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Compiler Construction/Run-time Considerations

Storage Management - Garbage Collector
In computing there are new tools that are waiting to emerge when developers can find the technology capable of supporting them. One good example is that 50 years ago John L. McCarthy came up with an idea to automatically reclaim the memory of objects that are no longer needed during the execution of Lisp programs. It was the origin of the garbage collection concept in computer science.

One of the tasks performed at runtime--as the compiled program is run by the user--is the management of allocated memory blocks, so as to minimize memory usage by deallocating blocks that will no longer be used. This is referred to as "garbage collection." Garbage collection is not available in all languages and compilers, but it is implemented in many of the most widely used. It takes much of the burden of memory management off of the programmer--when implemented correctly--and improves performance.

Ideally, garbage collectors (hereinafter GC) would remove every allocation that will never be used again. In practice, we can assume that if there is any way to reference to a block, it can be used again; if not, it would not be used (except accidentally). So GC works by retaining memory blocks that can be reachable by tracing every reference path at a moment and freeing the rest. For example,

function concat (string1, string2) {  new_string = alloc (string1.length + string2.length); copy (string1, new_string); copy (string2, new_string + string1.length); return new_string; }

This function returns a newly allocated memory block that is the concatenation of given two strings. Because the function only returns a new string and does not know how it is going to be used, it is the caller of this function that is responsible for freeing it, like:

var old = null var string = 'Writing compilers is ' if you have not finished this book { string := concat (string, 'anything but ') old := string } string := concat (string, 'fun!') if old != null { free (old) }

By letting GC free memory blocks when needed, the above can be simplified to:

string = 'Writing compilers is ' if you have not finished this book { string := concat (string, 'anything but ') } string := concat (string, 'fun!')

In practice, an execution path and reference dependencies can be far more complicated than the above; thus, it should be easy to imagine how GC would be a great help.

Brief History of Garbage Collection
In the Beginning, there was Static Allocation, and for a Time, it was Good. FORTRAN, circa the mid-1950's, had no garbage collection at all. Once a block of memory was allocated, it stayed allocated. The programmer could not deallocate it, even if he tried.

Circa 1958, Algol implemented a form of memory management called stack allocation. Following that, languages like C implemented heap allocation, which allows the programmer to arbitrarily allocate and de-allocate memory from the heap of available memory. In C, this is done with the call. While this allows very flexible memory allocation and de-allocation by the programmer, careless (mis)use often results in memory leaks and dangling pointers; programmer errors are not handled by the compiler or runtime environment.

In order to overcome the barriers of programmer error, we must either better instruct our programmers, or provide better systems for memory management. The string of non-accredited technical institutes that seem to have sprung up in the United States obviously do nothing to solve the former problem, so it is up to us to approach the latter.

Implementing Garbage Collection
There are two basic approaches to garbage collection: reference counting and batching (or tracing).

Reference Counting
In reference counting, a reference counter is associated with every allocated object. Whenever a reference is made to that object, the counter is incremented. When one dereferences that object, the counter is decremented. When the counter reaches 0, there are no existing references to this allocated object, so it can be safely removed (because there is no way for it to be accessed in the future).

When an object O is created, we also create a reference counter for O, calling it. At creation,. When we create a reference to O, we perform. When we destroy a reference to O,. When we decrement, we check to see if the counter is now 0. If it is, we free O.

To allow allocation of memory, we maintain a list of free memory blocks. When we allocate blocks, we remove them from the free list. When we deallocate, we add them to the list. Obviously, a major drawback of this method of allocation is fragmentation; having to defragment memory to allow allocation in the requested size can create inexplicable slowdowns for memory allocation. To prevent this, one can defragment the memory at some set interval, or perform more complex deallocation to keep memory contiguous, but the solution to this problem is likely to be non-trivial.

Reference counting has the primary advantages of being simple to implement, requiring no work at an interval, and thus avoiding the necessity to pause the current operating to sort out our garbage collection (which can lead to highly inexplicable pauses during seemingly ordinary operation).

However, reference counting has a number of severe limitations, namely it's inability to detect cyclical pointers (e.g. A references B and B references A, in which case neither A nor B will ever be collected), it's cost to pointer operations and use of space, and it's computational cost to allocation as a result of fragmentation. Some of these problems have solutions in the form of augmented reference counting, but some are best solved by using a different form of garbage collection.

Batching/Tracing
In batching, the heap is viewed as a directed graph, with pointers as edges between allocated space (which we view as nodes in the graph). To do garbage collection, we traverse the graph and mark each node we reach. If a node is not marked, it cannot be reached and can be safely deallocated.

Unlike reference counting, this method must be run at a specific time (reference counting, of course, is performed as a feature of ordinary allocation and deallocation). Garbage collection--when it must be performed at a specific time rather than simply as the program runs, as with reference counting--can be performed when storage is exhausted, before a segment of code that needs to run quickly is executed, or simply during idle time when there's nothing for the computer to do. The best way depends somewhat on the type of program running; for example, in Microsoft Word, there is enough idle time (while the user sits thinking up funny parenthetical statements) that the third option is best. On a real-time system, however, it might be better to do garbage collection always before certain code runs.

Batching relies on accurate identification of memory pointers. There are a number of issues here; something that appears to be an integer may be a pointer, while something that appears to be a pointer may not be. A language could be designed specifically to avoid this confusion, but many popular languages, such as C and C++, are not designed in such a way. The compiler could mark at compile-time as pointers anything used as a pointer, but again this requires extra overhead. Some of the same methodology could be applied at runtime, but with additional runtime costs. Regardless of how this is done, however, it is important to be conservative with pointer identification. When it doubt, it is best to assume something is a pointer than is not. After all, it is far better to have extra un-collected garbage than to eliminate blocks that will later be used.


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Child. Who changed this? It caused a really interesting feedback loop. 203.142.128.167 04:21, 16 Jun 2004 (UTC)

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

title: Template talk:Stub


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Serial Programming/Error Correction Methods

Introduction
There are 3 main types of handling errors:

 * acknowledge or retry (ACK-NAK).

 * "Forward Error Correction" (FEC)

 * Pretend It Never Happened

ACK-NAK
Each packet is checked by the receiver to make sure it is "good".

If it *is* good, the receiver (eventually) tells the sender that it came through OK -- it acknowledges (ACK) the packet.

All versions of ACK-NAK absolutely require Two Way Communication .

How does the receiver know it's good ?
The sender calculates a checksum or CRC for the entire packet (except for the footer), then appends it to the end of the packet (in the footer/trailer).

The typical CRC is 32 bits, often a Fletcher-32  checksum.

Aside: Note that the checksum or CRC are forms of hashing, ie, irreversibly shrinking data. Checksums and CRCs are weaker algorithms than "cryptographically strong" message authentication code algorithms such as MD5 or SHA variants. Cryptographically strong algorithms can detect errors better than checksums or CRCs, but they take more time to calculate.

Whenever the receiver receives a packet, the receiver calculates exactly the same checksum or CRC, then compares it to the one in the footer/trailer. If they match, the entire packet is (almost certainly) good, so the receiver sends an ACK.

When there's even the slightest question that the packet has any sort of error (which could be *either* in the actual data *or* in the header *or* in the checksum bits -- there's no way for the receiver to tell), the receiver discards it completely and (in most cases) pretends it never saw it.

If it's not good, the sender sends it again.

How does the sender know it wasn't good ?
It never got the ACK. (So either the packet was corrupted, *or* the ACK was corrupted -- there's no way for the sender to know).

"Stop-and-wait ARQ"
The simplest version of ACK-NAK is "Stop-and-wait ARQ".

The sender sends a packet, then waits a little for an ACK. As soon as it gets the ACK, it immediately sends the next packet. If the sender doesn't hear the ACK in time, it starts over from the beginning, sending the same packet again, until it does get an ACK.

The receiver waits for a packet. If the packet passes all the error-detection tests perfectly, the receiver transmits an ACK (acknowledgment) to the sender.

Subtleties: If the receiver receives the packet perfectly, but the ACK message is delayed too long, then the transmitter sends another copy of the message (a "communication echo"). Imagine the packet contained the message "deduct $11,000 from Fred's account.". When the receiver gets this second copy of the packet, what should it do? Certainly it should send an ACK (otherwise the transmitter will keep trying to send this packet over and over). Either or both of the following problems could occur:
 * The delayed first ACK could hit the transmitter after it transmits the second copy of the message, so it transmits the next packet. Then the second ACK hits the transmitter, tricking the transmitter into thinking that "next packet" has been successfully received, when it hasn't.
 * When the receiver gets 2 identical consecutive packets saying "deduct $11,000 from Fred's account", are these 2 legitimate independent transactions, and so it should deduct $22,000 from Fred's account? Or is it really just 1 transaction, with a bit of echo, and so should deduct a total of only $11,000 from Fred's account?

Both of these problems can be solved by adding a "sequence number". The transmitter keeps a count of how many independent packets it has transmitted to that receiver, and puts that sequence number in the header of each packet. But when it re-transmits a packet, it re-transmits that same identical packet with that same identical sequence number. Also, the receiver, rather than sending a generic "ACK" message, specifies which particular packet it is responding to by putting its sequence number in the ACK message. When there is a communication echo, the receiver sees the same sequence number, so ACKs that sequence number (again) but then discards and ignores the extra, redundant copy of a packet it already received. When the transmitter is sending a new packet that merely happens to contain the same data, the receiver sees a different sequence number, so it ACKs that new sequence number, and takes another $11,000 out of Fred's account. Poor Fred.

A 1-bit sequence number (alternating 1 - 0 - 1 - 0 for each new packet, and ACK1 ACK0 ACK1 ACK0 in response) is adequate for a stop-and-wait system. But as we will see, other ARQ protocols require a larger sequence number.

Subtleties: Some early protocols had the receiver send a NAK (negative acknowledgment) to the sender whenever a bad packet was received, and the sender would wait indefinitely until it received *either* an ACK *or* a NAK. This is a bad idea. Imagine what happens when (a) a little bit of noise made a bad packet, so the receiver sends the NAK back to the sender, but then (b) a little bit of noise made that NAK unrecognizable. Alternatively, imagine a shared-medium network with 1 sender and 2 receivers. What happens when a little noise messes up the "destination" field of the packet ?

With "Stop-and-wait ARQ", the sender and the receiver only needs to keep 1 packet in memory at a time.

streaming ARQ
The sender sends a packet, then the next packet, then the next, without waiting.

As it sends each packet, it puts a copy of that packet in a "window".

Each packet is consecutively numbered. (The sequence number must be at least large enough to uniquely identify every packet in the window).

... turn-around time ... bouncing off geostationary satellites ...

The receiver occasionally transmits an acknowledgment ("I got all packets up to 8980", "I got all packets up to 8990").

