text:
User:Twowheelappl/sandbox

My preferred choice is to update three articles: Zeppelin NT has a reference to Thermic welding as a method of joining plastic airship hulls. I believe this is a mistaken reference to some form of Plastic welding which I would like to correct. Further, I think the Thermic welding article should be improved to include the welding related information from the Thermite article (and potentially replace that section, or at least link to it) as well as updated to include some additional details relating to analysis of thermite joints (I have completed an initial literature search and have found some articles related to modeling thermite welds along with some alternative energy research involving thermite as an energy storage medium and thermal models of its use in that capacity.)

My second choice would be to update the Friction welding article (particularly Spin welding probably) to include information on friction hydropillar welding/friction taper plug welding. I have not done as much initial footwork on this one, though.Twowheelappl (talk) 11:04, 19 October 2017 (UTC)

Instructor Comments
Adding heat flow modeling of thermite or exothermic welds would to the existing Exothermic Welding article would be good.

Draft Edits For Peer Review (Thermite)
These changes have already been made to the thermite article:

Was:

"However, defects such as slag inclusions and voids (holes) are often present in such welded junctions and great care is needed to operate the process successfully. Care must also be taken to ensure that the rails remain straight, without resulting in dipped joints, which can cause wear on high speed and heavy axle load lines."

Is:

"However, defects such as slag inclusions and voids (holes) are often present in such welded junctions and great care is needed to operate the process successfully. Both Finite Element Analysis and experimental analysis of thermite rail welds has shown that weld gap is the most influential parameter affecting defect formation.  Shrinkage cavity formation and cold lap welding defects are reduced by increasing thermite temperature, increasing preheat, and increasing gap.  However, these all promote microporosity.  Care must also be taken to ensure that the rails remain straight, without resulting in dipped joints, which can cause wear on high speed and heavy axle load lines."

These changes have already been made to the thermite article:

Was:

"Because standard iron-thermite is difficult to ignite, burns with practically no flame and has a small radius of action, standard thermite is rarely used on its own as an incendiary composition. It is more usually employed with other ingredients added to increase its incendiary effects."

Is:

"Because standard iron-thermite is difficult to ignite, burns with practically no flame and has a small radius of action, standard thermite is rarely used on its own as an incendiary composition. Increases in gas content of the reaction products of a thermite increases the heat transfer rate (and therefore damage) of that particular thermite blend. It is more usually employed with other ingredients added to increase its incendiary effects."

Twowheelappl (talk) 22:13, 16 November 2017 (UTC)

Draft Edit for Peer Review (Zappelin_NT)
See https://en.wikipedia.org/wiki/Zeppelin_NT#Hull_and_structure Twowheelappl (talk) 22:13, 16 November 2017 (UTC)

Final Edit Incorporating Peer Review
I had considered increasing the content of heat transfer modeling information in my final edit, but most of the peer reviewers indicated that as the draft stands, it has a good balance between detail and accessibility. I did add one additional detail about the modeling. Other than that, only grammatical changes were made.

The article edits now read:

thermite:

"However, defects such as slag inclusions and voids (holes) are often present in such welded junctions and great care is needed to operate the process successfully. The numerical analysis of thermite welding of rails has been approached similar to casting cooling analysis.  Both this Finite Element Analysis and experimental analysis of thermite rail welds has shown that weld gap is the most influential parameter affecting defect formation.  Increasing weld gap has been shown to reduce shrinkage cavity formation and cold lap welding defects, and increasing preheat and thermite temperature further reduces these defects.  However, reducing these defects promotes a second form of defect: microporosity.  Care must also be taken to ensure that the rails remain straight, without resulting in dipped joints, which can cause wear on high speed and heavy axle load lines."

thermite:

"Because standard iron-thermite is difficult to ignite, burns with practically no flame and has a small radius of action, standard thermite is rarely used on its own as an incendiary composition. In general, an increases in the volume of gaseous reaction products of a thermite blend increases the heat transfer rate (and therefore damage) of that particular thermite blend. It is more usually employed with other ingredients added to increase its incendiary effects."

Twowheelappl (talk) 18:40, 9 December 2017 (UTC)

title: User:Twowheelappl/sandbox


text:
Talk:Origins (Judge Dredd story)

Suggestion
I think that once this is up and running we should get a picture (possibly a cover) and I'll be willing to do the plot. (Would you like it if I started "The Connecton"s plot?) J Smithy

 * Thats cool - keep it going. Once the full plot is revealed we can probbaly tighten things up but it seems fine for now. I'd guess that Origins proper will be launched with a big fanfare and a fancy cover so I suspect that cover will be the best one for going in here - but we'll have to wait and see. (Emperor 00:20, 19 August 2006 (UTC))

 * There's a cover by Jock next issue which seems pretty good but apparently there's gonna be another one by the same artist that'll run during Origins, that'll take some beating. I think we should put a cover up once Origins is finished (I'm sure there'll be a brilliant one for that). Or we could change them about when a better cover's up. Dunno, what do you think? Smithy 21:11, 26 August 2006 (UTC)

 * I suppose we keep a weather eye out and if something good and iconic crops up we can go for that. If something better appears then there shouldn't be a problem replacing it. (Emperor 14:14, 29 August 2006 (UTC))

 * I'd say that Brian Bolland cover to 1505 has to be the iconic image for Origins - at least for the conceivable future (Emperor 00:51, 14 September 2006 (UTC))

 * OK image now live what do you reckon? (Emperor 14:05, 18 September 2006 (UTC))

Works for me fine! If there's always a better cover, I suppose we could chage it but that seems to do the job! Smithy 20:10, 21 September 2006 (UTC)

 * FP seem to be suggesting this is the trade cover but they may just mocked it up for trade paperback they are already offering for pre-order  I still vote for the cover of 1505. (Emperor 23:35, 29 September 2006 (UTC))

 * Unless the trade cover changes, 1505 would seem to be the best choice. It's a strong image anyway, and Bolland is famous in comics in the UK and the US.Richard75 18:27, 6 November 2006 (UTC)

Origins
just wondering if it's a good idea to do the plot here as well. It's gonna be very long, right into next year. What do you think? Smithy 20:42, 7 September 2006 (UTC)

 * Well the Comics Project is fairly clear on what Wikipedia is and isn't and they recommend "concise summaries" of plot so I think things can be kept fairly tight or can be tightened up at a later date once the bigger picture emerges (see the other entries in the Judge Dredd storylines category for examples of he kind of summary we'd be looking for). We'd also be looking for comments, discussion and any other relevant informaiton that can add depth to this. I here there is an Alan Grant interview in the upcoming Redeye covering thigs that we might want to throw in to a trivia section - like who Robin Low is and who Judge Logan is based on. Adding nough plot to explain the scope of the story and some extra depth and references so people can take it on further (Emperor 00:51, 14 September 2006 (UTC))

WikiProject Comics B-Class Assesment required
This article needs the B-Class checklist filled in to remain a B-Class article for the Comics WikiProject. If the checklist is not filled in by 7th August this article will be re-assessed as C-Class. The checklist should be filled out referencing the guidance given at Version 1.0 Editorial Team/Assessment/B-Class criteria. For further details please contact the Comics WikiProject. Comics-awb (talk) 17:12, 31 July 2008 (UTC)

C-Class rated for Comics Project
As this B-Class article has yet to receive a review, it has been rated as C-Class. If you disagree and would like to request an assesment, please visit WikiProject_Comics/Assessment and list the article. Hiding T 16:38, 23 February 2009 (UTC)

Redirect as ATD
Okay, so yesterday, I redirected this to Judge Dredd based on the fact that just yesterday, another Dredd storyline article, Tour of Duty, finished AfD and was closed as redirect. Someone is now insisting on putting this one through AfD as well. I would argue that's needless and overburdensome, based on the result of the prior AfD. I've been down this road before and I don't think we need another round of, "But it's an obvious redirect!" "We just redirected another article like this one!" "Did you even think about redirect?" "Do you even ATD, bro?"

Have at me, inclusionists. Just Another Cringy Username (talk) 19:10, 2 September 2022 (UTC)

 * By that logic, you would just redirect every Judge Dredd article that exists to Judge Dredd without further discussion. But anyway, Origins seems to have several online reviews which Tour of Duty did not. For example:

 * https://www.denofgeek.com/movies/judge-dredd-origins-a-comic-that-should-be-a-movie/?amp
 * https://www.bigglasgowcomicpage.com/essential-judge-dredd-origins-review/
 * https://theslingsandarrows.com/judge-dredd-origins/
 * http://dreddreviews.blogspot.com/2012/12/origins.html
 * https://www.comicon.com/2021/08/06/preview-judge-dredd-origins-making-a-classic-into-an-essential/
 * Richard75 (talk) 20:59, 2 September 2022 (UTC)
 * Maybe not every article, but at least the ones which have had their notability challenged for two years now. And how many of those reviews would pass WP:RS? Just Another Cringy Username (talk) 23:17, 2 September 2022 (UTC)
 * Surprisingly, Den of Geek is RS per WP:VG/RS. But the other refs are literally the definition of self-published sources, one even has "blog" in the URL... IMHO, you could nominate this for AfD, this article is a WP:GNG failing, WP:ALLPLOT article that is almost WP:TNT. VickKiang (talk) 00:56, 3 September 2022 (UTC)
 * Yeah, I figured if any of these met RS, it would be that one.
 * I don't have a problem nominating it; I just thought it was an obvious redirect given the outcome of the Tour of Duty AfD. Seems needlessly bureaucratic to go through the whole process again. Just Another Cringy Username (talk) 01:25, 3 September 2022 (UTC)
 * @Richard75 @VickKiang So can I redo the redirect? Or are we going to have to do the full AfD? Just Another Cringy Username (talk) 01:55, 3 September 2022 (UTC)
 * As the redirect was disputed, IMHO let's do an AfD. VickKiang (talk) 01:57, 3 September 2022 (UTC)

source: wikimedia

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

title: Talk:Origins (Judge Dredd story)


text:
Harold Crooms

Harold Crooms (born October 24, 1979) is an American professional stock car racing driver. He competes in the CARS Super Late Model Tour and short track racing divisions, and has also raced in the NASCAR Xfinity Series.

Racing career
Crooms primarily races in the CARS Super Late Model Tour and short tracks in his home state of Florida.

In 2020, he tested an ARCA Menards Series car for MBM Motorsports at Daytona International Speedway; he was the 18th fastest of 38 participants on the first day of testing, followed by ranking 18th of 39 on the second. He made his NASCAR Xfinity Series debut for the team on the Daytona road course in August, where he finished 35th after a brake failure.

