text:
Compiler Construction/Glossary

Glossary
The glossary is intended to provide definitions of words or phrases which relate particularly to compiling, not to provide definitions of general computing jargon, for which a reference to Wikipedia may be more appropriate.

If the word or phrase you want is not here:
 * 1) Try looking it up on Wikipedia.
 * 2) Try searching the World Wide Web using Google or similar.

If you find the word or phrase, consider adding it to this glossary with explanation and/or references.

If you don't find the word or phrase, or can't understand what you did find, add the word or phrase itself to this glossary without an explanation, and leave it for someone else to explain.

When adding an item
When adding a word or phrase, please keep things more or less in alphabetical order to make it easier for others to find things.

The following guidelines will keep this glossary consistent (just follow the same style as some existing entry):
 * The key word or phrase should be:
 * preceded by a blank line,
 * in bold,
 * on a line by itself.
 * The actual definition/description should be indented.
 * Any cross-references to other items in this glossary should be in italic.
 * The alphabetical splits can be refined as required to try and keep the number of entries in each section below about 20.

Wikipedia

 * Addressing mode
 * Algol 60
 * Assembly/Assembler language
 * Tiny BASIC language
 * BASIC
 * BNF: Backus-Naur Form
 * Bootstrapping
 * C
 * C++
 * Chomsky
 * Chomsky hierarchy
 * Compilers
 * Computer architecture
 * CPU design
 * EBNF: Extended BNF
 * FORTRAN
 * High-level language
 * Low-level language
 * Interpreters
 * Pascal
 * PL/0 - Wirth's Toy/Example Compiler from A+DS=P
 * PL/1
 * Recursion
 * Side effect
 * Ada

A to I
checking for meaning
 * This is an alternative phrase for semantic analysis.

comment
 * Part of a source program which is intended for a human reader and which is ignored by the compiler. As a minimum, all variables and all routines should have explanatory comments.  A comment may attempt to explain nearby code, but problems ensue if the code is updated and the comment is not. Ideally, comments should tell WHAT code does, not HOW; the code should be self-explanatory.

compilation
 * The act of translation performed by a compiler.

compiler
 * A computer program that translates a computer program written in one computer language (called the source language) into an equivalent program written in another computer language (called the target language).

compile-time
 * The period during which a program is being compiled, as distinct from run-time when a program is actually running. Certain errors can be detected at compile-time (e.g. misspelt identifier) while other errors may not be detectable until run-time (e.g. division by 0).

 GNU
 * A recursive acronym "GNU's Not Unix". Independently written Unix lookalike, which can be combined with Linux kernel to produce GNU/Linux - a useful free operating system.

GPL
 * GNU General Public Licence. Much copyrighted free software, including GNU/Linux, is available under this licence. The general intention is to ensure that such software remains freely available for anyone to use and/or modify.

identifier
 * A name used in a program. Normally some (possibly unpronouncable) combination of letters and digits which starts with a letter.  Some programming languages may allow other characters such as underline '_' or hash '#'.  Identifiers may be defined by the programmer or they may be pre-defined by the language (e.g. 'sqrt' is often a pre-defined identifier for referring to a 'square-root' function).  Older programming languages used to limit identifiers to a maximum of 6 characters - the resulting abbreviations tended to make programs more difficult to understand.  Many programming languages allow identifiers of almost any length, though some may only take the first 32 characters into account.  There are two main styles of writing long identifiers: either  now_for_some_long_name  or  NowForSomeLongName.

implementation language
 * This is the programming language in which the compiler or interpreter is written. It might be the same as either the source language or the target language.

intermediate language
 * Some moderately low-level language used as an interface between the front-end and the back-end of a compiler.

interpreter
 * A computer program which examines a computer program written in some source language and carries out the actions required by that program more or less directly, without translating it into some other language. May well be slower than a compiled program, especially if there is a lot of calculation.

J to R
just-in-time compilation
 * A cross between a compiler and an interpreter. The source language is parsed in real-time and translated to machine language which is run immediately. The underlying machine runs the code whereas with an interpreter, it does the work itself.

keyword
 * Many programming languages reserve some identifiers as keywords for use when indicating the structure of a program, e.g. if is often used to indicate some conditional code. Languages such as Pascal/C/C++ have around 50 reserved keywords, Fortran doesn't have any, COBOL has around 300.

lexical analysis
 * This is an alternative name for scanning or tokenisation. The function of lexical analysis is to scan the source program (a sequence of characters arranged on lines) and convert it to a sequence of valid tokens.  Any comments are usually removed at this stage as well.  Syntax analysis is responsible for checking that it is a valid sequence.

machine language
 * This is the lowest level language. It consists of just binary digits. It was only ever used when computers were first invented to create the first compilers.

parsing
 * This is an alternative name for syntax analysis.

pass
 * The number of times that the source text has to be scanned or rescanned in order to compile the program. Due to limited main memory, some early compilers had a large number of passes (about 60).

pragma
 * This is sometimes referred to as a significant comment. The intention is that of passing directives to the compiler or the preprocessor at various points in the source program.  Such directives might indicate whether speed or space is more important, or if certain checks should be suppressed, or if a particular routine is to be timed, etc.

preprocessor
 * A program that takes text and performs lexical conversions on it. The conversions may include macro substitution, conditional inclusion, and inclusion of other files. This can be used to write platform-independent code by excluding source files that are not necessary on certain platforms without changing the build instructions.

reserved word
 * This is an alternative phrase for keyword.

run-time
 * The period when a program is running/being executed/doing some hopefully useful work, as distinct from compile-time when the program is being translated.

S to Z
scanning
 * This is an alternative word for lexical analysis or tokenisation.

semantic analysis
 * The function of semantic analysis is to check that the source program is meaningful. Note that a program can have a valid meaning and still be incorrect if it doesn't do what was really intended.

source language
 * The language accepted as input by a compiler, and translated/compiled into a target language. It is normally a high-level language, written using some mixture of English words and mathematical notation.

syntax analysis
 * This is an alternative name for parsing. The function of syntax analysis is to check that the source program is grammatically correct, i.e. that we have a valid sequence of tokens.  Checking that the source program actually means something is the job of semantic analysis.

target language
 * The language in which the output of a compiler is written. It is normally a low-level language such as assembler, written with somewhat cryptic abbreviations for machine instructions, but may instead be machine code for some actual or virtual computer.

token
 * A fundamental symbol as processed by syntax analysis. A token may be an identifier 'Result', a reserved keyword if, a compound symbol '<=', or a single character '+'.

tokenisation
 * This is an alternative word for lexical analysis or scanning. It means 'conversion to tokens'.

title: Compiler Construction/Glossary


text:
Talk:Van Dwelling/Cooking

This shouldn't be a book in itself. It appears to be a subpage of the Van Dwelling wikibook. If anything, "Cooking" should probably redirect to the "Cookbook" wikibook.

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

title: Talk:Van Dwelling/Cooking


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Physiotherapy Assessment/Paediatric

= Paediatric assessment =

The important thing to remember when assessing a very young child is that they probably don't want to be there (especially if they're in hospital). In fact, most adults don't want to be there either but at least they're able to recognise that it's probably in their best interests to attend. A child attends because somebody made them.

Play is going to be an important aspect of your assessment (as well as your treatment). If you can distract them with balls, trampolines, toys and wedges you'll get a lot further than with instructions, commands and regimes. Children want to climb up onto things. They want to throw balls and hang from beams. If you can make their trip to you a little more fun and a little less scary, you'll end up not only getting a better response to your treatment, but you may find yourself enjoying your job!

Balls
(Keywords)Throwing, kicking, rolling, sitting, bouncing. Postural control, strengthening, co-ordination.

Wedges
(Keywords)Wedges are good for stretching muscles, placing limbs into normal positions and props for climbing over, under and around.

Trampolines
There is some controversy over the use of mini-trompolines (Rebound therapy) in rehabilitation programmes for children. Research in the US, UK and New Zealand has shown that the number of A&E admissions after trompoline-related injury is staggeringly high. Most conclude with a statement calling for a complete ban on the sale of recreational trampolines. However, they also found that most injuries came about as a result of unsupervised activity, which hopefully would never occur in a physio gym. The use of trampolines has (perhaps surprisingly) been shown to be effective in assisting with sputum clearance, among other, more well-documented effects.

Mats
(Keywords)Rolling, tumbling, falling safely, clean.

title: Physiotherapy Assessment/Paediatric


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Python Programming/Tuples

A tuple in Python is much like a list except that it is immutable (unchangeable) once created. A tuple of hashable objects is hashable and thus suitable as a key in a dictionary and as a member of a set.

Overview
Tuples in Python at a glance:

Tuple notation
Tuples may be created directly or converted from lists. Generally, tuples are enclosed in parentheses.

A one item tuple is created by an item in parentheses followed by a comma:

Also, tuples will be created from items separated by commas.

Packing and Unpacking
You can also perform multiple assignment using tuples.

Note that either, or both sides of an assignment operator can consist of tuples.

The example above: article, noun, verb, adjective, direct_object = t is called "tuple unpacking" because the tuple t was unpacked and its values assigned to each of the variables on the left. "Tuple packing" is the reverse: t=article, noun, verb, adjective, direct_object. When unpacking a tuple, or performing multiple assignment, you must have the same number of variables being assigned to as values being assigned.

Operations on tuples
These are the same as for lists except that we may not assign to indices or slices, and there is no "append" operator.

For lists we would have had:

Tuple Attributes
Length: Finding the length of a tuple is the same as with lists; use the built in len method.

Conversions
Convert list to tuples using the built in tuple method.

Converting a tuple into a list using the built in list method to cast as a list:

Dictionaries can also be converted to tuples of tuples using the items method of dictionaries:

Uses of Tuples
Tuples can be used in place of lists where the number of items is known and small, for example when returning multiple values from a function. Many other languages require creating an object or container to return, but with Python's tuple assignment, multiple-value returns are easy:

This resulting tuple can be easily unpacked with the tuple assignment technique explained above:

Using List Comprehension to process Tuple elements
Occasionally, there is a need to manipulate the values contained within a tuple in order to create a new tuple. For example, if we wanted a way to double all of the values within a tuple, we can combine some of the above information in addition to list comprehension like this:

Exercises

 * 1) Create the list ['a', 'b', 'c'], then create a tuple from that list.
 * 2) Create the tuple ('a', 'b', 'c'), then create a list from that tuple. (Hint: the material needed to do this has been covered, but it's not entirely obvious)
 * 3) Make the following instantiations simultaneously: a = 'a', b=2, c='gamma'. (That is, in one line of code).
 * 4) Create a tuple containing just a single element which in turn contains the three elements 'a', 'b', and 'c'. Verify that the length is actually 1 by using the len function.

title: Python Programming/Tuples


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Python Programming/Dictionaries

A dictionary in Python is a collection of unordered values accessed by key rather than by index. The keys have to be hashable: integers, floating point numbers, strings, tuples, and, frozensets are hashable. Lists, dictionaries, and sets other than frozensets are not hashable. Dictionaries were available as early as in Python 1.4.

Overview
Dictionaries in Python at a glance:

Dictionary notation
Dictionaries may be created directly or converted from sequences. Dictionaries are enclosed in curly braces, {}

Also, dictionaries can be easily created by zipping two sequences.

Operations on Dictionaries
The operations on dictionaries are somewhat unique. Slicing is not supported, since the items have no intrinsic order.

Combining two Dictionaries
You can combine two dictionaries by using the update method of the primary dictionary. Note that the update method will merge existing elements if they conflict.

