What is electron hole? Isn't it just an absence of electron?
Why absence of electron is called hole?
Show that $\ker \hat{T} = \text{ann}(\text{range } T)$ This is an old exam problem: Let $V$ and $W$ be finite dimensional vector spaces over a field $F$ and let $T: V \to W$ be a linear transformation. Define $\hat{T}: W^* \to V^*$ by $(\hat{T}(f))(v)=f(T(v))$. Here, $U^*$ denotes the dual vector space $U$. Since $\hat{T}$ is a linear transformation prove $\ker \hat{T} = \text{ann}(\text{range } T)$ where $\text{ann}(Y) = \{g \in U^* : g(y)=0 \text{ for all } y \in Y \}$ for a subspace $Y$ of vector space $U$.  I'm really not sure how to start this problem. Any advice would be great! How to initialize all members of an array to the same value? Prove for any $a&gt;1$ there exist a natural number, $N$, such that $a^n&gt;n$ for all $n \ge N$  My attempt:  By definition, for $N$ to be a natural number $1\le N \le ∞$ for any integer $N$.  By induction, the base case $n=1$ is true because $a^1&gt;1$ which is true.  Now assume that $a^k&gt;k$ is true, then it is also true for $n= k+1$.  Hypothesis  $a^{k+1}&gt;k+1$  Now I am not sure if I am on the right track I wrote a regular expression which works well in a certain program (grep, sed, awk, perl, python, ruby, ksh, bash, zsh, find, emacs, vi, vim, gedit, …). But when I use it in a different program (or on a different unix variant), it stops matching. Why? Where do I put item tags on items inside a chest or your inventory? I am making a Minecraft Mini Game adventure map, and one of the levels has a /setblock command, I am using the command in a way to spawn an item in a chest (Wooden Button) I need the wooden button to have CanPlaceOn tag so it can be placed on a log. I have tried this command: setblock 294 5 -1251 minecraft:chest 0 replace {Items:[{Count:1,Slot:0,id:minecraft:wooden_button,CanPlaceOn:[log]}]}` How does one [easily] calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$, where $\mathrm{pop}(n)$ counts the number of bits '1' in the binary representation of $n$? Is there any trick to calculate the sum? From what I already have, it definitely converges. I already know how to do it on Mac OS X (plenty of instructions online).  But is it possible to do it on Ubuntu? If yes, how? Meaning of Conditional sentences with 'should' and 'happen to' I'm a bit familarized with conditional sentences. But the following sentences with 'should' and 'happen to' are quite confusing too me. They are:   If it should rain, I will stay at home. 2.If it should happen to rain, I'll stay at home   If it happens to rain, I'll stay at home.    What are the meaings of those sentences ? Do they all mean the same ? Thank you a lot in Advance for your answer. In the Avengers movie, why is Loki sweating and weak when he first arrives on Earth, in the SHIELD base? Calculating p-values for two tail test for population variance A sporting goods manufacturer claims that the variance of string tensions for any decent tennis racquet should be about 9 pounds. The string tensions of 18 randomly selected tennis racquets produced a variance of 8.13 pounds. Find the p-value to test the manufacturer's claim (assume the population is normally distributed).  I tried attempting the question as follows: ${H_o}: {\sigma ^2} = 9$ ${H_1}: {\sigma ^2} \ne 9$ We are also given the following: $n=18$ $s^2 = 8.13$ Test Statistics: $\chi^2 = {{(n-1)S^2}\over{\sigma^2}}$ Calculating Test Statistics under $H_o$ gives us: $\chi^2 = {{(18-1)8.13}\over 9} = 15.35667$  For calculating the p-value I proceeded as follows: $p-value = 2\min\{P[\chi^2&gt;15.35667],\ P[\chi^2&lt;15.35667]\}\ =2\times 0.430192 = 0.860384$  I am not certain if my method for calculating the p-value is correct or not. Can someone please tell if the above process is valid?  