NASCAR
(key) ( Bold – Pole position awarded by qualifying time. Italics – Pole position earned by points standings or practice time. * – Most laps led. )

title: Harold Crooms


text:
Wikipedia:WikiProject Spam/LinkReports/judibetqq.life

Links

 * judibetqq.life resolves to [//<IP_ADDRESS> <IP_ADDRESS>]
 * Link is not on the blacklist.
 * Link is not on the domainredlist.
 * Link is not on the Monitorlist.
 * None of the mentioned users is on the blacklist.
 * Link is not on the whitelist.
 * Link would be caught by rule \bjudibetqq\.life\b on the monitor list (Automonitor: reported to m:Spam blacklist (diff - )).
 * Link is globally blacklisted by \bjudibetqq\.life\b
 * Link would be caught by rule \bjudibetqq\.life\b on the monitor list (Automonitor: reported to m:Spam blacklist (diff - )).
 * Link is globally blacklisted by \bjudibetqq\.life\b

Users
No users found.

Additions
No additions recorded.


text:
Talk:All the Feels (album)

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 * Fitz And The Tantrums All The Feels.jpg

title: Talk:All the Feels (album)


text:
User:Ilya.Gubarev/Books/North West England

North West England

 * Overview
 * North West England

 * Greater Manchester
 * Greater Manchester

 * Manchester (city)
 * Manchester

 * Merseyside
 * Merseyside

 * Liverpool
 * Liverpool
 * Albert Dock
 * Anfield
 * The Cavern Club
 * Croxteth Hall
 * Gambier Terrace
 * Liverpool Cathedral
 * Liverpool Metropolitan Cathedral
 * Merseyside Maritime Museum
 * Queensway Tunnel
 * Kingsway Tunnel
 * Museum of Liverpool
 * World Museum
 * Walker Art Gallery
 * Pier Head
 * Speke Hall

 * Knowsley
 * Metropolitan Borough of Knowsley
 * Knowsley Hall

 * St. Helens
 * Metropolitan Borough of St Helens
 * Dream (sculpture)
 * North West Museum of Road Transport

 * Sefton
 * Metropolitan Borough of Sefton
 * Crosby Beach
 * Meols Hall
 * Southport Pier
 * Lord Street, Southport
 * Formby

 * Wirral
 * Metropolitan Borough of Wirral
 * Bidston Hill
 * Bidston Windmill
 * Birkenhead Park
 * Birkenhead Priory
 * Hamilton Square
 * Hilbre Island
 * Leasowe
 * Port Sunlight


text:
Portal:Real analysis

Department description
The goal of the department of real analysis is to give rigorous backing to the machinery of calculus and the real number system. By the end of this course, a student should know the real numbers like the back of their hand, and be more than ready to learn the generalizations of topology.

Department news

 * Wednesday, August 23, 2006 - Department founded!

Courses on Real Analysis

 * Introduction to Real Analysis
 * Ordered Sets and Fields
 * Sequences
 * Series
 * Limits

Offsite Courses

 * Analysis 1
 * Analysis 2

Other Resources
Analysis is a highly technical topic and is full of pitfalls and subtle counterexamples. You will need to read and complete the exercises in at least one of the following books in addition to using this site as a supportive community of peers and teachers.

Online Textbooks

 * Real analysis (Work in progress)
 * Mathematical Analysis I
 * Mathematical Analysis II
 * Basic Analysis: Introduction to Real Analysis

Hard-copy Textbooks

 * Lay, Steven R. (2005). Analysis With an Introduction to Proof Pearson Prentice Hall ISBN<PHONE_NUMBER>
 * Gelbaum & Olmsted. (2003). Counterexamples in Analysis Dover Publications. ISBN<PHONE_NUMBER>
 * Rudin, Walter. (1976). Principles of Mathematical Analysis McGraw-Hill Science/Engineering/Math. ISBN 007054235X

Wikipedia

 * Real analysis

Online courses
Might contain video lectures, course notes, homework, exam question papers and their solutions.

With video

 * 1) Real analysis, Francis Su, Harvey Mudd College, Spring 2010.
 * 2) Analysis Lectures, simpleMath, 2014.

Without video

 * 1) MA541 Real analysis, Bhaba K. Sarma, IIT Guwahati, 2014 – 2015.
 * 2) Analysis I, Oleg Zaboronski, University of Warwick, 2012-2013.
 * 3) Analysis II, Oleg Zaboronski, University of Warwick, 2008-2009.
 * 4) Analysis III, Oleg Zaboronski, University of Warwick, 2015-2016.
 * 5) MATH 6101-090: Foundations of Real Analysis, David C. Royster, University of Kentucky, Fall 2008.
 * 6) MATH 6101-090: Foundations of Real Analysis, David C. Royster, University of Kentucky, Fall 2006.
 * 7) Math 320 Analysis I, Huyi Hu, Michigan State University, Fall 2015.
 * 8) Math 320 Analysis I, Lawrence Roberts, Michigan State University, Spring 2009.
 * 9) Math 421 Analysis II, Richard Siefring, Michigan State University, Spring 2010.
 * 10) Math 140-Analysis, Lei Ni, University of California, San Diego, Winter 2010.
 * 11) Math 4111 Introduction to Analysis, Mohan Kumar, Washington University in St. Louis, Fall 2013.
 * 12) Math 140A Foundations of Real Analysis I, Todd Kemp, University of California, San Diego, Winter 2014.
 * 13) Math 320 (Real Analysis), Santiago Cañez, University of California, San Diego, Winter 2015.
 * 14) Math 401: Introduction to Real Analysis, Aissa Wade, Penn State University, Spring 2009.
 * 15) Math 401: Introduction to Analysis, Aissa Wade, Penn State University, Fall 2005.
 * 16) Math 125A: Real Analysis, John K. Hunter, University of California, Davis campus, Fall 2012.
 * 17) Math 171: Fundamental Concepts of Analysis, K. Soundararajan, Penn State University,  Spring 2010.

Active participants
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this department, you can list your name here (this can help small departments grow and the participants communicate better; for large departments a list of active participants is not needed).

Основы математического анализа

title: Portal:Real analysis


text:
Talk:Anna Elbakyan

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 * Anna Elbakyan.jpg

title: Talk:Anna Elbakyan


text:
User:Aggrippa Harmony

Aggrippa Harmony Unwobuesor born on 16th March 1995 better known by his nickname Bobby Ray born to Mr. And Mrs Aggrippa Okpara is an indigene of Okporowo Community of Ahoada East Local Government Area of Rivers State.

EARLY LIFE

Harmony grew up with Hus parents in Okporowi community where he helped his mom (Ihuoma Aggrippa) in farming activies.

title: User:Aggrippa Harmony


text:
User:AlyssaBeavers/sandbox

To go on the "Autoimmunity" page under "Nutrition and Autoimmune diseases" section

Research has demonstrated an association between low serum vitamin D and autoimmune diseases, including multiple sclerosis, type 1 diabetes, and Systemic Lupus Erythematosus (commonly referred to simply as lupus). However, since photosensitivity occurs in lupus, patients are advised to avoid sunlight which may be responsible for vitamin D deficiency seen in this disease. Polymorphisms in the vitamin D receptor gene are commonly found in people with autoimmune diseases, giving one potential mechanism for vitamin D's role in autoimmunity. There is mixed evidence on the effect of vitamin D supplementation in type 1 diabetes, lupus, and multiple sclerosis. Vitamin D plays a role in immune function by acting on T cells and natural killer cells.

title: User:AlyssaBeavers/sandbox


text:
Hermann Ruissel

Hermann Ruissel or Herman Ruissel (c. 1360 - c. 1420) was a medieval Parisian goldsmith who crafted jewelry for the King of France and other persons of high rank.

From 1385 to 1389 Ruissel crafted jewellery mainly for Philip the Bold. After 1389 he also executed works for Charles VI of France and his brother Louis I, Duke of Orléans and Louis's wife Valentina Visconti, Duchess of Orléans. From 1390, he held the court title of Valet de chambre to Charles VI.

Among Ruissel's early works is the Three Brothers, created in 1389 for Philip's eventual successor John the Fearless. The Brothers (now lost) were for a time part of the Crown Jewels of England. The piece consisted of three rectangular red spinels of 70 carats each in a triangular arrangement, separated by three round white pearls of 10-20 carats each, with another pearl suspended from the lowest spinel. The middle of the pendant was a deep blue diamond cut as a pyramid or octahedron, and weighing about 30 carats.

Charles VI commissioned 32 works by Ruissel. 17 of these commissions stipulated that a duplicate of the work be made for his younger brother, Louis I, Duke of Orléans, as a gift.

Ruissel made 18 gold buttons decorated with broom flowers and heraldic mottos for the 1396 marriage of Isabella of Valois to King Richard II of England.

In 1400, Ruissel was commissioned to make a large image of the Trinity for Charles, using jewels – scores of pearls, and also diamonds, rubies, and sapphires – taken from a brooch in the form of a white-enameled golden hart and large golden collar which Richard II had given to Charles, and which Charles had had disassembled and melted down shortly after Richard's 1400 deposition and death.

Ruissel also made the piece now known as the Calvary of King Matthias Corvinus, a crucifix executed in gold, white enamel, pearls, and other gems, made using the Ronde-bosse technique. Margaret III, Countess of Flanders, commissioned the work as a gift celebrating the New Year in 1403 for her husband, Phillip the Bold. It is now in the treasury of Esztergom Basilica in Hungary.

For Henry V of England Ruissel created a golden necklace with white enamel bears, which were one of Henry's heraldic devices. The necklace was later gifted to Sigismund of Luxembourg, King of Germany and Hungary, on the occasion of his visit to England in 1416.

title: Hermann Ruissel


text:
College Algebra/Grammar quiz

This is a self quiz for the text Algebra/Grammar  {Which of the following could be a variable? -1 -2 -3 +a
 * type=""}
 * 1 is a constant, because it represents a specific number
 * 2 is a constant, because it represents a specific number
 * 3 is a constant, because it represents a specific number
 * a can be a variable, because it does not represents a specific number

{If y=2x+4z-5k, which of the following does Y not depend on? -x +y -z -k
 * type=""}
 * y is dependent on x
 * y is not dependent on itself
 * y is dependent on z
 * y is dependent on k

{Which variable is not the independent variable in the following equation: x=2y+3z-5k +x -y -z -k
 * type=""}
 * x is a dependent variable
 * y is a independent variable
 * z is a independent variable
 * k is a independent variable

{Which monomial is like 3x? -3y -2z -z +x
 * type=""}
 * If two monomials are "like", they can be added/subtracted
 * If two monomials are "like", they can be added/subtracted
 * If two monomials are "like", they can be added/subtracted
 * x and 3x can be added and subtracted from each other

{Using the distributive property, which is equal to 5(3+2+10) -5(3)+2+10 -5(3)+3(2)+2(10) +5(3)+5(2)+5(10) -15
 * type=""}
 * 2 and 10 must be multiplied by something
 * 2 and 10 must be multiplied by 5
 * 15 must be multiplied by 5

{Which set is a subset of the set {1, 3, 5, 7} -{1, 2, 3} -{1, 3, 5, 7, 9} -{1, 2, 3, 4 ...} +{3, 7}
 * type=""}
 * 2 is not a member of the original set
 * 9 is not a member of the original set
 * 4 is not a member of the original set

{Which is a member of the set of natural numbers? +6/3 -2/3 -1/2 -2.1 Return to College Algebra
 * type=""}
 * 6/3=2

title: College Algebra/Grammar quiz


text:
Discrete Mathematics/Set theory

Introduction
Set Theory starts very simply: it examines whether an object belongs, or does not belong, to a set of objects which has been described in some non-ambiguous way. From this simple beginning, an increasingly complex (and useful!) series of ideas can be developed, which lead to notations and techniques with many varied applications.

