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User:Egm6341.s11.team4.BM/HW7

=Problem 7.1: Integrating the Logistic Equation Using Hermite-Simpson, Forward Euler, and Backward Euler=

This problem is given on Lecture Slide 40-1 with references to lecture slides 39-1 and lecture slides 40-1 and lecture slides 40-2.

Given
The logistic equation

$$ \frac{dx}{dt}=rx(1-\frac{x}{x_{max}}) $$

With the following conditions

$$ x_{max}=15 $$; $$ r=1.4 $$; $$ t=[0,20] $$

And the two initial conditions

$$ x_0=3< \frac{1}{2}x_{max} $$; $$ x_0=9 > \frac{1}{2}x_{max} $$

Find
1. The Integral over the time span using Hermite-Simpson and the step size $$h=2^{k} \hat{h} $$ for varying values of k.

2. The Integral over the time span using Forward Euler and the step size $$h=2^{k} \hat{h} $$ for varying values of k.

3. The Integral over the time span using Backward Euler and the step size $$h=2^{k} \hat{h} $$ for varying values of k.

Solution
All Code is included at the end of this problem. The Newton-Raphson method was used to solve the nonlinear problem in Hermite-Simpson. First, we will look at the case where the initial condition is $$ x_0=3 $$ then the case where $$ x_0=9 $$. Now, we we are to develop the forward and backward Euler integration schemes using the $$ x_{bar} $$ scheme introduced in class. It should be noted that we expect the explicit forward Euler scheme to produce unstable results and the implicit backward Euler scheme to be stable.

Forward Euler Development
Starting with the general forward Euler equation we will use the bar notation introduced in class.

$$ \bar{x}:=\frac{x}{x_{max}} $$

General Forward Euler Equation

$$ x_{i+1}=x_i (1+\frac{dx_i}{dt}h) $$

Which, for our equation, becomes

$$ \bar{x}_{i+1}=x_i(1+(r\bar{x}_i(1-\bar{x}_i))h) $$

Rearranging this, the equation becomes

$$ \bar{x}_{i+1}=hr\bar{x}_i(1-\bar{x}_i) +\bar{x}_i$$

Now our x vector can be obtained from this $$ \bar{x} $$ vector simply by multiplying through by the maximum value of x.

Backward Euler Development
Starting from the general Backward Euler equation.

$$ x_{i+1} = x_i + \frac{dx_{i+1}}{dt}h $$

$$ \bar{x}_{i+1}=\bar{x}_i + hr\bar{x}_{i+1}(1-\bar{x}_{i+1}) $$

Rearranging into a form that is second order in $$ \bar{x}_{i+1} $$.

$$ hr\bar{x}_{i+1}^{2}+(1-hr)\bar{x}_{i+1}-\bar{x}_i=0 $$

Now, employing the quadratic equation.

$$ \bar{x}_{i+1} = \frac{-(1-hr) \pm \sqrt{(1-hr)^2+4hr}} {2hr}$$

Once again, the true x vector can be found by multiplying the $$ \bar{x} $$ vector by the maximum value of x.

Both of these equations will be used when employing the Forward and Backward Euler integration.

Initial Condition X=3
Looking at figure 1 in the set of plots below, we see that the Hermite-Simpson integration method produces highly accurate results when plotted against the analytic solution. This accuracy maintains even as we increase the step size in accordance with the $$ h=2^k \hat{h} $$ rule. Similarly, as expected, the Forward Euler Integration scheme blows up as we increase the step size, ultimately providing results that are useless. The Backward Euler Integration scheme appears to be stable and provides results that rival the Hermite-Simpson integration scheme.

Initial Condition X=9
Looking at figure 2 in the set of plots below, we see similar results as obtained when the initial condition X=3 was used. First, we see that the Hermite-Simpson integration method once again produces highly accurate results when plotted against the analytic solution. This accuracy maintains even as we increase the step size in accordance with the $$ h=2^k \hat{h} $$ rule. Similarly the Forward Euler Integration scheme blows up as we increase the step size. Lastly, the Backward Euler Integration scheme is stable and provides results that rival the Hermite-Simpson integration scheme.

MATLAB Code
This problem was solved by Brendan Mahon

title: User:Egm6341.s11.team4.BM/HW7


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

title: File:Egm6344.s11.team4.BM.HS9.jpg


title: File:Egm6344.s11.team4.BM.FE9.jpg


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title: File:Egm6344.s11.team4.bm.HS3.jpg


title: File:Egm6344.s11.team4.bm.FE3.jpg


title: File:Egm6344.s11.team4.BM.BE3.jpg


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Good and Evil

GOOD AND EVIL This matter has been treated in depth in the Summa Theologica by Saint Thomas Aquinas. The Summa is an ancient yet timeless treatise, of which contemporary translations of the complete and shortened versions are readily available, concerning the purpose of human existence. The words: good, true and beautiful are synonyms having respectively the antonyms: evil or bad, false and ugly. The following is a modern philosophical essay on this, arguably the most important of subjects: Why we are here. First, the nature of what constitutes a human being and the manner in which a solitary human being interacts with other human beings and ceases to be alone needs explication. There is a dualism consisting of man's individuality - how he relates to him or her self and his or her societal relationship to others. The relationship of a human being to him or her self has a reflexive quality. The relationship of an individual to society, of which one is a part, has a reciprocal quality.

A reasonable hypothesis is that a human being consists of a material body and an immaterial soul or spirit. Evolutionary biologists - if that is the correct term - have evidence from mitochrondrial DNA that man has two primitive ancestors, one identifiable with the female principle and the other, the male principle, each originating from Africa. It would be a conjectural extrapolation to call them respectively, Eve and Adam. Since then, the human population has increased to about seven thousand million with a nett increase of one or two new human beings per second, despite disease, accident, war and artificial fertility control.

There is an immediate indispensable primary purpose for man's existence on Earth. That is to produce subjects - more human beings. This generally requires man to expend much energy to produce objects as an intermediary prior to the production of subjects. In concrete terms, objects are everything that man produces or is producing - food, clothing, building materials et c. as well as every idea and invention that promotes the progress of civilization - everything required for every human being to work towards living a contented life of moderation.

The soul is the direct creation of the creator of the universe - God. The body comes from the parents going back through generations. God has created man, soul and ultimately man's body, in his own image. Thus if we could see God, which is the blessing that the justified souls in heaven are able to do, then we would see a likeness of ourselves in God. There is a conjugacy of images. The proper place to commence new human life is the family, that is the natural unit of society. Each child has the natural right to be conceived exclusively from his or her two parents and born from his or her mother. The soul comes into being at conception, from which instant a new human being is now in the process of reaching maturity, followed by a gradual decline and then death. The age of the human being is measured from birth. The Biblical span of human life is given as six hundred and thirteen thousand and six hundred and twenty hours average, so there is no time to squander since time belongs to God who will demand an account from each human being of how every hour had been used. At death, the body dies, but the soul continues and will live for ever. Time wasting dissipative activities that are neither restful nor recreational but leave the person tense and nervous have therefore got to be avoided. Time is life and is much more valuable than money.

The purpose of having as many human being on Earth, that can reasonably accommodated at any one time, is for each human being to attain to Eternal Life. If, for a multiplicity of reasons, fewer human beings come into existence than could have done, then the purpose which is to give glory to God will have been only partially fulfilled. The previous statement, that God could fail to accomplish his divine will, might not be true since only God knows how many human beings he wills should come into existence during the lifetime of this planet Earth and he can make this absolute population come about, for he as the author of time, could never be pressed for time. There is much disagreement about how large the Earth's population should be and on the maximum population that could live on the Earth at any one time. They who assert that the Earth is overpopulated are calling into question their own right to exist.

Every human being is unique. This means that there never will be two or more human beings that are indistinguishable from one another, where the comparison is being made over the total human population counted over the whole of time. Even identical twins have differences that diverge with time as they mature with age.

The essay is based on the most simple of premises, that each person has been created for perfect happiness.

Perfect happiness has to at least satisfy the following four attributes.

The happiness that has been attained must be deserved in due equivalent measure. That is happiness has to be consistent with justice.

Perfect happiness cannot last for any finite time however measured and however large - that is happiness needs to last forever or remain in eternity.

Perfect happiness cannot be limited to one or to even several different places or have any finite space dimension but extend infinitely to and throughout the universe or if such exists, a multiverse.

In the definition of perfection of happiness, in which there is no defect, happiness has to be perfect in every aspect or have an infinite perfection in every one of an infinitude of dimensions.

If any or some or all of the above attributes are violated, then that is an evil.

To say that no trace of unhappiness can co-exist within in the state of perfect happiness is perfectly true but philosophically redundant. This is mentioned for the sake of clarity.

This perfect happiness will be found only in Heaven, a place but primarily a state where there is peace, joy and rest from travail. There can be no loneliness in Heaven, because love [charity] is brought to perfection. Each soul in Heaven will receive from God the appropriate measure of happiness according to their greater or lesser capacity, as the just reward for their good works in this life.

Boredom is a form of suffering and is therefore an evil. In the state of perfect happiness in Heaven, there can be no psychological distress, no bodily illness and no pain, for these are all evils.

This state of being can only to be reached primarily through suffering in this world - that is the willing acceptance of what unavoidably befalls each of us. The state of perfect happiness is also to be attained through what is termed by theologians: " Good Works " and prayer which is another name for communication with the divine deity, God who is to be discovered in the depths of the subconscious. It is a paradox that patient acceptance of everything that happens to us, the suffering that we cannot change, that God sends, is the most certain route to perfect happiness.

Patience, a moral virtue, is a form of another moral virtue, humility, the latter being an absolutely indispensable quality for the attainment of happiness. True happiness in this world may be described by the word " contentment " where often there is no pleasure involved what ever. Pleasures, provided that they are lawful, have the purpose of directing us to a desire for God.

A cause of suffering for human beings could be the imperfection of the material world and of course the material body. This may have been the intention of the creator, so that every suffering patiently endured would make that person deserving of a greater happiness after death, but certainly after the " Last Day " after which all things, including the now living human body, will be made [by the creator] perfect in Heaven. Much of the suffering - temporal evil - in this world results from the vissicitudes of life that come from life's rough stony path. True religion is a good insurance policy that a human being will not be destroyed spiritually by the difficulties and sufferings they will encounter in life, but rather that the hardships in their progress through life will be tempered. It is on the Last Day that Christ will come from Heaven in magnificent splendour to judge the nations. The most secret thoughts, words, actions and motivations or intentions of every individual who has ever lived will become known to everyone. This is the General Judgement. When a human being dies, their soul is then judged by Christ and will be assigned its appropriate place in Eternity. This is called the Particular Judgement. The Earth and the rest of the Universe exist because God has created the Universe to reflect his own glory. God has amongst other attributes these: He is pure spirit and exists everywhere in the Universe and throughout the exterior of the Universe, if such exists. He exists uncreated of himself from eternity. He is infinitely good. He is infinitely intelligent. God has all knowledge. He is infinitely powerful. He is infinitely wise. He is infinitely courteous for he does not control us but leaves us free, respecting our decisions. God is the absolute truth for God cannot deceive and God cannot be deceived. God is absolutely just. God is infinitely merciful. The attributes of God above, are mutually dependent.

