text:
Bahá'í Faith/Spouse Selection/Maturity

Questions to ask yourself and your potential spouse:
Do you behave responsibly as an adult?

Do reliable adults see you as physically, psychologically, socially and morally mature?

Do you have an understanding and comprehension of the realities of life, and the capacity for further development and understanding?

Can you handle difficult situations with maturity?

Introduction
Marriage is an important responsibility requiring a degree of maturity if it is to be successful. Family life, which normally comes soon after marriage, is an even greater responsibility requiring even greater maturation to be successful. Maturity can be considered from its physical, psychological, social and moral perspectives. Physical maturity is easy to discern and its standards generally accepted. Each higher level of maturity is more difficult to ascertain and there is less agreement on its characteristics. Ideas about the level of development needed for marriage differ from culture to culture and within cultures.

Different peoples and individuals mature at different times and in different ways. Maturation is a gradual and continuous process. Determining when you have reached a sufficient level of development for marriage is in itself a sign of maturity. Outside standards and references should be used to verify your judgments, as a symptom of immaturity is to believe you are more grown up than you really are. Seeking and being open to outside verification from reliable sources about your own development is a mark of adulthood.

The hallmark of adulthood is knowing and mastering your self. This insight and ability allow you to share yourself honestly and caringly with another, and to get to truly know and love them. The station of adulthood is that of true understanding and nobility. It is marked by a personal unity among our physical, psychological, social and moral faculties, as well as with our thoughts, feelings and actions.

Maturity has been much explored in each of these areas. It is well accepted that individuals go through fairly predictable and identifiable stages of growth. Each stage evolves out of the earlier ones and incorporates aspects of lower stages for its development. We can clearly see this process in physical development. The infant gradually grows to childhood and the adolescent is the outcome of the developed child. All of these less mature qualities evolve into the adult. The lines between the various levels of physical development we have identified are not clear. It is a gradual process that slowly takes on more characteristics of the latter while leaving behind those of the former.

As each subsequent level depends on the earlier levels, none can be skipped. You cannot go from infancy to adolescence or childhood to adulthood without passing through the intervening stages. This is important to understand that like physical growth, neither you nor your partner can be expected to leap ahead in development from psychological childhood to adulthood without going through the necessary growth and processes required.

Not only do these aspects of our nature go through developmental stages, they seem to be interrelated to one another. Psychological development grows out of and is related to physical development, as social does with psychological and moral with social. Though these can be seen as separate parts of our development, they depend on and affect the others. Examples of developmental theories in psychology are those proposed in cognitive, social-emotional and moral development. The different stages we go through in the process of becoming mature can be synthesized and summarized as follows: 1) pre-conventional, egocentric, impulsive; 2) conventional, ethnocentric, conformist; and 3) post-conventional, world-centric, autonomous. The higher the level of development the couples have reached, the better the chance for a happy and successful marriage.

It is possible to be developed in one area but not the others. The later and higher levels depend on and are affected by the lower levels. Many people who are physically mature lack psychological maturity and some who are psychologically and intellectually mature are limited in social development, and those developed in all the other areas may not possess well-developed moral-volitional capacities.

Physical-Material
Physical maturity is generally reached by the mid to late teens. Statistical data supports the hypothesis that the older you are when you get married, the more likely the marriage will be successful, that is, to last longer and be more satisfying to the couple. Physical age does not seem the important variable in these studies, rather the degree of psychological, social and moral maturity that generally comes with age. Some people achieve this stage at a young age, and some never seem to attain it though they are physically adults. The rule of thumb here might be to consider getting married at the youngest age at which you are psychologically, socially and morally mature enough to be successfully married, and to select a good spouse for marriage.

Certain physical and other skills associated with being an adult will be useful and needed by the couple to live a fruitful and productive life. The ability to earn a livelihood and to support others can be considered another mark of physical readiness for marriage. A person who has not adequately met their physical needs on their own, such as survival and safety, will have difficulty in meeting these basic needs in marriage. They will also be hampered in focusing on the higher order needs related to marriage and family.

Being able to control your physical desires, passions and impulses could be considered another aspect of physical maturity (see Chastity below), but both of these involve other aspects of being an adult as well. Just because someone has reached physical adulthood does not mean they are ready for marriage. Though this may be the first thing we look at, it is of the least importance. A physically healthy and attractive person may be disfigured and disgusting emotionally, intellectually, morally or otherwise and this may not be readily apparent.

Psychological-Intellectual
A sound mind requires a sound body in which to function. Psychological maturity is much more complex and harder to measure than physical maturity. It is also much more important. Psychologically we move through several stages related to various aspects of our character.

In cognitive development, Piaget found four generally accepted stages (scholars expand or divide them differently) with many sub stages that everyone goes through: sensorimotor, preoperational, concrete operational and formal operational. A person should be able to function at the higher stages that require logical use of symbols related to abstract concepts and a moving away from egocentric thinking. One may also consider the taxonomy of the cognitive domain as a standard for maturity. A more mature person would have greater facility using all the hierarchical levels of the cognitive domain: knowledge, comprehension, application, analysis, synthesis and evaluation. Each higher stage and level incorporates and transcends the lower one.

Self-regulation, the ability to set goals, self observe, self-assess and self-reinforce, is a key trait needed for successful life, learning and marriage. People with a high level of self-efficacy learn, grow and achieve more. They accept challenge, see themselves as capable, invest a lot of effort, use effective strategies to achieve their goals and persist in the face of difficulty. Assuming personal responsibility for your growth and development is part of maturity.

Maturity is highly related to self-understanding, which is related to intelligence, including its perceptual, emotional and other aspects. A sign of maturity is recognizing your limitations and your strengths--understanding yourself. It is also important to know your attitudes, values and perceptions. True understanding of self involves all aspects of self and can be aided through study of philosophy, psychology and religion. In the same way we are to know ourselves, we must also come to know and appreciate our spouse.

Social-Emotional
Knowing oneself is the foundation of any fruitful social relationship. Without self-knowledge, you will not be able to determine if the signs present in yourself or your relationship are pointing in the right direction. It is important to know how you deal with your emotions such as anger and disappointment.

Marriage is a social act and greatly affects society. Successfully fulfilling the roles of being a child, parent and grandparent, as well as sibling, cousin, spouse and community member are crucial to social development individually and collectively.

The ability to maintain healthy relations with others is essential in a good marriage. The capacity for kindness, compassion, friendship, companionship, fellowship and getting along with others are all indicators of maturity in this area. Marriage is about having all these qualities on a profound and ever-evolving level. The ability to collaborate, cooperate and consult with others to solve problems is a mark of maturity (see Consultation below).

A competitive attitude, especially in relationship with your spouse, is harmful. Being able to over look the shortcomings of others and focus on our own and to think not only of your self, but be concerned about the welfare of another is vital to a healthy relationship. Trust, consideration and ability to make and keep a firm commitment are foundational to a marriage.

Each party should be equipped in performing some socially redeemable service to society that can also contribute to the well-being of the marriage and family. Possessing civic virtues and the ability to contribute to the society in which one lives, as well as humanity as a whole, is a mark of social maturity.

Erikson's theory of social-emotional development is worth considering in helping to determine maturity and readiness for marriage. From this theory a person should have successfully dealt with the crisis to establish a healthy measure of trust, autonomy, initiative, accomplishment, and identity before they are ready for intimacy, the level at which a person develops the ability to give and receive love needed to begin making long-term commitments to relationships. This stage is followed by the stages of generativity, which relates to having a family, and integrity, an acceptance of the worth of one’s life.

Spiritual-Moral
Happiness and harmony in the marriage and family are directly related to the moral character and maturity of the individual. Every great wisdom and contemplative tradition recognizes that the foundation of all life is to be an ethical and good human being. The fundamental reality of an individual is recognized to be moral and spiritual. Marriage is essentially a sacred act that came into being and force through religion. The marriage bond is considered a divine institution, that promises blessings if carried out appropriately. Most civil laws regarding marriage have developed from religious traditions.

Morality is the most complex and subtlest of the signs of maturity, but also the most important. Moral maturity and a deep spiritual bond can sustain and nourish the marriage and the individuals in this world and may continue to do so long after the physical bond has been broken.

The superficial, material and trivial understanding of life that characterizes adolescence with its impetuosity and vehemence is superseded by a more comprehensive, spiritual and significant grasp of reality with its calmness and insight. A number of virtues are identified with spiritual maturity including patience, wisdom, forbearance, humility, selflessness and service (see Character below for a fuller description of these qualities). Being able to balance mercy with justice, freedom with responsibility and with excellence with humility symbolizes maturity.

Most religions would associate recognition of, acceptance of and obedience to a higher divine power as a mark of spiritual maturity. This is characterized as a spiritual transformation through a spiritual force that allows the person to transcend material concerns. Spiritual maturity results from following the spiritual practices found in the sacred scriptures, such as prayer, meditation, fasting, study and good actions. Having spiritual standards and adjusting your life to them bodes well for stable patterns of growth and relationships. Overcoming the ego and selfishness is one of the great challenges of adulthood, and marriage and family life are excellent laboratories for assisting in this process.

Achieving a successful marriage and spiritual maturity requires effort, sacrifice, wisdom and self-abnegation. Trustworthiness and truthfulness are the bedrock of all positive relationships and spirituality. Some agreeable level of equity and justice is necessary for a successful marriage. If either party unjustly and inequitably treats the other, disunity and distress will result. The pattern in most marriages, families and societies is that women are unfairly treated. Creating equitable relationships requires maturity of a high order. You and your partner being able to do this is an indication of maturity.

One may use the moral development theories to help determine moral maturity and readiness for marriage. Kohlberg’s (1984) theory of moral development has three stages (pre-conventional [egocentric], conventional [ethnocentric] and post conventional [world-centric]) and six levels or orientations (punishment-obedience, instrumental relativist, interpersonal concordance, authority and social-order maintaining, social-contract legalistic and universal ethical principle). Gilligan (1982) proposes that women make moral judgments based on care rather than justice found in Kohlberg’s theory, and suggests the following three stages: orientation to self (selfish), morality of care (care) and morality of nonviolence (universal care). Males and females seem to go through the same stages with different emphases on justice/rights and care/communion respectively.

title: Bahá'í Faith/Spouse Selection/Maturity


text:
Bahá'í Faith/Spouse Selection/References and Resources

Resources and References

 * Bergner, R. (Winter 2000). Love and Barriers to Love: An Analysis for Psychotherapists and Others. American Journal of Psychotherapy, Vol. 54, No. 1.
 * Campbell, V., & Bond, R. (1982). Evaluation of a character education curriculum. In D. McClelland (ed.), Education for values. New York: Irvington Publishers.
 * Danesh, H. (13 April 1991). "Marriage and Sexuality" An edited and abbreviated transcript of an talk found at http://bahai-library.org/talks/marriage.sexuality.html
 * Gilligan, C. (1982). In a different voice: Psychological theory and women's development. Cambridge, MA: Harvard University Press.
 * Huddleston, J. (1993). Perspectives, purposes, and brotherhood: A spiritual framework for a global society. In S. Bushrui, I. Ayman, & E. Laszlo, Transition to a global society (pp. 142-150). Oxford, England: Oneworld Publications Ltd.
 * Kohlberg, L. (1984). The psychology of moral development. San Francisco: Harper & Row.
 * Larson, J. (2000). Should we stay together. New York: Jossey Bass.
 * Warren, N. & Abraham, K. (2005). Falling in love for all the right reasons. New York: TimeWarner.

