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Psycholinguistics/Theories and Models of Language Acquisition

Language Acquisition- An Overview
Language acquisition is the process by which humans acquire the capacity to perceive, produce and use words to understand and communicate. It involves the picking up of diverse capacities including syntax, phonetics, and an extensive vocabulary. However, learning a first language is something that every normal child does successfully without much need for formal lessons. Language development is a complex and unique human quality but yet children seem to acquire language at a very rapid rate with most children's speech being relatively grammatical by age three (Crain & Lillo-Martin, 1999). Grammar, which is a set of mental rules that characterizes all of the sentences of a language, must be mastered in order to learn a language. Most children in a linguistic community seem to succeed in converging on a grammatical system equivalent to everyone else in the community with few wrong turns, which is quite remarkable considering the pitfalls and complexity of the system. By the time a child utters a first word, according to the Linguistic Society of America, he or she has already spent many months playing around with the sounds and intonations of language, but there is still no one point at which all children learn to talk. Children acquire language in stages and different children reach various stages at different times, although they have one thing in common and that is that typically developing children learning the same language will follow an almost identical pattern in the sequence of stages they go through. The stages usually consist of: Language acquisition is a complex and unique human quality for which there is still no theory that is able to completely explain how language is attained. However most of the concepts and theories we do have explaining how native languages are acquired go back to the approaches put forward by researchers such as Skinner, Chomsky, Piaget and others. Most of the modern theories we have today have incorporated aspects of these theories into their various findings.
 * cooing- 6 months- use phonemes from every language
 * babbling- 9 months- selectively use phonemes from their native language
 * one word utterances- 12 months- start using single words
 * telegraphic speech- 2 years- multi-word utterances that lack in function
 * normal speech- 5 years- almost normal developed speech

Behaviourist Theory

In 1957 a piece of literature appeared that would come to affect how we view language, human behaviour and language learning. B.F Skinner's Verbal Behaviour (1957) applied a functional analysis approach to analyze language behaviour in terms of their natural occurrence in response to environmental circumstances and the effects they have on human interactions. Skinner's behaviour learning approach relies on the components of classical conditioning, which involves unconditioned and conditioned stimuli, and operant conditioning but particularly the elements of operational conditioning. Operational conditioning refers to a method of learning that occurs through rewards and punishments for behaviour. Behaviour operates on the environment to bring about favorable consequences or avoid adverse ones. These same ideas of operant conditioning can also be applied to language acquisition because Skinner believed that language could be treated like any other kind of cognitive behaviour. According to the behaviourist theory, language learning is a process of habit formation that involves a period of trial and error where the child tries and fails to use correct language until it succeeds. Infants also have human role models in their environment that provide the stimuli and rewards required for operant conditioning. For example, if a child starts babblings, which resembles appropriate words, then his or her babbling will be rewarded by a parent or loved one by positive reinforcement such as a smile or clap. Since the babblings were rewarded, this reward reinforces further articulations of the same sort into groupings of syllables and words in a similar situation (Demirezen, 1988). Children also utter words because they cause adults to give them the things they want and they will only be given what they want once the adult has trained or shaped the child through reinforcement and rewards speech close to that of adult speech. Before long children will take on the imitation or modeling component of Skinner's theory of language acquisition in which children learn to speak by copying the utterances heard around them and by having their responses strengthened by the repetitions, corrections and other reactions that adults provide. However, before a child can begin to speak, they first start by listening to the sounds in their environment for the first years of their life. Gradually, the child learns to associate certain sounds with certain situations such as the sound of endearment a mother produces when feeding her child. These sounds then become pleasurable for the child on their own without being accompanied by food and eventually the child will attempt to imitate these sounds to invite the attention of his mother or another adult. If these sounds resemble that of adult language the mother will respond with reward and the operant conditioning process begins.

Innateness Theory
Noam Chomsky's innateness theory (or nativist theory) proposes that children have an inborn or innate faculty for language acquisition that is biologically determined. According to Goodluck (1991), nativists view language as a fundamental part of the human genome, as a trait that makes humans human, and its acquisition is a natural part of maturation. It seems that the human species has evolved a brain whose neural circuits contain linguistic information at birth and this natural predisposition to learn language is triggered by hearing speech. The child's brain is then able to interpret what she or he hears according to the underlying principles or structures it already contains (Linden, 2007). Chomsky has determined that being biologically prepared to acquire language regardless of setting is due to the child's language acquisition device (LAD), which is used as a mechanism for working out the rules of language. Chomsky believed that all human languages share common principles, such as all languages have verbs and nouns, and it was the child's task to establish how the specific language she or he hears expresses these underlying principles. For example, the LAD already contains the concept of verb tense and so by listening to word forms such as "worked" or "played,” the child will then form a hypothesis that the past tense of verbs are formed by adding the sound /d/,/t/ or /id/ to the base form. Yang (2006) also believes that children also initially possess, then subsequently develop, an innate understanding or hypothesis about grammar regardless of where they are raised. According to Chomsky, infants acquire grammar because it is a universal property of language, an inborn development, and has coined these fundamental grammatical ideas that all humans have as universal grammar (UG). Children under the age of three usually don't speak in full sentences and instead say things like "want cookie" but yet you would still not hear them say things like "want my" or "I cookie" because statements like this would break the syntactic structure of the phrase, a component of universal grammar. Another argument of the nativist or innate theory is that there is a critical period for language acquisition, which is a time frame during which environmental exposure is needed to stimulate an innate trait. Linguist Eric Lenneberg in 1964 postulated that the critical period of language acquisition ends around the age of 12 years. He believed that if no language was learned before then, it could never be learned in a normal and functional sense. It was termed the critical period hypothesis and since then there has been a few case examples of individuals being subject to such circumstances such as the girl known as Genie who raised in an abusive environment to age 13, which didn't allow her to develop language skills.

Cognitive Theory
Jean Piaget was a Swiss psychologist who was famous for his four stages of cognitive development for children, which included the development of language. However, children do not think like adults and so before they can begin to develop language they must first actively construct their own understanding of the world through their interactions with their environment. A child has to understand a concept before he or she can acquire the particular language which expresses that concept. For example, a child first becomes aware of a concept such as relative size and only afterward do they acquire the words and patterns to convey that concept. Essentially it is impossible for a young child to voice concepts that are unknown to them; therefore, once a child learns about their environment, then they can map language onto their prior experience. An infant's experience of a cat is that it meows, is furry and eats from a bowl in the kitchen; hence they develop the concept of cat first and then learn to map the word "kitty" onto that concept. Language is only one of the many human mental or cognitive activities, and many cognitivists believe that language emerges within the context of other general cognitive abilities like memory, attention and problem solving because it is a part of their broader intellectual development. However, according to Goodluck (1991), once language does emerge it is usually within certain stages and children go through these stages in a fixed order that is universal in all children. There is a consistent order of mastery of the most common function morphemes in a language and simple ideas are expressed earlier than more complex ones even if they are more grammatically complicated. Piaget's cognitive theory states that children's language reflects the development of their logical thinking and reasoning skills in stages, with each period having a specific name and age reference. There are four stages of Piaget's cognitive development theory, each involving a different aspect of language acquisition:

 * 1) Sensory-Motor Period- (birth to 2 years) Children are born with "action schemas" to "assimilate" information about the world such as sucking or grasping. During the sensory-motor period, children's language is "egocentric" and they talk either for themselves or for the pleasure of associating anyone who happens to be there with the activity of the moment
 * 2) Pre-Operational Period- (2 years to 7) Children's language makes rapid progress and the development of their "mental schema" lets them quickly "accommodate" new words and situations. Children's language becomes "symbolic" allowing them to talk beyond the "here and now" and to talk about things such as the past, future and feelings.
 * 3) Egocentrism- Involves "animism" which refers to young children's tendency to consider everything, including inanimate objects, as being alive. Language is considered egocentric because they see things purely from their own perspective.
 * 4) Operational Period- (7 to 11 years) and (11 years to adulthood) Piaget divides this period into two parts: the period of concrete operations and the period of formal operations. Language at this stage reveals the movement of their thinking from immature to mature and from illogical to logical. They are also able to "de-center" or view things from a perspective other than their own. It is at this point that children's language becomes "socialized" and includes things such as questions, answers, commands and criticisms.

