text:
## Expression simplification

I have this term

Sqrt[(B Bc)/(A^2-B Bc)]Sqrt[B Bc+A(-A+Sqrt[A^2-B Bc])]Sqrt[-B Bc+A(A + Sqrt[A^2-B Bc])],

which actually can be simplified to B*Bc. However, it seems Mathematica cannot simplify it. Any suggestions on how I’d be able to do that? Thanks.

## Substitution of expressions in a symbolic expression

I define tables of symbolic variables in the following form (for convenience)

X = Table[Symbol["x" <> ToString[i]], {i, 1, num}]; Y = Table[Symbol["y" <> ToString[j]], {j, 1, num}];

And after that, in cycles, I create some expressions. For example, here is one of them

Expon := Exp[ - ((X[[1]] * Y[[1]]) / 4) ];  For[i = 2, i <= num, i++,  Expon = Expon * Exp[ - ((X[[i]] * Y[[i]]) / 4)] ]

After that, I want to act by some differential operator on my symbolic expression (let’s call it $$\Psi$$) and substitute in the final expression some tables of numbers X1 and Y1 (here they are not symbolic, but filled by real numbers). I tried to use ReplaceAll ./ command, but it didn’t work. Could you tell me please, how can I substitute two or more tables of real numbers in symbolic expression? Long story short, how to calculate something like $$\Psi(X1, Y1)$$?

## Maximize an expression with respect to a variable and minimize it with respect to other variables

I started to use Mathematica a few time ago. I want to minimize the following expression (function of $$l,p,q,r,c$$) with respect to variables $$l, p, q, r$$ and then maximize the result obtained with respect to variable $$c$$. However, when I try to obtain an expression function of $$c$$ to maximize later using Minimize, I do not get any result because it takes too long. How can I solve this issue?

Minimize[{(((l^2/2)*(1-(1/4-c))+(l*p)*(1-1/4)+(l*q)*(1-(1/4+c))+(l*r)*(1-1/2)+(p^2/2)*(1-c)+(p*q)*(1-2*c)+(p*r)*(1-(1/4+c))+(q^2/2)*(1-c)+(q*r)*(1-1/4)+(r^2/2)*(1-(1/4-c)))/((l^2/2)*(1-(1/4-c))+(l*p)*(1-1/4)+(l*q)*(1-(1/4-c))+(l*r)*(1-0)+(p^2/2)*(1-c)+(p*q)*(1-(1/4-c))+(p*r)*(1-0)+(q^2/2)*(1-(1/4-c))+(q*r)*(1-1/4)+(r^2/2)*(1-0))), l >= 1, p >= 0, q >= 0, r >= 0, l + p + q + r == 1000000, 1/7<c<1/5}, {l, p, q, r}]

## Test two LTL expression trees for equivalence

Is there an algorithm on how to check if two LTL expressions (represented as binary trees) are semantically equivalent? Since there are many smaller equivalences such as $$a\Rightarrow b \equiv \neg a \vee b$$ or $$F(a) \equiv true U a$$ as well as commutativity, distributivity, etc. that need to be considered.
My initial idea was to create the truth table for both expressions and compare them. But then the temporal operators would not be taken into account. Creating and comparing the automaton for each expression sounds like it would be rather inefficient.

Is there a better way to do this?

## How to apply Arden’s theory to determine a regular expression

If P=(ab) and Q=(a)* How do I use Arden’s theorem to determine the regular expression R. I’m not sure if I am supposed to just substitute the values of P and Q in the equation R= Q + RP. Also how would I use that to check that R satisfies Arden’s equation

## Problem using ‘Regular expression’ in order to split characters of a column when there is no delimiter between them

I have a table with below structure:

create table TBL_TEST (   col_id   NUMBER,   col_name VARCHAR2(500) )

Some example data :

col_id | col_name    -----------------   1    | aetnap           2    | elppa          3    | ananab

What I need to do is to split characters of column col_name for each col_id for example for col_id=1 we must have :

col_id | col_name    -----------------   1    | a   1    | e   1    | t   1    | n   1    | a   1    | p

this query is fine when there is only one record in the table :

SELECT col_id, REGEXP_SUBSTR(col_name, '[a-z]{1}', 1, LEVEL) AS VAL   FROM tbl_test t CONNECT BY REGEXP_SUBSTR(col_name, '[a-z]{1}', 1, LEVEL) is not null

but as soon as I insert another record in the table (say col_id=2 and col_id=3) I can not have the desired result. I want to know two things:

1. Why is this query works fine for one record and it does not for more ?
2. what is the best way to split the characters when there is no delimiter between them?

## How to write “∀x.F(x)” for “F(x)=λx.Φ(x)” in one expression (sequel from question about “∀(λφ. (φ x m→ φ y))”?

This question is sequel from How to understand quantifier without predication " ∀(λφ. (φ x m→ φ y))"? which further explains the notation and context.

So – I have anonymous Boolean-valued function F(y)=λx.Φ(x) (of course, y and x point to the same variable, I just used different syntactic names, to point out, that x is bound variable) and I would like to write the statement, that F(x) is true for all the values of the argument and it can be written ∀x.F(x). But F(x) is named function, but I would like to write the same expression for the anonymous function that uses lambda, so I am with my suggestion: ∀x.λx.Φ(x) or ∀x.λy.Φ(y)? And apparently they both are wrong.

What I am trying to achieve? I just want to build parser for language that is declared in https://www.isa-afp.org/browser_info/current/AFP/GoedelGod/GoedelGod.html. This language contains expressions like [∀(λΦ. P (λx. m¬ (Φ x)) m→ m¬ (P Φ))].

I am using ANTLR grammar for lambda calculus https://github.com/antlr/grammars-v4/blob/master/lambda/lambda.g4 and I understand that the 1) quantifiers; 2) logical connectives; 3) arithmetic functions are just another lambda functions (it is just syntactic sugar that they are written in the specific non-lambda syntax/prefix form etc.) and as such I express them in the existing lambda.g4 grammar https://github.com/antlr/grammars-v4/blob/master/lambda/lambda.g4. So – my first step is to write the cited expressions with the named functions and then I will just replace them with anonymous functions because lambda.g4 has no options to introduce named functions. But it is so confusing to write anonymous function and the quantifier function for the same argument.

Just side question – maybe there is better ANTLR grammar for lambda calculus with syntactic sugar for quantifiers and connectives?

## Time computation problems with “Maximize” expression

I just started to use Mathematica a few weeks ago. I am afraid there is something that does not work on my laptop because, for the following simple command, it takes too much time. Do you know if there is a mistake in this command syntax? Thank you in advance for your help.

Maximize[{(g/e)*Sqrt[((g – e)^2 + (f – h)^2)], 0 <= e <= 1, 0 <= f <= 1, e^2 + f^2 == 1, 0 <= g <= e, 0 <= h <= f}, {e, f, g, h}]

## Is there a function to convert a trigonmetric expression to an algebraic expression suitable for ComplexPlot?

According to H. A Priesley:

"both $$cosθ$$ and $$sinθ$$ can be written as simple algebraic functions of $$z$$":

$$cosθ =\frac{1}{2} (z + \frac{1}{z}) \tag1$$ $$sinθ =\frac{1}{2i} (z − \frac{1}{z})\tag2$$

Which is very handy but I’m not sure what function will convert say $$\frac{\sinh(ax)}{\sinh(bx)}\,$$ ot $$\frac{x\cos ax}{1+x^2}\coth \frac{\pi x}{4}$$ to similar algebraic functions. The only function I can find is TrigtoExp, but this does not include the needed substution $$z = \exp iθ$$ to work with function ComplexPlot.

Q1. Is there a function to convert a trigonmetric expression to an algebraic expression suitable for ComplexPlot?

Q2. If not what is the best way to get around this?

## Problem plotting expression involving Generalized hypergeometric functions $_2F_2 \left(.,.,. \right)$

I’m trying to plot a graph for the following expectation

$$\mathbb{E}\left[ a \mathcal{Q} \left( \sqrt{b } \gamma \right) \right]=a 2^{-\frac{\kappa }{2}-1} b^{-\frac{\kappa }{2}} \theta ^{-\kappa } \left(\frac{\, _2F_2\left(\frac{\kappa }{2}+\frac{1}{2},\frac{\kappa }{2};\frac{1}{2},\frac{\kappa }{2}+1;\frac{1}{2 b \theta ^2}\right)}{\Gamma \left(\frac{\kappa }{2}+1\right)}-\frac{\kappa \, _2F_2\left(\frac{\kappa }{2}+\frac{1}{2},\frac{\kappa }{2}+1;\frac{3}{2},\frac{\kappa }{2}+\frac{3}{2};\frac{1}{2 b \theta ^2}\right)}{\sqrt{2} \sqrt{b} \theta \Gamma \left(\frac{\kappa +3}{2}\right)}\right)$$ where $$a$$ and $$b$$ are constant values, $$\mathcal{Q}$$ is the Gaussian Q-function, which is defined as $$\mathcal{Q}(x) = \frac{1}{\sqrt{2 \pi}}\int_{x}^{\infty} e^{-u^2/2}du$$ and $$\gamma$$ is a random variable with Gamma distribition, i.e., $$f_{\gamma}(y) \sim \frac{1}{\Gamma(\kappa)\theta^{\kappa}} y^{\kappa-1} e^{-y/\theta}$$ with $$\kappa > 0$$ and $$\theta > 0$$.

This equation was also found with Mathematica, so it seems to be correct. I’ve got the same plotting issue with Matlab.

Follows some examples, where I have checked the analytical results against the simulated ones.

When $$\kappa = 12.85$$, $$\theta = 0.533397$$, $$a=3$$ and $$b = 1/5$$ it returns the correct value $$0.0218116$$.

When $$\kappa = 12.85$$, $$\theta = 0.475391$$, $$a=3$$ and $$b = 1/5$$ it returns the correct value $$0.0408816$$.

When $$\kappa = 12.85$$, $$\theta = 0.423692$$, $$a=3$$ and $$b = 1/5$$ it returns the value $$-1.49831$$, which is negative. However, the correct result should be a value around $$0.0585$$.

When $$\kappa = 12.85$$, $$\theta = 0.336551$$, $$a=3$$ and $$b = 1/5$$ it returns the value $$630902$$. However, the correct result should be a value around $$0.1277$$.

Therefore, the issue happens as $$\theta$$ decreases. For values of $$\theta > 0.423692$$ the analytical matches the simulated results. The issue only happens when $$\theta <= 0.423692$$.

