I just read the formula to determining the no. of paths of length $m$. It was $\frac{n(n-1)\cdots(n-m)}{2}$ Do we have something for cycles too?I think it should be $\frac{n(n-1)\cdots (n-m+1)}{2}$. Does this make any sense? It is just a basic question so if you think it is not something to response then I humbly request you to not answer but please don't downvote this. As I this is my first course in graph theory! (In complete graphs) * Thanks
Hello everyone interested. Here's a seemingly simple problem.  Suppose $G$ is a complete, simple graph with vertices $\{1,2,..,n\}$, $n\geq k\geq 3$.  How many cycles of length $k$ can we have on $G$?  The statement seems simple but I have trouble putting everything together.  So I say fix a vertex and start forming cycles with length k. But I get nowhere... What is the proper approach to this problem since counting is not easy? Is there a relevant theorem that I am unaware of?  Thanks for any help...
Is there a way to set app-specific volume? How to show $\alpha\int_a^cf(x)dx+\beta \int_c^b f(x)dx =0$ means $f(x)=0$ everywhere $?$ Question :  Let $$f\ :\ [a,b]\rightarrow \mathbb R$$   be  continuous . And  there  exist  constants  $\alpha$  $\beta$  such  that  $\forall c\in [a,b]$  $$\alpha \int_a^c f(x)dx +\beta \int_c^b f(x)dx =0 $$   Prove  that  $f=0$  allover  $[a,b]$.  Attempt :(not much useful )   Since  the  equation  holds  for  all  $c\in [a,b]$   , if  take  $c=a$  then  we  have $$\beta \int_a^b f(x)dx =0 $$  And  taking  $c=b$  we  get $$\alpha \int_a^b f(x)dx=0 $$  See  if  both  $\alpha=0$  and  $\beta=0$  hold  then  there  is  nothing  to  do . If  only  one  of  them  is  $0$   and  the  other  is  non-zero  then  from  the  last  two  equations  we  get $$\int_a^b f(x)dx =0 $$ and  this  does   not  ensure   $f$'s  being  identically  $0$  on  $[a,b]$ (for  example : $f(x)=sin x$  and $a=0$,$b=2\pi $ ) .   So  our  only  possibility  is  that  both $\alpha \neq 0$  and $\beta \neq 0$  must  hold .  Now  I  cannot  proceed  any  further  from  this  point . Please  give  me  some   hints  as  to  what  to  do  next .  Thaanks. Did merging Black Holes in GW150914 give up entropy and information to the gravitational waves, since they lost 3 solar masses? Since the final Black Hole (BH) had 3 solar masses less of mass than the original binary BH, it seems the 2 BHs lost mass, and with it event surface area, entropy, and information. If that came from inside the BHs it seems that it would be a contradiction of what a BH is --'nothing escapes its event horizon'. Some may say that came from the rotational energy of the pair, but the new BH now seems to have less than the two original ones summed. So, did some of that escape the horizons? Cell identification by Body Cells How do the B-cells and T-cells recognize or distinguish a body cell from a foreign particle?  I suspect something to do with this causes Auto Immune Disorders. Can I get a rebate after using my HSA? Conversion Error setting value for 'null Converter' - Why do I need a Converter in JSF? The help pages in R assume I know what those numbers mean, but I don't. I'm trying to really intuitively understand every number here. I will just post the output and comment on what I found out. There might (will) be mistakes, as I'll just write what I assume. Mainly I'd like to know what the t-value in the coefficients mean, and why they print the residual standard error.   Call: lm(formula = iris$Sepal.Width ~ iris$Petal.Width)  Residuals:      Min       1Q   Median       3Q      Max  -1.09907 -0.23626 -0.01064  0.23345  1.17532    This is a 5-point-summary of the residuals (their mean is always 0, right?). The numbers can be used (I'm guessing here) to quickly see if there are any big outliers. Also you can already see it here if the residuals are far from normally distributed (they should be normally distributed).  Coefficients:                  Estimate Std. Error t value Pr(&gt;|t|)     (Intercept)       3.30843    0.06210  53.278  &lt; 2e-16 *** iris$Petal.Width -0.20936    0.04374  -4.786 4.07e-06 *** --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1    Estimates $\hat{\beta_i}$, computed by least squares regression. Also, the standard error is $\sigma_{\beta_i}$. I'd like to know how this is calculated.  I have no idea where the t-value and the corresponding p-value come from. I know $\hat{\beta}$ should be normal distributed, but how is the t-value calculated?  Residual standard error: 0.407 on 148 degrees of freedom   $\sqrt{ \frac{1}{n-p} \epsilon^T\epsilon }$, I guess. But why do we calculate that, and what does it tell us?  Multiple R-squared: 0.134,  Adjusted R-squared: 0.1282    $ R^2 = \frac{s_\hat{y}^2}{s_y^2} $, which is $ \frac{\sum_{i=1}^n (\hat{y_i}-\bar{y})^2}{\sum_{i=1}^n (y_i-\bar{y})^2} $. The ratio is close to 1 if the points lie on a straight line, and 0 if they are random.  