Showing an orthogonalisation process
Proving the two given vectors are orthogonal
Which is faster/best? SELECT * or SELECT column1, colum2, column3, etc I've heard that SELECT * is generally bad practice to use when writing SQL commands because it is more efficient to SELECT columns you specifically need.  If I need to SELECT every column in a table, should I use   SELECT * FROM TABLE   or   SELECT column1, colum2, column3, etc. FROM TABLE   Does the efficiency really matter in this case? I'd think SELECT * would be more optimal internally if you really need all of the data, but I'm saying this with no real understanding of database.  I'm curious to know what the best practice is in this case.  UPDATE: I probably should specify that the only situation where I would really want to do a SELECT * is when I'm selecting data from one table where I know all columns will always need to be retrieved, even when new columns are added.   Given the responses I've seen however, this still seems like a bad idea and SELECT * should never be used for a lot more technical reasons that I ever though about. Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$ Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$.   My try:  Let $\mathbb Z^n\cong \mathbb Z^m $. To show that $m=n$.  Case 1: Let $m&gt;n$. Now that $\mathbb Z^m$ has $m$ generators whereas $\mathbb Z^n$ has $n$  generators and an isomorphism takes a generator to generator ; that is the contradiction.  Please correct me if I am wrong.  The case  $m&lt;n$ also follows similarly. Why does an object with a glossy shader render in black? How big an RSA key is considered secure today? I think 1024 bit RSA keys were considered secure ~5 years ago, but I assume that's not true anymore. Can 2048 or 4096 keys still be relied upon, or have we gained too much computing power in the meanwhile?  Edit: Lets assume an appropriate padding strategy. Also, I'm asking both about security of signatures and security of data encryption. Object glitching/dissapearing in render due to parenting problems I have completed my animation but I realised when I render it that one of the propellers blades starts to glitch. On some frames it is in shot and some it is out of shot and this is really strange. These 'blades' are both parented to the same 'propeller' and the 'propeller' is parented to the middle ball. I have baked the animation through so this shouldn't be a problem. These propellers are passive rigid bodies. I have tried to figure out this problem and I just can't get to a solution. I've never had it where a rigid body object glitches out like that in the rendered animation, it's almost like it's jumping, which on the render these frames show the object as invisible. The video attached will show.   If anyone has experienced problems like this or knows an answer to my dilemma that would be greatly appreciated, thanks! How to represent kWh usage by year against average temperature? How do rays of light getting focused at a point create an image?  How are the information of object's shape and dimensions expressed as an image when light rays converge?  Is it understandable by solving wave equation for light? What are path and contour integrals? I am very confused on what path and contour integrals actually represent. I tried looking for an answer on Google but the only examples were related to Quantum Mechanics. For instance, real Riemann integrals can be thought as &quot;the area under the curve&quot;. What about path/contour integrals? Also, is there a difference between path and contour integrals, or are they the same thing? I've been thinking of contour integrals as path integrals where the path is a &quot;closed loop&quot;, but I'm not sure if that's the case. It's all so confusing, and not being able to visualise these concepts makes it feel like I'm just memorising formulas and theorems without any reason. I have been thinking about path integrals as integrating over a curve in three dimensions. For example, in an $xyz$ coordinate system (with $y$ pointing “up”), we have a path on the $x$-&quot;$z$&quot; plane and integrate over that path. That is, instead of integrating over and along &quot;$y=0$&quot;, we integrate over a certain path. But I don't know if this is a correct way to think of path integrals. This issue has been frustrating me a lot because I'm now studying theorems that involve path/contour integrals and I just can't understand what they really do or mean. By path integral I mean $$\int_{\gamma\vert_{[a,b]}}f(z)\hspace{0.2em}dz = \int_{a}^{b}f(\gamma(t))\cdot\gamma'(t)\hspace{0.2em}dt$$ where $f:U\subseteq\mathbb{C}\rightarrow\mathbb{C}$ is a function that satisfies all conditions for complex integration and $\gamma:[a,b]\rightarrow U\subseteq\mathbb{C}$ is at least a piece-wise regular path. Any help would be appreciated. I have a dictionary of values read from two fields in a database: a string field and a numeric field. The string field is unique, so that is the key of the dictionary.  I can sort on the keys, but how can I sort based on the values?  Note: I have read Stack Overflow question here  and probably could change my code to have a list of dictionaries, but since I do not really need a list of dictionaries I wanted to know if there is a simpler solution to sort either in ascending or descending order. Is any divergence-free curl-free vector field necessarily constant? Creating an index of quality from multiple variables to enable rank ordering Infinite Series $‎\sum\limits_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$ ‎If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then‎, $$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)‎.$$ ‎Since if we set $C:=\sum_{m=2}^{\infty}|a_m|&lt;\infty$‎, ‎then‎ $$\sum_{n=1}^{\infty}\sum_{m=2}^{\infty}|a_m\frac{1}{n^m}|\leq\sum_{n=1}^{\infty}\sum_{m=2}^{\infty}|a_m|\frac{1}{n^2}\leq C\sum_{n=1}^{\infty}\frac{1}{n^2}&lt;\infty‎$$ and ‎by Cauchy's double series theorem‎, ‎we can switch the order of summation‎: $$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=1}^{\infty}\sum_{m=2}^{\infty}a_m\frac{1}{n^m}=\sum_{m=2}^{\infty}a_m\sum_{n=1}^{\infty}\frac{1}{n^m}=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)‎.$$ This shows that $‎\sum_{n=2}^{\infty}\frac{\zeta(n)}{k^n}=\sum_{n=1}^{\infty}\frac{1}{kn(kn-1)}$.  My Questions:  1) It's obvious that $\sum_{n=1}^{\infty}\frac{1}{2n(2n-1)}=\log(2)$, but how can I evaluate $\sum_{n=1}^{\infty}\frac{1}{3n(3n-1)}$?  2) Is there another method to evaluate $\sum_{n=2}^{\infty}\frac{\zeta(n)}{k^n}$? How do you define an arbitrary sort sequence for views? How to reduce the size of VMware disk My Ubuntu image VMware disk has grown to 5GB physical file size. It's a dynamically growing disk with the maximum capacity of 8GB. I cleaned some stuff inside Linux distro and now inside it occupies 3GB. Obviously the physical file size didn't shrink back automatically. Is there a way (if possible free of charge) to shrink this disk back to ~3GB?  