Saturday, May 22, 2021

Recent Questions - Mathematics Stack Exchange

Recent Questions - Mathematics Stack Exchange

Projection of a Circle in 3D onto the $xy$-Plane

Posted: 22 May 2021 07:49 PM PDT

As shown in the figure, I want to determine the equation of a the ellipse formed by projecting a circle that lies in the plane $z=Ax+By$ onto the $xy$-plane. Projected Circle

PDE‘s stability

Posted: 22 May 2021 07:48 PM PDT

I am struggling with the following PDE's stability, my intuitive is to use the semigroup theory, but it seems that I can not even compute the eigenvalues, any other idea?

$$w_t=-aw-b^2\int_o^{2\pi}sin\theta B(t,\theta)d\theta \\ B_t=-w^*B_{\theta}-B^*_\theta w+\lambda B$$

Here a,b are constant and w is a function of t, B is a function of t and $\theta$, $w^*$ and $B^*_\theta$ are known functions.

Finding a non-piecewise function that maps all $\mathbb{Z}/{0}$ to $0$ and $0$ to $1$

Posted: 22 May 2021 07:47 PM PDT

I am trying to devise a function that maps $\mathbb{Z}/\{0\}$ to $0$ and maps $0$ to $1$ that is not a piecewise function.

If $\mathbb{Z}/\{0\}$ to $0$ and maps $0$ to $1$ is not possible then a function $\mathbb{N}/\{0\}$ to $0$ and maps $0$ to $1$ could possibly work too.

Any help is much appreciated (I am really stuck on this one), thanks in advance for the help!

(Context: This is for a program I am trying to write without if statements)

No Hamiltonian Cycle

Posted: 22 May 2021 07:39 PM PDT

I want to prove that a simple graph that is 3-regular and has edge chromatic index 4 does not have a Hamiltonian cycle.

After some research, I have found that the Petersen graph fits the criteria and can be seen as an example of a 3-regular, 4-edge-chromatic, simple graph that does not have a Hamiltonian cycle. However, I am having trouble generalizing the proof without using the Petersen graph. Any thoughts?

linear divided by square root of quadratic

Posted: 22 May 2021 07:34 PM PDT

I want to find the $x\in\mathbb{R}$ which maximizes $$f(x)=\frac{a+bx}{\sqrt{(x-c)^2+d^2}}$$ for given real numbers $a,b,c,d\neq0$.

Since this is a differentiable function, I do this by solving for $f'(x)=0$, which yields a single extremum $\displaystyle x_0=c+\frac{bd^2}{a+bc}$ (or none if $a+bc=0$). And $\displaystyle f''(x_0)=-\frac{d^2((a+bc)^2+b^2d^2)}{(a+bc)((x-c)^2+d^2)^{5/2}}$, so whether it is a maximum or minimum depends solely on whether $a+bc>0$ or $a+bc<0$.

While the computational part of this is pretty clear to me (and I've verified the computations in Maple), I'm struggling with the intuition. What is the interpretation of $a+bc$ here, and why is it so important that its sign determines the direction of the curve, or even makes $f$ monotonous exactly when $a+bc=0$? I feel like there must be some geometric/parabola interpretation here, but I fail to see it.

Closed Form for p>3 for Ratios of odd/even sums, from a Stand-Up Math video

Posted: 22 May 2021 07:25 PM PDT

I was watching this youtube vid by Stand-up Maths and it got me thinking about a general form for higher degrees. After a little dinking around I found them for degree 2 and 3. I am wondering if expressions for higher order terms is known? The video is [YouTube] https://youtu.be/E5-pgBnGyzw.

The equation I am using is: $$ \frac{\sum_{n=0}^{\infty}(2n+1)^p}{\sum_{n=0}^{\infty}(2n)^p} $$ and have found: $$ p=2,\ \ \ \frac{2n+3}{2n} $$ $$ p=3,\ \ \ \frac{2n^2+4n+1}{2n^2} $$ The distance between 2 points given by a parallel line connected to the minimums of the 2 polynomials for $p=3$ was a nice surprise. The distance for $p=2$ is $\frac{3\sqrt{5}}{5}$.

p=3 Graph

I would guess the $p=4...m$ will have equations of degree $m+1$. Thanks in advance for the input.

Open and Closed Sets Over Alexandroff Space

Posted: 22 May 2021 07:45 PM PDT

I am trying to proof this theorem. I already proved 1 and 2 but I'm struggling to prove 3.

Theorem 7

  1. For any two different elements $P$ and $Q$ of an Alexandroff space is $$ \begin{array}{l} P \in O Q \quad \Longrightarrow \quad Q \notin O P \quad \text { and } \\ P \in C Q \quad \Longrightarrow Q \notin C P . \end{array} $$
  2. $P \in C Q \quad \Longleftrightarrow Q \in O P$.
  3. $C P \subseteq C Q \quad \Longleftrightarrow \quad O Q \subseteq O P$

Where $OQ$ is the maximal neighbourhood of $Q$ and $OP$ the maximal neighbourhood of $P$ and $CP$ is the smallest closed set containing $P$. The same for $CQ$ respectively.