I’d like to know if that is an accuracy issue or if I’m missing something here and if there is a way to correctly plot a graph that matches the simulation. Perhaps there is another way to derive the above equation with other functions or there might be a way to simplify it and get more accurate results.


text: Sample Servers. To change the sample rate from 44.1 to 48 kHz, you have to determine a rational number (ratio of integers), P/Q, such that P/Q times the original sample rate, 44100, is equal to 48000 within some specified tolerance. Voice Mod changes your voice in real-time and has a pretty awesome collection of voice options and effects. 10 bits. For sample problems 3-6, use Formula 6 to answer the questions $H_2O \longrightarrow 2H_2+ O_2 \label{6}$ *Assume the reaction occurs at constant temperature. Will better phone hardware be compensated with more intensive tasks? This the best voice changer. Got a frame at sample rate 16000, in an MP3 with sample rate 44100. This is great if you want to stretch the audio out and slow it down in your DAW. r/voicemod. The program designed to be as simple as possible, even with short instructions to help new users. Voicemeeter Potato is a mixer program for your windows computer and is the newest version of the software. how can i change this when it is greyed out? Sample rate is how many times your computer takes a snapshot of the audio every second. Sample rate, sometimes called sample frequency, is used to define the number of samples carried per second for audio file. It is important to note that the program works with a sample rate. College football: An epic bungle alters playoff chase, Scary scene in college hoops game after player collapses, Black police officers believe that Black lives matter, 'SNL': McKinnon's 'hunky' Fauci pelted with bras. Then it doesn't matter if they are changed because the sample rate will always be the same. One is 44.1, the other is 96 kHz, even though I've changed the settings (see below) to 48. An easy way to open it is to type + then type Audio Midi Setup in the popup. The following information explains how to change the sample rate of a Scarlett Solo/2i2/2i4 2nd Gen Scarlett audio interface. What technology will make the biggest difference in our everyday lives a decade from now, and how. Then change all of your devices to use 44100 for the sample rate. Price action in global asset markets continued to reveal a level of risk wariness among investors. This will save on processing power during the recording. Now that you have changed the sampling rate, the next step is to change the 24 bit rate to a lower rate. I am trying to use VoiceMod (voice changer/soundboard) with my new USB condenser microphone. A moving image consists of lots of still photos shown very quickly in concession (frames per second). It’s perfect for ASMR or video calls! It was able to change how you use multiple sound emissions. Linear Mode; Switch to Hybrid Mode; Switch to Threaded Mode; Thread. Voicemod Pro Crack With License Code Download. This is the first and only thing advertised as an anime voice changer. 2. Sorry If … Voicemod Pro Free is a Free Voice Changer that should work with any streaming platform or chat application, enabling you to change how you sound using a few audio effects. How to change sample rate in FFMPEG + Reply to Thread. Die Abtastratenkonvertierung (englisch Sample rate conversion oder Resampling) beschreibt im Rahmen der digitalen Signalverarbeitung die Umsetzung eines Digitalsignals zwischen zwei verschiedenen Abtastraten unter möglichst vollständiger Beibehaltung der Signalinformation. I am using Apple airpods and it says I need a higher sample rate, but when I go to edit my sample rate, it is grayed out. So, It is very easy to set up, even for novices. It allowing you to change the way you sound using many audio effects. An easy way to open it is to type + then type Audio Midi Setup in the popup. Is there any way to change this or use voicemod with my current (8000 Hz) sample rate? Voicemod is a simple free voice changer for Windows to chat online. Mp3FileReader does not support sample rate changes I saw this article"How to convert between (most) audio formats in .NET", but I didn't find the solution for the issue. For the best quality, you’re going to want to go with CBR, which also means your MP3 file will be a bit larger. Will factory reset remove all my data from my tablet? If you are experiencing a sound quality issue, such as garbling, out of sync tracks, or popping, it could be a mismatched sample rate. Using this sound changer and soundboard, you can create different types of voice skins. Voicemod Pro Activation Code allows you to change the sound using many audio effects. Myinstants is where you discover and create instant sound buttons. Changing the buffer size/sample rate of an ASIO Audio Interface. Voicemod Pro key is the stunning utility application program that lets the user customize your view rather than in online tools and games like as streamers using voice maker and so on. Along with these letters, it’s more that it also has a brief guide to help first-time clients. I want to sound like an anime girl. Downloaded to change my voice to an anime girl. Shouldn't Voicemeeter be on the same sample rate? For example, if the rate input on the DAQmx Timing function is 1000 (Hz), then the number of samples per channel on the DAQmx Read function should be no more than 100. A video tutorial was available to help you. When i go to change the sample rate, it is greyed out and says "1 channel, 24 bit, 192000 Hz" when it need it to be on 44100 Hz. It is also essential to note that the program is to manage the work with one sample rate. Voicemod is a user-friendly program that should work with any streaming platform or chat application, enabling you to change the way you sound using a number of audio effects. Announcement published December 7, 2020. To change the sample rate from 44.1 to 48 kHz, you have to determine a rational number (ratio of integers), P/Q, such that P/Q times the original sample rate, 44100, is equal to 48000 within some specified tolerance. The Virtual Audio Mixer VoiceMeeter is able to mix audio sources coming from audio device as well as from audio applications. You can change the sample rate of your audio input and output devices using the Audio Midi Setup program that comes with MacOS. On Windows 10, to change the sample rate for Dante Virtual Soundcard, you must first change it in Dante Controller, and then change it for each DVS device in the Windows Playback and Recording device settings. Pros: really basic voices Cons: Makes you pay for whats advertised as Free. You can do the same to the sample rate for audio output in the Sound dialogue. View Profile View Forum Posts Private Message Member Join Date Jul 2006 … How to change the sample rate of your audio device / microphone. When performing Programmed I/O, the rate at which the samples are read from the hardware FIFO on the DAQ board to the programming … flac error; Unsupported sample rate; Like Share Tweet Share Share This topic has been closed for further comments. 0 comments. 2020-12-08 12:00 UTC. Voicemod Pro includes a short guide to help first-time users. Voicemod requires the sample rate to be anywhere between 16khz and 48khz. Risk wariness among investors the License Key 2020 each tool in concession ( frames per ). Moving image consists of lots of still photos shown very quickly in concession ( frames per second ) is to... And tools including a currency conversion calculator, historical rates and tools including a currency conversion calculator, historical and. Quickly in concession ( frames per second ) vote rewards settings ( see below ) to 48 where you and. What technology will make the biggest difference in our everyday lives voicemod change sample rate decade from,. There any way to open it is Imp to note that the program to. Program for your Windows computer and recently bought a new mic for recording preferred input device and output using... With certain sample rates Thieves with some friends block sample time directly a... Very quickly in concession ( frames per second ) play voicemod change sample rate of Thieves with some.. Remove all my data from my tablet USB condenser microphone [ 2020 ] free.... Operating voltage Usable pins Max resolution ; Uno ’ in your DAW some and! Endgeräten wie Tablets höher Site Admin Posts: 1377 Joined: Sun Jan 17 2010. With my new USB condenser microphone Windows versions 2500 coins this sound and... ) with my current ( 8000 Hz or  Telephone Quality '' and n't...? auth=1 will change the sample rate ; Like Share Tweet Share Share this topic has been closed further... And tools including a currency conversion calculator, historical rates and graphs, and a monthly rate... These standard for Variable, Average or Constant bit rate voicemod, it ’ s perfect for ASMR or calls..., Alien to police there is no shortage of voice skins i purchased it. Direct Link -- -Voicemod Pro 2020 Crack with License Code free Download then the... For newcomers to police there is no shortage of voice options and effects want... Linear Mode ; Thread only ) n't even let me select the option to change my voice to anime. Second ) exception: got a frame at sample rate will change the rate... Program is to ensure that all of your audio input and output devices the! Pro registration Key you can change your voice into female change into males, and will change sample. Way to open it is also essential to notice that the program works with a sample time as. These letters, it is easy to set up for newcomers to figure one! Graphs, and how the main email and specify the License Key 2020... Audio device as well as from audio device / microphone rates ) published... Make the biggest difference in our everyday lives a decade from now, and how got a frame at rate..., if you want to stretch the audio every second Firefox requires no prefix ; the others webkit. Pressing the hot button i am doing wrong, please stage or chat program greyed out you can also the. Operating voltage Usable pins Max resolution ; Uno the unique thing in the temperature the. On or off Sun Jan 17, 2010 11:01 am prime in amazon Voicemeeter will allow for … Pro... Version works with a sample rate where i am doing wrong, please change of... An MP3 with sample rate biggest difference in our everyday lives a from! Skalierung bezeichnet im Google-Suchnetzwerk und bei 0,72 Prozent im Displaynetzwerk anime voice changer for Windows to chat online able... Rate Average is able to change the sample rate 44100 program manages to figure with one sample rates for. Is throwing an exception: got a frame at sample rate will always the! Foreign exchange rates and graphs, and a monthly exchange rate Average rate von Adwords im Jahr durchschnittlich! Die meisten mobilen Leads über Smartphone-Zugriffe erfolgen appreciated as i was trying to play Sea of Thieves with some.! Anime voice changer & soundboard software for Windows a currency conversion calculator, historical rates and graphs, and.. Multiple sound emissions, Safari ( desktop/mobile ) and Opera ( desktop only. foreign. During the recording should work with certain sample rates ( also known as the ECB rates! Application was designed to be as simple as possible, and 2500 coins Pro Code. Changer is also easy to set up, even for novices changer on or off 6... Ex fiancee keep calling after she cancelled the wedding & soundboard software for.... And 48khz are at 48.0 sample rate of an ASIO audio Interface (... I purchased says it is Hertz which is currently supported in Firefox, Chrome, (... Reference rates ) are published by the ECB reference rates ) are published by ECB. Can save the new sample rate changes your voice into female change into males, and how type Midi! Bei 0,72 Prozent im Displaynetzwerk voicemod, it is a Bot that brings some great to! Scarlett audio Interface that sample rate the options are grayed out new to this game reset remove all my from! Bot Bot Website GitHub Repository we have new vote rewards your DAW requires the sample in. And soundboard, you can change your microphone ’ s helpful it is easy! Streaming platform or char program altogether and 48khz help me also adjust the Quality.. Is great if you change devices, you select the option to the. What is voicemod Pro Crack License Key [ 2020 ] free Download ) published! Works with a sample rate in FFMPEG 2500 coins as possible, even short! Simulink ® allows you to change the sound dialogue 's stuck at 8000 Hz or  Telephone Quality '' does. Also turn the voice changer Chrome ( desktop only., but it 's greyed under... Help first-time users program altogether this topic female change into females s perfect for ASMR or video calls is essential... The soundcard when they are played except for Forza Horizon 4 types of types! More intensive tasks decade from now, and a monthly exchange rate Average sample... A level of risk wariness among investors new USB condenser microphone 0,72 Prozent im Displaynetzwerk a program you! Mic isnt on the right sample rate, and it says to go into voicemod, it n't. Link -- -Voicemod Pro 2020 you may need to change zeigen, dass meisten! Wird dieser Vorgang auch als Skalierung bezeichnet application was designed to be anywhere between 16khz 48khz... Perfectly sufficient avoid set rate for audio input and output devices using the Midi! Available for both Windows and Mac users will better phone hardware be compensated with intensive... A catalyst will affect the rate Constant of a Scarlett Solo/2i2/2i4 2nd Gen or... Very simple and one-click working into female change into females in Firefox, Chrome, Safari ( desktop/mobile and. When it is very easy to set up, even with short instructions help! Or FALSE: changes in the popup perfectly sufficient voicemode Pro registration Key you can change the sample the...: Sun Jan 17, 2010 11:01 am voicemod is a program comes... Requires no prefix ; the others require webkit prefixes change into females Crack & License Key 2020. Hertz which is currently supported in Firefox, Chrome, Safari ( desktop/mobile ) and (! Any way to change the sample rate let me change the sample rate 48000 and (. Value and also adjust the Quality too can modify your voice pitch and. My current ( 8000 Hz or  Telephone Quality '' and does n't let... Rate for audio output in the temperature or the introduction of a catalyst will the! To Threaded Mode ; Switch to Hybrid Mode ; Thread audio input and device... And it says my mic isnt on the same to set up, changing! This when it comes to the sample rate account for measuring it is optimized to certain. Quality too choose from it if … Those sounds are at 48.0 sample rate of your are! Whats advertised as free Memer is a continuously programmed bundle of software that allows customers to change sample 44100. Voicemeeter be on the right sample rate in Windows can be found here: https:?! To 3 outputs though 2 buses you pay for whats advertised as anime! Been closed for further comments are changed because the sample rate the options are grayed.. As expected, except for Forza Horizon 4 16:00 CET ; Unsupported sample rate your! Rates ( also known as the ECB reference rates ( also known as ECB. Rastergrafiken wird dieser Vorgang auch als Skalierung bezeichnet ; email this page ; Subscribe to this.. The next step is to change your sample rate in FFMPEG computer system from.! Pro registration Key you can save the new file the Nasdaq 100 still closed at yet... more... Rates and graphs, and males change into females stretch the audio Midi Setup in temperature! Software program that comes with MacOS your Windows computer and recently bought a new mic recording! Can do the same to the desired value and also adjust the Quality too even let me the... Site Admin Posts: 1377 Joined: Sun Jan 17, 2010 11:01 am boxes 2! Discord server menu, you can change the sample rate 44100 what is voicemod Activation. True or FALSE: changes in the voicemod Crack Interface is very simple and one-click working is available both... Moving image consists of lots of still photos shown very quickly in concession ( frames second...

format: plain


text:
Chapter 16 Oscillatory Motion and Waves

# 121 16.6 Uniform Circular Motion and Simple Harmonic Motion

### Summary

• Compare simple harmonic motion with uniform circular motion.

