Reputation
54,816
Next tag badge:
98/100 score
33/20 answers
Badges
3 51 134
Newest
 Necromancer
Impact
~843k people reached

6h
comment The simplification of divided difference of cosine function
If your question is about how to manipulate a trigonometric expression, you shouldn't give it a title and tags that trick the reader into thinking you need help with limits.
1d
comment What is the meaning of $\exp(\,\cdot\,)$?
@kahen: One might go even further and read $e^z$ as multi-valued, being defined as $\exp(z \ln e)$.
2d
awarded  Necromancer
2d
comment Why do we have “another” definition for the kernel?
The isomorphism theorems for congruences.
Aug
1
answered How can I intuitively interpret this vector operation?
Aug
1
comment How can I obtain this division's limit without using derivatives?
@fibonatic: The question was how to solve the problem with the basic limits, not how to prove the basic limits. (and, IIRC, the usual pedagogy in introductory courses is that the derivatives of the trig functions are proven using the basic limits anyways, rather than the other way around)
Aug
1
comment How can I obtain this division's limit without using derivatives?
@Lemony: We aren't: we're using one of the "basic known limits" involving $\sin$.
Jul
31
answered Why formulate continuity in terms of pre-images instead of image?
Jul
29
comment Why can mathematical induction only be used with natural numbers?
Another variation: structural induction.
Jul
26
comment Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)
I don't see why it would be nonsurjective; to solve $(T-\lambda)f = g$, you can just solve for the leading term $f_0 = -\lambda^{-1} g_0$, and recursively solve for the remaining ones from $f_{n-1} - \lambda f_n = g_n$, and it's not "clear" that the result shouldn't be in the Hilbert space. On the space of all sequences, we can write the inverse as $S = -\lambda^{-1} \sum_{k=0}^{\infty} (T/\lambda)^{k}$. And I think for $|\lambda| > 1$ it maps square summable to square summable, since $$|(Sf)_n| \leq |\lambda|^{-1} \sum_{k=0}^n |f_{n-k} \lambda^{-k}| \leq \max_k(f_k) |\lambda - 1|^{-1}$$
Jul
26
comment What do sine, tan, cos actually mean?
$\sin$ is not a function of $\theta$. $\sin \theta$ is a function of $\theta$. $x,y,r,\theta$ are collectively dependent variables, so the statement $\sin \theta = y/r$ isn't as trivial as, say, defining $f(t) = \pi$, and by geometry it turns out not to be ill-defined like trying to define $g(x) = y$ would be in a context where $x$ and $y$ are independent (and so, e.g., it would require $g(0) = 0$ and $g(0) = 1$ as $y$ varies while keeping $x$ fixed)).
Jul
26
comment Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)
I think you only meant your final sentence in the case of $\lambda = 0$....
Jul
26
comment Why isn't every element of the spectrum an eigenvalue? (Where is the error in my proof?)
I once read a text that advised that intuition about and experience with finite dimensional linear algebra is misleading as often as it is helpful when considering infinite dimensional linear algebra.
Jul
26
comment Why the determinant of a matrix with the sum of each row's elements equal 0 is 0?
I think I would have hinted the opposite side: that multiplication by this vector encodes the information given.
Jul
25
comment explanation of notation in programming problem
But anyways, the entire reason I brought up this tangent is because it is misleading to the audience to flat out say "there is no such thing as the factorial of $-1$!".
Jul
25
comment explanation of notation in programming problem
@user21820: $\infty/\infty$ is not in the domain of division, so that formula is not defined at $\binom{-1}{-1}$ even if you wanted to use it there. Also, there is no logic in dismissing a generalization to all integer $r$ simply because it doesn't simultaneously generalize to all integer $n$: "don't let the perfect be the enemy of the good". Arbitrary integer $r$ even with nonnegative $n$ is is an important case, since the methods for doing more involved manipulations of summations are easier if you work with sums over all integers:e.g.$\sum_r\binom{n}{r}x^n$ not $\sum_{r=0}^n\binom{n}{r}x^n$
Jul
25
answered explanation of notation in programming problem
Jul
25
comment explanation of notation in programming problem
@user21820: It is also misleading to deny there is any such thing as $(-1)!$ :P. For a simple example of dividing by $(-1)!$, it lets us continue to use the usual formula to obtain $\binom{n}{-1} = \binom{n}{n+1} = 0$ when $n$ is a nonnegative integer.
Jul
25
comment explanation of notation in programming problem
@user21820: It is not unreasonable to define $(-n)! = \infty$ (projective infinity) when $n$ is a positive integer. This is consistent with the recursion ($0! = 0 \cdot (-1)!$ is indeterminate) and with $\Gamma(0) = \infty$. And most of the time you would even consider $(-1)!$ in a real problem either you want to know the result blows up, or you're dividing by $(-1)!$ and want to get $0$ as the result.
Jul
19
comment Confused about the Arrow Category
@Chilango: As I said, a pair $(h,k)$ is enough to identify a "morphism from $f$ to $g$"; it is not enough to identify a "morphism".