Intereting Posts

Compute $\phi_X(t)=E(e^{it^\top X})$ if $X\stackrel{d}{=}\mu + \xi AU$ with $AA^\top=\Sigma$
Surjective function into Hartogs number of a set
Solving the inequality $(x^2+3)/x\le 4$
How to evaluate $\lim\limits_{n\to +\infty}\frac{1}{n^2}\sum\limits_{i=1}^n \log \binom{n}{i} $
Find all the numbers $n$ such that $\frac{12n-6}{10n-3}$ can't be reduced.
Diagonalization of a projection
Centralizer of a specific permutation
Analytic continuation of the bounded holomorphic function
Calculate the sum of infinite series with general term $\dfrac{n^2}{2^n}$.
Integral is area under the graph
Prove that matrix is symmetric and positive definite given the fact that $A+iB$ is.
Which of these numbers is the biggest
$a_0 = 5/2$. $a_k = (a_{k-1})^{2} – 2$ for $k\geq1. \prod_{k=0}^{\infty}{\left(1-1/a_k\right)}.$
If $\langle a,b \rangle=0$, then $a,b$ are perpendicular?
$G \cong G \times H$ does not imply $H$ is trivial.

The problem is to show that the cone $ z^2= x^2+y^2$ is not an immersed smooth manifold in $\mathbb{R}^3$.

- Proving a subset is not a submanifold
- show that $\omega$ is exact if and only if the integral of $\omega$ over every p-cycle is 0
- Kähler form convention
- Show that the arc length of a curve is invariant under rigid transformation.
- Index notation.
- What exactly is a Kähler Manifold?
- if the curvature is constant and positive, then it is on the circunference
- horizontal vector in tangent bundle
- Why does $\frac{dq}{dt}$ not depend on $q$? Why does the calculus of variations work?
- Why is the image of a smooth embedding f: N \rightarrow M an embedded submanifold?

Consider the vector space generated by the velocity vectors at $0$ of all curves passing through the origin: it is of dimension $3$. If the cone were an inmersed submanifold, its dimension should therefore be $3$ and then it would have to have a non-empty interior.

(This is the idea of a proof: you should actually prove every statement I made!)

Recall that a function $f:\mathbb R^2 \to \mathbb R^3$ is called an *immersion* if $f$ is differentiable and the derivative $Df$ is injective at every point of $\mathbb R^2$. There are then a few problems here, the first being that $z$ is not a function of $x$ and $y$ (it has two values, $-\sqrt{(x^2 + y^2)}$ and $\sqrt{(x^2 + y^2)}$). Setting this aside, you may take the “upper half” of the cone $$z = \sqrt{(x^2 + y^2)}$$

so that now $z$ is a function of $x$ and $y$, however this is still not an immersion since $Dz$ is not defined at the point $(0,0)$. One way to see this is by writing down the definition (using a limit) of $Dz$ and showing that the limit doesn’t exist. However it is easier to see this geometrically. You can make a paper model: cut out a disk with some paper, then cut out a wedge (“pizza slice”) from the disk. Then take the disk with the sliced removed and glue along the edges where you cut out the wedge. You have constructed the image of the function $z$, and you can see that it has a singular nonsmooth point, being the vertex of the cone.

- Why are the solutions of polynomial equations so unconstrained over the quaternions?
- Is this an outer measure, if so can someone explain the motivation
- To confirm the Novikov's condition
- Must the (continuous) image of a null set be null?
- Does such localization of integral extension preserve inclusion?
- Passing from induction to $\infty$
- System of equations, limit points
- Every matrix can be written as a sum of unitary matrices?
- How to show $\mathcal{L}(\mathbb{R}) \otimes \mathcal{L}(\mathbb{R}) \subset \mathcal{L}(\mathbb{R^2})$?
- Using the Reflection Theorem
- Testing whether an element of a tensor product of modules is zero
- Do all polynomials of even degree start by decreasing as you plot from $-\infty$ upward?
- Class group and factorizations
- The meaning of implication in logic
- Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{lcm (n,k)} \rangle$