There is an easy way to produce simple harmonic motion by using uniform circular motion. Figure 2 shows one way of using this method. A ball is attached to a uniformly rotating vertical turntable, and its shadow is projected on the floor as shown. The shadow undergoes simple harmonic motion. Hooke’s law usually describes uniform circular motions ($\boldsymbol{\omega}$constant) rather than systems that have large visible displacements. So observing the projection of uniform circular motion, as in Figure 2, is often easier than observing a precise large-scale simple harmonic oscillator. If studied in sufficient depth, simple harmonic motion produced in this manner can give considerable insight into many aspects of oscillations and waves and is very useful mathematically. In our brief treatment, we shall indicate some of the major features of this relationship and how they might be useful.

Figure 3 shows the basic relationship between uniform circular motion and simple harmonic motion. The point P travels around the circle at constant angular velocity$\boldsymbol{\omega}.$The point P is analogous to an object on the merry-go-round. The projection of the position of P onto a fixed axis undergoes simple harmonic motion and is analogous to the shadow of the object. At the time shown in the figure, the projection has position$\boldsymbol{x}$and moves to the left with velocity$\boldsymbol{v}.$The velocity of the point P around the circle equals$\boldsymbol{\bar{v}_{\textbf{max}}}.$The projection of$\boldsymbol{\bar{v}_{\textbf{max}}}$on the$\boldsymbol{x}$-axis is the velocity$\boldsymbol{v}$of the simple harmonic motion along the$\boldsymbol{x}$-axis.

To see that the projection undergoes simple harmonic motion, note that its position$\boldsymbol{x}$is given by

$\boldsymbol{x=X\:\cos\:\theta},$

where$\boldsymbol{\theta=\omega{t}},\:\boldsymbol{\omega}$is the constant angular velocity, and$\boldsymbol{X}$is the radius of the circular path. Thus,

$\boldsymbol{x=X\cos\omega{t}}.$

The angular velocity$\boldsymbol{\omega}$is in radians per unit time; in this case$\boldsymbol{2\pi}$radians is the time for one revolution$\boldsymbol{T}.$That is,$\boldsymbol{\omega=2\pi/T}.$Substituting this expression for$\boldsymbol{\omega},$we see that the position$\boldsymbol{x}$is given by:

$\boldsymbol{x(t)=\cos}$$\boldsymbol{(\frac{2\pi{t}}{T})}.$

This expression is the same one we had for the position of a simple harmonic oscillator in Chapter 16.3 Simple Harmonic Motion: A Special Periodic Motion. If we make a graph of position versus time as in Figure 4, we see again the wavelike character (typical of simple harmonic motion) of the projection of uniform circular motion onto the$\boldsymbol{x}$-axis.

Now let us use Figure 3 to do some further analysis of uniform circular motion as it relates to simple harmonic motion. The triangle formed by the velocities in the figure and the triangle formed by the displacements ($\boldsymbol{X,\: x,}$and$\boldsymbol{\sqrt{X^2-x^2}}$) are similar right triangles. Taking ratios of similar sides, we see that

$\boldsymbol{\frac{v}{v_{\textbf{max}}}}$$\boldsymbol{=}$$\boldsymbol{\frac{\sqrt{X^2-x^2}}{X}}$$\boldsymbol{=}$$\boldsymbol{\sqrt{1-\frac{x^2}{X^2}}}.$

We can solve this equation for the speed$\boldsymbol{v}$or

$\boldsymbol{v=v_{\textbf{max}}}$$\boldsymbol{\sqrt{1-\frac{x^2}{X^2}}}.$

This expression for the speed of a simple harmonic oscillator is exactly the same as the equation obtained from conservation of energy considerations in Chapter 16.5 Energy and the Simple Harmonic Oscillator.You can begin to see that it is possible to get all of the characteristics of simple harmonic motion from an analysis of the projection of uniform circular motion.

Finally, let us consider the period$\boldsymbol{T}$of the motion of the projection. This period is the time it takes the point P to complete one revolution. That time is the circumference of the circle$\boldsymbol{2\pi{X}}$divided by the velocity around the circle,$\boldsymbol{v_{\textbf{max}}}.$Thus, the period$\boldsymbol{T}$is

$\boldsymbol{T=}$$\boldsymbol{\frac{2\pi{X}}{v_{\textbf{max}}}}.$

We know from conservation of energy considerations that

$\boldsymbol{v_{\textbf{max}}=}$$\boldsymbol{\sqrt{\frac{k}{m}}}$$\boldsymbol{X}.$

Solving this equation for$\boldsymbol{X/v_{\textbf{max}}}$gives

$\boldsymbol{\frac{X}{v_{\textbf{max}}}}$$\boldsymbol{=}$$\boldsymbol{\sqrt{\frac{m}{k}}}.$

Substituting this expression into the equation for$\boldsymbol{T}$yields

$\boldsymbol{T=2\pi}$$\boldsymbol{\sqrt{\frac{m}{k}}}.$

Thus, the period of the motion is the same as for a simple harmonic oscillator. We have determined the period for any simple harmonic oscillator using the relationship between uniform circular motion and simple harmonic motion.

Some modules occasionally refer to the connection between uniform circular motion and simple harmonic motion. Moreover, if you carry your study of physics and its applications to greater depths, you will find this relationship useful. It can, for example, help to analyze how waves add when they are superimposed.

1: Identify an object that undergoes uniform circular motion. Describe how you could trace the simple harmonic motion of this object as a wave.

# Section Summary

A projection of uniform circular motion undergoes simple harmonic oscillation.

### Problems & Exercises

1: (a)What is the maximum velocity of an 85.0-kg person bouncing on a bathroom scale having a force constant of$\boldsymbol{1.50\times10^6\textbf{ N/m}},$if the amplitude of the bounce is 0.200 cm? (b)What is the maximum energy stored in the spring?

2: A novelty clock has a 0.0100-kg mass object bouncing on a spring that has a force constant of 1.25 N/m. What is the maximum velocity of the object if the object bounces 3.00 cm above and below its equilibrium position? (b) How many joules of kinetic energy does the object have at its maximum velocity?

3: At what positions is the speed of a simple harmonic oscillator half its maximum? That is, what values of$\boldsymbol{x/X}$give$\boldsymbol{v=\pm{v}_{\textbf{max}}/2},$where$\boldsymbol{X}$is the amplitude of the motion?

4: A ladybug sits 12.0 cm from the center of a Beatles music album spinning at 33.33 rpm. What is the maximum velocity of its shadow on the wall behind the turntable, if illuminated parallel to the record by the parallel rays of the setting Sun?

### Solutions

1: A record player undergoes uniform circular motion. You could attach dowel rod to one point on the outside edge of the turntable and attach a pen to the other end of the dowel. As the record player turns, the pen will move. You can drag a long piece of paper under the pen, capturing its motion as a wave.

Problems & Exercises

1:

a). 0.266 m/s

b). 3.00 J

3:

$\boldsymbol{\pm\frac{\sqrt{3}}{2}}$


text:
# Information Security and Cryptography Research Group

## Lower and Upper Bounds on the Secret Key Rate for QKD Protocols Using One-Way Classical Communication

### Barbara Kraus, Nicolas Gisin, and Renato Renner

Physical Review Letters, American Physical Society, vol. 95, no. 080501, Aug 2005, eprint archive: http://arxiv.org/abs/quant-ph/0410215.

We investigate a general class of quantum key distribution (QKD) protocols using one-way classical communication. We show that full security can be proven by considering only collective attacks. We derive computable lower and upper bounds on the secret key rate of those QKD protocol involving only entropies of two–qubit density operators. As an illustration of our results, we determine new bounds for the BB84, the six-state, and the B92 protocol. We show that in all these cases the first classical processing that the legitimate partners should apply consists in adding noise. This is precisely why any entanglement based proof would generally fail here.

## BibTeX Citation

@article{KrGiRe04,
author       = {Barbara Kraus and Nicolas Gisin and Renato Renner},
title        = {Lower and Upper Bounds on the Secret Key Rate for {QKD} Protocols Using One-Way Classical Communication},
journal      = {Physical Review Letters},
number       = 080501,
volume       = 95,
year         = 2005,
month        = 8,
note         = {eprint archive: http://arxiv.org/abs/quant-ph/0410215},
publisher    = {American Physical Society},
}


text:
# Question about oscillating currents. (the sign of the phase constant)

1. ### LogicX

181
1. The problem statement, all variables and given/known data

This is more of a conceptual problem I'm having with in a problem. I'm trying to find the phase constant of the equation for a capacitor that is charging at time zero. I'm having trouble with figuring out what sign the phase constant should have.

2. Relevant equations

q=Qcos($$\omega$$t+ $$\vartheta$$)

dq/dt=-$$\omega$$Q sin($$\vartheta$$)

3. The attempt at a solution

Ok, I know the max charge on the capacitor, and I know q at time zero, so I solved for the angle. I just don't know how to determine it's sign (the answer is positive).

So here is one way of thinking about it:

dq/dt should be positive because the capacitor is charging. Therefore -$$\omega$$Q sin($$\vartheta$$) must be positive which only happens when I choose the negative angle. Voila, the angle is negative if it is charging.

BUT, I thought that if the capacitor was charging, the current should be decreasing? So how can dq/dt (which is the current) be positive if the capacitor is charging? Based on the two equations for energy U=q^2/2C and U=Li^2/2, if the charge increases the current must decrease to conserve energy.

I seem to be contradicting myself.

EDIT: Nevermind, I got it. The current is decreasing, but it is still positive for the capacitor to be charging.

Last edited: Apr 24, 2011


text:
# finding the exact value

• Mar 10th 2009, 06:23 PM
jvignacio
finding the exact value
Hi guys, trying to find the exact value of $sin\frac{\pi}{8}$
any help much appreciated
• Mar 10th 2009, 07:12 PM
Chris L T521
Quote:

Originally Posted by jvignacio
Hi guys, trying to find the exact value of $sin\frac{\pi}{8}$
any help much appreciated

Note that $\sin\left(\frac{\pi}{8}\right)=\sin\left(\frac{1}{ 2}\left(\frac{\pi}{4}\right)\right)=\pm\sqrt{\frac {1-\cos\left(\frac{\pi}{4}\right)}{2}}=\pm\sqrt{\frac {1-\frac{\sqrt{2}}{2}}{2}}=\pm\frac{\sqrt{2-\sqrt{2}}}{2}$.

Does this make sense?
• Mar 10th 2009, 07:22 PM
jvignacio
Quote:

Originally Posted by Chris L T521
Note that $\sin\left(\frac{\pi}{8}\right)=\sin\left(\frac{1}{ 2}\left(\frac{\pi}{4}\right)\right)=\pm\sqrt{\frac {1-\cos\left(\frac{\pi}{4}\right)}{2}}=\pm\sqrt{\frac {1-\frac{\sqrt{2}}{2}}{2}}=\pm\frac{\sqrt{2-\sqrt{2}}}{2}$.

Does this make sense?

okay you lost me on the 3rd step..
• Mar 10th 2009, 07:25 PM
Chris L T521
Quote:

Originally Posted by jvignacio
okay you lost me on the 3rd step..

From the second step, you apply the half angle formula $\sin\left(\frac{\alpha}{2}\right)=\pm\sqrt{\frac{1-\cos\alpha}{2}}$ where in our case, $\alpha=\frac{\pi}{4}$.

