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Classical Mechanics/Constrained

Physics - Classical Mechanics

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What are constrained systems?
In many cases in mechanics, the motion of bodies is constrained in some way: for example, a massive bead may be constrained to move along a bent wire of certain shape; a massive cylinder may be rolling along a surface (but not sliding or flying around); or two masses may be connected by a rigid stick of fixed length.

In each of these cases, there are forces acting on the constrained bodies. In the above examples, the wire produces a force on the bead, the plane acts by the force of friction on the cylinder, and the stick pulls or pushes on the two masses. These forces may vary in time and we do not know the magnitude of these forces in advance. We know, however, that these forces are at every time exactly such as to guarantee that the constraints hold. The bead would fly away if there were no forces acting on it, but the wire provides a force that keeps the bead in place. The two masses connected by a rigid stick experience a force from the stick that is exactly necessary to keep them at a constant distance from each other. (This is what it means that the stick is "rigid".)

In the Newtonian approach to mechanics, these systems are treated by introducing variables $$F_1, F_2, ...$$ representing the unknown forces, and solving the system of equations for the unknown forces and accelerations. This procedure might be complicated; moreover, we are not always interested in the magnitudes of these unknown forces.

In the Lagrangian approach, there are two straightforward ways to treat constrained systems:

 * The method of solving the constraints. In this method, we introduce the generalized coordinates in such a way that the constraints are automatically satisfied. For example, suppose a point mass is constrained to move along a circle of radius $$R$$. We might describe this situation by saying that the Cartesian coordinates $$x,y$$ are constrained to satisfy the equation $$x^2+y^2=R^2$$. Now we can introduce the angle $$\phi$$ as the "generalized coordinate" and express the Cartesian coordinates of the point mass as $$x=R\cos\phi,y=R\sin\phi$$. These coordinates solve the constraint for all $$\phi$$. The power of the Lagrangian approach is that any generalized coordinates are good enough; so we can now directly write the Lagrangian in terms of the function $$\phi(t)$$ and forget about the fact that the system is constrained. We shall automatically obtain the correct equations of motion.
 * The method of Lagrange multipliers. In this method, we do not try to introduce new generalized coordinates that solve the constraints. (This may be difficult; not all algebraic equations can be solved!) Instead, we formulate the variational problem in the presence of constraints: the correct trajectory $$q_i(t)$$ is such that the action functional has an extremum while the constraint equations are satisfied. For example, if a point mass is constrained to move along a circle of radius $$R$$, but is otherwise unforced, then the Lagrangian is $$L=\frac{m}{2}(\dot x^2+\dot y^2)$$ and the constraint is $$x^2+y^2=R^2$$. The correct trajectory $$x(t),y(t)$$ will be such that the integral $$\int L dt$$ has the minimum value while the constraint holds at every $$t$$. Thus we need to solve the problem of conditional minimization.

Conditional minimization and Lagrange multipliers
A conditional minimization problem can be solved by the method of Lagrange multipliers. One takes a different, specially modified Lagrangian that describes the fact that the system is constrained. The modified Lagrangian is equal to the normal Lagrangian plus special terms containing Lagrange multipliers. Let us now explain this method.

For simplicity, consider the minimization of a function $$F(x,y)$$ with respect to variables $$x,y$$, subject to the constraint that $$G(x,y)=0$$.

First recall how the problem would be solved without the constraint: the minimum (or, more generally, an extremum) of $$F$$ would be the point $$(x,y)$$ where the partial derivatives of $$F(x,y)$$ vanish: $$ \frac{\partial F(x,y)}{\partial x}=0, \quad \frac{\partial F(x,y)}{\partial y}=0. $$ This would give a system of two equations that determines the two unknowns $$x_*,y_*$$.

With the constraint, the above system of equations will not give the correct answer because the solution $$x_*,y_*$$ most probably will not satisfy the constraint: $$G(x_*,y_*)\neq 0$$. Let us look at the problem geometrically. The constraint $$G(x,y)=0$$ determines a curve or several curves in the $$(x,y)$$ plane; we are looking for the point on that curve where the function $$F$$ has an extremum. Let us imagine the level lines of the function $$F$$, i.e. the lines $$F(x,y)=A$$ for various values of the constant $$A$$. The constraint curve $$G(x,y)=0$$ may go across the level lines of $$F$$; it means that the value of $$F$$ changes along the curve. It is clear from this geometric consideration that the extremum of $$F$$ along the constraint curve will be the point where the constraint curve is tangent to some level line of $$F$$. A condition for two curves to be tangent is that their normal vectors are parallel. The normal vector to a curve $$G(x,y)=0$$ at a point $$(x,y)$$ has components $$(\partial G(x,y) /\partial x, \partial G(x,y)/\partial y)$$. The normal vector to the surface line $$F(x,y)=A$$ has components $$(\partial F(x,y) /\partial x, \partial F(x,y)/\partial y)$$. These two vectors are parallel if there exists a number $$\lambda$$ such that $$ \frac{\partial F(x,y)}{\partial x} = \lambda \frac{\partial G(x,y)}{\partial x}, \quad \frac{\partial F(x,y)}{\partial y} = \lambda \frac{\partial G(x,y)}{\partial y}. $$ This is, together with the constraint $$G(x,y)=0$$, a system of three equations that determines the three unknowns $$x,y,\lambda$$. In this way we can solve the conditional minimization problem.

Note that the equations are the same as for minimization of the function $$K(x,y,\lambda)\equiv F(x,y)-\lambda G(x,y)$$ with respect to the three variables $$x,y,\lambda$$ without any constraints. Therefore, the conditional minimization problem is equivalent to a normal minimization problem for the different function, $$K(x,y,\lambda)$$. This new function is built by adding the original function $$F$$ and the constraint $$G$$ multiplied by an extra variable $$\lambda$$. This variable is called the Lagrange multiplier.

Example of using Lagrange multipliers
Here is a worked example. Suppose we need to maximize the function $$F(x,y)=5x+12y$$ under the constraint $$x^2+y^2=1$$. First, we write the constraint in the form $$G(x,y)=0$$. For instance, we may take $$G(x,y)=1-x^2-y^2$$. (It does not matter how we choose the function $$G$$, as long as the constraint is equivalent to the equation $$G(x,y)=0$$. Then we make a new function $$K(x,y,\lambda)=F(x,y)-\lambda G(x,y)=5x+12y+\lambda(x^2+y^2-1).$$ Then we need to minimize this function with respect to the three variables $$x,y,\lambda$$. We obtain the system of equations: $$5+2\lambda x=0,\quad 12+2\lambda y=0,\quad x^2+y^2-1=0.$$ It is easy to solve these equations: $$x=\frac{5}{13}, \quad y=\frac{12}{13}, \quad \lambda =-\frac{13}{2}.$$ These are the required values of $$x$$ and $$y$$. The value of the Lagrange multiplier $$\lambda$$ is useless for us now (but it will be useful when we apply this method to problems in mechanics!).

General case
The general form of the constrained optimization problem is the following. We need to find an extremum (or all extrema) of a given function $$F(x_1, ..., x_n)$$, where $$\boldsymbol x\equiv \{x_1, ..., x_n\}$$ is an array of variables satisfying $$m$$ different constraints $$G_1(\boldsymbol x)=0, ..., G_m(\boldsymbol x)=0$$.

The geometric consideration that I showed you for the simple case (the example with functions $$F(x,y),G(x,y)$$ above) can be generalized to many dimensions and many constraints: one considers level surfaces of $$F$$ and surfaces given by the constraints. A constrained extremum will be at a point $$\boldsymbol x$$ if the level surface of $$F$$ are tangent to the constraint surface at that point. The constraint surface is an intersection of $$m$$ different surfaces $$G_j=0$$, each having its own normal vector $$\boldsymbol n_j = \partial G_j / \partial \boldsymbol x$$. It can be shown using elementary vector algebra (I omit the proof) that the normal vector $$\partial F/\partial \boldsymbol x$$ to the level surface of $$F$$ must be a linear combination of the $$m$$ normal vectors $$\boldsymbol n_j$$. Therefore, the conditions for the constrained extremum to be located at a point $$\boldsymbol x$$ are that (1) $$\boldsymbol x$$ must satisfy all the constraints and (2) that there should exist $$m$$ numbers $$\lambda_1, ..., \lambda_m$$ such that $$ \frac{\partial F}{\partial \boldsymbol x} = \lambda_1 \frac{\partial G_1}{\partial \boldsymbol x} + ... + \lambda_m \frac{\partial G_m}{\partial \boldsymbol x} $$

It is easy to see that these conditions are equivalent to the conditions for an extremum of a new function $$K(\boldsymbol x, \boldsymbol \lambda) \equiv F(\boldsymbol x) - \lambda_1 G_1(\boldsymbol x) -... -\lambda_m G_m(\boldsymbol x),$$ with respect to $$(m+n)$$ variables $$x_1, ..., x_n, \lambda_1, ..., \lambda_n$$, without any constraints.