Social Interactionist Theory
Vygotsky's social interaction theory incorporates nurture arguments in that children can be influenced by their environment as well as the language input children receive from their care-givers. Although the theories of Skinner, Chomsky and Piaget are all very different and very important in their own contexts, they don't necessarily take into account the fact that children don't encounter language in isolation. The child is a little linguist analyzing language from randomly encountered adult utterances. The interaction theory proposes that language exists for the purpose of communication and can only be learned in the context of interaction with adults and older children. It stresses the importance of the environment and culture in which the language is being learned during early childhood development because this social interaction is what first provides the child with the means of making sense of their own behaviour and how they think about the surrounding world. According to Williamson (2008), children can eventually use their own internal speech to direct their own behaviour in much the same way that their parents' speech once directed their behaviour. Speech to infants is marked by a slower rate, exaggerated intonation, high frequency, repetition, simple syntax and concrete vocabulary. This tailored articulation used by care-givers to young children to maximize phonemic contrasts and pronunciation of correct forms is known as child-directed speech (CDS). Vygotsky also developed the concepts of private speech which is when children must speak to themselves in a self guiding and directing way- initially out loud and later internally and the zone of proximal development which refers to the tasks a child is unable to complete alone but is able to complete with the assistance of an adult. The attention and time that a mother spends talking about topics that the child is already focused on highly correlates with early vocabulary size. In the early stages of a child`s life this is usually done through motherese or ``baby talk`` which may allow children to ``bootstrap`` their progress in language acquisition (Williamson, 2008). The mother and father also provide ritualized scenarios, such as having a bath or getting dressed, in which the phases of interaction are rapidly recognized and predicted by the infant. The utterances of the mother and father during the activities are ritualized and predictable so that the child is gradually moved to an active position where they take over the movements of the care-taker and eventually the ritualized language as well. Basically the care-giver is providing comprehensible contexts in which the child can acquire language (Mason, 2002). Another influential researcher of the interaction theory is Jerome Bruner who elaborated and revised the details of the theory over a number of years and also introduced the term Language Acquisition Support System (LASS), which refers to the child`s immediate adult entourage but in the fuller sense points to the child`s culture as a whole in which they are born. Adults adapt their behaviour towards children to construct a protected world in which the child is gradually inclined to take part in a growing number of scenarios and scripts and in this way the child is led gradually further and further into language. However, one must remember that although our social context provides support for language acquisition, it does not directly provide the knowledge that is necessary to acquire language; and this, perhaps, is where a child’s innate abilities come into play.

Usage-Based Theory
The usage-based theory of language suggests that children initially build up their language through very concrete constructions based around individual words or frames on the basis of the speech they hear and use. Basically this means, according to Tomasello (2003) the developer of the theory, that children learn language from their language experiences and a language structure emerges from language use. The usage-based theory takes constructions, which are direct form meaning pairings, to be the basic units of grammar and believe that children learn constructions by first mastering specific instances before going on to generalize and use the constructions productively with other lexical items. Constructions gradually become more general and more abstract during the third and fourth years of life and grammar emerges as the speakers of a language create linguistic constructions out of recurring sequences of symbols (Tomasello, 2003). Tomasello (2003) also emphasizes the effects of frequency of use on cognitive representations, as patterns that are repeated for communicative reasons seem to become automated and conventionalized. Research by Saxton (2010) indicates that the more often a linguistic form occurs in the input, the more often it is experienced by the child and the stronger the child's representation of it becomes. It will then be activated more easily when used on subsequent occasions. Therefore the child's mental representation is reinforced or increasingly entrenched and the more deeply entrenched a structure is, the more likely it becomes that this will form the basis of the child's speech output. Usage-based linguistics holds that language use shapes entrenchment through frequency repetitions of usage, but there are separable effects of token frequency and type frequency(Doughty & Long, 2003). According to Doughty and Long (2003), token frequency is how often in the input particular words or specific phrases appear and type frequency counts how many different lexical items a certain pattern or construction is applicable to. Linguistic forms with high token frequency will be learned early and lead to more strongly entrenched linguistic representations and seem to protect the child from error. Token frequency also has a strong influence on child learning; there is often a close relationship between adult input and child output (Saxton, 2010). Type frequency determines productivity because high type frequency ensures that a construction is used frequently, thus strengthening its representational schema and making it more accessible for further use with new items. Also the more items the category must cover, the more general are its criteria features, and the more likely it is to extend to new items (Doughty & Long, 2003). Another term coined in the usage-based theory is pre-emption which is an anti-frequency mechanism that suggests that children who experience a verb in a rare construction will avoid using that verb in a more common structure.

Optimality Theory
Optimality Theory (OT) was originally proposed by Prince and Smolensky (1993) and has subsequently been further developed by other researchers. OT suggests that the observed forms of language arise from the interaction between conflicting constraints and like other models of linguistics, contain an input and an output and a relation between the two. A constraint is a structural requirement that may be either satisfied or violated by an output form and a surface form. A constraint is considered optimal if it incurs the least serious violations of a set of constraints, taking into account their hierarchical ranking. In optimality theory, the essence of both language learning in general (learnability) and language acquisition (actual development children go through) entails the rankings of constraints from an initial state of the grammar to the language specific ranking of the target grammar (McCarthy, 2004). OT is a development of generative grammar, a theory sharing the quest for universal principles such as universal grammar but differs from the theory proposed by Chomsky because optimality theory believes that these universal constraints are violable (Kager,1999). Languages are able to differ in their ranking of constraints by giving priorities to some constraints over others. Language acquisition can be described as the process of adjusting the ranking of these constraints that are considered universal:

According to Archangeli & Langendoen (1997) these constraints include constraints governing aspects of phonology, such as syllabification constraints, constraints governing morphology and constraints that determine the correct syntactic properties of a language. There is also one family of constraints whose properties cut across all subdisciplinary domains, called the faithfulness constraints, which say that input and output are identical. Faithfulness is the general requirement for linguistic forms to be realized as close as possible to their lexical "basic forms" and violations of faithfulness lead to differences between input and output (Archangeli & Langendon, 1997). Another term coined by the optimality theory is markedness, which refers to the continuum that language-universal and language-specific properties rest on, with completely unmarked properties being those found in virtually all languages and extremely marked properties being found quite rarely. However markedness embodies universality in a "soft" sense, with violations of universality existing between languages.
 * GEN- takes an input and generates the list of possible outputs or candidates
 * EVAL- chooses the optimal candidate based on the constraints, and this candidate is the output
 * CON- provides the criteria, the form of strictly ordered violable constraints, used to decide between constraints

Native Language Magnet Model
Young children learn their mother tongue rapidly and effortlessly, following similar developmental paths regardless of culture. How infants accomplish this task has become the focus of debate especially for Patricia Kuhl who has developed the Native Language Magnet Model to help explain how infants at birth can hear all the phonetic distinctions used in the world's languages. According to Kuhl and colleagues (2005), to acquire a language, infants have to discover which phonetic distinctions will be utilized in the language of their culture and do so by discriminating among virtually all the phonetic units of the world's languages. During the first year of life, prior to the acquisition of word meaning, infants begin to perceive speech by forming perceptual maps of the speech they hear in their environment. Kuhl's (2005) research focused on the mechanism underlying the development transition from an infant’s universal phonetic capacity to native phonetic discrimination. They used ERP brain measure of infants' native and non-native speech perception in infancy to predict language in the 2nd and 3rd years of life. Although we still remain capable of discriminating non-native phonetic contrasts as we age, it is at a reduced level when compared with native contrasts. The idea that more than selection is involved in the development of phonetic perception has been clearly demonstrated by experimental findings showing that native language phonetic perception shows a significant improvement between 6 and 12 months of age. Previous studies had shown native language improvement after 12 months of age and before adulthood, but newer studies such as Kuhl's and colleagues has gone beyond selection in explaining developmental change in infants' perception of speech. The Native Language Magnet Model (NLM) proposed by Kuhl (1994, 2000) focuses on infants' native phonetic categories and how they could be structured through ambient language experience. The NLM specified three phases in development:

