text:
ACR/ACR&Universe

Cosmic Evolution
Which, by itself, means the evolution of everything within the universe. With other words it is the story of whatever happened, happens and will happen in the universe. Which, according to what we can make out today, produces all kind of forms of matter. By what we can see we also realize that there are forms which are too small for us to “see” and others for which we have no technology yet to know. From this two the ones we cannot see are the more troublesome. Because whatever is there, what is too small for us to see, but we know have to be there, are the ones on which the ones we can see, are build from. With other words we don’t know on what our world is build on. But how can we specify how things work if we do not even know from what it is build? Luckily, or unluckily, we have fantasy. Which means, we can take what we know or guess and put it together in any form we like or ‘fantasize’ and see what would that lead to. Nature uses the same tactic by using the chance of movement to bring forms of matter together (action) and let the natural principles (circumstances) deside what will evolve from its reaction. The big difference is that nature cannot use matter which is fictional. Which, by its nature, always produces a physical product which is the next step of the evolution and from which the next step can be evolved. But exactly this restriction enables humans, if they follow the same limitations, to reconstruct what has happen in the universe. Here is an attempt to do just that:

Evolution - growing naturally
Naturally, in this case, means that whatever happens is purely physical. There are no immediate or long term predefined purposes which guide what is happening, only the possibilities that a physical action can happen. And there are absolute no restrictions on what can happen. The facts, that the evolution of matter goes from small to big, and that there is no knowing change of principles, makes the 'reconstruction' of the development which has happened in the universe, as known today, possible. For this we need something which can evolve (grow) and the circumstances which makes it evolve. Therefore we start out with an endless space which is filled with an infinite amount of the smallest matter (mass) possible – which is named here the UrParticle [UP]. But this is enough. Matter, no matter how small, seem to have the possibility to bind with another matter to form something bigger. But since the smallest possible matter, a space filled with an uniform mass, almost certain rules out that it could produce a pulling force another natural way needed to be found to explain how single masses can stay together and grow. And to figure out how this is possible we need to combine what can be learned from the Newton Cradle and what else nature teaches us. And there is only one physical phenomenon which could do that - movement (energy). The universe is big enough so that we can use quite a big part to explain nature, without reaching its borders. Therefore for our purpose there is no exchange with any matter from outside of the universe. Which means we deal with a fixed amount of matter even if the universe is, for our feeling of size, endless. But we also always have to have in mind that, since there are no predefined purposes, there are no 'mistakes'. Everything is right, no matter what is the outcome. There is also never a 'starting over' - once it started it just goes on. Therefore whatever are the principles today were the principles at the assumed start and will be in the future.

Possible growing steps
Here are the basic steps on how matter could have evolved from its UP-state to galaxies or humans: Step 1: growth by size - 3 dimensional If matter is moving then there is the possibility that it collides (make physical contact) with another one. And there is the possibility that after a collision they stay together and grow (build) into a new form of matter. This is possible if the movements of the matters are in harmony, which means that their frequencies, directions, strength and phase match. They can be added without that they move the UPs apart. Since there is no pulling force active between the UPs at this stage it needs pressure of the environment to keep the UPs together. But probably both UPs had their movements changed since an exchange of movement (adaptation) took place. If they actually stay together then they now move in unison in a new combined form of movement. When there is enough empty space available in the environment then this movement can be a straight one with the appropriate direction and speed. But if the environment restricts or even prohibits free movement then the physical movement takes on an appropriate form or stops in which case the physical movement has to be passed on as a potential movement (potential energy). If none of the UPs can pass this potential movement on to the environment then it is passed on between the UPs which have collided and these may start to oscillate the potential movement between them. Since, in this case, this passing on of potential movement is directed toward each other it gives the impression that the UPs are drawn to each other. They physically stay together now on their own (no pressure needed). This rises the energy level of the now physically joint matter which can be termed that they are exited or charged or have a higher temperature. Therefore what is otherwise a one time happening, can become continuous while the two, now joint UPs, exchange a potential movement between each other. Specifically if energy is constantly supplied from the environment which restricts their movement and applies pressure on them. Now, the new form of matter does not physically move and actually occupies the smallest space it can. Since the physical movement before the collision needed lots of space, room is now left into which the environment (pressure) can bring other UPs from outside. Which, under the right conditions, could join into the energy exchange and therefore grow the matter again. E.g. into a ring like formation where the potential movement is passed on in a circle. With the possibility that this ring formations can grow into any size. If any matter in the ring is not physically touching the matter to which it is passing the energy then the passing on is interrupted (resistance). Therefore, on the next pass of energy (cycle), the inertia of the UP which has physical space available will transform the potential movement into a physical move, directed toward the next matter in line, until it touches the next matter and then pass on the remaining potential movement. Opening space in the ring will be closed consecutively by movements of matter, with the additonal effect of a shift in the phase, until no space is left. This passing on of energy in ring form provides the possibilities that any matter in this ring can be a member of more rings which pass energy through it. The amount of energies which can be passed through an UP in different directions at the same time is probably limited because at some point this energies will cancel each other out or add up to a form which is impossible to pass on. But the possibility is there. And if rings join then the matter can grow in different directions into matter which is build of interconnecting rings. In multiring formations energy is passed on in different direction which might impose some physical regulations on the form which can be build. This would include of where rings can be joint, how many rings can be interconnected and in which direction energy can be passed before the form falls apart again. Which means that the many passed on energies within the form must sum up to an equilibrium of energies which is characteristic for this form - an energy signature [ES]. Which means that if an energy is passed on to this formation it could be imprinted with a specific sequence of energies which are active within when it is passed through. Or the reaction will be specific to this sequence if it is acted on regarding to changes, redirection or split up (functionality). But this also provides the possibility that stable forms of matter can be build which react with always the same specific functions when the circumstances are the same (provides a specific feedback). Once an ES is reached through the building and arranging of rings then the growth of size of this form ends. The urparticles [UP] of matter have grown into an element matter [EM].

Step 2: growth by repetition - 1 dimensional The possibilities to grow does not end with the EM. In the overall shape of an EM there might be spots which enable to grow out rings which might not change the ES of the EM but provide the possibility to pass on energies or ES with some added control. Which might enable to grow another form there in consequence of the sequence of energy which is passed (ES). If EM connect direct together then the physical selection is limited. Both sides need to be connected to a place in their surface which is able to pass on the same kind of energy (movement). If they do not have that circumstances provided then they cannot hook up. And of course there has to be a chance that the EMs can come close enough and are right positioned to each other. Another possibility is that EMs have developed which can act as connecting centers which pass all sorts of energies in different directions without overloading them with its own. Since ESs are in essence a sequence of passed on energies they can be taken in at different connections (maybe one energy at a time), compared or handled otherwise and send out again in maybe one or more directions. Which would increase the intelligenz level of the manipulation of passed on energies or ESs. Which would provide the possibilities to handle EMs in symbolic form like a word. Forms of EM can be duplicated, complimented or compared without the need to now how it is physically build up. This has been translated into a sequence of energies. That this functionality is used by nature is shown through the C-atom in the amino group of an amino-acid (molecule) which can connect to four EMs. Which also shows the possibilities such Connecting EMs [CEM] provide. E.g. the immense amount of functionalities or data, which systems of EMs provide, which can be placed in any sequence to provide unlimited possibilities. This can be seen as the beginning of the development of segmented systems and the basis of functionalities which are developed to a much higher level and applied in a brain. Step 3: growth by repetition - 2 dimensional If growth has produced a chain of EMs and if they are connected by CEMs, then there is the possibility that the CEMs which form links to the CEMs of another chain. Or they can ‘grow’ a duplicate of the systems of EM, which they connect, out of the vacant connections. Which would form a blanket of chains or a 2 dimensional memory. Step 4: growth by building 3 dimensional systems If 1 or 2 dimensional shapes are physically moved (or only partially) by pressure of the environment than there are the possibilities that they could form any kind of 3 dimensional shapes. Some with functionality others with none. Step 5: growth by physical adaptation When EMs have grown into shapes which constitute a 3 dimensional physical shape then this physical shape can provide circumstances which has to be met before its energy-function can be performed. Like an EM-blanket forms a tube through which matter, charged with a specific energy, has to pass to get to the place where it triggers a specific reaction. This adapts the movement of the EM to the physical shape of the environment which in turn could change the shape or functionality of the travelling EM. A form of feedback from the environment which adds conditions to the action. A higher level of intelligenz. Step 6: growth by teaching When EMs have grown into a shape which is able to control the movement of other EMs then it has the possibility to control the growth of this matter. E.g. if a tube of EMs guide the movement of an EM along its path then it can make it connect to certain EMs which are part of the tube and imprint some parts of the ES on the passing EM. Which would create an EM system with a predefined set of functionality. Which means that the possibilities to react are now placed besides the possibilities to act and, with a constant source of energies, can keep on going to act/react to fullfil a predefined function. Which also means that the evolution is not any more a function of chance but now happens with a purpose. A next level of intelligenz. Step 1 to N: There are many steps which lead to the growing which nature has reached today and are most likely not specific at all. With a lot of fantasy many can be envisioned. But the ACR is not a theory of fantasy. Therefore the basics of the ‘Growth by movement’ has to be worked out first and then the steps of growth will either become obvious or have to be worked out individual as well.

