text:
User:GeorgeTok/Reflective Games Analysis

'Reflective Games Analysis By George Tokkos

Andrew McNeil’s ‘Kingmaker’ is a board game from the seventies, which is set in the 1400’s and based on the war of the roses between the House of Lancaster and the House of York. The game allows about 2-6 players, and when playing we had 6 players taking part. To play the game each player must build and control a party of nobles who through battles with other nobles, as well as some political diplomacy, try to kill off the other players groups of nobles in order to take power which is done by controlling at least one member of one of the two counterpart royal families. The board itself is a map of Britain in the 15th century, consisting of major cities with castles, large towns and ports, which are interconnected by roads, with the rest of the map occupied by open land which are organised into grids in order to count moves. To begin with each player is given a number of cards, and also receives resource cards each turn, which can be anything from another noble which can move, to a faction of archers which can be assigned to a players existing noble’s forces. There are also town cards that give control of a town to the player in possession of that card as well as title or office cards, for example the Archbishop of Canterbury card, or the Marshal of England card. The cards are pretty much self-explanatory as they have all the required information on them, and the pieces are small round cardboard pieces that have a picture of their particular noble’s coat of arms on them. The members of the royal families pieces are octagonal and carry either a red or white rose depending on the family as well as the name of the royal on them for example Margaret or Richard. This is the basic framework of the game although there are some other small aspects which effect the game like the random event cards which are drawn every round and can order certain nobles and royals to move, as well as ships which can transport royals and nobles. At first game play is rather slow, as players get familiar with the board, which was dated to the majority of us who had little knowledge of medieval England. The pieces at first appear small and difficult to handle however once this first hurdle was overcome the pace of the game picked up and the game became very enjoyable. Players began engaging in battle and the game soon became very competitive, with different approaches being taken by different players. Some jumped into battles and tried to kill off as many other players and nobles as possible, and others were more cautious, using their moves carefully, avoiding other players and trying to abduct the members of the royal families. The random event cards played an important part in how the game panned out, as one player frequently used ships to transport nobles, storms kept coming up which sent the ships back to port which essentially wasted that players go and time. One player appeared to be winning the game as they had control of a royal member however their baron was ordered to Plymouth, and then the Plague struck Plymouth and so that player lost all of their pieces and were out. This gifted me the opportunity to win the game, as I had the Archbishop of Canterbury, which is needed to crown a king or queen and I had a royal family member, so I hastily moved to Canterbury and crowned Margaret of York Queen of England and therefore won the game. The game was colourful, stimulating, engaging and thoroughly enjoyable. It was also educational, teaching players geography, history as well as the political processes of Medieval England. All round Kingmaker must be given top marks as it excellently portrays and recreates Medieval Politics and the battle for power, all of course in an pleasurable and fun way.

The 1955 Vietnam role-playing exercise recreated the events and scenarios running up to the Vietnam War, and allowed players to engage in the political processes, which either lead to war or peace. In order to play out the exercise, the participants were split into either pairs or threes and assigned them one of the countries or regions that were involved in the tense build up to the war. These consisted of, the Vietminh who were the native inhabitants of Vietnam and the communist threat to the west, French Indochina who were part of the French colonial empire in south east Asia and made up of a federation of three Vietnamese regions. China, USA, Russia, France and Britain made up the other countries. Each player or team, represented one of the countries and had there own goals, which are listed on their own exclusive achievement score sheet which score the player different points for different achievements, for example 200 points for being voted to chair the Geneva Peace Conference. These score sheets remain hidden and other players cannot see them, which make it difficult to determine one another’s true goals.

The game had three possible main outcomes, which were for the Vietminh to take control of the country with a communist government, French Indochina to take control of the country, or for peace and an agreement to be met where the country is shared. China and Russia, as communists naturally aid the Vietminh in their cause, while France and the USA aid the French Indochinese to support their colonial investment and to aid their fight against communism respectively. While the British simply push for peace in the region. The role-play requires players to simulate patriotism and to actively engage with other players in order to reach agreements, which can be announced or kept secret until the time is right. Players can only score points by interacting with other players in order to achieve an outcome that is on their score sheet. So players make deals, break promises, and even sign written notes to ensure they keep their word. After playing the game for sometime two struggles appear to take centre stage and these are the struggle for control of Vietnam between the Vietminh and the French Indochinese, and the other being the larger scale battle for world power between the USA and Russia. The game accurately portrayed the geographical scenarios and situations that would have been apparent at the time, as it was pretty easy for the Vietminh, China and Russia to make quick and effective decisions as they are all in the same vicinity, whereas the USA, France and Britain had to take time to confer and then had to meet with the French Indochinese to implement and seek approval on the ground for their plans. Also the score sheets allowed the communist nations to carry out swift aggressive moves more readily as they scored for them and these ran parallel with their long-term goals anyway. On the other hand the British and the USA had to take time as they sought little bloodshed and ultimately peace. The game although was all down to decisions made by the players, some random scenarios were thrown up by the narrator, which would pressure players into making decisions. This was healthy for the exercise as it in a way forced certain conclusions, which in turn could form or destroy allegiances and alliances. In the end of the game the players representing the Vietminh, bribed their way to chairing the Geneva Peace Conference, and had decided in secret along with Russia to sabotage all plans for peace and instead declare all out nuclear war, which won Russia points, as well as the Vietminh who scored for more than half of Vietnam being in communist hands. The exercise all in all was superbly organised and carried out to perfection, although, players became ultra-competitive to the point that they began to stray from their nations traditions which at times made the game unrealistic, as after all, all players are human and wanted to win the game. In summary the exercise was educational and taught the players a lot about the political, cultural and geographical tensions of that time period as well as how international relations between nations work, including ways to start and avoid war.

‘Modern Society’ is a card game, which quite literally is based on ‘Modern Society’ and all its Socio-economic/politics aspects from Education to Healthcare to Organic food production. The game itself is played by each of the prescribed 3-4 players being dealt cards that hold different social values in the 4 categories which are coloured blue, red, green and black. There is also a track point scoring measuring scale that sits in the middle of all players and depending on the effect of the event cards which are dealt every season (turn), determines how players score. Players can also obtain laws by trading in points, which allow them to either implement restrictions or remain safe from restrictions, which may obstruct players scoring points. The game itself is made up of a lot of pieces, which are not easily differentiated and small. This makes the game difficult to read and play with any real pace, which has an effect on the enjoyment that the game gives players. In terms of understanding the political processes that the game supposedly recreates, the cards are often very random for example there is a car, which springs an assassination all of a sudden which is rather unrealistic. It is at times difficult to have any kind of game plan or strategy for playing Modern Society because the game is ultimately based a lot on chance as the cards are dealt randomly which means the player acts on what their cards say, rather than cards being dealt as a result of how players have acted. There was often slight confusion when playing the game, as a lot of counting and measuring is required, with some players taking the opportunity to cheat amongst all the miscalculation. The game could be seen as accurate in its portrayal of the complexity of modern societies and displays the many components that must be maintained properly in order to uphold a functional society. Every move affects all players as the society is shared, this is original and realistic as we all share one habitat talking generally and broadly, for example when a player plays an event card which affects one of the categories it affects all players including the player who played the card, in this sense the game is accurate. However the game in my opinion is dull, lacklustre and boring at times. Its extremely repetitive as the game structure is very simple while at the same time the game mechanics are complicated and the playing process is repeated 3 times, and then the game is over.

Guy Debord’s ‘The Game Of War’ is a war game that places two armies of equal strength against each other with only one goal. Destroy your enemy. Players are split into two teams who play against each othe for victory. The board is a grid made up of 500 squares (20 x 25) some of which are occupied by permanent fittings, which act as terrain and cannot be moved, and also block communication. The armies consist of several pieces, which all have a different rating of speed and range which determines how far and at what pace they may move. Each piece also has defensive and offensive value that determines the extent to which they can damage an opponent’s piece and how much of an opponent’s attack they can withstand. Each team has the same amount of pieces and these consist of Infantry, Foot Artillery, Foot Transmission, Cavalry, Mounted Artillery, Mounted Transmission and Arsenals. There are also Forts, which do not belong to either army but can be occupied by either and add to the defence value of the pieces occupying it. The game begins with some parts of the armies already occupying forts. The rules are rather simple once grasped, but grasping them took some time. The board although mathematically symmetrical and straightforward looks very complicated once all the pieces are on the board, and with the pieces themselves being the same colour as the board and the mountainous terrain and although aesthetically pleasing to the eye players may have to look closely to see clearly. The rules of movement again are simple by definition however require some mathematic calculation to carry out properly.

