' = y + (u+1), so case (B) holds for x', using u+1.
 * If case (C) was true, y = x+v. If v=1, then y=x', so case (A) holds for x'.  Otherwise,
 * y = x+(v-1)' (lemma 3)
 * = (x+(v-1))' = x'+(v-1), so case (C) holds for x', using the number "v-1" that lemma 3 provides.

Definition:

If x=y+u for some u (case (B) above), we say "x > y". The number u is called "x-y". If y=x+v for some v (case (C) above), we say "x < y". The number v is called "y-x".

The trichotomy law then says that, for any x and y, exactly one of x=y, xy is true.

We define the operation of finding the number u or v, above, as "subtraction". We can subtract any smaller number from a larger one.

Everything that we have done is severely constrained by the fact that we are dealing only with positive nonzero numbers. When the full integers are defined, we will be able to subtract freely.

Multiplication
Definition: There is an operation of multiplication, denoted x•y, as follows:
 * For all x, x•1 = x                     (clause 1)
 * For all x and y, x•y' = x•y + x   (clause 2)

This is well-defined for all x and y, by induction on y: Clause 1 defines it for y=1 and all x, and clause 2 defines it for y' and all x, assuming it is defined for y and all x.

Lemma 4: 1•x = x•1 = x

Proof by induction on x: If x=1, this is obvious by clause 1. Otherwise, assume it is true for x. Then:
 * 1•x' = 1•x + 1 (clause 2)
 * = x•1 + 1 (induction)
 * = x + 1 = x' (clause 1)

Lemma 5: x'•y = x•y + y

Proof by induction on y: If y=1, this is obvious by lemma 4. Otherwise, assume it is true for y. Then:
 * x'•y' = x'•y + x' (clause 2)
 * = x•y + y + x' (induction)
 * = (x•y + x) + y' (various theorems about addition)
 * = x•y' + y' (clause 2)

Theorem 4 (commutative law of multiplication): x•y = y•x

Proof by induction on x: If x=1, this follows from lemma 4. Otherwise, assume it is true for x. Then:
 * x'•y = x•y + y (lemma 5)
 * = y•x + y (induction)
 * = y•x' (clause 2)

Theorem 5 (distributive law): x•(y+z) = x•y + x•z

Proof by induction on z, for all x and y:
 * If z=1, x•(y+1) = x•y + x = x•y + x•1 (clauses 1 and 2)

Now assume it is true for some z, and for all x and y:
 * x•(y+z') = x•((y+z)') = x•(y+z) + x (clause 2)
 * = x•y + x•z + x (induction)
 * = x•y + x•(z') (clause 2)

Theorem 6 (associative law of multiplication): (x•y)•z = x•(y•z)

Proof by induction on z, for all x and y: If z=1, this follows from lemma 4. Otherwise, assume it is true for z. Then, for all x and y:
 * (x•y)•z' = (x•y)•z + x•y (clause 2)
 * = x•(y•z) + x•y (induction)
 * = x•((y•z) + y) (distributive law)
 * = x•(y•z') (clause 2)

Continuing, we can define, and prove the properties of, all the other operations of integer, rational, real, and complex arithmetic. All from intuitive set theory and the Peano axioms.

title: Natural number


text: File:Intuitiveaddition.JPG

title: File:Intuitiveaddition.JPG


text:
User talk:Vorrakamol

My name is Vorrakamol Suebsirikul, also known as Kwanjai. I was born in Bangkok, Thailand; lived here for about 10 years. My family and I moved to the United States about 13 years ago. I came back to Thailand about two years ago, attending Mahidol University International College here in Salaya, majoring in TIM. The reason why I chose this major is because i love doing service jobs. It started from when I was 16 years old, my family owns a Thai restaurant in Chicago, IL. I used to work there since i was 16 years of age and I loved it! I like meeting new people and serving my customers. After graduating from MUIC, I would like to try (to discover myself more, to see what I really like) these following careers. First, flight attendant, restaurant business and/or hotel business.