If the receiver is expecting packet number 9007, but it receives a packet with an *earlier* number (that it had already received successfully), it transmits (or possibly re-transmits) a "I got all packets up to 9006" message.

When the sender receives an acknowledgment of any packet in the "window", it deletes that copy.

When the sender's window gets full, it waits a little, then tries re-sending the packets in the window starting with the oldest.

So when the sender suspects an error in some packet, it resend *all* packets starting with the erroneous packet. This guarantees that the receiver will (eventually) receive all packets in order.

Optionally, If the receiver is expecting packet number 9007, but it receives packet number 9008, it may transmit a negative acknowledge (NAK) for 9007, and ignores any higher packet numbers until it gets packet 9007.

When the sender receives a NAK for any packet in the window, it re-starts transmission with that packet (and keeps it in the window).

With "streaming ARQ", the sender needs to keep the entire window of packets in memory at a time. But the receiver still only needs to handle 1 packet at a time, and handles them in consecutive order.

(Some people think of "streaming" as one big packet the size of the window using "stop-and-wait" protocol, divided into smaller "sub-packets").

Selective Repeat ARQ
A selective repeat ARQ system is a kind of streaming ARQ.

But instead of the receiver only handling 1 packet at a time, and discarding all packets higher or lower than the one it is looking for, the receiver tries to keep a copy of all packets it receives in a window of its own, and negotiates with the sender to try to resend *only* the erroneous packets.

FEC
If you have only one-way communication, you are forced to use Forward Error Correction, sometimes called EDAC (Error Detection And Correction).

You transmit the data, then (instead of a CRC) you transmit "check bits" that are calculated from the data.

... NASA space probes ... compact disks ...

The simplest kind is "repeat the message".

If I send the same packet twice, and noise only corrupts one of them, *and* the receiver can tell which one was corrupted, then no data was lost. If I send the same packet 3 times, and noise corrupts any one of them, then the receiver can do "best 2 out of 3". The "check bits" are 2 copies of the data bits. In fact, noise could corrupt a little bit of *all three* of them, and you could still extract all the data -- align the 3 packets next to each other, and do "best 2 out of 3" for every bit. As long as there were only a few bits of noise in each packet, and the noise was in a different place in each packet, all the data can be recovered.

... (put picture here) ...

There are some very clever kinds of FEC (Hamming codes, Reed-Solomon codes) that can correct all kinds of common errors better than "best 2 out of 3", and only require the same number of "check bits" as there are data bits.

Pretend It Never Happened
A sender often streams audio and video live, in real-time.

What should a receiver do when a packet gets mangled ?

If it sends a message back to the sender, asking it to resend that packet, by the time the reply gets back, it's probably several video frames later. It's too late to use that information.

Rather than pausing the entire movie until the request makes a round-trip, it's far less jarring to the audience if the receiver silently accepts some signal degradation while drawing attention away from it. The receiver discards the mangled packet, does its best to guess the data that would have been in the packet, and continues on as if nothing had happened. For example, it might fill a space with nearby pixels' colors. A receiver employing a strategy like this should log how often it had to fill in gaps, so that the user can troubleshoot a connection that isn't capable of consistent exact reproduction.

combination
Even when they have 2-way communication, sometimes people use FEC anyway. That way small amounts of noise can be corrected at the receiver. If a packet is corrupted so badly that FEC cannot fix it, the protocol falls back on ACK-NAK retransmission (or on Pretend It Never Happened).


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Evolutionary Biology/Speciation

The biological species concept, which was first introduced by Ernst Mayr, defines a species as a population whose members have the potential to interbreed to produce viable, fertile offspring, but do not produce healthy offspring with other species.

Different Modes
There are different modes of speciation that can lead to the origin of species. We will focus on allopatric and sympatric speciation, but will also introduce two other modes: parapatric and peripatric. Gene flow, the loss or gain of alleles in a population due to the migration of fertile individuals or between gamete populations, is reduced in all of these speciation modes. A recent trend towards designating speciation events as ecological speciation or nonecological speciation is motivated by a desire to focus more on the process, rather than the geography, leading to speciation events. This dichotomy may be somewhat misleading, however, as empirical work indicates that ecological factors are important in driving speciation regardless of whether or not gene flow is present, leading many to question whether speciation is ever truly nonecological.

Allopatric Speciation
A geographic barrier is the predominant means by which allopatric speciation occurs.

How it works

 * 1) There are two populations that interact by interbreeding, which allows for gene flow (genetic exchange), within a broad geographic range. The resulting offspring are fertile because these populations are of the same species.
 * 2) A geographic barrier arises, isolating the two populations. The newly isolated populations now go through a period of isolation. Some of the individuals within a population might exhibit a trait more suitable for the new environment. This variant allows for differential survivorship and reproduction. Those individuals that carry the trait are more likely to survive in the new environment. Therefore, they will leave more offspring. Gradually, the variant trait will become dominant. Theoretically, this would happen with both of the newly isolated populations resulting in reproductive isolation.
 * 3) The geographic barrier ends. The two populations now have the opportunity to interbreed again. If the populations do not recognize each other as potential mates or cannot produce fertile offspring, speciation has occurred because the populations are reproductively isolated. The two populations have become two different species. If speciation has not occurred, the populations can still interbreed successfully to produce viable, fertile offspring.

An organism's mobility plays a significant role in allopatric specition. Two populations of birds would not be hindered by an emerging mountain range because of their ability to fly. The size and remoteness of a population also play a major role in speciation. There is a greater chance of a species forming in a small, isolated population because natural selection and genetic drift are more likely to alter the gene pool there.

These barriers are not limited to the more recognizable ones, mountain ranges and rivers, though. Populations can also become reproductively isolated by members of a population colonizing a new remote area without the presence of their parent population.

Sympatric Speciation
Speciation does not only occur when there is a geographic barrier separating two populations; Sympatric speciation occurs when divergent selection causes preferred mating with genetically similar individuals but not with the parent population, hence reproductively isolating one part of the population from another.

Sympatric Speciation in Plants
Plants frequently undergo this type of speciation. The most common type of sympatric speciation is a mutant condition called polyploidy, which is caused by a doubling of chromosomes during cell division. Autopolyploid mutations occur when an individual has more than two chromosome sets, all derived from a single species. A more common type of polyploidy species is an allopolyploid, which forms a hybrid from two species. Sympatric speciation generally occurs in plants through hybrids of related species and errors in cell division, producing fertile polyploids.

Sympatric Speciation in Animals
Sympatric speciation is less common in animals than plants, but still exists. Animals can become reproductively isolated by becoming dependent on different resources than those used by their parent species. It is possible for there to be a genetic change that selects for different resources. Sympatric speciation in animals, which isolates the species reproductively, results from preferences for different habitats, food, resources, and breeding partners.

One of the best examples of sympatric speciation in animals is with the 200 different species of cichlids that inhabit Lake Victoria in East Africa. Sympatric speciation has occurred here as a result of variations in available resources, as well as selective mating based on coloration. This idea was tested by a group of scientists at Holland’s University of Leiden using P.pundamilia (blue back) and P. nyererei (red back). Although females of the P. pundamilia species only mated with males of the same species in normal lighting, they mated with the P. nyererei males in monochromatic orange lighting which made the two species appear identical. The resulting hybrids between the two species were still fertile. Mate choice based on coloration led to sympatric speciation within the cichlid species in this case.

Parapatric Speciation
In the absence of a geographic barrier, parapatric speciation can also lead to the emergence of a new species. In this type of speciation, gene flow is reduced because individuals are reproducing with the mates closest to them. Therefore, mating is not random. The phenotypes are gradually divergent from the extreme forms, but are all equally present within the population. Gene flow is reduced within the population. Parapatric speciation cannot produce separate species in the absence of divergent selection.

Peripatric Speciation
This type of speciation correlates (?) with allopatric speciation. It occurs in a few members of the population, which exhibit a different appearance than the majority. Genetic drift plays the major role in speciation. This mode is the hardest to support with evidence.

Resources
Campbell, Reece. Biology, Sixth edition. Benjamin Cummings. 2001.

http://evolution.berkeley.edu/evosite/evo101/VC1dParapatric.shtml

http://www.mhhe.com/biosci/esp/2001_gbio/folder_structure/ev/m3/s2/index.htm

http://www.pbs.org/wgbh/evolution/library/05/2/l_052_05.html

title: Evolutionary Biology/Speciation


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File:Histogramofdice.png

Histogram of Dice Rolls for Statistics Course.

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Japanese/Contributor's Guide

This page aims to help potential contributors better understand the principles behind the current work and give ideas for how best to add new material.

Structure to follow
This book has been restructured so many times but there is an effort under way to merge some of the duplicate material.

A syllabus and lesson plan was created for the so-called "Practical Lessons". This may be useful for future contributors, but never turned out any lessons.

User:Retropunk sorted the JLPT 4 grammar list. This is now up at Sugu ni Hajimemashō where it is being sorted into a lesson plan.

Lessons
There are many ways to organise a lesson, this is one:
 * 1) Dialogue (by posters, conversations, or whatever.)
 * 2) Discussion of Vocabulary in Dialogue
 * 3) * Discuss various uses of vocabulary if necessary (e.g., politeness)
 * 4) * Optional links to dialogues for previously learned vocabulary
 * 5) Grammar
 * 6) * Discuss grammar points, giving more examples if necessary.
 * 7) * Link to previous grammar points (maybe by categories)
 * 8) Optional Quick Review
 * 9) Optional link to Wikiversity for practice work.

Reading material

 * 1) Text (poster, conversation, literary text or anything fitting the level)
 * 2) Vocabulary
 * 3) * List new vocabulary.
 * 4) * Discuss various uses of vocabulary where appropriate (e.g., politeness)
 * 5) Grammar
 * 6) * Discuss grammar points, giving more examples if necessary.
 * 7) * Link to previous grammar points (maybe by categories)
 * 8) Optional Quick Review
 * 9) Optional link to Wikiversity for practice work.

Romanisation
Learning the Japanese scripts will give the student useful insight into the language that the simple (and limited) romanisations cannot offer. Romanisation (rōmaji) should, therefore, only be used in introductory lessons to assist with the learning of the kana, and in the reference guides. Kana and kanji should be used in all subsequent lessons.

Please use the Wikipedia:Manual of Style for Japan-related articles. Specifically:

 * Consonants will be written using the Revised Hepburn system (s, sh, z, j, t, ts, ch, f)
 * "n" mora will be written n' before a vowel, or "y" (zen'aku, kon'yaku)

Examples:
 * しつ: shitsu
 * どうぞ: dōzo
 * えいご: eigo
 * せんぱい: senpai
 * せんや: sen'ya (as different from せにゃ senya)
 * : Tōkyō

The macrons (Ā,ā,Ē,ē,Ī,ī,Ō,ō,Ū and ū) are easily added through the edit interface by choosing "Romaji" in the drop-down list below the "Save page" button.

