I was playing with the concept of Interpolation in Python and ended up with this plot:  import numpy as np from scipy.interpolate import interp1d import matplotlib.pyplot as plt  x = np.linspace(0, 10, num=11, endpoint=True) cosx = np.cos(-x**2/9.0) linearInterpolateFunction = interp1d(x, cosx) cubicInterpoloateFunction = interp1d(x, cosx, kind='cubic')  xnew = np.linspace(0, 10, num=41, endpoint=True) plt.plot(x, cosx, 'o', xnew, linearInterpolateFunction(xnew), '-', xnew, cubicInterpoloateFunction(xnew), '--') plt.legend(['Original Data', 'Linear Interpolate', 'Cubic Interpolate'], loc='best')     The way Cubic Interpolation models the function made me think of nonlinear modeling approaches and nonlinear Regression.  Could you please explain to me what are the differences between the two (nonLinear Regression and Cubic Interpolation) and  and if I could use Cubic Interpolation for modeling my data? what are the pros and cons to this? If you could provide me with applicable examples, that'd be great.  by the way, I've seen  to my question before but does its example on why you better use Regression still count even when using Cubic Interpolation instead of Linear Interpolation ?  Thanks...
Explain briefly What is meant by interpolation.How is it related to the concept of regression?  interpolation is art of reading between the lines of a table and in elementary mathematics the term usually denotes the process of computing the intermediate values of a function from a set of given or tabular values of that function.  I can't give the answer of second question.
I am trying to prove the following identities:  a. $$\sum_{k=0}^n(-1)^k{n\choose k}^2 = \bigg\{^{0 \ \text{if k is odd}}_{(-1)^m{2m\choose m} \ \text{if n = 2m}}$$ b. $$\sum^k_{i=0} {n+i \choose i}={n+k+1\choose k}$$  For part $b$ I know I can use induction. It is clearly true when $k=0$ and then assuming for $k-1$ we get $$\sum^k_{i=0}{n+i\choose i}={n+k\choose k-1}+{n+k \choose k}={n+k+1\choose k} \ ...$$ How to customize the ubuntu gnome 3 desktop? This is maybe a stupid question, but how can I choose my own background image on ubuntu gnome 3?? I click on Settings -> Background -> then I click on the current background preview to change it but I see only the defaults images! No + symbol to add or something. On the 'Pictures' tab there is nothing to select, I get an empty white page? I am running ubuntu gnome 13.04 3.9.6-030906-generic.  Thanks! What is the advantage of using sudo apt-get autoremove over a cleaner app? Why does C# disallow readonly local variables? Having a friendly debate with a co-worker about this.  We have some thoughts about this, but wondering what the SO crowd thinks about this? I was held at the Polish borders when I attempted to enter by train. They took my fingerprints and sent me back with a written document. I respected the decision and returned immediately.   The document is in Polish concerning the decision taken, and I don't understand it. I am from Cameroon. My limited understanding from what was explained in English at the time was that the asylum status I requested was denied me. I only realised later that I asked the wrong authorities (border guards instead of immigration officers).  This was exactly one year and three months ago now. Do I still have access to any Schengen country? Which command controls the open file limits? python ImportError: No module named primes I have always wondered why diamonds shine. Can anyone tell me why? Permalinks: custom post type -> custom taxonomy -> post I am having trouble working with WordPress rewrite rules, and could use some help.  I have a custom post type called _shows_.  All shows have a single custom taxonomy category _show-category_. A _show_ will never have more than one _show-category_.  I would like my urls to route in this manner:  www.mysite.com/shows/  =&gt;  archive-shows.php  www.mysite.com/shows/%category%/ =&gt; taxonomy-show-category.php  www.mysite.com/shows/%category%/%postname%/ =&gt; single-shows.php   So as a real world example, lets say we have a _show-category_ "Foo" and a _show_ post titled "Bar" that has "Foo" as it's _show-category_. I would expect my WordPress app to look like this:  www.mysite.com/shows/foo/ =&gt; shows all posts under the foo category www.mysite.com/shows/foo/bar =&gt; shows the indivual post   I try to avoid plugins when possible, but am open to any solution. Crafty solutions to the following limit The following problem came up at dinner, I know some ways to solve it but they are quite ugly and as some wise man said: There is no place in the world for ugly mathematics.  These methods are using l'Hôpital, but that becomes quite hideous very quickly or by using series expansions.  So I'm looking for slick solutions to the following problem:     Compute $\displaystyle \lim_{x \to 0} \frac{\sin(\tan x) - \tan(\sin x)}{\arcsin(\arctan x) - \arctan(\arcsin x)}$.   I'm curious what you guys will make of this. I am using the algorithmic package to typeset an algorithm. I would like to have connecting lines between the beginning and end of a loop or conditional like in the algorithm2e package. Is there an easy way to accomplish this, or am I better off switching to algorithm2e? Prove that $$1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$$   $\bf{My\; Try::}$ Using Binomial expansion of  $$(1-x)^{-n} = 1+nx+\frac{n(n+1)}{2}x^2+\frac{n(n+1)(n+2)}{6}x^3+.......