Hey guys! If function $f$ is even and function $g$ is odd, then what would be $(f\circ g)(x)$? I have deduced that $(f\circ g)(x)$ would be even. Is this conclusion correct?
Let $g,g',\left(g'\right)^2-gg''$ are positive valued functions that exists on the domain $[1,\infty).$ Prove that $\sum _{n=1}^{\infty }\frac{g'\left(n\right)}{g\left(n\right)}$ converges iff $lim_{
n→∞}g(x)<∞.$
My attempt
We know from the Hypothesis that $g(x)>0, g'(x)>0, (g')^2>gg'', \forall ...
Then an equation is considered yucky if $Y < 10$ and each of these paths are not simply connected
An equation is considered very yucky if $1< Y < 5$ and is irreducible if $Y = 0$
The mathematical quagmire, is defined to be the set of all yucky equations
What remains to be explored, is how yucky each region of the quagmire is relative to each other (relative yuckiness)
and whether, given a quagmire restricted by some constraints (e.g. all equations involving exponentials and polynomials) whether there exists a unique path between the most yucky equation to the least yucky equation that composed of the same type of letters
@Sonal_sqrt if you can write anything in terms of prime factors in the ring or you have zero divisors, you cannot possibly find a euclidean division algorithm in it
the second one looks a lot more "yucky" (subjectively) than the original, yet the existence of that identity means the expression on the right cannot have a very small $Y$ value, hmmm
Yeah, forms are very often equivalent. I've actually graded differential equations quizzes where two people got seemingly wildly different answers to a problem, but closer inspection revealed that they were actually one and the same.
Ah, I see what you mean. So any function induces a type of algebraic object. You could have the only operation be the $f$. Then, provided the $f$ is binary, you have a magma
What has to be true about $f$ for the corresponding magma to be associative though? Like, $f(a,b)=a\cdot b+1$ isn't, provided we are pulling $a,b,\cdot,+$ from $\mathbb{Z}$
ah, I don't think there is a simplication of this for general $f$, but it definitely suggests the two arguments are symmetric somehow. Need to think
$f(a,f(b,c)) = f(f(b,c),a)$ is for commutativity, so that suggests $f(b,c) = a$ at some abstract line of symmetry thus there will be fixed points there. I forgot the geometric counterpart to associativity, need to check
apologies if this is trivial: if $z, \lambda_1, \lambda_2 > 0$, how many solutions for $z$ does $\lambda_1^2 z^{\lambda_2 z}+\lambda_2^2 z^{\lambda_1 z} = 0$ have?
@MJD Since you're the user who created (or at least started) the MathJax tutorial on meta, maybe you' would have some comments on this recent suggestion on meta: Organizing The MathJax Tutorial. (Basically, the post suggest to add some kind of ToC to the post.)
If needed, we can discuss this further in the Math Meta Chat. (This seems to be close on-topic in that room - I used this room simply because it was a place where I was able to ping you.)
I did not interact with the MathJax tutorial on this site too much, so I don't really have strong opinion on this.
Since this was mentioned in Math Meta Chat, perhaps you can ask for additional feedback there. Maybe some of the users who visit that room responds. (Although I am not sure that much feedback is needed here.)
The only thing that came to my mind was whether the OP is actually allowed to edit the post - considering that they have relatively low reputation. However, for CW-post this should not be on problem. (They would not be able to edit regular posts on meta.)
Hey guys! Will the function $f_{e}(x) = x^2$ statisfy the following requirement? $$$$ Assume that $f: [0, ∞) → ℝ$. $$$$ Define a function $f_{e}: ℝ → ℝ$ which is even and $f_{e}(x)$ = $f(x)$ for all $x≥0$.
Well, there is a bit to unpack, there. First, $x^2\neq f(x)^2$, and your description leads me to believe that you mean to say $f_e=f^2$ that being said, $f^2\neq f$ and so does not satisfy your requirement. Also, $(x\text{sin}(x))^2=x^2\text{sin}^2(x)\neq x^2\text{sin}(x^2)$
I have another question about increasing functions if you guys don't mind. Is the following conclusion that I made about this statement is true?$$$$ This is the statement:
If functions $f$ and $g$ are given to be increasing, then $fg$ would also be increasing on the interval $[a,b]$? This what I did to prove this statement is indeed true. $$$$ Let $x_1$ and $x_2$ be elements of the interval $[a,b]$. Using the defintion of increasing functions, I deduced that $f(x_1)$ < $f(x_2)$ and $g(x_1)$ < $g(x_2)$. So this can be rewritten as $(fg)(x_1)$ = $f(x_1) * g(x_1)$ < $f(x_2) * g(x_2)$ = $(f*g)(x_2)$. Would this be a valid approach?
@Random-15 No, you're computing $|z|$ incorrectly. That is, in fact, the unit circle centered at $0$. Remember that if $z=x+yj$, then $|z|^2 = x^2+y^2 = z\bar z$.
When changing variables in a double integral, is there any strategy for finding suitable transformations so that the non-rectangular domain of integration gets mapped to a rectangular domain? Say one is given the domain bounded by $x=y^2,x=3y^2,y=x^2,y=2x^2$.