Can we prove that the thermodynamic state of a system is completely determined by any two out of the three factors $P,V$ and $T$ ? (Without using statistical mechanics. Only using Thermodynamics) NB: I have not learnt the axiomatic formulation of Thermodynamics.
The state postulate is as follows: The state of a simple compressible system is completely specified by two independent, intensive properties. My first question is whether there is any justification for this postulate or whether it is simply an empirical law? That is, why is it not the case that 3 intensive variables are required to fix the state or even 4? If it is an empirical law with no justification and just happens to be a law our universe abides by, then why is it not one of the laws of thermodynamics as it clearly provides foundational value to the field of thermodynamics? My second question pertains to how we can know whether 2 intensive variables are in fact independant. Say temperature $T$ and specific volume $v$ for example. If we are given a thermodynamic problem, how would we know whether these are independent or not? or perhaps specific internal energy $u$ and pressure $P$. Are there certain combinations that are simply always independent of each other or does independence between intensive variables vary from problem to problem and require a full understanding of the problem at hand? Let us take a phase change for example. From my understanding of phase changes, I know that the boiling temperature is dependent on the pressure and hence I won't be able to use $T$ and $P$ as 2 variables to completely specify the state. But how would I know know what combinations of intensive variables will be independent in this case or any other case for that matter? Does it simply require rote memorization of all independent pairs of intensive variables for all thermodynamic problems or is there a more simple way to figure out what is independent and what isn't? I hope this question makes some sense and any help would be most appreciated!
Solvable Lie algebra with codimension 1 ideal There is an exercise in Humphreys's An Introduction to Lie Algebras and Representation Theory: "Any nilpotent Lie algebra contains a codimension 1 ideal".  The proof I am thinking of is the following.  Suppose the Lie algebra $L$ satisfies $L\neq[L,L]$. Then $\pi\colon L\to L/[L,L]$ is a projection onto an abelian algebra. Choose a codimension 1 subspace $I\subset L/[L,L]$. Then $I$ is an ideal and its preimage $\pi^{-1}(I)$ is necessarily an ideal in $L$, because any subspace containing the derived algebra is an ideal. Moreover, $\pi^{-1}(I)$ has codimension 1. $\square$  The only condition on $L$ is that $L\neq[L,L]$, which holds for solvable Lie algebras as well. Is this proof correct? How come Humphreys didn't set the exercise with "nilpotent" replaced by "solvable"? I need your help with references. I have 1 main reference section, and I want to include a second list of references as an appendix. For the main reference I use:  \newpage  \bibliographystyle{plainnat}  \bibliography{library}   The problem is that I don't know how to include the second list using a different bib file, and then calling the function \nocite{*} to show every sources inside the file.   I have the appendix part workout. I need your help to design the function to call the second bibliography, and that does't interfere with the main reference:   \newpage  \appendix  \section{Other Sources} \label{app:ReferencesB}   In addition, I will prefer that the word References is not shown after Other Sources.  Thank you so much!! When does the placement of a resistor matter? I've been a programmer for years but am very new to electronic engineering. So far I see people saying it doesn't matter where a resistor goes in your LED circuit as long as it is present, but it sounds like that's the nature of the basic LED circuit.  When does the placement of a resistor start to become important? Two logos in beamer after certain frame Example of a ring without Invariant Basis Property Sufficient and necessary conditions to get an infinite fiber $g^{-1}(w)$ Should I load an entire html page with AJAX? Java2D Performance Issues I'm having performance oddities with Java2D. I know of the sun.java2d.opengl VM parameter to enable 3D acceleration for 2D, but even using that has some weird issues.   Here are results of tests I ran:  Drawing a 25x18 map with 32x32 pixel tiles on a JComponent Image 1 = .bmp format, Image 2 = A .png format  Without -Dsun.java2d.opengl=true  120 FPS using .BMP image 1 13 FPS using .PNG image 2  With -Dsun.java2d.opengl=true  12 FPS using .BMP image 1 700 FPS using .PNG image 2  Without acceleration, I'm assuming some kind of transformation is taking place with every drawImage() I do in software, and is pulling down the FPS considerably in the case of .PNG. Why though, with acceleration, would the results switch (and PNG actually performs incredibly faster)?! Craziness!  .BMP Image 1 is translated to an image type of TYPE_INT_RGB. .PNG Image 2 is translated to an image type of TYPE_CUSTOM. In order to get consistent speed with and without opengl acceleration, I have to create a new BufferedImage with an image type of TYPE_INT_ARGB, and draw Image 1 or Image 2 to this new image.   