Let us then formulate the recipe to solve the problem of constrained optimization. We introduce an array $$\boldsymbol \lambda$$ of $$m$$ different Lagrange multipliers $$\boldsymbol \lambda \equiv \{\lambda_1, ..., \lambda_m\}$$ and build a new function $$K(\boldsymbol x, \boldsymbol \lambda) \equiv F(\boldsymbol x) - \lambda_1 G_1(\boldsymbol x) -... -\lambda_m G_m(\boldsymbol x).$$ We then find an extremum of this function with respect to the total set of $$(m+n)$$ variables $$x_1, ...,x_n, \lambda_1, ...,\lambda_m$$. In order to do that, we need to solve a system of $$(m+n)$$ equations: $$ \frac{\partial K}{\partial x_1}=0, ..., \frac{\partial K}{\partial x_n}=0, \frac{\partial K}{\partial \lambda_1}=0, ..., \frac{\partial K}{\partial \lambda_n}=0.$$ By solving these equations, we will obtain a set of values $$x_1, ..., x_n$$ which we are interested in. The values of the auxiliary variables $$\lambda_j$$ can be discarded.

Motion constrained to a surface
Let us now consider the Constrained Mechanical Problem Number One: A point mass is moving in a potential $$V(\boldsymbol x)$$ and, additionally, is constrained to move along a surface given by an equation $$G(\boldsymbol x)=0$$. (This can be physically realized by e.g. a mass point sliding without friction on top of a curved surface.)

According to the Lagrangian approach, we must find an extremum of the action, $$S=\int L(\boldsymbol x, \dot {\boldsymbol x})dt$$, under the condition that $$G(\boldsymbol x(t))=0$$ for all times $$t$$. We can apply the method of Lagrange multipliers. Note that we have, in effect, infinitely many constraints---one constraint for each moment of time $$t$$. Therefore, we need to introduce a set of infinitely many Lagrange multipliers, one Lagrange multiplier for each $$t$$. It is convenient to arrange this set of Lagrange multipliers into a function $$\lambda(t)$$.

According to the method of Lagrange multipliers, we need to build a "modified action" which is equal to old action minus the sum of all the constraints multiplied by their respective Lagrange multipliers. Therefore, the new action is $$\tilde S[\boldsymbol x(t), \lambda(t)] = \int L(\boldsymbol x, \dot {\boldsymbol x}) dt - \int \lambda(t) G(\boldsymbol x(t)) dt. $$ Solving the constrained optimization problem is equivalent to finding an extremum of the functional $$\tilde S$$ with respect to arbitrary $$\boldsymbol x(t)$$ and $$\lambda(t)$$.

It should now be clear how to approach the "Constrained Problem Number One" in principle. What remains is some technical work:

 * Deriving the Euler-Lagrange equations from the modified action $$\tilde S$$ and solving them. This will be a system of equations for the unknown functions $$\boldsymbol x(t)$$ and $$\lambda(t)$$.
 * Interpreting the function $$\lambda(t)$$. It will turn out that $$\lambda(t)$$ is related to the force that is needed to keep the point mass moving only along the surface $$G(\boldsymbol x)=0$$. So the Lagrange multipliers have a direct physical interpretation in this case. Namely, we shall show that the time-dependent force $$\boldsymbol F(t)$$ exerted by the surface is equal to $$-\lambda(t) \partial G/\partial \boldsymbol x(t)$$.

Example
A massive bead is set on a wire curved in the vertical plane (coordinates $$x,z$$) as a plot of the function $$z=C x^2$$, where $$C$$ is a given constant. The only external force is the gravitational field of the Earth. We would like to determine the equation of motion for the position of the bead.

Choose the constraint function as $$G(x,z)=z-C x^2$$. Then the modified Lagrangian is $$\tilde L = \frac{1}{2} (m\dot x^2+m\dot z^2)-mgz-\lambda(z-C x^2).$$ The Euler-Lagrange equations are derived in the standard way:
 * Variation w.r.t. $$x(t)$$ gives: $$m\ddot x(t) = \lambda(t) 2 C x(t)$$
 * Variation w.r.t. $$z(t)$$ gives: $$m\ddot z(t) = -mg - \lambda(t) $$
 * Variation w.r.t. $$\lambda(t)$$ gives: $$ 0 = z(t) - C x^2(t) $$

It is not easy to solve these equations by hand, but deriving them is straightforward and requires "no thinking", as physicists say. (That is, we simply follow general rules, and we do not need to make any special decisions or find special tricks for each particular situation.)

Exercise: There is an arbitrary choice in selecting the constraint function $$G$$. For example, we could have selected $$G(x,z)=2x^2-2z/C$$ or even $$G(x,y)=x-\sqrt{z/C}$$, and the constraint line $$G(x,z)=0$$ remains the same. Show that the equations of motion also remain the same, up to a change in the definition of $$\lambda(t)$$.

Lagrange multipliers and constraining forces
We see from the equation for $$z(t)$$ in the above example that $$-\lambda(t)$$ looks like the component of the normal force in the $$z$$ direction. So it is clear that the Lagrange multiplier $$\lambda$$ is somehow related to the unknown constraining force. We shall now derive this relationship in a more general case.

Consider again the Constrained Problem Number One. We solved it by using the modified Lagrangian $$\tilde L = L - \lambda(t)G(\boldsymbol x)$$. The term $$-\lambda(t)G(\boldsymbol x)$$ looks like an extra piece of potential energy, although it depends on time through the factor $$\lambda(t)$$). So this is a somewhat weird kind of potential energy, but let us examine it in more detail. As long as the point mass remains within the constraint surface, we have $$G(\boldsymbol x)=0$$ and this "extra potential energy" is equal to zero. But if the point mass could move a little bit off the constraint surface, say in a direction given by a small vector $$\delta \boldsymbol x$$, then this "extra potential energy" would change by $$ \delta V = \delta \boldsymbol x \cdot \frac{\partial (\lambda G)}{\partial \boldsymbol x}. $$ This looks like work done by a force. The force is directed orthogonally to the constraint surface and is equal to $$\boldsymbol F= -\lambda\frac{\partial G}{\partial \boldsymbol x}.$$ But we expect precisely this kind of force to act on the mass point by the constraining device.

Let us verify more formally that $$\boldsymbol F$$ is in fact the constraining force we were looking for. The Euler-Lagrange equations that follow from the Lagrangian $$\tilde L$$ are of the form "mass $$\cdot $$acceleration" $$ \equiv \frac{d}{dt}\frac{\partial L}{\partial \dot {\boldsymbol x}}= \frac{\partial L}{\partial \boldsymbol x} - \lambda(t) \frac{\partial G}{\partial \boldsymbol x}.$$ The term $$\partial L / \partial \boldsymbol x$$ describes the usual "free" forces due to potential energy in the original Lagrangian $$L$$. Now it is clear that the term $$-\lambda\frac{\partial G}{\partial \boldsymbol x}$$ describes the additional forces due to constraints.

Exercise: There is, of course, an arbitrary choice in defining the constraint function $$G$$. For instance, we may choose the constraint function as $$17G$$ or $$G^3$$ or some other function $$f(G)$$ instead of $$G$$. Show that the constraining force does not depend on the choice of the function $$G$$ (because $$\lambda$$ would change appropriately for every different choice of $$G$$!).

Exercise: Figure out how to compute the constraining force $$\boldsymbol F$$ if there are several independent constraints $$G_1(\boldsymbol x)$$, ..., $$G_n(\boldsymbol x)$$.