--Vorrakamol 04:14, 23 April 2010 (UTC)

title: User talk:Vorrakamol


text:
User talk:Flint28027

My name is Shanagun Sothisaovapark I’m 19 years old I’m from Bangkok Thailand. My favorite sport is basketball. I’m study in Marketing major. The reason why I’m study in marketing major because marketing is all around us to have a successful business you need to be able to hit the target group and analyze them. Hence, that’s why I’m chosen marketing major. I prefer to learn this course because I was interested about how computer worked and I want to know more about the computer system.

title: User talk:Flint28027


text:
User:Egm6341.s10.Team4.andy/HW7

= Problem 1: Linearization about Equilibrium Point $$\displaystyle \hat{x} = x_{max}$$ =

Given
Refer Lecture slide 38-4 for problem statement

The logistic equation for population dynamics is given by 38-3,

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

$$\displaystyle \dot{x}(t) = f(x) = rx \left(1-\frac{x}{x_{max}}\right) $$ $$
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

Find
Linearize Equation (1) about the equilibrium point $$\displaystyle \hat{x} = x_{max}$$.

Solution
With a small perturbation $$\displaystyle y $$, the value of $$\displaystyle x $$ is given by,

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

$$\displaystyle \begin{align} x &= \hat{x} + y \\ &= x_{max} + y \end{align} $$
 * }.
 * }.

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

$$\displaystyle \begin{align} \Rightarrow \frac{dx}{dt} &= \frac{d(x_{max} + y)}{dt} \\ &= \frac{dy}{dt} \end{align} $$
 * }.
 * }.

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

$$\displaystyle \begin{align} \Rightarrow \frac{dy}{dt} &= r(x_{max}+y) \left(1-\frac{(x_{max}+y)}{x_{max}}\right) \\ &= r(x_{max}+y) \left(-\frac{y}{x_{max}}\right) \\ &= -r(y+\frac{y^2}{x_{max}})

\end{align} $$
 * }.
 * }.

Since, $$\displaystyle y $$ is a small perturbation compared to $$\displaystyle x_{max} $$,

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

$$\displaystyle \begin{align} \Rightarrow \frac{dy}{dt} &= -r(y+\cancelto{0}{\frac{y^2}{x_{max}}})\\ &= -ry \end{align} $$
 * }.
 * }.

And so, the solution

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

$$\displaystyle \begin{align} \Rightarrow y = y_0 e^{-rt} \Rightarrow &y(t) \to 0 \ as\ t \to \infty \\ \Rightarrow &x(t) \to x_{max} \ as\ t \to \infty \end{align} $$
 * }.
 * }.

Author
Solved and typed by - Egm6341.s10.Team4.andy 06:16, 23 April 2010 (UTC)

= Problem 7: Logistic Equation: Inconsistent (Trapezoidal) Simpson's rule & Newton-Raphson =

Given
Refer Lecture slide 41-2 for problem statement

The logistic equation for population dynamics is given by 38-3,

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

$$\displaystyle \dot{x}(t) = f(x) = rx \left(1-\frac{x}{x_{max}}\right) $$ $$
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

Here, $$\displaystyle x_{max}=10;\ r = 1.2,\ $$ and $$\displaystyle t \in [0,10] $$

The analytical solution is given by,

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

$$\displaystyle x(t) = \frac{x_0x_{max}e^{rt}}{x_{max} + x_0(e^{rt}-1)} $$ $$
 * $$\displaystyle \longrightarrow (2)
 * }.
 * }.

Find
To solve logistic equation defined by Equation (1) for two initial conditions, $$\displaystyle x_0=2\ and\ x_0 = 7,\ $$, using Inconsistent(Trapezoidal) Simpson's rule given by,

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

$$\displaystyle z_{i+1} = z_{i} + \frac{h/2}{3} \left[f_i + 4f_{(i+1/2)} +f_{i+1}\right] $$ $$
 * $$\displaystyle \longrightarrow (3)
 * }.
 * }.

The Inconsistent rule (Trapezoidal) is given by,

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

$$\displaystyle z_{i+1/2} = \frac{1}{2} [z_{i} + z_{i+1}] $$
 * }.
 * }.

Solution
Consider the Equation (1) in which $$\displaystyle f_{(i+1/2)}$$ is a function of $$\displaystyle z_{i}$$ and $$\displaystyle z_{i+1}$$, where $$\displaystyle z_i, z_{i+1} \rightarrow$$ values of $$\displaystyle x$$ at $$\displaystyle t = t_i$$ and $$\displaystyle t = t_{i+1}$$.

Since, this is an initial value problem, $$\displaystyle z_i$$ is known. So, Equation (3) as a whole becomes a function of $$\displaystyle z_{i+1}$$ given by,

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

$$\displaystyle F(z_{i+1}) = 0 $$ $$
 * $$\displaystyle \longrightarrow (4)
 * }.
 * }.

