text:
Paideia High School/Fingerprints

The Language of Science" (pub. 1930).
 * Fingerprints is a Paideia Unit Plan based upon the chapter of this same name of Tobias Dantzig's book,

These guidelines address teachers for the purpose of guiding instruction. See Paideia Learning Plan for the student's point of view.

Column One
Column One teaching and learning should make up about 10% to 15% of the total scheduled instructional time. It is didactic in nature and uses teacher lectures, text books or other didactic instructional materials, and questioning appropriate to this mode of education. Teaching in this mode encompasses of three facets: Exoridium, Interpretation, and Erudition.

Exordium
The Exordium is the teacher's didactic introduction to the work that is the subject of a Paideia Unit Plan. This introduction consists of both an oral and physical (or electronic) presentation of the work. For longer works, the teacher may limit the oral presentation to key parts of the work. The teacher should read texts live distinctly, accurately, and intelligently. Other works should be orally presented similarly as appropriate to the type of work. In addition, the teacher should provide high quality audio and video recordings of works if possible. Students should have a consumable print copy of the work both electronically and in hard-copy if possible.

Oral Presentation

 * Because no oral performances were available, I recorded myself reading (set to background music comprised of several modern and jazz compositions, each of which is entitled "Fingerprints") this first chapter of Tobias Dantzig's book, "Number: The Language of Science." This recording was first played in its entirety for the students in a setting where their only task was to listen. The recording will be played a second time for the same students, at which point they will have a complete copy of the text before them and will be asked to follow along, underlining any words they are unfamiliar with and any passages they do not understand. As well, they will be encouraged to take margin notes to indicate any sections that are of particular interest, and to jot down any questions that arise throughout the reading.

Written Presentation

 * is available online as a viewable PDF at Google Books.
 * Available in Gateway to the Great Books
 * Also available in Danzig's book The Language of Science.''

Notable quotes within or pertaining to the text:
 * "This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands." - Albert Einstein
 * "The harmony of the universe knows only one musical form - the legato; while the symphony of number knows only its opposite - the staccato. All attempts to reconcile this discrepancy are based on the hope that an accelerated staccato may appear to our senses as a legato." - Tobias Dantzig
 * "Man is the measure of all things." - Protagoras

Interpretation
Textual Interpretation refers to the teacher's didactic analysis of a written work in terms of the four major questions a demanding reader should ask of a text. Using the term work instead of book, these questions are: (1) What is the text about as a whole? (2) What is being said in detail, and how? (3) Is the text true, in whole or part? and (4) What of it? The Exordium begins to answer the first question because it introduces the whole text both orally and in writing (although the oral presentation may be limited in the case of longer works). It is in the Interpretation stage of Column One instruction that the teacher didactically begins to thoroughly unlock the second question. The third and fourth questions are relevant to Interpretation, but question two receives most of the teacher's attention. It is only when the student begins Column Two activities that a fuller grasp of questions three and four begins to mature in the student's mind. Consistent with the purpose of Column One instruction, the teacher is simply introducing elements of proper interpretation for the student to build on during Column Two and Column Three learning.

These questions can be appropriately modified for analysis of a work in science, mathematics, or the fine arts. Analysis of these types of works proceeds in a manner analagous to that of a text. Inasmuch as imaginative literature can be considered a work of fine art, the analogies drawn in Adler's and Van Doren's How to Read a Book can serve as a guide to other types of works too.

Interpretation of  or
In textual interpretation, the teacher didactically presents the author's key terms in the stage of Interpretation of Words and Terms. In addition, the teacher anticipates unfamiliar general vocabulary. These words and terms are given didactically, either orally or in writing, or both, but the task of interpreting is a Column Two skill of learning that the teacher must coach. Consequently, the purpose of this Column One stage is merely to point out key terms and potentially unfamiliar terms. The student must learn how to interpret. Again, this planning can be adapted to other kinds of works.

Interpretation of Key Terms
The key terms are those few words or phrases the author uses in unique, special, or important ways.[3]. In the Column Two stage, students are coached both to find these key terms and to "unlock" them on their own. This skill is essential to analytical reading. In this Column One stage, however, the teacher points out a list of such terms. Students should understand that this listing is not necessarily exhaustive. In addition, the teacher provides students with a handful of these terms with definitions worked out in detail for consideration; it should consist of about three to five terms.

 * base of numeration
 * cardinal number
 * number sense
 * one-to-one-correspondence
 * ordered succession
 * ordinal number
 * tactile number sense
 * visual number sense

General Vocabulary
The teacher may anticipate general unfamiliar vocabulary and point out the more difficult words either orally or in writing or both. However, defining words using context, a dictionary, or a knowlegable friend or adult (including the teacher) is a Column Two skill that must be coached. Teachers must help students build good habits of knowing the meanings of words. Dictionaries, in both electronic and book form must be available, and students must be taught how to use them. They must also be taught how to ask for definitions--a perfectly acceptible life skill commonly used by demanding readers!

 * abstract
 * advent
 * ambiguity
 * anthropological
 * anthropomorphic
 * artifice
 * attribute
 * attrition
 * concrete
 * conjecture
 * devise
 * discriminate
 * economy
 * elegant
 * eminent
 * essence
 * etymology
 * exhausted
 * explicit
 * extraction
 * faculty
 * heterogeneous
 * homogeneous
 * impenetrable
 * inarticulate
 * kinship
 * latent
 * latter
 * metamorphosis
 * metaphysical
 * motley
 * mystic
 * numeration
 * obliterate
 * obsolete
 * parity
 * permeate
 * philology
 * physiological
 * plausible
 * precedence
 * preliminary
 * Providence
 * quaint
 * reckon
 * relic
 * rosary
 * rudimentary
 * ruse
 * scope
 * simultaneously
 * speculation
 * superfluous
 * supersede
 * supplanted
 * symmetric
 * tactile
 * Thimshian
 * typify
 * vintage
 * wrought

Note: If the teacher points out words in anticipation of their potential difficulty, this should be done in context by giving citations or electronically highlighting the words.

After students have read and marked passages and vocabulary, have them create a personal glossary, looking up marked words and notating the correct definitions for this usage. I choose to give an open note vocabulary quiz at this point. The rules were simple--student could use only their copy of the text (to look at context) and their glossary notes to complete the following quiz:

“Fingerprints” Vocabulary Quiz

1.	Ruse

2.	Faculty

3.	Corroborate

4.	Plausible

5.	Impenetrable

6.	Render

7.	Consolidated

8.	Heterogeneous

9.	Plurality

10.	Derive

11.	Exhausted

12.	Adept

13.	Supersede

14.	Tacit

15.	Permeate

16.	Surmise

17.	Philology

18.	Imperceptibly

19.	Attrition

20.	Obliterate (-ed)

21.	Rudimentary

22.	Conjecture

23.	Metamorphosis

24.	Anthropomorphic

25.	Suffices

26.	Providence

27.	Physiological

28.	Irreducible

29.	Ambiguity

30.	Proposition A.	A maneuver designed to trick or deceive

B.	To have merged two or more elements together into one

C.	To remove or destroy completely, so that no trace is left

D.	The quality of state of being uncertain

E.	Highly skilled and/or expert in performing a task

F.	Figure out through the use of logic

G.	All used up

H.	A process of change in which one thing becomes something that appears entirely different

I.	To be completely spread or diffused throughout a medium—as water is to a soaked sponge

J.	Is enough

K.	The study of languages and their development to better understand human culture

L.	A likely or believable on the surface

M.	To speculate or suppose based on little evidence

N.	A statement to be argued as true or false

O.	Change so subtle it almost can’t be noticed

P.	Cannot be moved or seen through

Q.	To make R.	Having to do with the physical body

S.	A grouping consisting of items that are significantly different from each other

T.	Divine guidance—the hand of God

U.	To back up evidence with additional evidence or testimony

V.	The ability to act or do

W.	Unspoken, but understood

X.	To take the place of something

Y.	Having human characteristics

Z.	Speculation or guessing

AA. The rate of loss

BB. Having a varied and large number or quantity

CC. Basic

DD. Cannot be transformed to a simpler condition

The goal here is not vocabulary building per se, but development of a sense of what words are both important enough and unfamiliar enough to warrant looking them up--which is a highly individualized situational decision. A secondary developmental goal is effective use of the dictionary in determining which meaning is relevant based on context.

Interpretation of Sentences
One point of didactic interpretation at the level of interpreting sentences is to use grammar to get at the author's meaning. As with interpreting words and terms, this level of interpretation is also a Column Two skill that must be coached. Consequently, at the Column One didactic level, a teacher should choose a handful of the most difficult sentences in the text for demonstration. The teacher will always unlock the grammar of a few important and more difficult sentences for students independent of whether these sentences are key premises to an argument. As appropriate to the text, a teacher should also consider demonstrating the grammar of sentences that work together as propositions in the author's most important arguments.

Another important point is to demonstrate the meter and prosody in both poetry and prose texts. This aim is in great danger of being completely overlooked or forgotten in a world where oral reading is not nearly as common as it once was. Nevertheless, great speechs often succeed in part because the author understands prosody. Lincoln's Gettysburg Address and King's I Have a Dream serve as striking examples. The teacher should select sentences or verses to demonstrate both meter and rhythm.

 * :
 * :
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Interpretation of Passages
At the level of passages, the full trivium (grammar, logic, and rhetoric) come into play. First, a teacher should choose one or two of the author's most important arguments for a demonstration of how to use logic as a key to interpreting a text. Next, the teacher should select several passages to demonstrate how they conform (or not) to rhetorical, poetical, and sylistic rules. These rules, of course, must be didactically taught as prerequisites to interpretation of texts.

 * :
 * :
 * :
 * <Passage Illustration Stylistic Rules>: <demonstration of the stylistic rules and their value in the success of the author's purpose>

Outline of Fingerprints

 * 1) Man possesses a rudimentary number sense.
 * a. Number sense precedes counting.
 * i. Only humans can count.

 * 2) Only a few other species possess a crude number sense akin to man.
 * a. Solitary wasps possess number sense (5, 12, 24 grubs).
 * i. Genus Eumenus (5 grubs/males – 10 grubs/females).
 * b. Many birds possess number sense.
 * i. The squire and the crow example.
 * c. Prospective arguments against points a and b above.
 * i. Species possessing number sense are exceedingly few.
 * 1. This argument stands.
 * ii.	Number sense in animals is limited in scope.
 * 1.	But number sense is also extremely limited in man!

 * 3) Scope of rudimentary number sense in man quite limited.
 * a. Human number sense is difficult to disentangle from counting.
 * b. Direct visual number sense rarely extends beyond four.
 * c. Tactile number sense is even more limited.
 * d. Anthropological evidence supports this argument.
 * i. Australian Aborigines.
 * ii. African Bushmen.
 * iii. English language studies corroborate.

 * 4) Rudimentary number sense is the precursor to the more sophisticated human concept of number.
 * a. Sophisticated number concept is a prerequisite of higher level thinking.
 * b. Counting truly sets us apart from all other species.

 * 5) The origins of counting.
 * a. Concrete preceded the abstract.
 * b. Heterogeneous (concrete) numbers → homogenous (abstract) number concept.
 * c. One-to-one correspondence → no counting required to compare two collections.
 * i. Matching/tallying are examples of one-to-one correspondence.
 * ii. Model collections afford comparisons with concrete collections.
 * iii. Cardinal number develops.
 * 1. Number words originally derived from concrete collections → abstract.
 * 2. Number words are based on one-to-one correspondence → no counting required.
 * iv. Ordinal number concept develops.
 * 1. Cardinal number words.
 * 2. Ordered succession.
 * 3. Phonetic scheme for repeating.
 * v. From systems of counting evolved basic arithmetic, multiplication, and all higher math.

 * 6) Finger counting provides pathway ←→ between cardinal and ordinal number.
 * a. Fingers raised simultaneously → cardinal number.
 * b. Fingers raised in succession → ordinal number.
 * c. Philological evidence highly supports this claim.
 * i. Five is frequently equated to “hand.”

