- #1

fresh_42

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**Rules:**

**a)**In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month.

**b)**It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.

**c)**If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.

**d)**You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.

**Referees:**

@QuantumQuest

@StoneTemplePython

@Infrared

@wrobel

@fresh_42

**Hints:**on demand

**Questions:**

**Basics:**

**1.**(solved by @Delta² ) Let [itex](x_n)[/itex] be a sequence of positive real numbers such that [itex]x_{n+1}\geq\dfrac{x_n+x_{n+2}}{2}[/itex] for each [itex]n[/itex]. Show that the sequence is (weakly) increasing, i.e. [itex]x_n\leq x_{n+1}[/itex] for each [itex]n[/itex].

**2.**(solved by @timetraveller123 ) Let [itex]p(x)[/itex] be a non-constant real polynomial. Suppose that there exists a real number [itex]a[/itex] such that [itex]p(a)\neq 0[/itex] and [itex]p'(a)=p''(a)=0\,.[/itex] Show that not all of the roots of [itex]p[/itex] are real.

**3.**(solved by @julian ) Find the area enclosed by the curve ##r^2 = a^2\cos 2\theta\,##.

**4.**(solved by @Buzz Bloom ) (You may use wolframalpha.com for calculations. It is biological nonsense, cp. post #4, but for the sake of the problem, we will make the following assumptions.) One tiny nocturnal and long-living beetle decided one night to climb a sequoia. The tree was exactly ##100\, m## high at this time. Every night the beetle made a distance of ##10\, cm##. The tree grew every day evenly ##20\, cm## along its entire length.

Did the beetle eventually reach the top of the tree? And if so, how many nights will he need at least?

**5.**(solved by @julian ) Show that ##\lim_{n \to \infty} \sqrt[n]{p_1a_1^n + p_2a_2^n + \cdots + p_ka_k^n} = max \{a_1, a_2, \cdots ,a_k\}## where ##p_1, p_2, \cdots, p_k \gt 0## and ##a_1, a_2, \cdots, a_k \geq 0##.

**6.**(solved by @julian ) Show that the sequence ##(a_n)## with ##a_1 = \sqrt[]{b}\; , \;b>0\,##, ##a_{n+1} = \sqrt[]{a_n + b}## for ##n = 1, 2, 3, \ldots## converges to the positive root of the equation ##x^2 - x - b = 0##.

**Combinatorics and Probabilities:**

**7.**(solved by @Jacob Nie ) How many times a day is it impossible to determine what time it is, if you have a clock with same length (identical looking) hour and minutes hands, supposing that we always know if it's morning or evening (i.e. we know whether it's am or pm).

**8.**For the class of ##n \times n## matrices whose entries, (if flattened and sorted) would be ##1, 2, 3, ..., n^2 -1 ,n^2## prove that there always exists two neighboring entries (in same row or same column) that must differ by at least ##n##.

**9.**There are ##r## sports 'enthusiasts' in a certain city. They are forming various teams to bet on upcoming events. A pair of people dominated last year, so there are new rules in place this year. The peculiar rules are:

each team must have an odd number of members

each and every 2 teams must have an even number of members in common.

For avoidance of doubt, nothing in the rules say a given player can only be on one team.

With these rules in place, is it possible to form more than ##r## teams?

Hint: Model these rules with matrix multiplication and select a suitable field.

**Calculus:**

**10. a)**(solved by @eys_physics ) Determine ##\int_1^\infty \frac{\log(x)}{x^3}\,dx\,.##

**10. b)**(solved by @bhobba ) Determine for which ##\alpha## the integral ##\int_0^\infty x^2\exp(-\alpha x)\,dx## converges.

