Find explicit formula and prove by induction
Guess the formula for $\sum\frac 1{(4n-3)(4n+1)}$ and prove by induction
Question  I calculated the classical heat capacity of a diatomic gas as $C_V = (9/2)Nk_B$, however the accepted value is $C_V = (7/2)Nk_B$.  I assumed the classical Hamiltonian of two identical atoms bound together as $$ H = \dfrac{1}{2m}( |\bar{p}_2|^2 + |\bar{p}_2|^2)+ \dfrac{\alpha}{2} |\bar{q}_1-\bar{q}_2|^2. $$ I calculated the partition function of $N$ particles as $$ Z = \left( \iiint_{-\infty}^{\infty} \iiint_{-\infty}^{\infty} \iiint_{-\infty}^{\infty} \iiint_{-\infty}^{\infty} e^{-\beta H} ~d^3q_1~d^3p_1~d^3q_2~d^3p_2 \right)^N \propto V^N T^{(9/2)N}. $$ I calcuated the heat capacity as $$ C_V = \dfrac{\partial }{\partial T} \left( k_B T^2 \dfrac{\partial \ln(Z)}{\partial T} \right) = \dfrac{9}{2}k_BN. $$  Why does the classical argument fail?  Classical Derivation  The partition function is \begin{align} Z &amp;=&amp; \left( \frac{1}{h^6} \int \mathrm{e}^{- \beta H(\bar{q}_1,\bar{q}_2,\bar{p}_1,\bar{p}_2)} ~d^{3}q_1 ~d^{3}q_2 ~d^{3}p_1   ~d^{3}p_2 \right)^N \\&amp;=&amp; \left( \frac{1}{h^6} \int \mathrm{e}^{- \beta ((|\bar{p}_1|^2+|\bar{p}_2|^2)/(2m)+\alpha |\bar{q}_1-\bar{q}_2|^2/2)} ~d^{3}q_1 ~d^{3}q_2 ~d^{3}p_1   ~d^{3}p_2 \right)^N \end{align} A useful gaussian integral \begin{align} \int_{-\infty}^{\infty} e^{-\gamma (x-x_0)^2}dx = \sqrt{\dfrac{\pi}{\gamma}} \end{align} The partition function can be evaluated using separated integrals \begin{align} \iiint_{-\infty}^{\infty} \mathrm{e}^{- \beta |\bar{p}_1|^2} ~d^{3}p_1 = \iiint_{-\infty}^{\infty} \mathrm{e}^{- \beta |\bar{p}_2|^2} ~d^{3}p_2 = \left(\sqrt{\dfrac{\pi}{\beta}}\right)^3 \end{align} and \begin{align} \iiint_{-\infty}^{\infty} \iiint_{-\infty}^{\infty} \mathrm{e}^{- \beta \alpha |\bar{q}_1-\bar{q}_2|^2/2 } ~d^{3}q_1 ~d^{3}q_2 = \left( \sqrt{\dfrac{\pi}{\beta \alpha/2}} \right)^3 \iiint_{-\infty}^{\infty}  ~d^{3}q_1 = \left( \sqrt{\dfrac{\pi}{\beta \alpha/2}} \right)^3 V \end{align} The last set of integrals are improper integrals. One has to take the limit as the space approaches infinite containment. In that limit, integrating one set of variables $d^3q_2$ approaches the limit of a finite Gaussian term, while the other $d^3q_1$ approaches the diverging value of the total volume of the gas.  The partition function is \begin{align} Z &amp;=&amp; \left( h^{-6} \left(\sqrt{\dfrac{\pi}{\beta}}\right)^3 \left(\sqrt{\dfrac{\pi}{\beta}}\right)^3 \left( \sqrt{\dfrac{\pi}{\beta \alpha/2}} \right)^3 V \right)^N \\&amp;=&amp; \left( h^{-6} \left(k_B T \pi\right)^{9/2} \left( \dfrac{2}{\alpha} \right)^{3/2} V \right)^N \\&amp;=&amp; \left( h^{-6} \left(k_B \pi\right)^{9/2} \left( \dfrac{2}{\alpha} \right)^{3/2} \right)^N V^N T^{9N/2} \end{align} I use Lasso logistic regression in order to identify a smaller subset of important variables. I start with N=51 (28/23) and 32 predictors.   So far it looks pretty promising, because I can identify four important predictors in my optimal model.   Now I would like to take those four predictors and examine them along with some control variables in a standard logistic regression.   My question is, does that analysis strategy make sense? Is there a better way to include controls or other variables that might be interesting?  For a better understanding:   Identify important variables via Lasso logistic regression Do further analysis including identified predictors and other control variables using standard logistic regression (using AIC to check model fit) Best compiler warning level for C/C++ compilers? Extreme learning machine: what's it all about? I've been thinking about, implementing and using the Extreme Learning Machine (ELM) paradigm for more than a year now, and the longer I do, the more I doubt that it is really a good thing. My opinion, however, seems to be in contrast with scientific community where -- when using citations and new publications as a measure -- it seems to be a hot topic.  The ELM has been introduced by  around 2003. The underlying idea is rather simple: start with a 2-layer artificial neural network and randomly assign the coefficients in the first layer. This, one transforms the non-linear optimization problem which is usually handled via backpropagation into a simple linear regression problem. More detailed, for $\mathbf x \in \mathbb R^D$, the model is  $$ f(\mathbf x) = \sum_{i=1}^{N_\text{hidden}} w_i  \, \sigma\left(v_{i0} + \sum_{k=1}^{D} v_{ik} x_k \right)\,.