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


text:
Bahá'í Faith/Spouse Selection/Understands and Values Love, Marriage, Family Life and Children

Questions to ask yourself and your potential spouse:
Do you have a realistic understanding and appreciation of love, marriage and family life?

How much will you sacrifice to make the marriage and family successful?

What are your goals for marriage and family life?

How do you define love, marriage, family and happiness? What do they look like?

How strong are the characteristics of and barriers to romantic love mentioned below in your relationship?

Introduction
It is a bad sign for a good marriage if either you or your potential spouse does not have a basic understanding of what it means to love, be married and have a family. Some people may not know how to love or have a desire to be married and have a family. It is important that both partners have similar plans for the future, including whether or not they want children. You will need to sacrifice traits that may not be helpful to your relationship and desires that may interfere with the unity of the family, but you should not have to sacrifice your good qualities and principles. This issue will need to be addressed positively if one expects to have a happy marriage and healthy family life.

We are given many unrealistic and harmful ideas and attitudes about love, marriage and family through our culture and the media. Of the many ideas that seem to be prevalent and growing in the world that seem especially harmful, one is the construct of romantic love with its many fantasies, often now including the idea that there is only one true “soul mate” for each person. Neither science nor spiritual traditions support these notions as true or healthy.

Love
Love is the foundation of any successful relationship and the essential beginning point of a marriage. There are many types of love. The love that should be between two people is considered the most central, special, important and sacred in this world in this text, next to love for God. Any love for others must be based on a healthy and sound love for yourself. This is not an egotistical love, but an acceptance, respect, appreciation and caring for oneself.

Raymond Bergner in “Love and Barriers to Love: An Analysis for Psychotherapists and Others” (American Journal of Psychotherapy, Vol. 54, No. 1, Winter 2000) presents a summary of the literature of the characteristics of romantic love and the most common barriers to being able to love. According to Bergner, the characteristics of romantic love are 1) an investment in the well-being of the other, 2) appreciation and admiration of the other, 3) sexual desire, 4) intimacy and inclusion, 5) commitment, 6) exclusivity, and 7) understanding. This giving of honor, value and centrality to another person is the type of relationship upon which marriage is generally based. It allows us to transcend ourselves and become better, happier persons.

There are several barriers to love that we can consider in our partner and ourselves. The failure to understand or appreciate love or to see or treat others as people are two major barriers to love. There are a number of limited, maladaptive and pathological conceptions of love. Some people conceive love as only a feeling, not as a complex relationship of which feelings are only a part. Others define love in materialistic terms, such as meeting selfish needs. We may also see others as commodities, tools to satisfy our desires or to play roles in our lives. Lack of maturity, sound development and ability to overlook others differences and faults are also barriers.

The romantic love, infatuation, passion, obsession, lust and other forms of superficial attachments common today will not generally stand the test of facing the realities of marriage and family. That which causes the “love” passes. Another lover comes along that one imagines will better meet one’s fantasies or desires. Today they are lovers; tomorrow they are discarded and unwanted. These types of love seem to predominate in Western society, epitomized by the consumer culture with its latest fads and fashions mentality.

Marriage
Marriage is a responsibility that should be governed by the noblest of intentions. Marriage requires maturity and self-sacrifice by both parties. The vital significance of marriage and family on several levels is recognized in many cultures throughout history. Families are perceived to be the foundation of many communities and civilizations.

People who are married have consistently higher quality of life indicators: they are healthier, happier, wealthier and live longer. This correlation is true even in less successful marriages, but begins to break down as the marriage breaks down. As with most things in life, moderation is preferable. Desperation to be married too much can lead one to make poor choices. Sometimes we are so anxious to achieve our goal, that we fail to discern accurately the signs leading to it, and follow instead our desires and passions in the false hopes it will come out right.

Family
Marriage and family life are enjoined in all religions, not only for its spiritual benefits to the individuals involved, but for its social and personal advantages as well. Some believe marriage and the family seems to be the best condition for raising children and constitute the foundation of a stable society. Marriage and family life constitute the basis of the perpetuation of the human race and civilization. Though marriage and family as divine institutions have been recognized by every major religion, the forms and guidance about successful marriages and families by these religions have evolved as humanity has evolved. Some of the practices enjoined in older religions are no longer suitable for today’s conditions.

Having specific, immediate and challenging goals for marriage and family will increase the likelihood of your success. Goals help direct attention, mobilize effort, increase persistence and motivate strategy development. Sharing and comparing your goals with your friends will facilitate your understanding, refining and accomplishing them. They will help you in both direct and indirect ways.

The more time and effort you put into your goals, the closer you will come to achieving them. It needs to be an active process of continual progress, not one of sitting and waiting for the right person to come along. Learn from your mistakes. The more effort one makes, the greater they will progress. Set high goals, developing yourself and acquiring more capacity and virtues on a daily basis. Effective monitoring and feedback help goal attainment. Use both internal and external assessment to get feedback. Each day assess yourself on how well you are doing, the progress you have made and still need to make and periodically get feedback from a respected and reliable source.

Children
One of the key aspects of family is related to children. How many children do you wish to have? How do you wish to raise them? What kind of people do you want your children to become?

Children constitute an important part of a marriage from both a scientific and religious point of view. Most religions consider procreation the primary and sacred purpose of marriage. Raising children to be good is seen as a sacred responsibilities of marriage by religion and a civic duty by society.

How you wish to raise your children is a matter that will deeply affect the family and should be agreed upon as much as is practical before marriage. Deep differences in this area can lead to tensions, divisions and disintegration in the family unit.


text:
Bahá'í Faith/Spouse Selection/Consultation

Questions to ask yourself and your potential spouse:
Do you consult often and well on matters?

Do those close to you and in whom you trust and respect support the marriage?

Do the individual and group subjective and objective data match?

Introduction
Consulting with one another, with family and friends, and possibly with professionals, is an important test of a good relationship. The people who should know you best and who should have your best interests at heart are your parents. It has become unfashionable to ask parents for permission to marry, but this practice has been prescribed by most religions and has been practiced in most cultures. All marriages suffer problems and need support, and the wisdom and support of your immediate families can be of great assistance. If they do not agree to the marriage, an important support will be missing, physically, psychologically, socially and spiritually.

Those who know you well and who love you often can see problems that you are overlooking in your desire to be married or in your state of love. If your family and friends do not approve or get along with your selected spouse, this is a sign that something is amiss. If your spouse does not approve of or get along with your family and friends, this is another sign of problems. The same is true for you and their family and friends. Consulting with family and friends, especially parents, is a symbol of your respect, love and gratitude for them, strengthens your bond with them and enlists their future aid and support.

In some countries the opposite condition occurs: the parents choose their children’s spouses and do not consult with them. The right to choose your spouse is important, but so is seeking your parents’ consent.

Consultation with others can make you more aware of your own strengths and problems, as well as those of your potential partner and the marriage. It can help you to be more certain that you are making a good decision. It will strengthen a good relationship and provide insight into issues for which you may be unaware. It also may raise questions that you can explore with yourself and your intended.

Some characteristics needed for consultation to be successful are love, humility, openness, frankness, detachment and objectivity. Sometimes these are difficult when we feel passionately about something or someone. These skills will be essential in solving problems in your relationship and creating unity, foundational factors for developinging and strengthening a happy and successful marriage. Being able to see your intended in consultation will give you some indication of how they will function in the marriage setting when you are required to consult on matters of mutual interest. It can demonstrate their ability to listen, share power, think through a problem, bring relevant facts to bear and arrive at a decision that observes the rights and prerogatives of all involved. A person who cannot patiently, carefully and dispassionately consider all aspects of a matter, will create difficulties when decisions have to be made. Consultation is a means of creating security, peace and tranquility in a family.

Every matter will have a subjective and objective dimension, as well as its individual and group dimensions. Consultation includes all dimensions. If you cannot agree on objective matters or if you see the world so differently that agreement is difficult, these are signs that achieving unity and harmony in the relationship will be difficult. Subjective methods of investigating truth, such as prayer, meditation, intuition, dreams and other sorts of guidance can assist in one’s search, but should be used in conjunction with more empirical, objective approaches.


text: Category:Learning theories in practice

title: Category:Learning theories in practice


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User:Yongtao-Li/HW2.9

=Problem 2.9: "Gauss-Legendre Quadrature and Runge's phenomenon"=

Given
Legendre polynomial

Find
A) Use (9.1) to generate $$ {P_5}(x) $$ and Matlab command "roots" to compute the roots of $$ {P_5}(x) $$ to check values in table on p7-5. Plot the roots on $$ [-1,+1] $$ using Matlab "plot" command.

B) Repeat the above for $$ {P_{10}}(x) $$ and observe the location of roots near end points -1 and +1.

Solution
A) Substitute n=5 in (9.1) and then can get $$ {P_5}(x) $$:

Use the following codes in Matlab :

p=[63 0 -70 0 15 0]/8 roots(p)

we can get the roots for $$ {P_5}(x) $$: ans = 0  -0.9062   -0.5385    0.9062    0.5385

The above answers are exactly the same as the data in table on p7-5.

Use the "plot" command we can get the distribution of roots on $$ [-1,+1] $$ as following:

B) Substitute n=10 in (9.1) and then can get $$ {P_{10}}(x) $$:

Use the following codes in Matlab :

p=[46189 0 -109395 0 90090 0 -30030 0 3465 0 -63]/256 roots(p)

we can get the roots for $$ {P_{10}}(x) $$: ans = -0.9739  -0.8651   -0.6794    0.9739    0.8651    0.6794   -0.4334    0.4334   -0.1489    0.1489

The above answers are exactly the same as the data in table on p7-5.