 * Phase 1- infants are capable of differentiating all the sounds of human speech and abilities are derived from their general auditory processing mechanisms rather than from a speech-specific mechanism
 * Phase 2- infants' sensitivity to the distributional properties of linguistic input produces phonetic representations. Experience accumulates and the representations most often activated begin to function as perceptual magnets for other members of the category
 * Phase 3- The perception termed perceptual magnet effect produces facilitation in native and a reduction in foreign language phonetic abilities

Recently Kuhl's research has initiated the revision of the NLM and expanded the model to include native language neural commitment, which explains effects of language experience on the brain. Native language neural commitment describes the brain's early coding of language and how it affects our subsequent abilities to learn the phonetic scheme of a new language. This is due to the fact that initial language exposure causes physical changes in neural tissue that reflects the statistical perceptual properties of language input (Kuhl 2005). The neural networks then become committed to the patterns of native language speech. Another finding by Kuhl (2008) that has expanded the Native Language Magnet Model has been the research indicating that both native and non-native performances at 7 months of age predicted future language abilities but in opposite directions. Better native phonetic perception at 7 months of age predicted accelerated language development at between 14 and 30 months whereas better non-native performance at 7 months predicted slower language development at 14 and 30 months. Results supported the view that the ability to discriminate non-native phonetic contrasts reflects the degree to which the brain remains in the initial state, open and uncommitted to native language speech patterns.

Learning Exercise A
Follow the link to get to the jeopardy challenge I created to test your knowledge on the theories of language acquisition. Try to answer correctly as many questions as you can, collecting jeopardy cash along the way until you meet the final jeopardy question at the end. Good luck!

Jeopardy Challenge
http://www.superteachertools.com/jeopardyx/jeopardy-review-game.php?gamefile=1299185662

Learning Exercise B
Answer the following essay questions to the best of your ability, using external sources if needed.

1. You are a new theorist in the field of psycholinguistics and are trying to determine which perspective you are willing to take on how individuals acquire language. Being the great researcher that you are, you want your opinions to be based on evidence-based knowledge. Analyze and pick a position based on new evidence from within the field of linguistics defending why a certain perspective or theory better explains language acquisition. Make sure to contrast your arguments against other theories or models and clearly support why other theorists should accept your view. If you want to be really ambitious you can even create your own theory or model to endorse your ideas, but make sure that you have evidence backing why you think your theory could hold up against any other.

Language acquisition has been a subject of debate for decades. It is so complex that there is really no single principle that can be applied to all. Its complexity has led to multiple theories and studies that has shaped our general perception and understanding towards it. Working in the Gulf for more than a decade exposed me to multicultural students and second language speakers. Undeniably, all language acquisition theories acknowledge individual differences. In my observation, generally, individual learns language through a combination of innate and cognitive abilities ( Innateness l cognitive theory ) and external factors (optimality theory). A child grows up with interaction with mother, parents and / or nanny. So their exposure to people continuously interacting with them plays a great role in their language acquisition.

Contrary to that Piaget’s theory, modern theories focuses on the amount of input / output in the child’s “little world”. As for me, one of factor that we should also focus are children born to multi-lingual parents. Because of migration, mixed marriages, children automatically becomes exposed to several language input / barriers. I have students who are native Arabic but cannot speak in their own language but can understand it but cannot converse with it. They are more oriented to English language. In my observation, a child will always have innate capability to learn language, the amount of exposure a child gets ( from environment ) will determine the extent of the usefulness of language a child acquired. One area of language that I would like to add and I believe more studies must ne done on this is the exposure to social media. Authenticity of the “carrier” of language must be clearly identified. With our fast phased life and continuous advancement in technology, we must acknowledge its effect on child’s growth, more so with their language acquisition. I am not pointing to its advantages or disadvantages but only its acceptance in our field of study of language acquisition. I prepared a model of this concept: This represent my position on how a child acquire language.

Input - immediate social factors like parents, nannies, social background,positive reinforcements

Individual - innate cognitive abilities

Technological Evolution - social media, access to gadgets

2. Recently someone you knew had a baby and with your ne found psycholinguistics knowledge you realize that you may have some advice to help with the baby's language acquisition when the time comes. Using what you know about the theories and models of language acquisition, what tips or guidance could you give this person to help her baby to achieve language acquisition? Specifically describe with examples if there are certain aspects of the theories or models of language acquisition that could support the infant in developing language skills.

With what I learn from the theories that I have studied and read, I will mention to her that her presence is very much vital in the child’s language acquisition. Development stages of the child also will affect his/her language. If her neighbor’s son speaks at 1 year old and hers has not yet, then she must understand it is different for every child. When this happens, then she can add more input that may come from other members of the family and other positive reinforcements. She can increase amount of input especially in the formative years. The goal is to create a positive learning environment to a child so he or she can develop confidence in what a child does. For me, a great amount of confidence is required for a child to be effective in acquiring language.

Conclusion
Language acquisition has been one of the central topics in cognitive science but has also been one of the most controversial. Languages are complex combinations of elegant principles and historical accidents, which is perhaps one of the reasons why there is no monolithic explanatory theory of language. The goal of language acquisition research is to describe how a child becomes competent to produce and understand language, select the proper processing strategies and achieve language "milestones." However, there are a range of theories of language acquisition that have been created but most of these theories cannot agree on the role that both nature and nurture play in language acquisition. The theories do have one thing in common though, and that is the fact that they all believe that language acquisition is the key aspect that distinguishes humans from other organisms and by understanding how different aspects of language are acquired we can better understand the main vehicle by which we communicate.


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User:Egm6341.s11.TEAM1.Sujin

=Spring 2011 Team 1 Homework Archived= Homework 1 Homework 2 Homework 3 Homework 4 Homework 5 Homework 6 Homework 7

Wiki Editing Help Link: Help:Editing Wiki Formula Editing Help Link: Help:Formula How to link the archived version of your HW report to the Report table. https://docs.google.com/Doc?docid=0AY5m1lfNbLroZGM4Mm5mZ2JfMTQzZDVkNnFkZGc&hl=en link to "ideal" reports http://en.wikiversity.org/wiki/User:Egm6341.s10/Report_table/Good_model_reports Link to class HW Report Table for Fall 2011 http://en.wikiversity.org/wiki/User:Egm6341.s11/Report_table

Homework 1.4

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

title: User:Egm6341.s11.TEAM1.Sujin


text:
User:EGM6341.s11.TEAM1.Yoon.HW1

Part (A)
Define m,M as below,

Consider,

Integrating each side of the non-equality, where, m and M are constants by definition of (3.1)

Deviding (3.3) by $$\displaystyle \frac{1}{\int^b_a w(x)dx}$$ yields

Applying Intermediate Value Theorem,

Thus,

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

$$\displaystyle \begin{align} \therefore \int_a^b w(x) \cdot f(x)dx = f(\xi ) \int_a^b w(x)dx \quad _{for} \quad w(x) \geqq 0 \end{align} $$
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style = | $\displaystyle \color{red}{\begin{align} (3.7) \end{align}}$|undefined
 * }
 * }

Part (B)
Define m,M as below,

Consider,

Integrating each side of the non-equality, where, m and M are constants by definition of (3.8)

Deviding (3.10) by $$\displaystyle \frac{1}{\int^b_a |w(x)|dx}$$ yields

Since $$\displaystyle w(x)<0 $$, $$\displaystyle |w(x)| $$ becomes $$\displaystyle - w(x) $$,