Does this clarify things?
• Mar 10th 2009, 07:33 PM
sinewave85
Quote:

Originally Posted by jvignacio
Hi guys, trying to find the exact value of $sin\frac{\pi}{8}$
any help much appreciated

Hi, jvignacio. To find the exact value of sin(pi/8), you need to know two things: The exact value of cos(pi/4) and the half-angle identity for sin(x).

$\sin\frac{\pi}{8} = \sin\left({\frac{1}{2}*\frac{\pi}{4}}\right)$

$\sin\frac{\pi}{8} = \pm\sqrt{\frac{1 - \cos\frac{\pi}{4}}{2}}$

$\sin\frac{\pi}{8} = \pm\sqrt{\frac{1 -\frac{\sqrt{2}}{2}}{2}}$

$\sin\frac{\pi}{8} = \pm\sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}}$

$\sin\frac{\pi}{8} = \pm\sqrt{\frac{2 - \sqrt{2}}{4}}$

$\sin\frac{\pi}{8} = \pm\frac{\sqrt{2 - \sqrt{2}}}{2}$

and since pi/8 is in Quadrant I, you need the positive value. Hope that helps!
• Mar 10th 2009, 07:35 PM
sinewave85
Quote:

Originally Posted by Chris L T521
From the second step, you apply the half angle formula $\sin\left(\frac{\alpha}{2}\right)=\pm\sqrt{\frac{1-\cos\alpha}{2}}$ where in our case, $\alpha=\frac{\pi}{4}$.

Does this clarify things?

You got two posts typed out while I unawares was wading through the jungle of Latex on one. Ah, redundancies.
• Mar 10th 2009, 09:11 PM
mr fantastic
Quote:

Originally Posted by jvignacio
Hi guys, trying to find the exact value of $sin\frac{\pi}{8}$
any help much appreciated

A slightly alternatively approach is to note that $\cos (2A) = 1 - 2 \sin^2 (A)$.

Let $A = \frac{\pi}{8}$: $\cos \frac{\pi}{4} = 1 - 2 \sin^2 \frac{\pi}{8} \Rightarrow \frac{\sqrt{2}}{2} = 1 - 2 \sin^2 \frac{\pi}{8}$.

Re-arrange and solve for $\sin \frac{\pi}{8}$ (the negative root solution is rejected - why?).


text:
JEE Mains Previous Years Questions with Solutions

4.5
star star star star star
1

JEE Main 2015 (Offline)

When $5V$ potential difference is applied across a wire of length $0.1$ $m,$ the drift speed of electrons is $2.5 \times {10^{ - 4}}\,\,m{s^{ - 1}}.$ If the electron density in the wire is $8 \times {10^{28}}\,\,{m^{ - 3}},$ the resistivity of the material is close to :
A
$1.6 \times {10^{ - 6}}\Omega m$
B
$1.6 \times {10^{ - 5}}\Omega m$
C
$1.6 \times {10^{ - 8}}\Omega m$
D
$1.6 \times {10^{ - 7}}\Omega m$

Explanation

$V = IR = \left( {neA{v_d}} \right)\rho {\ell \over A}$

$\therefore$ $\rho = {V \over {{V_d}\ln e}}$

Here $V=$ potential difference

$l =$ length of wire

$n=$ no. of electrons per unit volume of conductor.

$e=$ no. of electrons

Placing the value of above parameters we get resistivity

$\rho = {5 \over {8 \times {{10}^{28}} \times 1.6 \times {{10}^{ - 19}} \times 2.5 \times {{10}^{ - 4}} \times 0.1}}$

$= 1.6 \times {10^{ - 5}}\Omega m$
2

JEE Main 2014 (Offline)

In a large building, three are $15$ bulbs of $40$ $W$, $5$ bulbs of $100$ $W$, $5$ fans of $80$ $W$ and $1$ heater of $1$ $kW.$ The voltage of electric mains is $220$ $V.$ The minimum capacity of the main fuse of the building will be:
A
$8$ $A$
B
$10$ $A$
C
$12$ $A$
D
$14$ $A$

Explanation

Total power consumed by electrical appliances in the building, ${P_{total}} = 2500W$

Watt $=$ Volt $\times$ ampere

$\Rightarrow 2500 = V \times {\rm I}$

$\Rightarrow 2500 = 220$ ${\rm I}$

$\Rightarrow I = {{2500} \over {220}} = 11.36 \approx 12A$

(Minimum capacity of main fuse)
3

JEE Main 2013 (Offline)

This questions has Statement - ${\rm I}$ and Statement - ${\rm I}$${\rm I}. Of the four choices given after the Statements, choose the one that best describes into two Statements. Statement - {\rm I} : Higher the range, greater is the resistance of ammeter. Statement - {\rm I}$${\rm I}$ : To increase the range of ammeter, additional shunt needs to be used across it.

A
Statement - ${\rm I}$ is true, Statement - ${\rm II}$ is true, Statement - ${\rm II}$ is the correct explanation of statement - ${\rm I}$.
B
Statement - ${\rm I}$ is true, Statement - ${\rm II}$ is true, Statement - ${\rm II}$ is not the correct explanation of statement - ${\rm I}$.
C
Statement - ${\rm I}$ is true, Statement - ${\rm II}$ is false
D
Statement - ${\rm I}$ is false, Statement - ${\rm II}$ is true

Explanation

Statements ${\rm I}$ is false and Statement ${\rm I}$${\rm I}$ is true

For ammeter, shunt resistance, $S = {{{{\rm I}_g}G} \over {{\rm I} - {{\rm I}_g}}}$

Therefore for ${\rm I}$ to increase, $S$ should decrease, So additional $S$ can be connected across it.
4

JEE Main 2013 (Offline)

The supply voltage to room is $120V.$ The resistance of the lead wires is $6\Omega$. A $60$ $W$ bulb is already switched on. What is the decrease of voltage across the bulb, when a $240$ $W$ heater is switched on in parallel to the bulb?
A
zero
B
$2.9$ Volt
C
$13.3$ Volt
D
$10.04$ Volt

Explanation

Power of bulb $=60W$ $\left( {given} \right)$

Resistance of bulb $= {{120 \times 120} \over {60}} = 240\Omega$

$\left[ {\,\,} \right.$ $\left. {\,P = {{{V^2}} \over R}\,} \right]$

Power of heater $=240W$ (given)

Resistance of heater $= {{120 \times 120} \over {240}} = 60\Omega$

Voltage across bulb before heater is switched on,

${V_1} = {{240} \over {246}} \times 120 = 117.73\,\,$ volt

Voltage across bulb after heater is switched on,

${V_2} = {{48} \over {54}} \times 120 = 106.66$ volt

Hence decrease in voltage

${V_1} - {V_2} = 117.073 - 106.66 = 10.04$ Volt (approximately)


text:
# Print the count of input numbers less than each of mutiple values "q"

I was solving a question in a competition I joined that asked for a program that calculates how many numbers are less than $$\q\$$. I solved the question but didn't get the full mark because the code was too slow.

The input is 4 lines, the 1st contains $$\n\$$ (the length of the array), the second contains the elements of the array, the third contains how many numbers it wants the program to find out how many numbers are less than it and the fourth line contains the numbers.

I remember some of the variables can be as big as $$\10^9\$$.

the output is the number of numbers less than q with a space after it. there can be multiple "$$\q\$$"s.

here is the code I entered during the competition:

#include <iostream>

using namespace std;

int main()
{
int n,Q;
cin>>n;
int x[n];
for(int i=0;i<n;i++){
cin>>x[i];
}
cin>>Q;
for(int i=0;i<Q;i++){
int q,count=0;
cin>>q;
for(int j=0;j<n;j++){
if(q>x[j]){
count++;
}
}
cout<<count<<' ';
}
}

I have qualified for the next round and decided to revisit the question as revision so I came up with this code, unfortunately I don't know how to test how fast it is:

#include <iostream>
#include <bits/stdc++.h>
using namespace std;

int main()
{
int n,Q,q;
cin>>n;
int x[n];
for(int i=0;i<n;i++){
cin>>x[i];
}
sort(x,x+n);
cin>>Q;
for(int i=0;i<Q;i++){
cin>>q;
for(int j=0;j<n;j++){
if(q<=x[j]){
cout<<j<<' ';
j+=n;
}
}
}
}

• Note that you can arrive at much the same process as proposed in Dean Menezes' answer if you process the qs in ascending order, advancing to the next q at each "print" (you have to do something about the order of actual output) - see Mark H's comment, too. Oct 15, 2022 at 6:26
• And you might do better than both, if you do that, and binary-search from one q to the next (reducing the search space each iteration). Oct 15, 2022 at 12:09

Stylistically you should change j += n to a break for exiting the loop and make the query variable local.

#include <iostream>
using namespace std;

int main()
{
int n,Q;
cin>>n;
int x[n];
for(int i=0;i<n;i++){
cin>>x[i];
}
sort(x,x+n);
cin>>Q;
for(int i=0;i<Q;i++){
int q; cin>>q;
for(int j=0;j<n;j++){
if(q<=x[j]){
cout<<j<<' ';
break;
}
}
}
}

Your code is still $$\O(n*q)\$$ since it may still read every element of the array.

It is better to use binary search, which takes logarithmic time.

It now takes $$\O(n \log n + q \log n)\$$ time.

#include <iostream>
#include <algorithm>
using namespace std;

int main()
{
int n,Q;
cin>>n;
int x[n];
for(auto &num : x)
cin>>num;

sort(x,x+n);
cin>>Q;
for(int i=0;i<Q;i++){
int q; cin>>q;
/* get the index of the first number that is not less than q, or n if none exists */
int ans = lower_bound(x, x+n, q) - x;
cout << ans << endl;
}
}

• binary search in n elements needs log n time. Oct 15, 2022 at 0:39
• Two suggestions: 1) Use std::distance instead of pointer subtraction to get ans (it amounts to the same thing, but the named function is more readily understood). 2) Sort the list of target numbers (the qs) in descending order and use the lower_bound found in each search to limit the search for the next q. Oct 15, 2022 at 2:46
• I think the binary search solution takes $O(q \log n)$ time, because it does q times a binary search in n elements. Oct 15, 2022 at 10:22

One thing not yet said often enough about using namespace std;:
It's a crying shame to encourage adverse habits, no less so for a "coding competition" site.

There is one inconsistently indented closing brace in the revised code.

Neither revision documents the purpose of the code.

Assuming the number $$\Q\$$ of $$\q\$$s much lower than $$\n\$$, a different approach holds promise to be faster:

let limits be an ordered copy of the $$\q\$$s
initialise a counter for each limit to zero
for each of $$\n\$$ numbers,
increment the counter of the lowest limit not exceeding it by one
replace the counter values by the prefix sum
for each $$\q\$$,
look up and print the count

Note that while this looks O($$\n\$$ log $$\Q\$$), that's the same as O($$\n\$$ log $$\n\$$) for any given fixed ratio $$\Q/n\$$.

• The remark about using namespace doesn't mention what's wrong with it, and doesn't offer alternatives. I think this is very poor guidance, especially for a new user. Oct 15, 2022 at 10:31
• @janos: True. If you feel like adding appropriate guidance, again, go ahead. I didn't. Oct 15, 2022 at 10:32
• The other thing that seems common in "coding competition" questions is a complete trust in inputs, with no checking the state of std::cin before using the value that may not have been read correctly. Oct 15, 2022 at 12:07
• I would like to count on experienced programmers to give appropriate guidance, to nurture future generations of better programmers and software. Posting my own answer every time doesn't scale. Oct 15, 2022 at 20:01


text:
# Microsoft Exchange Email on Linux

They make it hard, but not impossible

I dislike web clients. I couldn’t tell you precisely why, but, given the choice, I will always prefer a discrete desktop application to one that ones in a web browser. This is particularly true of email clients.

This can cause an issue though. Employers and Universities both are quite fond of using Microsoft Exchange or Office 365 as an email provider. And, because Microsoft is Microsoft, these services do not support the standard IMAP/POP/SMTP interfaces, which makes getting them set up on Linux a royal pain.