Use the "plot" command we can get the distribution of roots on $$ [-1,+1] $$ as following:

From the above image, we can observe that quadrature nodes become more dense near the end points. That means the interpolation with high order polynomial may has more oscillation near the edge of interval region because the interpolation only exactly accurate at the quadrature nodes and this phenomenon is so called Runge's phenomenon.

title: User:Yongtao-Li/HW2.9


text:
User:Vanished user adhmfdfmykrdyr/Ottava logs

__NOINDEX__ [07:15]  I have morals [07:15]  You might not agree with them [07:15]  There is a fine line there. [07:15]  but it doesn't mean I have done anythting amoral [07:15]  No murder [07:15]  no stealing [07:15]  no rape [07:15]  nothing [07:15]  definitionally, you are way wrong [07:15]  You sound pretty immoral to me because of your lack of empathy. [07:15]  otherwise, every self centered person is a psychopath, which would mean most of the world is [07:16]  You've stated a willingness to get rid of everyone on Wikiversity to futher your own Wikiversity agenda. [07:16]  Hell, your ranting is very self centered [07:16]  It is motivated by a hurt ego [07:16]  In the context of a wiki, you're willing to committ genocide. [07:16]  because I didnt treat you like the princess you wish you were [07:16] <Ottava> If you want a sychophant, go find one [07:16] <Ottava> demanding everyone arbitrarily fulfill the role will get you no where [07:16] <LauraHale> I guess that makes me overly capable of empathy to think genocide is bad. [07:16] <Ottava> I'm not here to fulfill your psychological inadequacies. [07:16] <Ottava> You aren't capable of any empathy :) [07:16] <Ottava> You have proven tha [07:17] <Ottava> that [07:17] <Ottava> You are as self centered as I am [07:17] <LauraHale> And I have the logs of your statements to support my claim of Ottava believes in community genocide to support Ottava's world view. :) [07:17] <Ottava> tha is the amusing part [07:17] <Ottava> which community did I say to kill off? [07:17] <LauraHale> Wikiversity. [07:17] <LauraHale> Ottava's way or no way. [07:18] <LauraHale> Ottava was a bad boy. [07:18] <Ottava> Wikiversity removal would be genocide? [07:18] <Ottava> o.O [07:18] <LauraHale> Ottava lost his privileges because he was willing to kill the community to do his way. [07:18] <Ottava> Okay, now that is quite out of touch [07:18] <LauraHale> Wikiversity content =/= Wikiversity community. [07:18] <Ottava> Laura, do your parents know that you consider the disabanding of an online website the equivalent of Nazis burning Jews for their religion? [07:18] <Ottava> I think they fucked up in raising you.


text: File:Eml5526.s11.team4.ccook.h2.problem4.fig1.png


title: File:Eml6341.s11.team6.hw2.9-1.jpg


title: File:Eml6341.s11.team6.hw2.9-2.jpg


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User:Yongtao-Li/HW2.10

=Problem 2.10: "Proof of Simpson's Rule(simple)"=

Given
In the interval $$ \left[ {a,b} \right] $$ there are three nodes $$ {x_0} = a, {x_1} = \frac{2},  x{_2} = b $$ satisfy that

Find
Use (10.1) to find expression for $$ \left\{ \right\} $$ in terms $$ \left( {{x_i},f({x_i})} \right) $$.

Proof
From (10.1) we can have three equations for 3 unknown $$ \left\{ \right\} $$

$$ {c_2}{x_0}^2 + {c_1}{x_0} + {c_0} = f\left( \right) $$

$$ {c_2}{x_1}^2 + {c_1}{x_1} + {c_0} = f\left( \right) $$

$$ {c_2}{x_2}^2 + {c_1}{x_2} + {c_0} = f\left( \right) $$

And then we write them in matrix form:

where

$$ \left[ K \right] = \left[ {\begin{array}{ccccccccccccccc} 1&&{x_0^2} \\ 1&&{x_1^2} \\ 1&&{x_2^2} \end{array}} \right] $$

Because the determine of the matrix K is

$$ \det \left[ K \right] = - \left( {{x_0} - {x_1}} \right)\left( {{x_0} - {x_2}} \right)\left( {{x_1} - {x_2}} \right) $$

we then compute the inverse of matrix K is

$${\left[ K \right]^{ - 1}} = \left[ {\begin{array}{ccccccccccccccc} {\frac}&{ - \frac}&{\frac} \\ { - \frac}&{\frac}&{ - \frac} \\ {\frac{1}}&{ - \frac{1}}&{\frac{1}} \end{array}} \right]\; $$

Then we can get $$ \left\{ \right\} $$ by multiple $$  {\left[ K \right]^{ - 1}} $$ on both sides of (10.2):

Consider that $$ {x_0} = a, {x_1} = \frac{2},  x{_2} = b $$ we can rewrite $$ \left\{ \right\} $$:

If integrate (10,1) on $$ \left[ {a,b} \right] $$, then

(10.9) is exactly Simpson's Rule for numerical integration.

title: User:Yongtao-Li/HW2.10


text:
User:Eml5526.s11.team3.ushnish12/Homework 2

=--Problem 2.3--=

It is given that $$\underline{\omega} = \sum_{i=1}^{n} \beta _{i}\underline{a_{i}}$$, where {$$ \underline{a_{i}} $$} is an orthonormal basis vector.

Then (1) p. 8-1 is equivalent to: $$ \underline{w} * {P} (\underline{v}) = 0 $$ $$ \forall $$ $$ \underline{w} = \Sigma_{i} \beta_{i} \underline{a}_{i} $$ $$\in \mathbb{R}^n$$

We can also write

$$ \underline{w} *\underline {P} (\underline{v}) = 0$$ $$ \forall \left[ \beta_{1}, ..., \beta_{n} \right] \in \mathbb{R}^n$$

This can be represented as:
 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle \underline{\omega} = \beta _{1}\underline{a_1}+\beta _{2}\underline{a_2}+\beta _{3}\underline{a_3}+....\beta _{n}\underline{a_n}$$
 *
 *

$$\displaystyle (Eq. 1) $$

Choice 1
 * }
 * }

Let $$\beta_1 = 1$$ and  $$\beta _2 = \beta _3=...\beta _n=0$$

Therefore, equation (1) can be written as-

$$\underline{a_1}.\underline{P}(\underline{v})=0$$

Choice 2

Let $$\beta_2 = 1$$ and  $$\beta _1 = \beta _3=...\beta _n=0$$

Therefore, equation (1) can be written as-

$$\underline{a_2}.\underline{P}(\underline{v})=0$$

.

.

Choice n

Let $$\beta_n = 1$$ and  $$\beta _1 = \beta _2=...\beta _{n-1}=0$$

Therefore, equation (1) can be written as-

$$\underline{a_n}.\underline{P}(\underline{v})=0$$

Hence we can state that :

$$ \underline{a}_{i} * \underline{P} (\underline{v}) = 0 $$        for $$ \forall_{i} = 1, ..., n $$

=--Problem 2.5--=

Problem Statement
To show that

$$ \underline{b}_{i} * \underline{P} (\underline{v}) = 0 $$ where i=1, ..., n

is equivalent to

$$ \underline{w} * \underline{P} (\underline{v}) = 0 $$ where $$ \underline{w} = \Sigma_{i} \alpha_{i} \underline{b}_{i} $$

and $$ \forall \left[ \alpha_{1}, ..., \alpha_{n} \right] \in \mathbb{R}^n$$

Solution
It is given that-
 * {| style="width:100%" border="0" align="left"

|$$\displaystyle \underline{b}_{i} * \underline{P} (\underline{v}) = 0 $$  where   i = 1, ..., n $$
 * $$\displaystyle (Eq. 1)

 * }
 * }

Choice 1

Taking i=1, we get

$$ \underline{b}_{1} * \underline{P} (\underline{v}) = 0 $$

Multiplying both sides by $$\alpha _{1}$$ , we get-

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

|$$\displaystyle \alpha _{1}\underline{b}_{1} * \underline{P} (\underline{v}) = 0 $$ $$
 * $$\displaystyle (Eq. 2)

 * }
 * }

Choice 2

Taking i=2, we get

$$ \underline{b}_{2} * \underline{P} (\underline{v}) = 0 $$

Multiplying both sides by $$\alpha _{2}$$ , we get-

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

|$$\displaystyle \alpha _{2}\underline{b}_{2} * \underline{P} (\underline{v}) = 0 $$ $$
 * $$\displaystyle (Eq. 3)

 * }
 * }

......

......

......

Choice n

Taking i=n, we get

$$ \underline{b}_{n} * \underline{P} (\underline{v}) = 0 $$

Multiplying both sides by $$\alpha _{n}$$ , we get-

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

|$$\displaystyle \alpha _{n}\underline{b}_{n} * \underline{P} (\underline{v}) = 0 $$ $$
 * $$\displaystyle (Eq. 4)

 * }
 * }

Adding Equations 2 to 4, we get

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

|$$\displaystyle(\alpha _{1}\underline{b}_{1} + \alpha _{2}\underline{b}_{2} +\alpha _{n}\underline{b}_{n}$$) $$ *   \underline{P} (\underline{v}) = 0 $$ $$
 * $$\displaystyle (Eq. 5)

 * }
 * }

or we can write Eq(5) as -

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

|$$\displaystyle \sum_{i=1}^{n} \alpha _{i}\underline{b_{i}}$$* $$ \underline{P} (\underline{v}) = 0 $$ $$
 * $$\displaystyle (Eq. 6)

 * }
 * }

Now by definition,

$$\underline{\omega} = \sum_{i=1}^{n} \alpha _{i}\underline{b_{i}}$$ where $$ \forall \left[ \alpha_{1}, ..., \alpha_{n} \right] \in \mathbb{R}^n$$ and {$$\underline{b_{i}}$$} is an orthonormal basis vector.

So Eq.(6) can be written as -

$$ \underline{w} * \underline{P} (\underline{v}) = 0 $$


text:
Survey design/Workshop/Instructor notes

2012
Orientation Week, Wed 9:30-12:30

Introductions

 * 13:30-14:00

Pt 1 - Survey research

 * 14:00-14:45

Pt 2 - Survey design

 * 15:00-15:45

Pt 3 - Survey critique

 * 16:00-16:30

2011
Orientation Week, Tuesday 10:00 - 15:00

These notes provide a rough outline of how the f2f workshop was structured and delivered in 2011.

Instructor/participants
This was the third time Jtneill had provided the workshop. Participants were typically international and domestic PhD students from a wide variety of fields, including Business and Government and Health and Medical Sciences. Students were generally in the early stages of developing PhD proposals or in the process of ethics applications, survey design and data collection.

There were 20 enrolments (full) with about 12 shows before lunch and about 7 shows after lunch (in part due to a competing session about how to complete an ethics application - this clash should be avoided if possible). In future, the class could be over-enrolled e.g., to 30 to help accommodate more students. The room has 23 computers, but could accommodate even up to 30.

Materials
Some slides were culled and reorganised for 2011 (e.g., to include levels of measurement earlier - good). But more should be culled - basically all slides after the lunch break are unnecessary for the workshop.

A single-page handout with the workshop name, instructor contact details, timeline and links and logins would be useful in future.

It would also be desirable to contact all participants approximately 1 week beforehand to provide a link to the workshop website, links to recommended readings, and a request to bring along electronic or hard copies of any draft surveys.

Introductions

 * 10:00 - 10:30
 * 1) Have participants add name tags during introductions
 * 2) Each participant to share briefly the three things listed, but the facilitator should generally ask for some educative additional/elaborative material from each participant e.g.,
 * 3) Name, Institution
 * 4) Field, Topic, Research Question, Hypotheses
 * 5) Type of survey design
 * 6) Stage of development for survey questions
 * 7) Issues, Questions, Hopes/Goals for Workshop
 * 8) Create a whiteboard map of the room each person's name and survey topic and workshop goals.

