If you hover just above the event horizon of a black hole, would you see the future of the universe?
Does someone falling into a black hole see the end of the universe?
Right identity and Right inverse implies a group Let $(G, *)$ be a semi-group. Suppose   $ \exists e \in G$ such that $\forall a \in G,\  ae = a$;  $\forall a \in G, \exists a^{-1} \in G$ such that $aa^{-1} = e$.    How can we prove that $(G,*)$ is a group? Coordinate system for accurately calculating areas of polygons that cross UTM Zones? I have multiple polygons, all located offshore around the UK. I am trying to calculate the areas of these polygons in square kilometres, without splitting them by UTM zone.  Is it possible to do this accurately? I have been using World Mercator, but there are big distortions at this scale. If the polygons were limited to the east coast, say, I would use UTM31N. However, as they are all over the UK waters, I am unsure what to use.  Is there a coordinate system that covers this area? British National Grid does not extend far enough out to sea to be of use. Referring to correctly-styled algpseudocode procedure name in document Given an algorithm written in algpseudocode:  \documentclass{article} \usepackage{algorithm} \usepackage{algpseudocode} \begin{document} \begin{algorithm}   \caption{Part 1}   \begin{algorithmic}[1]     \Procedure {FooProc}{$G$}       \State ...     \EndProcedure   \end{algorithmic} \end{algorithm} \end{document}   How can I now refer to FooProc in my document (outside of the algorithm environment) such that it has the same typeset as in inside the algorithm environment? I have been wondering about the differences between model paramters and model hyperparameters, as well as what their categorization means for a learning problem.  Is the distinction between model parameters and hyperparameters only about reducing the complexity of the problem or are there implications for the 'model quality' to be considered?  Let's consider an example from (1, Eq. 2.43), a linear expansion of basis functions   $$f_\theta(x) = \sum_{m=1}^{M} \theta_m h_m(x), $$  where $h_m\:\forall m=1,...,M$ is some function of $x$, $\theta=[\theta_1,...,\theta_M]^\intercal$ is a vector of the parameters of the model and $M$ is the number of basis functions.   In my understanding, we would consider $\theta$ to contain the parameters of the model, while $M$ is a .  To quote the Wikipedia article of hyperparameter that I linked above:     In machine learning, a hyperparameter is a parameter whose value is   set before the learning process begins. By contrast, the values of   other parameters are derived via training.   It makes sense treating the values in $\theta$ as parameters that should be learned from the data, similar to the coefficients in a linear model - it is surely far more reliable  and  specifying them manually would be quite annoying, although it's not impossible. But what about $M$? It surely is easier to pick some integer value for it. But would it really be terrible to determine $M$ based on the data, too? Probably not, as that is what we would do if we search a grid of possible values for the best performing one.   Let's consider the above basis expansion model and choose Gaussian kernel functions to serve as basis functions. As the Kernel itself has parameters too  this makes the problem a bit more complicated.  $$f_\theta(x) = \sum_{m=1}^{M}\theta_m K_{\lambda_m}(\mu_m,x), $$ with $\mu_1,...,\mu_m$ being the location of the kernels and $\lambda_1,...,\lambda_m$ their scales/bandwidths.  Friedman et al. write in (1, p. 36)        "In general we would like the data to dictate them as well. Including   these as parameters changes the regression problem from a   straightforward linear problem to a combinatorially hard nonlinear   problem. In practice, shortcuts such as greedy algorithms or two stage processes are used."   I see that if we were to treat $M$, $\theta_1 ... \theta_M$, $\mu_1 ... \mu_M$ and $\lambda_1 ... \lambda_M$, as model parameters, this problem would be really complex, especially with the choice of $M$ determining the number of the other parameters. But what if we simplify the model definition - let us for example use the same scale/bandwidth for each kernel, that gives us only one parameter $\lambda$. Furthermore, let's say we specify $\mu_1 ... \mu_M$ based on some heuristic, which would mean we treat it as a hyperparameter.  This gives us   $$f_{\theta,\lambda}(x) = \sum_{m=1}^{M}\theta_m K_\lambda(\mu_m,x), $$  Besides increased complexity of the involved optimization problem, does treating $\lambda$ as a parameter rather than a hyperparameter have other negative or undesired effects?      (1) Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. The   elements of statistical learning. Vol. 1. No. 10. New York: Springer   series in statistics, 2001. Why I cannot unpin some directories from file explorer in windows 10 I use both GNOME and KDE SC. In GNOME, I use Docky but I do not use it with KDE. So, I would like Docky to start up only when I login into GNOME not KDE. In KDE, there is an option to make a program autostart only in KDE. But I cannot find any such option in GNOME.    How can I do this?    Thanks in advance. Are Canon EOS and Nikon DSLR lenses interchangeable? Can we use the lenses from, say a Canon EOS 550D kit (Canon EF-S 18-55mm) with a Nikon DSLR D5000 for example) ? Also, is the reverse possible ?  I'm looking at interchangeability between recent DSLR lenses only, and not with older lenses. Also, for "average" lenses, and not ultra wide/ultra zoom.  The reason for this question is that I'm trying to decide between a Nikon D5000 and a Canon EOS 500D. I like the Nikon better, but Canon is more popular here amongst people I can interchange lenses with :) Why, historically, do we multiply matrices as we do? Multiplication of matrices &mdash; taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product &mdash; is not a very intuitive operation: if you were to ask someone how to mutliply two matrices, he probably would not think of that method. Of course, it turns out to be very useful: matrix multiplication is precisely the operation that represents composition of transformations. But it's not intuitive. So my question is where it came from. Who thought of multiplying matrices in that way, and why? (Was it perhaps multiplication of a matrix and a vector first? If so, who thought of multiplying them in that way, and why?) My question is intact no matter whether matrix multiplication was done this way only after it was used as representation of composition of transformations, or whether, on the contrary, matrix multiplication came first. (Again, I'm not asking about the utility of multiplying matrices as we do: this is clear to me. I'm asking a question about history.) Proof that the irrational numbers are uncountable Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set $\mathbb{Q}$ of rational numbers is countable, but how would you show that the irrationals are uncountable? In Bishop's book "Pattern Classification and Machine Learning", it describes a technique for regularization in the context of neural networks. However, I don't understand a paragraph describing that during the training process, the number of degrees of freedom increases along with the model complexity. The relevant quote is the following:     An alternative to regularization as a way of controlling the effective   complexity of a network is the procedure of early stopping. The   training of nonlinear network models corresponds to an iterative   reduction of the error function defined with respect to a set of   training data. For many of the optimization algorithms used for   network training, such as conjugate gradients, the error is a   nonincreasing function of the iteration index. However, the error   measured with respect to independent data, generally called a   validation set, often shows a decrease at first, followed by an increase as the network starts to over-fit. Training can therefore be   stopped at the point of smallest error with respect to the validation   data set, as indicated in Figure 5.12, in order to obtain a network   having good generalization performance. The behaviour of the network   in this case is sometimes explained qualitatively in terms of the   effective number of degrees of freedom in the network, in which this   number starts out small and then to grows during the training process,   corresponding to a steady increase in the effective complexity of the   model.   It also says that the number of parameters grows during the course of training. I was assuming that by "parameters", it refers to the number of weights controlled by the network's hidden units. Maybe I'm wrong because the weights are prevented to increase in magnitude by the regularization process but they don't change in number. Could it be referring to the process of finding a good number of hidden units?  What's a degree of freedom in a neural network? What parameters increase during  training? A PCI Express 3.0 Video Card into a PCI-E 1.0 x16 slot Perhaps I'm trying to push my mobo too hard, but I think I have read that PCI-e 3.0 is back compatible to previous PCI-e specifications. But 3.0 into a 1.0, is it possibly asking too much?  