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### Author Topic: TT1 Problem 2 (morning)  (Read 4408 times)

#### Victor Ivrii

• Elder Member
• Posts: 2599
• Karma: 0
##### TT1 Problem 2 (morning)
« on: October 16, 2018, 05:28:13 AM »
(a)  Find Wronskian  $W(y_1,y_2)(x)$ of a fundamental set of solutions $y_1(x) , y_2(x)$ for ODE
\begin{equation*}
-y'' (e^x+e^{-x}+2) + y' (e^x-e^{-x}) +2y=0.
\end{equation*}
(b) Check that $y_1(x)=e^x+1$ is a solution and find another linearly independent solution.

(c) Write the general solution, and find solution such that ${y(0)=0, y'(0)=2}$.

#### Monika Dydynski

• Full Member
• Posts: 26
• Karma: 30
##### Re: TT1 Problem 2 (morning)
« Reply #1 on: October 16, 2018, 11:23:09 AM »
(a) Find Wronskian $W(y_1,y_2)(x)$ of a fundamental set of solutions $y_1(x)$, $y_2(x)$ for ODE

$$-y''(e^{x}+e^{-x}+2)+y'(e^{x}-e^{-x})+2y=0.$$

Dividing but sides by $-(e^{x}+e^{-x}+2)$, we get

$$L[y]=y''-y'\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}+2}-y\frac{2}{e^{x}+e^{-x}+2}=0,$$

where $p(x)=-\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}+2}$, and $q(t)=-\frac{2}{e^{x}+e^{-x}+2}$.

By Abel's Theorem,

\begin{align}W(y_1,y_2)(x)&=c\exp(\int-{p(x)dx})\\&=c\exp(\int\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}+2}dx)\\&=c (e^x+e^{-x}+2).\end{align}

Let $c=1 \Rightarrow W(y_1,y_2)(x)=e^x+e^{-x}+2$.

b) Check that $y_1(x)=e^{x}+1$ is a solution and find another linearly independent solution.

Since $y_1(x)=e^{x}+1 \Rightarrow y_1 '(x)=e^{x}$, and $y_1 ''(x)=e^{x}$

Plugging $y_1$, $y_1 '$, and $y_1 ''$ into the ODE, we have

\begin{align}-e^{x}(e^{x}+e^{-x}+2)+e^{x}(e^{x}-e^{-x})+2(e^{x}+1)&=0\\-2e^x-e^{2x}-2+2e^{x}+e^{2x}+2&=0\end{align}

$y_1(x)$ satisfies the ODE $\Rightarrow$ $y_1(x)$ is a solution.

Given $y_1(x)$, we can find another linearly independent solution.

We know from the definition of the Wronskian that
$$W(y_1,y_2)(x)=y_1y_2 '-y_1 'y_2=(e^x+1)y_2 '-y_2 e^x$$

Equating the two expressions for the Wronskian, we get

$$(e^x+1)y_2 '-y_2 e^x=e^x+e^{-x}+2$$

Dividing both sides by $e^x+1$, and multiplying by integrating factor $\mu=\frac{1}{e^x+1}$,

$$\frac{1}{e^x+1}y_2=\int{\frac{e^x+e^{-x}+2}{e^{2x}+2e^x+1}}dx+C$$
$$\frac{1}{e^x+1}y_2=-\frac{1}{e^x}$$
$$y_2=-\frac{e^x+1}{e^x}$$
$$y_2=-1-\frac{1}{e^{x}}$$

c) Write the general solution. Find solution such that $y(0)=0$, $y'(0)=2$

The general solution to the ODE is

$$y(x)=c_1 (e^x+1)+c_2(-1-\frac{1}{e^{x}}).$$

$\Rightarrow y'(x)=c_1e^x+c_2e^{-x}$

$$\cases{c_1-c_2=0\\c_1+c_2=2} \Rightarrow \cases{c_1=1\\c_2=1}$$

Thus, the solution that satisfies $y(0)=1$, $y'(0)=2$ is

$$y(x)=e^x-\frac{1}{e^{x}}.$$

#### Doris Zhuomin Jia

• Jr. Member
• Posts: 6
• Karma: 2
##### Re: TT1 Problem 2 (morning)
« Reply #2 on: October 16, 2018, 11:27:14 AM »
Hi, there is my solution for Problem 2

#### Mengmeng Shang

• Newbie
• Posts: 4
• Karma: 0
##### Re: TT1 Problem 2 (morning)
« Reply #3 on: October 16, 2018, 11:32:56 AM »
$y'' - \frac{e^x-e^{-x}}{e^x+e^{-x}+2} y' - \frac{2}{e^x+e^{-x}+2}y = 0$

$W = Ce^{-\int{-\frac{e^x-e^{-x}}{e^x+e^{-x}+2}}dx} = C{e^x+e^{-x}+2}$

$take C =1, W = e^x+e^{-x}+2$

#### Monika Dydynski

• Full Member
• Posts: 26
• Karma: 30
##### Re: TT1 Problem 2 (morning)
« Reply #4 on: October 16, 2018, 11:44:27 AM »
Hi, there is my solution for Problem 2

Hey Doris, you wrote the definition of the Wronskian incorrectly, $W(y_1,y_2)(x)\ne y_1(x)y_2’(x)+y_2(x)y_1'(x)$.

$$W(y_1,y_2)(x)=\begin{array}{|c c|}y_1(x)& y_2(x) \\ y_1’(x) & y_2’(x)\end{array}=y_1(x)y_2’(x)-y_2(x)y_1'(x).$$

Hope that helps!

#### Weina Zhu

• Jr. Member
• Posts: 6
• Karma: 1
##### Re: TT1 Problem 2 (morning)
« Reply #5 on: October 16, 2018, 01:34:56 PM »
I am trying to go the cosh~sinh way, but cannot continue when finding the second solution, can someone help me out of this?
Thanks in advance. I tried to see classmates' post, but they gave the answer directly;D

#### Samarth Agarwal

• Jr. Member
• Posts: 6
• Karma: 11
##### Re: TT1 Problem 2 (morning)
« Reply #6 on: October 16, 2018, 02:39:54 PM »
Integration of (cosh(x) + 1)/(e^x + 1)^2 wrt dx = -e^(-x)/2 + c
Therefore g = (e^x + 1)(-e^(-x)/2 + c)
« Last Edit: October 16, 2018, 02:41:25 PM by Samarth Agarwal »

#### Victor Ivrii

• Elder Member
• Posts: 2599
• Karma: 0
##### Re: TT1 Problem 2 (morning)
« Reply #7 on: October 18, 2018, 04:36:31 AM »
Monika did everything right, I also commend her comment to Doris' post.

Weina
You are right that equation is
$$-2 (\cosh(x)+1)y''+2\sinh(x)y'+2y=0$$
and we can rewrite it as
$$(\cosh(x)+1)y''-\sinh(x)y'-y=0$$
and then
$$W=C\exp\Bigl(\int \frac{\sinh(x)}{\cosh(x)+1}\,dx\Bigr) = C\exp\Bigl(\ln (\cosh(x)+1)\Bigr)=C_1(\cosh(x)+1).$$
But it is not another way, just another form.
« Last Edit: October 18, 2018, 04:42:28 AM by Victor Ivrii »


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# Tag Info

The Hamiltonian is time-reversal invariant: $c_{k\uparrow}\rightarrow c_{-k,\downarrow}, c_{k\downarrow}\rightarrow -c_{-k,\uparrow}$. You can check that explicitly. The ground state is also invariant, because Cooper pairs are all spin singlet. One of the significant implications of time-reversal symmetry for s-wave superconductors is the Anderson's ...

format: plain


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# Portal:Electronics

## The Electronics portal

Surface-mount electronic components

Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter. The identification of the electron in 1897, along with the invention of the vacuum tube, which could amplify and rectify small electrical signals, inaugurated the field of electronics and the electron age.

Electronics deals with electrical circuits that involve active electrical components such as vacuum tubes, transistors, diodes, integrated circuits, optoelectronics, and sensors, associated passive electrical components, and interconnection technologies. Commonly, electronic devices contain circuitry consisting primarily or exclusively of active semiconductors supplemented with passive elements; such a circuit is described as an electronic circuit.

The nonlinear behaviour of active components and their ability to control electron flows makes amplification of weak signals possible. Electronics is widely used in information processing, telecommunication, and signal processing. The ability of electronic devices to act as switches makes digital information-processing possible. Interconnection technologies such as circuit boards, electronics packaging technology, and other varied forms of communication infrastructure complete circuit functionality and transform the mixed components into a regular working system.