Exercises
Write a program that:
 * 1) Asks the user for a string, then creates the following dictionary. The values are the letters in the string, with the corresponding key being the place in the string. https://docs.python.org/2/tutorial/datastructures.html#looping-techniques
 * 2) Replaces the entry whose key is the integer 3, with the value "Pie".
 * 3) Asks the user for a string of digits, then prints out the values corresponding to those digits.

title: Python Programming/Dictionaries


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Project Management/PMBOK/Integration Management

Integration management is a collection of processes required to ensure that the various elements of the projects are properly coordinated. It involves making trade-offs among competing objectives and alternatives to meet or exceed stakeholder needs and expectations.

Comprised of:

 * Project plan development
 * Integrating and coordinating all project plans to create a consistent, coherent document

 * Project plan execution
 * Carrying out the project plan, according to the strategy, plan and activities as per the plan

 * Integrated change control
 * Coordinating changes across the project

Project plan development

Inputs Tools and techniques Outputs Other planning inputs Project planning methodology Project plan Historical information Stakeholder skills and knowledge Supporting details Organizational policies Project management information system

Constraints Earned value management

Assumptions

What is a plan? A scheme, program, or method worked out beforehand for the accomplishment of an objective: a plan of attack. A proposed or tentative project or course of action: had no plans for the evening. A systematic arrangement of elements or important parts; a configuration or outline: a seating plan; the plan of a story. A drawing or diagram made to scale showing the structure or arrangement of something. In perspective rendering, one of several imaginary planes perpendicular to the line of vision between the viewer and the object being depicted. A program or policy stipulating a service or benefit: a pension plan. Planned, Planning, Plans To formulate a scheme or program for the accomplishment, enactment, or attainment of: plan a campaign. To have as a specific aim or purpose; intend: They plan to buy a house. To draw or make a graphic representation of. What are the main purposes of a project plan? Guide the execution Document assumptions Planning decisions – alternatives Facilitate communication Identify key management reviews Baseline for progress measurement and control

Top 10 Reasons Projects Fail Inadequately trained and /or inexperienced project managers Failure to set and manage expectations Poor leadership at any and all levels Failure to adequately identify, document and track requirements Poor plans and planning processes Poor effort estimation Cultural and ethical misalignment Misalignment between the project team and the business or other organizations it serves Inadequate or misused methods Inadequate communication, including progress tracking and reporting source - gantthead.com

In any project, the good things for the project do not happen just like that, where as bad things can. For a project to progress and get over on time, within budget and with required quality, the project steps need to be planned well in advance, and who will do, what, when should be decided well in advance. For any team to be successful, the following 14 points are very important;

Application of Q12 practices in project planning

If the clan can assure the meeting of the following objectives, then it can be termed as a good plan Bring clarity into the work of each team member (Do I know what is expected me?) Ensure material, equipment, skills, knowledge availability to the team, to do their work properly (Do I have the material and equipment I need to do my work right?) Ensure that, as far as possible, the work allocated is matching to the team members liking and skill set (Do I have the opportunity to do what I do best every day?) Ensure better communication within the team, so that good work gets the recognition it deserves (In the last 7 days, have I received recognition or praise for good work?) Does my supervisor, or someone at work, seem to care about me as a person? Ensure, while doing the project work, the team member develops as a professional (Is there someone at work who encourages my development?) Ensure, a culture of open communication and every body gets their turn to share (At work do my opinions seem to count?) Ensure that the road-map is clear to all (Does the charter of my project make me feel, my job is important?) Facilitate individual excellence along with team excellence (Are my co-workers, committed to doing quality work?) Create an environment of trust (Do I have a best friend at work?) Ensure that, everyone gets proper feedback on their performance and the teams performance (In the last six months, has someone talked with me about my progress?) Ensure that every one advances in their career (This last year, have I had opportunities at work to learn and grow?)

Contents of a project plan

Project charter Project management approach Scope statement WBS (to the level of control) Responsibility chart / assignments Network diagram / major milestones Budget Schedule Resources Change control plan / resources Performance measurement baselines Management plans (scope, schedule, cost, quality, staffing, communications, risk response, procurement Subsidiary management plans •Scope management plan •Schedule management plan •Cost management plan •Quality management plan •Staffing management plan •Communications management plan •Risk response plan •Procurement management plan Inputs to project plan development Other planning inputs Statement of work Scope statement Contract copy etc Historical information Cost / effort / quality related information of similar past projects Risk databases of similar past projects Legal aspects of similar past projects Organizational capability baselines Industry databases  etc.... Organizational policies Policies for project management HR policy Quality policy etc.... Constraints

Factors that limit the project team's options

Most of the projects are operated under the triple constraints of scope / schedule / cost. when the scope increases the schedule increases along with cost. Hence there is always a tradeoff between these constraints. Other examples of constraints can be; legal policies of the land in which the project is executed geographical conditions cultural issues etc..

Assumptions Some of the examples of assumptions are; It is assumed that the test team is available to test the product, when the development is completed It is assumed that, training will be provided to those who are new to the technology, within two weeks of the start of the project It is assumed that the schools are closed during winter, hence the buildings can be utilized by the project team

Baseline Original project plan with the approved changes. A baseline is required to track the progress of the project from the approved plan. Can there be multiple baselines? the answer is 'yes'. Project baselines may be changed by formally approved changes, but the evolution of the baseline should be documented.

Project plan execution Inputs Tools and techniques Outputs project plan general management skills work results supporting details product skills and knowledge change requests organizational policies work authorization system

preventive action status review meetings

corrective action project review meetings

project management information system

organizational procedures

What are the supporting details?

estimation worksheets skills database contract statement of work (SOW) review reports project communication

Examples of organizational policies? quality policy change management policy recruitment policy

What is the difference between preventive action and corrective action? Prevention - proactive action to prevent something from happening Correction - rectifying something, after the problem has happened Preventive action - actions focused at preventing something from happening. Example : training, peer reviews, quality planning (proactive in nature) Corrective action - actions focused at correcting something, which has already happened (reactive in nature)

What are the general management skills required for project plan execution? Management - planning, organizing, executing, controlling People management Time management Emotional intelligence Initiative Leadership Communication skills - written, oral Essential technology awareness to efficiently communicate Confidence Organizational political awareness Product management skills and project execution It is not possible to manage a project without the basic technical knowledge about the project's product This do not mean, extensive technical knowledge about the product as required by the design personnel Means, minimum technical knowledge required to participate in the project discussions and decisions in a meaningful way What is a work authorization system? Decide what to do when and by whom An official go-ahead or no-go for the work Definition of necessary approvals and empowerment levels

Integrated change control Inputs Tools and techniques Outputs project plan change control system project plan updates performance reports configuration management corrective action change requests performance measurement lessons learned

additional planning

project planning

project management information system

Integrating the changes that results from all the control processes Taking corrective action Working with the change control board Configuration management Performing as per the defined change control procedures / system Managing changes Ensuring that everything is going as per the change management plans Controlling within the baselines Refining control limits, wherever required Collection of data Project control meetings Negotiating Communicating Resolving conflict Working within policies Conducting post mortems and capturing lessons learned Ensuring implementation of lessons learned in the subsequent phases of the project Issuing updates to the project plan What are the most commonly used performance reports? earned value analysis reports milestone charts risk reports etc

What is a change request? A formal document containing; description of the change required initiated by approved by date etc

PROJECT CHANGE REQUEST FORM

Project Name:

Project. No:

Request No:

Date:

Requested by:

Telephone:

PROPOSED CHANGE Short Name:

Description: (Attachments as necessary) Change is: External to Scope [ ]

Internal to Scope [ ] Requestor Signature:

APPROVED FOR EVALUATION Date:

Priority:

Project Manager: [ ] Yes [ ] No Signature:

Sponsor: [ ] Yes [ ] No Signature:

EVALUATION OF PROPOSED CHANGE Work Packages Impacted:

Estimated Project Impact: Effort Days:

Cost:

Schedule:

Cost Benefit:

RECOMMENDED ACTION: Implement [ ] Defer [ ] Reject [ ] Recommended Date of Implementation:

Recommended by:

Date:

APPROVED Work Package Leader:

Date:

Project Manager:

Date:

Sponsor:

Date:

Assigned to:

Accepted:

ACTION COMPLETED: Date:

Approved by:

CHANGE LOG FORM

Project Name:

Project. No:

Request Number Date Summary Description Action I / D / R Target Date Resolve Date

What is a change control system? A typical change control system will have the following components; change request form change approval process capture of change implementation information change control boards change communication system What is configuration management? In projects different work products gets generated during the course of the project and these evolve as the project progresses. One of the critical challenges in projects (especially product development) is to keep track of the versions of the work products going into any release. This is accomplished by a configuration management system. Typical components of a configuration management system are; Version numbering system Repository for storing / pointing to the work products and their interdependencies Check in forms / system Check out forms / systems What are the triggers for project plan updation? It can be anything under the sun! Some of the major causes are listed below; Project work deviates drastically from planned work due to; Improper requirements / changing requirements Improper design Insufficient skill sets Insufficient scheduling knowledge Insufficient tracking etc Project scope changes Non availability of critical resources on time Organizational priority changes Project plan updates require approval from sponsor and is normally done immediately after a project review (It is not something a project manager can do as and when he/she feels so. In that case it will affect project tracking due to  moving dates and the team also can take the planned completion dates casually, eventually making the project a bleeding project. What  do you mean by lessons learned in project management? The meaning of lessons learned is 'lessons learned'  in project management also. Since projects are temporary endeavors, it brings with it additional challenges like; The same individual contributor repeating the same mistake in the next project also The issues got addressed in one project, continue to happen in another project due to lack of communication The lessons learned forgotten by the time the team moves on to the next phase of the project


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User talk:Abrachan

Abrachan Pudussery B.E, M.S  has around two decades of experience in software development, quality management and project management. He started his career as a computer programmer and subsequently played the roles of systems analyst, project manager, senior project manager, head – QA and manager – Process Engineering. He worked for organizations such as HCL, FEC Singapore, OTE Muscat, Think Business Networks and Novell. At present he works as the Managing Director of Prologic Systems (Private) Limited, Bangalore, India.

He is a PMI certified project manager (PMP) and PRINCE2 practitioner, with around ten years experience in project management involving large teams. He has successfully managed several domestic and international projects which gave him vast exposure to pre-sales support, account management, software  size estimation, proposal preparation, contract administration, project management, quality management and multi cultural teams. He played active role in one ISO 9000 and two CMM implementations. He is a trained trainer of the QES - quality education system by Philip.B.Crosy and has trained more than two hundred software professionals on Crosby's methodology for quality improvement and mentored them through their corrective / preventive action projects resulting in improved ROI for their organizations.

He is a trainer / speaker  / researcher of project management, quality management, open source and iterative software development models. Co-authored and delivered a tutorial on iterative software project  management, at the SPM-ICON 2004, the international conference on practical project management, held at Bangalore. He presented the Pragati project at the Bangalore-linux conference on 3rd December 2004 which got presented subsequently at Brainshare Europe and Brainshare US. He architected the PMPdistilled project management training program, which benefitted around 840 project managers worldwide.

