Evaluate the limit $\lim_\limits{x\to-\infty} (1+\frac{1}{x})^{x^2}$
Computing limit of $(1+1/n)^{n^2}$
Mutable vs immutable objects I'm trying to get my head around mutable vs immutable objects. Using mutable objects gets a lot of bad press (e.g. returning an array of strings from a method) but I'm having trouble understanding what the negative impacts are of this. What are the best practices around using mutable objects? Should you avoid them whenever possible? Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional Suppose that we have a Hilbert Space $H$ and $M$ is a closed subspace of $H$. Let $T\colon H\rightarrow M$ be the orthogonal projection onto $M$.   I have to show that $T$ is compact iff $M$ is finite dimensional.  So if we assume that $M$ is finite dimensional then $\overline{T(B(0,1))}$ is a closed bounded set in a finite dim vector normed space and so it is compact. Which gives that $T$ is compact.  But I am unsure how to prove that if $T$ is compact then $M$ is finite dimensional?  Thanks for any help At the end of the game, how are medals awarded?   As a secondary, what is the relation between bronze silver and gold medals, and how much experience do they reward? I have a table and I am always getting the data by ordering a column. In my case, I  order by datenum. However, the selection is so slow because of the ordering.  Is it possible to order the table after datenum. This column is defined as unique. If  defined as an index, would the selection will be faster? I'm looking for a bijection from the closed interval [0,1] to the real line. I have already thought of $\tan(x-\frac{\pi}{2})$ and $-\cot\pi x$, but these functions aren't defined on 0 and 1.   Does anyone know how to find such a function and/or if it even exists?   Thanks in advance! How to group distinct values from a list using odata rest api Hi I have a list with two fields "Regions" &amp; "Countries" I am using a rest api to retrieve the list of just regions like so: http:mysitename/_api/web/lists/getByTitle('TaxTool')/items?$select=Region  In the results there are lots of duplicates and I was wondering how to return unique values. $S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective? $S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is  a continuous function.  I've already shown that $$T_{\mathrm{max}} := \mathrm{sup}\{ T(x) : x \in S^2\}  \: \: \mathrm{ and }\: \: T_{\mathrm{min}}:= \mathrm{inf}\{ T(x) : x \in S^2\}$$ are reached by $T$ with the argument that $S^2$ is closed and bounded and therefore after the Heine-Borel-Theorem compact, and therefore since T is continuous it assumes its max and min in $S^2$.   Now i have answer the following:     1) Is there a point $x_0 \in S^2$ with $T(x_0)=T(-x_0)$ ?      2) Is there a value $T \in(T_{\mathrm{min}}, T_{\mathrm{max}})$ which is only assumed at exactly one point of $S^2$ ?   I wanted to show that $T$ is injective to get that 1) is false and 2) is true but i didn't get anywhere. Any ideas? Thanks in advance! I have a login called user1. user1 can execute stored procedure [Test].[SP1]  EXEC [Test].[TestSortation] EXEC [TestSortation]   Both of the above commands work.  Now if I give the user1 the server role of sysadmin in addition to public. the later of the 2 statements above fails, and throws this error:     Could not find stored procedure 'TestSortation'.   I don't want to have to change every stored procedure in the database to have the schema name included with EXECute commands manually. Can someone explain to my why this is happening. Also, is there a way to get around this, automatically fix my stored procedures, have the schema default to [Test]? A friend of mine misplaced his passport while staying in Jeju island. Normally I would advise him to just travel to Seoul, get an emergency passport from his embassy and fly back home without issues, but I know that Jeju island has a special immigration regime.   Are there any immigration checks on the way from Jeju island to Seoul? If not, can one travel that route without a passport? How do I redirect a domain to another domain locally? My /etc/hosts file looks like this:  127.0.0.1       localhost 10.20.7.67      testsitealpha.dev othersite.dev   testsitebeta.dev   The first two work. The last one doesn't. Why not? How do I make it work? The reason I am doing this, is because I have a test server (othersite.com), which is on the local network, but it's ip can vary. e.g. 10.20.7.98 one day, 10.20.7.35 another, etc.  So, how do I make my system always resolve testsitebeta.dev to the same ip as othersite.dev? Why is energy in a wave proportional to amplitude squared? Why can't a single ReLU learn a ReLU? As a follow-up to  I simplified even more and tried to train a single ReLU (with random weight) to a single ReLU.  This is the simplest network there is, and yet half the time it fails to converge.  If the initial guess is in the same orientation as the target, it learns quickly and converges to the correct weight of 1:      If the initial guess is "backwards", it gets stuck at a weight of zero and never goes through it to the region of lower loss:        I don't understand why.  Shouldn't gradient descent easily follow the loss curve to the global minima?  