title: ACR/ACR&Universe


text:
ACR/Nature

Nature - its possibilities and restrictions
When 'Nature' is considered then it is usually meant everything what is happening throughout the universe without the interference of humans. Which makes it a prime suspect of the curiosity of humans and also a victim of human fantasy. Fantasy is very useful to keep natural evolutions going, as applied by nature, but can be very hindering or lead into the wrong direction if unchecked by reality. Therefore if fantasy is used to explain how nature works then it need to be taken into consideration what nature can do and what it can't do. Which is basically defined through the fact that nature is a physical phenomenon. Which means it is limited through its physical properties. This physical properties are all there is when times are considered when the abilities of humans still need to be evolved and are therefore far into the future. Which would be: but: Which is true for the past, the present and the future even when creatures with nonphysical abilities are able to interfere, like humans. Because humans are a part of the natural process.
 * There are physical masses and their movements
 * There are no goals
 * There are no mistakes
 * There are no limitations
 * There is no starting over

Natural Evolution
If the term evolution is considered for the phenomenon of something growing on itself then it can be used to describe what is happening through nature. Where the main character would be that changes are producing something bigger from what was before which is basically an adapted version of what was before. Which lead to the conclusion that there was sometime a stage when it had to start with something which was in its smallest version and in circumstances (environment) which enabled that changes could take place. Since basically everything we know is build out of a physical substance which we call matter it would be logic to assume that this something in its smallest form is matter. Which happens to have the property of mass and a function which is called inertia when it is moving. It also happens that the main reason, why some form of evolution takes place, seem to be movement. Therefore the circumstances which need to be present are that there are matters in their smallest form, as many as there could be, and they need to be moving in a space which can accommodate any changes which are happening to them. A universe filled with smallest possible particles of matter. And if the matter particles are also moving then the phenomenon of the natural evoluion takes place which can also be called life. How the universe, its filling of matter and their movement came into existence can only be guessed at the current stage of knowledge. But it can be a realistic starting point which, through evolution, can lead to the present stage of the universe. But the sequence of changes, called evolution, can only, since the starting situation is in its most primitive form, happen according to realistic principles. Which include the physical properties matter provides and the principles layed out above. Which would mainly separate them into the physical and possibility principles. The physical principles are: The most important possibility principle, which encompasses all others is: This principles lead to the, which seem to be the only, form of changes which can happen naturally: an action, which happens in the present of circumstances which lead to a reaction which can only be of one kind for this action and circumstances (ACR). And this, since the reaction is at the same time an action, have to happen in a sequence which never stops as long as the universe, filled with matter, exists.
 * There can only be a change which can be accommodated through matter
 * There can only be one change at a time
 * There is no predefined goal which needs to be achieved.

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

title: ACR/Nature


text:
Numerical Analysis/Newton's method exercises

In this section we are going to look at a method for approximating solutions to equations. We all know that equations need to be solved on occasion and in fact we’ve solved quite a few equations ourselves to this point. we were able to actually find the solutions, but it’s not always possible to do that exactly and/or do the work by hand. That is where this application comes into play. So, let’s see what this application is all about. Let’s suppose that we want to approximate the solution to f(x0) and let’s also suppose that we have somehow found an initial approximation to this solution say, x0. This initial approximation is probably not all that good and so we’d like to find a better approximation. This is easy enough to do. First we will get the tangent line to f(x) at x0.

y=f(x0)+f'(x0)(x-x0)

Example:

1)Find a nonzero root of equation x2-sin x using newton's method.

Solution:

Let us take our initial guess as .5

f'(x)=2x-cos x.

now we will use newton's formula to find x2=x1-f'(x1)f(x1)

x2=.5-sin(.5)2(.5)-cos(.5)=2.374

In this way we get x3=1.4702,x4=1.0594,x5=0.9057,x6=0.8727,x7=0.8766,x8=0.8767

2)Find an approximation to 51/2 to ten decimal places with starting point as 2.

51/2 is the root of f(x)=x2-5

so now using the formula,x2=x1-f'(x1)f(x1)

we get,x1=2

x2=2.25

x3=2.236111111111

x4=2.236067977915

x5=2.236067977499

x6=2.236067977499

Exercise

Use Newton's method to approxmate the zero's of function f(x) = x3+2x+2 Solution: 0.358

Algorithm Failures: In some cases the conditions on function necessary for convergence are satisfied, but the point chosen as the initial point is not in the interval where the method converges. In such cases a different method, such as bisection, should be used to obtain a better estimate for the zero to use as an initial point.

Example:Use x0=1 to find the approximation to the solution to f(x)=x1/3.

using Newton's Formula xn+1=-2xn.

x1=-2

x2=4

x3=-8

x4=16 ..

Because each step produces new approxmation that is far from the root, method fails.

Exercise

Find the roots of the polynomial f(x)=ex-2x with starting point as 1 Solution: Method fails

Quiz
{Newton's method for solving f(x) = 0 is} +xm+1=xm-f(xm)/f'(xm) -xm+1=xm+f(xm)/f'(xm) -xm+1=xm-f'(xm)/f(xm) -xm+1=xm-f(xm)/f''(xm)

{Newtons method is used to approximate the extrema of a function} -true +false

{Newton's method fails if f'(a) is equal to zero and a is used as an initial value in the iteration process} +true -false

{The square root of 346 using newtons method is} +18.6010752 -18.5010752 -18.7010752 -18.4010752

{Find the roots of the equation x2-4x+15 for x=2 as the initial point using newtons method} -5.316624 -1.316624 -3.667891 +Method fails


text:
ACR/ACR&Creature

The evolution of a living creature
To call the existance of a living creature an evolution might not seem appropriate. But when it is considered that the time, from when a living creature starts to grow until it dies, is a constant time of changes then it has many similarities with e.g. the cosmic evolution. The big difference would be that the changes which happen during the cosmic evolution are many at a much broader range which might never end while the evolution of a living creature can be separated in two big stages and it will end after the second one. The first stage is always the building of the body of the creature and the second is the function of this body (living) in an environment which might more or less change from moment to moment. Each step has its specific characteristic.

The creation of a living body
The creation of a living body is basically an action of repetition. Since an evolution cannot happen unless every action/circumstance/reaction [ACR] is remembered all the ACRs, which were happening while this creature was evolving, have been memorized. In blocks called 'genes'. Which can be build from one ACR or many ACRs which enables the genes to provide the right 'blueprint' for the body in question. Since a natural evolution does not have the tools to 'know' which ACR is unnecessary for the building of the last developed creature no stored ACR is erased. It might always be used for another version of the creature or even for another creature. That this system is used can be figured out by the way the building instructions, the genes, are arranged - in a simple sequencial way (Chromosomes). This enables specialized molecules (carriers of the action) to pass along the genes and select from them the right instructions (build up an action) in the right order which are needed to build the latest version of this specimen. The most important character of this stage is that everything needed for the proper function of the body itself is provided. This includes the parts of the body which are provided but are not fully physical or mentally functional yet. With other words the body is now an entity on its own and does not need any help from outside to live. Just the right circumstances which need to be satisfied by the environment.