He has just released his agile project management workshop doagile - appliedagile

More details of him are availble at his personal web site

title: User talk:Abrachan


text:
MATLAB Programming

Forewords
MATLAB is a mathematical software package which can be used for analysing and solving mathematical and engineering problems such as numerical computations, graphs plotting, programming and more. MATLAB treat all the variables in form of scalar, vector or matrices form. For this book of MATLAB Programming, it is aimed to give a new MALAB users have an overview of how to use the basic and intermediate level of MATLAB and of course to keep this free for students who might be using this book for their course. To start off, MATLAB is a scripting language (Note that I didn't mentioned it is programming language) for data analysis, think of it as a super-powered programmable calculator with useful default functions that are built specifically for math applications.

The name MATLAB itself is an abbreviation for "MATrix LABoratory". Originally, it was a simple interactive matrix calculator in early 1980's. Gradually, it evolves into programming language developed by MathWorks. It started out as a matrix programming language where linear algebra programming was simple. It can be run both under interactive sessions and as a batch job. While other programming languages mostly work with numbers one at a time, MATLAB is designed to operate primarily on whole matrices and arrays. All MATLAB variables are multidimensional arrays, no matter what type of data. A matrix is a two-dimensional array often used for linear algebra.

MATLAB are also designed to be equipped with toolboxes. A toolbox is a package of custom functions and/or classes which are generally for a specific topic (such as control system, signal processing or even deep learning).The toolboxes are either provided officially by Mathworks and require a expensive license, others can be downloaded for free from the File Exchange.

Finally, this book assumes that you have access to latest MATLAB version but if you didn't have, no worries. I believe that most of the examples are also can be run using Octave as the later tried to compatible matching the programming syntax of MATLAB (for most of time)

Click this link to try the free version of MATLAB: | Octave-Online.

Other Wikibooks
A number of other Wikibooks use MATLAB to teach their subjects. The following Wikibooks make use of MATLAB:

 * Signals and Systems
 * Control Systems

title: MATLAB Programming


text:
Cell Biology/Genes/Gene expression

Gene expression is the first stage of a process that decodes what the DNA holds in a cell. It is the expression of a gene that gives rise to a protein.

How does gene expression occur?

Genetic expression is a complex process. It is regulated by a series of mechanisms.

Gene expression begins with transcription of DNA, giving rise to messenger RNA (mRNA). This is performed by the enzyme RNA polymerase, which produces the mRNA. The mRNA in prokaryotes is coupled with several ribosomes which are responsible for translating proteins.

In eukaryotes, mRNA that is made from DNA is immature, and is called pre-mRNA. Pre-mRNA loses non-coding sections (called exons), maturing to mRNA. mRNA is coupled to ribosomes on Rough Endoplasmatic Reticle (RER) where translation happens. Translation is made when a new polypeptide is formed. The genetic code indeed says the order of pe polypeptides, but it doesn't give us a clue about it's tridimensional structure. Tridimensional structure is given by post-translational processes.

Translation occurs following transcription wherein the protein synthesis machinery gets into action and uses its tools to read out the message that the RNA holds.

There are some genes known to be without coding proteins. Yet, they work as regulation sequences in a cell. In this case, the sequences can enhance coding (called "enhancers") or they can inhibit (called "repressors"). When a protein is coupled with these genes, a substrate or hormone, they join together.

In multicellular organisms only certain cells will produce a certain type of protein; e.g.: Haemoglobin is encoded in every cell of a mammal organism (it includes humans), but only precursors of red blood cells are allowed to express it (red blood cells are not allowed to express it, because they lose their nucleus). However, the enhancers and repressors are present in every cell of a mammal.

Genetic Information

In nature, there is information found in all living cells. Different cultures have often studied this information and used various forms of recording techniques to display it. Ancient Egyptians, in particular, referred to this information and its records as "provider of attributes" and determined it ||| to mean several, and that was earlier in human history of recording something that was known about nature.

There were often other signs as well that accompanied Egyptian writings on the source of this "information key of life". Among them were double, water and wick of twisted flax. But the most central one, for modern science, of course, was the snake like determinative that meant a worm or serpent in the limit of life. This limit, water, was "N" meaning that something or someone is, the essence which would be referred to by the Greeks as "esse" or "ens", and in today's English terms, the "essence". In anthropology, the language of gene expression is rooted in the sources of knowledge that Odhiambo Siangla of Kenya has called "rieko" and Jeremy Narby of Switzeland has termed the "cosmic serpent". Both Siangla and Narby are not only experts in cultures but are trained in communication and expression. And from both the key has been the "three letter word". In the alphabet of the three letter word found in cell biology are the organic bases, which are adenine (A), guanine (G), cytosine (C) and thymine (T). It is the triplet recipe of these bases that make up the ‘dictionary’ we call in molecular biology genetic code.

The codal system enables the transmission of genetic information to be codified, which at the molecular level, is conveyed through genes.

What is gene ? A gene is a region of DNA that produces a functional RNA molecule. If a region of DNA is not functional, that region is not a transmissible form of information for protein synthesis. And because the information is not transmissible, it is not readily functional. There are various sizes of gene. The first recorded attempts to imagine the very small was the Horus Eye, which is also a pristine idea of limit. Today we talk about bases. The insulin gene, for example, has 1.7 x $$10^3$$, about 1700 nucleotides. There exists a receptor gene known as low-density lipoprotein (LDL). This protein has 4.5 x$$10^4$$ nucleotides. In terms of nucleotides this (LDL) approximates to 45,000 nucleotides. Now, with the dystrophic gene as another example, we find the nucleotides to be around 2.0 x$$10^8$$, approximately 200,000,000 nucleotides in number.

Now, the introns. It is the noncoding regions of DNA that are called introns meaning the “intervening sequences”. Introns make up a greater part of the nucleotide sequences of a gene. The coding regions are called exons to mean “expression sequences”. They constitute a minority of the nucleotide progression of a DNA and they instruct cellular workshops for the formation of proteins via amino acids.

Through proteins, the expression of genetic information is achieved. In particular are the enzymes. Even during the ancient time the enzymes were understood and utilized well. The enzymes catalyze the chemical reactions of anabolic kind, that is, the building of cellular food and those of catabolic type, the braking down of food. The two processes are collectively termed metabolism. What, further, can we add about proteins?

We can further say that proteins are concentration of heteropolymers manufactured from amino acids. There are 20 amino acids used in synthesizing natural proteins. It is clear that a protein may consist of many, in fact, several hundred amino acid sediments. It is essentially unlimited in number to speak about how many different proteins we can make from combinations of amino acids. Mathematics explains it well. There is therefore a diverse set of proteins whose forms and functions can be achieved by means of a coding system explained below.

Genetic information flows unidirectional, from DNA to protein and with messenger RNA (mRNA) as intermediate. First, DNA encodes genetic information into an RNA molecule. This is called transcription (TC) of the information. Then the information gets converted into proteins, being named here translation (TL). It is this concept of information current that is called the Central Dogma of molecular biology. The Central Dogma is the fundamental theme in our exploration on gene articulation.

In order to complete the picture, we can add two further aspects of information flow. We can add duplication of the genetic material, which occurs prior to cell division. And that a DNA, in this case, represents duplication process, —DNA transfer. Wherefore in this case it is known as DNA replication. But where some viruses have RNA instead of DNA as their genetic material, we speak about reverse transcription (RT). With this transcription, we get a DNA molecule as a copy of the viral RNA genome.

In other words, genetic information, whether historically traced world wide (Narby,1998) or particularly assigned to ancient Africa (Siangla,1997) involves gene expression. Both DNA and RNA are polynucleotides, where nucleotides are the monomer—building units, which are composed of three basic subunits called nitrogenous base, sugar, and phosphoric acid. Genetic information is contained in DNA. The genetic code in DNA expresses the connection between the polynucleotide alphabet of four bases and 20 amino acids. In one strand of the parental DNA molecule, there is a dictated amino acid sequence strictly for protein production.

We will discuss in the next few postings, a relatively detailed understanding of the polymerization of amino acids sequence as directed by base sequences of messenger RNA.

At the moment, though, let us note that protein synthesis is an expression of genetic information. Protein synthesis is the cellular procedure, as we have said, of making proteins and involves two main processes: Transcription and Translation. The two processes mean that the direction of the synthesis is from DNA to RNA and then from RNA to protein respectively. Is this true to all organisms?

Yes. With a few exceptions, which are in mitochondria, and as stated above, some viruses become exceptions to this order because in their genetic material, they have RNA instead of DNA as their initial information source. However it is true that in all organisms, methods that relate the nucleotide sequence in messenger RNA to the amino acid sequence in proteins (genetic code proper) are the same. For in the given exceptions there occurs reverse transcription (RT). With that viral example of transcription noted, we get DNA molecular information being copied from the genome of viral RNA.

Building on this clue that is provided by transcription processes, we can readily see that a three-nucleotide sense codon denotes each amino acid. For example, UUU specifies phenylalanine, UCU specifies serine and GCA specifies alanine. But UAC and UAU both specify tyrosine. We will speak more about this tyrosine when expanding cell biology in the study of melanin.

Here now are other ways to see the remaining three properties of the genetic code. One is the contiguous property. With this property the codons do not overlap and at the same time they do not separated by spacers. The other is degenerate property in which there is more than one codon for some amino acids as exemplified by tyrosine in the above paragraph. And finally, there is the unambiguous property. With this genetic code of unambiguity, each codon specifies only one amino acid.

title: Cell Biology/Genes/Gene expression


text:
Introduction to Astrophysics

Astrophysics is the study of the physics of the universe. Various objects, such as stars, planets, and galaxies, are studied under the scope of astrophysics. Since astrophysics is a very broad subject, many disciplines of physics are typically applied.

Historical context

 * 1) Early Views
 * 2) Aristotle
 * 3) /Nicolas Copernicus/
 * 4) /Galileo/
 * 5) Johannes Kepler
 * 6) Isaac Newton
 * 7) /Albert Einstein/
 * 8) /Stephen Hawking/

Basics

 * 1) Stellar Luminosity
 * 2) Stellar Classification
 * 3) Hertzsprung‐Russell Diagram

Stellar Types
Real Hypothetical
 * 1) /Light Stars/
 * 2) /Heavy Stars/
 * 3) /Main Sequence Stars/
 * 4) /Red Giants/
 * 5) /White Dwarfs/
 * 6) /Black holes/
 * 7) /Neutron Stars/

The science of Astronomy relies on universality of Physical laws. "Same in the heavens as on earth". The world of stellar objects presents us with stupendously larger scales of agglomeration of matter and energy, at which to apply the known laws. Thus we are led to new predictions of entities that have never been observed before. Likewise some species of elementary particles and certain states of matter never observed on Earth need to be postulated in order to accord with observations. Some objects and phenomena falling in such categories and awaiting confirmation or further observational details are in this list. Eventually we expect them to be confirmed in all detail. If they contradict our predictions or conflict with phenomena known on Earth, well that would be even more exciting because that would be a new discovery awaiting explanation.
 * 1) /Black Dwarfs/
 * 2) /Quark Stars/
 * 3) /Protostars/
 * 4) /Dark Matter/
 * 5) /Dark Energy/
 * 6) /Vacuum Energy/

Stellar Formation & Evolution

 * 1) /Interstellar Medium/
 * 2) /Phases of Stellar Evolution/
 * 3) /Degenerate Stars/
 * 4) /Supernovae/
 * 5) /Compact Stellar & Relativistic Objects/
 * 6) /Close Binary Systems/

Stellar Structure

 * 1) /Hydrostatic Equilibrium/
 * 2) /Nuclear Reactions/
 * 3) /Energy Transport/

Star clusters and galaxies

 * 1) /Star Cluster/
 * 2) /Galaxies:Shapes and Evolution/

Quasars and Pulsars

 * 1) /Quasars/
 * 2) /Pulsars/

Appendices

 * A: Laws and Formulae
 * B: Astronomical Constants

title: Introduction to Astrophysics


text: Serbian

title: Serbian


text:
Robotics/Computer Control/The Interface

This chapter covers the hardware and its interface used to control the robot.
 * 1) /Computers/
 * 2) Single Board Computers and multichip modules/
 * 3) /Microcontrollers/
 * 4) /Remote Control/ (wired)
 * 5) ../../Communication Sensors/ (wireless)
 * 6) /Networks/

title: Robotics/Computer Control/The Interface


text:
Template:Browsebar

Browse: Shelves | Categories | Alphabetical Listing | By Completion Status | By Reading Level

title: Template:Browsebar


text:
Introduction to Astrophysics/Laws and Formulae

Kepler's Laws

 * 1) The orbit of each planet is an ellipse which has the sun at one of its foci.
 * 2) Each planet moves in such a way that the line joining it to the sun sweeps out equal areas in equal times.
 * 3) The squares of the periods of revolution of the planets about the Sun are proportional to the cubes of their mean distance from it.