Definition: Set
The present definition of a set may sound very vague. A set can be defined as an unordered collection of entities that are related because they obey a certain rule.

'Entities' may be anything, literally: numbers, people, shapes, cities, bits of text, ... etc

The key fact about the 'rule' they all obey is that it must be well-defined. In other words, it must describe clearly what the entities obey. If the entities we're talking about are words, for example, a well-defined rule is:

X is English

A rule which is not well-defined (and therefore couldn't be used to define a set) might be:

X is hard to spell

 * where X is any word

Elements
An entity that belongs to a given set is called an element of that set. For example: Henry VIII is an element of the set of Kings of England. : kings of England {Henry VIII}

Set Notation
$$\{ -3, -2, -1, 0, 1, 2, 3 \}$$
 * To list the elements of a set, we enclose them in curly brackets, separated by commas. For example:

$$\{ \text{ integers between } -3 \text{ and } 3 \text{ inclusive} \}$$
 * The elements of a set may also be described verbally:

$$\{x | x \text{ is an integer and } |x| < 4 \}$$
 * The set builder notation may be used to describe sets that are too tedious to list explicitly. To denote any particular set, we use the letter $$x$$:

$$\{x | x \in \mathbb{Z}, |x| < 4 \}$$
 * or equivalently

$$\text{dog } \in \{ \text{quadrupeds}\}$$ $$\text{Washington DC} \notin \{\text{European capital cities}\}$$
 * The symbols $$\in$$ and $$\notin$$ denote the inclusion and exclusion of element, respectively:

$$\{2, 3, 5, 7, 11, \dots \}$$
 * Sets can contain an infinite number of elements, such as the set of prime numbers. Ellipses are used to denote the infinite continuation of a pattern:

Note that the use of ellipses may cause ambiguities, the set above may be taken as the set of integers individible by 4, for example.

 * Sets will usually be denoted using upper case letters: $$A$$, $$B$$, ...
 * Elements will usually be denoted using lower case letters: $$x$$, $$y$$, ...

The universal set
The set of all the entities in the current context is called the universal set, or simply the universe. It is denoted by $$U$$.

The context may be a homework exercise, for example, where the Universal set is limited to the particular entities under its consideration. Also, it may be any arbitrary problem, where we clearly know where it is applied.

The empty set
The set containing no elements at all is called the null set, or empty set. It is denoted by a pair of empty braces: $$\{ \}$$ or by the symbol $$\varnothing$$.

It may seem odd to define a set that contains no elements. Bear in mind, however, that one may be looking for solutions to a problem where it isn't clear at the outset whether or not such solutions even exist. If it turns out that there isn't a solution, then the set of solutions is empty.

For example:

 * If $$U = \{\text{words in the English language}\}$$ then $$\{\text{words with more than 50 letters}\} = \varnothing$$.
 * If $$U = \{\text{whole numbers}\}$$ then $$\{x | x^2 = 10\} = \varnothing$$.

Operations on the empty set
Operations performed on the empty set (as a set of things to be operated upon) can also be confusing. (Such operations are nullary operations.) For example, the sum of the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product). This may seem odd, since there are no elements of the empty set, so how could it matter whether they are added or multiplied (since “they” do not exist)? Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, notice that zero is the identity element for addition, and one is the identity element for multiplication.

Special numerical sets
Several sets are used so often, they are given special symbols.

Natural numbers
The 'counting' numbers (or whole numbers) starting at 1, are called the natural numbers. This set is sometimes denoted by N. So N = {1, 2, 3, ...}

Note that, when we write this set by hand, we can't write in bold type so we write an N in blackboard bold font: $$\mathbb{N}$$

Integers
All whole numbers, positive, negative and zero form the set of integers. It is sometimes denoted by Z. So Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

In blackboard bold, it looks like this: $$\mathbb{Z}$$

Real numbers
If we expand the set of integers to include all decimal numbers, we form the set of real numbers. The set of reals is sometimes denoted by R.

A real number may have a finite number of digits after the decimal point (e.g. 3.625), or an infinite number of decimal digits. In the case of an infinite number of digits, these digits may:

 * recur; e.g. 8.127127127...

 * ... or they may not recur; e.g. 3.141592653...

In blackboard bold: $$\mathbb{R}$$

Rational numbers
Those real numbers whose decimal digits are finite in number, or which recur, are called rational numbers. The set of rationals is sometimes denoted by the letter Q.

A rational number can always be written as exact fraction p/q; where p and q are integers. If q equals 1, the fraction is just the integer p. Note that q may NOT equal zero as the value is then undefined.
 * For example: 0.5, -17, 2/17, 82.01, 3.282828... are all rational numbers.

In blackboard bold: $$\mathbb{Q}$$

Irrational numbers
If a number can't be represented exactly by a fraction p/q, it is said to be irrational.
 * Examples include: √2, √3, π.

Set Theory Exercise 1
Click the link for Set Theory Exercise 1

Relationships between Sets
We’ll now look at various ways in which sets may be related to one another.

Equality
Two sets $$A$$ and $$B$$ are said to be equal if and only if they have exactly the same elements. In this case, we simply write:

$$A = B$$

Note two further facts about equal sets:

 * The order in which elements are listed does not matter.

 * If an element is listed more than once, any repeat occurrences are ignored.

So, for example, the following sets are all equal:

$$\{1, 2, 3\} = \{3, 2, 1\} = \{1, 1, 2, 3, 2, 2\}$$

(You may wonder why one would ever come to write a set like $$\{1, 1, 2, 3, 2, 2\}$$. You may recall that when we defined the empty set we noted that there may be no solutions to a particular problem - hence the need for an empty set.  Well, here we may be trying several different approaches to solving a problem, some of which in fact lead us to the same solution.  When we come to consider the distinct solutions, however, any such repetitions would be ignored.)

Subsets
If all the elements of a set $$A$$ are also elements of a set $$B$$, then we say that $$A$$ is a subset of $$B$$ and we write:

$$A \subseteq B$$

For example:

In the examples below:

If $$T = \{2, 4, 6, 8, 10\}$$ and $$E = \{\text{even integers}\}$$, then $$T \subseteq E$$ If $$A = \{\text{alphanumeric characters}\}$$ and $$P = \{\text{printable characters}\}$$, then $$A \subseteq P$$ If $$Q = \{\text{quadrilaterals}\}$$ and $$F = \{\text{plane figures bounded by four straight lines}\}$$, then $$Q \subseteq F$$

Notice that $$A \subseteq B$$ does not imply that $$B$$ must necessarily contain extra elements that are not in $$A$$; the two sets could be equal – as indeed $$Q$$ and $$F$$ are above. However, if, in addition, $$B$$ does contain at least one element that isn’t in $$A$$, then we say that $$A$$ is a proper subset of $$B$$. In such a case we would write:

$$A \subset B$$

In the examples above:

$$E$$ contains ... -4, -2, 0, 2, 4, 6, 8, 10, 12, 14, ... , so $$T \subset E$$

$$P$$ contains $, ;, &, ..., so $$A \subset P$$

But $$Q$$ and $$F$$ are just different ways of saying the same thing, so $$Q = F$$

The use of $$\subset$$ and $$\subseteq$$; is clearly analogous to the use of < and ≤ when comparing two numbers.

Notice also that every set is a subset of the universal set, and the empty set is a subset of every set.

(You might be curious about this last statement: how can the empty set be a subset of anything, when it doesn’t contain any elements? The point here is that for every set $$A$$, the empty set doesn’t contain any elements that aren't in $$A$$.  So $$\oslash \subseteq A$$ for all sets $$A$$.)

Finally, note that if $$A \subseteq B \text{ and } B \subseteq A \text{ then } A$$ and $$B$$ must contain exactly the same elements, and are therefore equal. In other words:

$$A \subseteq B \text{ and } B \subseteq A \text{ then } A = B$$

Disjoint
Two sets are said to be disjoint if they have no elements in common. For example:

If $$A = \{\text{even numbers}\}$$ and $$B = \{1, 3, 5, 11, 19\}$$, then $$A$$ and $$B$$ are disjoint sets

Venn Diagrams
A Venn diagram can be a useful way of illustrating relationships between sets.

In a Venn diagram:

 * The universal set is represented by a rectangle. Points inside the rectangle represent elements that are in the universal set; points outside represent things not in the universal set.  You can think of this rectangle, then, as a 'fence' keeping unwanted things out - and concentrating our attention on the things we're talking about.

 * Other sets are represented by loops, usually oval or circular in shape, drawn inside the rectangle. Again, points inside a given loop represent elements in the set it represents; points outside represent things not in the set.

On the left, the sets A and B are disjoint, because the loops don't overlap.

On the right A is a subset of B, because the loop representing set A is entirely enclosed by loop B.

Venn diagrams: Worked Examples
Example 1

Fig. 3 represents a Venn diagram showing two sets A and B, in the general case where nothing is known about any relationships between the sets.

Note that the rectangle representing the universal set is divided into four regions, labelled i, ii, iii and iv.

What can be said about the sets A and B if it turns out that:

 * (a) region ii is empty?

 * (b) region iii is empty?

(a) If region ii is empty, then A contains no elements that are not in B. So A is a subset of B, and the diagram should be re-drawn like Fig 2 above.

(b) If region iii is empty, then A and B have no elements in common and are therefore disjoint. The diagram should then be re-drawn like Fig 1 above.

Example 2

 * (a) Draw a Venn diagram to represent three sets A, B and C, in the general case where nothing is known about possible relationships between the sets.

 * (b) Into how many regions is the rectangle representing U divided now?

 * (c) Discuss the relationships between the sets A, B and C, when various combinations of these regions are empty.

(a) The diagram in Fig. 4 shows the general case of three sets where nothing is known about any possible relationships between them.

(b) The rectangle representing U is now divided into 8 regions, indicated by the Roman numerals i to viii.

