Showing an orthogonalisation process Can anyone show that:  $\mathbf{a_{\perp}}=\mathbf{a}-\frac{\mathbf{x}\mathbf{x}^T}{\mathbf{x}^T\mathbf{x}}\mathbf{a}$,  $\mathbf{a}\in\mathbb{R}^N$, $\mathbf{x}=(1,1,\dots,1)^T\in\mathbb{R}^N$  results in $\mathbf{a}$ becoming orthogonal to $\mathbf{x}$?  Thank you for your help.
Proving the two given vectors are orthogonal I am given the vectors $\mathbf{w}, \mathbf{v}, \mathbf{u}$ in $\mathbb{R}^n$ such that $\mathbf{u} \neq 0$ and $\mathbf{w} = \mathbf{v} - \frac{\mathbf{u}\,\bullet\,\mathbf{v}}{\|\mathbf{u}\|^2}\bullet\mathbf{u}$. I am asked to show that the vector $\mathbf{w}$ is orthogonal to $\mathbf{u}$.     So far, I have written out the definition of orthogonal: two vectors are orthogonal if and only if their dot product is zero.   So what we need to prove is $\mathbf{w}\bullet\mathbf{u} = 0$ where $\mathbf{w}\bullet\mathbf{u}$ is defined as $\mathbf{w}^T\bullet\mathbf{u}$.   However, I have been stuck on this problem for about an hour and haven't made any significant progress from here. How do we go about proving that the vectors are orthogonal?
I would like to prove that the equation $ 3^x+4^x=5^x $ has only one real solution ($x=2$)  I tried to study the function $ f(x)=5^x-4^x-3^x $ (in order to use the intermediate value theorem) but I am not able to find the sign of $ f&#39;(x)= \ln(5)\times5^x-\ln(4)\times4^x-\ln(3)\times3^x $ and I can't see any other method to solve this exercise... How do I disable "you need to format this disk" message in Windows Vista/7? I've started using TrueCrypt to encrypt a partition on an external USB hard drive. I've opted for using a raw partition as opposed to a file container.  But here's the annoyance. Whenever I connect this drive to a computer running Vista or Windows 7, I get the message popup "You need to format the disk in drive J: before you can use it".   The default option is "Format disk", and I'm afraid one of these days I'll accidentally trigger it.  Is there any way to disable this message? What temperature is achieved in focus point by 5000 flat 1x1cm mirrors onto a satellite dish? There is this  ("R5800 Solar Death Ray") where teenager built a 5000 mirror device which concetrates the solar rays which is showing the potency of the mosaic method of concentrating sunlight using elliptical dish. It can fire the wood, boil the water in seconds, melt an aluminum, glass or even burn the whole in concrere or rock.  What is the approximate maximum temperature (could be a range) is achieved by this mirror device in its focus point on sunny day?       The R5800 solar concentrator, replaced by the new R23k, is made from   an ordinary fiberglass satellite dish. It is covered in about 5800   3/8" (~1cm) mirror tiles. When properly aligned, it can generate a   spot the size of a dime with an intensity of 5000 times normal   daylight. This intensity of light is more than enough to melt steel,   vaporize aluminum, boil concrete, turn dirt into lava, and obliterate   any organic material in an instant. It stands at 5'9" and is 42"   across.   Technical details &amp; calculations:     Calculations: Area of dish aperture (elliptical)= (pi x 102cm x   73cm)/4 = 5848 sq cm       Area of focal point (circular)= pi(0.6cm)^2 = 1.14 sq cm      Dish Type= 0.7m x 1m offset focus elliptical.      Concentration Power= 5000x      Output Power Estimate = 560 watts How to know what program is listening on a given port? I suspect a program is listening on port 8000 on my machine.  When I run the following command, I get this error:  &gt; python -m SimpleHTTPServer # Lots of python error socket.error: [Errno 98] Address already in use   If I use another port (8000 is the default), the web server runs fine.  If I run wget localhost:8000 from the command line, it returns 404 Not Found.  What can I do (or what tools are available) to find what program is listening on port 8000, and from there where that program is configured? How to set an image as a background for Frame in Swing GUI of java? How to rate philosophy essays? I might end up teaching philosophy in Finnish lukio/gymnasium. I will not get an opportunity to practice teaching it - I do get to practice teaching mathematics a fair deal, though. (The situation is somewhat unfortunate.) I have taken subject studies in philosophy at university. They amount to around 60 credits (ECTS).  