$$  So we get $$nx=\frac{1}{3}$$ and $$\frac{nx(nx+x)}{2}=\frac{1}{3}\cdot \frac{3}{6}$$  We get $$\frac{1}{3}\left(\frac{1}{3}+x\right)=\frac{1}{3}\Rightarrow x=\frac{2}{3}$$  So we get $$n=\frac{1}{2}$$  So our series sum is $$(1-x)^{-n} = \left(1-\frac{2}{3}\right)^{-\frac{1}{2}} = \sqrt{3}$$  Although I know that this is the simplest proof, can we solve it any other way someting like defining $a_{n}$ and then use Telescopic sum.  Thanks. How can wifi penetrate through walls when visible light can't? I did search the question on Physics S.E considering it would be previously asked. I found this  But in this question , the answers are w.r.t to or in comparison with microwaves , their absorption and certain other things. I didn't find a sort of general answer that could be the answer to the question. So the question is - wifi or radio waves  reach us through concrete walls . They also reach us through the ceiling (if some one is using it in the flat above ours ). Even through the air they travel such a lot , bending around corners or doors . Now I would not compare them to microwaves (because I don't want the answer in terms of properties of the material but physics). Visible light which is so much powerful than them can't penetrate black opaque paper leave alone the walls. Same is true for gamma rays (penetration through a very thick wall). So why then radio waves being so very less powerful than light  waves are able to travel through walls? There should be a general concept as to why the radio waves are able to pass through walls but microwaves or light waves cannot ! A linked question is also that sound travels much faster in solids(walls) but is not audible in it though it is in air. After reading @BillN's answer, it would be really helpful if any one could explain it in terms molecular resonance or crystalline structure or electrical conductivity or how does molecular resonance or crystalline structure or electrical conductivity cause this. Fingerprint activation in Dell Vostro 14 in Ubuntu 18.04 I have Dell Vostro 14 with OS Ubuntu 18.04.  I tried   sudo apt install -y fprintd libpam-fprintd   but not able to use fingerprint reader of my laptop. The Ubuntu installer does not allow me to install on NTFS partitions, but certain circumstances requires me to do so. Is it possible? Import error when opening empty notebook Custom Settings and fields I have created the custom setting with the hierarchy type    I have added some fields to the custom settings  phone field is added to the custom settings  custom setting Name:Order_Configuration__c  Phone :12345 - one profile  phone:456798 - another profile   Is it possible to access the different values for different profiles in custom settings using apex trigger code Getting "experimental" true displacement to work This thing came up in a combinatorics course I am taking.  Choose a fixed set of primes $p_1,p_2,\dots,p_k$ and let $A_n$ be number of integers in $\{1,2,\dots,n\}$ which are not divisible by any of the $p_i$'s. $A_n$ is given by $ n - \sum_{1\leq i_1 \leq k} \lfloor \frac{n}{p_{i_1}} \rfloor + \sum_{1\leq i_1 &lt; i_2 &lt; k} \lfloor \frac{n}{p_{i_1}p_{i_2}}\rfloor - \dots $. Now, if $n$ is divisible by each of the $p_i$'s then we have the simpler expression : $A_n = n \prod_{i=1}^{k}(1-\frac{1}{p_i})$(added later : note I am not assuming $n = \prod_{i=1}^{k}p_i$ here, this holds true whenever $n = c \prod_{i=1}^{k} p_i $ for some integer c, since each $\lfloor \frac{n}{p_{i_1} \dots p_{i_j}} \rfloor = \frac{n}{p_{i_1} \dots p_{i_j}}\ )$.  Another student pointed out that even if $n$ does not have $\prod_{i=1}^{k} p_i$ as a factor, the approximation $B_n = n \prod_{i=1}^{k}(1-\frac{1}{p_i})$ to $A_n$ is quite close in some specific cases. It is easy to see that $\lim_{n\to\infty}\frac{A_n - B_n}{n}=0$ as $\lim_{n\to\infty}\frac{1}{n}\lfloor \frac{n}{p_{i_1}p_{i_2}\dots p_{i_j}} \rfloor \to \frac{1}{p_{i_1}p_{i_2} \dots p_{i_j}},$ however that is not strong enough to imply that $A_n - B_n$ will be always close. Is there any way to analyze how well this approximation will perform in general? I am interested in the worst case for small to moderately sized $n$.  Just to give a feel of how close $A_n$ and $B_n$ can get, assuming the set of primes is $\{2,3,7\}$ we have (assuming my program is correct):  n     A(n)     B(n) 17    5    4.86   27    8    7.71   37    11      10.57   107   31      30.57   1111  318     317.43   3001  858     857.43   4007  1145    1144.86   5000  1429    1428.57 . "Hot" tab in is badly skewed and misleading - time to replace it with actual hot questions? Each site got a &quot;hot&quot; tab, where its contents are calculated based on . However take a look on the screenshots below and see for yourself that &quot;hot&quot; got nothing to do with many of the questions listed there. :    :    As you can see, there are questions with negative score, no answers, and even closed. I won't be surprised to see spam in there as well. This can't be any worse. What I suggest, instead of fixing the formula or patching it, simply have the hot tab show the same questions as the Hot Network Questions sidebar. (from the specific site only, of course.) Since the hot questions &quot;pool&quot; (as seen ) is limited to 100 for the whole network, either expand it or have the per-site list show questions with less hotness points - it would still be better than what we have now.