**10. c)**(solved by @Keith_McClary ) Find a sequence of functions ##f_n\, : \,\mathbb{R}\longrightarrow \mathbb{R}\, , \,n\in \mathbb{N}## such that $$\sum_\mathbb{N}\int_\mathbb{R}f_n(x)\,dx \neq \int_\mathbb{R}\left(\sum_\mathbb{N}f_n(x) \right) \,dx$$

**10. d)**(solved by @Delta2 ) Find a family of functions ##f_r\, : \,\mathbb{R} \longrightarrow \mathbb{R}\, , \,r>0## such that

$$

\lim_{r \to 0}\int_\mathbb{R}f_r(x)\,dx \neq \int_\mathbb{R} \lim_{r \to 0} f_r(x) \,dx

$$

**10. e)**(resolved in https://www.physicsforums.com/threa...cannot-always-be-swapped.960617/#post-6092583) Find an example for which

$$

\dfrac{d}{dx}\int_\mathbb{R}f(x,y)\,dy \neq \int_\mathbb{R}\dfrac{\partial}{\partial x}f(x,y)\,dy

$$

**11.**(solved by @Delta² ) Let ##f##, ##g##: ##\mathbb{R} \rightarrow \mathbb{R}## be two functions with ##f\,''(x) + f\,'(x)g(x) - f(x) = 0##. Show that if ##f(a) = f(b) = 0## then ##f(x) = 0## for all ##x\in [a,b]##.

**12. a)**Let ##f\, : \,\mathbb{R}^2 \longrightarrow \mathbb{R}## be defined as

$$

f(x,y) =

\begin{cases}

1 & \text{if } x \geq 0 \text{ and }x \leq y < x+1\\

-1 & \text{if } x \geq 0 \text{ and }x+1 \leq y < x+2\\

0 & \text{elsewhere }

\end{cases}

$$

Now calculate ##\int_\mathbb{R}\left[\int_\mathbb{R}f(x,y)\,d\lambda(x) \right] \,d\lambda(y)## and ##\int_\mathbb{R}\left[\int_\mathbb{R}f(x,y)\,d\lambda(y) \right]\,d\lambda(x)\,,## and why isn't it a contradiction to Fubini's theorem.

**12. b)**(solved by @Keith_McClary ) Show that the integral

$$

\int_A \dfrac{1}{x^2+y}\,d\lambda(x,y)

$$

with ##A=(0,1)\times (0,1)\subseteq \mathbb{R}^2## is finite.

**13.**Let ##f\, : \,(0,1)\longrightarrow \mathbb{R}## be Lebesgue integrable and $$

Y := \{\,(x_1,x_2)\in\mathbb{R}^2\,|\,x_1,x_2\geq 0\, , \,x_1+x_2\leq 1\,\}

$$

Show that for any ##\alpha_1\, , \,\alpha_2 > 0##

$$

\int_Y f(x_1+x_2)x_1^{\alpha_1}x_2^{\alpha_2}\,d\lambda(x_1,x_2) = \left[\int_0^1 f(u)u^{\alpha_1+\alpha_2+1}\,d\lambda(u) \right]\cdot \left[\int_0^1 v^{\alpha_1}(1-v)^{\alpha_2}\,d\lambda(v) \right]

$$

Hint: Consider ##\phi\, : \,(0,1)^2\longrightarrow \mathbb{R}^2## with ##\phi(u,v)=(vu,(1-v)u)\,.## and apply Fubini's theorem.

**14.**(solved by @Delta² ) Let ##f## be a differentiable function in ##\mathbb{R}##. If ##f'## is invertible and ##(f')^{-1}## is differentiable in ##\mathbb{R}##, show that ##[I_A (f')^{-1} - f \circ [(f')^{-1} ]]' = (f')^{-1}## where ##I_A## with ##I_A(x) = x## is the identity function ##I_A : A \to A##

Hint: Let ##y = f'(x)## then ##(f')^{-1}(y) = x##. By differentiating, we can get to a useful result including the second derivative of ##f(x)##. Next, we can utilize ##[f\circ (f')^{-1}]'## and incorporate the identity function ##I_A##.

**Linear Algebra:**

**15.**Given the Heisenberg algebra $$\mathcal{H}=\left\{\,\begin{bmatrix} 0&x&z\\0&0&y\\0&0&0 \end{bmatrix}\,\right\}=\langle X,Y,Z\,:\,[X,Y]=Z \rangle $$ and $$\mathfrak{A(\mathcal{H})}=\{\,\alpha\, : \,\mathcal{H}\longrightarrow \mathcal{H}\, : \,[\alpha(X),Y]=[\alpha(Y),X]\,\forall\,X,Y\in \mathcal{H}\,\} $$

