Sunday, August 8, 2021

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange

Centroid and midpoints cyclic then prove that $AC^2 + BC^2 = 2AB^2.$

Posted: 08 Aug 2021 08:44 PM PDT

Let $ABC$ be a non-equilateral triangle with integer sides. Let $D$ and $E$ be respectively the mid-points of $BC$ and $CA$ ; let $G$ be the centroid of $\Delta{ABC}$. Suppose, $D$, $C$, $E$, $G$ are concyclic.Prove that $AC^2 + BC^2 = 2AB^2.$

My progress:

enter image description here

  • By angle chase we get, $$\angle GBA=\angle GED=\angle GCD\implies (CGB)~~\text{is tangent to}~~AB.$$
  • Similarly, we get $$\angle GAB=\angle GDE=\angle GCE\implies (CGA)~~\text{is tangent to}~~AB.$$
  • By power of point, we have $$\frac{AB^2}{4}=FA^2=FB^2=FG\cdot FC=\frac{1}{3}FG^2. $$

"Something does not depend on a particular coordinate system". What is the meaning of this sentence?

Posted: 08 Aug 2021 08:34 PM PDT

I am reading "Calculus Metric Version 7th Edition International Edition" by James Stewart.

On p.879, Stewart wrote as follows:

"It is often useful to parametrize a curve with respect to arc length because arc length arises naturally from the shape of the curve and does not depend on a particular coordinate system."

I cannot understand why arc length does not depend on a particular coordinate system.
I cannot understand even what the sentence "something does not depend on a particular coordinate system" means precisely.

What is a curve?
I think a curve itself depends on a particular coordinate system.
When we define a curve, we always use a particular coordinate system.

The shape of the convex function

Posted: 08 Aug 2021 08:23 PM PDT

In the textbook, convex function is defined as follow. A function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. But, I can't imagine the shape of the graph. Why does a convex function refer to a function that is in the shape of a cup, and a concave function is in the shape of a cap? (by wikipedia)

Show that $det_i(A)$ is linear

Posted: 08 Aug 2021 08:15 PM PDT

Let $A=(C_1,C_2,...,C_n)$ a matrix $nXn$

With columns $(C_1,C_2,...,C_n)$

If the column $C_k$ is the sum of other two columns of $A$ $C_k=C^´_k+C^{''}_k$ show that

$det_i(C_1,C_2,...,C^´_k+C^{''}_k,...,C_n)=det_i(C_1,C_2,...,C^{'}_k,...,C_n)+det_i(C_1,C_2,...,C^{''}_k,...,C_n)$

where $det_i(A)= \sum_{j=1}^{n}a_{ij}det(A_{ij}) $ and $i$ is a fixed row

i was trying by induction on $n$ but I get nowhere any hint? thank you!

"Artin Theorem" Counterexample

Posted: 08 Aug 2021 08:20 PM PDT

In the Galois Course on Coursera "Artin's Theorem" (I could not find a reference so I don't know if this is standard terminology) is given:

Let $F$ be a field and $G$ a subgroup of $\text{Aut} (F)$ with $|G|=n$. Then $|F:F^G|=n$ and $G=\text{Gal}(F/F^G)$ where $F^G$ is the subfield fixed by $G$.

But consider the subgroup $H$ of $\text{Aut} (\mathbb{C})$ generated by the element which sends $z \mapsto -z$.

Then $|H|=2$, but $\mathbb{C}^H=\{0\} $ so $|\mathbb{C}:\mathbb{C} ^H|\neq 2$.

Where did I go wrong?

Probability of median of 3 values drawn from $U(0, 1)$ being less than $c$?

Posted: 08 Aug 2021 08:34 PM PDT

Draw $x_1, x_2, x_3$ from $U(0, 1)$ what is the probability that the median of $x_1, x_2, x_3$ is $< c$?

The approach that I had in mind is to find the CDF of the second order statistic (the median in this case), differentiate it to get the PDF, and then integrate from 0 to $c$ to get the probability. But is there a more clever approach given that there are only 3 values here?

