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2d
comment Set of rational numbers bounded between two irrationals is a closed set?
"Closed" in which ambient space X and for which topology on X?
2d
comment Convergence of a product of two infinite sums
If you placed correctly the absolute values in (1), this is indeed an identity.
2d
comment Stability of an equilibrium point in 6D based on eigenvalues
By definition, the index of a critical point is the dimension of the negative-definite submatrix of the hessian matrix calculated at that point. In your case, this dimension is obviously 3. (What you will achieve thanks to this information kind of escapes me, but...)
2d
comment Find the value of :$\lim_{\Delta t \rightarrow 0^+} \frac{\epsilon}{\sqrt{16\pi D (\Delta t)^3}}e^{-\epsilon^2/(4D(\Delta t))}$
For every positive $a$ and every $b$, $x^be^{a/x}\to+\infty$ when $x\to0^+$. For every $a\ne0$, $1-e^{-ax}\sim ax$ when $x\to0$ hence $(1-e^{-ax})^2/x\sim a^2x\to0$ when $x\to0$. Are these what you are asking?
2d
comment Limit of the succession $\lim_{n \to{+}\infty}{n\sin(n\pi)}$
Sequences converge (or not) to zero. Limits are (or are not) zero. But limits do not converge to anything.
2d
comment Stability of an equilibrium point in 6D based on eigenvalues
Clearly there are 3 unstable directions and 3 stable directions. What other kind of description are you expecting?
2d
comment Stability of an equilibrium point in 6D based on eigenvalues
Six eigenvalues in 3D? Sure about that?
2d
comment How to solve an initial value problem consisting of a matrix?
See en.wikipedia.org/wiki/Matrix_differential_equation sections 5 and 6. (Note: The Edit to your question does not make it eligible for reopening. You are still basically proposing a naked exercise, saying you have no idea to solve it. This is the definition of a "No personal input" question and the five personal comments you posted (are not the way to ask for reopening and) will not change this fact.)
2d
comment Logistic model - solution verification
Your solution yields $X(0)=X_0/(1+X_0)$ hence it cannot hold. The other suggestion is the correct solution.
2d
comment Periodic orbits of a dynamical system
"To rule out the existence of a limit cycle" would prove difficult since every solution passing by some $u>0$, $v>0$, is a cycle with implicit equation $$u^bv^a=ce^{u+v},$$ for some $0<c\leqslant b^aa^be^{-a-b}$. Keywords: Lotka-Volterra system.
2d
revised Formula for smallest multiple of given number, whose every digit is 1
edited title
2d
revised If $\sum a_n$ is convergent, is $\sum\frac{a_n}{n}$ absolutely convergent?
added 6 characters in body; edited title
2d
revised Limit of $\left(e^2 \frac{(1+x)^{1/x}}{(1+x²)^{1/x²}}\right)^{1/x}$ when $x\to0$
edited title
2d
comment On a Probability notation - $\mathbb{E}[X(.)|\mathcal{F}]_G$
No. And I would not recommend it, unless it is explained in detail.
2d
comment What is “Bourbaki's style in mathematics”?
@B.Pasternak Did you read the article?
2d
revised Evaluate the integral $\iint \operatorname{curl}(yi+2j)\cdot n \, d\sigma $
added 24 characters in body; edited title
2d
revised Proof of $\sum\limits_{k=1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90}$ using the Fourier series of $|x|$
edited tags; edited title
2d
comment Proof of $\sum\limits_{k=1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90}$ using the Fourier series of $|x|$
Fully expand the square $$\left( \frac {\pi}{2} - \frac {4}{\pi} \sum_{j=1}^{\infty} \frac {\cos((2j-1)x)}{(2j-1)^2} \right)^2$$ and use the identities $$\int_{-\pi}^\pi\cos^2((2j-1)x)dx=\pi$$ and, for $\ell\ne j$, $$\int_{-\pi}^\pi\cos((2j-1)x)\cos((2\ell-1)x)dx=0.$$
2d
comment Probability- random variables
You already posted this one hour ago. Why delete and repost?
2d
comment I think I invented an algorithm and now I need a website to check if it already exists.
You already posted this one hour ago. Why delete and repost?