Since ##\mathfrak{A(\mathcal{H})}## is a Lie algebra and $$[X,\alpha]=[\operatorname{ad}(X),\alpha]=\alpha(X)\circ \alpha - \alpha \circ \operatorname{ad(X)}$$ a Lie multiplication, we can define

\begin{align*}

\mathcal{H}_0 &:= \mathcal{H}\\

\mathcal{H}_{n+1} &:= \mathcal{H}_n \ltimes \mathfrak{A(\mathcal{H}_n)}

\end{align*}

and get a series of subalgebras $$\mathcal{H}_0 \leq \mathcal{H}_1 \leq \mathcal{H}_2 \leq \ldots$$

Show that

##\mathfrak{sl}(2)<\mathcal{H}_n## is a proper subalgebra for all ##n\ge 1##

##\dim \mathcal{H}_{n} \ge 3 \cdot (2^{n+1}-1)## for all ##n\ge 0##, i.e. the series is infinite and doesn't get stationaryl

As a counterexample, if we started with ##\mathcal{H}=\mathfrak{su}(2)\text{ or }\mathfrak{su}(3)## we would get ##\mathcal{H}_n=\mathcal{H}_0## and we were stationary right from the start, which can easily be seen by solving the corresponding system of linear equations.

Hint: The multiplication in ##\mathcal{H}_n## is given by $$

[(X,\alpha),(Y,\beta)]=([X,Y],[\alpha,\beta]+[\operatorname{ad}(X),\beta]-[\operatorname{ad}(Y),\alpha])$$ However, it is not really needed here. Calculate ##\mathfrak{A}(\mathcal{H})=\mathfrak{A}(\mathcal{H}_0)## and find a copy of ##\mathfrak{sl}(2)## in it, i.e. a ##2 \times 2## block with zero trace. Then note that all ##\mathcal{H}_n## have a central element (= commutes with all others), and consider its implication for ##\mathfrak{A}(\mathcal{H}_n)\,.## Proceed by induction.

**16.**(solved by @julian ) Consider the Lie algebra of skew-Hermitian ##2\times 2## matrices ##\mathfrak{g}:=\mathfrak{su}(2,\mathbb{C})## and the Pauli matrices (note that Pauli matrices are not a basis!)

$$

\sigma_1=\begin{bmatrix}0&1\\1&0\end{bmatrix}\, , \,\sigma_2=\begin{bmatrix}0&-i\\i&0\end{bmatrix}\, , \,\sigma_3=\begin{bmatrix}1&0\\0&-1\end{bmatrix}

$$

Now we define an operation on ##V:=\mathbb{C}_2[x,y]##, the vector space of all complex polynomials of degree less than three in the variables ##x,y## by

\begin{align*}

\varphi(\alpha_1\sigma_1 +\alpha_2\sigma_2+\alpha_3\sigma_3)&.(a_0+a_1x+a_2x^2+a_3y+a_4y^2+a_5xy)= \\

&= x(-i \alpha_1 a_3 +\alpha_2 a_3 - \alpha_3 a_1 )+\\

&+ x^2(2i\alpha_1 a_5 +2 \alpha_2 a_5 + 2\alpha_3 a_2 )+\\

&+ y(-i\alpha_1 a_1 -\alpha_2 a_1 +\alpha_3 a_3 )+\\

&+ y^2(2i\alpha_1 a_5 -2\alpha_2 a_5 -2\alpha_3 a_4 )+\\

&+ xy(-i\alpha_1 a_2 -i\alpha_1 a_4 +\alpha_2 a_2 -\alpha_2 a_4 )

\end{align*}

Show that

**16. a)**an adjusted ##\varphi## defines a representation of ##\mathfrak{su}(2,\mathbb{C})## on ##\mathbb{C}_2[x,y]##

**16. b)**Determine its irreducible components.

**16. c)**Compute a vector of maximal weight for each of the components.

Hint: This is an easy example of a ##\mathfrak{su}(2,\mathbb{C})## representation which shall demonstrate how the ladder up and down operators actually work. Choose ##(1,x,y,x^2,xy,y^2)## as ordered basis for the representation space ##V=\mathbb{C}_2[x,y]## and verify ##[\varphi(\alpha_1,\alpha_2,\alpha_3),\varphi(\alpha'_1,\alpha'_2,\alpha'_3)]=\varphi([(\alpha_1,\alpha_2,\alpha_3),(\alpha'_1,\alpha'_2,\alpha'_3)])## with the adjusted transformation