Electrical and electromechanical science and technology deals with the generation, distribution, switching, storage, and conversion of electrical energy to and from other energy forms (using wires, motors, generators, batteries, switches, relays, transformers, resistors, and other passive components). This distinction started around 1906 with the invention by Lee De Forest of the triode, which made electrical amplification of weak radio signals and audio signals possible with a non-mechanical device. Until 1950 this field was called "radio technology" because its principal application was the design and theory of radio transmitters, receivers, and vacuum tubes.

most electronic devices use semiconductor components to perform electron control.  The study of semiconductor devices and related technology is considered a branch of solid-state physics, whereas the design and construction of electronic circuits to solve practical problems come under electronics engineering.  This article focuses on engineering aspects of electronics.

## Selected biography

Joseph Henry (December 17, 1797 – May 13, 1878) was a Scottish-American scientist. During his lifetime, he was considered one of the greatest American scientists since Benjamin Franklin. While building electromagnets, he discovered the electromagnetic phenomenon of self-inductance. He also discovered mutual inductance independently of Faraday, though Faraday was the first to publish his results. His work on the electromagnetic relay was the basis of the electrical telegraph, jointly invented by Samuel Morse and Charles Wheatstone. The SI unit of inductance, the henry, is named after him.

## Selected design

Credit: User:Ilmari Karonen
A diagram of a Möbius resistor showing the current flow.

## News

August 14, 2014

512K Day arrives, surpassing some routers capacity, breaking the internet. More...

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The European Organization for Nuclear Research (CERN) said that repairing the Large Hadron Collider (LHC) will cost up to 16.6 million or US\$21 million. More...

April 30, 2008

HP Labs announces the creation of a Memristor, the fourth basic element of electronic circuits with the Resistor, Capacitor, and Inductor.

December 4, 2007

On the third day of the 2007 Taipei IT Month in Taiwan yesterday, notebook computers and desktop computers built with AMD's Phenom processor and Intel Penryn processor openly battled for the consumer-market after each company launched their quad core processors. More...

February 27, 2007

The new South Pole Telescope has recently collected its first light in a long-term project to learn about the nature of dark energy. More...

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## Consumer showcase

Roomba is a robotic vacuum cleaner made and sold by iRobot. It is marketed as a Robotic Floorvac. The Roomba was first released in 2002 with updates and new models released in 2003, 2004, 2005 and 2006. As of May 2006, over 2 million units have been sold, making it the most successful domestic robot so far. The unit is a disc, thirteen inches (34 cm) in diameter and less than four inches (9 cm) high. A large contact-sensing bumper is mounted on the front half of the unit, with an infrared sensor at its top front center.

## Selected article

Inductance is a measure of the amount of magnetic flux produced for a given electric current. The term was coined by Oliver Heaviside in February 1886. The SI unit of inductance is the henry (symbol: H), in honour of Joseph Henry. The symbol L is used for inductance, possibly in honour of the physicist Heinrich Lenz.

The inductance has the following relationship:

${\displaystyle L={\frac {\Phi }{i}}}$

where; L is the inductance in henrys, i is the current in amperes, Φ is the magnetic flux in webers. Strictly speaking, the quantity just defined is called self-inductance, because the magnetic field is created solely by the conductor that carries the current.

When a conductor is coiled upon itself N number of times around the same axis (forming a solenoid), the current required to produce a given amount of flux is reduced by a factor of N compared to a single turn of wire. Thus, the inductance of a coil of wire of N turns is given by:

${\displaystyle L={\frac {\lambda }{i}}=N{\frac {\Phi }{i}}}$

where, ${\displaystyle \lambda }$ is the total 'flux linkage'.

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# totient valence function

Given an integer $n$, count how many integers $m$ in the set $\{n+1,n+2,\ldots n^{2}\}$ satisfy $\phi(m)=n$. This is the of $n$, usually labelled $N_{\phi}(n)$. (The only two special cases are 2 and 6, for which one has to look a little beyound $n^{2}$).

Robert Carmichael conjectured that $N_{\phi}(n)=1$ never. Two sequences in Sloane’s OEIS that list numbers with higher totient valences than preceding numbers are A007374 and A097942.

Title totient valence function TotientValenceFunction 2013-03-22 15:50:57 2013-03-22 15:50:57 CompositeFan (12809) CompositeFan (12809) 5 CompositeFan (12809) Definition msc 11A25


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How do you factor 66+11 using GCF?

Dec 6, 2016

Factors of $66 + 11$ are $11 \times 7$

Explanation:

As $66$ is a multiple of $11$, GCF of $11$ and $66$ is $11$.

Hence we can write $66 + 11$ as

$11 \times 6 + 11 \times 1$ and now using distributive property of numbers

this becomes $11 \times \left(6 + 1\right) = 11 \times 7$

Hence factors of $66 + 11$ are $11 \times 7$.


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# Darstellungstheorietage and Nikolaus Conference 2021

Speaker: Laura Voggesberger (Kaiserslautern)

Title: An Introduction to Nilpotent Pieces

Abstract:

Let $$G$$ be a connected reductive algebraic group over an algebraically closed field $$k$$, and let Lie$$(G)$$ be its associated Lie algebra. In 2011, Lusztig defined a partition of the unipotent variety of $$G$$. This partition is very useful when working with representations of $$G$$. Equivalently, one can consider certain subsets of the nilpotent variety of $$g$$ called pieces. This approach appears in Lusztig’s article. The pieces for the exceptional groups of type $$G_2$$, $$F_4$$, $$E_6$$, $$E_7$$ and $$E_8$$ in bad characteristic have not yet been determined. In this talk, I will give an introduction to both definitions of the nilpotent pieces and present a solution to this problem for groups of type $$G_2$$ and $$F_4$$.


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Skip to main content

# 6.1: Add and Subtract Polynomials

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$


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## Thomas' Calculus 13th Edition

Given: $a_{n+1}=(a_n)^{n+1}$ and $a_1=\dfrac{1}{2}$; and $n=1,2,3,....n$ Here, we can see that $a_2=(a_1)^{1+1}=(\dfrac{1}{2})^{2} \\ a_3=(a_2)^{2+1}=(\dfrac{1}{2})^{3}.....$ and $a_n=(\dfrac{1}{2})^{n!}$ $\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}(\dfrac{1}{2})^{n!}$ Next, we need to apply the direct comparison test. $\Sigma_{n=1}^\infty (\dfrac{1}{2})^{n!} \leq \Sigma_{n=1}^\infty(\dfrac{1}{2})^n$ we can see that the series $\Sigma_{n=1}^\infty(\dfrac{1}{2})^n$ converges Therefore, the series converges.


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Journal of Prime Research in Mathematics

# Generalized Identities and Inequalities of Čebyšev and Ky Fan Type for ∇−convex function

Faraz Mehmood$$^{a,*}$$, Asif R. Khan$$^b$$

$$^a$$Department of Mathematics, Dawood University of Engineering and Technology, New M. A Jinnah Road, Karachi-74800, Pakistan.

$$^b$$Department of Mathematics, University of Karachi, Karachi, Pakistan.

Correspondence should be addressed to: faraz.mehmood@duet.edu.pk

### Abstract

In the present article we establish three generalizations, first generalization is related to discrete Čebyšev identity for function of higher order ∇ divided difference with two independent variables and give its special case as a sequence of higher order ∇ divided difference. Moreover, we deduce results of discrete inequality of Čebyšev involving higher order ∇−convex function. The second and third generalizations are for integral Čebyšev and integral Ky Fan identities for function of higher order derivatives with two independent variables and discuss its inequalities using ∇−convex function. Generalized results give similar results of Pěcari´c’s article [23] and recapture some established results.

#### Keywords:

convex function, Čebyšev’s inequality, Ky Fan’s inequality.


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# Biblioteca

Música » Guns N' Roses »

## Since I Don't Have You

33 execuções | Ir para página da faixa

De Quinta 7 de abril de 2011 a Sábado 7 de abril de 2012 Todo o tempo

Faixas (33)
Faixa Álbum Duração Data
Since I Don't Have You 4:19 Mar 3 2012, 2h43
Since I Don't Have You 4:19 Fev 15 2012, 12h39
Since I Don't Have You 4:19 Fev 10 2012, 13h06
Since I Don't Have You 4:19 Fev 9 2012, 10h47
Since I Don't Have You 4:19 Jan 31 2012, 15h05
Since I Don't Have You 4:19 Jan 31 2012, 15h05
Since I Don't Have You 4:19 Jan 27 2012, 13h48
Since I Don't Have You 4:19 Jan 13 2012, 12h40
Since I Don't Have You 4:19 Dez 30 2011, 10h40
Since I Don't Have You 4:19 Dez 16 2011, 11h04
Since I Don't Have You 4:19 Dez 6 2011, 2h38
Since I Don't Have You 4:19 Dez 2 2011, 13h40
Since I Don't Have You 4:19 Nov 19 2011, 12h39
Since I Don't Have You 4:19 Nov 18 2011, 0h43
Since I Don't Have You 4:19 Nov 12 2011, 12h52
Since I Don't Have You 4:19 Nov 8 2011, 21h44
Since I Don't Have You 4:19 Nov 5 2011, 10h50
Since I Don't Have You 4:19 Nov 1 2011, 21h25
Since I Don't Have You 4:19 Out 22 2011, 10h23
Since I Don't Have You 4:19 Out 20 2011, 2h35
Since I Don't Have You 4:19 Out 13 2011, 21h18
Since I Don't Have You 4:19 Out 1 2011, 13h16
Since I Don't Have You 4:19 Set 23 2011, 22h26
Since I Don't Have You 4:19 Set 17 2011, 15h55
Since I Don't Have You 4:19 Set 16 2011, 16h51
Since I Don't Have You 4:19 Set 1 2011, 2h27
Since I Don't Have You 4:19 Ago 25 2011, 2h30
Since I Don't Have You 4:19 Ago 23 2011, 19h22
Since I Don't Have You 4:19 Ago 22 2011, 21h32
Since I Don't Have You 4:19 Ago 19 2011, 17h33
Since I Don't Have You 4:19 Ago 4 2011, 21h00
Since I Don't Have You 4:19 Ago 2 2011, 21h09
Since I Don't Have You 4:19 Jul 20 2011, 15h52


