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Proving a function is bounded Hi i was wondering if  anyone could help me with my revision This is a question of a past paper i'm stuck on.  Let $$f:[a,b]\to \Re$$ be Riemann intergrable. Part (a) is to prove that f is bounded and part (b) is to Give an example of a bounded function f that is not Riemann integrable  my attempt for part (a) is since f is Riemann intergrable then it is continuous on [a,b] which means that it must also be bounded? or is this incorrect i find this topic rather difficult  For part (b) i dont really have a clue i cant recall any function like the one they ask for Is there a possibility to search when the embedded txt file was created in the word document? I'm working in Word 2010 and the extension of the document is simple: .doc  It was inserted using the Word options:  Insert -&gt; Object -&gt; Create from file -&gt; Insert as icon   Thank you, I already tried the right click and properties option, but the problem with it, that it shows only the current date for all of the 3 options (created/modified/accessed).           I also tried to get the timestamp but it is inactive when I'm on the txt object.   Sorry not to mention it before: Thank you all of you here for the help and support it is really appreciated. Order of general- and special linear groups over finite fields. A problem about an $R$-module that is both injective and projective. Let $R$ be a domain that is not a field, and let $M$ be an $R$-module that is both injective and projective. Prove that $M= \left \{ 0 \right \}$.   This is exercise 7.52 of Rotman's Advanced Modern Algebra. Using theorems before exercises, because $M$ is injective and $R$ is a domain, I conclude that $$\forall m\in M ,\forall r\in R\ (r\neq 0) ,\exists {m}'\in M \Rightarrow m=r{m}'$$ and also because $M$ is projective there is a surjective $\psi$ from free $R$-module $F$ with basis $\left \{ e_{i} \right \}_{i\in I}$ to $M$ and thus we can conclude that for every $m\in M$ we have $$m=\sum r_{i}\Psi (e_{i})$$ now I don't know how should I use these together.  The idea of what is happening or a suggestion or a hint will be great. Proof Verification: Converse of Intermediate Value Theorem Why does the area of a bar in a histogram represent the frequency of that particular class In a bar-graph, the height of the bars represent the frequency of a particular category but it is counter-intuitive that in a histogram, instead of the height of the bar(which represents the frequency density), the area of the bar represents the frequency of the particular class. Why is this so? A ring without the Invariant Basis Number property I was reviewing my homework and it seems I overlooked something crucial while proving some ring has no Invariant Basis Number property. This is exercise VI.1.12 in Aluffi's Algebra: Chapter 0  The setup: $V$ is a $k$-vector space and let $R = \mathrm{End}_{k}(V)$.   Prove that $\mathrm{End}_{k}(V\oplus V) \cong R^4$ as an $R$-module Prove that $R$ doesn't satisfy the IBN property if $V = k^{\oplus \mathbb N}$.   For the first, I used to the fact that $V \oplus V$ is both the product and coproduct (in $k$-Vect) of $V$ with itself to get the isomorphism. What I just realized is I only showed that the two are isomorphic as groups not $R$-modules. So what would be the $R$-module structure on $\mathrm{End}_{k}(V \oplus V)$?   For the second, I used the fact that $V = k^{\oplus \mathbb N}$ implies $V \cong V \oplus V$ which in turn implies $R = \mathrm{End}_{k}(V) \cong \mathrm{End}_{k}(V \oplus V)$. Again, I just realized that I only showed the latter two are isomorphic as groups.   It may be obvious (and maybe why my professor let it pass?) but I can't come up with a good $R$-module structure that makes the two group isomorphisms $R$-linear.  Edit:  Explicitly, these are the isomorphisms I'm dealing with. Let $\pi_j, i_j$ be the natural projection/inclusion maps of the $j$-th factor resp. and $\psi: k^{\oplus \mathbb N} \oplus k^{\oplus \mathbb N} \to k^{\oplus \mathbb N}$ the isomorphism given by $\psi(e_i, 0)=e_{2i-1}$ and $\psi(0, e_i)=e_{2i}$.  Then the first isomorphism $\mathrm{End}_k(V \oplus V)\to R^4$ is given by $\varphi \mapsto (\pi_1\varphi i_1,\pi_2\varphi i_1,\pi_1\varphi i_2,\pi_2\varphi i_2)$  The second isomorphism $R \to \mathrm{End}_k(V \oplus V)$ is given by $\alpha \mapsto \psi^{-1} \alpha \psi$  The composition doesn't seem to be $R$-linear if I use the obvious $R$-module structure on $R$ and $R^4$. Show that the series $\sum(\exp(\frac{(-1)^n}{n})-1)$converges, but not absolutely. Show that the series converges, but not absolutely. $\sum_{n=1}^{\infty}( $exp$(\frac{(-1)^n}{n})-1)$.   