Suppose I give the students an exam, which asks them to write an essay or two. After that my task is to grade the exams in a fair way and comment them in such a way that the students can learn from the feedback.  -  Suppose the task was to write an essay on the problem of universals - nominalism and realism, say. Let us say 6 points is the perfect score for this task.  I would expect the students to define nominalism and realism (2 points, maybe?) and explain the problem (2 points). Remaining two points for taking some stance and arguing for it - realism is more reasonable because so and so. Partial credits for demonstrating only partial understanding.  -  Am I missing some important aspects of philosophical thinking? Should I emphasize the different parts of an answer in a different way? I suspect my perspective is influenced by fairly extensive mathematics education - is there something particular I should watch out for? Over the years I've written some libraries. These libraries sometimes depend on each other.  As a result, the "tex" directory is structured as follows:  tex ├── library |   ├── library1.sty |   ├── library2.sty |   └── library3.sty ├── project1 |   ├── master.tex |   ├── chapter1.tex |   └── chapter2.tex ├── project2 |   └── article.tex └── project3     └── paper.tex   The libraries depend on each other. As a result, in every library that depends on another library, this is part of the header:  %library2.sty \ProvidesPackage{library2} \RequirePackage{../library/library1}   In other words, the library is imported from the "root of every project". This is not a good design choice. What if someone copies the libraries in the folder of the root of a project, or what if the folder "library" is renamed...  It is however reasonable the "active directory" still remains in the project root.  What can be done to resolve such library dependencies? On Windows 7, is there a command prompt line that can start cmd as an administrator? On Linux, we can use  sudo bash   to run a shell as the superuser.  On Windows 7, we can use     Start menu -> (in search box, type) cmd -> Right click on search result and choose Run as Administrator   to run a command prompt as an administrator. Is there also a command prompt line that is like Linux's sudo or su so as to start a shell as an administrator? I know mathematically the answer to this question is yes, and it's very obvious to see that the dimensions of a ratio cancel out, leaving behind a mathematically dimensionless quantity. However, I've been writing a  (the specifics of which are out of scope), which has me thinking about the problem because I decided to handle angle units as dimensioned quantities, which seemed natural to enable the unit conversion with degrees. The overall purpose of the library is to disallow operations that don't make sense because they violate the rules of dimensional analysis, e.g. adding a length quantity to an area quantity, and thus provide some built-in sanity checking to the computation. Treating radians as units made sense because of some of the properties that dimensioned quantities seemed to me to have:  The sum and difference of two quantities with the same dimension have the same physical meaning as both quantities separately. Quantities with the same dimension are meaningfully comparable to each other, and not meaningfully comparable (directly) to quantities with different dimensions. Dimensions may have different units that are scalar multiple (sometimes with a datum shift).  