One approach that I just thought of is to consider the complement, the probability that the median is greater than $c$. This means exactly 2 of $x_1, x_2, x_3$ are $\geq c$ or all 3 are greater than $c$.

The latter occurs with probability $\left(1 - c \right)^3$. The former occurs with probability $\binom{3}{2} \left(1 - c \right)^2 c$.

So the probability that median is less than $c$ should be $$ 1 - \left(1 - c\right)^3 - \binom{3}{2} \left(1 - c\right)^2 c $$ ?

How to show that the minimal polynomial and the characteristic polynomial has the same set of roots?

Posted: 08 Aug 2021 08:12 PM PDT

my thought is that the we can use minimal polynomial to write down the rational canonical form of the matrix and then we can use this canonical form to write down the characteristic polynomial and then we deduce that the minimal polynomial and the characteristic polynomial share the same set of roots

can someone help explain the t interval in this series

Posted: 08 Aug 2021 08:11 PM PDT

look here

Why do I often find the interval t in Euler's polynomial is often different, some say |t|<π, some say |t|<2π. Why is that, at first I thought |t| this has no effect in the series since this Euler series departs from 1/cosh(t). Can anyone explain this?

https://www.google.com/url?sa=t&source=web&rct=j&url=https://ijpam.eu/contents/2012-78-1/5/5.pdf&ved=2ahUKEwirlYjO-KLyAhVfILcAHRIDBdoQFnoECAMQAg&usg=AOvVaw1I4JMEZmXWbxyYUNYSSyPv

Condition of existence of bounded solution of partial differential equation

Posted: 08 Aug 2021 08:10 PM PDT

Given the following partial differential equation \begin{align} \left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma-1}{r} \sin\psi \frac{\partial}{\partial\psi} + \frac{\beta}{2} \frac{\partial^2}{\partial\psi^2} \right) T(r, \psi) = -1. \end{align} $T(r, \psi): [0,1]\times[0,2\pi] \to \mathbb{R}^+$ is a real function defined on the domain of unit disk with $r$ and $\psi$ being, respectively, the radial and polar angle coordinates. $\beta$ and $\gamma$ are real positive parameters. The boundary condition is Dirichlet at the edge of unit disk, i.e., $T(r, \psi)|_{r=1} = 0$. And $T(r,\psi)$ is periodic with respect to polar angle $\psi$, i.e., $T(r, \psi+2\pi) = T(r, \psi)$.

The question is what is the condition of $\gamma$ and $\beta$ to have bounded solution of $T(r, \phi)$ with $T(r,\phi)|_{r=0} < \infty$.

difficult determinant problem

Posted: 08 Aug 2021 08:03 PM PDT

I am having difficulty with finding the determinant of this matrix:

\begin{pmatrix} x_1y_1&1+x_1y_2&...&1+x_1y_n\\ 1+x_2y_1&x_2y_2&...&1+x_2y_n\\ ...&...&...&...\\ 1+x_ny_1&1+x_ny_2&...&x_ny_n\\ \end{pmatrix}

I considered a related matrix \begin{pmatrix} 1+x_1y_1&1+x_1y_2&...&1+x_1y_n\\ 1+x_2y_1&1+x_2y_2&...&1+x_2y_n\\ ...&...&...&...\\ 1+x_ny_1&1+x_ny_2&...&1+x_ny_n\\ \end{pmatrix} and I was able to show that the determinant of this matrix is 0. Then I tried to see how this can be applied to find the determinant of the first matrix, but I haven't found any connection.

Any help is appreciated.

Find the genus for these compact surfaces

Posted: 08 Aug 2021 08:02 PM PDT

Why the number of the handles are 3 and 5 respectively?

I can't find the genus of the two surfaces. Is there any specific method for finding the genus?

enter image description here

enter image description here

P.s) Addendum for the second figure.