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## What’s For Dinner: Potato Leek Chowder

This week, we got new golden potatoes, corn and leeks in our Bountiful Baskets! So I decided, since it was raining on Tuesday, to make a potato leek chowder for dinner. We roasted the corn on the grill and removed it from the cobs on Saturday. Then it was put into the freezer in 2-cup servings to use later. So for this recipe, I just pulled out and defrosted the amount I needed.

## What’s For Dinner: Jamaican Me Hungry Sweet Potatoes

How many of you think of sweet potatoes as that sweet, marshmallow topped dish served at Thanksgiving? Or, perhaps you’ve had them baked into a pie resembling pumpkin pie? If you’re lucky, you’ve had them baked just like a regular potato, or even better, made into air fries! They are really good for you, rich in fiber, beta carotene (the darker orange the flesh is, the more they have) and Vitamin B6.

## What’s For Dinner: Coleslaw Vegan Style

Coleslaw is quick and easy to make… Even if you’re starting the dressing from scratch! It can be a side dish, or eaten as a snack. It’s great for picnics and tailgate parties. But how do you make a vegan dressing? Start with a commercial Vegenaise? Not me! This is how I make vegan coleslaw dressing. Here’s my setup. Equipment Blender Ingredients 8 oz soft (not silken!) tofu 4 tablespoons apple cider vinegar

## In The Kitchen: Make Your Own Rice Milk

Aren’t these some of the prettiest milk bottles you’ve ever seen? I got these at the Container Store for about $4 each. The covers are plastic lined, and they hold a quart. (You can also purchase the covers separately.) If you need a lot of them, they are also available on Amazon, Quattro Stagioni Milk Bottle, Set of 12. A case sells for about$46, a bit of a savings.

## What’s For Snack: Kale Chips

Do you eat Kale? Well, you should! But most people either don’t know what to do with it, or don’t like it. One of the Bountiful Basket volunteers told DH that she feeds it to her rabbits! Wow! Her rabbits are eating healthier than she is! Kale is loaded with nutrition that your body needs. Three and a half ounces of raw kale contains 13621 IU of Vitamin A, 72 mg of Calcium, 10 mg of Magnesium and 228 mg of Potassium!

## What’s For Dinner: Baked Falafel

DH is crazy about falafel! At one time, in the past, we were going to any restaurant he could find that served it, just so he could try their version. I learned a couple of things along the way: I don’t like it when it is seasoned with cheap, yellow curry powder (aka McCormick.) I actually don’t like _anythin_g seasoned with that. I also don’t like them when they are heavy and greasy, without a kind of fluffy interior.

## What’s For Dessert: Raspberry Chocolate Pie

Note: The subtitle for this one should be “Why I need a Vitamix!” The combination of chocolate and raspberry is one of my all-time favorites. I love the sweet, tartness raspberries add to the chocolate. This is a dark, rich chocolate ganache, combined with a cranberry/raspberry juice, done vegan. Instead of the dairy cream used in a traditional ganache, we’re using cashew creme. The crust is the basic nut crust recipe that I published yesterday.

## In The Kitchen: Basic Nut Crust

This is a basic, no bake, nut and grain crust for no bake pies. It’s easy to make, and easy to alter depending on the flavors you like and any allergies you may have. Here’s the setup for the one I’m making today. Equipment Food Processor 9-inch pie plate Ingredients 6 Medjool dates, pitted, cut in half 1 cup walnuts 1⁄2 cup pecans 1⁄4 cup almonds 2 tablespoons golden flax meal

## What’s For Dinner: Brussels Sprouts

What is your family’s reaction when these show up at your house? I bet I can guess! “Yuck!” “I’m not eating THAT!” “I HATE those!” Not at our house! When these show up in my Bountiful Baskets I jump for joy. I’m so happy when the time of year arrives that these are available! This didn’t used to be the case. I had a friend one time that often used the phrase, “Take (food of your choice) and ruin it by (method of your choice).

## Saucy Tricks: Vegan Tzatziki

I love tzatziki! We used to eat at Greek restraunts all the time. I would always make sure what I ordered came with it, or I would order it on the side. One of our old favorites was Pita Jungle, where we would often get the Mediterranean Platter. However, now we don’t go there now. Half of this platter contains 1810 mg of sodium! If you ate the whole thing yourself, you would end up consuming 1240 calories and a whopping 3620 mg of sodium.


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# Derivative of inner product of function of matrices

I am working on an optimization problem where I have to find derivate of $⟨F(X),WF(X)Z⟩$ with respect to X. Here $⟨,⟩$is a standard inner product (Frobenius dot product), W & Z are a constant matrix, and $f$ a function of a matrix, whose output is also a matrix. I have previously asked a slightly different version of the same question, below is the solution for it \begin{align} \phi &= \langle F,WF \rangle = \langle W,FF^T \rangle\\ \frac{\partial \phi}{\partial X} &= \langle W,dF F^T+FdF^T\rangle\\ &=\langle W+W^T,dF F^T\rangle\\ &= \langle (W+W^T)F,\frac{\partial F}{\partial X}\rangle \end{align} I am working on slightly modified problem, but I have not been able to come up with such compact answer. Here is link Derivative of inner product of matrix-valued functions of matrices to my previous question

• Thank you for your comment, I just made the changes. – Dushyant Sahoo Jan 10 '18 at 17:22
• One difficulty is that the derivative of $F$ with respect to $X$ is a 4-dimensional matrix. – Rodrigo de Azevedo Jan 10 '18 at 17:47
• I thought for simplicity, initially, $X$ can be taken a scaler. This could give a good idea how the solution behaves. – Dushyant Sahoo Jan 10 '18 at 18:06

For ease of typing, I'll use the product notation $$A:B=\langle A,B\rangle$$ Then consider the derivative with respect to a single element of the independent matrix, $X_{ij}$, which is a scalar quantity. \eqalign{ \phi &= F:WFZ \cr d\phi &= dF:WFZ + F:W\,dF\,Z \cr &= (WFZ + W^TFZ^T):dF \cr } For ease of typing (once again), I've used the notation $$dF = \frac{\partial F}{\partial X_{ij}}$$ To obtain the full matrix result, sum over all of the $\{i,j\}$ indices.
Let $D_\alpha(\epsilon)$ be the derivative of $f$ at $\alpha$. Then the derivative of the composite function $g$ at $\alpha$ as a function of $\epsilon$ ought to be $\newcommand{\ang}[1]{\left\langle{#1}\right\rangle}\ang{D_\alpha(\epsilon),Wf(\alpha)Z}+\ang{f(\alpha),WD_\alpha(\epsilon)Z}$. By the product rule or something. Frankly I can't be bothered to put in effort, when you've already asked an almost identical question, and haven't shown any work in the question you wrote above.


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## How to calculate π

How would you calculate $\pi$? I remember being sent as a boy of ten into the schoolyard to find circular objects to measure. Our attempts to wrap a tape around a dustbin lid clearly did not represent the optimal method.

Perhaps you know a few digits of $\pi$. Starting from scratch, how would you find more? We know $\pi$ is irrational, so it cannot be expressed as the ratio of two numbers. Worse, $\pi$ is transcendental, so it is not the solution of any polynomial equation with integer coefficients.

There is a passage of the Bible (1 Kings 7:23) which refers to “a molten sea” which is “ten cubits from the one brim to the other” and “a line of thirty cubits did compass it about”. This figure of $\pi=3$ did not represent the state of the art, with more accurate values being recorded much earlier in Egypt and Mesopotamia.

A famous approximation is attributed to Archimedes (287-212 BC), who worked using only geometric methods. Archimedes took a circle and inscribed within it a polygon. The perimeter of the polygon, which is contained within the circle, is smaller than the circumference of the circle itself. Then he superscribed a polygon outside the circle. The perimeter of this second, larger polygon is larger than the circumference of the circle itself. If you think about increasing the number of sides on a polygon you will notice that the more sides a polygon has, the closer it gets to the shape of a circle. By increasing the number of sides on his two polygons and considering their perimeters, Archimedes could define two series, one increasing and one decreasing, both of which have $\pi$ in their limit.

