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| bio | website | |
|---|---|---|
| location | Baltimore, MD | |
| age | 23 | |
| visits | member for | 2 years, 2 months |
| seen | 22 mins ago | |
| stats | profile views | 198 |
Graduate student at Johns Hopkins University. Interests include Mathematical Physics, Particle Theory, Geometry (combinatorial, differential, and algebraic), Representation Theory, and Algebraic Topology.
I haven't been very active here recently. I'm currently pretty busy with research and teaching responsibilities, so I don't intend to answer any questions here unless they are somehow related to what I'm studying. This is honestly not much of a loss at all, as there are many capable users here who provide answers far better than what I am capable of.
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May 7 |
awarded | Caucus |
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May 1 |
accepted | Existence of an entire function with algebraically independent derivatives |
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Apr 24 |
comment |
Nice examples of groups which are not obviously groups @wim The point is that the states of the 3x3x3 cube, as they would be observed by someone actually solving the cube (with pieces unlabeled), do form a group naturally (as a quotient of the group of moves of the cube), while the states of the 4x4x4 cube do not. The "invisible differences" of center permutations and orientations are not so important from a mathematical perspective, but from the perspective of someone solving the cube it is a big difference, as the puzzle now requires an additional step to solve, and would typically be considered as a completely different puzzle. |
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Mar 20 |
awarded | Yearling |
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Feb 27 |
awarded | Custodian |
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Feb 27 |
reviewed | Looks Good How does $({{n/e})^n} / ({({n/{2e}})^n})$ simplify to $2^n$ (MIT OpenCourseware 6.006) |
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Jan 9 |
awarded | Good Answer |
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Nov 2 |
answered | Can we express $\sin 1^\circ$ in a real closed, not repetitive, radical forms? |
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Oct 18 |
comment |
How is the error term in the Taylor expansion of a function arrived at? Here's a sketch of a proof of Taylor's theorem in case anyone else is having trouble understanding. $f(x)=f(a)+ \int_a^x f'(t) dt$. Integrating by parts $n$ times gives $f(x)=f(a)+(x-a)f'(a)+ \cdots + \frac{f^{n}(a)}{n!}(x-a)^n + \int_a^x \frac{f^{n+1}(t)}{(n+1)!}(x-t)^{n} dt$. Now it's easy to see that this integral is at most (in absolute value) $|\frac{\max_t f^{n+1}(t)}{(n+1)!} (x-a)^{n+1}|$ (where the maximum is for $t$ in the inerval $[a,x]$), and with just a little bit of messing around we can get to the form you've provided (namely by applying the intermediate value theorem on this). |
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Oct 18 |
comment |
How is the error term in the Taylor expansion of a function arrived at? All the inequalities should be fixed. Sorry about that. The $n+1$ isn't arbitrary. For $f(x)=x$, for example, if you take the 0th-order Taylor polynomial, it's just $a$. But it's not possible, for any $x \ne a$, to write $f(x)=a+\frac{f^2(b)}{2!}(x-a)^2$, or any higher power, because $f^2(b)=0$ regardless of $b$, so the right hand side will always be $a$ and the left hand side is $x$. Of course, for some functions you might be able to do what you are saying, but not in general. As for why you can get it to work with $n+1$, the proof of Taylor's theorem itself is a pretty good explanation... |
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Oct 18 |
revised |
How is the error term in the Taylor expansion of a function arrived at? edited body |
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Oct 17 |
answered | How is the error term in the Taylor expansion of a function arrived at? |
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Oct 17 |
awarded | Nice Answer |
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Oct 7 |
answered | Commutator relationship proof $[A,B^2] = 2B[A,B]$ |
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Oct 7 |
comment |
The real line with its additive group is a topological group? If you want to prove that the additive group of real numbers is a topological group, the relevant map is the addition map $(g_1,g_2) \mapsto g_1 + g_2$. The real numbers under multiplication are not even a group, because 0 is not invertible. |
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Oct 6 |
revised |
Can vectors be inverted? Edited to match edit of OP |
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Oct 6 |
answered | Can vectors be inverted? |
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Oct 6 |
comment |
Can vectors be inverted? Do you mean for $C$ to be a scalar (or a 1x1 matrix)? That's the only way in which the right hand side product will be defined and of the same size as the left hand side. |
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Sep 28 |
answered | Can a subgroup generated by two finite subgroups be infinite? |
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Sep 25 |
answered | Example of a union of subfields that is not a field |