The Arsenals are what arm all pieces of the army, and they do this via communication, which is also the case with real armies of the past and present, making the game very realistic in this sense. Communication is in effect when there is a clear path on the board between the arsenals, which cannot move, and the pieces. However the transmission pieces can reflect communication allowing player’s armed forces to move around the board as long as they maintain communication via the transmission pieces. A breakdown in communication effectively means your forces are powerless and cannot attack because in reality they would have no ammunition or food. Again this is realistic. When playing the game each army is allowed a few moves each go, and so the opposing team opted to go for a basic and less risky approach, and go for straightforward military action which was to attack at the front, simple and directly. Our team on the other hand attempted to utilise the range of our cavalry, and came up with an intricate plan to travel around the mountainous terrain and destroy the defenceless enemy arsenals and therefore disarm their forces. When deep into game play, it was very enjoyable to confer with your teammates in a secretive way while your opponents sat just across the board a few feet away. Players exchanged notes containing military tactics, and some players assumed leadership of their team, all realistic again. The game realistically recreates the setting of warfare, and its requirement of strategic and tactical concern is most probably identical to real warfare in the past and really allows players to relate to the pressure that someone like Napoleon for example would have experienced. All in all Guy Debords’ Game Of War is a gripping and realistic simulation of war and completely serves its purpose.


text:
User:Markclaysius/Books/Computer Science

 * Concepts
 * Development Environment Setup
 * School:Computer science

 * COMPUTER PROGRAMMING
 * Topic:Computer programming

 * Computing Fundamentals
 * Topic:Computing fundamentals

 * Introduction to Computer Science
 * Introduction to Computer Science
 * History of computing
 * Basics of computer architecture
 * Personal computers
 * What is an Algorithm
 * How to go about implementing a problem solution as a program
 * Development environment and language principles
 * Types and variables
 * Operators and expressions
 * Control structures
 * Procedures and functions
 * Arrays, strings and records
 * Basic I/O

 * Introduction to Programming
 * Introduction to Programming
 * Introduction to Programming/About Programming
 * Introduction to Programming/Programming Languages
 * Introduction to Programming/Organization part 1
 * Introduction to Programming/Variables
 * Introduction to Programming/Variables part 2
 * Introduction to Programming/Variables part 3
 * Introduction to Programming/Control structures
 * Introduction to Programming/Subprograms
 * Introduction to Programming/Scope

 * Introduction to Programming Logic
 * Introduction to Programming Logic
 * Introduction to Programming Logic/Key Terms
 * Introduction to Programming Logic/Variables
 * Introduction to Programming Logic/Lesson 2: Conditionals and Logic
 * Introduction to Programming Logic/Looping
 * Introduction to Programming Logic/Lesson 4: Code Organization Techniques

 * Hungarian Notation
 * CamelCase

 * Procedure Oriented Programming
 * Programming: BASIC
 * Object-Oriented Programming
 * Programming: Python
 * Topic:Python
 * Introduction to and setup of Python
 * Basic Python data types
 * Python operators
 * Basic Python functions
 * Truth and flow control in Python
 * Object-Oriented Python
 * Python libraries

 * Programming: Ruby
 * Topic:Ruby
 * Introduction to Ruby

 * Programming: Java
 * Introduction to Programming in Java
 * Introduction to Programming in Java/Introduction to Java
 * Introduction to Programming in Java/Integer variables
 * Introduction to Programming in Java/Boolean Variables
 * Introduction to Programming in Java/Strings
 * Introduction to Programming in Java/Java Classes
 * Introduction to Programming in Java/Inheritance
 * Introduction to Programming in Java/Arrays
 * Introduction to Programming in Java/Swing

 * Programming: Java Learning
 * Learning Java
 * Introduction to Java
 * Basic Java Language
 * Java Decision Structures
 * Java Objects and Classes
 * Java error handling
 * Important Java classes
 * Java programming tips
 * Java/Fundamental exercises
 * Java error handling II
 * Java/Inheritance
 * Java/Classes II
 * Swing and AWT
 * Java applets
 * Java File IO
 * Java/Security
 * Java/Networking
 * Java/Intermediate Exercises
 * Java/Multithreading
 * Java/Networking II
 * Java/Advanced Swing

 * Programming: C++
 * C++
 * Introduction to C++
 * Variables and User Input
 * C++/Simple Math
 * Conditional Statements
 * Loops (C++)
 * Functions and Recursion
 * More Functions
 * C++ pointers
 * Classes and Inheritance
 * Templates in C++, part 1
 * Templates in C++, part 2
 * The STL
 * C++/STL Algorithms

 * Programming: HTML
 * HTML
 * Introduction to HTML
 * WYSIWYG vs. Manual Coding in HTML
 * Web Design/What is HTML
 * Basic formatting in HTML
 * Fonts and colours in HTML
 * Links in HTML
 * Graphical Content and Multimedia
 * Lists in HTML
 * Tables in HTML
 * Frames in HTML
 * Forms in HTML
 * Web Design/HTML Challenges

 * DATABASE
 * Database: MySQL
 * Database management systems
 * Introduction to Databases
 * Introduction to SQL
 * Introduction to PHP

 * OPERATING SYSTEM
 * NETWORKING
 * Topic:Operating systems

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


text:
User:Markclaysius/Books/Computer Science 2

 * Legacy Operating Systems
 * Existing Operating Systems
 * Implementation
 * Concepts
 * School:Computer science

 * COMPUTER PROGRAMMING
 * Topic:Computer programming

 * Computing Fundamentals
 * Topic:Computing fundamentals

 * Introduction to Computer Science
 * Introduction to Computer Science
 * History of computing
 * Basics of computer architecture
 * Personal computers
 * What is an Algorithm
 * How to go about implementing a problem solution as a program
 * Development environment and language principles
 * Types and variables
 * Operators and expressions
 * Control structures
 * Procedures and functions
 * Arrays, strings and records
 * Basic I/O

 * Introduction to Programming
 * Introduction to Programming
 * Introduction to Programming/About Programming
 * Introduction to Programming/Programming Languages
 * Introduction to Programming/Organization part 1
 * Introduction to Programming/Variables
 * Introduction to Programming/Variables part 2
 * Introduction to Programming/Variables part 3
 * Introduction to Programming/Control structures
 * Introduction to Programming/Subprograms
 * Introduction to Programming/Scope

 * Introduction to Programming Logic
 * Introduction to Programming Logic
 * Introduction to Programming Logic/Key Terms
 * Introduction to Programming Logic/Variables
 * Introduction to Programming Logic/Lesson 2: Conditionals and Logic
 * Introduction to Programming Logic/Looping
 * Introduction to Programming Logic/Lesson 4: Code Organization Techniques

 * Hungarian Notation
 * CamelCase

 * Procedure Oriented Programming
 * Programming: BASIC
 * Object-Oriented Programming
 * Programming: Python
 * Topic:Python
 * Introduction to and setup of Python
 * Basic Python data types
 * Python operators
 * Basic Python functions
 * Truth and flow control in Python
 * Object-Oriented Python
 * Python libraries

 * Programming: Ruby
 * Topic:Ruby
 * Introduction to Ruby

 * Programming: Java
 * Introduction to Programming in Java
 * Introduction to Programming in Java/Introduction to Java
 * Introduction to Programming in Java/Integer variables
 * Introduction to Programming in Java/Boolean Variables
 * Introduction to Programming in Java/Strings
 * Introduction to Programming in Java/Java Classes
 * Introduction to Programming in Java/Inheritance
 * Introduction to Programming in Java/Arrays
 * Introduction to Programming in Java/Swing

 * Programming: Java Learning
 * Learning Java
 * Introduction to Java
 * Basic Java Language
 * Java Decision Structures
 * Java Objects and Classes
 * Java error handling
 * Important Java classes
 * Java programming tips
 * Java/Fundamental exercises
 * Java error handling II
 * Java/Inheritance
 * Java/Classes II
 * Swing and AWT
 * Java applets
 * Java File IO
 * Java/Security
 * Java/Networking
 * Java/Intermediate Exercises
 * Java/Multithreading
 * Java/Networking II
 * Java/Advanced Swing

 * Programming: C++
 * C++
 * Introduction to C++
 * Variables and User Input
 * C++/Simple Math
 * Conditional Statements
 * Loops (C++)
 * Functions and Recursion
 * More Functions
 * C++ pointers
 * Classes and Inheritance
 * Templates in C++, part 1
 * Templates in C++, part 2
 * The STL
 * C++/STL Algorithms

 * Programming: HTML
 * HTML
 * Introduction to HTML
 * WYSIWYG vs. Manual Coding in HTML
 * Web Design/What is HTML
 * Basic formatting in HTML
 * Fonts and colours in HTML
 * Links in HTML
 * Graphical Content and Multimedia
 * Lists in HTML
 * Tables in HTML
 * Frames in HTML
 * Forms in HTML
 * Web Design/HTML Challenges

 * DATABASE
 * Database: MySQL
 * Database management systems
 * Introduction to Databases
 * Introduction to SQL
 * Introduction to PHP

 * Concepts
 * Development Environment Setup
 * OPERATING SYSTEM
 * NETWORKING
 * Topic:Operating systems
 * School:Computer science

 * COMPUTER PROGRAMMING
 * Topic:Computer programming

 * Computing Fundamentals
 * Topic:Computing fundamentals

 * Introduction to Computer Science
 * Introduction to Computer Science
 * History of computing
 * Basics of computer architecture
 * Personal computers
 * What is an Algorithm
 * How to go about implementing a problem solution as a program
 * Development environment and language principles
 * Types and variables
 * Operators and expressions
 * Control structures
 * Procedures and functions
 * Arrays, strings and records

 * Introduction to Programming
 * Introduction to Programming
 * Introduction to Programming/About Programming
 * Introduction to Programming/Programming Languages
 * Introduction to Programming/Organization part 1
 * Introduction to Programming/Variables
 * Introduction to Programming/Variables part 2
 * Introduction to Programming/Variables part 3
 * Introduction to Programming/Control structures
 * Introduction to Programming/Subprograms
 * Introduction to Programming/Scope