The words " good " and " evil " are not all together self-evident in their meaning but require an explicit explanation. There is much that is good and much that is evil in this world. Theologians call the good on this Earth " temporal good ". Eternal good and eternal evil are the two final states after death and they are mutually exclusive states. Eternal good is the absolute absence of eternal evil and eternal evil is the absolute absence of eternal good. God has created us for perfect happiness to be with him, ultimately in Heaven for ever, after we die. " Good " is that which promotes the attainment of this goal which is eternal union with the creator God. In contrast, " evil " delays, frustrates or can even destroy the possibility of the soul from ever reaching the purpose [Heaven] for which that soul has been created. Separation of a soul from God forever would be an infinite evil. The greatest suffering of lost souls is the pain of loss. The infinite good is to gaze on the face of God in eternity - called by theologians, the beatific vision. Evil may arise from the soul itself, or from others or from the environment. Moral evil is the only objective harm. Natural disasters - earthquakes, tsunamis, droughts et c. cause much suffering but only the evil of refusing the infinite good, which is God really deserves to be called " evil ". The former, seen from God's viewpoint must be regarded as temporary inconveniences. Moreover, God has given mankind the intelligence and imagination to prevent the worst effects of natural disasters from causing loss of life and destruction of property. War is very much a man-made disaster that is completely preventable using God's help. No one ever chooses that which appears to him or her as evil, but sometimes prefers evil where this is disguised as good. Evil seen in its true light is extremely unattractive. Human beings are drawn inexorably towards the good, real or apparent. Every other inferior loyalty should be subsumed into a sovereign supreme loyalty to God. Thus terrorism, either private or state sponsored, is always an evil. Moral evil always has pride, the antithesis of humility, as the antecedant. The pride or hubris mentioned here should not be confused with " pride " in one's appearance and grooming, which is the virtue of showing respect for one's self, for the occasion and for others.

Living " the moral life " would never be easy. The " E " word - that is Emotion often interferes either constructively or destructively in the quest to live " the good or moral life ". Many people have " good " genes giving them a flying start but poverty does not necessarily imply the adjunct of a poor mentality. Poverty, such that the basics of life are just too scarce, is an evil making " moral living " very difficult. At the other extreme, an excess of personal wealth might easily spiral towards [toward] corruption and consequent moral evil. As is said in the Bible: " Enough is enough. " - so generally, there is an optimum for personal wealth.

If none of the above is true then some other explanation is needed to account for human existence. The Universe could not have come into existence by itself. There appears to be no other reasonable explanation than the above in however many different ways this has been expressed. According to Occam's razor, a philosophical concept, the explanation requiring the least number of initial assumptions is the correct one. There has to be at least one assumption. In the above explanation for man's existence, there is the one starting or initial assumption - that each human being has in him or her the unquenchable desire for inexhaustible wellbeing or perfect happiness. Everyone knows that this is true.

In everyone's life there is from time to time a choice each individual has to make between good and evil. Such situations are innumerable, for example where a person in a position of trust as a treasurer in a corporation who handles large sums of money daily. By deciding on the good or right course of action, that soul is preparing well for what has been called " a happy death " - that is a good transition from this life to a life of perfect happiness in the next. The only true happiness or " good " on this Earth is " doing God's will ", that is following an upright conscience. Doing " the right thing " does not always give pleasure and on the contrary very often is quite disagreeable but certainly promotes happiness. In order to reach what has been called by theologians " the beatific vision " each soul needs God's help - this is the province of the theology of salvation in which Jesus Christ is absolutely central. In the theology of salvation, it is taught that Christ has divine nature and human nature all at once. This union of the nature of the infinite God and the nature of finite man is called by theologians " the hypostatic union ".

In regard to God's judgement of each soul, this occurs instantaneously at death. Then each soul will be rewarded or punished as appropriate. The actual state of the soul at death will determine the place the soul will occupy in eternity. Theologians say that a soul in " a state of grace " at death will go to Heaven. A soul that has in a very serious or grave matter offended the justice of God, with full knowledge and full consent, without repentance before or just prior to death will be consigned to Hell, where the body will also go on the Last Day. After death, no further change is possible in the state of the soul. This is now fixed for Eternity.

The perfect eschatalogical " good " is therefore the attainment of eternal happiness in Heaven, an irreversible state and an infinite good - the infinite good. By contrast, eschatalogical " evil " is the antithesis of this happiness and is described in one word - Hell, the other eternal and irreversible state and is definitively the infinite evil.

The Ten Commandments - Decalogue - in the Biblical Book of Exodus present a minimum code of moral conduct. The positive precepts are only implicit in the Decalogue and need to be made explicit in all their clarity. Good has been equated with beauty and truth. The temporal good in this context is the promotion of beautiful music, art, artifacts, architecture and environment for these all enhance the common good and give joy, another name for happiness. Temporal evil is characterized by a cult of ugliness, where by some unwritten " law " every building in the neighbourhood [neighborhood] has to be an eyesore and everyone is depressed by this.

Waste is an economic evil often brought about by allowing political expedients to take precedence over solidly established scientific principles or basing decisions on dubious science. The sequestration of millions of tonnes of carbon dioxide in to the lower geological strata, thus treating carbon dioxide as a form of atmospheric pollution, could well be a case in point. That is an incurring of a totally uncalled for and unnecessary use of resources to support belief in an unproven scientific theory. Dubious science because many presumably reputable scientists assert that increasing atmospheric carbon dioxide will cause catastrophic heating of the Earth, causing the planet to become a little cooler than the planet Venus but many other equally reputable scientists believe the exact opposite, that carbon dioxide in the atmosphere will all mostly be removed by the natural but mysterious process of photosynthesis. This means that carbon capture and storage is wrong in principle because of the above disagreement amongst scientists, since it has not been established with any certainty that this action is an efficient way of preventing the occurrence of a climatic change - planetary temperature increase, that might not be happening in any case. It is therefore a wasteful use of resources that unnecessarily increases the cost of electricity generation, this falling on the poorer members of society the most making them yet poorer. This is true irrespective of how carbon dioxide is increasing in the Earth's atmosphere. Geo-engineering designed to lower the average temperature of the Earth might just be another example of spiritual pride, that is hubris, again wasting the resources of the Earth.

Every form of mutilation of the body is an evil because it is defacing the image of God which is beautiful, especially in the countenance or human face. The voluntary subcutaneous injection of indelible inks and piercings is either a presumption of improving that which is already perfect - God's image - or it is a defacement of this image. Mutilation in the name of justice, such as marking prisoners, as above with identification numbers or punishment by amputation or emasculation are evils, for this denies human beings - even criminals or political prisoners - their human dignity. Female circumcision, correctly called genital mutilation is a prevailant evil perpetrated in the name of culture that urgently needs erradication. This particular form of mutilation diminishes or destroys the due happiness of the married state and so consequently is a great evil. Amputation for an exclusively therapeutic purpose and procedures for the correction of gross abnormalities is an entirely different matter.

The Bible mentions four sins, evils that are so heinous that direct divine retribution is called for. These are wilful murder, the sin of Sodom, oppression of the poor and defrauding labourers [laborers] of their wages. These are all more or less self explanatory. Abortion, with the exception of " Difficult Cases " can be recognised as a form of human sacrifice and is a great evil. Even in the difficult " save the mother or save the child cases " abortion cannot be described as being good. Every child, once conceived has the absolute right to see the light of day. This even applies to a child who is destined as an adult to carry out abortions. A mother who wants rid of her unborn child is only able to do this because the grandmother of the unborn child, in her day, did not want to get rid of her child. Family immolation, comparatively rare but still far too common a forensic event in this age, occurs where a husband or wife takes revenge, the presumed motive, by destroying the greatest blessing of marriage, their children. Apart from the consummate cruelty, this is an action of unbelievable stupidity. The commandment of the Decalogue fobidding murder implies that no life should be destroyed wantonly but all life is to be treated with due respect and in particular human life from conception until the natural end of life, after the allotted span.

In view of the above, explanations of the meaning of good and evil that deny the existence of an all powerful uncreated creator, that is atheism, have to be regarded as devoid of validity. Given that the above explanation is correct, then every form of practical atheism is to be regarded plainly as rediculous. The very fact of the cosmological constant is so precise in the " fined tuned Universe " is very remarkable, pointing to a statistical improbability that the universe came about by " chance " whatever that is. In any scientific explanation of the Universe's origin, it is proper to leave God out of it until well beyond the point where science has reached the stage of unknowability. Then perhaps, if the end of the road has been reached for science and further progress has become genuinely impossible, the existence of a creator God might by default, be the only explanation. There should be a partition between theology and science or in the words of the astrophysicist Georges Lemaitre, father of cosmology, a wall of separation should always exist between sacred theology and natural science. If an all powerful creator did not exist, then not only is it very probable that we would not exist but the premise cited above could only be hypothetically fulfilled. An uncreated being having infinite power and infinite intellect is needed to create a human being and thence to create perfect happiness for that individual human being. The declaration so often made that God does not exist - period, constitutes a non-explanation for man's beginning and his or her final end.

The existence of an infinite number of universes only serves to support the view that everything has been created by an infinitely powerful being and does not in the least affect the above explanation for man's presence in this particular universe. In the Biblical book of Genesis there is written down that God rested on the seventh day. This implies that God is still creating out of nothing since there is a further succession of six working days after the Sabbath, even for God.

All of the foregoing explanation might of course be entirely false, but if that is the case, then any distinctly different " true " explanation must by Occam's philosophical razor principle have just one inital assumption for there cannot be fewer, as does the above explanation and yet still preserve the requirement that each person has an intrinsic need for happiness that has a perfection satisfying the four above minimum attributes. The true explanation would be unique. If such a simpler explanation could never be found, then practically such an explanation does not exist, and only the original explanation, presented above, for human existence on Earth remains, this being the true version. Facile or simplistic explanations that assert that perfect happiness is unattainble or that man has not been created to be perfectly happy - not in this world but in the next - must necessarily be false, for they deny the truth of the initial premise.

