Replacing a expression in another expression with a Variable
Can I simplify an expression into form which uses my own definitions?
3 Statements of axiom of choice are equivalent This winter I started to study much harder the set theory and especially the axiom of choice. Unfortunately, I have problems with solving the next exercise:  Prove that the 3 statements of the axiom of choice are equivalent :  1) For any non-empty collection $X$ of pairwise disjoint non-empty sets, there exists a choice set.  2) For any non-empty collection $X$ there is a choice function.  3) For any non-empty set $X$, there exists a function $f:P(X)\setminus\{\varnothing\}\to X$ so that for any non-empty set $A\subseteq X$, $f(A) \in A$. What is tail recursion? Whilst starting to learn lisp, I've come across the term tail-recursive. What does it mean exactly? Footnote in tabular environment Just as background, I should say I am a mathematics grad student who is trying to learn some physics. I've been reading "The Theoretical Minimum" by Susskind and Hrabovsky and on page 134, they introduce  transformations. Here's the first example they use:  Consider a particle moving in the x,y plane under the influence of a potential, $V$, which depends only on the radius, with Lagrangian:   $L=\frac{m}{2}(\dot{x}^2+\dot{y}^2)-V(x^2+y^2)$  This is clearly invariant under rotations:  $x \rightarrow x\cos \theta + y\sin \theta$  $y \rightarrow -x\sin \theta + y\cos \theta$  All well and good. Now they say "consider what happens ... when the angle $\theta$ is replaced by an infinitesimal angle $\delta$." Already I could say "What the heck is $\delta$ really?", but I'm willing to play along with my intuition. Since $\delta$ is infinitesimal, we work to first order and say   $\cos \delta=1$ and $\sin \delta= \delta$.  Plugging this into our rotation formulas above, we obtain:  $x \rightarrow x+y\delta$  $y \rightarrow y-x\delta$  By differentiating, we see that:  $\dot{x} \rightarrow \dot{x}+\dot{y}\delta$  $\dot{y} \rightarrow \dot{y}-\dot{x}\delta$  Plugging these into the Lagrangian and ignoring terms higher than first order, we see that the Lagrangian is invariant under this transformation.  My main problem with all of this is that I don't understand what the physical nature of an infinitesimal transformation actually is. All I got from the above was that if you do this formal calculation by following rules like "only work to first order in $\delta$," then the Lagrangian is invariant. This is in contrast to the case where we have an actual transformation, like a rotation, where there is no question about what is going on physically.  I would also like to know how all of this relates to rigorous mathematics. In mathematics, I can't recall ever using infinitesimals in an argument or calculation, so it would be useful if there was some way of formulating the above in terms of limits/ derivatives/ differential forms (for example). I sense a connection to Lie Algebras, as the infinitesimal version of the rotation is $(I+A)$ where $I$ is the identity matrix and $A$ is an element of the Lie Algebra of $SO(2)$.   Here are some questions whose answers I believe may be useful to me (feel free to answer some or all):  -What is an infinitesimal quantity like $\delta$ to the physicist?   -Why do physicists argue using infinitesimals rather than "standard" calculus?  -What is the physical meaning of infinitesimal transformation? How does it relate to Lie Algebras?   -Is there a rigorous theoretical apparatus for justifying the computations shown above?  -What is meant by the Lagrangian being invariant under infinitesimal transformations?  If any of the questions seem too vague, please say so. Thanks in advance for your insights! Is the product of symmetric positive semidefinite matrices positive definite? Why would spacetime curvature cause gravity? Square roots -- positive and negative It is perhaps a bit embarrassing that while doing higher-level math, I have forgot some more fundamental concepts. I would like to ask whether the square root of a number includes both the positive and the negative square roots.  I know that for an equation $x^2 = 9$, the solution is $x = \pm 3$. But if simply given $\sqrt{9}$, does one assume that to mean only the positive root? And when simply talking about the square root of a number in general, would one be referring to both roots or just the positive one, when neither is specified? How to change the starting reference number in Natbib in a given bibunit? I'm using bibunit and Natbib packages with a numerical numbering.   \begin{bibunit}[plainnat]     \nocite{ref1,ref2}     \putbib[mybiblio] \end{bibunit}  \begin{bibunit}[plainnat]     \nocite{ref3,ref4}     \putbib[mybiblio]  \end{bibunit}   How can I set the starting number for ref3 and ref4 to be different from 1 (say 15), without affecting the starting number for ref1 and ref 2? The desired result is the following:   [1] ref 1 [2] ref 2 [15] ref3 [16] ref4 This is a follow-up question to the discussion about the finite axiom of choice .  