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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

**CauchySchwarz inequality**

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**All_Is_Number****Member**- Registered: 2006-07-10
- Posts: 258

ganesh wrote:

Newton's laws of motion.First law:- Every body continues in its state of rest or uniform motion unless compleeled by an external force to change its position.

Interestingly, this is *not* Newton's first law of motion. It is actually a restatement of his second law, *F*=*ma*.

Strangely enough, Newton made the oft repeated error when he stated the first law. However, in the pages of *Principia* immediately preceding his statement of the first law, he described the law as something substantially different.

The first law should be (and was *described* by Newton as):

*There exists a frame of reference in which an object continues in its state of rest or uniform motion unless a net external force is applied to that object.*

In fact, without this first law, the second law need not hold.

*You can shear a sheep many times but skin him only once.*

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

**The Sylow theorems** (group theory)

1. If *G* is a group of order *p[sup]n[/sup]k* (where *p* is a prime, *n* and *k* are positive integers and *p* is coprime with *k*) then *G* has a subgroup of order *p[sup]n[/sup]*, called a Sylow *p*-subgroup.

2. If *P* and *Q* are Sylow *p*-subgroups of *G*, there exists *g* ∊ *G* such that *gPg*[sup]−1[/sup] = *Q*.

3. The number of Sylow *p*-subgroups must divide the order of *G* and be congruent to 1 modulo *p*.

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

**Plancks radiation law**

The intensity per unit wavelength of the radiation of a black body of absolute temperature is given by

[align=center]

[/align]where is Plancks constant, is the speed of light in a vacuum, and is Boltzmanns constant.

From this, both **Stefans law** (which states that the intensity of the radiation of a black body is proportional to the 4[sup]th[/sup] power of its absolute temperature) and **Wiens displacement law** (which states that the maximum wavelength of the radiation of a black body is inversely proportional to its absolute temperature) can be derived.

To derive Stefans law, hold

constant and evaluate . For Wiens displacement law, set .**http://z8.invisionfree.com/DYK/index.php?showtopic=867**

*Last edited by JaneFairfax (2009-09-24 03:51:09)*

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

**Three functions**

**Note**:*All three of the above mentioned functions are multiplicative,which means that:*

**Euler's formula**

**Note**:

*Last edited by anonimnystefy (2012-03-16 01:07:37)*

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

**Lowest Common Multiple Identity**

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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