Can anyone please show me how to solve this problem? I'm stuck $$\sum_{k=1}^n k = \frac{n(n+1)}{2}$$ $$ and $$ $$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k\right)^2$$  To prove by induction, it has to be as following: 1) Base Case: Show that P(n) is true. 2) Induction Hypothesis: Assume P(k) is true. for $k\in\Bbb N$  3) Induction Step: Show that P(k + $1$) is also true.
How can I prove that     $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$    for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.  Thanks
Which graphics cards are supported by the new AMDGPU driver in Ubuntu 16.04? The new open source AMDGPU driver pre-installed in Ubuntu 16.04 should support all the "Volcanic Islands" cards out-of-the-box. This means that AMD Radeon R9 285, R9 380/380X and R9 Fury/FuryX graphics cards will be supported out-of-the-box if Ubuntu 16.04 LTS is installed.  However I read that the open source AMDGPU driver has an experimental support for the "Sea Islands" graphics cards (HD7790, R7 260, R9 290, R7 360, R9 390). Is this support enabled by default on Ubuntu 16.04, or will these graphics cards use the Radeon driver?  Thanks Xml configuration versus Annotation based configuration In a few large projects i have been working on lately it seems to become increasingly important to choose one or the other (XML or Annotation). As projects grow, consistency is very important for maintainability.   My questions are: what are the advantages of XML-based configuration over Annotation-based configuration and what are the advantages of Annotation-based configuration over XML-based configuration? When to use LinkedList over ArrayList in Java? Reapplying for a UK Standard Visitor visa after a refusal Weird artifacts in my UV mapped texture Any ideas what's causing those chunks of geometric artifacts?  Argh. Understanding reference counting with Cocoa and Objective-C I'm just beginning to have a look at Objective-C and Cocoa with a view to playing with the iPhone SDK. I'm reasonably comfortable with C's malloc and free concept, but Cocoa's references counting scheme has me rather confused. I'm told it's very elegant once you understand it, but I'm just not over the hump yet.  How do release, retain and autorelease work and what are the conventions about their use?  (Or failing that, what did you read which helped you get it?) Show that a positive operator on a complex Hilbert space is self-adjoint Let $(\mathcal{H}, (\cdot, \cdot))$ be a complex Hilbert space, and $A :  \mathcal{H} \to \mathcal{H}$ a positive, bounded operator ($A$ being positive means $(Ax,x) \ge 0$ for all $x \in \mathcal{H}$). Prove that $A$ is self-adjoint. That is, prove that $(Ax,y) = (x, Ay)$ for all $x,y \in \mathcal{H}$. Here's what I have so far. Because $A$ is positive we have $\mathbb{R} \ni (Ax,x) = \overline{(x,Ax)} = (x,Ax)$, all $x \in \mathcal{H}$. Next, I have seen some hints that tell me to apply the polarization identity: $$(x,y) = \frac{1}{4}((\lVert x+y \rVert^2 + \lVert x-y \rVert^2) - i(\lVert x + iy \rVert^2 - \lVert x - iy \rVert^2)),$$ where of course the norm is defined by $\lVert \cdot \rVert^2 = (\cdot, \cdot)$. So my guess is that I need to start with the expressions: $$(Ax,y) = \frac{1}{4}((\lVert Ax+y \rVert^2 + \lVert Ax-y \rVert^2) - i(\lVert Ax + iy \rVert^2 - \lVert Ax - iy \rVert^2)),$$ $$(x,Ay) = \frac{1}{4}((\lVert x+Ay \rVert^2 + \lVert x-Ay \rVert^2) - i(\lVert x + iAy \rVert^2 - \lVert x - iAy \rVert^2)),$$ and somehow show they are equal. But here is where I have gotten stuck. Hints or solutions are greatly appreciated. Let $A$ be an $n \times n$ matrix with real enteries. Define $\langle x,y\rangle_A = \langle Ax,Ay\rangle, \quad x,y \in \mathbb R^n$ , where $\langle,\rangle$ is a standard inner product on $\mathbb R^n$. Then  $\langle x,y\rangle_A$ is an inner product iff  1)  $\ker A = \{0\}$.  2) $\operatorname{rank} A= n$.  3) all eigenvalues of $A$ are positive.  4) all eigen value of $A$ are non negative.  Clearly if $\ker A \neq \{0\}$, then there exist $x \neq 0$ such that $Ax =0$, then $\langle Ax, Ax\rangle_A = 0$, then $\langle ,\rangle_A$ is not inner product, so $\ker A = \{0\}$, thus rank of $A$ is $n$.  I am unable to check the 3) and 4) option. Please help to check the other option. Thank you I want to exclude certain local folders with Google Drive but I *keep* them on the computer Using Google Drive on Windows (although I think the same question applies to OS/X) I want to exclude certain local folders but keep them on the computer.  When I go to Drive preferences and deselect specific folders, it wants to remove them from the local PC and keep them only in the cloud.  This doesn't work for me.  I feel like I could do this in earlier versions of the Google Drive client but I cannot recall for sure.  Is this possible? From  by Richard Brualdi  We have a chess master. He has 11 weeks to prepare for a competition so he decides that he will practice everyday by playing at least 1 game a day. To make sure that he's able to take a couple of breaks, he also decides that he wont play more than 12 games per week.  My question is, how do you prove/dis-prove that there will be a succession of (consecutive) days in which he will play k games, $k \geq  22$?  Specifically, I don't quite understand the use of the pigeonhole principle here.  Thanks in advance! :D  EDIT  I'll try to clarify what I'm asking:  The goal is to prove that for every possible practice schedule that the chess master makes, there will always be a consecutive sequence of days such that he plays exactly k games.  