 * 7) Man’s articulate fingers a prerequisite for various realms of higher thinking.
 * a. Precursor to calculation.
 * i. Finger counting.
 * ii. Finger arithmetic.
 * iii. Finger multiplication.
 * iv. Finger calculators.
 * b. Math the basis of all exact sciences.
 * c. Exact sciences → man’s material and intellectual progress.

 * 8) Number language predates written history by thousands of years.
 * a. Number words extraordinarily linguistically stable.
 * i. Yet with exception of “five” – no trace of original meaning discernible.
 * 1. This due to change over time of original concrete object words.

 * 9) Ten fingers the foundation of our base 10 number system → a physiological accident!
 * a. Base 10 by far most widespread.
 * i. Highly linguistically documented.
 * b. Remaining bases predominantly 5 and 20 → fingers and toes.
 * i. Romans.
 * ii. Axtecs.
 * iii. Mayans.
 * c. Binary the only other evidenced alternative.
 * i. Economy of symbols.
 * ii. Most fundamental system.
 * 1. Logic is binary based.
 * 2. Computers are binary based.
 * 3. Can everything be reduced to yes-no/off-on?
 * d. Other bases more practical.
 * i. Base 12 provides great number of divisors (2,3,4,6).
 * ii. Base 11 dramatically reduces redundant expression of rational numbers.
 * iii. Base 10 neither prime nor highly divisible (2,5)

 * 10) Man is the measure of all things (Protagorus).

Erudition
Erudition refers to all manner of background information assumed by or necessary to understanding or fully appreciating the text.

<Note that these categories are presented alphabetically for ease of organization>

Anthropological References
Cultural Anthropology
 * Auvergne was a historic province in south central France. It was originally the feudal domain of the Counts of Auvergne. It is now the geographical and cultural area that corresponds to the former province.
 * Ancient Aztecs: The Aztec people were certain ethnic groups of central Mexico, particularly those groups who spoke the Nahuatl language and who dominated large parts of Mesoamerica in the 14th, 15th and 16th centuries, a period referred to as the late post-classic period in Mesoamerican chronology.
 * Emperor of China refers to any sovereign of Imperial China reigning between the founding of China, united by the King of Qin in 221 BCE, and the fall of Yuan Shikai's Empire of China in 1916. When referred to as the Son of Heaven (Chinese: 天子 tiānzǐ), a title that predates the Qin unification, the Emperor was recognized as the ruler of "All under heaven" (i.e., the world). In practice not every Emperor held supreme power, though this was most often the case.
 * Maya Indians: The Maya people constitute a diverse range of the Native American people of southern Mexico and northern Central America. The overarching term "Maya" is a convenient collective designation to include the peoples of the region who share some degree of cultural and linguistic heritage; however, the term embraces many distinct populations, societies, and ethnic groups, who each have their own particular traditions, cultures, and historical identity.
 * The Roman Empire (Latin: imperium romanum) was the post-Republican period of the ancient Roman civilization, characterised by an autocratic form of government and large territorial holdings in Europe and around the Mediterranean. The term is used to describe the Roman state during and after the time of the first emperor, Augustus.

Linguistic Anthropology
 * Philology is the study of language in written historical sources; as such it is a combination of literary studies, history and linguistics. Classical philology is the philology of the Greek, Latin and Sanskrit languages. Classical philology is historically primary, originating in European Renaissance Humanism, but was soon joined by philologies of other languages both European (Germanic, Celtic, Slavistics, etc.) and non-European (Sanskrit, Persian, Arabic, Chinese, etc.). Indo-European studies involves the philology of all Indo-European languages as comparative studies.
 * Australian Aborigines are those people regarded as indigenous to the Australian continent. In the High Court of Australia, Australian Aborigines have been specifically identified as a group of people who share, in common, biological ancestry back to the original occupants of the continent.
 * Bushmen of South Africa: The indigenous people of southern Africa, whose territory spans most areas of South Africa, Zimbabwe, Lesotho, Mozambique, Swaziland, Botswana, Namibia, and Angola, are variously referred to as Bushmen, San, Sho, Barwa, Kung, or Khwe. These people were traditionally hunter-gatherers, part of the Khoisan group and are related to the traditionally pastoral Khoikhoi.
 * The Greek language, an independent branch of the Indo-European family of languages, is the language of the Greeks. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history; other systems, such as Linear B and the Cypriot syllabary, were previously used. The alphabet arose from the Phoenician script, and was in turn the basis of the Latin, Cyrillic, Coptic, and many other writing systems.
 * The Mongolian language is the official language of Mongolia and the best-known member of the Mongolic language family. The number of speakers across all its dialects may be 5.2 million, including the vast majority of the residents of Mongolia and many of the Mongolian residents of the Inner Mongolia autonomous region of China.
 * Old German (Old High German) refers to the earliest stage of the German language and it conventionally covers the period from around 500 to 1050. Coherent written texts do not appear until the second half of the 8th century, and some treat the period before 750 as 'prehistoric' and date the start of Old High German proper to 750 for this reason. There are, however, a number of Elder Futhark inscriptions dating to the 6th century (notably the Pforzen buckle), as well as single words and many names found in Latin texts predating the 8th century.
 * Semitic: In linguistics and ethnology, Semitic (from the Biblical "Shem", Hebrew: שם, translated as "name", Arabic: ساميّ) was first used to refer to a language family of largely Middle Eastern origin, now called the Semitic languages. This family includes the ancient and modern forms of Akkadian, Amharic, Arabic, Aramaic, Ge'ez, Hebrew, Maltese, Phoenician, Tigre and Tigrinya among others. As language studies are interwoven with cultural studies, the term also came to describe the extended cultures and ethnicities, as well as the history of these varied peoples as associated by close geographic and linguistic distribution.
 * Thimshian language: The Tsimshian are an indigenous people of the Pacific Northwest Coast. Tsimshian translates to Inside the Skeena River. Their communities are in British Columbia and Alaska, around Terrace and Prince Rupert and the southernmost corner of Alaska on Annette Island. There are approximately 10,000 Tsimshian. Their culture is matrilineal with a societal structure based on a clan system. Early anthropologists and linguistics grouped Gitxsan and Nisga'a as Tsimshian because of linguistic affinities. Under this terminology they were referred to as Coast Tsimshian, even though some communities were not coastal. The three peoples identify as separate nations. There are many other ways to spell the name, such as Tsimpshean, Tsimshean, Tsimpshian, and others.

Biographical References

 * About the author: Tobias Dantzig
 * List of Citations
 * Comte de Buffon, Georges-Louis Leclerc (7 September 1707 – 16 April 1788) was a French naturalist, mathematician, cosmologist, and encyclopedic author. His works influenced the next two generations of naturalists, including Jean-Baptiste Lamarck and Georges Cuvier. Buffon published thirty-six quarto volumes of his Histoire naturelle during his lifetime; with additional volumes based on his notes and further research being published in the two decades following his death. It has been said that "Truly, Buffon was the father of all thought in natural history in the second half of the 18th century."
 * Curr, Edward Micklethwaite published many reports and several books throughout his career, including Pure Saddle Horses in 1863, an account of his travels through Europe and the Middle East in the early 1850s, and The Australian Race: Its Origins, Languages, Customs in 1886. However, his most widely known work is Recollections of Squatting in Victoria, which was first published in 1883 but republished as an abridged version in 1965, recounting Curr's experiences managing his father's properties in northern Victoria forty years earlier, including his interactions with the local Aboriginal Australians.
 * Grimaldi, Francesco Maria born April 2, 1618 in Bologna (Italy) and died on December 28, 1663 in Bologna, was an Italian Jesuit priest, mathematician and physicist who taught at the Jesuit college in Bologna. See Catholic Encyclopedia article for Francesco Maria Grimaldi.
 * Laplace, Pierre-Simon was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the so-called Bayesian interpretation of probability was mainly developed by Laplace.
 * Leibniz, Gottfried occupies a prominent place in the history of mathematics and the history of philosophy. He developed the infinitesimal calculus independently of Isaac Newton, and Leibniz's mathematical notation has been widely used ever since it was published. He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685[4] and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is at the foundation of virtually all digital computers.
 * Protagoras (ca. 490 BC – 420 BC) was a pre-Socratic Greek philosopher and is numbered as one of the sophists by Plato. In his dialogue Protagoras, Plato credits him with having invented the role of the professional sophist or teacher of virtue. He is also believed to have created a major controversy during ancient times through his statement that "man is the measure of all things." This idea was very revolutionary for the time and contrasting to other philosophical doctrines that claimed the universe was based on something objective, outside the human influence.
 * Russell, Bertrand is generally credited with being one of the founders of analytic philosophy. He was deeply impressed by Gottfried Leibniz (1646–1716) and wrote on every major area of philosophy except aesthetics. He was particularly prolific in the field of metaphysics, the logic and the philosophy of mathematics, the philosophy of language, ethics and epistemology. When Brand Blanshard asked Russell why he didn't write on aesthetics, Russell replied that he didn't know anything about it, "but that is not a very good excuse, for my friends tell me it has not deterred me from writing on other subjects."

Geographical References

 * <links to relevant maps--delete if not used>
 * <information relevant to geographical references--include citations--delete if not used>

Mathematical References

 * Cardinal number: In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number – the number of elements in the set.
 * Finger counting, or dactylonomy, is the art of counting along one's fingers. Though marginalized in modern societies by Arabic numerals, formerly different systems flourished in many cultures, including educated methods far more sophisticated than the one-by-one finger count taught today in preschool education. Finger counting can also serve as a form of manual communication, particularly in marketplace trading and also in games such as morra. Finger counting is studied by ethnomathematics.
 * Number Sense: In mathematics education, number sense can refer to "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations." Many other definitions exist, but are similar to the one given. Some definitions emphasize an ability to work outside of the traditionally taught algorithms, e.g., "a well organised conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms". The term "number sense" involves several concepts of magnitude, ranking, comparison, measurement, rounding, percents, and estimation
 * One-to-one correspondence: The notion of one-to-one correspondence is so fundamental to counting that we don't even think about it. When we count out a deck of cards, we say, 1, 2, 3, ..., 52, and as we say each number we lay down a card. Each number corresponds to a card. Technically, we can say that we have put the cards in the deck and the numbers from 1 to 52 in a one-to-one correspondence with each other. If we count out a second deck and in the same way and count to exactly 52 when we lay out the cards, we know that both decks have the same number of cards. This means that we could pair up the cards, one from each deck, and not have any cards left over when all the pairs had been made. By putting the cards in each deck in a one-to-one correspondence with each other, we are sure that both decks have the same number of cards. These things are so obvious they seem silly. However, if we want to know the size of an unknown quantity, but the counting task is tricky, we can try to put the unknown quantity in one-to-one correspondence with some known quantity. This is the strategy that Georg Cantor used to compare different sizes of infinity.
 * Ordered succession
 * Ordinal number: In linguistics, ordinal numbers are the words representing the rank of a number with respect to some order, in particular order or position (i.e. first, second, third, etc.). Its use may refer to size, importance, chronology, etc. They are adjectives. They are different from the cardinal numbers (one, two, three, etc.) referring to the quantity.
 * Reckoning

Philosophical References

 * Metaphysics is a branch of philosophy dealing with theories of existence and knowledge. A person who studies metaphysics is referred to as a metaphysician. Metaphysicians are concerned with explaining the fundamental nature of being and the world. The metaphysician attempts to clarify the fundamental notions by which people understand the world, including existence, the definition of object, property, space, time, causality, and possibility. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:
 * "What is there?"
 * "What is it like?"

