text: Last edited: Jan 19, 2013. The Alt code shortcut for the Square root symbol is Alt+251 or 221A, Alt+X. A simple way to add a square root symbol is the shortcut key and the shortcut is Alt + 251. For MS Word, place the curser at the desired location; for Excel, activate the cell that will contain the symbol. When you want to type square root, cube root and fourth root symbols on your documents then the easy way is to use alt code shortcuts. If you have a Mac keyboard, press "option" and "v" at the same time instead. The meaning of mathematical symbols of square root has been present throughout the years, already in Babylonian times a clay tablet from the 18th century BC, shows that said civilization knew the square root of two as well as a calculation algorithm. And for this, we have two different ways which we can use. You could use this method for other math symbols as well. I have a standard American keyboard and was wondering how to type a square root symbol. 40 px. To insert this symbol (using the Alt code), follow these three simple steps: Place the insertion pointer at where you want to insert the symbol. Tap a symbol to view an alternative … Step 1: Click at the point where you wish to add the square root symbol. Bold. and as most of us wait for Apple to push out a the iPhone 5, a large amount of us have renewed our cell phone contracts and now have decided to get a Blackberry, an HTC, an LG or a Samsung Android … Hold the option or alt key and type 221C to produce fourth root symbol ∜. This is how the squared symbol will appear in your text. The shortcut key for Square Root symbol is 221A, Alt+X (type 221A and then press Alt + X). I tried \surd, but did not consider all my expression below this symbol. Apple Footer. To easily get the square root symbol from the dialog, select. However, in this section, I’ll show you the easiest approaches. Question … https://www.youtube.com/channel/UCmV5uZQcAXUW7s4j7rM0POg?sub_confirmation=1 How to Type Square Root Symbol in Excel Or you can go to emojis and use a tick symbol  - not exactly a square root … You can open the message window or just the notes app. For an Android, this is where you will find the Squared symbol. Once you calculate a square root from a number, the next thing you can do is to add a square root symbol. Many times when you place your mouse in the answer box you will see another "window" that will pop up that will allow you to click on the radical sign. Phones don't really come with instruction manuals anymore, and learning how to use your phone effectively can be a big task. For an Android, this is where you will find the Squared symbol. … This will open the other options for this key on the keyboard. This is basically an adding button for you. There are several ways to get any symbol in Word including the square root symbol. On your Android phone, you’re not limited to typing only the symbols you see on the alphabetic keyboard. How to use your Android calculator tablet phone mobileDont forget to like our video's, Its what makes this more worthwhile. So the iPhone 4S is right here. https://www.youtube.com/channel/UCmV5uZQcAXUW7s4j7rM0POg?sub_confirmation=1How to Type Numbers With the Square Root Symbol in Word Press and hold the Alt key on your keyboard. Like how ² is alt + 253 and is alt + 254. thanks and DON'T SAY TO COPY AND PASTE. I then "chop out" the first hat - the position of the character to be superscripted is now 4, the position of the other hats is now 7 and 11. You can also find u-221A, u*221A, un+221A, u221A, u=221A or c+221A. Whilst on Mac, it is Option+V. Using squared symbol on an android can be super easy as it has the symbol built-in the phones number keys which can be accessed through the following steps. Square root of some number "A" is a number "X" such that "X" multiplied by itself would be "A". The alt code (character code) for the square root symbol is 251. Place the curser at where you want to type the symbol. As you hold on to the Alt key, press 251 (the square root symbol’s Alt code) on the numeric keypad at the right-hand side of the keyboard. If you have any further questions, please let me know in the comments. Just how we clicked the numbers keypad on the phone, similarly, we will do the same for a Windows phone. Open the numeric keypad now, and long-press the number two until it shows you more hidden options for this key. Alternatively, for MS Word users, type the character code (221A), then press Alt+X to convert this code into the symbol. - Eric In this section, I’ll show you how to insert the square root symbol in Word using the insert equation feature. why? On Mac, press Option+V on the keyboard. Return to the app in which you need the symbol, tap and hold down on the screen and select "Paste." Use the Enter key when finished to continue typing the rest of your equation. Tap on this next. Posted on Dec 13, 2016 9:22 AM Reply I have this question too (1853) I have this question too Me too (1853) Me too. This site contains user submitted content, comments and opinions and is for informational purposes only. Keep scrolling down on the screen until you find the tab for ‘Keyboard’. This is how you may insert the Square root symbol into Word. Just as in Microsoft Word, you can Insert the square root symbol into Excel through several methods. With this method, you can type the symbol anywhere you want to enter text including Word, Excel, and PowerPoint. Cube Root; This is how the square symbol will look like. and once you release the Alt key, and the symbol appears. This code point first appeared in version 1.1 of the Unicode® Standard and belongs to the "Mathematical Operators" block which goes from 0x2200 to 0x22FF. If you are on Mac, use Option+V shortcut to type the square root symbol in your Word document. Most cell phones have calculators today, but iPhone offers a full-function scientific calculator too. Tap on the ‘+’ symbol in the top right corner of the screen that now appears. The space for ‘Phrase’ is where you would simply copy and paste any symbol (in this case the squared symbol which I copied from a message sent on an App). The shortcut for square root symbol on mac is Option + V. The simplest way to type the square root symbol on Mac is to use the keyboard shortcut which is Option + V. Thus, to type the square root symbol on Mac: The square root symbol should be inserted for you. Now Option-V pauses, allowing me to accent the next character with a ˇ . You can use the ‘^’ to show the reader that whatever you will write after this will be a power to that number. This tool is very convenient to help you preview the symbol, including viewing the details of the symbol display and the effect displayed on the web page. A simple way to add a square root symbol is the shortcut key and the shortcut is Alt + 251. Learn how to type the square root symbol using the equation tool in Microsoft Word 2010 You can try this on your iPhone. More Less. 4 years ago. 1.Press +=> (where ever you want to insert square root, you will get equation box, and then) . From this dialog, search for the symbol you wish to insert. Find the square, cube or fourth root symbols to insert on Pages, Keynote and Numbers. To type the square root symbol in Word using the keyboard, first ensure that your Num Lock is turned on. Go to where you want to type the square root symbol. If you are using Windows go to character under the Accessories Menu. Squared symbols can be used for a lot of reasons when texting. Select Symbol and then More Symbols. This will open the other options for this key on the keyboard. Best Way To Root My Device. Since Android 5000! For instance, I put ^^ in under shortcut. Copy and paste the Square symbol or use the unicode decimal, hex number or html entity in social websites, in your blog or in a document. If the number is a fraction, before filling in the numbers, first click on the box in the square root symbol, then select the “Design” menu again, this time select the “fraction” menu tab then click again on the “Stacked fraction” option, it will form a fraction. I … Tap and hold down the symbol until it is highlighted and tap "Copy." Apple may provide or recommend responses as a possible solution based on the information provided; every potential issue may involve several factors not detailed in the … the symbol’s alt code) on the numeric keypad; then release the Alt key. To put the square root symbol, you must enter the following key combination: Option + V key Square root symbol in word. Specify the cell that will contain the symbol. … 6.Go in symbol. You can also spell it with u 221A unicode, u plus 221A, uncode 221A or unicode + 221A. Next, click "symbol" … How to get the square root sign inside a legend? By clicking this, you can add any symbol as you like. If you have a Mac keyboard, press "option" and "v" at the same time instead. Square root symbol on Mac. For example, if you were take the square root of 4 you could say: sqrt(4)= 2. Type "squared symbol" into the Safari search box. Then whilst pressing down the Alt key, type the symbol’s Alt code on the numeric keypad which is 251. When i try to download apps from the google play store it says, "your app will be downloaded shortly." Option-V used to create the square root symbol. Copy the Cube Root in the above table (it can be automatically copied with a mouse click) and paste it in word, Or . At the same time, press 2, 5, then 1. The square root symbol shortcut in Mac is Option + V. For Mac users, to type the square root symbol in Excel, specify the cell that will contain the symbol; then press Option+V on your keyboard. Oct 17, 2014 6:04 PM Reply Helpful (1) Thread reply - more options. To open the scientific calculator, turn your iPhone to landscape view. You are in the right place if you want to type the square root symbol. To access these special keyboards, tap the symbol or numeric key, such as the ?1J key. The mathematical square root symbol is called radical. This section will focus on that. If you have a Windows keyboard, hold down the "alt" key. But only third and fourth level chooser keys and unicode hex codes can produce mathematical square root text symbol. Step 2: Click the Symbols button at the right end of the … https://www.youtube.com/channel/UCmV5uZQcAXUW7s4j7rM0POg?sub_confirmation=1How to Type Numbers With the Square Root Symbol in Word (If you have locked your iPhone in Portrait view, this won’t work until you unlock it: swipe up from the bottom of the screen to open the command center and tap the Orientation Lock button.) Whether you are using Mac or the Windows PC, you’ll learn the techniques to easily type or insert this symbol anywhere in your PC including apps like Microsoft Word, Excel, PowerPoint, etc. Press and hold Alt (left Alt only on an international keyboard) while typing 2 5 1 on the numeric keypad. How do you find the square root on a iPhone calculator. You will write down a shortcut for it so that whenever you type the shortcut, you will get the phrase instead. Tap the various buttons on the Calculator app screen to input your equation. Square Symbol Preview Variations. 4.press Alt+ equalto Then type /sqrt. To insert a square root, just type \sqrt and then the Enter or Tab key to insert the square root symbol. 0 0. This should give you the symbol. Additionally, you always just enter the text "sqrt." Open the keyboard for your phone, and open the numeric keypad to see all the numbers on your phone’s screen. The Android Phones and Windows Phone might work in the same way when talking about how to use square symbols. When long clicking on some symbols/characters on your soft keyboard you will see the key pop-up expanding to give you an additional choice of symbols, then drag and release your finger to print them. Facebook Twitter. Inserting the squared symbol on your Android smartphone is relatively easy and straightforward. Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. 3.Go to Insert. Every positive number "A" has two square roots: positive and negative ±√a . When you open the settings for Keyboard, you will find the tab for ‘Text Replacement’. The number and variety of special character keyboards varies from phone to phone. Many thanks, More Less. The US keyboard layout still gives me a square root symbol. At this moment, only the Mathematical Operators symbols including the Square root symbol will display in the visible area of the Symbols dialog box. Question: Q: how do you find a square root on an iPhone calculator More Less. Major love for reading, but writing is what keeps me going. It could be Microsoft Word, Excel or even on a browser. Then go to your phone's Settings menu and tap "General," "Keyboard," "Text Replacement" and "+." Posted on Oct 17, 2014 5:46 PM Reply I have this question too (537) I have this question too Me too (537) Me too. Some symbols have not … Next, click "symbol" … Some buttons, such as the square root must come before you type a value, as illustrated in the figure. Joined: Dec 4, 2013 Messages: 5 Thanks Received: 0. You can also use a Cube symbol by long-pressing the number 3. To type a square root in Microsoft Word without using keyboard shortcuts, click the "Insert" button at the top of the screen. You need to hold down the Alt key as you type 251 on the numeric keypad. This application is completely safe and woks with over 7000 Android devices, making it the one-stop shop for people who want to root the Android phones. If you are an iPhone user, here is what you need to do. On the other hand, if you want to use it in anything other than Quora, copy the Unicode value and paste it: √. Copy Square root symbol from google! square root √ U+221A copy and paste. To insert the squared sign, just long-press the number 2 and it will insert the superscript ². Square root or principal square root symbol √ does not have 2 on the root. The sound playback (e.g. If you select the font you are using and scroll down (about two thirds of the way down) you should find the square root symbol (along with a va Or you can highlight the square root symbol, click [Ctrl] C, go to your document and click [Ctrl] V … With this tool, you can adjust the size, color, italic, and bold of Square Root(symbol). How to make a Square Root Symbol in a text document on your computer: For word processors, you can: Copy and paste the following square root symbol into your document: √ Highlight the square root sign with your mouse, right click, select copy, go to your document and right click your mouse again and select paste. This final option will have you searching through a menu of symbols and selecting the square root symbol. Double-click on the symbol to insert it into your Excel worksheet. For the Renaissance, the mathematicians of the epic are asked to define the square root of a negative number, this is how complex numbers arise, providing even … If you are looking for how to type numbers with square root symbol in word,Then here is the simple method to do this.Watch this video. On windows, the square root symbol shortcut is Alt+125. For instance, 6^2, means, 6 to the power 2. Toggle navigation Unicode® Symbol. Dream to publish my own novels someday. 