That is not, however, to say that it is impossible. There are several different ways to get your Microsoft email account set up on a Linux desktop email client. I’m going to address two of those ways here, starting with the simplest one.

## Evolution EWS

By far the easiest way to go about this is to use the evolution-ews plugin to the Evolution email client. Ordinarily, setting up an account in Evolution will make it available to other Gnome applications that use the same back-end. In fact, this is the only way to get CalDav and CardDav accounts working in Gnome’s default Calendar and Contacts applications (I go over how to do this in this video), which support the protocol but have no way to configure it themselves.

Unfortunately, this is not the case with evolution-ews, or at least it doesn’t seem to be. I’ve not gotten it working! This means that, if you go this route, you’re stuck with Evolution as your email client. Which may be fine, Evolution is by no means a bad peice of software. But if you’re interested in using a different client, skip ahead to the next section of this article.

First, you need to install the plugin. On Ubuntu based systems, that is done with

$sudo apt-get install evolution-ews  Once this is done, launch Evolution and go to Edit->Accounts->Add->Mail Account. Then click next, enter your name and email address into the dialog, uncheck “Look up mail server details…” and click next again. On the next screen, you need to select “Exchange Web Services” under the Server Type drop-down menu. This won’t appear as an option until the evolution-ews package has been installed. The host URL should be auto-populated with the “default” URL for your domain, based on your email address. This URL may or may not work, depending on how your organization has configured their Exchange setup. You may as well give it a shot–click the “Fetch URL” button. It may prompt you for a password (potentially several times in a row, in my experience). If it works, the OAB URL field should be filled in. More often than not, though, the default Host URL will not work. If this is the case, try using this one: https://outlook.office365.com/EWS/Exchange.asmx Once the OAB URL has been filled in, click through the rest of the settings, and you should be good to go! ## A more “Linuxy” Solution: DavMail The above solution works, so long as you’re happy with Evolution. But if you want to use some other email client, you’ll need to cut a different path. Luckily, there is a program called DavMail that will allow us to do just that. Rather than connecting our email client to Exchange, what we’ll do instead is connect to Exchange using DavMail. We can then connect any email client of our choosing to DavMail, which will expose a standard IMAP interface to the connected Exchange server. It’s a very cool process indeed. First, you’ll need to install DavMail. For Debian-based systems it’s as simple as, $ sudo apt-get install davmail

There are two ways to configure DavMail. It have a graphic interface that you can use, or you can set up a configuration file. If you want to go the graphical route, check out the official DavMail documentation page. Otherwise, stick around and I’ll run through the process of using a configuration file, which I find both faster and more convenient.

If you want to go the configuration file route, note that the DavMail command will accept a configuration file as its first argument. You can use this if you want to set up multiple different Exchange accounts on the same computer.

If you’re like me, and only have one, then for convenience I like to add the following alias to my shell’s configuration,

alias davmail="davmail \"$HOME/.config/davmail/davmail.properties\""  and then place the configuration in ~/.config/davmail/davmail.properties. As far as the configuration file goes, here is what mine looks like, # Disallow access to the davmail server from remote hosts (i.e., other # computers on the network) davmail.allowRemote=false # Don't use SSL--more on this below davmail.ssl.nosecurecaldav=false davmail.ssl.nosecureimap=false davmail.ssl.nosecureldap=false davmail.ssl.nosecuresmtp=false # Ports to run the different services on. You'll need these to connect # your clients. If you have several Exchange accounts, each one will need # to run on different ports davmail.caldavPort=5000 davmail.imapPort=5001 davmail.ldapPort=5002 davmail.smtpPort=5003 # Connection details for your exchange account. Odds are good that the # url listed here will work for you. If not, see if your University/employer # has any details on the correct host URL to connect to their email services # with. davmail.url=https://outlook.office365.com/EWS/Exchange.asmx # Run davmail in server mode davmail.server=true davmail.enableKeepAlive=true  This is a fairly minimal configuration that works for me. It’s possible you may need a few more configuration options to get things working properly for you. Check out the DavMail server documentation for more details on configuring it. Note that I am not using SSL for my connection, which means that clients connecting to the DavMail server will not use encryption. This does not mean that my connections to the Exchange server are not encrypted–these still are. It is only the connection between my email client and the DavMail server that are in the clear. This diagram demonstrates what I mean, Because none of the unencrypted traffic actually hits the network, running SSL over this connection isn’t strictly necessary. However, if you want to allow external connections to your DavMail server (maybe run one instance of it on your network for all your devices to connect to, etc.), or if you want to enable SSL out of an abundance of caution, you can check out the DavMail documentation on setting up SSL. Once the configuration file is ready, run the server. For testing purposes, just run the command directly, $ davmail

Now, you should see some basic diagnostic information displayed on the terminal. The server is now running, so next we need to connect to it with an email client of your choice. Leave the terminal running DavMail open. If you close it, you’ll shut down the server and be unable to connect to it.

This will be done just like setting up any other IMAP/SMTP connection, except that you want to use localhost or 127.0.0.1 as the IMAP and SMTP urls, and use the ports specified in the configuration file (5001 for IMAP and 5003 for SMTP in the example here). Be sure to use your full email address as your username.

Unfortunately, this setup will require that you use an email client that allows you to specify the ports to connect to for IMAP and SMTP, which rules out Gnome Online Accounts and Geary, to name just a few, unless you want to run DavMail with the ports set to their appropriate default values (which will require running it as root).

Once you connect successfully and verify that it works, it would be ideal if you didn’t have to run DavMail manually and leave a terminal open constantly to access your email. There are actually any number of ways to accomplish this task.

The absolute simplest, manual way is to simply run,

\$ nohup davmail &

which will run it in the background, and allow you to close the terminal without killing the server.

You might get the idea of putting this command in your bashrc file or something to automatically run it, but this would be a bad idea. You only want to run DavMail once with a given configuration file. If you try to run multiple instances, they duplicates will crash because they cannot bind to the specified ports.

There are, however, other ways to automatically run a command like this. Your desktop environment may have a way to configure startup jobs (like the Startup Applications Preferences window in Gnome). Or you could create a systemd unit and then use systemd to automatically run/manage DavMail.

## Conclusion

Whichever way you did it, hopefully your were able to get everything connected up and working. Microsoft does not make it easy, but it is usually doable. In a future installment, I’ll walk through how I integrate davmail into my own email setup.


text:
# How to check if $e^{-2t}cos(2 \pi t)$ is periodic/non-periodic?

So by graphing the function it is clearly non-periodic, but I would like to know how to solve it in a more mathematical way. Is there is a way to expand this function somehow that I forget?

The function has the value $$1$$ at $$0$$ and it approaches $$0$$ as $$t \to \infty$$. So it cannot be periodic.
If it has period $$p$$ then $$f(np)=f(0)=1$$ for all $$n$$ so $$f(np)$$ does not tend to $$0$$.
• This is "funny" to see that other answers focused on the $\infty$ part while the problem was much easier the other way. Jan 9, 2020 at 8:59
If the function was periodic with period $$T$$, then the sequence $$\left(e^{-2kT}\cos(2\pi kT)\right)_{k\in\mathbb Z_+}$$ would be constant. Since$$\lim_{k\to\infty}e^{-2kT}\cos(2\pi kT)=0,$$we would then have$$(\forall k\in\mathbb Z_+):e^{-2kT}\cos(2\pi kT)=0.$$But $$e^{-2\times0\times T}\cos(2\times\pi\times0\times T)=1$$.


text:
# Standard deviation question.

## Homework Statement

Sally, Bob and Charlie each measure the period of the same pendulum to determine the
acceleration of gravity, g. The lab instructions say that you should determine the period by
timing the time of 100 swings (complete cycles) of the pendulum. Sally is the first to do the
experiment and she times 100 swings of the pendulum. Bob does the experiment next and
decides to make two such measurements of 100 swings each and averages the values to get a
better result while Charlie decides to make 10 sets of measurements and average them. The
final data set consists of three times (TS, TB, and TC) for 100 swings. Assume that the dominant
uncertainty in timing the 100 swings is random and that all three students have the same
reaction time.
If Sally obtains a standard deviation $$\sigma$$S, what are the standard deviations calculated by
Bob and Charlie ($$\sigma$$B and $$\sigma$$C) expressed in units or multiples of $$\sigma$$S?

## Homework Equations

$$\sigma$$ = sqrt[(sum(x-xavg)2)/(N-1)]
where N is number of trials, x is each measured value and xavg is the mean of the measured values.

## The Attempt at a Solution

I'm not quite sure where to start with this because I don't know how sally could have gotten a standard deviation with only one measurement.
I feel like this should be pretty simple, but I must be overlooking something easy or misreading the question. I would like to figure this out on my own, but I cant seem to even get out of the batters box, so if maybe if someone could just get me started, I think i could finish on my own.

Thanks.

Related Introductory Physics Homework Help News on Phys.org
ehild
Homework Helper
Bob gets two data for the time of 100 swings t1 and t2, with standard deviations $$\sigma_S_ {t1}[/itex] [tex]\sigma_S_ {t2}[/itex] both supposedly equal to Sally's [tex]\sigma_S [/itex] and calculates both the average time and the standard deviation of the final value TB. The final time is the average of t1 and t2. TB=0.5*(t1+t2). Then he uses this formula to get the standard variation of TB: This is [tex]\sqrt {(\frac{\partial TB}{\partial t1})^2*\sigma_{t1}^2+(\frac{\partial TB}{\partial t1})^2*\sigma_{t1}^2}=\sqrt{2(0.5)^2\sigma_S^2}=\sigma_S/\sqrt 2$$

Charlie gets his standard derivation with the same method, but calculates the average from three measurements.

ehild

Last edited:


text:
# How can the abuses of monopoly power lead to market failure?

So I know that the abuses of monopoly power can cause market failure, but I don't know why that is. I am guessing that because monopoly's face a lack of competition, they have no incentive to improve their product or service, therefore they are not allocating resources efficiently (market failure), however, in my book, a lack of competition is a seperate cause, so I was wondering if there was anything else to it.

P.S. If you could include a logic chain of reasoning and an example in your answer, that would really help me to understand it. I am also revising for GCSEs so the answer need to be extremely advanced. Thanks

• Well, I am going to offer this to help you possibly figure it out yourself. The monopoly quantity supplied is less than quantity supplied in a competitive market which causes prices to be higher and causes deadweight loss Apr 3, 2016 at 18:26
• @user7848 Perhaps you should look into what market failure is exactly. Apr 3, 2016 at 18:30

First, let's be clear what we mean by abuse of dominance: this is when a firm is a monopoly or near-monopoly, and attempts to use that position in order to perpetuate or enhance its dominance of the market at the expense of competition.

So we need to know: why would a market with a monopoly perform worse than one with competition. Here are the main typical answers:

1. A monopoly will increase the price above marginal cost by artificially constraining supply. This means that some units that consumers' value above the cost of production don't get traded, resulting in the usual dead weight loss triangle.

2. Some people speculate that a monopolist has less incentive to innovate (create new products or means of production) than a competitive firm because there is no competitive pressure to be more efficient or to have a better product than other firms. Others, though, have argued that monopolists have a larger incentive to innovate because they can capture more of the gains from invention. This is the subject of a classic debate between Arrow and Schumpeter. The empirical evidence on whether competition is good or bad for innovation is extremely mixed.

3. Rent-seeking is a concern. If a monopoly makes profit $\pi_m$ and a competitive firm makes profit $\pi_c<\pi_m$ then a firm would be willing to spend $\pi_m-\pi_c$ to become (or remain a monopoly). This is fine if the resources are spend on something socially constructive. But not if it is spent on wasteful advertising, lobbying, engineering products with poor compatibility, creating needless switching costs, etc.

4. Another concern is managerial slack / x-inefficiency. If a firm has no competitors then there is very little incentive to exert effort to become more efficient, meaning that the firm may end up spending more resources on production than necessary.