 * Introduction to Programming Logic
 * Introduction to Programming Logic
 * Introduction to Programming Logic/Key Terms
 * Introduction to Programming Logic/Variables
 * Introduction to Programming Logic/Lesson 2: Conditionals and Logic
 * Introduction to Programming Logic/Looping
 * Introduction to Programming Logic/Lesson 4: Code Organization Techniques

 * Hungarian Notation
 * CamelCase

 * Procedure Oriented Programming
 * Programming: BASIC
 * Object-Oriented Programming
 * Programming: Python
 * Topic:Python
 * Introduction to and setup of Python
 * Basic Python data types
 * Python operators
 * Basic Python functions
 * Truth and flow control in Python
 * Object-Oriented Python
 * Python libraries

 * Programming: Ruby
 * Topic:Ruby
 * Introduction to Ruby

 * Programming: Java
 * Introduction to Programming in Java
 * Introduction to Programming in Java/Introduction to Java
 * Introduction to Programming in Java/Integer variables
 * Introduction to Programming in Java/Boolean Variables
 * Introduction to Programming in Java/Strings
 * Introduction to Programming in Java/Java Classes
 * Introduction to Programming in Java/Inheritance
 * Introduction to Programming in Java/Arrays
 * Introduction to Programming in Java/Swing

 * Programming: Java Learning
 * Learning Java
 * Introduction to Java
 * Basic Java Language
 * Java Decision Structures
 * Java Objects and Classes
 * Java error handling
 * Important Java classes
 * Java programming tips
 * Java/Fundamental exercises
 * Java error handling II
 * Java/Inheritance
 * Java/Classes II
 * Swing and AWT
 * Java applets
 * Java File IO
 * Java/Security
 * Java/Networking
 * Java/Intermediate Exercises
 * Java/Multithreading
 * Java/Networking II
 * Java/Advanced Swing

 * Programming: C++
 * C++
 * Introduction to C++
 * Variables and User Input
 * C++/Simple Math
 * Conditional Statements
 * Loops (C++)
 * Functions and Recursion
 * More Functions
 * C++ pointers
 * Classes and Inheritance
 * Templates in C++, part 1
 * Templates in C++, part 2
 * The STL
 * C++/STL Algorithms

 * Programming: HTML
 * HTML
 * Introduction to HTML
 * WYSIWYG vs. Manual Coding in HTML
 * Web Design/What is HTML
 * Basic formatting in HTML
 * Fonts and colours in HTML
 * Links in HTML
 * Graphical Content and Multimedia
 * Lists in HTML
 * Tables in HTML
 * Frames in HTML
 * Forms in HTML
 * Web Design/HTML Challenges

 * DATABASE
 * Database: MySQL
 * Database management systems
 * Introduction to Databases
 * Introduction to SQL
 * Introduction to PHP

 * NETWORKING
 * OPERATING SYSTEM
 * Basic I/O
 * Topic:Operating systems

 * Concepts
 * Development Environment Setup
 * School:Computer science

 * COMPUTER PROGRAMMING
 * Topic:Computer programming

 * Computing Fundamentals
 * Topic:Computing fundamentals

 * Introduction to Computer Science
 * Introduction to Computer Science
 * History of computing
 * Basics of computer architecture
 * Personal computers
 * What is an Algorithm
 * How to go about implementing a problem solution as a program
 * Development environment and language principles
 * Types and variables
 * Operators and expressions
 * Control structures
 * Procedures and functions
 * Arrays, strings and records
 * Basic I/O

 * Introduction to Programming
 * Introduction to Programming
 * Introduction to Programming/About Programming
 * Introduction to Programming/Programming Languages
 * Introduction to Programming/Organization part 1
 * Introduction to Programming/Variables
 * Introduction to Programming/Variables part 2
 * Introduction to Programming/Variables part 3
 * Introduction to Programming/Control structures
 * Introduction to Programming/Subprograms
 * Introduction to Programming/Scope

 * Introduction to Programming Logic
 * Introduction to Programming Logic
 * Introduction to Programming Logic/Key Terms
 * Introduction to Programming Logic/Variables
 * Introduction to Programming Logic/Lesson 2: Conditionals and Logic
 * Introduction to Programming Logic/Looping
 * Introduction to Programming Logic/Lesson 4: Code Organization Techniques

 * Hungarian Notation
 * CamelCase

 * Procedure Oriented Programming
 * Programming: BASIC
 * Object-Oriented Programming
 * Programming: Python
 * Topic:Python
 * Introduction to and setup of Python
 * Basic Python data types
 * Python operators
 * Basic Python functions
 * Truth and flow control in Python
 * Object-Oriented Python
 * Python libraries

 * Programming: Ruby
 * Topic:Ruby
 * Introduction to Ruby

 * Programming: Java
 * Introduction to Programming in Java
 * Introduction to Programming in Java/Introduction to Java
 * Introduction to Programming in Java/Integer variables
 * Introduction to Programming in Java/Boolean Variables
 * Introduction to Programming in Java/Strings
 * Introduction to Programming in Java/Java Classes
 * Introduction to Programming in Java/Inheritance
 * Introduction to Programming in Java/Arrays
 * Introduction to Programming in Java/Swing

 * Programming: Java Learning
 * Learning Java
 * Introduction to Java
 * Basic Java Language
 * Java Decision Structures
 * Java Objects and Classes
 * Java error handling
 * Important Java classes
 * Java programming tips
 * Java/Fundamental exercises
 * Java error handling II
 * Java/Inheritance
 * Java/Classes II
 * Swing and AWT
 * Java applets
 * Java File IO
 * Java/Security
 * Java/Networking
 * Java/Intermediate Exercises
 * Java/Multithreading
 * Java/Networking II
 * Java/Advanced Swing

 * Programming: C++
 * C++
 * Introduction to C++
 * Variables and User Input
 * C++/Simple Math
 * Conditional Statements
 * Loops (C++)
 * Functions and Recursion
 * More Functions
 * C++ pointers
 * Classes and Inheritance
 * Templates in C++, part 1
 * Templates in C++, part 2
 * The STL
 * C++/STL Algorithms

 * Programming: HTML
 * HTML
 * Introduction to HTML
 * WYSIWYG vs. Manual Coding in HTML
 * Web Design/What is HTML
 * Basic formatting in HTML
 * Fonts and colours in HTML
 * Links in HTML
 * Graphical Content and Multimedia
 * Lists in HTML
 * Tables in HTML
 * Frames in HTML
 * Forms in HTML
 * Web Design/HTML Challenges

 * DATABASE
 * Database: MySQL
 * Database management systems
 * Introduction to Databases
 * Introduction to SQL
 * Introduction to PHP

 * OPERATING SYSTEM
 * NETWORKING
 * Topic:Operating systems

 * Concepts
 * Development Environment Setup
 * School:Computer science

 * COMPUTER PROGRAMMING
 * Topic:Computer programming

 * Computing Fundamentals
 * Topic:Computing fundamentals

 * Introduction to Computer Science
 * Introduction to Computer Science
 * History of computing
 * Basics of computer architecture
 * Personal computers
 * What is an Algorithm
 * How to go about implementing a problem solution as a program
 * Development environment and language principles
 * Types and variables
 * Operators and expressions
 * Control structures
 * Procedures and functions
 * Arrays, strings and records
 * Basic I/O

 * Introduction to Programming
 * Introduction to Programming
 * Introduction to Programming/About Programming
 * Introduction to Programming/Programming Languages
 * Introduction to Programming/Organization part 1
 * Introduction to Programming/Variables
 * Introduction to Programming/Variables part 2
 * Introduction to Programming/Variables part 3
 * Introduction to Programming/Control structures
 * Introduction to Programming/Subprograms
 * Introduction to Programming/Scope

 * Introduction to Programming Logic
 * Introduction to Programming Logic
 * Introduction to Programming Logic/Key Terms
 * Introduction to Programming Logic/Variables
 * Introduction to Programming Logic/Lesson 2: Conditionals and Logic
 * Introduction to Programming Logic/Looping
 * Introduction to Programming Logic/Lesson 4: Code Organization Techniques

 * Hungarian Notation
 * CamelCase

 * Procedure Oriented Programming
 * Programming: BASIC
 * Object-Oriented Programming
 * Programming: Python
 * Topic:Python
 * Introduction to and setup of Python
 * Basic Python data types
 * Python operators
 * Basic Python functions
 * Truth and flow control in Python
 * Object-Oriented Python
 * Python libraries

 * Programming: Ruby
 * Topic:Ruby
 * Introduction to Ruby

 * Programming: Java
 * Introduction to Programming in Java
 * Introduction to Programming in Java/Introduction to Java
 * Introduction to Programming in Java/Integer variables
 * Introduction to Programming in Java/Boolean Variables
 * Introduction to Programming in Java/Strings
 * Introduction to Programming in Java/Java Classes
 * Introduction to Programming in Java/Inheritance
 * Introduction to Programming in Java/Arrays
 * Introduction to Programming in Java/Swing