My Try:  Let $a_n=$exp$(\frac{(-1)^n}{n})-1$. I was going to use alternating series test because the sequence $\{a_n\}$ is alternating. But $\{|a_n|\}$ is not decreasing to $0$. So, I am stuck now. Can anybody please give me a hint? I am trying to add smoke to a cigarette. I've created a Cylinder, hit space and chose quick smoke while this sphere was selected. It created a transparent bigger sphere and it is emitting smoke in solid and texture mode. But when I try it out in render mode, it won't show the smoke for some reason. Here is my setup:    The cylinder has a basic white material. Thanks! I am having trouble adding an image picker to a block form defined in a custom module.  If I create a custom block type I can define an image field as entity_reference and in the block's form view tell it to use entity_browser and media_entity_browser.  This provides me with an picker with which I can select my image:    I am trying to recreate this in a block provided by my custom module, without much success  // /src/Plugin/Block/MyCustomBlock.php  function blockForm($form, FormStateInterface $form_state)  {     ...      $form['image'] = [       '#type' =&gt; 'entity_browser',       '#entity_browser' =&gt; 'media_entity_browser',       '#title' =&gt; $this-&gt;t('Image'),       '#default_value' =&gt; $this-&gt;configuration['image'] ?? null,       '#weight' =&gt; '0',     ];      ... }   This gives me the following.  Note the lack of the bounding box, title and option to edit remove items:     Exporting the configuration files for the working form view gets me this:  // core.entity_form_display.block_content.lead_block.default.yml      ...  field_image:     weight: 1     settings:       entity_browser: media_entity_browser       field_widget_display: rendered_entity       field_widget_edit: true       field_widget_remove: true       selection_mode: selection_append       field_widget_display_settings:         view_mode: default       open: false     third_party_settings: {  }     type: entity_browser_entity_reference     region: content   ...   But if I try and use 'type' =&gt; 'entity_browser_entity_reference' in my blockForm function the field disappears altogether.  Looking into various core files I can see that entity_browser is a @FormElement and entity_browser_entity_reference is a @FieldWidget but I don't really understand the difference.  As you might guess I am fairly new to Drupal and I feel like I am missing something fundamental.  I (think I) understand that custom block types are actually nodes internally and that they are subject to different rules to blocks which I define in my custom modules, but I don't fully grasp what the differences here are.  Is is it possible to use the entity browser in my custom module block form in the same way as it is used in the form which I build through the admin's custom block type builder?  Thank you. I have some survey response data that are inconsistent and I'm not sure what is the best way to handle them.  This is the nature of the inconsistency:    Have you ever had a food poisoning episode that you think might have been caused by the food in a particular restaurant ? a) Yes b) No Did any of these episodes prevent you from eating at that restaurant again?  a) Does not apply, never have had a food poisoning episode b) Yes c) No Following these episodes, did you come back to your normal condition after 24 hours?  a) Does not apply, never have had a food poisoning episode b) Yes c) No   I believe that the intent of the designers of the survey was that anyone answering "No" to 1) should answer "Does not apply" to 2) and 3), but that's not how it worked. A large group of respondents picked "No" for 1) and "No" for at least one of 2) or 3).    What is the best way to deal with this inconsistency?    Where are some good references (books, papers, websites, etc.) to look at discussions and solutions for issues of this kind, and data cleaning in general? Merging banking and casual style in moderncv (2) Dwellers return automatically after finding a low amount of items in Wasteland I have a strange thing happening - dwellers I send to explore wasteland are automatically returning to vault after finding a little amount of items (4 or 5).   The exploration log says: "That's as much as I can carry. I better head back and show the Overseer what I found".  