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3h
comment Derivative of product of Vector Transpose and Vector
For Q2, I don't know if there's any general principle at work: just write $C(x) = (C_1(x),...,C_n(x))$ and work out by hand that the last two expressions are equal to each other. Although I agree that the transpose should still be there.
15h
comment Finding all real numbers x such that $x \lceil x \lceil x \lceil x \rceil \rceil \rceil = 88$
Your idea doesn't make sense to me: you can't have both $x=y$ and $x<y$.
15h
comment equally spaced on circle question
One common strategy in such problems (which might or might not work here, but it's worth a try) is to show that if $\{x_1,\dots,x_N\}$ are not equally spaced, then there's some related sequence $\{y_1,\dots,y_N\}$ such that $f_{s,N}(y_1,\dots,y_N) > f_{s,N}(x_1,\dots,x_N)$. For example, if the two gaps in $x_1,x_2,x_3$ aren't the same size, then set $y_j=x_j$ for all $j\ne2$ and $y_2=(x_1+x_3)/2$.
1d
comment Inner product inequalities with a diagonal matrix defining the inner product
I explored the specific case where $x=(-1,2,2)$, $y=(2,-1,2)$, and $z=(2,2,-1)$, and labeled the diagonal entries of $A$ as $1,b,c$ (scaling so that $a=1$ is wlog). The set of $(b,c)$ values satisfying your four inequalities is a football shape with endpoints at $(0,0)$ and $(1,1)$. In this case at least, both $x^TAy<x^TAz$ and $x^TA^{-1}y > z^TA^{-1}z$ turned out to be the exact same region - the half of the football under the diagonal line from $(0,0)$ to $(1,1)$.
2d
revised The sum of two continuous periodic functions is periodic if and only if the ratio of their periods is rational?
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2d
comment Probability of getting the same vector result
Half-baked idea for an upper bound: the places where the coordinates of $f(v)$ change from $0$ to $1$, or vice versa, seem to constrain $v$ reasonably strongly: the two sums on either side of the split are close to equal. Perhaps one can sort vectors according to the places where these bit changes occur?
2d
comment Probability of getting the same vector result
Observation: If $v$ and $w$ have all $1$s in their first halves, and the same number of $1$s in their second halves, then $f(v)=f(w)$. This gives an improved lower bound of around $2^{-n/2}$ for the probability in question - maybe something like $2^{-n/2}/n$.
Jul
26
answered Pairs of integers with gcd equal to a given number
Jul
26
comment A lower bound for an arithmetic function
Under the further assumption that $\phi(N) \sim N$, then your lower bound is already $\sim N\tau(N)$, since $C_A(N) \ge \phi(N) - (N-\#A) \sim N$. And the same upper bound $\sim N\tau(N)$ follows from $C_A(d) \le C_A(1) = \#A \sim N$.
Jul
26
comment A lower bound for an arithmetic function
Note that for highly composite $N$ it is possible for $C_A(N)=0$.
Jul
26
comment (Putnam) Let $f:[1,3] \rightarrow \mathbb{R}$ such that $-1 \leq f(x) \leq 1 $ for all x and
What happens if you consider $f(x)=1$ for $1\le x\le2$, and $f(x)=-1$ for $2\le x\le 3$?
Jul
24
answered Lim sup/inf of average value
Jul
23
comment Smoothing a function
If you're asking what the relationship between the two functions would be, then you should describe a precise construction of the second function from the first.
Jul
23
comment Smoothing a function
One method of changing a function into an infinitely differentiable function is to replace it by the function that's identically zero. I doubt that's what you have in mind; what relationship do you desire between the original function and its replacement?
Jul
23
revised Smoothing a function
added 12 characters in body
Jul
22
comment Intuition behind generating continuous random valiables
"Probability 0" is not the same thing as "never happens". Indeed, your observation (sampling a real number at random uniformly from an interval) is the prototypical example illustrating the difference.
Jul
22
answered If $f(x)=o(\log^{(k)}(x))$ for all $k$, can $f$ diverges?
Jul
22
comment If $f(x)=o(\log^{(k)}(x))$ for all $k$, can $f$ diverges?
I find $\log^k x$ a highly misleading notation; to me it is standard for $(\log x)^k$. I think $\log_k x$ is preferable (even though a different notation clashes with that), or at least $\log^{(k)}x$ (although that can mean derivatives as well).
Jul
21
answered Probability of being the millionth customer (What Would You Do?)
Jul
21
answered Help with little-oh given $f(n) = n^\epsilon$ and $g(n) = (\lg n)^4$