The second case is the there are 6 holes for each surface of the cube.

Plus this holes penetrated the center of the cube.

My attempt was to find using polar coordinates but it not worked.. Help me!

Posted: 08 Aug 2021 07:52 PM PDT

$\mathtt{Prove\:\:that\:\:,if\:\:axes\:\:be\:\:oblique\:\:and\:\:inclined\:\:at\:\:an\:\:angle\:\:\omega\:\:, the\:\:equation\:\:of\:\:normal\:\:to\:\:y=f(x)\:\:at\:\:(x,y)\:\:is\\(Y-y)\bigg(cos(\omega)+\dfrac{dy}{dx}\bigg)+(X-x)\bigg(1+cos(\omega)\dfrac{dy}{dx}\bigg)=0}$

Euler's partition method. How does someone use it?

Posted: 08 Aug 2021 07:52 PM PDT

I came across partitions recently and am not very much informed about it but I have a question regarding Euler's method for this. I came to know about this formula from a YouTube video so, it may not be the full equation.

Symbols: P(n) -> Partition of n
$\pi$ -> Product
n -> Any number

$\sum_{}^\infty (P(n)*x^n) = \prod\limits_{k=1}^\infty (\frac{1}{1-x^k})$

I can't understand:

  1. What does 'x' mean in this expression
  2. How can someone calculate partition with this.

The video I learnt this from is this.



Note: I am not informed regarding this subject so, I may have the question all wrong. I only want to know how this works. Today, I don't think this formula is used today after Ramanujan's work. I just want to know how the people before used it.

Find Exact Solution of an equation involving e^x

Posted: 08 Aug 2021 07:47 PM PDT

I simplified an equation down to this:

e^x(x+1) = 1/2

Then I am perplexed on how I can get an exact solution out of this. I can see graphically (split it on either side) and see where they cross which is around -0.3 or so.

I thought to use the identity Ln(AB)=Ln(A)+Ln(B) to help, but it won't do much as I will still have x and Ln(x) in the equation: x+Ln(x+1)=Ln(1/2) Which is still not intuitive. (if I apply Lawn of differences to this it will take me right back to where I started of course!)

Any cool tricks I can apply to help find an exact solution. Much appreciated.

A special Matrix decomposition

Posted: 08 Aug 2021 07:41 PM PDT

Let $W\in \mathbb{R}^{n \times n}$ is a matrix.

I want to factorize $W=A^T A$. $$W$$ has got zero trace and it has got equal number of positive and negative eigen values.

Can we do such a factorisation

Find the volume of the region which lies inside both $x^2+y^2=r^2$ and $y^2+z^2=r^2$

Posted: 08 Aug 2021 07:57 PM PDT

Find the region inside two cylinders $x^2+z^2 \le r^2$ and $y^2+z^2 \le r^2$.
I attempted this question by combining the two inequalities $x^2+2z^2+y^2 \le 2r^2$ and I got the bounds for $z$ which is $-\sqrt{{(2r^2-x^2-y^2)}/{2}} \le z \le \sqrt{{(2r^2-x^2-y^2)}/{2}}$.
And I found the bound on $y$ to be $-\sqrt{(2r^2-x^2)} \le y \le \sqrt{(2r^2-x^2)}$, the bound on x is $0 \le x \le r$.
I am not sure if the above attempt is correct, if not, can anyone provide the correct way to find the bounds on z and y?

Random walks and generating functions confusion/ seems like a contradiction. In search of intuition.