Newton's gravitational law
$$F = G\frac{m_1 m_2}{r^2}$$

Black body radiation
Wein’s displacement law(only for black bodies)

λmax = W/T

Wein’s constant(W)= 2.90 × 10−3 m K

T= temperature of the black body

Stefan's Law
$$E = {\sigma}{T^4}$$

Apparent Magnitude
$$m = -2.5 \log_{10} I + K$$

Pogson's Formula
$$m_1 - m_2 = 2.5 \log_{10} \left(\frac{I_2}{I_1}\right)$$

Absolute Magnitude
$$m - M = 5 \log_{10} \left(\frac{d}{10}\right)$$


text:
Category:Subject:Science

ar:تصنيف:علوم az:Kateqoriya:Elm zh-min-nan:Category:Kho-ha̍k ca:Categoria:Ciència el:Κατηγορία:Επιστήμη es:Categoría:Ciencias eo:Kategorio:Udk:5 fr:Catégorie:Classe 5 - Mathématique. Sciences exactes et naturelles ko:분류:자연과학 hy:Կատեգորիա:Գիտություն it:Categoria:Scienze nl:Categorie:Exacte wetenschap ja:カテゴリ:科学 oc:Categoria:Sciéncias ru:Категория:Наука vi:Thể loại:Khoa học zh:Category:科学

title: Category:Subject:Science


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Category:Main page

This category is for the Main Page and its sub-versions.

da:Kategori:Forside eo:Kategorio:Ĉefpaĝo fr:Catégorie:Accueil

title: Category:Main page


text:
Category:Subject:Humanities

cy:Categori:Dyniaethau ko:분류:인문학 ja:カテゴリ:人文科学

title: Category:Subject:Humanities


text:
Category:Subject:Music

an:Categoría:Mosica br:Rummad:Sonerezh fo:Bólkur:Tónleikur io:Category:Muziko os:Категори:Музыкæ kw:Category:Ylow lmo:Category:Müsica mt:Category:Mużika mo:Category:Музикэ nap:Categoria:Museca nn:Kategori:Musikk scn:Category:Musica fiu-vro:Category:Muusiga zh-yue:Category:音樂

da:Kategori:Musik de:Kategorie:Musik es:Categoría:Música fr:Catégorie:Musique it:Categoria:Musica he:קטגוריה:מוזיקה hu:Kategória:Zene nl:Categorie:Muziek ja:カテゴリ:音楽 pl:Kategoria:Muzyka pt:Categoria:Música ru:Категория:Музыка sr:Категорија:Музика vi:Thể loại:Âm nhạc

title: Category:Subject:Music


text: Category:Subject:Dance

title: Category:Subject:Dance


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Category:Subject:Religion

fr:Catégorie:Religion ko:분류:종교 tr:Kategori:Dinler

title: Category:Subject:Religion


text:
Talk:Linear Algebra/Linear Transformations

I added a concrete example of a nonlinear transformation S early on:

"For example, the transformation S (whose input and output are both vetors in R2) defined by

$$S\mathbf{x} = S\begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix}  xy\\ cos(y)\end{pmatrix}$$

is non-linear (for example, it does not fix the origin). Linear transformations are easy to study, and any transformation is nearly linear if one looks at it closely enough, i.e. zooms in on it. Hence one can shed light on complicated nonlinear transformations by studying easier, linear ones."

hopefully this is a good idea? i found that when i first learned about linear transformations i had little motivation as to why we study them. do apologise if i'm messing someone's book up! rmijic

Some TODOs
This page needs some notes on null space/column space and rank/nullity. (Nullity is already briefly mentioned.)

The Point
The thing that makes this article so important is that it describes in relatively readable terms why matrices exist and what matrix multiply is for. Prior to reading this I had wondered for years why anyone would invent something as convoluted as matrix math, despite having taken a course in linear algebra. One of the major problems with math articles in wiki* and elsewhere is that the articles are almost always written on a level far higher than the subject matter. They are written by people familiar with the material for people who have already learned it and just need a refresher (undoubtedly this is usually unintentional). Thus it is a very steep learning curve for those trying to learn new material.

My point is simply this: If you are going to edit or add onto this sort of article, please remember to make your additions couched in the simplest terms you can without diluting the material. As Einstein said: "Make everything as simple as possible, but not simpler." Anuran 15:44, 1 April 2006 (UTC)

preserves identity
"preserves identity: T0 = 0" .... um surely this is preserving the null vector? non-trivial linear transformations do not preserve the identity, this might be confusing? Sorry, too far out of my field to correct personally.

Isomorphism
It has been written that $$T$$ is an isomorphism if it is bijective and an inverse exists (among other conditions).

But if $$T$$ is bijective, then its inverse does exist, doesn't it?


text: User talk:66.36.141.243

title: User talk:66.36.141.243


text:
Celestia/Development

Celestia's source code is freely available on GitHub. Only official Celestia developers can upload modified code to the GitHub server, but anyone can download the code and modify it for their own purposes. If you would like to contribute code to the project, please discuss it on the Celestia Web Forum in its Celestia Development Forum.

A sociological comment:
 * As with most software projects with established developer communities and protocols, a potential contributor to Celestia will have to undergo a form of apprenticeship. The program is large and complex and its underlying design philosophy has to be understood. It also takes a while to gain the trust of the people currently working on the program. By themselves, programming skills aren't enough. Too often individuals unfamiliar with the program or with the people involved have suggested significant changes and have become discouraged because their ideas weren't incorporated immediately. Don't let this happen to you. Read the forum. Read the developers mailing list. Participate in discussions. Contribute ideas. Try out things with the code on your own. Be patient. It'll take a while.

The following sections describe how to obtain the code and modify it for specific operating systems.

 * GNU/Linux and *BSD
 * macOS
 * Windows

title: Celestia/Development


text:
Celestia/Development/Win32 platform

Celestia development options using Windows-32 software packages:

Subversion (svn)
Celestia's source code is available on a Subversion (SVN) repository hosted on SourceForge.

Several Subversion clients are available for Windows. A very popular one is TortoiseSVN, a Windows shell extension that lets you use SVN commands right from Windows Explorer. A command line version is available with Cygwin.

To get the latest Celestia source using the command line client, cd to the desired directory and then issue the command

svn co https://celestia.svn.sourceforge.net/svnroot/celestia/trunk

With Tortoise SVN, select 'Checkout' from the Windows Explorer popup menu, and then enter this URL:

https://celestia.svn.sourceforge.net/svnroot/celestia/trunk

Microsoft C++ & SDK tools for Windows XP and Windows 7
In order to build software for Windows, you need both an SDK (Software Development Kit) and a compiler. The SDK provides libraries and include files needed for building for a particular version of Windows. The compiler converts text files into binary programs, linking them to the SDK libraries. Microsoft provides free versions of their SDKs and compilers, although with limited features, in order to encourage development of software for use with their operating systems. They also sell enhanced versions for use by commercial software development organizations.

VS2010
Currently, there is no Visual Studio 2010 solution checked into the subversion repository. If you have VS2010, you will have several unsupported steps to perform, in addition to the steps outlined for previous versions of Visual Studio.
 * First, VS2010 will convert the previous version's project files into VS2010 format. These converted versions will have mismatching $(TargetFile) and linker output file definitions. These will have to be manually reconciled. One choice is to set the General project property "Target Name" to "$(ProjectName)" and the linker property "Output File" to "$(OutDir)$(TargetName)$(TargetExt)". This will allow you to build, but it will not run.
 * Second, and most significantly, the static libraries zlib and libpng in the repository are compiled to work only with the redistributable runtime dlls for VS2008, namely msvcm90.dll, msvcp90.dll and msvcr90.dll. These runtimes are incompatible with VS2010, so you will have to obtain zlib and libpng (and the debug versions) that will run with 2010. The easiest way to do this may be to grab the sources from sourceforge, build them yourself, put the new libs in windows\lib\x86, and change the entries in the linker's "Additional Dependencies" property.
 * Finally, you must ensure that the celestia.exe and the dlls in windows\dll are copied into $(ProjectDir), and that the Debugging "Command" property is set to "$(ProjectDir)\$(TargetFileName)" and the "Working Directory" property is set to $(ProjectDir).

There may be other steps to perform, depending upon the versions of zlib and libpng that you obtain.

VS2008
In December 2007, Microsoft released Visual C++ 2008 Express Edition. As of Celestia version 1.6.0, this is the only version of the compiler recommended for building Celestia. Note that Visual C++ 2008 will not run on Windows 98. For this reason only, you may want to use Visual Studio 2005. If you have Windows 2000, XP, Vista or 7, compiling is much easier with Visual C++ 2008.

 * If you do not already have Microsoft Visual C++ 2008, you may download it for free from here:

 * http://msdn.microsoft.com/en-us/express/future/bb421473

When installing VS C++ 2008 SP1 using the Web installation, it automatically downloads and installs all of the correct SDK libraries, too. This happens under both Windows XP, 32bit, and Windows 7, 64bit. The SDK does not need to be downloaded or installed separately.

 * If you also need the Windows SDK, it is a free download from Microsoft:

 * http://www.microsoft.com/downloads/details.aspx?FamilyId=E6E1C3DF-A74F-4207-8586-711EBE331CDC&displaylang=en

Step by step instructions for Visual C++ 2008 Express Edition
These instructions work for both Windows XP and Windows 7.

This process assumes that you've followed the steps above, checked out the source from SVN, and installed the Visual C++ compiler and Windows SDK.

 * 1) Copy iconv.dll, intl.dll and lua5.1.dll from trunk\celestia\windows\dll\x86 into  trunk\celestia
 * 2) * Celestia is a 32bit x86 application. There are no x64 libraries.
 * 3) Open celestia.sln in Microsoft Visual C++.Net 2008 and allow the updater to work.
 * 4) * In other words, select (double-click on) celestia.sln in the directory trunk\celestia
 * 5) Set the build type to "Release". (The project provided from svn has it set to "Debug")
 * 6) * Specifically, in the line of icons across the top of the VisualStudio window, where it initially says "Debug", open its dropdown menu and select "Release"
 * 7) Build!
 * 8) * Open the menu "Build" and select "Rebuild Solution"

VS2005
''Recommended only for users of Windows 98/ME, where the Visual C++ 2008 is not supported. If you are running XP, Vista or 7, you should use the much easier process described in the previous section.''