Part 1: Survey research

 * 10:30 - 11:15
 * 1) See slides

Break

 * 11:15 to 11:30

Part 2: Survey design

 * 11:30 to 13:15
 * 1) See slides

Lunch

 * 13:15 to 14:00
 * 1) An extra 15 mins would be better
 * 2) Copy one or more draft surveys for critique from class

Part 3: Survey critique

 * 14:00 to 15:00
 * 1) Distribute surveys
 * 2) 15 mins to critique
 * 3) 45 mins to discuss

Part 4: Questions/Discussion/Evaluation

 * 15:00 to 15:30

title: Survey design/Workshop/Instructor notes


text:
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text:
Paideia High School/Cosmic View/Glossary

Anopheles: The members of the Anopheles genus of mosquito which can carry the parasite which causes malaria.

Asteroid: A naturally occurring solid object, smaller than a planet, orbiting a star. See further the article in Wikipedia: Asteroid

Bacilli: A taxonomic class, within phylum Firmicutes - the aerobic rod-shaped bacteria. See further the article in Wikipedia: Bacilli

Bacterium: A single celled organism with no nucleus. Singular form of bacteria.

Comet: A celestial body consisting mainly of ice, dust and gas in an (usually very eccentric) orbit around the Sun and having a "tail" of matter blown back from it by the solar wind as it approaches the Sun. See further the article in Wikipedia: Comet

Constellation: An arbitrary formation of stars perceived as a figure or pattern; Any of the 88 officially recognized regions of the sky, including all stars and celestial bodies in the region.

Corium: Also known as dermis.

Cyclops: A one-eyed creature of any species.

Ellipse: An oval; A closed curve, the locus of a point such that the sum of the distances from that point to two other fixed points (called the foci of the ellipse) is constant; equivalently, the conic section that is the intersection of a cone with a plane that does not intersect the base of the cone. See further the article in Wikipedia: Ellipse

Epidermis: The outer, protective layer of the skin of vertebrates, covering the dermis

Exponent: The power to which a number, symbol or expression is to be raised. See further the article in Wikipedia: Exponentiation

Flagellum: Singular form of flagella. A flagellum is a tail-like projection that protrudes from the cell body of certain prokaryotic and eukaryotic cells, and functions in locomotion (the ability to move).

Galaxy: A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas dust, and an important but poorly understood component tentatively dubbed dark matter.

Helix: Something twisted or spiral.

Keratin: A family of fibrous structural proteins.

Light year: A unit of length (abbreviation ly; equal to just under 10 trillion kilometres (1016 metres)) equal to the distance light travels in one year; used to measure extremely large distances.

Malaria: A disease spread by mosquito, in which a protozoan, Plasmodium, multiplies in blood every few days. See further the article in Wikipedia: Malaria

Malpighian layer: The deepest layer of epidermis bordering the dermis.

Meteorite: A metallic or stony object or body that is the remains of a meteor.

Mite: A minute arachnid, of the order Acarina, of which there are many species; as, the cheese mite, sugar mite, harvest mite, etc. See further the article in Wikipedia: Mite

Nebula: A cloud of gas and dust in space.

Orbit: A circular or elliptical path of one object around another object.

Provincicial

Panorama: An unbroken view of an entire surrounding area. A picture or series of pictures representing a continuous scene.

Papillae: Plural for papilla, a lump shaped anatomically structure.

Proboscis: An elongated tube from the head or connected to the mouth, of an animal. The tubular feeding and sucking organ of certain invertebrates like insects, worms and molluscs.

Protrude: To extend from, above or beyond a surface or boundary; to bulge outward; to stick out.

Sinusoid: Also known as a sine wave. A mathematical function that describes a smooth repetitive oscillation. A wave that constantly maintains its wave shape.

Spherical: A sphere (from Greek σφαῖρα—sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.

Umbra: A shadow.

Vicissitude: Regular change or succession from one thing to another, or one part of a cycle to the next; alternation; mutual succession; interchange.


text:
User talk:Bmsweet

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title: User talk:Bmsweet


text:
User talk:MarkDilley

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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) 22:13, 31 January 2011 (UTC)

you have mail. John Vandenberg (chat) 20:55, 1 February 2011 (UTC)
 * thanks John!

title: User talk:MarkDilley


text:
Paideia High School/Letter from a Birmingham Jail

Letter From a Birmingham Jail

Background:

Jim Crow Laws in the segregated South

Jim Crow Laws

Starting in the 1890s, states throughout the South passed laws designed to prevent Black citizens from improving their status or achieving equality. These statutes, which together were known as Jim Crow, were in place and enforced until the 1950s and 60s. Here is a sampling of those laws, grouped by topic.

EDUCATION

Florida: The schools for white children and the schools for negro children shall be conducted separately.

Kentucky: The children of white and colored races committed to reform schools shall be kept entirely separate from each other.

Mississippi: Separate schools shall be maintained for the children of the white and colored races.

Mississippi: Separate free schools shall be established for the education of children of African descent; and it shall be unlawful for any colored child to attend any white school, or any white child to attend a colored school.

New Mexico: Separate rooms shall be provided for the teaching of pupils of African descent, and such pupils may not be admitted to the school rooms occupied and used by pupils of Caucasian or other descent.

North Carolina: School textbooks shall not be interchangeable between the white and colored schools, but shall continue to be used by the race first using them.

ENTERTAINMENT

Alabama: It shall be unlawful to conduct a restaurant or other place for the serving of food in the city, at which white and colored people are served in the same room, unless such white and colored persons are effectually separated by a solid partition extending from the floor upward to a distance of seven feet or higher, and unless a separate entrance from the street is provided.

Alabama: It shall be unlawful for a negro and white person to play together or in company with each other at any game of pool or billiards.

Alabama: Every employer of white or negro males shall provide for such white or negro males reasonably accessible and separate toilet facilities.

Georgia: All persons licensed to conduct a restaurant, shall serve either white people exclusively or colored people exclusively and shall not sell to the two races within the same room or under the same license.

Georgia: It shall be unlawful for any amateur white baseball team to play on any vacant lot or baseball diamond within two blocks of a playground devoted to the Negro race, and it shall be unlawful for any amateur colored baseball team to play baseball within two blocks of any playground devoted to the white race.

Georgia: All persons licensed to conduct the business of selling beer or wine...shall serve either white people exclusively or colored people exclusively and shall not sell to the two races within the same room at any time.

Louisiana: All circuses, shows, and tent exhibitions, to which the attendance of more than one race is invited shall provide not less than two ticket offices and not less than two entrances.

Virginia: Any public hall, theatre, opera house, motion picture show or place of public entertainment which is attended by both white and colored persons shall separate the white race and the colored race.

FREEDOM OF SPEECH

Mississippi: Any person guilty of printing, publishing or circulating matter urging or presenting arguments in favor of social equality or of intermarriage between whites and negroes, shall be guilty of a misdemeanor.

HEALTH CARE

Alabama: No person or corporation shall require any white female nurse to nurse in wards or rooms in hospitals, either public or private, in which negro men are placed.

Louisiana: The board of trustees shall maintain a separate building, on separate grounds, for the admission, care, instruction, and support of all blind persons of the colored or black race.

Mississippi: There shall be maintained by the governing authorities of every hospital maintained by the state for treatment of white and colored patients separate entrances for white and colored patients and visitors, and such entrances shall be used by the race only for which they are prepared.

HOUSING

Louisiana: Any person...who shall rent any part of any such building to a negro person or a negro family when such building is already in whole or in part in occupancy by a white person or white family shall be guilty of a misdemeanor.

Mississippi: The prison warden shall see that the white convicts shall have separate apartments for both eating and sleeping from the negro convicts.

LIBRARIES

Texas: Negroes are to be served through a separate branch or branches of the county free library, which shall be administered by a custodian of the negro race under the supervision of the county librarian.

North Carolina: The state librarian is directed to fit up and maintain a separate place for the use of the colored people who may come to the library for the purpose of reading books or periodicals.

MARRIAGE

Arizona: The marriage of a person of Caucasian blood with a Negro shall be null and void.

Florida: All marriages between a white person and a negro, or between a white person and a person of negro descent to the fourth generation inclusive, are hereby forever prohibited.

Florida: Any negro man and white woman, or any white man and negro woman, who are not married to each other, who habitually live in and occupy in the nighttime the same room, shall each be punished by imprisonment not exceeding 12 months, or by fine not exceeding five hundred dollars.

Maryland: All marriages between a white person and a negro, or between a white person and a person of negro descent, to the third generation, inclusive…are forever prohibited, and shall be void.

Mississippi: The marriage of a white person with a negro or mulatto or person who shall have one-eighth or more of negro blood, shall be unlawful and void.

Wyoming: All marriages of white persons with Negroes, Mulattos, Mongolians, or Malaya hereafter contracted in the State of Wyoming are, and shall be, illegal and void.

SERVICES

Georgia: No colored barber shall serve as a barber to white women or girls.

Georgia: The officer in charge shall not bury, or allow to be buried, any colored persons upon ground set apart or used for the burial of white persons.

TRANSPORTATION

Alabama: All passenger stations in this state operated by any motor transportation company shall have separate waiting rooms or space and separate ticket windows for the white and colored races.

Alabama: The conductor of each passenger train is authorized and required to assign each passenger to the car or the division of the car, when it is divided by a partition, designated for the race to which such passenger belongs. Maryland: All railroad companies are hereby required to provide separate cars or coaches for the travel and transportation of the white and colored passengers.

WORK

Oklahoma: The baths and lockers for the negroes shall be separate from the white race, but may be in the same building. (Mining companies)

For more Jim Crow laws, visit the Martin Luther King, Jr., National Historic Site. Downloaded from: http://americanradioworks.publicradio.org/features/remembering/laws.html 1/31/11

Other links and general information: http://americanradioworks.publicradio.org/features/remembering/danger.html


text:
Paideia High School/Der Erlkönig

Exordium
WHO rides there so late through the night dark and drear? The father it is, with his infant so dear; He holdeth the boy tightly clasp'd in his arm, He holdeth him safely, he keepeth him warm.

"My son, wherefore seek'st thou thy face thus to hide?" "Look, father, the Erl-King is close by our side! Dost see not the Erl-King, with crown and with train?" "My son, 'tis the mist rising over the plain."

"Oh, come, thou dear infant! oh come thou with me! Full many a game I will play there with thee; On my strand, lovely flowers their blossoms unfold, My mother shall grace thee with garments of gold."

"My father, my father, and dost thou not hear The words that the Erl-King now breathes in mine ear?" "Be calm, dearest child, 'tis thy fancy deceives; 'Tis the sad wind that sighs through the withering leaves."

"Wilt go, then, dear infant, wilt go with me there? My daughters shall tend thee with sisterly care My daughters by night their glad festival keep, They'll dance thee, and rock thee, and sing thee to sleep."