My attempt:

Suppose that $C P \subseteq C Q$. As both are the intersection of the closed sets that contain $P$ and $Q$ respectively, then both $P$ and $Q$ are in $CQ$. By 1, we know that if $P \in CQ$ then $Q \notin C P$. That is, $Q \in X - CP$, which is an open set that contains $Q$ but do not contain $P$. Because of that, $P \notin OQ$.

That is what I thought it will be useful, but I can't find a way to get to the conclusion. Thanks for reading. Any help will be appreciated. :)

Fourier: $f(x) = x$ with $0 \leq x \leq 2\pi$ (Exercise almost solved)

Posted: 22 May 2021 07:43 PM PDT

Question: fourier series is: $f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inx}$, where $c_n = \frac{1}{2\pi}\int_{0}^{2\pi}f(x)e^{inx}dx$

Find the fourier series for $f(x) = x$ in $0 \leq x \leq 2\pi$. Manipulate the answer to get:

$$f(x) = \pi + \sum_{n = 1}^{\infty}\frac{1}{n}\sin(nx).$$

Consider the case with $n = 0$.

My answer

I used $e^{inx} = \cos(nx) + i\sin(nx)$ and I calculated $c_n$, I found: $\frac{1}{2\pi} ( \int_{0}^{2\pi}x\cos(nx)dx + i\int_{0}^{2\pi}x\sin(nx)dx).$

But, $x\cos(nx)$ it's odd, so is equal to 0.

I calculated the integral of the sin and arrived at: $-i*1/n$.

So

\begin{align} f(x) &= \sum_{n=-\infty}^{\infty} c_n e^{inx}\\ & = \sum_{n=-\infty}^{\infty} -\frac{i}{n} e^{inx}\\ &= \sum_{n=-\infty}^{\infty} -\frac{i}{n}(\cos(nx) + i\sin(nx)\\ & = \sum_{n=-\infty}^{\infty} -\frac{i}{n}\cos(nx) +\sum_{n=-\infty}^{\infty}\frac{\sin(nx)}{n}. \end{align}

But I am not managing to simplify for:

$$f(x) = \pi + \sum_{n = 1}^{\infty}\frac{1}{n}\sin(nx).$$

Can anyone help me with this last step?

There should be possibly infinite ordered triplets for the following question.

Posted: 22 May 2021 07:53 PM PDT

If L.C.M of three natural numbers $a, b, c$ is $p^2q^2r^2$(where $p, q, r$ are different prime numbers) such that sum of all possible triplets (a,b,c) are given by $m^n$ (where $m$ and $n$ are prime), then find the value of ($m-4n$)

This is how solved.
As the L.C.M of the given numbers is $p^2q^2r^2$, this means that none of the numbers a,b,c can have $p^0 orp^1,q^0orq^1,r^0 or r^1$ as their factors.
But they can have $p,q,r$ as their factors in powers of two or higher.
For example if $a=p^2q^5r^7$,$b=p^7q^2r^3$ and $c=p^9q^8r^2$ then the L.C.M of a,b,c is $p^2q^2r^2$.
Hence there should be infinite solution i.e. infinite ordered triplets.
But the author has provided with the following solution:
L.C.M. $p^2$ can be obtained from the numbers a, b, c in the following combinations. $$p^2 p^2 p^2 ⇒ 1 arrangement$$
$$p^2 p^1 p^2 ⇒ 3 arrangement$$ $$p^2 p^1 p^1 ⇒ 1 arrangement$$ $$p^2 p^2 p^0 ⇒ 3 arrangement$$ $$p^2 p^1 p^0 ⇒ 6 arrangement$$ $$p^2 p^0 p^0 ⇒ 3 arrangement$$ $$———————————––— 19 arrangement$$

So if L.C.M. is $p^2q^2r^2$ total combinations are = $19^3$ = $m^n$ ⇒ m – 4n = 7.
This is where I am confused and unable to comprehend as why the solution is provided in such a way, because in the above arrangements L.C.M will not be $p^2q^2r^2$.

Counter-example to intersection of maximal ideals

Posted: 22 May 2021 07:48 PM PDT

I was reading Ulrich Gortz's Algebraic geometry and saw the following result:

uhdbf

I know the first equality is true for any commutative rings.

My question is: Is it true that if $A$ is NOT a finitely generated $K-$algebra, for a field $K$, then, $$rad\, \mathfrak{a}\not= \bigcap_{\substack{\mathfrak{a}\subseteq\mathfrak{m}\subset A \\ \text{maximal ideals}}} \mathfrak{m}$$ If yes, then what is a counter-example?