Religious References

 * Agnosticism is the view that the truth value of certain claims—especially claims about the existence or non-existence of any deity, but also other religious and metaphysical claims—is unknown or unknowable. Agnosticism can be defined in various ways, and is sometimes used to indicate doubt or a skeptical approach to questions. In some senses, agnosticism is a stance about the similarities or differences between belief and knowledge, rather than about any specific claim or belief.
 * Catholicism is a broad term for the body of the Catholic faith, its theologies and doctrines, its liturgical, ethical, spiritual, and behavioral characteristics, as well as a religious people as a whole. Jesuits are men who are members of a religious order called The Society of Jesus (Latin: Societas Iesu, S.J., SJ, or SI), who follow the teachings of the Catholic Church.
 * Providence In theology, divine providence, or simply providence, is God's activity in the world. By implication, it is also a title of God. A distinction is usually made between "general providence" which refers to God's continuous upholding the existence and natural order of the universe, and "special providence" which refers to God's extraordinary intervention in the life of people

Scientific References

 * Solitary Wasp: The general life style of a solitary wasp consists of a lone female mating and then preparing and provisioning one or more nests each containing one or more cells with food for her young. The egg hatches and the larvae consumes the supplied food without ever leaving the cell. After pupation the new adults emerge and seek a mate after which the males (being shorter lived in most species) die and the females go on to restart the cycle.
 * Genus Eumenus: Many insects also have a well-developed number instinct, especially among the solitary wasps. The mother wasp lays her eggs in individual cells and provides each egg with a number of live caterpillars on which the young feed when hatched. Some species of wasp always provide five, others 12, and others as high as 24 caterpillars per cell. The number of caterpillars is different among species, but it is always the same for each sex of the eggs. The male solitary wasp in the genus Eumenus is smaller than the female, so the mother supplies him with only five caterpillars; the larger female receives ten caterpillars in her cell. In other words, she can distinguish between both the numbers five and ten in the caterpillars she is providing and which cell contains a male and which contains a female.

Column Two
Column Two teaching consists of coaching. This mode of instruction aims at helping students to form habitual skills in the language, scientific, and fine arts. Thus, a teacher must correct students as they practice listening, speaking, reading, writing, observing with the senses unaided, observing with the aid of scientific apparatus, measuring, estimating, calculating, and exercising dexterity in the musical and visual arts. Each of these arts in turn rely upon the aquisition of fine and gross motor, imagination, and memory skills. These rules for developing Paideia Unit Plans address teachers. For their counterpart written for students, see Paideia Learning Plan.

Column Two learning comprises 65% to 75% of scheduled learning time. Its chief charactaristic is student activity. Students must be practicing some skill or skills while the teacher corrects him or her. While athletic coaching is an obvious example of this type of instruction, debate coaching, directing a drama, art instruction, and piano lessons also represent coaching. In order to coach well, a teacher must have a repertoire of activities carefully designed to exercise desired skills. Additionally, the teacher must have a clear idea of how to correct the skills as students practice them to ensure their habitual formation.

Mnuemonic Skills
<consider this category for every unit>

Listening Skills
After listening to the audio of the text for vocabulary and meaning, students are offered the opportunity to listen to it one more time for accuracy. They are able to earn points for the number of reading errors against the text they can successfully locate on a third listening.

Speaking Skills

 * <imitation of quality oral readings--should be a part of every unit>
 * <recitations from memory--should be some part of every unit>
 * <oral presentations of written work--should be a part of most units>
 * <class discussions--should be a part of every unit>
 * <recitation of the rules of reading from Adler's and Van Doren's How to Read a Book or similar rules>

Outline of Fingerprints

 * 1) Man possesses a rudimentary number sense.
 * a. Number sense precedes counting.
 * i. Only humans can count.

 * 2) Only a few other species possess a crude number sense akin to man.
 * a. Solitary wasps possess number sense (5, 12, 24 grubs).
 * i. Genus Eumenus (5 grubs/males – 10 grubs/females).
 * b. Many birds possess number sense.
 * i. The squire and the crow example.
 * c. Prospective arguments against points a and b above.
 * i. Species possessing number sense are exceedingly few.
 * 1. This argument stands.
 * ii.	Number sense in animals is limited in scope.
 * 1.	But number sense is also extremely limited in man!

 * 3) Scope of rudimentary number sense in man quite limited.
 * a. Human number sense is difficult to disentangle from counting.
 * b. Direct visual number sense rarely extends beyond four.
 * c. Tactile number sense is even more limited.
 * d. Anthropological evidence supports this argument.
 * i. Australian Aborigines.
 * ii. African Bushmen.
 * iii. English language studies corroborate.

 * 4) Rudimentary number sense is the precursor to the more sophisticated human concept of number.
 * a. Sophisticated number concept is a prerequisite of higher level thinking.
 * b. Counting truly sets us apart from all other species.

 * 5) The origins of counting.
 * a. Concrete preceded the abstract.
 * b. Heterogeneous (concrete) numbers → homogenous (abstract) number concept.
 * c. One-to-one correspondence → no counting required to compare two collections.
 * i. Matching/tallying are examples of one-to-one correspondence.
 * ii. Model collections afford comparisons with concrete collections.
 * iii. Cardinal number develops.
 * 1. Number words originally derived from concrete collections → abstract.
 * 2. Number words are based on one-to-one correspondence → no counting required.
 * iv. Ordinal number concept develops.
 * 1. Cardinal number words.
 * 2. Ordered succession.
 * 3. Phonetic scheme for repeating.
 * v. From systems of counting evolved basic arithmetic, multiplication, and all higher math.

 * 6) Finger counting provides pathway ←→ between cardinal and ordinal number.
 * a. Fingers raised simultaneously → cardinal number.
 * b. Fingers raised in succession → ordinal number.
 * c. Philological evidence highly supports this claim.
 * i. Five is frequently equated to “hand.”

 * 7) Man’s articulate fingers a prerequisite for various realms of higher thinking.
 * a. Precursor to calculation.
 * i. Finger counting.
 * ii. Finger arithmetic.
 * iii. Finger multiplication.
 * iv. Finger calculators.
 * b. Math the basis of all exact sciences.
 * c. Exact sciences → man’s material and intellectual progress.

 * 8) Number language predates written history by thousands of years.
 * a. Number words extraordinarily linguistically stable.
 * i. Yet with exception of “five” – no trace of original meaning discernible.
 * 1. This due to change over time of original concrete object words.

 * 9) Ten fingers the foundation of our base 10 number system → a physiological accident!
 * a. Base 10 by far most widespread.
 * i. Highly linguistically documented.
 * b. Remaining bases predominantly 5 and 20 → fingers and toes.
 * i. Romans.
 * ii. Axtecs.
 * iii. Mayans.
 * c. Binary the only other evidenced alternative.
 * i. Economy of symbols.
 * ii. Most fundamental system.
 * 1. Logic is binary based.
 * 2. Computers are binary based.
 * 3. Can everything be reduced to yes-no/off-on?
 * d. Other bases more practical.
 * i. Base 12 provides great number of divisors (2,3,4,6).
 * ii. Base 11 dramatically reduces redundant expression of rational numbers.
 * iii. Base 10 neither prime nor highly divisible (2,5)

 * 10) Man is the measure of all things (Protagorus).

The following questions are then a good point to begin class discussion of the actual content of the text. These questions are intended to lead students to an understanding of the question, "What does this piece specifically say?" and thus prepare them to better answer the companion questions, "What is this piece, as a whole, about?" and "What of it?":

“Fingerprints” by Dantzig 1.	What is “number sense” as Dantzig uses the term?

2.	Give an example from the text of an animal demonstrating number sense, and explain in your own words HOW that example demonstrates the animal having number sense:

3.	In paragraph nine, who is the author referring to when he speaks of “. . .those savages. . .”?

4.	What skill is it that the author contends has allowed us to move beyond our number sense to be able to really use numbers to understand the universe?

5.	What is the difference between cardinal and ordinal numbers?

6.	What are the “two principles which permeate all. . .realms of thought. . .”?

7.	In your own words, explain what each of these two principles (from question number six) mean:

8.	What specific type of counting always precedes or accompanies any counting system?

9.	What word is the most probable origin for the word for the number “five” in most languages?

10.	What does the “base” number of a counting system mean?

11.	What is the base number for the vast majority of counting systems around the world and throughout history?

12.	Why is that (the base identified in question eleven) the base most human cultures use?

13.	What evidence does Buffon give in support of the practical usefulness of a base 12 system?

14.	In what way does our counting system support the idea that: “Man is the measure of all things”?

Have students answer the questions individually, and turn them in for individual scoring. Hand the papers back, and go over them in class, calling on students who have given particularly appropo, insightful, or interesting answers to share theirs in class, and as an instructor, share what distinguished those answers. Students may correct their papers and turn them in as "redos" after this activity.

Writing Skills

 * <Choose phrases from text and express them in different ways>
 * <Reconstruct a previously disarranged passage from the text>
 * <Compose verses or lines in imitation of the author>
 * <Change a passage or poem of one kind into another kind>
 * <Imitate a passage>
 * <Write a composition imitating the author>
 * <Translate sentences or passages into Latin or another language>

Visual Number Sense
Students will be asked to observe screen shots of sets of dots flashed momentarily before their eyes via the overhead projector. Some of the sets will be random groupings of white dots against a black background, while others will present random groups of different colored dots, and yet others will present dots in more recognizable patterns, for example on the face of one or more game die. Students will be asked to estimate/count/calculate how many of the given objects exist in specific screen shots. The purpose is for students to get a sense of how symmetric pattern reading, counting, and/or calculating augment their visual number sense to a great degree.

Students will record and turn in their own data throughout the experiment, and analyze the results in a write-up portion of the lab.

Aural number sense
Students will also listen to prerecorded clips of combinations of tones produced by a tone generator, and will be asked to identify the number of tones blended in the individual clips. They will then listen to the same combinations of tones, but this time with each tone added to the mix in rapid succession such that their counting skills will be enabled in determining the number of tones in a given clip.

Students will record and turn in their own data throughout the experiment, and analyze the results in a write-up portion of the lab.

title: Paideia High School/Fingerprints


text:
User:Gpstone

This is a test for my first post in Wikiversity.

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

title: User:Gpstone


text: File:Eml5526.s11.team4.simko.h3.problem7.fig2.gif


text:
Preschool Language and Skills/1

 * Created: 2011 01 25 | Percent completed: [[Image:100%.png]]

Reinforcer Assessment Ages 5 and under
Look at the list below. Not all students will like items on the list. You must be a reinforcer detective. Some reinforcers are obscure such as vacuum cleaning or playing with an old answering machine.

Using parents, siblings, teachers, or aides, check the list of items/activities the child likes or seems to find fun or reinforcing. Items may not be played with or used appropriately but the student might simply be attracted to them. Before starting any behavioral program, ensure you have at least a few reinforcers.

Do not continue to lesson 2 until this is completed

Print this page and put a checkmark next each item the student likes as defined by requesting or initiating engagement with the item or event. If possible, enter a specific description (brand, type, name) of the item.

