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22682
bio website mathproblems123.wordpress.com
location Chambery, France
age 26
visits member for 3 years, 7 months
seen 3 hours ago

PHD - interests: PDE, Free boundaries, Shape Optimization


Sep
15
comment basic question involving topology and the Hausdorff distance
If you can read a bit in French, I recommend springer.com/mathematics/book/978-3-540-26211-4 In chapter 2 there is a detailed discussion of different types of convergence of sets. There is an exercice which resembles your question. If you prove that $\Omega_n$ converges to $\Omega$ in the Hausdorff distance, then the convexity should imply the convergence of the boundaries.
Aug
15
comment Families of Square Roots of Identity Matrices
You can tackle the general case using reduction (diagonalization).
Aug
9
comment Distance between triangle's centroid and incenter, given coordinates of vertices
The sides of the triangle can be immediately calculated. After that, the coordinates of $G$ are just the averages of the coordinates of the vertices. For the incenter the formula is simple: you can find it here mathworld.wolfram.com/Incenter.html In my opinion there's no simpler way.
Jul
17
comment How prove $A^2=0$,if $AB-BA=A$
@Kelenner: I'm sorry. I didn't make all the computations.
Jul
17
comment How to find the determinant of this matrix
@user143993: This last equivalence you wrote is the definition of dependent vectors.
Jul
17
comment How to find the determinant of this matrix
@user143993: The statement you wrote is not right. The determinant is zero if and only if the columns (or the lines) are related. You can prove it like this: if the determinant is not zero, then any relation between the columns implies that the coefficients are zero (because the system has the unique solution of zeros). Therefore the columns are not related. Conversely, if the determinant is zero, then the system doesn't have a unique solution, i.e. has a non zero solution, i.e. you can find a relation between the column vectors.
Jul
17
comment How to find the determinant of this matrix
No, there is no $f$ here. The equation still defines a conic.
Jul
17
comment How prove $A^2=0$,if $AB-BA=A$
@Kelenner: I think it is $2^kA^k=A^kB-BA^k$. To prove that $A$ is nilpotent it is enough to prove that $tr(A^k)=0$ for each $k$, which you did. After that you can deduce that all eigenvalues are zero using a Vandermonde system.
Jul
17
comment Disconnecting using totally disconnected sets
This was already asked here: math.stackexchange.com/questions/31667/… and here: mathoverflow.net/questions/55718/…
Jul
13
comment Quaternion Group as Permutation Group
@angryavian: The phrase: "I couldn't figure out how to do it" says the opposite. The wolfram citation may be wrong, since it contradicts Cayley's theorem.
Jul
13
comment Quaternion Group as Permutation Group
See what the application $f(x)=ax$ does with the elements of the group. That is the permutation corresponding to $a$. Do this for every $a$ to find the permutation representartion.
Jul
9
comment Clarification: Suppose that $(v_1, \ldots ,v_n)$ is a basis of $V$ and $(w_1, \ldots ,w_n)$ is the basis of $W$ …
Yes, I get what you mean, but if you got this result from a book, you should search for the DEFINITION of M. $M$ is not arbitrary. The idea is that to every two bases you associate a matrix, and it is natural to think of the matrix associated to a linear application $T$, especially if in the proof you can see $M(T)$. Look for the definition of that matrix, which must be in there somewhere. $M(T)$ cannot be just an arbitrary matrix, because then the application you consider does not need to be linear and invertible.
Jul
9
comment Clarification: Suppose that $(v_1, \ldots ,v_n)$ is a basis of $V$ and $(w_1, \ldots ,w_n)$ is the basis of $W$ …
You could look at the definitions before this proposition and figure out what $M$ is...
Jul
8
comment Evaluate $ \int_{0}^{\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan x)}\,\mathrm dx$
What is the source of this problem?
Jul
8
comment Square root of a $3\times3$ matrix
If the matrix is not positive definite, it does not have a square root over $\Bbb{R}$...
Jul
8
comment Analyzing a particular type of functions
How do you define continuity or differentiability for functions defined on $\Bbb{Z}$? You need to be able to move in a neighborhood of a point $x_0$ to speak of differentiability.
Jul
7
comment Geometrical proof of the existence of square roots
Yes, thank you for the correction.
Jun
9
comment Taylor series of $\sin(x)$ converges uniformly on $[-\pi,\pi]$?
Take the absolute value of the remainder and note that the $N$-th derivative of $\sin$ is bounded.
May
13
comment Proving that a polynomial $|P(x)|=e^x$ has a solution
@GinKin: You can prove that the exponential grows faster. It is true that in this case we have an indetermination, but $P(x)/e^x$ goes to $0$, so we can write $e^x-P(x)=e^x(1-P(x)/e^x)$ which is not indeterminate anymore.
Apr
20
comment Question about Hahn-Banach theorem
@luozhenghua: You can edit your answer to correct the mistakes instead of writing them into comments.