I feel like this has something to do with violating Hilbert nullstellensatz.

Is taking the integral from 1 to 3 of any degree 2 polynomial injective and surjective?

Posted: 22 May 2021 07:49 PM PDT

enter image description here

I really have no idea where to start with this one.

How does a vector-valued functions inherit properties from its components?

Posted: 22 May 2021 07:14 PM PDT

Let $f: \mathbb{R}^n \to \mathbb{R}^m$ be a vector-valued function.

My question is: how does $f$ inherit its property from its components? and if a property holds for all $f_i$, does that necessarily mean that $f$ satisfies said property as well?

For example: is it true that if all $f_i$, $i \in \{1, \ldots, n\}$, the components of $f = (f_i)$ satisfies property $P$, then $f$ satisfies property $P$ as well?

If every $f_i$ is differentiable, then $f$ is differentiable.

If every $f_i$ is continuous, then $f$ is continuous.

If every $f_i$ is Lipschitz, then $f$ is Lipschitz.

Of course, taking the intersection of the domains for which properties hold into account.

Also for properties that makes sense for vector-valued functions (this excludes things like convexity)

Does this always hold?

What are some good books I can use to learn Algebra 1?

Posted: 22 May 2021 07:42 PM PDT

I would like some suggestions on some good books to learn Algebra 1. This is my first time learning the topic but I already know some of the topics (ex. Linear Equations and Real Numbers). Does anyone have any suggestions? Thank you.

Edit: By Algebra 1 I mean the subject that's taught in high school usually in either 8th or 9th grade.

Choosing an element of $W_x$ in a computable way

Posted: 22 May 2021 07:13 PM PDT

If $W_x$ denotes the domain of the program with number $x$, the question is:

Is there a partial computable function $f$ such that if $W_x$ is not empty then $f(x)\in W_x$, and otherwise, $f(x)$ is undefined?

My attempt is to take $f(x)=x$ if $x\in W_x$ and otherwise, $f$ is undefined. But I don't know how to deal with the non-empty condition.

sandwich theorem

Posted: 22 May 2021 07:44 PM PDT

Does sandwich theorem hold for divergent sequences? Given a divergent sequence $\{v_n\},$ can we find 2 sequences $\{u_n\}$ and $\{w_n\},$ such that $$\{u_n\} < \{v_n\} < \{w_n\},$$ and if $\{u_n\}$ and $\{w_n\}$ diverges to L = ±∞, then can we conclude $\{v_n\}$ also diverges to L = ±∞?

How do I calculate the following double integral and evaluate the limits of $\theta$?

Posted: 22 May 2021 07:49 PM PDT

Let R be a circular disc in $\Bbb R^2$ of a radius $a$ and center $(a,0)$. What is $$\int\int_R\ \frac{1}{x}dxdy$$

Used following equations:

$x^2 + y^2 = r^2\\ x = r \cos(\theta)\\ y = r \sin(\theta)$

The steps I followed :

$(x-a)^2 + (y)^2 = a^2\\ x^2 - 2xa + a^2 + y^2 = a^2\\ r^2 - 2(r\cos(\theta))a = 0 \\ r = 0 , ~ r = 2a\cos(\theta)$

$$\begin{align*}&= \int _{-\pi/2}^{\pi/2}\int _0^{2a\cos(\theta)}\frac{1}{r\cos(\theta)}~r~dr~d{\theta}\\& = \int _{-\pi/2}^{\pi/2} \frac{1}{\cos(\theta)}\int_0^{2a\cos(\theta)}dr~d\theta\\& =\int _{-\pi/2}^{\pi/2} 2a \space d\theta\\&= 2\pi a\end{align*}$$

I understand that for $r = 2a\cos(\theta),\theta$ goes between $0\to\pi/2$ for the upper part of the circle and between $\pi/2\to \pi$ for lower part of circle. As $r = -2a$, for $\theta = \pi $ so we discard $\pi$ and instead go between $0\to-\pi/2$ (For the lower part of the circle). Making the limits $ -\pi/2 \le\theta\le \pi/2$ . Needed a bit more elaboration on choice of $\theta$ and verification on my steps. If you guys could help out.