(c) Various combinations of empty regions are possible. In each case, the Venn diagram can be re-drawn so that empty regions are no longer included. For example:
 * If region ii is empty, the loop representing A should be made smaller, and moved inside B and C to eliminate region ii.

 * If regions ii, iii and iv are empty, make A and B smaller, and move them so that they are both inside C (thus eliminating all three of these regions), but do so in such a way that they still overlap each other (thus retaining region vi).

 * If regions iii and vi are empty, 'pull apart' loops A and B to eliminate these regions, but keep each loop overlapping loop C.

 * ...and so on. Drawing Venn diagrams for each of the above examples is left as an exercise for the reader.

Example 3

The following sets are defined:

 * U = {1, 2, 3, …, 10}
 * A = {2, 3, 7, 8, 9}
 * B = {2, 8}
 * C = {4, 6, 7, 10}

Using the two-stage technique described below, draw a Venn diagram to represent these sets, marking all the elements in the appropriate regions.

The technique is as follows:

 * Draw a 'general' 3-set Venn diagram, like the one in Example 2.

 * Go through the elements of the universal set one at a time, once only, entering each one into the appropriate region of the diagram.

 * Re-draw the diagram, if necessary, moving loops inside one another or apart to eliminate any empty regions.

Don't begin by entering the elements of set A, then set B, then C – you'll risk missing elements out or including them twice!

Solution

After drawing the three empty loops in a diagram looking like Fig. 4 (but without the Roman numerals!), go through each of the ten elements in U - the numbers 1 to 10 - asking each one three questions; like this:

First element: 1
 * Are you in A? No
 * Are you in B? No
 * Are you in C? No

A 'no' to all three questions means that the number 1 is outside all three loops. So write it in the appropriate region (region number i in Fig. 4).

Second element: 2
 * Are you in A? Yes
 * Are you in B? Yes
 * Are you in C? No

Yes, yes, no: so the number 2 is inside A and B but outside C. Goes in region iii then.

...and so on, with elements 3 to 10.

The resulting diagram looks like Fig. 5.

The final stage is to examine the diagram for empty regions - in this case the regions we called iv, vi and vii in Fig. 4 - and then re-draw the diagram to eliminate these regions. When we've done so, we shall clearly see the relationships between the three sets.

So we need to:
 * pull B and C apart, since they don't have any elements in common.

 * push B inside A since it doesn't have any elements outside A.

The finished result is shown in Fig. 6.

The regions in a Venn Diagram and Truth Tables
Perhaps you've realized that adding an additional set to a Venn diagram doubles the number of regions into which the rectangle representing the universal set is divided. This gives us a very simple pattern, as follows:

 * With one set loop, there will be just two regions: the inside of the loop and its outside.

 * With two set loops, there'll be four regions.

 * With three loops, there'll be eight regions.

 * ...and so on.

It's not hard to see why this should be so. Each new loop we add to the diagram divides each existing region into two, thus doubling the number of regions altogether.

But there's another way of looking at this, and it's this. In the solution to Example 3 above, we asked three questions of each element: Are you in A? Are you in B? and Are you in C? Now there are obviously two possible answers to each of these questions: yes and no. When we combine the answers to three questions like this, one after the other, there are then 23 = 8 possible sets of answers altogether. Each of these eight possible combinations of answers corresponds to a different region on the Venn diagram.

The complete set of answers resembles very closely a Truth Table - an important concept in Logic, which deals with statements which may be true or false. The table on the right shows the eight possible combinations of answers for 3 sets A, B and C.

You'll find it helpful to study the patterns of Y's and N's in each column.

 * As you read down column C, the letter changes on every row: Y, N, Y, N, Y, N, Y, N

 * Reading down column B, the letters change on every other row: Y, Y, N, N, Y, Y, N, N

 * Reading down column A, the letters change every four rows: Y, Y, Y, Y, N, N, N, N

Set Theory Exercise 2
Click link for Set Theory Exercise 2

Operations on Sets
Just as we can combine two numbers to form a third number, with operations like 'add', 'subtract', 'multiply' and 'divide', so we can combine two sets to form a third set in various ways. We'll begin by looking again at the Venn diagram which shows two sets A and B in a general position, where we don't have any information about how they may be related.

The first two columns in the table on the right show the four sets of possible answers to the questions Are you in A? and Are you in B? for two sets A and B; the Roman numerals in the third column show the corresponding region in the Venn diagram in Fig. 7.

Intersection
Region iii, where the two loops overlap (the region corresponding to 'Y' followed by 'Y'), is called the intersection of the sets A and B. It is denoted by A ∩ B. So we can define intersection as follows:

 * The intersection of two sets A and B, written A ∩ B, is the set of elements that are in A and in B.

(Note that in symbolic logic, a similar symbol, $$\scriptstyle \wedge$$, is used to connect two logical propositions with the AND operator.)

For example, if A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, then A ∩ B = {2, 4}.

We can say, then, that we have combined two sets to form a third set using the operation of intersection.

Union
In a similar way we can define the union of two sets as follows:
 * The union of two sets A and B, written A ∪ B, is the set of elements that are in A or in B (or both).

The union, then, is represented by regions ii, iii and iv in Fig. 7.

(Again, in logic a similar symbol, $$\scriptstyle \vee$$, is used to connect two propositions with the OR operator.)

 * So, for example, {1, 2, 3, 4} ∪ {2, 4, 6, 8} = {1, 2, 3, 4, 6, 8}.

You'll see, then, that in order to get into the intersection, an element must answer 'Yes' to both questions, whereas to get into the union, either answer may be 'Yes'.

The ∪ symbol looks like the first letter of 'Union' and like a cup that will hold a lot of items. The ∩ symbol looks like a spilled cup that won't hold a lot of items, or possibly the letter 'n', for i'n'tersection. Take care not to confuse the two.

Difference

 * The difference of two sets A and B (also known as the set-theoretic difference of A and B, or the relative complement of B in A) is the set of elements that are in A but not in B.

This is written A - B, or sometimes A \ B.

The elements in the difference, then, are the ones that answer 'Yes' to the first question Are you in A?, but 'No' to the second Are you in B?. This combination of answers is on row 2 of the above table, and corresponds to region ii in Fig.7.

 * For example, if A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, then A - B = {1, 3}.

Complement
So far, we have considered operations in which two sets combine to form a third: binary operations. Now we look at a unary operation - one that involves just one set.

Clearly, this is the set of elements that answer 'No' to the question Are you in A?.
 * The set of elements that are not in a set A is called the complement of A. It is written A′ (or sometimes AC, or $$\bar{A}$$).

 * For example, if U = N and A = {odd numbers}, then A′ = {even numbers}.

Notice the spelling of the word complement: its literal meaning is 'a complementary item or items'; in other words, 'that which completes'. So if we already have the elements of A, the complement of A is the set that completes the universal set.

1. Commutative

 * $$A \cup B = B \cup A$$
 * $$A \cap B = B \cap A$$

2. Associative

 * $$A \cup (B \cup C) = (A \cup B) \cup C$$
 * $$A \cap (B \cap C) = (A \cap B) \cap C$$

3. Distributive

 * $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$
 * $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$

4. Special properties of complements

 * $$(A')' = A$$
 * $$U'=\phi$$
 * $$\phi'=U$$
 * $$A\cap B' = A - B$$

5. De Morgan's Law

 * $$(A \cap B)'= A' \cup B'$$
 * $$(A \cup B)' = A' \cap B'$$.

Cardinality
Finally, in this section on Set Operations we look at an operation on a set that yields not another set, but an integer.

 * The cardinality of a finite set A, written | A | (sometimes #(A) or n(A)), is the number of (distinct) elements in A. So, for example:
 * If A = {lower case letters of the alphabet}, | A | = 26.

Generalized set operations
If we want to denote the intersection or union of n sets, A1, A2, ..., An (where we may not know the value of n) then the following generalized set notation may be useful:

 * A1 ∩ A2 ∩ ... ∩ An = $$\scriptstyle \bigcap_{i=1}^{n}$$ Ai

 * A1 ∪ A2 ∪ ... ∪ An = $$\scriptstyle \bigcup_{i=1}^{n}$$ Ai

In the symbol $$\scriptstyle \bigcap_{i=1}^{n}$$ Ai, then, i is a variable that takes values from 1 to n, to indicate the repeated intersection of all the sets A1 to An.

Set Theory Exercise 3
Click link for Set Theory Exercise 3

title: Discrete Mathematics/Set theory


text:
Talk:Media literacy

Course instructor's notes
This project was developed by students enrolled in a course offered by Renee Hobbs, Professor at Temple University School of Communications and Theater, in the Fall of 2007 entitled Theory and Practice of Media Literacy Education.

Dear BTMM 5310 Students:

Here are some comments on your initial work:

1. Please be sure to add your scholar's citation to the list at the bottom of the top page of the wiki. We'll keep using reference lists at the botton of each page. Some of the scholars you read and summarized are not on the list. Please fix this.

2. Let's carefully distinguish between the chapters in Masterman we read (Teaching the Media, 1985) and Masterman's earlier 1980 book (as critiqued by Alvarado). This is not clear in some of the new writing posted. Please fix.

3. I've moved around the pages to take better advantage of hyperlinking this semester, and also to solve the problem of length (many shorter pages are less intimidating.) You'll see now there are a number of smaller pages. Let's try to take advantage of shorter pages and hyperlinking, where it makes sense.

4. If you would like to create new pages (or hyperlinks to existing pages), just put two brackets in front of and in back of the name for the new page, like this Intellectual Traditions in Media Literacy. When you save the page, you will see the new page name in red. Clicking on it takes to you that page, to add content.

5. Also, let's practice using the "summary" line at the bottom. (I forget to do this all the time, but it's useful when doing collaborative writing.) You don't have to use it for small changes, but if you use it when you add content, it will help us keep track of our contributions...and make it clear where we have disagreements, points of contention, etc.