 * Windows Platform SDK: Web Install

 * http://www.microsoft.com/downloads/details.aspx?familyid=0BAF2B35-C656-4969-ACE8-E4C0C0716ADB&displaylang=en

 * Windows Platform SDK: disk image

 * http://www.microsoft.com/downloads/details.aspx?FamilyID=e15438ac-60be-41bd-aa14-7f1e0f19ca0d&DisplayLang=en

 * Visual Studio 2005 Express C++: Web install

 * http://www.microsoft.com/express/2005/download/default.aspx

 * Visual Studio 2005 Express C++: disk image

 * http://msdn2.microsoft.com/en-us/express/aa718401.aspx

Celestia libraries for VS2005
The libraries in the Celestia repository were built with VS2008, and will not work on Windows 98/ME:

Celestia Libraries for Visual Studio 2005 Express:

 * http://www.shatters.net/~claurel/celestia/winbuild/

 * As of October, 2007, use
 * cspice.lib
 * winbuild-2.zip
 * winlibs-2.zip

 * Several source code files have been added to Celestia's svn repository since winbuild-2.zip was created. Insert these lines into the OBJS section of celestia/src/celengine/engine.mak if you're using makerelease.bat to build from the current subversion repository:

$(INTDIR)\axisarrow.obj \ $(INTDIR)\customrotation.obj \ $(INTDIR)\frametree.obj \ $(INTDIR)\precession.obj \ $(INTDIR)\scriptrotation.obj \ $(INTDIR)\timeline.obj \ $(INTDIR)\timelinephase.obj \

 * The SPICE include files are included in winlibs-2.zip.

 * No binary DLLs (Dynamically Loaded Libraries) are included in these Zip files, so you'll have to get them from one of the precompiled Celestia installation kits. As of January 26, 2008, use the DLLs which are included in Celestia v1.5.0. The official release of Celestia v1.5.0 is available on SourceForge.

The version of Celestia that's included in the Windows Celestia distribution kits is compiled using the script makerelease.bat. The older VS2003 versions of the build scripts are still what are in the cvs archive on SourceForge. The makefiles and .bat scripts which have been updated for VS2005/VS2008 are included in winbuild-2.zip [Remark: There is no makerelease.bat in winbuild-2.zip], which is in the VS2005 winbuild directory shown above.

Don't forget that you'll have to configure the PATH, LIB and INCLUDE environment variables according to your particular development environment so that the build scripts will find the Visual Studio and Celestia programs, include files and libraries.

Here's one example .BAT script for defining the search lists. It probably won't work for you: you probably have put the files in different directories. Change it appropriately.

Set INCLUDE=C:\Program Files\Microsoft Platform SDK for Windows Server 2003 R2\include;%INCLUDE% Set INCLUDE=C:\Program Files\Microsoft Platform SDK for Windows Server 2003 R2\include\mfc;%INCLUDE% Set INCLUDE=C:\Program Files\Microsoft Visual Studio 8\VC\include;%INCLUDE% Set INCLUDE=C:\cvs\Celestia\celestia\inc;%INCLUDE% Set PATH=C:\Program Files\Microsoft Platform SDK for Windows Server 2003 R2\Bin;%PATH% Set PATH=C:\Program Files\Microsoft Visual Studio 8\Common7\IDE;%PATH% Set PATH=C:\Program Files\Microsoft Visual Studio 8\VC\bin;%PATH% Set LIB=C:\Program Files\Microsoft Platform SDK for Windows Server 2003 R2\Lib;%LIB% Set LIB=C:\Program Files\Microsoft Visual Studio 8\VC\lib;%LIB% Set LIB=C:\cvs\Celestia\celestia\lib;%LIB%

title: Celestia/Development/Win32 platform


text:
Talk:Introduction to Philosophy/Logic/Modal Logic

I don't like this writing style very much. The content seems empty and little more than little thought-provoking limericks you might expect to find in a "deep thoughts" calendar. This is not what logic is. It seems more like a college students attempt to convince the world that he has accomplished something in his life.

Toodles, mccp, Eli Foley


text:
Celestia/Development/Win32 platform/Tortoise CVS

This page is obsolete. Celestia code development now uses Subversion (SVN). CVS no longer works. For details of building Celestia under Windows, please read the page Windows XP 32bit & Windows 7, 64bit

Acquiring Celestia CVS with TortoiseCVS (by Wcomer)

'''Celestia no longer uses CVS. Use ToroiseSVN instead.'''

 * 1) Download the latest version here: (1-2 min, 3.7 MB) http://www.tortoisecvs.org/download.shtml
 * 2) Follow the TortoiseCVS installation instructions. Note – you may be asked to reboot your machine but in my experience this wasn’t necessary.
 * 3) Create a new folder on your hard drive. Mine is called “C:\TortoiseCVS”. This will be your CVS repository
 * 4) Right-click your CVS repository folder and select “CVS Checkout”. If you do not see this option, then you must reboot your machine and try again.
 * 5) Enter “:pserver:anonymous@celestia.cvs.sourceforge.net:/cvsroot/celestia” as your CVSROOT. This should cause several other fields to fill in:
 * 6) Protocol - “Password server (:pserver:)”;
 * 7) Server – “celestia.cvs.sourceforge.net”;
 * 8) Repository folder – “/cvsroot/celestia”;
 * 9) User name – “anonymous”.
 * 10) Enter “celestia” as your Module.
 * 11) Press OK. Now the CVS client will attempt to download the Celestia source code. Expect the download time to be about 20 minutes mostly because the CVS server is a bit slow. If it is successful then there will be a subfolder called “celestia” within your CVS repository which will contain the current Celestia CVS archive.
 * 12) The checkout process may timeout, especially if you started it during a time of heavy traffic (that's late in the evening if accessing from Europe). You can continue the checkout at any time. Tortoise will restart with the incompletely checked out directory.
 * 13) If you normally use a dialup modem, you may safely do the initial checkout at an external broadband access and then copy the complete CVS directory to your local computer. Tortoise marks all CVS directories and will recognize the copied one instantly. Subsequent checkouts are usually only some dozen kB in size, and can be easily accessed from a dialup line.

Also, download the Windows-specific shared libraries that aren't included with Celestia on SourcForge. The libraries for use with Visual Studio 2003 are at http://shatters.net/celestia/files/winlibs-new.zip Unzip these files and put them in a directory with the other library files.


text:
Topology/Metric Spaces

Before we begin
Before we discuss topological spaces in their full generality, we will first turn our attention to a special type of topological space, a metric space. This abstraction has a huge and useful family of special cases, and it therefore deserves special attention. Also, the abstraction is picturesque and accessible; it will subsequently lead us to the full abstraction of a topological space.

Definition
A metric space is a Cartesian pair $$(X,d)$$ where $$X$$ is a non-empty set and $$d:X\times X\to[0,\infty)$$, is a function which is called the metric which satisfies the requirement that for all $$a,b,c\in X$$: Note that some authors do not require metric spaces to be non-empty. We annotate $$(X,d)$$ when we talk of a metric space $$X$$ with the metric $$d$$.
 * 1) $$d(a,b)=0$$ if and only if $$a = b$$
 * 2) $$d(a,b)=d(b,a)$$ (symmetry)
 * 3) $$d(a,c)\le d(a,b)+d(b,c)$$ (triangle inequality)

Examples

 * An important example is the discrete metric. It may be defined on any non-empty set X as follows
 * $$d(x,y)=\begin{cases}1&:x\ne y\\0&:x=y\end{cases}$$

 * On the set of real numbers $$\R$$, define $$d(x,y)=|x-y|$$ (The absolute distance between $$x$$ and $$y$$).  To prove that this is indeed a metric space, we must show that $$d$$ is really a metric. To begin with, $$d(x,y)=|x-y|\ge0$$ for any real numbers $$x$$ and $$y$$.
 * $$d(x,y)=|x-y|=0\iff(x-y)=0\lor-(x-y)=0\iff x=y$$
 * $$d(x,y)=|x-y|=|y-x|=d(y,x)$$
 * $$d(x,z)=|x-z|=|x-y+y-z|=\bigr|(x-y)+(y-z)\bigr|\le|x-y|+|y-z|=d(x,y)+d(y,z)$$
 * On the plane $$\R^2$$ as the space, and let $$d\bigl((x_1,y_1),(x_2,y_2)\bigr)=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$.
 * This is the euclidean distance between $$(x_1,y_1)$$ and $$(x_2,y_2)$$).

 * We can generalize the two preceding examples. Let $$V$$ be a normed vector space (over $$\R$$ or $$\Complex$$). We can define the metric to be: $$d(x,y)=\|x-y\|$$. Thus every normed vector space is a metric space.
 * For the vector space $$\Complex^n$$ we have an interesting norm. Let $$x=(x_1,\ldots,x_n)$$ and $$y=(y_1,\ldots,y_n)$$ two vectors of $$\Complex^n$$. We define the p-norm: $$\|x\|_p=\left(\sum_{i=1}^n|x_i|^p\right)^\frac1p$$. For each $$p$$-norm there is a metric based on it. Interesting cases of $$p$$ are:
 * $$p=1$$. The metric is $$d(x,y)=\|x-y\|_1=\sum_{i=1}^n|x_i-y_i|$$
 * $$p=2$$. The metric is good-old Euclid metric $$d(x,y)=\|x-y\|_2=\sqrt{\sum_{i=1}^n|x_i-y_i|^2}$$
 * $$p=\infty$$. This is a bit surprising: $$d(x,y)=\|x-y\|_\infty=\max_{i=1\ldots n}\bigl\{|x_i-y_i|\bigr\}$$ As an exercise, you can prove that $$\lim_{p\to\infty}\|x\|_p=\max_{i=1\ldots n}\{|x_i|\}$$ thus justifying the definition of $$\|\cdot\|_\infty$$.
 * The great-circle distance between two points on a sphere is a metric.
 * The Hilbert space is a metric space on the space of infinite sequences $$\{a_k\}$$ such that $$\sum_{i=1}^\infty a_k^2$$ converges, with a metric $$d\bigl(\{a_i\},\{b_i\}\bigr)=\sqrt{\sum_{i=1}^\infty(a_i-b_i)^2}$$.
 * The concept of the Erdős number suggests a metric on the set of all mathematicians. Take $$x,y$$ to be two mathematicians, and define $$d(x,y)$$ as 0 if $$x,y$$ are the same person; 1 if $$x,y$$ have co-authored a paper; $$n$$ if the shortest sequence $$(\{x,a_1\},\{a_1,a_2\},\ldots,\{a_{n-1},y\})$$, where each step pairs two people who have co-authored a paper, is of length $$n$$; or $$\infty$$ if $$x\ne y$$ and no such sequence exists. This metric is easily generalized to any reflexive relation (or undirected graph, which is the same thing).  Note that if we instead defined $$d(x,y)$$ as the sum of the Erdős numbers of $$x,y$$, then $$d$$ would not be a metric, as it would not satisfy $$d(x,y)=0\iff x=y$$. For example, if $$x=y$$ = Stanisław Ulam, then $$d(x,y)=2$$.

Note
Throughout this chapter we will be referring to metric spaces. Every metric space comes with a metric function. Because of this, the metric function might not be mentioned explicitly. There are several reasons:
 * We don't want to make the text too blurry.
 * We don't have anything special to say about it.
 * The space has a "natural" metric. E.g. the "natural" metric for $$\R^n,\Complex^n$$ is the euclidean metric $$d_2$$.