Student name: _________________________________________

Date: _________________________________________________

title: Preschool Language and Skills/1


text: File:Egm5526.s11.team2.oztekin.HW3.11.n=7.png


title: File:Nm1.s11.mtg19.djvu


text: File:Eml5526.s11.team4.simko.h3.problem7.fig3.gif


title: File:Nm1.s11.team5.HW3.P4.fig1.png


text:
User:Egm5526.s11.team-2.langpm

Weak form
This can be found on page 49 in the text. Find $$ u(x) \!$$ among the smooth functions that satisfy $$ u(l)=\overline{u} $$ such that

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

$$  \displaystyle \int_0^l\frac{dw}{dx}AE\frac{du}{dx}\,dx=\Big (w(x)A\overline{t}\Big )\Big \vert_{x=0}+\int_0^lw(x)b\,dx \qquad \forall w(x) \quad \text{with}\quad w(l)=0 $$     (1)
 * style="width:95%" |
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 *
 * }

Equivalence Proof from Text
To start, they used integration by parts
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$$  \displaystyle \int_0^lw(x)\frac{d}{dx} \left (AE\frac{du}{dx} \right ) \,dx = \left ( w(x)AE\frac{du}{dx} \right ) \Bigg \vert_0^l-\int_0^l\frac{dw}{dx}AE\frac{du}{dx}\,dx $$     (2) Substituting Eq. 1 into 2 we get
 * style="width:95%" |
 * style="width:95%" |
 *
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \int_0^lw(x)\frac{d}{dx} \left (AE\frac{du}{dx} \right ) \,dx = \left ( w(x)AE\frac{du}{dx} \right ) \Bigg \vert_0^l-\Big (w(x)A\overline{t}\Big )\Big \vert_{x=0}+\int_0^lw(x)b\,dx $$     (3) Because we specified that $$w(x=l)=0 \!$$ we have the following
 * style="width:95%" |
 * style="width:95%" |
 *
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \int_0^lw(x)\frac{d}{dx} \left (AE\frac{du}{dx} \right ) \,dx+\int_0^lw(x)b\,dx =- \left ( w(x)AE\frac{du}{dx} \right ) \Bigg \vert_{x=0}-\Big (w(x)A\overline{t}\Big )\Big \vert_{x=0} $$     (4)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \int_0^lw(x)\left [ \frac{d}{dx} \left (AE\frac{du}{dx} \right )+b \right ]\,dx +w(x)A\left ( E\frac{du}{dx}+\overline{t} \right )\Bigg \vert_{x=0}=0 $$     (5) Now, the authors select the form of $$ w(x) \!$$ to be
 * style="width:95%" |
 * style="width:95%" |
 *
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle w(x) = \psi(x) \left[\frac{d}{dx} \left(AE\frac{du}{dx} \right ) +b \right ] $$     (6) prescribing the form of $$ \psi(x) \!$$ to be
 * style="width:95%" |
 * style="width:95%" |
 *
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle 0< \psi(x) \quad \text{for} \quad 0
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \psi(x) = x(l-x) $$
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Setting $$ w(x) \!$$ and $$ \psi(x) \!$$ to the above forms and plugging into Eq. 5 we get the following
 * {| style="width:100%" border="0"

$$  \displaystyle \int_0^l\psi(x)\left [ \frac{d}{dx} \left (AE\frac{du}{dx} \right )+b \right ]^2\,dx +w(x)A\left ( E\frac{du}{dx}+\overline{t} \right )\Bigg \vert_{x=0}=0 $$     (8)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

My Question
Now, I think this is were I am getting lost. Here is what I can make from it... Due to the way we defined $$\psi(x),\!$$ the integral in Eq.8 must be zero. This is because we assumed a particular form for $$ w(x) \!$$ which contrains Eq.5's intregral expression to zero (being that $$ \psi(x=0)=\psi(x=l)=0 \!$$ which will result when evaluating at the limits as some type of expression of the form $$ 0 - 0 \!$$).
 * {| style="width:100%" border="0"

$$  \displaystyle \therefore w(x)A\left ( E\frac{du}{dx}+\overline{t} \right )\Bigg \vert_{x=0}=0 $$     (9) Now the text goes on to redefine $$ w(x) \!$$ to allow for the natural condition to be satisfied through Eq.9. They define $$ w(x) \!$$ as
 * style="width:95%" |
 * style="width:95%" |
 *
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle w(x) = \begin{cases} 1, & x=0\\ 0, & x=l \end{cases} $$ I can't see how they make this jump and I have been thinking about this for a few days now off and on. I thought that Eq. 8 is a particular formulation of Eq. 5, which came from the authors prescribing the form of $$ w(x) \!$$ in Eq. 6. Wouldn't this therefore not allow us the freedom to again choose $$ w(x) \!$$ to be whatever we want it to be and apply it to the particular case of Eq. 5 (Eq. 9), which is already dependent on $$ w(x) \!$$ being of the form of Eq. 6? Because it seems to me that our abilities to formulate Eq. 9 should depend on choosing the particular form of $$ w(x) \!$$ to match Eq. 6. Am I miss understanding the term "arbitrary"? Please Help.
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

title: User:Egm5526.s11.team-2.langpm


title: File:Fe1.s11.tm7.HW3.fig3.2.1.jpg


text:
User:Egm6341.s11.team5.shin/HW3

= Problem 3 - Plotting $$e_n^L(f,x), f_n^L(x), l_{i,n}(x), q_{n+1}(x)$$ = From (meeting 15 page 1)

Given
1. $$\displaystyle f(x)=\frac{e^x-1}{x} $$ on $$\displaystyle [-1,+1] $$, $$\displaystyle n=4 $$
 * $$\displaystyle t=5 $$, $$\displaystyle x=0.75 $$

2. $$\displaystyle f(x)=\frac{1}{1+4x^2} $$ on $$\displaystyle [-5,+5] $$, $$\displaystyle n=8 $$
 * $$\displaystyle t=4.5 $$, $$\displaystyle x=3 $$

Find
1.
 * Plot $$\displaystyle x \mbox{ vs. } f(x)$$, $$\displaystyle f_n^L(x)$$, $$\displaystyle e_n^L(x)$$, $$\displaystyle l_{i,n}(x)$$, $$\displaystyle q_{n+1}(x), n=4$$.
 * Plot $$\displaystyle l_{2,4}(\cdot)$$

2.
 * Plot $$\displaystyle x \mbox{ vs. } f(x)$$, $$\displaystyle f_n^L(x)$$, $$\displaystyle e_n^L(x)$$, $$\displaystyle l_{i,n}(x)$$, $$\displaystyle q_{n+1}(x), n=8$$.
 * Plot $$\displaystyle l_{3,8}(\cdot)$$

Addition
 * Find $$\displaystyle e_4^L(f,t)$$ and $$\displaystyle e_4^L(f,x)$$
 * Find $$\displaystyle e_8^L(f,t)$$ and $$\displaystyle e_8^L(f,x)$$

Solution
1. $$\displaystyle f(x)=\frac{e^x-1}{x}, x \in [-1,+1] $$
 * From the class notes (p.11-3), the Lagrangian interpolation error is defined as
 * {| style="width:100%" border="0" align="left"

e_4^L(f;x)=f(x)-f_4^L(x) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (3-1)
 * }
 * ,where
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

f_4^L(x) = \sum_{i=0}^4 l_{i,4}(x)f(x_i) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (3-2)
 * }
 * Also, the Lagrangian interpolation functions are defined as
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{i,4}(x)=\prod_{j=0}^4 \frac{x-x_j}{x_i-x_j} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (3-3)
 * }
 * From the class notes (p.11-3), $$\displaystyle q_{5} $$ can be computed using the equation below.
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

q_{5}(x)=\prod_{i=0}^{4}(x-x_i) $$ $$ Plot:
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (3-4)
 * }
 * }
 * Nm1.s11.team5.HW3.fig3a.png]
 * Nm1.s11.team5.HW3.fig3b.png]
 * Nm1.s11.team5.HW3.fig3c.png]
 * Nm1.s11.team5.HW3.fig3d.png]

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

Addition: Find $$\displaystyle e_4^L(f,t)$$ and $$\displaystyle e_4^L(f,x)$$, $$\displaystyle t=5, x=0.75 $$
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }
 * Using Eq (3-1),(3-2) and (3-3), $$\displaystyle e_4^L(f,t)$$ and $$\displaystyle e_4^L(f,x)$$ can be computed.
 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} e_4^L(f,t)&=11.0239 \\ e_4^L(f,x)&=-1.6274 \times 10^{-4} \end{align} $$ Matlab code:
 * style="width:50%; padding:10px; border:2px solid #8888aa" align="center" |
 * style="width:50%; padding:10px; border:2px solid #8888aa" align="center" |
 * }
 * }
 * }
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }

2. $$\displaystyle f(x)=\frac{1}{1+4x^2}, x \in [-5,+5] $$
 * From the class notes (p.11-3), the Lagrangian interpolation error is defined as
 * {| style="width:100%" border="0" align="left"

e_8^L(f;x)=f(x)-f_8^L(x) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (3-5)
 * }
 * ,where
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

f_8^L(x) = \sum_{i=0}^8 l_{i,8}(x)f(x_i) $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (3-6)
 * }
 * Also, the Lagrangian interpolation functions are defined as
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{i,8}(x)=\prod_{j=0}^8 \frac{x-x_j}{x_i-x_j} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (3-7)
 * }
 * From the class notes (p.11-3), $$\displaystyle q_{9} $$ can be computed using the equation below.
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

q_{9}(x)=\prod_{i=0}^{8}(x-x_i) $$ $$ Plot:
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (3-8)
 * }
 * }
 * Nm1.s11.team5.HW3.fig3e.png]
 * Nm1.s11.team5.HW3.fig3f.png]
 * Nm1.s11.team5.HW3.fig3g.png]
 * Nm1.s11.team5.HW3.fig3h.png]

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

Addition: Find $$\displaystyle e_8^L(f,t)$$ and $$\displaystyle e_8^L(f,x)$$, $$\displaystyle t=4.5, x=3 $$
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }
 * Using Eq (3-5),(3-6) and (3-7), $$\displaystyle e_8^L(f,t)$$ and $$\displaystyle e_8^L(f,x)$$ can be computed.
 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} e_8^L(f,t)&=1.784 \\ e_8^L(f,x)&=-0.3786 \end{align} $$ Matlab code:
 * style="width:50%; padding:10px; border:2px solid #8888aa" align="center" |
 * style="width:50%; padding:10px; border:2px solid #8888aa" align="center" |
 * }
 * }
 * }
 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }

= Problem 4 - Area of a Bifolium = From (meeting 15 page 2)

Given
The bifolium: $$\displaystyle r(\theta)=2sin(\theta)cos^{2}(\theta)$$, where $$\displaystyle \theta=[0,\pi]$$.

Find
1. Do literature search to find history and application, if any, of this classic curve. 2. Find the area of one folium,with an accuracy of $$10^{-6}$$ by:
 * 2.1 Composite trapezoidal
 * 2.2 Composite Simpson
 * 2.3 Area of triangles

Solution
The bifolium has the following mathematical equation.
 * {| style="width:100%" border="0" align="left"

r(t)=2\sin(t)\cos^2(t) \mbox{, } t \in [0, \pi] $$ $$ In this problem, we're only interested in computing the area in one leaf, so $$\displaystyle t $$ will have the different range.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (4-1)
 * }
 * }
 * $$\displaystyle t \in [0, \frac{\pi}{2}]$$

In order to apply composite Trapezoidal rule and composite Simpson's rule, we need to have the equation expressed in the Cartesian coordinate system. We have the following properties to obtain the equation defined in the Cartesian coordinate system.
 * {| style="width:100%" border="0" align="left"

\begin{align} \sin(t)&=\frac{y}{r} \\ \cos(t)&=\frac{x}{r} \\ r&=\sqrt{x^2+y^2} \end{align} $$ $$ Substitute Eq(4-2) to Eq(4-1) then, we have the following equation expressed in terms of $$\displaystyle x,y $$.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (4-2)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

(x^2+y^2)^2=2x^2y $$ $$ Before jumping into the calculation of area, it should be noted that the values that $$\displaystyle x $$ can take on are different from $$\displaystyle t \in [0,\frac{\pi}{2}]$$. But, it can be found out simply by running the following code. Matlab code:
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (4-3)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

As a result, $$\displaystyle x $$ is in the range of the following.
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }
 * $$\displaystyle x \in [0.0, 0.6495] $$

In order to obtain $$\displaystyle y $$ corresponding to $$\displaystyle x $$ in the range mentioned above, the following equation needs to be solved. The equation below is derived from Eq(4-3) by expanding out Eq(4-3) and expressing in terms of $$\displaystyle y $$.
 * {| style="width:100%" border="0" align="left"

y^4+2x^2y^2-2x^2y+x^4=0 $$ $$ Eq(4-4) is the 4th-order polynomial equation expressed in terms of $$\displaystyle y $$. In this equation, the terms expressed in $$\displaystyle x $$ will be considered as coefficients. Eq(4-4) can be solved using MATLAB command 'roots'. Area computed using Wolfram Alpha The area of one leaf in bifolium is $$\displaystyle \frac{\pi}{16} $$. (Refer to Wolfram Alpha) 1. Composite Trapezoidal rule
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (4-4)
 * }
 * }
 * From the class notes (p.7-4), the composite Trapezoidal rule is defined as below.
 * {| style="width:100%" border="0" align="left"

I_n=h(\dfrac{1}{2}f(x_0)+f(x_1)+f(x_2)+...+f(x_{n-1})+\dfrac{1}{2}f(x_{n})) \mbox{, }h=\dfrac{b-a}{n} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (4-5)
 * }
 * The area can be computed by subtracting the integration on the upper curve by the integration on the lower curve on one leaf.
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

AREA=I_{upper}-I_{lower} $$ $$ Matlab code:
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (4-6)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

Result:
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }
 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} n&=10000 \\ Area&=0.196349420478339 \\ Accuracy&=1.203710234265465 \times 10^{-7} \end{align} $$ 2. Composite Simpson's rule
 * style="width:50%; padding:10px; border:2px solid #8888aa" align="center" |
 * style="width:50%; padding:10px; border:2px solid #8888aa" align="center" |
 * }
 * }
 * }
 * From the class notes (p.7-4), the composite Simpson's rule is defined as below.
 * {| style="width:100%" border="0" align="left"

I_n=\dfrac{h}{3}(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n))\mbox{, }h=\frac{b-a}{n} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (4-7)
 * }
 * The area can be computed in the same way as in Eq(4-6).
 * The area can be computed in the same way as in Eq(4-6).