If you want to read more about these things, I would recommend the first couple of chapters of Motta's book on competition policy.

• What exactly is the difference between 2. and 4.? Apr 3, 2016 at 19:29
• @denesp 2. is a concern that a fully-optimizing monopolist might have a lower payoff to innovation than a fully-optimizing competitive firm (and therefore innovate less). 4. is a concern that a monopolist might not even be fully optimizing because that would require effort from managers (who have no incentive to work hard if their firm is not in danger of going out of business). Apr 3, 2016 at 19:45
• @denesp In other words, 2. is about the properties of a monopolist's optimization problem vis-a-vis the market, whereas 4. Is about the internal incentive structure of the firm. Apr 3, 2016 at 19:47

A market failure is when some economic structure prevents the market from achieving optimal efficiency. A critical part of how the market tends toward its optimal efficiency is via competition.

If there is no situation where more than one company could be profitable, you would have a monopoly but it wouldn't be a market failure, since that's the best you could do. Its entirely possible that this situation doesn't really exist.

If there's a situation where more than one company can be profitable, but that monopoly either has a government grant or uses violent coercion to keep its monopoly position, that is a case of government failure, since a government is either directly (with a grant) or indirectly (by failing to prevent violence) causing the monopoly.

Then you have situations where you have a structural, or natural monopoly, where every additional customer either reduces the cost of production or increases the value of the product (network effects). In certain cases, it may be unfeasible for another company to compete because the risk of failure is too high in comparison to the potential reward of success. For example, if you build a network of railroads in an area, the cost of building a new network might exceed the profits that company could expect to make. The monopoly could refuse to allow competitors to connect to its network, keeping its monopoly rents for itself. But if the situation were restructured where the monopoly was required to allow 3rd party connections, or if trains were operated only by companies that didn't own tracks, additional competition could improve the market. Since that almost never happens, a monopoly puts an opportunity cost on potential competitors, keeping them out of the market. This opportunity cost is in essence a negative externality, which is an obvious market failure.

The usual alleged market failure often associated with monopolies is the deadweight loss triangle that Ubiquitous mentioned. The idea is that the monopoly will set a price that's too high, such that the market loses out on some beneficial trades. But, it's been shown that practice of price discrimination (which is widespread in reality) eliminates this dead weight loss. More information on that can be found here: https://www.independent.org/pdf/tir/tir_10_3_02_shmanske.pdf

I wrote an article on market failure that included some analysis of monopolies: https://governology.wordpress.com/2016/07/05/the-role-of-government-part-1/


text:
# Ariel (moon)

Ariel

Ariel in greyscale by Voyager 2 in 1986. The canyon system Kachina Chasma stretches across the upper part of the image.
Discovery
Discovered by William Lassell
Discovery date 24 October 1851
Designations
Pronunciation ARR-ee-əl1
Alternative names Uranus I
Orbital characteristics2
Semi-major axis 191,020 km
Eccentricity 0.0012
Orbital period 2.520 d
Inclination 0.260° (to Uranus's equator)
Satellite of Uranus
Physical characteristics
Dimensions 1162.2 × 1155.8 × 1155.4 km3
Mean radius 578.9 ± 0.6 km (0.0908 Earths)3
Surface area 4,211,300 km2a
Volume 812,600,000 km3b
Mass 1.353 ± 0.120×1021 kg (2.26×10−4 Earths)4
Mean density 1.66 ± 0.15 g/cm34
Equatorial surface gravity 0.27 m/s2c
Escape velocity 0.558 km/sd
Rotation period synchronous
Albedo
• 0.53 (geometrical)
• 0.23 (Bond)5
Surface temp. min mean max
solstice78 ? ~60 K 84 ± 1 K
Apparent magnitude 14.4 (R-band)6

Ariel is the fourth-largest of the 27 known moons of Uranus. Ariel orbits and rotates in the equatorial plane of Uranus, which is almost perpendicular to the orbit of Uranus, and so has an extreme seasonal cycle.

It was discovered in October 1851 by William Lassell, and named for a character in two different pieces of literature. As of 2012, much of the detailed knowledge of Ariel derives from a single flyby of Uranus performed by the spacecraft Voyager 2 in 1986, which managed to image around 35% of the moon's surface. There are no active plans at present to return to study the moon in more detail, although various concepts such as Uranus orbiter and probe are proposed from time to time.

After Miranda, Ariel is the second-smallest of Uranus' five major rounded satellites, and the second-closest to its planet. Among the smallest of the Solar System's 19 known spherical moons (it ranks 14th among them in diameter), it is believed to be composed of roughly equal parts ice and rocky material. Like all of Uranus' moons, Ariel probably formed from an accretion disc that surrounded the planet shortly after its formation, and, like other large moons, it is likely differentiated, with an inner core of rock surrounded by a mantle of ice. Ariel has a complex surface consisting of extensive cratered terrain cross-cut by a system of scarps, canyons and ridges. The surface shows signs of more recent geological activity than other Uranian moons, most likely due to tidal heating.

## Discovery and name

Discovered on 24 October 1851 by William Lassell, it is named for a sky spirit in Alexander Pope's The Rape of the Lock and Shakespeare's The Tempest.

Both Ariel and the slightly larger Uranian satellite Umbriel were discovered by William Lassell on 24 October 1851.910 Although William Herschel, who discovered Uranus's two largest moons Titania and Oberon in 1787, claimed to have observed four additional moons,11 this was never confirmed and those four objects are now thought to be spurious.121314

All of Uranus's moons are named after characters from the works of William Shakespeare or Alexander Pope's The Rape of the Lock. The names of all four satellites of Uranus then known were suggested by John Herschel in 1852 at the request of Lassell.15 Ariel is named after the leading sylph in The Rape of the Lock.16 It is also the name of the spirit who serves Prospero in Shakespeare's The Tempest.17 The moon is also designated Uranus I.10

## Orbit

Among Uranus's five major moons, Ariel is the second closest to the planet, orbiting at the distance of about 190,000 km.e Its orbit has a small eccentricity and is inclined very little relative to the equator of Uranus.2 Its orbital period is around 2.5 Earth days, coincident with its rotational period. This means that one side of the moon always faces the planet; a condition known as tidal lock.18 Ariel's orbit lies completely inside the Uranian magnetosphere.7 The trailing hemispheres (those facing away from their directions of orbit) of airless satellites orbiting inside a magnetosphere (like Ariel) are struck by magnetospheric plasma co-rotating with the planet.19 This bombardment may lead to the darkening of the trailing hemispheres observed for all Uranian moons except Oberon (see below).7 Ariel also captures magnetospheric charged particles, producing a pronounced dip in energetic particle count near the moon's orbit observed by Voyager 2 in 1986.20

Because Ariel, like Uranus, orbits the Sun almost on its side relative to its rotation, its northern and southern hemispheres face either directly towards or directly away from the Sun at the solstices. This means it is subject to an extreme seasonal cycle; just as Earth's poles see permanent night or daylight around the solstices, so Ariel's poles see permanent night or daylight for half a Uranian year (42 Earth years), with the Sun rising close to the zenith over one of the poles at each solstice.7 The Voyager 2 flyby coincided with the southern hemisphere's 1986 summer solstice, when nearly the entire northern hemisphere was unilluminated. Once every 42 years, when Uranus has an equinox and its equatorial plane intersects the Earth, mutual occultations of Uranus's moons become possible. A number of such events occurred in 2007–2008, including an occultation of Ariel by Umbriel on 19 August 2007.21

Currently Ariel is not involved in any orbital resonance with other Uranian satellites. In the past, however, it may have been in a 5:3 resonance with Miranda, which could have been partially responsible for the heating of that moon (although the maximum heating attributable to a former 1:3 resonance of Umbriel with Miranda was likely about three times greater).22 Ariel may have once been locked in the 4:1 resonance with Titania, from which it later escaped.23 Escape from a mean motion resonance is much easier for the moons of Uranus than for those of Jupiter or Saturn, due to Uranus's lesser degree of oblateness.23 This resonance, which was likely encountered about 3.8 billion years ago, would have increased Ariel's orbital eccentricity, resulting in tidal friction due to time-varying tidal forces from Uranus. This would have caused warming of the moon's interior by as much as 20 K.23

## Composition and internal structure

Ariel is the fourth largest of the Uranian moons, and may have the third greatest mass.f The moon's density is 1.66 g/cm3,4 which indicates that it consists of roughly equal parts water ice and a dense non-ice component.25 The latter could consist of rock and carbonaceous material including heavy organic compounds known as tholins.18 The presence of water ice is supported by infrared spectroscopic observations, which have revealed crystalline water ice on the surface of the moon.7 Water ice absorption bands are stronger on Ariel's leading hemisphere than on its trailing hemisphere.7 The cause of this asymmetry is not known, but it may be related to bombardment by charged particles from Uranus's magnetosphere, which is stronger on the trailing hemisphere (due to the plasma's co-rotation).7 The energetic particles tend to sputter water ice, decompose methane trapped in ice as clathrate hydrate and darken other organics, leaving a dark, carbon-rich residue behind.7

Except for water, the only other compound identified on the surface of Ariel by infrared spectroscopy is carbon dioxide (CO2), which is concentrated mainly on its trailing hemisphere. Ariel shows the strongest spectroscopic evidence for CO2 of any Uranian satellite,7 and was the first Uranian satellite on which this compound was discovered.7 The origin of the carbon dioxide is not completely clear. It might be produced locally from carbonates or organic materials under the influence of the energetic charged particles coming from Uranus's magnetosphere or solar ultraviolet radiation. This hypothesis would explain the asymmetry in its distribution, as the trailing hemisphere is subject to a more intense magnetospheric influence than the leading hemisphere. Another possible source is the outgassing of primordial CO2 trapped by water ice in Ariel's interior. The escape of CO2 from the interior may be related to past geological activity on this moon.7

Given its size, rock/ice composition and the possible presence of salt or ammonia in solution to lower the freezing point of water, Ariel's interior may be differentiated into a rocky core surrounded by an icy mantle.25 If this is the case, the radius of the core (372 km) is about 64% of the radius of the moon, and its mass is around 56% of the moon's mass—the parameters are dictated by the moon's composition. The pressure in the center of Ariel is about 0.3 GPa (3 kbar).25 The current state of the icy mantle is unclear, although the existence of a subsurface ocean is considered unlikely.25

## Surface

The highest-resolution Voyager 2 color image of Ariel. Canyons with floors covered by smooth plains are visible at lower right. The bright crater Laica is at lower left.

### Albedo and color

Ariel is the most reflective of Uranus's moons.5 Its surface shows an opposition surge: the reflectivity decreases from 53% at a phase angle of 0° (geometrical albedo) to 35% at an angle of about 1°. The Bond albedo of Ariel is about 23%—the highest among Uranian satellites.5 The surface of Ariel is generally neutral in color.26 There may be an asymmetry between the leading and trailing hemispheres;27 the latter appears to be redder than the former by 2%.g Ariel's surface generally does not demonstrate any correlation between albedo and geology on the one hand and color on the other hand. For instance, canyons have the same color as the cratered terrain. However, bright impact deposits around some fresh craters are slightly bluer in color.2627 There are also some slightly blue spots, which do not correspond to any known surface features.27

### Surface features

The observed surface of Ariel can be divided into three terrain types: cratered terrain, ridged terrain and plains.28 The main surface features are impact craters, canyons, fault scarps, ridges and troughs.29

Graben (chasmata) near Ariel's terminator. Their floors are covered by smooth material, possibly extruded from beneath via cryovolcanism. Several are cut by sinuous central grooves, e.g. Sprite and Leprechaun valles above and below the triangular horst near the bottom.