The Live-Test of the finished body
But what the first stage cannot know is into what kind of environment this body will be born. Therefore certain functionalities can only be provided physically but cannot be prepared for what they will actually do. One sample for this is the speech of a human. He will have a mouth to create sounds and ears to hear the sounds which his mouth can make. But what kind of sounds will be needed/used in the environment, into which the human is born, cannot be predicted. She will also have some limbs which have the ablity to perform specific movements but what movements this limbs will perform together cannot be predicted. And she will have a pair of eyes to 'see' her environment, but what will she see? When a creature is ready to function in an environment then it needs to start a process which adapts its body to this environment. Depending on the circumstances, which are usually simmilar for certain levels of creatures, this process can last for just a few seconds to years. This process is basically a learning process and done in a play like fashion. Its task is e.g. to coordinate the movements of the limbs to the body when the body needs to be moved. Or to coordinate the movements of the limbs to the picture of the environment, presented trough the eyes, to move something within the environment. But this kind of tasks cannot be learned right away. There are other preliminary tasks which need to be learned before the main tasks can be accomplished. E.g. the eyes need to learn to 'see' first. Muscles need to be exercised to gain the necessary strenght and agility. And while this is being done the progress has to be recorded to enable the refinements necessary. For this the first stage equips the body with senses and a brain. The senses can be separated into the senses needed to assure the proper function of the body (unconsious) and the one which are always 'noticable' (concious). The senses are the abilities to measure bodily functions, to translates this measurments into electrical or chemical pulses (information). This information is then send through the nervous system to a central place where they can be stored, called up and used as needed. This place is the brain. The brain needs to be able to mark and store sequences of elements of maybe electrical or chemical types. This sequences can then be recalled to be compared, changed or used to form another sequence of the same elements or to be assimilated with another type of elements. E.g elements which are provided by other senses. But this sequences of elements cannot be provided for certain tasks, which the body needs to evolve, by the first stage of the evolution of the body. Therefore for this tasks the brain has no 'information' ready when the body starts to live in the environment into which it was born. Every part of this tasks: physical, functional and controls need to be evolved (established and learned) first. An evolution which lasts for the whole duration of the life of the body. The way the brain accomplishes all the tasks is by separating the information it gets and stores into the concious and the unconcious information. The concious informations are the ones it gets moment by moment trough the 'noticeable' senses which includes also a picture of the environment in which the body is in.

title: ACR/ACR&Creature


text:
ACR/ACR&Genes

The role of genes
Genes have been found to be the 'blueprints' for the living body. Which means they hold the information necessary to create the right cells in the right formation to perform a specific function which, together with other functions, enable the living body to live. For this the genes must be available when the body is build. And they must be available in such a form that the information, which they hold, can be aquired from them at the right time in the right sequence. To enable all this nature places them in every smallest building block which are used, cells, which ensures that there is always a set of genes available. It also places them in a special, although very simple, form, sequential like on a string. This ensures that any gene can be found wherever it is on this string. And the necessary information can be gathered any way needed.

The natural use of genes
Genes could be used in a straight forward, simple fashion: one cell type - one gene; one formation of cells - one gene; one function - one gene. But nature has found a better, more flexible way. It combines all possibilities, which are related, into one gene and let the circumstances deside which of them is appropriate at each request. What that means is best shown through the following sample which was presented by a NOVA show on TV: Two scientist found the genes, which are responsible for the creation of the eyes, in two very different species. Which had naturally also very different eyes. To find out more about the genes they removed the 'eye' gene on one specimen and replaced it with the 'eye' gene of the other. And then let the body develop. To their surprises the specimen, whith the 'wrong' eye-gene, developed its natural eyes anyway. It would not have developed an eye if there was no eye-gene. To conclude in the ACR fashion:
 * Action:
 * The development of the body. This could be short described that at this stage a specific molecule travels along the chromosomes, select the genes which hold the 'blueprints' necessary to build the cells which form an eye, get the blueprints and bring them to the place in the body where the eye will be build.

 * Circumstances:
 * the information which is embedded in the specific molecule which enables it to select the right genes and collect the right blueprints from them.

 * Reaction:
 * The building of the right eye in the right place of this specific body.

Conclusion: Eye-genes have the blueprints for any eye already build. And the body, defined by the chromosomes, equippes the specific molecules with the necessary circumstances before it reaches the eye-gene. Which ensures that always the right eye is build for the body. This would also suggest that genes cannot simply be assumed to be good for only one specific building process but for many possibilities which need to be specified before the gene is consulted.

The interpretation of ‘mutations’ in genes
When the subject of the evolution of humans are studied than one of the big mysteries which stands in the way is that the sequence of DNAs (also known as nucleotides A,C,G and T), is not passed on from generation to generation in the same order than it was received by the donating generation (some might have been replaced with a different type). Which means that the new generation, which is in development will not function quite in the same way as the donating generation has. This seems to be a logical use of the genes to enable the genes to absorb necessary changes of the specimen to adapt to a changing environment. What would enable a different explanation of this effect is the way this replacement of DNAs is interpreted. On one hand it could be seen as the faulty copying of the sequence (meaning it is random and unwanted), on the other hand it could be that the changes happen during the lifetime of the donating parent and are now passed on in an controlled fashion which will have a controlled outcome. If interpreted in the first way (as mutations) then all the data received through scientific means would be rendered useless, because no cause and therefore outcome could be established. While the latter would fit much better in the way the natural evolution can be explained. As a process which has started randomly, does not “know” what is happening and has no “goal” on its own, but has the ability to evolve e.g. matter from a single matter to very complicated formations of matter without loosing its functionality. In order to do this it is restricted into a simple form of operation, like the sequence of a process described through action, circumstances and reaction (ACR). Which enables the possibility of a change and also the discarding of the change if it does not fulfil the requirements of the adaptation (selection). Which is the only way a non-conscious (non-intelligent) system can keep on functioning. And which would also enable scientists to include POV, which he could not if he would deal with ‘unwanted mutations’. E.g. while 'mutations' could be happening at a predetermined happening and therefore could suggest a certain regularity, life induced changes would be completely unpredictable. On the other hand the reaction to life induced changes could hint on what the change in the environment has been and therefore lead to other conclusions.

title: ACR/ACR&Genes


text:
ACR/ACR&Physics

Energy = Movement
When the Author tried to figure an explanation of how evolution is possible then the obvious answer was through energy. But he could not find a real answer to what energy was. The only way to include power in happenings was to use the word 'energy' which could be exchanged by everybody in his mind with what he 'knows' energy is or in mathematical formulas by expressing it through an abstract number. But I don't know of any detailed explanation of every energy by itself. To separate different forms of energies the words work, heat, movement, potential, kinetic and others are used to give a better description. Even each one can be expressed mathematically through its symptoms its creates the only one, which seem to have the possibility to separate each one individually, is movement. Each movement can be defined through its mass, speed, direction, strength and occurrence. Some of it are interdependent and can therefore be left out when another one is present. But to use energy as an answer on how something physical, which cannot see, hear or feel like a human, know with what energy it is confronted with (to which it can react specific) there has to be a set of physical properties which make it possible to distinguish between energies. And the properties which describe each movement individually fit also energy. Therefore, it seems save to exchange energy (abstract) with movement (real) where traditionally energy has been used for an explanation.

The build up of matter
When it was discovered that the atom is not a solid matter but build of other particles then the question of how the atom is constructed needed to be answered. There were two models which were in consideration. The 'plum pudding' and the 'nuclear' model. Short, a space filled with matter and a mostly empty space. To get the answer, in 1911 Ratherford made a test. He shot alpha particles into a very thin gold plate and concluded his answer from the effect it had. Most alpha particles went right through and only some where deflected in different direction. He figured that since most went through the atom must be empty and the one which were deflected struck the nucleus. Or so he and other people thought. Apparently he had not seen a Newton Cradle in action. Otherwise he might have needed to consider that the alpha particles passed on a characteristic energy to the gold plate, which was passed through and catapulted like particles out on the other side. Which would have been another possibility and the test could not have been answered. Since the passed on energy can also be diverted in different direction the reflections could have been explained. ('Fundamentals Of Chemistry', ISBN 0-673-07876-0).

The Newton Cradle [NC]
What the Newton Cradle demonstrates is that an energy can be passed or being transformed through moving or stationary matter. Since the type of action, which is performed on one end, is decisive on what kind of reaction is evoked on the other end. The conclusion can be made that the energy, traveling through the matters, also carries some information. Which leads to the possibility that any energy, passed on or transformed can be overloaded with energy inherent in the matter through which it travels. Which gives it the possibility to be used as a form of communication – passing on information. A communication which only needs forms of matter to send or receive. A possibility which the ACR tries to exploid. Sample and possible prove: Since the matters, which form the newton cradle, are not made of one single mass is it almost certain that there are energies active within this matter which hold the matters physically together. Therefore, what could be assumed is that the energy which is passed through might have picked up other ‘energy’ information on its way through the matters. Energies, which might reveal the natural frequencies the matters in the line have, according to their size and build up. And if this natural frequencies are already known then it might also reveal the position of that matter in the line. The ACR uses this relevant action to explain how the world could have been built with passing energies from matter to matter.

title: ACR/ACR&Physics


text: User:Ha98574/Railroad/Korea/Korea Metropolian Subway Gaya Line


text: User:Ha98574/Railroad/Korea/Korea Metropolian Subway Gaya Line/Template


text: User:Ha98574/Railroad/Korea/Korea Metropolian Subway Ganggyeong Line


text: User:Ha98574/Railroad/Korea/Korea Metropolian Subway Ganggyeong Line/Template


text:
Sport Performance Analysis

Sport performance analysis is the science of observing and recording events and actions during competition and training environments. The content in Sport Performance Analysis examines techniques used to assess and monitor sporting performance using real-time and lapsed-time coding systems. The 3 main notational coding methods (time-motion, tactical and technical) are considered with reference to analysis during match-play and training activities. Effective video capturing techniques across a range of sports and the appropriate use of video in the coaching process will be explored. The implementation of commercially available coding systems will be demonstrated along with the use of innovative technologies relating to performance analysis and sport analytics. Sport Performance Analysis provides fundamental insight into a rapidly growing area of sport science.