$$

\varphi(\alpha_1,\alpha_2,\alpha_3):=\varphi(\alpha_1\cdot (i\sigma_1),\alpha_2\cdot (i\sigma_2),\alpha_3\cdot (i\sigma_3))

$$

and decompose ##V## into three invariant subspaces. To determine the weights, consider the ##\mathbb{C}-##basis $$H=\sigma_3,X=\dfrac{1}{2}\sigma_1+\dfrac{1}{2}i\sigma_2,Y=\dfrac{1}{2}\sigma_1-\dfrac{1}{2}i\sigma_2$$

**Abstract Algebra:**

**17. a)**Let [itex]n[/itex] be a positive integer. Let [itex]a_1,\ldots,a_k[/itex] be (positive) factors of [itex]n[/itex] such that [itex]\gcd(a_1,\ldots,a_k,n)=1[/itex]. How many solutions [itex](x_1,\ldots,x_k)[/itex] does the equation [itex]a_1x_1+\ldots+a_kx_k\equiv 0\mod n[/itex] have subject to the restriction that [itex]0\leq x_i<n/a_i[/itex] for each [itex]i[/itex]?

**17. b)**How does the solution change if [itex]\gcd(a_1,\ldots,a_k,n)=d>1[/itex]?

**18.**A function ##|\,.\,|\, : \,\mathbb{F}\longrightarrow \mathbb{R}_{\geq 0}## on a field ##\mathbb{F}## is called a value function if

\begin{align*}

&|x|=0 \Longleftrightarrow x=0 \\

&|xy| = |x|\;|y|\\

&|x+y| \leq |x|+|y|

\end{align*}

It is called Archimedean, if for any two elements ##a,b\,\,(a\neq 0)## there is a natural number ##n## such that ##|na|>|b|\,.## We consider the rational numbers. The usual absolute value

$$

|x| = \begin{cases} x &\text{ if }x\geq 0 \\ -x &\text{ if }x<0\end{cases}

$$

is Archimedean, whereas the trivial value

$$

|x|_0 = \begin{cases} 0 &\text{ if }x = 0 \\ 1 &\text{ if }x\neq 0\end{cases}

$$

is not.

Determine all non-trivial and non-Archimedean value functions on ##\mathbb{Q}\,.##

Hint: This is indeed a bit tricky. Since ##|\,.\,|## is non-Archimedean, there are elements ##a,b## with ##|n|<\frac{|b|}{|a|}## for all ##n\in \mathbb{N}\,.## If ##|n| > 1## for a natural number, then ##|n^k|=|n|^k## goes to infinity and cannot be bounded. Thus ##|n|\leq 1## for all ##n\in \mathbb{N}\,.## Note that ##|.|## is non-trivial. Pick a smallest natural number ##n_0=ab## and investigate it.

**19.**Let [itex]f(x)\in\mathbb{Q}[x][/itex] be a degree 5 polynomial with splitting field [itex]K[/itex]. Suppose that there is a unique extension [itex]F/\mathbb{Q}[/itex] such that [itex]F\subset K[/itex] and [itex][K:F]=3[/itex]. Show that [itex]f(x)[/itex] is divisible by a degree 3 irreducible element of [itex]\mathbb{Q}[x][/itex].

**20.**(solved by @julian ) Let's consider complex functions in one variable and especially the involutions

$$

\mathcal{I}=\{\, z\stackrel{p}{\mapsto} z\; , \; z\stackrel{q}{\mapsto} -z\; , \;z\stackrel{r}{\mapsto} z^{-1}\; , \;z\stackrel{s}{\mapsto}-z^{-1}\,\}

$$

We also consider the two functions $$\mathcal{J}=\{\,z\stackrel{u}{\longmapsto}\frac{1}{2}(-1+i \sqrt{3})z\; , \;z\stackrel{v}{\longmapsto}-\frac{1}{2}(1+i \sqrt{3})z\,\}$$

and the set ##\mathcal{F}## of functions which we get, if we combine any of them: ##\mathcal{F}=\langle\mathcal{I},\mathcal{J} \rangle## by consecutive applications. We now define for ##\mathcal{K}\in \{\mathcal{I},\mathcal{J}\}## a relation on ##\mathcal{F}## by

$$

f(z) \sim_\mathcal{K} g(z)\, :\Longleftrightarrow \, (\forall \,h_1\in \mathcal{K})\,(\exists\,h_2\in \mathcal{K})\,: f(h_1(z))=g(h_2(z))

$$

**20. a)**Show that ##\sim_\mathcal{K}## defines an equivalence relation.

**20. b)**Show that ##\mathcal{F}/\sim_\mathcal{I}## admits a group structure on its equivalence classes by consecutive application.