"My father, my father, and dost thou not see, How the Erl-King his daughters has brought here for me?" "My darling, my darling, I see it aright, 'Tis the aged grey willows deceiving thy sight."

"I love thee, I'm charm'd by thy beauty, dear boy! And if thou'rt unwilling, then force I'll employ." "My father, my father, he seizes me fast, Full sorely the Erl-King has hurt me at last."

The father now gallops, with terror half wild, He grasps in his arms the poor shuddering child; He reaches his courtyard with toil and with dread,-- The child in his arms finds he motionless, dead.

  Ballady Göthovy/Král duchů Goetheho vybrané básně/Král kouzelný Erlkönig Le Roi des Aulnes Keijuinkuningas Elzenkoning Król Olszyn Лесной царь (Гёте/Жуковский) Лесной царь (Гёте/Фет) Лесной царь (Гёте/Григорьев)

Interpretation
Interpretation content and notes here.

Erudition
The Alder King

Der Erlkönig

title: Paideia High School/Der Erlkönig


text:
Explicit binary remainder matrices

Research Project: SQUARE NON-SINGULAR MATRICES CONTAINING BINARY REMAINDERS AS ELEMENTS ASSOCIATED WITH TWO RAISED TO THE APPROPRIATE BINARY PLACE VALUE. The Division Algorithm with divisor two [2] is usually written: Q[n, N] = 2.Q[N - 1] + r{N - 1]   Equation 1. Q[n, N] is an uneven integer defined by Q[n, N] = 2.n + 1 where n = 1, 2, ... and Q[N] is identifiable with Q[N, n]. The quotient is an uneven integer, provided that the remainder is given by: r[N - 1] = [i]**{1 + Q[n, N]} The inequalities: 2**{N} < Q[n, N] < 2**{N + 1} are required to be satisfied by Q[n, N], thereby setting a lower and upper limit on [n]. The least value of Q[n, N] is [2**{N} + 1], for example, from which n[minimum] can be found. The greatest value of Q[N, n] is [2^{N + 1} - 1] from which n[maximum] can be found. [noting that the symbols: ** and: ^ both mean the same thing, exponentiation] In the context, Q[N] can be identified with Q[n, N] If Equation 1. is rearranged to: Q[N - 1] = 1/2[Q[n, N] - [i]**{1 + Q[n, N]}]  Equation 2. and iterated:       Q[N - 2] = 1/2[Q[N - 1] - [i]**{1 + Q[N - 1]}] ..............................................                        Q[1] = 2.Q[0] + r[0]  This is the final equation. that is needed to produce square matrices. There are [N + 1] uneven integers from the iteration above: Q[n, N], Q[N - 1], Q[N - 2], .... Q[1], Q[0] and if these are set explicitly in binary, then they fall naturally into a square non-singular matrix. In explicit binary: Q[n, N] = b[N].2**{N} + b[N - 1].2**{N - 1] + ..2**{k}.b[k] .. + b[0] Q[N - 1] =    0             b[N].2**{N - 1} + b[N - 1].2**{N - 2}        + b[0] ................................................................................         Q[0] =     0                 0               0                        + b[0]

Calling the matrix of binary digits B[n, N, [N + 1]], set the elements: b[1, 1] = 2**{N}, b[1, 2] = b[N - 1].2**{N - 1} ... b[1, N], b[1, N] = b[1].2**{1}, [N + 1]] = b[0] b[2, 1] = 0, b[2, 2] = b[N].2**{N - 1}, ... b[2, N] = b[2].2**{1}, b[2, [N + 1]] = b[0] the pattern continues to give the final row: bN + 1], 1] = 0, ...b[[N + 1], 2] = 0, b[[N + 1], N] = 0 b[[N + 1], [N + 1 = b[0]. EXAMPLE: Q[12, 4] = 53, n = 26, N = 5, and in explicit binary is: 32 + 16 + 0 + 4 + 0 + 1 The matrix of binary digits is:   B[26, 5, 6] is: 32      16      0      4        0       1                   0      16      8      0        2       1                   0       0      8      4        0       1                   0       0      0      4        2       1                   0       0      0      0        2       1                   0       0      0      0        0       1

Since Q[n, N] is uneven, b[0] = 1 and b[N] = 1 is the most significant digit. Substituting: r[N - k] = 2.b[k] - 1 into the expression for Q[n, N] gives Q[n, N] in terms of its remainders. The remainders r[N] = 1 and r[0] = 1, because of the corresponding values of b[0]and b[N]. The binary digit b[0] = 1, the least significant digit, because Q[N] is uneven and b[N] = 1 and is the most significant binary digit. Q[n, N] = 2**{N} + 2**{N - 1} + r[1].2**{N - 2} + r[2].2**{N - 3} + ... + r[N - 1] In the above example, r[1], r[2], r[3], and r[4] are required. r[4] = [i]**{1 + Q[27, 5]}, r[3] = [i]**{1 + Q[4]}, r[2] = [1]**{1 + Q[3]}, r[1] = 3. Subtituting these values into Q[27, 5] = 53 in the above equation in terms of remainders produces the first row of D[n, N, N + 1], the dual matrix of remainders, having [N + 1] rows and [N + 1] columns. There are no elements of value zero in this matrix of remainders. The remaining elements can be inferred from the pattern of simple examples.

In the above example: D[26, 5, 6] is: 32       16        8      - 4        2     - 1          32      - 16        8        4      - 2       1          32      - 16      - 8        4        2     - 1          32      - 16      - 8      - 4        2       1          32      - 16      - 8      - 4      - 2       1          32      - 16      - 8      - 4      - 2     - 1

The algebraic sums of the rows of D[26, 5, 6] are found to correspond to Q[27, 5], Q[4], Q[3], Q[2], Q[1], Q[0]. There are general expressions for the elements of D[n, N, [N + 1]], that is dual to B[n, N]. Each matrix D has been found to be non-singular in simple actual instances and is non-singular in all cases, as can be easily proved. Each uneven integer Q[n, N] is uniquely represented by a matrix B[n, N] and its dual matrix D[n, N, [N + 1]]. The subscript: [N + 1] that is the order of a square matrix can be discarded, since the order is always: [N + 1] PROOF OF NON-SINGULARITY OF D[N, n]. In the example above, D[5, 26] that represents Q[5, 26] which is the integer 53, D[5, 26] is seen to be non-singular by the following process on the determinant: |D[5, 26]|: Row 1 of the determinant of D is left unaltered and written down. Row 1 is multiplied by - 1 and added to row 2 of the original determinant and this new row 2 is written down. The original row 2 is multiplied by - 1 and added to row 3 of the original determinant so producing the new row 3 which ia again written down. This is continued until the original row 4, multiplied by - 1 is added to the final row which is row 6. The new determinant is:

|D[5,26]| := 32        16        8       - 4        2        - 1 0     - 32        0         8      - 4          2                   0         0     - 16         0        4        - 2                   0         0        0       - 8        0          2                   0         0        0         0      - 4          0                   0         0        0         0        0        - 2

This determinant has evidently a non-zero value and therefore the corresponding original matrix: D[5, 26] is non-singular.

THE GENERAL CASE: D[N, n] The elements of D[N, n] are given by the consecutive terms of the consecutive equations that represent in sequence: Q[N, n], Q[N - 1], ... Q[0] in terms of the binary remainders: r[N], r[N - 1], ... r[1], r[0]. As stated above, r[0] = 1 and r[N] = 1. The remainders: r[N - 1], ... r[1] have values of either + 1 or - 1 with 2^{N - 1} variations. The equations are: Q[N] = 2^{N} + 2^{N - 1} + r[1].2^{N - 2} + r[2].2^{N - 3} + ... + r[N - 1] Q[N - 1] = 2^{N} - 2^{N - 1}    +  2^{N - 2} + r[1].2^{N - 3} + r[2].2^{N - 4} + ... + r[N - 2] .....   Q[0] = 2^{N} - 2^{N - 1} - 2^{N - 2} ... -                                               - 1

There [N + 1] equations, above, having each [N + 1] terms, producing the [N + 1] row and [N + 1] column matrix: D[N, n]. The addition of the negative of an above row to particular row gives a determinant having non-zero elements on the principal diagonal and a triangle of zeros below the principal diagonal. THE SIMPLIFIED EIGENVALUE DETERMINANT: |D[N, n, x]| The primitive starting eigenvalue determinant is processed as above in the proof of non-singularity to give a much simpler form. Two examples may serve to indicate the structure of the simplified determinant of D. Example [i] For Q[1, 1]] = 3, |D[1, 1, x]| is: |[2 - x]      1   | |  2     [- 1 - x]| The above is the form of the primitive determinant in [x] of the matrix D[1, 1] Leaving row 1 unchanged but adding - 1 times row 1 to row 2 gives the simplified determinant [but not really simpler in the case of Q[1, 1] = 3], but simpler for all successive instances.

|D[1, 1, x]| becomes: |[2 - x]          1   | |  x        [- 2 - x]|

Example [ii] For Q[4, 15] = 31, row 1 is left unaltered but every successive row down to the last but one is multiplied by - 1 and added to the row below that row. The primitive |D[4, 15, x]| is:   [16 - x]            8            4           2          1 16        [- 8 - x]          4           2          1 16            - 8      [- 4 - x]         2          1 16            - 8          - 4      [- 2 - x]       1 16            - 8          - 4         - 2     [- 1 - x]

Using the process to prove non-singularity to the primitive determinant above gives the simpler determinant: [16 - x]         8              4            2           1 x       [- 16 - x]          0            0           0 0            x         [- 8 - x]         0           0 0            0              x       [- 4 - x]        0 0            0              0            x       [- 2 - x]

The elements of the simplified determinant below the non-principal diagonal comprising x's are all zeros for every case where Q > 3 and in general the elements above the principal diagonal but below the first row are not necessarily zero.

The matter of interest in the first instance here is to compute the amplitudes and arguments of the eigenvalues and analyse the results for Q = [1], 3, 5, 7, 9, ... so as to gain an insight into prime numbers and the prime factors of non-prime uneven integers. 2.99.204.18 22:38, 31 January 2011 (UTC)SHAWWPG19410425 (talk) 18:27, 16 June 2012 (UTC)

title: Explicit binary remainder matrices


text:
University of Canberra/RCC2011/Let's reverse-brainstorm: Why wikis don't work

 * 1) people have different goals & in a wiki people have to collaborate & they do it better on their own
 * 2) too hard to use (info architecture, structuring)
 * 3) we don’t have the skills
 * 4) ugly
 * 5) Unreliable technology
 * 6) People won’t agree
 * 7) it is public, the permanency frightens people & functions
 * 8) WE have enough info out there (google)
 * 9) too much time, too busy, sb else can do it
 * 10) wikis are messy
 * 11) what’s the point if it gets changed anyway
 * 12) lack of facilitation
 * 13) people want control over content, they want power
 * 14) you challenge people’s world view in a wiki
 * 15) Stepping backwards (ie. categories, they are really sophisticated taxonomies out there)
 * 16) Tech is poor
 * 17) lack of good tagging
 * 18) readability
 * 19) vandalism
 * 20) wordpress is much easier

What is a wiki?
Core features to a wiki: participation – you meet sb via an edit, complete history about the changes => accountability, creates a sense of ownership
 * 1) primacy of authorship (people want their own spaces)
 * 2) Solo-isolation of content
 * 3) People don’t get it, wikis are full of pdfs
 * 4) You can’t see how it works compared to blogs, twitter. You instantly see how it works
 * 5) lack of trust (in content) & legitimacy in large orgs
 * 6) fear

Why do wikis work?