Posted: 08 Aug 2021 08:37 PM PDT

Let $0<p=1-q,r<1$ be fixed probabilities, and $X_i,i=1,2,\dots$ be i.i.d. random variables with the following distribution: $$ X_i= \begin{cases} 1,&\text{with probability }rp,\\ 0,&\text{with probability }1-r,\\ -1,&\text{with probability }rq. \end{cases} $$ Define $$ S_0=0,\quad S_n=\sum_{i=1}^nX_i\quad(n>0). $$ This is a random walk that only moves with probability $r$, and stays put with probability $1-r$ at each step. Define the hitting time of level $1$ as usual: $$ \tau^+:=\inf\{n>0\:S_n=1\}. $$

Prove, for $0<s<1$, $ P^+(s)=\frac{1-(1-r)s-\sqrt{((1-r)s-1)^2-4r^2pqs^2}}{2rqs}. (1) $

Give a probabilistic interpretation of the form of $\eqref{eq1}$ of $P^+(s)$ of $\tau^+$ using the Geometric$(r)$ generating function $$ Q(s)=\frac{rs}{1-(1-r)s}. $$

I have successfully proved equation $\eqref{eq1}$, but I am stuck for the second part. To me, I would expect that if we set $q=0$, that would basically turn $\tau^+$ into a Geometric$(r)$ variable, since it stays at level $0$ with probability $1-r$, and it hits level $1$ with probability $r$, and it is the waiting time until we hit level $1$. However, when I substitute $q=0,p=1$ into my expression for $P^+(s)$, it does not return what I want it to. It seems like the reasonable way to geto to $Q(s)$ from $P^+(s)$ is to actually substitute in $q=1,p=0$, and then to take the reciprocal of this function. However, this does not make any intuitive sense to me, and I can't figure out what this would represent in terms of the random walk hitting level $1$. Why would we take the reciprocal? Why would we set it so that the random walk can only go down, rather than setting it so that it can only go up? Can anybody make sense of this? It seems like a contradiction between the two generating functions. Can anybody motivate why I would substitute in $q=1,p=0$ and then take the reciprocal? The reciprocal part seems especially confusing to me, because $$ \mathbb{E}\left(\frac1{s^{\tau^+}}\right)=\mathbb{E}(s^{-\tau^+}) $$ would seem to suggest to me that reciprocating the generating function is possibly related to considering the negative of that random variable, but how could we have the negative of a waiting time? Any help would be sincerely appreciated!

Solving 3rd degree inequality

Posted: 08 Aug 2021 08:32 PM PDT

I'm working on a CS problem and I'm trying to lower the problems complexity from $O(n)$ to $O(1)$. After some calculations the problem comes down to solving a 3rd degree Polynomial. The problem I ran into, and the reason I'm asking for help is that I can't factor it to solve for $n$. It looks like this :

$$n(n+1)(2n+1)\le k$$

$n,k$ both positive integers. $k$ is a constant, it will be known at compile time. I am trying to find the biggest $n$ that satisfies this inequality. I know its a high school problem and it's embarrassing but I spent too much time on it and I need to move on.

Correct verb for modulo function?

Posted: 08 Aug 2021 08:02 PM PDT

I found a similar question asked about a decade ago in the English stack exchange, but there were no definitive answers, so I hope this is ok to ask here, where people may have more relevant experience.

When writing, what is the correct verb form for a modulo function?

For example: "function f(x) takes a logarithm of base 5, multiplies by 2, adds 3, subtracts 5, and.... modulos by 9(???)"; or "we then modulo 10 by 5"

The best I can come up with is "performs a modulo function with quotient 9", but I don't know if that's correct, and it sounds like there should be a simpler word or phrase.

On continuity of an improper integral

Posted: 08 Aug 2021 07:55 PM PDT

I am stuck in the middle of a problem which requires to prove that the following function defined on $[0,1]$ is continuous at $x = 1^-$.