Wikiversity notes

 * Note also: Media Literacy. --McCormack 09:22, 31 May 2008 (UTC)
 * The above resource now appears here as Media literacy/notes. --CQ 14:44, 6 September 2009 (UTC)

Rights of the Child and access to the mass media
Hi,

As far as mass media and education are concerned, shouldn't it be mentioned that the Convention on the Rights of the Child is promoting children access to the mass media in these words : Article 17

States Parties recognize the important function performed by the mass media and shall ensure that the child has access to information and material from a diversity of national and international sources, especially those aimed at the promotion of his or her social, spiritual and moral well-being and physical and mental health. To this end, States Parties shall: (a) Encourage the mass media to disseminate information and material of social and cultural benefit to the child and in accordance with the spirit of article 29; (b) Encourage international co-operation in the production, exchange and dissemination of such information and material from a diversity of cultural, national and international sources; (c) Encourage the production and dissemination of children's books; (d) Encourage the mass media to have particular regard to the linguistic needs of the child who belongs to a minority group or who is indigenous; (e) Encourage the development of appropriate guidelines for the protection of the child from information and material injurious to his or her well-being, bearing in mind the provisions of articles 13 and 18. This may also balance the view of media as just something inevitable we have to cope with and at best educate about. Astirmays (discuss • contribs) 05:33, 11 September 2013 (UTC)

Additional resources on this topic
Additional resources on this topic are on-line at: https://www.redbanklibrary.org/program-resource-pages/fake-news It may be useful to scan these materials searching for additional resources or viewpoints that can improve this (already excellent) resource. Thanks! --Lbeaumont (discuss • contribs) 13:29, 10 October 2020 (UTC)

Free Speech and Personal Responsibility
Freedom of speech is broadly protected by the US supreme court. The right to free speech must be balanced by our personal responsibility to know how we know. While it is true that media communications are constructed, it is also true that objective reality exists. Furthermore, reality is our common ground. Media consumers benefit by being able to discern more and less accurate portrayals of our shared reality. I contend that this principle is fundamental to media literacy and encourage highlighting this principle in the course. Thanks! --Lbeaumont (discuss • contribs) 14:56, 10 October 2020 (UTC)

title: Talk:Media literacy


text:
Category talk:Villages in Illinois

Hello, I'm working on adding reference to the census information from the city data website, any help would be appreciated. Regards — The Sunshine Man 15:49, 16 May 2007 (UTC)

title: Category talk:Villages in Illinois


text:
Waves

Waves
Waves are the transfer of energy from one place to another in the form of a periodic disturbance. For example, when a stone is thrown onto a water surface, water molecules on the surface move up and down. As a water molecule moves it collides with the nearest water molecule and transfers energy to that molecule, thus setting it in motion.

Transverse and Longitudinal waves
There are two types of waves: Transverse and Longitudinal Waves.:
 * 1) Transverse Waves are waves that vibrate perpendicular to the direction of travel of the wave. Examples of Transverse Waves include, but are certainly not limited, to: Electro-Magnetic Waves*, Sine and Cosine Waves,and the wave of a string.
 * 2) Longitudinal Waves are waves that vibrate parallel to the direction of travel of the wave. They are also commonly referred to as Pressure or Compression Waves. Longitudinal waves include sound waves, (Although you wouldn't know it if you looked at sound waves on an osciliscope!) and the waves of a spring or slinky pulled back and forth

 * Electromagnetic Waves can travel through a vacuum. These waves are unique because they do not need a medium of travel unlike all other waves. Light and heat are examples of electromagnetic waves. Electromagnetic Waves travel at the speed of light and have no mass.

Equation for Velocity and Wavelength
In this equation: It can be understood as how frequently a certain distance is transversed.
 * f is the frequency (The number of peaks to pass a point in a given time, usually in Hz)
 * λ (The Greek letter lambda) is the wavelength (usually measured in meters)
 * V is the resulting velocity in m/s (meters per second). Wave speed is equal to the frequency times the wave length.

Medium
A medium is a substance through which a wave can travel. For example: Sound waves need a medium. It travels by the vibration of liquid,solid and gas.

Although some soundtrack wave can travel without a medium. For example: Visible light

Resources

 * Waves in plasmas

title: Waves


text:
US History/Road to Revolution

The French and Indian War
(The following text is from Wikipedia)

The French and Indian War (1754–1763) was the North American chapter of the Seven Years' War. The name refers to the two main enemies of the British, the royal French forces and the various American Indian forces allied with them. This conflict, the fourth such colonial war between the kingdoms of France and Great Britain, resulted in the British conquest of all of New France east of the Mississippi River, as well as Spanish Florida. France ceded control of French Louisiana west of the Mississippi to its Spanish ally, to compensate it for its loss of Florida. By the end of this war France kept only the tiny islands of Saint Pierre and Miquelon north of the Caribbean. These colonies today still allow France access to the Grand Banks.

In Great Britain and France, the North American theatre of the Seven Years' War war usually has no special name, and so the entire worldwide conflict is known as the Seven Years' War (or the Guerre de sept ans). The "Seven Years" refers to events in Europe, from the official declaration of war in 1756 to the signing of the peace treaty in 1763. These dates do not correspond with the actual fighting in North America, where the fighting between the two colonial powers was largely concluded in six years, from the Jumonville Glen skirmish in 1754 to the capture of Montreal in 1760.

Elsewhere the conflict is known by several names. In British North America, wars were often named after the sitting British monarch, such as King William's War or Queen Anne's War. Because there had already been a King George's War in the 1740s, British colonists named the second war in King George's reign after their opponents, and thus it became known as the French and Indian War. This traditional name remains standard in the United States, although it obscures the fact that American Indians fought on both sides of the conflict. American historians generally use the traditional name or the European title (the Seven Years' War), and have also invented other, less frequently used names for the war, including the Fourth Intercolonial War and the Great War for the Empire. Canadian francophones and English speakers both refer to it as the Seven Years' War (Guerre de Sept Ans) or the War of the Conquest (Guerre de la Conquête), as the war in which New France was conquered by the British and became part of the British Empire. This war was also known as the Forgotten War.

Reasons for war
The French and Indian War began less than a decade after France and Great Britain had fought on opposing sides in the European War of the Austrian Succession (1740–1748). One cause for the conflict was territorial expansion. Newfoundland's Grand Banks were fertile fishing grounds and coveted by both sides. Both sides also wanted to expand their territories for trapping furs to trade, and for other pursuits that aided their economic interests. Both the British and the French used trading posts and forts to claim the Ohio Country, the vast territory between the Appalachian Mountains and the Mississippi River, from the Great Lakes to the Gulf of Mexico. English claims resulted from royal grants with no definite western boundaries. La Salle had claimed the Mississippi River for France: its drainage area includes the Ohio River Valley. Both Great Britain and France took advantage of Native American factions to secure these claims, to protect their territories, and to keep the other from growing too strong.

A second cause was political & religious ideology. The English Protestant colonists feared papal influence in North America. New France was administered by French governors and Roman Catholic hierarchy. French missionaries included Armand de La Richardie. English history was told as a story of freedom from Catholic (i.e., foreign) influence. French control over North America represented a threat to Great Britain. In their turn, the French feared English anti-Catholicism, in a time when Catholics were still being persecuted under English law.

Céloron's expedition
In June 1747 the Governor-General of New France, the Marquis de la Jonquière, ordered Pierre-Joseph Céloron to lead an expedition to the Ohio Country to remove British influence from the area. Céloron was also to confirm allegiance of the Native Americans in the Ohio territory to the French crown.

Céloron's expedition consisted of 213 soldiers of the Troupes de la marine (French Marines) transported by twenty-three canoes. The expedition left Lachine on June 15, 1749, and two days later reached Fort Frontenac. It then continued along the shoreline of present-day Lake Erie. At Chautauqua Portage (Barcelona, New York), it moved inland to the Allegheny River.

The troop headed south to the Ohio River at present-day Pittsburgh, where Céloron buried lead plates engraved with the French claim to the Ohio Country. Whenever British merchants or fur-traders were encountered by the French, they were informed of the illegality of being on French territory and told to leave the Ohio Country. When the expedition arrived at Logstown, the Native Americans there informed Céloron that they owned the Ohio Country, and they would trade with the British, despite anything the French said.

Céloron continued the expedition. At its farthest point south, it reached the junction between the Ohio River and the Miami River, just south of the village of Pickawillany. Here lived the old Chief of the Miami tribe, whom Céloron called "Old Britain." When Céloron arrived at Pickawillany, he informed the elderly Chief of "dire consequences" of continuing to trade with the British. "Old Britain" ignored the warning. After this meeting, Céloron and his expedition began the trip home, reaching Montreal only on November 10, 1749. In his report, Céloron wrote: "All I can say is that the Natives of these localities are very badly disposed towards the French, and are entirely devoted to the English. I don't know in what way they could be brought back."

Langlade's expedition
On March 17, 1752, Governor-General de la Jonquière died. His temporary replacement was Charles le Moyne de Longueuil. It was not until July 1, 1752 that Ange Duquense de Menneville arrived in New France to take over the post.

In the spring of 1752, Longueuil dispatched an expedition to the Ohio River area. The expedition was led by Charles Michel de Langlade, an officer in the Troupes de la marine. Langlade was given 300 men, some French-Canadians, and others members of the Ottawa tribe. His objective was to punish the Miami of Pickawillany for continuing to trade with the British. At dawn on June 21, 1752, the war party attacked the British trading center at Pickawillany, killing fourteen people of the Miami nation, including "Old Britain." The expedition then returned home.

Marin's expedition
In the spring of 1754, Paul Marin de la Malgue was given command of a 2,000 man force of Troupes de la Marine and Aboriginals. His orders were to protect the Ohio from the British. Marin followed the route that Céloron had mapped out four years before. However, where Céloron had buried lead plates, Marin was constructing and garrisoning forts. The first fort that was constructed by Paul Marin was at Presque Isle (Erie, Pennsylvania) on Lake Erie's south shore. He then had a road built to the headwaters of Rivière aux Boeuf (now known as Waterford, Pennsylvania). Marin then constructed a second fort at Le Boeuf, designed to guard the headwaters of the Rivière aux Boeuf.

Tanaghrisson's proclamation
On September 3, 1753, Tanaghrisson (d. 1754), Chief of the Mingo, arrived at Fort Le Boeuf. One tradition states that Tanaghrisson hated the French because they had killed and eaten his father. Tanaghrisson told Marin, "I shall strike . . .", threatening the French. The show of force by the French had alarmed the Iroquois in the area. They sent Mohawk runners to William Johnson's manor in Upper New York. Johnson, known to the Iroquois as "Warraghiggey", meaning "He who does big business", had become a respected member of the Iroquois Confederacy in the area. In 1746, Johnson was made a colonel of the Iroquois, and later a colonel of the Western New York Militia.

At Albany, New York, there was a meeting between Governor Clinton of New York and Chief Hendrick, as well as other officials from a handful of American colonies. Chief Hendrick insisted that the British abide by their obligations and block French expansion. When an unsatisfactory response was offered by Clinton, Chief Hendrick proclaimed that the "Covenant Chain", a long-standing friendly relationship between the Iroquois Confederacy and the British Crown, was broken.