Objectives

 * 1) Demonstrate a sound knowledge of performance analysis principles in sport
 * 2) An ability to conduct performance analysis research relevant to sport
 * 3) An understanding of meaningful feedback / information in sport

Topics

 * Background to performance analysis in sport
 * The role of a performance analyst in sport
 * Defining key performance indicators in sport using flow charts
 * Performance analysis techniques including time-motion, tactical and technical analysis
 * Dynamic systems and momentum in performance analysis
 * Reliability and validity in performance analysis
 * Designing coding systems to analyse sporting performance
 * Monitoring training performance in sport
 * Performance analysis of officials and umpires in sport
 * Providing meaningful feedback to coaches

Resources

 * Hughes, M., and Franks, I. M. (2008). The essentials of performance analysis: an introduction. New York, NY: Routledge

Assignments

 * Research proposal that will outline why and how the research project will be conducted (research presented in the research poster). The research proposal must contain a literature review and outline the methods to be used in completing the research project. The major research project must address a gap in the published performance analysis research and the research proposal should identify the need for such research.
 * Research poster that presents the findings of the research project. The research poster must be of conference quality and contain an Introduction, Methods, Results, Conclusion and Reference section. An Abstract (no more than 250 words) must be submitted separate to the research poster.
 * Research progression requires weekly updates regarding the completion of the research project.

Coordinator
Keane Wheeler

title: Sport Performance Analysis


text:
Introduction to Sport Biomechanics

Biomechanics is a prominent sport science that explains the mechanical aspects of human movement. Introduction to Sport Biomechanics demonstrates the fundamental mechanical factors associated with human performance in sport. The description of gross movement patterns as well the forces associated with such movement will be explored with respect to linear and angular motion. Newton's Laws of motion will also be explored with specific application to a sporting context.

The basic principles of mechanical engineering will then be discussed in relation to sporting equipment design and the structure of the human body. This leads to an in-depth examination of gait biomechanics including walking, running, sprinting and changing directions (agility) relevant to sport. The kinetic link principles (effective force summation) will then be explored across various types of sports including those involving throwing, kicking and hitting.

Finally, basic fluid dynamic principles are considered including concepts of drag and propulsion in sports such as rowing, swimming and motor racing. Introduction to Sport Biomechanics provides a sound base of knowledge to then complete Applied Sport Biomechanics.

Objectives

 * 1) Demonstrate a thorough understanding of mechanical factors that influence human movement
 * 2) Demonstrate an applied knowledge of musculoskeletal biomechanics relevant to sport
 * 3) Ability to conduct qualitative technique analysis and provide meaningful feedback to improve sporting performance

Topics

 * Background to sport biomechanics
 * Describing motion including forms, linear, angular and projectile
 * Levers and torque associated with human movement
 * Application of Newton's Laws of Motion in sport
 * Basic Engineering principles and the application to orthopaedic biomechanics
 * Functional anatomy including spine, upper and lower extremity biomechanics
 * Gait biomechanics including walking, running and sport-specific movement patterns
 * Kinetic link principle applied to effective force summation in throwing, hitting and kicking sports
 * Fluid dynamic principles including concepts of drag and propulsion

Resources

 * Hall, S.J. Basic Biomechanics (2006). Boston: McGraw-Hill.

 * /Resources/ - A range of resources to help with understanding the topics, completing the assignments, and submitting for assessment.

Assignments

 * Examination assessing application of biomechanical concepts and principles in sport
 * Presentation involving a critical review of the scientific literature surrounding an area of interest in biomechanics
 * Portfolio demonstrating an ability to conduct and present biomechanics research effectively

Coordinator
Keane Wheeler

title: Introduction to Sport Biomechanics


text:
Introduction to Sport Coaching

Sport coaching requires sound understanding of athlete development and effective communication strategies. Introduction to Sport Coaching provides fundamental understanding of the coaching and officiating process. A background to coaching will be presented initially and will consider the history of sport coaching and the requirements of sport coaching, including coaching ethics, code of conduct and the various leadership / management styles of a coach. Effective group management techniques will then be demonstrated in various practical sporting contexts. The principles of training will then be considered including the 5 S's of training and the use of game sense training in relevant sports. Sport coaching will then be examined with respect to specific population groups, including children, older adults, females, athletes with a disability and athletes from various cultural backgrounds. Finally, the future of sport coaching will be considered. The concepts explored in Introduction to Sport Coaching are expanded in Sport Strength and Conditioning and Assessment of Sport Coaching.

Objectives

 * 1) Apply a basic coaching principles and plans as well as group management and time strategies to teach sport skills
 * 2) Demonstrate an understanding of the training requirements for various sports and different populations
 * 3) Conduct and interpret the results from a range of fitness and sport-specific tests

Topics

 * Background to sport coaching including accreditation and history
 * Risk management and ethics in sport coaching
 * The role of a coach including leadership and management in sport
 * Group management techniques in sport coaching
 * Application of training principles and designing training plans
 * Long-term athlete development models and the application in sport coaching
 * Coaching various populations groups including children, older adults, female athletes, athletes with a disability and athletes from different cultures
 * Future of sport coaching

Resources

 * ASC, Beginning Coaching, Australian Sports Commission

Assignments

 * Australian Sports Commission online coach development courses including beginning coaching principles, introductory level officiating, Play by the Rules Coaches and Officials.
 * Coaching practical to assess the effective delivery of a sport skills session.
 * Examination to assess the application of concepts presented in Introduction to Sport Coaching.

Coordinator
Keane Wheeler

title: Introduction to Sport Coaching


text:
Assessment of Sport Coaching

Assessing the effectiveness of sport coaching techniques is essential to enhance communication between athletes and their coaches. Assessment of Sport Coaching provides an in-depth understanding of the coach evaluation process using an evidence-based approach. Sport coaching will be examined using reflective and analytical skills to improve performance. The importance of feedback / feed-forward to improving coaching performance will be discussed with reference to appropriate methods to receive such information. The use of video analysis of coaching behaviours as well as implementing a coach mentoring program will be explored along with methods of coach goal-setting and self-reflection. Athlete learning styles are then considered with respect to the coach as an effective presenter and facilitator. Finally, the role of a head coach as a member of a high performance athlete support team is explored along with the other members of a high performance structure (e.g. high performance manager, assistant coaches, administrators and development officers). Assessment of Sport Coaching provides an individualised development program appropriate for coaches in a high performance setting.

Key Outcomes

 * 1) Demonstrated ability to research and present current evidence-based information relevant to sport coaching
 * 2) Demonstrated proficiency in assessing coaching performance including self-evaluation
 * 3) Demonstrated ability to provide meaningful feedback to coaches through a mentoring program

Topics

 * Purpose of assessment in sport coaching
 * Appropriate coach assessment techniques and the role of video analysis
 * Providing feedback to coaches through effective communication
 * Developing a coach assessment tool and using a team to provide feedback
 * Review coaching performance and the provision of information
 * Mentoring strategies to improve sport coaching performance
 * Setting goals as a sport coach
 * Mentor-coach communication
 * Coach self-reflection
 * Effective coaching techniques including the coach as a presenter
 * Designing training programs appropriate to athlete learning styles
 * Planning and designing presentations in sport coaching
 * Appropriate use of learning aids in sport coaching
 * Facilitating coaching workshops
 * Coach communication with a high performance support team and managers
 * The coach as a high performance manager
 * The coach as a professional

Resources

 * Assessor training participant manual (2006), Australian Sports Commission.
 * Presenter/Facilitator Manual (2009), Australian Sports Commission.

Assignments

 * Research seminar that critically reviews the literature surrounding an area of interest relevant to coaching.
 * Coaching practical session and self-reflection
 * Coaching log book and practical tasks

Coordinator
Keane Wheeler

title: Assessment of Sport Coaching


text:
Numerical Analysis/Exercises and quiz

In this section we are going to look at a method for approximating solutions to equations. We all know that equations need to be solved on occasion and in fact we’ve solved quite a few equations ourselves to this point. we were able to actually find the solutions, but it’s not always possible to do that exactly and/or do the work by hand. That is where this application comes into play. So, let’s see what this application is all about. Let’s suppose that we want to approximate the solution to f(x0) and let’s also suppose that we have somehow found an initial approximation to this solution say, x0. This initial approximation is probably not all that good and so we’d like to find a better approximation. This is easy enough to do. First we will get the tangent line to f(x) at x0.

y=f(x0)+f'(x0)(x-x0).