Applying Intermediate Value Theorem,

Thus,

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

$$\displaystyle \begin{align} \therefore \int_a^b w(x) \cdot f(x)dx = f(\xi ) \int_a^b w(x)dx \quad _{for} \quad w(x) < 0 \end{align} $$
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style="width:10%; padding:10px; border:2px solid #8888aa" |
 * style = | $\displaystyle \color{red}{\begin{align} (3.15) \end{align}}$|undefined
 * }
 * }

title: User:EGM6341.s11.TEAM1.Yoon.HW1


text:
User:EGM6341.s11.TEAM1.Sujin

Given
for $$ x\epsilon\ [0,1] $$ and

for $$x \epsilon\ [0,1] $$

Find
1) Plot $$ f(x) \ $$ and $$ g(x) \ $$ 2) Solve for:

Solution
Author EGM6341.s11.TEAM1.JANG

title: User:EGM6341.s11.TEAM1.Sujin


text:
User:Sujin/EGM 6341.Sujin

Given
for $$ x \epsilon\ [0,1] $$ and for $$ x \epsilon\ [0,1] $$

Find
Plot $$ f(x) \ $$ and $$ g(x) \ $$ Solve for:

Solution
Author

title: User:Sujin/EGM 6341.Sujin


text:
User talk:Sujin

 Hello Sujin, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

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MaintenanceBot (discuss • contribs) 03:53, 11 November 2013 (UTC)

title: User talk:Sujin


text:
User:Egm5526.s11.team2.oztekin/Sandbox

 That is a comment

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

$$  \displaystyle \eta :=  \left(      \frac{\nu^3}{\epsilon}   \right)^{1/4} $$     (1.2)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

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

$$  \displaystyle {x^2} = 1 $$     (1.2)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

$$\displaystyle {x^2} = 1$$

title: User:Egm5526.s11.team2.oztekin/Sandbox


text:
User talk:Egm5526.s11.team2.oztekin

 Hello Egm5526.s11.team2.oztekin, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. All users are expected to abide by our Privacy policy, Civility policy, and the Terms of Use while at Wikiversity.

To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
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 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
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You don't need to be an educator to edit. You only need to be bold to contribute and to experiment with the sandbox or your userpage. See you around Wikiversity! --Ottava Rima (talk) 02:12, 18 January 2011 (UTC)

title: User talk:Egm5526.s11.team2.oztekin


title: File:Function.1.A.pdf


text:
User:Alyssampanciocco

Ireland

Country Demographics: - In 2008 6,197,100 people. - Ireland is currently divided into the Republic of Ireland which is 5/6 of the country and Northern Ireland which is 1/6 of the country and part of the United Kingdom. - Of the approximate 6.2 million people living in the country, 4.5 are in                       the Republic and 1.8 are in Northern. Geographical Information: - Ireland is seperated from Great Brittain by the Irish Sea. It is also surrounded by the Celtic Sea and Atlantic Ocean. - It is the 3rd largest island in Europe. - The key mountain is Carrantoohil which is the highest point at 3,046 feet above sea level. - The country which is in Northern/Western Europe has a 600 mile coastline and an area of 32,595.1 square miles. - The largest city in the Country is Dublin containing 1mil. people. - To the east of the island is Great Brittain and the longest river is                            is the River Shannon which is 240 miles long. - The mild oceanic climate has frequent rain, however inland is warm in                            the summer and cooler in the winter. Ethnic Groups: -Those living in Ireland are either Irish or Ulster Scots and the largest religion is Christianity. 73% are Roman Catholic and the rest Protestant and 4% have none. A small ammount are also Jewish. - Less than 10% of the country speaks Irish and the rest speak English which is the countries official language. Economic Development: -Ireland began circulating the Euro in 2002. -GDP growth averaged 6% in 1995-2007, but the economy has dropped by over 3%                    in 2008, about 8% in 2009, and about 1% in 2010. - The countries unemployment rate is 13.7 percent. - The country grows turnips, barley potatoes, beets, wheat, dairy and beef. - The main industries are silver, lead, steel, zinc, aluminum, food product, glass and crystal. - The key exports are machinery and equipment, computers, chemicals, pharmecuticals, live animals and animal products. - The key imports are chemicals, petroleum, textiles, machinery and clothing. -The GDP per capita is $37, 600 in 2010. Health and Education: - The infant mortality rate is 3.89 deaths and 1,000 live births. - The life expectancy is for males 77.86 years and for females 82.41 years. - The average amount of education is up to 18 years and the literacy rate for both males and females each is 99 percent.

History of Nation: "1800. British rule over Ireland, present since the 12th century, is extended to the entire country by the 17th and 18th centuries and further centralized with the Act of Union in 1800 (whereby no parliament sat in Dublin anymore).

1870s. Strong national movement emerges in Ireland. The national political movement in favor of "home rule" succeeds in incorporating both members of the Anglo-Irish aristocracy and peasant famers who seek land reform. But resistance on the part of conservative British governments and the strong will of the Protestant population of the northern province—Ulster—to remain in the union delays home rule.

1914-18. A more radical stream of nationalism begins.

1919-21. Guerrilla-style war for independence ensues. The Unionist population of Northern Ireland remains adamant that no granting of either home rule or independence to the island should include them.

1922. The Anglo-Irish treaty gives 26 of the 32 counties of Ireland independence from the United Kingdom with some symbolic restrictions, such as the retention of the crown as head of state. The remaining 6 counties in the north of the island remain part of the UK.

1923. Those for and against the treaty fight a civil war over the spoils of government and some over the retention of symbolic links with Britain, which ends in the capitulation of the anti-treaty forces, who then form the political party Fianna Fáil in 1926.

1925. Partition of the island into Eire and Northern Island is informally made permanent.

1938. More than a decade of politically provoked and disastrous "economic war" with Britain ends.

1940. Ireland declares itself neutral in World War II.

1949. Although informally a republic since 1937, Ireland is formally declared a republic.

1950s. Emigration increases rapidly, and rural poverty becomes widespread.

1960s. The inward looking, tariff-centered economic policies are rejected in favor of an open policy, but the state still plays a huge role in the economy.

1970s. High government spending increases the national debt to unsustainable levels and sparks off high inflation. The oil crisis of 1979 also hits the country hard.

1973. Ireland joins the European Economic Community, along with Britain and Denmark.

1980s. High inflation and unemployment levels alongside income tax that reach over 65 percent.

1987. Ireland endorses the Single European Act, which establishes the common European market. The first social partnership agreements of the 1980s negotiate a plan for national economic recovery.

1990s. Tighter fiscal policies, trade and enterprise-friendly economic policies, and social partnership agreements, alongside other factors such as the long-term benefits of EU transfers, facilitate a turnaround in the economic fortunes of the country.

1991. EU countries sign the Maastricht Treaty, which formalizes the plan for European Monetary Union and agrees on the ground rules for entry into EMU.

1994-98. Following the paramilitary cease-fire in Northern Ireland and long negotiations, a peace process results in political agreements between Britain, Ireland, and Northern Ireland.

1995-96. The economy shows strong growth and a significant increase in employment opportunities.

1998. Ireland endorses the Amsterdam Treaty, which extends EU co-ordination of social and security policy and enlargement.

1999. EMU is introduced and the European Central Bank takes over monetary powers in Ireland."

title: User:Alyssampanciocco


text:
User:Egm6341.s11.team4.BM/HW1

= Problem 1: Limits and Taylor Series=

Given
Refer to Lecture slide for the problem statement

 * {| style="width:70%" border="0" align="center"

f(x) = \frac{e^x - 1}{x} $$ $$
 * $$\displaystyle
 * $$\displaystyle

 * }.
 * }.

Find
1. Find the limit of the function as x approaches zero.