True or Perfect happiness in this world - temporal happiness - is having the reasonablly moral certainty that we are expeditiously working towards [toward] our salvation, right up to the end of our life, so that when we die - at a time and place and in a manner unknown to us - we will immediately, or after a time of purification, then enter Heaven.

Suicide is an evil because this action is an offence [offense], the theological term being " sin ", that violates the sovereignty of God who is the author of life and death and this would most certainly place salvation in jeopardy. Encouraging or assisting someone to commit suicide would be an evil thing to do.

People who have commited suicide - a tragedy - did so in order to escape what they found was unbearable unhappiness. Their fallacy is that they have assumed that the depression, previously called by the name melancholia, or mental anguish could never vanish and their suffering would be lifelong without hope of remission, whereas they might have found perfect happiness in later years, in the measure that can be found on Earth if only they had exercised the moral virtue of patient endurance or fortitude. Should someone be destined never to find relief from clinical depression throughout their life, this situation could never justify suicide or euthanasia. The happiness of this world which is synonymous with pleasure, with the exception of the true happiness cited above, for seventy years average, even if not occasionally punctuated by periods of misery, does not qualify to be called perfect happiness. This has to be deferred to the next world. Any happiness here, in this world, is always unexpected and ever a transient.

Here it is useful to comment on the ancient Semitic writings that have been collected into one book - the [Holy] Bible. This book has never gone out of print ever since the invention of printing and many more or less reliable translations in almost every language spoken on the planet are available that contain a greater or lesser number of the original books. Extra books are included or excluded depending on which brand of Christianity is purveying a particular version, there being seventy-three books in total in one particular edition comprising forty-six books in the old dispensation and twenty-seven books in the new dispensation. Readers should beware, for the Bible is capable of interpretion in many distinctly different ways. It has been said that basing their lives on a particular understanding [interpretation] of the Bible, some have sought to establish a heavenly Jerusalem on Earth, only to be bitterly disappointed. Clearly then, the Bible needs an authoritative interpretation, if this Book of books is to be of any value other than just a work of world class literature. SHAWWPG19410425 (talk) 14:02, 18 August 2012 (UTC) SHAWWPG19410425 (talk) 15:10, 17 August 2012 (UTC)SHAWWPG19410425 (talk) 12:14, 2 September 2012 (UTC)78.146.188.9 21:40, 4 September 2012 (UTC)87.113.154.173 13:11, 5 September 2012 (UTC)87.113.154.173 13:17, 5 September 2012 (UTC)SHAWWPG19410425 (talk) 13:55, 8 September 2012 (UTC)SHAWWPG19410425 (talk) 13:59, 8 September 2012 (UTC)SHAWWPG19410425 (talk) 21:12, 10 September 2012 (UTC)SHAWWPG19410425 (talk) 10:35, 17 September 2012 (UTC)SHAWWPG19410425 (talk) 20:33, 18 September 2012 (UTC)SHAWWPG19410425 (talk) 20:49, 18 September 2012 (UTC)SHAWWPG19410425 (talk) 18:01, 27 September 2012 (UTC)SHAWWPG19410425 (talk) 18:26, 27 September 2012 (UTC)SHAWWPG19410425 (talk) 21:12, 5 October 2012 (UTC)SHAWWPG19410425 (talk) 11:24, 6 October 2012 (UTC)

title: Good and Evil


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User:Egm6341.s11.team4/HW7

=HW 7.0 problem: redo 5.2 and 5.4 cauchy noise=

Problem Statement
Refer to lecture notes [[media:nm1.s11.mtg25.djvu|25-2]] and [[media:nm1.s11.mtg25.djvu|26-1]] for detailed description 4). Run LSSM with Cauchy Random Noise

Problem Statement
Refer to lecture slide [[media:nm1.s11.mtg29.djvu|mtg-29]] for the problem statement. c).For u=0.5 Cauchy noise and c=$$1.5*{{C}_{cr}}$$, plot $$\underline$$ For u=0.5 Cauchy noise and c=$$1.5*{{C}_{cr}}$$, plot $$\underline$$

This problem was solved by shengfeng yang

=Problem 7.1: Integrating the Logistic Equation Using Hermite-Simpson, Forward Euler, and Backward Euler=

This problem is given on Lecture Slide 40-1 with references to lecture slides 39-1 and lecture slides 40-1 and lecture slides 40-2.

Given
The logistic equation

$$ \frac{dx}{dt}=rx(1-\frac{x}{x_{max}}) $$

With the following conditions

$$ x_{max}=15 $$; $$ r=1.4 $$; $$ t=[0,20] $$

And the two initial conditions

$$ x_0=3< \frac{1}{2}x_{max} $$; $$ x_0=9 > \frac{1}{2}x_{max} $$

Find
1. The Integral over the time span using Hermite-Simpson and the step size $$h=2^{k} \hat{h} $$ for varying values of k.

2. The Integral over the time span using Forward Euler and the step size $$h=2^{k} \hat{h} $$ for varying values of k.

3. The Integral over the time span using Backward Euler and the step size $$h=2^{k} \hat{h} $$ for varying values of k.

Solution
All Code is included at the end of this problem. The Newton-Raphson method was used to solve the nonlinear problem in Hermite-Simpson. First, we will look at the case where the initial condition is $$ x_0=3 $$ then the case where $$ x_0=9 $$. Now, we we are to develop the forward and backward Euler integration schemes using the $$ x_{bar} $$ scheme introduced in class. It should be noted that we expect the explicit forward Euler scheme to produce unstable results and the implicit backward Euler scheme to be stable.

Forward Euler Development
Starting with the general forward Euler equation we will use the bar notation introduced in class.

$$ \bar{x}:=\frac{x}{x_{max}} $$

General Forward Euler Equation

$$ x_{i+1}=x_i (1+\frac{dx_i}{dt}h) $$

Which, for our equation, becomes

$$ \bar{x}_{i+1}=x_i(1+(r\bar{x}_i(1-\bar{x}_i))h) $$

Rearranging this, the equation becomes

$$ \bar{x}_{i+1}=hr\bar{x}_i(1-\bar{x}_i) +\bar{x}_i$$

Now our x vector can be obtained from this $$ \bar{x} $$ vector simply by multiplying through by the maximum value of x.

Backward Euler Development
Starting from the general Backward Euler equation.

$$ x_{i+1} = x_i + \frac{dx_{i+1}}{dt}h $$

$$ \bar{x}_{i+1}=\bar{x}_i + hr\bar{x}_{i+1}(1-\bar{x}_{i+1}) $$

Rearranging into a form that is second order in $$ \bar{x}_{i+1} $$.

$$ hr\bar{x}_{i+1}^{2}+(1-hr)\bar{x}_{i+1}-\bar{x}_i=0 $$

Now, employing the quadratic equation.

$$ \bar{x}_{i+1} = \frac{-(1-hr) \pm \sqrt{(1-hr)^2+4hr}} {2hr}$$

Once again, the true x vector can be found by multiplying the $$ \bar{x} $$ vector by the maximum value of x.

Both of these equations will be used when employing the Forward and Backward Euler integration.

Exact Solution
Starting with the logistic equation and separating the variables, it becomes

 * $$\displaystyle \frac{1}{x(1-\frac{x}{x_{max}})}\;dx=r\;dt$$

Now, multiply the left-hand side of the equation by $$\frac{x_{max}}{x_{max}}$$  to simplify the denominator

 * $$\displaystyle \frac{x_{max}}{x(x_{max}-x)}\;dx=r\;dt$$

The left-hand side can now be deconstructed into the partial fractions

 * $$\displaystyle \left[\frac{1}{x}+\frac{1}{x_{max}-x}\right]dx=r\;dt$$

Integrating both sides yields

 * $$\displaystyle ln|x|-ln|x_{max}-x|-ln|x_0|+ln|x_{max}-x_0|=r(t-t_0)$$

Taking the inverse natural log of both sides

 * $$\displaystyle \frac{x(x_{max}-x_0)}{x_0(x_{max}-x)}=e^{r(t-t_0)}$$

Now algebraically solving for x

 * $$\displaystyle x(x_{max}-x_0)=x_0(x_{max}-x)e^{r(t-t_0)}$$

 * $$\displaystyle x(x_{max}-x_0)=x_0x_{max}e^{r(t-t_0)}-x_0xe^{r(t-t_0)}$$

 * $$\displaystyle x(x_{max}-x_0+x_0e^{r(t-t_0)})=x_0x_{max}e^{r(t-t_0)}$$

 * $$\displaystyle x = \frac{x_0x_{max}e^{r(t-t_0)}}{x_{max}-x_0+x_0e^{(t-t_0)}}$$

Initial Condition X=3
Looking at figure 1 in the set of plots below, we see that the Hermite-Simpson integration method produces highly accurate results when plotted against the analytic solution. This accuracy maintains even as we increase the step size in accordance with the $$ h=2^k \hat{h} $$ rule. Similarly, as expected, the Forward Euler Integration scheme blows up as we increase the step size, ultimately providing results that are useless. The Backward Euler Integration scheme appears to be stable and provides results that rival the Hermite-Simpson integration scheme.

Initial Condition X=9
Looking at figure 2 in the set of plots below, we see similar results as obtained when the initial condition X=3 was used. First, we see that the Hermite-Simpson integration method once again produces highly accurate results when plotted against the analytic solution. This accuracy maintains even as we increase the step size in accordance with the $$ h=2^k \hat{h} $$ rule. Similarly the Forward Euler Integration scheme blows up as we increase the step size. Lastly, the Backward Euler Integration scheme is stable and provides results that rival the Hermite-Simpson integration scheme.

MATLAB Code
This problem was solved by Brendan Mahon

Given
The parameters for this problem are given in both the problem statement in the lectures as well as Subchan and Zbikowski's paper Computational optimal control of the terminal bunt manoeuvre—Part 1: Minimum altitude case.

From Subchan and Zbikowski's paper we have the following:

Equations of Motion:

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

(2.1)
 * $$\displaystyle \dot \gamma = \frac{T - D}{mV}sin(\alpha) + \frac{L}{mV}cos(\alpha) - \frac{g cos(\gamma)}{V}$$
 *
 *
 * }
 * }

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

(2.2)
 * $$\displaystyle \dot V = \frac{T - D}{m}cos(\alpha) - \frac{L}{m}sin(\alpha) - g sin(\gamma)$$
 *
 *
 * }
 * }

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

(2.3)
 * $$\displaystyle \dot x = V cos(\gamma)$$
 *
 *
 * }
 * }

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

(2.4)
 * $$\displaystyle \dot y = V sin(\gamma)$$
 *
 *
 * }
 * }

And,

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

\begin{bmatrix} \gamma \\ v \\ x \\ y \end{bmatrix} $$
 * $$\mathbf{z} =
 * $$\mathbf{z} =
 * }
 * }

Where γ, V, x and y are state variables defined such that γ is the flight path angle, V is the speed, x is the horizontal position and y is the altitude.