**20. c)**Show that ##\mathcal{F}/\sim_\mathcal{J}## does not admit a group structure on its equivalence classes by consecutive applications.

Hint: Determine the groups ##F=\langle \mathcal{F}\rangle\, , \,I=\langle \mathcal{I} \rangle\, , \,J=\langle \mathcal{J} \rangle## and what distinguishes ##F/I## from ##F/J\,.##

**Topology:**

**21.**(solved by @julian ) A covering space ##\tilde{X} ## of ##X## is a topological space together with a continuous surjective map ##p\, : \,\tilde{X} \longrightarrow X\,,## such that for every ##x \in X## there is an open neighborhood ##U\subseteq X## of ##x,## such that ##p^{-1}(U)\subseteq \tilde{X}## is a union of pairwise disjoint open sets ##V_\iota## each of which is homeomorphically mapped onto ##U## by ##p##. A deck transformation with respect to ##p## is a homeomorphism ##h\, : \,\tilde{X} \longrightarrow \tilde{X}## with ##p \circ h=p\,.## Let ##\mathcal{D}(p)## be the set of all deck transformations with respect to ##p##.

Show that ##\mathcal{D}(p) ## is a group.

If ##\tilde{X}## is a connected Hausdorff space and ##h \in \mathcal{D}(p)## with ##h(\tilde{x})=\tilde{x}## for some point ##\tilde{x}\in \tilde{X}\,.## then ##h=\operatorname{id}_{\tilde{X}}\,.##

Hint: Show that ##\mathcal{D}(p)## is closed under inversion and multiplication. Then consider ##A:=\{\,\tilde{x}\in \tilde{X}\, : \,h(\tilde{x})=\tilde{x}\,\}\,.##

**22.**We define an equivalence relation on the topological two-dimensional unit sphere ##\mathbb{S}^2\subseteq \mathbb{R}^3## by ##x \sim y \Longleftrightarrow x \in \{\,\pm y\,\}## and the projection ##q\, : \,\mathbb{S}^2 \longrightarrow \mathbb{S}^2/\sim \,.## Furthermore we consider the homeomorphism ##\tau \, : \,\mathbb{S}^2 \longrightarrow \mathbb{S}^2## defined by ##\tau (x)=-x\,.## Note that for ##A \subseteq \mathbb{S}^2## we have ##q^{-1}(q(A))= A \cup \tau(A)\,.## Show that

##q## is open and closed.

##\mathbb{S}^2/\sim ## is compact, i.e. Hausdorff and covering compact.

Let ##U_x=\{\,y\in \mathbb{S}^2\,:\,||y-x||<1\,\}## be an open neighborhood of ##x \in \mathbb{S}^2\,.## Show that ##U_x \cap U_{-x} = \emptyset \; , \;U_{-x}=\tau(U_x)\; , \;q(U_x)=q(U_{-x})## and ##q|_{U_{x}}## is injective. Conclude that ##q## is a covering.

Hint: For (a) consider ##O\cup \tau(O)## for an open set ##O\,.## For (b) use part (a) and that ##\mathbb{S}^2## is Hausdorff. For (c) a covering map is a local homeomorphism, so we need continuity, openess, closedness and bijectivity. Again use the previous parts.