Constraints involving velocities
So far we considered only constraints that are expressed by functions of coordinates, such as $$G(x,y)=0$$. This form of the constraints covers a wide range of applications. However, there exist important cases where physical constraints cannot be expressed in this way. For example, the motion of a massive ball that is rolling on a surface without sliding, or the motion of a skater who is sliding on ice, can be described only using complicated constraints that involve velocities and coordinates at the same time. Such constraints, which are not equivalent to a simple function of coordinates, are called nonintegrable or nonholonomic constraints, whereas the constraints of the type we considered are called integrable or holonomic.

One would think that nonholonomic constraints could be simply added to the Lagrangian with Lagrange multipliers. It turns out, however, that the result is not the correct equations of motion for the problem! The main reason is that velocities $$\dot {\boldsymbol x}$$ are not varied independently from the coordinates $$\boldsymbol x$$, so the standard procedure involving the Lagrange multipliers is not the correct way to implement nonholonomic constraints. A special theory (based on the so-called Appell equation) was developed to derive the equations of motion for systems with nonholonomic constraints. However, this theory is beyond the scope of the minimal standard course.

title: Classical Mechanics/Constrained


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Educational Technology Innovation and Impact/Edutainment/BBC Jam

BBC Jam – Learn, explore and create with the BBC’s fun learning service

In January 2006 the BBC launched a new digital curriculum service known as BBC Jam. It is a free on-line service aimed at five to sixteen year olds and can be accessed either at home or at school. The service aims to offer a learner centred approach to learners least likely to succeed through more traditional learning environments. It aims to do this by providing an innovative combination of curriculum-based activities, video games, audio and animation and includes opportunities for learners to write teen soap storylines in French, and create audio, video and games. Content in maths, English, science, geography, French and business studies is available and materials will build over the next two years. However the launch has not been without its problems. There was concerted opposition to the proposal for a new online curriculum from commercial companies leading to a judicial review sought by educational software companies who argued that the use of state money to fund the BBC's plans would be illegal under European law. In early 2003 an out of court settlement was reached. In 2005, the online content was piloted in hundreds of UK schools. The BBC intends to provide a service that complements the existing marketplace for education services and will exclude teacher focused provision, concentrating more on learner centred activity establishing a dialogue directly with learners rather than teachers and parents. This Learner centred approach is designed to provide learning resources that motivate the learner and make a real difference, both to their personal development and to their understanding of a topic. This approach will provide a complementary educational experience to other forms of learning, and the BBC will exclude covering those areas where it feels that it will be difficult to offer a new or distinctive approach. The Department for Culture and Media and Sport has placed conditions on the range and scope of the learning topics the BBC Digital Curriculum Service can cover and in general the Department requires the service to stimulate, support and reflect the diversity of the UK, to innovate continually and to promote technological and pedagogical experimentation and to be distinctive from, and complementary to, services provided by the commercial sector. Additionally issues of access will need to be addressed and many subjects will have Welsh language versions of the content, and some subjects will also be made available as Scottish Gaelic or Irish Gaelic versions.

References http://www.bbc.co.uk/info/policies/digital_curriculum.shtml (accessed02/05/06) http://www.ltscotland.org.uk/mfle/creativeteaching/BBCdigitalcurriculum.asp (accessed 02/05/06) http://ferl.becta.org.uk/display.cfm?resID=13871 (accessed 02/05/06) http://news.bbc.co.uk/1/hi/education/4655292.stm (accessed 02/05/06)

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


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Wikijunior:Languages/Hungarian

What writing system(s) does this language use?
Hungarian uses the Latin alphabet, just like English. However, the alphabet is slightly different. Hungarian uses two kinds of diacritics: the Acute accent (´) indicates long vowels, the umlaut (¨) modifies the pronunciation of sounds O and U, like in German and the double acute accent (˝) is used for long vowels with umlaut (Ő is pronounced as a long Ö).

The digraphs and trigraphs Cs, Dz, Dzs, Gy, Ly, Ny, Sz, Ty and Zs count as one letter. The letters Q, W, X and Y are mostly used in borrowed words, only W is common in older family names.

How many people speak this language?
There are about 15 million people speaking Hungarian but only about 9.5-10 million of these live in Hungary (click here to learn about Hungary). The others in the surrounding countries and all around the world, most of them in Romania and Slovakia.

What is the history of this language?
Hungarian is a Uralic language like Finnish and Estonian.

Before 1000 AD Hungarian had a different writing system called Old Hungarian script. It looks like this: . This reads rovásírás.

One of the first text written in Hungarian is the founding declaration of the Benedictine Abbey of Tihany from 1055.

Who are some famous authors or poets in this language?
Imre Kertész, who won a Nobel Prize in literature, Sándor Petőfi, Géza Gárdonyi, Ferenc Molnár, click here to see more...

title: Wikijunior:Languages/Hungarian


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A History of Japan: From Mythology to Nationhood/Introduction

Introduction to Japanese History: Geography
Japan today is a modern democracy and economy comparable to the European and American model of politico-economics. A wealthy nation of more than 100 million people that is extensively involved in world wide commerce, it is now common to see Japanese tourists throughout the world and Japan itself now features as a popular destination for Western tourists. Japanese products from high-tech companies like Sony are to be found all over the world and have played their part in enriching society and culture – one only has to think of the once ubiquitous Walkman of the 1980s. This has not always been the case and throughout most of its history Japan practised an "isolationist" policy that allowed trade and contact with foreigners but restricted their movement and influence to small enclaves on the coast.

Japan is located on an archipelago of mountainous, volcanic islands in the northwestern corner of the Pacific Ocean. Before the Spanish began sending ships between Mexico and the Philippines late in the 16th century there was almost no naval activity in the Pacific ocean that had the capacity to disturb Japan in any way. To the Japanese, the Pacific was a boundary of their world, not a route to somewhere else. They never had to give any thought to what might be on the other sides.

To the south, there is an almost continuous chain of islands extending from Tanegashima, just off the southern end of Kyushu, down through and beyond the Ryukyu Islands all the way to Taiwan. There is no doubt that there has been movement of people up and down this chain and the people of the Ryukyu's have spoken a dialect of Japanese for a long time. However, the islands are simply too small to have ever become important. The Ryukyu Islands long had no official contact with Japan, but dealt rather with China for several centuries. They were taken under Japanese control in the 17th century to make sure that they did not fall into the hands of the Spanish or Portuguese, but were not incorporated into the Japanese state at that time, being a private possession of the Daimyo of Satsuma.

To the north, Japan was bounded by the large island of Hokkaido. This also was never considered a part of Japan before the Russians appeared on the scene and the Japanese took steps to establish a presence there in order to keep them out. The native peoples of Hokkaido were neither numerous enough nor technologically advanced enough to cause the Japanese any concern.

Japanese trade during its early history was mainly with Korea and China. The Japanese islands are laid out in such a way that from the Chinese point of view they are largely hidden behind Korea. It is about 500 miles across the East China Sea from Shanghai to Nagasaki. In early days that was an extremely dangerous voyage (shipbuilding technology being primitive) and communication was usually by way of Korea and the Shantung Peninsula, reducing the longest water crossing to 200 miles or so. Korea was a nervous country, being relatively small and living next to dangerous neighbors in China and Manchuria, and through much of its history it discouraged active contact. This policy applied to Japan as well. The periods of time during which Japan and Korea officially ignored each other are significantly longer than the ones during which they had frequent contact. Fishermen and others crossed all the time, of course, but without noticeable impact on higher culture.

All of this makes Japan an unusual place. It really is different from any other place, even Korea and China, in architecture, clothing styles, food, literature, music and dance, social customs, weapons and armor, political habits, and religion. Despite modern business suits and McDonalds, it remains different today, as anyone who spends significant time there will quickly discover.

About 75% of Japan is heavily forested mountain land that is thinly populated. Almost everyone lives in a restricted number of relatively flat lands close to the coast, all of which are separated from each other by mountains, which descend toward the sea in many places. Land transportation has always been difficult and until railroad lines began to be constructed, wheeled transport was not commonly used. Bulky goods were transported along the coast by boat or by foot. Generally, only soldiers rode horses. There were only a few places near cities where there were bridges over rivers. People waded shallow streams and ferried across deep rivers.

There are three main islands and a large number of small ones. One ancient name for the country was "the 88 islands". The biggest island is Honshu (which means mainland), the next in size is Kyushu (which means 9 provinces), and the third is Shikoku (which means 4 provinces).