Furigana
The lessons make extensive use of furigana. Please use the following templates to insert furigana:

Conversations
For conversations with Japanese and English side by side, use Japanese conversation.

Patterns and examples
Sentence patterns and examples can be highlighted using the Japanese pattern and Japanese example templates.

Related modules
Linking to related modules makes it easier for readers to refer to past lessons (e.g. to brush up on topics), vocabulary lists (e.g. when practicing new sentence patterns) or other useful content (such as the ../Verb conjugation table/). The Japanese related template can be used to consistently style these links.

Vocabulary pages
Please place vocabulary lists on subpages of Japanese/Vocabulary and use Japanese vocabulary entry to structure them. The can be used to conveniently link to the list from lesson pages.

Stroke order
There is a project on the Wikimedia Commons to upload images and animations of the stroke order for characters. There are directions on how to contribute. It's easy with the use of some free programs. Please contribute so that we'll have a standardized reference for our Wikibook users.

title: Japanese/Contributor's Guide


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Evolutionary Biology/Carl Linnaeus

< Evolution before Darwin

Carolus Linnaeus (1707-1778) was a Swedish botanist and taxonomist. One of his principle contributions was the taxonomic system of binomial nomenclature. He turned away from his love of botany in order to pursue the medical field (?). While studying in the Netherlands, he published his first work, entitled Systema Naturae (1725). In this work, he introduced his idea of three (?) main kingdoms created from the use of his classification system. His systema naturae was a continuous work that ended up in multiple volumes as the concepts of classification were modified with the growing number of species of plants and animals.

Scientific Thoughts
The scientific thinking of Carl Linnaeus began with natural theology. Because of his religions beliefs, he believed that it was possible to find an understanding of God by looking at His creations. In part of Systema Naturae, he explained how he thought that the divine order of the creations of God would possibly be revealed by studying nature and its origins.

Linnaeus' love of botany brought his first taxonomic categories to be those of plants. He based each group of plants on the number and arrangement of the plant's reproductive organs. This did not work as well as Linnaeus planned because it created a system that was very unnatural in the sense that it did not compare many other similarities and differences that are important to categorising plants. He first separated the plants into a large group (?), and then used smaller categories to further diversify the plants. He gave each plant two names which is the origin of binomial nomenclature. This form of classifying has been modified throughout the years and is still used today.

Binomial nomenclature consists of the two Latin names given to an organism. The first is the name of the genus to which the organism belongs, also referred to as the generic epithet. Genera are further divided into species, identified by the specific epithet (the second part of the name). The combination of these two category names creates the scientific name of organisms. Even though Linnaeus was not (?) the first to use the binomial naming system, he was the first to use it frequently enough for the system to be accepted and used more often by scientists everywhere.

Linnaeus' thoughts on evolution are very different from the modern day theories. He believed that species were immutable. Even though Linnaeus believed in immutability, he did believe that the creation of new species was possible, but that it is limited. (?)

Linnaeus was the father of taxonomic and gave us the binomial system of naming and classifying organisms.

title: Evolutionary Biology/Carl Linnaeus


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User:Spammy

Just a filler user page for the time being. I am a student at Florida State University currently working on my undergraduate degree in biomedical engineering. I recently switched majors from physics to my current field, which unfortunately added an extra two years onto my degree. Hopefully I will finish my PhD work by 2010 :/ More information to come.

title: User:Spammy


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Evolutionary Biology/James Hutton

Hutton (1726–1797) is best known for his theory of gradualism. Hutton was a geologist who left the medical field in order to travel and study the earth's surface. When Hutton's ideas began to circulate, geology was not a subject that was given much attention. The focus of the time was mineralogy.

In 1785, Hutton published his paper titled ‘Theory of the Earth.’ In this paper, he explained how the study of geology confines itself as the study of the material makeup of the earth. He suggested many explanations and origins for many geological occurrences ranging from molten rock formations to the origins of materials found on sea floors.

Hutton began his paper with his studies of atmospheric changes in a section he termed ‘Theory of Rain.’ In this section of his paper, Hutton went into detail of how moisture accumulates in the atmosphere until enough has gathered to form a dense moisture filled cloud with visible precipitate in the form of rain.

It was not until the third version of the Theory of the Earth was published when Hutton introduced his theory of gradualism. This is where he recognized that change did occur, but that change was the gradual culmination of slow processes happening over great amounts of time. By using this idea, it is possible to observe the earth today and make assumptions about the past since change happened over such a long period of time, the earth today must be relevant to the earth of the past.

title: Evolutionary Biology/James Hutton


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Evolutionary Biology/Jean-Baptiste Lamarck

< Evolution before Darwin

Jean-Baptiste Lamarck (1774 – 1829) was a French botanist who proposed two ideas that had great impact in the theory of evolution. Lamarck did not believe that a species could become extinct. Instead, he saw the idea of extinction as every member of a species evolving into another species. He believed that change was brought about through use and disuse and inheritance of acquired characteristics. This was the first time that a mechanism was proposed in order to explain how a change in a species occurred.

The first idea of use and disuse is only the first part of Lamarck’s revolutionary thought. It is referred to as the 'First Law' of his publication, Philosophie Zoologique. . The use or disuse theory explains that the parts of an organism that the organism uses most will undergo hypertrophy and will become more developed. Hypertrophy is the growth of a specific organ or tissue. So by this idea, the more an organism uses a part of its body, the larger and more developed that part will become. The parts of the organism that are not used as much undergo atrophy and begin to degrade from lack of use. This means that the part of the organism will get decrease in size due to the lack of use for that particular part of the organism. This is the beginning of the idea of inheritance of acquired traits.

The inheritance of acquired traits, also known as his 'Second Law' in Philosophie Zoologique, is the idea that the traits of an organism that undergo hypertrophy will be inherited by the next generation. This hypertrophy suggests an adaptation to the environment that can be passed on to the next generation through the genetic code of the organism.

Even though Lamarck made astounding progress throughout his life, this theory of evolution has been revised by following studies. The Lamarckian view of evolution is seen as a tendency for a species to reach perfection where it is actually simply adapting to their environment in a single generation. We now know that the inheritance of acquired characteristics is not an actual mechanism of evolution.

Darwin saw that Lamarck had some very good ideas that could lead to evolution. Many of Lamarck’s ideas can be seen in Darwin’s Origin of Species. Lamarck was a believer in immutability. Darwin used this idea to further his research on how species were indeed mutable. Darwin also used Lamarck’s inheritance of acquired characteristics as support for his natural selection idea. He used Lamarck’s basic structure, but changed the theory Lamarck used into the actual mechanism of natural selection. Instead of saying that the traits one organism acquires during its lifetime to better its success, Darwin said that these characteristics would eventually reveal themselves in offspring and later generations instead of in the present organism. These were just some of the ideas Darwin modified while writing the Origin of Species.


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Introduction to Philosophy/Empiricism

Empiricism is a philosophical idea based on the idea that all reliable knowledge about the world is gained in the process of experience. Famous empiricists were Hume, Locke and Berkley, basing themselves on ideas already postulated by Aristotle, that we are born with the Tabula Rasa which receives throughout our lives information on which it can base all thinking and knowing. The movement of empiricism was in part a countermovement to what Descartes and Spinoza had proposed as rationalism. Knowledge derived from experience is called a posteriori.

title: Introduction to Philosophy/Empiricism


text:
Introduction to Philosophy/Rationalism

Rationalism is a philosophical movement declaring that the most certain form of knowledge is derived from reason and that our senses are not reliable submitters of information about the outer world. The most famous rationalists were Descartes, Spinoza and Leibniz, creating theories of knowledge totally independent from any experience. The senses, according to the rationalists, can not tell us real truth, for example: we could dream that we are the king of the world and not truly be, but the angles of a triangle still sum up to 180 degrees. Knowledge from reason alone and not experience is known as a priori. Plato is considered by many to be the main precursor to rationalism, whereas his student, Aristotle, is often considered to be a precursor to empiricism.

title: Introduction to Philosophy/Rationalism


text:
Evolutionary Biology/Thomas Malthus

In 1798, he published the Principle of Population where he made the observations that the human race would be likely to overproduce if the population size was not kept under control.

Malthus then focused his studies on the human race. His calculations and theories produced an idea that the human population would increase geometrically while the food supply and natural resources would only increase arithmetically. This is a potential explanation for the predicted poverty and famine. He concluded that as more offspring are born, a more competitive nature would arise. As more offspring come into the population, fewer resources will be available for the population. This has the potential for competition between organisms for survival due to lack of resources. This competitive nature would be necessary for survival of individuals within a large population size unable to be supported by the environment. He believed that this uncontrollable population size would eventually be the cause of famine and poverty among humans. His reasoning behind this idea was divine intervention. He believed that this would be the punishment for man if he became too lazy.

Malthus’ Principle of Population caused Darwin to rethink many issues while coming up with his theory of natural selection. Malthus’ work made Darwin realize the importance of overpopulation and how it was necessary to have variability in different populations. Darwin also used Malthus’ ideas to use competition as well as the survival in numbers idea to come up with his full idea of natural selection.

title: Evolutionary Biology/Thomas Malthus


text:
Evolutionary Biology/Georges Cuvier

Georges Cuvier (1769 – 1832) is known for two topics that are very important to modern science. First, he is considered the founder of the study of functional anatomy. By studying the makeup of the body and its arrangements, he was able to make certain assumptions about body parts by simply examining others.

Cuvier was very interested in Linnaeus’ classification system. After extensive study, he modified the system. The system of taxonomy now included the realm of fossils. He was the first to realise that fossils were the organic remains of extinct animals. Because of his work with fossils, he was given the title of Father of Paleontology. Cuvier was also a believer of immutability of characteristics which rejected evolutionary theory as well as Lamarck’s idea of inheritance of acquired characteristics.

The most important contribution of Cuvier was his theory of catastrophism. Through his fossil study, he came to notice that some layers (strata) differed from other layers. He also observed that certain fossils were not present in all layers and some completely disappeared after a certain level. This gave Cuvier the idea that the reason for this observation was mass extinction. By adding these two ideas together, he came to his idea of catastrophism: the idea that the boundaries between the strata correspond with catastrophes that occurred throughout time that wiped out entire species. This was his explanation of why certain fossils sometimes disappeared after being found in some strata.

Cuvier’s theory of catastrophism is important to evolution because it gave other scientists a basis to work from. Many other discoveries and ideas come from some theory backing up or going against catastrophism. This theory is also import today with the ongoing investigation of the Chicxulub crater off of the Yucatán peninsula. This crater is a geologic finding potentially reinforces this theory making catastrophism a possible extinction mechanism of the dinosaurs.

title: Evolutionary Biology/Georges Cuvier


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Introduction to Philosophy/Why Study Philosophy?

We can hone our analytic and problem-solving abilities, and improve the way we think about thinking, by studying fundamental questions. It is often the reasoning and answers other people have come up with when faced with similar questions that can help most. This is useful in both academic and practical applications.

