Saturday, June 25, 2022

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange

Rarefaction wave

Posted: 25 Jun 2022 09:55 AM PDT

Consider the IVP $$u_t+uu_t=0$$ Subject to the initial condition;$$u(x,0)=u_0(x)=\begin{cases} 0 & x<0 \\ \frac{x}{\epsilon} & 0<x<\epsilon\\ 1 & x>\epsilon \end{cases}$$ Find the solution and send $\epsilon \rightarrow 0^+ $, to get the rarefaction wave.

Find the order of the following elements in $Inn(S_5)$

Posted: 25 Jun 2022 09:54 AM PDT

Find the order of the following elements in Inn($S_5$).
a) $\phi_{(1243)}$

So the elements in Inn($S_5$) are functions $\phi_{(1243)}: S_5 \to S_5$ via $x \to (1243)x(1243)^{-1}$. So the order of this function is the smallest integer $n$ such that $\phi^n = \epsilon$ (the identity map).

I'm confused and don't really understand what I need to do here.

Ratio of heights of a sphere,over and under water

Posted: 25 Jun 2022 09:51 AM PDT

When a sphere is floating in the water, only 10 % of the volume is above the surface, while the rest is below. I need to calculate the relationship between height for the part above, and below the water.

enter image description here

h is height of the sphere above the surface, and H below

Push down quantifiers and simplification of a FOL formula

Posted: 25 Jun 2022 09:50 AM PDT

I have this formula:

$\forall x,y(R(x,y) \wedge \exists z R(x,z))\qquad(1)$

I remember that it is correct to push down the universal quantifier in this way:

$\forall x(\forall y R(x,y) \wedge \exists z R(x,z))\qquad(2)$

I also think the first conjunct of $(1)$ implies the second one, then $(1)$ can also be semplified in just

$\forall x,y(R(x,y))\qquad(3)$

My problem is that I don't think an analogous implication would hold for $(2)$, then I'm not able to reach $(3)$ passing from $(2)$ (while, if the three formulas are equivalent, it should be possible).

Where am I wrong? I think the error is in the implication leading to $(3)$, but I don't see exactly why.

Show that if $f(x,y)\geq a(x^2+y^2)$ for some $a > 0$, $f$ is smooth and $f(0,0) = 0$, than Gaussian curvature at $(0,0)$ positive

Posted: 25 Jun 2022 09:46 AM PDT

There is a question in Differential Geometry which I am not sure how to solve:

For $f:\mathbb{R}^{2}\to\mathbb{R}$, show that if $f(x,y) \geq a(x^2+y^2)$ for some constant $a > 0$, $f$ is a smooth function, and $f(0,0) = 0$, then Gaussian curvature at $(0,0)$ is positive.

I have tried to explain geometrically that the surface of $f$ looks like a sphere at $(0,0)$ and its a minimum point, because the Gaussian curvature is the determinant of the Hessian function at critical point, therefore it need to be positive.

But I am not sure if this solution is formal enough (because it might be zero as well).

I also tried to solve it with Taylor series of $2$ variables.

I will really appreciate it if you can help.

simplify this:- $\arctan\frac{\cos x-\sin x}{\cos x+\sin x}$, where $0<x<\pi$

Posted: 25 Jun 2022 09:43 AM PDT

I tried to solve this

$$\arctan\frac{\cos x-\sin x}{\cos x+\sin x}=\arctan\frac{1-\tan x}{1+\tan x}=\arctan\frac{\tan\frac{\pi}{4}-\tan x}{1+\tan\frac{\pi}{4}\tan x}=\arctan\tan(\frac{\pi}{4}-x)=\frac{\pi}{4}-x$$

But suddenly I realised $x$ can be $\frac{\pi}{2}$ where $\cos x=0$ and I can't divide it by $\cos x$.

suddenly this simple question make me clueless what to do.

Please help to solve this, and How to overcome such problems in future?

Definition of a limit using epsilon delta

Posted: 25 Jun 2022 09:34 AM PDT

I am a little confused regarding the definition of a limit using $\varepsilon$,$\delta$.First of all I would like to know if the full understanding of the limit idea requires any prior knowledge about $\varepsilon ,\delta$ or is it a notion that first appears in the definition of limits.Second I am confused about what epsilon depicts. Does it depict the total of all positive real numbers? In addition to that why is $\delta $ expressed as a function of $\varepsilon$ ?

Equivalence between $G$-bundles and certain associated bundles, for infinite-dimensional $G$?