 * {| style="width:70%" border="0" align="center"

\lim_{x \rightarrow 0}f(x)=\frac{e^{x}-1}{x} $$
 * $$\displaystyle

 * }.
 * }.

2. Plot the function over the range from zero to one inclusive
 * {| style="width:70%" border="0" align="center"

f(x)=\frac{e^{x}-1}{x},\ x\in[0,1] $$
 * $$\displaystyle

 * }.

Solution
1. Function Limit

Before employing the Taylor series method of evaluating the limit for the function as x approaches zero, we will use the traditional L'Hopital's rule.

 * {| style="width:70%" border="0" align="center"

\lim_{x \rightarrow 0}\frac{e^{x}-1}{x} $$ Now, seeing that both the numerator and the denominator trend towards zero as x approaches zero, we will take the derivatives of both the numerator and the denominator.
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

\lim_{x \rightarrow 0}\frac{e^{x}}{1} $$
 * $$\displaystyle
 * $$\displaystyle

Evaluating the numerator at x equals zero we find
 * }.
 * }.

 * {| style="width:70%" border="0" align="center"

$$\displaystyle \frac{1}{1}=1 $$ $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$\displaystyle
 * style= |
 * }.
 * }.

L'Hopital's rule clearly tells us that the limit of the function approaches one. We will now seek a similar answer using the Taylor series expansion method as demonstrated in class.

We will begin with the Taylor series expansion for $$\displaystyle e^x $$ about zero.

 * {| style="width:70%" border="0" align="center"

\begin{align} e^x \approx \sum_{i=0}^n \frac{x^i}{i!} \\ \end{align} $$
 * $$\displaystyle

 * }.
 * }.

This Taylor series expansion to the n-th order can be displayed so simply since $$\displaystyle e^x $$ evaluated at zero is one and every derivative of the function $$\displaystyle e^x $$ is the function itself. As n goes to infinity this function becomes an equality.

Furthermore, we can evaluate the first term in the summation to get

 * {| style="width:70%" border="0" align="center"

\begin{align} e^x \approx 1+\sum_{i=1}^n \frac{x^i}{i!} \\ \end{align} $$
 * $$\displaystyle

 * }.
 * }.

We can algebraically rearrange this equation to get

 * {| style="width:70%" border="0" align="center"

\begin{align} e^x -1 \approx \sum_{i=1}^n \frac{x^i}{i!} \\ \end{align} $$
 * $$\displaystyle

 * }.
 * }.

Now we have the Taylor expansion of our numerator. To find the Taylor series expansion of our original function we simply need to divide both sides of the above equation by x.

 * {| style="width:70%" border="0" align="center"

\begin{align} \frac{e^x -1}{x} \approx \sum_{i=1}^n \frac{x^{i-1}}{i!} \\ \end{align} $$
 * $$\displaystyle

 * }.
 * }.

The left hand side of this equation is our function $$\displaystyle f(x)$$. The right hand side is a Taylor series approximation of the function. For n=$$\infty$$ this can be expressed as

 * {| style="width:70%" border="0" align="center"

\begin{align} \frac{e^x -1}{x} &= \sum_{i=1}^\infty \frac{x^{i-1}}{i!} \\ \end{align} $$
 * $$\displaystyle

 * }.
 * }.

Inspecting the Taylor series approximation we can see that, when evaluated at $$x=0$$, the numerator will be zero for every i except i=1. Mathematically, this can be expressed by pulling out the term i=1 from our summation, leaving the summation from 2 to infinity.

 * {| style="width:70%" border="0" align="center"

\begin{align} \lim_{x \rightarrow 0} \frac{e^x -1}{x} &= \frac{0^{0}}{0!}+\cancelto{0}{\sum_{i=2}^\infty \frac{0^{i-1}}{i!} }
 * $$\displaystyle

\end{align} $$

 * }.
 * }.

This leads us to our final conclusion (Equation 2), consistent with our expectations via L'Hopital's rule, that

 * {| style="width:70%" border="0" align="center"

$$\displaystyle \lim_{x \rightarrow 0} f(x) = \frac{e^x - 1}{x}=1 $$ $$
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * $$\displaystyle
 * style= |
 * }.
 * }.

2. Function Plot

The function f(x) is then plotted over the span [0,1] to graphically make clear how the function approaches 1 as x approaches 0. Two functions are plotted for this purpose. First, we will plot the function itself. This function, due to a discontinuity, is not evaluated at 0. Next, we will plot our Taylor series approximation using a value of n=30. Though this value is chosen arbitrarily to minimize computational expense, it is shown in problem 2 that this selection is sufficiently large to make the approximation error negligible. The Taylor series expansion, expressing no discontinuities, is evaluated over the entirety of the span [0,1].

Matlab Code:

Plots:

The evaluation of the original function.

The evaluation of the Taylor series approximation of the function.

Both the Taylor series approximation and the function itself clearly show that the value of the function as x approaches zero is one.

= Contributing Team Members =

Egm6341.s11.Team4.BM 10:27, 17 January 2011 (UTC)

= References =

[] I have relied heavily on Team 4's HW1 report from last year as a stylistic guide. I followed the Taylor Series expansion for the function as derived in class. In order to improve upon their representation of this information, I added the comparison method using L'Hopital's rule and removed the ellipses notation in favor of a closed form summation notation.

title: User:Egm6341.s11.team4.BM/HW1


text:
User talk:Eml5526.s11.team01.fercas

Hi team 01 =Problem 6.1: Quadratic Lagrangian Element Basis Function (QLEBF)= Solving a ordinary differential equation using Quadratic Lagrangian Element Basis Function (QLEBF). From Mtg.31

Given
The partial differential equation is:

The boundary conditions are:

$$ u\left(0\right) = 4$$

$$\frac{du}{dx}\left(1\right) = \frac{12}{5} $$

Find

 * 1. For nel = 2, compute $$\tilde{K}=\sum_{1}^{2}\tilde{K}^e$$ with $$\tilde{K}^e$$ given by equation (1) p.30-5
 * 2. Compute $$\tilde{k}^e$$ and $$L^e$$ for e = 1,2

 * 3. Compute $$\tilde{K}^e=L^{eT}k^eL^e$$ for e = 1,2. Compare to 1)
 * 4. Plot all the QLEBF for nel = 3
 * 5. Plot the exact solution, the numerical solution, and the error as a function of the number of elements

For nel = 2, compute $$\tilde{K}=\sum_{1}^{2}\tilde{K}^e$$ with $$\tilde{K}^e$$ given by equation (1) p.30-5
To calculate the stiffness matrix of the element we need the weak form; but, the weak form was already obtained in home work 5 which can be reach at http://en.wikiversity.org/wiki/User:Eml5526.s11.team01/Hwk5 Then, here we only copy that equation.

Now we use QLEBF to interpolate the solution $$U^h$$ and the weight funtion w. The aproximated soution can be interpolated as:

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

$$U^h=\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}u_1+\frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)}u_2+\frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}u_3 $$
 * $$\displaystyle (6.1.2) $$
 * }
 * }

Where $$u_{1}, u_{2}, u_{3}$$ are the nodal values of the funtion at the nodes 1, 2, and 3 respectively. The last equation can be written as:

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

$$U^h=\left [ N \right ]\left [ u \right ]^{e} $$
 * $$\displaystyle (6.1.3) $$
 * }
 * }

where $$\left [ u \right ]^{e}=\begin{bmatrix} u_1\\ u_2\\

u_3\end{bmatrix} $$

In the same way we can interpolate the weight funtion w to obtain:

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

$$w=\left [ N \right ]\left [ w \right ]^{e} $$
 * $$\displaystyle (6.1.4) $$
 * }
 * }

where $$\left [ w \right ]^{e}$$ is the vector with the nodal values of w

The derivative of $$U^h$$ and w are:

