For given $a \in \mathbb{R}$ there exists unique continuous function $f:\mathbb{R} \to \mathbb{R}$ that satiafy $f(x+y)=f(x)f(y)$ for $x,y \in \mathbb{R}$ and $f(1)=a$.   These theorems were discussed in my mathematical-modeling class and were given to us to prove them.
Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$.   $g(0)=1$. If $a=g(1)$,then $a&gt;0$ and $g(r)&gt;a^r$ for all $r$ in $\mathbb Q$.   Show that the function is strictly increasing if $g(1)$ is greater than $1$, constant if $g(1)$ is equal to $1$ or strictly decreasing if $g(1)$ is between zero and one, when $g$ is continuous.
Doesn't Radagast care about trees? In the book The Two Towers, Treebeard refers to Gandalf as "the only wizard that really cares about trees".  But in Fellowship of the Ring, we met the wizard Radagast the Brown, whose special focus is on animals, especially birds.  Of course, animals aren't trees, but it is difficult to imagine a person loving animals without also caring deeply about trees - especially when that person loves birds most of all, and so many birds are so  dependent upon trees for their survival.  More generally, Radagast seems like an all-around nature lover, and this too suggests that he "really cares about trees".  This leads me to imagine several possibilities:   Treebeard doesn't know Radagast (this seems unlikely - they have both been around for thousands of years, so surely they have bumped into each other at some point)   Treebeard forgot about Radagast, possibly because he was annoyed at Saruman and a bit angry at the time    Radagast actually doesn't care about trees    Is there some other alternative I have overlooked, or is one of the options I mentioned correct?  Or does Tolkien never address this apparent problem? The suggested edits count at top is incorrect. Multiple browser refreshes did not clear it. Approving/Rejecting review decreases the count, but as you can see the count is incorrect. Question on the stability of the solar system One of the pertinent questions about many body systems that causes me much wonder is why the solar system is so stable for billions of years. I came across the idea of "resonance" and albeit an useful concept, it hardly explains the long stability of the solar system. Normally an N body problem with inverse square mutual interaction is an example of a chaotic system. Is there any real progress about this stability issue? Is just after your PhD a good time to switch your research area? I am finishing my PhD soon. The area in which I pursued my PhD is very challenging as I was only able to publish couple of papers even after spending many years. I want to switch into other research are due these reasons   to attract more students  to be comfortable to do research (along with teaching, I should be able to do research)   Even though I like the research I am doing right now, I seem to be isolated in my university. If ten students join my university for a PhD, hardly one student takes this area. My research area is not like machine learning or deep learning etc, so researchers of my domain are also not that much interested in my work.   The thing is I can work different research area but: Is this the right time? I am heading to a post-doc and learning a new area will probably take several months. Although my switch is not 180 degree, I am still worried. I am trying to create an spritz app. Everything was working fine, but since yesterday I keep getting this error:     ./spritz: line 176: syntax error: unexpected end of file   I have checked the script file and everything seems perfect. I am confused, I have an if statement at last and it looks correct! Here is the last portion:  #checks if speed is 150 157 if [[ $2 -eq 150 ]]; 158 then 159 starttime=$SECONDS 160      FS=$'\n' 161      for j in `grep --color=always -iP '\b[^aeiou\s]*[aeiou][^aeiou\s]*\K[aeiou]' $1`; 162      do 163            #Reads the text file in the centre of the screen 164            echo "                                                    ___________________" 165            echo "                                                             $j"; 166            echo "                                                    ___________________" 167            echo "                                                                               Speed 150 wpm" 168            sleep  0.9; 169            clear; 170       done 171 endtime=$(($SECONDS - $starttime)) 172            echo "You read $words_read words in $endtime seconds!" 173       exit 8 174 fi   What could cause that error? I'm trying to wrap my head around the math/physics of how illuminance on the film/sensor plane remains equal for a given f-number regardless of the focal length of the lens.  Since a longer lens will have a larger entrance pupil at a given f-stop than a shorter lens at the same f-stop, how can the light falling on the film/sensor be equivalent?  Empirically, I know this to be true.  Say I take an incident meter reading (not reflected/TTL) off my subject, and get f/5.6 at 1/100s, I'll get proper exposure at the point at which i took the meter reading with that setting whether I use a 50mm lens or a 200mm lens (obviously composition will be different).    Wikipedia's  reads:     A 100 mm focal length f/4 lens has an entrance pupil diameter of 25   mm. A 200 mm focal length f/4 lens has an entrance pupil diameter of   50 mm. The 200 mm lens's entrance pupil has four times the area of the   100 mm lens's entrance pupil, and thus collects four times as much   light from each object in the lens's field of view. But compared to   the 100 mm lens, the 200 mm lens projects an image of each object   twice as high and twice as wide, covering four times the area, and so   both lenses produce the same illuminance at the focal plane when   imaging a scene of a given luminance.   I'm not sure I agree with the explanation on wikipedia.  The film/sensor doesn't care how big the image is that falls outside the bounds of the sensor; the film/sensor "sees" what it sees, and that's it. Is there a way to make a true toon shader material in Cycles?  