When the country was first politically unified, the government laid out a series of routes (it would be a stretch to call them roads) which, because of the mountainous terrain, established natural regions. It is good to become familiar with these because you will see these names all of the time. The Inland Sea (between Honshu and Shikoku) side of Honshu was called the Sanyodo (route on the sunny side of the mountains) and the Japan Sea route on the other side of Honshu was the Sanindo (the shady side). The mountains between these are particularly rough, so these really are almost different worlds. Honshu north and east of the ancient capital area (modern Kyoto, Osaka, and Nara) is wider, and so there were three routes, one on the Japan Sea coast, Hokurikudo (northern route), and two on the Pacific Ocean side of the main mountains, the Tokaido (eastern ocean route) and the Tosando (eastern mountain route), which meet each other north of Mt Fuji and near modern Tokyo. Other terms that are widely used are Tohoku (meaning northeast) for the far northeastern part of the country, Kanto (east of the barrier) for the large plain surrounding modern Tokyo, Kansai (west of the barrier) for the capital area, which is also called Kinai (inner region) or Kinki (nearby region). The area between the Kanto and the Kansai is often called Chubu (middle part).

Administrative Units
There are three different schemes for dividing Japan into administrative units. In the late 7th and early 8th centuries, the government set up a system that created 66 provinces. A province was given a local government structure that included a governor who was assigned by the capital. Even after that system collapsed, the feudal regimes of the medieval period continued to use the province as a convenient geographical descriptor, and the system was not formally abolished until the 19th century. Then the government established a new system with different units known as prefectures. There are 47 prefectures, and their names and boundaries are unrelated to the former province system.

In between, the Tokugawa Shogunate had its own system which was entirely different. It is hard to describe briefly because it was constantly changing with the ebb and flow of politics. The government directly ruled about 1/3 of the country, including all of the big cities. There was no formal division of this area into units. The rest of the country was assigned to quasi-feudal vassals as privately held domains (called han). The larger han were directly administered by their owners who collected their own taxes and maintained their own military forces. The smaller han were run by the central government and the owner was entitled to a fixed annual revenue paid out of the government treasury. The biggest han were all located on the periphery of the country and might be as big as a prefecture.

The government liked to keep the vassals who had han nervous and obedient. Han were constantly being resized. New han were abolished and established over the years. Han lords (daimyo) were frequently forced to give up one han and move to another on the other side of the country. It is therefore difficult to create a map of han. It would require a large book of maps to show all of the changes in the system.

There are good articles in Wikipedia on and. There is also a that is self-described as incomplete. However, it includes the major ones. I strongly recommend that anyone who is interested in historical topics should buy a modern tourist guidebook to Japan as a source of good maps and geographical information, although everything will be shown and described in terms of the modern prefectures. A large proportion of the tourist highlights are things that have survived from earlier historical periods and these will be described and pictured.

There is also a list of Prefectures on the page "Japan Today".

Pronunciation
In the 19th century a system was devised for rendering Japanese syllabic script into our alphabet by a missionary named Hepburn. This became widely used and you will find it in many older books. There is a description in Wikipedia under. It was replaced almost universally in the English speaking world by a new system that makes it particularly easy for English speaking people to pronounce Japanese, at the cost of leaving certain things out. This is confusingly called the "Modified Hepburn System," though it is very different from the original. I believe that it was mainly created by E.O. Reischauer, historian and once the US ambassador to Japan. There is a nearly identical system, which was officially adopted by the Japanese government. However, the Japanese do not romanize Japanese often, roman letters (romaji in Japanese) being normally used only for foreign words and phrases, so you will see the American "modified Hepburn" system most frequently.

All of the sounds in Japanese that are also in English and the spelling is regular. All consonants are the same. The letter "i" is pronounced as it would be in Italian or Spanish, which is to say like the "e" in me. The "e" is pronounced as in merry. The sound we mostly use "i" for is spelled "ai" in Japanese. The letter "o" is always the "o" in row (a boat), and the "u" is always like shoe. The letter "a" by itself is as in father.

If you listen to Japanese pronunciation, you will hear same vowels pronounced two different ways. One variant is short or quick, and the other the sound is held much longer. Tokyo sounds more like To-o-kyo-o. Technically, one is supposed to indicate this by using a macron to produce "ō" and "ū" when they are lengthened, but this causes so much trouble in practice, that it is almost never done. The official Japanese system uses the circumflex accent ^ instead of the macron. That is about the only thing that is different.

For those who know Japanese, it is relatively easy to know when to use long and when to use short vowels. That is because this issue came into the language with the adoption of many Chinese loanwords. There are a lot more sounds in Chinese than in Japanese. The word that the Japanese represent as "chu" with a long "uu," for example, is pronounced as "jung" in modern Chinese. The Japanese could not handle the "ng" sound and the long "u" is the result. The Japanese actually spell this out in their syllabary, writing the equivalent of "Toukyou" or "chuu". The original Hepburn notation captured this distinction. This article is only referring to systems intended for English speakers. There are many others in the world and they are designed with the spelling conventions of a different language in mind.

Writing
In Chapter 9 of the first volume of the Cambridge History of Japan Edwin Cranston observes that the modern Japanese have what is probably the most complicated writing system in the world. Fortunately, it is not really necessary to know anything about it in order to explore Japanese history. There is an article on it in Wikipedia, and a wikibook (for learners) at Japanese.

The main complicating factor is that in nearly all cases Chinese characters are used for the very large number of Chinese loan words in Japanese. These are pronounced according to Japanese renderings of the original Chinese. Nearly every character also has one or more additional readings which are native Japanese words, and characters are routinely used in normal writing with these readings also.

Since Japanese is an inflected language (one in which words are changed to reflect differences in grammatical usage) and Chinese characters don't inflect, an inflected word is written using a Chinese character to establish the main meaning and then the inflected ending is spelled out using Japanese-developed characters that represent one of 51 different syllables. There are two independent versions of this "syllabary", one with "curvy" characters (hiragana) and one with angular ones (katakana). The first is used most commonly, these days, with the second used in ways similar to the ways we use bold-face or italic characters, or for spelling foreign words. Both can therefore be used in the same block of text. Often foreign words are converted to sounds that the Japanese can say and are spelled out using katakana, but it is also not uncommon to simply plug in the original roman letters. One block of text can therefore use four different writing systems. Complicated indeed, you don't need to know this but it is good to understand the difficulties the translator faces.

Dates
Both Korea and Japan adopted the Chinese calendar system and retained it until they switched to the western system in recent times. The Chinese system was a complex one that was a mixture of lunar and solar elements. There are good articles on this in Wikipedia, including details of both the Chinese and Japanese variants. The everyday calendar was a lunar one which would inevitably fall out of synchronization with the solar calendar. This was handled by making adjustments on the same general principle as our "leap year". An extra month was added whenever necessary to prevent new year's day from deviating too much from its "home position" in the solar calendar. This is because the lunar month was set at 30 days, which means 360 days to the year. An extra month was not added every year, and the rules for deciding when to do it are the main difference between the Chinese and Japanese versions of the system. Most people will be aware that the Chinese lunar new year occurs later than our modern new year, and that it occurs on different dates from year to year.

There are names for the months in Japanese but they are not used in official documents, even today. It is simply first month, second month, and so on. It is possible to convert dates from the lunar calendar to the modern calendar, but it would be a lot of trouble and no one bothers. If you see a date in historical writing that says 15th day of the 3rd month, it will be a lunar calendar date and it will probably not fall in March. After the Japanese changed to the modern calendar, authors will usually write March 15th (though the Japanese still say 3rd month).

The exact date is not really so important unless you want to organize a celebration of the thousandth anniversary of some famous event. A more serious matter for the reader of history is counting years. The Romans dated everything by the names of the two consuls for the year. You would need to have a complete list (one was on display in the Forum) in order to arrange things in their proper order. The Chinese were not quite as bad, but almost. They very early developed the custom of proclaiming an auspicious "reign title," something like "perfect harmony" and then dating things in terms of that. If "perfect harmony" then suffered some catastrophe, like a dangerous rebellion, the court might very well decide to change its luck by coming up with a new name. This could happen at any time during the year. So, the 14th day of the 5th month of the 6th year of "perfect harmony" could be followed by the 15th day of the 5th month of the first year of "glorious dawn." Later on, the historians would clean things up by retroactively using "glorious dawn" for the whole year, but the government offices would still be full of files containing documents with the now imaginary 6th year of "perfect harmony" written all over them. And, as in the Roman case, you need to have a list of all the reign names to keep track. This system was started by the Han dynasty in 163 BC.