Posted: 25 Jun 2022 09:47 AM PDT

From Case 1 below, which I think I understand, I am trying to generalize to Cases 2 and 3. I was wondering if the claims below make sense and are valid?

Let $X$ be a smooth manifold.

Case 1: finite-dimensional linear representations.

Let $G$ be a Lie group, let $(\xi,V)$ be a smooth finite-dimensional real linear representation of $G$.

Suppose

  1. $\xi$ is faithful; and

  2. $G$ equals the joint stabilizer (in $\mathrm{GL}(V)$) of some (finite) list of mixed tensors $\alpha$ on $V$. (Cf. e.g. this post.)

Let $D$ denote the category whose objects are smooth real vector bundles on $X$ having $\mathrm{rank}=\mathrm{dim}\xi$ and equipped with mixed tensor fields $\alpha$ as for the ones on $V$ above, and with morphisms the vector bundle isomorphisms which preserve the $\alpha$.

Let $C$ denote the category of smooth principal $G$-bundles on $X$.

Then the $\xi$-associated vector bundle construction $P \mapsto P_\xi$ should induce an equivalence of groupoids $C \xrightarrow{\sim} D$.

Case 2: Smooth actions, $G$ still finite-dimensional.

Let $G$ be a (still finite-dimensional) Lie group, let $(\xi,A)$ be a smooth manifold $A$ equipped with a smooth action $\xi$ by $G$.

Suppose

  1. $\xi$ is faithful; and

  2. $G$ equals the joint stabilizer in $\mathrm{Diff}(A)$ of some (finite) list of mixed tensor fields $\alpha$ on $A$. (E.g., the isometries of a Riemannian manifold are known to form a finite-dimensional group.)

Let $D$ denote the category whose objects are smooth $A$-fiber bundles on $X$ equipped with vertical mixed tensor fields $\alpha$ as for the ones on $A$ above, and with morphisms the $A$-fiber bundle isomorphisms which preserve the $\alpha$.

Let $C$ denote the category of smooth principal $G$-bundles on $X$.

Then the $\xi$-associated bundle construction $P \mapsto P_\xi$ should induce an equivalence of groupoids $C \xrightarrow{\sim} D$.

Case 3: Infinite-dimensional smooth actions.

Let $G$ be some sort of (possibly) infinite-dimensional Lie group, let $(\xi,A)$ be a smooth manifold $A$ equipped with a smooth action $\xi$ by $G$.

Suppose

  1. $\xi$ is faithful; and

  2. $G$ equals the joint stabilizer in $\mathrm{Diff}(A)$ of some (finite) list of mixed tensor fields $\alpha$ on $A$. (E.g., symplectomorphisms of a symplectic manifold, or auto-diffeomorphisms preserving some almost-complex structure.)

Let $D$ denote the category whose objects are smooth $A$-fiber bundles on $X$ equipped with vertical mixed tensor fields $\alpha$ as for the ones on $A$ above, and with morphisms the $A$-fiber bundle isomorphisms which preserve the $\alpha$.

Let $C$ denote the category of smooth principal $G$-bundles on $X$.

Then the $\xi$-associated bundle construction $P \mapsto P_\xi$ should induce an equivalence of groupoids $C \xrightarrow{\sim} D$.

Evaluating the value of $\lim{R\to\infty} \int_{C_R} \frac {e^{iz}}{z}dz$

Posted: 25 Jun 2022 09:33 AM PDT

I'm having some trouble with the following exercise:

Let $C_R$ be a curve in the complex plane parametizied as: $\gamma(t)= Re^{it},0\leq t\leq\pi$. Let $$I(R):=\int_{C_R} \frac {e^{iz}}{z}dz$$What is the value of $\lim_{R\to \infty} I(R)$?

I tried two things:

1 - Evaluating the integral using the definition of line integral:

$$I(R)=i\int_0^\pi \frac {e^{iRe^{^{it}}}}{Re^{it}}Re^{it}dt=i\int_0^\pi e^{iRe^{it}} dt$$

But I couldn't continue from here.

2 - My second approach was using the fact that:

$$\left|\int_\gamma f\right| \leq Kl(\gamma)$$

Where $K=\sup\{|f(z)|,z\in C_R\}$

In this particular case, $l(\gamma) = R\pi$ and I got that: $$\left|\frac {e^{iz}}{z}\right| \leq \frac{1}{R}$$

So I got that: $$|I(R)|\leq \pi$$

and this doesn't allow me to conclude anything when $R\to \infty$

How can I solve this exercise?