 * Programming: Java Learning
 * Learning Java
 * Introduction to Java
 * Basic Java Language
 * Java Decision Structures
 * Java Objects and Classes
 * Java error handling
 * Important Java classes
 * Java programming tips
 * Java/Fundamental exercises
 * Java error handling II
 * Java/Inheritance
 * Java/Classes II
 * Swing and AWT
 * Java applets
 * Java File IO
 * Java/Security
 * Java/Networking
 * Java/Intermediate Exercises
 * Java/Multithreading
 * Java/Networking II
 * Java/Advanced Swing

 * Programming: C++
 * C++
 * Introduction to C++
 * Variables and User Input
 * C++/Simple Math
 * Conditional Statements
 * Loops (C++)
 * Functions and Recursion
 * More Functions
 * C++ pointers
 * Classes and Inheritance
 * Templates in C++, part 1
 * Templates in C++, part 2
 * The STL
 * C++/STL Algorithms

 * Programming: HTML
 * HTML
 * Introduction to HTML
 * WYSIWYG vs. Manual Coding in HTML
 * Web Design/What is HTML
 * Basic formatting in HTML
 * Fonts and colours in HTML
 * Links in HTML
 * Graphical Content and Multimedia
 * Lists in HTML
 * Tables in HTML
 * Frames in HTML
 * Forms in HTML
 * Web Design/HTML Challenges

 * DATABASE
 * Database: MySQL
 * Database management systems
 * Introduction to Databases
 * Introduction to SQL
 * Introduction to PHP

 * NETWORKING
 * OPERATING SYSTEM
 * Development Environment Setup
 * Topic:Operating systems

 * Concepts
 * Development Environment Setup
 * School:Computer science

 * COMPUTER PROGRAMMING
 * Topic:Computer programming

 * Computing Fundamentals
 * Topic:Computing fundamentals

 * Introduction to Computer Science
 * Introduction to Computer Science
 * History of computing
 * Basics of computer architecture
 * Personal computers
 * What is an Algorithm
 * How to go about implementing a problem solution as a program
 * Development environment and language principles
 * Types and variables
 * Operators and expressions
 * Control structures
 * Procedures and functions
 * Arrays, strings and records
 * Basic I/O

 * Introduction to Programming
 * Introduction to Programming
 * Introduction to Programming/About Programming
 * Introduction to Programming/Programming Languages
 * Introduction to Programming/Organization part 1
 * Introduction to Programming/Variables
 * Introduction to Programming/Variables part 2
 * Introduction to Programming/Variables part 3
 * Introduction to Programming/Control structures
 * Introduction to Programming/Subprograms
 * Introduction to Programming/Scope

 * Introduction to Programming Logic
 * Introduction to Programming Logic
 * Introduction to Programming Logic/Key Terms
 * Introduction to Programming Logic/Variables
 * Introduction to Programming Logic/Lesson 2: Conditionals and Logic
 * Introduction to Programming Logic/Looping
 * Introduction to Programming Logic/Lesson 4: Code Organization Techniques

 * Hungarian Notation
 * CamelCase

 * Procedure Oriented Programming
 * Programming: BASIC
 * Object-Oriented Programming
 * Programming: Python
 * Topic:Python
 * Introduction to and setup of Python
 * Basic Python data types
 * Python operators
 * Basic Python functions
 * Truth and flow control in Python
 * Object-Oriented Python
 * Python libraries

 * Programming: Ruby
 * Topic:Ruby
 * Introduction to Ruby

 * Programming: Java
 * Introduction to Programming in Java
 * Introduction to Programming in Java/Introduction to Java
 * Introduction to Programming in Java/Integer variables
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 * Programming: Java Learning
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 * Programming: C++
 * C++
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 * Programming: HTML
 * HTML
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 * Web Design/What is HTML
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 * Lists in HTML
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 * Web Design/HTML Challenges

 * DATABASE
 * Database: MySQL
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 * OPERATING SYSTEM
 * NETWORKING
 * Topic:Operating systems


text:
User:Markclaysius/Books/Computer Science Course Study Book

Computer Science

 * School:Computer science

 * COMPUTER PROGRAMMING
 * Topic:Computer programming

 * Computing Fundamentals
 * Topic:Computing fundamentals

 * Introduction to Computer Science
 * Introduction to Computer Science
 * History of computing
 * Basics of computer architecture
 * Personal computers
 * What is an Algorithm
 * How to go about implementing a problem solution as a program
 * Development environment and language principles
 * Types and variables
 * Operators and expressions
 * Control structures
 * Procedures and functions
 * Arrays, strings and records
 * Basic I/O

 * Introduction to Programming
 * Introduction to Programming
 * Introduction to Programming/About Programming
 * Introduction to Programming/Programming Languages
 * Introduction to Programming/Organization part 1
 * Introduction to Programming/Variables
 * Introduction to Programming/Variables part 2
 * Introduction to Programming/Variables part 3
 * Introduction to Programming/Control structures
 * Introduction to Programming/Subprograms
 * Introduction to Programming/Scope

 * Introduction to Programming Logic
 * Introduction to Programming Logic
 * Introduction to Programming Logic/Key Terms
 * Introduction to Programming Logic/Variables
 * Introduction to Programming Logic/Lesson 2: Conditionals and Logic
 * Introduction to Programming Logic/Looping
 * Introduction to Programming Logic/Lesson 4: Code Organization Techniques

 * Hungarian Notation
 * CamelCase

 * Procedure Oriented Programming
 * Programming: BASIC
 * Object-Oriented Programming
 * Programming: Python
 * Topic:Python
 * Introduction to and setup of Python
 * Basic Python data types
 * Python operators
 * Basic Python functions
 * Truth and flow control in Python
 * Object-Oriented Python
 * Python libraries

 * Programming: Ruby
 * Topic:Ruby
 * Introduction to Ruby

 * Programming: Java
 * Introduction to Programming in Java
 * Introduction to Programming in Java/Introduction to Java
 * Introduction to Programming in Java/Integer variables
 * Introduction to Programming in Java/Boolean Variables
 * Introduction to Programming in Java/Strings
 * Introduction to Programming in Java/Java Classes
 * Introduction to Programming in Java/Inheritance
 * Introduction to Programming in Java/Arrays
 * Introduction to Programming in Java/Swing

 * Programming: Java Learning
 * Learning Java
 * Introduction to Java
 * Basic Java Language
 * Java Decision Structures
 * Java Objects and Classes
 * Java error handling
 * Important Java classes
 * Java programming tips
 * Java/Fundamental exercises
 * Java error handling II
 * Java/Inheritance
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 * Swing and AWT
 * Java applets
 * Java File IO
 * Java/Security
 * Java/Networking
 * Java/Intermediate Exercises
 * Java/Multithreading
 * Java/Networking II
 * Java/Advanced Swing

 * Programming: C++
 * C++
 * Introduction to C++
 * Variables and User Input
 * C++/Simple Math
 * Conditional Statements
 * Loops (C++)
 * Functions and Recursion
 * More Functions
 * C++ pointers
 * Classes and Inheritance
 * Templates in C++, part 1
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 * The STL
 * C++/STL Algorithms

 * Programming: HTML
 * HTML
 * Introduction to HTML
 * WYSIWYG vs. Manual Coding in HTML
 * Web Design/What is HTML
 * Basic formatting in HTML
 * Fonts and colours in HTML
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 * Graphical Content and Multimedia
 * Lists in HTML
 * Tables in HTML
 * Frames in HTML
 * Forms in HTML
 * Web Design/HTML Challenges

 * DATABASE
 * Database: MySQL
 * Database management systems
 * Introduction to Databases
 * Introduction to SQL
 * Introduction to PHP

 * OPERATING SYSTEM
 * Topic:Operating systems

 * Development Environment Setup
 * Operating Systems/Development Environment

 * Concepts
 * Operating Systems/Concepts
 * Operating Systems/Kernel Models
 * Operating Systems/Process Management
 * Operating Systems/Process and Thread
 * Operating Systems/Concurrency
 * Operating Systems/Mutual Exclusion
 * Operating Systems/Deadlocks
 * Operating Systems/Memory Management
 * Operating Systems/Scheduling

 * Implementation
 * Operating Systems/x86
 * Operating Systems/PPC
 * Operating Systems/Booting
 * Operating Systems/basic x86 bootsector

 * Existing Operating Systems
 * Operating Systems/Unix
 * Operating Systems/Linux
 * Operating Systems/OS X
 * Operating Systems/Windows

 * Legacy Operating Systems
 * NETWORKING
 * Introduction to Networking
 * Topic:TCP/IP Fundamentals
 * Topic:Application Layer
 * Topic:Transport Layer
 * Topic:Internet Layer
 * Topic:Link Layer
 * Topic:Local Area Networks
 * Topic:Mobile Networks
 * Topic:Network security
 * Topic:Network administration
 * Topic:Pursuing network certification


text:
Talk:Complex socio-ecological systems/Critique of Resilience

Critique of Social-Ecological Resilience.

Comments.