1 decade ago. Fill in the Column Box with the desired number; To enter a number, click on the box and fill in the number you want to enter. At the same time, press 2, 5, then 1. Now whenever I type the shortcut ‘^2’ on the keyboard, I see the symbol as a suggestion writes under the space for typing. Say the hats were at positions 4, 8 and 12. Hi dp_lucky007! Then release the Alt key. An iPhone does not have an inbuilt symbol for square or cube symbols. I chop out the second hat, the character to superscript is now at 7 and the hat has moved to 10. In the "Phrase" field, paste the square root symbol. Those same shortcuts work when in an … Using a Shortcut Key . You can safely add this character in your html code with the entity: √ You can use the u+221A copy pc button below. This should give you the symbol. Save it next. Select Symbol and then More Symbols. Place the cursor where you want to type the square root symbol. When A 2= B then A is the square root of B indicated as √B = A. This should give you the symbol. but then it never downloads. Does this "irola" \$59 tablet have a glass screen or plastic???? If you use the symbol regularly, add this copy as a keyboard shortcut. How to type Cube Root in word? If you are trying to type it on Quora, enter the math mode and type [code]\sqrt{x} [/code]You will see $\sqrt{x}$. You need to hold down the Alt key as you type 251 on the numeric keypad. Any ideas what keyboard combination will create the square root symbol - √ ? For positive "A", the principal square root can also be written in … The ‘Shortcut’ is where you will write down how you can access this phrase without using a tab for it. Then just type the number or expression you want to include under the square root. Copy the Square Root in the above table (it can be automatically copied with a mouse click) and paste it in word, Or Select the Insert tab. The square root symbol shortcut in Windows is Alt+254 (on the numeric keypad). How to type Square Root Symbol [√] + Shortcut, Alt Code to type Square Root symbol [√] in Windows, How to type square root symbol on Mac – shortcut, Typing square Root Symbol in Microsoft Word, Typing square Root Symbol in Microsoft Excel, You may also like the following Tutorials. You can also use a Cube symbol by long-pressing the number 3. If you have a Windows keyboard, hold down the "alt" key. 1 1. shamim. Here, four different symbol keyboards are shown. Link to this Post; Question marked as Solved User profile for user: Heather Hollick Heather Hollick User level: Level 1 (108 points) iOS Apps Speciality level out of ten: 1. Either go to “Math Symbols” or search for “root” using the search box. Step 2: Type \sqrt with a space after the “t”. Most Android phones feature alternative character keyboards. Below are the steps to type the square root symbol in Windows PC using the Alt Code: The square root symbol will immediately be inserted as soon as you let go of the Alt key. The Android platform supports unicode and all its characters, however the soft keyboard that comes with your Android ROM probably won't carry all your desired symbols. the symbol’s alt code) on the numeric keypad; then release the Alt key. However, if you are using a laptop without the numeric keypad, click here to learn how to use the hidden numeric keypad on laptops. For Windows users, to type the square root symbol in Excel using a keyboard shortcut, first click on the cell to contain the symbol; press and hold the Alt key and press 251 (i.e. Look for a site that shows the symbol and open it. Long-press the calculator’s text (or results) to cut or copy the results. square root: block: Mathematical Operators (Math_Operators) common typos: u+221A, u+122A: There are alternative spelling that can be found in the wild for the unicode character 221A like u 221A, (u+221A) or u +221A. Alt Code Shortcut for Square Root Symbol. Hold down the Alt key and type 251 on … Thousands of new, high-quality pictures added every day. You can follow the steps as mentioned below to write down the squared or cubed symbol on your phone. You do not have t worry about corruption of data when … How to Type Square Root Symbol on Android Phone How to Type Squared on Phone How to Type Squared Symbol on Android Phone How to Type Subscript in Word How to Type Subscript on Mac How to Type the Squared Symbol (²) on Your Computer How to Write Co2 in Word How to Write Co2 With Small 2 How to Write Power of 2 in Word How to Write Square in Word How to Write Square in Word in … Check your system docs for a notebook keyboard, or for an OS other than Windows. Easy u+221A copy paste: ∙ U+2219 | U+221B … Square Root (Symbol/sign/mark) Preview and HTML-code. Choose the one that you require. Expand Collapse. The square is one of the most common mathematical operations, but how do you write the squared symbol on your computer or mobile phone? At least one symbol keyboard is available, though you may find multiple symbol keyboards, special numeric keypads, and even emoji keyboards. It is not on your keyboard, but a good representation of a square root is: sqrt(of some #). Aside from using the square root symbol shortcut, you can also use this mouse-based method to insert the square root symbol in to your Excel worksheets. The above steps will work fine if you are using a keyboard with the numeric keypad. Our tips and how-to articles will set you on the path to Android expertise. Select the … If you have a Mac computer, its procedure is different. I then read each character in the string in turn and note the location of the ^ ("hats")'s in each. With this tool, you can adjust the size, color, italic, and bold of Cube Root(symbol). 5 years ago. Now press the square root symbol shortcut for mac –. 2 6. drea. It is a versatile tool which comes with many other features for your phone. Stephen L . So for example if we want to write the square root of 16, it would look like this: The square root symbol shortcut in Mac is Option + V. For Mac users, to type the square root symbol in Excel, specify the cell … Select the Insert tab. #1. Now, how do you use these shortcuts to get the symbol? The root symbol √ will be entered when you release the Alt key. Use the parentheses to determine which part of a long equation gets calculated first. Square root symbol on samsung galaxy phone keyboard. To type a square root, first click where you want to insert the symbol. Add a SQUARE Root Symbol by Applying Custom Formatting . BaseColumns; CalendarContract.AttendeesColumns; CalendarContract.CalendarAlertsColumns; CalendarContract.CalendarCacheColumns; CalendarContract.CalendarColumns 251) using the numeric keypad, then release the Alt key. However, the square root (but not cube or 4th root) symbol is available as Alt+251. To switch keyboards, locate the special symbols, illustrated here. Without any further ado, below are the instructions to insert the square root symbol in Word: This action should display the Symbol’s dialog box. Top Android Phones List For Christmas 2011 . How to Type Square Root Symbol on Android Phone How to Type Squared on Phone How to Type Squared Symbol on Android Phone How to Type Subscript in Word How to Type Subscript on Mac How to Type the Squared Symbol (²) on Your Computer How to Write Co2 in Word How to Write Co2 With Small 2 How to Write Power of 2 in Word How to Write Square in Word How to Write Square in Word in … Here are the shortcuts for square root in Windows and Mac … Now, while the square symbol is shown as a symbol in all the three kinds of phones, there is another way of squaring your text. Save my name, email, and website in this browser for the next time I comment. How to Type Squared Symbol on Windows, iPhone and on Android, Rumours Suggest iPhone 13 Pro Lineup To Get 120Hz LTPO Panels, New Cyberpunk 2077 Mod Makes NPC Movement More Realistic, No Ties Between Violent Video Games And Aggressive Behavior, Finds Decade-Long Study, Microsoft To Introduce Modern Standby: Instantaneous Wake Up For Windows 10X & Windows 10 Devices, ASUS To Launch Refreshed ROG Strix Laptops With AMD Ryzen 9 5900HX, NVIDIA GeForce RTX 3080, and 2K 165Hz Screen At CES 2021. While you can find an easy shortcut on both, Android and Windows phone, but for an iPhone you will have to make a shortcut to use it on your phone. Open Character Viewer by pressing “Control + Command + Space”. The square symbol on the Android phone. Number and symbol keyboards. To find the symbol easily, at the bottom right area of the dialog, in the from drop-down menu, select Unicode (hex). The information above is just a quick way to perform the task. Search for the square root symbol from scrolling up or down using the scroll bar. This video will show you how to square root a number on a casio scientific calculator. Without any further ado, let’s get started! If you need to insert a Cube Root in a web page, please copy the HTML-code corresponding to the Cube Root in the above table. 5.0 "Lollipop" this symbol notifies, that there's an active root session. This is how the square symbol will look like. For example to take the square root of 25, you would enter sqrt(25). Now when i type ^^ on my phone/Ipad it auto corrects to the radical. Square Symbol Color Italic Symbol Black Square red Black Square orange Black Square pink Black Square green Black Square royal blue Black Square purple White Square red White Square orange White Square pink … To type the square root sign, there really isn't a way to do so using the keyboard. Size: Color: Click to pick color: More: Italic. But by doing ^1/2 or ^0.5 is easiest way to signify the square root. Tap "Shortcuts," the "+" sign and … It doesn’t involve using the keyboard. 5.Click on Insert from the top menu. 2.To insert Square root symbol (√) in Word, type 221A and then press Alt+X to convert the alt code into a square root symbol. I read each square (or textbox) text into a string. Windows Phone Will Gavin. In this article, we will teach you how to type the squared symbol on your computer or smartphone, be it on a word processor like Microsoft Word or in your emails and online conversations. Since I am an android user, I would like to start with my phone. #2. For Windows users, to type the square root symbol in Excel using a keyboard shortcut, first click on the cell to contain the symbol; press and hold the Alt key and press 251 (i.e. music) is supported by DolbyAudio (currently only the HTC One M9 as a replacement for BeatsAudio). How To Make The Square Root Symbol Effective And Free from harm Root-AndroidOnline.com. Step 1: Click the Insert tab at the top of the window. These are the steps involved to add the square root symbol anywhere in your computer. The square root symbol should appear in the equation field after you type \sqrt and press on the spacebar key. Like many mathematical symbols, it’s hard to find in most keyboards. The equation feature in Word is very helpful when you need to insert lots of mathematical symbols into your project. HTML-code. More Less. Android Root, is an application developed by Dr. Fone, with which in just a single click, you can use to root your Motorola. Using the insert symbol dialog is a mouse way approach to inserting the square root symbol in Word. Using the alt code is one of the simplest methods you can use to type the square root symbol in Windows. Find square root symbol stock images in HD and millions of other royalty-free stock photos, illustrations and vectors in the Shutterstock collection. #13 solution101, Jan 19, 2013. Again, at the top right area of the dialog, select Mathematical Operators in the Subset drop-down list. Cube Root (Symbol/sign/mark) Preview and HTML-code. The symbol informs you, that the device, as well as the mobile network with which you're connected, supports Voice over LTE (VoLTE). To type a square root, first click where you want to insert the symbol. To type the square root symbol in Word on your keyboard, press down the Alt key and type the Square Root symbol alt code (i.e. For ‘ keyboard ’ your Excel worksheet 2 as showing on your keyboard. key! Use square symbols, press  option '' and  v '' at the top of the screen that appears! The above steps will work fine if you have a glass screen plastic. Relatively easy and straightforward into Word plastic???????????... Is not on your keyboard. a Menu of symbols and selecting the square root is! Messages: 5 thanks Received: 0 once you release the Alt code on! Option will have you searching through a Menu of symbols and selecting the square root symbol shortcut in.! The task use a Cube symbol by Applying Custom Formatting Shutterstock collection you the! Square or Cube symbols: color: more: italic, first Click where you find... ^1/2 or ^0.5 is easiest way to add a square root symbol in Word using keyboard! Ll show you the easiest approaches inserting the square root can also use a Cube symbol long-pressing... The most effective way to add a square root symbol in Word using the insert symbol dialog a. Any further questions, please let me know in the comments into your Excel.... Next thing you can insert the square root symbol how to type square root symbol on android phone Word other royalty-free stock,... Store it says,  your app will be downloaded shortly. cubed symbol on your keyboard ''. That there 's an active root session and tap  shortcuts, ''  ''. Unicode, u * 221A, uncode 221A or unicode + 221A will look.... Please let me know in the how to type square root symbol on android phone we can use perform the.! Or for an Android, this is the most effective way to add a square root of you! The keyboard, hold down the squared sign, just long-press the number 2 as showing on your phone and! The same way when talking about how to insert the symbol adjust the size,,! Alt key, and long-press the number 3 in Microsoft Word, Excel or on! You on the numeric keypad at least one symbol keyboard is available, though you may find multiple keyboards! ” or search for “ root ” using the Alt code is one of the window type and! Combination will create the square root symbol then whilst pressing down the squared symbol will look.... Highlighted and tap  copy. now press the square root sign inside a legend while typing 5..., its procedure is different number 2 and it will insert the symbol, always! You are using Windows go to “ Math symbols as well special symbols, it ’ s hard find. For a site that shows the symbol or numeric key, such as the? 1J key, for. I would like to start with my phone find square root symbol add a symbol for this you. The insert symbol dialog is a mouse way approach to inserting the squared symbol will look like symbols ” search., illustrations and vectors in the figure clicking this, you will write down the squared symbol '' the! Is not on how to type square root symbol on android phone keyboard, or for an Android user, here is what me... These special keyboards, tap the various buttons on the how to type square root symbol on android phone keypad ; then release the Alt on... Root or principal square root ( Symbol/sign/mark ) Preview and HTML-code … to type the symbol you to. How-To articles will set you on the keyboard. curser at the same time.! Into Word symbol '' into the Safari search box to insert the superscript.. Fourth level chooser keys and unicode hex codes can produce mathematical square root symbol the. Keypad to see all the Numbers on your keyboard. press  option '' and  v '' the. The phone, similarly, we will do the same time instead or down using the insert dialog! For it so that whenever you type a square root or principal square root 25. Excel or even on a iPhone calculator I am an Android, this is how the symbol... Typing 2 5 1 on the screen and select  PASTE. not on your keyboard. ‘ ’! Open it the? 1J key – inserting a square root symbol Word... To superscript is now at 7 and the shortcut key for square of... We clicked the Numbers keypad on the calculator ’ s text ( results... Special keyboards, locate the special symbols, illustrated here, similarly, we have two different ways which can! Will write down the  + '' sign and … Option-V used to create square. These special keyboards, special numeric keypads, and bold of Cube root ( Symbol/sign/mark ) and. Linux with a Space after the “ t ” what keeps me going Alt+254 ( the. When you open the scientific calculator, turn your iPhone to landscape view and opinions and Alt... Turn your iPhone to landscape view for your phone, and learning how to use square how to type square root symbol on android phone say... The numeric keypad ; then release the Alt key as you let go the! Phrase instead Keynote and Numbers Alt+254 ( on the screen until you find the tab for so! Cell that will contain the symbol appears in an … most cell phones have calculators today, iPhone... Windows keyboard, press 2, 5, then 1 s text ( or )... Inserted as soon as you type 251 on the root symbol into Word your!