 * 1) creative process
 * 2) Once sb gets over fear, people start participating
 * 3) Trainspotting phenomenon
 * 4) Wikis harness accuracy
 * 5) low entry barrier, 2 secs to edit
 * 6) I can create a structure that I want

Sidetrack Wikipedia: Why Wikipedia is going to fail?
The correct question is if it is going to fail before it is fixed?
 * 1) Achievements: WP build a comprehensive database about academic journals
 * 2) Since wp, I am frustrated that I can’t edit Sydney Morning Herald online
 * 3) Success of WP was good PR


text:
User:EML5526.S11.Team3.vnarayanan/Homework Problems/Problem 2.7/Solution/2.7.1

$$ \underline{\Gamma}(F) = \begin{bmatrix}

1 &\frac{1}{2} & \frac{1}{3}  & \frac{1}{4}  &\frac{1}{5} \\

\frac{1}{2} & \frac{1}{3} & \frac{1}{4}  & \frac{1}{5}  & \frac{1}{6} \\

\frac{1}{3} &\frac{1}{4} &\frac{1}{5}  &\frac{1}{6}  &\frac{1}{7} \\

\frac{1}{4} & \frac{1}{5} &\frac{1}{6} &\frac{1}{7}  &\frac{1}{8} \\

\frac{1}{5} &\frac{1}{6} &\frac{1}{7}  &\frac{1}{8}  &\frac{1}{9} \end{bmatrix}$$


text:
University of Canberra/RCC2011/WikiCulture

Discussion of the culture at Wikipedia.

There is also a core of people at Wikipedia that make new people welcome.

Important to develop the culture online: purpose, sharing

Start developing a certain culture at wikipedia.

What is the scope of Wikipedia?

Is it about the quality of the articles?

Is there an umbrella wiki culture out there?

There needs to be a culture of forking.

Example: open source software

We don't need wikipedia for a neutral voice. We need them to give us a perspetive on a topic.

Defining scope and principles are important.


text:
University of Canberra/RCC2011/All about Appropedia

Few rules, let it emerge

Extending the likes of Wikipedia. WP has no How-To, so Apropedia does that.

http://www.appropedia.org/

Appropedia channel on YouTube

A YouTube video about Appropedia; not part of this session, but good background material.


text:
User:Yongtao-Li/FE-HW2.9

=Problem 2.9: "Example of WRF"

Given
General one dimensional Model 1.0/Data set 2 on P12-1:

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

Consider $$ {b_j}\left( x \right) = \cos \left( {jx + \phi } \right) $$ and $$ \phi = \frac{1}{4}\pi $$ or $$ \phi  = \frac{3}{4}\pi $$

Find
A) Let n=2, What is the ndof of this problem?

B) Find 2 equations that enforce boundary conditions for $$ {u^h}\left( x \right) = \sum\limits_{j = 0}^n $$

C) Find 1 more equation to solve for $$ {\mathbf{d}} = {\left\{ \right\}_{3 \times 1}} $$ by projecting the residue $$ {\mathbf{P}}\left(  \right) $$ on a basis function $$ {b_k}(x)   with  k = 0,1,2  $$ s.t. the additional equation is linear independence from the above 2 equations

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 $$ u_n^h\left( x \right) \ vs. \ u\left( x \right) $$ ($$ u\left( x \right) $$ is the exact solution)

G) Repeat (1) to (6) for n=4 and n=6

H) Compare $$ u_n^h\left( {x = 0.5} \right)forn = 2,4,6 $$. 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) Because n begins from 0, so the degree of freedom is 3 when n=2.

B) From (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 (9.3) 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

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

E)-H) In order to simplify the computation procedure, the following Fortran code based on (9.8), (9.9) and (9.10) was used to calculate the results, when n=2,4,6 and $$\phi = \frac{1}{4}\pi,\frac{3}{4}\pi$$:

program main use lib implicit none integer :: n,i,j,k,m,l real,allocatable :: KK,d,F,b real :: fi(2),pi,x(1001),y(1001),yh(1001),e(3,2) n=6; pi=3.1415926; fi(1)=pi/4.0; fi(2)=pi*3.0/4.0 open (unit=11, file="egm6611.s11.team6.hw2.9.1.plt", status="replace") open (unit=12, file="egm6611.s11.team6.hw2.9.2.plt", status="replace") open (unit=13, file="egm6611.s11.team6.hw2.9.3.plt", status="replace") do l=2,n,2 write(*,*) "When the degree of freedom is ", l+1 do i=1,2 allocate(KK(l+1,l+1),d(l+1),F(l+1),b(2,l+1)) do j=0,l b(1,j+1)=cos(j*1.0+fi(i)) b(2,j+1)=-1.0*j*sin(j*0.0+fi(i)) end do     do k=0,l do j=0,l if (k==j) then KK(k+1,j+1)=-1.0*k/2.0*(2.0*k-sin(2*fi(i))+sin(2*fi(i)+2*k)) else KK(k+1,j+1)=2*j*j*j*(cos(fi(i))*sin(fi(i))-cos(fi(i)+k)*sin(fi(i)+j))-k*(cos(fi(i))*sin(fi(i))-cos(fi(i)+j)*sin(fi(i)+k)))/(j*j-k*k)           end if            if (k==0) then               F(k+1)=-3.0*cos(fi(i))            else               F(k+1)=-3.0*(sin(fi(i)+k)-sin(fi(i)))/k            end if         end do      end do      write(*,*) "when phi= ",fi(i)      do j=1,l+1      KK(1,j)=b(1,j)      KK(2,j)=b(2,j)      end do      F(1)=0     F(2)=-4     call invert(KK)     do k=1,l+1        d(k)=0.0        do j=1,l+1           d(k)=d(k)+KK(K,j)*F(j)        end do     end do     write(*,*) "solution vector d"     write(*,*) d     x=0.0; y=0.0; yh=0.0     do m=1,1001     x(m)=(m-1)/1000.0     y(m)=-3.0/4.0*x(m)*x(m)-4.0*x(m)+19.0/4.0     do k=1,l+1        yh(m)=yh(m)+d(k)*cos((k-1)*x(m)+fi(i))     end do   end do   do k=1,l+1      e(l/2,i)=e(l/2,i)+d(k)*cos((k-1)*0.5+fi(i)) end do  e(l/2,i)=(-3.0/4.0*0.5*0.5-4.0*0.5+19.0/4.0)-e(l/2,i) if (i==1) then write(11,*) "zone" do m=1,1001 write(11,"(3F12.4)") x(m),y(m),yh(m) end do  else write(12,*) "zone" do m=1,1001 write(12,"(3F12.4)") x(m),y(m),yh(m) end do  end if   deallocate(KK,d,F,b) end do !for i=1-2 end do !for n=1-8 write(13,"(A,2F12.4)") "2",e(1,1),e(1,2) write(13,"(A,2F12.4)") "4",e(2,1),e(2,2) write(13,"(A,2F12.4)") "6",e(3,1),e(3,2) end program

module lib implicit none contains !  subroutine invert(matrix) ! invert a small square matrix onto itself implicit none real,intent(in out)::matrix integer::i,k,n; real::con ; n= ubound(matrix,1) do k=1,n con=matrix(k,k); matrix(k,k)=1. matrix(k,:)=matrix(k,:)/con do i=1,n if(i/=k) then con=matrix(i,k); matrix(i,k)=0.0 matrix(i,:)=matrix(i,:) - matrix(k,:)*con end if     end do   end do   return end subroutine invert end module lib

Run this code and then get the output result for all the situations as following:

When the degree of freedom is           3 when phi=  0.7853981 solution vector d     1.142170       7.209867     -0.7765067 when phi=   2.356194 solution vector d    -17.24394       13.96153      -4.152339 When the degree of freedom is           5 when phi=  0.7853981 solution vector d    -2.044705       13.62609      -7.081253       2.811028     -0.5599513 when phi=   2.356194 solution vector d    -16.73928       16.00889      -7.965427       2.459494     -0.4499143 When the degree of freedom is           7 when phi=  0.7853981 solution vector d    -2.067435       14.06402      -8.531017       4.714233      -1.883228 0.4795609    -5.8918476E-02 when phi=   2.356194 solution vector d    -19.44370       26.57353      -25.56540       18.85248      -9.806686 3.187393    -0.5081997

For n=2,4,6, $$ \phi = \frac{1}{4}\pi $$, we compare the results with exact solution in the following image:

For n=2,4,6, $$ \phi = \frac{3}{4}\pi $$, we compare the results with exact solution in the following image:

In both the images, we can observe that the results become more close compared to exact solution when n increases.

For different shift phases, we compare the how the error's convergence situation in the following image:

It seems that both the shift phase can make a convergence around n=6.

title: User:Yongtao-Li/FE-HW2.9


text:
University of Canberra/RCC2011/All about Wikitravel

The wikiTravel session was attended by several people including Mark Dilley, Laura Hale, John Vandenberg, Jutta, Jani Patokallio and Ken Hoetmer.

wikiTravel is owned by a used car dealership. They are not focused on updating the existing software, developing improvements or building community. They care that they make money on the Google ads. This means they have little stake in the development of the community on the wiki.

This session ended up becoming a question of could wikiTravel fork and asking a broader question of what is needed to successfully fork.

One opinion in this session was that you needed four successful components to fork: A tech person, a person to network and rally the community around the need to move, a person to handle promotion and SEO, and another person to help with content development at the new location.

One of the forks that was looked at was wikiFur. It successfully left Wikia. The issue with wikiFur is that they actually controlled the domain. Wikis like Uncyclopedia cannot easily fork because they do not control the domain. In the case of wikiTravel, forking would also be difficult because they don't control the domain or the trademark. The community would need to effectively rebrand. This is possible as the German and Italians have created a successful fork of wikiTravel in their own language, called Wikivoyage. (see here for their account of this fork)

We tried to come up with examples of successful forks. One successful fork mentioned was Conservapedia.

wikiHow was discussed. The founder, Jack Herrick, has told his community that they have a right to fork if they want to.

A lot of people don't fork because they are scared to do so. Success is clearly defined and wrapped up in their current project. Leaving is very scary.