The choice of essential questions is itself also interesting, as well as the answers - both questions posed, and answers given by particular philosophers depend on the time or era in which they lived.

Often, philosophers will go into professional fields such as law, medicine, and government. However, philosophy is about a great deal more than just academic education, analytical skills, and improving one's prospects for employment. It is not to be confused with history, semantics or debating (though all these elements are part of the study).

The study of philosophy is best undertaken seriously as an attempt to understand oneself and the universe, and even to reconcile the two. While a few years of reading up on philosophical history will certainly improve the quality of one's thinking, philosophy truly becomes worthwhile when it changes our lives. A thorough examination of internal and external life can dramatically change the way we see the world, casting a clearer light on phenomena that we were once immersed in. In this way, the study of philosophy is the study of our lives.


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Introduction to Philosophy/Pragmatism

Pragmatism is a philosophy that was developed and passed on by Peirce, Quine, James, and others.

The Correspondence Theory of Truth, which was held by Aristotle among others, holds that when we say something is true, this means that my understanding of reality corresponds with reality. So if I say that the Sun revolves around the Earth, it is not true because this does not correspond to reality. On the other hand, a Pragmatist would say that if it "works" for you, if it does not make your life problematic and has no significant impact on your life, then it is "true" in the sense that it works; it's pragmatic. This is similar to the perspectivist view adopted by Nietzsche.

See William James (1842-1910):

http://www.connect.net/ron/james.html

title: Introduction to Philosophy/Pragmatism


text:
User talk:212.117.81.29

I've moved your additions to Learning vi:Advanced tasks as fine-grained screen manipulation is really an advanced thing - a vi editor can get by without it. Basic tasks is really about getting to grips with how to edit. Later we can introduce things to make a vi editor's life easier, and so on. Dysprosia 12:22, 16 Jun 2004 (UTC)

title: User talk:212.117.81.29


text:
German/Level I/Freizeit/Answers

Introduction - Lesson 1 - Lesson 2 - Lesson 3

../Problems/ - ../Answers/ - ../Test/ - ../Test Answers/

Lesson 2 ~ Freizeit Answers

Sports and Activities
Soccer - Fußball

9-Pin Bowling - Kegeln

Continue Lesson>>

Spielen, Machen, and Other Verbs
1. Does he play 2. We do/make 3. They play soccer. 4. We play football, and he plays, too. 5. Do you guys play tennis? 6. She doesn't play volleyball. 7. Do you play 8. She plays

Continue Lesson>>

Other Verbs and Their Conjugations
1. wir, sie (pl), Sie 2. er, sie (sing), es, ihr 3. du, er, sie (sing), es, ihr 4. du 5. wir, sie (pl), Sie 6. er, sie (sing), es 7. wir, sie (pl), Sie 8. er, sie (sing), es, ihr 9. ich 10. du, er, sie (sing), es 11. du 12. er, sie (sing), es, ihr 13. ich 14. wir, sie (pl), Sie 15. du 16. er, sie (sing), es 17. wir, sie (pl), Sie 18. wir, sie (pl), Sie 19. ihr 20. du 21. ich 22. er, sie (sing), es 23. wir, sie (pl), Sie 24. du 25. ihr 26. ihr 27. wir, sie (pl), Sie 28. ich 29. er, sie (sing), es, ihr 30. ich

Continue Lesson>>

Two More Verb Forms

 * 1) Wir spielen Baseball, und sie schauen Fernsehen.
 * 2) Spielt ihr Fußball?
 * 3) Was machst du?
 * 4) Helena spielt nicht Tennis, und Markus spielt nicht Volleyball.
 * 5) Ich spiele nicht Football oder Fußball.
 * 6) Thomas und Hans spielen Basketball.
 * 7) Wir spielen Kegeln, aber er schaut Filme.
 * 8) Bruno macht Hausaufgaben und spielt Football.
 * 9) Christine macht nicht Hausaufgaben, aber sie spielt Volleyball.
 * 10) Wir spielen Schach.

Extra Credit: ich gehe; du gehst; er/sie/es/ihr geht; wir/sie/Sie gehen. Whether you did it or not, you will have to know this for the test.

Continue Lesson>>

Expressing likes and dislikes
A. Was spielst du gern? B. Ich spiele gern Football, aber ihr spielt nicht Football. Was spielt ihr? A. Wir spielen gern Volleyball und Fußball. Sie spielen Football. B. Gut! Tschüß! A. Bis später!

A. Ich heiße Markus. Wie heißt ihr? B. Ich heiße Thomas, und er heißt Hans. A. Was spielt ihr gern? C. Wir spielen Tennis gern, aber Fußball lieber. Aber am liebsten spielen wir Football. A. Spielt ihr Kegeln? B. Nein. Wir spielen nicht Kegeln.

Continue Lesson>>

Numbers

 * 1) dreißig
 * 2) zweiundvierzig
 * 3) neun
 * 4) siebenundachtzig
 * 5) eins
 * 6) dreizehn
 * 7) zwanzig
 * 8) sechzehn
 * 9) fünf
 * 10) einundfünfzig
 * 11) vier
 * 12) siebenundsechzig
 * 13) siebzehn
 * 14) neunundvierzig
 * 15) siebzig
 * 16) einhundertdreiundsechzig
 * 17) achthundertzwölf
 * 18) dreihundertachtundvierzig
 * 19) neunhundertfünfundsiebzig
 * 20) siebenhundertvier
 * 21) fünfhundertachtzig
 * 22) sechshunderteins
 * 23) fünfhundertsechsundneunzig
 * 24) vierhundertdreiundzwanzig
 * 25) zweihundertneunundfünfzig
 * 26) achthunderteinunddreißig
 * 27) sechshundertsiebenundvierzig
 * 28) neunhundertdrei
 * 29) zweihunderteinundfünfzig
 * 30) vierhundertfünfundsiebzig

Continue Lesson>>

Exact form

 * 1) Achtzehn Uhr Acht
 * 2) Zehn Uhr Sechsundzwanzig
 * 3) Sechzehn Uhr Neunundfünfzig
 * 4) Ein Uhr Zweiunddreißig
 * 5) Einundzwanzig Uhr vierundvierzig
 * 6) Null Uhr Neun (not Zwölf Uhr Neun = 12:09 p.m. This is because midnight is 12 a.m. and noon is 12 p.m.)
 * 7) Siebzehn Uhr Zwölf
 * 8) Acht Uhr Siebenundvierzig
 * 9) Dreiundzwanzig Uhr Zweiundfünfzig
 * 10) Sieben Uhr Dreißig

A. Wie spät ist es? B. Es ist Fünfzehn Uhr sechsunddreißig.

Continue Lesson>>

Before/After the Hour

 * 1) Viertel vor Sieben
 * 2) Sechsundzwanzig vor Fünf (oder Vier nach halb Fünf)
 * 3) Vier vor Neun
 * 4) Sechs nach Eins
 * 5) Viertel nach Drei
 * 6) Acht vor Eins
 * 7) Zehn nach Zehn
 * 8) Vierundzwanzig nach Neun (oder Sechs bis halb Zehn)
 * 9) Achtzehn vor Drei
 * 10) Halb Neun (not Halb Acht)

Continue Lesson>>

Saying When You Do Something
A. Spielst du Baseball? B. Ja, ich spiele Baseball. A. Wann spielst du Baseball? B. Um Achtzehn Uhr dreißig. A. Prima! Ich spiele auch Baseball um halb sieben. B. Bis dann! A. Tschüß!

Continue Lesson>>

Times of Day

 * 1) heute Morgen
 * 2) morgen Nacht
 * 3) vorgestern
 * 4) gestern Abend
 * 5) morgen
 * 6) heute Nachmittag
 * 7) gestern
 * 8) heute :)
 * 9) gestern Nachmittag
 * 10) morgen frühe (oder morgen Morgen)
 * 11) vorgestern Nacht
 * 12) morgen Abend
 * 13) heute Abend
 * 14) übermorgen Nachmittag
 * 15) morgen Nachmittag
 * 16) heute
 * 17) gestern Morgen
 * 18) übermorgen
 * 19) gestern Nacht
 * 20) heute Nacht
 * 21) vorvorgestern

Continue Lesson>>

Days and Months
1) Dienstag 2) im März 3) Sonntagnachmittag 4) am Donnerstag 5) Februar 6) Freitagnacht 7) Mittwoch 8) im Dezember 9) Mittwochmorgen 10) am Samstag 11) Oktober 12) Dienstagabend 13) Freitag 14) im August 15) Samstagnachmittag 16) am Montag 17) Juni 18) Donnerstagmorgen 19) Sonntag 20) im Januar 21) Samstagnacht 22) am Mittwoch 23) Juli 24) Sonntagmorgen 25) Donnerstag 26) im November 27) Montagabend 28) am Freitag 29) Mai 30) Donnerstagnachmittag

Continue Lesson>>

Dates

 * 1) vierter Juli (but am vierten Juli = on July, 4th,der vierte Juli = the 4th of July. The same applies to all dates)
 * 2) siebenundzwanzigster August
 * 3) erster Januar
 * 4) sechzehnter Mai
 * 5) fünfundzwanzigster Dezember
 * 6) neunundzwanzigster Februar
 * 7) elfter September
 * 8) einunddreißigster Oktober
 * 9) zwölfter März
 * 10) siebter November
 * 11) fünfter April
 * 12) zehnter Juni

Continue Lesson>>

Birthdays
A. Guten Tag, Frau Schneider. B. Hallo, Lisa. A. Wann haben Sie Geburtstag? B. Am einundzwanzigsten Februar. Und wann hast du Geburtstag? A. Ich habe am achten Juli Geburtstag.

A. Ich habe am vierzehnten Mai Geburtstag. Wann habt ihr Geburtstag? B. Ich habe auch im Mai Geburtstag! Ich habe am dritten Mai Geburtstag. C. Und ich habe am siebenundzwanzigsten September Geburtstag.

A. Guten Morgen. Wie geht's? B. Sehr gut. Heute habe ich Geburtstag! A. Alles Gute zum Geburtstag!

Continue Lesson>>

Seasons

 * 1) Der Zwanzigste Juli ist im Sommer.
 * 2) Der Achte März ist im Frühling.
 * 3) Der Vierzehnte Januar ist im Winter.
 * 4) Der Zweite Oktober ist im Herbst.
 * 5) Der Sechsundzwanzigste August ist im Sommer.
 * 6) Der Elfte September ist im Herbst.
 * 7) Der Neunundzwanzigste Februar ist im Winter.
 * 8) Der Fünfte Mai ist im Frühling.
 * 9) Der Siebzehnte November ist im Herbst.
 * 10) Der Dreißigste April ist im Frühling.

