given the function $       f:\mathbb{R}\to\mathbb{R}, f(x)=\left\{\begin{array}{0} 0, &amp; x\leq 0 \\          1, &amp; x&gt;0\end{array}\right. .   $ then $       F:\mathbb{R}\to\mathbb{R}, F(x)=\left\{\begin{array}{0} 0, &amp; x\leq 0 \\          x, &amp; x&gt;0\end{array}\right. .   $ is an antiderivative, since $F'(x) = f$. Some fellow students doubt this result. What do you think: a) is $F$ an antiderivative and b) if not does $f$ has an antiderivative and c) what is $F$? Thanks!!
Let $$ f(x) =    \begin{cases}\begin{align}     1\quad&amp;\text{ if }\; 0\leq x \leq 1 \\     2 \quad&amp;\text{ if }\; 1&lt;x \leq 2 \\   \end{align}\end{cases}$$  Then $\int^2_0 f(x)dx=3$ and an anti derivative of f(x) is $$F(x)=\begin{cases}\begin{align}     x\quad&amp;\text{ if }\; 0\leq x \leq 1 \\     2x \quad&amp;\text{ if }\; 1&lt;x \leq 2 \\   \end{align}\end{cases}$$ But, $F(2)-F(0)=4-0=4 \neq 3.$ Why has this happened?  Find an anti-derivative $G(x)$ of $f(x)$ such that $G(2)-G(0)=3$, the correct answer.  My attempt: $F(2)-F(0)=4 \neq 3$ because we need to break it up into two parts like this: $\int^2_1 2x dx + \int^1_0 x dx = x^2|^2_0 + \frac{1}{2} x^2|^1_0 =4-1+\frac{1}{2}-0=3.5$ but this is not right either. :(  For the "Find an anti-derivative $G(x)$..."I'm completely lost but perhaps if we solve the first part then I'll understand what it's asking for there.  Any help would be appreciated.
Storing hard crusted bread I'm currently baking a batch of sourdough bread. My last batch turned out perfectly with a hard crust and chewy inside. My problem is how to store the bread such that the crust stays hard. If I put it in a plastic bag the crust goes soft in a matter of hours and I'm afraid that the bread will turn dry if I leave it without some kind of protection. Note that I'm not looking for a long term storage solution.  What is the best way to store my bread, such that the crust stays hard and the inside stays chewy? Find out if any files were exported from my MacBook Can “due to” and “because of ” be used interchangeably? Is it fine to use due to in place of because of ? How about the other way around? Are any of these sentences ungrammatical?   He was lost because of the storm. He was lost due to the storm. He lost his way due to the storm. He  lost his way because of the storm. How to fix the MBR for Windows 7? I have installed Ubuntu as dualboot with Windows 7. Now I'm no longer using Ubuntu so i removed the Ubuntu partition and added the unallocated space to my Windows 7 partition. But after I restart my computer it goes directly to the error:   error: unknown filesystem grub rescue &gt;   I don't have the original Windows CD to fix the MBR. Is there a way to fix this error without using the Windows CD? Why doesn't Steam start? When I try to start Steam nothing happens. Reinstall doesn't help.  $ steam Running Steam on ubuntu 13.04 64-bit STEAM_RUNTIME is enabled automatically Installing breakpad exception handler for appid(steam)/version(1378500910_client) Installing breakpad exception handler for appid(steam)/version(1378500910_client) Installing breakpad exception handler for appid(steam)/version(1378500910_client) unlinked 0 orphaned pipes removing stale semaphore last operated on by process 7278 with name 0eBlobRegistryMutex_DC7EBB4AEA41D3C6C74E5EFFD6757CEC removing stale semaphore last operated on by process 7278 with name 0eBlobRegistrySignal_DC7EBB4AEA41D3C6C74E5EFFD6757CEC removing stale semaphore last operated on by process 7278 with name 0emSteamEngineInstance removing stale semaphore last operated on by process 7278 with name 0eSteamEngineLock Gtk-Message: Failed to load module "overlay-scrollbar" Installing breakpad exception handler for appid(steam)/version(1378500910_client) [0913/182910:WARNING:proxy_service.cc(958)] PAC support disabled because there is no system implementation   How to fix it? Are there any subtle differences between "somebody" and "someone", or can they be used completely interchangeably? Similarly, can you imagine a situation in which you would prefer "anybody" to "anyone" or vice versa? Best way to remove file extension from a string? So, I'm using a script I've made to convert videos to the webm format. A certain program calls the script, sending %f which is the full, absolute file name of the video, like this:  converter.sh %f   where %f has two possible extensions, .avi or .mpg.  # /bin/bash ffmpeg -i $1 `dirname $1``basename $1 avi`webm &amp;&amp; rm $1   It currently works perfectly when $1 contains a .avi file, because basename removes the .avi extension. When $1 ends with .mpg instead, the result is .mpgwebm.  How can I modify that script to be able to receive those two possible different formats?    Resuming: If $1 is /somedir/video.avi, the script should do:  ffmpeg -i /somedir/video.avi /somedir/video.webm   And if $1 is /somedir/video.mpg, the script should do:  ffmpeg -i /somedir/video.mpg /somedir/video.webm   I know this might be fool for some people, but I'm kind of new with the bash. Processing time/speed for Mosaic to New Raster I have a question on speeding up “Mosaic to New Raster” or finding an alternative.  After completing a suitability analysis we are trying to define movement corridors.  We used a modified CircuitScape that left 4 folders, each containing 1080 raster tiles across our study area.  We now need combine all 1080 rasters in each folder then combine these 4 combined rasters into a final raster which will be analyzed for paths.  I tried a python script that just wouldn’t work so I went on to Model Builder.  It iterates through the grids in the folder, collects the values (raster names), then mosaics all to a new raster.  It has been running for 5 days now and the results window indicates it is working.  I’m hoping it will conclude in the next day or 2.  But, we still have 3 more folders, each with 1080 rasters, that need to be mosaicked.  So….is there a way to speed up Mosaic to New Raster or is there an alternative I could use to produce the final grid?    The cell size is 270 meters, 540 x 540 columns and rows, 1 band, 32 bit floating. Batch renaming files Intuition for uniform continuity of a function on $\mathbb{R}$ How to visualize a value resulted from math node? How to visualize a value resulted from math node,or just to know it. I tried the way of scaling this value to the scene size and then pick the color and get the value but I want more effective way to do this. Confused AUC ROC score I am working on binary classification problem, I try to evaluate the performance of some classification algorithms (LR,Decission Tree , Random forest ...). I am using  a cross validation technique (to avoid over-fitting) with AUC ROC as scoring function to compare the performance of the algorithms, but I am getting a weird results with Random forest and AdbBoost, I have a perfect AUC_ROC score (i.e. =1) despite the fact that the recall(TPR) and FPR of this algorithms are different from 1 and 0 respectively .  def FPR(y_true, y_pred):     tn, fp, fn, tp = confusion_matrix(y_true, y_pred).ravel()     result = fp / (fp+tn)     return result def FNR(y_true, y_pred):     tn, fp, fn, tp = confusion_matrix(y_true, y_pred).ravel()     result = fn / (tp+fn)     return result FPR_scorer = make_scorer(FPR) FNR_scorer = make_scorer(FNR)     def get_CrossValResults2(model,cv_rst,bestIndx):     best=pd.DataFrame.from_dict(cv_rst).iloc[bestIndx]     roc=&quot;{:.12f}&quot;.format(best['mean_test_roc_auc'])     acc =&quot;{:.0%}&quot;.format(best['mean_test_accuracy'])     prec =&quot;{:.0%}&quot;.format(best['mean_test_precision'])     rec =&quot;{:.0%}&quot;.format( best['mean_test_recall'])     f1 =&quot;{:.0%}&quot;.format(best['mean_test_f1'])     r2=&quot;{:.2f}&quot;.format(best['mean_test_r2'])     g_mean=&quot;{:.2f}&quot;.format(best['mean_test_gmean'])     pr_auc=&quot;{:.8f}&quot;.format(best['mean_test_pr'])     fnr=&quot;{:.0%}&quot;.format(best['mean_test_fnr'])     fpr=&quot;{:.0%}&quot;.format(best['mean_test_fpr'])     rst = pd.DataFrame([[ model, acc,prec,rec,fpr,fnr,f1,roc,pr_auc,g_mean,r2]],columns = ['Model', 'Accuracy', 'Precision', 'Recall','FPR','FNR', 'F1-Score','ROC_auc','PR_auc','gmean','r2'])     