We find the root $$\displaystyle z_{i+1}$$ of Equation (4) using Newton-Raphson method 40-3,

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

$$\displaystyle z_{i+1}^{(k+1)} = z_{i+1}^{(k)} - \left[\frac{dF(z_{i+1}^{(k)})}{dz}\right]^{-1} F(z_{i+1}^{(k)}) $$ $$
 * $$\displaystyle \longrightarrow (5)
 * }.
 * }.

The Newton-Raphson iteration, starting with an initial guess as $$\displaystyle z_{i+1}^{0} = z_{i}$$, is stopped once the absolute tolerance

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

$$\displaystyle \begin{align} AbsTol &= ||z_{i+1}^{(k+1)} - z_{i+1}^{(k)}|| \\ &\le 10^{-6} \end{align} $$
 * }.
 * }.

is satisfied.

In the below Matlab code, for each of the h values $$\displaystyle x_{Actual} $$ is the vector calculated from Equation (3) in order to calculate the error max (for reference), $$\displaystyle e_{Max} = || x_{Actual} - x ||_{\infty} $$ where $$\displaystyle x $$ is calculated from Inconsistent Simpson's rule - Newton Raphson Method described above, for the given interval $$\displaystyle t \in [0,10] $$.

 Matlab Code:

$$\displaystyle x_0 = 2 $$ :- A comparison on $$\displaystyle h $$ values:

The plot for $$\displaystyle h \approxeq 0.001 $$:

$$\displaystyle x_0 = 7 $$ :- A comparison on $$\displaystyle h $$ values:

The plot for $$\displaystyle h \approxeq 0.001 $$:

Author
Solved and typed by - Egm6341.s10.Team4.andy 05:49, 23 April 2010 (UTC)

= Problem 13: Chebychev - Change of Variable =

Given
Refer Lecture slide 42-2 for problem statement

The integral for a function $$\displaystyle f(x)$$ is,

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

$$\displaystyle I = \int_{-1}^{+1}f(x)dx $$ $$
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.

between the interval $$\displaystyle x \in [-1,1]$$

Find
With a change of variable $$\displaystyle x = cos\theta$$, show that the integral in Equation (1) is,

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

$$\displaystyle I = \int_{0}^{\pi} f(cos\theta)sin\theta d\theta $$
 * }.
 * }.

Solution
With a change of variable $$\displaystyle x = cos\theta \Rightarrow dx = -sin\theta d\theta ; cos(\pi) = x = -1 \Rightarrow \theta = \pi; cos(0) = x = 1 \Rightarrow \theta = 0$$ Equation (1) is,

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

$$\displaystyle \begin{align}

I &= \int_{-1}^{1} f(x)dx \\ &= \int_{\pi}^{0} f(cos\theta) \left(-sin\theta d\theta \right) \\ &= - \int_{\pi}^{0}f(cos\theta) sin\theta d\theta \\ &= \int_{0}^{\pi}f(cos\theta) sin\theta d\theta \\

\end{align} $$
 * }.
 * }.

Author
Solved and typed by - Egm6341.s10.Team4.andy 06:32, 23 April 2010 (UTC)

title: User:Egm6341.s10.Team4.andy/HW7


title: File:42.1.jpg


text:
User talk:Kittipak

My name is Kittipak Pakjamsai. My nickname is K. I am TIM major. I am going to take a core course soon. I want to take this course because of computer is important in our life.

Question 1: How is cloud computing different than using a normal computer?

Answer for Q.1: There are four characteristics for cloud computing. First, cloud computing can be accessed by anyone and any computer. Secondly, data and information in cloud computing can be stored safely in wide network. Thirdly, when spaces of normal computers are full, cloud computing helps users to store their data and information into the internet or network. Finally, cloud computing can always be used without lacking of speed. However, on the other hands, there are four characteristics of normal computer to compare with cloud computing. First, normal computers can be accessed by anyone but only with the computer which storing the data. Second, data and information in normal computer can be easily lost if the computer turn off or having problems. Thirdly, there are limits of memories in normal computers. Finally, if there are too much data in normal computers, the processing will be slowed down. It also can be easily hanked.

Question 2: How could you turn a classroom into a cloud computing classroom?