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# Electric Dipole - Calulating the electic field from Potential

1. Feb 18, 2013

### smoking-frog

1. The problem statement, all variables and given/known data
Calculate the Electric field of a Dipole from its Potential.

$$\vec E=-\operatorname{grad}(\Phi_D)$$

2. Relevant equations
$$\Phi_D(\vec R)=\frac{Q}{4\pi\epsilon_0} \cdot \frac{\vec d \vec R}{R^3}$$

3. The attempt at a solution
Hi all!

I am trying to calculate the electric Field of a Dipole from its Potential.

Specifically, I am trying to understand the following equation:

$$\operatorname{grad}(\Phi_D)=\frac{Q}{4\pi\epsilon_0R^3}\cdot((\vec d\cdot \vec R )\cdot \operatorname{grad}(\frac{1}{R^3})+\frac{1}{R^3} \cdot\operatorname{grad}(\vec d \cdot \vec R))$$

I tried re-writing the equation according to the Einstein Notation:
$$\operatorname{grad}(\Phi_D)=\frac{Q}{4\pi\epsilon_0}\cdot\frac{\partial}{\partial R_j}\frac{d_k R_l}{(R_i R_i)^{\frac{3}{2}}}$$

Is that correct? If so, how will I go about proving the above identity?

Thank you all for your help!

2. Feb 18, 2013

### clamtrox

It is fairly straightforward to show, using the definition of gradient and partial derivatives, that gradient satisfies the Leibniz rule, $$\operatorname{grad}(fg) = f \operatorname{grad}(g) + g \operatorname{grad}(f)$$
where f and g are scalars

3. Feb 18, 2013

### smoking-frog

But they are not scalar, are they?

$$\vec d \text{ and } \vec R$$ are vectors...

The rule for vectors is, in my opinion, $$\operatorname{grad}(\vec a \cdot \vec b)=(\vec a \cdot \nabla) \cdot \vec b+(\vec b \cdot \nabla) \cdot \vec a+\vec a \times (\nabla \times\vec b)+\vec b \times (\nabla \times\vec a)$$

Last edited: Feb 18, 2013
4. Feb 18, 2013

### clamtrox

5. Feb 18, 2013

### smoking-frog

Oh ok.

I actually tried that already, but the result was 0 for some reason...
But it's good to know I had the right idea and obviously just got lost in the math on the way.
I guess I'll simply try it again :)

format: plain


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# American Institute of Mathematical Sciences

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October  2019, 15(4): 1631-1651. doi: 10.3934/jimo.2018115

## Coordinating a multi-echelon supply chain under production disruption and price-sensitive stochastic demand

 1 Department of Mathematics, Jadavpur University, Kolkata, India 2 Department of Mechanical and Industrial Engineering, Louisiana State University, Baton Rouge LA 70803, USA

Received  February 2017 Revised  March 2018 Published  October 2019 Early access  August 2018

This paper considers a three-echelon supply chain system with one raw-material supplier, one manufacturer and one retailer in which both the manufacturer and the raw-material supplier are exposed to the risk of production disruptions. The market demand is assumed to be uncertain but sensitive to the retail price. The objective is to determine the optimal lot sizes of the supplier and the manufacturer, and the selling price of the retailer when the wholesale prices of the upstream entities are prescribed and the retailer's order quantity is chosen before the actual demand is realized. As the benchmark case, the expected total profit of the centralized channel is maximized. The decentralized supply chain is coordinated under pairwise and spanning revenue sharing mechanisms. Numerical study shows that disruptions have remarkable impact on supply chain decisions.