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

Result:
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }
 * {| style="width:100%" border="0" align="left"

$$\displaystyle \begin{align} n&=10000 \\ Area&=0.196349471145946 \\ Accuracy&=6.9703415817024 \times 10^{-8} \end{align} $$
 * style="width:50%; padding:10px; border:2px solid #8888aa" align="center" |
 * style="width:50%; padding:10px; border:2px solid #8888aa" align="center" |
 * }
 * }
 * }

3. Sum of Triangles The area on the bifolium can be computed by summing all areas of triangle divided in uniform angle. Matlab code:
 * {| style="width:100%" border="0" align="left"

Result:
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$$\displaystyle \begin{align} n&=10000 \\ Area&=0.196349527930081 \\ Accuracy&=1.291928136692988 \times 10^{-8} \end{align} $$
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 * }
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 * }

title: User:Egm6341.s11.team5.shin/HW3


text:
User:Eml5526.s11.team4.yang/hw3

=Homework 3.4-Show$$\displaystyle A\ne B$$, when $$\displaystyle i\ne j$$=

Refer to lecture slide for the problem statement.

Given

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

$$ \displaystyle
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A={{b}_{i}}({{a}_{2}}^{'}{{b}_{j}}^{'}+{{a}_{2}}{{b}_{j}}^{''})

$$     (Eq 1)
 *
 * }
 * {| style="width:100%" border="0"

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

B={{b}_{j}}({{a}_{2}}^{'}{{b}_{i}}^{'}+{{a}_{2}}{{b}_{i}}^{''})

$$     (Eq 2)
 *
 * }

Objectives
Show$$\displaystyle A\ne B$$, when $$\displaystyle i\ne j$$
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$$ \displaystyle
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{{b}_{i}}(x)=\cos (ix)

$$     (Eq 3)
 *
 * }
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$$ \displaystyle
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{{b}_{j}}(x)=\cos (jx)

$$     (Eq 4)
 *
 * }

Solutions
Substituting Eqs. (3),(4) into Eqs. (1) and (2) yields:
 * {| style="width:100%" border="0"

$$ \displaystyle
 * style="width:95%" |
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\begin{align} & A={{b}_{i}}({{a}_{2}}^{'}{{b}_{j}}^{'}+{{a}_{2}}{{b}_{j}}^{''})=\cos (ix)[{{a}_{2}}^{'}\centerdot (-j)\sin (jx)+{{a}_{2}}\centerdot {{j}^{2}}\cos (jx)] \\ & \begin{matrix} {} & = \\ \end{matrix}{{a}_{2}}\centerdot {{j}^{2}}\cos (ix)\cos (jx)-{{a}_{2}}^{'}\centerdot j\sin (jx)\cos (ix) \\ \end{align}

$$     (Eq 5)
 *
 * }
 * {| style="width:100%" border="0"

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

\begin{align} & B={{b}_{j}}({{a}_{2}}^{'}{{b}_{i}}^{'}+{{a}_{2}}{{b}_{i}}^{''})=\cos (jx)[{{a}_{2}}^{'}\centerdot (-i)\sin (ix)+{{a}_{2}}\centerdot {{i}^{2}}\cos (ix)] \\ & \begin{matrix} {} & = \\ \end{matrix}{{a}_{2}}\centerdot {{i}^{2}}\cos (ix)\cos (jx)-{{a}_{2}}^{'}\centerdot i\sin (ix)\cos (jx) \\ \end{align}

$$     (Eq 6) When $$\displaystyle i\ne j$$, it is obvious that:
 *
 * }
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$$ \displaystyle
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{{j}^{2}}\cos (ix)\cos (jx)\ne {{i}^{2}}\cos (ix)\cos (jx)

$$     (Eq 7)
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
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j\sin (jx)\cos (ix)\ne i\sin (ix)\cos (jx)

$$     (Eq 8) When $$\displaystyle {{a}_{2}}^{'}\ne 0$$ or $$\displaystyle {{a}_{2}}\ne 0$$, it shows '''$$\displaystyle A\ne B$$. '''
 *
 * }

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

 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

Thus, if $$\displaystyle {{a}_{2}}^{'}$$ and  $$\displaystyle {{a}_{2}}$$ are not all zero, $$\displaystyle A\ne B$$ ( $$\displaystyle i\ne j$$).
 *
 * }

Specifically, we can take i=1 and j=2 for example, then Eqs. (5) and (6) become:
 * {| style="width:100%" border="0"

$$ \displaystyle
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A=4{{a}_{2}}\centerdot \cos (x)\cos (2x)-2{{a}_{2}}^{'}\centerdot \sin (2x)\cos (x)

$$     (Eq 9)
 *
 * }
 * {| style="width:100%" border="0"

$$ \displaystyle
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B={{a}_{2}}\cos (x)\cos (2x)-{{a}_{2}}^{'}\sin (x)\cos (2x)

$$     (Eq 10) When $$\displaystyle {{a}_{2}}^{'}\ne 0$$ or $$\displaystyle {{a}_{2}}\ne 0$$, it shows '''$$\displaystyle A\ne B$$. '''
 *
 * }

The equation $$\displaystyle A=B$$is valid only $$\displaystyle {{a}_{2}}^{'}=0$$ and $$\displaystyle {{a}_{2}}=0$$.

title: User:Eml5526.s11.team4.yang/hw3


text:
User:Egm5526.s11.team4.zou/HW3

= Problem 3.5 - Equavalent stiffness of a spring=

The following problem is problem 2.2 as taken from Fish and Belytschko's 'A first course in Finite Elements'

Given:
The geometry of bar of thickness $$\displaystyle t$$ with a centered square hole shown in figure 1

Find:
Prove that the equivalent stiffness of a spring aligned in x direction is

Where $$\displaystyle E$$ is the Young's modulus and $$\displaystyle t$$ is the width of the bar.

Solution:
First divided the system into four parts (1,2,3,4)shown as below

Every part can be equivalently viewed as a spring:

Where the equivalent spring stiffness of part1 is:

And equivalent spring stiffness of part4:$$\displaystyle {{K}_{4}}=\frac=\frac{Eat}{(l/10)}=\frac{10Eat}{l}={{K}_{1}}$$

Similarily:

The equivalent spring stiffness of part3 is equals to the equivalent spring stiffness of part3

Since $$\displaystyle {{K}_{2}}$$and $$\displaystyle {{K}_{3}}$$are parallel–connected and$$\displaystyle {{K}_{2}}={{K}_{3}}$$

So the equivalent spring stiffness of this parallelly–connected is$$\displaystyle {{K}_{2-3}}={{K}_{2}}+{{K}_{3}}=2{{K}_{2}}=\frac{5Eat}{l}$$

Then the system can be drawn like the following:

Since $$\displaystyle {{K}_{1}}$$, $$\displaystyle {{K}_{2-3}}$$and $$\displaystyle {{K}_{4}}$$are series–connected and$$\displaystyle {{K}_{1}}={{K}_{4}}$$

So the equivalent spring stiffness of this series–connected is

Substituting $$\displaystyle {{K}_{1}}$$, $$\displaystyle {{K}_{2-3}}$$and$$\displaystyle {{K}_{4}}$$yield:

Problem Statement
Consider a trial (candidate) solution of the form $$\displaystyle u(x)={{\alpha }_{0}}+{{\alpha }_{1}}(x-3)+{{\alpha }_{2}}{{(x-3)}^{2}}$$ and a weight function of the same form. Obtain a solution to the weak form in Problem 3.1. Check the equilibrium equation in the strong form in Problem 3.1; is it satisfied?

Check the natural boundary condition; is it satisfied?

Problem 3.1 Statement
From problem 3.1 in Fish and Belytschko :

Show that the weak form of

is given by

Acquiring the solution to the weak form
Consider general quadratic trial solution to the weak form,

Since the unknown solution and the weight function should have the same basis functions, we assume the weight function to be,

Plug Eq 10.3 into essential boundary condition, we have

The weak form requires that \omega (3)=0, which means,

Substitute Eq 10.3 ~ Eq 10.6 into the weak form Eq 10.2,

Since $$\displaystyle {{\beta }_{1}}$$ and $$\displaystyle {{\beta }_{2}}$$ are arbitrary, the coefficient of them must be zero, which yields,

Solve Eqs 10.8 we have,

Conbine Eqs 10.9 with Eq 10.5 we have the solution to weak form $$\displaystyle u(x)$$,

where $$\displaystyle I=\int_{1}^{3}{EAdx}$$, $$\displaystyle J=\int_{1}^{3}{(x-3)EAdx}$$, $$\displaystyle K=\int_{1}^{3}{{{(x-3)}^{2}}EAdx}$$ respectively.

Check the validity of the quadratic solution
Looking back to the strong form in Eq 10.1, we know that the quadradic solution will reduce to a constant after twice derivation. On the other hand the properties of $$\displaystyle E$$ and $$\displaystyle A$$ are unknown to us. Thus it is unlikely that the equality in Eq 10.1 should be satisfied. Hence we can almost conclude that any quadratic solution to this problem will fail to satisfy the strong form.

Moreover, since the weak form has absolve the natural boundary condition into one equation, the "error" produced by the solution can affect the equality of the natural boundary condition. Thus we have this possibility that the natural boundary condition is NOT satisfied.

We can illustrate this point by assuming $$\displaystyle E$$ and $$\displaystyle A$$ to be constants. In this case, $$\displaystyle I=2AE$$, $$\displaystyle J=-2AE$$, $$\displaystyle K=\frac{8}{3}AE$$, substitute them into Eq 10.10 we have,

Thus, natural B.C. becomes:

It is obvious that the trial solution don't satisfy the natural boundary condition. It is observed that the trial solution is only valid when the area A is infinite.

Secondly, adopting the trial solution, strong form becomes:

It is obvious that the trial solution don't exactly satisfy strong form.

And the strong form is valid only at $$\displaystyle x=2$$.

