User:Mathemagician83

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Mathemagician83 is a graduate student at the University of California San Diego in mathematical physics (the chemistry and biochemistry department).

My interests:

• Numerical solution methods for PDEs
• Linear algebra applied to functional calculus
• Calculus of variations
• Graph theory and general combinatorics, especially probabilistic counting problems
• Non-commutative geometry
• Gröbner bases
• Writing code to implement all of the above

These are my interests, and while I am no expert in many of those areas, I am fairly confident in some.

I am an atheist. I respect logic, because logic is all we mortals have, and because it has served science very well for hundreds of years, resulting in significant improvement in quality of life for humans. Aside from obvious logical reasoning which applies, I am forced to point out that religion, particularly the great monotheistic religions, primarily represented by Christianity, have brought almost entirely death and destruction to the world. I will for personal reasons heretofore refer to the Bible as "The Great Book of Contradictions" if ever such a reference is cynical enough to warrant itself. While many religious people will argue that Christianity produces "good" people, I argue that Christianity coerces compliance rather than allowing it to be genuine. Ask yourself this: Would you rather be loved by someone who believes he or she must love you because that's what "god wants," or someone who loves you with a love exempt of coercion?

I think the answer is clear. I'll take the route free of pretence and coercion. Unfortunately, however, the one person I have ever known to be so genuine was taken from me and killed.

Various Math

Here are some nice notes I write from time to time on various topics in applied math and physics.

Gröbner Bases

$S(f,g)={\frac {L}{\mathrm {lt} (f)}}f-{\frac {L}{\mathrm {lt} (g)}}g$

$L=\mathrm {LCM} (\mathrm {lt} (f),\mathrm {lt} (g))$

$G=\left\{g_{1},\dots ,g_{t}\right\}$

$I=\left\{f:f=u_{1}f_{1}+\cdots +u_{s}f_{s},u_{i}\in k[x_{1},\dots ,x_{n}]\right\}$

A reduction process similar to polynomial division can be used to determine membership in the ideal, $I$.

$f{\overset {G}{\longrightarrow }}_{+}r$

The primary accomplishment of Gröbner bases is to implement the following logic. If $G$ is a basis of the ideal, $I$, then

$f{\overset {G}{\longrightarrow }}_{+}0\iff f\in I$.

For any basis of $F\subset I$ it is true that

$f{\overset {F}{\longrightarrow }}_{+}0\Longrightarrow f\in I$,

but the converse is not generally true. Only for particular bases, Gröbner bases, is the converse also true, as implied above.

The most direct way to demonstrate this is to take a polynomial, $f\in I$, write it as a linear combination of a basis, $F=\left\{f_{1},\dots ,f_{s}\right\}\subset I$.

$f=u_{1}f_{1}+\cdots +u_{s}f_{s}$

If it can be written in this form, then we can see that dividing in series by $u_{1},\dots ,u_{s}$ results in the following series of remainders

$f{\overset {u_{1}}{\longrightarrow }}u_{2}f_{2}+\cdots +u_{s}f_{s}{\overset {u_{2}}{\longrightarrow }}u_{3}f_{3}+\cdots +u_{s}f_{s}$

So at first blush it seems that if a polynomial is in the ideal, then by any basis, $F\subset I$, it should reduce to remainder zero. However, this is not the case because linear combinations of $f\in F$ with coefficients in $k[x_{1},\dots ,x_{n}]$ may result in cancelled power products, thereby potentially destroying divisibility of the containing polynomial by any of the terms whose power products cancelled.

--Mathemagician83 (talk) 06:00, 20 July 2009 (UTC)

Circular Convolution

$\displaystyle {\frac {d}{dx}}\int _{a}^{x}f(x')\,dx'=$