Then we get the formula xn+1=xn-f(xn)/f'(xn).

Example:

1)Find a nonzero root of equation x2-sin x using newton's method.

Solution:

Let us take our initial guess as .5

f'(x)=2x-cos x.

now we will use newton's formula to find x2

x2=.5-sin(.5)2(.5)-cos(.5)=2.374

In this way we get x3=1.4702,x4=1.0594,x5=0.9057,x6=0.8727,x7=0.8766,x8=0.8767

2)Find an approximation to 51/2 to ten decimal places with starting point as 2.

51/2 is the root of f(x)=x2-5

so now using the formula,

we get,x1=2

x2=2.25

x3=2.236111111111

x4=2.236067977915

x5=2.236067977499

x6=2.236067977499

Exercise

Use Newton's method to approxmate the zero's of function f(x) = x3+2x+2 Solution: 0.358

Method Fails: In some cases the conditions on function necessary for convergence are satisfied, but the point chosen as the initial point is not in the interval where the method converges. In such cases a different method, such as bisection, should be used to obtain a better estimate for the zero to use as an initial point.

Example:Use x0=1 to find the approximation to the solution to f(x)=x1/3.

using Newton's Formula xn+1=-2xn.

x1=-2

x2=4

x3=-8

x4=16 ..

Because each step produces new approximation that is far from the root, method fails.

Exercise

Find the roots of the polynomial f(x)=ex-2x with starting point as 1 Solution: Method fails

Quiz
{Newton's method for solving f(x) = 0 is} +xm+1=xm-f(xm)/f'(xm) -xm+1=xm+f(xm)/f'(xm) -xm+1=xm-f'(xm)/f(xm) -xm+1=xm-f(xm)/f''(xm)

{Newtons method is used to approximate the extrema of a function} -true +false

{Newton's method fails if f'(a) is equal to zero and a is used as an initial value in the iteration process} +true -false

{The square root of 346 using newtons method is} +18.6010752 -18.5010752 -18.7010752 -18.4010752

{Find the roots of the equation x2-4x+15 for x=2 as the initial point using newtons method} -5.316624 -1.316624 -3.667891 +Method fails

title: Numerical Analysis/Exercises and quiz


text:
Sport Strength and Conditioning

Strength and conditioning is an integral part of any athletic development program across all sports. Sport Strength and Conditioning considers the various methods used to physically prepare athletes in sport. Training methods used to enhance the capacity of the 3 energy systems (phosphate, lactic and aerobic) is considered in the context of periodisation and long-term athlete development. Appropriate training modalities will be presented for aerobic, strength, power, speed and agility capacities. Effective prehabilitation and rehabilitation methods will then be explored with respect to injury prevention and the purpose of warm-up and cool-downs and different types of flexibility sessions. Taping and measuring loading / peaking index during training phases will be demonstrated along with participation in various recovery strategies. Finally, designing long-term athlete development programs for athletes of diverse physical requirements is considered in the context of strength and conditioning. Sport Strength and Conditioning provides an in-depth understanding of athletic physical preparation including appropriate testing and training methods.

Key Outcomes

 * 1) Demonstrated understanding of appropriate preparation and development of athletes using an evidence-based approach
 * 2) Ability to coach core lifts in sport strength and conditioning
 * 3) Ability to incorporate an appropriate strength and conditioning program in a long-term athlete development program

Topics

 * Background to sport strength and conditioning
 * Background to athlete assessment and conducting a needs analysis
 * Periodisation in sport strength and conditioning
 * Energy system training
 * Aerobic training methods
 * Strength development
 * Power development
 * Speed and agility
 * Flexibility and mobility
 * The role of warm-ups and cool-downs as a sport trainer
 * Recovery methods
 * Tapering and the role of the strength and conditioning coach
 * Rehabilitation and injury prevention in strength and conditioning
 * Strength and conditioning and special populations
 * Future of strength and conditioning

Resources

 * National Strength and Conditioning Association,(2008), Essentials of strength training and conditioning, Baechle, T. R. and Earle R.W. (Ed.), Champaign, IL: Human Kinetics.
 * Fleck, S. J. and Kraemer, W. J., (2004), Designing resistance training programs (3rd edn), Champaign, IL: Human Kinetics.
 * Wilmore, J. H., Costill, D.L. and Kenney, W. H., (2008), Physiology of sport and exercise (4th edn), Champaign, IL: Human Kinetics.

Assignments

 * Coaching log book and practical tasks including coaching the core lists relevant to strength and conditioning
 * Periodisation and planning assessment for a chosen sport including needs analysis, background research, training goals and testing, yearly plan, training programs appropriate to yearly plan, justification of training methods, monitoring of program and references
 * Examination of course material

Coordinator
Keane Wheeler

title: Sport Strength and Conditioning


text:
Numerical Analysis/Quiz

{What is the order of Runge Kutta 4th order method} + O(h4) - O(h3) - O(h2) - none

{Which of the following methods shows stability for stiff equation} -Both Rk2 & Multi step methods -Rk2 method -Multi Step Method +Both Methods are unstable for stiff equation

{Order of Euler's Method is less than multi step method} -True +False

title: Numerical Analysis/Quiz


text:
Introduction to Sport Psychology

Sport psychology involves the study of the psychological factors associated with participation and performance in sport. Introduction to Sport Psychology provides a fundamental understanding of how the various aspects of psychology can be applied to sport participation. Assessment of personality types will be discussed relating to sport participation. This will be expanded to discuss motivation and leadership contribution to sport participation as well as the relationship between anxiety and arousal with respect to enhancing sporting performance. Psychological skills training will then be demonstrated, including goal setting, team dynamics, imagery, positive self-in relation to achieving peak sporting performance. Introduction to Sport Psychology provides a greater understanding of the psychological processes of individual athletes and team dynamics to enhance sporting performance.

Key Outcomes

 * 1) Demonstrate an ability to collect and interpret sport psychology data
 * 2) Ability to incorporate sport psychology in an athlete development program
 * 3) Critically evaluate current sport psychology research and present this in a meaningful way to coaches

Topics

 * Background to sport psychology
 * Background to Psychological Skills Training
 * Personality and skill acquisition in sport
 * Motivation
 * Exercise behaviour and adherence
 * Stress and anxiety in sport
 * Arousal regulation in sport
 * Self-confidence and sporting performance
 * Imagery and concentration in sport
 * Social skills and team dynamics
 * Team and individual leadership
 * Coaching communication

Resources

 * Weinberg, R. S., & Gould, D. (Eds.). (2007). Foundations of sport and exercise psychology (4th ed.). Champaign, Ill: Human Kinetics.
 * Morris, J., & Summers, J. (Eds.). (2004). Sport psychology: Theory, applications and issues (2nd ed.). Brisbane: John Wiley and Sons.
 * Williams, J.M. (Ed.). (2006). Applied sport psychology: Personal growth to peak experience(5th ed.). Sydney: McGraw-Hill.
 * Burton, L. J. (2007). An interactive approach to writing essays and research reports in psychology (2nd ed.). Milton, Qld, Australia: John Wiley & Sons.
 * Morris, T, Spittle, M & Watt (2005) Imagery In sport. Sydney. Human Kinetics.

Assignments

 * Examination of course material
 * Research report that presents the results of a sport psychology experiment conducted during the course.

Coordinator
Keane Wheeler

title: Introduction to Sport Psychology


text:
Applied Sport Biomechanics

Sport Biomechanists work with high performance athletes and sporting teams to improve skills and abilities. Applied Sport Biomechanics provides an in-depth understanding of biomechanical principles in sport and the appropriate methodologies used to provide feedback to athletes. The importance of qualitative observations is explored initially with reference to providing meaningful information to coaches and athletes. The techniques for recording and analysing sporting skills are then demonstrated and starts with basic videography and biomechanical analysis using freeware available on the internet.

The use of more advanced video assessment techniques are then explored through optoelectric systems and high speed cameras. Quantitative biomechanical data collection techniques including force plates, pressure analysis, accelerometers and EMG are demonstrated with reference to improving sporting performance and injury prevention. Finally, the technological advances relevant to biomechanics are explored. Applied Sport Biomechanics promotes a working knowledge of the equipment and methods used to assess human mechanics in sport.

