What does it mean by a battery is of 1.5 Volt ? Does it mean that it would take 1.5 Joules of energy to transfer -1 columb of charge from the negative end to the positive end of the battery ?  What is the reason for it to take 1.5 Volt of energy to transfer the charge - is it needed to do work to move the charge against the electric field generated by the positive and negative sides of the battery ? Then would the Volt increase if the distance between the positive and the negative sides are increased (as it would have to do more work against the field)?  Then if you join a broken piece of wire to one side and a broken piece of wire to another side of the battery, why the potential difference between the two ends of the battery is still 1.5 Volt ?
So this is a question with which I have been stuck for a while and it all revolves around the concept of volt.  I have posted a LOT of questions on physics forums while trying to understand the concept of volt and potential difference. This is hopefully going to be the last one.  What do you exactly mean when you say that a battery is 2.5 V. I know that it says that the potential difference between the 2 terminals of the battery is 2.5 V  but then what is the potential of the 2 ends of the battery? Is there a unit of potential and not just potential difference?  Also volt = work/charge but how can work/charge express the difference of potentials across the 2 terminals?
I already know that if groups $G,H,K$ are finitely generated abelian groups, following is true.  If $G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$. I prove this by uniquness of factorization of finitely generated abelian groups.  My questions are   If group $G$ is finite,$G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$? Can you give me a easiest proof and it can be proved by projection function on external direct product? If $G,H,K$ are groups.  If $G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$. This statement is false. What is a counterexample?   *$\times$ is external direct product example of not surjective isometry on a bounded not closed set. Is it possible to save frequently used expressions in QGIS? Is there any way to save frequently used expressions in the selection or field calculator dialogs? Obviously I can cut and paste them into a file where I might save them, but I mean within the context of QGIS like you can do in ArcMap. Why do I have blurred vision and how do I fix it? I know there are some similar posts on adding review queues to the mobile web page. However, this is regarding the Stack Exchange Android app. I really would like to review posts and edits from the Android app, however this isn't included in the app yet. I'm not sure how it would be implemented; maybe there could be a button or something for review. Could we please see this? Why do I get "Permission denied (publickey)" when trying to SSH from local Ubuntu to a Amazon EC2 server? I have an instance of an application running in the cloud on Amazon EC2 instance, and I need to connect it from my local Ubuntu. It works fine on one of local ubuntu and also laptop. I got message "Permission denied (publickey)" when trying to access SSH to EC2 on another local Ubuntu. It's so strange to me.  I'm thinking some sort of problems with security settings on the Amazon EC2 which has limited IPs access to one instance or certificate may need to regenerate.  Does anyone know a solution? Package pgfplots Error: CRITICAL: shader=interp: got unsupported pdf shading type '0' For the set of data , the following code can't compile showing the error     Package pgfplots Error: CRITICAL: shader=interp: got unsupported pdf   shading type '0'. This may corrupt your pdf!.   \documentclass[border= 5mm]{standalone} \usepackage{pgfplots} \pgfplotsset{compat=newest} \begin{document}     \begin{tikzpicture}     \begin{axis}[unbounded coords=jump, view={70}{40}]     \addplot3 [surf,shader=interp]      table [x index={0}, y index={1}, col sep=tab] {mySurfaceData.txt};     \end{axis}     \end{tikzpicture} \end{document}    So, what went wrong to cause this?  P.S. the original data without NaN is available , where I would like to skip plotting those points of z=1. I'm building an HTML UI with some text elements, such as tab names, which look bad when selected. Unfortunately, it's very easy for a user to double-click a tab name, which selects it by default in many browsers.  I might be able to solve this with a JavaScript trick (I'd like to see those answers, too) -- but I'm really hoping there's something in CSS/HTML directly that works across all browsers. I have a WordPress install on a dedicated server with decent memory (upwards of 8GB).  The WordPress install includes many plugins - 50 active plugins. Most important to note is the  plugin to make WordPress multilingual. The theme is .  The stack is LAMP - CentOS 6.3, Apache 2.4.4, Percona MySQL, PHP-FPM.  I have been quite stupid, and didn't set up a test or staging server.  Everything was running fine till I updated to WordPress 3.8, and also updated several plugins at the same time.  I didn't pay attention at the time, but all my widgets got messed up, and so did any sort of navigation etc.  