Also from Subchan and Zbikowski we have:

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

 * $$\displaystyle D(h,V,\alpha) = \frac{1}{2}C_d \rho V^2 S_{ref}, \,\,\, C_d = A_1 \alpha^2 + A_2\alpha + A_3$$
 * }
 * }
 * }

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

 * $$\displaystyle L(h,V,\alpha) = \frac{1}{2}C_l \rho V^2 S_{ref}, \,\,\, C_l = B_1\alpha + B_2$$
 * }
 * }
 * }

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

 * $$\displaystyle \rho = C_1y^2 + C_2y + C_3$$
 * }
 * }
 * }

Where,

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

\begin{align} & m = 1005 \; kg \\ & g = 9.81 \; m/s^2 \\ & S_{ref} = 0.3376 \; m^2 \\ & A_{1} = -1.9431 \\ & A_{2} = -0.1499 \\ & A_{3} = 0.2359 \\ & B_{1} = 21.9 \\ & B_{2} = 0 \\ & C_{1} = 3.312 \times 10^{-9} \; kg/m^5 \\ & C_{2} = -1.142 \times 10^{-4} \; kg/m^4 \\ & C_{3} = 1.224 \; kg/m^3 \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

The initial conditions are given as:

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

\begin{align} & \gamma_0 = 0 \, rad\\ & V_0 = 272 \, m/s \\ & x_0 = 0 \, m \\ & y_0 = 30 \, m \\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

From the problem statement in the lecture notes we get the profiles for the two control inputs thrust, T(t) and angle of attack α(t):

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

T(t) = \begin{cases} 6000 \; N, & \mbox{for }0 \leq t < 27 \; \mbox{and} \; 33 \leq t \leq 40 \\ 1000 \; N, & \mbox{for }27 \leq t < 33 \end{cases} $$     (2.5)
 * $$\displaystyle
 * $$\displaystyle
 *

 * }
 * }

For part 1:

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

\alpha(t) = \begin{cases} 0.03 \; rad, & \mbox{for }0 \leq t < 21 \\ \frac{0.13-0.09}{27-21}(t-21) + 0.09 \; rad, & \mbox{for }21 \leq t < 27 \\ \frac{-0.2+0.13}{33-27}(t-27) - 0.13 \; rad, & \mbox{for }27 \leq t < 33 \\ \frac{-0.13+0.2}{40-33}(t-33) - 0.2\; rad, & \mbox{for }33 \leq t \leq 40 \end{cases} $$     (2.6)
 * $$\displaystyle
 * $$\displaystyle
 *
 * }
 * }

For part 2:

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

\alpha(t) = \begin{cases} 0.03 \; rad, & \mbox{for }0 \leq t < 21 \\ \frac{0.13-0.03}{27-21}(t-21) + 0.03 \; rad, & \mbox{for }21 \leq t < 27 \\ \frac{-0.2+0.13}{33-27}(t-27) - 0.13 \; rad, & \mbox{for }27 \leq t < 33 \\ - 0.2\; rad, & \mbox{for }33 \leq t \leq 40 \end{cases} $$     (2.7)
 * $$\displaystyle
 * $$\displaystyle
 *
 * }
 * }

Objective
a.) Solve for z(t), for t contained on [0,40] such that successive numerical solutions are of the order 10-6, then plot the state variables as a function of time and compare them to the solution using the MatLab function "ode45".  Also compute

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

 * $$\displaystyle J_1 = \int_0^{t_f} y(t)dt \,\,\, s.t. \,\, y(t_f) = 0$$
 * }
 * }
 * }

Use given alpha for part 1.

b.) Solve for z(t), for t contained on [0,40] such that successive numerical solutions are of the order 10-6, then plot the state variables as a function of time and compare them to the solution using the MatLab function "ode45".

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

 * $$\displaystyle J_2 = \int_0^{t_f} y(t)dt \,\,\, s.t. \,\, y(t_f) = 0$$
 * }
 * }
 * }

Use given alpha for part 2.

Solution
Both parts of the problem will be solved using the same technique. The only difference will be the alpha function input into the Matlab code.

From equation (1) on Lecture Slide [[media:nm1.s11.mtg38.djvu|38-3]] we can derive:

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

$$     (2.8)
 * $$\displaystyle \mathbf{z}_{i+1} = \mathbf{z}_{i} + \frac{h/2}{3}\left[\mathbf{f}(\mathbf{z}_{i})+4\mathbf{f}(\mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}))+\mathbf{f}(\mathbf{z}_{i+1})\right]
 * $$\displaystyle \mathbf{z}_{i+1} = \mathbf{z}_{i} + \frac{h/2}{3}\left[\mathbf{f}(\mathbf{z}_{i})+4\mathbf{f}(\mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}))+\mathbf{f}(\mathbf{z}_{i+1})\right]
 *
 * }
 * }

Where:

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

= \frac{1}{2}\left(\mathbf{z}_{i} + \mathbf{z}_{i+1}\right) + \frac{h}{8}\left(\mathbf{f}(\mathbf{z}_{i}) - \mathbf{f}(\mathbf{z}_{i+1})\right) $$     (2.9)
 * $$\displaystyle \mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}) = \mathbf{z}_{i+1/2}
 * $$\displaystyle \mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}) = \mathbf{z}_{i+1/2}
 *
 * }
 * }

F(zi+1) can then be defined by subtracting the right side of eqn. (2.8) from the left side of eqn. (2.8).

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

$$     (2.10)
 * $$\displaystyle \mathbf{F}(\mathbf{z}_{i+1}) = \mathbf{z}_{i+1} - \mathbf{z}_{i} + \frac{h/2}{3}\left[\mathbf{f}(\mathbf{z}_{i})+4\mathbf{f}(\mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}))+\mathbf{f}(\mathbf{z}_{i+1})\right] = \mathbf{0}
 * $$\displaystyle \mathbf{F}(\mathbf{z}_{i+1}) = \mathbf{z}_{i+1} - \mathbf{z}_{i} + \frac{h/2}{3}\left[\mathbf{f}(\mathbf{z}_{i})+4\mathbf{f}(\mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}))+\mathbf{f}(\mathbf{z}_{i+1})\right] = \mathbf{0}
 *
 * }
 * }

From Lecture Slide [[media:nm1.s11.mtg39.djvu|39-1]] we get the following:

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

$$     (2.11)
 * $$\displaystyle \mathbf{z}_{i+1}^{(k+1)} = \mathbf{z}_{i+1}^{(k)} - \left[\frac{\partial \mathbf{F}(\mathbf{z}_{i+1}^{(k)})}{\partial \mathbf{z}}\right]^{-1}\mathbf{F}(\mathbf{z}_{i+1}^{(k)})
 * $$\displaystyle \mathbf{z}_{i+1}^{(k+1)} = \mathbf{z}_{i+1}^{(k)} - \left[\frac{\partial \mathbf{F}(\mathbf{z}_{i+1}^{(k)})}{\partial \mathbf{z}}\right]^{-1}\mathbf{F}(\mathbf{z}_{i+1}^{(k)})
 *
 * }
 * }

In order to solve the problem we need to calculate the Jacobian as follows:

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

$$     (2.12) The values in the Jacobian can be seen in the MatLab code below.  "Main Script"
 * $$\displaystyle \frac{\partial \mathbf{F}}{\partial \mathbf{z}} = \left[\frac{\partial F_i}{\partial z_j}\right]
 * $$\displaystyle \frac{\partial \mathbf{F}}{\partial \mathbf{z}} = \left[\frac{\partial F_i}{\partial z_j}\right]
 *
 * }
 * }

 Function "zdot"

 Function "dfdz"

Function "nonlinode1" and "nonlinode2" to be used with the ode45 function.  Function "nonlinode1"   Function "nonlinode2"

a.) Part 1: Initial values for alpha

b.) Part 2: Modified values for alpha

c.) Compute $$\ J_1$$, $$\ J_2$$ and compare.
For two different input values, calculate $$ J = \int_{t=0}^{t_f}y(t) dt, y(t_f) = 0 $$ The integration is calculated numerically by using "trapz" function of matlab, and the algorithm is in the above "main script" m-file. For input value of s10. $$ \ J_1 = 16803.19 $$ For input value of s11. $$ \ J_2 = 9024.61 $$ $$ \ J_1 > J_2 $$ So, using modified input makes better results.

Given
Refer to Lecture note S10 41-2 for more detailed problem statement

The logistic equation for population dynamics is given in S10 38-3,

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

$$\displaystyle \dot{x}(t) = f(x) = rx \left(1-\frac{x}{x_{max}}\right) $$ $$
 *
$$\displaystyle
 * }.
 * }.

Where, $$\displaystyle x_{max}=10;\ r = 1.2,\ $$ and $$\displaystyle t \in [0,10] $$

And the analytical solution is given by,

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

$$\displaystyle x(t) = \frac{x_0x_{max}e^{rt}}{x_{max} + x_0(e^{rt}-1)} $$ $$
 *
$$\displaystyle
 * }.
 * }.

Find
Integrate the population function using the inconsistent Trapezoidal-Simpson Algorithm given by,

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

$$\displaystyle z_{i+1/2} = \frac{1}{2} [z_{i} + z_{i+1}] $$
 * }.
 * }.

Two initial conditions, $$\displaystyle x_0=2\ and\ x_0 = 7,\ $$.

Solution
The Hermite-Simpson algorithm behaves better than inconsistent Trapezoidal-Simpson algorithm, that the error is much smaller comparing to inconsistent Trapezoidal-Simpson algorithm. Below are the plots for comparing Hermite-Simpson algorithm with Trapezoidal-Simpson algorithm.

Reference
S10.Team4

=Signatures=

Brendan Mahon 16:53, 17 April 2011 (UTC)

shengfeng yang 19:14 20 April 2011 (UTC)

William Kurth 03:18, 21 April 2011 (UTC) Taeho Kim 00:45, 21 April 2011 (UTC)

Erle Fields 05:05, 21 April 2011 (UTC)

Hailong Chen 10:58 21 April 2011 (EST)

title: User:Egm6341.s11.team4/HW7


title: File:Nm1.s11.team1.HW4.fig5.2.3NEW.png


text:
User:Egm6341.s11.team1.arm/HW7

Problem 2: Solve parameters for Equation of Motion
Referenced the work of Team 3, Spring 2010.