Archimedes took this as far as polygons of 96 sides and came up with an upper limit for $\pi$ of $\frac{22}{7} \approx 3.142857$ and a lower limit of $\frac{223}{71} \approx 3.140845$. Because these coincide in the first three digits, we can conclude that $\pi$ begins $3.14 \ldots$

Others, following the method of Archimedes or developing it independently, extended to polygons with greater numbers of sides. This method culminated with Ludolph Van Ceulen who, around 1600, produced an approximation for $\pi$ to 35 decimal places. Those 35 digits were inscribed on Van Ceulen’s tombstone.

More modern approximations involve calculating the terms of an infinite series involving $\pi$, which can be rearranged to find an approximation for $\pi$. For example, the following was discovered by James Gregory:

$\frac{pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\ldots$

Using such a formula, one can generate $\pi$ to as many decimal places as necessary by taking more terms. However, this series converges very slowly. For example, 10,000 terms are needed to get just four decimal places. John Machin found a series that converges much more quickly to $\frac{pi}{4}$:

$\frac{\pi}{4}=4 \arctan{\frac{1}{5}}-\arctan{\frac{1}{239}}$

which uses a more general James Gregory result:

$\arctan{x}=x-\frac{x^3}{3}+\frac{x^5}{5}- \ldots (-1\leq x \leq 1)$

Machin used this formula to calculate one hundred digits of $\pi$ in 1706, the year William Jones introduced the modern use of the Greek letter π.

Computers enter the story in 1947 when D. F. Ferguson of the Royal Naval College in England calculated $\pi$ to 710 digits with a desk calculator. In 1950 George W. Reitwiesner published more than 2000 digits of $\pi$ in the journal Mathematics of Computation. These digits were calculated on the ENIAC computer at the suggestion of John von Neumann, who was interested to obtain a statistical measure of the randomness of the distribution of the digits (whether digits occur equally often in $\pi$ remains an open question).  Reitwiesner and his team used the same formula as Machin and ran ENIAC for around 70 hours, including the time taken to handle the punch cards.

More recently, series are used which operate on similar principles but converge even more quickly. The latest computer record used the 1989 Chudnovsky algorithm, which is based on the series:

$\frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}.!$

In 2011 Shigeru Kondo set this record on a custom-built home computer using y-cruncher, a program written by Alexander Yee, to calculate $\pi$ to ten trillion digits.


text:
Ring2DKernel¶

class astropy.convolution.Ring2DKernel(radius_in, width, **kwargs)[source]

2D Ring filter kernel.

The Ring filter kernel is the difference between two Tophat kernels of different width. This kernel is useful for, e.g., background estimation.

Parameters

Inner radius of the ring kernel.

widthnumber

Width of the ring kernel.

modestr, optional
One of the following discretization modes:
• ‘center’ (default)

Discretize model by taking the value at the center of the bin.

• ‘linear_interp’

Discretize model by performing a bilinear interpolation between the values at the corners of the bin.

• ‘oversample’

Discretize model by taking the average on an oversampled grid.

• ‘integrate’

Discretize model by integrating the model over the bin.

factornumber, optional

Factor of oversampling. Default factor = 10.

Examples

Kernel response:

import matplotlib.pyplot as plt
from astropy.convolution import Ring2DKernel
ring_2D_kernel = Ring2DKernel(9, 8)
plt.imshow(ring_2D_kernel, interpolation='none', origin='lower')
plt.xlabel('x [pixels]')
plt.ylabel('y [pixels]')
plt.colorbar()
plt.show()

(png, svg, pdf)


text:
eTheses Repository

# Proton-proton collisions at the Large Hadron Collider's ALICE Experiment: diffraction and high multiplicity

Matthews, Zoe Louise (2011)
Ph.D. thesis, University of Birmingham.

 PDF (9Mb)

## Abstract

Diffraction in pp collisions contributes approximately 30 % of the inelastic cross section. Its influence on the pseudorapidity density is not well constrained at high energy. A method to estimate the contributing fractions of diffractive events to the inelastic cross section has been developed, and the fractions are measured in the ALICE detector at 900 GeV (7 TeV) to be f$$_D$$=0.278$$\pm$$0.055 (f$$_D$$=0.28$$\pm$$0.054) respectively. These results are compatible with recent ATLAS and ALICE measurements. Bjorken’s energy density relation suggests that, in high multiplicity pp collisions at the LHC, an environment comparable to A-A collisions at RHIC could be produced. Such events are of great interest to the ALICE Collaboration. Constraints on the running conditions have been established for obtaining a high multiplicity pp data sample using the ALICE detector’s multiplicity trigger. A model independent method to separate a multiplicity distribution from ‘pile-up’ contributions has been developed, and used in connection with other findings to establish a suitable threshold for a multiplicity trigger. It has been demonstrated data obtained under these conditions for 3 months can be used to conduct early strangeness analyses with multiplicities of over 5 times the mean. These findings have resulted in over 16 million high multiplicity events being obtained to date.

Type of Work: Ph.D. thesis. Evans, David Colleges (2008 onwards) > College of Engineering & Physical Sciences Particle Physics Group, School of Physics and Astronomy QC PhysicsT Technology (General) University of Birmingham 3055
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.

Repository Staff Only: item control page


text:
# Re: [LON-CAPA-users] Algebra Expression Rendering in LON-CAPA

Hi,

It’s \sum

as in

\sum_{n=1}^{\infty}

- Gerd.

> On Jan 13, 2017, at 10:29 AM, Harding, Gene L <glhar...@purdue.edu> wrote:
>
> Hi,
>
> I am having trouble rendering an algebraic expression with a summation sign
> (capital Sigma). I want an expression that looks like this, using variables
> for the numeric portions of the expression:
> <image003.jpg>
> I am able to render the integration sign using this code:
> $function = "$numer/$denom [pi + int($sum_numer/($sum_denom_n_coeff n - >$sum_denom_const),1,3)*sin($w nt)]"; > …providing an expression that looks like this: > <image004.png> > > But when I try to render the summation sign using this code: >$function = "$numer/$denom  [pi + sum($sum_numer/($sum_denom_n_coeff n -
> $sum_denom_const),n,1,3)*sin($w nt)]";
> …I get the following:
> <image005.png>
>
> I have spent a couple of hours looking through the Maxima Manual and other
> references, as well as some examples I have found online, but have not been
> able to figure out the syntax for rendering the summation sign. Can anyone
> offer advice on how to do this?
>
> Also, although not so important, if someone can tell me how to replace the
> outer parentheses with square brackets, and/or how to display the numeric
> part of the sine argument with commas (e.g., 90,000 vs 90000), I would really
> appreciate it.
>
> Thanks,
>
> Gene L. Harding, PE
> Associate Professor of ECET
> 574-520-4190
> Purdue University-South Bend
>
> _______________________________________________
> LON-CAPA-users mailing list
> LON-CAPA-users@mail.lon-capa.org
> http://mail.lon-capa.org/mailman/listinfo/lon-capa-users

--
Gerd Kortemeyer, Ph.D.
Associate Professor of Physics
Director, LON-CAPA Project
Michigan State University
http://www.msu.edu/user/kortemey/

_______________________________________________
LON-CAPA-users mailing list
LON-CAPA-users@mail.lon-capa.org
http://mail.lon-capa.org/mailman/listinfo/lon-capa-users


text:
Science Forums

# The Actual Solution To The Ehrenfest Paradox (Relativistic Spinning Disk)

## Recommended Posts

19 hours ago, Jaro said:

I am almost convinced by the arguments in this post. I say almost because there is one troubling feature that I cannot resolve using the same arguments. And it is this:

If all the clocks on the spinning wheel are synchronized with each other, and they are synchronized (though ticking slower) with clocks in the stationary frame, then the coordinates representing constant time for both rotating and stationary observers are lines in the plane perpendicular to the time coordinate. However, suppose that the short rods the wheel is composed of were to decouple from the spokes and fly out in a straight line. To an observer in the stationary frame, who has not seen these rods be part of a rotating structure a moment go (all he sees is a rod moving in a straight line), the rods would now be Lorenz contracted and two clocks attached to the end of each rod would not be synchronized to him.

So, my issue is the apparently discontinuous transition from one coordinate system in which all observers agree on lengths and the synchronicity of clocks to one where they do not.

Thank you for addressing this point.

Hi Jaro,

There is only 1 observer and many emission points so It's easiest to think of the points on a ring/rim and the 'spokes' as the relevant length contraction to a particular point on the ring/rim as distances in the Gron Fig 9 part C plot (this is not a frame) which can be measured in time or distance (traveled by light in that time). Also, all of the points on the ring/wheel frame are synchronized at the axle and this axle is part of the carriage frame which is parallel with the road frame where the observer is situated.