There are a lot of complications on the technical end with trying to fork, such as bring the history of all the articles with it the history of the chat. It would probably take a good technical person three weeks to figure out and implement a technical solution to moving all the content over and trying to get the user database possibly moved over and trying to address the issue such as media needing to be copied over.


text:
University of Canberra/RCC2011/Wikimedia Australia session

A few items briefly touched on during this session were confidential.

Attendees: John Vandenberg, Bilby, LauraHale, SatuSuro, Jutta, .. ?

General things discussed included:

 * How Wikimedia Australia determines what projects to fund;
 * process and existing proposals
 * The structure of Wikimedia Australia's organization;
 * Wikimedia Australia's fundraising efforts;
 * Running Wiki Academies;
 * How to address the needs of diverse populations in Australia's wiki, collaborative learning, history, educational communities.

Beyond that, there was the discussion of Wikimedia Australia's role in helping to run Recent Changes Camp, why they chose to be involved and how they will be involved going forward. Their role in future wiki conferences was also discussed.

The scope of Wikimedia Australia's mission was also clarified and an explanation was provided as to how this differed from the goals and scope of Wikimedia Foundation.


text:
University of Canberra/RCC2011/Going forward: Future wiki conferences

RecentChangesCamp 2011 Canberra closed with a discussion on what we will do going forward as a result of this, how we were empowered and how we can share our learning. This involved discussing several ideas:

 * Trying to have more Wiki Academies around Australia;
 * Having a one day RecentChangesCamp in New Zealand, possibly in June or July of 2011. The focus would be on attracting a group more interested in research than in education, but all people interested in wikis would still invited. (The premise was also that RCC in Australia would be more education oriented.  The USA would be more commercial wiki users oriented.) The goal would be to build the infrastructure community network wise so that a three day RecentChangesCamp could be held in New Zealand in 2012.  People at the conference thought Wellington would be a good location to have it as it is centrally located.  It would not take more than three weeks to organize and promote once a venue was found.  Laura Hale volunteered to facilitate in order to insure continuity.
 * A discussion was had about trying to reach out to MPs to help push the idea of using collaborative practices as part of open governance. This might be realized either through a formal conference or possibly through Wiki Academies.
 * A conference proposal was put forth about having a standard academic wiki conference with a focus on doing research on best practices on wikis. Methodology would be really important.  Paper presentation would be open to everyone and it would be structured more like a post-graduate conference.  People would have the opportunity to have their research subjected to a double blind peer-review.  This conference would likely be held in Adelaide as the proposal was put forward by some one from Adelaide.  It would also introduce people to ethics review. An attempt could be made to try the peer reviewed practice on Wikiversity as a test to how feasible this was.  If that doesn't occur on Wikiversity, it could be tried on a locked wiki.  The conference would be about Critical improvement, not criticism.
 * An event in Newcastle around This Is Not Art-festival. Possibly a one day RCC type event, as many of the people who attend are very creative who might be extremely willing to engage in the wiki community if they are approached.

Beyond the planning of events such as these, we discussed how we were all ambassadors for wikis and the collaborative learning community. When we returned home, we were encouraged to continue our activism and to share what we learned.


title: File:Fe1.s11.team1.hw2.fig1.1.jpg


title: File:Fe1.s11.team1.hw2.fig1.2.jpg


text:
University of Canberra/RCC2011/Wikis for fun

This session involved a lot of drawing on the walls in order to talk about wiki structure. The video linked below provides a summary of what was discussed in that session.


title: File:Fe1.s11.team1.hw2.fig4.1.jpg


title: File:Fe1.s11.team1.hw2.fig9.2.jpg


title: File:Fe1.s11.team1.HW2.fig9.1.jpg


title: File:Fe1.s11.team1.HW2.fig9.4.jpg


text:
User:Eml5526.sp11.team1.adam/HW2

=Homework 2=

=Problem 7= Consider the family $$F=\begin{bmatrix}x,x^2,x^3,x^4 \end{bmatrix} $$

Is $$ F $$ an othogonal family evaluated at $$\Omega=\begin{bmatrix}0,1 \end{bmatrix} $$?

From Lecture Slide Mtg.10

Solution
Let's start by calculating $$\Gamma (F)$$

$$(F)=\int_{0}^{1} \begin{bmatrix} 1,x,x^2,x^3,x^4\\ x,x^2,x^3,x^4,x^5\\ x^2,x^3,x^4,x^5,x^6\\ x^3,x^4,x^5,x^6,x^7\\ x^4,x^5,x^6,x^7,x^8\\ \end{bmatrix}dx $$

Performing the integration we ogtain:

$$\Gamma(F)= \begin{bmatrix} x,\frac{x^2}{2},\frac{x^3}{3},\frac{x^4}{4},\frac{x^5}{5}\\ \frac{x^2}{2},\frac{x^3}{3},\frac{x^4}{4},\frac{x^5}{5},\frac{x^6}{6}\\ \frac{x^3}{3},\frac{x^4}{4},\frac{x^5}{5},\frac{x^6}{6},\frac{x^7}{7}\\ \frac{x^4}{4},\frac{x^5}{5},\frac{x^6}{6},\frac{x^7}{7}\frac{x^8}{8}\\ \frac{x^5}{5},\frac{x^6}{6},\frac{x^7}{7}\frac{x^8}{8},\frac{x^9}{9}\\ \end{bmatrix}$$

When the latter equation is evaluated at (0,1) we obtain:

$$\Gamma(F)= \begin{bmatrix} 1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}\\ \frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6}\\ \frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7}\\ \frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7}\frac{1}{8}\\ \frac{1}{5},\frac{1}{6},\frac{1}{7}\frac{1}{8},\frac{1}{9}\\ \end{bmatrix}$$

Where in can clearly be observed that $$\Gamma(F)$$ is not orthogonal

Contributing Members
=Problem 8= Show that equation (1) and equation (2) on page 4 Meeting 10 are equilavent.

From Lecture Slide Mtg.11

Given
Let's define our equations:

Equation (1) on page 4 Meeting 10

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

$$\int_{\Omega }{}w^{h}(x)P(U^{h}(x))dx=0;\forall w^{h}(x)$$
 * $$\displaystyle (Eq.8.1) $$
 * }
 * }

and Equation (2) on page 4 Meeting 10

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

$$\int_{\Omega}{} b_{i}(x)P(U^{h}(x))dx=0 $$
 * $$\displaystyle (Eq.8.2) $$
 * }
 * }

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

$$w^{h}(x)=\sum_{i=1}^{n}c_{i}b_{i}(x)$$
 * $$\displaystyle (Eq.8.3) $$
 * }
 * }

Solution
If we chose $$c_{i}$$ to be orthogonal, we have:

$$c_{1}=\begin{bmatrix}1,0,...,0 \end{bmatrix} $$

$$c_{2}=\begin{bmatrix}0,1,...,0 \end{bmatrix} $$

$$c_{3}=\begin{bmatrix}0,0,...,1 \end{bmatrix} $$

When we multiply $$c_{i}$$ times $$b_{i}$$ we observe that multiply by $$c$$ is equivalent to multiplying by the identity matrix, therefore replacing $$w^{f}(x)$$ in $$eq. 8.1$$ we have:

$$\int_{\Omega}^{ }w^{h}(x)P(U^{h}(x))dx=\int_{\Omega}^{ }\sum_{i=1}^{n}c_{i}b_{i}(x)P(U^{h}(x))dx=\int_{\Omega}^{ }I.b.P(U^{h}(x))dx=\int_{\Omega}^{ }b_{i}P(U^{h}(x))dx$$

title: User:Eml5526.sp11.team1.adam/HW2


title: File:Nm1.s11.team5.HW2.fig12a.png


text:
User:EML5526.S11.Team1.HS/HW2

=HW2=

Problem 1
Derive heat problem equation from Lecture Slide

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

$$ \frac{\partial}{\partial x}\left(A \left(x\right) k\left(x\right) \frac{\partial u}{\partial x}\right) + f \left(x,t \right) = A \left(x \right) \rho c \frac{\partial u}{\partial t} $$
 * $$\displaystyle (Eq. 2.1) $$
 * }
 * }

Solution
Fig. 1 Shows a bar of cross section A(x), with conductivity k(x), density ρ, specific heat c, with an internal heat source (or sink) f(x,t). In the bar the heat is transferred only by conduction in x direction. The temperature has been called u(x,t)

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

The partial differential equation for this problem can be obtained doing an energy balance in an infinitesimal element of the bar. Fig. 2 shows an infinitesimal element of length dx; heat entering the body on the left hand side is $$q\left(x \right)$$ and heat going out the body on the right hand side is $$q \left(x + dx \right)$$

Doing the energy balance we obtain:
 * {| style="width:100%" border="0"

$$ q \left(x \right) A \left(x \right) - q \left(x + dx \right) A \left(x+dx \right) + fdx = \frac{dE}{dt}$$
 * $$\displaystyle (Eq. 2.2) $$
 * }
 * }

Where $$ \frac{dE}{dt}$$ is the change of energy with respect to time which can be calculated in terms of the temperature as:
 * {| style="width:100%" border="0"

$$ \frac{dE}{dt} = A d x \rho c \frac{\partial u}{\partial t} $$
 * $$\displaystyle (Eq. 2.3) $$
 * }
 * }

The heat flow and the cross area at the right hand side of the element can be approximated by:
 * {| style="width:100%" border="0"

$$ q \left(x + dx \right) = q \left(x \right) + \frac{\partial q}{\partial x} dx $$
 * $$\displaystyle (Eq. 2.4) $$
 * }
 * }

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

$$ A \left(x + dx \right) = A \left(x \right) + \frac{\partial A}{\partial x} dx $$
 * $$\displaystyle (Eq. 2.5) $$
 * }
 * }

Replacing on Eq.2.2 we obtain:
 * {| style="width:100%" border="0"

$$ qA - \left(q + \frac{\partial q}{\partial x}dx \right)\left(A + \frac{\partial A}{\partial x} dx \right) + fdx = A d x \rho c \frac{\partial u}{\partial x} $$
 * $$\displaystyle (Eq. 2.6) $$
 * }
 * }

Doing the product we get:
 * {| style="width:100%" border="0"

$$ qA - qA - q \frac{\partial A}{\partial x} dx - A \frac{\partial q}{\partial x} dx - \frac{\partial A}{\partial x} dx \frac{\partial q}{\partial x} dx + fdx = A d x \rho c \frac{\partial u}{\partial x} $$
 * $$\displaystyle (Eq. 2.7) $$
 * }
 * }

Neglecting the second order term we have:
 * {| style="width:100%" border="0"

$$ -q \frac{\partial A}{\partial x} dx - A \frac{\partial q}{\partial x} dx + fdx =Adx \rho c \frac{\partial u}{\partial x}$$
 * $$\displaystyle (Eq. 2.8) $$
 * }
 * }

The first two terms in the latter equation are the derivative of a product; therefore:
 * {| style="width:100%" border="0"

$$ - \frac{\partial}{\partial x} \left(Aq \right) dx + fdx = Adx \rho c \frac{\partial u}{\partial x} $$
 * $$\displaystyle (Eq. 2.9) $$
 * }
 * }