All About Fractions [closed]

Posted: 22 May 2021 07:28 PM PDT

$1.$ What is the informal (or intuitive) and formal definition of fraction and all kinds of fractions$?$

$2.$ Can we say that "a fraction is a number or expression of the form $\dfrac{a}{b}$ or $-\dfrac{a}{b}$ or ${\large c}\dfrac{{\small a}}{\small{b}}$ or $-{\large c}\dfrac{{\small a}}{{\small b}}$ or ${\small\circ}\dfrac{{\large a}_{\small 1}}{{\large b}_{\small 1}}\times{\small\circ}\dfrac{{\large a}_{\small 2}}{{\large b}_{\small 2}}\times{\small\circ}\dfrac{{\large a}_{\small 3}}{{\large b}_{\small 3}}\times\cdots\times{\small\circ}\dfrac{{\large a}_{\small n}}{{\large b}_{\small n}}$ where ${\large a},{\large b},{\large c},{\large a}_{\small 1},{\large a}_{\small 2},{\large a}_{\small 3},\dots,{\large a}_{\small n},{\large b}_{\small 1},{\large b}_{\small 2},{\large b}_{\small 3},\dots,{\large b}_{\small n}$ can be any number or expression but ${\large b},{\large b}_{\small 1},{\large b}_{\small 2},{\large b}_{\small 3},\dots,{\large b}_{\small n}$ can't be zero$,{\large c}\dfrac{{\small a}}{\small{b}}$ & $-{\large c}\dfrac{{\small a}}{{\small b}}$ are the forms of mixed fraction$,{\large n}\in{\mathbb{Z}}_{>0}$ and ${\small\circ}$ is the sign of any fraction of the form ${\small\circ}\dfrac{{\large a}_{\small n}}{{\large b}_{\small n}}$ which can be $+$ or $-"?$ Does fraction include the numbers like $-\dfrac{-6}{-3},\dfrac{1}{-2\sqrt{e}}, -\dfrac{-\sqrt{5}}{-3}, \dfrac{\pi}{e},$$\dfrac{-\pi e}{\sqrt{2}}$$,-\dfrac{-\sqrt[5]{99}}{-\sqrt[7]{4}},\dfrac{\varphi^3-\varphi^6+7}{-\ln e^3},$$\dfrac{1.73}{-1000},$$-\dfrac{-0.66}{1.45},$$\dfrac{6i}{6i},$$\dfrac{-2i}{0.91},$$\dfrac{0}{-1+2i},$$-\dfrac{0-\sqrt[11]{9}i}{-\sqrt{7}+8i},$$-\dfrac{4-\sqrt[3]{ei}}{-\sqrt{-\pi i^2}+2.5i},$$-400\%,\varphi\%,$$-1.67\times10^{-5},$$\dfrac{-1}{-2}\times-\dfrac{4}{-7}\times\dfrac{-5}{8},$$\left(\dfrac{6-9.8i}{-\sqrt{8}+3i}\right)^2,$${\large 3}\dfrac{{\small -3}}{{\small 1}},$$-{\large 9.81}\dfrac{{\small \pi}}{{\small i}},$${\large 2}\dfrac{{\small 5}}{{\small 3}}\times\dfrac{1}{4},$$\dfrac{7}{9}\times\left(6.67\times10^{-11}\right),$$\dfrac{1}{3}\times50\%$ and the expressions like ${\large x}\dfrac{\small{\sqrt{x^3}}}{\small{5x^2}},$$\dfrac{x^\pi}{y^{i\varphi}},$$\dfrac{2x^y}{3y^x},$$\sqrt[4]{\dfrac{1-x^2}{1+x^2}},$$\dfrac{x^5}{y^2}\times\dfrac{z^9}{x-21},$$\dfrac{{\large\int} \dfrac{1}{y^2}\: \mathrm{d}y}{404},$$\dfrac{\sin x}{\log x},$$\dfrac{| x |}{x},$$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$?$ Which kind of fractions do they belong to$?$

$3.$ How can we express the set of all fractions in set-builder notation and roster notation$?$

$4.$ Is there any special symbol which is used to denote the set of all fractions$?$ Can I use the symbol $\mathbb{F}$ to denote this set$?$

$5.$ What was the historical background of fraction$?$

Prove that $G$ is Eulerian if and only if every block of $G$ is Eulerian.

Posted: 22 May 2021 07:40 PM PDT

Prove that $G$ is Eulerian if and only if every block of $G$ is Eulerian.

Attempt:

If G is Eulerian, then G has a partition into cycles edge-disjoint. I don't know how to apply it, though.

If every block is Eulerian, then every vertex in $G$ which is the edge-disjoint union of its blocks has even degree.

How many ways can a total of 8 players be paired in a group of 5 such that no two players fall in the same group more than 2 times?

Posted: 22 May 2021 07:41 PM PDT

I want to know different ways by which n players can be paired in groups of m such that no two players fall in the same group more than two times.

Example:

Input: [1, 2, 3, 4, 5, 6, 7, 8]

Output: {1.2.3.4.5, 1.2.3.6.7, 1.4.5.6.7, 2.4.6.8, 2.5.7.8, 3.4.7.8, 3.5.6.8}

I was able to compute all the permutations possible. But I need to filter that out such that no two players fall in the same group more than 2 times.