For $0\leq x \leq 1 $, define: $$ R(x) = \int\limits_0^1 \left(\frac{2}{\sqrt{1-xt^2}\sqrt{1-t^2}}-\frac{x}{1-xt}\right)dt.$$

Then show that $R(x)$ is continuous at $x=1^{-}$, that is $$\lim\limits_{x\to 1^-}R(x) = R(1) = \int\limits_{0}^{1}\left(\frac{2}{1-t^2}-\frac{1}{1-t}\right) = \int\limits_{0}^{1} \frac{dt}{1+t} = \ln 2.$$

My all attempts till now were futile. My most rigorous attempt was to first consider $$R(x) - R(1) = \int\limits_{0}^{1} \left(\frac{2}{\sqrt{1-xt^2}\sqrt{1-t^2}}-\frac{x}{1-xt} - \frac{1}{1+t} \right)dt = \int\limits_{0}^{1} \left(\frac{2}{\sqrt{1-xt^2}\sqrt{1-t^2}}-\frac{(1+x)\sqrt{1-t}}{(1-xt)\sqrt{1+t}\sqrt{1-t2}} \right)dt,$$ and then try to express the function inside the integral as product of two functions $g(t) F(x,t)$, such that $\int\limits_{0}^{1} g(t)dt$ converges and $F(x,t)$ is continuous on $[0,1] \times [0,1]$. I chose $$g(t) = \frac{1}{\sqrt[4]{(1-xt)^3}}$$ and $$F(x,t) = \frac{2\sqrt[4]{1-t^2}}{\sqrt{1-xt^2}} - \frac{(1+x)\sqrt[4]{1-t^2}\sqrt{1-t}}{(1-xt)\sqrt{1+t}}.$$ Unfortunately the above $F(x,t)$ is not continuous at $(1,1)$. For whatever $F(x,t)$ and $g(t)$ I am choosing, I am failing either to show that $F(x,t)$ is continuous or $g(t)$ converges.

The main result I am aiming at is: $$ 2K(\sqrt{x}) \sim -\log(1-x) + C,$$ for some constant $C$. And $K(x)$ is the elliptic integral of first kind.

Any help is highly appreciated.

Minimum number of handshakes in the party

Posted: 08 Aug 2021 07:50 PM PDT

In a party of $n$ people, certain pairs of people shake hands with each other. In any group of $k$ people, there exists at least one person who shakes hands with all other persons in that group. What is the minimum number of handshakes that can take place at the party?

The problem is inspired from a contest question which I want to solve in general. Here are my attempts in solving the problem:

There are $\binom n k$ possible groups. In each group, the minimum number of handshakes is $k-1$. So, at first I thought the answer should be $(k-1)\binom n k$. But I realized that the maximum number of handshakes is $\binom n 2$. So, I was highly over counting.

Then, I tried for small cases. For $n=4$ and $k=3$, we have $4$ people $A,B,C,D$. And there are $4$ groups of $3$. We take the first group $(A,B,C)$ where $A$ shakes hands with $B$ and $C$. Then in the groups $(A,B,D)$ and $(A,C,D)$, $B$ and $D$ shake hands with $D$. So, we have a total of $4$ handshakes as minimum in this case. From here, I think it's not easy to find the minimum for even small cases.

So, how to solve the problem?

To prove Inequality in Hausdorff metric [closed]

Posted: 08 Aug 2021 08:22 PM PDT

on which conditions of $\beta$ the following inequality holds
$\max\limits_{x\in A}(\min\limits_{y\in B}\beta (d(x,y))) \leq \beta (\max\limits_{x\in A}(\min\limits_{y\in B} d(x,y)))$,
Where $\beta:\mathbb R^{+} \to [0,1) $ and $\beta (t_n) \to 1$ implies $ t_n \to 0 $ as n tends to infinity , and A and B are compact subsets of metric space.

How to strengthen $ h(2h(x))=h(x)+x $ to force $h$ to be linear?

Posted: 08 Aug 2021 07:36 PM PDT

Let $h:\mathbb{R}\to\mathbb{R}$ be an injective function such that $$ h(2h(x))=h(x)+x $$ for all $x\in\mathbb{R}$, and $h(0)=0$. What would be an as mild as possible extra condition such that the only solution of the equation $h(x)=-x/2$ is $x=0$? Note that the function $x\mapsto -x/2$ satisfies the conditions, so somehow we need to fully exclude this function.