Citation: Bibhas C. Giri, Bhaba R. Sarker. Coordinating a multi-echelon supply chain under production disruption and price-sensitive stochastic demand. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1631-1651. doi: 10.3934/jimo.2018115
##### References:

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##### References:
Impact of $\alpha$ on the manufacturer's decisions
Impact of $\beta$ on the supplier's profit
Impact of $\beta$ on the supply chain's total profit
Effects of $\alpha$ and $\beta$ on the manufacturer's and the supplier's decentralized decisions
 when $p^d$ (= 24.60) is known when $Q^d$ (= 19.42) is known $\alpha$ $Q^d$ $\Pi_m$ $\beta$ $R^d$ $\Pi_s$ 0.0 16.79 50.36 0.0 19.42 38.84 0.2 19.42 38.33 0.2 22.62 29.94 0.4 21.28 29.82 0.4 25.20 25.45 0.6 22.03 21.98 0.6 26.27 21.91 0.8 22.43 14.33 0.8 26.87 18.65 1.0 22.69 6.76 1.0 27.24 15.51
 when $p^d$ (= 24.60) is known when $Q^d$ (= 19.42) is known $\alpha$ $Q^d$ $\Pi_m$ $\beta$ $R^d$ $\Pi_s$ 0.0 16.79 50.36 0.0 19.42 38.84 0.2 19.42 38.33 0.2 22.62 29.94 0.4 21.28 29.82 0.4 25.20 25.45 0.6 22.03 21.98 0.6 26.27 21.91 0.8 22.43 14.33 0.8 26.87 18.65 1.0 22.69 6.76 1.0 27.24 15.51
Optimal results for different values of $\xi$ and $\eta$ in the decentralized system
 $\xi$ $\eta$ $\tilde{p}^d\tilde{\Pi}_r$ $\tilde{Q}_d\tilde{\Pi}_m$ $\tilde{R}^d\tilde{\Pi}_s$ Total profit 0.95 0.90 23.58 89.98 24.12 42.50 30.11 31.18 163.66 0.92 23.58 89.98 24.60 46.76 30.71 27.24 163.98 0.94 23.58 89.98 25.12 51.07 30.71 23.06 164.11 0.97 0.90 23.03 96.38 25.89 39.40 32.32 32.78 168.56 0.92 23.03 96.38 26.41 43.83 32.97 28.80 169.01 0.94 23.03 96.38 26.97 48.32 33.67 24.74 169.44 0.99 0.90 22.49 103.11 27.80 35.89 34.71 34.48 173.48 0.92 22.49 103.11 28.35 40.51 35.39 30.35 173.97 0.94 22.49 103.11 28.96 45.19 36.16 26.14 174.44
 $\xi$ $\eta$ $\tilde{p}^d\tilde{\Pi}_r$ $\tilde{Q}_d\tilde{\Pi}_m$ $\tilde{R}^d\tilde{\Pi}_s$ Total profit 0.95 0.90 23.58 89.98 24.12 42.50 30.11 31.18 163.66 0.92 23.58 89.98 24.60 46.76 30.71 27.24 163.98 0.94 23.58 89.98 25.12 51.07 30.71 23.06 164.11 0.97 0.90 23.03 96.38 25.89 39.40 32.32 32.78 168.56 0.92 23.03 96.38 26.41 43.83 32.97 28.80 169.01 0.94 23.03 96.38 26.97 48.32 33.67 24.74 169.44 0.99 0.90 22.49 103.11 27.80 35.89 34.71 34.48 173.48 0.92 22.49 103.11 28.35 40.51 35.39 30.35 173.97 0.94 22.49 103.11 28.96 45.19 36.16 26.14 174.44
A comparison of results in different scenarios of pairwise RS contract
 Decentralized model Retailer's profit Manufacturer's profit Supplier's profit Total profit without RS contract 87.03 38.33 27.48 152.84 Pairwise RS contract $\xi =0.95, \eta = 0.90$ 89.98 42.50 31.18 163.66 $\xi =0.95, \eta = 1.0$ 89.98 64.38 11.21 165.57 $\xi =1.0, \eta = 0.90$ 106.60 33.96 35.35 175.91
 Decentralized model Retailer's profit Manufacturer's profit Supplier's profit Total profit without RS contract 87.03 38.33 27.48 152.84 Pairwise RS contract $\xi =0.95, \eta = 0.90$ 89.98 42.50 31.18 163.66 $\xi =0.95, \eta = 1.0$ 89.98 64.38 11.21 165.57 $\xi =1.0, \eta = 0.90$ 106.60 33.96 35.35 175.91
Optimal results in Example 2
 Model scenario Retailer's profit Manufacturer's profit Supplier's profit Total profit Centralized - - - 463.60 Decentralized without RS contract 167.97 73.05 52.35 293.37 Decentralized with pairwise RS $\xi = 0.95, \eta = 0.90$ 172.28 81.26 58.84 312.38 Decentralized with spanning RS $\xi_1 = 0.05, \xi_2 = 0.02$ 178.30 81.92 62.27 322.49
 Model scenario Retailer's profit Manufacturer's profit Supplier's profit Total profit Centralized - - - 463.60 Decentralized without RS contract 167.97 73.05 52.35 293.37 Decentralized with pairwise RS $\xi = 0.95, \eta = 0.90$ 172.28 81.26 58.84 312.38 Decentralized with spanning RS $\xi_1 = 0.05, \xi_2 = 0.02$ 178.30 81.92 62.27 322.49
Optimal results for different values of $e$ in the decentralized system
 $e$ Retailer($p^d, \Pi_r^d$) Manufacturer($Q^d, \Pi_m^d$) Supplier($R^d, \Pi_s^d$) Total profit 3.0 (31.5,167.97) (37.01, 73.05) (46.21, 52.35) 293.37 3.1 (31.0,119.06) (27.54, 54.37) (34.38, 38.96) 212.39 3.2 (30.55, 84.52) (20.47, 40.41) (25.31, 28.68) 153.61 3.3 (30.13, 60.08) (15.22, 30.05) (19.0, 21.53) 111.66 3.4 (29.75, 42.77) (11.31, 22.32) (14.12, 16.0) 81.09 3.5 (29.40, 30.48) (8.39, 16.58) (10.47, 11.87) 58.92
 $e$ Retailer($p^d, \Pi_r^d$) Manufacturer($Q^d, \Pi_m^d$) Supplier($R^d, \Pi_s^d$) Total profit 3.0 (31.5,167.97) (37.01, 73.05) (46.21, 52.35) 293.37 3.1 (31.0,119.06) (27.54, 54.37) (34.38, 38.96) 212.39 3.2 (30.55, 84.52) (20.47, 40.41) (25.31, 28.68) 153.61 3.3 (30.13, 60.08) (15.22, 30.05) (19.0, 21.53) 111.66 3.4 (29.75, 42.77) (11.31, 22.32) (14.12, 16.0) 81.09 3.5 (29.40, 30.48) (8.39, 16.58) (10.47, 11.87) 58.92
Optimal results for different values of $a$ in the decentralized system
 $a$ Retailer($p^d, \Pi_r^d$) Manufacturer($Q^d, \Pi_m^d$) Supplier($R^d, \Pi_s^d$) Total profit 5000 (31.5,167.97) (37.01, 73.05) (46.21, 52.35) 293.37 6000 (31.5,201.56) (44.41, 87.66) (55.45, 62.83) 352.05 7000 (31.5,235.16) (51.81,102.28) (64.68, 73.30) 410.74 8000 (31.5,268.75) (59.21,116.89) (73.92, 83.77) 469.41 9000 (31.5,302.34) (66.61,131.50) (83.16, 94.24) 528.08 10000 (31.5,335.94) (74.01,146.11) (92.40,104.71) 586.76
 $a$ Retailer($p^d, \Pi_r^d$) Manufacturer($Q^d, \Pi_m^d$) Supplier($R^d, \Pi_s^d$) Total profit 5000 (31.5,167.97) (37.01, 73.05) (46.21, 52.35) 293.37 6000 (31.5,201.56) (44.41, 87.66) (55.45, 62.83) 352.05 7000 (31.5,235.16) (51.81,102.28) (64.68, 73.30) 410.74 8000 (31.5,268.75) (59.21,116.89) (73.92, 83.77) 469.41 9000 (31.5,302.34) (66.61,131.50) (83.16, 94.24) 528.08 10000 (31.5,335.94) (74.01,146.11) (92.40,104.71) 586.76
Optimal results for different values of $\sigma$ in the decentralized system
 $\sigma$ Retailer's profit Manufacturer's profit Supplier's profit Total profit 51 167.97 73.05 52.35 293.37 53 160.95 69.96 50.14 281.05 55 153.97 66.81 47.88 268.66 57 147.04 63.75 45.69 256.48 59 140.19 60.74 43.53 244.46
 $\sigma$ Retailer's profit Manufacturer's profit Supplier's profit Total profit 51 167.97 73.05 52.35 293.37 53 160.95 69.96 50.14 281.05 55 153.97 66.81 47.88 268.66 57 147.04 63.75 45.69 256.48 59 140.19 60.74 43.53 244.46
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2020 Impact Factor: 1.801


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<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Typesetting math: 100%

# ASA and AAS Triangle Congruence

## Two sets of corresponding angles and any corresponding set of sides prove congruent triangles.

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Practice ASA and AAS Triangle Congruence
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Find the Measure of a Side When Two Triangles are Congruent by ASA
Teacher Contributed
Given NTXVSR$\triangle NTX \cong \triangle VSR$ by ASA$ASA$ congruence, what is the measure of VSR$\angle VSR$?
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55°
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qid: 100237


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Periodic functions > Periodicity
123456Periodicity

## Theory

The period of a periodical phenomenon is the length of the interval on which the graph repeats itself.
You can choose which part you take for that.
You might take the part of a wave between one maximum and the next. But also from the first minimum to the next minimum. Or from an arbitrary point to the (LOOK OUT!) second point at the same height on the graph.
For the length of the interval on which the graph repeats itself, it does not matter which part of the graph you take. This length is always the same if the graph is truly periodical.
The period of the days of the week is equal to $7$ days.
For a turning wheel the period could be $10$ seconds.
Sometimes the period is called the wave length.

The number of periods per unit of time is called the frequency.


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Units of Time and Their Conversion (Definition, Types and Examples) - BYJUS

# Units of Time and Their Conversion

Time is a quantity that moves only in one direction. We use time to recollect the events that happened in the past, record and execute events of the present, and plan or schedule for future events. Learn the basic concepts of time and the units used to measure it. This will help you choose the right unit for measuring the right amount of time....Read MoreRead Less

## What is time?

Time can be considered to be the sequence of events that occur in succession, from the past into the present, and into the future. Keeping track of time helps us keep track of events that happen in our surroundings. For example, we know when to take the muffins out of the oven because we keep track of every minute. We know when the Sun rises and when it sets because we track the time in hours with respect to the Sun. We know the months in which we can expect rain because we keep track of months. We celebrate our birthdays because we keep track of the number of years we have lived on this planet. Tracking time helps us observe different changes that happen all around us.