Both papers bring me again the idea of how wrong I am in bring my focus to resilience, and not to the framework of complex systems. Resilience per se revels one of the characteristics of the system, capacity to resist a changes. But complex systems theory brings us a whole new world of opportunities to study how to survive the ruptures/ thresholds/ trends in the actual system. Davidson paper shows it well decoupling ecological from social analytical perspectives. For her point of view the conception of resilience, and complex system also, need to be reinterpreted over social perspective, mainly because human have important characteristics that differentiate from others ecological structures (power, individual behaviors, agencies). As consequence, the idea of use the same ecological framework for social is “nagging”. I like the contraction example of the use of characteristic of system complexity to explain resilience in both study fields. But I also like her proposition that complex system theory could, and should be applied to social science. Just to cite one in my comments, I like the idea of look at how the hierarchical cycles of panarchy works to social systems. The feedback effects of fast and small hierarchical system over the slow and large cycle. I would like to use this framework in my governance study of climate change. Ricmel


text:
User talk:Ashong200010

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User talk:Heismark~enwikiversity

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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.

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title: User talk:Heismark~enwikiversity


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User talk:Valekvespa

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text:
Wikipedia/WikiBasics

We begin WikipediaAcademy by acquainting you first as a user and later as an editor.

The url for English Wikipedia is

http:\\en.wikipedia.org\Main_Page.htm

which takes you to the Main Page of the English Wikipedia.

BOX The prefix "en" in the url is the <code> for English language. By replacing "en" with the code for another language, you can reach the Main Page of the wikipedia in that language, if it exists. For example, replacing "mr" in place of "en" will take you to the Main Page of Marathi Wikipedia.

There are wikipedias in 278 languages and wikipedias exist for all major languages so chances are that you would find a wikipedia in the language you desire. /BOX

Let us acquaint ourselves with the Main Page.

At the top left corner is the Wikipedia logo. Clicking on it will return you to the Main Page anytime. In line with the logo is the welcome to Wikipedia Banner.

Below the logo is the dashboard with links which forms the left border column of the web page. The logo, banner and dashboard with its links are to be found on all pages of Wikipedia.

In the center is the content area. In the Main Page, the Content area has four panels. the top left panel display's Today's featured article. Featured articles are articles of the highest quality that can be found on Wikipedia. Each day, a blurb from a Featured Article is displayed in this panel with a link for //read more//.

The title of the article can be found

title: Wikipedia/WikiBasics


text:
User talk:AshLin

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text:
Complex analytic function

A complex function is a function that takes a complex argument and yields a complex result. The simplest examples of such functions are the common real functions that can be defined by a power series. Just use the same power series, while allowing the function argument to be complex.

If a complex function is differentiable as a complex function (on an open subset of $$\mathbb{C}$$), then it is said to be a complex analytic function or a holomorphic function (on that open subset of $$\mathbb{C}$$).

Differentiability as a complex function is defined (in the usual way) as a limit at a point:
 * $$f'(x) = \lim_{x\to X}\frac{f(x)-f(X)}{x-X}\,$$

but the epsilon-delta definition of the limit has to be interpreted very carefully. When we say $$f'(x) = D\,$$, that is,
 * $$D = \lim_{x\to X}\frac{f(x)-f(X)}{x-X}\,$$

we mean that, for every real $$\varepsilon > 0$$, there is a real $$\delta > 0$$ (yes, $$\varepsilon$$ and $$\delta$$ are real) such that
 * whenever $$x$$ is any complex number with $$0 < |x-X| < \delta,\,$$
 * $$\left|\frac{f(x)-f(X)}{x-X} - D\right| < \varepsilon\,$$

This looks just like the usual "real" definition, except that the absolute values are taken on complex numbers. (The absolute value of a complex number is just its modulus, that is, its distance from the origin. It is a real number.)

Now $$0 < |x-X| < \delta,\,$$ means that $$x$$ can be any complex number within the circle of radius $$\delta$$, centered on $$X$$. This means that, when taking limits of complex numbers,
 * $$\lim_{x\to X}\,$$

$$x$$ can approach $$X$$ from any direction in the complex plane, not just from the left or right as with real limits.

This extra requirement on the limit that defines a complex derivative places strong constraints on differentiability. It is what makes these functions so powerful. A differentiable complex function is also called a complex analytic function or a holomorphic function. Among the remarkable properties of such functions are:
 * If it is differentiable, its derivative is also differentiable, so it is differentiable any number of times.
 * At any point in an open region in which a function is differentiable, its power series (Taylor series) about that point converges to the function.
 * This is actually the more general "official" definition of "analytic"—that the power series converges to the function. Functions on the real numbers can have this convergence property too—they are called "real analytic functions".  But there are real functions that are differentiable but not real-analytic, such as
 * $$e^{-1/x^2}\,$$
 * Complex functions don't have this problem. If it is complex differentiable, it is complex analytic.

 * The real and imaginary parts of a complex analytic function obey the Cauchy-Riemann Equations. If we represent the complex function $$f(z)\,$$ by real-valued functions $$u$$ and $$v$$: $$f(z) = u + i v\,$$, and furthermore represent those as functions of two real arguments which are the real and imaginary parts of $$z$$: $$f(z) = u(x, y) + i v(x, y)\,$$, we have:
 * $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$   and    $$\frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$$

 * If a complex analytic function doesn't go to infinity anywhere in the complex plane, even at infinity, that function is constant. This remarkable result is Liouville's theorem.
 * When a function goes to infinity at some point, that point is called a singularity. Singularities are often of a special type called a pole.
 * The real function
 * $$\frac{1}{x^2+1}\,$$
 * is not constant, and doesn't go to infinity for any real arguments. But it does go to infinity (it has poles) at $$\pm i$$.

 * A complex analytic function is completely determined by its values on any line segment anywhere on the complex plane. So, for example, if we know that a function matches the exponential function just on the real line, we know its value everywhere.  That function is the "complex exponential".  Similar extensions can be made for other real-analytic functions.  That is, there are natural definitions for the complex sine, arctangent, logarithm, etc.
 * The power series for a complex analytic function converges (to the function itself, of course) on the interior of a circle around the point at which it is evaluated. That circle is the circle of convergence, with the radius of convergence.  Outside of that circle, the series diverges.  That doesn't mean the function isn't defined beyond that circle, just that that particular series fails.  One can usually expand a power series around some other point to get the value of the function elsewhere.  This is called analytic continuation.
 * The circle of convergence for a power series expansion just touches the nearest singularity. That is, the circle is the largest circle that does not contain any singularities.
 * This gives us insight into something about the real power series for the real arctangent function. A little calculation will show that, along the imaginary axis, $$\tan^{-1}(iz) = i \tanh^{-1}(z)\,$$.
 * But the $$tanh^{-1}\,$$ function has singularities at $$\pm 1\,$$, because $$\tanh(\infty) = 1\,$$ and $$\tanh(-\infty) = -1\,$$.
 * Therefore, the $$tan^{-1}\,$$ function has singularities at $$\pm i\,$$, so the radius of convergence of its power series around the origin is 1. Even restricting one's attention to real numbers, the power series
 * $$\tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} ...\,$$
 * diverges beyond x=1.

 * Revisiting the complex generalization of the exponential function, its power series is:
 * $$\exp(z) = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \frac{z^4}{4!} + \frac{z^5}{5!} ...\,$$

 * Setting $$z = i \theta\,$$, we have
 * $$\exp(i\theta) = 1 + i \theta + i^2\ \frac{\theta^2}{2!} + i^3\ \frac{\theta^3}{3!} + i^4\ \frac{\theta^4}{4!} + i^5\ \frac{\theta^5}{5!} ...\,$$

 * $$= \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} ... \right) + i \left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!}\right) ...\,$$
 * $$= \cos\theta + i \sin\theta\,$$  from the known power series for the sine and cosine functions.  So
 * $$ e^{i\theta} = \cos\theta + i \sin\theta\,$$

 * and Euler's famous formula, so mystifying when first seen:
 * $$ e^{i\pi} = -1\,$$

title: Complex analytic function


text:
User:EGM6341.s11.TEAM1.Sujin.HW6

Problem 6: Reproduce Kessler's Results using $$ (p_{2i}, p_{2i+1}) \ $$, where $$ i=1,2,3 \ $$. Start with the best of S10 to understand Kessler's code and reproduce its result
 Referenced from S10,Egm6341 Team1 HW5 Problem#10 to understand Kessler's Matlab code:|S10,Team1(HW5), but revised it to obtain correct P's and C's with s11 values

Given
 Kessler's Matlab code:

Reproduce Kessler's results
From eqn.6.1, 6.2 and 6.3, we can have following table;

By using the Kessler's Matlab code, n=3, following is obtained;

In problem.5, we obtained $$C_3, C_5 \ $$ and $$ C_7 \ $$ using eqn.6.1, 6.2, and 6.3;

And also, we can compute $$ p_2(1) \ $$ through $$ p_7(1) \ $$ using equations in the first above table, and from lecture note [[media:Nm1.s11.mtg32.djvu|(3) 32-1]], we know that $$ p_{2i+1}(1)=0 \ $$;

Let's compare these c's and p(1)'s with that from Kessler's code in the above second and third table.

$$\Rightarrow \;$$ c's are totally same with each result, but p(1)'s are different. So, it is needed to revise Kessler's code in order to generate correct p(1)'s. To be more specific, function [nsum, dsum]=fracsum(n,d) works correctly for updating cn and cd variables with proper values, but has flaws to generate correct coefficients, p(1)'s, which is considered as a sum of input vectors in this function.