Key Outcomes

 * 1) Demonstrated understanding of sport biomechanics assessment techniques
 * 2) Demonstrated experience analysing various sporting skills using appropriate biomechanical methods
 * 3) Ability to conduct effective biomechanical research

Topics

 * Background to sport biomechanics
 * Conducting qualitative biomechanical assessments
 * Fundamentals of video analysis in biomechanics
 * Kinematic analysis using video and coaching tools
 * Automatic tracking systems in biomechanics
 * Force and pressure analysis
 * Accelerometers and linear encoders
 * Filtering and measurement error in biomechanics
 * Isokinetic dynamometry
 * EMG analysis
 * Computer modelling and silhouette tracking
 * Injury biomechanics including pre and rehabilitation analysis
 * Ergonomics and postural analysis
 * Future of biomechanical analysis

Resources

 * Grimshaw, P., Lees, A., Fowler, N. and Burden, A. (2007). Sport and Exercise Biomechanics. Taylor and Francis, London.

Assignments

 * Analysis of digitizing reliability
 * Qualitative analysis of a selected sport skill
 * Research report based on kinetic data
 * Examination of course material

Coordinator
Keane Wheeler

title: Applied Sport Biomechanics


text:
Numerical Analysis/ODE Exercises

Euler's Method
1) Solve differential equation y' = 3t2y in [0,1] using Euler's method with n=10,y(0)=1. Solution: We need to find the solution ODE y' = 3t2y using Euler's method.

yn+1=yn+hf(tn,yn)

We divide time span with number points to find the step size h.

h=tmax-tmin/n=1-0/10=0.1

y1=y0+hf(t0,y0)

=1.000000+.1x3x02x1

=1.000000

y2=y1+hf(t1,y1)

=1.000000+.1x3x0.12x1.000000

=1.003000

y3=y2+hf(t2,y2)

=1.003000+.1x3x0.22x1.003000

=1.0150360

y4=y3+hf(t3,y3)

=1.01503600+.1x3x0.32x1.0150360

=1.0424420

y5=y4+hf(t4,y4)

=1.0424420+.1x3x0.42x1.0424420

=1.0924792

y6=y5+hf(t5,y5)

=1.0924792+.1x3x0.52x1.0924792

=1.1744151

y7=y6+hf(t6,y6)

=1.1744151+.1x3x0.62x1.1744151

=1.3012520

y8=y7+hf(t7,y7)

=1.3012520+.1x3x0.72x1.3012520

=1.4925360

y9=y8+hf(t8,y8)

=1.4925360+.1x3x0.82x1.4925360

=1.7791029

y10=y9+hf(t9,y9)

=1.77910290+.1x3x0.92x1.7791029

=2.2114249

Runge-Kutta Second Order Method (RK2)
2) Solve differential equation y' = 2ty in [0,1] using RK2 method yn+1=yn+h/4(k1+3K2) where k1=f(tn,yn),k2=f(tn+2/3h,yn+2/3hk1)with n=5,y(0)=1.

Solution: We need to find the solution ODE y' = 2ty using RK second order method.

yn+1=yn+h/4(k1+3K2)

k1=f(tn,yn)

k2=f(tn+2/3h,yn+2/3hk1)

y1=y0+h/4(k1+3K2)

k1=f(t0,y0)

=2x0.0x1

=0

k2=f(t0+2/3h,y0+2/3hk1)

=2x2/3x0.2x1

=0.26667

y1=y0+h/4(k1+3K2)

=1+0.2/4(0+3x0.26667)

=1.04

y2=y1+h/4(k1+3K2)

k1=f(t1,y1)

=2x0.2x1.04

=0.416

k2=f(t1+2/3h,y1+2/3hk1)

=f(0.2+0.13333,1.04+0.0554667)

=0.730304

y2=y1+h/4(k1+3K2)

=1.04+.2/4(0.416+3x0.730304)

=1.17035

y3=y2+h/4(k1+3K2)

k1=f(t2,y2)

=2x0.4x1.17035

=0.93648

k2=f(t2+2/3h,y2+2/3hk1)

=f(0.4+2/3(0.2),1.17035+2/3(0.2)(0.93648))

=1.34316

y3=y2+h/4(k1+3K2)

=1.17035+0.2/4(0.93648+3(1.34318))

=1.41865

y4=y3+h/4(k1+3K2)

k1=f(t3,y3)

=f(0.6,1.41865)

= 1.70238

k2=f(t3+2/3h,y3+2/3xhxk1)

=f(.6+.133333,1.41865+2/3(0.2)(1.70238))

=1.64563

y4=y3+h/4(k1+3K2)

=1.41865+.2/4(1.72038+3(1.64563))

=1.75061

y5=y4+h/4(k1+3Kundefined)

k1=f(t4,y4)

=f(0.8,1.75061)

= 2.800098

k2=f(t4+2/3h,y4+2/3hk1)

=f(0.8+.133333,1.75061+2/3(0.2)(2.800098))

=3.96472

y5=y4+h/4(k1+3K2)

=1.75061+.2/4(2.800098+3(3.96472))

=2.48532

y6=y5+h/4(k1+3K2)

k1=f(t5,y5)

=f(1,2.48532)

= 4.97064

k2=f(t5+2/3h,y5+2/3hk1)

=f(1+.133333,2.48532+2/3(0.2)(4.97064))

=7.1356

y6=y5+h/4(k1+3K2)

=2.48532+.2/4(4.97064+3(7.1356))

=3.80420

Adams-Bashforth Two Step
3) Solve differential equation y' = 2ty in [0,1] using Two step Adams Bashforth method with n=5,y(0)=1.

Solution: We need to find the solution ODE y' = 2ty using Two step Adams Bashforth method.

yn+2 = yn+1 + 3/2 x hf(tn+1,yn+1)-1/2 x hf(tn,yn)

Calculating h=1-0/5=0.2

We need two approxmations for calculating y2

Calculating y1 using RK second order method.

y1=1.04

y2 = y1 + 3/2 x hf(t1,y1)-1/2 x hf(t0,y0)

=1.04+3/2(.2)f(0.2,1.04)-1/2(0.2)f(0.0,1)

=1.1648

y3 = y2 + 3/2 x hf(t2,y2)-1/2 x hf(t1,y1)

=1.1648+3/2(.2)f(0.4,1.1648)-1/2(0.2)f(0.2,1.04)

=1.40275

y4 = y3 + 3/2 x hf(t2,y2)-1/2 x hf(t2,y2)

=1.40275+3/2(.2)f(0.6,1.40275)-1/2(0.2)f(0.4,1.1648)

=1.81456

y5 = y4 + 3/2 x hf(t4,y4)-1/2 x hf(t3,y3)

=1.81456+3/2(.2)f(0.8,1.81456)-1/2(0.2)f(0.6,1.40275)

=2.21333

y6 = y5 + 3/2 x hf(t5,y5)-1/2 x hf(t4,y4)

=2.21333+3/2(.2)f(1,2.21333)-1/2(0.2)f(0.8,1.81456)

=3.25098

Runge-Kutta Fourth Order (RK4)
4) Solve the ODE $$y'=2ty$$ on the interval $$[0,1]$$, using three steps of the Runge-Kutta 4 method with $$n=2$$ and $$y(0)=1$$.

Solution: The RK4 method is given by

where
 * $$k_1 = hf(t_, y_n),$$
 * $$k_2 = hf(t_n + h/2, y_n + k_1/2),$$
 * $$k_3 = hf(t_n + h/2, y_n + k_2/2),$$ and
 * $$k_4 = hf(t_n + h/2, y_n + k_3).$$

The step size $$h$$, is found using the interval $$[0,1]$$ and $$n=2,$$ and is given by

$$ h = \frac{(1 - 0)}{n} = \frac{(1 - 0)}{2} = 0.5. $$

And we know that $$y_0 = y(0) = 1,$$ so now we can now use the RK4 formula from Eqn.($$) to find each following iteration.

$$y_1:$$

We must first solve for k1, k2, k3, and k4.

k1 = (.5)(2)(0)(1) = 0;

k2 = (.5)(2)(.25)(1) = .25;

k3 = (.5)(.25)(1.125) = .140625;

k4 = (.5)(.5)(1.140625) = .28515625.

Applying these k, and y0 to Eqn.($$), we find

y1 = 1 + 1/6(0 + (2)(.25) + (2)(.140625) + .28515625) = 1.177734375.

$$y_2:$$

Similarly, we find k1, k2, k3, and k4.

k1 = (.5)(.5)(1.177734375) = .294433593;

k2 = (.5)(.75)(1.324951172) = .496856689

k3 = (.5)(.75)(1.42616272) = .534811019;

k4 = (.5)(1)(1.712545394) = .856272697.