As this is a wiki, if for some reason you think the metric is worth mentioning, you can alter the text if it seems unclear (if you are sure you know what you are doing) or report it in the talk page.

Motivation
The open ball is the building block of metric space topology. We shall define intuitive topological definitions through it (that will later be converted to the real topological definition), and convert (again, intuitively) calculus definitions of properties (like convergence and continuity) to their topological definition. We shall try to show how many of the definitions of metric spaces can be written also in the "language of open balls". Then we can instantly transform the definitions to topological definitions.

Definition
Given a metric space $$(X,d)$$ an open ball with radius $$r$$ around $$p$$ is defined as the set
 * $$B_r(p)=\bigl\{x\in X:d(x,p)<r\bigr\},(r\in\R^+)$$

Intuitively it is all the points in the space, that are less than $$r$$ distance from a certain point $$p$$.

Examples
Why is this called a ball? Let's look at the case of $$\R^3$$:
 * $$d\bigl((x_1,x_2,x_3),(y_1,y_2,y_3)\bigr)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2}$$

Therefore $$B_r\bigl((0,0,0)\bigr)$$ is exactly $$x_1^2+x_2^2+x_3^2<r^2$$ – The ball with $$(0,0,0)$$ at center, of radius $$r$$. In $$\R^3$$ the ball is called open, because it does not contain the sphere ($$x_1^2+x_2^2+x_3^2=r^2$$).

The Unit ball is a ball of radius 1. Lets view some examples of the $$B_1\bigl((0,0)\bigr)$$ unit ball of $$\R^2$$ with different $$p$$-norm induced metrics. The unit ball of $$\R^2$$ with the norm $$\|\cdot\|_p$$ is:
 * $$B_1\bigl((0,0)\bigr)=\Big\{(x,y)\in\R^2:d\bigl((x,y),(0,0)\bigr)<1\Big\}=\Big\{(x,y):\bigl\|(x,y)-(0,0)\bigr\|_p<1\Big\}=\Big\{(x,y):\|(x,y)\|_p<1\Big\}=\Big\{(x,y):\sqrt[p]{|x|^p+|y|^p}<1\Big\}$$

 * The metric induced by $$\|\cdot\|_1$$ in that case, the unit ball is: $$|x|+|y|<1$$
 * The metric induced by $$\|\cdot\|_2$$ in that case, the unit ball is: $$\sqrt{x^2+y^2}<1$$
 * The metric induced by $$\|\cdot\|_\infty$$ in that case, the unit ball is: $$\max\bigl\{|x|,|y|\bigr\}<1$$

As we have just seen, the unit ball does not have to look like a real ball. In fact sometimes the unit ball can be one dot:
 * The discrete metric, The unit ball is $$B_1\bigl((0,0)\bigr)=\Big\{d\bigl((0,0),(x,y)\bigr)<1\Big\}=\Big\{d\bigl((0,0),(x,y)\bigr)=0\Big\}=\{(0,0)\}$$

Definitions
Definition: We say that x is an interior point of A iff there is an $$\epsilon>0$$ such that: $$B_\epsilon(x)\sube A$$. This intuitively means, that x is really 'inside' A - because it is contained in a ball inside A - it is not near the boundary of A.

Illustration: Definition: The interior of a set A is the set of all the interior points of A. The interior of a set A is marked $$\operatorname{int}(A)$$. Useful notations: $$\operatorname{int}(S)=\{x\in S|x\text{ is an interior point of }S\}$$ and $$\operatorname{int}(S)=\cup \{A\subseteq S|A\text{ is open }\!\!\}\!\!\text{ }$$.

Properties
Some basic properties of int (For any sets A,B):
 * $$\operatorname{int}(A) \subseteq A$$
 * $$\operatorname{int}(\operatorname{int}(A)) = \operatorname{int}(A)\,$$
 * $$\operatorname{int}(A \cap B) = \operatorname{int}(A) \cap \operatorname{int}(B)$$
 * $$A \subseteq B \Rightarrow \operatorname{int}(A) \subseteq \operatorname{int}(B)$$

Proof of the first: We need to show that: $$ x\in \operatorname{int}(A) \implies x \in A$$. But that's easy! by definition, we have that $$x\in B_\epsilon(x)\subseteq A$$ and therefore $$x\in A$$

Proof of the second: In order to show that $$\operatorname{int}(\operatorname{int}(A)) = \operatorname{int}(A)\,$$, we need to show that  $$\operatorname{int}(\operatorname{int}(A)) \subseteq \operatorname{int}(A)$$ and  $$\operatorname{int}(\operatorname{int}(A)) \supseteq \operatorname{int}(A)$$.  The " $$\subseteq$$" direction is already proved: if for any set A, $$\operatorname{int}(A) \subseteq A$$, then by taking $$\operatorname{int}(A)$$ as the set in question, we get $$\operatorname{int}(\operatorname{int}(A)) \subseteq \operatorname{int}(A)$$. The " $$\supseteq$$" direction: let $$x \in \operatorname{int}(A)$$. We need to show that $$x \in \operatorname{int}(\operatorname{int}(A))$$.  If $$x\in \operatorname{int}(A)$$ then there is a ball $$B_\epsilon(x) \subseteq A$$. Now, every point y, in the ball $$B_\frac{\epsilon}{2}(x) $$ an internal point to A (inside $$\operatorname{int}(A)$$), because there is a ball around it, inside A: $$y \in B_\frac{\epsilon}{2}(x) \implies B_\frac{\epsilon}{2}(y) \subset B_\epsilon(x) \subset A$$. We have that $$x\in B_\frac{\epsilon}{2}(x) \subset \operatorname{int}(A)$$ (because every point in it is inside $$\operatorname{int}(A)$$) and by definition $$x \in \operatorname{int}(\operatorname{int}(A))$$.
 Hint: To understand better, draw to yourself $$x, B_\epsilon(x), B_\frac{\epsilon}{2}(x), y, B_\frac{\epsilon}{2}(y)$$.

Proof of the rest is left to the reader.

Reminder

 * [a, b] : all the points x, such that $$a \leq x \leq b $$
 * (a, b) : all the points x, such that $$a < x < b $$

Example
For the metric space $$\mathbb{R}$$ (the line), we have:
 * $$int([a,b]) = (a,b)$$
 * $$int((a,b]) = (a,b)$$
 * $$int([a,b)) = (a,b)$$
 * $$int((a,b)) = (a,b)$$

Let's prove the first example ($$int([a,b]) = (a,b)$$). Let $$x\in (a,b)$$ (that is: $$a < x < b $$) we'll show that $$x$$ is an internal point.
 Let $$\epsilon = \min\{x - a, b - x \}$$. Note that $$x + \epsilon \leq x + b - x = b$$ and $$x - \epsilon \geq x - x + a = a$$. Therefore $$ B_\epsilon(x) = (x - \epsilon, x + \epsilon) \subset (a,b)$$.
 We have shown now that every point x in $$(a,b)$$ is an internal point. Now what about the points $$a,b$$ ? let's show that they are not internal points. If $$a$$ was an internal point of $$[a,b]$$, there would be a ball $$B_\epsilon(a) \subset [a,b]$$. But that would mean, that the point $$ a - \frac{\epsilon}{2}$$ is inside $$[a, b]$$. but because $$ a - \frac{\epsilon}{2} < a $$ that is a contradiction. We show similarly that b is not an internal point.
 To conclude, the set $$(a,b)$$ contains all the internal points of $$[a,b]$$. And we can mark $$int([a,b]) = (a,b)$$

Definition
A set is said to be open in a metric space if it equals its interior ($$ A = Int(A)$$). When we encounter topological spaces, we will generalize this definition of open. However, this definition of open in metric spaces is the same as that as if we regard our metric space as a topological space.

Properties:
 * 1) The empty-set is an open set (by definition: $$int(\emptyset)=\emptyset$$).
 * 2) An open ball is an open set.
 * 3) For any set B, int(B) is an open set. This is easy to see because: int(int(B))=int(B).
 * 4) If A,B are open, then $$A\cap B$$ is open.  Hence finite intersections of open sets are open.
 * 5) If $${A_i: i \in I}$$ (for any set if indexes I) are open, then their union $$\cup_{i\in I} A_i$$ is open.

Proof of 2:
 Let $$B_r(x)$$ be an open ball. Let $$y \in B_r(x)$$. Then $$y \in B_{r-d(x,y)}(y) \subseteq B_r(x)$$.
 In the following drawing, the green line is $$d(x,y)$$ and the brown line is $$r-d(x,y)$$. We have found a ball to contain $$y$$ inside $$B_r(x)$$.

Proof of 4:
 A, B are open. we need to prove that $$int(A \cap B) = A\cap B$$. Because of the first propriety of int, we only need to show that $$int(A \cap B) \supseteq A\cap B$$, which means $$\forall x\in A\cap B:x\in int(A \cap B)$$. Let $$x \in A \cap B$$. We know also, that $$x \in int(A), x \in int(B)$$ from the premises A, B are open and $$x \in A, x \in B$$. That means that there are balls: $$B_{{\epsilon}_1}(x) \subset A, B_{{\epsilon}_2}(x) \subset B$$. Let $$\epsilon = \min\{{\epsilon_1, \epsilon_2}\}$$, we have that $$B_{\epsilon}(x) \subset A, B_{\epsilon}(x) \subset B \Rightarrow  B_{\epsilon}(x) \subset A\cap B $$. By the definition of an internal point we have that $$x\in int(A \cap B)$$ ($$B_{\epsilon}(x)$$ is the required ball).

Interestingly, this property does not hold necessarily for an infinite intersection of open sets. To see an example on the real line, let $$A_n=\{(-1/n,1/n)\}$$. We then see that $$\cap^\infty_{i=1}A_i=\{0\}$$ which is closed.

Proof of 5:
 Proving that the union of open sets is open, is rather trivial: let $${A_i: i \in I}$$ (for any set if indexes I) be a set of open sets. we need to prove that $$int(\cup_{i\in I} A_i) \supseteq \cup_{i\in I} A_i$$: If $$x\in A_i$$ then it has a ball $$B_\epsilon(x) \subset A_i \subseteq \cup_{i\in I} A_i$$. The same ball that made a point an internal point in $$A_i$$ will make it internal in $$\cup_{i\in I} A_i$$.

Proposition: A set is open, if and only if it is a union of open-balls.
 Proof: Let A be an open set. by definition, if $$x\in A$$ there there a ball $$B_{\epsilon_x}(x) \subseteq A$$. We can then compose A: $$A = \cup_{x\in A}B_{\epsilon_x}(x)$$. The equality is true because: $$ \cup_{x\in A}B_{\epsilon_x}(x) \subseteq A$$ because $$ \forall x \in A: B_{\epsilon_x}(x) \subseteq A$$. $$ \cup_{x\in A}B_{\epsilon_x}(x) \supseteq A$$ in each ball we have the element $$x$$ and we unite balls of all the elements of $$A$$.
 On the other hand, a union of open balls is an open set, because every union of open sets is open.

Examples
Proof: Let $$U$$ be a set. we need to show, that if $$x\in U$$ then $$x$$ is an internal point. Lets use the ball around $$x$$ with radius $$\frac{1}{2}$$. We have $$B_\frac{1}{2}(x) = \{y\mid d(x,y) < \frac{1}{2}\} = \{x\} \subseteq U$$. Therefore $$x$$ is an internal point.
 * As we have seen, every open ball is an open set.
 * For every space $$X$$ with the discrete metric, every set is open.
 * The space $$\mathbb{R}$$ with the regular metric. Every open segment $$(a,b)$$ is an open set. The proof of that is similar to the proof that $$int([a,b]) = (a,b)$$, that we have already seen.