Example code:  from tensorflow.keras.models import Sequential from tensorflow.keras.layers import Dense, ReLU from tensorflow import keras import numpy as np import matplotlib.pyplot as plt  batch = 1000   def tests():     while True:         test = np.random.randn(batch)          # Generate ReLU test case         X = test         Y = test.copy()         Y[Y &lt; 0] = 0          yield X, Y   model = Sequential([Dense(1, input_dim=1, activation=None, use_bias=False)]) model.add(ReLU()) model.set_weights([[[-10]]])  model.compile(loss='mean_squared_error', optimizer='sgd')   class LossHistory(keras.callbacks.Callback):     def on_train_begin(self, logs={}):         self.losses = []         self.weights = []         self.n = 0         self.n += 1      def on_epoch_end(self, batch, logs={}):         self.losses.append(logs.get('loss'))         w = model.get_weights()         self.weights.append([x.flatten()[0] for x in w])         self.n += 1   history = LossHistory()  model.fit_generator(tests(), steps_per_epoch=100, epochs=20,                     callbacks=[history])  fig, (ax1, ax2) = plt.subplots(2, 1, True, num='Learning')  ax1.set_title('ReLU learning ReLU') ax1.semilogy(history.losses) ax1.set_ylabel('Loss') ax1.grid(True, which="both") ax1.margins(0, 0.05)  ax2.plot(history.weights) ax2.set_ylabel('Weight') ax2.set_xlabel('Epoch') ax2.grid(True, which="both") ax2.margins(0, 0.05)  plt.tight_layout() plt.show()     Similar things happen if I add bias: 2D loss function is smooth and simple, but if the relu starts upside down, it circles around and gets stuck (red starting points), and doesn't follow the gradient down to the minimum (like it does for blue starting points):    Similar things happen if I add output weight and bias, too.  (It will flip left-to-right, or down-to-up, but not both.) I am needing to auto increment a field based on groups within a feature class. I have 8 plots within a given polygon and I need to assign them an ID from 1-8 for each set of plots within each polygon. The polygon would have its own unique ID number to be used to group the plots.  I assume it would be an alteration of this:  rec=0 def autoIncrement():  global rec  pStart = 1   pInterval = 1   if (rec == 0):    rec = pStart   else:    rec = rec + pInterval   return rec Date and time panel is greyed out, how do I restore it? Date and time is showing in the login screen (unity-greeter) but not when I login to my session.  When I go to date and time settings the panel is all grey :    Any idea how to recover from this ? I often find that I can zoom in to a point, but then cannot zoom any more, even when there is a lot of room to zoom to the meshes on the scene.   This is particularly annoying in the following scenario:  I often land up left clicking on the scene and the 3D cursor shifts there. I use Shift+C to shift the 3D cursor back to the origin. However, this also centers the view on the cursor and shifts the view to show all objects, and sometimes this zooms things way out &ndash; at this point, if I try to zoom in, I can zoom in to a point (using the pinch/zoom on trackpad), but then it stops zooming in.   How do I zoom in after that? Is there anything else I am doing wrong in this case? I remember reading this as a kid and it starts out with:   a boy in his home village where the other kids are mean to him during sports and they heat up their home with floating logs or something.   The kid is being hunted by some reaper character and that reaper finds him at a place called "The Edge" or something similar.   He buys a glove covered in an expensive dust that purifies things and is really rare.  In one book they had to climb down the edge for some reason.  There was another part where people only communicated telepathically and when sneaking into somewhere he only thought of light.  One of the main things is the floating ships and how he is really good at flying them and he assembles a team for a mission or something.  There is like float powder and stuff.  I read this close to 10 years ago. Which is correct: "has died" or "died"? To me, using Present Perfect form means the event can occur again. So, saying      someone has died   may not be grammatically correct.  Also, I noticed (it might be just coincidence):     passed away   is used more often than     has passed away   Is using Present Perfect correct here? My storage was not updated g++ undefined reference with a shared library I have tried using glm::rotate and such. glm::translate works fine, but rotate doesn't work (if I rotate before/after a translation).  Here is my current setup:  glm::vec4 Off = glm::vec4(0, 0, 0, 1); glm::vec4 result = Util::createTransform(offset) * Off;  for (int i = 0; i &lt; Tvertex.size(); i++) {     Tvertex[i] += glm::vec3(result.x, result.y, result.z);     Tcolors[i] = currentColor; }   And here is my Util::createTransform:  glm::mat4 Util::createTransform(glm::vec3 pos) {     glm::mat4 trans = glm::mat4(1.0);     trans = glm::translate(trans, pos);     return trans; }  glm::mat4 Util::createTransform(glm::vec3 pos, glm::vec3 rot) {     glm::mat4 trans = glm::mat4(1.0);     trans = glm::rotate(trans, glm::radians(rot.x), glm::vec3(1, 0, 0));     trans = glm::rotate(trans, glm::radians(rot.y), glm::vec3(0, 1, 0));     trans = glm::rotate(trans, glm::radians(rot.z), glm::vec3(0, 0, 1));     trans = glm::translate(trans, pos);     return trans; }