Here's, in order of importance, what would be ideal to find in such a shader:   Must use Cycles Must be an actual shader Must not rely on lighting Must be independent of camera position Must react directly to lighting rather than manually having to change it to emulate it Should be easily made into a node-group Should have the following configurability:   Main color Darkness color Shadow depth (how far the shadow goes from the unlit side)    Below, I'll be listing some already-found methods and their flaws:    Cycles's Toon BDSF:  Source: Blender  Result:    Node setup:    Cons:   Not a clear-cut two-color shading Reacts to diffusion of surrounding objects      Mask &amp; Ramp:  Source:   Result:    Node setup (Compositor):    Cons:   Very grainy Does not support colour textures Not easy to change shadow depth in a nodegroup Not a material, requires compositor Requires use of object indexes     Shadeless Fresnel:  Source:   Result:    Node setup:    Cons:   Uses camera's direction Ignores lighting Not easy to change shadow depth in a nodegroup     Shadeless Normal:  Source:   Result:    Node setup:    Cons:   Ignores lighting Direction of shadow is manually changed Not easy to change shadow depth in a nodegroup   However, this one is easily the best one. It's pretty great, and would work still images very well.    That's all I've got. Does anyone else have any methods that fall under most of the criteria I listed above? Also note that OSL is totally fine, if it works. Query standard input of lightning-input I want to query standard HTML textarea which is wrapped by lightning-textarea. How can I get it?  &lt;lightning-textarea name=&quot;secret_field&quot; label=&quot;Secret&quot; value={secretFieldValue}&gt;&lt;/lightning-textarea&gt; 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) How to add ground to DC circuit? I'm just a dabbler in electronic circuits and I'm really still kind of a noob. I am working on a project called the "Atari Punk Console" (Originally known as the "Stepped Tone Generator") but there are some parts of the wiring schematic that I don't understand, specifically the parts with the ground symbol.  Here is the schematic:   See those two ground symbols? I don't understand how to properly connect the circuit to ground at these points. How can I do this and be sure that I did it properly. I don't understand grounding at all so any pointers or clarification is also welcome. Thanks in advance. Convert char to int in C# I have a char in c#:  char foo = '2';   Now I want to get the 2 into an int.  I find that Convert.ToInt32 returns the actual decimal value of the char and not the number 2.  The following will work:  int bar = Convert.ToInt32(new string(foo, 1));   int.parse only works on strings as well.    Is there no native function in C# to go from a char to int without making it a string?  I know this is trivial but it just seems odd that there's nothing native to directly make the conversion. Central Moments of Symmetric Distributions I am trying to show that the central moment of a symmetric distribution:     $${\bf f}_x{\bf (a+x)} = {\bf f}_x{\bf(a-x)}$$ is zero for odd numbers. So for instance the third central moment $${\bf E[(X-u)^3] = 0}.$$ I started by trying to show that $${\bf E[(X-u)^3] = E[X^3] -3uE[X^2] + 3u^2E[X] - u^3}.$$ I am not sure where to go from here, any suggestions? Is there a better way to go about proving this? Google street view URL Hi I have the address of a property and my application can launch a browser to go to . If a good match is found it will take it directly there. Is there any thing I can append to the url to make it switch to streetview without having the exact coordinates? I dont't want to code any javascript or flash. How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$? This question came from the prelim exam I took last month. I have a proof that seems a bit unwieldy to me (posted as an answer), so I'm opening it up to ask if there are other ways of showing this.     Let $x$ be any positive real number, and define a sequence $\{a_n\}$ by   $$ a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} $$   where $[x]$ is the largest integer less than or equal to $x$. Prove that $\displaystyle{\lim_{n \to \infty} a_n = x/2}$. Reading Excel files from C# Display date/time in user's locale format and time offset I want the server to always serve dates in UTC in the HTML, and have JavaScript on the client site convert it to the user's local timezone.  Bonus if I can output in the user's locale date format. What is serial voting and how does it affect me? I just noticed that I lost a bunch of points from my reputation score, and I used the "reputation" tab on my user profile page to try and track down the cause.  During my investigation, I noticed there was an unusual event of type "reversal". In the normal place of a question title, it says "voting corrected".     What does this mean, and what caused it?  Did I do something wrong? Why did I lose all of that reputation? Is the system punishing me for leaving too many good answers?  Is there any way to earn it back? My Ubuntu is running fsck on every bootup On every bootup it's the same:  /dev/sda1: clean, 908443/38690816 files, 44176803/154733312 blocks   Is it some kind of option Ubuntu uses to ensure filesystem consistency or is there something wrong with my HDD? fsck takes up to 30s while booting and so about triples the time needed otherwise.  Full output (partly in German):  Begin: Loading essential drivers ... done. Begin: Running /scripts/init-premount ... done. Begin: Mounting root file system ... Begin: Running /scripts/local-top ... done. Begin: Running /scripts/local-premount ... done. Begin: Running /scripts/local-bottom ... done. done. Begin: Running /scripts/init-bottom ... done. fsck von util-linux 2.20.1 /dev/sda1: sauber, 908443/38690816 Dateien, 44176803/154733312 Blöcke udevd[623]: unknown key 'SYSFS{idVendor}' in /lib/udev/rules.d/45-libticables.rules:6  udevd[623]: invalid rule '/lib/udev/rules.d/45-libticables.rules:6'   * Starting mDNS/DNS-SD daemon                                                 [ OK ]  * Starting Reload cups, upon starting avahi-daemon to make sure remote queues are populated                                                                   [ OK ]  * Starting configure network device security                                  [ OK ]  * Starting bluetooth daemon                                                   [ OK ]  ####* Starting all other stuff This came up in  , but I can't see good documentation or examples that explain how the feature works. Show that $\frac 1{1+x+y^{-1}}+\frac1{1+y+z^{-1}}+\frac1{1+z+x^{-1}}=1$ if $xyz=1$ If $x.y.z=1$ show that $\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}=1$  My attempt - L.H.S$=\dfrac 1{1+x+y^{-1}}+\dfrac1{1+y+z^{-1}}+\dfrac1{1+z+x^{-1}}$  $=\dfrac y{y+xy+1}+\dfrac z{z+yz+1}+\dfrac x{x+zx+1}$  $=\dfrac{yz}{yz+xyz+z}+\dfrac z{z+yz+1}+\dfrac x{x+zx+1}$  What to do next?