For example, both the extra condition $h(x)\leq x\ \forall x$ as well as the extra condition $h(x)\geq x\ \forall x$ force $h(x)=x\ \forall x$, and thus $0$ is the only solution to the equation in question. But what about other conditions as e.g. monotone or surjective?

Finding the discriminant of a quadratic equation from the given information on the roots of a quadratic equation

Posted: 08 Aug 2021 08:02 PM PDT

I recently came accross an old question that I solved during my school days. Which is

If $\alpha, \beta$ are two real roots of a quadratic equation $ ax^2+bx+c=0 $ and $\alpha+\beta, \alpha^2+\beta^2, \alpha^3+\beta^3$ are in GP, then which of the following is correct?

a) $\Delta\neq0$

b) $b\Delta=0$

c) $c\Delta=0$

d) $\Delta=0$

After seeing the question, I immediately realised that $\alpha+\beta=\frac{-b}{a}$ and $\alpha\beta=\frac{c}{a}$.

And since $\alpha+\beta, \alpha^2+\beta^2, \alpha^3+\beta^3$ are in GP, I got the following equation, I wrote them as $(\alpha^2+\beta^2)^2=(\alpha+\beta)(\alpha^3+\beta^3)$ -> call eqn $i$.

I rewrote the above equation as $$[(\alpha+\beta)^2-2\alpha\beta]^2=(\alpha+\beta)[(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)]$$

I combined the above two and simplified further which resulted in $ac(b^2-4ac)=0$. And since $a$ can not be zero and $\Delta=b^2-4ac$, I concluded that $c\Delta=0$ and option c is correct.

However I later realised that eqn $i$ can expanded as follows,

$$\alpha^4+\beta^4+2\alpha^2\beta^2=\alpha^4+\beta^4+\alpha\beta^3+\beta\alpha^3$$.

Which will ultimately result in saying that $(\alpha-\beta)^2=0$ and therefore $\alpha=\beta$. If the roots are equal, the discriminant ($\Delta$) has to be zero which means option d is more correct. But most online websites only marked option c as the correct answer.

So which one is the correct answer really? Are they both correct? Or Am I missing something here?

How does the surface area of a rigid square piece of a metal traced on a sphere compare with the area of that piece of metal?

Posted: 08 Aug 2021 08:05 PM PDT

My main question is in Scenario 2. I don't know how to answer this main question.
Scenario 1 asks a similar question that I am able to answer, that led me to think of my main question.

Scenarios 3 and 4 ask similar questions to my main question, that I attempted to try to help me answer my main question.

Scenario One -- laying down a flexible piece of paper onto an sphere, and tracing the resulting shape

Suppose I have a 1cm x 1cm piece of paper, and a sphere that is somewhat larger (for example, a sphere about the size of an orange).

Suppose I lay the square piece of paper down on the orange, and I flatten it as much as I can. It is impossible to do this in a way where all of the piece of paper will touch the orange; some parts might be crumpled "into the air" where air is in between the paper and the orange; and some parts might be lying close to the orange, but paper is in between that part and the orange.

Suppose I then use a maker to trace a shape onto the orange, by tracing around where the edges the piece of paper touch the orange.

  • A question: how does the surface area of this traced-out shape compare with the area of the square piece of paper?
  • My answer: Intuitively, I can visualize that the surface area of the shape traced onto the orange will be less than the area of the square piece of paper, because only some points of paper are directly touching the orange; there are other "extra" points of paper that are in the air or lying on top of other points of paper. These "extra" points of paper are not contributing to the shape that is traced out, therefore the shape that is traced out must be less than the area of the square piece of paper.

Scenario Two -- rolling a rigid piece of metal onto a sphere, and tracing the resulting shape

Suppose, now, that I have a rigid square piece of material (such as a piece of metal) that cannot be bent, that is 1cm x 1cm, and a sphere that is about the size of an orange.