##### Share on other sites

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#### Popular Days

On 7/21/2020 at 3:36 PM, AnssiH said:

I just recently stumbled upon a Quora question regarding Ehrenfest Paradox, and noticed that people were giving completely wrong answers to it.

Basically the paradox is this; In terms of Special Relativity, how does spinning disk work in special relativity, if the circumference of the rotating disk undergoes length-contraction (since it's parallel to motion) while its radius does not (since it's perpendicular to motion), and this would imply that $\frac{circumference}{diameter} \neq \pi$.

I checked bunch of similar questions of the same topic, and can't find a single person giving the correct answer on Quora. Instead I find all sorts of face-palm inducing nonsense like;

• Only the atoms length contract, but the space between them does not.
• The disk would tear into smaller pieces along the radius to give shorter total circumference.
• The disk would implode under pressure from the shrinking circumference.
• Centrifugal forces would counteract Lorentz contraction.
• There's no strong enough material to build such disk because of Born rigidity and elasticity, thus no paradox.
• You need to use General Relativity to solve the paradox.
• The geometry of the spinning wheel is non-euclidean. Just accept it.

And bunch of other answers going completely off on a tangent on topics like people inside a spinning train setting their clocks. Basically every single answer I can find tells me the author probably holds serious misconceptions about Special Relativity itself.

Okay, it's Quora so I shouldn't expect too much, but still I would have expected that at least someone would have given the solution to to something this simple, instead of seeing bunch of people with credentials compete with silly answers. Some of those people are citing their own book about the topic while giving a terrible answer... I mean I'm not that smart, but I solved this problem in my head, while driving. It's really that simple if you actually already understand Special Relativity properly.

What really surprised me was when I went on to check how does Wikipedia see this, and it also doesn't explain the proper solution. There is only one passing mention of the correct solution (kind of, possibly, can't really tell) in the "Brief History" section... with no actual explanation. I guess this is why no one in Quora also knows the answer, but still I'm quite dumbfounded to realize that the actual solution is apparently not very well known at all. I can't find any article actually explaining the correct solution.

Looking at all the bad answers, it seems to me that that there are few different reasons why most people get this so wrong.

One is that many people think about length contraction as something that happens to objects, when more accurately it's what happens to your coordinate system when you change your perspective from one inertial frame to another, and follow Einstein convention for isotropic C. If you think it happens to "objects" because they "move", you might be inclined to bring up stuff like "atoms shrink by space between them does not", and that is completely wrong perspective.

Second is that many authors start to analyze realistic materials and Born rigidity, which to some people perhaps seem like a way out of the paradox in some convoluted way. But that is also a complete red herring. The paradox is a thought experiment, and it has got nothing to do with realistic materials. It's about geometry in terms of special relativity, which ought to produce self-consistent results regardless of inertial frame. Solving Ehrenfest paradox by bringing up realistic materials and centrifugal forces is like solving twin paradox with "planet earth cannot produce enough energy to actually run that experiment".

Third reason is that a mathematical analysis in the framework of special relativity is easiest to do by making certain approximations, which are exactly the approximations leading into wrong answers. That misleading approximation is the idea of placing a number of straight rods along the circumference of the disk, and this approximation is exactly what will give you wrong answers. That's right, Einstein's own analysis is also flawed for the same reason, even though it led into insights that led into General Relativity.

Why that approximation produces a critically wrong answer, and what is the correct answer? I'll explain in a bit...

The correct perspective

First, just to convince the reader that this problem is in fact fully solvable in terms of Special Relativity without any hocus pocus about elastic materials, please be aware that the frame transformation from one inertial frame to another can be conceived as a sort of rotating / scaling of events in spacetime.

Like this;
https://en.wikipedia.org/wiki/File:Lorentz_transform_of_world_line.gif

The dots in that animation represent events as plotted on a spacetime diagram, and the "squishing" of the whole structure represents frame transformation from one inertial frame to another. Some events get pushed "towards the future" and some events get pushed "towards the past". Nothing actually happens to "objects" just because we choose to plot them in a different inertial frame; it's just about how we must plot events, if we are to assume isotropic C, and if we are to remain self-consistent in our mapping between frames.

It really is a good idea to view Special Relativity simply as self-consistent frame transformation rules, and you start seeing that the whole question of length contraction is not about how different observers "see things", or how they "measure things", or "what happens to objects", but rather about how the universe must be plotted in spacetime diagrams when assuming different notions of simultaneity.

In a nutshell, if we switch from one inertial frame representation to another - assuming unique simultaneity to each frame - we must plot the world state "ahead" of us as pushing towards the future (things that had not yet happened in old frame, have already happened in new frame), and conversely the world state "behind" us as pushing towards the past (something that had already happened in old frame, has not yet happened in new frame). Analyzing moving objects like this is what leads into the concept of "length contraction".

Since we are effectively molding the spacetime diagram around, but preserving the same exact light-like connections between events (the causality - the order of connected events - remains unchanged), it should be pretty easy for anyone to see that if it is possible to represent a spinning disk as a "set of events" in one frame, and it would have to also transform along with all the other events in self-consistent manner to any different frame without hiccups. From this perspective, the actual question behind the paradox is simple; how would the spinning disk plot onto a spacetime diagram in terms of different notions of simultaneity?

Even if you can't instantly figure out the exact solution, you should be able to already convince yourself that there is an exact solution out there which would just mold the (events making up the) spinning disk in consistent manner, along with everything else around the situation. What that exact solution is - let's get to it.

The common error

Once the above is understood correctly, next it should be pretty easy to see how the "rigid rods along the circumference" analysis leads you down the wrong path, and at the same time get an glimpse of the correct solution.

• First, imagine a wheel-of-fortune, with pins sticking out from the outer circumference.
• Then we take a spoked wheel (a bicycle wheel), just proper size to snuggly fit inside the pins of the wheel-of-fortune.
• Last, let's enclose the whole two-disk setup inside a box with a snug fit.

The purpose of this setup is to signal us if we are doing something inconsistent with our transformation - if the inner wheel fits inside the larger wheel, and if both wheels fit inside the box in one inertial frame, this must be so in all inertial frames. If it's not, we have performed an error in our analysis.

Now let's take two straight rods - A and B - of exactly the same length, and tie rod A between two spokes of the inner wheel, so that both ends of the rod sit exactly on the circumference of the wheel. Don't worry about how good knots you can make - it's just a thought experiment about geometry!

Now let's set the inner wheel (and inner wheel only) spinning at a relativistic speed in a lab frame.

Since we have rod A spinning along, let's think about what happens if we shoot rod B along an inertial frame so that its path and speed co-incides exactly with the spinning rod A. To make discussion easier, let's look at the setup from the perspective where the rods will co-incide at the "bottom" part of the wheel.

At first glance it might seem like those two rods could be setup to become momentarily stationary in relation to each other, and thus their lengths would have to exactly match. But this would be an error. This is essentially the mistake that still exists in most commonly presented solutions (just see the Wikipedia article to find one example).

The rod that is attached to the spinning wheel is - obviously - never moving in straight line; it is rotating. It's front end is always moving in different direction than its back end (each end is moving parallel to the part of the circumference it touches). So, the first question is, how would we actually plot this situation in the frame of rod B? Remember, this is just about plotting events in terms of the SR convention of simultaneity.

If we plot the external box of the whole setup, in terms of the inertial frame of rod B, it's easy to propose relativistic speeds where the entire box gets plotted as length contracted to shorter length than rod B. The (non-rotating) wheel-of-fortune inside the box must also be mapped inside the box in every frame, and similarly squashed in the direction of motion - snuggly fitting inside the box. And the rotating bicycle wheel must fit also inside the pins of the wheel-of-fortune. It will get plotted also as snuggly fitting inside the wheel-of-fortune. Note though, the spokes will be plotted as curved because it is actively rotating and we are mapping it by a tilted simultaneity plane - this is just the flipside of the coin same coin that makes us map it as squashed.

Basically the internal configuration of our setup cannot change based on what inertial frame we map it from - rod A does not suddenly poke through the walls of the box just because we choose to plot the situation in different inertial frame. If we think it does, we are making an error in our analysis, or using invalid frame transformation. Basically it would imply an inconsistent change in the configuration of our system (some objects transforming in different ways than others - clearly invalid)

If we investigate a moment where the exact middle points of the rods meet in the same inertial frame, and we choose to plot this in terms of rod B's simultaneity, then the "front" end of rod A (in terms of direction of rotation) has already passed the "front" end of rod B (in terms of direction of motion of rod B in lab frame) some time ago. To be more accurate, since it's attached to a rotating wheel, it is also plotted as curving "upwards", already moving away from rod B. That's because this is essentially a temporal transformation, and we have transformed the world state "ahead" if us towards the "future".