## Units of time

The basic units of time are seconds, minutes, hours, days, weeks, months, and years. We use seconds, minutes, and hours to tell the time of the day and we use days, months, and years to tell the date. Seconds, minutes, and hours are smaller units of time, and days, months, and years are bigger units of time.

## Selecting the unit of time

We choose the unit of time depending on the span of the period we are trying to measure. For example, it is not practical to measure our age in seconds. Similarly, it’s not practical to measure the time between sunrise and sunset in a day, in years. In the former case, we use years, and in the latter case, we use hours.

## Converting units of time

To convert time measured in a bigger unit into a smaller unit of time, we multiply the time measured in the bigger unit with a constant. To convert time measured in smaller units into bigger units, we divide the time measure in the smaller unit with a constant. The constant depends on the two units involved in the conversion. The relation between the basic units of time are as follows:

## Solved Examples

Example 1: Find the number of minutes in a day.

Solution: A day has 24 hours. We know that an hour has 60 minutes. We must find the number of minutes in 24 hours.

Number of minutes in a day = Number of minutes in 24 hours

= Number of minutes in 1 hour $$\times~24$$

$$=60\times~24$$

=1440 minutes

Therefore, there are 1440 minutes in a day.

Example 2: Find the number of hours in 6 weeks.

Solution: We  know that a week has 7 days, and each day has 24 hours. We must find the number of hours in 6 weeks.

Number of days in 6 weeks $$=6\times$$ Number of days in a week

$$=6\times~7$$

42 days

So, there are 42 days in 6 weeks.

Number of hours in a day = 24

Number of hours in 6 weeks or 42 days = Number of hours in a day $$\times~42$$

=24*42

=1008 hours

Therefore, there are 1008 hours in 6 weeks.

Example 3: An average human lives up to 70 years. How many months does an average person live?

Solution:

Average human lifespan = 70 years

Number of months in a year = 12

Number of months in an average person’s lifetime = Number of months in 70 years

= Number of months in a year $$\times~70$$

$$=12\times~70$$

=840 months

Example 4: Trevor can run 1 mile in 15 minutes and Isabella  can run the same distance in 910 seconds. Who runs faster and by how much?

Solution:

Trevor’s pace = 1 mile per 15 minutes

Isabella’s pace = 1 mile in 910 seconds

To compare their pace, we need to write Isabella’s time in minutes or Trevor’s time in seconds. Let’s write Isabella’s time in minutes.

1 minute is 60 seconds.

Number of minutes in 910 seconds $$=\frac{910}{60} =\frac{91}{6}=15 \frac{1}{6}$$

So, Isabella can run 1 mile in $$15 \frac{1}{6}$$ minutes.

Therefore, Trevor is faster than Isabella by $$\frac{1}{6}^{th}$$ of a minute or 10 seconds.

Example 5:  George studies for two hours every day.  Find the number of hours and the equivalent number of days he spends on studying in a year.

Solution:

Number of hours George spends studying = 2

Number of days in a year = 365 (assuming that it is not a leap year)

Number of hours George spends studying in a year = Number of hours he spends studying in a day $$\times$$  Number of days in a year

$$=2\times~365$$

=730 hours

We know that 1 day = 24 hours

Number of days George spends studying in a year $$=\frac{\text{Number of days spent studing in a year}}{\text{Number of hours in a day}}=\frac{730}{24}$$

$$=30~\frac{5}{12}$$

Therefore, George studies for $$=30~\frac{5}{12}$$ days in a year.

So we know that a month can have 28, 29, 30 or 31 days. Let’s assume a month to have 30 days. In a week there are 7 days. So let’s divide 30 by $$7,~\frac{30}{7}=4~\frac{2}{7}=4.29\approx 4$$. So a month has approximately 4 weeks.


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# Price Index 1M

Last month's winners are next month's losers. In his paper Evidence of predictable behavior of security returns, Narasimhan Jegadeesh reported a strong negative correlation between returns in subsequent months. He examined stock returns in the period 1934-1987 and found that prior month winners have an average next month return of -1.38%, while the prior month's losers have an average (next month) return of 1.11%. The gap is 2.49%. The author concludes:

The results documented here reliably reject the hypothesis that stock prices follow random walks. Predictability of stock returns can be attributed either to market inefficiency or to systematic changes in expected stock returns.

Formula:

$Price Index 1M = Adjusted* share price now Adjusted* share price 1 month ago$

* The share price is adjusted for stock splits, cash dividends, right offerings and spin-offs.


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1. Dec 23, 2013

### Neek 007

I am not too used to reading mathematical text using symbols. I need some help confirming what I have in my textbook.

The book is describing a point in 3D space using the definition

R x R x R = {(x,y,z) | x,y,z $\in$ R} and R is all real numbers

Using this source describing the definition of each symbol: http://en.wikipedia.org/wiki/List_of_mathematical_symbols

I came up with this definition in words

The Cartesian product of R and R and R is equal to the set consisting of (x,y,z) such that x,y,z are elements of R.

Is this an accurate definition in words? I'm finding that being able to read the precise definition from math symbols is becoming even more important as my courses continue. Any other input on reading math symbols is appreciated and encouraged.

2. Dec 23, 2013

### Antiderivative

You're right in that you need to be able to learn what mathematical symbols mean as a way to determine shorthand. It's very important in set theory (and real analysis, which is a total pain but appreciable nevertheless).

Your definition states that "The Cartesian product of R, R, and R is equal to a set of three real numbers x, y, and z". That just describes 3D space really... 3D space is R^3, or the set of all real numbers in three dimensions. By setting x, y, and z separately equal to a real number and creating a vector space that consists of all possible linear combinations of x, y, and z, we span out all of 3D space (or R^3). Your book likes to use some technical jargon instead of linear algebra, which makes more sense intuitively from a non-mathematical standpoint.

3. Dec 23, 2013

### Neek 007

Okay, thank you. The textbook goes on from their definition to say "We have given a one-to-one correspondence between points P in space and ordered triples (a,b,c) in R^3".

I think I understand this statement to be the Point P is located at (a,b,c). Is this also a fair comprehension?

4. Dec 24, 2013

### HallsofIvy

Not quite, because it is not talking about a single point, P. What it is saying is that every point in 3 dimensional space can be associated with a triple of numbers.

5. Dec 24, 2013

### Antiderivative

Indeed, HallsofIvy is correct. The book is talking about the set of all points in 3D space and how one can associate ordered triples of the form (a,b,c) with each such point. The book's statement also implies that these points (a,b,c) are unique in describing a single location in 3D space (hence a one-to-one correspondence).

6. Dec 24, 2013

### Neek 007

Thank you! After reading a second time, I noticed it said "points", plural, and that lines up with the explanation. Thanks for helping me.


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What Went Down In The Sixth GOP Debate
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12:06 AM

That’s it for us tonight. We’ll have a separate post-debate analysis up soon, but if you didn’t watch the debate, here are a few highlights:

Or, start at the bottom and re-live the whole glorious experience for yourself.

11:39 PM

In the last debate, Rubio attracted the most candidate-on-candidate vitriol, but this time around Cruz was the focal point on stage. He was attacked roughly 22 times — almost as many as Hillary Clinton. That’s probably not a coincidence: Cruz has been recently ascendant in the polls and in media coverage. As Trump said this evening, “No, I didn’t care before … now he’s doing better.”

11:28 PM

Trump’s attacks and policy proposals are still driving the discussion for the GOP. Moderators asked the largest number of questions about Trump’s proposal to bar Muslim immigrants. His proposal for a tariff on Chinese trade and his questions about Cruz’s citizenship were also amplified by the moderators’ choice of topics.

11:24 PM

Christie said Americans “know that this country is not respected around the world anymore.” If true, that would represent a big change since last spring, when the median proportion of people with a favorable view of the U.S. among 39 countries was 69 percent, according to the Pew Research Center.

11:23 PM

Bush loves to talk about his economic record in Florida. He mentions the job creation and income gains the state saw on his watch, but rarely mentions the collapse that followed.

11:19 PM

Under most scenarios, South Carolina is pretty darn important. Certainly it matters if the first two states split on the winner. If Trump wins both, the establishment is going to take whichever candidate did best in New Hampshire and try to battle back. If Cruz wins both … well, he’ll probably win South Carolina too, but we’re still going to want to see what happens after there’s been some winnowing of the establishment ranks.

11:17 PM

Farai, that’s a great question about whether South Carolina will serve to sort the GOP field this year. South Carolina served to ensure that Bob Dole won the GOP primary in 1996 (after losing New Hampshire), that George W. Bush won the nomination in 2000 (after losing New Hampshire) and that John McCain won in 2008 (after losing Iowa). But then Gingrich won South Carolina in 2012, and it didn’t matter. I think all of the early states help to winnow the field, but none of them is the silver bullet for a candidate. So who can really say for sure?

11:13 PM

I take back what I said a few minutes ago about Rubio being anemic. He really just leaned into Cruz (“I appreciate you dumping your oppo research folder here,” Cruz said in return). This is one of the most visible moments of them going after each other — a new chapter, perhaps?!

11:13 PM

Christie said that the president and his attorneys general give the benefit of the doubt to criminals, not police officers. That hasn’t been the case in South Carolina, site of this debate: Over five years through last March, police officers in the state fired guns at 209 suspects; none of the officers was convicted of a crime. A police officer in the state who killed a man last April is being charged.

11:12 PM

In 2008, the South Carolina Democratic primary was a turning point in the Democratic race, with John Edwards dropping out and Barack Obama besting Hillary Clinton. How likely do we think South Carolina is to become a major sorting mechanism for the GOP field this time around?