Theorem
In any metric space X, the following three statements hold:
 * 1) The union of any number of open sets is open.
 * Proof: Let $$C$$ be a collection of open sets, and let
 * $$x \in \cup C$$. Then there exists a $$U \in C$$ such that $$x \in U$$.
 * So there exists an $$\epsilon>0$$ such that $$B_\epsilon(x) \subseteq U$$. Therefore
 * $$B_\epsilon(x) \subseteq \cup C$$.
 * 2) The intersection of a finite number of open sets is open.
 * Proof: Let $$x \in \cap C$$, where $$C$$ is a finite collection of open sets.
 * So $$x \in U$$ for each $$U \in C$$. Let $$C = {U_1,U_2,...,U_n}$$. For each $$i=1,2,3,...,n$$, there exists an $$\epsilon_i > 0$$ such that $$B_\epsilon(x) \subseteq U_i$$. Let $$\epsilon = min_i$${$$\epsilon_i$$}. Therefore $$ \epsilon>0$$ and $$B_\epsilon(x) \subseteq \cap C$$.
 * 3) The empty set and X are both open.

Theorem
In any metric space X, the following statements hold:
 * 1) The intersection of any number of closed sets is closed.
 * 2) The union of a finite number of closed sets is closed.

Definition
First, Lets translate the calculus definition of convergence, to the "language" of metric spaces: We say that a sequence $$x_n$$ converges to $$x$$ if for every $$\epsilon > 0$$ exists $$N$$ that for each $$ n^* > N$$ the following holds: $$d(x_{n^*},x) < \epsilon$$.
 Equivalently, we can define converges using Open-balls: A sequence $$x_n$$ converges to $$x$$ If for every $$\epsilon > 0$$ exists $$N$$ that for each $$ n^* > N$$ the following holds: $$x_{n^*} \in B_\epsilon(x)$$.

The latter definition uses the "language" of open-balls, But we can do better - We can remove the $$\epsilon$$ from the definition of convergence, thus making the definition more topological. Let's define that $$x_n$$ converges to $$x$$ (and mark $$x_n \rightarrow x$$), if for every ball $$B$$ around $$x$$ , exists $$N_B$$ that for each $$ n^* > N_B$$ the following holds: $$x_{n^*} \in B(x)$$. $$x$$ is called the limit of the sequence.

The definitions are all the same, but the latter uses topological terms, and can be easily converted to a topological definition later.

Properties

 * If a sequence has a limit, it has only one limit. Proof Let a sequence $$x_n$$ have two limits, $$x\,$$ and $$x^\prime$$. If they are not the same, we must have $$0<d(x,x^\prime)$$. Let $$\epsilon$$ be smaller than this distance. Now for some $$N$$, for all $$n>N$$, it must be the case that both $$x_n \in B_{\epsilon / 2}(x)$$ and $$x_n \in B_{\epsilon / 2}(x^\prime)$$ by virtue of the fact $$x\,$$ and $$x^\prime$$ are limits. But this is impossible; the two balls are separate. Therefore the limits are coincident, that is, the sequence has only one limit.
 * If $$x_n \rightarrow x$$, then almost by definition we get that $$d(x_n, x) \rightarrow 0$$. ($$d(x_n, x)$$ Is the sequence of distances).

Examples
The space $$\mathbb{R}^k$$ For any p-norm induced metric, when $$p\geq 1$$. Let $$\vec{x_n} = (x_{n,1},x_{n,2},\cdots, x_{n,k})$$. and let $$\vec{x} = (x_{1},x_{2},\cdots, x_{k})$$.
 Then, $$\vec{x_n} \rightarrow \vec{x}$$ If and only if $$ \forall i, 1\leq i \leq k: x_{n,i} \rightarrow x_{i}$$.
 * In $$\mathbb{R}$$ with the natural metric, The series $$x_n = \frac{1}{n}$$ converges to $$0$$. And we note it as follows: $$\frac{1}{n}\rightarrow 0$$
 * Any space, with the discrete metric. A series $$x_n$$ converges, only if it is eventually constant. In other words: $$x_n\rightarrow x$$ If and only if, We can find $$N$$ that for each $$ n^* > N$$, $$x_{n^*} = x$$
 * An example you might already know:

Uniform Convergence
A sequence of functions $$\{ f_n \}$$ is said to be uniformly convergent on a set $$S$$ if for any $$\epsilon>0$$, there exists an $$N$$ such that when $$a$$ and $$b$$ are both greater than $$N$$, then $$d(f_a(x),f_b(x)) < \epsilon$$ for any $$x \in S$$.

Closure
Definition: The point $$p$$ is called a point of closure of a set $$A$$ if there exists a sequence $$a_n, \forall n, a_n \in A$$, such that $$a_n \rightarrow p$$.

In other words, the point $$p$$ is a point of closure of a set $$A$$ if there exists a sequence in $$A$$ that converges on $$p$$. Note that $$p$$ is not necessarily an element of the set $$A$$.

An equivalent definition using balls: The point $$p$$ is called a point of closure of a set $$A$$ if for every open ball $$B$$ containing $$p$$, we have $$B \cap A \neq \emptyset$$. In other words, every open ball containing $$p$$ contains at least one point in $$A$$ that is distinct from $$p$$.
 The proof is left as an exercise.

Intuitively, a point of closure is arbitrarily "close" to the set $$A$$. It is so close, that we can find a sequence in the set that converges to any point of closure of the set.

Example: Let A be the segment $$[0,1) \in \mathbb{R}$$, The point $$p = 1$$ is not in $$A$$, but it is a point of closure: Let $$a_n = 1 - \frac{1}{n}$$. $$a_n \in A$$ ($$n > 0$$, and therefore $$a_n = 1 - \frac{1}{n} < 1$$) and $$ a_n \rightarrow 1 $$ (that's because $$ \frac{1}{n} \rightarrow 0$$).

Definition: The closure of a set $$A \subseteq X$$ $$({X},d)$$, is the set of all points of closure. The closure of a set A is marked $$\bar{A}$$ or $$Cl(A)$$.

Note that $$A \subseteq \bar{A}$$. a quick proof: For every $$x \in A$$, Let $$(a_n = x)\forall n$$.

Examples
For the metric space $$\mathbb{R}$$ (the line), and let $$a,b \in \mathbb{R}$$ we have:
 * $$Cl([a,b]) = [a,b]$$
 * $$Cl((a,b]) = [a,b]$$
 * $$Cl([a,b)) = [a,b]$$
 * $$Cl((a,b)) = [a,b]$$

Closed set
Definition: A set $$A \subseteq X$$ is closed in $${X}\,$$ if $$A = Cl(A)$$.
 Meaning: A set is closed, if it contains all its point of closure.

An equivalent definition is: A set $$A \subseteq X$$ is closed in $${X}\,$$ If for every point $$p \in A$$, and for every Ball $$B, p \in B$$, then $$B \cap A \neq \emptyset$$.
 The proof of this definition comes directly from the former definition and the definition of convergence.

Properties
Some basic properties of Cl (For any sets $$A,B$$):

This union can therefore not be a closed subset of the real numbers.
 * $$A \subseteq Cl(A)$$
 * $$Cl(Cl(A))=Cl(A)$$
 * $$Cl(A \cup B) = Cl(A)\cup Cl(B)$$
 * $$A $$ is closed iff $$A = Cl(A)$$
 * While the above implies that the union of finitely many closed sets is also a closed set, the same does not necessarily hold true for the union of infinitely many closed sets. To see an example on the real line, let $$A_n=\{[-1+\frac{1}{n},1-\frac{1}{n}]\}$$. We see that $$\cup_{i=1}^\infty A_i=(-1,1)$$ fails to contain its points of closure, $$\pm 1.$$

The proofs are left to the reader as exercises. Hint for number 5: recall that $$Cl(A)=\cap \{A\subseteq S|S\text{ is closed }\!\!\}\!\!\text{ }$$.

Open vs Closed
That is, an open set approaches its boundary but does not include it; whereas a closed set includes every point it approaches. These two properties may seem mutually exclusive, but they are not:

 * In any metric space $$(X, d)$$, the set $$X$$ is both open and closed.

 * In any space with a discrete metric, every set is both open and closed.

 * In $$\R$$, under the regular metric, the only sets that are both open and closed are $$\R$$ and $$\emptyset$$. However, some sets are neither open nor closed.  For example, a half-open range like $$[0, 1)$$ is neither open nor closed.  As another example, the set of rationals is not open because an open ball around a rational number contains irrationals; and it is not closed because there are sequences of rational numbers that converge to irrational numbers (such as the various infinite series that converge to $$\pi$$).

Complementary set
A Reminder/Definition: Let $$A$$ be a set in the space $$X$$. We define the complement of $$A$$, $$A^c$$ to be $$X \setminus A$$.

A Quick example: let $$X = [0,1]; A = [0,\frac{1}{2}]$$. Then $$A^c = (\frac{1}{2},1]$$.

The plot continues...
A very important Proposition: Let $$A$$ be a set in the space $$(X,d)$$. Then, A is open iff $$A^c$$ is closed.
 Proof: ($$\Rightarrow$$) For the first part, we assume that A is an open set. We shall show that $$ A^c = Cl(A^c)$$. It is enough to show that $$Cl(A^c) \subseteq A^c$$ because of the properties of closure. Let $$p \in Cl(A^c)$$ (we will show that $$p \in A^c$$).
 for every ball $$B, p\in B$$ we have, by definition that (*)$$B \cap A^c \neq \emptyset$$. If the point is not in $$A^c$$ then $$p \in A$$. $$A$$ is open and therefore, there is a ball $$B$$, such that: $$p \in B \subseteq A$$, that means that $$B \cap A^c = \emptyset$$, contradicting (*).
 ($$\Leftarrow$$) On the other hand, Lets a assume that $$A^c$$ is closed, and show that $$A$$ is open. Let $$p$$ be a point in $$A$$ (we will show that $$p \in int(A)$$). If $$p$$ is not in $$int(A)$$ then for every ball $$B, p \in B$$ we have that $$B \nsubseteq A$$. That means that $$B \cap A^c \neq \emptyset$$. And by definition of closure point $$p$$ is a closure point of $$A^c$$ so we can say that $$p \in Cl(A^c)$$. $$A^c$$ is closed, and therefore $$p \in A^c = Cl(A^c)$$ That contradicts the assumption that $$p \in A$$

Note that, as mentioned earlier, a set can still be both open and closed!

On $$\mathbb{R}$$
The following is an important theorem characterizing open and closed sets on $$\mathbb{R}$$.

Theorem: An open set $$O$$ in $$\mathbb{R}$$ is the union of countably many disjoint open intervals.