And conversely, the world state behind us is plotted as pushing towards the past; the rear ends of the rods have not yet met. And since the rod is constantly rotating, the rear end of rod A is also plotted as curving "upwards", and moving towards rod B.

This is why, if you plot down the shape of the spokes of the wheel from the perspective of rod B, the end result looks like this;

https://en.wikipedia.org/wiki/File:Relativistic_wheels.gif

This is simply a result of plotting the events making up the "supposed world state" as transformed as per the Einstein convention of clock synchronization. A convention for plotting data. Nothing more, nothing less.

The error almost everyone makes is that they view length contraction as something actually occurring to objects themselves. This makes them more prone to assume that it is a good enough approximation to just take rod A as momentarily occupying some inertial frame. If you assume this, and repeat the above thought experiment, you end up in a situation where rod A and rod B can share the inertial frame, and thus are plotted as the same length. Since rod B can be plotted as longer than the entire box, sharing a momentary inertial frame would mean that rod A could momentarily become longer than the box encapsulating the whole contraption, poking a hole through it!

Also, since it was attached to the spokes of the rotating wheel, you'd have to plot the wheel also as having a larger length between two spokes than can be made to fit inside the box wheel-of-fortune, or inside the box. Obviously this result would mean your analysis is completely invalid, plain and simple.

And make no mistake about this - the same error happens no matter how short measurement rods you use. Shorter rods have smaller error, but you can always propose a speed where the error becomes obvious. And with smaller rods there's more of them so the end result of any full analysis is exactly as invalid. Basically you can't have an entire rod in a single inertial frame, while also being attached to the spinning wheel. These are mutually exclusive circumstances.

The same error exists in Einstein's co-rotating observer thought experiment, albeit in more subtle manner. But the point is, the co-rotating observer cannot have a measurement rod in any single inertial frame if that rod is to be also attached to the rotating wheel. The approximation necessary to imagine that situation will always make the analysis invalid for the same reasons as described above.

The correct solution

First clue to understanding how this situation really gets plotted correctly is this; Rod B only shares inertial frame with an infinitesimally thin slice of the spinning wheel. This is true by the very definition of "spinning". Also from the perspective of the lab frame (where the hub of the wheel is stationary), each infinitesimal slice of the spinning wheel is sitting in a different inertial frame, and does not have any "length" assignable to any single inertial frame. This is a simple mathematical fact arising from the very definitions behind special relativity and "spinning wheel".

Second key to understanding this is also associated with properly understanding length contraction as coordinate transformation. Remember when I said "if we switch from one inertial frame representation to another, we must plot the world state "ahead" of us as pushing towards the future, and conversely the world state "behind" us as pushing towards the past.". Note what happens in-between; the world state in the infinitesimal slice exactly perpendicular to the motion does not transform at all! This is btw also why the spokes at the bottom and at the top of the spinning wheel were plotted as straight in the relativistic wheel visualization above. (And I can show why the spokes are plotted as curved with another thought experiment too if anyone is interested)

This leads into the simple fact that, in the above experiment, at the moment when the middle part of rod A and rod B meet, a non-rotating observer sitting at the hub of the wheel co-incides with this infinitesimal plane that is cutting through the wheel, and that observer will agree with simultaneity of all events that co-incide with that infinitesimal plane.

We could repeat the same experiment in any direction, and get the same result, because the wheel is symmetrical. Thus we can see how the observer in the middle will in fact agree with the notion of simultaneity of any infinitesimal slice of the wheel in any direction. Which is the same as saying, the inertial frame where the hub of the spinning wheel is stationary, will agree with the simultaneity notion of any infinitesimal slice of the spinning wheel. Thus, Lorentz length contraction never comes into play - the wheel circumference does not Lorentz contract at all.

So getting back to the original Ehrenfest Paradox, the correct solution is simply to realize that the definitions of Special Relativity do not in any shape or form even imply that the circumference of the wheel becomes Lorentz contracted for a non-rotating observer. Almost everyone instantly assume that it does, because - again - most everyone tend to think about length contraction as something happening to objects as oppose to being a transform to our coordinate system when we plot data down in self-consistent manner. This misconception is so pervasive that it's included in the very opening statement of the paradox without considering its validity. And this misconception makes people prone to analyze the situation by approximating it with measurement rods, always bringing in the exact error I just described, and coming up with all sorts of silly answers.

To summarize;
The non-rotating observer (at rest with the hub of the spinning wheel) actually shares simultaneity with every single infinitesimal part of the spinning wheel in exactly the same way the moving observer shares simultaneity with one infinitesimal slice of the wheel when passing by. Because each "inertial slice" of the wheel is infinitesimally thin (by definition), no part of the wheel actually occupies any length in any single inertial frame, and thus Lorentz contraction never comes into play.

If this still sounds like a strange claim to you, you are forgetting where length contraction comes from. It comes from dynamic notion of simultaneity, and only applies to how we plot spatial distances between two events.

So the TL;DR solution is, the spinning wheel circumference does not length contract at all. Nothing in Special Relativity implies that it does, other than a set of pervasive misconceptions and invalid simplifications, whose application just lead into erratic and inconsistent conclusions. Proper application of Special Relativity will show that the spinning disk actually has got exactly the same geometry as the non-spinning disk!

And as it turns out, all of the "commonly accepted" (or maybe there isn't one) solutions I can find are invalid. Contrary to Wikipedia, the geometry of the spinning disk is still exactly euclidean. As in, $\frac{circumference}{diameter} = \pi$ also for relativistic spinning disk. The only way to get the commonly cited result of $\frac{circumference}{diameter} = \pi \sqrt{1- (\omega R )^2 / c^2 }$ is to make the error in approximation I've explained above! And yes Einstein also made this same error.

Do note that this solution is all about how geometry gets plotted in terms of special relativity - it's not about how to set a wheel in rotation or about realistic materials. This solution simply arises from how objects get plotted into different inertial frames in self-consistent manner, following exactly the definitions of Special Relativity, and thus it is also exactly the correct solution to the original Ehrenfest Paradox.

Sorry about the length of this. I didn't want to just state what the correct solution is without explaining it in sufficient detail to give everyone a chance to convince themselves about this. Because it seems like the misconception here is so common that the actual end result probably just sound immediately wrong to most people until they think it through themselves.

I am going to revisit this because I feel your knowledge is worth reading in a clear consciousness... Been a long week so to speak

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# 300 Followers Problem.

Algebra Level 4

Over all sets of natural numbers $$(k_1 , k_2 , k_3,...., k_{100})$$ , how many possible values are there of

$\large\left(\prod_{i=1}^{100}k_{i}\right)^{\left ( \sum_{i=1}^{100}k_{i} \right )} \pmod{ k_{100} }$

×


text:
## IACR News item: 27 January 2023

###### Senyang Huang, Rui Qi Sim, Chitchanok Chuengsatiansup, Qian Guo, Thomas Johansson
ePrint Report
In this paper, we present the first chosen-ciphertext (CC) cache-timing attacks on the reference implementation of HQC. We build a cache-timing based distinguisher for implementing a plaintext-checking (PC) oracle. The PC oracle uses side-channel information to check if a given ciphertext decrypts to a given message. This is done by identifying a vulnerability during the generating process of two vectors in the reference implementation of HQC. We also propose a new method of using PC oracles for chosen-ciphertext side-channel attacks against HQC, which may have independent interest.

We show a general proof-of-concept attack, where we use the Flush&Reload technique and also derive, in more detail, a practical attack on an HQC execution on Intel SGX, where the Prime&Probe technique is used. We show the exact path to do key recovery by explaining the detailed steps, using the PC oracle. In both scenarios, the new attack requires $53,857$ traces on average with much fewer PC oracle calls than the timing attack of Guo et al. CHES 2022 on an HQC implementation.

Additional news items may be found on the IACR news page.


text:
# Newton's laws of motion

Newton's First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica.
Classical mechanics
$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \vec{v})$
Newton's Second Law
History of ...
Formulations
Newtonian mechanics
Lagrangian mechanics
Hamiltonian mechanics

Newton's laws of motion are three physical laws that form the basis for classical mechanics, directly relating the forces acting on a body to the motion of the body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, first published on July 5, 1687.[1] Newton used them to explain and investigate the motion of many physical objects and systems.[2] For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion.

First law
There exists a set of inertial reference frames relative to which all particles with no net force acting on them will move without change in their velocity. This law is often simplified as "A body persists its state of rest or of uniform motion unless acted upon by an external unbalanced force." Newton's first law is often referred to as the law of inertia. So in other words "every object in motion will stay in motion until acted upon by an outside force."
Second law
Observed from an inertial reference frame, the net force on a particle of constant mass is proportional to the time rate of change of its linear momentum: F = d(mv)/dt. This law is often stated as, "Force equals mass times acceleration (F = ma)": the net force on an object is equal to the mass of the object multiplied by its acceleration.
Third law
Whenever a particle A exerts a force on another particle B, B simultaneously exerts a force on A with the same magnitude in the opposite direction. The strong form of the law further postulates that these two forces act along the same line. This law is often simplified into the sentence, "To every action there is an equal and opposite reaction."