Before today they successfully carried up to 30 items and never returned to vault on their own.  I don't know is it relevant or not, but I have a lot of free storage space available.  What caused this change? What can I do to prevent this? The Wifi is working fine but no internet connection in Windows 8 "Available jobs to/for them" First of all, English is not my first language. I have a question, maybe a basic one, about this phrase:     The situation highlights the mismatch between some areas of training and available jobs to/for them.   Should it be "available jobs to them" or "available jobs for them"?  Sorry if the question is really basic. Fuzzy string search library in Java I'm looking for a high performance Java library for fuzzy string search.  There are numerous algorithms to find similar strings, Levenshtein distance, Daitch-Mokotoff Soundex, n-grams etc.  What Java implementations exists? Pros and cons for them? I'm aware of Lucene, any other solution or Lucene is best?  I found these, does anyone have experience with them? Why is electric field of an infinite plate constant at all points? I know from Gauss law, it is $\vec{E}=\dfrac{\sigma}{2 \epsilon_0}(\hat{n})$ at all points. But it doesn't make sense because of the inverse square nature of electric field which suggests if you move further away from the plane, electric field must reduce.   Then why is electric field of an infinite plate constant at all points? Best technique to create oldschool (fake 3D) racing game? What would be a good approach to develop the render system for an oldschool type racing game that uses a pseudo 3D scenery, like for example Outrun or Lotus Esprit Turbo Challenge? There's an endless scrolling road and scenery graphics are placed like billboard items, etc. I think you get the idea.    Are these developed in a similar way like  or is there a different technique behind it? Does the technique has a specific name I could search for on the web? I need to develop something like that in Flash. The $n$ complex $n$th roots of a complex number $z$ Suppose $z$ is a nonzero complex number, so $z=re^{i\theta}$. Show that there are only $n$ distinct complex $n$-th roots, given by $r^{1/n}e^{i(2\pi k+\theta)/n}$ for $0\leq k\leq n-1$.   My proof: Let $z=re^{i\theta}\in\mathbb{C}$ and $a\in\mathbb{C}$ s.t. $a^{n}=z$. Let $a=\rho e^{i\varphi}$. Then, $a^{n}=(\rho e^{i\varphi})^{n}$. Setting $\rho^{n}e^{i\varphi n}=re^{i\theta}$ and rewriting in polar form, we get $$\rho^{n}(\cos(n\varphi)+i\sin(n\varphi))=r(\cos(θ)+i\sin(\theta)).$$ Thus $\rho^{n}=r$ $\rightarrow$ $\rho=r^{1/n}$ and $n\varphi=\theta\rightarrow\varphi=\theta/n$. Thus the roots of $a\in\mathbb{C}$ have the form $$r^{1/n}(\cos(\tfrac{\theta}{n})+i\sin(\tfrac{\theta}{n}))=r^{1/n}(\cos(\tfrac{\theta}{n}+2\pi j)+i\sin(\tfrac{\theta}{n}+2\pi j))$$ for some $j\in\mathbb{Z}$ due to the periodicity of trig functions. Then, $$r^{1/n}(\cos(\tfrac{\theta}{n}+2\pi j)+i\sin(\tfrac{\theta}{n}+2\pi j))=r^{1/n}e^{i\left(2\pi j+\tfrac{\theta}{n}\right)}=r^{1/n}e^{i(\theta+2\pi k)/n}$$ for $0\leq k\leq n-1 ,j=nk\in\mathbb{Z} $.  I feel like there could be a more concise way to put this. How did Montague Apparate into the Hogwarts bathroom? In year 5, Fred and George jam Montague into the Vanishing Cabinet. He's found weeks later in a toilet.      ‘He never managed to get all the words out,’ said Fred, ‘due to   the fact that we forced him head-first into that Vanishing Cabinet   on the first floor.’      […]      ‘They’ve found Montague, sir, he’s turned up jammed inside a toilet   on the fourth floor.’   ‘How did he get in there?’ demanded Snape.   ‘I don’t know, sir, he’s a bit confused.’   In year 6 Malfoy expands on that.      ‘I had to mend that broken Vanishing Cabinet that no one’s used for years. The one Montague got lost in last year.’      […]      ‘The other’s in Borgin and Burkes,’ said Malfoy, ‘and they make a kind of passage between them. Montague told me that when he was stuck in the Hogwarts one, he was trapped in limbo but sometimes he could hear what was going on at school, and sometimes what was going on in the shop, as if the Cabinet was travelling between them, but he couldn't make anyone hear him ... in the end he managed to Apparate out, even though he’d never passed his test. He nearly died doing it.   What happened here? Did he Apparate out into Hogwarts, Apparate into Borgin and Burkes and somehow wander back to Hogwarts and get stuck into a toilet?