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Feb
14
comment Connected graph with minimum distance
Can you do any part of this? E.g., can you find a tree with $7$ vertices, and decide which type it is?
Feb
14
comment $V_1, V_2$ subspaces of $V$. Show that there is a basis $\mathcal B$ for V such that $\mathcal B \cap (V_1 \cup V_2)= \emptyset$
Yes. ${}{}{}{}$
Feb
14
comment Lifting isomorphisms of fields to automorphisms of polynomial rings
Suppose $L$ is the rationals, $\alpha=2$, $\beta=3$, so $q(t)=t-2$, $p(t)=t-3$. Is there an automorphism $\psi$ of ${\bf Q}[x]$ such that $\psi(t-2)=t-3$?
Feb
14
comment Do we have a general form for this integral?
Why the dirichlet-series tag?
Feb
14
answered Finding a congruent matrix
Feb
14
comment For every irrational $\alpha$, the set $\{a+b\alpha: a,b\in \mathbb{Z}\}$ is dense in $\mathbb R$
@Matt, the link is actually to a proof of a much stronger result, namely, to the uniform distribution (modulo $1$) of the sequence of multiples of an irrational.
Feb
14
awarded  complex-analysis
Feb
14
comment Finding a branch of complex logarithm $\log(z)$ with parabola branch cut
I take it you are familiar with the principal branch of the logarithm, with branch cut the nonpositive reals. Think how my description works for that case. Start at $(r,0)$, go counterclockwise until you reach $z$. If you reach $z$ before crossing the branch cut, then $z$ is in the upper half plane, and you can take $\log z=\log r+i\theta$. If you cross the nonpositive reals before you reach $z$, then $z$ is in the lower half-plane, and you have to subtract $2\pi$ from the argument. It's the same for the parabola, or any other branch cut (unless the cut winds around the origin).
Feb
14
revised What is mathematical research like?
debunked
Feb
14
comment Finding a congruent matrix
I know what it means to find a matrix that is similar to another; I don't know what it means to find a matrix that is congruent to another. Also, the eigenvalues are $\pm1$, which are there no matter what field you are working in (although things may be a little tricky in fields of characteristic $2$).
Feb
14
comment Finding a branch of complex logarithm $\log(z)$ with parabola branch cut
DJ, you accepted this answer earlier, then unaccepted it. That is, of course, your prerogative, and you don't owe anyone an explanation --- but if there is something particular that you find unsatisfactory in the answer, maybe I can improve it, if you let me know.
Feb
14
comment Finding a congruent matrix
You diagonalize a matrix by finding eigenvalues and eigenvectors, not by using elementary row operations.
Feb
14
comment An Exercise of Finite Groups
It's not hard to find out what cyclic group it is by typing "Fibonacci group" into the web, but I'm not finding any sources that give a proof of the answer.
Feb
14
comment An Exercise of Finite Groups
@Maisam, the notation means the group is generated by those $5$ elements, not that those are its only elements.
Feb
14
comment I am having trouble using the addition formula to derive the identity from this problems
Once you have corrected the statement of the problem, can you write out what happens when you use the addition formula for the cosine function?
Feb
14
comment Possibility of Unboundedness in Least Squares Minimization
Probably I'm misunderstanding the question, but the $2$-norm is always non-negative, as is its square, so the term you have written down is bounded below by zero.
Feb
14
comment loss-of-significance error
But that's what bob is saying. In the original form, you are subtracting two large quantities, and that's where you lose significant figures.
Feb
14
comment Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$
Found it, 2003 A3. Dan Bernstein's solution is at cr.yp.to/papers/putnam2003.ps and two or three solutions are at amc.maa.org/a-activities/a7-problems/putnam/-pdf/2003s.pdf
Feb
14
comment Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$
@Zilliput, maybe forumgeom.fau.edu/POLYA/ProblemCenter/POLYA035.html and forumgeom.fau.edu/POLYA/ProblemCenter/TSPOLYA035A3.pdf work better.
Feb
13
answered Condition for mapping linearly independent vectors to linearly independent vectors