Difficulty understanding a concept used

Posted: 25 Jun 2022 09:15 AM PDT

Remark $1.3$ of the pdf here states that

There are three monic quadratic factors with $2^{\frac{1}{4}}$ as a root, but only one of them,$X^2 − 2^{\frac{1}{2}}$, has coefficients in $\mathbb{Q}(2^{\frac{1}{4}})$.

I could not get why other factors cannot be minimal polynomial of $2^{\frac{1}{4}}$ and why should the coefficents must lie in $\mathbb{Q}(2^{\frac{1}{4}})$

Convergence of $\sum \frac{b_n}{n}$ Where $b_n = 1, -1, -1, 1, 1, 1, -1, -1,-1,-1,1,1,1,1,1,....$

Posted: 25 Jun 2022 09:42 AM PDT

How can I show convergence or divergence of the following sequence?

$$\sum \frac{b_n}{n}$$

Where $b_n = 1, -1, -1, 1, 1, 1, -1, -1,-1,-1,1,1,1,1,1,....$

Im not sure how to use any of the standart theorems to show convergence or divergence.

Any attempts to use comparison test, or finding an upper bound failed.

Any ideas?

Condition number in norm 2 of a Jordan block

Posted: 25 Jun 2022 09:07 AM PDT

Let $J$ be an $n \times n$ Jordan block for the eigenvalue $\lambda$. We know that the condition number in norm 2 of $J$ is $\mu_2(J)= \lVert J \rVert_2 \lVert J^{-1}\rVert_2$ where $\lVert J \rVert _2= \sqrt{\rho(J^HJ)}$ and $\rho$ is the spectral radius, that is defined as $\rho(A)=\max _i |\lambda_i|$ where $\{\lambda _i\}_{i=1}^n$ are the eigenvalues of $A$. I tried to calculate it directly but this seems to go nowhere since we have that : $$\mu_2(J)=\sqrt{\rho(J^HJ)\rho((J^H)^{-1}J^{-1})}$$ Is it possible to calculate it reasonably? Can I estimate it?

Expressing a finite function as a infinite series

Posted: 25 Jun 2022 09:54 AM PDT

I am searching for a method to express a real valued function as an infinite series, is it possible? If so what kind of functions can be written as an infinite series? Can i express any function as a converging infinite series? What are the limitations?Can i express any function $f(x)=y$ as an infinite series and if i am able to do so how accurate will the sum of first few terms for approximating the function to a few decimal places?

Caption: I dont have any knowledge in the rigor of infinite series and I am just taking my calc 2 course so forgive me if this is a illogical question.

Linear programming problem ( Assignment Problem)

Posted: 25 Jun 2022 09:29 AM PDT

Let $C_{ij} \geq 2$ be the cost of the $(i, j)$th cell of an assignment problem. If a new cost matrix is generated by the elements $C_{ij}^*=(1/2)^*C_{ij+1}$, then what i can say about optimal assignment plan & cost of assignment.

Show the following relation related to densities

Posted: 25 Jun 2022 09:30 AM PDT

Let $X$ be a set and define the upper and lower uniform densities as follow: $$M(X) = \lim_{r \rightarrow \infty} \max_{x \in \mathbb{R}} \frac{|X \cap[x,x+r]|}{r},$$ and $$m(X) = \lim_{r \rightarrow \infty} \min_{x \in \mathbb{R}} \frac{|X \cap[x,x+r]|}{r}.$$ If $X \subset d \mathbb{Z}$ show that $M(X)+m(d\mathbb{Z} \setminus X) = 1/d$. What I can show is that $M(d\mathbb{Z}) = m(d\mathbb{Z})= 1/d$ without too much trouble. But I cannot seem to find a way to get the desired result. Does anyone have an idea on how to approach this?

Summing even and odd terms of binomial coefficients, with increasing powers of 2 and alternating signs

Posted: 25 Jun 2022 09:33 AM PDT

For a complex number $z$, let $(1+z)^{15}=a_{0}+a_{1} z+\cdots+a_{15} z^{15}$. The value of $$ \left(a_{0}-4 a_{2}+16 a_{4}+\cdots-2^{14} a_{14}\right)^{2}+\left(2 a_{1}-8 a_{3}+32 a_{5} \cdots-2^{15} a_{15}\right)^{2 \mid} $$

While I was entirely stumped by this question, I found some similar questions on here, specifically A sum involving binomial coefficients, powers and alternating signs. This did not equip me to be able to solve the above question, though, since this problem drops odd terms and even terms, alternates signs, AND has a factor of increasing powers of 2, not to mention squares the result and then sums them.