If the angle is treated as a dimension, my 3 made up properties are satisfied, and everything &quot;makes sense&quot; to me. I can't help thinking that the fact that radians are a ratio of lengths (SI defines them as m/m) is actually critically important, even though the length is cancelled out. For example, though radians and steradians are both dimensionless, it would be a logical error to take their sum. I also can't see how a ratio of something like (kg/kg) could be described as an &quot;angle&quot;. This seems to imply to me that not all dimensionless units are compatible, which seems analogous to how units with different dimensions are not compatible. And if not all dimensionless units are compatible, then the dimensionless &quot;dimension&quot; would violate made-up property #1 and cause me a lot of confusion. However, treating radians as having dimension also has a lot of issues, because now your trig functions have to be written in terms of $\cos(\text{angleUnit}) = \text{dimensionless unit}$ even though they are analytic functions (although I'm not convinced that's bad). Small-angle assumptions in this scheme would be defined as performing implicit unit conversions, which is logical given our trig function definitions but incompatible with how many functions are defined, especially since many authors neglect to mention they are making those assumptions. So I guess my question is: are all dimensionless quantities, but specifically angle quantities, really compatible with all other dimensionless quantities? And if not, don't they actually have dimension or at least the properties of dimension? Explicitly calling a default method in Java Using C++ library in C code Equivalent statement of AC or not? Here's a corollary of AC stating that: if there exists an onto mapping $f:A\to B$, then there exists a 1-1 mapping $g:B\to A$. What I'm confused about is that whether or not  AC can be derived from this proposition, in return? I have tried but haven't found a proof. Any help would be appreciated. 'Some index files failed to download' while updating I'm trouble with exercise 1.8 of Carroll's Space-Time and Geometry:     If $\partial_\nu T^{\mu \nu} = Q^\mu$, what physically does the spatial vector $Q^i$ represent? Use the dust energy momentum tensor to make your case.    First of all, since the momentum energy tensor of dust $T^{\mu \nu}=\rho U^\mu U^\nu$ is symmetric, one has the energy-momentum conservation law $$ \partial_\nu T^{\mu \nu} = \partial_\nu T^{\nu \mu}  =0.$$ A vector $Q$ is constant in any frame related by a Lorentz transformation, while this isn't the case for the components $Q^\mu$. But here $Q=0$ and hence $Q^\mu=0$ also for every component $\mu =0,\cdots,4$?   Even if a straightforwardly apply the equation: $$ Q^i = \partial_\nu T^{i \nu} = (\partial_\nu \rho) U^i U^\nu + \rho(\partial_\nu U^i)  U^\nu + \rho U^i(\partial_\nu  U^\nu ),$$ doesn't seems to yield something...   Is my first reasoning right ? Otherwise: can someone give me a hint ?  Thanks How to center text vertically within a textbox in Photoshop It's pretty annoying that you can't center text vertically in Ps automatically, like you can in Indd. It's even more annoying that the bounding box of text can't be 'collapsed' or 'fit to text', like in Indd.  I could convert the text to shape, but then I lose coloring. I could convert to smart object, but then its not vector anymore, for when I export to pdf.  Is there a work around anyone knows of that would make it easier to center text vertically? Adjusting space between array rows and columns Shine a light into a superconductor A type-I superconductor can expel almost all magnetic flux (below some critical value $H_c$) from its interior when superconducting. Light as we know is an electromagnetic wave. So what would happen if we shine a light deep into a type-I superconductor? Suppose the magnetic field of the light is significantly lower than $H_c$.  And what about type-II superconductors? How to install and configure Wine? Teleporting a player from the overworld to the end Is there any way to teleport a player from the overworld to the end?   Or a way of teleporting a player to a random enderman in the end? I'm trying to parse json response.. but keep getting this error      Invalid conversion from runtime type List ANY> to Map String,ANY>   Map&lt;String, Object&gt; MapLv3 = new Map&lt;String, Object&gt;();      Integer x2 = 0; for (Object oh : CallsList){     MapLv3.put(String.valueof(x2), oh);     x2++; } index = x2; System.debug('Map Lv 3 ==&gt;'+MapLv3);  Map&lt;String, Map &lt;String, Object&gt;&gt; Map3 = new Map&lt;String, Map &lt;String, Object&gt;&gt;(); for (Integer a2 = 0; a2 &lt; index; a2++){     Map3.put(String.valueof(a2),(Map&lt;String,Object&gt;) MapLv3.get(String.valueof(a2))); } System.debug('Segment ==&gt;'+Map3.get('0').get('segments'));