So for example, if k = 22, how do you prove that no matter what his schedule is like, he will always play exactly k games in r days, $1 \leq r\leq  77$.  My main question is: what is the general method for proving different values of k? Are there values of k (from 1 to 132 inclusive) that are impossible to obtain?  Note that he does not necessarily have to play 132 games within 77 days! Also, keep in mind the bounds of k. What does f-stop mean? Why is the Cantor function not absolutely continuous? Is there an easy way to see that the Cantor function is not absolutely continuous that directly uses the definition of absolutely continuous? I am trying to write a condition into a SQL WHERE clause that forces the column comparison to a variable to be compared in lowercase. However the addExpression function isn't accomplishing this (since that is putting the expression in the field selection, not the where clause.  Here is what I have tried:  $category = 'mobile wifi'; $query = db_select('taxonomy_term_data', 'ttd')     -&gt;fields('ttd', array('tid')); $query-&gt;innerJoin('taxonomy_vocabulary', 'tv', 'ttd.vid = tv.vid'); $query-&gt;addExpression("LOWER(ttd.name) = $category"); $result = $query-&gt;condition('machine_name', 'images_cat', '=')     -&gt;execute()     -&gt;fetchAssoc();   And this:  $category = 'mobile wifi'; $query = db_select('taxonomy_term_data', 'ttd')     -&gt;fields('ttd', array('tid')); $query-&gt;innerJoin('taxonomy_vocabulary', 'tv', 'ttd.vid = tv.vid'); $result = $query-&gt;condition('machine_name', 'images_cat', '=')     -&gt;condition('LOWER(ttd.name)', $category, '=')     -&gt;execute()     -&gt;fetchAssoc();   The second one becomes invalid and the first one causes the problem I mentioned above. Any thoughts or suggestions?  Thanks, Patrick Two definitions of $\limsup$ Here are two equivalent definitions of $\limsup_{n\rightarrow\infty} a_n$:   Let $u_n=\sup\{a_n, a_{n+1}, a_{n+2},\ldots\}$. Then $$\limsup_{n\rightarrow\infty} a_n = \lim_{n\rightarrow\infty} u_n =    \lim_{n\rightarrow\infty}\left(\sup\{a_n,a_{n+1},\ldots\}\right)$$ Let $E$ be the set of all subsequential limits of $\{a_n\}$. Then $$\limsup_{n\rightarrow\infty} a_n = \sup E$$   I'm curious as to which one people usually learn first, or which one people find more intuitive. Baby Rudin only has the second definition, but doesn't mention the first as far as I know.  As far as intuition, I think of the second definition as "the biggest limit point." I'm not sure how to think of the first definition though. I can see that $u_n \ge u_{n+1}$ for all $n$, but I'm not sure how to interpret the limit of those $u_n$ conceptually.  Are there any other ways to conceptualize $\limsup$? How to connect Nokia Lumia 730 to Ubuntu 15.04? My Nokia Lumia 730 (WP8.1) is not getting detected on Ubuntu 15.04.  Can anybody tell me how to fix this? Different ways to execute a shell script There are several ways to execute a script. The ones I know are: /path/to/script # using the path (absolute or relative) . script        # using the . (dot) source script   # using the `source` command  Are there any other way? What are the differences between them? Are there situations that I must use one and not another? What is the difference between "I've been to China" and "I've gone to China"? Both are in the present perfect, but one uses the verb 'to go' and the other 'to be'. Is there a rule for this?  Is there any chance that the differences between "I have been" vs. "I have gone", are differences in English dialect (I don't know if this is the right word).  Edit Could I say that North American English speakers use "I have gone", and maybe U.K speakers say "I've been"? Or vice versa. When is it useful to include the code in a thesis? I am writing my master of science thesis and I am not sure whether I should add some of the code I have written: they are not proper programs, but more like input files to be fed to a commercial program (Nastran) very popular in my field.  My concerns are related to the fact that I am not making any discovery, just applying the things I have learned reading the manual. Still, I feel the code will make the effort I put into the thesis to also learn this language clearer. Online calculators such as  (view page source) use the below formulas to get meters per degree. I understand in general how distance per degree varies depending on latitude location, but I do not understand how that translates to the below. More specifically, where do the constants, the 3 "cos" terms in each formula, and coefficients (2, 4, 6; 3, and 5) for "lat" come from?      // Set up "Constants"     m1 = 111132.92;     // latitude calculation term 1     m2 = -559.82;       // latitude calculation term 2     m3 = 1.175;         // latitude calculation term 3     m4 = -0.0023;       // latitude calculation term 4     p1 = 111412.84;     // longitude calculation term 1     p2 = -93.5;         // longitude calculation term 2     p3 = 0.118;         // longitude calculation term 3      // Calculate the length of a degree of latitude and longitude in meters     latlen = m1 + (m2 * Math.cos(2 * lat)) + (m3 * Math.cos(4 * lat)) +             (m4 * Math.cos(6 * lat));     longlen = (p1 * Math.cos(lat)) + (p2 * Math.cos(3 * lat)) +                 (p3 * Math.cos(5 * lat)); In a ring homomorphism we always have $f(1)=1$?