Every dynasty used its own titles (often recycling names already used one or more times). Taking only important dynasties, there are just under 400 of them. Initially the Korean kingdoms simply used the Chinese titles, but they soon enough started inventing their own, as did the Japanese. This is where the January problem comes in. In Japan the first year of Kansei "lenient government" corresponds approximately to 1789. If I write Kansei first year, first month, first day, that is a purely Japanese date. If I try to make things clearer by saying 1789, first month, first day, you might now be thinking something very different. What I will do in my essays is use 1789 most of the time because it should not be necessary for a reader to have a list of all of the year names. However, this will be a lunar date and the first day of the year would not be January first, but sometime in the spring. If I should have some reason to use more precise dates, then I will say Kansei 1 (1789). Should it ever come up, there is a program on the Web called "nengocalc". "Nengo" is the name for a reign title. Nengocalc permits you to reliably convert back and forth from traditional dates to modern dates. The Japanese still use reign titles, by the way, but they now follow the convention started by the Chinese in the Ming dynasty where each emperor uses only one. If I have counted correctly the official Japanese total is now 247, including a period in the 14th century when there were rival emperors using different titles simultaneously. There were also "unofficial" titles used at various times in the past. The reign title associated with the current emperor is Heisei. Heisei 1 was 1989. After his death the emperor will be known as "the Heisei Emperor." The reigning emperor is always referred to as Tenno Heika, "his majesty the emperor." Only foreigners would ever call him Emperor Akihito.

Names
Japanese, Korean, and Chinese names are always written family name first. Writers in English frequently reverse them, but this opens the way to confusion and I prefer to keep the proper order at all times. Mao is the family name and Tse-tung the personal name.

In China the Ch'in dynasty began a convention that continued down to the end of the dynastic system whereby an emperor was given a reign name that had no relation to the name he had borne previously to his enthronement. This name was usually awarded only after his death, and it was often descriptive. For example, the most militaristic of the Han emperors is named Han Wu-ti, meaning "the martial emperor of the Han dynasty." A milder mannered Han emperor was Wen-ti, the scholarly or cultured emperor. For the Ming and Ch'ing dynasties the emperor's posthumous name was the same as the reign title used when he was alive. That has also become the rule in Japan but applies only to the last four emperors, the Meiji Emperor, the Taisho Emperor, the Showa Emperor, and the reigning Heisei Emperor.

Prior to that each emperor was given a posthumous name separately from the system of reign titles. This custom was started late in the 8th century but all earlier emperors were given reign names at that point and historians routinely use them because they are short and simple and at no time is it considered proper to identify the living emperor by using his ordinary name.


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User:Jntg4~enwikibooks

Soon to be a User of all the Wiki's.

title: User:Jntg4~enwikibooks


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The Art of Cities

The Art of Cities  A panoramic view of downtown Chicago.

Cities, the central fixtures of our very civilization, are home to a vast portion of the world's population. They bring together people, commerce and industry. Since ancient times, people have built settlements - simply because they are far better for business than isolated, singular homes. But what are the planning, design and architectural considerations which need to be taken into account to make them truly work? What would the ideal suburb look like? How far should the nearest train station be? Easy-to-navigate grids or interesting and endlessly variable cul-de-sacs and curved streets?

These and similar concerns have vexed city planners for centuries, and the results have been as numerous as they have been amazingly varied. An endless variation on the city theme: central business districts with high-rise glass temples of commerce, leafy suburbs just a subway ride from the city and endless highways which snake from district to district.

This book aims to explain some of the thousands of tribulations regarding city design, from major points like street layouts to seemingly trivial concerns such as urban beautification. The topic is vast, but the study of where we live is one that is perhaps universally important.

Part One: Zoning

 * What is Zoning?
 * Major Zoning Models

Part Two: Transportation

 * Roads and Streets
 * Highways
 * Public Transport
 * Airports and Seaports
 * The Street Plan

Part Three: Civic Buildings

 * Types of Civic Buildings
 * Clustering
 * Positioning

Part Four: Suburbia

 * The Rise of Suburbia
 * What Suburbs Are Not
 * Positioning
 * The Pricing Problem

Part Five: Beautification

 * Why is Beautification Needed?
 * Necessities
 * Luxuries
 * Styles

title: The Art of Cities


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Category:Brassicas

Various types of brassicas

title: Category:Brassicas


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The Art of Cities/Zoning

What is Zoning?
Zoning is the term (chiefly utilised in North America) used to the describe the pre-assigned "function" of an area of land by local governments. Zoning allows authorities to designate specific areas which may be used for a particular purpose. For those of you familiar with the PC game SimCity, you will know all about RCI. RCI, short for Residential, Commercial and Industrial, is an abbreviation used to describe the three major zoning categories.

Zoning is the principal reason that you don't find many shopping malls on sleepy cul-de-sacs or mansions beside factories. Using zones forces the three categories together, for organisational purposes as well as common sense, and creates neighbourhoods, commercial districts and industrial estates.

Here are the three RCI zoning categories explained in a little more detail:

 * Residential refers to dwellings, either one-family residences or larger buildings such as apartment buildings.

 * Commercial describes businesses, almost always offices or retail outlets (shops). Of the three categories, commercial developments are those most likely to be exempt from zoning and allowed to be built alone. This is simply because having all the Wal-Marts or all the petrol stations in one place doesn't make business sense.

However, this page is about the positioning of these zones, their sizes and their impact based on the first two factors.

Major Zoning Models
In the 1920s, a man named Earnest Burgess invented the first true zone model. A zone model is essentially a plan of a common style of zoning, indicating the areas where RCI developments are located and their respective wealth. This not only showed where the buildings were, but also how the wealth was distributed, and explained which zones ended up close to each other as a result of low income. His model showed the major trends in zoning and its impact, all with one simple diagram: the Concentric Ring Model.

The Concentric Ring Model shows that in the centre of most developments is a central business district (CBD), and surrounding that (often separated by a ring of undeveloped land) is a manufacturing sector, under the industrial "umbrella". Of course, not many people want to live near factories, so after that comes low wealth residential development. As one moves further from the central area of activity, the income of residents increases - they have been grouped by the zones. Here is the model on the right, and the legend for all of the diagrams on this page is to the left.

Please note that this is a graphical representation only and does not indicate a perfectly circular street plan!

In the late 1930s, economist Homer Hoyt revised the above model to account for modern transport arteries, and realised that a more realistic zoning model evolving in many cities was not one of relative uniformity but of a more irregular nature, with businesses flowering around transport routes and residences flocking to business. It was this trend which led to the second major zone model in use today.

The Sector Model (shown left) features the same CBD as the Concentric Ring Model, but this time it is completely encompassed by a "poor" residential area, with two opposing branches. To one side of these branches is the industrial area dealing with manufacturing. Note how this undesirable "neighbour" is separated from the higher-end homes by the low wealth areas, so the residential areas 2 and 3 are only next to other houses.

By the end of World War II, many large cities had evolved to take on a new form. While retaining their original CBDs (and relatively similar low-medium-high wealth "bands" radiating from it), they had grown to such a size that one centralised, all-purpose business centre was impractical for several reasons: the lengthy commute from outlying areas, the lack of a focal point for local commerce and the increase in "heavy" industry - highly polluting, particularly undesirable factories. Therefore, the Sector Model was adapted to create the final, and most globally accurate, zone model.