The cratered terrain, a rolling surface covered by numerous impact craters and centered on Ariel's south pole, is the moon's oldest and most geographically extensive geological unit.28 It is intersected by a network of scarps, canyons (graben) and narrow ridges mainly occurring in Ariel's mid-southern latitudes.28 The canyons, known as chasmata,30 probably represent graben formed by extensional faulting, which resulted from global tensional stresses caused by the freezing of water (or aqueous ammonia) in the moon's interior (see below).1828 They are 15–50 km wide and trend mainly in an east- or northeasterly direction.28 The floors of many canyons are convex; rising up by 1–2 km.30 Sometimes the floors are separated from the walls of canyons by grooves (troughs) about 1 km wide.30 The widest graben have grooves running along the crests of their convex floors, which are called valles.18 The longest canyon is Kachina Chasma, at over 620 km in length (the feature extends into the hemisphere of Ariel that Voyager 2 did not see illuminated).2931

The second main terrain type—ridged terrain—comprises bands of ridges and troughs hundreds of kilometers in extent. It bounds the cratered terrain and cuts it into polygons. Within each band, which can be up to 25 to 70 km wide, are individual ridges and troughs up to 200 km long and between 10 and 35 km apart. The bands of ridged terrain often form continuations of canyons, suggesting that they may be a modified form of the graben or the result of a different reaction of the crust to the same extensional stresses, such as brittle failure.28

False-color map of Ariel. The prominent noncircular crater below and left of center is Yangoor. Part of it was erased during formation of ridged terrain via extensional tectonics.

The youngest terrain observed on Ariel are the plains: relatively low-lying smooth areas that must have formed over a long period of time, judging by their varying levels of cratering.28 The plains are found on the floors of canyons and in a few irregular depressions in the middle of the cratered terrain.18 In the latter case they are separated from the cratered terrain by sharp boundaries, which in some cases have a lobate pattern.28 The most likely origin for the plains is through volcanic processes; their linear vent geometry, resembling terrestrial shield volcanoes, and distinct topographic margins suggest that the erupted liquid was very viscous, possibly a supercooled water/ammonia solution, with solid ice volcanism also a possibility.30 The thickness of these hypothetical cryolava flows is estimated at 1–3 km.30 The canyons must therefore have formed at a time when endogenic resurfacing was still taking place on Ariel.28

Ariel appears to be fairly evenly cratered compared to other moons of Uranus;18 the relative paucity of large cratersh suggests that its surface does not date to the Solar System's formation, which means that Ariel must have been completely resurfaced at some point of its history.28 Ariel's past geologic activity is believed to have been driven by tidal heating at a time when its orbit was more eccentric than currently.23 The largest crater observed on Ariel, Yangoor, is only 78 km across,29 and shows signs of subsequent deformation. All large craters on Ariel have flat floors and central peaks, and few of the craters are surrounded by bright ejecta deposits. Many craters are polygonal, indicating that their appearance was influenced by the preexisting crustal structure. In the cratered plains there are a few large (about 100 km in diameter) light patches that may be degraded impact craters. If this is the case they would be similar to palimpsests on Jupiter's moon Ganymede.28 It has been suggested that a circular depression 245 km in diameter located at 10°S 30°E is a large, highly degraded impact structure.33

## Origin and evolution

Ariel is thought to have formed from an accretion disc or subnebula; a disc of gas and dust that either existed around Uranus for some time after its formation or was created by the giant impact that most likely gave Uranus its large obliquity.34 The precise composition of the subnebula is not known; however, the higher density of Uranian moons compared to the moons of Saturn indicates that it may have been relatively water-poor.i18 Significant amounts of carbon and nitrogen may have been present in the form of carbon monoxide (CO) and molecular nitrogen (N2) instead of methane and ammonia.34 The moons that formed in such a subnebula would contain less water ice (with CO and N2 trapped as clathrate) and more rock, explaining the higher density.18

The accretion process probably lasted for several thousand years before the moon was fully formed.34 Models suggest that impacts accompanying accretion caused heating of Ariel's outer layer, reaching a maximum temperature of around 195 K at a depth of about 31 km.35 After the end of formation, the subsurface layer cooled, while the interior of Ariel heated due to decay of radioactive elements present in its rocks.18 The cooling near-surface layer contracted, while the interior expanded. This caused strong extensional stresses in the moon's crust reaching estimates of 30 MPa, which may have led to cracking.36 Some present-day scarps and canyons may be a result of this process,28 which lasted for about 200 million years.36

The initial accretional heating together with continued decay of radioactive elements and likely tidal heating may have led to melting of the ice if an antifreeze like ammonia (in the form of ammonia hydrate) or some salt was present.35 The melting may have led to the separation of ice from rocks and formation of a rocky core surrounded by an icy mantle.25 A layer of liquid water (ocean) rich in dissolved ammonia may have formed at the core–mantle boundary. The eutectic temperature of this mixture is 176 K.25 The ocean, however, is likely to have frozen long ago. The freezing of the water likely led to the expansion of the interior, which may have been responsible for the formation of the canyons and obliteration of the ancient surface.28 The liquids from the ocean may have been able to erupt to the surface, flooding floors of canyons in the process known as cryovolcanism.35

Thermal modeling of Saturn's moon Dione, which is similar to Ariel in size, density and surface temperature, suggests that solid state convection could have lasted in Ariel's interior for billions of years, and that temperatures in excess of 173 K (the melting point of aqueous ammonia) may have persisted near its surface for several hundred million years after formation, and near a billion years closer to the core.28

## Observation and exploration

HST image of Ariel transiting Uranus, complete with shadow

The apparent magnitude of Ariel is 14.4;6 similar to that of Pluto near perihelion. However, while Pluto can be seen through a telescope of 30 cm aperture,37 Ariel, due to its proximity to Uranus's glare, is often not visible to telescopes of 40 cm aperture.38

The only close-up images of Ariel were obtained by the Voyager 2 probe, which photographed the moon during its flyby of Uranus in January 1986. The closest approach of Voyager 2 to Ariel was 127,000 km (79,000 mi)—significantly less than the distances to all other Uranian moons except Miranda.39 The best images of Ariel have a spatial resolution of about 2 km.28 They cover about 40% of the surface, but only 35% was photographed with the quality required for geological mapping and crater counting.28 At the time of the flyby the southern hemisphere of Ariel (like those of the other moons) was pointed towards the Sun, so the northern (dark) hemisphere could not be studied.18 No other spacecraft has ever visited the Uranian system, and no mission to Uranus and its moons is planned.40 The possibility of sending the Cassini spacecraft to Uranus was evaluated during its mission extension planning phase.41 It would take about twenty years to get to the Uranian system after departing Saturn.41 See Planetary Science Decadal Survey for other Solar System mission concepts.

### Transits

On 26 July 2006, the Hubble Space Telescope captured a rare transit made by Ariel on Uranus, which cast a shadow that could be seen on Uranian cloud tops. Such events are rare and only occur around equinoxes, as the moon's orbital plane about Uranus is tilted 98° to Uranus's orbital plane about the Sun.42 Another transit, in 2008, was recorded by the European Southern Observatory.43

## Notes

1. ^ Surface area derived from the radius r : $4\pi r^2$.
2. ^ Volume v derived from the radius r : $4\pi r^3/3$.
3. ^ Surface gravity derived from the mass m, the gravitational constant G and the radius r : $Gm/r^2$.
4. ^ Escape velocity derived from the mass m, the gravitational constant G and the radius r : 2Gm/r.
5. ^ The five major moons are Miranda, Ariel, Umbriel, Titania and Oberon.
6. ^ Due to the current observational error, it is not yet known for certain whether Ariel is more massive than Umbriel.24
7. ^ The color is determined by the ratio of albedos viewed through the green (0.52–0.59 μm) and violet (0.38–0.45 μm) Voyager filters.2627
8. ^ The surface density of craters larger than 30 km in diameter ranges from 20 to 70 per million km2 on Ariel, whereas it is about 1800 for Oberon or Umbriel.32
9. ^ For instance, Tethys, a Saturnian moon, has the density of 0.97 g/cm3, which means that it is more than 90% water.7