Followings are (1) new function, [nsum, dsum]=re_fracsum(n,d) for newpn and newpd variables and (2) partially changed function, [c,p]=traperror(n).;

Finally, correct coefficients, p(1)'s are obtained from changed Kessler's code as following; , $$\begin{align}\;p_1(1)=p_3(1)=p_5(1)=p_7(1)=0\end{align}$$

Show that $$ z_{i+\frac{1}{2}} = z \left ( s=\frac{1}{2} \right ) = \frac{1}{2}(z_i+z_{i+1})+\frac{h}{8}(f_i-f_{i+1}) $$
From lecture note [[media:Nm1.s11.mtg37.djvu|p.37-3]], we know coefficients, $$\displaystyle\begin{align}c_i's\end{align}$$.

From eqn.6.4, each $$\displaystyle C_i $$ term can be replaced with $$\displaystyle Z $$ term.

From lecture note, [[media:Nm1.s11.mtg37.djvu|(6) p.37-1]] and [[media:Nm1.s11.mtg38.djvu|(4) p.38-1]], we know $$\displaystyle\dot{Z_i}=Z_i'\frac{1}{h}=f_i$$.

Thus,

Show that $$ Z'_{i+1/2}=Z'(s=1/2)=-\frac{3}{2}(Z_i-Z_{i+1})-\frac{1}{4}(Z'_i+Z'_{i+1}) $$
From lecture note [[media:Nm1.s11.mtg37.djvu|(1) p.37-2]], we know that

Using eqn.6.4, matrix, each $$\displaystyle C $$ can be represented in $$\displaystyle Z $$ term.

Thus,

Problem 9: Prove matrix $$ \underline{A} \ $$ multiplied with matrix $$ \left \{ c_j \right \} \ $$ gives the matrix $$ \begin{Bmatrix} z_i \\ z_i' \\ z_{i+1} \\ z_{i+1}' \end{Bmatrix} \ $$
 Solved without assistance

Verify Matrix $$ A^{-1} $$
In lecture note [[media:Nm1.s11.mtg37.djvu|p.37-2]], A matrix is defined as following;

By using matlab command 'inv', we can easily obtain inverse matrix

Matlabcode:

Identify basis functions $$ \overline{N} \ _i(s) $$ for $$ i=1,2,3,4 $$
In lecture note [[media:Nm1.s11.mtg38.djvu|(2) p.38-1]] and [[media:Nm1.s11.mtg37.djvu|(1) p.37-1]], $$\displaystyle Z(s)$$ is defined as following ; for i=1, 2, 3, and 5 and from lecture note [[media:Nm1.s11.mtg37.djvu|(2) p.37-3]], we know

From eqn.6.4 matrix, we have following equations;

Now, we can expand eqn.9.2, and find $$ \overline{N}_i,$$, for i=1,2,3,4, in terms of s

Thus,

Plot $$\displaystyle N_i(s) $$ for i=1,2,3,4

Matlab Code:

title: User:EGM6341.s11.TEAM1.Sujin.HW6


text:
User talk:Bronsonprofindo

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title: User talk:Bronsonprofindo


text: File:Tm7 hw6 9 1.jpg

title: File:Tm7 hw6 9 1.jpg


text: File:Tm7 hw6 9 2.jpg

title: File:Tm7 hw6 9 2.jpg


text: File:Tm7 hw6 9 3.jpg

title: File:Tm7 hw6 9 3.jpg


text:
User:Egm6341.s11.team4.hylon/hw6

=Problem 6.2 Derive Expression for Higher Order Trapezoidal Rule Error=

Problem Statement
Refer to lecture notes [[media:nm1.s11.mtg31.djvu|31-3]] for more detailed description.

Objective
Derive
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & E_{n}^{T}=I-{{T}_{0}}(n) \\ & =\sum\limits_{r=1}^{l}{{{h}^{2r}}{{{\bar{d}}}_{2r}}[{{f}^{2r-1}}(b)-{{f}^{2r-1}}(a)]-{{(\frac{h}{2})}^{2r}}\sum\limits_{k=0}^{n-1}{\int\limits_^{{{P}_{2l}}({{t}_{k}}(x)){{f}^{(2l)}}(x)dx}}} \\ \end{align}

$$ And :{| style="width:100%" border="0" $$\displaystyle
 * }
 * style="width:95%" |
 * style="width:95%" |

{{\bar{d}}_{2r}}=\frac{{{P}_{2r}}(1)}

$$
 * }.

Solution
In steps 2ab, we already got
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & E=[{{P}_{2}}(t){{g}^{(1)}}(t)]_{-1}^{+1}-\int\limits_{-1}^{+1}{{{P}_{2}}(t){{g}^{(2)}}(t)dt} \\ & {{P}_{2}}(t)=-\frac{2!}+\frac{1}{6} \\ & {{P}_{3}}(t)=-\frac{3!}+\frac{1}{6}t \\ \end{align}

$$ Substitute $$ g(t)=f(x(t))$$, $${{g}^{(i)}}(t)={{(\frac{h}{2})}^{i}}{{f}^{(i)}}(x(t))$$, $$x=\frac{h}{2}t+\frac{{{x}_{k}}+{{x}_{k+1}}}{2} $$ and $$dx=\frac{h}{2}dt $$ into above equation, we have
 * }
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & E=[{{P}_{2}}(t){{g}^{(1)}}(t)]_{-1}^{+1}-\underset{-1}{\overset{+1}{\mathop \int }}\,{{P}_{2}}(t){{g}^{(2)}}(t)dt \\ & =[{{P}_{2}}(+1){{(\frac{h}{2})}^{1}}{{f}^{(1)}}({{x}_{k+1}})-{{P}_{2}}(-1){{(\frac{h}{2})}^{1}}{{f}^{(1)}}({{x}_{k}})]-\underset{\overset{\mathop \int }}\,{{P}_{2}}(\frac{x-{{x}_{k+1}}+x-{{x}_{k}}}{h}){{(\frac{h}{2})}^{2}}{{f}^{(2)}}(x)\frac{2}{h}dx \\ & ={{(\frac{h}{2})}^{1}}{{P}_{2}}(1)[{{f}^{(1)}}({{x}_{k+1}})-{{f}^{(1)}}({{x}_{k}})]-{{(\frac{h}{2})}^{1}}\underset{\overset{\mathop \int }}\,{{P}_{2}}(\frac{x-{{x}_{k+1}}+x-{{x}_{k}}}{h}){{f}^{(2)}}(x)dx \\ \end{align}

$$ Hence,
 * }
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & E_{1}^{T}=\frac{h}{2}\sum\limits_{k=0}^{0}{E} \\ & ={{(\frac{h}{2})}^{2}}{{P}_{2}}(1)\sum\limits_{k=0}^{0}{[{{f}^{(1)}}({{x}_{k+1}})-{{f}^{(1)}}({{x}_{k}})]}-{{(\frac{h}{2})}^{2}}\sum\limits_{k=0}^{0}{\underset{\overset{\mathop \int }}\,{{P}_{2}}(\frac{x-{{x}_{k+1}}+x-{{x}_{k}}}{h}){{f}^{(2)}}(x)dx} \\ & ={{(\frac{h}{2})}^{2}}{{P}_{2}}(1)[{{f}^{(1)}}(b)-{{f}^{(1)}}(a)]-{{(\frac{h}{2})}^{2}}\sum\limits_{k=0}^{0}{\underset{\overset{\mathop \int }}\,{{P}_{2}}(\frac{x-{{x}_{k+1}}+x-{{x}_{k}}}{h}){{f}^{(2)}}(x)dx} \\ \end{align}

$$ And in step 3ab, we also obtained
 * }
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & E=[{{P}_{2}}(t){{g}^{(1)}}(t)+{{P}_{4}}(t){{g}^{(3)}}(t)]_{-1}^{+1}-\int\limits_{-1}^{+1}{{{P}_{4}}(t){{g}^{(4)}}(t)dt} \\ & {{P}_{4}}(t)=-\frac{4!}+\frac{1}{6}\frac{2!}-\frac{7}{360} \\ & {{P}_{5}}(t)=-\frac{5!}+\frac{1}{6}\frac{3!}-\frac{7}{360}t \\ \end{align}

$$ Using the same technique for step2ab, we have
 * }
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & E=[{{P}_{2}}(t){{g}^{(1)}}(t)+{{P}_{4}}(t){{g}^{(3)}}(t)]_{-1}^{+1}-\underset{-1}{\overset{+1}{\mathop \int }}\,{{P}_{4}}(t){{g}^{(4)}}(t)dt \\ & ={{(\frac{h}{2})}^{1}}{{P}_{2}}(1)[{{f}^{(1)}}({{x}_{k+1}})-{{f}^{(1)}}({{x}_{k}})]+{{(\frac{h}{2})}^{3}}{{P}_{4}}(1)[{{f}^{(3)}}({{x}_{k+1}})-{{f}^{(3)}}({{x}_{k}})]-\underset{\overset{\mathop \int }}\,{{P}_{4}}(\frac{x-{{x}_{k+1}}+x-{{x}_{k}}}{h}){{(\frac{h}{2})}^{4}}{{f}^{(4)}}(x)\frac{2}{h}dx \\ & ={{(\frac{h}{2})}^{1}}{{P}_{2}}(1)[{{f}^{(1)}}({{x}_{k+1}})-{{f}^{(1)}}({{x}_{k}})]+{{(\frac{h}{2})}^{3}}{{P}_{4}}(1)[{{f}^{(3)}}({{x}_{k+1}})-{{f}^{(3)}}({{x}_{k}})]-{{(\frac{h}{2})}^{3}}\underset{\overset{\mathop \int }}\,{{P}_{4}}(\frac{x-{{x}_{k+1}}+x-{{x}_{k}}}{h}){{f}^{(4)}}(x)dx \\ \end{align}