$$  Now, only the $w_i$ are adjusted (in order to minimize squared-error-loss), whereas the $v_{ik}$'s are all chosen randomly. As a compensation for the loss in degrees-of-freedom, the usual suggestion is to use a rather large number of hidden nodes (i.e. free parameters $w_i$).  From another perspective (not the one usually promoted in the literature, which comes from the neural network side), the whole procedure is simply linear regression, but one where you choose your basis functions $\phi$ randomly, for example  $$ \phi_i(\mathbf x) = \sigma\left(v_{i0} + \sum_{k=1}^{D} v_{ik} x_k \right)\,.$$  (Many other choices beside the sigmoid are possible for the random functions. For instance, the same principle has also been applied using radial basis functions.)  From this viewpoint, the whole method becomes almost too simplistic, and this is also the point where I start to doubt that the method is really a good one (... whereas its scientific marketing certainly is). So, here are my questions:   The idea to raster the input space using random basis functions is, in my opinion, good for low dimensions. In high dimensions, I think it is just not possible to find a good choice using random selection with a reasonable number of basisfunctions. Therefore, does the ELM degrade in high-dimensions (due to the curse of dimensionality)? Do you know of experimental results supporting/contradicting this opinion? In the linked paper there is only one 27-dimensional regression data set (PYRIM) where the method performs similar to SVMs (whereas I would rather like to see a comparison to a backpropagation ANN) More generally, I would like to here your comments about the ELM method. Binomial confidence interval estimation - why is it not symmetric? I have used the following r code to estimate the confidence intervals of a binomial proportion because I understand that that substitutes for a "power calculation" when designing receiver operating characteristic curve designs looking at detection of diseases in a population.  n is 150, and the disease, we believe, is 25% prevalent in the population. I have calculated the values for 75% sensitivity and 90% specificity (because that's what people seem to do).      binom.test(c(29,9), p=0.75, alternative=c("t"), conf.level=0.95)      binom.test(c(100, 12), p=0.90, alternative=c("t"), conf.level=0.95)   I have also visited this site:    Which is a java page which calculates binomial confidence intervals, and it gives the same answer.  Anyway, after that lengthy set-up, I want to ask why the confidence intervals are not symmetric, e.g. sensitivity is      95 percent confidence interval:    0.5975876 0.8855583      sample estimate probability: 0.7631579    Sorry if this is a basic question, but everywhere I look seems to suggest that they will be symmetric, and a colleague of mine seems to think they will be too. How should I submit bug reports and feature requests? Inspired by  I became curious about what is the current preferred mechanism for submitting bug reports and feature requests to Apple. I'm aware that they have the internal RDAR bug tracking system, and that there is an  project to crowdsource the tracking of issues, but I'm at a loss as to the preferred channel to submit something.    Is there a public-facing portal for submitting issues? If I multiclass into 2 or more spellcasting classes, how do I determine my known/prepared spells? If I multiclass into 2 or more classes/subclasses that can cast spells, how do I determine what spells I know or can prepare?  For instance, if I am playing a multiclassed character with 1 level in warlock, 2 levels in bard, and 3 levels in paladin, how do I determine which spells I know/have prepared?    Related: How to calculate $\int \frac{x^2 }{(x^2+1)^2} dx$? I'm trying  to calculate $\int \frac{x^2 }{(x^2+1)^2} dx$ by using the formula: $$ \int udv = uv -\int vdu $$  I supposed that $u=x$ s.t $du=dx$, and also that $dv=\frac{x}{(x^2+1)^2}dx$, but I couldn't calculate the last integral. what is the tick here?           the answer must be: $ \frac{1}{2}arctan\ x - \frac{x}{2(1+x^2)} $ + C Today I was thinking about composition of functions.  