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May
15
comment show that $C^a([0,1]) \subset C^b([0,1])$ for $0<b \le a \le 1$?
Yes, this is the correct argument. However the notation should be $C^a([0,1]) \subset C^b([0,1])$, not $C^a([0,1]) \in C^b([0,1])$
May
10
awarded  Taxonomist
May
7
comment $f(x)$ is a polynomial satisfying $2 + f(x)f(y)=f(x)+f(y)+f(xy)$, find $f(f(2)$), given $f(2)=5.$
How is $g(x)$ defined? And what is $g(2)$?
May
3
comment Convergence of $\sum\limits_{n=1}^{\infty}\left (1-n\sin \frac{1}{n}\right)^\alpha$ for parameter $\alpha$
Here is a hint: $\sin x \approx x - \frac{x^3}{6} + O(x^5)$ for small $x$. Now work out what this means for $1 - n \sin \frac{1}{n}$.
May
3
comment How to prove equality $K(x, K(x)) = K(x) + O(1) $?
Is $K$ a function of one or of two arguments?
Apr
27
comment Stability of equilibriums
Can you linearize the system about (1,1) or about (-1,-1)?
Apr
27
comment Congruence Property of Monotone Operators
More precisely, there are $x \ne y$ such that $Ax = Ay$, if $A$ does not have full rank.
Apr
27
answered Congruence Property of Monotone Operators
Apr
22
comment The convergence of probability for $X_nY_n$ and $X_n/Y_n$
The statement about $X/Y$ is not correct without additional assumptions on $X$ and $Y$. Take $X_n = 1$ a.s., $Y_n = \frac{1}{n}$ a.s. and thus $X = 1$ a.s., $Y = 0$ a.s.
Apr
20
comment $f_1,f_2 \in L^q(\mu)$ and $\int_\mathcal{X}f_1gd\mu = \int_\mathcal{X}f_2gd\mu$ for all $g \in L^p(\mu)$ implies $f_1=f_2$ a.e.
Sorry - this was supposed to be $g = sgn(f_1-f_2)|f_1-f_2|^{p-1}$. Check whether this $g$ is in $L^q$.
Apr
20
comment $f_1,f_2 \in L^q(\mu)$ and $\int_\mathcal{X}f_1gd\mu = \int_\mathcal{X}f_2gd\mu$ for all $g \in L^p(\mu)$ implies $f_1=f_2$ a.e.
Just use this $g$ and work out $ \int_X (f_1-f_2)g$. What is the result?
Apr
20
comment Find all the primes that satisfy $p \mid 2^p - 1$
Looks good to me too. Now write it down properly.
Apr
20
comment $f_1,f_2 \in L^q(\mu)$ and $\int_\mathcal{X}f_1gd\mu = \int_\mathcal{X}f_2gd\mu$ for all $g \in L^p(\mu)$ implies $f_1=f_2$ a.e.
Try $g = sign((f_1-f_2) \cdot |f_1-f_2|^{1/(p-1)}$.
Apr
19
awarded  Yearling
Apr
15
comment Is a holomorphic function with zero derivative on a close connected set constant?
@Justin - the statement is true if the set has more than one point.
Apr
14
comment Is a holomorphic function with zero derivative on a close connected set constant?
This is because the set could be a single point. Take $f(z) = z^2$, where $f' = 0$ on $\{0\}$.
Apr
14
answered Numerically solving a steady state equation (diffusion reaction with monod kinetics)
Apr
11
comment What is $\lim_{n\to\infty} \frac{n}{\ln(n)} (n^{1/n}-1)$?
Set $h = \ln n/n$ for a moment. Can you rewrite the entire expression in terms of $h$? What happens to $h$ as $n \to \infty$?
Apr
10
answered Finding what $p$ does the series: $\sum_{n=1}^{\infty}\frac{\ln n}{n^p}$ converges
Apr
8
answered A trouble with the existence of an $C_{0}^{\infty}$- function $v,v>0$ such that $\int_{\Omega}hv^{\alpha}>0$