$$ Hence,
 * }
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & E_{2}^{T}=\frac{h}{2}\sum\limits_{k=0}^{1}{E} \\ & ={{(\frac{h}{2})}^{2}}{{P}_{2}}(1)\sum\limits_{k=0}^{1}{[{{f}^{(1)}}({{x}_{k+1}})-{{f}^{(1)}}({{x}_{k}})]}+{{(\frac{h}{2})}^{4}}{{P}_{4}}(1)\sum\limits_{k=0}^{1}{[{{f}^{(3)}}({{x}_{k+1}})-{{f}^{(3)}}({{x}_{k}})]}-{{(\frac{h}{2})}^{4}}\sum\limits_{k=0}^{1}{\underset{\overset{\mathop \int }}\,{{P}_{4}}(\frac{x-{{x}_{k+1}}+x-{{x}_{k}}}{h}){{f}^{(4)}}(x)dx} \\ & ={{(\frac{h}{2})}^{2}}{{P}_{2}}(1)[{{f}^{(1)}}(b)-{{f}^{(1)}}(a)]+{{(\frac{h}{2})}^{4}}{{P}_{4}}(1)[{{f}^{(3)}}(b)-{{f}^{(3)}}(a)]-{{(\frac{h}{2})}^{4}}\sum\limits_{k=0}^{1}{\underset{\overset{\mathop \int }}\,{{P}_{4}}(\frac{x-{{x}_{k+1}}+x-{{x}_{k}}}{h}){{f}^{(4)}}(x)dx} \\ & =\sum\limits_{r=1}^{2}{{{(\frac{h}{2})}^{2r}}{{P}_{2r}}(1)[{{f}^{(2r-1)}}(b)-{{f}^{(2r-1)}}(a)]-{{(\frac{h}{2})}^{4}}\sum\limits_{k=0}^{1}{\underset{\overset{\mathop \int }}\,{{P}_{4}}(\frac{x-{{x}_{k+1}}+x-{{x}_{k}}}{h}){{f}^{(4)}}(x)dx}} \\ \end{align}

$$ Using the same technique for step $$n$$ab, we can obtain
 * }
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:92%; padding:10px; border:2px solid #ff0000" |
 * style="width:92%; padding:10px; border:2px solid #ff0000" |

\begin{align} & E_{n}^{T}=\frac{h}{2}\sum\limits_{k=0}^{n-1}{E} \\ & =\sum\limits_{r=1}^{l}{{{(\frac{h}{2})}^{2r}}{{P}_{2r}}(1)[{{f}^{(2r-1)}}(b)-{{f}^{(2r-1)}}(a)]-{{(\frac{h}{2})}^{2l}}\sum\limits_{k=0}^{1}{\underset{\overset{\mathop \int }}\,{{P}_{2l}}(\frac{x-{{x}_{k+1}}+x-{{x}_{k}}}{h}){{f}^{(2l)}}(x)dx}} \\ \end{align}

$$ This is also the higher order trapezoidal rule error theorem.
 * }

=Problem 6.3 Verify Bernoulli Number and Comparison with HOTRE=

Problem Statement
Refer to lecture notes [[media:nm1.s11.mtg31.djvu|31-3]] for more detailed description.

Given
Bernoulli numbers:

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

$$\displaystyle \frac{x}{{{e}^{x}}-1}=\sum\limits_{r=0}^{\infty }{\underbrace{\frac{(2r)!}}_{-{{{\bar{d}}}_{2r}}}{{x}^{2r}}} $$
 * style="width:95%" |
 * style="width:95%" |
 * }

1).
Verify $${{\bar{d}}_{2}},{{\bar{d}}_{4}},{{\bar{d}}_{6}}$$

2).
Find $${{\bar{d}}_{8}},{{\bar{d}}_{10}}$$ using above given expression, compare with $${{\bar{d}}_{2r}}=\frac{{{P}_{2r}}(1)}$$, where $${{P}_{2r}}(1)$$ is the polynomial of HOTRE evaluated at $$x=1$$.

Solution
Using Taylor Series to expand $$\frac{1}{{{e}^{x}}-1}$$ at $${{x}_{0}}=0$$ we have
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\frac{1}{{{e}^{x}}-1}=\frac{1}{x}-\frac{1}{2}+\frac{x}{12}-\frac{720}+\frac{30240}-\frac{1209600}+\frac{4700160}-\frac{691{{x}^{11}}}{130767436800}+\cdots

$$
 * }

WolframAlpha

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

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & \frac{x}{{{e}^{x}}-1}=1-\frac{x}{2}+\frac{12}-\frac{720}+\frac{30240}-\frac{1209600}+\frac{4700160}-\frac{691{{x}^{12}}}{130767436800}+\cdots \\ & ={{B}_{0}}+{{B}_{1}}x+\frac{2!}{{x}^{2}}+\frac{4!}{{x}^{4}}+\frac{6!}{{x}^{6}}+\frac{8!}{{x}^{8}}+\frac{10!}{{x}^{10}}+\frac{12!}{{x}^{12}}+\cdots \\ \end{align}

$$ Comparing above equation, we obtain Bernoulli numbers:
 * }
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & {{B}_{2}}=\frac{1}{6} \\ & {{B}_{4}}=-\frac{1}{30} \\ & {{B}_{6}}=\frac{1}{42} \\ & {{B}_{8}}=-\frac{1}{30} \\ & {{B}_{10}}=\frac{5}{66} \\ & \vdots \\ \end{align}

$$ Hence we have :{| style="width:100%" border="0" $$\displaystyle
 * }
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & {{{\bar{d}}}_{2}}=-\frac{1}{12} \\ & {{{\bar{d}}}_{4}}=\frac{1}{720} \\ & {{{\bar{d}}}_{6}}=-\frac{1}{30240} \\ & {{{\bar{d}}}_{8}}=\frac{1}{1209600} \\ & {{{\bar{d}}}_{10}}=-\frac{1}{\text{47900160}} \\ \end{align}

$$ From higher order trapezoidal rule error theorem, we already know that
 * }
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:95%" |
 * style="width:95%" |

\begin{align} & {{P}_{2}}=-\frac{2!}+\frac{1}{6} \\ & {{P}_{4}}=-\frac{4!}+\frac{1}{6}\frac{2!}-\frac{7}{360} \\ & {{P}_{6}}=-\frac{6!}+\frac{1}{6}\frac{4!}-\frac{7}{360}\frac{2!}+\frac{31}{15120} \\ & {{P}_{8}}=-\frac{8!}+\frac{1}{6}\frac{6!}-\frac{7}{360}\frac{4!}+\frac{31}{15120}\frac{2!}-\frac{5}{23811} \\ & {{P}_{10}}=-\frac{10!}+\frac{1}{6}\frac{8!}-\frac{7}{360}\frac{6!}+\frac{31}{15120}\frac{4!}-\frac{5}{23811}\frac{2!}+\frac{11}{171629} \\ \end{align}

$$ So, we can get
 * }
 * {| style="width:100%" border="0"

$$\displaystyle
 * style="width:92%; padding:10px; border:2px solid #ff0000" |
 * style="width:92%; padding:10px; border:2px solid #ff0000" |

\begin{align} & {{{\bar{d}}}_{2}}=\frac{{{P}_{2}}(1)}=\frac{-\frac{1}{2!}+\frac{1}{6}}{4}=-\frac{1}{12} \\ & {{{\bar{d}}}_{4}}=\frac{{{P}_{4}}(1)}=\frac{-\frac{1}{4!}+\frac{1}{6}\frac{1}{2!}-\frac{7}{360}}{16}=\frac{1}{720} \\ & {{{\bar{d}}}_{6}}=\frac{{{P}_{6}}(1)}=\frac{-\frac{1}{6!}+\frac{1}{6}\frac{1}{4!}-\frac{7}{360}\frac{1}{2!}+\frac{31}{15120}}{64}=-\frac{1}{30240} \\ & {{{\bar{d}}}_{8}}=\frac{{{P}_{8}}(1)}=\frac{-\frac{1}{8!}+\frac{1}{6}\frac{1}{6!}-\frac{7}{360}\frac{1}{4!}+\frac{31}{15120}\frac{1}{2!}-\frac{5}{23811}}{256}=\frac{1}{1209600} \\ & {{{\bar{d}}}_{10}}=\frac{{{P}_{10}}(1)}=\frac{-\frac{1}{10!}+\frac{1}{6}\frac{1}{8!}-\frac{7}{360}\frac{1}{6!}+\frac{31}{15120}\frac{1}{4!}-\frac{5}{23811}\frac{1}{2!}+\frac{11}{171629}}{1024}=-\frac{1}{\text{47900160}} \\ \end{align}

$$
 * }

title: User:Egm6341.s11.team4.hylon/hw6


text:
User:Knagy~enwikiversity

Welcome to my userpage. I appreciate anyone correcting spelling or grammar errors. Please browse around (but don't vandalize) my userpage.

