We often learn the structure “Let’s do something”, but why it there an apostrophe-s after let and before the verb?  Why does we need ’s in this structure?   Does ’s means is or does it mean was?
The verb let means  “allow”, “permit”, “not prevent or forbid”, “pass, go or come”  and it's used with an object and the bare infinitive.         Are you going to let me drive or not?     Don't let him off the hook.     Before we let our children surf the Internet, they have to do their homework.     She lets the cats out before leaving.   He let us into the house.      The full form let us can be used similarly        Let us know as soon as possible.      Please, let us help you.   They will never let us forget.      None of the above can be contracted; however, when let us is used for making a suggestion; giving self-encouragement; expressing a consequence or plea,  it is often contracted to let's      Let's go out   Let's have a party   Let's see what happens   Let's stand together in this emergency   Let's not forget those who sacrificed their lives     Questions   I believe that let + us is the only instance where this type of contraction occurs. Is there an explanation as to why verbs such as (i) give + us, (ii) get + us or (iii) take + us were not similarly contracted? When was the apostrophe used to replace the omitted letter "u" in let us?  What is the origin of let's?
Book recommendations My website used to have sitelinks and now it doesn't. It's very possible that it's due to changing the website to a sidebar design instead of having an "interstitial" type landing page which limited the number of choices, but I'm not sure.  Here is how sitelinks might look for a site:    What are some things that I can do to improve my chances of getting sitelinks? I have a service (kado-demo.service) that a normal user (kado) should be able to restart without entering any password.   I created /etc/sudoers.d/user_restart and added  kado ALL=NOPASSWD: /bin/systemctl restart kado-demo.service   Permissions:  -r--r-----   1 root root   585 Mar  5 10:47 user_restart   As user k, I now should be able to execute the given command, but this happens:  /bin/systemctl restart kado-demo.service  ==== AUTHENTICATING FOR org.freedesktop.systemd1.manage-units ===  Authentication is required to restart 'kado-demo.service'.   Multiple identities can be used for authentication:   (...)   Which is the same result as before I added the file.   The /etc/sudoers contains  #includedir /etc/sudoers.d   And I even get the same result when I add content of the user_restart file directly to /etc/sudoers.  I'm running out of ideas... I have lost my passport here in Beograd. Can I travel just with my Czech ID card from Belgrade to Budapest? How do you know when to wean? What signs can a baby show to indicate he/she is ready for weaning?  What does a parent look for to determine the right timing for this for baby, mom and family? Are there any clones/alternatives for running a Stack Exchange style Q&A site? Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$ After reading , the most popular answer use the identity $$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1}.$$  What's the name of this identity? Is it the identity of the  modified.  How can we prove it? I tried by induction, but without success. Can we also prove it algebraically?  Thanks for your help.    EDIT 01 : This identity is known as the  because, on Pascal's triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed. Is this a capacitor start motor and can I start it with a microwave capacitor I salvaged an old motor out of a air conditioner and I want to know if it's a capacitor start motor if I can start it with the capacitor I have and how to start it.  I included pictures of the motor and capacitor btw i have never worked with capacitive start motors so i need details on exactly how to wire it up with out ruining it Distribution of Dark Matter around galaxies It is well-known that measurements of the velocity profile in galaxies are not compatible with Newtonian laws. A way to circumvent the problem is to assume that galaxies are surrounded by a spherical halo of Dark Matter. The density of mass of this Dark Matter can then be computed from the velocity profile with Newton's laws.   My question is the following: the formation of a galaxy involves essentially only the gravitational interaction. According to Newton (and General Relativity too), the trajectory of a particle does not depend on its mass. As a consequence, I do not understand how usual matter and Dark Matter could have followed different trajectories leading to such different mass distributions nowadays. Can somebody explain this (apparent) paradox ? Use grep to find content in files and move them if they match The Frobenius Coin Problem I am asked to prove that:       For integers $n, x,y &gt; 0$, where $x,y$ are relatively prime,        every $n \ge (x-1) (y-1)$ can be expressed as $xa + yb$, with nonnegative integers $a,b \ge0$.   