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Software engineer and long-time dabbler in mathematics; 11th in the Putnams forever ago but I've long since atrophied.


8h
comment homomorphism $\phi$ : $\mathbb Z$ $\rightarrow$ $\mathbb Z$ using $\phi(n)$=$2n$
While it's somewhat implicit in the question, it would be better to write $(\mathbb{Z}, +)$ here to make it explicit that (a) this is a group homomorphism and not e.g. a mapping with respect to some larger structure such as a ring;and (b) that it's the usual group operation on $\mathbb{Z}$ and not something a bit more exotic.
2d
comment Inverse of $f(x)=x^n+x$ on $[0,\infty)$
@columbus8myhw No - if it did then the quintic $y^n+y+d=0$ would be solvable for all rational $d$, but it's not too hard to produce examples of this form which are unsolvable (see en.wikipedia.org/wiki/… for an example).
Aug
28
comment Can we rotate a 3D lattice of deformed spheres?
The edits over the last couple of days have served to make this question much less suitable for math.SE, as they've introduced wholly inappropriate amounts of wild speculation and theory into what is supposed to be a largely objective site Q&A site. I strongly encourage you to revert them.
Aug
27
comment Algebraic process to find numbers so that $xy=45$ and $x+y=18$
Regarding the first paragraph (and in particular the use of the rational root test), I have a hard time thinking of circumstances in which the use of 'number' with no qualifiers (as in the question) would imply rational but not already imply integer; this seems superfluous to me. (And of course, if you don't know that $x$ and $y$ are rational you can't apply the RRT in the first place.)
Aug
26
answered In which Fields, does $x^n-x$ have a multiple zero?
Aug
25
comment A tight lower bound for the entropy of the XOR of two random variables
Doesn't this fall out of the information-theoretic characterization instantly? Since $U=X\oplus(X\oplus U)$, if $H(X\oplus U)$ weren't that large then $U$ wouldn't have full entropy. The same should be true of any function $f(X,U)$ for which there's a function $g()$ such that $U=g(X, f(X,U))$.
Aug
25
comment How to mathematically determine if the magnitude of a cross product is up/down(positive/negative?)?
@user3618509 as suggested in Phonon's answer, torque is a vector quantity; it can't be positive or negative per se. You can speak about the sign of its components (or, for instance, you can ask 'will this work 'for' or 'against' the body's current angular momentum, i.e. is $\tau\cdot L$ greater than or less than zero), but you can't say whether torque itself is positive or negative.
Aug
23
comment Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?
@Did True - actually, there are several other ways that this could go wrong. I retract my previous comment entirely.
Aug
23
comment Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?
Since $\frac{d\tan x}{dx}=\sec^2x\geq 1$ for all $x$, no fixed point can be attractive (and it seems immensely unlikely that we could hit one 'naturally'). This suggests that the only other possibility is an attracting cycle, and I suspect that might be ruled out with tighter analysis.
Aug
22
revised An estimate involving gaps in a subsemigroup of $(\mathbb N,+)$
Fixed the argument in the second paragraph to apply to both cases.
Aug
22
answered An estimate involving gaps in a subsemigroup of $(\mathbb N,+)$
Aug
22
comment On the phrase “identify the graph…”
Is a definition of $Q$ provided in the text? That would go a long way. My suspicion is that by the graph of $\psi$ they mean the set $\{t\times \psi(t)\ : t\in\mathbb{P}^1\}$ $=\{\langle x,y\rangle\times\langle\psi(x),\psi(y)\rangle :\langle x,y\rangle\in\mathbb{P}^1\}$. This is the usual definition of a graph of a function $f():D\mapsto D$ as a subset of $D^2$, though you'll then need to carefully map that down to the projective variety.
Aug
22
comment How to find ${\large\int}_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$
@Shahar They ebb and flow - there was a swath of them about a year ago too, IIRC.
Aug
21
comment Matrix with entries from $1$ to $16$, each occuring once, and determinant $40800$
@String You can compute the determinant a fair bit more quickly than that - not blazingly fast, but faster enough, esp. since all the entries are integers. You also have enough freedom with row and column swaps to put e.g. 1 in the top-left corner, leaving only 15! possibilities. And 15! is within striking distance; not small, but not unfathomably huge. I would bet that with some smart branch-and-bound techniques one could find the largest possible $5\times5$ determinant this way.
Aug
21
comment Convergence of the series $\sum_{n=1}^{\infty}\frac{1+n!}{(1+n)!}$
@labbhattacharjee Even that isn't needed; since all terms are positive we have the nonconvergent $\sum_n\frac1{n+1}$ as a lower bound.
Aug
21
answered If $x^2 a x=a^{-1}$, then $a$ has a cube root.
Aug
21
comment If $x^2 a x=a^{-1}$, then $a$ has a cube root.
Just follow the hint: compute $(xax)^3$ using what you know about $x^2ax$. You should find that it simplifies substantially...
Aug
21
comment How prove $\frac{\sqrt{2}}{3}n^2<\sum_{k=1}^{n^2-1}\sqrt{1-\frac{\sqrt{k}}{n}}<\sqrt{2}n^2$
The form isn't perfect for it, but you're right - this looks suspiciously like a Riemann sum for an integral and I would expect that you can massage things to a point where you can treat it as such.
Aug
20
comment A sum of difference of floors
@pam you're missing something, I think - the two quantities you're taking the floor of are only $\frac1h\leq 1$ apart, whatever the value of $h$, so the difference between their floors will never be more than 1.
Aug
20
answered Automorphisms of $\langle \mathbb{N}, \cdot \rangle$