Continue Lesson>>

Periods of Time

 * 1) from 8:00 till 9:40
 * 2) from Friday till Sunday
 * 3) I play baseball from 5 p.m. until 7 p.m. on Tuesday.
 * 4) They play volleyball from 11:30 till 2.
 * 5) Do you play on Saturday from 10:15 till 11:45?
 * 6) We do homework from 10 p.m. today till 1:30 a.m. tomorrow.
 * 7) from noon on Wednesday till midnight on Friday
 * 8) from May 3rd till June 24th
 * 9) You guys don't play on Thursday and Friday from 3 p.m. until 5 p.m.
 * 10) Do you (formal) play soccer on Tuesday from 8 p.m. till 9:30 p.m.?

Continue Lesson>>

A Number of Times

 * q
 * k
 * p
 * m
 * e
 * t
 * a
 * o
 * i
 * l
 * r
 * b
 * h
 * c
 * j

Continue Lesson>>

Often Adverbs

 * 1) Ich spiele immer im Herbst Fußball.
 * 2) Manchmal lesen sie am Freitag.
 * 3) Wir schauen nie Fernsehen.
 * 4) Ihr schwimmt oft, aber ihr macht nur manchmal Hausaufgaben.
 * 5) Wir spielen oft Tennis, und wir spielen immer Baseball.
 * 6) Er macht nie Sport.
 * 7) Ich spiele nicht nur Volleyball. Manchmal im Winter spiele ich Basketball.
 * 8) Spielst du meistens Football?
 * 9) Wie oft schreibt sie?
 * 10) Ich spiele nur im Frühling Fußball, nicht im Herbst.

Continue Lesson>>

Time-Related Words

 * 1) Ich habe am Wochenende, aber nicht am Samstag von Vierzehn Uhr dreißig bis Siebzehn Uhr, Freizeit.
 * 2) Ich habe am Morgen von Neun bis halb Elf Zeit.
 * 3) Ich habe manchmal am Wochenende Zeit.
 * 4) Ich habe um halb Elf am Samstag Freizeit.
 * 5) Ich habe am sechsundzwanzigsten April um Sechzehn Uhr zwanzig Zeit.
 * 6) Ich habe am Freitagnachmittag Freizeit.
 * 7) Ich habe am Sonntag, von Siebzehn Uhr fünfundvierzig bis Mitnacht, Freizeit.
 * 8) Ich habe von Viertel nach Zehn bis Zwanzig vor Sechzehn Zeit.
 * 9) Ich habe am Dienstag, Mittwoch, Freitag, und Sonntag Freizeit.
 * 10) Ich habe nie Zeit.

Back to Lesson 2>>

Freizeit/Answers

title: German/Level I/Freizeit/Answers


text:
User talk:Spammy

Drop a line if you want to talk about a book/section.

title: User talk:Spammy


text:
Topology

General Topology is based solely on set theory and concerns itself with structures of sets. It is at its core a generalization of the concept of distance, though this will not be immediately apparent for the novice student. Topology generalizes many distance-related concepts, such as continuity, compactness, and convergence.

Before You Begin
In order to make things easier for you as a reader, as well as for the writers, you will be expected to be familiar with a few topics before beginning.

 * Real analysis
 * Continuous Functions
 * Sequences & Series, Convergence & Divergence
 * /Set Theory/
 * Set Operations: Union, Intersection, Complement, De Morgan's laws, etc.
 * Order Relations: Ordered Sets, Equivalence relations, Lattices.
 * Functions: Definition and Properties of Functions
 * Cardinality: Finite, Countable, and Uncountable sets
 * Zorn's Lemma and the Axiom of Choice
 * Mathematical Logic & Proofs
 * Mathematics is all about proofs. One of the goals of this book is to improve your skills in doing proofs, but you will not learn any of the basics here.

Motivation and Preliminaries

 * Chapter 1.1 /Introduction/
 * Chapter 1.2 /History/
 * Chapter 1.3 /Basic Concepts Set Theory/

Introduction to Topology

 * Chapter 2.2.1 /Metric Spaces/
 * Chapter 2.2.2 /Topological Spaces/
 * Chapter 2.2.3 /Bases/
 * Chapter 2.2.4 /Points in Sets/
 * Chapter 2.2.5 /Sequences/
 * Chapter 2.2.6 /Subspaces/
 * Chapter 2.2.7 /Order/
 * Chapter 2.2.8 /Order Topology/
 * Chapter 2.2.9 /Product Spaces/
 * Chapter 2.2.10 /Quotient Spaces/
 * Chapter 2.2.11 /Continuity and Homeomorphisms/

Properties of Topological Spaces

 * Chapter 2.3.1 /Separation Axioms/
 * Chapter 2.3.2 /Connectedness/
 * Chapter 2.3.3 /Path Connectedness/
 * Chapter 2.3.4 /Compactness/
 * Chapter 2.3.5 /Comb Space/
 * Chapter 2.3.6 /Local Connectedness/
 * Chapter 2.3.7 /Linear Continuum/
 * Chapter 2.3.8 /Countability/
 * Chapter 2.3.9 /Cantor Space/
 * Chapter 2.3.10 /Completeness/ - not a topological property
 * Chapter 2.3.11 /Completion/
 * Chapter 2.3.12 /Perfect map/ - optional section which is challenging

Vector Spaces

 * Chapter 3.1 /Vector Spaces/
 * Chapter 3.2 /Morphisms/
 * Chapter 3.3 /Convexity/
 * Chapter 3.4 /Hahn-Banach Theorem/
 * Chapter 3.5 /Topological Vector Spaces/
 * Chapter 3.6 /Euclidean Spaces/
 * Chapter 3.7 /Normed Vector Spaces/
 * Chapter 3.8 /Banach Spaces/
 * Chapter 3.9 /Hilbert Spaces/

Homotopy

 * Chapter 4.1 /Free group and presentation of a group/
 * Chapter 4.2 /Deformation Retract/
 * Chapter 4.3 /Homotopy/
 * Chapter 4.4 /The fundamental group/
 * Chapter 4.5 /Induced homomorphism/

Polytopes

 * Chapter 5.1 /Simplicial complexes/
 * Chapter 5.2 /Barycentric Coordinates/
 * Chapter 5.3 /Geometric Complexes/
 * Chapter 5.4 /Barycentric Subdivision/
 * Chapter 5.5 /Simplical Mappings/
 * Chapter 5.6 /Imbedding Theorem/

Homology

 * Chapter 6.1 /Exact Sequences/
 * Chapter 6.2 /Homology Groups/
 * Chapter 6.3 /Singular Homology/
 * Chapter 6.4 /Relative Homology/
 * Chapter 6.5 /Mayer-Vietoris Sequence/
 * Chapter 6.6 /Excision Theorem/
 * Chapter 6.7 /Eilenberg–Steenrod axioms/
 * Chapter 6.8 /Relative Homotopy/
 * Chapter 6.9 /Vietoris Homology/

Cohomology

 * Chapter 7.1 /Cohomology/
 * Chapter 7.2 /Cohomology Product/
 * Chapter 7.3 /Cap-Product/
 * Chapter 7.4 /Relative Cohomology/
 * Chapter 7.5 /Induced Homeomorphism/
 * Chapter 7.6 /Čech Cohomology/

Advanced Methods

 * Chapter 8.1 /Poincaré Duality/
 * Chapter 8.2 /Spectral Sequences/

Differential Topology

 * Chapter 9.1 /Manifolds/
 * Chapter 9.1.1 Categories of Manifolds
 * Chapter 9.2 /Tangent Spaces/
 * Chapter 9.3 /Vector Bundles/

Appendices

 * A /Further Reading/
 * B /Index/

Question & Answer
Have a question? Why not ask the very textbook that you are learning from?

1. What is the difference between topology, algebra and analysis?

 * Topology is a generalization of analysis and geometry. It comes in many flavors: point-set topology, manifold topology and algebraic topology, to name a few. All topology generalizes concepts from analysis dealing with space such as continuity of functions, connectedness of a space, open and closed sets, (etc.). Algebraic topology attributes algebraic structures (groups, rings etc.) to families of topological spaces to distinguish topological differences in those families. Manifold topology works with spaces that are locally the same as Euclidean space, i.e., surfaces. Often manifolds are equipped with extra structure, such as smooth, PL, symplectic, etc. A naive description of topology is that it identifies those qualities of a space that do not change under twisting and stretching of that space. As such, it is popularly referred to as "rubber sheet geometry." In reality topology does far more than this, in fact providing a rigorous foundation under all branches of mathematics dealing with "spaces."

 * Algebra deals with the structure of sets under various operations with particular properties. Commonly studied algebraic objects include Groups, Rings and Field. One of the major results from Algebra include Galois Theory, which eventually shows that there is no general solution to quintic polynomial equations by radicals. Also important results from Algebra are the Fundamental Theorem of Algebra (which says that, in the Field of Complex numbers, every non constant polynomial has at least one root), Group Classification, and much more.

 * Analysis (or specifically real analysis) on the other hand deals with the real numbers $$\mathbb{R}$$ and the standard topology and algebraic structure of $$\mathbb{R}$$. Analysis provides rigorous proofs for the definitions of derivatives and integrals, as well as treatments of sequences and limits. One can, in some sense, view it as a rigorous treatment of the Calculus.

2. How are the concepts of base and open cover related? It seems that every base is an open cover, but not every open cover is a base. But, why are both concepts needed?

 * The terms base and open cover are not evidently related. Every base is an open cover which is probably the main relation. Take a second countable topological space for instance (second countable means that the space has a countable base for its topology). Such a space satisfies the property that every open cover has a countable sub-cover. To prove this we use the countability of the base. Basically, for any open cover, we choose for each element of the space, an element of the open cover containing it and hence a basis element contained in that element of open cover. Therefore, for any open cover, we can generate a open cover of basis elements that is an 'open refinement' (see Wikipedia for definition). From here we can get properties of open covers from properties of the base. If the base is countable, we can generate a countable open cover from the original cover.

The reason we have both definitions is because these two things have different properties. The most useful fact about a base is that it determines the topology. A basis must have "arbitrarily small" sets, that is, any open set contains a basis element. On the other hand, an open cover does not determine the topology at all. It can be used to build things such as partitions of unity, and often draws on the compactness property. Topology Expert (talk) 04:17, 8 June 2008 (UTC)

3. What is a homology?

 * While homotopy is what one would want to study at first, it turns out that most questions in homotopy theory are rather hard. If we "replace" the spheres of homotopy theory with simplices we can extract similar information about "holes" in the space (often what one is interested in), we get a much more computable sequence of groups. The Hurewicz theorem even gives us that in some cases the homotopy groups can be calculated via the homology groups.