Here are the results running with that:  Without -Dsun.java2d.opengl=true  120 FPS using .BMP image 1 120 FPS using .PNG image 2  With -Dsun.java2d.opengl=true  700 FPS using .BMP image 1 700 FPS using .PNG image 2  My real question is, can I assume that TYPE_INT_ARGB will be the native image type for all systems and platforms? I'm assuming this value could be different. Is there some way for me to get the native value so that I can always create new BufferedImages for maximum performance?   Thanks in advance... How to receive sms from our android app? Optional argument to \item I am typesetting an ordered list where the entries may appear out of order, but should be assumed in order otherwise. I don't want to have to jump through \setcounter hoops to do this. That is, I want something like  \begin{enumerate}   \item First   \item Second   \item[8] Eighth!   \item Ninth!   \item[4] Fourth \end{enumerate}   to give the following Good output, instead of the Bad output:    A minimum working example follows.  \documentclass{article} \usepackage{enumitem} % preferably something compatible with this guy \begin{document}  Good:  \begin{enumerate}   \item[1.] First   \item[2.] Second   \item[8.] Eighth!   \item[9.] Ninth!   \item[4.] Fourth \end{enumerate}  Bad:  \begin{enumerate}   \item First   \item Second   \item[8] Eighth!   \item Ninth!   \item[4] Fourth \end{enumerate}  \end{document}   It should be noted that I don't want to have to specify the period afterwards, in case my list using a different numbering, e.g. Roman. Compatibility with enumitem is preferred, but not required.  What is the correct way of going about this? Is there a way to reach a `protected` member of another object from a derived type? class MyBase {     protected object PropertyOfBase { get; set; } }  class MyType : MyBase {     void MyMethod(MyBase parameter)     {         // I am looking for:         object p = parameter.PropertyOfBase;  // error CS1540: Cannot access protected member 'MyBase.PropertyOfBase' via a qualifier of type 'MyBase'; the qualifier must be of type 'MyType' (or derived from it)     } }   Is there a way to get a protected property of a parameter of a type from an extending type without reflection? Since the extending class knows of the property through its base type, it would make sense if possible. Custom attributes - Yea or nay? I did not know that you could find an answer for that. However, I can only use Excel so far to do it.  How to calculate 3.6! by hand? Rounding integers to nearest multiple of 10 Allow User to input HTML in ASP.NET MVC - ValidateInput or AllowHtml This is solely a reference request. I have heard a few versions of the following theorem:     If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ whenever the expectations are finite, then $X,Y$ are independent.   And there is a similar version for characteristic functions. Could anyone provide me a serious reference which proves one or both of these theorems? How do I call legacy PHP files? I am unable to call PHP files inside the Drupal directory. My task is maintaining staff's old pages on the new platform. With pages written on clear HTML, there is no problem: I place files in a folder I created (domain.org/sites/default/files/share/staff_folder).  In sites/default/files/.htaccess I added DirectoryIndex share and now the link to the domain.org/sites/default/files/share/staff_folder/index.html can be used. When I import pages written in PHP, e.g. calling domain.org/sites/default/files/share/staff2_folder/index.php, I get a 403 error:     Forbidden      You don't have permission to access /sites/default/files/share/staff2_folder/index.php on this server.   While /sites/default/files/share/staff2_folder/index.html still works. Trying to remember the name of a grade-school book about an alien in a cereal box When I was a kid (Grade school, Elementary, can't remember the exact grade, but it was between 1992 and 1999) I read a book about a kid who found an alien in his cereal.  This alien informed him that he was there to defeat an evil threat to Earth, that took the form of the school's most popular student.  The alien came with a gun that did nothing, and the alien in the popular kid disguise had the power to make the kid's fingernails bleed by damaging his nerve endings.  In the end, it was revealed that the evil alien had a huge craving for, and vulnerability towards, pizza flakes, and was defeated by gorging himself on them and then exploding. How to draw commutative diagrams? Can anyone help me in giving commands for commutative diagrams in LaTeX ? It doesn't seem to appear properly in the LaTeX guides that I have. A command for a simple triangular diagram with arrows associated to maps would be good enough, or maybe some appropriate reference. What happens to the Matrix if Neo takes the blue pill? If Neo defies his fate and refuses to be removed from the Matrix (by taking the blue pill) what happens next?   Would another "The One" get selected by The Architect or The Oracle and is there anything to suggest that the current instance of The Matrix actually could have multiple "The One"'s (perhaps one of the 'potentials') in the event that the Chosen One manages to get himself killed or simply refuses to play ball?