Suppose I use this piece of metal to trace a shape onto the orange in the following way:

  • I start by placing the centre point of the square onto the orange, such that it's the only point touching the orange.
  • Then I smoothly roll the square piece of metal along the orange, making a line from the centre of the square piece of metal, to the midpoint of one the edges of the piece of metal. I then mark a dot onto the orange, with a marker, where the midpoint of the edge of the square touches the orange. I then undo this rolling motion, so that the centre point square piece of metal is once again touching the orange.
  • Similarly, I mark the midpoints of the other three edges of the square (ie, starting from the centre, and rolling it to those midpoints). And similarly, I mark the corners of the square. And then, I mark many, many other points along the edge of the square.
  • Finally, I can join all the points marked with some sort of smooth (possibly curved?) line.

Main Question: A shape now is "traced" onto the orange. Is its surface area less than, equal, or greater than 1 square centimetre?

I have trouble even making a guess! (I welcome any observations that may help me visualize certain properties of this scenario, even if they don't provide a mathematically rigorous explanation. I especially welcome explanations that use high-school (or earlier) level mathematics.)

Indeed, I can't even make a guess about what shape is traced! Are the corners of the traced shaped right angles? If looking directly down at the shape (ie from a bird's-eye-view, as if your eye, the centre of the shape, and the centre of the sphere, made a straight line), would the shape look like a square? I can't visualize this!

Similar Questions I asked myself, as attempts to gain an understanding of the Main Question

Scenario Three -- tracing a rigid square onto a cylinder

Suppose a trace a shape onto a cylinder in the following (obvious) way:

  • I lay a cylinder flat on a table (ie, so one of it's circular faces is resting on the table).
  • I then place one edge of the square so that it is perpendicular to the table, against the side of the cylinder, and I draw a line along that edge, with marker, onto the cylinder.
  • Then I smoothly roll the square towards the other vertical edge of the square. As I smoothly roll the square, the horizontal edges of the square will "roll along" the cylinder. As they do so, I mark the places where they contact the cylinder.
  • Finally, I draw a vertical line onto the cylinder, when I reach the square's other vertical edge.

I am somewhat confident that the surface area of the resulting shape would equal one square centimetre:

  • It seems that (as I'm rolling the square along the cylinder) if I take any two points on the square that are vertical to each other, the distance between these two points on the square will be equal to the distance between the points where they each touch the cylinder (as the square is being rolled onto the cylinder).
  • Also, suppose I consider any two points that are horizontal to each other on the square, and I mark where these points touch the cylinder (as the square is being rolled), using purple ink on the cylinder. If I measure the distance between the two points on the square, and if I measure the distance an ant would walk along the curved surface of the cylinder between the two points in purple ink that I marked on the cylinder, my guess is that these two distances would be the same.

Considering this, it feels like all points of the square will be "placed" onto the cylinder's surface without any distortion/warping/stretching. That's why I am making a guess that the surface area of the resulting shape traced onto the cylinder will equal the surface area of the square piece of metal.

Scenario 4 -- Tracing a square onto a cone

I am not confident in my guess here. My guess is that the shape traced onto the curved part of the cone will have a surface area equal to the area of the square.

However, I have trouble even visualizing the tracing process.

Suppose I start with the cone resting on the table (ie, with the circular face of the cone flat on the table). I note that I can choose a line on the curved surface of the cone, that runs from the tip of the cone down to any point on the edge of its circular face. I then start by placing the square onto the cone, such that the midpoint of two opposite sides of the square align with the "line" on the cone that I chose earlier. I can then do an obvious "rolling" motion of the square, once going right, and once going left, to trace out a shape onto the cylinder.

But I am unsure if I can visualize the shape created:

  • For example, when rolling the square, starting with the starting placement described above, and then rolling right, there will be a line traced out (onto the cone) by the upper horizontal edge of the square, and a line traced out by the lower horizontal edge of the square. Will these lines that are traced out be parallel to the table? My intuition is saying "no"; my guess is that that the lines will slant downwards towards the table, as you roll the square from the middle of the square to its right edge .. but I'm not sure.
  • Similarly, the vertical(ish) line created by the vertical right edge of the square .. will it be "vertical" (ie, aligned with a line that runs from the tip of the cone, down to a point the edge of the cone's circular face)? I cannot visualize this well enough to even make a guess.