I enabled error_log for a bit, i.e.  ini_set('log_errors',TRUE); ini_set('error_reporting', E_ALL); ini_set('error_log', dirname(__FILE__) . '/error_log.txt');   This resulted in an immense file - basically the error_log grows by about a GB every 4 minutes or so.  The primary error I found in the error log was database server disconnect, better known as      MySQL server has gone away   The first priority was to patch the problem. So I installed  plugin. This mostly, though not completely solved the issue. Many of the widgets came back.  However, the special (non-English) characters are all messed up (converted to question marks). I tried to fix this by opening wp-mysql-ping.php and commenting out   if ( !DB_CHARSET &amp;&amp; version_compare(mysql_get_server_info($this-&gt;dbh), '4.1.0', '&gt;='))                                     {                                         $this-&gt;query("SET NAMES '" . DB_CHARSET . "'");                                     }   Didn't help.  Besides, the plugin is not a permanent or complete solution. So leaving the site in this messed up state, I started working on other possibilities:  As suggested at , I edited my.cnf to add  wait_timeout = 60   Didn't help.  Also added caching to my.cnf  query_cache_type = 1 query_cache_limit = 8M query_cache_size = 512M   Didn't help.  (and yes, I restarted mysql every time I edited my.cnf)  Also tried the method : In wp-db.php before  $this-&gt;select( $this-&gt;dbname, $this-&gt;dbh );   I added  $this-&gt;query("set session wait_timeout=600" );   Didn't help.  I'm at my wit's end. Please help! Filling cells with sequential numbers in an Excel (2003) macro I need to fill an excel column with a sequential series, in this case from -500 to 1000.  I've got a macro to do it, but it takes a lot of lines for something that seems like it should be a single function [something like FillRange(A2:A1502, -500, 1000, 1)].  But if that function exists, I can't find it.  Is the following as simple and elegant as it gets?  'Draw X axis scale Cells(1, 1).Value = "mV" Cells(2, 1).Value = -500 Cells(3, 1).Value = -499 Cells(4, 1).Value = -498  Dim selection1 As Range, selection2 As Range  Set selection1 = Sheet1.Range("A2:A4") Set selection2 = Sheet1.Range("A2:A1502")  selection1.AutoFill Destination:=selection2 Tangent bundle : is a manifold I have studied what a differentiable manifold is, and what a tangent space at a given point is, and read the proof that its dimension is equal to the dimension of the manifold. Here a tangent vector is a derivation on the germs of differentiable functions at a point.  Now I am trying to see that the tangent bundle is indeed a differentiable manifold. I get that for an open set $U$ in Eucliden space, it takes the form $U \times \mathbb R^n$ with the vector $\sum a_i x_i$ giving the tangent vector $\sum a_i \partial /\partial x_i$. Now the book says that it is "obvious" that these glue up and give a differentiable manifold.  I may be stupid; but I will be grateful if somebody can hold my hand through this "obvious" statement. Cannot successfully source .bashrc from a shell script Normally we can source ~/.bashrc file using this command  source ~/.bashrc   but if I write this in a shell script and execute it, nothing happens. Why? Is there any way to do this?  My script:  #!/bin/bash chmod a+x ~/.bashrc source ~/.bashrc   Also tried . (dot) instead of source. Same result. Largest Mass Diffraction Does metric signature affect the stress energy tensor? If one were to derive the stress-energy tensor for a metric with $(+,-,-,-)$ signature would it be different from the stress-energy tensor derived from the same metric but with $(-,+,+,+)$ signature? I have a database which contains 5 tables. Each table contains 24 rows and each row contains 4 columns.  I want to display these records in Excel sheet. The heading of each table is the name of the table, but I am unable to merge the columns for heading.  Please help me. I’m currently in the process of reconsidering how I do bibliography management. It seems to be most sensible to use biblatex with biber. I’d like to know what tools are available on which I can build a solution that works for me. I’m interested in everything from full-blown bibliography management suites to pretty-printers for .bib files; both as traditional programs and online services.  Tools in question should fulfill the following criteria:   Unicode support. Support for the additional entry types and fields that biblatex supports. Support for biber’s improved cross-referencing. Rolle's Theorem and the Mean Value Theorem Let $f\colon \mathbb{R} \to \mathbb{R}$ have derivatives of all orders. Suppose that $a &lt; b$ and that $f(a)=f(b)=f'(a)=f'(b) =0$. Prove that $f'''(c) = 0$ for some $c$ in $(a,b)$. High Memory Usage on Linux Server How strong of magnetic field would noticibly attract a person? There is iron in our blood, which is magnetic. Roughly how strong would a magnet have to be to induce a noticeable attraction? It would be nice to know this for several distances. Also, do electromagnets that strong exist? Setting default values (conditional assignment) In Ruby you can easily set a default value for a variable   x ||= "default"   The above statement will set the value of x to "default" if x is nil or false  Is there a similar shortcut in PHP or do I have to use the longer form:  $x = (isset($x))? $x : "default";   Are there any easier ways to handle this in PHP?