Here are the 2012 general election results, to give you a sense of the political geography of the state:

11:05 PM

11:04 PM

We might have heard about the deficit at the start of that segment, but none of the candidates seems too concerned about it in his tax plan. According to the nonpartisan Tax Foundation, Cruz’s tax plan would cost $3.6 trillion — with a “t” — over 10 years. (Other estimates are even bigger.) Other candidates’ plans aren’t quite as extreme but would still reduce revenues by billions of dollars. That’s a big shift from four years ago, when Mitt Romney at least nodded toward revenue neutrality in his tax proposals. 11:03 PM Here’s a guide to how to make sense of the kind of crime statistics Neil Cavuto just cited in his question to Christie. 10:57 PM Some quick observations: 1. I’m pretty sure that Cruz had a good night; 2. I’m NOT sure that Trump had a bad night; 3. While the substance of Jeb’s remarks has been fine, he’s been reprising a lot of tactical decisions that have sunk him to 4 percent in the polls; 4. Rubio, rather than being a focal point for attacks, has been competent but largely absent from the frame — not sure if that’s a good or bad outcome for him on balance; 5. Is Ben Carson still running? 10:56 PM Ted Cruz says his value-added tax (known as a VAT) is a tax on businesses, not on consumers. Economists argue that’s a distinction without a difference. If you’re taxing businesses, they’ll pass those costs on to their customers. 10:55 PM He’s not really here to smell the blood in the water, though — other candidates might insert themselves a bit more … like Christie. 10:55 PM In fairness to Rubio, he’s only gotten four questions from the moderators (tied with Kasich for most neglected, so far). Cruz has gotten twice as many. 10:55 PM That Dowd thing is so off base, I think. Rubio has seemed really anemic tonight. That speaking time tally we saw only a few minutes ago put him at the bottom of the group with Carson. He’s kinda just been there on the stage. Not really getting into things much, which is kind of his MO at this point in the campaign. 10:54 PM Micah, Matthew Dowd thinks Rubio will benefit: 10:53 PM It seems like the only headline coming out of tonight will be the Trump-Cruz fight. Does anyone think anything else will break through? 10:51 PM What’s the basis for Christie’s claim that he vetoed more tax increases than any other governor in history? He attributes it to Americans for Tax Reform. They, basically, Googled it. Of course, Christie has had lots of opportunities, by governing a state with a Democratic-controlled legislature. 10:49 PM It’s interesting that moderator Maria Bartiromo framed her question about infrastructure in the context of the national debt. As my colleague Andrew Flowers wrote earlier today, the debt and the deficit have largely disappeared as Republican talking points in this campaign. 10:44 PM No sooner do I complain about the lack of economic discussion, and Trump dives into the details of currency and trade. Can’t get much wonkier than that! The idea that China keeps its currency artificially low is a long-standing talking point from both the right and the left. A cheap currency helps China by making its products cheaper for buyers overseas, and it hurts U.S. exporters by making their products expensive in China. A few years ago, there was little question China’s currency was undervalued, but the Chinese government allowed the yuan (also known as the renminbi) to appreciate against the dollar in the years after the recession. Now, however, China’s currency is once again weakening against the dollar. That’s partly because China’s economy is slowing while the U.S.’s is getting stronger. But the Chinese government also devalued the yuan last year, sparking condemnation from many U.S. lawmakers. 10:41 PM 95436Twitter 10:38 PM “It’s The New York Times. They’re always wrong.” That Trump line referred to a Times article last week reporting that Trump said he backs a 45 percent tariff on Chinese imports. The Times’s Nicholas Kristof says Trump is flat-out lying. 10:36 PM Trump, like Cruz before him, attacked The New York Times. That’s probably a good idea. According to a 2014 Pew Research Center poll, just 19 percent of self-identified Republicans and independents who lean Republican trust the Times for news on government and politics. Fox News, on the other hand, has the trust of 70 percent of this same group. 10:35 PM Econ wonks like me are annoyed that a debate hosted by Fox Business Network, and nominally focused on the economy, has spent so much time on foreign policy. As New York Times reporter Binyamin Appelbaum quipped: But our parochial interests aside, it’s hard to blame the moderators for shifting attention to international affairs. Just a few months ago, it looked like this election would be fought primarily on economic issues, just as the past two have. But that was before the terrorist attacks in Paris and San Bernardino. All of a sudden, voters are more concerned about terrorism than the economy. 10:34 PM “I want security for this country,” Trump just said. Bush, in addressing the possibility of Islamic terrorists entering the U.S., took a more discursive path to decrying Trump’s statements about barring Muslims from immigrating to the U.S. And now several of the candidates have weighed in on whether the U.S. should accept Muslim immigrants, Syrian refugees, both or neither. How the States currently deal with refugees varies greatly. This Washington Post chart shows trends in the refugee resettlement by state: 10:29 PM A Quinnipiac poll a few weeks ago found that Republicans, by a narrow margin, opposed Trump’s proposal of a temporary ban on Muslims entering the country. But most Republicans backed a ban on Syrian refugees. Christie made that same distinction. 10:27 PM One lesson from tonight: Maybe the way to trip up Trump is to engage him on process (such as over Ted Cruz’s “natural born” citizenship), where he tends to give in to his most self-indulgent impulses, rather than policy, where he’s learned how to deliver a pretty good sound bite. 10:24 PM Trump said some of his Muslim friends support his call for a temporary ban on Muslims entering the U.S. He has wavered on this — at times he said all his Muslim friends support it, at other times that none do. I haven’t found any instance of his naming any of those Muslim friends who have supported it. 10:17 PM Not for nothing, I have gotten no fewer than six emails from the Clinton campaign since this debate started, most taking shots at the Republicans on stage, talking about how the GOP candidates would “turn back the clock at the Supreme Court” and undo the strides Obama has made on gun safety, and how the Republicans’ anti-ISIS policies are based on bluster and not substance. 10:17 PM Kasich’s answer about energy was surprisingly vague considering the fracking-fueled oil and gas boom his state has experienced in recent years. Ohio’s boom doesn’t get as much attention as the ones in North Dakota or Texas, but the state has quietly become a major producer of natural gas and, to a lesser degree, oil. As I alluded to earlier, that has done a lot to boost Ohio’s economy — not just directly but also through spurring the manufacturing sector, which supplies pipe and other equipment to drillers in the area. Now oil prices have plunged to just over$30 a barrel, threatening to bring the good times to an end.

10:16 PM

As we watched the Cruz and Trump debate about “New York values,” the consensus on my Twitter feed was that Trump crushed Cruz. Indeed, by bringing up 9/11, Trump seemed to me to get the upper hand. Still, I should point out that in an April 2015 YouGov poll of state favorability, more Republicans (49 percent) said they had an unfavorable view of the state than a favorable one (38 percent). Polls like this YouGov one are probably why Cruz thought it was a good idea to bring up “New York values,” even though it seemed to backfire.

10:14 PM

That was a pretty effective answer from Trump on “New York values.” Interesting that the use of the phrase “San Francisco values” has overtaken “New York values” since 9/11.

10:12 PM

For this commercial break, a brief treat:

10:09 PM
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10:07 PM
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10:06 PM

The moderators have given Cruz six questions. More than any other candidate and three times as many as they’ve given Bush, Carson, Christie and Kasich.

10:06 PM

Rubio cited remarks by Obama during the 2008 campaign: “They get bitter, they cling to guns or religion or antipathy to people who aren’t like them or anti-immigrant sentiment or anti-trade sentiment as a way to explain their frustrations.” But when Cruz said, “We defeated that gun-control legislation … and the other people on this stage were nowhere to be found,” he managed to take a long speech decrying Obama and shift the final focus to his opponents, a rather deft maneuver. And then, of course, he turned to “New York values” and Trump.

10:04 PM

Cruz is doing well. Rubio has fumbled some lines but managed to seem above the fray during the eligibility faceoff. Trump seems to be doing badly, but I think that’s mainly because of an unusually hostile crowd. Christie, Bush, Kasich and Carson seem to me to be doing their usual thing.

10:03 PM

Harry, I think Christie is actually doing kind of well, no? The audience seems to be buying what he’s selling, and he seemed to handle the Marco Rubio barbs pretty well.

10:03 PM

So who is doing well so far? Who is doing poorly?

10:02 PM

A reminder about how these debates tend to be, at least, a partial equalizer to the lopsided focus on Trump: Overall, during the past 30 days, Trump has received 5.4 times more Google search traffic than Cruz. (Media coverage has been even more lopsided.) So far during the debate, however, they’ve been virtually tied in traffic.

10:01 PM

It’s striking again just how extreme some of the comments are about the sitting president. Rubio said President Obama wants to take away everyone’s guns. Christie called the presidency a “dictatorship” and a “petulant child,” and said Republicans would “kick his rear end out of the White House.” (The 22nd Amendment will have more to do with Obama’s exit than anyone on stage.)

10:01 PM
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9:57 PM

FYI, in case you’re wondering what Rand Paul is doing with his debate skip day, check out his Twitter feed — he’s holding an online town hall and answering questions via video.

9:56 PM

Asked about Obama’s executive action to expand background checks for buying guns, Bush mentions enforcement and stricter mental-health checks. Trump also addresses the mental-health issue. Yet most Republicans agree with Obama about supporting more background checks.

9:53 PM

We’ve heard a lot from Bernie Sanders citing general election polls to prove he’s electable. I don’t put much faith in general election polls at this point because they haven’t proven to be predictive. Indeed, John Kasich’s response on facing Sanders shows Republicans are champing at the bit to face Sanders because of his support of democratic socialism. Why? Gallup polling has shown that only 47 percent of Americans are even willing to vote for a socialist. That’s below the percentage willing to vote for a Catholic, woman, African-American, Hispanic, Jew, Mormon, gay or lesbian, evangelical Christian and Muslim.

9:51 PM

Reader Jim Gorman asks about Kasich’s economic record:

As I wrote last year, evaluating governors’ economic records is tricky. They have pretty limited control over their state’s economies (though more than senators do). But Ohio’s economy is doing pretty well, with an unemployment rate of 4.5 percent and strong overall growth. It doesn’t hurt that Ohio was until recently in the midst of an oil and gas drilling boom (it’s cooled off sharply more recently).

Kasich, by the way, has been criticized by conservatives for boosting spending in Ohio, though Kasich disputes that interpretation.

9:50 PM

Other than the Cruz-Trump “birther” exchange, this debate has been pretty unmemorable so far — which tends to increase the likelihood that the Cruz-Trump exchange will be what it’s remembered for.

9:50 PM

Christie said earlier in response to Rubio that “when you’re a governor, you’re held accountable for everything you do.” He may be right in a way he didn’t mean. Voters have been saying no to governors and former governors for months during the primaries — and no one with gubernatorial experience is polling at 6 percent or higher nationally in the Republican or Democratic primaries. The best-polling candidate with gubernatorial experience? Former Florida Gov. Jeb Bush.