Answer for Q.2: To form a cloud computing classroom, both teachers and students need to have their own account in the same website. Then they can share all content, opinion, homework ifrom class in the webpage. All students are able to access and have more time to study all things happening in their classes.

title: User talk:Kittipak


text:
User talk:Babygrac3

 * Peung.jpg

My name is Najpapasr Siritananan. I am called by my friends as "Peung." I'm currently studying at Mahidol University International College, majoring in Travel Industry Management. My birthday is on 15th March 1990. I'm now 20 years old. I like travelling and listening to music.

Homework (27/04/10)

1.How is cloud computing different than using a normal computer?

- Cloud computing is a public utility that people use to share their information and resources to the computer. It is more convenient than using a normal computer because normal computer have a limit space to store the information. On the other hand, cloud computing is the best place that people can store their information and anything they want. However, it also help peoples' lives because they can access the data anywhere.

2. How could you turn a classroom into a cloud computing classroom?

- Cloud computing can help students and teacher have well connection. If the teacher want to teach something to students, they can post it on the computer. This way will help students understand more about the lesson. Also, when they want to turn in their homework, they can put it to the computer and the teacher can check it. However, it will create a less chance of misunderstanding between students and teachers

Homework (06/05/10) 1. What is the site you selected

- www.mog.com

2. What is the site's purpose and why is it considered a web 2.0 website?

- It helps people to get in touch both offline and online. It helps people who signed in for the website to write blog or short diaries, uploading their photos to other. It is considered a web 2.0 as other can also share their information.

3. What is your favorite website and why?

- My most favorite site is Facebook. I like Facebook because of it helps me keeping in touch with my friends around the world easily although it is sometimes time-consuming.

Homework (02/07/10) 1. What is the best computer sepc?

- Motherboard: The Motherboard's FSB (Front Side Bus) speed is an important indicator of a Computer's speed. It is a bi-directional bus (i.e., interface) that carries all electronic signals to and from the CPU to the components - Hard Disk, Main Memory, etc. The faster the FSB, the faster your Computer! Today's fastest FSBs come at speeds above 1 GHz!

Processor: Currently, the following Intel Processors are available:

Intel Core i3 Processor, with the 540 and the 530 Product Numbers

Intel Core i5 Processor, with a variety of Product Numbers

Intel Core 17 Processor, again with a variety of Product Numbers available under it

Hard Disk: Current technologies here are ISA and SATA, with Hard Disk speeds being 5400 RPM, 7200 RPM and 10000 RPM. Of the two technologies, SATA is faster and costlier. Today's Hard Disks come with capacities of 320 GB and even 500 GB!

Main Memory: An article about Memory technologies would take up an entire website! EDO, SDRAM, PC100, PC133, PC800, RDRAM, DDR, DRAM, etc., are memory technologies past and present.

As far as your purchase decision is concerned, the higher the speed (Memory speed is internally measured in nanoseconds, and the speed it communicates to the FSB is measured in MHz or GHz) of the Memory, the faster your computer. Current Memory technology is DDR SDRAM and speeds are 667 MHz to 1 GHz and above

Again, the more Memory you have, the more applications you can run and the faster you can run them. How much Memory you have is measured in MBs or GBs. If all you run are simple applications (e-Mail, Browser, Word Processor, etc.) on Windows XP, even 512 MB Memory is decent. However, if you run larger applications (like Design, Mathematical/ Engineering/ Statistical applications or high-resolution Games), you will need at least 2 GB Memory. Today's Computers come with 2, 3, even 4 GB Memory!

Graphics and Sound: All Motherboards today come with built-in Graphics and Sound. This was not the case earlier. However, the Graphics and Sound that is built into the motherboard is very basic.

Most of the graphics-intensive programs - AutoCAD, CorelDRAW and the like, would run much faster with an add-on Graphics card with dedicated memory. In fact, many of today's graphics-intensive games would not even start playing, on the built-in graphics or sound card! This is because before starting up the game, the game would look for a dedicated Graphics Card. If it does not find any, it would not start up! So if you plan to run more that a Word processor, e-Mail, Browser and such stuff, a dedicated Graphics Card such as nVIDIA or Radeon, with 512 MB or even 1 GB of dedicated Video RAM is a must!

Again, if you are into music editing/ making movies/ audio CDs/ DVDs on your Computer, you must go in for a dedicated Sound Card. Sound Blaster has remained the market leaders and preferred brand in the sound cards segment for many many years now!