The Multiple Nuclei Model (to the right) shows the typical original layout: CBD, neighboured by manufacturing industries, themselves next to the low-wealth residential zones. Next to that are the medium-wealth homes, and finally the wealthy residences. However, several new aspects have emerged which were not present in the first two models:
 * A zone of heavy industry (area 7) on the fringes of the poorer residential areas. When the industries became larger, they were pushed as far from the central areas as possible, settling in the low-wealth zones, where potential employees were abundant and they would not overly damage the economy by eating into the wealthy areas.
 * In area 8, a zone appeared which had rarely existed before. It is an outlying business district. When cities became so large that districts were themselves comparable to cities in size, it was clear that a 2-hour daily commute to get to places of work or retail would not suffice, so offices became decentralised: while the CBD still thrived with big multi-nationals, more local businesses shifted their headquarters to outlying areas. New commercial centres were formed, feeding the needs of their individual areas. These specialised, more local "focuses" of communities have become almost global concepts; a standard fixture of most metropolitan areas.
 * Perhaps the most important modern zone in any city is the suburb, shown at area 9. Suburbs, which are discussed in greater detail elsewhere in this book, are the ultimate solution to alleviating the pressures of the city: leave it! These very town-like communities are almost completely residential and are usually separate from the main populated area, thrown apart from it by a belt of green, undeveloped land. This almost entirely post-war fixture of cities has reduced the strain both on the city and its inhabitants.
 * It is important to note that these models differ in different regions of the world. Wealthy residential areas are usually located on the outlying areas of a city in Anglo-America, whereas in Europe more wealthy areas of a city lie in the center, closer to the CBD.

title: The Art of Cities/Zoning


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Talk:A-level Mathematics/OCR/C2/Dividing and Factoring Polynomials

i think there is a mistake; the question starts with the divisor being $$x^2 + 3x - 1$$ and the answer is correct with this divisor. a simple mistake, they have put $$x^2 + 3x +1$$ on a few of the lines. somewhat confusing and i cannot edit the picture myself hope this helps
 * Think I've fixed it now :) Aleksev 20:42, 3 May 2006 (UTC)

There's another mistake - in the example of dividing polynomials, the remainder comes out as -18x+15


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User:JoshEnglish

I'm Josh. I have contributed to a few pages, and hope to write more information in the WikiBooks sections, especially those concerning writing.

title: User:JoshEnglish


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Wikijunior:Big Book of Fun Science Experiments/Soap powered boat

Simple Steps
Make a boat (flat as in the one shown in the picture above). Make sure to not to make the slit too big.

Cut a small piece of a bar of soap (enough to fit in the slit shown in the picture above).

Place the piece of soap in the slit of your boat.

Now, get a large bowl of water (or anything that collects water that has a large space for your boat to move around). (Use ordinary water).

Place your boat on the surface of the water and watch your boat move itself forward!

What causes this?
Water is made up of tiny particles smaller than dust called molecules. These molecules are really close to each other at the water's surface and so cause a sort of tension. (You see this tension when insects or leaves float on the surface of the water. This tension looks like if the water is covered with a clear plastic).

When the soap touches the surface of the water, it breaks this tension by moving the molecules away from each other. This causes large spaces between the molecules near the soap.

Water molecules from under the water's surface rush upwards to fill the spaces to recreate the tension. This causes a push from the area of the boat where the soap is. Since the soap is at the back of your boat, it causes a push from there (like someone pushing a car from the back).

(This is also the principle of how soap gets your hands cleaner than when you wash them with water alone. When you wash your dirty hands with water alone, the water's surface tension blocks any more water from touching your hands. This causes only a little bit of dirt to be washed away. When you add soap, the soap breaks this surface tension and allows water to continue to touch your hands and so more dirt gets washed away easily.)


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User talk:200.89.144.140

Please stop adding nonsense to Wikibooks. It is considered vandalism. If you would like to experiment, use the sandbox. Thank you.

title: User talk:200.89.144.140


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User talk:Peeps147

Image:Chrom.gif
Why did you gave Creative Commons license to an image found in the Internet? --Derbeth talk 21:12, 3 May 2006 (UTC)

title: User talk:Peeps147


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User talk:Lee5347

Image:Array.jpg
You cannot just copy an image from the Internet. See rules on upload page and proposed Image use policy. --Derbeth talk 21:15, 3 May 2006 (UTC)

title: User talk:Lee5347


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User talk:Krs5925

Image Copyright Problem: Image:UreaStruct.png
Thanks for uploading Image:UreaStruct.png. The Wikimedia Foundation is very careful about the images included in Wikibooks because of copyright law. We need you to specify two things on the image description page:
 * The copyright holder, and
 * The copyright status

The copyright holder is usually the creator. If the creator was paid to make this image, then their employer may be the copyright holder. If several people collaborated, then there may be more than one copyright holder. If you created this image, then you are the copyright holder.

Because of the large number of images on Wikibooks, we've sorted them using image copyright tags. Just find the right tag corresponding to the copyright status of this image, and paste it onto the image description page like this:.

There are 2 basic ways to licence an image on Wikibooks:
 * 1) An open content licence
 * Public Domain

 * The copyright holder gets the best protection of his work by licencing their work under an open content license like the Creative Commons Attribution-ShareAlike licence. If you have the express permission of the copyright holder to licence their work under the above licence, use the image copyright tag: cc-by-sa-2.5.  The GNU Free Documentation License is another choice for licencing one's work.  Again, if you have the express permission of the copyright holder, use the tag: GFDL.

 * The copyright holder can also release his work into the public domain, see here for images released into the public domain.
 * Images from certain sources are automatically release into the Public Domain. This is true for most governments like the federal United States government. However not all governments release their work into the public domain, such as the UK government .  Non-free licence governments are listed here.

For any other sources of for more information see Image copyright tags. Please remember that if you don't tag your images, they will be deleted.

 * P.S. If you have uploaded other images without including copyright tags, please go back and tag them.  Also, please tag all images that you upload in the future.

If you have any questions, just leave a message on my talk page. Thanks again. Derbeth talk 21:17, 3 May 2006 (UTC)

title: User talk:Krs5925


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Talk:Signals and Systems/Introduction

LTI
The introduction page starts discussing LTI systems without even expanding saying what 'LTI' stands for.

System ID
This material is duplicated from other material already found in this book. I am storing the text here for now. --Whiteknight (talk) (current) 21:30, 3 May 2006 (UTC)

 * Memory
 * A system with memory is any system where the output depends on more than just the current value of the input. For example, a system that averages the values of the last five seconds of input and gives that as its output demonstrates memory. Confusingly, a system that can predict the future (say, a function that gives the output you are going to give it in 2 seconds) also demonstrates memory.

 * Invertability:
 * An invertible system is any system where the original signal can be reproduced from the systems output.

 * Causality
 * A causal system is a system that depends only on the current and previous inputs. So a system that gets values from the future may have memory, but it is not causual.

 * Stability
 * A stable system will produce a finite output if the input is finite. An unstable system can create an infinitely big answer, for some finite input.

 * Time Invariance
 * A time invariant system is a system where the same input gives the same output, regardless of how long the system's been running for. An example of a time invariant system is an amplifier.

 * Linearity
 * A linear system scales the output in the same way as the input and is additive.

\forall x_1(t) \mapsto y_1(t), x_2(t) \mapsto y_2(t) \Longrightarrow a*x_1(t)+b*x_2(t) \mapsto a*y_1(t)+b*y_2(t) $$
 * Thus a linear systems responds with zero when the input is zero.

title: Talk:Signals and Systems/Introduction


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Educational Technology Innovation and Impact/Why use Technology in Education/Logical-Mathematical Intelligence

Logical Mathematical Intelligence

Logical Mathematical Intelligence is one of the most securely documented intelligences. Piaget [1] describes the intelligence as the ability to handle objects, moving to the ordering of the objects and grows into the ability to think formally about relationships without objects. One of the simplest forms of the application of logical mathematical intelligence is counting.

The use of this intelligence is demonstrated in the form of problem solving but not necessarily mathematical problems; it involves the act of selecting key points of information in a logical systematic sequence. Thus the investigation and analysis of data leading to results can assist in underpinning the understanding, not just taking the information on face value [2].

The use of Logical Mathematical Intelligence in the form of the usage of reasoning skills to unravel a mystery has been the trade mark of many of the worlds leading detectives. Other professions [3] that demonstrate the ability to use Logical Mathematical Intelligence are physicists, computer programmers and accountants, famous people with a high level of Logical Mathematical intelligence include; Albert Einstein and Bill Gates.

Many authors demonstrate their ability to make best use of Logical Mathematical Intelligence when composing a novel. The structure of the novel is planned, research is conducted and data collected and analysed before presenting in a logical sequence often being changed or re-written to suit the writers and readers needs.