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User talk:Knagy~enwikiversity

Hello, and welcome to my talk page! Please leave your comments, suggestions, and opinions below. Also, please leave your username and date so I can reply or thank you. Enjoy my user page!

Welcome!
 Hello Knagy, 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. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! Jtneill - Talk - c 10:23, 18 April 2011 (UTC)

Mascot welcome
 Woof! My name is Jack Russell. I am a dog and a Wikiversity mascot. I am pretty new around Wikiversity. Perhaps we can learn together! Tail wag...

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Encyclopedic articles?
Hi Knagy, I'm wondering about your intentions for University of Kansas and Kansas Jayhawks? These pages appear to me to be copies of University of Kansas and Kansas Jayhawks. Are these pages intended as copies or for developing something different here at Wikiversity? Sincerely, James. -- Jtneill - Talk - c 10:23, 18 April 2011 (UTC)
 * Hi Knagy, Thanks for the reply on my talk page. A couple of things to consider:
 * If you are wanting to use Wikipedia article text, then it is better to import rather than copy (that way the editing history is retained).
 * I think it would help the Wikiversity community if you could explain what the intention of the pages is - if its just replication of Wikipedia, then this isn't necessary (e.g., just link instead to Kansas Jayhawks. From a quick google search it is apparent that the text of these articles already appears in many places - so as it stands there doesn't seem be any particular value-add for having this encyclopedic content on WV. Or is this perhaps an attempt at saturating search bots with KU content and not really about building learning material on Wikiversity? -- Jtneill - Talk - c 05:07, 19 April 2011 (UTC)

Hi Knagy, I think a lot of this discussion would be better on the University of Kansas talk page. As with Jtneill, I would like to be a bit clearer about what your intention is in developing this page. A link to the wikipedia article would be really useful, but as mentioned above, what learning material is being proposed. - please respond on the article talk pages, as this would facilitate a more inclusive discussion. By the way are you familiar with the Campus Ambassador programme. It would be really cool if there was a campus ambassador at the University of Kansas! Do you know whetehr this has happened?Leutha 21:19, 19 April 2011 (UTC)

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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.

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User:Tristan.lear

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baboon males often approach each other and gently grasp each other's penises. Primatologists refer to this behavior as "diddling." A tension reducing gesture, akin to the human handshake, it reciprocally assures that neither party has aggressive intentions. Ancient Romans would conclude a transaction by placing their testicles in each other's palm while mutually reciting an oath - the ultimate expression of trust. The words "testimony" and "testament" both derive from this ancient practice.

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text:
User:EGM6341.s11.TEAM1.Yoon.HW6

2
We know,

At $$\displaystyle t = \pm 1$$,

We also know,

And,

At $$\displaystyle t = \pm 1$$,

Functions $$\displaystyle p_{3}(t), p_{5}(t), p_{7}(t), \cdots, $$ are odd functions.

Since functions $$\displaystyle p_{2}(t), p_{4}(t), p_{6}(t), \cdots, $$ are made zero at $$\displaystyle t = \pm 1$$, Equation (3) becomes,

Since $$\displaystyle p_{(2r-1)}(t) $$ are odd functions, we know $$\displaystyle p_{(2r-1)}(1) + p_{(2r-1)}(-1) = 0 $$.

And so $$\displaystyle E $$ becomes,

Using the below relationship

In Equation (4) $$\displaystyle g_k(t)$$can be transferred to $$\displaystyle f(x_k)$$ as shown below:

And so, Equation (4) becomes,

Substituting Equation (5) in Equation (1),

In the above error equation $$\displaystyle \beta \neq f^{(2r-2)}(x_n) - f^{(2r-2)}(x_0)$$, but it contains terms that involve all the points from $$\displaystyle x_0 \ to\ x_n $$.

But instead of canceling out terms that involve odd order derivatives of $$\displaystyle g(t)$$, if we had canceled only terms that involve even order derivatives of $$\displaystyle g(t)$$ then the term $$\displaystyle \beta $$ would involve only first and last points, i.e., $$\displaystyle x_0 = a $$ and $$\displaystyle x_n = b $$, as we have got in Problem 5 above.

1
Part.a

Above result is exactly same with $$\displaystyle \bar{d_{2r}} = \frac{P_{2r}(1)}{2^{2r}}$$

Result table

Part.b Since $$\displaystyle \bar{d_{2r}} = \frac{P_{2r}(1)}{2^{2r}}$$,

Result table

Part.c

Then, based on the relation between $$\displaystyle \bar{d_{2r}} $$ and Bernoulli numbers, we can obtain $$\displaystyle \bar{d_{2}}, \,\bar{d_{4}}, \,\bar{d_{6}}$$ as follows.

Since $$\displaystyle B_{2}=\frac{1}{6}, \, B_{4}=-\frac{1}{30}, \, B_{6}=\frac{1}{42}, B_{8}=-\frac{1}{30}, \, B_{10}=\frac{5}{66}$$,

Result table

--EGM6341.s11.TEAM1.Yoon 17:01, 3 April 2011 (UTC)

Given
The results calculated from the $$ xcoth \,$$ method in Problem 6 are: $$\overline{d_{2}}=\frac{-1}{12}, \overline{d_{4}}=\frac{1}{720}, \overline{d_{6}}=\frac{-1}{30240}, \overline{d_{8}}=\frac{1}{1209600}, \overline{d_{10}}=\frac{-1}{47900160} \,$$ Using the $$ \frac{P_{2r}(1)}{2^{2r}}\,$$ method, remember that: $$\overline{d_{2}}=\frac{P_2^{(1)}}{2^2}\,$$ $$\overline{d_{4}}=\frac{P_4^{(1)}}{2^4}\,$$ $$\overline{d_{6}}=\frac{P_6^{(1)}}{2^6}\,$$ $$\overline{d_{8}}=\frac{P_8^{(1)}}{2^8}\,$$ $$\overline{d_{10}}=\frac{P_{10}^{(1)}}{2^{10}}\,$$

Solution
$$ P_2=C_1\frac{t^2}{2!}+C_3\,$$ $$ P_4=C_1\frac{t^4}{4!}+C_3\frac{t^2}{2!}+C_5\,$$ $$ P_6=C_1\frac{t^6}{6!}+C_3\frac{t^4}{4!}+C_5\frac{t^2}{2!}+C_7\,$$ $$ P_8=C_1\frac{t^8}{8!}+C_3\frac{t^6}{6!}+C_5\frac{t^4}{4!}+C_7\frac{t^2}{2!}+C_9\,$$ $$ P_{10}=C_1\frac{t^{10}}{10!}+C_3\frac{t^8}{8!}+C_5\frac{t^6}{6!}+C_7\frac{t^4}{4!}+C_9\frac{t^2}{2!}+C_{11}\,$$ where $$ C_1=-1, C_3=\frac{1}{6}, C_5=\frac{-7}{360}, C_7=\frac{31}{15120}, C_9=\frac{1}{4725}, C_{11}=\frac{73}{3421440} \,$$
 * Note: Coefficients $$ C_7, C_9, C_{11} \,$$ can be found using the same method as in Problem 1.

$$ P_2(1)=(-1)(\frac{1}{2})+\frac{1}{6}=\frac{-1}{3}\Rightarrow\overline{d_{2}}=(\frac{-1}{3})(\frac{1}{4})=\frac{-1}{12}\,$$ $$ P_4(1)=(-1)(\frac{1}{24})+(\frac{1}{6})(\frac{1}{2})-\frac{7}{360}=\frac{1}{45}\Rightarrow\overline{d_{4}}=(\frac{1}{45})(\frac{1}{16})=\frac{1}{720}\,$$ $$ P_6(1)=(-1)(\frac{1}{720})+(\frac{1}{6})(\frac{1}{24})+(\frac{-7}{360})(\frac{1}{2})+\frac{31}{15120}=\frac{-2}{945}\Rightarrow\overline{d_{6}}=(\frac{-2}{945})(\frac{1}{64})=\frac{-1}{30240}\,$$ $$ P_8(1)=(-1)(\frac{1}{40320})+(\frac{1}{6})(\frac{1}{720})+(\frac{-7}{360})(\frac{1}{24})+(\frac{31}{15120})(\frac{1}{2})-\frac{127}{604800}=\frac{1}{4725}\Rightarrow\overline{d_{8}}=(\frac{1}{4725})(\frac{1}{256})=\frac{1}{1209600}\,$$ $$ P_{10}(1)=(-1)(\frac{1}{3628800})+(\frac{1}{6})(\frac{1}{40320})+(\frac{-7}{360})(\frac{1}{720})+(\frac{31}{15120})(\frac{1}{24})-(\frac{127}{604800})(\frac{1}{2})+\frac{73}{3421440}=\frac{-2}{93555}\Rightarrow\overline{d_{8}}=(\frac{-2}{93555})(\frac{1}{1024})=\frac{-1}{47900160}\,$$

Solved by: -Egm6341.s10.team1.andrewdugan 08:17, 24 March 2010 (UTC)

title: User:EGM6341.s11.TEAM1.Yoon.HW6