Dinwiddie's reaction
Governor Robert Dinwiddie of Virginia found himself in a predicament. Many merchants had invested heavily in fur trading in Ohio. If the French made good on their claim to the Ohio Country and drove out the British, then the Virginian merchants would lose a lot of money. Dinwiddie could not possibly allow the loss of the Ohio Country to France. In October 1753 he wrote a letter to the commander of the French forces in the Ohio Country, Jacques Legardeur de Saint-Pierre, demanding an immediate French withdrawal. To deliver it he delegated Major George Washington of the Virginia militia. Major Washington left for Fort Le Boeuf on the 31st of October, along with his interpreter Jacob Van Braam and several other men.

A few days later, Washington and his party arrived at Wills Creek (Cumberland, Maryland). Here Washington enlisted the help of Christopher Gist, a surveyor who was familiar with the area. They arrived at Logstown on November 24, 1753. At Logstown, Washington met with Tanaghrisson, who was angry over the French military encroachment upon his land. Washington convinced Tanaghrisson to accompany his small group to Fort Le Boeuf.

On December 12, 1753, Washington and his men reached Fort Le Boeuf. Commander Saint-Pierre invited Washington to dine with him that evening. Over dinner, Washington presented Saint-Pierre with the letter from Dinwiddie. Saint-Pierre was civil in his response, saying, "As to the Summons you send me to retire, I do not think myself obliged to obey it." The French explained to Washington that France's claim to the region was superior to that of the British, as René-Robert Cavelier, Sieur de a Salle (1643–1687) had explored the Ohio Country nearly a century earlier.

Washington's party left Fort Le Boeuf early on December 16, 1753. By January 16, 1754, they had arrived in Williamsburg, Virginia. In his report, Washington stated, "The French had swept south." They had constructed and garrisoned forts at Presque Isle, Le Boeuf and Venango.

War
The French and Indian War was the last of four major colonial wars between the British, the French, and their Native American allies. Unlike the previous three wars, the French and Indian War began on North American soil and then spread to Europe, where Britain officially declared war on France on May 15, 1756, marking the beginnings of the Seven Years' War in Europe. Native Americans fought for both sides, but primarily alongside the French (with one exception being the Iroquois Confederacy, which sided with the American colonies and Britain). The first major event of the war was in 1754. Lieutenant Colonel George Washington, then twenty-one years of age, was sent to negotiate boundaries with the French, who did not give up their forts. Washington led a group of Virginian (colonial) troops to confront the French at Fort Duquesne (present day Pittsburgh). Washington discovered the French troops at the Battle of Jumonville Glen (about six miles or ten kilometers North-West of soon-to-be-established Fort Necessity). In the ensuing skirmish, a French Officer, Joseph Coulon de Jumonville, was killed. Washington pulled back a few miles and established Fort Necessity. The French forced Washington and his men to retreat. Meanwhile, the Albany Congress was taking place as means to discuss further action.

Edward Braddock led a campaign against the French at Fort Duquesne in 1755. Washington was again among the British and colonial troops. Braddock employed European tactics -- bold, linear marches and firing formations -- and employed heavy cannon. This led to disaster at the Monongahela. The French and natives were heavily outmanned and outgunned. But they used superior tactics, taking cover behind trees and bushes to gun down and rout the British. Braddock was killed. Despite four close calls, Washington escaped unharmed and led the survivors in retreat. This stunning British defeat heralded a string of major French victories over the next few years, at Fort Oswego, Fort William Henry, Fort Duquesne, and Carillon, where veteran Montcalm famously defeated five times his number. The sole British successes in the early years of the war came in 1755, at the Battle of Lake George, which secured the Hudson Valley; and in the taking of Fort Beauséjour (which protected the Nova Scotia frontier) by Lieutenant Colonel Robert Monckton. A consequence of this last battle was the subsequent forced deportation of the Acadian population of Nova Scotia and the Beaubassin region of Acadia.

In 1756 William Pitt became Secretary of State of Great Britain. His leadership, and France's continued neglect of the North-American theater, eventually turned the tide in favor of the British. The French were driven from many frontier posts such as Fort Niagara, and the key Fortress Louisbourg fell to the British in 1758. In 1759, the Battle of the Plains of Abraham gave Quebec City to the British, who had to withstand a siege there after the Battle of Sainte-Foy a year later. In September of 1760, Pierre François de Rigaud, Marquis de Vaudreuil-Cavagnal, the King's Governor of New France, negotiated a surrender with British General Jeffrey Amherst. General Amherst granted Vaudreuil's request that any French residents who chose to remain in the colony would be given freedom to continue worshiping in their Roman Catholic tradition, continued ownership of their property, and the right to remain undisturbed in their homes. The British provided medical treatment for the sick and wounded French soldiers, and French regular troops were returned to France aboard British ships with an agreement that they were not to serve again in the present war.

Summary of the war in America In 1752 the French and their Native allies raided a trading outpost sited at modern day Cleveland rid the area of Pennsylvanians. In 1754 General George Washington attacked French soldiers and then became trapped in his poorly built, Fort Necessity in Great Meadows Pennsylvania and more than one-third of Washingtons's men shortly became casualties. Twenty-two year old Washington and his men surrendered and were allowed to leave back to Virginia. In July 1755, a few miles south of Fort Duquensne in Pennsylvania, the combined forces of French and Natives attacked British colonial troops that were preparing a to assault the fort. The aftermath that ensued would result in a British defeat and General Edward Braddock would be killed. Once London heard of this Britain declared war upon France and formally began the seven years war. After this the British feared that France would attempt to retake Nova Scotia and that the 12,000 French Nova Scotians would break their neutrality, so the British military forced around seven thousand French Nova Scotians from their homeland. This was history's first large-scale modern deportation, the French would be sent from Louisiana to the Caribbean and families would become torn apart. In July of 1758 The British had recaptured the fort at Loiusberg winning control of the St. Lawerence River. This would cut the major French supply route and open up more supply lines for the British. In the fall of 1758 the Shawnee and Delaware Natives accepted peace offerings from the British and the French abandoned Fort Duquesne. A decisive attack would happen in the fall of 1759 when General James Wolfe's forces defeated the French on the Plains of Abraham and thus taking Quebec. A year after this event the British would capture Montreal and the North American stage of the war would be over.

Outcome
Though most of the North American fighting ended on September 8, 1760, when the Marquis de Vaudreuil surrendered Montreal — and effectively all of Canada — to Britain (one notable late battle allowed the capture of Spanish Havana by British and colonial forces in 1762), the war officially ended with the signing of the Treaty of Paris on February 10, 1763. The treaty sealed France's loss of all its North American possessions east of the Mississippi except for Saint Pierre and Miquelon islands off Newfoundland. All of Canada was ceded to Britain. France regained the Caribbean islands of Guadeloupe and Martinique, which had been occupied by the British. The economic value of these islands to France was greater than that of Canada at the time, because of their rich sugar crops; and the islands were easier to defend. However, the British were happy to take New France: defense was not an issue, and they had many sources of sugar. Spain gained Louisiana, including New Orleans, in compensation for its loss of Florida to the British.

French Canada contained approximately 65,000 French-speaking Roman Catholic residents. Early in the war, in 1755, the British had expelled French settlers from Acadia. (Some of these eventually fled to Louisiana, creating the Cajun population.) Now at peace, and eager to secure control of its hard-won colony, Great Britain made concessions to its newly conquered subjects with the Quebec Act of 1774. The history of the Seven Years' War, particularly the siege of Québec and the death of British Brigadier General James Wolfe, generated a vast number of ballads, broadsides, images, maps and other printed materials, which testify to how this event continued to capture the imagination of the British public long after Wolfe's death in 1759. The European theatre of the war was settled by the Treaty of Hubertusburg on February 15, 1763. The war changed economic, political, and social relations between Britain and its colonies. It plunged Britain into debt, which the Crown chose to pay off with tax money from its colonies. These taxes contributed to the beginning the American Revolutionary War.

Battles and expeditions

 * United States

 * Battle of Jumonville Glen (May 28, 1754)
 * Battle of Fort Necessity, aka the Battle of Great Meadows (July 3, 1754)
 * Braddock Expedition (Battle of the Monongahela aka Battle of the Wilderness) (July 9, 1755)
 * Kittanning Expedition (climax September 8, 1756)
 * Battle of Fort Duquesne (September 14, 1758)
 * Battle of Fort Ligonier (October 12, 1758)
 * Forbes Expedition (climax November 25, 1758)

 * Province of New York
 * Battle of Lake George (1755)
 * Battle of Fort Oswego (August, 1756)
 * Battle on Snowshoes (January 21, 1757)
 * Battle of Fort Bull (March 27, 1756)
 * Battle of Sabbath Day Point (July 26, 1757)
 * Battle of Fort William Henry (August 9, 1757)
 * Attack on German Flatts (1757) (November 12, 1757)
 * Battle of Carillon (July 8, 1758)
 * Battle of Ticonderoga (1759)
 * Battle of La Belle-Famille (July 24, 1759)
 * Battle of Fort Niagara (1759)
 * Battle of the Thousand Islands, 16-25 August, 1760

 * West Virginia
 * Battle of Great Cacapon (April 18, 1756)

 * Canada
 * New Brunswick
 * Battle of Fort Beauséjour (June 16, 1755)

 * Nova Scotia
 * Battle of Louisburg (July 27, 1758)

 * Ontario
 * Battle of Fort Frontenac (August 25, 1758)
 * Battle of the Thousand Islands, 16-25 August, 1760

 * Quebec
 * Battle of Beauport (July 31, 1759)
 * Battle of the Plains of Abraham (September 13, 1759)
 * Battle of Sainte-Foy (April 28, 1760)
 * Battle of Restigouche, July 3-8, (1760)

 * Newfoundland
 * Battle of Signal Hill September 15, 1762

Proclamation of 1763
(The following text is taken from the Wikipedia article)

The Royal Proclamation of 1763 was issued October 7, 1763 by George III following Great Britain's acquisition of French territory in North America after the end of the Seven Years' War. The purpose of the proclamation was to make sure Britain could control its new territory for its The Proclamation in essence forbade Americans from settling or buying land west of the Appalachians. Colonists were angry because many already had land in that area. Additionally, the Proclamation gave the Crown a monopoly in land bought from Native Americans.

Native land
In the fall of 1763, a royal decree was issued that prohibited the North American colonists from establishing or maintaining settlements west of an imaginary line running down the crest of the Appalachian Mountains. The proclamation acknowledged that Native Americans owned the lands on which they were then residing and white settlers in the area were to be removed.

However, provision was made to allow specially licensed individuals and entities to operate fur trading ventures in the proscribed area. There were two motivations for this policy:

To avoid warfare with the Indians. Neither side evidenced any affection for the tribes, but Indian conflicts were very expensive, and the British hadn't yet deployed enough soldiers in the West to keep the peace. Some Indians welcomed this policy, believing that separation from the colonies would allow them to resume their traditions. Others realized that the proclamation would, at best, only provide breathing room before the next onslaught of settlers.