Thanks in advance! How can I retrieve data within my widget I have a Widget which I am creating but my quesiton is how do I access my custom table which I created via my plugin. I am attempting to fill the dropdown from the table called tfp which contains the column liters which I wish to popluate the values into the drop down   class my_plugin extends WP_Widget {  // constructor function my_plugin() {         parent::WP_Widget(false, $name = __('Thomsons Fuels Widget', 'wp_widget_plugin') );  }  // widget form creation function form($instance) {  //Check values if( $instance) {  $title = esc_attr($instance['title']);  $dropdown= esc_attr($instance['dropdown']);  $textarea = $instance['textarea'];  } else {  $title = '';  $textarea = ''; } ?&gt;   &lt;p&gt; &lt;label for="&lt;?php echo $this-&gt;get_field_id('dropdown'); ?&gt;"&gt; &lt;?php _e('Liters',   'wp_widget_plugin'); ?&gt;&lt;/label&gt; &lt;select&gt; &lt;option value="volvo"&gt;Volvo&lt;/option&gt; &lt;/select&gt;  &lt;/p&gt;   How I created the table in the plugin   global $wpdb; global $custom_table_example_db_version;  $table_name = $wpdb-&gt;prefix . 'tfp'; // do not forget about tables prefix  // sql to create your table // NOTICE that: // 1. each field MUST be in separate line // 2. There must be two spaces between PRIMARY KEY and its name //    Like this: PRIMARY KEY[space][space](id) // otherwise dbDelta will not work $sql = "CREATE TABLE " . $table_name . " (   id int(11) NOT NULL AUTO_INCREMENT,   liters VARCHAR(100) NOT NULL,   price VARCHAR(100) NOT NULL,      postcodearea varchar(20) NULL,         dft varchar(20) NULL,   PRIMARY KEY  (id) );";  // we do not execute sql directly // we are calling dbDelta which cant migrate database require_once(ABSPATH . 'wp-admin/includes/upgrade.php'); dbDelta($sql);  // save current database version for later use (on upgrade) add_option('custom_table_example_db_version', $custom_table_example_db_version); This works just fine:      protected void txtTest_Load(object sender, EventArgs e)     {         if (sender is TextBox) {...}      }   Is there a way to check if sender is NOT a TextBox, some kind of an equivalent of != for "is"?  Please, don't suggest moving the logic to ELSE{} :) Prove $(x+y)(y+z)(x+z)≤\frac8{27}(x+y+z)^3$ The entire exercise was like $$8xyz≤(x+y)(y+z)(x+z)≤\frac8{27}(x+y+z)^3\quad x,y,z&gt;0$$  and I managed to solve (demonstrate) the first inequality (from the left) using $$\frac{x+y}2≥\sqrt{xy}$$ but I can't solve the second one. I tried several ideas but I haven't succeeded. Thanks a lot. How to safely remove proprietary Intel driver I recently installed the proprietary Intel drivers for my Toshiba Satellite A105 s4284 running Ubuntu 14.10. Nouveau was working just fine, I was just attempting to make it so that I was able to run CS:S (no need for explanation on that.)  Long story short, I went and downloaded the Intel Graphic Drivers for Linux* from . After downloading and installing the .deb, I ran intel-linux-graphics-installer which opened up a window that downloaded &amp; installed the open Intel drivers. I then rebooted, and it's slow. Very slow, especially on 3D effects and when I hit Alt+Tab.  Now, I don't know if the previously installed Nouveau drivers are conflicting with these new Intel drivers (if that's the issue, what should I do about fixing it?) What I do know is that my system isn't behaving right and I want it to (obviously.) If removing the Nouveau drivers isn't the solution, how can I safely remove the just-installed Intel drivers?  Does anyone have any suggestions? A single row background colour Is there a way to see who voted on your posts? Is there a word for happiness made possible by a tragic situation? How to prove this inequality $ x + \frac{1}{x} \geq 2 $ When a user has the  privilege, they can click on the post score to see the individual counts for upvotes and downvotes.  When a user doesn't have this privilege this functionality isn't present, except after loading newly added answers without refreshing the page. After clicking the "new answers to this question" link the click functionality to view vote counts is added to all posts, though actually clicking on it results in a dialog stating that an error occurred (I'd assume due to not having the required reputation).  May be related to the fix for this previous bug report: What does "Could not find or load main class" mean?