Explanation of why doesn't heat flow spontaneously from a colder to a hotter body The second law of thermodynamics says that heat cannot spontaneously flow from a colder to a hotter reservoir but only with the expenditure of mechanical energy. This is taken as a postulate or law in thermodynamics. Is there a deeper reason (probably from statistical mechanics) for why this is so?
How do you prove the second law of thermodynamics from statistical mechanics? How do you prove the  from statistical mechanics? To prove entropy will only increase with time? How to prove? Please guide.
What is an intuitive explanation for how PCA turns from a geometric problem (with distances) to a linear algebra problem (with eigenvectors)? I've read a lot about PCA, including various tutorials and questions (such as , , , and ).   The geometric problem that PCA is trying to optimize is clear to me: PCA tries to find the first principal component by minimizing the reconstruction (projection) error, which simultaneously maximizes the variance of the projected data.    When I first read that, I immediately thought of something like linear regression; maybe you can solve it using gradient descent if needed.  However, then my mind was blown when I read that the optimization problem is solved by using linear algebra and finding eigenvectors and eigenvalues. I simply do not understand how this use of linear algebra comes into play.  So my question is: How can PCA turn from a geometric optimization problem to a linear algebra problem? Can someone provide an intuitive explanation?  I am not looking for an answer like  that says "When you solve the mathematical problem of PCA, it ends up being equivalent to finding the eigenvalues and eigenvectors of the covariance matrix." Please explain why eigenvectors come out to be the principal components and why the eigenvalues come out to be variance of the data projected onto them  I am a software engineer and not a mathematician, by the way.  Note: the figure above was taken and modified from . Can one construct a non-measurable set without Axiom of choice? Is axiom of choice required to show the existence of non-measurable sets? Is there a Lebesgue non-measurable set that can be constructed without axiom of choice?  Related question on MO says it is consistent: SQL Server Escape an Underscore How do I escape the underscore character?  I am writing something like the following where clause and want to be able to find actual entries with _d at the end.  Where Username Like '%_d' Laptop w/o battery US bound flight? Is being a right triangle both necessary and sufficient for the Pythagorean Theorem to hold? I recently encountered a Stack Overflow question (since closed) in which the OP was testing for whether a triangle was right by whether or not it "met" the criteria of the Pythagorean Theorem (i.e. whether or not the square of the hypotenuse is equal to the square of the two sides). The code was to this effect:  public void Test(int a, int b, int c) {    if ((c * c) == ((a * a) + (b * b)) {      System.out.println("This is a right triangle");    }    else {      System.out.println("This is not a right triangle");    } }   (There are obviously some other problems with this code, like the fact that it doesn't validate the input to make sure that the inputs are positive and the fact that it only accepts integers).  The question was asking about something completely different in the code and never directly addressed the test, but I got to thinking: is this a valid test for whether a triangle is a right triangle?  Obviously, the Pythagorean Theorem states that, for all right triangles, $a^2 + b^2 = c^2$ (where $a$ and $b$ are sides and $c$ is the hypotenuse). This will hold for all right triangles, so being a right triangle is a sufficient condition for the Pythagorean Theorem to hold.  Is it also a necessary condition? I.e. if $c^2 = a^2 + b^2$ for some arbitrary triangle, is the triangle necessarily a right triangle? Or are there counterexamples? What happens to the semiconductor in an LED when it is given voltage that it cannot handle? Does iOS have a firewall? Why do I need to type `./` before executing a program in the current directory? While executing a C program, a.out, using the Ubuntu terminal, why do I always need to type ./ before a.out, instead of just writing a.out? Is there solution for this? How to set Oracle's Java as the default Java in Ubuntu? How to determine the maximum number to pass to make -j option? Convergence of the $\sum_{n=1}^{\infty}n^3b_n^3$ where $b_n=\int_{-\pi}^{\pi}f(t)\sin (nt)\ dt$. What books are recommended for learning calculus on my own? I recently graduated with a degree in bachelor of science with a focus interactive and multimedia design. I had to opportunity to take 1 C++ course and 1 HTML course. I was also only required to take one math class, introduction to calculus (and it was hardly calculus, it was more of a math class for designers)  Since graduating I have pursued a career in web development. I'm considering going back for a MS in computer science or MS in web technologies.   