To concentrate colonial settlements on the seaboard where they could be active parts of the British mercantile system. British trade officials took it as a first priority to populate the recently secured areas of Canada and Florida (referring to the Treaty of Paris), where colonists could reasonably be expected to trade with the mother country. Settlers living west of the Appalachians would be highly self-sufficient and have little opportunity to trade with English merchants.

The reaction of colonial land speculators and frontiersmen was immediate and negative. They believed their fight in the recent war had been "rewarded" by the creation of a vast restricted native reserve in the lands they coveted. Most concluded that the proclamation was only a temporary measure: a number ignored it entirely and moved into the prohibited area. Almost from its inception, the proclamation was modified to suit the needs of influential people with interests in the American West, both high British officials and colonial leaders.

Beginning in 1764, portions of the Proclamation Line were adjusted westward to accommodate speculative interests. Later, in 1768, the first Treaty of Fort Stanwix formally recognized the surrender of transmontane lands claimed by the Iroquois.

The Proclamation of 1763 was a well-intentioned measure. Pontiac’s Rebellion had inflicted a terrible toll on the frontier settlements in North America and the British government acted prudently by attempting to avoid such conflict in the foreseeable future.

The colonists, however, were not appreciative and regarded the new policy as an infringement of their basic rights. The fact that western expansion was halted at roughly the same time that other restrictive measures were being implemented, made the colonists increasingly suspicious

Almost immediately, many British colonists and land speculators objected to the proclamation boundary, since there were already many settlements beyond the line (some of which had been temporarily evacuated during Pontiac's War), as well as many existing land claims yet to be settled. Indeed, the proclamation itself called for lands to be granted to British soldiers who had served in the Seven Years' War. Prominent American colonists joined with land speculators in Britain to lobby the government to move the line further west. As a result, the boundary line was adjusted in a series of treaties with Native Americans. The Treaty of Fort Stanwix and the Treaty of Hard Labor (both 1768) and the Treaty of Lochaber (1770) opened much of what is now West Virginia and Kentucky to British settlement.

Organization of new colonies
Besides regulating colonial expansion, the proclamation dealt with the management of newly ceded French colonies. It established government for four areas: Quebec, West Florida, East Florida, and Grenada. All of these were granted the ability to elect general assemblies under a royally appointed governor or a high council, which could then create laws and ordinances specific to the area in agreement with British and colonial laws. In the meantime, the new colonies enjoyed the same rights as native-born Englishmen, something that British colonists had been fighting over for years. An even bigger affront to the British colonies was the establishment of both civil and criminal courts complete with the right to appeal--but those charged with violating the Stamp or Sugar Act were to be tried in admiralty court, where the defendant was considered guilty until he or she could prove his or her innocence.

Legacy
The influence of the Royal Proclamation of 1763 on the coming of the American Revolutionary War (1775–1783) has been variously interpreted. Many historians argue that the proclamation ceased to be a major source of tension after 1768, since the aforementioned treaties opened up extensive lands for settlement. Others have argued that colonial resentment of the proclamation contributed to the growing divide between the colonies and the Mother Country.

In the United States, the Royal Proclamation of 1763 ended with the American Revolutionary War, because Great Britain ceded the land in question to the United States in the Treaty of Paris (1783). Afterwards, the U.S. government also faced difficulties in preventing frontier violence, and eventually adopted policies similar to those of the Royal Proclamation. The first in a series of Indian Intercourse Acts was passed in 1790, prohibiting unregulated trade and travel in Native American lands. Additionally, the U.S. Supreme Court case Johnson v. M'Intosh (1823) established that only the U.S. government, and not private individuals, could purchase land from Native Americans.

The Royal Proclamation continued to govern the cession of aboriginal land in British North America, especially Upper Canada and Rupert's Land. The proclamation forms the basis of land claims of aboriginal peoples in Canada – First Nations, Inuit, and Metis. The Royal Proclamation of 1763 is thus mentioned in Section Twenty-five of the Canadian Charter of Rights and Freedoms.

The Stamp Act and other Laws
In 1764, George Grenville became the British Chancellor of the Exchequer (minister of finance). He allowed customs officers to obtain general writs of assistance, which allowed officers to search random houses for smuggled goods. Grenville thought that if profits from smuggled goods could be directed towards Britain, the money could help pay off debts. Colonists were horrified that they could be searched without warrant at any given moment. Also in 1764, with persuasion from Grenville, Parliament began to impose several taxes on the colonists. The Sugar Act of 1764 reduced the taxes imposed by the Molasses Act, but at the same time strengthened the collection of the taxes. It also provided that British judges, and not juries, would try cases involving that Act.

The next year, Parliament passed the Quartering Act, which required the colonies to provide room and board for British soldiers stationed in North America; the soldiers would serve various purposes, chiefly to enforce the previously passed acts of Parliament.

Following the Quartering Act, Parliament passed one of the most infamous pieces of legislation: the Stamp Act. Previously, Parliament imposed only external taxes on imports. But the Stamp Act provided the first internal tax on the colonists, requiring that a tax stamp be applied to books, newspapers, pamphlets, legal documents, playing cards, and dice. The legislature of Massachusetts requested a conference on the Stamp Act; the Stamp Act Congress met in October that year, petitioning the King and Parliament to repeal the act before it went into effect at the end of the month, crying "taxation without representation."

The act faced vehement opposition throughout the colonies. Merchants threatened to boycott British products. Thousands of New Yorkers rioted near the location where the stamps were stored. In Boston, the Sons of Liberty, a violent group led by radical statesman Samuel Adams, destroyed the home of Lieutenant Governor Thomas Hutchinson. Parliament did indeed repeal the Stamp Act, but additionally passed the Declaratory Act, which stated that Great Britain retained the power to tax the colonists, even without substantive representation.

Believing that the colonists only objected to internal taxes, Chancellor of the Exchequer Charles Townshend proposed bills that would later become the Townshend Acts. The Acts, passed in 1767, taxed imports of tea, glass, paint, lead, and even paper. The colonial merchants again threatened to boycott the taxed products, reducing the profits of British merchants, who in turn petitioned Parliament to repeal the Townshend Acts. Parliament eventually agreed to repeal much of the Townshend legislation. But Parliament refused to remove the tax on tea, implying that the British retained the authority to tax the colonies despite a lack of representation.

In 1773, Parliament passed the Tea Act, which exempted the British East India Company from the Townshend taxes. Thus, the East India Company gained a great advantage over other companies when selling tea in the colonies. The colonists who resented the advantages given to British companies dumped British tea overboard in the Boston Tea Party in December of 1773.

The Boston Tea Party

In retaliation for the Boston Tea Party, Parliament passed the Coercive Acts, which were in the colonies known as the Intolerable Acts. Parliament reduced the power of the Massachusetts legislature and closed the port of Boston. Also, the Quartering Act was extended to require private individuals to lodge soldiers. Furthermore, Parliament allowed royal officials accused of crimes to be tried by a British, rather than a colonial, jury.

First Continental Congress
In order to debate a response to the Intolerable Acts, all American colonies except for Georgia sent delegates to the First Continental Congress at Philadelphia. The Congress met in September 1774 and issued a Declaration of Rights and Grievances. When the Congress adjourned, it stipulated another Congress would meet if King George III did not meet the demands of the Declaration. When the Second Congress did meet, the military hostilities of the Revolutionary War had already begun, and the issue of Independence, rather than a redress of grievances, dominated the debates.

Education
Literacy grew for both men and women during the 18th century. In New England and the Middle States, more middle-class girls were sent to school. However, as Science and the requirements for citizenship became more a part of education, girls were excluded from learning these topics.

Higher education continued to develop, with the 1746 opening of The College of New Jersey (later known as Princeton), and King's College (now Columbia) in 1754. All of these universities were meant exclusively for White men, though some of the colleges experimented by admitting Native Americans. In the public schools, vocational education expanded.

Though what was lost by failing to educate the underclasses was incalculable, we can gauge the lost possibilities through such individuals as Benjamin Banneker and Phillis Wheatley. Mr. Banneker, a self-educated free African-American, observed the stars, wrote his own almanac, and was one of the surveyors of what would later become the District of Columbia. Miss Wheatley, an African-born slave educated and freed by her mistress, wrote a remarkable volume of poems published in the year 1773. Most of these had been published in The Newport Mercury, edited by Benjamin Franklin's brother James.

Questions For Review
1. What were the reasons for the French and Indian War?

2. What was the strategy of General Braddock against the French at Fort Duquesne? What was the strategy of the defending French and Indian forces?

3. Examine the succession of acts imposed upon the American Colonists in the wake of the war, beginning with the Sugar Act. What was the intended purpose of each act? What was its actual effect?

title: US History/Road to Revolution


text:
Discrete Mathematics/Graph theory

Introduction
A graph is a mathematical way of representing the concept of a "network".

A network has points, connected by lines. In a graph, we have special names for these. We call these points vertices (sometimes also called nodes), and the lines, edges.

Here is an example graph. The edges are red, the vertices, black.

In the graph, $$v_1, v_2, v_3, v_4$$ are vertices, and $$e_1, e_2, e_3, e_4, e_5$$ are edges.

Definitions of graph
There are several roughly equivalent definitions of a graph. Set theory is frequently used to define graphs. Most commonly, a graph $$G$$ is defined as an ordered pair $$G=(V,E)$$, where $$V=\{v_1,\ldots,v_n\}$$ is called the graph's vertex-set and $$E=\{e_1,\ldots,e_m\}\subset\{\{x,y\}|x,y\in V\}$$ is called the graph's edge-set. Given a graph $$G$$, we often denote the vertex—set by $$V(G)$$ and the edge—set by $$E(G)$$. To visualize a graph as described above, we draw $$n$$ dots corresponding to vertices $$v_1,\ldots, v_n$$. Then, for all $$i,j\in\{1,\ldots,n\}$$ we draw a line between the dots corresponding to vertices $$v_i, v_j$$ if and only if there exists an edge $$\{v_i,v_j\}\in E$$. Note that the placement of the dots is generally unimportant; many different pictures can represent the same graph.

Alternately, using the graph above as a guide, we can define a graph as an ordered triple $$G=(V,E,f)$$:
 * a set of vertices, commonly called V
 * a set of edges, commonly called E
 * a relation $$f: E \rightarrow \{\{x,y\}|x,y\in V\}$$ that maps to each edge a set of endpoints, known as the edge-endpoint relation. We say an edge $$e\in E$$ is incident to a vertex $$v\in V$$ iff $$v\in f(e)$$.

In the above example,
 * V={v1, v2, v3, v4}
 * E={e1, e2, e3, e4, e5}
 * f such that e1 maps to {v1, v2}, e2 maps to {v1, v3}, e3 maps to {v1, v4}, e4 maps to {v2, v4}, and e5 maps to {v3, v4}.