Proof: Let $$x\in O$$. Let $$a=\sup\{t|t\notin O, t<x\}$$ and let $$b=\inf\{t|t\notin O, t>x\}$$. There exists an open ball $$(x - \epsilon, x + \epsilon)$$ such that $$(x - \epsilon, x + \epsilon) \subseteq O$$ because $$O$$ is open. Thus, a≤x-ε and b≥x+ε. Thus, x ∈(a,b). The set O contains all elements of (a,b) since if a number is greater than a, and less than x but is not within O, then a would not be the supremum of {t|t∉O, t<x}. Similarly, if there is a number is less than b and greater than x, but is not within O, then b would not be the infimum of {t|t∉O, t>x}. Thus, O also contains (a,x) and (x,b) and so O contains (a,b). If y≠x and y∈(a,b), then the interval constructed from this element as above would be the same. If yy} would also be less than a because there is a number between y and a which is not within O. Similarly if y>b, then sup{t|t∉O, t<y} would also be greater than b because there is a number between y and b which is not within O. Thus, all possible open intervals constructed from the above process are disjoint. The union of all such open intervals constructed from an element x is thus O, and so O is a union of disjoint open intervals. Because the rational numbers is dense in R, there is a rational number within each open interval, and since the rational numbers is countable, the open intervals themselves are also countable.

Examples of closed sets

 * 1) In any metric space, a singleton $$\{x\}$$ is closed. To see why, consider the open set, $$\{x\}^c$$. Let $$y \in \{x\}^c$$. Then $$y \neq x$$, so $$d(y,x) > 0$$. Let $$\epsilon = \frac{1}{2}d(y,x)$$. Then $$B_\epsilon(y) \subseteq \{x\}^c$$. So $$\{x\}^c$$ is open, and hence $$\{x\}$$ is closed.
 * 2) In any metric space, every finite set $$T = \{x_1,x_2,...,x_n\}$$ is closed. To see why, observe that $$T^c = \Big[\bigcup\{x_i\}\Big]^c = \bigcap\{x_i\}^c$$ is open, so $$T$$ is closed.
 * 3) Closed intervals [a,b] are closed.
 * 4) Cantor Set Consider the interval [0,1] and call it C0. Let A1 be equal {0, $$\tfrac{2}{3}$$} and let dn = $$(\tfrac{1}{3})^{n}$$. Let An+1 be equal to the set An∪{x|x=a+2dn, a∈An}. Let Cn be $$\textstyle \bigcup_{a\in A_n}$${[a,a+dn]}, which is the finite union of closed sets, and is thus closed. Then the intersection $$\textstyle \bigcap_{i=1}^\infty {C_i}$$ is called the Cantor set and is closed.

Exercises

 * 1) Prove that a point x has a sequence of points within X converging to x if and only if all balls containing x contain at least one element within X.
 * 2) In $$\mathbb{R}$$ the only sets that are both open and closed are the empty set, and the entire set. This is not the case when you look at $$\mathbb{Q}\subseteq\mathbb{R}$$. Give an example of a set which is both open and closed in $$\mathbb{Q}$$.
 * 3) Let $$A$$ be a set in the space $$x$$. Prove the following:
 * 4) $$Cl(A) = Int(A^c)^c$$
 * 5) $$Int(A) = Cl(A^c)^c$$

Definition
Let's recall the idea of continuity of functions. Continuity means, intuitively, that you can draw a function on a paper, without lifting your pen from it. Continuity is important in topology. But let's start in the beginning:

The classic delta-epsilon definition: Let $$(X,d),(Y,e)$$ be spaces. A function $$f : X \rightarrow Y $$ is continuous at a point $$x$$ if for all $$\epsilon_x > 0$$ there exists a $$\delta_{\epsilon_x} > 0$$ such that: for all $$x_1$$ such that $$d(x,x_1) < \delta_{\epsilon_x} $$, we have that $$e(f(x), f(x_1)) < \epsilon_x$$.

Let's rephrase the definition to use balls: A function $$f : X \rightarrow Y $$ is continuous at a point $$x$$ if for all $$\epsilon_x > 0$$ there exists $$\delta_{\epsilon_x} > 0$$ such that the following holds: for every $$x_1$$ such that $$x_1 \in B_{\delta_{\epsilon_x}} (x) $$ we have that $$ f(x_1) \in B_{\epsilon_{x}}(f(x))$$. Or more simply: $$ f(B_{\delta_{\epsilon_x}}(x)) \subseteq B_{\epsilon_{x}}(f(x))$$

Looks better already! But we can do more.

Definitions:
 * A function is continuous in a set S if it is continuous at every point in S.
 * A function is continuous if it is continuous in its entire domain.

Proposition: A function $$f : X \rightarrow Y $$ is continuous, by the definition above $$\Leftrightarrow$$ for every open set $$U$$ in $$Y$$, The inverse image of $$U$$, $$f^{-1}(U)$$, is open in $$X$$. That is, the inverse image of every open set in $$Y$$ is open in $$X$$.
 Note that $$f$$ does not have to be surjective or bijective for $$f^{-1}$$ to be well defined. The notation $$f^{-1}$$ simply means $$f^{-1}(U) = \{x \in X: f(x) \in U\}$$.

Proof: First, let's assume that a function $$f$$ is continuous by definition (The $$\Rightarrow$$ direction). We need to show that for every open set $$U$$, $$f^{-1}(U)$$ is open.

Let $$U\subseteq Y$$ be an open set. Let $$x \in f^{-1}(U)$$. $$f(x)$$ is in $$U$$ and because $$U$$ is open, we can find and $$\epsilon_x$$, such that $$B_{\epsilon_x}(f(x)) \subseteq U$$. Because f is continuous, for that $$\epsilon_x$$, we can find a $$\delta_{\epsilon_x} > 0$$ such that $$ f(B_{\delta_{\epsilon_x}}(x)) \subseteq B_{\epsilon_{x}}(f(x)) \subseteq U$$. that means that $$B_{\delta_{\epsilon_x}}(x) \subseteq f^{-1}(U)$$, and therefore, $$x$$ is an internal point. This is true for every $$x$$ - meaning that all the points in $$f^{-1}(U)$$ are internal points, and by definition, $$f^{-1}(U)$$ is open.

($$\Leftarrow$$)On the other hand, let's assume that for a function $$f$$ for every open set $$U \in Y$$, $$f^{-1}(U)$$ is open in $$X$$. We need to show that $$f$$ is continuous.

For every $$x\in X$$ and for every $$\epsilon_x > 0$$, The set $$B_{\epsilon_x}(f(x))$$ is open in $$Y$$. Therefore the set $$V = f^{-1}(B_{\epsilon_x}(f(x)))$$ is open in $$X$$. Note that $$x\in V$$. Because $$V$$ is open, that means that we can find a $$\delta_{\epsilon_x}$$ such that $$B_{\delta_{\epsilon_x}}(x) \subseteq V$$, and we have that $$ f(B_{\delta_{\epsilon_x}}(x)) \subseteq B_{\epsilon_{x}}(f(x))$$.

The last proof gave us an additional definition we will use for continuity for the rest of this book. The beauty of this new definition is that it only uses open-sets, and there for can be applied to spaces without a metric, so we now have two equivalent definitions which we can use for continuity.

Examples

 * Let $$f$$ be any function from any space $$(X,d)$$, to any space $$(Y,e)$$, were $$d$$ is the discrete metric. Then $$f$$ is continuous. Why? For every open set $$U$$, the set $$f^{-1}(U)$$ is open, because every set is open in a space with the discrete metric.
 * Let $$f:\mathbb{R}\rightarrow\mathbb{R}; f(x)=x$$ The identity function. $$f$$ is continuous: The source of every open set is itself, and therefore open.

Exercise

 * 1) Prove that a function $$f : X \rightarrow Y $$ is continuous $$\Leftrightarrow$$ for every closed set $$U$$ in $$Y$$, The inverse image of $$U$$, $$f^{-1}(U)$$, is closed in $$X$$.

Uniform Continuity
In a metric space X, function from X to a metric space Y is uniformly continuous if for all $$\epsilon$$, there exists a $$\delta$$ such that for all $$x_1,x_2\in X$$, $$d(x_1,x_2)<\delta$$ implies that $$d(f(x_1),f(x_2))<\epsilon$$.

Isometry
An isometry is a surjective mapping $$f: X \rightarrow Y$$, where $$(X, \delta )$$ and $$(Y , \rho )$$ are metric spaces and for all $$a, b \in X$$, $$\delta (a, b) = \rho (f(a), f(b))$$.

In this case, $$(X, \delta )$$ and $$(Y, \rho )$$ are said to be isometric.

Note that the injectivity of $$f$$ follows from the property of preserving distance:

 * $$f(a)=f(b)$$
 * $$\implies\rho(f(a),f(b))=0$$
 * $$\implies\delta(a,b)=0$$
 * $$\implies a=b$$

So an isometry is necessarily bijective.

Exercises

 * 1) Show that a set is a metric open set iff it is a (possibly infinite) union of open balls.
 * 2) Show that the discrete metric is in fact a metric.

title: Topology/Metric Spaces


text:
Guide to Unix/BSD

Berkeley Software Distribution or BSD is a flavor of UNIX that was developed at the University of California, Berkeley. Today, the BSD or *BSD systems are free Unix-like operating systems based on the university's distribution. To honor this, these systems put BSD at the end of their names. The *BSD systems are DragonFly BSD, FreeBSD, NetBSD, and OpenBSD.

Contents

 * /Introduction/ how *BSD compares with other Unix-likes
 * /External Links/ learn more about *BSD

Distributions

 * /FreeBSD/: focuses on speed and performance.
 * /Mac OS X/: Darwin and BSD
 * /NetBSD/: focuses on portability
 * /OpenBSD/: focuses on code correctness (and thus security) and easy to security
 * /Historical/: BSD operating systems that have been unmaintained or abandoned.

Other Wikibooks

 * A Neutral Look at Operating Systems has a comparison of the *BSD systems.
 * Mac OS X Tiger

title: Guide to Unix/BSD


text: Canadian History/Prime Ministers

title: Canadian History/Prime Ministers


text:
Accountancy

Accounting is the process by which financial information about a business is recorded, classified, summarized, interpreted, and communicated. Accounting is the language of business.

Categories

 * 1) Financial accounting is a major branch of accounting that describes the collection, recording, and extraction of financial information, in order to periodically summarize profit and loss, the balance sheet, and cash flow in accordance with legal, professional, and capital market requirements. The main task is to prepare regular financial reports.
 * 2) Managerial accounting is the branch of accounting for internal use, to which end an organization provides information only accessible to its insiders in order to assist decision-makers. The mission of managerial accounting is to utilize, inform, and control the functions of management.

Contents

 * 1) Principles of accounting
 * 2) Introduction to accountancy
 * 3) /The Accounting Equation/
 * 4) Double entry
 * 5) /Debtors/ (an asset)
 * 6) /Creditors/ (a liability)
 * 7) /Prepayments/ (an asset)
 * 8) /Accruals/ (debt or credit)
 * 9) /Depreciation/ (an expense)

Proposed additional sections

 * Accounting standards
 * /Assets/
 * /Non-current assets/
 * /Liabilities/
 * /Cash/
 * /Receivables/
 * /Investments/
 * Books of prime entry
 * Year end adjustments
 * The trial balance
 * /Journals/
 * /Provisions/
 * Capital and reserves
 * /Disclosures/
 * Balance sheet
 * Income statement
 * Cash flow statement
 * Accounting policies
 * /GAAP/
 * /IFRS/
 * /Income and Expenditure Recognition/
 * /Cost Flow Assumptions/
 * /Single Entry System/
 * Basic Bookkeeping
 * Financial ratios
 * /GST/

Glossary and Appendices

 * Glossary
 * Professional bodies
 * Notes to authors
 * /Templates/
 * Quick reference

Other pages

 * /Amortization/
 * Cost accounting
 * Managerial accounting

Resources

 * Accounting I and II - Concise review with practice set and quizzes.

title: Accountancy