Optical Drive: Today, you have a choice of Optical Drives. Here are the choices you have:

CD-ROM Drive (Read CDs)

Combo Drives (Read and Write CDs, Read DVDs)

Super Drive (Read and Write CDs as well as DVDs)

Blu-Ray Drives (Read and Write CDs, DVDs as well as the new Blu-Ray Disks)

A plain vanilla CD-ROM Drive is now passé, while a Blu-Ray Disk may be a costly luxury at the moment, for, its prices will continue to fall. At the moment, the best bet is definitely to go in for a Super Drive that will let you play as well as record on both CDs as well as DVDs

Operating System: Lastly - and equally importantly, your choice of Operating System will also affect your specifications. While Windows XP would run quite well with 512 MB Memory, you will require a minimum 1 GB, if your choice is Windows Vista Business: we recommend 2 GB RAM if you are choosing a machine running Vista Home Premium, Vista Business or Vista Ultimate. Interestingly, the latest version of Windows, Windows 7, requires less in terms of Memory, than its predecessor, Windows Vista! However, we recommend that you still go for 2 GB Memory, if your new Machine has Windows 7 on it

The next section of this article presents you with the specifications for the current Intel Processors. You may like to look at this, even though a table laying out specifications of one model after the other is hardly a help, as far as taking your PC decision is concerned! Therefore if you want to cut through to our recommended PC Specifications, click here!

The rest of this article guides you through to help you choose current value-for-money Desktop/ Notebook specifications, as well as next-higher specifications. These specifications will be updated, as and when trends change.

title: User talk:Babygrac3


text:
User talk:Tassawat

Homework 1 Introduction My Name is Tassawat Wangsupakitkosol. My nickname is sugus. I am still study in Mahidol University International College. I take TIM course. My birthday is on 17 Feb 1990. I am twenty years old this year. I like to eat pizza. I love to play football very much. I am good at listening and speaking.

Homework 2		                                                                              Tassawat Wangsupakitkosol(5180718) Phutich Paisalsak(5180663) Computer Essential Section 1

1.	How is cloud computing different than using a normal computer?

Cloud computing is a technology that uses the internet as central share network. It also allows other users to access the network. Users can save as many works as they want on the network. These help them to have more space for other important applications in storage. You can see if your friends are working. Moreover, using cloud computing helps users get new friends. In addition, it is convenient to do work. A good example of cloud computing is yahoo, g-mail, and Amazon. Cloud computing only needs the internet connection. Normal computing does not have internet access. Presently you can access to internet almost everywhere in the central business district, which makes life become more convenient.

2.	How could you turn a classroom into a cloud computing classroom?

First of all, all the students in the classroom need to register network like, Amazon, Yahoo, and G-mail etc. These networks are connecting together. Moreover, students will have to plan their schedule, which uses to set for work as a team. Furthermore, students will use their ID number to access the network which is safe and personal. For example, Wikiversity offers us for more comfortable work. Instructor can post anything on Wikiversity and students can read them. They can share an assignment with others.

title: User talk:Tassawat


text:
User:Egm6341.s10.team4.anandankala/HW7-9

=Parameterization of ellipse=

Given
$$\displaystyle x=a cost; y=b sint.$$

Prove That
$$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2}=1$$

Solution
We have $$\displaystyle x=acost; y=bsint.$$

$$\displaystyle x=a cost \Rightarrow x^2=a^2(cost)^2$$

$$\displaystyle y=b sint \Rightarrow y^2=b^2(sint)^2$$

$$\displaystyle \Rightarrow \frac{x^2}{a^2}=(cost)^2 and \frac{y^2}{b^2}=(sint)^2 $$

$$\displaystyle \Rightarrow \frac{x^2}{a^2}+ \frac{y^2}{b^2}=(cost)^2+(sint)^2 $$

$$\displaystyle \Rightarrow \frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1 $$


text: File:HW7 prob7-1.jpg

title: File:HW7 prob7-1.jpg


text:
User:Egm6341.s10.team4.anandankala/HW7-15

=Integral Evaluation=

Given
$$\displaystyle f(cos\theta)= \frac{a_0}{2} + \sum_{k=1}^{\infty} a_kcos(k\theta)$$

$$\displaystyle I= \int_{-1}^{1}f(x)dx= \int_{0}^{\pi}f(cos\theta)sin\theta d\theta$$

Prove that
$$\displaystyle I= a_0 + \sum_{j=1}^{\infty} \frac{2a_{2j}}{1-(2j)^2}$$