In the given interpretation mass, acceleration, momentum, and (most importantly) force are assumed to be externally defined quantities. This is the most common, but not the only interpretation: one can consider the laws to be a definition of these quantities. Notice that the second law only holds when the observation is made from an inertial reference frame, and since an inertial reference frame is defined by the first law, asking a proof of the first law from the second law is a logical fallacy. At speeds approaching the speed of light the effects of special relativity must be taken into account.[3]

### Newton's first law: law of inertia

Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare. Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed. [4]

Newton's first law is also called the law of inertia. In a simplified form, it states that if the vector sum of all forces (also known as the net force) acting on an object is zero, then the state of motion of the object does not change. In particular: Newton's first law: An object at rest remains at rest and an object in motion will remain in motion unless acted on by an unbalanced force.

• An object that is not moving will not move until a net force acts upon it.
• An object that is moving will not change its velocity (accelerate) until a net force acts upon it.

The first point needs no comment, but the second seems to violate everyday experience. A hockey puck sliding along a table doesn't move forever; rather, it slows and eventually comes to a stop. According to Newton's laws, though, the hockey puck does not stop of its own accord, but because of a force applied in the opposite direction to the direction of motion. That force is easily identified as a frictional force between the table and the puck. In the absence of such a force, as approximated by an air hockey table or ice rink, the puck's motion would not slow.

There are no perfect demonstrations of the law, as friction usually causes a force to act on a moving body, and even in outer space gravitational forces act and cannot be shielded against, but the law serves to emphasize the elementary causes of changes in an object's state of motion.

The above treatment of Newton's first law is an over-simplification, though. A more sophisticated approach to the law of inertia is given by:

There is a class of frames of reference (called inertial frames) relative to which the motion of a particle not subject to forces is a straight line. [5]

Newton placed the law of inertia first to establish frames of reference for which the other laws are applicable (see Gailili & Tseitlin[6], or Woodhouse[5]). Such frames are called inertial frames.

To understand why the laws are restricted to inertial frames, consider a ball at rest within an accelerating body: an airplane on a runway will suffice for this example. From the perspective of anyone within the airplane (that is, from the airplane's frame of reference when put in technical terms) the ball will appear to move backwards as the plane accelerates forwards (the same feeling as being pushed back into your seat as the plane accelerates). This motion appears to contradict Newton's second law as, from the point of view of the passengers, there appears to be no force acting on the ball that would cause it to move. The reason why there is in fact no contradiction to the second law is because Newton's second law (without modification) is not applicable in this situation: Newton's first law does not apply because the stationary ball does not remain stationary. Thus, it is important to establish whether the various laws are applicable or not, inasmuch as they are not applicable in all situations.[7]

#### History of the Law of Inertia

Newton's first law is a restatement of what Galileo had already described and Newton gave credit to Galileo. It differs from Aristotle's view that all objects have a natural place in the universe. Aristotle believed that heavy objects like rocks wanted to be at rest on the Earth and that light objects like smoke wanted to be at rest in the sky and the stars wanted to remain in the heavens. However, a key difference between Galileo's idea and Aristotle's is that Galileo realized that force acting on a body determines acceleration, not velocity. This insight leads to Newton's First Law—no force means no acceleration, and hence the body will maintain its velocity.

The law of inertia apparently occurred to several different natural philosophers and scientists independently. The inertia of motion was described in the 3rd century BC by the Chinese philosopher Mo Tzu, and in the 11th century by the Muslim scientists, Alhazen[8] and Avicenna.[9] The 17th century philosopher René Descartes also formulated the law, although he did not perform any experiments to confirm it.

### Newton's second law

Explanation of Newton's second law by NASA employee.

Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur. The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.

In Motte's 1729 translation (from Newton's Latin), the second law of motion reads:[10]

LAW II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. — If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

Using modern symbolic notation, Newton's second law can be written as a vector differential equation:

$\mathbf F_{\text{net}} = {\mathrm{d}(m \mathbf v) \over \mathrm{d}t} = m \ \frac{d \mathbf v}{dt}$

where F is the force vector, m is the mass of the body, v is the velocity vector and t is time.

The product of the mass and velocity is the momentum of the object (which Newton himself called "quantity of motion"). Therefore, this equation expresses the physical relationship between force and momentum for systems of constant relativistic mass. The equation implies that, under zero net force, the momentum of a system is constant; however, any mass that enters or leaves the system will cause a change in system momentum that is not the result of an external force. This equation does not hold in such cases. See open systems.

It should be noted that, as is consistent with the law of inertia, the time derivative of the momentum is non-zero when the momentum changes direction, even if there is no change in its magnitude. See time derivative.[11]

Since the mass of the system is constant, this differential equation can be rewritten in its simpler and more familiar form:

$\mathbf F = m \mathbf a$

where

$\mathbf a = \frac{\mathrm{d} \mathbf v}{\mathrm{d}t}$

is the acceleration.

A verbal equivalent of this is "the acceleration of an object is proportional to the force applied, and inversely proportional to the mass of the object". In general, at slow speeds (slow relative to the speed of light), the relationship between momentum and velocity is approximately linear. Nearly all speeds within the human experience fall within this category. At higher speeds, however, this approximation becomes increasingly inaccurate and the theory of special relativity must be applied.

#### Impulse

The term impulse is closely related to the second law, and historically speaking is closer to the original meaning of the law.[12]The meaning of an impulse is as follows:[13][14]

An impulse occurs when a force F acts over an interval of time Δt and is given by $\int_{\Delta t} \mathbf F \,\mathrm{d}t$.

The words motive force were used by Newton to describe "impulse" and motion to describe momentum; consequently, a historically closer reading of the second law describes the relation between impulse and change of momentum. That is, a mathematical rendering of the original wording resembles a finite difference version of the second law, such as

$\mathbf{I} = \Delta\mathbf{p} = m\Delta\mathbf{v}$

where I is the impulse, Δp is the change in momentum, m is the mass, and Δv is the change in velocity.

The analysis of collisions and impacts uses the impulse concept.[15]

#### Relativity

Taking special relativity into consideration, the law of resultant force can be put in terms of acceleration as follows:[16]

\begin{align} \mathbf F &= \frac {\mathrm{d} \mathbf {p} }{\mathrm{d} t} = \frac {\mathrm{d} (m \mathbf {v}) }{\mathrm{d} t} \\ &= m \frac {\mathrm{d}\mathbf{v}} {\mathrm{d}t} + \frac {\mathrm{d} m }{\mathrm{d} t}\mathbf {v} \\ &= m \frac {\mathrm{d}\mathbf{v}} {\mathrm{d}t} + \frac{1}{c^2} \frac {\mathrm{d} E }{\mathrm{d} t}\mathbf{v} \\ &= m \frac {\mathrm{d}\mathbf{v}} {\mathrm{d}t} + \frac { \mathbf{F} \cdot \mathbf{v} }{c^2}\mathbf{v} \end{align}

where the famous result for the energy E = mc2 is used (where c is the speed of light in free space). Note that this is an approximate formula, where γ in the exact expression of relativistic energy, E = γmc2, is approximately one. The relation

$\frac {\mathrm{d}E} {\mathrm{d}t} = \mathbf F \cdot \mathbf v$

describes the work done by the force per unit time. Here F·v is the vector dot product.

This equation can be rearranged to form the modified force law

$m \frac {\mathrm{d}\mathbf{v}}{\mathrm{d}t} = \mathbf F - \frac {\mathbf F \cdot \mathbf v}{c^2} \mathbf v \text{,}$

which shows that although the change of momentum is in the direction of the force, in general the acceleration of the mass is not in the direction of the force. However, when the speed of the moving body is much lower than the speed of light, the equation above reduces to the familiar F=ma.

#### Open systems

So-called variable mass systems that are not closed systems, like a rocket burning fuel and ejecting spent gases, can not be directly treated by making mass a function of time in the second law.[17][18] The reasoning, given in An Introduction to Mechanics by Kleppner and Kolenkow and other modern texts, is that Newton's second law applies fundamentally to particles. In classical mechanics, particles by definition have constant mass. In case of well-defined systems of particles, Newton's law can be extended by summing over all the particles in the system. In this case, we have to refer all vectors to the center of mass. Applying the second law to extended objects implicitly assumes the object to be a well-defined collection of particles. However, 'variable mass' systems like a rocket or a leaking bucket do not consist of a set number of particles. They are not well-defined systems. Therefore Newton's second law can not be applied to them directly. The naïve application of F = dp/dt will usually result in wrong answers in such cases. However, applying the conservation of momentum to a complete system (such as a rocket and fuel, or a bucket and leaked water) will give unambiguously correct answers.