Square and Quarter Circle

Posted: 22 May 2021 07:13 PM PDT

$ABCD$ is a square of side $18$ cm. $F$ is a point inside the square, such that $BCF$ forms an equilateral triangle. $CFA$ is a quarter circle with centre $B$. $E$ is the point on $AB$ such that the area of the region $EFC$ is half that of the area of the quarter circle.

Find the length of $AE$ (in cm).

enter image description here

Rigorous proof of time dilation (using only 1 spatial dimension).

Posted: 22 May 2021 07:44 PM PDT

$\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\mc}{\mathcal}$ $\newcommand{\R}{\mathbf R}$ $\newcommand{\ST}{\mathbf S}$

Introduction

The purpose of this post is under the theorem of time dilation in special relativity. Before even stating the theorem, I wanted to make precise the fundamentals so that one can write a rigorous proof. Following is my attempt to come up with an axiomatization of special relativity. I was not able to follow the thought experiments in physics books (though I was able to appreciate them). More precisely, I would not like to use trains and lightning bolts in my arguments since they are not precise mathematical objects.

Also, I have seen a purely mathematical treatment in Gregory Naber's Geometry of Minkowski Spacetime but I was not able to follow the motivation behind the postulation of a bilinear form of a certain index on spacetime. I would like to anchor thing to "speed of light is constant in all inertial frames" and, I believe, the following achieves this. Basically, I would like to have a "well-motivated" development of the subject. Of course, this is inherently a subjective matter. (Here is my previous attempt at formalizing special relativity).

The problem I am facing is mentioned in the last section. Briefly, I am stuck at proving time dilation without having at least two spatial dimensions. Further, the axiom that I right now can come up with to exploit the extra dimension is ugly. More details below.

I have not given the proofs of any of the theorems below for two reasons. One is that this post is already very long and the second is that the proofs of these theorems are routine given the definitions.

Lastly, I hope the motivations of the axioms and definitions are clear to the reader and I will make every effort to clarify any of them if needed.

Axiomatization

We postulate that spacetime is a $2$-dimensional oriented affine space, which we denote as $\ST$, and whose elements are called events, equipped with a collection $\mc P$ of $1$-dimensional affine subspaces, whose elements are called photons.

A particle is any $1$-dimensional affine subspace of $\ST$. Thus each photon is a particle.

Axiom 1. There exist at least two non-parallel photons.

A frame of reference is an orientation preserving affine isomorphism $T:\R^2\to \ST$. The idea is that any frame of reference assigns coordinates to spacetime. We think of the $x$-axis of $\R^2$ as the "spatial axis" and the $y$-axis as the "temporal axis." More formally, given a frame of reference $T:\R^2\to \ST$ and an event $p$ in $\ST$, we define the spatial coordinate of $p$ as measured by $T$ as the $x$-coordinate of $T^{-1}(p)$, and the temporal coordinate of $p$ as measured by $T$ as the $y$-coordinate of $T^{-1}(p)$. The observer corresponding to $T$ is defined as the image of the $y$-axis under $T$. The idea here is that the events to which $T$ assigns $0$ as the spatial coordinate are the ones "experienced by $T$."

Let $T$ be a frame of reference. We say that two events $p$ and $q$, we say that $p$ and $q$ are simultaneous with respect to to $T$ if $p$ and $q$ have the same temporal coordinates with respect to $T$. Let $\alpha$ be a particle. The velocity of $\alpha$ as measured by $T$ is defined as the slope of the line $T^{-1}(\alpha)$. The magnitude of velocity is called speed.

A frame of reference $T$ is said to be admissible if satisfies the following two conditions.

  • Each photon has speed $1$ with respect to it.

Axiom 2. There exists an admissible frame of reference.

Axiom 3. If a particle has speed $1$ with respect to an admissible frame of reference then the particle is a photon.

Theorem 1. Let $\alpha$ and $\beta$ be two photons which are not parallel. Then every particle parallel to $\alpha$ or $\beta$ is a photon, and every photon is parallel to $\alpha$ or $\beta$. Further, through every event there pass exactly two photons.

Now let $S$ and $T$ be two admissible frames of references and assume that $T(0) = S(0)$ for simplicity of the discussion. Let $v_1 = T^{-1}(Se_1)$ and $v_2 = T^{-1}(Se_2)$. Let $\ell_i$ be the line passing through the origin and $v_i$. If $m_1$ and $m_2$ be the two photons passing through $p$ and let $r_i = T^{-1}(m_i)$. Since each photon has speed $1$ with respect to $T$, we have $$ r_1 = \set{(x, y)\in \R^2:\ x=y} \quad \text{ and }\quad r_2 = \set{(x, y)\in \R^2:\ x+y=0} $$ Now using the fact that each photon has speed $1$ with respect to $S$, and he fact that both $S$ and $T$ are orientation preserving, we see that $r_1$ passes through $v_1+v_2$ and $r_2$ passes through $v_1-v_2$. From this it follows that the angle which $v_1$ makes with the $x$-axis is same as the angle which $v_2$ makes with the $y$-axis. (The following lemma is not exactly true, for in the figure followed by the lemma we could very well replace $v_1$ by $-v_1$ and $v_2$ by $-v_2$. I still think it is true enough and hence I do not remove it.)