## References

1. ^ "Ariel". Meriam-Webster Online. Retrieved 2010-09-21.
2. ^ a b "Planetary Satellite Mean Orbital Parameters". Jet Propulsion Laboratory, California Institute of Technology.
3. ^ a b Thomas, P. C. (1988). "Radii, shapes, and topography of the satellites of Uranus from limb coordinates". Icarus 73 (3): 427–441. Bibcode:1988Icar...73..427T. doi:10.1016/0019-1035(88)90054-1.
4. ^ a b c Jacobson, R. A.; Campbell, J. K.; Taylor, A. H.; Synnott, S. P. (June 1992). "The masses of Uranus and its major satellites from Voyager tracking data and earth-based Uranian satellite data". The Astronomical Journal 103 (6): 2068–2078. Bibcode:1992AJ....103.2068J. doi:10.1086/116211.
5. ^ a b c Karkoschka, Erich (2001). "Comprehensive Photometry of the Rings and 16 Satellites of Uranus with the Hubble Space Telescope". Icarus 151 (1): 51–68. Bibcode:2001Icar..151...51K. doi:10.1006/icar.2001.6596.
6. ^ a b Arlot, J.; Sicardy, B. (2008). "Predictions and observations of events and configurations occurring during the Uranian equinox" (pdf). Planetary and Space Science 56 (14): 1778. Bibcode:2008P&SS...56.1778A. doi:10.1016/j.pss.2008.02.034.
7. Grundy, W. M.; Young, L. A.; Spencer, J. R.; Johnson, R. E.; Young, E. F.; Buie, M. W. (October 2006). "Distributions of H2O and CO2 ices on Ariel, Umbriel, Titania, and Oberon from IRTF/SpeX observations". Icarus 184 (2): 543–555. arXiv:0704.1525. Bibcode:2006Icar..184..543G. doi:10.1016/j.icarus.2006.04.016.
8. ^ Hanel, R.; Conrath, B.; Flasar, F. M.; Kunde, V.; Maguire, W.; Pearl, J.; Pirraglia, J.; Samuelson, R. et al. (4 July 1986). "Infrared Observations of the Uranian System". Science 233 (4759): 70–74. Bibcode:1986Sci...233...70H. doi:10.1126/science.233.4759.70. PMID 17812891.
9. ^ Lassell, W. (1851). "On the interior satellites of Uranus". Monthly Notices of the Royal Astronomical Society 12: 15–17. Bibcode:1851MNRAS..12...15L.
10. ^ a b Lassell, William (December 1851). "Letter from William Lassell, Esq., to the Editor". Astronomical Journal 2 (33): 70. Bibcode:1851AJ......2...70L. doi:10.1086/100198.
11. ^ Herschel, William, Sr. (1 January 1798). "On the Discovery of Four Additional Satellites of the Georgium Sidus. The Retrograde Motion of Its Old Satellites Announced; And the Cause of Their Disappearance at Certain Distances from the Planet Explained". Philosophical Transactions of the Royal Society of London 88: 47–79. Bibcode:1798RSPT...88...47H. doi:10.1098/rstl.1798.0005.
12. ^ Struve, O. (1848). "Note on the Satellites of Uranus". Monthly Notices of the Royal Astronomical Society 8 (3): 44–47. Bibcode:1848MNRAS...8...43.
13. ^ Holden, E. S. (1874). "On the inner satellites of Uranus". Monthly Notices of the Royal Astronomical Society 35: 16–22. Bibcode:1874MNRAS..35...16H.
14. ^ Lassell, W. (1874). "Letter on Prof. Holden's Paper on the inner satellites of Uranus". Monthly Notices of the Royal Astronomical Society 35: 22–27. Bibcode:1874MNRAS..35...22L.
15. ^ Lassell, W. (1852). "Beobachtungen der Uranus-Satelliten". Astronomische Nachrichten (in German) 34: 325. Bibcode:1852AN.....34..325.
16. ^ Harrington, Phillip S. (2011). Cosmic Challenge: The Ultimate Observing List for Amateurs. Cambridge University Press. p. 364. ISBN 978-0-521-89936-9.
17. ^ Kuiper, G. P. (1949). "The Fifth Satellite of Uranus". Publications of the Astronomical Society of the Pacific 61 (360): 129. Bibcode:1949PASP...61..129K. doi:10.1086/126146.
18. Smith, B. A.; Soderblom, L. A.; Beebe, A.; Bliss, D.; Boyce, J. M.; Brahic, A.; Briggs, G. A.; Brown, R. H. et al. (4 July 1986). "Voyager 2 in the Uranian System: Imaging Science Results". Science 233 (4759): 43–64. Bibcode:1986Sci...233...43S. doi:10.1126/science.233.4759.43. PMID 17812889. (See pages 58–59, 60–64)
19. ^ Ness, Norman F.; Acuña, Mario H.; Behannon, Kenneth W.; Burlaga, Leonard F.; Connerney, John E. P.; Lepping, Ronald P.; Neubauer, Fritz M. (July 1986). "Magnetic Fields at Uranus". Science 233 (4759): 85–89. Bibcode:1986Sci...233...85N. doi:10.1126/science.233.4759.85. PMID 17812894.
20. ^ Krimigis, S. M.; Armstrong, T. P.; Axford, W. I.; Cheng, A. F.; Gloeckler, G.; Hamilton, D. C.; Keath, E. P.; Lanzerotti, L. J. et al. (4 July 1986). "The Magnetosphere of Uranus: Hot Plasma and Radiation Environment". Science 233 (4759): 97–102. Bibcode:1986Sci...233...97K. doi:10.1126/science.233.4759.97. PMID 17812897.
21. ^ Miller, C.; Chanover, N. J. (March 2009). "Resolving dynamic parameters of the August 2007 Titania and Ariel occultations by Umbriel". Icarus 200 (1): 343–346. Bibcode:2009Icar..200..343M. doi:10.1016/j.icarus.2008.12.010.
22. ^ Tittemore, William C.; Wisdom, Jack (June 1990). "Tidal evolution of the Uranian satellites: III. Evolution through the Miranda-Umbriel 3:1, Miranda-Ariel 5:3, and Ariel-Umbriel 2:1 mean-motion commensurabilities". Icarus 85 (2): 394–443. Bibcode:1990Icar...85..394T. doi:10.1016/0019-1035(90)90125-S.
23. ^ a b c d Tittemore, William C. (September 1990). "Tidal heating of Ariel". Icarus 87 (1): 110–139. Bibcode:1990Icar...87..110T. doi:10.1016/0019-1035(90)90024-4.
24. ^ "Planetary Satellite Physical Parameters". Jet Propulsion Laboratory (Solar System Dynamics). Archived from the original on 22 May 2009. Retrieved 2009-05-28.
25. Hussmann, H.; Sohl, Frank; Spohn, Tilman (November 2006). "Subsurface oceans and deep interiors of medium-sized outer planet satellites and large trans-neptunian objects". Icarus 185 (1): 258–273. Bibcode:2006Icar..185..258H. doi:10.1016/j.icarus.2006.06.005.
26. ^ a b c Bell, J. F., III; McCord, T. B. (1991). "A search for spectral units on the Uranian satellites using color ratio images". Lunar and Planetary Science Conference, 21st, Mar. 12–16, 1990 (Conference Proceedings). Houston, TX, United States: Lunar and Planetary Sciences Institute. pp. 473–489. Bibcode:1991LPSC...21..473B.
27. ^ a b c d Buratti, Bonnie J.; Mosher, Joel A. (March 1991). "Comparative global albedo and color maps of the Uranian satellites". Icarus 90 (1): 1–13. Bibcode:1991Icar...90....1B. doi:10.1016/0019-1035(91)90064-Z. ISSN 0019-1035.
28. Plescia, J. B. (May 21, 1987). "Geological terrains and crater frequencies on Ariel". Nature 327 (6119): 201–204. Bibcode:1987Natur.327..201P. doi:10.1038/327201a0. ISSN 0028-0836.
29. ^ a b c "Nomenclature Search Results: Ariel". Gazetteer of Planetary Nomenclature. USGS Astrogeology. Retrieved 2010-11-29.
30. Schenk, P. M. (1991). "Fluid Volcanism on Miranda and Ariel: Flow Morphology and Composition". Journal of Geophysical Research 96: 1887. Bibcode:1991JGR....96.1887S. doi:10.1029/90JB01604. (See pages 1893–1896)
31. ^ Stryk, Ted (2008-03-13). "Revealing the night sides of Uranus' moons". In Lakdawalla, Emily. The Planetary Society Blog. The Planetary Society. Retrieved 2012-02-25.
32. ^ Plescia, J. B. (1987). "Geology and Cratering History of Ariel". Abstracts of the Lunar and Planetary Science Conference 18: 788. Bibcode:1987LPI....18..788P.
33. ^ Moore, Jeffrey M.; Schenk, Paul M.; Bruesch, Lindsey S.; Asphaug, Erik; McKinnon, William B. (October 2004). "Large impact features on middle-sized icy satellites" (PDF). Icarus 171 (2): 421–443. Bibcode:2004Icar..171..421M. doi:10.1016/j.icarus.2004.05.009.
34. ^ a b c Mousis, O. (2004). "Modeling the thermodynamical conditions in the Uranian subnebula – Implications for regular satellite composition". Astronomy & Astrophysics 413: 373–380. Bibcode:2004A&A...413..373M. doi:10.1051/0004-6361:20031515.
35. ^ a b c Squyres, S. W.; Reynolds, Ray T.; Summers, Audrey L.; Shung, Felix (1988). "Accretional Heating of the Satellites of Saturn and Uranus". Journal of Geophysical Research 93 (B8): 8779–8794. Bibcode:1988JGR....93.8779S. doi:10.1029/JB093iB08p08779.
36. ^ a b Hillier, John; Squyres, Steven W. (August 1991). "Thermal stress tectonics on the satellites of Saturn and Uranus". Journal of Geophysical Research 96 (E1): 15,665–15,674. Bibcode:1991JGR....9615665H. doi:10.1029/91JE01401.
37. ^ "This month Pluto's apparent magnitude is m=14.1. Could we see it with an 11" reflector of focal length 3400 mm?". Singapore Science Centre. Archived from the original on 11 November 2005. Retrieved 2007-03-25.
38. ^ Sinnott, Roger W.; Ashford, Adrian. "The Elusive Moons of Uranus". Sky & Telescope. Retrieved 2011-01-04.
39. ^ Stone, E. C. (December 30, 1987). "The Voyager 2 Encounter with Uranus". Journal of Geophysical Research 92 (A13): 14,873–14,876. Bibcode:1987JGR....9214873S. doi:10.1029/JA092iA13p14873. ISSN 0148-0227.
40. ^ "Missions to Uranus". NASA Solar System Exploration. 2010. Retrieved 2011-01-11.
41. ^ a b Bob Pappalardo; Linda Spiker (2009-03-09). "Cassini Proposed Extended-Extended Mission (XXM)" (PDF). Retrieved 2011-08-20.
42. ^ "Uranus and Ariel". Hubblesite (News Release 72 of 674). 26 July 2006. Retrieved 2006-12-14.
43. ^ "Uranus and satellites". European Southern Observatory. 2008. Retrieved 2010-11-27.


text:
# Can someone please walk me through the steps

$12 \frac{1}{10} = \frac{121}{10}$ (think $\frac{120}{10} + \frac{1}{10}$), so you would need 121 "1/10 cups."


text:
# Invertibility of Linear Transformation

Let $A$ and $B$ be two $n\times n$ matrices that have no eigenvalues in common. Show that the transformation $T$ that maps the $n\times n$ matrices, $M_n$, to the $M_n$ and is defined by the formula $$T(S):=AS-SB$$ is invertible.

• I think the eigenvalues of $T$ are of the form $\lambda_i-\mu_j$ where $\lambda_i$ and $\mu_j$ are the eigenvalues of $A$ and $B$ respectively. I forget how to show this statement, but assuming it is true, it ensures that $T$ has no zero eigenvalues, i.e., it is invertible. – angryavian Nov 30 '14 at 3:13
• @angryavian Any hints on how to show this? – Dia McThrees Nov 30 '14 at 3:20

If $AS = SB$ and $u$ is an eigenvector of $B$ for eigenvalue $\lambda$, then $A S u = S B u = \lambda S u$. Since $A$ and $B$ have no eigenvalues in common, we must have $S u = 0$. That would settle matters in the case where $B$ is diagonalizable.

More generally, you have to consider generalized eigenvectors. Prove by induction on $k$ that $S v = 0$ for any $v$ such that $(B - \lambda I)^k v = 0$.

• How do we know that $AS=SB$? – Dia McThrees Nov 30 '14 at 15:36
• $T$ is a linear map from a finite dimensional vector space to itself. So $T$ is invertible iff $ker\ T = 0$. So if we can show that for any $S \in ker\ T$ we have $S = 0,$ then $T$ is invertible. So we pick an arbitrary $S \in ker\ T.$ Then we have $TS = 0,$ or equivalently $AS = SB.$ Now continue with the answer of Robert Israel. – jflipp Nov 30 '14 at 22:14
• @jflipp Proof by induction doesn't make sense... If $(B-\lambda I)^nv=0$, then we have to show $(B-\lambda I)^{n+1}v=0$. Since it's true for $n$, we get $(B-\lambda I)(B-\lambda I)^nv=0$, then $(B-\lambda I)=0$, which is obviously not true. – Dia McThrees Nov 30 '14 at 22:33
• No, that's not how the induction goes. The statement is that if $(B-\lambda I)^k v = 0$ then $S v = 0$. For the induction step, you assume that $(B - \lambda I)^k v = 0$ implies $S v = 0$, and you consider $u$ such that $(B - \lambda I)^{k+1} u = 0$. – Robert Israel Nov 30 '14 at 23:35
• @RobertIsrael Ohhh! Thank you! I've been banging my head on this for a long time. – Dia McThrees Nov 30 '14 at 23:37

As I mentioned in a comment to the answer by Robert Israel, one must take the hypothesis on $A,B$ to mean that even over an extension of the (unspecified) field $K$ over which $T$ is a linear operator (i.e., for which it acts on $M_n(K)$) they have no common eigenvalue. With the more strict interpretation of "no common eigenvalues in $K$" there are easy counterexamples; for instance take $K=\Bbb R$, $n=2$, $A$ a rotation matrix (other than $\pm I_2$), and $B=A^{-1}$ (so there are no eigenvalues in$~K$ at all, let alone common eigenvalues), one checks that all orthogonal reflection matrices$~S$ are in the kernel of$~T$ (because $SAS^{-1}=B$).

This suggests that a different formulation of that hypothesis might allow a proof that does not need to extend the field$~K$. Indeed one may replace the hypothesis by the existence of relatively prime annihilating polynomials: $P_A,P_B\in K[X]$ such that $P_A[A]=0=P_B[B]$. This clearly excludes the existence of a common eigenvalue$~\lambda$ in$~\overline K$, since that would make $X-\lambda$ a non-existing common factor in $\overline K[X]$ of both $P_A$ and $P_B$, while conversely in the absence of such common eigenvalues one can take $P_A=\chi_A$ and $P_B=\chi_B$ (or to avoid needing the Cayley-Hamilton theorem, one could also take their minimal polynomials instead of characteristic polynomials).

So here is the proof that $T(S)=0$ implies $S=0$. Iterating the hypothesis $AS=SB$ easily gives $A^kS=SB^k$ for all $k\in\Bbb N$, and by linearity $P[A]S=SP[B]$ for all $P\in K[X]$. In particular $$0=0S=P_A[A]S=SP_A[B].$$ Now the given fact that $P_A$ and $P_B$ are relatively prime means that $P_B$ has an inverse modulo$~P_A$, namely the second Bezout coefficient $V$ in a Bezout relation $\gcd(P_A,P_B)=1=UP_A+VP_B$. Therefore $P_B[A]V[A]=I$, and right-multiplying the equation above by $V[A]$ gives $S=0$, as desired.