**Especially for Physicists:**

**23.**(solved by @PeroK ) On the occasion of the centenary of Emmy Noether's theorem.

This example requires some introduction for all members who aren't familiar with the matter, so let me first give some background information.

\begin{align*}

x &\longmapsto x^* := x +\varepsilon \psi(t,x,\dot{x})+O(\varepsilon^2)\\

t &\longmapsto t^* := t +\varepsilon \varphi(t,x,\dot{x})+O(\varepsilon^2)

\end{align*}

and we compare $$ S=S(x(t))=\int \mathcal{L}(t,x,\dot{x})\,dt\text{ and }S^*=S(x^*(t^*))=\int \mathcal{L}(t^*,x^*,\dot{x}^*)\,dt^* $$

Since the functional ##S## determines the law of motion of the particle, $$S=S^*$$ means, that the action on this particle is unchanged, i.e. invariant under these transformations, and especially

\begin{equation*}

\dfrac{\partial S}{\partial \varepsilon}=0 \quad\text{ resp. }\quad \left. \dfrac{d}{d\varepsilon}\right|_{\varepsilon =0}\left(\mathcal{L}\left(t^*,x^*,\dot{x}^*\right)\cdot \dfrac{dt^*}{dt} \right) = 0 ~~(*)

\end{equation*}

Emmy Noether showed exactly hundred years ago, that under these circumstances (invariance), there is a conserved quantity ##Q##. ##Q## is called the Noether charge. $$S=S^* \Longrightarrow \left. \dfrac{d}{d\varepsilon}\right|_{\varepsilon =0}\left(\mathcal{L}\left(t^*,x^*,\dot{x}^*\right)\cdot \dfrac{dt^*}{dt} \right) = 0 \Longrightarrow \dfrac{d}{dt}Q(t,x,\dot{x})=0$$

with $$Q=Q(t,x,\dot{x}):= \sum_{i=1}^N \dfrac{\partial \mathcal{L}}{\partial \dot{x}_i}\,\psi_i + \left(\mathcal{L}-\sum_{i=1}^N \dfrac{\partial \mathcal{L}}{\partial \dot{x}_i}\,\dot{x}_i\right)\varphi = \text{ constant}$$

The general way to proceed is:

A. Determine the functions ##\psi,\varphi##, i.e. the transformations, which are considered.

B. Check the symmetry by equation (*).

C. If the symmetry condition holds, then compute the conservation quantity ##Q## with ##\mathcal{L},\psi,\varphi\,.##

Hint: The Lagrange function with ##\vec{r}=(x,y,z,t)=(x_1,x_2,x_3,t)## of this problem is $$ \mathcal{L}=T-U=\dfrac{m}{2}\,\dot{\vec{r}}\,^2-\dfrac{U_0}{\vec{r\,}^{2}} $$

a) Give a reason why the energy of the particle is conserved, and what is its energy?

b) Consider the following transformations with infinitesimal ##\varepsilon##

$$\vec{r} \longmapsto \vec{r}\,^*=(1+\varepsilon)\,\vec{r}\,\, , \,\,t\longmapsto t^*=(1+\varepsilon)^2\,t$$

and verify the condition (*) to E. Noether's theorem.

c) Compute the corresponding Noether charge ##Q## and evaluate ##Q## for ##t=0##.

**24.**Solve and describe the solution step by step in quadrature a Lagrangian differential equation with Lagrangian

$$L(t,x,\dot x)=\frac{1}{2}\dot x^2-\frac{t}{x^4}.$$

**25.**We consider the vector field ##X\, : \,\mathbb{R}\longrightarrow \mathbb{R}^2## given by ##X(p) := \left(p,\begin{pmatrix} 1\\0 \end{pmatrix}\right)\,.##

Compute the derivative ##d\phi\, : \,T\mathbb{R}^2\longrightarrow T\mathbb{R}^3## of the stereographic projection to the north pole, i.e. plane to sphere with ##\phi(0,0)=(0,0,-1)##, and describe the tangent bundle ##T\mathbb{S}^2## of ##\mathbb{S}^2##. Show that position vectors and tangent vectors are orthogonal.

Compute the vector field ##d\phi(X)## on ##\mathbb{S}^2##. How is it related to the curves ##\gamma(t)=\phi(t,y_0)\,?##

Is ##d\phi(X)## a continuous vector field on ##\mathbb{S}^2## without zeros?

Hint: The stereographic projection to the north pole is given by

\begin{align*}

\phi\, &: \, \mathbb{R}^2\longrightarrow \mathbb{R}^3\\

\phi(x,y)&=\dfrac{1}{x^2+y^2+1} \begin{bmatrix}

2x\\2y\\x^2+y^2-1

\end{bmatrix}

\end{align*}

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