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comment Prove that any projection on a normed linear on a subspace satisfies $\|I-P\|\geq 1$
$0$ is not invertible, unless it's a $0$-dimensional space.
comment Polygons in Polygon
I presume you're assuming the original polygon is convex.
answered Royden Real Analysis, Chapter $3$ Proposition $9$
revised Numerical methods for nonlinear wave equation
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comment Numerical methods for nonlinear wave equation
AFAIK, the nonlinearity and initial/boundary conditions are arbitrary (of course you might run into numerical issues if they are too wild).
comment Matrix diagonalization takes infinitely many operations?
Presumably they're talking about Householder transformations: constructing this matrix involves taking square roots but not solving any polynomials of higher degree than $2$. With a finite number of such operations, it's impossible in general to solve a general polynomial of degree $3$ or higher, and thus to diagonalize a matrix whose characteristic polynomial is such.
answered Specific question about investment returns that needs someone smart to answer it!
comment If $A \in \mathrm{GL}_n(R)$, $A^l=1$, what do we know about $\mathrm{char}(A)$, the characteristic polynomial of $A$?
When you say $K$-algebra, exactly what are you assuming? Unital, I guess; associative? commutative?
answered Numerical methods for nonlinear wave equation
comment Use standard error of mean or population distribution?
What does this have to do with the sample mean? The condition is that exactly two have marks over $70$. That says nothing about the sample mean. If the question was about the sum of those students' marks, you might look at the distribution of the sample mean.
comment When does a matrix game and the sign flipped matrix game have the same nash equilibria?
For zero-sum games, how does changing the sign "reverse the players"? It can change the nature of the optimal strategies completely. For example $$ \pmatrix{1 & -1 & -2\cr -1 & 1 & -2\cr 2 & 2 & 0\cr}$$ has a saddle point, but for $$ \pmatrix{-1 & 1 & 2\cr 1 & -1 & 2\cr -2 & -2 & 0\cr}$$ the optimal strategies are mixtures $(1/2, 1/2, 0)$.
answered Is it generally true that $P(X_1,\ldots,X_n \mid \theta) = P(X_1 \mid \theta)\cdots P(X_n \mid \theta)$ if $X_i$'s are independent?
answered Can you solve $y'+x+e^y=0$ by series expansion?
answered Bounded function - Proving $f(x)=0$ for all $x$
revised Recovering an analytic function
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answered Recovering an analytic function
comment Given the roots of polynomial over finite field, what is the count of its distinct nonzero coefficients?
As long as the cardinality of the field is much larger than $n$, I see no reason that the coefficients can't all be distinct and nonzero.
comment Determine the smallest disc in which all the eigen values of a given matrix lie
Gershgorin only gives a bound. It's not necessarily the best bound. The question is poorly written. As stated, the correct answer is "None of the above": the smallest is approximately $|z - .495567| < 3.833136671$.
comment Given the roots of polynomial over finite field, what is the count of its distinct nonzero coefficients?
Which finite field is it? In general I doubt that there's very much you can say: you just have to look at which elementary symmetric polynomials of the roots happen to be $0$.
comment Substitution $t=x^{2}$ in indefinite integrals
Of course it's explicit. But you have to be careful of whether $x > 0$ or $x < 0$. If you're substituting $t = x^2$ in an integral where $x > 0$, then $x = \sqrt{t}$ and $dx = dt/(2 \sqrt{t})$. If $x < 0$, then $x = -\sqrt{t}$ and $dx = - dt/(2 \sqrt{t})$.