How should I approach this? I have very limited knowledge in number theory. Computing $\lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}$. I have a problem with the calculation of the following limit. \begin{equation} \lim_{n\rightarrow \infty} \frac{1+\sqrt{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n} \end{equation} I do not know where to start! Thank you very much Looking at the following dataset:   Date        Visits   Carts      carts       Orders                       Created   converted    Created 2011-11-11    12277     161        9          36   2011-11-12    11871     93         5          19     2011-11-13    13072     107        8          8      2011-11-14    13594     112        4          34     2011-11-15    12741     129        8          43     2011-11-16    15491     261        16         57  2011-11-17    13418     186        17         42       I've been asked to plot this on a graph, using the Date has the X-Axis and the rest of the data on the Y-Axis. The problem is that the scale of the data is dramatically different. where Visits are in the thousands and Orders Created are in the low tens, the data doesn't plot well on a graph.     I was wondering what a statistician would do in this scenario, I could divide the the Visits by a 1000 and then put in the description (Visits (K)), but then I start to have the same problem with Carts Created, as they are in the hundreds and everything else is in the low tens.  What kind of thing is done in this scenario? Show that the Euler characteristic of a chain complex is equal to the Euler characteristic of its homology 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.) How to assign from a function which returns more than one value? I'm having trouble with \includegraphics I'm using MacTeX Firstly and fore-mostly I am a complete new comer to the world of LaTeX and so please do bear with me. I will try to be as accurate as I can but if you need any more info please let me know.  All I want is to figure out how to import a graphic into LaTeX (MacTeX).  I am using the following in my code:  \documentclass[10pt,a4paper,draft]{report} \usepackage{graphicx}   Then in my document section I'm using   \begin{figure} \includegraphics[width = 50mm]{Name.jpg} \caption{Label} \end{figure}   Once I compile the document, the area for the picture has been allocated but there is no image, I have tried .png and .pdf files also.  I know that this is a very simple question and I hope there is a simple solution but I cannot find anything on the net about this (which may mean I'm being dim).  Also I'm using Latexian but I have tried 4 different editors and it doesn't work in any. Applying graduated renderer in PyQGIS? Just trying to run the following code already given in the pyqgis cookbook to create graduated colors symbology (as in example with 2 classes)  I define the source info for the shapefile but when I run execute the code within my test-plugin, it just adds the layer without classifiying. Just adds as usual, all polylines appear same.  Any solution or working code/method?  link to code in cookbook   from qgis.core import  (QgsVectorLayer, QgsMapLayerRegistry, QgsGraduatedSymbolRendererV2, QgsSymbolV2,QgsRendererRangeV2)  myVectorLayer = QgsVectorLayer('C:/my_test_shape_file.shp', 'test_shp_file', 'ogr')  myTargetField = 'target_field' myRangeList = [] myOpacity = 1  # Make our first symbol and range... myMin = 0.0 myMax = 50.0 myLabel = 'Group 1' myColour = QtGui.QColor('#ffee00') mySymbol1 = QgsSymbolV2.defaultSymbol(myVectorLayer.geometryType()) mySymbol1.setColor(myColour) mySymbol1.setAlpha(myOpacity) myRange1 = QgsRendererRangeV2(myMin, myMax, mySymbol1, myLabel) myRangeList.append(myRange1)  #now make another symbol and range... myMin = 50.1 myMax = 100 myLabel = 'Group 2' myColour = QtGui.QColor('#00eeff') mySymbol2 = QgsSymbolV2.defaultSymbol(myVectorLayer.geometryType()) mySymbol2.setColor(myColour) mySymbol2.setAlpha(myOpacity) myRange2 = QgsRendererRangeV2( myMin, myMax, mySymbol2, myLabel) myRangeList.append(myRange2)  myRenderer = QgsGraduatedSymbolRendererV2('', myRangeList) myRenderer.setMode(QgsGraduatedSymbolRendererV2.EqualInterval) myRenderer.setClassAttribute(myTargetField) myVectorLayer.setRendererV2(myRenderer) QgsMapLayerRegistry.instance().addMapLayer(myVectorLayer)      @Kelly's code works perfectly.   However I noticed that both the code (one I have typed in my first message and yours in the 2nd message) DO NOT work in QGIS v1.7.3 but WORKS in QGIS v1.8.0. I think this was a bug(?) that already resolved in v1.8.0  And one more question;  Do you have any sample code for (natural breaks, Jenks) classification of the "numeric_attribute_field" based on given number of classes (i.e. no of classes will be a variable in the code, say "n", and I will pass that from the plugin GUI i.e. n = spinBox.value() ) Can White Walkers talk? Through all the five seasons of Game of Thrones we had never seen a White Walker talking.  Does this mean they can't talk human language (e.g. they have their own language) or we will never see them talking (i.e. they can't talk at all)? Is it true that a language is regular? Is it true that for every regular language $L \subseteq \{0,1\}^*$, language $\big\{ w^{|w|} \big| w \in L \big\}$ is regular? I don't know how to prove that.Could you give me a hint? Thank you