Thomas Andrews
Reputation
83,941
96/100 score
 5h comment Is a polynomial always reducible? More generally $p(q(x))-p(r(x))$ is always divisible by $q(x)-r(x)$. (Roots is a good heuristic, but since $p$ might have repeated roots, it is possible that $p(p(x)+x)$ has fewer instances of the root.) 12h comment Finite Dimensional Vector Spaces and Completeness First of all, you'd want to give a metric on the vector space - there isn't a single metric that can be found just knowing that a space is an abstract vector space over the reals. Secondly, you'll need your field to be a complete field, first. 12h comment Quadratic Sieve Algorithm: Why is $(x − \lfloor \sqrt{n} \rfloor)^2 ≡ n ($mod $p)$? Since the author only appears to use that formula to show thatn $n$ is a quadratic residue, modulo $p$, it would appear that the minus sign in the paper was a typo. 15h comment Limit as n to infinity of sum to n — changing upper bound to infinity? @a.s. But it is never a function of $n$ hen written as $\sum_n x_i$. It is some sequence in some space, whether real numbers or vectors or functions. 16h comment Limit as n to infinity of sum to n — changing upper bound to infinity? When we write $x_i$, we always mean, unless otherwise stated, that $x_i$ only depends on $i$. That is the notion of "series." There is no other meaning. That problem isn't of the form $\lim_{n\to\infty}\sum_{i=1}^n x_i$ by its very nature, since there isn't a single sequence $x_1,x_2,\dots$. 16h comment Limit as n to infinity of sum to n — changing upper bound to infinity? What is $n$ when you define $x_i=\frac{1}{n}(n-1)$? AGain, you aren't making sense. Do you mean $x_n=\frac{1}{n}(n-1)$? 16h comment What is the Laplace transform of $d/dt(f * g)$ in terms of $F(s)$ and $G(s)$? You are misapplying FTC if you get $d/dt(f*g)=f(0)g(t)$. 16h comment Limit as n to infinity of sum to n — changing upper bound to infinity? Wait, that edit is very confused. Can you give an example of $x_n$ if it doesn't depends on $n$? Are they random variables? Usually random variables are written as capital letters: $X_i$. 16h comment Limit as n to infinity of sum to n — changing upper bound to infinity? That's the definiton of $\sum_{i=1}^\infty x_i$, so yes, it is true in general. 17h comment What is the Laplace transform of $d/dt(f * g)$ in terms of $F(s)$ and $G(s)$? The Laplace Transform of the derivative $h'$ is $sH-h(0)$, according to Wikipedia. So that should be $sF(s)G(s)-(f*g)(0)$. 19h comment Why is $\sin(x^{2})$ similar to $\sin(x) \cdot x$? For example, we can say $\sin(x^2)=x\sin(x) + O(x^4)=x^2+O(x^4)$ 19h comment Why is $\sin(x^{2})$ similar to $\sin(x) \cdot x$? Topology doesn't have a notion of distance or order, so it won't answer this. But what you can show is that for some $\delta_1>0$ and $C_1>0$ we have that \$|\sin(x^2)-(x^2-x^6/6)|