# All Questions

3 views

10 views

### Is $H^2(\Omega)$ compactly embedded on $H_0^1(\Omega)$?

Considering $\Omega$ bounded and $\partial \Omega$ smooth. I already know that $H^2(\Omega)$ is continuously embedded on $H_0^1(\Omega)$, thus if I take a bounded sequence in $H^2(\Omega)$ it is also ...
23 views

### Big-O vs. Best Big-O

Is there a difference between the method to find a big-O function and the method to find the best big-O function. Take for example the following function: $f(n) = 1 + 2 + 3 + ... + n$ It is easy to ...
13 views

13 views

### Fibre of a local homeomorphism can be covered by disjoint open sets.

Let $f\colon X\rightarrow Y$ be an open local homeomorphism and $y\in Y$. Do there exist pairwise disjoint open neighborhoods $U_x$ for $x\in f^{-1}(y)$? If not, what would be mild topological ...
10 views

### Integral curves on immersed submanifold

An exercise of the book "Introduction to smooth manifolds - John M. Lee" asks to prove that if $S$ is a closed embedded submanifold of a manifold $M$, and $X$ is a vector field on $M$ tangent to $S$, ...
23 views

### An open interval as a union of closed intervals.

For $a<b, a,b\in\Bbb R$ $$(a,b)=\bigcup_{0<\delta<(b-a)/2} I_{\delta} \quad I_{\delta}:=[a+\delta,b-\delta]$$ Clearly the RHS is an (uncountable) infinite sum of closed intervals. I ...
4 views

### Generalized eigenvalue problem equivalence

I would like to show that there exists a Laplacian matrix $H$ for every Laplacian matrix $L$ such that $$Lx = \lambda D x \equiv Lx = \lambda Hx,$$ where $$D=\sum_{i=1}^n d_ie_ie_i^\top,$$$e_i$ is the ...
15 views

### Relationship between vectors in convex quadrilateral

We have the triangle $ABC$. $M$ is the center of the line segment $BC$. $D$ is a point in the triangle's plane so that $ABDC$ is a convex quadrilateral. $N$ is center of the line segment $AD$. ...
13 views

### probability class 12

Three groups of children contain 3 girl and 1 boy;2 girls and 2 boys;and 1 girl and 3 boys. One child is selected at random from each group.find the chance that the three children selected comprise 1 ...
13 views

### Existence of differentiable functions on $\mathbb R$ whose derivative is constant on the complement of uncountable set but not everywhere

Let $A$ be a countable subset of the set of real numbers and $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f'$ is constant on $\mathbb R \setminus A$ , then I know that $f'$ is ...
5 views

### Usage of Phase Portrait of a system of 2 linear first order ODEs

Let's say have a linear system $\frac{\mathrm{d}\underline{y}}{\mathrm{d}t} = A\cdot \underline{y}$, let say 2 dimensional, and I have $\lambda_1,\lambda_2$ eigenvalues of $A$ and ...
7 views

### How to drive the membership function in fuzzy clustering nean

I am learning about Fuzzy c-means (FCM) which is a method of clustering which allows one piece of data to belong to two or more clusters. This method (developed by Dunn in 1973 ) is frequently used in ...
50 views

### Integral of cos(1/x) dx

Is the following integral expression correct (neglecting the constant of integration)? $$\int\cos\left(\frac{1}{x}\right)dx = x^2\sin\left(2x\right)$$ When I take the derivative, it returns to the ...
35 views

20 views

### Homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$

Let $\phi_1$ and $\phi_2$ be two injective ring homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$. Show that there exists a $g\in GL_2(\mathbb{R})$ such that $\phi_2(x) = g\phi_1(x)g^{-1}$ for all ...