I feel my math has suffered dramatically as I have always been behind most others on my math studies (middle school through college). I want to learn calculus as I feel it's an integral part of any CS degree.   Does anyone have any recommendations on any great books that I can read through and teach my self a bit of calculus or other advanced math topics?  Thanks for any advice Windows 10 non-persistent installation (live ISO) We've got a Windows 10 installation running in an environment where it can occasionally lose power and restart. However, this sometimes leads to corruption of files.  I've had a look at Windows "To-Go" but this seems to be persistent. And I've also checked out virtualisation, but that would require us to run a non-persistent Windows 10, in another Windows 10 environment.  Is it possible to install Windows 10 as a "non-persistent live CD"? So that a user will have a seemingly fresh Windows 10, but doesn't require to re-install it at each startup? How to turn hundreds of text URLs in Excel into clickable hyperlinks? I have a  and I connected it to my servo () per these details:   PIC pin 1 to positive pin 8 to negative pin 2 to servo red wire (signal) servo brown wire to negative servo orange to positive (servo has three colors: brown, red and orange wires)   I programmed PIC with this code:      #include &lt;xc.h&gt;     #define _XTAL_FREQ 20000000     #pragma config WDTE=OFF , BOREN=OFF , PWRTE=ON , MCLRE=OFF , FOSC=INTRCIO      void main(){     TRISIO = 0b0000000;     GPIO = 0b000000;         while(1==1){             GPIO = 0b000000;             __delay_ms(1);             GPIO = 0b111111;             __delay_ms(19);         }  }   But servo does not move a bit. What is the problem?  Edit: i found out it rotate servo arm very fast to its end(the point it couldn't rotate any more then it sound like tick tick tick .....)  i changed my code to this(to reverse it back):              GPIO = 0b000000;             __delay_ms(1);             GPIO = 0b111111;             __delay_ms(19);             GPIO = 0b000000;             __delay_ms(2);             GPIO = 0b111111;             __delay_ms(18);   but it behave the same. At the moment the most frustrating part of my brewday is getting the cold wort into the fermenter. Here is my process. I have a 20 litre saucepan that i use as a kettle that does not have a tap. At flame out I add my immerseion chiller that is plumbed into my kitchen sink tap. When I have reached temperature I remove the chiller and whirlpool and leave it to rest to 30 minutes. I am use a siphon to rack directly into the plastic bucket fermenter.  The problem is that the siphon always gets clogged with the hops after i have only transfered about 20% of the wort. I have also tried carefully pouring the wort through a seive, but this sort of depfeats the point of the whirlpool and also the seive got blocked with hops as well. All in all i end up wasting a lot of the wort because i am afraid of getting a lot trub in the fermenter.  What are the best processes to use with my current equpment to get the wort into the fermenter? Group theoretic meaning of natural isomorphisms between certain functors So imagine you have a group $G$ and we consider the set of group homomorphisms from $\mathbb{Z}$ to  $G$ specified by  $\forall g$ $\in G$ $\exists$   $\phi(1)=g$. Each of these homomorphisms is in 1-1 relation with the elements of $G$.  Consider groups for a moment in the context of category theory, viewing the groups as one object categories and thereof group homomorphism as functors. What would a natural isomorphism mean in group theory terms for G?  My thoughts are that the natural transformations equate to maps between homomorphisms and since the homomorphisms are in 1-1 relation with the elements of G, then a natural transformation is a map between group elements, i.e another element of the group. I'm not entirely convinced by this.   Anyway continuing my reasoning a natural isomorphism is a map between the homomorphisms that has an inverse, the thing is in relation to the elements of G this is every element? Furthermore I'm looking to find an equivalence class defined by this natural isomorphism, is this just a collection of the elements that are each others inverses?  Thanks in advance for any assistance. In the case of languages with a C-like syntax, we declare the main() method to return an int or float value (or void).  Is it possible to declare a non-void return type from main() in Java?  If not, then why not?  Does this mean that a Java program doesn't return any value to the OS? Make hyperref take pdfinfo from \title and \author I once read about an option for hyperref that automatically takes the info provided via the standard \author and \title, and puts it into the pdfinfo. Now I want to implement it in my template and can't find it anywhere in the hyperref manual nor anywhere else.  Did I dream about this or can someone help me out here?  (A similar, but more general question: ) Is the Star Wars Universe just one galaxy?