I don't see how the sum can even be expressed as a binomial to somehow use calculus, nor do I see any algebraic manipulations; I've only managed to embarrassingly restate the problem statement as

$$ \left(\sum_{k=0}^{7} 2^{2 k}({ }^{15} C_{2 k})(-1)^k\right)^{2}+\left(\sum_{k=0}^{7} 2^{2 k+1}\left({ }^{15} C_{2 k+1}\right)(-1)^k\right)^{2} $$

$f(x)>0, f''(x)>0$. Prove: $\int_a^b f(x) dx > (b-a) f(\frac{a+b}{2})$

Posted: 25 Jun 2022 09:20 AM PDT

On $[a,b]$ function $f$ is differentiable for arbitrary order, and $f(x)>0, f''(x)>0$. Prove: $\int_a^b f(x) dx > (b-a) f(\frac{a+b}{2})$.

I first try Taylor expansion at $x_0=\frac{a+b}{2}$, and drop higher order term ($(x-x_0)^3$ terms). But this works only locally at the neighborhood of $x_0$. How to prove this inequality on a finite interval? Thank you.

Prove an inequality with Jensen's inequality and his companion.

Posted: 25 Jun 2022 09:44 AM PDT

Problem :

Let $f(x)$ be continuous convex increasing on $(0,\infty)$ such that $f(\alpha)=0$ ; $x,a,\alpha>0$ with the constraint :

$$\infty>\alpha\geq x\geq \frac{xf'\left(x\right)+af'\left(a\right)}{f'\left(x\right)+f'\left(a\right)}\geq \frac{\left(x+a\right)}{2}\geq a $$

Where $\forall n\ge 2$ :$$f(x)=\sum_{i=0}^{n}a_ix^i$$ And $a_0 \ge a_1 \ge ... \ge a_n>0$

Then we have for $a,x,\alpha$ sufficiently large :

$$f\left(\frac{\left(x+a\right)}{2}\right)+f\left(\frac{\left(xf'\left(x\right)+af'\left(a\right)\right)}{f'\left(x\right)+f'\left(a\right)}\right)-f\left(a\right)-f\left(x\right)+f(\alpha)\geq f(\alpha)$$

My attempt :

We use majorization (with an alternative criteria see Aops page) as first step:

$$\alpha=\alpha$$

And :

$$\alpha+x\geq \alpha+\frac{xf'\left(x\right)+af'\left(a\right)}{f'\left(x\right)+f'\left(a\right)}$$

Finally : $$\frac{x+a}{2}+\frac{xf'\left(x\right)+af'\left(a\right)}{f'\left(x\right)+f'\left(a\right)}-a-x\leq 0$$

Wich is wrong .

How to prove it ?

Integral involving the eror function

Posted: 25 Jun 2022 09:06 AM PDT

I would like to know if the following integral has an explicit expression or not. $$F(r)=\int_1^r e^{t^2/2} \operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)dt$$

where the error function inside is defined as

$$\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)=\frac{1}{\sqrt{\pi}}\int_0^t e^{-x^2/2}dx.$$

Thus, $$F(r) = \frac{1}{\sqrt{\pi}}\int_{1}^{r}\int_0^{t}e^{\frac{t^2-x^2}{2}}dt dx.$$

Are their any known explicit expressions, series expansion or asymptotic known for such integrals?

Let $G$ be a group of p- power roots of unity in $C$, where p is a prime. Then show that G is isomorphic to a proper quotient of G itself.

Posted: 25 Jun 2022 09:43 AM PDT

$G$={$z$ $\in$ $C$ : $z^{p^n}=1$} , where p be any prime.

let $f:G \to G$ such that $f(z)=z^p$ ,where z $\in$ G ,is group homomorphism.

$Kerf$={$z\in G : f(z)=1$} $\implies$ {$z \in G : z^p=1$}.

I don't know how to prove it further .

Negation of Cauchy Criterion

Posted: 25 Jun 2022 09:23 AM PDT

What is the negation of the Cauchy criterion for sequences? I initially believed that the negation of the Cauchy criterion for a sequence ${p_n}$ in $\mathbb{R}$ is

$\exists \epsilon>0:\forall N\in \mathbb{N}, \exists (n\geq N \lor m\geq N):|p_n - p_m| \geq \epsilon$

due to the effect of DeMorgan's Law on the part $\forall(n \geq N \land m\geq N)$ ... of the Cauchy criterion statement.