I need this to fit the image on the DVD. How to select a clustering method? How to validate a cluster solution (to warrant the method choice)? One of the biggest issue with cluster analysis is that we may happen to have to derive different conclusion when base on different clustering methods used  (including different linkage methods in hierarchical clustering).  I would like to know your opinion on this - which method will you select, and how. One might say "the best method of clustering is which gives you the right answer"; but I may question in response that cluster analysis is supposed to be an unsupervised technique - so how do I know which method or linkage is the right answer?  In general: is a clustering alone robust enough to rely on? Or we need a second method and get a shared result to be based on both?  My question is not only about possible ways to validate / evaluate clustering performance, but is broader - on what basis do we select/prefer one clustering method/algorithm over another one. Also, is there common warnings that we should look around when we are selecting a method to cluster our data?  I know that it is very general question and very difficult to answer. I only would like to know if you have any comment or any advise or any suggestion for me to learn more about this. Bold name in bib when having a prefix in last name What is serial voting and how does it affect me? I have installed Ubuntu on my laptop and now I want to remove it and the Grub Loader. How I can do it without the Windows installation disk (I cannot use the Recovery Console). I see that there a number of questions (, , , , ) on best practices, none of them answer the issue I am going to talk about.  We all need to prepare large documents (or slides) from time-to-time. And once you have reached a certain number of pages, you find that compiling the whole document takes a substantial time, which you need to do to check that the part you have written is error free, specially if it contains a number of images.   To circumvent the situation, I have used the following practices over years to keep compile time at minimum.   Divide the document in to many chapters: The chapters are put in separate files. Only the one I am working one is uncommented.  \documentclass{book}  \begin{document}  %\input {chapterone.tex} \input {chaptertwo.tex} %\input {chapterthree.tex} %\input {chapterfour.tex}  \end{document}    The above technique can also be used for \include.   Keep a preamble replica of the main file in a temporary file (build file): I use this technique when I am preparing a presentation with many slides, say a hundred. Prepare a main file, copy it to another build file. Delete everything between  \begin{document} and \end{document} in the build file. Prepare a slide (or a group of slides) in the build file. Cut and paste the contents between \begin{document} and \end{document} in the build file to relevant location in the main file only when satisfied with the output (image locations, overfull boxes etc.)     We can hear your inputs about the best practices you practice when a you are preparing a document or presentation from scratch.    Now, I have failed to find a good answer in the case when you have finished the whole document or presentation and now you need tweaking the small things, may be edit a text here, add an image there. What will be your best practice?  Perhaps, again keeping only one chapter uncommented will be usable here. But what about large presentations? May be adding a \end{document} just after the point-of-change will be beneficial, but only if you can put that before midpoint of the file.  Commenting a part of the file will also work, but that is error-prone, at least for me. Although the merits of stepwise model selection has been discussed previously, it is becoming unclear to me what exactly is "stepwise model selection" or "stepwise regression". I thought I understood it, but not so sure anymore.  My understanding is that these two terms are synonymous (at least in a regression context), and that they refer to the selection of the best set of predictor variables in an "optimal" or "best" model, given the data. (You can find the Wikipedia page , and another potentially useful overview .)    Based on several previous threads (for example here: ), it appears that stepwise model selection is considered a cardinal sin. And yet, it seems to be used all the time, including by what seem to be well respected statisticians. Or am I mixing up the terminology?  My main questions are:   By "stepwise model selection" or "stepwise regression", do we mean: A) doing sequential hypothesis testing such as likelihood ratio tests or looking at p-values? (There is a related post here: ) Is this what is meant by it and why it is bad? Or B) do we also consider selection based on AIC (or similar information criterion) to be equally bad? From the answer at , it appears that this too is criticized. On the other hand, Whittingham et al. (2006; )1 seems to suggest that variable selection based on information-theoretic (IT) approach is different from stepwise selection (and seems to be a valid approach)...?  And this is the source of all my confusion.  To follow up, if AIC based selection does fall under "stepwise" and is considered inappropriate, then here are additional questions: If this approach is wrong, why is it taught in textbooks, university courses, etc.? Is all that plain wrong? What are good alternatives for selecting which variables should remain in the model? I have come across recommendations to use cross-validation and training-test datasets, and LASSO.   I think everyone can agree that indiscriminately throwing all possible variables into a model and then doing stepwise selection is problematic. Of course, some sane judgement should guide what goes in initially. But what if we already start with a limited number of possible predictor variables based on some (say biological) knowledge, and all these predictors may well be explaining our response? Would this approach of model selection still be flawed? I also acknowledge that selection of the "best" model might not be appropriate if AIC values among different models are very similar (and multi-model inference may be applied in such cases). But is the underlying issue of using AIC-based stepwise selection still problematic?  If we are looking to see which variables seem to explain the response and in what way, why is  this approach wrong, since we know "all models are wrong, but some are useful"?   1. Whittingham, M.J., Stephens, P.A., Bradbury, R.B., &amp; Freckleton, R.P. (2006). Why do we still use stepwise modelling in ecology and behaviour? Journal of Animal Ecology, 75, pp. 1182–1189.