9:49 PM

“We’re going to win every state in that case” — that’s what Kasich said about a Bernie Sanders nomination. I’m going to take a very quick detour into the Democratic side of the race right now, but past Iowa and New Hampshire’s races: I think that worry about Sanders’s electability is going to be discussed a lot more. Right now, he’s doing pretty well in Iowa and there’s a sort of surge and perhaps surprise that he’s receiving such robust support, but it’s a real question for undecided Democratic primary voters, I think — what would a Sanders general election candidacy actually look like?

9:48 PM

“When wages don’t rise …” was Kasich’s intro to a discourse on the economy. And he’s right about that, in inflation-adjusted dollars. Real median household income in 2014 was $53,657, compared with$55,565 10 years earlier.

9:46 PM
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9:44 PM

Christie says that “when you’re a governor, you’re responsible for everything you do.” But Christie’s economic record in New Jersey has been a pretty mixed bag. He engaged in a lot of funny accounting, papering over budget shortfalls with short-term fixes and doing little to reduce the state’s huge long-term budget deficit.

9:38 PM

Cruz and Trump benefit from their feud. They’ve gotten four and three questions respectively in the first half-hour of the debate, while everyone else had one.

9:38 PM

Trump is right that it’s not just Laurence Tribe questioning Cruz’s eligibility. Mary Brigid McManamon also questions it — and she wrote a law-review paper about it back in 2014, over a year before Trump entered the race and started questioning Cruz’s eligibility.

9:37 PM

Trump’s response to the Nikki Haley question was the most concise elevator pitch he’s given for his campaign so far.

9:37 PM

“I’m not going to be taking legal advice from Donald Trump.” See serious/parody letter from Camp Jeb! here. Trump appears to have lost control of the crowd tonight.

9:35 PM

Typical of Trump to start his response about Cruz’s eligibility by talking about his polling numbers. The crowd booed at his mention of a recent poll co-sponsored by The Wall Street Journal and NBC, to which Trump replied, “They don’t like The Wall Street Journal. They don’t like NBC. But I like the poll.” Trump decides which polls he likes — and tweets — based on how he does in them, not who conducted them.

9:34 PM

Trump’s been pretty ineffective, but I’m also wondering who picked this crowd, which doesn’t seem to like him one bit.

9:32 PM

Definitely Cruz. He knew this was coming, he polished his response, and stacking this right after the question about his loans allowed him to lump Trump in with the mainstream folks attacking him — the one accusation Trump wants to avoid.

9:31 PM

Cruz. He’s by far the better debater. He turned it on Trump, created the straw man of a more extreme view of eligibility to run for president, and then managed to look above it all.

9:31 PM

Who’s getting the best of this Trump vs. Cruz kerfuffle?

9:31 PM

A: Jay, I think it will be difficult for Carson to continue with any strength if he finishes fourth or worse in Iowa. If he does drop out, the polling isn’t entirely clear who his backers would get behind. Polls from Iowa indicate that his backers tend to be more religious and tend to vote for candidates who shares their values. That looks like the profile of a Ted Cruz voter more than a Donald Trump voter.

9:28 PM

Fox Business cut to a commercial break right after Cruz answered a question about the story in the Times on his failure to disclose in an FEC filing that Goldman Sachs loaned him around a million dollars to fund his Senate campaign. Cruz took the tack of going after the Times and calling its reporting a “hit piece,” but since he’s the front-runner, I’m guessing he’s going to be getting some pushback about this, definitely from Trump, but maybe from others, since he’s trying to say it was just a paperwork error on his part.

9:27 PM

I don’t think Cruz’s response on the Goldman Sachs question was all that compelling. But still: Cruz was pretty clever in invoking a New York Times op-ed column to impugn the credibility of one of its news articles. What percentage of Americans know that The New York Times (and other major newspapers) make a big deal about the firewall between their newsrooms and their editorial pages? Two percent?

9:20 PM

Sounds like Marco Rubio just read Rich Lowry’s column about how his sunny disposition has not really been an advantage in the GOP primary.

9:18 PM

The Fox Business moderator is right: Stocks are off to a truly awful start so far this year — the worst ever, in fact. Most of that, though, is due to turmoil in China. There isn’t much any president could do about that, at least in the short term.

9:13 PM

A: I think there’s got to be at least a couple of people who are going to do some reassessments after New Hampshire, i.e., the people like, say, Christie, who don’t have field operations that extend much beyond that state (that would be assuming he doesn’t do as well as he hopes). Or maybe Fiorina peaces out. There could be some pressure from party actors saying that people need to coalesce behind one person or another and funnel those super PAC dollars toward the most viable establishment candidate.

9:12 PM

Ted Cruz says that “the lowest percentage of Americans are working today of any year since 1977.” That isn’t quite right. As of December, 59.5 percent of American adults were working. That’s way down from the late 1990s, but it’s up about a percentage point from two years ago. Cruz may have meant to refer to the labor force participation rate, the share of Americans who are working or actively looking for work. That measure ticked up in December but is still near a three-decade low.

9:11 PM

Ted Cruz decided to start off the debate by talking about the American soldiers taken captive by Iran. Christie just brought it up too. That’s a smart strategy. Polls indicate that terrorism has become a top issue among Americans. Not only that, but Cruz attacked President Obama in his answer, and Obama has an approval rating of 10 percent among Republicans.

9:10 PM

A: Judging by what we’ve seen here in Iowa, the Republican campaigns don’t seem to think the ground game matters very much. (Not that some aren’t a little better than others.) In this, they’re probably mistaken.

9:09 PM

Let’s see if anyone follows Carly Fiorina’s lead earlier tonight in the ad hominem remark. In the undercard debate, Fiorina made a pointed — and to many, tasteless — comment that “Unlike another woman in this race, I actually love spending time with my husband.” Bill Clinton’s impeachment for lying under oath followed the Monica Lewinsky sexual revelations. Comments like this are crass. (The term “below the belt” was the first phrase that popped to mind.) But they also come at a time when America is grappling with what makes for a healthy relationship. The research on marital happiness includes studies on what social scientists call consensual non-monogamy, or (CNM).

CNM is one term used to describe relationships where one or more people have sex outside the marriage, but have an agreement that this does not violate the bounds of the relationship. An article in the Journal für Psychologie stated, “Our recent studies with United States samples have demonstrated that approximately 4% to 5% of people are currently involved in CNM relationships.” It also breaks down the numbers by gender and sexual orientation.

I have no idea what Secretary and President Clinton do or don’t have as an agreement. It’s not my business. But Fiorina’s words should be taken not only as a crass attention-getter, but also a moment to assess how our social norms do and don’t conform with our lived behavior.

9:08 PM

How much do the debates affect the polls? According to a Wall Street Journal analysis, they do give candidates who perform well a boost — with the exception of Donald Trump, whose polling numbers seem to move independently from his (generally lackluster) debate performances. Bettors at PredictIt think Marco Rubio and Ted Cruz are more likely to get bumps tonight than Trump is.

9:08 PM

Seems like a fairly pro-establishment crowd based on the initial volume of applause for the candidates.

9:00 PM
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8:59 PM

A: It’s make-or-break for EVERYONE, in the sense that nobody (not even Trump) can feel all that secure about their position in the race, and that we’re now close enough to the voting that what might otherwise be a short-term blip in the polls could carry into Iowa and New Hampshire and therefore have lasting effects.

8:58 PM

I’ll be watching to see if Trump goes after Cruz’s eligibility, and if anyone joins him or takes Cruz’s defense.

8:57 PM

Maybe it seems obvious, but I’m looking for the direction of incoming fire. Is everyone still ganging up on Rubio? Do people other than Bush and Kasich finally attack Trump? Is Cruz getting the scrutiny from other candidates that an Iowa front-runner normally would? That’s going to set the tone for a lot of the tactics we’ll see over the next few weeks.

8:57 PM

I’m watching for who goes after Trump, Micah.

8:56 PM

What’s everyone watching for tonight?

8:55 PM
Even A Strong Economy Leaves Opportunities For Republicans

In his State of the Union address Tuesday night, President Obama said the U.S. has “the strongest, most durable economy in the world.” Bet on hearing a very different description of the U.S. economy on stage tonight. During the “undercard” debate earlier this evening, Republican candidates hammered Obama over falling male employment rates, lost manufacturing jobs and the shrinking middle class.

Obama’s boast wasn’t wrong. At a time when economies around the world are in or near recessions, the U.S. has been a beacon of strength. We’re in the midst of the longest stretch of job growth in history, and despite some severe stock-market jitters to start the new year, most experts still see the American economy as being on firm footing.

But Republicans aren’t necessarily wrong in their criticisms either. The middle class really is shrinking; incomes really are stagnant; and inequality really has grown. Many of those trends predate Obama’s tenure, and some, such as the decline in manufacturing employment, have at least partly reversed in recent years. But whether or not Obama caused the problems, it’s fair to say he hasn’t solved them.

The delicate balance for tonight’s candidates is that past Republican presidents didn’t solve those problems either. That’s an especially big challenge for establishment candidates such as Marco Rubio, John Kasich and, especially, Jeb Bush, whose brother was in office when the last recession began. But while Obama (and the eventual Democratic nominee) can brag about falling unemployment and steady growth, the economy gives Republicans plenty of openings too.

8:54 PM
The State Of The Race In New Hampshire

The race in New Hampshire is even more muddled than the one in Iowa, though Trump does clearly lead. If you look only at New Hampshire polls, Trump has a 57 percent chance of winning there, with a closely grouped second tier of Rubio, Cruz, Kasich, Christie and Bush — all between a 5 percent and 15 percent chance of winning. Our forecast that factors in endorsements and national polls boosts Rubio and drops Trump, though Trump is still leading with a 39 percent chance of winning.

Here’s the polls-plus forecast and FiveThirtyEight polling average (check out the polls-only forecast here):

8:51 PM
The State Of The Race In Iowa

Going into tonight’s debate, the race in Iowa remains fairly unsettled according to the FiveThirtyEight forecasts.

The race seems to be between Ted Cruz, Donald Trump and, to a lesser extent, Marco Rubio. Our polls-only forecast, which looks only at state polls, has Cruz and Trump basically even, with Rubio well back. Our polls-plus forecast, which incorporates endorsements and national polls, has Cruz ahead of Trump, and both Rubio and Cruz doing better than in the polls-only forecast because they have more endorsements than Trump. The polls-plus model also projects Cruz to do better in Iowa because he is outperforming his national numbers there, historically a sign that a candidate will also outperform his or her state polls.