The term "complete system" in the last sentence refers to a larger, constant mass system that accounts for all the particles. This effectively means that F = dp/dt is true only for constant mass systems. Expressed in the form F = ma, however, it correctly describes the motion of any particle or system regardless of varying mass, provided that the resultant force F is taken to include the thrust of any mass entering or leaving the system.

### Newton's third law: law of reciprocal actions

Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi. For a force there is always an equal and opposite reaction: or the forces of two bodies on each other are always equal and are directed in opposite directions.

Newton's third law. The skaters' forces on each other are equal in magnitude, but act in opposite directions.
Explanation of Newton's third law by NASA employee.

A more direct translation is:

LAW III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. — Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinges upon another, and by its force changes the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say, if the bodies are not hindered by any other impediments. For, as the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.

In the above, as usual, motion is Newton's name for momentum, hence his careful distinction between motion and velocity.

The Third Law means that all forces are interactions, and thus that there is no such thing as a unidirectional force. If body A exerts a force on body B, simultaneously, body B exerts a force of the same magnitude body A, both forces acting along the same line. As shown in the diagram opposite, the skaters' forces on each other are equal in magnitude, but act in opposite directions. Although the forces are equal, the accelerations are not: the less massive skater will have a greater acceleration due to Newton's second law. It is important to note that the action and reaction act on different objects and do not cancel each other out. The two forces in Newton's third law are of the same type (e.g., if the road exerts a forward frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third law predicts for the tires pushing backward on the road).

Newton used the third law to derive the law of conservation of momentum;[19] however from a deeper perspective, conservation of momentum is the more fundamental idea (derived via Noether's theorem from Galilean invariance), and holds in cases where Newton's third law appears to fail, for instance when force fields as well as particles carry momentum, and in quantum mechanics.

## Importance and range of validity

Newton's laws were verified by experiment and observation for over 200 years, and they are excellent approximations at the scales and speeds of everyday life. Newton's laws of motion, together with his law of universal gravitation and the mathematical techniques of calculus, provided for the first time a unified quantitative explanation for a wide range of physical phenomena.

These three laws hold to a good approximation for macroscopic objects under everyday conditions. However, Newton's laws (combined with Universal Gravitation and Classical Electrodynamics) are inappropriate for use in certain circumstances, most notably at very small scales, very high speeds (in special relativity, the Lorentz factor must be included in the expression for momentum along with rest mass and velocity) or very strong gravitational fields. Therefore, the laws cannot be used to explain phenomena such as conduction of electricity in a semiconductor, optical properties of substances, errors in non-relativistically corrected GPS systems and superconductivity. Explanation of these phenomena requires more sophisticated physical theory, including General Relativity and Relativistic Quantum Mechanics.

In quantum mechanics concepts such as force, momentum, and position are defined by linear operators that operate on the quantum state; at speeds that are much lower than the speed of light, Newton's laws are just as exact for these operators as they are for classical objects. At speeds comparable to the speed of light, the second law holds in the original form F = dp/dt, which says that the force is the derivative of the momentum of the object with respect to time, but some of the newer versions of the second law (such as the constant mass approximation above) do not hold at relativistic velocities.

## Relationship to the conservation laws

In modern physics, the laws of conservation of momentum, energy, and angular momentum are of more general validity than Newton's laws, since they apply to both light and matter, and to both classical and non-classical physics.

This can be stated simply, "Momentum, energy and angular momentum cannot be created or destroyed."

Because force is the time derivative of momentum, the concept of force is redundant and subordinate to the conservation of momentum, and is not used in fundamental theories (e.g. quantum mechanics, quantum electrodynamics, general relativity, etc.). The standard model explains in detail how the three fundamental forces known as gauge forces originate out of exchange by virtual particles. Other forces such as gravity and fermionic degeneracy pressure also arise from the momentum conservation. Indeed, the conservation of 4-momentum in inertial motion via curved space-time results in what we call gravitational force in general relativity theory. Application of space derivative (which is a momentum operator in quantum mechanics) to overlaping wave functions of pair of fermions (particles with semi-integer spin) results in shifts of maxima of compound wavefunction away from each other, which is observable as "repulsion" of fermions.

Newton stated the third law within a world-view that assumed instantaneous action at a distance between material particles. However, he was prepared for philosophical criticism of this action at a distance, and it was in this context that he stated the famous phrase "I feign no hypotheses". In modern physics, action at a distance has been completely eliminated, except for subtle effects involving quantum entanglement. However in modern engineering in all practical applications involving the motion of vehicles and satellites, the concept of action at a distance is used extensively.

Conservation of energy was discovered nearly two centuries after Newton's lifetime, the long delay occurring because of the difficulty in understanding the role of microscopic and invisible forms of energy such as heat and infra-red light.

## References and notes

1. ^ See the Principia on line at Andrew Motte Translation
2. ^ Andrew Motte translation of Newton's Principia (1687) Axioms or Laws of Motion
3. ^ In the second law, m must be treated as the relatistic mass, producing the relativistic expression for momentum, and the third law must be modified to allow for the finite signal propagation speed between distant interacting particles.
4. ^ Isaac Newton, The Principia, A new translation by I.B. Cohen and A. Whitman, University of California press, Berkeley 1999.
5. ^ a b NMJ Woodhouse (2003). Special relativity. London/Berlin: Springer. p. 6. ISBN 1-85233-426-6.
6. ^ Galili, I. & Tseitlin, M. (2003), "Newton's first law: text, translations, interpretations, and physics education.", Science and Education 12 (1): 45–73
7. ^ On a more technical note, although Newton's laws are not applicable on non-inertial frames of reference, such as the accelerating airplane, they can be made to do so with the introduction of a "fictitious force" acting on the entire system: basically, by introducing a force that quantifies the anomalous motion of objects within that system (such as the ball moving without an apparent influence in the example above
8. ^ Abdus Salam (1984), "Islam and Science". In C. H. Lai (1987), Ideals and Realities: Selected Essays of Abdus Salam, 2nd ed., World Scientific, Singapore, p. 179-213.
9. ^ Fernando Espinoza (2005). "An analysis of the historical development of ideas about motion and its implications for teaching", Physics Education 40 (2), p. 141.
10. ^ According to Maxwell in Matter and Motion, Newton meant by motion "the quantity of matter moved as well as the rate at which it travels" and by impressed force he meant "the time during which the force acts as well as the intensity of the force". See Harman and Shapiro,cited below.
11. ^ The use of algebraic expressions became popular during the 18th century, after Newton's death, while vector notation dates to the late 19th century. The Principia expresses mathematical theorems in words and consistently uses geometrical rather than algebraic proofs.
12. ^ I Bernard Cohen (Peter M. Harman & Alan E. Shapiro, Eds) (2002). The investigation of difficult things : essays on Newton and the history of the exact sciences in honour of D.T. Whiteside. Cambridge UK: Cambridge University Press. p. 353. ISBN 052189266X.
13. ^ Hannah, J, Hillier, M J, Applied Mechanics, p221, Pitman Paperbacks, 1971
14. ^ Raymond A. Serway, Jerry S. Faughn (2006). College Physics. Pacific Grove CA: Thompson-Brooks/Cole. p. 161. ISBN 0534997244.
15. ^ WJ Stronge (2004). Impact mechanics. Cambridge UK: Cambridge University Press. p. 12 ff. ISBN 0521602890.
16. ^ C Møller (1976). The Theory of Relativity (Second Edition ed.). Oxford UK: Oxford University Press. p. 70-75. ISBN 019560539X.
17. ^ Halliday; Resnick. Physics. 1. pp. 199. "It is important to note that we cannot derive a general expression for Newton's second law for variable mass systems by treating the mass in F = dP/dt = d(Mv) as a variable. [...] We can use F = dP/dt to analyze variable mass systems only if we apply it to an entire system of constant mass having parts among which there is an interchange of mass."  [Emphasis as in the original]
18. ^ Kleppner; Kolenkow. An Introduction to Mechanics. pp. 133–134. "Recall that F = dP/dt was established for a system composed of a certain set of particles...it is essential to deal with the same set of particles throughout the time interval...Consequently, the mass of the system can not change during the time of interest."
19. ^ Newton, Principia, Corollary III to the laws of motion


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Home > CC3MN > Chapter 6 Unit 10 > Lesson INT1: 6.4.2 > Problem6-143

6-143.

Solve each equation below for the indicated variable. Show all steps.

1. Solve for $x$: $2x + 22 = 12$

$x = −5$

1. Solve for $y$: $2x − y = 3$

Add $y$ to both sides of the equation.

How can you move $3$ to isolate $y$ on the right side of the equation?

1. Solve for $x$: $2x + 15 = 2x − 15$

If you subtract $2x$ from both sides of the equation, what kind of statement is left? Is there any value of $x$ that will make this equation true?

1. Solve for $y$: $6x + 2y = 10$

$y = 5 − 3x$