Discussion
We note that the Example3.3 in the textbook also used quadratic solution to solve a similar strong form(thus similar weak form). Though the quadratic solution fails to satisfy the strong form, it does satisfy the natural boundary condition(see figure 3.7(b) in the book). However, a mistake in the book is found: The third equation in page 57 is not valid, the term $$\displaystyle \int_{0}^{2}{({{\beta }_{1}}x+{{\beta }_{2}}{{x}^{2}})}10dx$$ should be $$\displaystyle \int_{0}^{2}{({{\beta }_{1}}x+{{\beta }_{2}}{{x}^{2}})}10xdx$$. Since the following conclusion in the book is based on that equation, it's reasonable to doubt the fact that their quadratic solution doesn't violate the natural b.c.

title: User:Egm5526.s11.team4.zou/HW3


text:
User:Eml5526.s11.team6.joglekar/hwk2

My Homework
Problem 2.4

=Prove= $$ \int\frac{x^2}{1+x}\, dx=\frac{x^2}{2}-x+log(1+x)+k $$

Solution
Let $$ x^2=u $$

$$ 2x dx=du $$

$$ \frac{1}{1+x}=v $$

$$\frac{(-1)}{(1+x)^2} dx=dv $$

$$ x=w $$

$$dx=dw$$

Rule for Integration by parts:
 * $$\int u v\, dw = u v w - \int u w\, dv - \int v w\, du.\!$$

Therefore $$ \int\frac{x^2}{1+x}\, dx=\ (x^2)*\frac{1}{1+x}*x - \int (x^2)*x*\frac{(-1)}{(1+x)^2}dx- \int\frac{1}{1+x}*x*2x dx $$ $$\int\frac{x^2}{1+x}\, dx=\frac{x^3}{1+x} + \int \frac{x^3}{(1+x)^2}dx - \int\frac{2x^2}{1+x}dx $$

Now Let $$\int\frac{x^2}{1+x}\, dx= I $$

Thus $$\int\frac{x^2}{1+x}\, dx=\frac{x^3}{1+x} + \int \frac{x^3}{(1+x)^2}dx -2I $$

$$3I=\frac{x^3}{1+x} + \int \frac{x^3}{(1+x)^2}dx $$

$$\int \frac{x^3}{(1+x)^2}dx=I_1$$

$$3I=\frac{x^3}{1+x} + I_1$$

$$To find:I_1: $$

let

$$1+x=a $$

$$x=a-1 $$

$$x^3=(a-1)^3 $$

$$\frac{1}{1+x}=\frac{1}{a} $$

Therefore

$$ I_1 =\int\frac{(a-1)^3}{a^2} da $$

$$ I_1=\int\frac{a^3-3a(a-1)-1}{a^2} da $$

$$ I_1=\int a-3\frac{(a-1)}{a}-\frac{1}{a^2}da $$

$$ I_1=\int a da-3\int (1-\frac{1}{a}) da-\int\frac{1}{a^2}du $$

$$ I_1=\frac{a^2}{2}-3a+3log(a)+\frac{1}{a}$$

but a=1+x

Therefore

$$ I_1=\frac{(1+x)^2}{2}-3(1+x)+3log(1+x)+\frac{1}{1+x} $$

Substituting back

$$3I=\frac{x^3}{1+x} + \frac{(1+x)^2}{2}-3(1+x)+3log(1+x)+\frac{1}{1+x} $$

$$3I=\frac{x^3+1}{1+x} + \frac{(1+x)^2}{2}-3(1+x)+3log(1+x) $$

$$I=\frac{x^3+1}{3(1+x)}+\frac{(1+x)^2}{6}- (1+x)+log(1+x) $$

$$I=\frac{(x+1)(x^2-x+1)}{3(x+1)}+\frac{x^2}{6}+\frac{2x}{6}+\frac{1}{6}-1-x+log(1+x) $$

$$I=\frac{x^2}{3}-\frac{x}{3}+\frac{1}{3}+\frac{x^2}{6}+\frac{2x}{6}+\frac{1}{6}-1-x+log(1+x) $$

$$I=\frac{x^2}{2}-x+log(1+x)-\frac{(-1}{2} $$

$$I=\frac{x^2}{2}-x+log(1+x)+k $$

2.4.5
Given:

$$\frac{d}{dx}[(2+3x)*\frac{du}{dx}]+5x = 0 for all x - >0,1< $$

$$ u(1)=4, -\frac{du}{dx} (x=0) = 6 $$

To find:

Expression for u(x):

Solution:

$$ \frac{d}{dx}[(2+3x)\frac{du}{dx}]+5x=0 $$

$$ \frac{d}{dx}[(2+3x)\frac{du}{dx}]= - 5X $$

Integrating with respect to x

$$ (2+3x) \frac{du}{dx}= \frac{-5x^2}{2}+k1 $$

$$ \frac{du}{dx}= \frac{-5x^2}{2(2+3x)}+\frac{k1}{2+3x} $$

Integrating again with respect to x

$$ u(x)= \frac{-5}{2}[\frac{2*log(3x+2)+3x(x-4)}{3^3*2}]+k1[\frac{ln(2+3x)}{3}]+k2 $$

To find the constants, we use the given boundary conditions:

$$ \frac{du}{dx}= \frac{-5*x^2}{2(2+3x)}+\frac{k1}{(2+3x)} $$

$$ but \frac{-du}{dx}(x=0)=6 $$

Substituting x=1 in the above equation,

k1= -12

Substituting k1=-12 and simplying, we get:

$$ u(x)= \frac{-118}{27}*log(2+3x)-\frac{5*x^2}{12}+\frac{5}{9}x+k2 $$

Also u(x)=4 at x=1

Substituting x=1 in the expression for u(x);

k2= 10.896

Thus final expression for u(x) is

$$ u(x)=\frac {-118}{27}*log(2+3x)-5*\frac{x^2}{12}+\frac{5}{9}x+10.896 $$

title: User:Eml5526.s11.team6.joglekar/hwk2


text:
User talk:Deane kennedy

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title: User talk:Deane kennedy


text:
User:Eml5526.s11.team3.akj/Homework 3

Problem Statement 3.1
(Do Problem 2.9 with polynomial basis functions with $$k=1$$: $$\{{{(x+k)}^{j}},j=0,1,2,\cdots \}$$. 1.)	Find two equations that enforce boundary conditions for $${{u}^{h}}(x)$$ 2.)	Find one more equation to solve for $$d=\{{{d}_{j}}\}$$ by projecting the residue $$P({{u}^{h}})$$ on a basis function $${{b}_{k}}(x)$$ with k = 0, 1, 2 such that the additional equation is linearly independent from the above two equations in 1.) 3.)	Display the three equations in matrix form: $$Kd=F$$ and comment on the symmetric properties of $$K$$. 4.)	Solve for $$d$$. 5.)	Construct $${{u}^{h}}(x)$$ and plot $${{u}^{h}}(x)$$ vs $$u(x)$$ which is the exact solution. 6.)	Repeat 1-5 for n = 4 and n = 6. 7.)	Computer $$u_{n}^{h}(x=0.5)$$ for n = 2, 4, 6 Error $${{e}_{n}}(0.5)=u(0.5)-{{u}^{h}}(0.5)$$ Plot $${{e}_{n}}(0.5)$$ vs n (convergence)

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

$$\displaystyle
 * style="width:95%" |
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u(1)=0 and \frac{du}{dx}(0)=-4

$$
 * }

Solution
1) For $$n=2$$  :

for $$n=2$$  the basis function is

$$\displaystyle

\left\{ {{b}_{i}}\left( x \right)={{(x+1)}^{i}},i=0,1,2 \right\}

$$ then, $$b_{0}(x)=1,b_{1}(x)=(x+1),b_{2}(x)=(x+1)^{2}$$

the main aim of this problem is to solve the PDE given by Eq(1) by discrete Weighted Residual form method, in the method we assume the solution $$u(x)$$  as  $$u^{h}(x)= \sum_{j=0}^{n}d_{j}b_{j}(x)$$

for $$n=2$$ ,  $$u^{h}(x)= \sum_{j=0}^{2}d_{j}b_{j}(x)$$

i.e.

given essential boundary condition as $$u(1)=0$$

hence the first equation which enforces boundary condition is $$ d_{0}1+d_{1}(1+1)+d_{2}(1+1)^{2}=0 $$

Essential boundary condition is

given natural boundary condition as $$\frac{du}{dx}(0)=-4$$

natual boundary condition written as $$d_{0}\frac{db_{0}}{dx}+d_{1}\frac{db_{1}}{dx}+d_{2}\frac{db_{2}}{dx}=-4$$ at $$x=0$$

i.e  $$ d_{0}0+d_{1}(1)+d_{2}2(0+1)^{2}=0 $$

Natural boundary condition is

2) We have to find one more equation to solve for $$d=\{{{d}_{j}}\}$$  by projecting the residue  $$P({{u}^{h}})$$  on a basis function  $${{b}_{k}}(x)$$  with k = 0, 1, 2 such that the additional equation is linearly independent from the above two equations in 1.)

hence we project the residue $$P({{u}^{h}})$$  on a basis function  $${{b}_{1}}(x)$$  to obtain the third equation which is linearly independent to the two boundary conditions obtained.

we get,

given $$a_{2}=2$$,  $$f(x)=3$$  and since we are projecting on first basis function  $$b_{0}(x)=1$$

hence Eq (3.4) becomes $$2d_{0}\int_{0}^{1}\frac{\mathrm{d}^{2} (1)}{\mathrm{d} x^{2}}dx+2d_{1}\int_{0}^{1}\frac{\mathrm{d}^{2} (x+1)}{\mathrm{d} x^{2}}dx+2d_{2}\int_{0}^{1}\frac{\mathrm{d}^{2} (x+1)^{2}}{\mathrm{d} x^{2}}dx=-3\int_{0}^{1}dx$$

the third equation is

3) Eq.(3.2), (3.3) and (3.5) can be written in Matrix form as

Hence

Clearly $$\mathbf{K}$$  is non-Symmetric matrix

4) Using Matlab, we can easily get

5) Hence  $$u^{h}(x)=8-\frac{5}{2}(x+1)-\frac{3}{4}(x+1)^{2}$$

which on simplification gives

the solution of PDE Eq(3.1) gives the exact solution $$u(x)$$

Eq(3.1)is $$\frac{\mathrm{d}^{2}u }{\mathrm{d} x^{2}}=\frac{-3}{2}$$

integrating the above equation once we get

but from essential BC, $$c=-4$$

hence integrating again Eq(3.8) and applying Natural B.C we get exact solution

Matlab Code:

6) To compute $$u^{h}(x=0.5)$$   for n = 2 Substituting   $$x=0.5$$  in Eq(3.7)

we get,

$$u(0.5)=2.5625$$

Hence error $${{e}_{n}}(0.5)=2.5625-2.5625=0$$

Plot $${{e}_{n}}(0.5)$$ vs n (convergence)

For $$n=4$$  :

For $$n=4$$  and  $$n=6$$  the computation becomes difficult hence Matlab is used fro computational purpose.

1) for $$n=4$$  the basis function is

$$\displaystyle \left\{ {{b}_{i}}\left( x \right)={{(x+1)}^{i}},i=0,1,2,3,4 \right\} $$ then, $$b_{0}(x)=1,b_{1}(x)=(x+1),b_{2}(x)=(x+1)^{2},b_{3}(x)=(x+1)^{3},b_{4}(x)=(x+1)^{4}$$

Two equations that enforce boundary conditions are

Essential boundary condition

and Natural boundary condition is

2)

Additional equations to solve for $$d=\{{{d}_{j}}\}$$  by projecting the residue  $$P({{u}^{h}})$$  on the basis function  $${{b}_{k}}(x)$$  with k = 0, 1, 2 such that the additional equations are linearly independent from the above two equations in 1.)

3) Equations in matrix form

Hence

Clearly $$\mathbf{K}$$  is non-Symmetric matrix

4) Using Matlab, we can easily get

5) Hence  $$u^{h}(x)=8-\frac{5}{2}(x+1)-\frac{3}{4}(x+1)^{2}$$

which on simplification gives

and exact solution

Matlab Code:

Note: It is observed that on solving for $$n=4$$  we got  $$d_{3}=0$$  and  $$d_{4}=0$$  hence the plot for  $$n=2$$  and  $$n=4$$  are same

6) Since  $$d_{3}=0$$  and  $$d_{4}=0$$  for  $$n=4$$  hence  $$u^{h}(x)$$  and  $$u(x)$$  will be same for  $$n=2$$  and  $$n=4$$  hence

and $$u(0.5)=2.5625$$

Hence error $${{e}_{n}}(0.5)=2.5625-2.5625=0$$

For $$n=6$$  :

1) for $$n=6$$  the basis function is

$$\displaystyle \left\{ {{b}_{i}}\left( x \right)={{(x+1)}^{i}},i=0,1,2,3,4,5,6 \right\} $$

then, $$b_{0}(x)=1,b_{1}(x)=(x+1),b_{2}(x)=(x+1)^{2},b_{3}(x)=(x+1)^{3},b_{4}(x)=(x+1)^{4},b_{5}(x)=(x+1)^{5},b_{6}(x)=(x+1)^{6}$$

Two equations that enforce boundary conditions are

Essential boundary condition

and Natural boundary condition is

2)

Additional equations to solve for $$d=\{{{d}_{j}}\}$$  by projecting the residue  $$P({{u}^{h}})$$  on the basis function  $${{b}_{k}}(x)$$  with k = 0, 1, 2,3,4 such that the additional equations are linearly independent from the above two equations in 1.)