General Topology

 * Aleksandrov; Combinatorial Topology (1956)
 * Baker; Introduction to Topology (1991)
 * Dixmier; General Topology (1984)
 * Engelking; General Topology (1977)
 * Munkres; Topology (2000)
 * James; Topological and Uniform Spaces (1987)
 * Jänich; Topology (1984)
 * Kuratowski; Introduction to Set Theory and Topology (1961)
 * Kuratowski; Topology (1966)
 * Roseman; Elementary Topology (1999)
 * Seebach, Steen; Counterexamples in Topology (1978)
 * Willard; General Topology (1970)

Algebraic Topology

 * Marvin Greenberg and John Harper; Algebraic Topology (1981)
 * Allen Hatcher, Algebraic Topology (2002)
 * Hu, Sze-tsen, Cohomology Theory (1968)
 * Hu, Sze-tsen, Homology Theory (1966)
 * Hu, Sze-tsen, Homotopy Theory (1959)
 * Albert T. Lundell and Stephen Weingram, The Topology of CW Complexes (1969)
 * Joerg Mayer, Algebraic Topology (1972)
 * James Munkres, Elements of Algebraic Topology (1984)
 * Joseph J. Rotman, An Introduction to Algebraic Topology (1988)
 * Edwin Spanier, Algebraic Topology (1966)

de:Mathematik: Topologie Topologie 位相幾何学 Topologia ogólna

title: Topology


text:
Talk:Calculus/Ordinary differential equations

Section Move
I believe it would be a good idea to compile all of the sections of differential equations under a seperate module. Having its own section on the mathematics bookshelf would help with the overall organization of the mathematics section. There are so many different topics in differential equations that I feel the move is justified.

Spammy 16-06-04

That would have some advantages, but differential equations also clearly come under the calculus heading, and should be fully covered there, which means leaving those pages where they are.

Fortunately, Wikibooks has more flexibility than paper books. I see two obvious options.

 * We could copy the ODE coverage from this wikibook to a new ODE book, then expand on it in the new book to cover much beyond this book's scope, while leaving this book's coverage largely unaltered (since its ODE coverage is no more than I'd expect for a book of its declared scope.) Presumably PDE's would be treated in the same way.

 * We could regard the Calculus book as a multiple volume series, making the mathematics bookshelf an hierarchical list (which would certainly help with its organisation). Trying to fit all the maths books in at the same level obscures the dependance relations between the subjects, and multi-volume series are reasonably common for textbooks.

 * Do this, and the current Calculus book would be left with only links to its subsidary books, possible interleaving chapters from different books in a suggested order.

I prefer the second option. Anyone else got an opinion? Carandol 18:35, 16 Jun 2004 (UTC)

Just wanted to add that there is a Wikibook on differtial equation. Would save time and effort, if work on that book was coordinated from here. Would be better still if that effort was integrated into this, as Carando above suggests

 * I agree with all of the above. Differential Equations do not belong in an introductory calculus textbook, except as a small reference for further reading. 128.120.218.130 (talk) 22:14, 26 August 2009 (UTC)

 * I too agree that a comprehensive exposition of diff eqs belongs elsewhere. However, there is a little bit of it in Stewart and it is usually briefly touched in third-semester calculus (at the couple of schools I've been at).  So perhaps we should keep a bit. Tiled (talk) 17:26, 25 November 2009 (UTC)

Explain the old style differential notation
It would be useful to have an explanation of the old style of writing differential equations where the dx and dy already appear as separate quantities, such as y - sin^2(x) dx + sin(x)dy = 0. The article shows treating separable equations by using dx and dy separately, but older texts also present non-separable equations with the differentials apart.

Tashirosgt (discuss • contribs) 06:23, 4 April 2011 (UTC)


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Topology/Basic Concepts Set Theory

This chapter concisely describes the basic set theory concepts used throughout this book—not as a comprehensive guide, but as a list of material the reader should be familiar with and the related notation. Readers desiring a more in-depth understanding of set theory should read the Set Theory Wikibook.

Basic Definitions
The empty set is denoted by symbol $$\varnothing$$. A finite set consisting of elements $$x_1, x_2, \ldots, x_n$$ is denoted $$\{x_1, x_2, \ldots, x_n\}$$. Set theorists commonly, albeit sloppily, do not distinguish strictly between a singleton set $$\{x\}$$ and its single element $$x$$.

For a more in depth understanding of how elements of sets relate to each other, we must first define a few terms. Let A and B denote two sets. Note that $$\left|P(A)\right|=2^{\left|A\right|}$$.
 * The union of A and B, denoted $$A\bigcup{B}$$, is the set of all x that belong to either A or B (or both).
 * The intersection of A and B, denoted $$A\bigcap{B}$$, is the set of all x that belong to both A and B.
 * The difference of A and B, denoted $$A\backslash B$$ or $$A-B$$, is the set of all $$x\in A$$ such that $$x\notin B$$.
 * In contexts where there is a set containing "everything," usually denoted U, the complement of A, denoted $$A^c$$, is $$U\backslash A$$.
 * The symmetric difference of A and B, denoted $$A\Delta B$$, is defined by $$A\Delta B=(A\backslash B)\bigcup{(B\backslash A)}$$.
 * A is a subset of B, denoted $$A\subseteq B$$, if and only if every element in $$A$$ also belongs to $$B$$. In other words, when $$\forall x\in A:x\in B$$. A key property of these sets is that $$A=B$$ if and only if $$A\subseteq B$$ and $$B\subseteq A$$.
 * A is a proper subset of B, denoted $$A\subsetneq B$$, if and only if $$A\subseteq B$$ and $$A \ne B$$. (We do not use the notation $$A\subset B$$, as the meaning is not always consistent.)
 * The cardinality of A, denoted $$\left|A\right|$$, is the number of elements in A.
 * Examples:
 * $$\left|\left\{1,2,3,4,5\right\}\right|=5$$
 * $$\left|\varnothing\right|=0$$
 * $$\left|\left\{\varnothing\right\}\right|=1$$
 * The power set of A, denoted $$P(A)$$, is the set of all subsets of A.
 * Examples:
 * $$P(\varnothing)=\left\{\varnothing\right\}$$
 * $$P(\left\{x\right\})=\left\{\varnothing,\left\{x\right\}\right\}$$
 * $$P(\left\{x,y\right\})=\left\{\varnothing,\left\{x\right\},\left\{y\right\},\left\{x,y\right\}\right\}$$

Ordered n-tuples are denoted $$(x_1,x_2,\ldots,x_n)$$. For two ordered sets $$X=(x_1,x_2,\ldots,x_n)$$ and $$Y=(y_1,y_2,\ldots,y_n)$$, we have $$X=Y$$ if and only if $$\forall i \in \mathbb{N}, 1 \le i \le n:x_i = y_i$$.

N-tuples can be defined in terms of sets. For example, the ordered pair $$\langle x,y\rangle$$ was defined by Kazimierz Kuratowski as $$\left(x,y\right):=\left\{\{x\},\{x,y\}\right\}$$. Now n-tuples are defined as
 * $$(x_1, x_2,\ldots, x_n)\ :=\ \{\langle 1,x_1\rangle ,\langle 2,x_2 \rangle ,\ldots,\langle n,x_n\rangle \}.$$

We now can use this notion of ordered pairs to discuss the Cartesian Product of two sets. The Cartesian Product of A and B, denoted $$A\otimes B$$, is the set of all possible ordered pairs where the first element comes from A and the second from B; that is,
 * $$A\otimes B=\left\{ (a,b)~\left| ~a\in A,~b\in B \right. \right\}$$.

Now that we have defined Cartesian Products, we can turn to the notions of binary relations and functions. We say a set R is a binary relation from A to B if $$R\subseteq A\otimes B$$. If $$(x,y)\in R$$, it is customary to write xRy. If R is a relation, then the set of all x which are in relation R with some y is called the domain of R, denoted domR. The set of all y such that, for some x, x is in relation R with y is called the range of R, denoted ranR. A binary relation F is called a function if every element x in its domain has exactly one element y in its range such that xFy. Also, if F is a function, the typical notation is $$F(x)=y$$ instead of xFy.

There are a few special types of functions we should discuss. A function $$F:A\to B$$ is said to be onto a set B, or a surjective function from A to B, if ran$$F=B$$. A function F is said to be one-to-one or injective if $$a_{1}\in \text{dom }F,~a_{2}\in \text{dom }F,\text{ and}~a_{1}\ne a_{2}$$ implies $$F(a_{1})\ne F(a_{2})$$. A function that is both injective and surjective is called bijective.

Exercises
If you can successfully answer the following problems, you are ready to study topology! Please take the time to solve these problems.

 * 1) Prove that the empty set is a subset of every set.
 * 2) Consider the set $$A_n=(-n,n)$$ for each n in the set of natural numbers. Does the union over all $$A_n$$ (for n in the set of natural numbers) equal $$\mathbb R$$ (the set of all real numbers)? Justify your answer.
 * 3) Using $$A_n$$ from above, prove that no finite subset of $$A_n$$ has the property that the union of this finite subset equals $$\mathbb R$$. Once you study topology, you will see that this constitutes a proof that $$\mathbb R$$ is not compact.

title: Topology/Basic Concepts Set Theory


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Evolutionary Biology/Charles Lyell

< Evolution before Darwin

Charles Lyell lived from 1797 to 1875. Throughout his life, his first love was geology. By using his studies of the Earth, Lyell was one of the first scientists to try and create a time scale for the history of the earth.

His book titled Principles of Geology is thought to be one of the most important scientific books ever written because it proposed a way to study the forces that changed the geology of the earth by observing geology of the modern world. This directly contrasted the theories of catastrophism and supernatural occurrences. In this book, he introduced his theory of uniformitarianism. This is the idea that by using observations of current natural processes, we can predict how processes occurred in the past. In order to do this, we must accept that changes in nature occurring millions of years ago are similar to the changes that occur today. This idea of a uniform process of change is the basis of the theory of uniformitarianism. He also believed that each of these processes followed the already known natural laws. He thought that the changes the earth went through would continue on the same path throughout the rest of the existence of the earth. This theory is very important in the aspect that it can possible explain occurrences like mountain building, glaciation and volcano creation.

Darwin was very impressed by Lyell’s geologic findings in Principles of Geology. He studied Lyell’s work as he traveled on the Beagle. As he explored the coastlines of South America, Lyell’s book had a great influence on how Darwin looked at the geologic aspects of the land. In turn, many of Darwin’s experiences and findings reinforced many ideas put forth by Lyell in Principles of Geology.

title: Evolutionary Biology/Charles Lyell


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Teaching Assistant in France Survival Guide

Teaching assistants in France  are non-nationals granted temporary work visas to teach their native languages in French public schools. English, Spanish, Italian and German language assistants are the most common.

This guidebook serves current, potential, and past teaching assistants in France. (For those of us who have already served our terms, helping those who follow can be therapeutic.) It strives to assemble in one place, and improve upon, information useful to assistants — perhaps to any foreigner living in the Hexagon.

