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6h
comment What is the intent of this problem, disguised as an eigenvalue - eigenvector problem?
Actually it could be a single second eigenvalue with multiplicity $2$. That would be the case if $a=b=c$. In fact the discriminant of the characteristic polynomial turns out to be $36 (a^2 + b^2 + c^2 - ab - ac - bc)(ab + ac + bc)$. There are three distinct eigenvalues when this is nonzero.
8h
comment For a symmetric matrix $X$, is $A^T X^{-1} A$ symmetric for any $A$?
Of course you do have to assume that $X^{-1}$ exists.
8h
comment Digital Roots of Square Numbers
@mathlove Thanks for catching that.
10h
comment What is the definition of a Critical Point?
"Might" is too vague to use in definitions.
10h
comment What is the definition of a Critical Point?
@MichaelHoppe No, that's a really bad idea, leading to considerable confusion. There are critical points that are not minima or maxima.
1d
comment Analytical result for element-wise vector division?
Vector-matrix multiplication? But elementwise division is of no help for that.
1d
comment Conditional Expectation for IIDs
What is $E[X+Y \mid X + Y]$?
1d
comment Help with Euler Equations
The second derivative part is a bit tricky. You want to think of $\dfrac{d^2y}{dt^2} = \dfrac{d}{dt} \dfrac{dy}{dt}$. You already know how to write the "inner" $dy/dt$ in terms of $dy/dx$ and $t$. Use the product rule on this, and then again $\dfrac{d}{dt} = \dfrac{dx}{dt} \dfrac{d}{dx}$.
1d
comment Help with Euler Equations
It's not a nonlinearity: the Euler equation is as linear as they come. What it removes is the presence of non-constant coefficients.
1d
comment Interesting Combinatorics question relating the coefficients of variables in Pascal's Triangle
If $b \ge 1$, it suffices to have $a \ge 3 b$.
2d
comment System of equations to solve this nested radical.
The point is that there isn't anything special about this. You have a target number $1.74793$ and a function $f(A)$, and you found a value of $A$ such that $f(A) = 1.74793$.
2d
comment System of equations to solve this nested radical.
If $f$ is a continuous function on $[a,b]$ with $f(a) < c < f(b)$, the Intermediate Value Theorem says there is some $x \in (a,b)$ with $f(x) = c$. You seem to have found that $x$ in the case $f(A) = \sqrt{1/A} \sqrt{A + 2 \sqrt{A + 2\sqrt{A + \ldots}}}$ and $c = 1.75793$.
2d
comment Does this function achieve a maximum or minimum?
That's part of it.
2d
comment On Period of Linear Recurring Sequences modulo $P^e$
Don't confuse the integers mod $p^e$ with the finite field $\mathbb F_{p^e}$. They are quite different.
2d
comment Interesting Combinatorics question relating the coefficients of variables in Pascal's Triangle
Because if $b$ is small and $a$ is large, ${a \choose b} > {a+1 \choose b-1}$.
2d
comment Interesting Combinatorics question relating the coefficients of variables in Pascal's Triangle
What's the question? You can't prove something that isn't true.
Jul
25
comment Can we embed unital Banach algebras into semi-simple ones?
I meant nilpotent in the algebraic sense. So if $B$ contains a nonzero nilpotent element, you can't embed it in an abelian semi-simple Banach algebra.
Jul
24
comment How to solve the paradox in the negation to definition of limit?
Yes, the book made a mistake. They are not equivalent. For example, it might be that $f(u_n) - f(v_n) = 1$ for even $n$ and $0$ for odd $n$. Then with $\epsilon = 1$, $\forall N \in {\mathbb N}: \exists n \ge N: |f(u_n) - f(v_n)| \ge \epsilon$ is true, but the book's version with $\forall n \in \mathbb N$ is false.
Jul
24
comment Replacing pinv with inv in MATLAB
There's no way around the fact that your system has rank $< n$. But if you adopt the criterion that you want ${\bf A x}$ to be as close as possible to $\bf y$ and (subject to that) ${\bf x}$ should be as close as possible to $0$, then you do have a unique answer, which is exactly what the pseudoinverse gives you.
Jul
24
comment Replacing pinv with inv in MATLAB
Neither does $[{\bf A};{\bf b}]$ have an inverse. But both have a pseudoinverse, and pinv is happy to compute it.