Children’s games have been created and refined over the centuries to assist in the development of Logical Mathematical Intelligence. These include mathematical puzzles, strategic war games even word association type games all of which can be found throughout the civilised world.

The most recent example that has attained almost euphoric status is a mathematical puzzle designed to stimulate Logical Mathematical Intelligence called ‘Sudoku’[4]. The aim of the puzzle is to enter numbers 1 trough to 9 in a grid. The grid is usually 9 x 9 in size separated into smaller grids of 3 x 3 also called regions. The rules that govern the placement of the individual numbers are that you must only have one occurrence of a number in any given column, row and region. Some numbers are already placed in the grid these are called the givens. The more difficult puzzles require the use of Logical Mathematical Intelligence and can seldom be accomplished by scanning alone.

References:

[1]Gardner, H. (1983) Frames of Mind: The Theory of Multiple Intelligences. Basic Books - New York.

[2] www.longleat.net/ggrow/7in/logical.html (accessed 02/05/2006)

[3] www.inspiring-breakthrough.co.uk/learning-styles/mathematical-logical-learning.htm (accessed 02/05/2006)

[4]www.en.wikipedia.org/wiki/Sudoku (accessed 02/05/2006)


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Educational Technology Innovation and Impact/Edutainment/Educational Geography Games

There is a vast array of Educational Geography Games available to both parents and educationalist aimed at developing and improving the geographical knowledge and awareness of individuals. Games have been created to satisfy the need to make learning fun, in a subject area that at times can become dry or bland games can reinvigorate the learning experience. Games have the advantage of turning repetitive activity that is vital to memory retention into a competitive activity with the possibility of experiencing the winning sensation whilst learning.

The games can provide a fun learning environment for all ages from baby through all the key stages leading to adulthood. Games have been designed to be extremely flexible and can cover complex or simple topics. The games can provide a learning environment that covers specific geographic locations to enhance the learning at a localised level or can be broad enough to cover the countries of the world.

Card games range from the more traditional games like ‘snap’ where two pictures or words that are the same produces a winning combination, to the more modern and probably more complex card game of ‘Top Trumps’. This game of cards comprises multiple ranges of data a covering a range of attributes of a geographical location. Players compare information at random with the highest or best value winning the hand.

Board games invariably contain the element of travel across a country or region taking account of key locations, places of interest and climate. Some more strategic board games involve the capture of territory and enable the player to understand what the economic drivers are within individual countries or areas.

Puzzles can be in picture format or words. Pictures can be used to associate wildlife, plants and landscape to countries of origin. Word searches can provide the learner with a test of memory or logic.

Touch ‘ n ‘ Learn systems can be used from the earliest age to help associate places with names. Young children can make good use of this type of equipment to build up a basic understanding of geography.

Interactive maps allow learners to find out in great detail about specific areas of the globe. Often these are supported by computer software that contains a database of information that can produce very detailed facts to help inform the learning process.

Finally on-line games give the learner the opportunity to pit their skills and knowledge against another player anywhere in the world. A good example of this can be found at Geosense

References:

http://www.educationallearninggames.com/geography-games-us-states-2.asp (accessed 02/05/2006)

http://www.geosense.net (accessed 02/05/2006)


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User:Mdd4696

Hello, I'm Mdd4696.

title: User:Mdd4696


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Communication Systems/Wave Propagation

This page will discuss some of the fundamental basics of EM wave propagation.

Electromagnetic Spectrum

 * [[Image:cont_emspec2.jpg]]

Radio Waves
Maxwell first predicted the existence of electromagnetic waves in the 19th century. He came to this conclusion by careful examination of the equations describing electric and magnetic phenomenon. It was left up to Hertz to create these waves, and Marconi to exploit them.

In spite of one hundred years of study, exactly what radio waves are and why they exist, remain somewhat of a mystery.

Electromagnetic waves in free space, or TEM waves, consist of electric and magnetic fields, each at right angles to each other and the direction of propagation.

 * [[Image:tem_wave.gif]]

The relationship between wavelength and frequency is given by:

 * $$c = \lambda f$$

 * where c is the speed of light (approximately 300,000 m/s in vacuum), f is the frequency of the wave, and λ is the wavelength of the wave.

Radio waves can be reflected and refracted in a manner similar to light. They are affected by the ground terrain, atmosphere and other objects.

Maxwell’s equations state that a time varying magnetic field produces an electric field and a time varying electric field produces a magnetic field. This is kind of a chicken and egg situation.

Radio waves propagate outward from an antenna, at the speed of light. The exact nature of these waves is determined by the transmission medium. In free space, they travel in straight lines, whereas in the atmosphere, they generally travel in a curved path. In a confined or guided medium, radio waves do not propagate in the TEM mode, but rather in a TE or TM mode.

Radio waves interact with objects in three principle ways:

 * Reflection – A radio wave bounces off an object larger than its wavelength.
 * Diffraction – Waves bend around objects.
 * Scattering – A radiowave bounces off an object smaller than its wavelength.

Because of these complex interactions, radio wave propagation is often examined in three distinct regions in order to simplify the analysis:

 * Surface (or ground) waves are located very near the earth’s surface.
 * Space waves occur in the lower atmosphere (troposphere).
 * Sky waves occur in the upper atmosphere (ionosphere).

The boundaries between these regions are somewhat fuzzy. In many cases, it is not possible to examine surface waves without considering space waves.

 * [[Image:electron_density_altitude.gif]]

Surface Waves
These are the principle waves used in AM, FM and TV broadcast. Objects such as buildings, hills, ground conductivity, etc. have a significant impact on their strength. Surface waves are usually vertically polarized with the electric field lines in contact with the earth.

 * [[Image:surface_wave.gif]]

Refraction
Because of refraction, the radio horizon is larger than the optical horizon by about 4/3. The typical maximum direct wave transmission distance (in km) is dependent on the height of the transmitting and receiving antennas (in meters):

 * $$ d_{\max } \approx \sqrt {17h_t }  + \sqrt {17h_r } \quad {\rm{km}}$$

However, the atmospheric conditions can have a dramatic effect on the amount of refraction.

 * [[Image:refraction_distance.gif]]

Super Refraction
In super refraction, the rays bend more than normal thus shortening the radio horizon. This phenomenon occurs when temperature increases but moisture decreases with height. Paradoxically, in some cases, the radio wave can travel over enormous distances. It can be reflected by the earth, rebroadcast and super refracted again.

Sub refraction
In sub refraction, the rays bend less than normal. This phenomenon occurs when temperature decreases but moisture increases with height. In extreme cases, the radio signal may be refracted out into space.

Space Waves
These waves occur within the lower 20 km of the atmosphere, and are comprised of a direct and reflected wave. The radio waves having high frequencies are basically called as space waves. These waves have the ability to propagate through atmosphere, from transmitter antenna to receiver antenna. These waves can travel directly or can travel after reflecting from earth’s surface to the troposphere surface of earth. So, it is also called as Tropospherical Propagation. In the diagram of medium wave propagation, c shows the space wave propagation. Basically the technique of space wave propagation is used in bands having very high frequencies. E.g. V.H.F. band, U.H.F band etc. At such higher frequencies the other wave propagation techniques like sky wave propagation, ground wave propagation can’t work. Only space wave propagation is left which can handle frequency waves of higher frequencies. The other name of space wave propagation is line of sight propagation. There are some limitations of space wave propagation.

 * 1) These waves are limited to the curvature of the earth.
 * 2) These waves have line of sight propagation, means their propagation is along the line of sight distance.

The line of sight distance is that exact distance at which both the sender and receiver antenna are in sight of each other. So, from the above line it is clear that if we want to increase the transmission distance then this can be done by simply extending the heights of both the sender as well as the receiver antenna. This type of propagation is used basically in radar and television communication.

The frequency range for television signals is nearly 80 to 200 MHz. These waves are not reflected by the ionosphere of the earth. The property of following the earth’s curvature is also missing in these waves. So, for the propagation of television signal, geostationary satellites are used. The satellites complete the task of reflecting television signals towards earth. If we need greater transmission then we have to build extremely tall antennas.

Direct Wave
This is generally a line of sight transmission, however, because of atmospheric refraction the range extends slightly beyond the horizon.

Ground Reflected Wave
Radio waves may strike the earth, and bounce off. The strength of the reflection depends on local conditions. The received radio signal can cancel out if the direct and reflected waves arrive with the same relative strength and 180o out of phase with each other.