However, I am unsure if this negation is correct (if, for example, there only exists $n\geq N : |p_n - p_m| \geq \epsilon$, what is the significance of $m$ in $|p_n - p_m|$?).

Probability of selecting object from an urn

Posted: 25 Jun 2022 09:47 AM PDT

Let's say I have an urn with 10 unique objects, and I choose 3 objects from it (each choice is made without replacement). Then the probability of choosing any one object is 3/10. I calculated this probability by summing the probability the object is chosen on the first pick + probability chosen on second pick + probability it's chosen on the third pick. I'm wondering if there is a simpler/more intuitive way to arrive at this probability though.

One idea I had is that for any random permutation of the 10 objects, we can split the permutation into first a group of 3 and second a group of 7 objects. There's then a 3/10 probability that a given object is in the first group (first set of 3 objects chosen).

Limit does not preserve strict inequality

Posted: 25 Jun 2022 09:30 AM PDT

When I was reading Rudin's proof of L'Hospital's Rule, I was confused with the following two lines. Since my question is not directly related to the proof itself but more on the limiting behavior of functions, I will only mention the part I don't understand. The set up is that functions $f,g$ are continuous on $[a,b]$ and differentiable on $(a,b)$, the limit of $\frac{f'(t)}{g'(t)}$ exists and quals to $A$, which could be anything on the extended real line.

  1. Given that $$\frac{f(x) - f(y)}{g(x) - g(y)} =\frac{f'(t)}{g'(t)} < r,$$ where we can treat $x$ as the variable, $y$ and $t$ as fixed, then as $x\to a$, Rudin wrote $$\frac{f(x)}{f(y)}\le r. $$

  2. Given that $$\frac{f(x)}{g(x)}< r - \frac{g(y)}{g(x)}+\frac{f(y)}{f(x)},$$ if we let $x\to a$, then $$\frac{f(x)}{g(x)} < r .$$

Notice the first one changes to $\le$ after taking the limit while the second one remains $<$. May I ask if there is a general rule on this? Could it be related to limsup or liminf?

New update: I was actually misquoted the second part - Rudin wrote $\frac{f(x)}{g(x)}< q$ directly.

Differentiating a function inside a function.

Posted: 25 Jun 2022 09:21 AM PDT

Let's say that there's a function $f(x) = g(x^2 + 5x)$, and $f'(0) = 10$. What would $g'(0)$ be here?

Because this is a function composition I tried using the chain rule in this way -

(I took the inside function $g(x)$ as u)

\begin{align} & \ \ \ \ \ \ \ f'(x) = \frac{dg}{du}.\frac{du}{dx}\\[1em] & \Rightarrow f'(x) = g'(x^2+5x).(2x + 5)\\[1em] & \Rightarrow f'(0) = g'(0).(2*0 + 5)\\[1em] & \Rightarrow 10 = g'(0).(2*0 + 5)\\[1em] & \Rightarrow g'(0) = 2 \end{align}

Nonnegative random variable whose characteristic function is differentiable at 0 has first moment

Posted: 25 Jun 2022 09:53 AM PDT

I'm trying to prove the following claim:

Let $X$ be a nonnegative random variable, and let $\phi(t) = \mathbb E\left[e^{itX}\right]$ be its characteristic function. Suppose $\phi$ is differentiable at $t=0$. Then $\mathbb E[X] < \infty$.

This is a component of an exercise in my probability theory textbook (Achim Klenke, "Probability Theory: A Comprehensive Course", Exercise 15.4.4(iii)).

This question has also been asked here, and there's an answer that supposedly gives a counterexample, but I'm not convinced: the counterexample is a symmetric random variable instead of nonnegative, and it's not clear to me how the characteristic function of a a symmetric random variable relates to the corresponding nonnegative random variable (i.e. its absolute value).