Lemma 2. Let $S$ and $T$ be two admissible frames. Write $v_i = T^{-1}(Se_i)$ for $i=1, 2$. Let $l_i = \set{tv_i:\ t\in \R}$, $m_1$ denote the $x$-axis, and $m_2$ denote the $y$-axis.

  • The orientation of $(v_1, v_2)$ is same as the orientation of $(e_1, e_2)$.
  • The line joining $v_1$ and $v_2$ is parallel to $\set{(x, y)\in \R^2:\ x+y=0}$, and the line joining the origin and $v_1+v_2$ is parallel to $\set{(x, y)\in \R^2:\ x=y}$.
  • The length of the two vectors $v_1$ and $v_2$ are equal.
  • The signed angles $\angle(m_1, l_1)$ and $\angle(m_2, l_2)$ are equal.

enter image description here

Given two frames $S$ and $T$, we define the velocity of $S$ with respect to $T$ as the velocity of the observer corresponding to $S$ with respect to $T$.

Theorem 3. Let $S$ and $T$ be two admissible frames. Then the velocity of $S$ with respect to $T$ is same as the negative of the velocity of $T$ with respect to $S$.

Question

Now suppose $S$ and $T$ are two admissible frames with $S$ having speed $v$ with respect to $T$. It is immediately clear that if $p$ and $q$ are two events that are observed to occur in the same place by $T$, then the temporal difference between $p$ and $q$ is measured differently in the two frames. We would like to quantify this difference. The problem then essentially asks to find out the length of $v_2$ (or $v_1$) in the figure above, since the dilation factor is nothing but $XO/BO$. However, the axioms so far cannot be used to derive the length of $v_2$.

In Einstein's thought experiment we have a train housing a light clock which fires a photon perpendicular to the motion of the train and it is tacitly assumed that the dimension perpendicular to the motion of the train `behave the same with respect to both the observes.' I am able to make this thought experiment precise but with a very ugly axiom (that directions perpendicular to relative motion are unaffected). Also, it is unsettling that one needs an extra spatial dimension to establish this factor of time dilation.

EDIT: Andreas Blass had pointed out in the comments that the temporal order of two events may be different for different observers. This amounted to an error in my formalism and resulted in this edit.

If $\Omega$ is locally Lipschitz, then $\Omega = \bigcup_{k = 1}^N \Omega_k$ for $\Omega_k$ star shaped with respect to an open ball $B_k$.

Posted: 22 May 2021 07:56 PM PDT

I am reading Galdi's Introduction to the mathematical theory of Navier Stokes equations and there is an argument which comes up quite often that I really don't understand.

In many theorems of Chapter $3$, we prove the existence of solutions of a problem defined on an open set $\Omega \subset \mathbb R^n$ that statisfies the following condition: $$\Omega = \bigcup_{k = 1}^N \Omega_k$$ where each $\Omega_k$ is star shaped with respect to some open ball $B_k$ with $\overline{B}_k \subset \Omega_k$.

Then, in the next chapters, he uses these results but for $\Omega$ bounded and locally Lipschitz. Therefore, it seems to me that a bounded locally Lipschitz open set should satisfy the above condition, but I really have no idea how to show that.. Clearly $\overline{\Omega}$ is compact, so we can cover it by a finite number of balls, but then how do we prove that they are star-shaped with respect to some other balls $B_k$ ? Any idea ?

Given distance between cars, how to calculate the relative position and identify if the car is on my left or right?

Posted: 22 May 2021 07:29 PM PDT

I am doing decentralized control of vehicles. Assume I am a car with $2$ sensors, sensor $A$ in front of me and sensor $B$ behind. Using these sensors I can measure the distance from me to other vehicles $C,D$ and $E$. Apart from that I don't have any other information and I cannot communicate with other cars.

Objective:

  1. Calculate the distance between the other cars and their relative position.
  2. To figure out if the other cars are behind or in front of me.
  3. To figure out if the other cars are on my left or right.

Given length of $AC$, I know that finding relative position of car $C$ is a right triangle problem. $AC$ is hypotenuse, I need to find this right triangle's legs. All possible right triangles lie in circle with diameter $AC$. However, without defining axis, it is not clear which tringle to choose. So using only sensor $A$ we cannot find the relative position.

enter image description here

Now we draw another circle with diameter $BC$. And now the two circles will intersect at $J$ and that intersection is a point we need to calculate the relative position. Moreover, the intersection of these circles lie on the extension of line $AB$. This is very good, we can use line $AB$ as $y$ axis. Then to find a relative position of car $C$ we need to solve two equations with two unknowns:

\begin{align} AJ^2+JC^2=AC^2\\ (AJ+AB)^2+JC^2=BC^2 \end{align}

where we assume $AB$ is known in priori.