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

$$\frac{dU^h}{dx}=\begin{bmatrix} \frac{2x-x_2-x_3}{\left (x_1-x_2 \right )\left (  x_1-x_3\right )} & \frac{2x-x_1-x_3}{\left (x_2-x_1  \right )\left (  x_2-x_3\right )}  & \frac{2x-x_1-x_2}{\left (x_3-x_1  \right )\left (  x_3-x_2\right )} \end{bmatrix}\left [ u \right ]^e $$ The latter equation can be written as:
 * $$\displaystyle (6.1.5) $$
 * }
 * }

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

$$\frac{dU^{h}}{dx}=\left[B\right]\left [ u \right ]^{e} $$
 * $$\displaystyle (6.1.6) $$
 * }
 * }

and in similar way w can be written as:

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

$$\frac{dw}{dx}=\left[B\right]\left [ w \right ]^{e} =\left [ B \right ]^T\left [ w \right ]^{eT}$$ Replacing equations (6.1.3), (6.1.4), (6.1.6), and (6.1.7) in the weak form equation (5.7.4) we obtaine:
 * $$\displaystyle (6.1.7) $$
 * }
 * }

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

$$\left [ w \right ]^{eT}\int_{0}^{l}\left [ B \right ]^{T}\left ( 2+3x \right )\left [ B \right ]dx\left [ u \right ]^{e}=5\left [ w\right ]^{eT}\int_{0}^{l}\left [ B \right ]^{T}x\times dx-30w^{1}$$
 * $$\displaystyle (6.1.8) $$
 * }
 * }

The last equation is the weak form for any element; that is why the integration was done from 0 to l where l is the lenght of the element. Also in the last equation the factor $$w^{1}$$ in the nodal value of w at x = 1; therefore that factor is present in the weak form for the last element. Cancelling the factor $$\left [ w \right ]^{eT}$$ in equation (6.1.8)we obtain an equation of the form:$$Kd=f$$; therefore, we can extract from that equation the stiffness matrix which is:

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

$$k^{e}=\int_{0}^{l}\left [ B \right ]^{T}\left ( 2+3x \right )\left [ B \right ]dx$$
 * $$\displaystyle (6.1.9) $$
 * }
 * }

We developed a Matlab code (shown below) to calculate the global stiffness matrix $$\tilde{K}$$ using equation (1) p.30-5. The obtained matrix was:

Compute $$\tilde{k}^e$$ and $$L^e$$ for e = 1,2
Using the weak form developed above we calculate the stiffness matrix of the elements 1 and 2. As each elemet has 3 degrees of freedom and the total number of degrees of freedom is 7, the Location matrix is 3X5. The stiffness matrix are:

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

$$k^{1}=\begin{bmatrix} 10.83 & -12.66 &1.83 \\ -12.66 & 29.33 & -16.66\\ 1.83 & -16.66 & 14.83 \end{bmatrix} $$
 * $$\displaystyle (6.1.11) $$
 * }
 * }

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

$$k^{2}=\begin{bmatrix} 17.83 & -20.66 &2.83 \\ -20.66 & 45.33 & -24.66\\ 2.83 & -24.66 & 21.83 \end{bmatrix} $$ And the location matrix are:
 * $$\displaystyle (6.1.12) $$
 * }
 * }

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

$$L^{2}=\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & \\ 0 & 1 & 0 &0 & 0 & 0 & \\ 0 & 0 & 1 & 0 &0  & 0 & \end{bmatrix} $$
 * $$\displaystyle (6.1.13) $$
 * }
 * }

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

$$L^{2}=\begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 & \\ 0 & 0 & 0 &0 & 1 & 0 & \\ 0 & 0 & 0 & 0 &0  & 1 & \end{bmatrix} $$
 * $$\displaystyle (6.1.14) $$
 * }
 * }

Now the global matrix are calculated by using equation (2) p.31-1 which is repeated here:

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

$$\tilde{k}^{e}=L^{eT}k^{e}L^{e} $$
 * $$\displaystyle (6.1.15) $$
 * }
 * }

Therefore, for the element 1 whe have:

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

$$\tilde{k}^{1}=\begin{bmatrix} 1 & 0 &0 \\ 0 & 1 &0 \\ 0 & 0 &1 \\ 0 &0 &0 \\ 0 & 0 &0 \\ 0 & 0 & 0\\  0& 0 & 0 \end{bmatrix}\begin{bmatrix} 10.83 & -12.66 & 1.83\\ -12.66 & 29.33 &-16.66 \\ 1.83 & -16.66 & 14.83 \end{bmatrix}\begin{bmatrix} 1 & 0 & 0& 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix}=\begin{bmatrix} 10.83 & -12.66 & 1.83 & 0 & 0\\ -12.66 & 29.33 & -16.66 & 0 &0 \\ 1.83& -16.66 & 14.84 & 0 & 0\\  0& 0 & 0 & 0 & 0\\  0& 0 & 0 &0  & 0 \end{bmatrix} $$
 * $$\displaystyle (6.1.16) $$
 * }
 * }

In similar way for the elment 2 we have:

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

$$\tilde{k}^{2}=\begin{bmatrix} 0 & 0 &0 \\ 0 & 0 &0 \\ 0 & 0 &0 \\ 0 &0 &0 \\ 1 & 0 &0 \\ 0 & 1 & 0\\  0& 0 & 1 \end{bmatrix}\begin{bmatrix} 17.83 & -20.66 & 2.83\\ -20.66 & 45.33 &-24.66 \\ 2.83 & -24.66 & 21.83 \end{bmatrix}\begin{bmatrix} 0 & 0 & 0& 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 &0 \\ 0& 0& 17.83 & -20.66 & 2.83\\  0& 0 & -20.66 & 45.33 & -24.66\\  0& 0 &2.830 &-24.66  & 21.83 \end{bmatrix}$$
 * $$\displaystyle (6.1.17) $$
 * }
 * }

Performing the sum we obtain:

We can observ that this equation is equal to equation (6.1.10) because it corresponds to the same system with the same weak form. The location matrix is only a methodology for asembly the global matrix and therefore it can not change the final result.

Plot all the QLEBF for nel = 3
Qubic Lagrange Element Basis Function for every element are given by:

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

$$L_{1}=\frac{(x-x_2)(x-x_3)}{(x_1-x_2)(x_1-x_3)}; L_{2} =\frac{(x-x_1)(x-x_3)}{(x_2-x_1)(x_2-x_3)}; L_{3} = \frac{(x-x_1)(x-x_2)}{(x_3-x_1)(x_3-x_2)}u_3 $$
 * $$\displaystyle (6.1.19) $$
 * }
 * }

Where $$x_{1}, x_{2}, and x_{3}$$, are the coordinates of the nodes 1, 2, and 3 respectively. This funtions are repeated on all the elements. Figure 6.1.1 shows the QLEBF for 3 elements, where it can be observed that every basis funtion has the value of 1 at the respective node and 0 on the others, of course the basis function are valid only inside their respective element. In the Figure, Li,j corresponds to j the basis funtion of the element i

Plot the exact solution, the numerical solution, and the error as a function of the number of elements
The analitical solution was developed before in other homeworks; therefore,here we only copy it

Figures 6.1.2 shows the analitical and numerical solution obtained with different number of elements. Although good aproximation was achieved even whit low number of elements, the convergence with a tolerance of $$10^{-6}$$ was not achieved even with more than 100 elements. With a tolerance of $$10^{-4}$$ convergence was achieved with 54 elements. Figure 6.1.3 shows the error as a function of the number of elements. In that figure, it can be observed that initially the error decreases fast but from aproximately 25 elements the relation between the number of elements and the error in logaritmic sacale tend to be linear and to achieve a tolerance of $$10^{-6}$$ more than 200 elements could be needed; but, is probalby that with higher number of elements the error decrease even slower because with more element the round-off error increases.