What I've tried: Klenke has a proof of the existence of $2n^\textrm{th}$ moments when $\phi^{(2n)}(0)$ exists for some $n \geq 1$, but I've been having trouble adapting the proof to the first derivative case. I tried considering a symmetric random variable $Y$ for which $Y^2 = X$ (and thus $\mathbb E[Y^2] = \mathbb E[X]$). If I were able to prove $\phi_Y''(0)$ exists, then I'd be done. But letting $f(x) = x^2$, and noting that $f_* \mathbb P_Y = \mathbb P_X$, I ended up computing: $$ \phi_Y(t) = \int_{\mathbb R} e^{ity} \mathbb P_Y[dy] = \mathbb P[Y=0] + 2\int_{(0,\infty)} \cos\left(t\sqrt x\right) \mathbb P_X[dx] $$ and it's not at all obvious to me that this map should be differentiable (let alone twice differentiable).

Any suggestions/places to find the answer?

Integral with binomial to a power $\int\frac{1}{(x^4+1)^2}dx$

Posted: 25 Jun 2022 09:41 AM PDT

I have to solve the following integral:

$$\int\frac{1}{(x^4+1)^2}dx$$

I tried expanding it and then by partial fractions but I ended with a ton of terms and messed up. I also tried getting the roots of the binomial for the partial fractions but I got complex roots and got stuck. Is there a trick for this kind of integral or some kind of helpful substitution? Thanks.

EDIT:

I did the following:

Let $x^2=\tan\theta$, then $x = \sqrt{\tan\theta}$ and $dx=\frac{\sec^2\theta}{2x}d\theta$

Then:

$$I=\int\frac{1}{(x^4+1)^2}dx = \int\frac{1}{(\tan^2\theta+1)^2} \frac{\sec^2\theta}{2x}d\theta=\int\frac{1}{\sec^4\theta} \frac{\sec^2\theta}{2x}d\theta$$

$$I=\frac{1}{2}\int{\frac{1}{\sec^2\theta \sqrt{\tan\theta}}}d\theta$$.

After this I don't know how to proceed.

Constrained optimization : Contour lines and Lagrange's multiplier

Posted: 25 Jun 2022 09:08 AM PDT

Basically the core of Lagrange's multiplier says that the solution to a constrained optimization occurs when the contour line of the function being maximized/minimized is tangential to the constraint curve.

I am not able to convince myself of above statement.

Consider below diagram wherein 'f' is some function being maximized/minimized and g(x,y) is the constraint. the contour lines of f and g(x,y) are plotted.

enter image description here

Clearly the contour line which corresponds to local maximum of f is not tangential to g i.e. the gradient vector of f and g don't align at the intersections of f=5 and g.

I am not looking for a rigorous proof or something, just want to get a basic understanding of why gradient of f and g should point in same direction at the local maxima/minima of f?

Does invariance of dimension also hold for closed domains?

Posted: 25 Jun 2022 09:07 AM PDT

Brouwer's theorem of invariance of dimension under homeomorphism holds for non-empty open subsets of $\mathbb{R}^{n}$:

Let $\emptyset\neq U$ open $\subseteq\mathbb{R}^{n}$ and $\emptyset\neq V$ open $\subseteq\mathbb{R}^{m}$. If $U$ and $V$ are homeomorphic, then $n=m$.

A domain is any non-empty connected open set in a topological space. The closure of a domain is called a closed domain. (Encyclopedia of Mathematics: Domain)

Does the following theorem hold also:

Let $\emptyset\neq U \subset \mathbb{R}^{n}$ be a closed domain in $\mathbb{R}^{n}$, and let $\emptyset\neq V \subset \mathbb{R}^{m}$ be a closed domain in $\mathbb{R}^{m}$. If $U$ and $V$ are homeomorphic, then $n=m$.

What is the difference between a Limit and Derivative?

Posted: 25 Jun 2022 09:30 AM PDT

When you get the derivative of a point, arn't you just getting the limit at that point?

I'm not quite sure why they need to be named differently when they seem to be doing the same thing.

Another (non-homological) proof of the invariance of dimension

Posted: 25 Jun 2022 09:12 AM PDT

the only proof I know of that $\mathbb{R}^n$ cannot be homeomorphic to $\mathbb{R}^{n+1}$ requires quite a lot of mathematical apparatus, namely singular homology theory with its exact sequences and pretty abstract setting, as well as some previous (though admittedly not too difficult) calculations of the homology groups of spheres to be used after "excising" a point in $\mathbb{R}^n$.

Is anybody aware of an alternate route? Original research papers are not a problem, as long as they don't involve even more elaborate frameworks or some arcane theory, since eluding those is precisely the point. (To the algebraic-theorists around: it's not that I have any particular bad feelings about homology groups, just curiosity ;-)

Thanks,
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