Difference between fermi energy level in metals and semiconductors It is not exactly a question its a kind of confusion ... Here my confusion starts... what is meant by Fermi energy level?? The only definition that i found was the energy level at which the probability of finding of electron is 50%..  First question is whether this definition applicable to all the materials or only for metals or only for semiconductors??  But in semi conductors the Fermi energy level is in mid of the valency band and the conduction band...   So my second question was in semi conductors whether the Fermi energy level is the maximum energy that the electron in that material is having or it is a particular energy that 50% of the electrons are having the energy more than that level ??  sorry if i am messing up the topic i am keeping all the confusions that are having in my mind in to words.  Last question was in if electron is in the highest energy level of the conduction band what does it mean ?? electron is having the energy only equal to the energy level????
What's the difference between Fermi Energy and Fermi Level? I'm a bit confused about the difference between these two concepts. According to Wikipedia the  and  are closely related concepts. From my understanding, the Fermi energy is the highest occupied energy level of a system in absolute zero? Is that correct? Then what's the difference between Fermi energy and Fermi level?
Does the sign of scores or of loadings in PCA or FA have a meaning? May I reverse the sign? I performed principal component analysis (PCA) with R using two different functions (prcomp and princomp) and observed that the PCA scores differed in sign. How can it be?  Consider this:  set.seed(999) prcomp(data.frame(1:10,rnorm(10)))$x              PC1        PC2  [1,] -4.508620 -0.2567655  [2,] -3.373772 -1.1369417  [3,] -2.679669  1.0903445  [4,] -1.615837  0.7108631  [5,] -0.548879  0.3093389  [6,]  0.481756  0.1639112  [7,]  1.656178 -0.9952875  [8,]  2.560345 -0.2490548  [9,]  3.508442  0.1874520 [10,]  4.520055  0.1761397  set.seed(999) princomp(data.frame(1:10,rnorm(10)))$scores          Comp.1     Comp.2  [1,]  4.508620  0.2567655  [2,]  3.373772  1.1369417  [3,]  2.679669 -1.0903445  [4,]  1.615837 -0.7108631  [5,]  0.548879 -0.3093389  [6,] -0.481756 -0.1639112  [7,] -1.656178  0.9952875  [8,] -2.560345  0.2490548  [9,] -3.508442 -0.1874520 [10,] -4.520055 -0.1761397   Why do the signs (+/-) differ for the two analyses? If I was then using principal components PC1 and PC2 as predictors in a regression, i.e. lm(y ~ PC1 + PC2), this would completely change my understanding of the effect of the two variables on y depending on which method I used! How could I then say that PC1 has e.g. a positive effect on y and PC2 has e.g. a negative effect on y?    In addition: If the sign of PCA components is meaningless, is this true for factor analysis (FA) as well? Is it acceptable to flip (reverse) the sign of individual PCA/FA component scores (or of loadings, as a column of loading matrix)? Ackermann Function primitive recursive I am reading the wikipedia page on ackermann's function,   And I am having trouble understanding WHY ackermann's function is an example of a function which is not primitive recursive.  I understand that the ackermann function terminates, and thus is a total function. So why is it not primitive recursive? The only information that i can find on the wikipedia page is      [Ackermann's function] grows faster than any primitive recursive function and is therefore   not primitive recursive   Which isn't a good example of why it is not primitive recursive.  Could anyone explain to me how exactly ackermann's function is NOT primitive recursive please? In Obi-Wan's Order 66 warning transmission why didn't he tell the surviving Jedi about the Sith, including Palpatine's true identity? If you've seen the premiere episode of Rebels, you will have seen Obi-Wan's warning message that he broadcast to Jedi throughout the galaxy in RotS. In this message he speaks cryptically &amp; warns of a "dark shadow" falling &amp; to avoid the temple &amp; await a "new hope".   Wouldn't it have been useful to inform the Jedi that their mortal enemy the Sith had returned? And specifically that Palpatine himself was a Sith? You'd assume that the majority of the Jedi who were not in the loop of the Council would be unaware of the existence of the Sith. Yoda mentions a few times that they should keep it under wraps. I think now would've been the time to reveal the truth. The more info the surviving Jedi have the better. Who knows if the likes of Kanan knows anything about the Sith's return! I find Obi-Wan's omission of this information bizarre. Is there a tool to encrypt a file or directory? What's the most popular way to encrypt individual files or folders? Did Stephen Baxter or Arthur C. Clarke's estate ever comment on continuing "A Time Odyssey"? 2007 book "Firstborn", the third and last book of "" trilogy by Stephen Baxter and Arthur C. Clarke - an "orthoquel" to "2001: A Space Odyssey" - ends with a cliffhanger which obviously implies continuation:      Myra said, “My God.Charlie? ”  Bisesa stared. “Who is ‘we’?”  “We call ourselves the Lastborn. We are at war. We are losing.” She held out her hands. “Please. Come with me now.”  Bisesa and Myra, still hugging each other, reached out their free hands. Their fingertips touched Charlie’s.  A clash of cymbals.   However, sadly, Sir Arthur died in 2008, with no further books forthcoming.  Did Stephen Baxter  or Arthur C. Clarke's estate ever comment on continuing "A Time Odyssey" after 2008?  Wikipedia has no info:     The 2008 Gollancz edition of the most recent book describes itself on the cover as the "Conclusion" of the series, but the plot is not concluded in the book. The immediate threat is averted but the enemy is not defeated, suggesting that further novels in the series were intended. With the death of Clarke, one of the two authors, it is not clear whether these will be written. How do I secure my house against intruders? What are some suggestions for securing your house from an intruder? Right now we have a GE Simon XT alarm system, but it doesn't cover breaking a window. We put some PVC pipes in the tracks of the lower floor windows as well. What other suggestions do you have?  EDIT: I also would like to know what structural changes I should be looking at. A simple one is to ensure that the deadbolt goes into a stud (they don't always!). Any other suggestions like this would be great. I want to do:  cat update_via_sed.sh | sed 's/old_name/new_name/' &gt; new_update_via_sed.sh   in my program.  But I want to use variables, e.g.  old_run='old_name_952' new_run='old_name_953'   I have tried using them but the substitution doesn't happen (no error). I have tried:  cat update_via_sed.sh | sed 's/old_run/new_run/' cat update_via_sed.sh | sed 's/$old_run/$new_run/' cat update_via_sed.sh | sed 's/${old_run}/${new_run}/' What happened to Startups SE? I really enjoyed that site. I've only posted on it once, but I enjoyed seeing other people's posts and appreciated reading about entrepreneurial dilemmas. This is Exercise 3.1.A. in      Suppose that $\pi: X\rightarrow Y$ is a continuous map of differentiable manifolds. Show that $\pi$ is differentiable if differentialble functions pull back to differentiable functions, i.e., if pullback by $\pi$ gives mpa $\mathcal{O}_Y\rightarrow \pi_*\mathcal{O}_X$.   Let $f: V\subseteq Y\rightarrow \mathbb{R}$ be a differentiable function on an open subset of $Y$. Let $(U,\phi)$ be a chart of $X$, $(V,\psi)$ be a chart of $Y$. Then $f\circ\pi(\phi^{-1})$ is differentiable and $f\circ\psi^{-1}$ is differentiable. We need to show that $\psi\circ\pi\circ\phi^{-1}$ is differentiable. I don't know how to prove this.   Can we say that since $(U,\psi\circ\pi)$ constructs a chart of $X$, by the compatibility, $\psi\circ\pi\circ\phi^{-1}$ is differentiable? This seems true but this does not use the fact that $f$ is differentiable.  Sorry I have no background of Differentiable Manifolds. Any help would be appreciated.  I also saw this post: , but I don't understand how $\gamma$ is smooth in the answer. $\gamma=\beta\circ\rho$, where $\rho$ is defined to be smooth, but $\beta$ is not. Every time I try to create a table using MySQL Workbench 6.3 CE it adds "broken" comments at the end of the script.  Example:  I am trying to create a table called "user". This is what the table should look like:   However when I click the apply button, the script generates incorrectly:  CREATE TABLE `user` (   `userid` INT NOT NULL AUTO_INCREMENT COMMENT '',   `record_status` INT NOT NULL DEFAULT 1 COMMENT '',   `createdby` VARCHAR(75) NOT NULL COMMENT '',   `createddate` DATETIME NOT NULL COMMENT '',   `updatedby` VARCHAR(75) NULL COMMENT '',   `updateddate` DATETIME NULL COMMENT '',   `username` INT NOT NULL COMMENT '',   `password` VARCHAR(75) NULL COMMENT '',   `firstname` VARCHAR(75) NULL COMMENT '',   `lastname` VARCHAR(75) NULL COMMENT '',   `email` VARCHAR(75) NULL COMMENT '',   `facebook_userid` VARCHAR(75) NULL COMMENT '',   `email_verified` VARCHAR(75) NULL COMMENT '',   PRIMARY KEY (`userid`, `username`)  COMMENT '');   The comment in this line should not be there, so the code should be:   PRIMARY KEY (`userid`, `username`));   and the code executes when I paste it into an editor and remove this comment. However I create massive databases daily. (Each with +- 15 tables). I can't keep doing this for each table as it is time consuming.  How can I disable this feature that adds the comments to the script? I don't need them in my databases Let $\mathbf{FinVec}$ denote the category of finite-dimensional real vector spaces and linear maps, let $F\colon\mathbf{FinVec}\rightarrow\mathbf{FinVec}$ be a functor. For each pair of objects $V,W$, the Hom-set $\operatorname{Hom}(V,W)$ is itself a finite-dimensional vector space and we equip it with the unique structure of a topological space/of a smooth manifold that is compatible with its vector space structure. The composition maps are then continuous/smooth. In this way, we view $\mathbf{FinVec}$ as enriched over the category $\mathbf{Top}$/$\mathbf{Diff}$ respectively. It then makes sense to ask whether the functor $F$ is enriched over $\mathbf{Top}$/$\mathbf{Diff}$. That is