We can do this for all cars on our right, i.e., $C$ and $D$. We know their relative position and using this information, we can even calculate the distance between them. However, we need to be careful, we need to know if they are in front of us or behind to be able to calculate the distance between them. But that can be checked comparing $AC$ vs $BC$. Since $AC<BC$, we know that car $C$ is on front, and since $AD>BD$, car $D$ is behind. Now we have achieved objectives $1$ and $2$.

However, I have a problem with a left side. I can calculate the car's $E$ relative position, but I don't know if it is on my right or left. So I cannot calculate distance $CE$ or $DE$ correctly, there is always two possible answers. Axis $y$ is well defined, comparing the distance from $A$ and $B$ to other cars, we can know if the other car is in $+y$ or $-y$ side. Axis $x$ is not well defined, from distance information we are not able to identify if the car is on our right or left. Is there any smart way to achieve the last objective without adding any additional sensors?

EDIT: I have tried to place the sensors $A$ and $B$ asymmetrically, I don't think anything will change. There is still confusion between right and left. In the picture below, if $AC=AD$ and $BC=BD$, there still will be two possible answers.

enter image description here

Integral over the inverse of the generalized regularized incomplete gamma function over its domain with all 3 arguments equated.Non-integral form hard

Posted: 22 May 2021 07:51 PM PDT

I said before that the upper bound was one. As I do not know the complex behavior of this function, it would be even harder to integrate past the real domain. Please try the three "argumented" inverse generalized incomplete gamma function integral which seems to diverge for whatever reason over the unit interval. By the way here is the graph for the inverse function of the integrand and:$$\mathrm{\mathit{Q_2}=\int_0^βQ^{-1}(x,x,x)dx= \int_0^β Q^{-1}\left(x,\frac{Γ(x,x)}{Γ(x)}-x\right)dx= \int_0^β Q^{-1}\left(x,Q(x,x)-x\right)dx≈.221182...:}$$enter image description here

Here is one of my sources of this function here. I very strongly suggest it to learn more about this function.

Here is also a proposed asymptotic x-value to this function. This is the graph of the reciprocal integrand. Let this aforementioned value be called β:enter image description here

This is the inspiration for this problem and the reason for the subscript of 2:

Reference Problem

First we need to figure out the value of the endpoint for the pure real valued $\mathrm{Q^{-1}(x,x,x)}$ denoted by β to resemble the b value in the upper bound of the integral. Then it is needed to have this integral to put into exact form, without any approximations, unless they lead to an exact form, as a non-integral form. It can be an "open" form, it does not need to be closed. If you simply use the definition of the Riemann Sum or the nth derivative of this function, unless they lead to an insightful alternate non-integral form of this constant, then I will only upvote, maybe, your answer, but not check it. It has to be exact without approximations.

Related

If you integrate this function, it is probably non-elementary, you do not have to find this value of β. The sum representations of this function are ugly and you need to solve a proportion. My only idea would be to use the Riemann Sum definition and generalized taylor series to integrate immediately, or after one step. The derivatives of this function are ugly and are undefined at x=0, unless a limit is taken maybe. I hope this improper limit does not diverge. It has some weird behavior with the β value being $.25991\le β<.25992\ $according to these calculations.

If I try to up the number of digits of evaluate accuracy and/or the number of places for x, the input, then the site does not know if that value is part of the domain as it returns the same input and/ or a different decimal approximation. One of these two results oscillates providing ambiguous results on the value of β defined above. Taking a simple average of the bounds above yields these interesting possible closed forms for β.

Evaluating $\mathrm{Q^{-1}(.259915,. 259915,. 259915)≈703.9}$. This shows that this function evaluated at β is $\infty$.The result does not have to be, but it still must be exact. Please leave me feedback and corrections!

Progress: Evaluating the integrand for certain approximations of β on Wolfram Alpha, I have tested to see if the value returned has the function with the three arguments, meaning this is not in the domain of this function for real x, or an actual number, which means that we are getting closer to the actual value of β which then means that it can be evaluated and is thus in the domain. Here is the value I have for beta using this method:β=$.259913533314486931…\ $. Here are some of these possible closed forms for β.

I hope this amazing result is of help. None of these seem likely though as they are too complicated, but who said this would look nice anyways? Evaluating this function at this value gets 38.489... according to Wolfram Alpha.