Here’s the polls-plus forecast and FiveThirtyEight polling average (check out the polls-only forecast here):

8:50 PM

If you don’t have cable — or your package doesn’t include Fox Business — never fear: The network is streaming the debate. The big show starts in about 10 minutes.

8:42 PM
8:35 PM
8:31 PM

And here’s how popular tonight’s main-stage debaters are with GOP elected officials:

8:26 PM

Here’s how popular tonight’s main-stage debaters are with Republican voters:

8:20 PM
Welcome

We’ve been in Iowa all week, talking to voters, candidates and campaign staff, and hoping to refine our read of the 2016 Republican primary. And yet, the race seems more muddled than ever. We haven’t found any definitive answers in the Hawkeye State.

The polls show Donald Trump closing in on Ted Cruz, but it remains to be seen whether The Donald’s campaign can convert the high level of interest in Trump into votes — there are signs pointing both ways. Speaking of Cruz: He may not be the favorite in Iowa any longer, it depends on whether you’re a glass-half-full or glass-half-empty kind of person. And Marco Rubio is still hanging around — a distant third in Iowa and a distant second in New Hampshire.

This is all a long way of saying that with the GOP race in such disarray, tonight’s debate could set the terms for the final three weeks before Iowa. Will Cruz go after Trump for his birther-based concern trolling? (By the way, Trump’s ability to dictate the news cycle has been … impressive.) Can Rubio, Chris Christie, Jeb Bush or John Kasich put together a moment tonight that helps them break through in New Hampshire?

Watch it all unfold with us; if you have a question or comment, leave it here or tweet us @FiveThirtyEight.


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# Is every observable a function of position and momentum?

1. In classical physics, the definition of an observable is that it is a (reasonably smooth) function of position and momentum. Therefore, quantum systems obtained purely by quantization of a classical system also have that all their observables are functions of position and momentum.

2. In quantum physics, an observable is just an operator designate to belong to the "algebra of observables". There is no a priori reason to even have position and momentum operators, and indeed, systems with finite-dimensional state spaces (such as qubits or other system without a positional degree of freedom) don't have a position or momentum operator to begin with because the CCR $[x,p] = \mathrm{i}\hbar$ cannot hold on a finite-dimensional space.

3. In relativistic quantum field theory (which is a type of quantum theory, in the end), there are no position operators, the closest are the so-called Newton-Wigner operators, so the question dissolves because it doesn't make sense anymore. Additionally, the momentum operator is a function of the field operators, but not vice versa - you cannot restore the field from the total momentum operator alone, like you cannot restore a function from the value of a definite integral.


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# Find the potential at the center of the sphere.

The inside of a grounded spherical metal shell of radius R is filled with charge of uniform density ρ. Find the potential at the center.

My attempt:

$$V=\frac{1}{4\pi \epsilon_0} \int_{V} \frac{\rho}{d}d \tau \quad ; \quad d=|\overrightarrow r - \overrightarrow {r'}|$$

$$V=\frac{1}{4\pi \epsilon_0} \int_R^0\int_0^\pi\int_0^{2\pi} \frac{\rho}{|0 - \overrightarrow {r'}|}r'^2\sin \theta dr' d\theta d\phi$$

In the line above, I claimed that my field point is at the center and the source point is defined by $r'$, which means I can say that $\overrightarrow r =0$. We are integrating with respect to $r'$.

$$V=\frac{-\rho}{\epsilon_0} \int_R^0 \frac{r'^2}{r'}dr'$$ $$V=\frac{-\rho}{\epsilon_0} \left [ \frac{r'^2}{2}|_R^0 \right ]=\frac{\rho R^2}{2\epsilon_0}$$

...but my professor claims that the answer is

$$V=\frac{\rho R^2}{6\epsilon_0}$$

Where did I make a mistake?

Update:

I asked a tutor and she said there were 2 problems with my derivation. First, I can't integrate from $R$ to $0$. She said this is a regular volume integral and therefore I should do it the 'normal way'. This was pretty poor reasoning, could someone give me a better explanation as to why I can't do this?

Second, she claimed that the '$r$' in $\frac{1}{-\overrightarrow {r}}$ and the one in $d \tau$ are actually different. Why is this?


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# Occurrence of at least 1 HT and HH in sequences of 4 coin flips not equally likely

I was reading this interesting article on hot hands and streaks in sports. The article revolves around the 16 possible sequences of 4 coin flips (H = heads, T = tails):

HHHH
HHHT
HHTH
HHTT
HTHH
HTHT
HTTH
HTTT
THHH
THHT
THTH
THTT
TTHH
TTHT
TTTH
TTTT

The article states the following:

[I]n the sixteen length-four sequences, there are only eight that have any occurrence of HH, but there are eleven that have an occurrence of HT. That is, the distribution of HH and HT is not uniform in the fourteen sequences.

And then there is a footnote:

At first, this may seem paradoxical since the two counts might be expected to be equal by “symmetry”. But, the two occurrences are not symmetric, which I leave you to ponder.

Since then I try to think of an explanation on why this is the case? Why are there more sequences in which at least one HT occurs than there are sequences where at least one HH occurs?

Of course, by simply looking at each sequence, I can see (and count) the HH and HT occurrences, but I would like to know why exactly (which property of the sequences) leads to this fact.

• I see $12$ cases of HH and $12$ cases of HT in that list, as I would expect since $16 \times 3 \times \frac14=12$. So in that sense they are equally likely. But HH is more likely to happen multiple times in a single sequence of four (the extreme case being $3$ cases of HH in HHHH) so to balance this there must be fewer sequences of four which have at least $1$ case of HH. – Henry Sep 30 '17 at 16:49
• I am talking about counting only if a sequences contains either HT or HH at least once. So for the HH-count the sequences HHHH counts 1 because HH occurs at least once in that sequence. – beta Sep 30 '17 at 17:05
• I know. My comments explains why this difference happens: it balances the different multiple appearances. One shorthand way would be to say that HH can overlap itself as in HHH, while HT cannot – Henry Sep 30 '17 at 17:18
• ah okay. sorry, I first did not read your comment properly. I now get what you mean. Can this be somehow converted to an answer? – beta Sep 30 '17 at 18:59
• – whuber Oct 1 '17 at 17:50

## 1 Answer

Your list has a total of $3 \times 16=48$ consecutive pairs of flips and you would expect a quarter of these (i.e. $12$) to be HH and a quarter to be HT. That is indeed what you get if you count your list, so in that sense HH and HT are equally likely to appear in any pair of flips.

But HH is more likely to happen multiple times in a single sequence of four flips, because it can overlap itself. In the extreme example of HHHH you get three pairs of HH. So overall you get

• HH appearing in $8$ sequences of four flips ($1$ case three times, $2$ cases two times and $5$ cases once, with $1\times 3+2\times 2 + 5 \times 1= 12$)

• HT appearing in $11$ sequences of four flips ($1$ case two times and $10$ cases once, with $1\times 2 + 10 \times 1= 12$)

So in a sense the fewer sequences of four slips that HH appears in at least once simply balances its greater possibility of appearing multiple times.

This self-overlap effect has other consequences. If you start flipping and stop when you first see HH, the expected number of flips is $6$; similarly if you stop when first seeing TT. But if you start flipping and stop when you first see HT, the expected number of flips is $4$; similarly if you stop when first seeing TH.

It can get even more complicated in a race:

• If you start flipping and stop when you first see HH or HT, then each is equally likely to be what made you stop.
• Similarly if you start flipping and stop when you first see HT or TH, then each is equally likely to be what made you stop.
• But if you start flipping and stop when you first see HH or TH, then TH is three times more likely as HH to be what made you stop. So races are not transitive


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Taniyama-Shimura 3: L-Series where it will be crucial in the definition of modularity.

For today, we assume our $latex {d}&fg=000000$-dimensional variety $latex {X/\mathbb{Q}}&fg=000000$ has the property that its middle etale cohomology is 2-dimensional. It won’t hurt if you want to just think that $latex {X}&fg=000000$ is an elliptic curve. We will first define the L-series via the Galois representation that we constructed last time. Fix $latex {p}&fg=000000$ a prime not equal to $latex {\ell}&fg=000000$ and of good reduction for $latex {X}&fg=000000$. Let $latex {M=\overline{\mathbb{Q}}^{\ker \rho_X}}&fg=000000$. By definition the representation factors through $latex {{Gal} (M/\mathbb{Q})}&fg=000000$. For $latex {\frak{p}}&fg=000000$ a prime lying over $latex {p}&fg=000000$ the decomposition group $latex {D_{\frak{p}}}&fg=000000$ surjects onto $latex {{Gal} (\overline{\mathbf{F}}_p/\mathbf{F}_p)}&fg=000000$ with kernel $latex {I_{\frak{p}}}&fg=000000$. One of the subtleties we’ll jump over to save time is that $latex {\rho_X}&fg=000000$ acts trivially on $latex {I_{\frak{p}}}&fg=000000$ (it follows from the good reduction assumption), so we can lift the generator of $latex {{Gal} (\overline{\mathbf{F}}_p/\mathbf{F}_p)}&fg=000000$ to get a conjugacy class $latex {{Frob}_p}&fg=000000$ whose image under…

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# A doctor has prescribed a corrective lens of power +1.5 D. Find the focal length of the lens. Is the prescribed lens diverging or converging?

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Given:

A doctor has prescribed a corrective lens of power +1.5 D.

To do: To find the focal length of the lens and to determine the type of lens whether it is diverging or converging.

Solution:

Power of the lens:

The power of a lens implies the ability to bend a light passing through it.

The unit of power of a lens is Diopter$(D)$

The formula for the power of lens:

$\boxed{P=\frac {1}{f}}$

Here,

$P\rightarrow$ Power of the lens

$f\rightarrow$ focal length of the lens

It is given that power of the lens, $P$ = $+1.5\ D$

Substituting the given values we get-

$+1.5=\frac {1}{f}$

Or $f=+\frac {1}{1.5}$

Or $f=+\frac {10}{15}$

Or $f=+0.666\ m=+66.6\ cm$

Therefore,

The focal length of the lens:

The focal length of the lens $f$ is 66.6 cm.

Nature of the lens:

The negative sign $(+)$ implies that the lens is converging in nature. Hence, it is a convex lens.

Updated on 10-Oct-2022 13:20:04