Using the Fourier’s law, the heat flow can be expressed as function of temperature. For an unidirectional case we have:
 * {| style="width:100%" border="0"

$$ q\left (x \right) = -k \left(x \right) \frac{\partial u}{\partial x} $$
 * $$\displaystyle (Eq. 2.10) $$
 * }
 * }

Where: k(x) is the conductivity of the material. Replacing equation Eq.2.9 in Eq.2.10 we have:
 * {| style="width:100%" border="0"

$$ \frac{\partial}{\partial x} \left( Ak \frac{\partial u}{\partial x} \right) + f \left(x,t \right) = A \rho c \frac{\partial u}{\partial t} $$
 * $$\displaystyle (Eq. 2.11) $$
 * }
 * }

Problem 4
Show that
 * {| style="width:100%" border="0"

$$ \int \frac{x^2}{1+x} dx = \frac{x^2}{2} - x + ln \left(1+x \right) + k $$
 * $$\displaystyle (Eq. 4.1) $$
 * }
 * }

part 1
Prove that
 * {| style="width:100%" border="0"

$$ \int lnxdx = xlnx -x $$
 * $$\displaystyle (Eq. 4.2) $$
 * }
 * }

In order to integrate by parts we choose $$ f \left(x \right) = lnx $$ and $$ g' \left(x \right) = 1 $$

Now we apply the formulae for integration by parts which is:

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

$$ \int f \left( x \right) g' \left( x \right)dx = f \left(x \right) g \left(x \right) - \int g \left(x \right) f' \left(x \right)dx $$
 * $$\displaystyle (Eq. 4.3) $$
 * }
 * }

Where $$ f' \left(x \right) = \frac{1}{x}, g \left(x \right) = x $$

Substituting these values into Eq.4.3 we get:

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

$$ \int lnxdx = xlnx - \int \frac{x}{x} dx = xlnx -x $$
 * $$\displaystyle (Eq. 4.4) $$
 * }
 * }

part 2
Prove that
 * {| style="width:100%" border="0"

$$ \int xlnxdx = \frac{1}{2} x^2 \left[lnx - \frac{1}{2} \right] $$
 * $$\displaystyle (Eq. 4.5) $$
 * }
 * }

In order to integrate by parts we choose: $$ f \left(x \right) = lnx $$ and $$ g' \left(x \right) = x $$

Where $$ f' \left(x \right) = \frac{1}{x}, g \left(x \right) = \frac{x^2}{2} $$

Substituting these values into Eq.4.3 we get:

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

$$ \int xlnxdx = \frac{x^2}{2} lnx - \int \frac{x^2}{2} \frac{1}{x} dx = \frac{x^2}{2} lnx - \int \frac{x}{2}dx = \frac{x^2}{2} lnx - \frac{x^2}{4} $$
 *
$$\displaystyle (Eq. 4.6) $$
 * }
 * }

Factoring out $$ \frac{1}{2} x^2 $$ we get:

part 3
Find
 * {| style="width:100%" border="0"

$$ \int \frac{x^2}{1+cx} dx $$
 *
$$\displaystyle (Eq. 4.7) $$
 * }
 * }

The integral can also be written as follows:

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

$$ \int \frac{x^2}{1+cx} dx = \frac{1}{c} \int \left[x - \frac{x}{1+cx} \right]dx = \frac{1}{c} \left[\int xdx - \int \frac{x}{1+cx} dx \right] $$
 *
$$\displaystyle (Eq. 4.8) $$
 * }
 * }

To solve this integral, we made the next change of variables:
 * {| style="width:100%" border="0"

$$ y = 1+cx \Rightarrow  x = \frac{y-1}{c}  \Rightarrow  dy = cdx $$
 *
$$\displaystyle (Eq. 4.9) $$
 * }
 * }

Substituting these values into Eq.4.8 we get:
 * {| style="width:100%" border="0"

$$ \int \frac{ \left(y -1 \right)^2}{c^2 y} \frac{dy}{c} $$
 *
$$\displaystyle (Eq. 4.10) $$
 * }
 * }

Which can be expanded to:
 * {| style="width:100%" border="0"

$$ \frac{1}{c^3} \int \frac{y^2 - 2y +1}{y} dy = \frac{1}{c^3} \int \left(y -2+ \frac{1}{y} \right) dy = \frac{1}{c^3} \left( \frac{y^2}{2} -2y+lny \right) $$
 *
$$\displaystyle (Eq. 4.11) $$
 * }
 * }

Finally, replacing y as a function of x we have:
 * {| style="width:100%" border="0"

$$ \frac{1}{c^3} \left[ \frac{ \left(1+cx \right)^2}{2} - 2 \left(1+cx \right) + ln \left(1+cx \right) \right] $$
 *
$$\displaystyle (Eq. 4.12) $$
 * }
 * }

Choosing $$ c = 1 $$
 * {| style="width:100%" border="0"

$$ \int \frac{x^2}{1+x} dx = \frac{x^2}{2} - x - \frac{3}{2} + ln \left(1+x \right) $$
 *
$$\displaystyle (Eq. 4.13) $$
 * }
 * }

Wolfram Alpha Proof for Part 3

part4
Find
 * {| style="width:100%" border="0"

$$ \int \frac{x^2}{a+bx} dx $$
 *
$$\displaystyle (Eq. 4.14) $$
 * }
 * }

We start by changing the variables with:

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

$$ y = a +bx \Rightarrow  x = \frac{y - a}{b}  \Rightarrow  dx = \frac{dy}{b}$$
 *
$$\displaystyle (Eq. 4.15) $$
 * }
 * }

Replacing these values into Eq.4.14 we get:

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

$$ \int \frac{ \left( y-a \right)^2}{b^2 y} \frac{dy}{b} = \frac{1}{b^3} \int \left(y -2a+ \frac{a^2}{y} \right) dy = \frac{1}{b^3} \left( \frac{y^2}{2} - 2ay+ a^2 ln y \right) $$
 *
$$\displaystyle (Eq. 4.16) $$
 * }
 * }

Changing the variables we get:

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

$$ \frac{1}{b^3} \left( \frac{ \left(a+bx \right)^2}{2} - 2a \left( a+bx \right) + a^2 ln \left(a+bx \right) \right) $$
 *
$$\displaystyle (Eq. 4.17) $$
 * }
 * }

Choosing $$ a = b =1 $$ we have:
 * {| style="width:100%" border="0"

$$ \int \frac{x^2}{1+x} dx = \frac{ \left(1+x \right)^2}{2} - 2 \left(1+x \right) + ln \left(1+x \right) $$
 *
$$\displaystyle (Eq. 4.18) $$
 * }
 * }

The result is the same as Eq.4.13

Wolfram Alpha Proof for Part 4

Part 5
Find the exact solution $$ u \left(x \right) $$ for the problem (3) p. 9-2 on Mtg.9

The problem is solving the differential equation:

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

$$ \frac{d}{dx} \left[ \left( 2+3x \right) \frac{du}{dx} \right] +5x = 0  \forall x \in \left]0,1 \right[  $$
 *
$$\displaystyle (Eq. 4.19) $$
 * }
 * }

With the boundary conditions:

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

$$ u \left(1 \right) = 4 ~,~ - \frac{du}{dx} \left(x = 0 \right) = 6 $$
 *
$$\displaystyle (Eq. 4.20) $$
 * }
 * }

The differential equation can be written:
 * {| style="width:100%" border="0"

$$ \frac{d}{dx} \left[ \left( 2+3x \right) \frac{du}{dx} \right] = -5x $$
 *
$$\displaystyle (Eq. 4.21) $$
 * }
 * }

Now integrate:

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

$$ \left[ \left( 2+3x \right) \frac{du}{dx} \right] = -5 \frac{x^2}{2} + k $$
 *
$$\displaystyle (Eq. 4.22) $$
 * }
 * }

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

$$ \frac{du}{dx}= -2.5 \frac{x^2}{\left(2 +3x \right)} + \frac{k}{\left( 2+3x \right)} $$
 *
$$\displaystyle (Eq. 4.23) $$
 * }
 * }

Now the solution of $$ u \left( x \right) $$ is:

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

$$ u \left( x \right) = -2.5 \int \frac{x^2}{ \left(2+3x \right)} dx + k \int \frac{dx}{ \left(2+3x \right)} $$
 *
$$\displaystyle (Eq. 4.24) $$
 * }
 * }

The first integral can be solved using Eq.4.17 with  $$  a = 2 ~,~ b=3 $$   and the second integral can be solved

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

$$ \int \frac{dx}{ \left(2+3x \right)} = \frac{1}{3} ln \left(2+3x \right) + q $$
 *
$$\displaystyle (Eq. 4.25) $$
 * }
 * }

Therefore the total solution is:
 * {| style="width:100%" border="0"

$$ u \left(x \right) = -2.5 \frac{1}{3^3} \left[ \frac{ \left(2+ 3x \right)^2}{2} - 2 \times 2 \left(2+3x \right) + 2^2 ln \left( 2+3x \right) \right] + \frac{k}{3} ln \left(2+3x \right) +q $$
 *
$$\displaystyle (Eq. 4.26) $$
 * }
 * }

After simplifying everything we obtain:

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

$$ u \left(x \right) = \frac{-5}{12}x^2 + \frac{30}{54}x + pln \left(2+3x \right) + r $$
 *
$$\displaystyle (Eq. 4.27) $$
 * }
 * }

Where $$ p $$ and $$ r $$ are constants and can be solved using the boundary conditions (Eq.4.20)

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

$$ u \left(1 \right) = 4 = \frac{-5}{12} + \frac{30}{54} + pln \left(5 \right) + r $$
 *
$$\displaystyle (Eq. 4.28) $$
 * }
 * }

The derivative of u is:

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

$$ \frac{du}{dx} = \frac{-10}{12}x + \frac{30}{54} + \frac{3p}{\left(2+3x \right)} $$
 *
$$\displaystyle (Eq. 4.29) $$
 * }
 * }

Therefore:

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

$$ \frac{du}{dx} \left(x=0 \right) = \frac{30}{54} + \frac{3p}{2} = -6  $$
 *
$$\displaystyle (Eq. 4.30) $$
 * }
 * }

Solving for p then gives:
 * {| style="width:100%" border="0"

$$ p = - \frac{118}{27} $$
 *
$$\displaystyle (Eq. 4.31) $$
 * }
 * }

Replacing p into Eq.4.28 gives:

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

$$ r = 10.895 $$
 *
$$\displaystyle (Eq. 4.32) $$
 * }
 * }

The complete solution is:

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

$$ u \left(x \right) = \frac{-5}{12}x^2 + \frac{30}{54}x - \frac{118}{27} ln \left(2+3x \right) + 10.895  $$
 *
$$\displaystyle (Eq. 4.33) $$
 * }
 * }

title: User:EML5526.S11.Team1.HS/HW2


title: File:Nm1.s11.team5.HW2.fig12b.png