I'm thinking about buying an AMD . My mobo is an Asus p5g41-m le Package babel Error: Unknown option `francais' How to solve following equation: $x^x = C$ as $x, C \in \mathbb{R}_{&gt;0}$. I tried to use exponent and such that: $x\ln x = \ln C$. Quaternions, Rotations and Real numbers I haven't really formally studied Algebra at anywhere near this level, but I was told about the existence of Quaternions a few years ago and I find them really cool. I also like how pure quaternions are analogous to cross products in $R^3$, and that it gives me a way of doing it algebraically rather than relying on hand rules.  Over the course of using them it's my understanding that multiplying an element by another, both imaginary numbers, rotates it in some manner so as to be orthogonal to both, similar to rotations of $\frac{\pi}{2}$ in the space with one real and one imaginary axis when multiplied by $i$, which is why it's useful for cross products as it will rotate it to be orthogonal to a plane spanned by linear combinations of those two vectors [afaik].   What I can't intuit though is why for any pure quaternion multiplication, there's a negative real part [the scalar product] if it's meant to correspond to a rotation within $R^3$ and the space of pure Quaternions is Isomorphic to $R^3$. Similarly I don't understand why $i^2 = j^2 = k^2 = $ A negative real. Calculate sums of inverses of binomial coefficients I made chili and it's too hot! If I cook it longer will the heat calm down? Unmarshal a single element list fails Find all values of a for which $\{v_1,v_2,v_3\}$ form a basis for ${\bf R}^3$ Checking if file is checked out through CSOM Is there a way to check if a file is checked out by another user? I am going trhough a list file by file and would like to avoid the "file is opened by another user" error. Something like this would be nice:  Guid id = new Guid(slideLibrary); // ID of Slide-Library. List list = ctx.Web.Lists.GetById(id); ListItemCollection items = list.GetItems(CamlQuery.CreateAllItemsQuery()); ctx.Load(items); // Loading all the fields System.Security.SecureString pwd = new System.Security.SecureString(); foreach (char c in "pwd".ToCharArray()) pwd.AppendChar(c);  ctx.Credentials = new SharePointOnlineCredentials(login, pwd); ctx.ExecuteQuery();  // Iterating through all the entries. foreach (var item in items) {     String currentID = item["ID"].ToString(); // Getting the ID of the current item.      if (!item.IsCheckedOut()) // This is the part I am looking for.     {        // Do stuff     }     else     {        Console.WriteLine("Item is blocked, skipping it.")     } }   I have yet not been able to find something fitting. What formula should be used to determine "hot" questions? Right now  is fairly broken -- it's a simple descending sort by views. As Joel said , it is &quot;a self-fulfilling prophecy.&quot; But this is not intentional, it's only because we haven't had time to improve it yet! As I sit down to write a better algorithm, I thought you guys might have some insight into what would make a good formula to determine the &quot;hot&quot; questions -- over the last 24 hours, the last week, the last month, and all time. Here are the variables we have to work with:  Votes [aka Score] Views Answers Answer Votes [aka Score] Whether the question has an accepted answer or not Time question was originally asked Time of last activity on question Reputation of asker Reputation(s) of answerers  Joel cited the reddit formula as an example, but I think our site is a little different than just a basic link submission site. Plus, we have more variables and data to work with. I found a  describing a few formulas: Y Combinator Hacker News  (p - 1) / (t + 2)^1.5 p = votes (points) from users t = time since submission in hours  Reddit  t = (time of entry post) - (Dec 8, 2005) x = upvotes - downvotes y = {1 if x &gt; 0, 0 if x = 0, -1 if x &lt; 0) z = {1 if x &lt; 0, otherwise x} log(z) + (y * t)/45000  Thoughts? Ideas? I don't want to overthink this; I just want the &quot;hot&quot; questions to be relevant and useful to us. So the simpler the algorithm the better! Based on my analysis of the above and the comments so far, here's the second version of what I have implemented so far. This might suck. I don't know:  (log(Qviews)*4) + ((Qanswers * Qscore)/5) + sum(Ascores) -------------------------------------------------------- ((QageInHours+1) - ((QageInHours - Qupdated)/2)) ^ 1.5  Note that accepted answers weight not at all in hotness. This is intentional, as I feel accepted answers are a fine social contract, but not a good data point for question or answer quality. This algorithm will heavily favor questions with LOTS of answers, as the sum(Ascores) are now included -- one assumes if there are lots of answers, there will be a lot more voting on the answers, too. Update: Note that this formula is what is used on the  linked from the homepage of each site. It is not the formula used to determine the  list. See .