Improvements: -Added display of absolute error to user can monitor progress of convergence -Code restructured, variables renamed, and additional comments and to make code easier to follow and inputs easier to change -Improved parts of code to reduce time for absolute error to converge to $$10^{-6}$$ -->Old code time: Longer than 1 hour -->This code time: 107 sec

Given
Equations of motion for an aircraft:

where

and

Physical parameters were taken from the Subchan and Zbikowski paper:

Initial conditions were taken from the Subchan and Zbikowski paper:

Values for the controls are given in [[media:Egm6341.s10.mtg41.djvu|S10 Mtg 41]]: $$ \underline{u} (t)= \left \lfloor T(t), \alpha\ (t) \right \rfloor \ $$

Find: z and J
Solve to find the matrix $$z \ $$ for the bunt maneuver using S10 and S11 values, as well as values for $$ J \ $$ such that:

$$ J= \int_{0}^{t_f} y(t)\, dt \ $$ such that $$ y(t_f)=0 \ $$

Solve: Using S10 values
The following MATLAB code was developed to solve for $$\underline{z} \ $$ using Newton-Raphson and Hermite-Simpson:

The solution was confirmed using the MATLAB command ode45:

Solve: Using S11 values
The different values of $$\alpha \ $$ used for the S11 solution are shown below:

The following MATLAB code executes the model using these new values of $$\alpha \ $$:

Solve: Compare J values
The following MATLAB code was used to use the MATLAB trapz command to integrate the output of Parts 1 and 2 and determine $$ J_1 \ $$ and $$ J_2 \ $$.

$$J_1 = 16860 \ $$ $$J_2 = 10824 \ $$

Meaning that since $$ J_2 < J_1 \ $$, the S11 values provide a better solution.

title: User:Egm6341.s11.team1.arm/HW7


text:
Communiversity

A communiversity is an initiative designed to bring together a university and a community.

 * The History of Communiversity – Where Teachers Learn and Participants Teach, from University of Missouri-Kansas City

The Port Huron Statement
The Port Huron Statement was a document published by the Students for a Democratic Society in June 1962.

Dialectics of Liberation Congress
The Dialectics of Liberation Congress in London between 15 and 30 July 1967:
 * Joe Berke, the congress and radical education

title: Communiversity


text:
User:Egm6341.s11.team4.fields/HW7

HW7.2

title: User:Egm6341.s11.team4.fields/HW7


text:
File:HW7 2(Gamma t).png

= Summary=

title: File:HW7 2(Gamma t).png


text:
File:HW7 2(Y X).png

= Summary=

title: File:HW7 2(Y X).png


text:
File:HW7 2(Y t).png

= Summary=

title: File:HW7 2(Y t).png


text:
File:HW7 2(X t).png

= Summary=

title: File:HW7 2(X t).png


text:
File:HW7 2(V t).png

= Summary=

title: File:HW7 2(V t).png


text:
User:Egm6341.s11.team4.fields/HW7.2

=Problem 7.2: Optimal Solution of Aircraft Bunt Maneuver =

Refer to lecture slides [[media:nm1.s11.mtg40.djvu|40-1]], [[media:egm6341.s10.mtg40.djvu|40-3]] and [[media:egm6341.s10.mtg41.djvu|41-1]] for the problem statement.

Given
The parameters for this problem are given in both the problem statement in the lectures as well as Subchan and Zbikowski's paper Computational optimal control of the terminal bunt manoeuvre—Part 1: Minimum altitude case.

From Subchan and Zbikowski's paper we have the following:

Equations of Motion:

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

(2.1)
 * $$\displaystyle \dot \gamma = \frac{T - D}{mV}sin(\alpha) + \frac{L}{mV}cos(\alpha) - \frac{g cos(\gamma)}{V}$$
 *
 *
 * }
 * }

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

(2.2)
 * $$\displaystyle \dot V = \frac{T - D}{m}cos(\alpha) - \frac{L}{m}sin(\alpha) - g sin(\gamma)$$
 *
 *
 * }
 * }

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

(2.3)
 * $$\displaystyle \dot x = V cos(\gamma)$$
 *
 *
 * }
 * }

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

(2.4)
 * $$\displaystyle \dot y = V sin(\gamma)$$
 *
 *
 * }
 * }

And,

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

\begin{bmatrix} \gamma \\ v \\ x \\ y \end{bmatrix} $$
 * $$\mathbf{z} =
 * $$\mathbf{z} =
 * }
 * }

Where γ, V, x and y are state variables defined such that γ is the flight path angle, V is the speed, x is the horizontal position and y is the altitude.

Also from Subchan and Zbikowski we have:

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

 * $$\displaystyle D(h,V,\alpha) = \frac{1}{2}C_d \rho V^2 S_{ref}, \,\,\, C_d = A_1 \alpha^2 + A_2\alpha + A_3$$
 * }
 * }
 * }

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

 * $$\displaystyle L(h,V,\alpha) = \frac{1}{2}C_l \rho V^2 S_{ref}, \,\,\, C_l = B_1\alpha + B_2$$
 * }
 * }
 * }

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

 * $$\displaystyle \rho = C_1y^2 + C_2y + C_3$$
 * }
 * }
 * }

Where,

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

\begin{align} & m = 1005 \; kg \\ & g = 9.81 \; m/s^2 \\ & S_{ref} = 0.3376 \; m^2 \\ & A_{1} = -1.9431 \\ & A_{2} = -0.1499 \\ & A_{3} = 0.2359 \\ & B_{1} = 21.9 \\ & B_{2} = 0 \\ & C_{1} = 3.312 \times 10^{-9} \; kg/m^5 \\ & C_{2} = -1.142 \times 10^{-4} \; kg/m^4 \\ & C_{3} = 1.224 \; kg/m^3 \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

The initial conditions are given as:

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

\begin{align} & \gamma_0 = 0 \, rad\\ & V_0 = 272 \, m/s \\ & x_0 = 0 \, m \\ & y_0 = 30 \, m \\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

From the problem statement in the lecture notes we get the profiles for the two control inputs thrust, T(t) and angle of attack α(t):

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

T(t) = \begin{cases} 6000 \; N, & \mbox{for }0 \leq t < 27 \; \mbox{and} \; 33 \leq t \leq 40 \\ 1000 \; N, & \mbox{for }27 \leq t < 33 \end{cases} $$     (2.5)
 * $$\displaystyle
 * $$\displaystyle
 *

 * }
 * }

For part 1:

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

\alpha(t) = \begin{cases} 0.03 \; rad, & \mbox{for }0 \leq t < 21 \\ \frac{0.13-0.09}{27-21}(t-21) + 0.09 \; rad, & \mbox{for }21 \leq t < 27 \\ \frac{-0.2+0.13}{33-27}(t-27) - 0.13 \; rad, & \mbox{for }27 \leq t < 33 \\ \frac{-0.13+0.2}{40-33}(t-33) - 0.2\; rad, & \mbox{for }33 \leq t \leq 40 \end{cases} $$     (2.6)
 * $$\displaystyle
 * $$\displaystyle
 *
 * }
 * }

For part 2:

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

\alpha(t) = \begin{cases} 0.03 \; rad, & \mbox{for }0 \leq t < 21 \\ \frac{0.13-0.03}{27-21}(t-21) + 0.03 \; rad, & \mbox{for }21 \leq t < 27 \\ \frac{-0.2+0.13}{33-27}(t-27) - 0.13 \; rad, & \mbox{for }27 \leq t < 33 \\ - 0.2\; rad, & \mbox{for }33 \leq t \leq 40 \end{cases} $$     (2.7)
 * $$\displaystyle
 * $$\displaystyle
 *
 * }
 * }

Objective
a.) Solve for z(t), for t contained on [0,40] such that successive numerical solutions are of the order 10-6, then plot the state variables as a function of time and compare them to the solution using the MatLab function "ode45".  Also compute

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

 * $$\displaystyle J_1 = \int_0^{t_f} y(t)dt \,\,\, s.t. \,\, y(t_f) = 0$$
 * }
 * }
 * }

Use given alpha for part 1.

b.) Solve for z(t), for t contained on [0,40] such that successive numerical solutions are of the order 10-6, then plot the state variables as a function of time and compare them to the solution using the MatLab function "ode45".

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

 * $$\displaystyle J_2 = \int_0^{t_f} y(t)dt \,\,\, s.t. \,\, y(t_f) = 0$$
 * }
 * }
 * }

Use given alpha for part 2.

Solution
Both parts of the problem will be solved using the same technique. The only difference will be the alpha function input into the Matlab code.

From equation (1) on Lecture Slide [[media:nm1.s11.mtg38.djvu|38-3]] we can derive:

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

$$     (2.8)
 * $$\displaystyle \mathbf{z}_{i+1} = \mathbf{z}_{i} + \frac{h/2}{3}\left[\mathbf{f}(\mathbf{z}_{i})+4\mathbf{f}(\mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}))+\mathbf{f}(\mathbf{z}_{i+1})\right]
 * $$\displaystyle \mathbf{z}_{i+1} = \mathbf{z}_{i} + \frac{h/2}{3}\left[\mathbf{f}(\mathbf{z}_{i})+4\mathbf{f}(\mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}))+\mathbf{f}(\mathbf{z}_{i+1})\right]
 *
 * }
 * }

Where:

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

= \frac{1}{2}\left(\mathbf{z}_{i} + \mathbf{z}_{i+1}\right) + \frac{h}{8}\left(\mathbf{f}(\mathbf{z}_{i}) - \mathbf{f}(\mathbf{z}_{i+1})\right) $$     (2.9)
 * $$\displaystyle \mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}) = \mathbf{z}_{i+1/2}
 * $$\displaystyle \mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}) = \mathbf{z}_{i+1/2}
 *
 * }
 * }

F(zi+1) can then be defined by subtracting the right side of eqn. (2.8) from the left side of eqn. (2.8).