just to ask whether the induced map $F\colon\operatorname{Hom}(V,W)\rightarrow\operatorname{Hom}(FV,FW)$ is continuous/smooth for all $V,W$. (Irrelevant to the rest of the question, but the reason this is a useful notion is that it is the natural hypothesis assuring that the functor induces another functor on the category of topological/smooth vector bundles, applying the original functor fiberwise.) In the case $V=W$, by functoriality, we can restrict $F$ to a group homomorphism $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$. If the functor is continuous/smooth, so is this map. Note that $\operatorname{GL}(V)$ is a Lie group. It is a general result that a measurable group homomorphism between Lie groups is automatically smooth. This begs the following questions:  What's an argument for the existence of functors that aren't continuous? Can we construct one? Given the above, it either happens that the restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$ are continuous and continuity fails elsewhere, or the induced maps will be non-measurable, so that the construction will necessarily have to involve choice to some extent. Are there continuous functors that aren't smooth? Given the above, a continuous functor will have smooth restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$, but smoothness may fail elsewhere. Today I saw a sentence:  She surely can do it.  What does that mean? How does it differ from:  She certainly can do it. Surely she can do it.  This slight difference in meaning of those sentences I am not able to notice. Could you please help me? Thanks in advance! What is the intuition behind the unit normal vector being the derivative of the unit tangent vector? I've seen the math, but... It just doesn't make sense to me. How is the slope going to point perpendicular to the vector that is clearly a straight line not going in that direction? The Aleph numbers and infinity in calculus. I'm currently working through the book Heisenberg's Quantum Mechanics (Razavy, 2010), and am reading the chapter on classical mechanics. I'm interested in part of their derivative of a generalized Lorentz force via a velocity-dependent potential.   I understand the generalized force   $$F_i = -\frac{\partial V}{\partial x_i} + \frac{d}{dt}\left(\frac{\partial V}{\partial v_i}\right)$$  that they derive from a Lagrangian of the form $L = \frac{1}{2}m|\vec v|^2 - V(\vec r,\vec v,t)$.  However, in the next (critical) step of the derivation, the author cites a theorem from Helmholtz saying      ...according to Helmholtz, for the existence of the Lagrangian, such a generalized force can be at most a linear function of acceleration, and it must satisfy the Helmholtz identities.   The three Helmholtz identities are then listed as:  $$\frac{\partial F_i}{\partial \dot{v_j}} = \frac{\partial F_j}{\partial \dot{v_i}},$$  $$\frac{\partial F_i}{\partial v_j} +  \frac{\partial F_j}{\partial v_i} = \frac{d}{dt}\left(\frac{\partial F_i}{\partial \dot{v_j}} +  \frac{\partial F_j}{\partial \dot{v_i}}\right),$$  $$\frac{\partial F_i}{\partial x_j} -  \frac{\partial F_j}{\partial x_i} = \frac{1}{2}\frac{d}{dt}\left(\frac{\partial F_i}{\partial v_j} - \frac{\partial F_j}{\partial v_i}\right).$$  I'm trying to understand where this theorem comes from. Razavy cited a 1887 paper by Helmholtz. I was able to find , but it is in German, so I could not verify whether or not it proved the theorem. Additionally, I could not find it in any recent literature. I searched online and in Goldstein's Classical Mechanics.   The only similar concept that I can find is in the  where we have three equations known as Helmholtz conditions. Are these two concepts one in the same? If so, how should I interpret the function $\Phi$ and the matrix $g_{ij}$ that appear in the Helmholtz conditions I found online?   If the cited theorem from Razavy does not relate from the inverse Lagrangian problem, could I have some help finding the right direction? . It is a mathematical expression/sequence/set of numbers with the arrow exactly under any one of the numbers (with or without n= 0). How to do it in latex? Any of the up arrow types (single line or double line) shown in the figure is okay. Number after list name I'm sure this is a really simple setting I'm having trouble finding. If I delete a web part then add it again, there is a number in brackets after the title that goes up by 1 each time. How do I get rid of this? How to show $\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$? Let $a\geq 0$ and $ b\geq 0$. Prove that $\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$. [Hint: Use the identity $(a^n -b^n)=(a-b)(\sum_{i=0}^{n-1}a^ib^{n-1-i})$]    I need some help! I cannot do it even with the hint... :( How to ignore the first line of data when processing CSV data? Suppose I have a paper with the following text:     Roses are red, violets are blue, sugar is sweet, and so are you [3],[4].   As one can see, the author got the information from two other references. If I want to add this information in my thesis, do I also have to reference [3] and [4] or can I just reference this particular paper, I got the information from?