If $$f$$ is not injective — that is, if $$\exists e,e'\in E$$ such that $$e\neq e', f(e) = f(e')$$ — then we say that $$G$$ is a multigraph and we call any such edges $$e,e'\in E$$ multiple edges. Further, we call edges $$e \in E$$ such that $$|f(e)|=1$$ loops. Graphs without multiple edges or loops are known as simple graphs.

Graphs can, conceivably, be infinite as well, and thus we place no bounds on the sets V and E. We will not look at infinite graphs here.

Directions, Weights, and Flows
We define a directed graph as a graph such that $$f$$ maps into the set of ordered pairs $$\{(x,y)|x,y\in V\}$$ rather than into the family of two-element sets $$\{\{x,y\}|x,y\in V\}$$. We can think of an edge $$e\in E$$ such that $$f(e)=(x,y)$$ as 'pointing' from $$x$$ to $$y$$. As such we would say that $$x$$ is the tail of edge $$e$$ and that $$y$$ is the head. This is one of the vagaries of graph theory notation, though. We could just as easily think of $$x$$ as the head and $$y$$ as the tail. To represent a directed graph, we can draw a picture as described and shown above, but place arrows on every edge corresponding to its direction.

In general, a weight on a graph $$G$$ is some function $$c: E(G)\rightarrow \mathbb R$$.

A flow $$(G,c)$$ is a directed graph $$G=(V,E,f)$$ paired with a weight function such that the weight "going into" any vertex is the same amount as the weight "going out" of that vertex. To make this more formal, define sets Then, formally stated, our requirement on the weight function is $$\sum_{e\in f^+(v)} c(e)=\sum_{e\in f^-(v)} c(e),\; \forall v\in V(G).$$
 * $$f^+(v)=\{e\in E(G)|f(e)=(v,x), x\in V(G)\},\; \forall v\in V(G)$$
 * $$f^-(v)=\{e\in E(G)|f(e)=(x,v), x\in V(G)\},\; \forall v\in V(G)$$

Algebraic Graph Theory
While set theory is frequently used when discussing graphs, other approaches can simplify certain operations. A set can be defined using an adjacency matrix $$A$$ where element $$a_{i,j}$$ is a 1, if there is an edge between vertex i and vertex j and 0 otherwise.

Special Graphs
Some graphs occur frequently enough in graph theory that they deserve special mention. One such graphs is the complete graph on n vertices, often denoted by Kn. This graph consists of n vertices, with each vertex connected to every other vertex, and every pair of vertices joined by exactly one edge. Another such graph is the cycle graph on n vertices, for n at least 3. This graph is denoted Cn and defined by V := {1,2,..,n} and E := {{1,2},{2,3}, ..., {n-1,n},{n,1}}. Even easier is the null graph on n vertices, denoted Nn; it has n vertices and no edges! Note that N1 = K1 and C3 = K3.

Some Terms
Two vertices are said to be adjacent if there is an edge joining them. The word incident has two meanings:

 * An edge e is said to be incident to a vertex v if v is an endpoint of e.
 * Two edges are also incident to each other if both are incident to the same vertex.

Two graphs G and H are said to be isomorphic if there is a one-to-one function from (or, if you prefer, one-to-one correspondence between) the vertex set of G to the vertex set of H such that two vertices in G are adjacent if and only if their images in H are adjacent. (Technically, the multiplicity of the edges must also be preserved, but our definition suffices for simple graphs.)

Subgraphs
A subgraph is a concept akin to the subset. A subgraph has a subset of the vertex set V, a subset of the edge set E, and each edge's endpoints in the larger graph has the same edges in the subgraph. A

A subgraph $$H$$ of $$G$$ is generated by the vertices {$$a, b, c,...$$}$$\in H$$ if the edge set of $$H$$ consists of all edges in the edge set of $$G$$ that joins the vertices in $$H=$${$$a, b, c, ...$$}.

A path is a sequence of edges $$$$ such that ei is adjacent to ei+1 for all i from 1 to N-1. Two vertices are said to be connected if there is a path connecting them.

Trees and Bipartite Graphs
A tree is a graph that is (i) connected, and (ii) has no cycles. Equivalently, a tree is a connected graph with exactly $$n-1$$ edges, where there are $$n$$ nodes in the tree.

A Bipartite graph is a graph whose nodes can be partitioned into two disjoint sets U and W such that every edge in the graph is incident to one node in U and one node in W. A tree is a bipartite graph.

A complete bipartite graph is a bipartite graph in which each node in U is connected to every node in W; a complete bipartite graph in which U has $$n$$ vertices and V has $$m$$ vertices is denoted $$K_{n,m}$$.

Adjacent,Incident,End Vertices

Self loops,Parallel edges,Degree of Vertex

Pendant Vertex : Vertex Degree one "Pendant Vertex" Isolated Vertex : Vertex Degree zero "Isolated Vertex"

Hamiltonian and Eulerian Paths
Hamiltonian Cycles: A Hamiltonian Cycle received its name from Sir William Hamilton who first studied the travelling salesman problem. A Hamiltonian cycle is a path that visits every vertex once and only once i.e. it is a walk, in which no edge is repeated (a trail) and therefore a trail in which no vertex is repeated (a path). Note also it is a cycle, the last vertex is joined to the first.

A graph is said to be Eulerian if it is possible to traverse each edge once and only once, i.e. it has no odd vertices or it has an even number of odd vertices (semi-Eulerian). This has implications for the Königsberg problem. It may be easier to imagine this as if it is possible to trace the edges of a graph with a pencil without lifting the pencil off the paper or going over any lines.

Planar Graphs
A planar graph is an undirected graph that can be drawn on the plane or on a sphere in such a way that no two edges cross, where an edge $$e = (u,v)$$ is drawn as a continuous curve (it need not be a straight line) from u to v.

Kuratowski proved a remarkable fact about planar graphs: A graph is planar if and only if it does not contain a subgraph homeomorphic to $$K_5$$ or to $$K_{3,3}$$. (Two graphs are said to be homeomorphic if we can shrink some components of each into single nodes and end up with identical graphs. Informally, this means that non-planar-ness is caused by only two things—namely, having the structure of $$K_5$$ or $$K_{3,3}$$ within the graph).

Coloring Graphs
A graph is said to be planar if it can be drawn on a plane in such way that no edges cross one another except of course for meeting at vertices

Each term, the Schedules Office in some university must assign a time slot for each final exam. This is not easy, because some students are taking several classes with finals, and a student can take only one test during a particular time slot. The Schedules Office wants to avoid all conflicts, but to make the exam period as short as possible.

We can recast this scheduling problem as a question about coloring the vertices of a graph. Create a vertex for each course with a final exam. Put an edge between two vertices if some student is taking both courses. For example, the scheduling graph might look like this: Next, identify each time slot with a color. For example, Monday morning is red, Monday afternoon is blue, Tuesday morning is green, etc.

Assigning an exam to a time slot is now equivalent to coloring the corresponding vertex. The main constraint is that adjacent vertices must get different colors; otherwise, some student has two exams at the same time. Furthermore, in order to keep the exam period short, we should try to color all the vertices using as few different colors as possible. For our example graph, three colors suffice: red, green, blue.

The coloring corresponds to giving one final on Monday morning (red), two Monday afternoon (blue), and two Tuesday morning (green)...

K Coloring
Many other resource allocation problems boil down to coloring some graph. In general, a graph G is kcolorable if each vertex can be assigned one of k colors so that adjacent vertices get different colors. The smallest sufficient number of colors is called the chromatic number of G. The chromatic number of a graph is generally difficult to compute, but the following theorem provides an upper bound:

Theorem 1. A graph with maximum degree at most k is (k + 1)colorable.

Proof. We use induction on the number of vertices in the graph, which we denote by n. Let P(n) be the proposition that an nvertex graph with maximum degree at most k is (k + 1)colorable. A 1 vertex graph has maximum degree 0 and is 1colorable, so P(1) is true.

Now assume that P(n) is true, and let G be an (n + 1)vertex graph with maximum degree at most k. Remove a vertex v, leaving an nvertex graph G. The maximum degree of G is at most k, and so G is (k + 1)colorable by our assumption P(n). Now add back vertex v. We can assign v a color different from all adjacent vertices, since v has degree at most k and k + 1 colors are available. Therefore, G is (k + 1)colorable. The theorem follows by induction.

Weighted Graphs
A weighted graph associates a  label (weight) with every edge in the graph. Weights are usually real numbers, and often represent a "cost" associated with the edge, either in terms of the entity that is being modeled, or an optimization problem that is being solved.

title: Discrete Mathematics/Graph theory


text:
Henry I, Duke of Mödling

Henry of Mödling (1158 - 1223) known as the Elder was the "Duke" of Mödling from 1177 to 1223. He was the second son of Henry II, Duke of Austria. His lands spanned from Liesing to Piesting and Bruck an der Leitha. His first known use of the title 'Duke of Mödling' was from 1205.

Life
Henry was knighted with his brother Leopold V in 1174.

On the death of their father, Henry II, Duke of Austria, he received the lands of Mödling from his brother, Leopold V, Duke of Austria around 1177. The castle of the same name became his Seat of Power.

He went to Vienna on the 25 August 1190 to see his brother off to join the Third Crusade, while he traveled to Italy to act as a representative of the Babenbergs and to attend the coronation of Henry VI, Holy Roman Emperor in Spring 1191.

He is known to have taken part in the German Crusade of 1197 with his nephew, Frederick I of Austria and Wolfger, Bishop of Passau. By 6 June 1197, he was at the Imperial court encampment at Castro St. Giovanni and a few days later, at the same time with Duke Frederick in Linaria. When they left Apulia to reach Sicily, they were encouraged further, after two ships sunk at sea, with the Abbots of Werd and Beuern. The Emperor now took full command and instructed Conrad, Archbishop of Mainz & Chancellor of Germany to lead the fleet over the Mediterranean. They sailed in early September 1197 and arrived in Acre a few weeks later.

His branch of the Babenberg line did not focus on matters of war, but became mostly interested in arts. His surname is somewhat confusing, as Mödling was not a separate duchy, but a subordinate-duchy ruled by the Babenbergs in Vienna. His territory, under his reign, experienced a marginal prosperity. During his government, he built Mödling castle in the 12th century. The most prominent guest was the minstrel Walther von der Vogelweide in 1219.

He married Richeza of Bohemia († 19 April 1182 ), daughter of Vladislaus II, Duke of Bohemia and had one son, Henry III, Duke of Mödling called the Younger.

He was buried with his brother, Leopold V of Austria and his nephew, Frederick I of Austria, at Heiligenkreuz Abbey.

title: Henry I, Duke of Mödling


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