3) Equations in matrix form

Hence

Clearly $$\mathbf{K}$$  is non-Symmetric matrix

4) Using Matlab, we can easily get

5) Hence  $$u^{h}(x)=8-\frac{5}{2}(x+1)-\frac{3}{4}(x+1)^{2}$$

which on simplification gives

and exact solution

Plot of $$u^{h}(x)vs u(x) for n=2,n=4,n=6$$

the approximate $$u^{h}(x)$$  and exact solution  $$ u(x) $$  for n=2 ,n=4 and n=6 are found to be same, which is also evident from the plot where all the plots overlap on one another

Matlab Code:

Note: It is observed that on solving for $$n=6$$  we got  $$d_{3}=0$$ ,  $$d_{4}=0$$  ,  $$d_{5}=0$$  and  $$d_{6}=0$$  hence the plot for  $$n=4$$  and  $$n=6$$  are same

6) Since  $$d_{3}=0$$ ,  $$d_{4}=0$$  ,  $$d_{5}=0$$  and  $$d_{6}=0$$  for  $$n=6$$  hence  $$u^{h}(x)$$  and  $$u(x)$$  will be same for  $$n=4$$  and  $$n=6$$  hence

and $$u(0.5)=2.5625$$

Hence error $${{e}_{n}}(0.5)=2.5625-2.5625=0$$

Plot showing error between approximate and actual solution at x=0.5

since we got both approximate and actual solution to be the same hence the error is zero through out

Solution
This is the special case of Problem 3.1 where k=0 hence the basis functions will be $$\displaystyle

\left\{ {{b}_{i}}\left( x \right)={{x}^{i}},i=0,1,2 \right\}

$$

1) For $$n=2$$  : for $$n=2$$  the basis function is  $$b_{0}(x)=1,b_{1}(x)=x,b_{2}(x)=x^{2}$$

applying the boundary conditions as mentioned in Problem 3.1 the two equations satisfying boundary conditions will be

Essential boundary condition is

and

Natural boundary condition is

2) One more equation which is independent to the BC's and which is obtained by projecting the residue $$P({{u}^{h}})$$  on a basis function  $${{b}_{k}}(x)$$  with k = 0, 1, 2

hence the third equation is

3) Matrix form

Hence

Clearly $$\mathbf{K}$$  is non-Symmetric matrix

4) Using Matlab, we can easily get

5) Hence

exact solution as solved in Problem 3.1

The approximate solution happens to match completely with the exact solution

Matlab Code:

6)  $$u^{h}(x)$$  obtained in part 5 is same as the one in Problem 3.1 hence  $$u^{h}(x=0.5)$$  will be same as one found in Problem 3.1

and error is $${{e}_{n}}(0.5)=2.5625-2.5625=0$$

For $$n=4$$  : 1)

for $$n=4$$  the basis function is  $$b_{0}(x)=1,b_{1}(x)=x,b_{2}(x)=x^{2},b_{3}(x)=x^{3},b_{4}(x)=x^{4}$$

Two equations that enforce boundary conditions are Essential boundary condition is

and

Natural boundary condition is

2) Additional equations to solve for $$d=\{{{d}_{j}}\}$$  by projecting the residue  $$P({{u}^{h}})$$  on the basis function  $${{b}_{k}}(x)$$  with k = 0, 1, 2 such that the additional equations are linearly independent from the above two equations in 1.)

3) Matrix form

Hence

Clearly $$\mathbf{K}$$  is non-Symmetric matrix

4) Using Matlab, we can easily get

5) Hence

exact solution

The approximate solution happens to match completely with the exact solution

Matlab Code:

6)

and error is $${{e}_{n}}(0.5)=2.5625-2.5625=0$$

For $$n=6$$  : 1)

for $$n=6$$  the basis function is  $$b_{0}(x)=1,b_{1}(x)=x,b_{2}(x)=x^{2},b_{3}(x)=x^{3},b_{4}(x)=x^{4},b_{5}(x)=x^{5},b_{6}(x)=x^{6}$$

Two equations that enforce boundary conditions are Essential boundary condition is

and

Natural boundary condition is

2) Additional equations to solve for $$d=\{{{d}_{j}}\}$$  by projecting the residue  $$P({{u}^{h}})$$  on the basis function  $${{b}_{k}}(x)$$  with k = 0, 1, 2,3,4 such that the additional equations are linearly independent from the above two equations in 1.)

3) Matrix form

Hence

Clearly $$\mathbf{K}$$  is non-Symmetric matrix

4) Using Matlab, we can easily get

5) Hence

exact solution

The approximate solution happens to match completely with the exact solution

Plot of $$u^{h}(x)vs u(x) for n=2,n=4,n=6$$

the approximate $$u^{h}(x)$$  and exact solution  $$ u(x) $$  for n=2 ,n=4 and n=6 are found to be same, which is also evident from the plot where all the plots overlap on one another Matlab Code:

6)

and error is $${{e}_{n}}(0.5)=2.5625-2.5625=0$$

Plot showing error between approximate and actual solution at x=0.5

since we got both approximate and actual solution to be the same hence the error is zero through out

Solution
For $$n=2$$  : the basis functions will be $$\displaystyle

\left\{ {{b}_{i}}\left( x \right)={1,cos(ix+\frac{pi}{4})},i=0,1,2 \right\}

$$

1)

for $$n=2$$  the basis function is  $$b_{0}(x)=1,b_{1}(x)=cos(x+\frac{pi}{4}),b_{2}(x)=sin(x+\frac{pi}{4})$$

applying the boundary conditions as mentioned in Problem 3.1 the two equations satisfying boundary conditions will be

Essential boundary condition is

and

Natural boundary condition is

2) One more equation which is independent to the BC's and which is obtained by projecting the residue $$P({{u}^{h}})$$  on a basis function  $${{b}_{k}}(x)$$  with k = 0, 1, 2

hence the third equation is

3) Matrix form

Hence

Clearly $$\mathbf{K}$$  is non-Symmetric matrix

4) Using Matlab, we can easily get

5) Hence

exact solution as solved in Problem 3.1

6)

and error is $${{e}_{n}}(0.5)=2.3394-2.5625=-0.2231$$

For $$n=4$$  : the basis functions will be $$\displaystyle

\left\{ {{b}_{i}}\left( x \right)={1,cos(ix+\frac{pi}{4})},i=0,1,2,3,4 \right\}

$$

1)

for $$n=4$$  the basis function is  $$b_{0}(x)=1,b_{1}(x)=cos(x+\frac{pi}{4}),b_{2}(x)=sin(x+\frac{pi}{4}),b_{3}(x)=cos(2x+\frac{pi}{4}),b_{4}(x)=sin(2x+\frac{pi}{4})$$

applying the boundary conditions as mentioned in Problem 3.1 the two equations satisfying boundary conditions will be

Essential boundary condition is

and

Natural boundary condition is

2) Additional equations which are independent to the BC's and which are obtained by projecting the residue $$P({{u}^{h}})$$  on a basis function  $${{b}_{k}}(x)$$  with k = 0, 1, 2

hence the third equation is

3) Matrix form

Hence

Clearly $$\mathbf{K}$$  is non-Symmetric matrix

4) Using Matlab, we can easily get

5) Hence

exact solution as solved in Problem 3.1

6)

and error is $${{e}_{n}}(0.5)=2.5629-2.5625=0.0004$$

For $$n=6$$  : the basis functions will be $$\displaystyle

\left\{ {{b}_{i}}\left( x \right)={1,cos(ix+\frac{pi}{4})},i=0,1,2,3,4,5,6 \right\}

$$

1)

for $$n=6$$  the basis function is  $$b_{0}(x)=1,b_{1}(x)=cos(x+\frac{pi}{4}),b_{2}(x)=sin(x+\frac{pi}{4}),b_{3}(x)=cos(2x+\frac{pi}{4}),b_{4}(x)=sin(2x+\frac{pi}{4}),b_{5}(x)=cos(3x+\frac{pi}{4}),b_{6}(x)=sin(3x+\frac{pi}{4})$$

applying the boundary conditions as mentioned in Problem 3.1 the two equations satisfying boundary conditions will be

Essential boundary condition is

and

Natural boundary condition is

2) Additional equations which are independent to the BC's and which are obtained by projecting the residue $$P({{u}^{h}})$$  on a basis function  $${{b}_{k}}(x)$$  with k = 0, 1, 2

hence the third equation is

3) Matrix form

Hence

Clearly $$\mathbf{K}$$  is non-Symmetric matrix

4) Using Matlab, we can easily get

5) Hence

exact solution as solved in Problem 3.1

6)

and error is $${{e}_{n}}(0.5)=2.5625-2.5625=0$$

title: User:Eml5526.s11.team3.akj/Homework 3


text:
User:Eml5526.s11.team6.joglekar/hwk3

Given
General one dimensional Model 1.0/Data set 2 on page 1 of Mtg 12:

where $${a_2} = 2$$,  f = 3

Consider $$ {b_j}\left( x \right) = (x+k)^j $$ and k=1 to avoid $${b_j}'\left( 0 \right)=0 $$

Find
A) If n=2, What is the D.O.F. of this problem?

B) Find two equations that enforce boundary conditions for $$ {u^h}\left( x \right) = \sum\limits_{j = 0}^n $$

C) Find one more equation to solve for $$ {\mathbf{d}} = {\left\{ \right\}_{3 \times 1}} $$ by projecting the residue $$ {\mathbf{P}}\left(  \right) $$ on a basis function $$ b_k\left( x \right)     $$ with  k = 0,1,2 such that the additional equation is linear independent from the equations found in the solution of the (B)

D) Display 3 equations in matrix form $$ {\mathbf{Kd}} = {\mathbf{F}} $$ and observe symmetric property of $$ {\mathbf{K}} $$

E) Solve for $$ {\mathbf{d}} $$

F) Construct $$ u_n^h\left( x \right) $$ and plot $$ u_n^h\left( x \right) \ vs \ u\left( x \right) $$ ....(where $$ u\left( x \right) $$ is the exact solution)

G) Repeat (A) to (F) for n=4 and n=6

H) Compare $$ u_n^h\left( {x = 0.5} \right)   for n = 2,4,6 $$

If error $$ {e_n}\left( {0.5} \right) = u\left( {0.5} \right) - {u^h}\left( {0.5} \right) $$ ,Plot $$ {e_n}\left( {0.5} \right) \ vs \ n $$

A
A) Since for this problem,'n' starts from 0;

D.O.F.=n+1

D.O.F.=3

B
B) From (Eq 3.1.2) we can get $$ \sum\limits_{j = 0}^2 {{d_j}{b_j}\left( 1 \right)} = 0 $$

Then rewrite the above equation in matrix form:

where

From (3.1.3) we can get $$ \sum\limits_{j = 0}^2 {{d_j}b_j^{\left( 1 \right)}\left( 0 \right)} =  - 4 $$

Then rewrite the above equation in matrix form:

where

Equations 3.1.4 and 3.1.6 are the two equations that enforce the boundary conditions for $$ {u}^h\left( x \right) $$

c
C) Because the residue $$ P = \frac{d}\left( {{a_2}\frac{d}{b_j}\left( x \right)} \right) - f\left( x \right)$$, after projecting the residue we get:

$$ \sum\limits_{j = 0}^2 {\left\{ {\int_0^1 {{b_k}\left( x \right)\left[ {\frac{d}\left( {{a_2}\frac{d}{b_j}\left( x \right)} \right)} \right]dx} } \right\}} {d_j} = - \int_0^1 {{b_k}\left( x \right)f\left( x \right)dx} $$

Then rewrite the above equation in matrix form:

where

$$ {K_{kj}} = 2\int_0^1 {{b_k}\left( x \right)b_j^{\left( 2 \right)}\left( x \right)dx} $$

$$         = 2\int_0^1 (x+1)^k. j.(j-1).(x+1)^{j-2} dx $$

It is clear that matrix K is not symmetric.