Horizontally polarized waves are reflected with almost the same intensity but with a 180o phase reversal.

Vertically polarized waves generally reflect less than half of the incident energy. If the angle of incidence is greater than 10o there is very little change in phase angle.

Sky Waves
These waves head out to space but are reflected or refracted back by the ionosphere. The height of the ionosphere ranges from 50 to 1,000 km.

Radio waves are refracted by the ionized gas created by solar radiation. The amount of ionization depends on the time of day, season and the position in the 11-year sun spot cycle. The specific radio frequency refracted is a function of electron density and launch angle.

A communication channel thousands of kilometers long can be established by successive reflections at the earth’s surface and in the upper atmosphere. This ionospheric propagation takes place mainly in the HF band.

The ionosphere is composed of several layers, which vary according to the time of day. Each layer has different propagation characteristics:

 * D layer – This layer occurs only during the day at altitudes of 60 to 90 km. High absorption takes place at frequencies up to 7 MHz.
 * E layer – This layer occurs at altitudes of 100 to 125 km. In the summer, dense ionization clouds can form for short periods. These clouds called sporadic E can refract radio signals in the VHF spectrum. This phenomenon allows amateur radio operators to communicate over enormous distances.
 * F layer - This single nighttime layer splits into two layers (F1 and F2) during the day. The F1 layer forms at about 200 km and F2 at about 400 km. The F2 layer propagates most HF short-wave transmissions.

Because radio signals can take many paths to the receiver, multipath fading can occur. If the signals arrive in phase, the result is a stronger signal. If they arrive out of phase with each other, they tend to cancel.

Deep fading, lasting from minutes to hours over a wide frequency range, can occur when solar flares increase the ionization in the D layer.

The useful transmission band ranges between the LUF (lowest usable frequency) and MUF (maximum usable frequency). Frequencies above the MUF are refracted into space. Below the LUF, radio frequencies suffer severe absorption. If a signal is near either of these two extremes, it may be subject to fading.

Meteors create ionization trails that reflect radio waves. Although these trails exist for only a few seconds, they have been successfully used in communications systems spanning 1500 km.

The Aurora Borealis or Northern Lights cause random reflection in the 3 - 5 MHz region. Aurora causes signal flutter at 100 Hz to 2000 Hz thus making voice transmission impossible.

Fading and Interference
Radio signals may vary in intensity for many reasons.

Flat Earth Reflections (Horizontal Polarization)
There are at least two possible paths for radio waves to travel when the antennas are near the earth: direct path and reflected path. These two signals interact in a very complex manner. However, ignoring polarization and assuming a flat earth can produce some interesting mathematical descriptions.

 * [[Image:direct_relected_wave.gif]]

 * p1 = direct wave path length

 * p2 = reflected wave path length

 * $$\Delta$$p = p2 - p1 difference in path lengths

 * d = distance

From the geometry we can observe:

 * $$p_1^2 = \left( {h_r  - h_t } \right)^2  + d^2$$

 * $$p_2^2 = \left( {h_r  + h_t } \right)^2  + d^2$$

 * $$p_2^2 - p_1^2  = \left( {h_r  - h_t } \right)^2  + d^2  - \left( {h_r  + h_t } \right)^2  - d^2  = 4h_r h_t $$

 * $$\left( {p_2 - p_1 } \right)\left( {p_2  + p_1 } \right) = 4h_r h_t $$

But$$\Delta p = \left( {p_2 - p_1 } \right)$$ and $$d \approx p_1  \approx p_2 $$

 * $$\Delta p2d \approx 4h_r h_t $$ therefore $$\Delta p \approx \frac{d}$$

If the difference in the two paths $$\Delta$$p, is 1/2 $$\lambda$$ long, the two signals tend to cancel. If $$\Delta$$p is equal to $$\lambda$$, the two signals tend to reinforce. The path difference $$\Delta$$p therefore corresponds to a phase angle change of:

 * $$\varphi _p = \frac{\lambda }\Delta p = \frac$$

The resultant received signal is the sum of the two components. The situation is unfortunately made more complex by the fact that the phase integrity of the reflected wave is not maintained at the point of reflection.

If we limit the examination of reflected waves to the horizontally polarized situation, we obtain the following geometry:

 * [[Image:vector_sum.gif]]

Applying the cosine rule to this diagram, we obtain a resultant signal of:

 * $$E_r = E_1 \sqrt {2\left( {1 - \cos \varphi _p } \right)}  = 2E_1 \sin \left( {\frac{2}} \right)$$

The signal strength of the direct wave is the unit distance value divided by the distance: $$E_r = \frac{d}$$ Therefore, the received signal can be written as:

 * $$E_r = \frac{d}\sin \left( {\frac} \right)$$

For small angles this can be approximated by:

 * $$E_r \approx \frac{d}\frac = E_0 \frac$$

Multipath Fading
The received signal is generally a combination of many signals, each coming over a different path. The phase and amplitude of each component are related to the nature of the path. These signals combine in a very complex manner. Some multipath fading effects are characterized by delay spread, Rayleigh and Ricean fading, doppler shifting, etc. Fading is the most significant phenomenon causing signal degradation. There are several different categories of fading:

 * Flat fading: the entire pass band of interest is affected equally (also known as narrow or amplitude varying channels).
 * Frequency selective fading: certain frequency components are affected more than others (also known as wideband channels). This phenomenon tends to introduce inter-symbol interference.
 * Slow fading: the channel characteristics vary at less than the baud rate.
 * Fast fading: the channel characteristics vary faster than the baud rate.

Time Dispersion
Time dispersion occurs when signals arrive at different times. Signals traveling at the speed of light move about 1 foot in 1 nanosecond. This spreading tends to limit the bit rate over RF links.

Rayleigh Fading
The Rayleigh distribution can be used to describe the statistical variations of a flat fading channel. Generally, the strength of the received signal falls off as the inverse square of the distance between the transmitter and receiver. However, in cellular systems, the antennas are pointed slightly down and the signal falls of more quickly.

 * [[Image:Rayleigh_fading_doppler_10Hz.gif|300px]]

Ricean Fading
The Ricean distribution is used to describe the statistical variations of signals with a strong direct or line-of-sight component and numerous weaker reflected ones. This can happen in any multipath environment such as inside buildings or in an urban center.

A received signal is generally comprised of several signals, each taking a slightly different path. Since some may add constructively in-phase and others out of phase, the overall signal strength may vary by 40 dB or more if the receiver is moved even a very short distance.

Doppler Shift
A frequency shift is caused by the relative motion of the transmitter and receiver, or any object that reflects/refracts signal. This movement creates random frequency modulation. Doppler frequency shift is either positive or negative depending on whether the transmitter is moving towards or away from the receiver. This Doppler frequency shift is given by:

 * $$f_d = \frac{c}f_c $$

vm is the relative motion of the transmitter with respect to the receiver, c is the speed of light and fc is the transmitted frequency. In the multipath environment, the relative movement of each path is generally different. Thus, the signal is spread over a band of frequencies. This is known as the Doppler spread.

Atmospheric Diffraction
Radio waves cannot penetrate very far into most objects. Consequently, there is often a shadow zone behind objects such as buildings,hills, etc.

The radio shadow zone does not have a very sharp cutoff due to spherical spreading, also called Huygens’ principle. Each point on a wavefront acts as it were a point source radiating along the propagation path. The overall wavefront is the vector sum of all the point sources or wavelets. The wavelet magnitude is proportional to $$1 + \cos \theta $$ where $$\theta$$ is measured from the direction of propagation. The amplitude is maximum in the direction of propagation and zero in the reverse direction.

Reflection
Reflection normally occurs due to the surface of earth or building & hills which have large dimension relative to the wavelength of the propagation waves. The reflected wave changes the incident angle.

There is similarity b/w the reflection of light by a conducting medium. In both cases, angle of reflection is equal to angle of incidence. The equality of the angles of reflection & incidence follows the second law of reflection for light.

Diffraction
Diffraction occurs in beams of light or waves when they become spread out as a result of passing through a narrow slit. Maximum diffraction occurs when the slit through which the wave passes through is equal to the wavelength of the wave. Diffraction will result in constructive and destructive interference.

title: Communication Systems/Wave Propagation