Applying these k, and y1 to Eqn.($$), we find

y2 = 1.177734375 + 1/6(.294433593 + (2)(.496856689) + (2)(.534811019) + .856272697) = 1.713407993

$$y_3:$$

Repeating in a similar fashion, we find y3.

k1 = (.5)(1)(1.713407993) = .856703996;

k2 = (.5)(1.25)(2.141759991) = 1.338599995;

k3 = (.5)(1.25)(2.38270799) = 1.489192494;

k4 = (.5)(1.5)(3.202600487) = 2.401950365.

Applying these k, and y2 to Eqn.($$), we find

y3 = 1.713407993 + 1/6(.856703996 + (2)(1.338599995) = (2)(1.489192494) + 2.401950365) = 3.19911455,

with $$y_3 = 3.19911455$$ as our final answer.

title: Numerical Analysis/ODE Exercises


text:
User:Charles Jeffrey Danoff/Collaborative Lesson Planning Spring 2010 Notes

''Many of the ideas I had about Collaborative Lesson Planning I thought were original may have been for me, but not for humans. I'd like to find those things and build upon them.''

I hope to teach Collaborative Lesson Planning again in the next round of P2PU courses in January, 2010. Might also want to take this course: Composing free and open online educational resources 2011 offered at the same time.

Get ideas from
Unlocking the promise of open educational resources and its signers, incl.
 * User:Leighblackall:
 * Going naked - Openism and freedom in academia
 * Composing educational resources
 * UCI OpenCourseWare
 * Online Instructor Training
 * The Open University OpenLearn Learning Space
 * Creating open educational resources
 * LeMill
 * Learning resources and copyright
 * Composing free and open online educational resources
 * Composing free and open online educational resources/Critique and feedback
 * Hewlett Foundation page on [http://www.hewlett.org/programs/education-program/open-educational-resources Open Educational Resources.
 * WikiEducator
 * OER Handbook
 * Richard Baraniuk
 * TED Talk via COERLL
 * professors from kyoto to cambridge building an electrical engineering textbook
 * create, mix, rip, burn
 * move pub to comm authored materials that're modular, or published to individual class
 * So, what we have to do is get it right from the very beginning and what we have to do is find an intellectual property framework that makes sharing safe, and makes it easily understandable, and the inspiration here is taken from open-source software, things like Linux and the GPL.
 * anyone develop their own peer-review process
 * How to Learn Something for Nothing via Richard B.
 * Cape Town Open Education Declaration:
 * Dr Panagiota Alevizou
 * Open Education Resources: An Overview for Teachers
 * Teachers w/o borders
 * http://www.teacherswithoutborders.org/resources/open-educational-resources
 * Open Educational Resources
 * Why remix an Open Educational Resource?
 * Free to Learn Guide
 * connexions
 * http://siyavula.org.za/downloads/

Ready for Students to Start Next Course

 * Review/case-study of round 1 of collab lesson planning
 * intro to concept written out w/ref's to a lot of the stuff above

ideas

 * direct the course more towards tefl teachers?

cf.

 * For Exposure, Universities Put Courses on the Web By, D.D. GUTTENPLAN


text:
User talk:PaulJRichardson

 Hello PaulJRichardson, 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. To get started, you may

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Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:17, 9 November 2010 (UTC)

title: User talk:PaulJRichardson


text:
User:ViralStorm~enwikiversity

I am a newcomer/lurker who is partaking of Wikiversity articles in an attempt to conduct self-guided studies.

My name is Richard Belleville, 22, from British Columbia, Canada. Some of my interests include computing, language, mathematics, mythology, philosophy, psychology, and some other varying disciplines of science.

title: User:ViralStorm~enwikiversity


text:
User talk:Mahmoudasem

 Hello Mahmoudasem, 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. To get started, you may

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title: User talk:Mahmoudasem


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User talk:NVSBLmonster

 Hello NVSBLmonster, 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. To get started, you may

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title: User talk:NVSBLmonster


text:
User:JenniferLockett

Jennifer Lockett is a teacher of Ancient and Medieval History at Trinity Valley School in Fort Worth, Texas. She received her graduate degree in archaeology at the University of California at Los Angeles and did her undergraduate work at the University of Miami (with a junior year abroad at the University of Edinburgh in Scotland). She is an avid student of the potential of technology in the classroom. You can check out her webpage and many podcasts at IndianaJen.com or follow her on twitter @JenniferLockett

title: User:JenniferLockett


text:
User talk:JenniferLockett

 Hello JenniferLockett, 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. To get started, you may

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title: User talk:JenniferLockett


text:
Fixed Point Iteration

What is Fixed Point Iteration?
http://en.wikipedia.org/wiki/Fixed_point_iteration

Algorithm
To find a solution to p = g(p) given an initial approximation p0.

INPUT: Initial approximation p0; tolerance TOL; maximum number of iterations N0. OUTPUT: approximate solution p or message of failure.

Step 1: Set i =1

Step 2: While $$i \leq No$$ Steps 3 - 6

Step 3: Set p = g(p0)

Step 4: If $$|p - po| \leq TOL$$ then OUTPUT (p); (The procedure was successful). STOP.

Step 5: Set i = i + 1

Step 6: Set p0 = p. (Update p0)

Step 7: OUTPUT ('The method failed after N0 iterations'). STOP.

Examples
1. Find square root of 2 accurate till third decimal (10-3). The initial point is 2.

Answer.

 * We know that,

 * x0 = 2

 * xn+1 = 0.5 * ( x0 + (2/ x0))

 * x1 = 0.5 * (2 + (2/2)) = 1.5

 * x2 = 0.5 * (1.5 + (2/1.5)) = 1.416

 * x3 = 0.5 * (1.416 + (2/1.416)) = 1.4142

 * We found out the square root of 2 accurate till 4th decimal in 3 steps.

As implemented in python
The following python code implements the functionality of this section.

The two statements under "# Original estimate of square root (initial approximation.)" provide a pretty good estimate of initial approximation of $$p_0.$$ Unfortunately they are computationally intensive and add significant time to completion of function

The code within the  loop is both very simple and very fast. $$$$ $$$$$$$$$$$$

Tolerance  is determined by the desired precision of the set by the code that invokes this function. The following optional code is valid in debugging mode. It provides information about what happens during the  loop and verifies that the calculated square root is in fact within tolerance. Compared with python's method  this code is slow for small values of precision. However, for numbers containing 550 decimal digits or more, this code (without debugging) is a little faster than python's method.

sqrt(2) with 1,000 places of decimals
Result was produced with only 12 passes through loop.

2. Consider the iteration pn+1 = g(p0) when the function g(x) = 1 + x - x2/4 is used. The fixed points can be found by solving equations x = g(x). The two solutions (fixed points of g) are x = -2 and x = 2. The derivative of the function is g'(x) = 1 - x/2, and there are only two cases to consider

 * Case 1 (P = -2):
 * Start with P0 = -2.05
 * We get
 * P1 = -2.100625
 * P2 = -2.20378135
 * ..... $$\infty$$
 * Therefore, if |g'(x)| > 1 then sequence will not converge to P = -2

 * Case 2 (P = 2):
 * Start with P0 = 1.6
 * We get
 * P1 = 1.96
 * P2 = 1.9996
 * ..... 2
 * Therefore, if |g'(x)| < 1 then sequence will converge to P = 2

Applications
Two methods in which Fixed point technique is used:

1. Newton Raphson Method

 * Formula
 * $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$

 * where,
 * xn - initial point
 * f(xn) is the value of the function at that point
 * f'(xn) is the value of the differentiated function at that point.

 * Plug all these values into the above equation to get xn+1. It becomes the next initial point. Repeat until you get a point within an acceptable degree of error

2. Secant Method

 * Used to avoid differentiated form in Newton Raphson's method. Only problem is you need two initial points for this method (xn and xn-1)

 * Formula
 * $$x_n=x_{n-1}-f(x_{n-1})\frac{x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}$$

 * Similar to Newton Raphson's method plug in all values to generate next approximation.