D
In order to simplify the computation, the following fortran code was used based on equations (3.1.8),(3.1.9) and (3.1.10) for n=1-10

Using the above codes we can get solutions for all the possible inputs as follows:

The exact solution of eq.3.1.1 by enforcing the boundary condition is

$$ u(x)= -\frac{3x^2}{2}-8x+\frac{19}{2} $$

The results are compared with the exact solution as following:

From the above figure it can be seen that the approximate solution almost equals to exact solution for n=4 and above.

The above figure shows the value of error function with respect to value of n.

From the figure it is clear that error function follows random path for lower values on n and for higher values error function decreases with increase in n.

Solution can be considered to be converging after n=8.

Given
General one dimensional Model 1.0/Data set 2 on page 1 of Mtg 12:

where $${a_2} = 2$$,  f = 3

Consider $$ {b_j}\left( x \right) = (x+k)^j $$ and k=0

A
A) Since for this problem,'n' starts from 0;

D.O.F.=n+1

D.O.F.=3

B) From (Eq 2.9.2) we can get $$ \sum\limits_{j = 0}^2 {{d_j}{b_j}\left( 1 \right)} = 0 $$

Then rewrite the above equation in matrix form:

where

From (3.1.3) we can get $$ \sum\limits_{j = 0}^2 {{d_j}b_j^{\left( 1 \right)}\left( 0 \right)} =  - 4 $$

Then rewrite the above equation in matrix form:

where

Equations 3.1.4 and 3.1.6 are the two equations that enforce the boundary conditions for $$ {u}^h\left( x \right) $$

Integration can be obtained using following matlab code:

Given
General one dimensional Model 1.0/Data set 2 on page 1 of Mtg 12:

where $${a_2} = 2$$,  f = 3

Consider $$ {b_j}\left( x \right) = 1+{a_j}cos(jx)+{b_j}sin(jx) $$

To find
A) If n=2, What is the D.O.F. of this problem?

B) Find two equations that enforce boundary conditions for $$ {u^h}\left( x \right) = \sum\limits_{j = 0}^n $$

C) Find one more equation to solve for $$ {\mathbf{d}} = {\left\{ \right\}_{3 \times 1}} $$ by projecting the residue $$ {\mathbf{P}}\left(  \right) $$ on a basis function $$ b_k\left( x \right)     $$ with  k = 0,1,2 such that the additional equation is linear independent from the equations found in the solution of the (B)

D) Display 3 equations in matrix form $$ {\mathbf{Kd}} = {\mathbf{F}} $$ and observe symmetric property of $$ {\mathbf{K}} $$

E) Solve for $$ {\mathbf{d}} $$

F) Construct $$ u_n^h\left( x \right) $$ and plot $$ u_n^h\left( x \right) \ vs \ u\left( x \right) $$ ....(where $$ u\left( x \right) $$ is the exact solution)

G) Repeat (A) to (F) for n=4 and n=6

H) Compare $$ u_n^h\left( {x = 0.5} \right)   for n = 2,4,6 $$

If error $$ {e_n}\left( {0.5} \right) = u\left( {0.5} \right) - {u^h}\left( {0.5} \right) $$ ,Plot $$ {e_n}\left( {0.5} \right) \ vs \ n $$

Solution
A) Since for this problem,'n' starts from 0;

D.O.F.=n+1

D.O.F.=3

B) From (Eq 2.9.2) we can get $$ \sum\limits_{j = 0}^2 {{d_j}{b_j}\left( 1 \right)} = 0 $$

Then rewrite the above equation in matrix form:

where

From (3.1.3) we can get $$ \sum\limits_{j = 0}^2 {{d_j}b_j^{\left( 1 \right)}\left( 0 \right)} =  - 4 $$

Then rewrite the above equation in matrix form:

where

Equations 3.1.4 and 3.1.6 are the two equations that enforce the boundary conditions for $$ {u}^h\left( x \right) $$

Following matlab code is used to compute integration

The above figure compares the exact solution and the approximated solution

It can be seen that at n=5, approximate solution almost equals exact solution

The above figure compares error between the exact and approximate solution at x=0.5

title: User:Eml5526.s11.team6.joglekar/hwk3


text:
User:Egm6341.s11.team4.kurth/hw3

=Problem 3.2: Investigating the Values of G(x) at All Nodes $$x_i$$= Refer to lecture slide [[media:Nm1.s11.mtg14.djvu|14-3]] for problem statement.

Given
Note: shorthand notation was used, however it is implied that these equations refer to the Lagrange interpolation of a continuous function $$\displaystyle f$$ which has $$(n+1)$$ continuous derivatives on $$\displaystyle I_t := \mathcal{H}(t,x_0,x_1,...,x_n)$$ as defined on lecture slides [[media:Nm1.s11.mtg11.djvu|11-2,11-3]].

Objective
Show that

Solution
First we will rearrange the second term in (2.1) as follows

From equation (2) on lecture slide [[media:Nm1.s11.mtg11.djvu|11-3]]:

where $$\displaystyle \xi \in I_t$$.

Rearranging (2.4) and substituting it back into (2.3) yields

From equation (3) on lecture slide [[media:Nm1.s11.mtg11.djvu|11-3]]

Upon inspection it can easily be seen that

From (2.5) and (2.7)

Solved by William Kurth.

title: User:Egm6341.s11.team4.kurth/hw3


text:
Social Media/Sport

Sport and Social Media

Contact: Leigh Blackall leighblackall@gmail.com 0404561009

Unit description and learning outcomes
This course is intended for people who want to know more about the uses, issues and examples of social media used in sport and society. It is an introductory course that will focus on the development of skills as well as critical appreciation of issues arising as a result of widespread use of social media.

Learning objectives:

 * 1) Describe different types of social media tools and applications*
 * 2) Compare different types of social media tools and applications*
 * 3) Setup and use a popular social media tools and applications*
 * 4) Use collaborative document authoring software and processes
 * 5) Use and demonstrate understanding of copyright and media formats
 * 6) Demonstrate awareness of issues in social media in sport and wider society

* As per the Australian National Training Framework's ICAWEB201A Use social media tools for collaboration and engagement

Generic skills

 * 1) Communication skills
 * 2) Working independently and with others
 * 3) Professionalism and social responsibility

Prerequisites and/or co-requisites
No Prerequisites and/or co-requisites, although very complementary with Sport sponsorship and media

Delivery mode

 * 1) This unit involves face to face lectures and tutorials at the University of Canberra.
 * 2) Online participation in part or in full, is optional and can be done via the website.

Lectures

 * 1) Orientation to the unit, assignments and assessment criteria
 * 2) The pros and cons of social media so far
 * 3) Privacy, data, economic models and other basis’ for social media services
 * 4) Cultures and trends in large social media networks
 * 5) Open source and participatory economics, mashup, sampling and re appropriation culture
 * 6) Proposals and activism for alternative social media networks
 * 7) Sport Fans and Social Media
 * 8) Coaches and Players with Social Media
 * 9) Social Media and Sport Controversies
 * 10) The Law Regarding Sport and Social Media
 * 11) Social Media in Sport Marketing and Sponsorship
 * 12) Sport Organisations and Social Media

Tutorials

 * 1) Get ready - Orientation to the unit, assignments and assessment criteria
 * 2) Set up a weblog - Set up a personal weblog for future tutorial exercises
 * 3) Editing a Mediawiki - Using and editing Wikipedia, Wikiversity and Wikibooks
 * 4) Set up an RSS reader - Track and manage content with Google Reader
 * 5) Bookmarking and tagging - Bookmarking and collaborative resource sharing using Delicious
 * 6) Photos online - Source and share photos and video on Flickr, Picasa and Wikimedia Commons
 * 7) Student presentations
 * 8) Student presentations
 * 9) Student presentations
 * 10) Student presentations
 * 11) Student presentations
 * 12) Student presentations

Lists of required texts/readings

 * 1) Sport and Social Media Unit
 * 2) Social media generic course outline
 * 3) Social media course website

Materials and equipment

 * 1) If you have a laptop or similar portable wireless Internet device, please make sure you are able to log on to the University Wireless and bring it to lectures and tutorials.
 * 2) If you do not have a portable, you will need regular access to a computer and the Internet for this Unit.

Unit website

 * Socially Constructed Media blog - Announcements, notes, demonstrations and exercises for this course will be posted here.

Assessment overview
To pass this unit, you must pass all 3 assessments.

Special assessment requirements

 * 1) To pass this Unit, you must gain a pass of 50% or above for each of the three assignments.
 * 2) If special circumstances require you to seek an extension, or a resubmission, please contact your lecturer before the due date of the assignment.

Academic Integrity

 * 1) Students have a responsibility to uphold University standards on ethical scholarship. Good scholarship involves building on the work of others and use of others’ work must be acknowledged with proper attribution made. Cheating, plagiarism, and falsification of data are dishonest practices which contravene academic values.

Workload

 * 1) The amount of time you will need to spend on study in this unit will depend on a number of factors including your prior knowledge, learning skill level and learning style.
 * 2) In planning your time commitments you should note that for a 3cp unit the total notional workload over the semester or term is assumed to be 150 hours. These hours include time spent in classes.
 * 3) The total workload for units of different credit point value should vary proportionally. For example, for a 6cp unit the total notional workload over a semester or term is assumed to be 300 hours.

Special needs

 * 1) Students who need assistance in undertaking the unit because of disability or other circumstances should inform their Unit Convener or UC AccessAbility (formerly the Disabilities Office) as soon as possible so the necessary arrangements can be made.

Attendance requirements

 * 1) This unit is an open and networked educational unit, meaning all content and instruction used in lectures and tutorials is available on the course website, making it possible for people at a distance to study online.

Withdrawal

 * 1) If you are planning to withdraw please discuss with your unit convener. Please see withdrawal of units for further information on deadlines for withdrawal.

Required IT skills

 * 1) It is not a requirement, but having a working knowledge of computers, the Internet and basic media such as images and video, is an advantage in this unit.

Costs

 * 1) There are no extra costs in this unit.

Student Feedback

 * 1) All students enrolled in this unit will have an opportunity to provide anonymous feedback on the unit at the end of the Semester via the Unit Satisfaction Survey (USS) which will be presented to you on OSIS.
 * 2) Your lecturer or tutor may also invite you to provide more detailed feedback on their teaching through an anonymous questionnaire.

Authority of this Unit Outline
Special conditions apply to Unit Conveners changing unit requirements, or introducing new requirements, from those set out in the Unit Outline. These conditions apply to changes on Section 2 (Academic content), Section 5 (Assessment), or other matters requiring student compliance such as attendance on field trips. Changes may only be made:

(a) if the Head of Discipline and a majority of students have agreed in writing;

(b) if written advice of the change is then provided on the unit site in the learning management system. If this is not possible, written advice of the change must be then forwarded to each student enrolled in the unit at their registered address; and

(c) in addition to (a) and (b), under other circumstances specified in the relevant Unit Outline.

Any change to the information contained in Section 2 (Academic content), and Section 5 (Assessment) of this document, will only be made by the Unit Convener if the written agreement of Head of Discipline and a majority of students has been obtained; and if written advice of the change is then provided on the unit site in the learning management system. If this is not possible, written advice of the change must be then forwarded to each student enrolled in the unit at their registered term address. Any individual student who believes him/herself to be disadvantaged by a change is encouraged to discuss the matter with the Unit Convener.

title: Social Media/Sport