return rst        cross_val_rst = pd.DataFrame(columns = ['Model', 'Accuracy', 'Precision', 'Recall','FPR','FNR', 'F1-Score','ROC_auc','PR_auc','gmean','r2'])              scoring = {'accuracy':'accuracy','recall':'recall','precision':'precision','fpr':FPR_scorer,'fnr':FNR_scorer,'f1':'f1' ,'roc_auc':'roc_auc','pr':'average_precision','gmean':Gmean_scorer,'r2':'r2'}     param_grid = {'n_estimators': [200],               'max_depth': [80,90],               'min_samples_leaf': [2,3, 4],             'min_samples_split': [2,5,12],             'criterion': [ 'gini'],              'class_weight' : [class_weights], 'n_jobs' : [-1]}  clf = GridSearchCV(RandomForestClassifier(class_weight=class_weights), param_grid, cv=kfold,scoring=scoring,refit=refit)#Fit the model bestmodel = clf.fit(X,Y)       cross_val_rst = cross_val_rst.append(get_CrossValResults2(model='Random Forrest',bestIndx=bestmodel.best_index_,cv_rst=bestmodel.cv_results_),ignore_index=True) Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$ Can someone give an explanation using the definition of convergence in partial sum to show how the above infinite sum converges to 1?  Thanks List selectors for Objective-C object I have an object, and I want to list all the selectors to which it responds. It feels like this should be perfectly possible, but I'm having trouble finding the APIs. We have a few cases of chat accounts remaining without valid parent account:  and  for example. What should mods do when that happens:  delete the chat account notify the SE team so they can reattach or clean it up leave it as is: it's a normal situation How can one derive Schrödinger equation? Say, for example, you and a group of people were all sailing out in the ocean and something happens...then you say,  "I guess we are all in the same boat"  You are literally in the same boat with everyone else, and your current situation is the same as everyone else.  Now such a phrase can be applied literally and figuratively. I was wondering if there was a word that would mean both literally and figuratively. I know this is a tad bit of an oxymoron, but I was curious to see if there was such an "all encompassing" word for this. One of the mysteries that my numerics class left me with are "stiff systems". Our prof didn't really manage to explain what they are. Now that the class (and the test) are over, I'm still curious.   The most concise definition about "stiff systems" I've found so far is: "There is no universally accepted definition of stiffness". So, if there is none, I don't expect one.   I'd be thankful about information on both of the following points:   From what kind of (real-world) problems do stiff systems arise? What's so "stiff" about them? What's the good thing - or the bad thing - that I'd worry about when I have to numerically solve such a real-world problem? How to divide the money based on the below conditions? I've noticed the following happen a few times:   A user posts a question. The question is closed as a duplicate and the usual "Possible Duplicate" links are added at the top. The user promptly removes the duplicate links, even though the question is still closed as a duplicate. (excessive use of list items =P)   Now, I could understand this behavior if there was some dispute about whether or not the question was a duplicate, and the OP were to remove the links and post reasons why they disagree with the closing. However, I've noticed this happening without any attempt to defend the question (such as ).  In my opinion, no one should be allowed to remove the "Possible Duplicate" links as long as the question is still closed as a duplicate. The links should remain for the benefit of the community, so they can navigate to the duplicate questions and either post answers there or read them over and judge if the closing was actually fair. Once/if the question is reopened the links could be edited out if one so chooses.  I guess this is part discussion, part feature request:  Could the "Possible Duplicate" links remain un-editable for as long as the question is still closed?