Appendix
 Matlab Code:

title: User talk:Eml5526.s11.team01.fercas


text:
User:Eml5526.s11.team01.fercas

This Fernando Casanova of team 01

Useful Links - Course Wiki
Eml5526 Course Wiki

Lecture Plan
Lecture Plan

Homework - Report Table
Report table

Group Page
Team 1 Page

Homework Figures
[| Homework Figures]

title: User:Eml5526.s11.team01.fercas


text: File:Moviola URS schematic.JPG

title: File:Moviola URS schematic.JPG


text:
User:Eml5526.s11.team7.flater

Sçrατçh Ærεα

EML 5526, Spring 2011
 * Homepage
 * Course Wiki
 * Lecture plan
 * Homework report table (with comments)
 * Team 7 Page

EGM6611, Spring 2011
 * HW3
 * HW4

Stress Triaxiality
 * $$ \chi = \frac{\text{mean stress}}{\text{Mises equivalent stress}} = \frac{\Sigma_m}{\Sigma_{eq}} = \frac{\frac{1}{3}\left(\Sigma_1 + \Sigma_2 + \Sigma_3 \right)}{\sqrt{\frac{1}{2} \left[ (\Sigma_1 - \Sigma_2)^2 + (\Sigma_2 - \Sigma_3)^2 + (\Sigma_3 - \Sigma_1)^2 \right]}}$$

title: User:Eml5526.s11.team7.flater


text:
Talk:HiFi

This resource is not named in the best way. Invisible learning resources aren't terribly useful.

A project is being described. How about Vacuum tube amplifier repair project, linked from EE_Introduction_to_Electronics?

The organization and naming systems on Wikiversity need a lot of work. --Abd 18:31, 18 January 2011 (UTC)

 * This is a constructivst educational approach; where constructivism sounds just like what it is: construction.  By constructing this amp, there comes confidence to master the actual electronic engineering.  I am attempting to show that this, as the only native approach to Internet education, is the best.

 * This way lots of highly experienced people bring information from their individual projects to help construct the knowledge. Already I have accessed someone who can prove that tubes are better than solidstate--this is individually significant, and proof of why constructivism is the native knowledge construction of  the Internet, Web, and Wiki!  There is also a lot of other valuable data here that could not have come any other way, and there will be more from these, and future, sources.

 * Though thanks for your suggestion :)
 * --John[[Image:bessa66.png|12px]]Bessatalk 15:56, 3 February 2011 (UTC)

 * John, You might also want to contact Eric Barbour on Wikipedia Review. He owns Metasonix, which makes vacuum tube amplifiers for audiophiles.  —Moulton 16:11, 3 February 2011 (UTC)

Reverb Regulation with Polyonic Feedback
John, have you ever come across a reverb regulator that exploited "polyonic feedback"?

These might have been used in the early days of audio tape recording, where print-through on wire-recorders or thin high-speed tapes created an unwanted synthetic echo or reverb.

But they might also have been used by Les Paul to intentionally add precision band-shaped reverb.

Moulton 15:37, 3 February 2011 (UTC)

 * I know nothing about electrical engineering and especially about valves. That is what this is all about. --John[[Image:bessa66.png|12px]]Bessatalk 15:51, 3 February 2011 (UTC)

title: Talk:HiFi


text:
File:Str.Mat.L3a.ex1.jpg

Summary
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text:
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Summary
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File:Str.Mat.L3a.ex1.fbd2.JPG

Summary
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title: File:Str.Mat.L3a.ex1.fbd2.JPG


text:
User:Mabuabsdd/Aqib need progress percents, (Aqib promise)

Need progresses

Deletion:5%

Warnings:77%

Progress requested as: 82%

Good, But need help


text:
User:EML5526.S11.Team1.HS

= My World: "The Engineering World" =

Hi, My name is Harrison Sheffield and am currently at the University of Florida studying Mechanical Engineering

Useful Links - Course Wiki
Eml5526 Course Wiki

Lecture Plan
Lecture Plan

Homework - Report Table
Report table

Group Page
Team 1 Page

Homework Figures
Homework Figures

HW Drafts

 * HW2
 * HW3
 * HW4
 * HW5
 * HW6
 * HW7

HW Assignments

 * Homework 1
 * Homework 2
 * Homework 3

title: User:EML5526.S11.Team1.HS


text:
User:Egm6341.s11.team2/hwk1

=Problem 1.1: The Limit and Plot of a Given Function f(x)=

From the lecture slide Mtg 3-1

Problem Statement
Given:

i) Determine $$ \lim_{x \to 0}f(x) $$ and

ii) Plot $$ f\left(x\right) $$

Solution
i) Finding the limit of $$\displaystyle f(x) $$

Since it is possible to determine the limit of both the numerator and denominator independently as $$\displaystyle x $$ tends to $$\displaystyle 0 $$ as shown below and It is therefore clear that we have indeterminate form of type $$\displaystyle \frac{0}{0} $$. Limits of functions having this indeterminate form are easily found using L'Hôpital's Rule. Thus using L'Hôpital's Rule we find that,

ii) Plot of f(x)

Shown below is the plot of $$\displaystyle f(x) $$ on the interval $$\displaystyle [0,1] $$ which graphically verifies our solution in part i) of the limit of $$\displaystyle f(x) $$ as $$\displaystyle x $$ tends to $$\displaystyle 0 $$.

 * Hw1bplot.png]

=Problem 1.2: Determination of Taylor Polynomial and Remainder=

From the lecture slide Mtg 3-4

Given: Taylor's Theorem
Taylor's Theorem

Let $$\displaystyle f(x) $$ be a function with derivatives $$\displaystyle f^{(1)},f^{(2)},...,f^{(n)} $$ that are continuous on $$\displaystyle I:=[a,b] $$ and where $$\displaystyle f^{(n+1)} $$ exist on $$\displaystyle (a,b) $$.

If $$\displaystyle x_{0} \in I $$, then for any point $$\displaystyle x $$ in $$\displaystyle I $$ there exists a point $$\displaystyle \xi $$ between $$\displaystyle x $$ and $$\displaystyle x_{0} $$ such that

Where $$\displaystyle p_{n}(x) $$ is called the Taylor Polynomial for $$\displaystyle f $$ at $$\displaystyle x_{0} $$

and $$\displaystyle R_{n+1}(x) $$ is the remainder of $$\displaystyle f $$ at $$\displaystyle x_{0} $$

Find
The Taylor Polynomial $$\displaystyle {p_n}(x) $$ and remainder $$\displaystyle {R_{n + 1}}(x) $$ at $$\displaystyle {x_0}=0 $$ for

i) $$ \displaystyle g(x) = {e^x} $$

ii) $$ \displaystyle f(x) = \frac{x} $$

Solution
i) For $$ \displaystyle g(x) = {e^x} $$ all derivative are identically equal to the original function. Evaluating these derivatives for $$\displaystyle n=1,2,...,n,n+1 $$ at $$\displaystyle x_{0} = 0 $$ results in the following

Then by the definition of the Taylor Polynomial $$\displaystyle p_{n}(x) $$ provided in (2.2) we can write

This can be more concisely written in summation notation as

Using the definition of the remainder $$\displaystyle R_{n+1}(x) $$ shown in (2.3) and the $$\displaystyle n+1 $$-th derivative found in (2.4) we can write

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$$ \displaystyle \therefore {e^x} = {p_n}(x) + {R_{n + 1}}(x) = 1 + \sum\limits_{i = 1}^\infty {\frac}  + \frac{e^\xi } $$     (2.10)
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ii) Next we will consider $$\displaystyle f(x) $$.