Suppose we have a countable collection of non-empty sets $\{A_1, A_2, A_3,\cdots\}$ Reasoning as indicated in that discussion, we can prove the existence of a choice function $c_n$ such that $c_n(i)$ belongs to $A_i$ for $i = 1,\cdots,n$  Again reasoning as suggested and using induction, we can prove the existence of a function $C$ defined over $\{1, 2, \cdots\}$ such that $C(n)$ is a choice function defined for $i=1,\cdots,n$ with $C(n)(i)$ an element of A_i. We can require in addition that $C(n)(i)$ and $C(n-1)(i)$ are equal for $i=1,\cdots,n-1$.  Now it would it seem that we can prove the countable axiom of choice by taking the union of $C(1)$, $C(2)$, $\cdots$  The question is as follows: why is that proof wrong? I suspect the answer may have something to do with the axiom of union. Is it now considered acceptable to follow a proper noun ... or any noun ... with a pronoun? E.g.      The President he issued a new executive order. In mathematics, "A or B" includes "A and B". Does "either" mean "A or B but not (A and B)" or does it include the possibility of "A and B"? The context might be mathematics, formal logic or ordinary language. 80's/90's movie: man and woman driving through a red desert planet I am trying to identify a sci-fi movie from the 80's or 90's, with the following parameters:  Setting:    Scenes of a red desert planet   Culture:    Either people roam fairly solo and try to acquire resources to survive, or you live in a weird society where if a stranger is captured he needs to play in a game of fight to the death.  Mobility:      One foil character has a giant battle-bot-looking tank as his form of transportation through the desert. Protagonist has a car that has a plastic half bubble on top to look out from.   Robot:      The protagonist runs around in the beginning of the film with a female robot/android whose face is melted off after it has acid touch its face.(This scene happens next to the protagonist's rover)  Love interest:      Somehow protagonist comes across female protagonist in desert. She smells really bad and is hungry. They sleep outside, and he tells her to sleep away from him. He uses a thermal blanket(the first I ever saw in my life). She wakes up next to him, smelling horrible. He bathes her in some water hole they find(as if water were not scarce and they could just waste water.)   Pupa monsters:      Scene where they are in a building and they are trying to get out. They come across a room that is white, in which multiple giant, white, snot, pupa are hanging securely from the ceiling. Through curiosity, they disturb them, and now they have to get out of that room fast.  Boss Antagonist:      Pervert bionic man, that is actually attached to a giant robotic arm, as only form of movement. He is in black and has a pale bald head as his most human looking part. $X$ and $Y$ are independent rv having pdf $f(t)=\frac{1}{\pi} \frac{1}{1+t^2}$ determine pdf of $Z=\frac{X+Y}{3}$ Identifying pin numbers from PCB Selecting the right crank length What is the best way to determine the correct crank length for my height and more specifically my leg length? A  not only grants me a Sentinel Raven, but also the handy ability to see through its eyes.  Unlike the telepathic connection with a regular familiar, I don't have to use an action to do so.    If I am blinded, but my Sentinel Raven is not (and within telepathic range), can I use my raven's sight to cast spells that require me to see my target? Would this negate any disadvantage resulting from me being blinded? Or would I still suffer disadvantage, for example, because I have to cast from my raven's perspective and not my own point of view? Varying behavior for possible loss of precision Help with a proof. Countable sets. Let $G$ be a non abelian group of order $p^3$, with $p$ prime.  I'm proving that $Z(G)$ (its center) is of order $p$. I already know how to do it by saying that its order can't be $p^3$, nor 1, and if it's $p^2$ then $G/Z(G)$ is cyclic then G is abelian, Abs!  But I've been reading another way of proving it and the author supposes that $|Z(G)|=p^2$. Then, she says: "$\exists\ x$ outside the center of $G$, of order $p$." Could someone give me a hint? Thanks! Can you transfer a Windows 7 XP-mode VHD to a new PC without the need for reactivation? At the moment I am running Vista 32 bit and use VPC 2007 a lot. I have several Windows 2000 Virtual Machines set up with complex software configurations. For me, one of the advantages of working with VPCs is that a Virtual Machine can easily be copied to another PC (like a notebook). Also, when I move to a new PC, I can immediately work with my existing Virtual Machines without the need to reinstall all applications.   I am thinking about buying a new pc with Windows 7 64 bit pro, and using the XP-Mode Virtual Machine that comes with Windows 7. The advantages would be support for more memory (so being able to run more Virtual Machines simultaneously) and having a licenced XP Virtual Machine.  I would like to know if you can transfer a VHD that is based on the XP-mode VHD to another machine without Activation issues. The other machine will of course, also be running Windows 7. I would guess that this would not be a problem.   However, I did read about a guy who had to reinstall his Windows 7 because of a harddisk failure, and could not access his XP-mode VHD anymore because it demanded re-activation.  So can you transfer a Windows 7 XP-mode VHD to a new PC without the need for reactivation?