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

$$     (2.10)
 * $$\displaystyle \mathbf{F}(\mathbf{z}_{i+1}) = \mathbf{z}_{i+1} - \mathbf{z}_{i} + \frac{h/2}{3}\left[\mathbf{f}(\mathbf{z}_{i})+4\mathbf{f}(\mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}))+\mathbf{f}(\mathbf{z}_{i+1})\right] = \mathbf{0}
 * $$\displaystyle \mathbf{F}(\mathbf{z}_{i+1}) = \mathbf{z}_{i+1} - \mathbf{z}_{i} + \frac{h/2}{3}\left[\mathbf{f}(\mathbf{z}_{i})+4\mathbf{f}(\mathbf{g}(\mathbf{z}_{i},\mathbf{z}_{i+1}))+\mathbf{f}(\mathbf{z}_{i+1})\right] = \mathbf{0}
 *
 * }
 * }

From Lecture Slide [[media:nm1.s11.mtg39.djvu|39-1]] we get the following:

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

$$     (2.11)
 * $$\displaystyle \mathbf{z}_{i+1}^{(k+1)} = \mathbf{z}_{i+1}^{(k)} - \left[\frac{\partial \mathbf{F}(\mathbf{z}_{i+1}^{(k)})}{\partial \mathbf{z}}\right]^{-1}\mathbf{F}(\mathbf{z}_{i+1}^{(k)})
 * $$\displaystyle \mathbf{z}_{i+1}^{(k+1)} = \mathbf{z}_{i+1}^{(k)} - \left[\frac{\partial \mathbf{F}(\mathbf{z}_{i+1}^{(k)})}{\partial \mathbf{z}}\right]^{-1}\mathbf{F}(\mathbf{z}_{i+1}^{(k)})
 *
 * }
 * }

In order to solve the problem we need to calculate the Jacobian as follows:

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

$$     (2.12)
 * $$\displaystyle \frac{\partial \mathbf{F}}{\partial \mathbf{z}} = \left[\frac{\partial F_i}{\partial z_j}\right]
 * $$\displaystyle \frac{\partial \mathbf{F}}{\partial \mathbf{z}} = \left[\frac{\partial F_i}{\partial z_j}\right]
 *
 * }
 * }

The values in the Jacobian can be seen in the MatLab code below.

Function "nonlinode" to be used with the ode45 function.

a.) Part 1: Initial values for alpha

b.) Part 2: Modified values for alpha

title: User:Egm6341.s11.team4.fields/HW7.2


text:
File:HW7 2T.png

= Summary=

title: File:HW7 2T.png


text:
File:HW7 2Alpha1.png

= Summary=

title: File:HW7 2Alpha1.png


text:
File:HW7 2Alpha2.png

= Summary=

title: File:HW7 2Alpha2.png


text: File:NM1 HW7 3 Error InitPop3 HS.jpg

title: File:NM1 HW7 3 Error InitPop3 HS.jpg


text: File:NM1 HW7 3 Error InitPop9 HS.jpg

title: File:NM1 HW7 3 Error InitPop9 HS.jpg


text:
General life satisfaction

General life satisfaction is a social science construct which is a closely related to quality of life. It is the overall degree to which one is content with one's life and its circumstances.

title: General life satisfaction


text:
General health and well-being

Jtneill stub Link to any subpages this page might have

Learning resources

 * General health
 * Well-being
 * University student satisfaction

Wikipedia

 * Positive psychology
 * Qigong

title: General health and well-being


text:
User talk:StuRat/ideal firm size

The ideal firm size is the theoretically most competitive size for any company, in a given industry, at a given time; which corresponds with the highest possible per-unit profit.

Discussion
If only diseconomies of scale were considered, then the ideal firm size would be one worker. However, economies of scale also apply, which state that large firms can have lower per-unit costs due to buying at bulk discounts (components, insurance, real estate, advertising, etc.) and can also limit competition by buying out competitors, setting proprietary industry standards (like Microsoft Windows), etc. If only these "economies of scale" applied, then the ideal firm size would be infinitely large. However, since both apply, the firm must not be too small or too large, to be an ideal competitor.

Variation in ideal firm size by industry
The "diseconomies of scale" do not tend to vary widely by industry, but "economies of scale" do. An auto maker can buy millions of tons of steel for use in forming engine blocks and warehouse it indefinitely, if this will get a better price. A florist can't buy millions of tons of fresh flowers to sell, or they will wither before they are sold. This results in disparate interpretations of economies of scales for different types of organizations.

Variation in ideal firm size over time
Note that the ideal size of a firm may also change over time, as industry and market conditions change. If a supplier finds a way to manufacture small batches of a major component at prices comparable to the large batch price, this will help small firms more than large firms. The same "diseconomies of scale" still apply, however, so now the ideal firm size will be smaller.

Effects of agricultural, industrial, and service-based economies on ideal firm size
Also note that the basis of an economy will affect the ideal firm size. An agricultural society will tend to have small firms, as agriculture has a limited economy of scale (especially in the absence of refrigeration and other storage methods that allow for long term storage of large batches). An industrial society will tend to have large firms, as industry has a substantial economy of scale. A service-based economy will again favor smaller firms, as services have a limited economy of scale. There will, of course, be exceptions, such as Microsoft, which is a huge services company. This success may be attributed to getting a "jump start" on the competition with the Windows Operating System. One could predict that as Linux and other competing operating systems gain market share, this initial advantage will disappear, and the ideal firm size law will once again apply, causing Microsoft to contract, as industrial giants like GM have in the past. One could also predict that as the US economy shifts to a service-based economy from an industrial one, small and mid-sized firms, like Google, will be more successful than large companies, like Microsoft.

title: User talk:StuRat/ideal firm size


text:
KOIIA/Communication and Identities in Institutional Arenas - Part I/Final paper gme

Course KOIIA I – Final paper: an introduction
The purpose of the empirical study is to offer some possibilities of reflection on the relationship between Computer Mediated Communication (CMC) and language learning in online virtual institutional environments, with an analysis of the interaction in the virtual classroom as a learning place with no physical presence that will contribute towards an understanding of the complexity of students’ languaging activity in online synchronous environments. Examples of virtual institutional settings are videoconferencing programs that create a virtual classroom, an online environment where students can meet synchronously using multimodal resources (whiteboard, text chat, audio and video), and communicate with each other, with the teacher and the educational institution which is providing the course. We are facing a new kind of communication with new didactical challenges. The following types of empirical data will be used in the present study:

 * 1) Recorded online seminars (as screen recordings) in a videoconferencing environment. The seminars are conducted by the students, without teacher presence or intervention.
 * 2) Textual artifacts provided in advance by the teacher, with exercises and questions that are to be attended to during the student-led seminar

Taking sociocultural points of departure and using Conversation Analysis as an analytical tool to study students’ interaction, the main purpose of the study is to describe and understand what is going on when students engage in CMC in institutional settings, their agency in sense making, i.e. what is going on when they interact with each other engaging in different kinds of communicative activities and using the tools that are at hand in the multimodal environment. Linell discusses about the relation between agency and structure, or between language use and langue system, and argues that we do not talk because we know the system, but rather rules and conventions in speech communities get standardized through the fact that: “[…] we are always immersed in languaging, in and through which we develop habits of using and reusing elements of language and routines of enacting communicative projects through linguistic and other semiotic means” (2009: 56). The focus of the study is in the relationship between students’ peers interaction in the attempt to attend to the issues offered in the teacher’s material as a default route for the online session:

 * How are students relating to the textual artifacts provided by the language course, such as language exercises, questions and other forms of scaffolding, within the virtual environment?
 * How is the material content perceived by the students in the context of the online seminar and in relation to the main purpose of their meeting, which is to enhance students’ communication using the target language?
 * How is the material provided by the teacher affecting students’ interaction and their doing of identity?

To attempt to define identity is a rather complicated enterprise. Identity means to be the same as something else, but this way of considering it implies the notion of a static and unchangeable nature of identity. Even when acknowledging the fact that an individual’s identity is not just about one easily defined label, to consider identity as an addition of several labels or identities is not an adequate solution to the issue either. The notion of intersectionality shows a possible direction on how identity(-ies) can be analyzed as something negotiable, created in the interaction with an audience. In this sense it is more fruitful to relate to identity as a verb, rather than a noun, and where social practices constitute the loci where the doing of identity takes place and is performed in the participation in communities of practice (Wenger, 1998). For further considerations on this subject, see also my other reflections on Intersectionality and Identities and their enactments in human life in Wikiversity. It is from these theoretical points of departure that issues of identities and negotiation of meaning will be addressed in the present study.


title: File:Correlation.1.A.pdf


text:
File:HW7.2 Alpha1 Alpha2.png

= Summary=

title: File:HW7.2 Alpha1 Alpha2.png


text:
File:HW7.2 Thrust.png

= Summary=

title: File:HW7.2 Thrust.png


title: File:Nm1.s11.team1.HW7.fig7.2.1.png


title: File:Nm1.s11.team1.HW7.fig7.2.2.png


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User talk:Rhowardrn

 Hello Rhowardrn, 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.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

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! --Abd 17:36, 18 April 2011 (UTC)

title: User talk:Rhowardrn


title: File:Nm1.s11.team1.HW7.fig7.2.3.png


title: File:Nm1.s11.team1.HW7.fig7.2.4.png


text:
Talk:Artificial intelligence

Is this meant to be more of a history, or is it intended to look at the science (so far) behind it all? What did you have in mind for this course? ( 19:06, 18 April 2011 65.60.81.66 )

Replace this with Redirect AI
This seems to be abandoned and does not contain much content. The only relevant linked content can be found at Artificial_intelligence/prospects and I suggest that this page be replaced with a redirect to the existing, slightly better maintained page on AI. If there are no objections I'll come back in a while and do it.--7yl4r 17:59, 15 November 2011 (UTC)

title: Talk:Artificial intelligence


text:
Instructional design/Online engagement

Introduction | Module 1 | Module 2| Exercise Next

Introduction
A challenge for any online training is student engagement. In an in-classroom training, an instructor is able to adapt his training to suit his student’s needs by asking questions, getting students to discuss a topic or responding to a student’s answer. Whereas with online training, those elements are lacking. How does an Instructional Designer (ID) overcome that?

In the early years of online training, it consisted of just uploading the in-classroom content online with little or no thought for student engagement. With the advances of technology and this generation’s tech saviness, we as IDs have many tools at our disposal to enhance student engagement and be on the same level or higher than their in-classroom counterparts.

So what can you do to improve student engagement? The answer, by giving the student a learning environment that they can apply what they already know to what you are teaching them.

Lesson Goal
This lesson is meant for Instructional designers who create online training on topics that usually aren’t engaging to the student. They will be able taught to use interactive design methods to create online deliverables that will be both engaging and cost effective. The end result shall be a learning environment that will encourage student interest and engagement.