Solution
$$\displaystyle I= \int_{-1}^{1}f(x)dx= \int_{0}^{\pi}f(cos\theta)sin\theta d\theta$$

$$\displaystyle \Rightarrow I= \int_{0}^{\pi}(\frac{a_0}{2}+ \sum_{k=1}^{\infty}a_kcos(k\theta))sin\theta d\theta$$

$$\displaystyle \Rightarrow I= \int_{0}^{\pi} \frac{a_0}{2} sin\theta d\theta + \sum_{k=1}^{\infty}\int_{0}^{\pi} a_kcos(k\theta))sin\theta d\theta $$

$$\displaystyle \Rightarrow I= \dfrac{a_0}{2} [cos\theta]_{\pi}^{0}+ \sum_{k=1}^{\infty}\int_{0}^{\pi} \dfrac{a_k}{2} [sin((1+k)\theta) +sin((1-k)\theta)]d\theta $$

$$\displaystyle \Rightarrow I= a_0+ \sum_{k=1}^{\infty}\dfrac{a_k}{2} [\int_{0}^{\pi} sin((1+k)\theta)d\theta+\int_{0}^{\pi} sin((1-k)\theta)d\theta] $$

$$\displaystyle \Rightarrow I= a_0+ \sum_{k=1}^{\infty} \dfrac{a_k}{2} [[\dfrac{cos((1+k)\theta)}{(1+k)}]_{\pi}^{0} +[\dfrac{cos((1-k)\theta)}{(1-k)}]_{\pi}^{0}]$$

Since all the cosine terms for odd values of k become zero. The above expression is modified as below.

$$\displaystyle \Rightarrow I= a_0+ \sum_{j=1}^{\infty} \dfrac{a_{2j}}{2} [[\dfrac{cos((1+2j)\theta)}{(1+2j)}]_{\pi}^{0} +[\dfrac{cos((1-2j)\theta)}{(1-2j)}]_{\pi}^{0}]$$

$$\displaystyle \Rightarrow I= a_0+ \sum_{j=1}^{\infty} \dfrac{a_{2j}}{2} \dfrac{2}{(1+2j)}]+[\dfrac{2}{(1-2j)}$$

$$\displaystyle \Rightarrow I= a_0+ \sum_{j=1}^{\infty} a_{2j}\dfrac{1}{(1+2j)}]+[\dfrac{1}{(1-2j)}$$

$$\displaystyle \Rightarrow I= a_0+ \sum_{j=1}^{\infty} \dfrac{2a_{2j}}{1-(2j)^2}.$$

Hence the proof.


title: File:HW7-prob7-2.jpg


text:
User talk:Minkkk

~* My name is Haruetai Limudompunth. ~* My ID. is 5180020. ~* My nickname is Mink. ~* I am a sophomore of Mahidol University International College. ~* I am studying in TIM major.

See you in class :D

Homework #2 :

 * 1. How is cloud computing different than using a normal computer?

Answer : Cloud computing is a public utility that people use to share their information and resources to the computer. It is more convenient than using a normal computer because normal computer have a limit space to store the information. On the other hand, cloud computing is the best place that people can store their information and anything they want. However, it also help peoples' lives because they can access the data anywhere.

 * 2. How could you turn a classroom into a cloud computing classroom?

Answer : Cloud computing can help students and teacher have well connection. If the teacher want to teach something to students, they can post it on the computer. This way will help students understand more about the lesson. Also, when they want to turn in their homework, they can put it to the computer and the teacher can check it. However, it will create a less chance of misunderstanding between students and teachers.

title: User talk:Minkkk


text:
User talk:Jaded

Pornpawit Obsuwan. My short name is Jaded. I am 20 years. I love to learn outside of the class about the business and food. I am a second year student choosing TIM as a major. This is the first semester since being here that I know the reason and my mind to study. I promise to myself to work hard. I am looking forward to dropping out of school for 1 years before getting back again.

How is cloud computing different than using a normal computer? The cloud computer is very convenient for all pepople because when we do our projects and and assignmentsm, we can maintain it in the servers. Moreover, files and datas are not lost when our computers are broken. We can get out datas anywhere that provides internet to us. On the other hand, normal computers are unconvenient for all people because we have to save a data on a computer, and we need to use a thumb drive or a floppy drive to save them. Furethermore, when our computers are error and broken, we can not bring our work back, and noone reposible for it. How could you turn a classroom into a cloud computing classroom? We need to register the some websites such as google, yahoo, or wiki. Therefore, when teachers assign homework, or when we have to send homework, we can put it on that servers. We will not lost datas

title: User talk:Jaded