In this guidebook you will learn about being an assistant: the application and acceptance process, the paperwork, teaching and eventually leaving. You will also learn about traveling on a budget in Europe. If any of its information has fallen out of date or is lacking, please do contribute.

Currently, much of the information in this guide is geared towards American English assistants. Being a Wiki, we hope that it will grow to accommodate everyone in the program, eventually spanning languages.

Add your name and contact info to the Teaching Assistant Directory (for all languages) if you'd like to get in touch with other assistants in your académie, or perhaps e-mail former assistants for advice.

Some teaching assistants in the program do not go to metropolitan France at all; they go to overseas departments such as Guadeloupe. If you're one of them, you'll find this guide needs a whole new chapter: please write it!

A few good links for more information on the Teaching Assistantship program are:
 * Assistants in France Forums
 * FTAP Paperwork Guide - What To Do and When To Do It
 * Guide for English Language Assistants in France
 * Assistants Etrangers en France from CIEP
 * French Culture - Assistantship Program
 * The Assistants d'Anglais Program
 * Assistants Etrangers Webring


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General Relativity/What is a tensor?

< General Relativity

A tensor is a powerful abstract entity. And while its abstractness makes it a somewhat difficult thing to describe, we can begin to get a feel for what a tensor is through a non-abstract example.

Suppose you are sailing. The wind is coming from a certain direction and can be described as a vector, a directional quantity. Now there are many ways to represent this vector. For example, you can represent it as a speed from a certain direction ($$\vec w$$). Alternatively, you can break it up into components and describe the vector as a combination of a certain amount of wind from the east and another amount from the south ($$\vec w'$$). But despite the different ways of describing this vector, there is still this underlying abstract thing — the wind speed from a certain direction.

Now there is another important vector — the force that the wind produces when it hits the sail. If the direction of the force were always the same as the direction of the wind, then we could represent the relationship with a scalar, which just multiplies the wind vector by a constant factor to get the force vector.

However, life isn't that easy because the force is not always in the direction of the wind. In fact it usually isn't. So we can't represent the relationship between the wind and the force by a simple scalar. However, there is one important useful fact that we can use — the relationship between the wind speed and the force on the sail is (approximately at low speeds) linear. That is, if you double the speed of the wind, you double the force. The function that computes the force from the wind is a linear operator. The fact that this operator is linear lets us represent it in terms of a matrix, relative to a given basis:

$$T = \begin{pmatrix} x & y \\ z & v \end{pmatrix}$$

Note that just like you can change the way you represent the speed of the wind($$\vec w$$ and $$\vec w'$$), and the force it produces ($$\vec F$$ and $$\vec F'$$), you can change the way you represent the operator that connects the two. In fact, whenever you change the representations of the wind and the forces, you will have to change the matrix in order to talk about the same situation. However, just like a vector is an abstract thing that can represent the wind or the force that the wind produces when it hits the sail, there is another kind of abstract thing that you can use to represent the relation between these things. In symbols

$$\vec F = T \cdot \vec w$$ $$\vec F' = T' \cdot \vec w'$$

That thing in the middle, that T, is an example of a tensor. It is defined by the way in which $$\vec F$$ relates to $$\vec w$$. Tensors are really abstract things, but we now can begin to see the power of this abstractness. For one thing you can do algebra with tensors. So, say instead of one sail T, you have two sails T and U. We can represent the total force

$$\vec F = T \cdot \vec w + U \cdot \vec w$$

Now because T and U are matrices and are linear we can combine them to form a new tensor V.

$$\vec F = (T + U) \cdot \vec w$$

$$\vec F = V \cdot \vec w \ \mathrm{where} \ V = T + U\,$$

We can also multiply two tensors together. We have the force that the wind produces on the ship. Now the force that the ship produces on the water, when wind is acting on ship with force $$F_{wind}$$ can be represent also as a tensor (the same way we did with wind, sail and ship):

$$\vec F_{ocean} = W \cdot \vec F_{wind}$$

$$\vec F_{ocean} = W \cdot V \cdot \vec w$$

$$\vec F_{ocean} = X \cdot \vec w$$

where tensor $$X = W \cdot V = W \cdot (T + U)\,$$ characterizes ship and both sails.

So what is a tensor?
Now that we have seen an example of a tensor, we can be more explicit about what a tensor is. A tensor is a linear function. In the case that we described, a tensor takes a vector and turns it into another vector. We can be more general and talk about tensors that turn a vector into a scalar or a scalar into a vector.

At this point, you should get some sense as to why tensors are important in general relativity. General relativity is all about matter changing the way that distances work. How do you find a distance? Well you take a vector and put it into a function. If the distance is short enough the function is (approximately) linear, and you can describe it as a tensor.

Special tensors
One special tensor is called the Kronecker delta tensor. It is just the identity matrix.

An identity is like 0 for normal addition (adding 0 to a number keeps it the same) and 1 for multiplication (multiplying a number by 1 gives the number itself).

The identity matrix is often represented by δij, where

 * $$\delta^i_j = \left\{

\begin{matrix} 1 & \mbox{if} & i = j \\ 0 & \mbox{if} & i \ne j \end{matrix} \right. $$

Actually, the fact that $$\delta^i_j$$ is actually a tensor (rather than just a symbol that has two indices) is due to the fact that a mixed tensor that has this as its components in any one coordinate frame, will have the same components in any frame! Technically, we call such entities numerically invariant tensors.

Applications of tensors
Tensors are used in many places.

Apart from general relativity, tensors are used extensively in Continuum Mechanics. Tensors can be used to specify the stress at any point in the continuum. Such tensors are called stress tensors. The stress at any small surface (a tiny region around a point) in the continuum can be obtained by the matrix multiplication of this stress tensor and the unit vector (column vector) that is normal to the surface.

General relativity in one page
There are two things that can be described with tensors.

One is the curvature. A tensor describes how much things are bent. Using this tensor you can do things like calculate distances and angles, and figure out the shortest path through an area of space.

The second is stress-energy which is roughly how much energy and momentum exists in a particular location and what direction it is flowing in.

The basic equation of general relativity relates these two tensors. That's the basic idea. Everything else involves just getting used to doing the math.

Exercises
(Note: What I'd really like to do here is to have people add their answers to an answers page where they can see other people's answers. So feel free to add your answers HERE)

1) Describe an example of a tensor.

2) Suppose you have a situation in which the response of the sail to the wind is non-linear. How can you describe this in terms of tensors?

3) Can pressure be expressed as tensor?

title: General Relativity/What is a tensor?


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Talk:Guide to Social Activity/Appearance

Body Language
Perhaps the body language part should be moved to a section about strategies when on dates? It just seems to me that it is out of place.

A question...
Why is this article supposed not to be gender specific?
 * very good question. Especially since the title notes that it's for guys. Jordanmills 15:21, 19 September 2006 (UTC)
 * I disagree that it needs to be gender neutral at all. Necromonger Wekeepwhatwekill 22:44, 29 August 2018 (UTC)


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Statistics/Introduction/Need To Know

Statistics is a diverse subject and thus the mathematics that are required depend on the kind of statistics we are studying. A strong background in linear algebra is needed for most multivariate statistics, but is not necessary for introductory statistics. A background in Calculus is useful no matter what branch of statistics is being studied.

At a bare minimum the student should have a grasp of basic concepts taught in Algebra and be comfortable with "moving things around" and solving for an unknown. Most of the statistics here will derive from a few basic things that the reader should become acquainted with.

Absolute Value
$$|x| \equiv \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}$$

If the number is zero or positive, then the absolute value of the number is simply the same number. If the number is negative, then take away the negative sign to get the absolute value.

Examples

 * 42| = 42
 * -5| = 5
 * 2.21| = 2.21

Factorials
A factorial is a calculation that gets used a lot in probability. It is defined only for integers greater-than-or-equal-to zero as:

$$ n! \equiv \begin{cases} n \cdot (n-1)!, & n \ge 1 \\ 1, & n = 0 \end{cases} $$

Examples
In short, this means that:

Summation
The summation (also known as a series) is used more than almost any other technique in statistics. It is a method of representing addition over lots of values without putting + after +. We represent summation using a big uppercase sigma: ∑.

Examples
Very often in statistics we will sum a list of related variables:

$$	\sum_{i=0}^n x_i = x_0 + x_1 + x_2 + \cdots + x_n $$

Here we are adding all the x variables (which will hopefully all have values by the time we calculate this). The expression below the ∑ (i=0, in this case) represents the index variable and what its starting value is (i with a starting value of 0) while the number above the ∑ represents the number that the variable will increment to (stepping by 1, so i = 0, 1, 2, 3, and then 4). Another example:

$$ \sum_{i=1}^4 2i = 2(1) + 2(2) + 2(3) + 2(4) = 2 + 4 + 6 + 8 = 20 $$

Notice that we would get the same value by moving the 2 outside of the summation (perform the summation and then multiply by 2, rather than multiplying each component of the summation by 2).

Infinite series
There is no reason, of course, that a series has to count on any determined, or even finite value—it can keep going without end. These series are called "infinite series" and sometimes they can even converge to a finite value, eventually becoming equal to that value as the number of items in your series approaches infinity (∞).

Examples
This example is the famous geometric series. Note both that the series goes to ∞ (infinity, that means it does not stop) and that it is only valid for certain values of the variable r. This means that if r is between the values of -1 and 1 (-1 < r < 1) then the summation will get closer to (i.e., converge on) 1 / 1-r the further you take the series out.

Linear Approximation
Let us say that you are looking at a table of values, such as the one above. You want to approximate (get a good estimate of) the values at 63, but you do not have those values on your table. A good solution here is use a linear approximation to get a value which is probably close to the one that you really want, without having to go through all of the trouble of calculating the extra step in the table.

$$ f\left(x_i\right) \approx \frac{f\left(x_{\lceil i \rceil}\right) - f\left(x_{\lfloor i \rfloor}\right)}{x_{\lceil i \rceil} - x_{\lfloor i \rfloor}} \cdot \left(x_i - x_{\lfloor i \rfloor}\right) + f\left(x_{\lfloor i \rfloor}\right) $$

This is just the equation for a line applied to the table of data. xi represents the data point you want to know about, $$x_{\lfloor i \rfloor}$$ is the known data point beneath the one you want to know about, and $$x_{\lceil i \rceil}$$ is the known data point above the one you want to know about.

Examples
Find the value at 63 for the 0.05 column, using the values on the table above.