Using the Taylor Polynomial and Remainder for $$\displaystyle e^{x} $$ shown in (2.10), we will subtract $$\displaystyle 1 $$ from each side to get

Dividing (2.11) through by $$\displaystyle x $$ results in the desired function f(x) for which we wish to obtain the Taylor Polynomial $$\displaystyle p_{n}(x) $$ and remainder $$\displaystyle R_{n+1}(x) $$ results in

So in summary, the function $$\displaystyle f(x) $$ has the Taylor Polynomial $$\displaystyle p_{n}(x) $$ and remainder $$\displaystyle R_{n+1}(x) $$
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$$  \displaystyle {p_n}(x)= \sum\limits_{i = 1}^\infty {\frac} $$
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$$  \displaystyle {R_{n + 1}}(x)=\frac{e^\xi } $$

where $$ \xi \in [{x_0},x] $$ (2.14)
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=Problem 1.3: Prove the IMVT=

''' This problem was solved without referring to S10 homework. '''

From the lecture slide Mtg 4-3

Prove: The Integral Mean Value Theorem (IMVT)
Prove the Integral Mean Value Theorem (IMVT) for

a) $$\displaystyle w\left(x\right) \geq 0, \quad\forall x \in \left[a,b\right]$$ and

b) $$\displaystyle w\left(x\right) \neq 0, \quad\forall x \in \left[a,b\right]$$

using same technique used in lecture 4 to prove IVMT for $$\displaystyle w\left(x\right) = 1 $$.

Required
Integral Mean Value Theorem:

Let $$\displaystyle w(x)$$ be a nonnegative integrable function defined on $$\displaystyle [a,b] $$

If $$\displaystyle f(x) $$ is a continuous function defined on $$\displaystyle [a,b] $$, then

for some $$\displaystyle \xi $$ such that

a) $$\displaystyle w\left(x\right) \geq 0, \quad\forall x \in \left[a,b\right]$$
Suppose $$\displaystyle \exist $$ an $$\displaystyle m $$ and $$\displaystyle n $$ such that

This implies that $$\displaystyle f(x) $$ is bounded below by $$\displaystyle m $$ and bounded above by $$\displaystyle M $$,

For a given weight function

and by simply multiplying the inequality in (3.5) by $$\displaystyle w(x) $$ one obtains,

A corollary to the Fundamental Theorem of Calculus ensures that, upon integration of (3.7) the following relation is obtained

$$\displaystyle \Rightarrow $$

The definite integral,

is just some numerical value therefore enabling one to divide (3.9) through by (3.10) to get

Furthermore, the Intermediate Value Theorem ensures for a continuous function $$\displaystyle f(x) $$ on the interval $$\displaystyle x \in [a,b] $$ that if there exists a value $$\displaystyle Z $$ on the interval $$\displaystyle [m,M] $$ then there is at least one such point $$\displaystyle \xi $$ for which,

Recognizing that $$\displaystyle Z $$ in (3.11) is

it then follows from (3.11) and (3.13) that

b) $$\displaystyle w\left(x\right) \neq 0, \quad\forall x \in \left[a,b\right]$$
This can really be broken down into two cases $$\displaystyle w(x) > 0 $$ and $$\displaystyle w(x) < 0 $$

Case 1: $$\displaystyle w(x)>0 $$

Observe that $$\displaystyle w(x) > 0 $$ is really just a more restricted case than $$\displaystyle w(x) \geq 0 $$ since it does not include $$\displaystyle w(x) = 0 $$. Because of this it is not necessary to reprove what has already been proven in the first part. Therefore by virtue of the case where $$\displaystyle w(x) \geq 0 $$ the case for $$\displaystyle w(x) > 0 $$ can be reasonably inferred from part a.

Case 2: $$\displaystyle w(x)<0 $$

Consider $$\displaystyle w\left(x\right) f\left(x\right) $$ continuos on $$\displaystyle\left[a,b\right] .$$

Let $$\displaystyle m $$ and $$\displaystyle M $$ be what was previously defined in (3.4). Multiplying (3.5) by $$\displaystyle |w\left(x\right)| $$

Integrating (3.15) over $$\displaystyle \left[a,b\right] $$

$$\displaystyle m $$ and $$\displaystyle M $$ are both constants and can be pulled out of the integrals.

Let

Divide (3.18) by $$\displaystyle \alpha $$

By the Intermediate Value Theorem (IVT), $$\displaystyle \exists \xi  \in \left[a,b\right] $$ such that $$\displaystyle f\left(\xi\right)=Z. $$

Since $$ \displaystyle w\left(x\right) < 0 ,$$

Therefore,

= Problem 1.4 Infinity Norm of Functions =

From the lecture slide Mtg 5-3

Problem Statement
Given:

and

i) Find $$\displaystyle {\left\| f \right\|_\infty }$$, $$\displaystyle {\left\| g \right\|_\infty }$$ and $$\displaystyle {\left\| {f - g} \right\|_\infty }$$.

ii) Plot $$\displaystyle {f(x)}$$, $$\displaystyle {g(x)}$$ and $$\displaystyle f(x)-{g(x)}$$

iii) Then establish a relationship between the infinity norms of the two functions $$\displaystyle f(x) $$ and $$\displaystyle g(x) $$.

Solution
i)

Taking the first derivative of $$\displaystyle f(x) $$ and $$\displaystyle g(x) $$ indicates that both function are increasing on the interval $$\displaystyle [0,1] $$ and

Similar analysis shows that the difference of the functions $$\displaystyle f(x) $$ and $$\displaystyle g(x) $$ results in an increasing behavior as seen here

Which means that $$\displaystyle {f(x)}$$ and $$\displaystyle {g(x)}$$ and therefore $$\displaystyle {f(x) - g(x)}$$ all achieve their maximal value at $$\displaystyle x=1 $$. Hence
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$$ {\left\| f \right\|_\infty } = \max \left| {f(x)} \right| = {\left. {\left| {f(x)} \right|} \right|_{x = 1}} = 8 $$     (4.6)
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$$ {\left\| g \right\|_\infty } = \max \left| {g(x)} \right| = {\left. {\left| {g(x)} \right|} \right|_{x = 1}} = \frac{1}{2}(e - \frac{1}{e}) \approx 1.17520 $$     (4.7)
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$$ {\left\| {f - g} \right\|_\infty } = \max \left| {f(x) - g(x)} \right| = {\left. {\left| {f(x) - g(x)} \right|} \right|_{x = 1}} = 8 - \frac{1}{2}(e - \frac{1}{e}) \approx 6.82479 $$     (4.8)
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ii) Using the following Fortran code

program main real,allocatable::x,f,g,fg integer::i,n real::e=2.71828 open (unit=10, file="input", status="old") read(10,*)n allocate(x(n+1),f(n+1),g(n+1),fg(n+1)) do i=1,n+1 x(i)=-1.0+2.0/n*(i-1) f(i)=3.0*x(i)+5*x(i)**3 g(i)=0.5*(e**x(i)-e**(-1.0*x(i))) fg(i)=f(i)-g(i) end do  open (unit=11, file="hw1.4.plt", status="replace") do i=1,n+1 write(11,"(4F12.4)") x(i),f(i),g(i) end do   end program main

We can easily get the image of $$\displaystyle {f(x)}$$, $$\displaystyle {g(x)}$$ and $$\displaystyle {f(x)-g(x)}$$ as following

iii) Concluding remark on the relationship between the infinity norms of the given functions

We can see from above that,

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$$ \displaystyle {\left\| {f - g} \right\|_\infty } = 6.8248 $$

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$$ \displaystyle {\left\| f \right\|_\infty } - {\left\| g \right\|_\infty } = 8 - 1.17520 = 6.8248 $$

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thus in this case we have,

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$$ \displaystyle {\left\| {f - g} \right\|_\infty } = {\left\| f \right\|_\infty } - {\left\| g \right\|_\infty } $$     (4.9)
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In this case we can't prove the assertion made in Lecture slide 5-3. The relationship shown in (4.9) is a special case and it arises from the monotonically increasing behavior of $$\displaystyle f(x)$$ and $$\displaystyle g(x)$$. Because both functions are monotonically increasing on $$\displaystyle [0,1] $$ then the difference between the two will also be monotonically increasing and thus $$\displaystyle {f(x) - g(x)}$$ has an infinity norm at $$\displaystyle x = 1 $$ on this domain.

= References =

=Contributing Members=

title: User:Egm6341.s11.team2/hwk1