Fourier transform interpretation

Posted: 08 Aug 2021 08:39 PM PDT

There's an intuitive interpretation of the inverse Fourier Transform with a system of connected wheels spinning in the complex plane. Then the signal (a real one, like usual sine for example) is the the real part of position of a point attached to the system in time. This is pretty easily explained in below article.

https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/#From_Smoothie_to_Recipe

I think I understood the concept quite well, however I puzzle over some symmetric intuitive interpretion solely for Fourier Transform (FT) not the inverse. In other words: I know how to reproduce a signal knowing it's FT, but I struggling to imagine how a particular signal can be decomposed into its compononets.

To further explain I will stick to the below convention:

$$f(t) = \frac{1}{\sqrt{2\pi}} \int^{\infty}_{-\infty} F(\omega) e^{i \omega t} \mathrm{d}\omega $$ $$F(\omega) = \frac{1}{\sqrt{2\pi}} \int^{\infty}_{-\infty} f(t) e^{-i \omega t} \mathrm{d}t$$ where $f(t)$ is a signal and $F(\omega)$ its FT.

If we take a look at the equations we can see, that the only difference is the sign in the exponent. It suggests, that one should think of it as spinning backwards through the wheels system with each point of a particular signal. However, now we integrate over frequency. Choosing an arbitrary $\omega$ we can think of moving points with the angular speed of $\omega$. My idea is that if chosen $\omega$ matches a frequency of some compononent of the considered signal, then all the contribution to the integral value comes from that compononent, the not-mathing frequencies somehow cancel.

Let now consider a sine wave. Following the above idea we choose the matching frequency. We know the integral does not exist as it oscilates between -1 and 1 as sine itself. At least if we don't divide it into its positive and negative part. Then we get two delta functions as we should - a negative and a positive one.

There's where my questions arise.

Why is it like that? Why the rest of a signal cancels when integrating at an arbitrary frequency? What about for example constant wave then? Maybe I should look at such a sine wave as a corresponding complex pulse? Should we look differently at odd and even functions or decompose into positive and negative parts?

That bothers me for so long I eventually decided to search here for help.

I would appreciate some deeper explanation as it stays unclear to me. It would be great if it would follow my presented idea.

I've also asked this question one of my professors. He gave me a different approach with matrix diagonalization and spectral theorem, if that's a better way to understand it I would be also grateful for hints or explanation.

Also, maybe you can point me to an appropriate book or paper regarding this topic?

What is a "supplementary subspace"?

Posted: 08 Aug 2021 08:03 PM PDT

Let $Q$ be a quadratic form of vector space $V$ over a field $k$ with characteristic $\neq 2 $, $V^{0}$ be its orthogonal complement.

If $U$ is a supplementary subspace of $V^0$ in $V$, then $V = U \oplus V^0$ .

What does supplementaty subspace mean in above proposition? This is s a proposition in Serre's A Course in Arithmetic, and it only says that it's clear to prove, but never defines what a supplementary subspace is.

Probability - Tales Game

Posted: 08 Aug 2021 08:41 PM PDT

A dice game played by two players is like this: each player throw two dice and sum theier results; that is the number of points the player scored. Whoever scores more, wins.

One additional detail is that if the numbers of both dices of a player are equal, the player can roll the two dice again and the sum of these points will be added to the previous sum - and so on, indefinitly.

  A) A player has k points. Calculate his probability of victory.    B) A group of friends decided to play the same game with n players.         Find the winning probability for a player who scored k points.

I've tried for some time to do this, but it seems impossible to me. I don't know much of this kind of probability. Does anyone know a way to solve this?
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