First we confirm on the above table that we need to approximate the value. If we know it exactly, then there really is no need to approximate it. As it stands this is going to rest on the table somewhere between 60 and 70. Everything else we can get from the table:

$$ f(63) \approx \frac{f(70) - f(60)}{70 - 60} \cdot (63 - 60) + f(60) = \frac{1.66691  - 1.67065}{10} \cdot 3 + 1.67065 = 1.669528 $$

Using software, we calculate the actual value of f(63) to be 1.669402, a difference of around 0.00013. Close enough for our purposes.

title: Statistics/Introduction/Need To Know


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Modern Physics/Introduction to Special Relativity

At first glance, special relativity seems weird. A number of rules which we use to understand the world no longer apply. These rules include the rule that velocities can be added to each other, the rule that time and space are different things, the rule that matter and energy are separate, the rules for momentum. Because we can no longer rely on our normal intuition to figure things out, things can seem bizarre and confusing.

Fortunately, the apparent oddness of special relativity does not arise from deep complexity. Underneath the surface, there are some very beautiful simple symmetries and rules, and these symmetries and rules allow us to relate special relativity with the physics with which we are familiar, with only a few minor amendments.

Nevertheless, some of the ideas of special relativity seem so contrary to the way that we naively expect that the universe ought to behave and to our everyday experience that a perfectly natural reaction is that special relativity could be not possibly be correct.

However, numerous experiments have demonstrated that it's not special relativity that is incorrect, but rather our low speed intuition about how things should work. After all, we don't normally deal with things moving at close to the speed of light, and there is no particular reason to believe that our low speed thinking applies at high speed. And experiments clearly indicate that they don't. Furthermore, it turns out that many ordinary phenomena really are the result of relativity. Magnetism, for example, is a purely relativistic phenomenon that we see daily but which can't be properly explained without relativity.

But coming to this realization was difficult for physicists. One way of intuiting special relativity is take the journey that physicists did. We shall start with the ordinary everyday universe of classical Newtonian physics, which we studied in the previous two books, and then gradually travel out to the places which are odd. We are helped in this journey by the fact that the road has already been travelled by generations of physicists who gradually came to the realization that the world was stranger than they had imagined.

Therefore, the approach we shall take is to first go down a short historical survey which illustrates why special relativity is necessary. We shall then talk about the great conceptual breakthrough that Albert Einstein made, and then from the simple axioms that he figured out we shall see the subtle consequences that follow. We shall see too how straightforward special relativity really is, how simple at its heart.


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Evolutionary Biology/Darwin's Life and Impact

Charles Darwin (1809-1882) was one of the most influential individuals in the history of science. Although he published a number of important works during his lifetime, he is best known for The Origin of Species (1859). Although the idea that species evolve was not a new one (see Evolution before Darwin), it was not widely accepted in Darwin's time. Aside from the enormous body of evidence that species in fact do change over time, in Origin, Darwin fully developed natural selection as the first plausible mechanism regarding how they change. A great deal of Darwin's success arose from the details of his life, travels, and the circle of influential individuals with whom he associated.

Early life
Robert Darwin (1766-1848) sent his son Charles to medical school at Edinburgh University when he was only 16 years old. Charles found it boring, and withdrew at age 18, after which his father sent him to Christ College in Cambridge to study for the clergy. While at Christ College, Darwin became interested in botany and insects, and he attended several botany lectures. One of Darwin's teachers, John Stevens Henslow, launched Darwin's interest in geology. Professor Henslow was known to teach his students to make careful observations of natural phenomena. While enrolled at Christ College, Darwin found himself unfit for a life as a minister.

HMS Beagle
The most important experience of Darwin's life was his trip aboard HMS Beagle as a naturalist, a person familiar with natural history, particularly in zoology and botany. Darwin was not the top choice as a naturalist aboard the Beagle; he was very fortunate to even receive this opportunity. Robert FitzRoy, the captain of the Beagle, initially asked Henslow, Darwin's professor, to be the naturalist of the ship, but Henslow's wife discouraged him. Then, Henslow's brother-in-law was approached to replace Henslow, but he too rejected the job. Finally, Henslow recommended Darwin as a well-qualified replacement. Captain FitzRoy was originally cold to the idea of having Darwin on the voyage, but he ultimately decided that he would allow Darwin on the journey. Darwin's position at first was the captains travel companion, since FitzRoy had already given a surgeon named McKormick the naturalist position. McKormick, though, left the position soon after the Beagle departed England, leaving Darwin as the naturalist.

The Beagle left England in 1831, and it returned in 1836. The overall purpose of the voyage was to map uncharted areas of the South American coastline. While aboard the Beagle, Darwin read Charles Lyells Principles of Geology (1830). Lyell was a famous geologist and eventually became a lifelong friend of Darwin. He developed the theory of uniformitarianism, which holds that the Earth is very old, and that geological changes are due to slow, continuous processes. These processes, such as erosion and deposition, are constant. Lyell's uniformitarianism view opposed the generally-accepted catastrophism view proposed by Georges Cuvier. Catastrophism is the theory that changes in the earth are due to natural disasters.

Later Life and Sickness
A month after returning to England, Darwin proposed to Emma Wedgwood, his first cousin. On January 30, 1839, they got married. Darwin became sick around this time. He may have been infected by Trypanosoma cruzi, a parasite prevalent in both Central and South America. The infection causes Chagas' disease. Darwin showed many of the symptoms related to Chagas disease: flatulence, vomiting, palpitations, and insomnia. Whatever sickness Darwin had, it caused him to stay near home for much of the rest of his life. Darwin was not even able to attend his father's funeral due to the severity of his illness. However, this sickness may have given him more time to read, think, and write. He inherited a lot of money, and his wife brought a significant amount of money into the marriage, affording him additional time towards his studies.

Basis of Evolution
From recovered notes, historians know that Darwin saw unique species of plants and animals in South America, such as finches and iguanas. He noticed that organisms differed more as a function of geographic distribution than as a function of climate: the temperate South American animals more closely resembled the tropical South American animals than the temperate European animals. Also, some characteristics of the fossils Darwin observed in South America paralleled some contemporary South American organismal features. While on the Galápagos Islands, Darwin had collected many species of finches.

After returning to England, Darwin gave this collection of finches to John Gould, a man who studied birds. Gould measured the differences in beak sizes of the finches, and his measurements showed great variances. The sample included small, medium, and large beaks. Darwin began to ponder how and why this diversity in the finches' beak structure took place. He imagined that these changes within populations may better adapt them to their local environment. He soon realized that adaptation and speciation were closely related. A new species may develop as a result of slow, gradual adaptations, just as a big change in the Earth could occur from slow continuous processes. Darwin thus began to formulate his theory of natural selection as the main process in evolution. Almost a half century earlier, Robert Malthus had written An Essay on the Principles of Population (1798). This essay explained that human growth rate is geometric whereas food production is only arithmetic. Continuation of geometric human growth in this fashion leaves a huge discrepancy between food and population. Darwin read this in 1838, and he pictured this discrepancy as a struggle for existence. In most cases, the struggle for existence is not an obvious battle, and some may not even regard it as a struggle. What actually takes place is an inheritance of favorable characteristics. "Favorable characteristics" cause individuals to survive better. Since they survive better, individuals that inherit favorable characteristics will gradually represent greater numbers in the population. Darwin saw that eventually favorable characteristics would replace the unfavorable ones, and a new species would arise.

Natural Theory and God
Before Darwin's publications, natural theology dominated. Natural theology is the idea that God's plan can be known through human reason. According to this logic, adaptation of any given organism is evidence that God intended the adaptation for a specific function. Knowing that his theory would contradict the beliefs of natural theology and would cause anger once published, Darwin procrastinated publication.

Besides coming from religious individuals, strong beliefs came from scientific minds. Those who did believe in evolution around Darwin's time thought that it occurred by the inheritance of acquired characteristics, a theory known as Lamarckism (proposed by Jean-Baptiste Lamarck). Lamarckism claims that the parts of the body that are used frequently during an individuals lifetime are exaggerated in the offspring. This, however, was later proven false through genetics. The theory of "Inheritance of Acquired Characters" as it is called by scientific community was proved false by August Weismann, a German evolutionary theorist. Weismann cut the tails of lab rats over many generations but could not find any change in the length of the tails of new-born rats. He also concluded that piercing of ear pinna, wearing of tight clothes for a slimmer waist by women of middle ages, all carried out over generations doesn't change any of those characters during birth. The changes an individual makes to his body in his lifetime do not alter his sex cells. Only changes that affect the "Germ Cells" or sex cells of an individual in lifetime are inherited by the next generation.

Origin of Species and Publication
Throughout his life, Darwin corresponded with scholarly groups, friends, and colleagues. Lyell knew of Darwin's ideas regarding natural selection, and suggested he publish them quickly. Not long after, Darwin received a letter from Alfred Russell Wallace, another British naturalist working in Southeast Asia. Wallace wanted Darwin to review his work before publication. As Darwin began to read, he was shocked to find that Wallace's theory was almost exactly the same as his own. Initially, Darwin worked with Wallace, and in 1858, they publicly announced to the Linnaean Society of London the theory of natural selection. Because Darwin had begun writing notes on his theory of natural selection on September 28, 1838, well before Wallace, and he had much more thoroughly explained it, he is credited for the theory of natural selection.

In 1859, the first edition of The Origin of Species was published under the name: On The Origin of Species by Means of Natural Selection or the Preservation of Favoured Races in the Struggle for Life. His theory of natural selection states that the individuals best adapted to their environment will produce the most offspring. This will result in an increase in the proportion of individuals with the favorable genotypes. Although this increase in favorable genotypes is the cause for natural selection, genetics was not well-understood until Gregor Mendel's paper, written in 1865, was re-discovered in 1900.

In The Origin of Species, Darwin did not relate his theory of natural selection to humans because he knew it would cause even more outrage among his friends and family. In addition to The Origin of Species, Darwin wrote 19 other books, and he addressed human evolution in two of his publications late in his life: The Descent Of Man, And Selection In Relation To Sex (1871) and Expression Of Emotions In Man And Animals (1872).

Effects
Darwin's observations, experiments, and theories affect many scientific fields, including ecology and geology. Even modern day pesticide companies take Darwin's theory of natural selection into account when developing a mixture of chemicals targeted for a certain pest. Knowing that natural selection will favor a pest with resistant genes and cause this chemically resistant pest to grow in proportion, developers periodically change the chemical composition.

Darwin revolutionized the world of science with his publication of The Origin. His theory came to affect the entire culture because it was seen to question the very underpinnings of the worldview. The debate continues even today as is seen with the creationism versus evolution controversy in some school systems. Darwin's theory of natural selection unified biology, and he unalterably changed the world. The "Theory of Evolution" had profound effect on all other branches of science as well. It was for the first time during the course of scientific history, that science was separated from the concept of god/religion and was purely judged on the merits of experiment, observation and inference. In a way Darwin's theory provided complete freedom to science.

Resources
http://www.terindell.com/asylum/jason/darwin.html

http://www.cartage.org.lb/en/themes/Biographies/MainBiographies/D/darwincharles/2.html

http://www.csuchico.edu/~curban/Darwin/DarwinSem-S95.html

Campbell, Reece. Biology, Sixth edition. Benjamin Cummings. 2001.