Exercises
If 'f' is continuous and 'x' is a fixed point on 'f' then what is 'f(x)'? Solution: f(x) = x

Find the square root of 0.5 using fixed point iteration? Initial point 0.1 and tolerance 10-3

Calculating x1 i.e. 1st Iteration Solution: x1 = 2.55

Calculating x2 i.e. 2nd Iteration Solution: x2 = 1.373

Calculating x3 i.e. 3rd Iteration Solution: x3 = 0.8685

Calculating x4 i.e. 4th Iteration Solution: x4 = 0.722

Calculating x5 i.e. 5th Iteration Solution: x5 = 0.707

In above problem, how many iterations or steps are needed to achieve an accuracy of 10-8 Solution: Iteration: 6

Quizzes
{Is there any possibility that the fixed point isn't unique -Yes +No
 * type=""}

{ Assuming g ε C[a,b], If the range of the mapping y = g(x) satisfies y ε [a,b], then} +g has a fixed point in [a,b] -g has no fixed point in [a,b] -Depends on some other condition

{ If |g'(x)| > 1, then the iteration xn+1 = g (x) produces a sequence } +that diverges away from P -that converges towards P -Depends

{The fixed point iteration will diverge unless x0 = 0. +True -False
 * type=""}

{Consider the following fixed point iteration; $$f(x)=\frac 12\left(\frac ax + x\right)$$. Rate (order) of convergence of this iteration is +Quadratic -Linear -Cubic
 * type=""}

title: Fixed Point Iteration


text: Digital Libraries/development sign up

title: Digital Libraries/development sign up


text:
Numerical Analysis/stability of RK methods/Quizzes

Quizes
{ the A-stability is characterized by: - all the points in the left half of the complex plane. - it a special case of the absolute stability. - inludes all $$\{ z \in \mathbb{C} | \mathrm{Re}(z) < 0 \}$$, + all of the above
 * type=""}

{Euler method is: - A-stable + not A-stable
 * type=""}

{the absolute stability region for RK4 is greater than the A-stability region for the same method:
 * type=""}

+ false - True - True for certain values of $$ b_j$$.

{A numerical method is stable if : + small change in the initial condition will produce small change in subsequent steps. - small change in the initial condition will produce huge change in subsequent steps. - big change in the initial conditions produce oscillatory solution at the end.
 * type=""}

{$$ \lambda =1$$ is always a root of the characteristic polynomial of the multistep method:
 * type=""}

+ True - False - usually not the case

{the root of the C.P. can be real or complex,and the method still be stable: + True - False
 * type=""}

{A multi- step method is strongly stable if : - $$ \lambda =1$$ is the only root of magnitude 1. - all other roots has magnitude <1 + the first and the second sentence - just one root inside the unit circle if it has more than one.
 * type=""}

{if more than one root has magnitude equal to 1, and the others are less than one, the method is - strongly stable. - A-stable + weakly stable. - Unstable.
 * type=""}

{The numerical method is unstable if - $$ \lambda > 1$$ for at least one root - $$ Re(\lambda) > 1$$ for at least one root - $$ Re(\lambda) \leqslant 1$$ for at least one root + the firs and the second sentences.
 * type=""}

{the absolute stabity region for the explicit Euler's method is + unit cicle in the complex plane, its center shifted to the left, by one unit. - unit cicle in the complex plane, its center shifted to the rigt, by one unit. - the whole left side of the complex plane. - the method is unstable.
 * type=""}

{ None of the RK methods is A-Stable: + True - False
 * type=""}

{All explicit methods are : - A-Stable + Not A-Stable. - Unstable. - None of the above
 * type=""}

{implicit multistep methods are A-stable if the have order at most -1 +2 -3 -they are always A-stable.
 * type=""}


text:
Numerical Analysis/stability of RK methods/Exercises

Ex:1
find the stability function for RK2 which is given by: $$ y_{n+1}=y_{n}+hf(t+\frac {h}{2},y_{n}+\frac {hf(t_n,y_n)}{2})$$

Solution: applying this method to the test equation $$ y'=\lambda y$$
 * we get $$ \begin{align}

y_{n+1} &=y_n+h \lambda y_n+\frac {(h \lambda)^2}{2}y_n \\ \Rightarrow \end{align}$$ the stability polynomial $$ \phi(z)=1+h \lambda +\frac {(h \lambda)^2}{2}$$

Ex:2
find the absolute stability region for RK2. Solution: by setting $$ z = h \lambda $$
 * $$ \Rightarrow $$ the abs.stability region is given by $$\{z\in \mathbb{C}||1+ h \lambda + \frac {(h \lambda)^2}{2}| < 1 \}$$

Ex:3
find the characteristic polynomial for RK2. Solution: it is $$ r^{i+1}=(1+z+\frac {z^2}{2})r^i$$ divide both sides of the equation by $$ r^i;r \ne 0$$ you get $$ r^{i+1}=(1+z+\frac {z^2}{2})r^i$$
 * $$ \Rightarrow \phi(r,z)=r-1+z+\frac {z^2}{2}$$

Ex:4
is RK2 stable, if it is what type of stability. Solution: you get $$ r^{i+1}=(1+z+\frac {z^2}{2})r^i$$ by setting z=0,
 * $$ \Rightarrow r=1$$
 * so the method is strongly stable since r=1, is the only root, and has a value of 1.

Ex:5
determine the stabilityfo Back Ward Euler method which is given by: $$ y_{n+1}=y_{n}+ hf(t_{n+1},y_{n+1})$$

Solution: applying this method to the test equation $$ y'=\lambda y$$
 * we get $$ \begin{align}

y_{n+1} &=\frac {1}{1-h \lambda}y_n \\ \Rightarrow &=\frac {1}{1-z}y_n; z=h\lambda\\ \end{align}$$
 * let $$ G(z)=\frac {1}{1-z}$$
 * since G(z) approaches 0, ans Re(z) approaches infinity,

then B.E.M is L-stable.


text:
User:Zngga

Zngga 19:25, 9 November 2010 (UTC)

title: User:Zngga


text:
User talk:Joeriz68

 Hello Joeriz68, 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. To get started, you may

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title: User talk:Joeriz68


text:
User talk:JosepVirgili

 Hello JosepVirgili, 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. To get started, you may

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title: User talk:JosepVirgili


text: User:Wangen

title: User:Wangen


text: File:2010 11 09 15 00 14.djvu

title: File:2010 11 09 15 00 14.djvu


text:
User:Arided/MathCourses

Here are outlines of the courses I have in mind:

 * 1) Short Precalculus (including "learning how to learn mathematics")
 * 2) Short Calculus
 * 3) Short Advanced Calculus (i.e. Calc II and/or Calc III in 6 weeks!)
 * 4) Mathematics for computer programmers (probably focusing on "Concrete Mathematics")
 * 5) Mathematics for game designers (a completely new course in early development phase!)
 * 6) Short statistics

The selection of topics is based on a very informal assessment of the initial interests of participants in the "DIY Math" course at P2PU this past autumn. One of the reasons for planning several courses at once is that I'd rather not leave anyone out. (It's also fine to add to the above list.)

I think these more structured courses can be supported by:

 * DIY Math, a drop-in area for any mathematical discussion (including planning work related to these and other ongoing mathematics courses) -- a sort of ongoing "departmental seminar".
 * Math Designers, a course/project related to the design of mathematics courses suited to the peer-based learning context. This can take the six courses I outlined above can be taken as a starting point, but it would be great to develop them further.  I hope that people from PlanetMath may want to participate in this, as it would be an excellent way to improve the quality of the PlanetMath corpus (check completeness, introduce pedagogy, etc.).

It is my view that a "course facilitator" should generally NOT have to act as a "teacher surrogate", but in fact, that courses should be co-facilitated by their participants. To the extent possible, the course material can also be made suitable for self-directed study. At the same time, it is very useful to be able to ask and answer questions (both about the content and also about "meta" how-to-learn-mathematics stuff). The drop-in area should be useful for meta-discussions.

In my experience, open-ended discussions do not replace a structured learning environment -- as for how a given individual will want to structure their learning experiences, that will depend on various factors. Feel free to add more courses to the list :)

How is this supposed to work?
As a general model, if a given subject can be clumped into 6 pieces like the ones in "Short Precalculus", that seems like a decent start. The "depth" of a given course should mirror how much time people want to put in per week. One hour? Ten hours? Mostly, though, I think we'd just look for ways to cluster all known mathematics in bunches of six (without too much overlap at the top level of the overall outline)! Some overlap is OK, e.g. the course for game designers might feature some bits and pieces from some of the other courses. As I've indicated in the Short Advanced Calculus course, non-overlapping topics can also be merged to increase the depth (and time-intensity) of a given course.

What are the real challenges?
So far in my limited experience, it seems like "follow-through" is the main challenge. Hopefully the structured design sketched above should help a lot. I think it would also be good if the system kept track of the kinds of courses people might like to take, and if people were required to ante up a certain level of commitment before a given course would run. Ideally the facilitator shouldn't have to chase people to make them do work. I've said more about these topics in http://metameso.org/~joe/docs/probation-report-final.pdf in case you want details (see the "Pilot Study" section).

All that in six weeks?!
For any given course, I think it is likely to be better to set ambitious goals and get people to commit to them up front. If someone wants to go slowly through a topic, that's OK with me! (See the alternative pacings for Short Advanced Calculus.) My working guess is that people will do better in the courses if they know they are going to have to work hard.

title: User:Arided/MathCourses