Key Learning Objectives
At the completion of this Wikiversity module you will be able to:
 * 1) Identify key points of a topic.
 * 2) Identify a scenario where key points can be taught.
 * 3) Identify improvements to scenario to what was learnt.
 * 4) Create an enhanced assignment.

title: Instructional design/Online engagement


text:
Tax preparation

Hello, I have no idea how to start this, but I really want it to exist, so I'm just going to spill a bunch of suggestions. I personally was looking for a how-to sort of deal for personal income tax preparation. Something that goes over both the methods of doing so, and the tools/materials/information/resources needed to do so. (Courses could be listed according to region and governmental level in some sort of index, I suppose) I was thinking of this as maybe like an 'Accounting for Laymen' sort of thing. I looked at the Accounting page, and it is crazy intimidating.

A course could be organized like this:

(Government Type) ex: Country/state/province/county/city/etc. (Income tax rate(s)) ex: Graduated or Flat. If Graduated, list in table form. Is there a level, below which, no tax is collected at all? (Due date(s)) ex: Collected how often? When? (Collecting Body) ex: Who does the collecting? How do they accept payment? What form does the payment need to take? (Required Materials) -Information- ex:  What information must you have before beginning the process of personal income tax preparation? -Forms- ex: What forms are going to be necessary for the process? -Fees- ex: Will there be fees? For what? How much? -Materials/Resources- ex: What kind of programs exist that can aid in this process? These should be listed in a table format that compares their relative advantages/disadvantages, such as cost, frequency of updates and accuracy(I do not know if there is a rating for this). What kind of aid might someone enlist in this process? These shold be listed in a table format as well. (Methods) ex: This section would go over the methods of preparing personal income tax. What is taxable income? What isn't? What counts as a tax exemption? Are there government programs/incentives that could be taken advantage of? How? I have no idea how to organize this section.

I'd be very grateful for any help you wikidemics can provide.

As far as learning projects go, I'm thinking a set of info/papers/account info/etc (whatever's needed) could be created for a John Doe type guy, or for several John Doe type guys. Learners would do his/her/their taxes and submit them for critiques. These would be updated when laws are changed, along with the course contents themselves. What do you think?

title: Tax preparation


text:
Instructional design/Module 1

Module 1 - Key Points: Introduction
The following video will guide teach you how to discern the key points from any topic you are developing training for. If you wish to follow the script as you view the module, click the notes tab on the top left hand corner of the screen.

Key Points Presentation

A downloadable copy of the presentation is available below.

Key Points PPT

Steps to identifying key points

 * 1) Decide if you will find an SME or study to be the SME.
 * 2) Identify the key points to be taught.
 * 3) Questions to ask:
 * Who is your audience?
 * What parts of the topic does the student need to know?
 * Which key points need to be emphasized?
 * Does the student need to understand the why, how, what behind a key point?
 * Does the student need to know how to apply that knowledge in different situations or is it more procedural?
 * What are the student learning time constraints?
 * What are the available deliverable formats?

title: Instructional design/Module 1


text:
Instructional design/Module 2

Module 2 – Enhanced Assignments
The disengaged online student is a problem that all online IDs face. The following video module will guide to enhance your online training development. If you wish to follow the script as you view the module, click the notes tab on the top left hand corner of the screen.

title: Instructional design/Module 2


text:
Instructional design/Online engagement/Key Points

Introduction | Module 1 | Module 2| Exercise Next

Module 1 - Key Points: Introduction
The following video will guide teach you how to discern the key points from any topic you are developing training for. If you wish to follow the script as you view the module, click the notes tab on the top left hand corner of the screen.

Key Points Presentation

A downloadable copy of the presentation is available below.

Key Points PPT

Steps to identifying key points

 * 1) Decide if you will find an SME or study to be the SME.
 * 2) Identify the key points to be taught.
 * 3) Questions to ask:
 * Who is your audience?
 * What parts of the topic does the student need to know?
 * Which key points need to be emphasized?
 * Does the student need to understand the why, how, what behind a key point?
 * Does the student need to know how to apply that knowledge in different situations or is it more procedural?
 * What are the student learning time constraints?
 * What are the available deliverable formats?


text:
Instructional design/Online engagement/Enhanced Assignments

Introduction | Module 1 | Module 2| Exercise Next

Module 2 – Enhanced Assignments
The disengaged online student is a problem that all online IDs face. The following video module will guide to enhance your online training development. If you wish to follow the script as you view the module, click the notes tab on the top left hand corner of the screen.

Key Points Presentation

A downloadable copy of the presentation is available below.

Key Points PPT

Steps to creating an enhanced engagement assignment

 * 1) Identify which key points needs to be included in the learning problem.
 * 2) Allow students to create their own learning problem.
 * 3) Allow students apply that new knowledge to their learning problem.


text:
Instructional design/Exercise

Introduction | Module 1 | Module 2| Exercise

Module 3 – Exercise
Now that you’ve completed module 1 & 2, let’s practice what you’ve learnt. Choose a topic and list out key points that you think a student should know about it and create an enhanced assignment from module 2. Add to the below table by clicking the edit button to the top right of the page.

title: Instructional design/Exercise


text:
User:Tacirci

File:Flag of Turkey.svg File:Flag-map of Turkey.svg|<<<<>>>>Türkei<<<<>>>> File:Flag of Turkey.gif </gallery

title: User:Tacirci


text:
User:Egm6341.s11.team4.yang/hw7

=HW 7.0 problem: redo 5.2 and 5.4 cauchy noise=

Problem Statement
Refer to lecture notes [[media:nm1.s11.mtg25.djvu|25-2]] and [[media:nm1.s11.mtg25.djvu|26-1]] for detailed description 4). Run LSSM with Cauchy Random Noise

Problem Statement
Refer to lecture slide [[media:nm1.s11.mtg29.djvu|mtg-29]] for the problem statement. c).For u=0.5 Cauchy noise and c=$$1.5*{{C}_{cr}}$$, plot $$\underline$$ For u=0.5 Cauchy noise and c=$$1.5*{{C}_{cr}}$$, plot $$\underline$$

This problem was solved by shengfeng yang

title: User:Egm6341.s11.team4.yang/hw7


title: File:5.2-7.0.jpg


title: File:5.2-7.0=1.jpg


title: File:5.2-7.0=1.5.jpg


title: File:5.4-7.0.jpg


text:
User talk:Fspgrundtvig

 Hello Fspgrundtvig, 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.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.
 * Explore Wikiversity with the links to your left.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.
 * Follow Wikiversity on twitter (http://twitter.com/Wikiversity) and identi.ca (http://identi.ca/group/wikiversity).

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! --Leutha 09:54, 19 April 2011 (UTC)

Create User page
Hi Frands, I've added material about creating a Userpage a new page and a little more to the ASGELE portal. I would be interested in some feedback.Leutha 12:35, 3 July 2011 (UTC)

Response
Hi, I have responded on my talk page.Leutha 12:46, 17 October 2011 (UTC)

Links page
Hi, here's a link to the page I did: User:Leutha/gaming node. Leutha 14:51, 15 December 2011 (UTC)

Categories
Good to chat yesterday. The European Parliament Financial Transaction Tax simulation is automatically placed in the Portal:Simulation and Gaming Archive, under Policy making within the EU, because the section on Accessing Games and Simulations in the Simulation and Gaming Archive automatically lists items in categories, and when expanded, sub categories.Leutha (talk) 12:54, 22 November 2012 (UTC)

title: User talk:Fspgrundtvig


text:
Red Herrings (simulation)

Introduction
This is a simulation exercise, in which you will take the role of one of three interest groups concerned with fishing in the North Sea – regulators, scientists and fishers. The task is for regulators to decide what to do about cod quotas in the North Sea. Quotas, allocated by EU regulators, are maximum amounts of fish which a fishing fleet is allowed to catch each year. Should quotas remain the same? Should they be increased to allow fishermen to make a living? Or should a total ban of North Sea cod fishing be put in place?

Learning outcomes
Those participating in this simulation will
 * consider the issue of a fishing ban from one of three perspectives.
 * interact with other teams who have differing priorities.
 * the fishermen and scientists willdevelop communication skills in terms of formulating a view point and expressing it to a group of people with an opposing viewpoint and the group of regulators who have a neutral point of view.
 * The regulators will have to consider the arguments and come up with a way forward which both the fishermen and scientists find acceptable.
 * Ths simulation also enables participants to develop negotiating skills

Resources
To print, click on resource and then click on Full resolution . Provide printouts in each row to the people playing the relevant role:

title: Red Herrings (simulation)


text:
File:Solarpaneel.JPG

Licensing
for school project

title: File:Solarpaneel.JPG


text:
File:Schakelingx.JPG

Licensing
for school project

title: File:Schakelingx.JPG


text:
File:Schakeling2.JPG

Licensing
for school project

title: File:Schakeling2.JPG


text:
File:Sankeydiagr.JPG

Licensing
for school project

title: File:Sankeydiagr.JPG


text:
User talk:C breezy

hey baby girl (:

title: User talk:C breezy


text:
Semiconductor Devices Lecture Aid

Solid and Semiconductor

 * Crystal ([[Media:Crystal.H.1.20170311.pdf |H1.pdf]])

Energy Band

 * Energy Band ([[Media:EnergyBand.1.A.20150923.pdf |pdf]])

PN Junction

 * Conduction.1.A ([[Media:Conduction.1.A.pdf |pdf]])
 * PN Junction.1.A ([[Media:PN Junction.1.A.pdf |pdf]])

PN Junction Note
 * PN Junction Note ([[Media:PNJunction.H.1.20170403.pdf |H1.pdf]])

BJT
Bijunction Transistor

MOSFET
go to [ Electrical_%26_Computer_Engineering_Studies ]
 * MOSFET.1.A ([[Media:MOSFET.1.A.pdf |pdf]])

title: Semiconductor Devices Lecture Aid


title: File:Conduction.1.A.pdf


text: File:PN Junction.1.A.pdf

title: File:PN Junction.1.A.pdf


text:
Talk:Chemical Equilibrium

Are curly brackets { } supposed to represent concentration (square) brackets [ ]? I have not seen this notation before.

title: Talk:Chemical Equilibrium


title: File:Translations.1.A.pdf


text:
User:EGM6341.s11.TEAM1.Sujin.HW7

Problem 3: Solving the Logistic Equation Using Inconsistent Trap-Simpson's rule & Newton-Raphson
 Solved without assistance

Given
Refer Lecture slide 41-1 for problem statement

The logistic equation for population dynamics is given by

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

$$\displaystyle \dot{x}(t) = f(x) = rx \left(1-\frac{x}{x_{max}}\right) $$ $$
 * $$\displaystyle

(7.1)
 *
 * }

Here, $$\displaystyle x_{max}=15;\ r = 1.4,\ $$ and $$\displaystyle t \in [0,20] $$