What is the adjusted R-squared?  F-statistic: 22.91 on 1 and 148 DF,  p-value: 4.073e-06    F and p for the whole model, not only for single $\beta_i$s as previous. The F value is $ \frac{s^2_{\hat{y}}}{\sum\epsilon_i} $. The bigger it grows, the more unlikely it is that the $\beta$'s do not have any effect at all. Why are these estimates to the German tank problem different? Suppose that I observe $k=4$ tanks with serial numbers $2,6,7,14$.   What is the best estimate for the total number of tanks $n$?   I assume the observations are drawn from a discrete uniform distribution with the interval $[1,n]$. I know that for a $[0,1]$ interval the expected maximum draw $m$ for $k$ draws is  $1 - (1/(1+k))$. So I estimate $\frac {k}{k+1}$$(n-1)≈$ $m$, rearranged so $n≈$ $\frac {k+1 }{k}$$m+1$.  But the frequentist estimate from  is defined as:  $n ≈ m-1 + $$\frac {m}{k}$  I suspect there is some flaw in the way I have extrapolated from one interval to another, but I would welcome an explanation of why I have gone wrong! Proving that free modules are flat (without appealing projective modules) Suppose $R\neq 0$ is a commutative ring with $1$. Let $M$ be a free $R$-module. I would like to prove that $M$ is a flat $R$-module. Everywhere I have looked (mostly online) this is proved by first proving that every free module is projective, and then proving that every projective module is flat. Unfortunately, Atiyah &amp; Macdonald's "Introduction to Commutative Algebra" (Chapter 2) does not discuss projective modules. But the result that every free module is flat comes very handy in the exercises.   So my question is,     Is it possible to prove that every free module is flat just by   definitions and without appealing to projective modules?   Thanks! Can we know the fundamental nature of space and time? How to show bibliography alphabetically with biblatex? How to find files containing two strings in different lines How can I get textures on edge of walls like in Super Metroid and Aquaria? Games like Super Metroid and Aquaria present the terrain with the other facing parts having rocks and stuff while deeper behind them (i.e. underground) there's different detail or just black.      I would like to do something similar using polygons. Terrain is created in my current level as a set of overlapping square boxes. I'm not sure if this rendering method will work such a system for creating terrain but if anyone has ideas I'd love to hear them. Otherwise I'd like to know how I should re-write the terrain rendering system so it actually works to draw terrain in this manner... How to achieve this 3D "Card" effect How do I create this type of effect with an image? I'm wondering how to get the thick edges and the tilted look. Is there an online tool that will convert a flat image to look like this? I have a set of polygons representing administrative districts.  The districts were digitized by hand, and there are small spaces (slivers) between polygons where the polygons should be touching. How can I have the polygons snap together, removing the spaces? Prove by mathematical induction that $2n ≤ 2^n$, for all integer $n≥1$? I need to prove  $2n \leq 2^n$, for all integer $n≥1$ by mathematical induction?  This is how I prove this:  Prove:$2n ≤ 2^n$, for all integer $n≥1$  Proof: $2+4+6+...+2n=2^n$  $i.)$ Let $P(n)=1         P(1): 2(1)=2^1\implies 2=2$.        Hence, $P(1)$ is true.  $ii.)$  Assume that $P(n)$ is true for $n=k$, i.e, $2+4+6+...+2k=2^k$, and prove that $P(n)$ is also true for $n=k+1$, i.e, $2+4+6...+2(k+1)=2^{(k+1)}$  from the assumption add $2(k+1)$ on both sides so we have   $2+4+6...2k+2(k+1)=2^k+2(k+1)$  I'm confused with $2^k+2(k+1)$, I don't know how to make $2^k$ be equivalent to $2^{k+1}$.  I feel i'm doing something wrong.   Any help would be appreciated! Adding "noise" to overlapping X,Y coordinates so no longer in EXACT same place? I have a series of points representing animal observations. Many of them overlap, having the exact same coordinates, and this is causing me trouble when I want to run some analytical tools which requite no identical points. I was wondering if there's any tool that can add some "noise" to the coordinates, that's to say, adding some random decimal values at the end in order to obtain slightly different coordinates which are still close enough in space so as to not really modify the observation. I've heard about such a tool, but I can't find it, at least in ArcGis 10. Maybe there's another software that can do it? Thanks! java.lang.ClassNotFoundException: org.apache.struts2.dispatcher.ng.filter.StrutsPrepareAndExecuterFilter How do I get a Ralink RT3290 wireless card working? I want to do something like String.Format("[{0}, {1}, {2}]", 1, 2, 3) which returns:  [1, 2, 3]   How do I do this in Python?