Proving monotonicity of a function $g$

Posted: 22 May 2021 07:30 PM PDT

Let $r\ge 1$. For $r-1\leq x \le r+1$ we define $f(x)=\arccos\left(\frac{x^2 + r^2 - 1}{2 r x}\right)$. Now let $g:[1,\infty)\to\mathbb{R}$ be given by $$g(r)=\frac{\int_{r-1}^{r+1}{r(f(x))^2\,dx}}{\int_{r-1}^{r+1}{ f(x)\,dx}}.$$ I want to show that $g$ is a monotonically decreasing function.

Notes

  1. Numerically, this looks to be almost certainly true.
  2. I have already proved that $g$ has some interesting properties; for instance $g(1)=\pi-2$ and $\lim_{r\to \infty} g(r)=\frac{8}{3\pi}$. But ideally I'd like to show that the function decreases monotonically between these two values.
  3. I've tried looking at the derivative - but it seems too ghastly to be useful!

Stirling number relation

Posted: 22 May 2021 07:46 PM PDT

Let $S(k,n)$ denote the Stirling number, i.e. the number of ways to partition $k$ distinguishable objects into $n$ indistinguishable blocks. Then consider $S(n+r,n)$ for some fixed $r$, how can I show that $S(n+r,n)$ is a polynomial in $n$ of degree $2r$?

How to simplify $\frac{\sqrt{2}+2(\cos 20^\circ+\cos 25^\circ)}{\sin \left(90-\frac{45}{2}\right)\sin 55^\circ \sin 57.5^\circ}$?

Posted: 22 May 2021 07:24 PM PDT

The problem is as follows:

Simplify the following expression:

$B=\frac{\sqrt{2}+2(\cos 20^\circ+\cos 25^\circ)}{\sin \left(90-\frac{45}{2}\right)\sin 55^\circ \sin 57.5^\circ}$

The alternatives given in my book are as follows:

$\begin{array}{ll} 1.&7.5\\ 2.&6\\ 3.&8\\ 4.&5\\ \end{array}$

I'm not sure how to proceed here because the division seems kind of complicated to simplify.

But I could spot that suspicioulsy $\sqrt{2}= \csc 45^\circ$

and also $2= \csc 30^\circ$

But I don't know if these would come into play in solving this problem as it is challenging. Perhaps does it exist a way to solve this without much fuss?.

I could also spot that:

$\sin \left(90-\frac{45}{2}\right)= \cos \frac{45}{2}$

$\sin 55^\circ = \cos 35^\circ$

and

$\sin 57.5^\circ= \cos 32.5^\circ = \sin \frac{65}{2}^\circ$

The rest I presume that involves the simplifcation of the expresion using sum to product formulas. But I got stuck with those. Can someone help me here?.

Does the set of real numbers with bounded partial quotients have positive measure?

Posted: 22 May 2021 07:10 PM PDT

We say a real number $x$ has bounded partial quotients if its continued fraction expansion $[a_0; a_1, a_2 \cdots]$ is bounded by some constant $M=M(x)$. The set $A$ consisting of those numbers whose partial quotients are bounded forms a dense, uncountable subset of $\mathbb{R}$ which includes the algebraic numbers of degree $\leq 2$. It appears that it is an open problem whether or not $A$ contains any algebraic numbers of degree $>2$.

Question : Is it known whether or not the set $A$ has measure zero?

The reason this question is interesting has to do with Diophantine approximation. We say a real number $x$ is badly-approximable if there exists a positive constant $C=C(x)$ such that $$\left|x-\frac{p}{q}\right|>\frac{C}{q^2}$$

for all rational $p/q \neq x$. Here, the name fits, since for any irrational $x$, $|x-p/q|<\frac{1}{q^2}$ for infinitely many pairs $(p,q)$, and badly-approximable numbers are precisely those for which we cannot do better than this, i.e., merely scaling the numerator by a certain constant $C$ ruins everything.

It turns out that the property defining the set $A$ discussed earlier (i.e., bounded partial quotients) is in fact completely equivalent to this property of being badly-approximable. Thus asking about the size of $A$ is essentially asking about the size of the set of worst-approximable real numbers.

Why would the projection function open and not closed?

Posted: 22 May 2021 07:34 PM PDT

Given $p: \mathbb{R} \times \mathbb{R} \to \mathbb{R}, (x,y) \mapsto x$

Why is $p$ open but not closed?

Shouldn't it be the case $p$ is closed not open because it sends a singleton to singleton, and all singletons in Euclidean spaces are closed?

Projection of a point on a plane

Posted: 22 May 2021 07:42 PM PDT

I have a homework problem which I cannot solve.

Equation of a hyperplane is given as

$$f(x) = w^tx + b $$

Given $f(x) = 0$

show that the projection of a point xa on the plane is :

$$x_p = x_a - \frac{|f(x_a)|w}{\| w\|^2}$$

I was thinking of taking a unit vector lying on the plane and taking the dot product. But I am not getting the result I am looking for

Could anyone point me into the right direction ? Thanks.
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