The analytical solution is given by the S10 lecture note38-3 as following;

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

$$\displaystyle x(t) = \frac{x_0x_{max}e^{rt}}{x_{max} + x_0(e^{rt}-1)} $$ $$
 * $$\displaystyle

(7.2)
 *
 * }

Logistic equation defined by eqn 7.1 for two initial conditions, $$\displaystyle x_0=3\ and\ x_0 = 9,\ $$, can be solved by using Inconsistent(trapez.) Simpson's rule given by the s10 lecture note 38-3,

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

$$\displaystyle z_{i+1} = z_{i} + \frac{h/2}{3} \left[f_i + 4f_{(i+1/2)} +f_{i+1}\right] $$ $$
 * $$\displaystyle

(7.3)
 *
 * }

The Inconsistent rule (Trapezoidal) is given by,

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

$$\displaystyle z_{i+1/2} = \frac{1}{2} [z_{i} + z_{i+1}] $$

(7.4)
 *
 * }

Solution
By considering the eqn 7.1 in which $$\displaystyle f_{(i+1/2)}$$ is a function of $$\displaystyle z_{i}$$ and $$\displaystyle z_{i+1}$$, where $$\displaystyle z_i, z_{i+1} \rightarrow$$ values of $$\displaystyle x$$ at $$\displaystyle t = t_i$$ and $$\displaystyle t = t_{i+1}$$. This is an initial value problem, $$\displaystyle z_i$$ is known. So, eqn 7.3 as a whole becomes a function of $$\displaystyle z_{i+1}$$ given by,

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

$$\displaystyle F(z_{i+1}) = 0 $$ $$
 * $$\displaystyle

(7.5) $$\displaystyle z_{i+1}$$ of eqn 7.5 can be found by using Newton-Raphson method
 *
 * }
 * {| style="width:100%" border="0" align="left"

$$\displaystyle z_{i+1}^{(k+1)} = z_{i+1}^{(k)} - \left[\frac{dF(z_{i+1}^{(k)})}{dz}\right]^{-1} F(z_{i+1}^{(k)}) $$ $$
 * $$\displaystyle

(7.6) The Newton-Raphson iteration, starting with an initial guess as $$\displaystyle z_{i+1}^{0} = z_{i}$$, is stopped once the absolute tolerance reaches appropriate criterion.
 *
 * }

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

$$\displaystyle \begin{align} AbsTol &= ||z_{i+1}^{(k+1)} - z_{i+1}^{(k)}|| \le 10^{-6} \end{align} $$
 * }
 * }

$$\displaystyle x_0 = 3 $$ :- A comparison on $$\displaystyle h $$ values:

$$\displaystyle x_0 = 9 $$ :- A comparison on $$\displaystyle h $$ values:

title: User:EGM6341.s11.TEAM1.Sujin.HW7


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text:
Introduction to Italian/Lesson 13

I mesi dell'anno - Months of the year
gennaio - January febbraio - February marzo - March aprile - April maggio - May giugno - June luglio - July agosto - August settembre - September ottobre - October novembre - November dicembre - December

Remember: the months in Italian are usually written in lower case (but they can be written in upper case, too, though it happens more rarely)

I giorni della settimana - Days of the week
lunedì - Monday martedì - Tuesday mercoledì - Wednesday giovedì - Thursday venerdì - Friday sabato - Saturday domenica - Sunday

Le stagioni - Seasons
Primavera - Spring Estate - Summer Autunno - Autumn Inverno - Winter

I colori - Colors
rosso - red verde - green giallo - yellow rosa - pink nero - black bianco - white blu - blue marrone - brown viola - purple arancione - orange azzurro or celeste - light blue grigio - grey

Le Regioni d'Italia – Regions of Italy

 * (l')Abruzzo/(gli) Abruzzi
 * (la) Basilicata
 * (la) Calabria
 * (la) Campania
 * (l')Emilia-Romagna
 * (l')Emilia
 * (la) Romagna
 * (il) Friuli-Venezia Giulia
 * (il) Friuli
 * (la) Venezia Giulia – Giulian Venice
 * (il) Lazio
 * (la) Liguria
 * (la) Lombardia – Lombardy
 * (le) Marche
 * (il) Molise
 * (la) Puglia – Apulia
 * (la) Sardegna – Sardinia
 * (la) Sicilia – Sicily
 * (il) Trentino-Alto Adige – Südtirol
 * (il) Trentino
 * (l')Alto Adige
 * (il) Veneto
 * (la) Valle d’Aosta – Aosta Valley

title: Introduction to Italian/Lesson 13


text: File:File-Egm6342.s11.team3.hw7.sskim.png


text:
User:GFHund

Ich bin Gerhard Hund (* 1932 in Leipzig, Deutschland), ein Mathematiker (1955) und Schachspieler (1948).--GFHund 03:41, 20 April 2011 (UTC)

title: User:GFHund


title: File:Nm1.s11.team1.HW7.fig1.png


title: File:Nm1.s11.team1.HW7.fig2.png


text:
Rogers 19 point theory

This project is to interpret Carl Rogers' 19 point personality theory, which is the objective phenomenological field theory that is the basis of American humanistic psychology. Many humanists oppose the theory because it is objective as they insist that Rogerian therapy is subjective as it is about cleints' personal, and hence, subjective experience.

The paradox in therapy is that highly objectified research has provided the most humanistic approach separately in Europe and the New World. In other words, American existentialism is homegrown. The battle has been between a complex model of highly-aggregated experiential information VS the minutely-focused "slow boat" experimental model taken by the behaviorists.

Rogers' theory is highly beneficial because it creates groundwork for psychotherapy as a way for people to get on track irrespective of their problems. As it is organic, it does not look at a person as having abnormalities, but as an organism that is attempting to self-actualize the best way it can. Specifically, if an organism is allowed to interrelate to socially evolve naturally, especially during infancy, everything should turn out well irrespective of "vulnerabilities" such as birth defects. In the cases where early interrelation has been maligned, then the path to health is to restore the interrelatedness that should have been established during early growth.

The problem is that the theory looks like he wrote it on the subway during rush hour. I created a Rogers 19 point theory class. I took all the important words, and I believe they can be reassembled in any way to have meaning, and I think, anyway, it might be fun and therapeutic. (So far, people have tried to interpret the theory line by line, which is, of course, not conceptual.)

Absolutely key to Rogers' theory, or model, is the idea of "self and experience" where "process" connects the two. When the process that connects self and experience is broken, a person is said to have "fragile process." In humanistic terms, fragile process describes the very difficult to treat personality disorders (from the DSM), such as Narcissistic and Borderline Personality Disorders.

When the connecting process between experience and self is fundamentally broken, we become disassociated, which describes PDs. This creates, for instance, loneliness in a crowd--a way to describe modern cities. So I think that the fragile process that has evolved from Rogers' initial model is applicable to all as we all get disassociated at some time or other, and it isn't fun.

The topic of process has been exciting for a long time: David Hume thought process was God, Freud equated it to ego.

Reference material
Here is recently written background material on Rogers' model to help with the exercise:
 * "Self, experience, and the connecting process" in Rogers' personality theory (APA style)
 * Rogers 19 point theory of personality
 * How field theory works in Gestalt
 * Rogers' conditions for therapeutic change from his theories
 * "A Gestalt" as a functional object

The Game
Link up these words and phrases taken from the 19 point model into a few normal sentences (here, if you want to):

 * -individuals
 * -organisms
 * -exist
 * -continually changing world of experience
 * -phenomenal field
 * -centre
 * -reacts to the field as it is experienced and perceived
 * -perceptual field is "reality" for the individual
 * -reacts
 * -organized whole
 * -phenomenal field
 * -portion of the total perceptual field
 * -differentiated as the self
 * -result
 * -interaction
 * -environment
 * -evolutional interaction with others
 * -structure of the self is formed
 * -fluid
 * -consistent conceptual pattern of perceptions of characteristics and relationships of the "I" or the "me", together with values attached to these concepts
 * -basic tendency
 * -striving
 * -actualize, maintain and enhance the experiencing organism
 * -vantage point
 * -understanding
 * -internal frame of reference
 * -individual
 * -behavior
 * -goal-directed
 * -attempt
 * -satisfy
 * -needs as experienced
 * -field
 * -perceived
 * -emotion accompanies
 * -behavior
 * -kind
 * -emotion
 * -related
 * -perceived significance
 * -behavior
 * -maintenance
 * -enhancement
 * -values experienced directly
 * -interjected or taken over from others
 * -perceived
 * -distorted as if experienced directly
 * -experiences occur
 * -life
 * -self
 * -ignored
 * -no perceived relationship
 * -self structure
 * -denied symbolization
 * -distorted symbolization
 * -experience is inconsistent with the structure of the self
 * -ways of behaving
 * -adopted
 * -consistent
 * -concept of self
 * -behavior
 * -may be brought about
 * -organic experiences
 * -needs
 * -not been symbolized
 * -may be inconsistent
 * -structure of the self
 * -not "owned"
 * -psychological adjustment exists
 * -concept of the self
 * -all the sensory and visceral experiences of the organism
 * -assimilated
 * -symbolic level
 * -consistent relationship with the concept of self
 * -psychological maladjustment
 * -organism denies awareness
 * -significant sensory and visceral experiences
 * -not symbolized and organized into the gestalt of the self structure
 * -basic
 * -potential psychological tension
 * -experience which is inconsistent
 * -organization of the structure of the self
 * -perceived as a threat
 * -more
 * -perceptions
 * -more rigidly the self structure is organized to maintain itself
 * -conditions
 * -absence of threat to the self structure
 * -experiences
 * -inconsistent with it
 * -perceived
 * -examined
 * -structure of self revised
 * -assimilate
 * -include
 * -experiences
 * -individual perceives and accepts
 * -consistent
 * -integrated system all
 * -sensory and visceral experiences
 * -more understanding of others and is more accepting of others as separate individuals
 * -perceives and accepts into his self structure more
 * -organic experiences
 * -replacing
 * -present value system
 * -based extensively
 * -introjections
 * -distortedly symbolized
 * -continuing organismic valuing process

title: Rogers 19 point theory


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College graduates have brought us the Enron collapse, the BP oil spill, the Chernobyl disaster, subprime mortgages along with the housing foreclosures and near financial collapse that followed, and loss of the Aral Sea. Perhaps knowledge is not enough. Perhaps we need to make wiser choices. --Lbeaumont 11:33, 20 April 2011 (UTC)

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hi how are you what are you up to on this fine sunny day my freind