It has nice properties, its always associative, there is an identity, and if we restrict to bijective functions then we have an inverse.  But then I thought about commutativity.  My first intuition was that bijective self maps of a space should commute but then I saw some counter-examples. The symmetric group is only abelian if $n \le 2$ so clearly there need to be more restrictions on functions than bijectivity for them to commute.  The only examples I could think of were boring things like multiplying by a constant or maximal tori of groups like $O(n)$ (maybe less boring).  My question: In a euclidean space, what are (edit) some nice characterizations of sets of functions that commute? What about in a more general space?   Bonus: Is this notion of commutativity important anywhere in analysis? In my native language, double negations "cancel out". I have been trying to learn English using movies and such things, but double negation confuses me. I see instances when double negation cancels out, and some when they do not.  For example, "I haven't got no money" is interpreted as "I have no money." "There wasn't anyone who didn't enjoy the party" is interpreted as "Everyone enjoyed the party."  Is this context dependent? If so, how does one decide which interpretation is correct? It has been a long time since I have used calculus, and I am trying to understand how the mean and variance of the Laplace distribution with pdf $$f(x|\mu,\sigma) = \dfrac{1}{2 \sigma}\,e^{-\Large\frac{|x-\mu |}\sigma}$$ are derived.  I know that E$[X] = \mu$ and Var$[X] = 2\sigma^2$, but I don't understand how that happens.   Many thanks. "An hour" or "a hour" Solution of limit $\lim\limits_{x\to 0} \frac {e^{-1/x^2}}{x} $ Small question, I'm trying to solve this limit but I just can't wrap my head around this problem.  $$\lim_{x\to 0} \frac {e^{-1/x^2}}{x} $$  L'Hopital just seems to make it messier.   It's probably pretty simple - I'd like to hear what I'm missing. $\gcd(a,b)\!=\!1\!=\!\gcd(a,c)\Rightarrow\gcd(a,bc)\!=\!1$ [coprimes to $\,a\,$ are product closed] How do I go about proving this?  If $\gcd(a,b)=1$ and $\gcd(a,c)=1$, then $\gcd(a,bc)=1$. I'm very confused with gcd proofs. When ssh'ing, how can I set an environment variable on the server that changes from session to session? $GL_n(\mathbb F_q)$ has an element of order $q^n-1$ For fixed prime power $q$ show that the general linear group $GL_n(\mathbb F_q)$ of invertible matrices with entries in the finite field $\mathbb F_q$ has an element of order $q^n-1$.   I tried to show this question with showing diagonal matrix but i can't find element directly  competible with order i think i am on wrong way please give me clue ? As the question says, I need to edit my /proc/acpi/wakeup file, but when I open it with any text editor, it shows blank file. cat command, on the other hand, displays correct file contents in terminal. When I navigate to that file in file manager, it says it's size is 0 bytes. I'm using Xubuntu 15.04. I'm thoroughly confused. Let $(a_n)_{n \in \mathbb{N}}$ be a convergent sequence with limit $a \in \mathbb{R}$. Show that the arithmetic mean given by: $$s_n:= \frac{1}{n}\sum_{i=1}^n a_i \tag{A.M.} $$ also converges to $a$.  I have read:  but unfortunately the answers there don't help me much because I don't understand their substitutions and most of it all, why their substitutions seem to work.   What I know from the problem is that since $a_n$ is convergent and $\epsilon &gt;0$ is given, I can say that:   $$\exists N_1 \in \mathbb{N}, \forall n \geq N_1: |a_n-a|&lt;\epsilon_1 $$ I also know that since $a_n$ is convergent, it is bound, so $(a_n) &lt; M, \ \forall n \in \mathbb{N}$   I need to show that: $$\exists N_2 \in \mathbb{N}, \forall n \geq N_2: |s_n-a|&lt; \epsilon_2$$ I started as follows: $$ \left|s_n-a \right| = \left|\frac{1}{n}\sum_{i=1}^na_i -a\right|= \left|\frac{1}{n}\sum_{i=1}^m(a_i-a)+\frac{1}{n}\sum_{i=m+1}^n(a_i-a) \right| \\ \leq \frac{1}{n} \sum_{i=1}^m|(a_i-a)|+\frac{1}{n}\sum_{i=m+1}^n|a_i-a|$$ I believe to understand that the left sum after the $\leq$ is finite, bound and doesn't depend on the values that $n$ takes on. However, I don't understand where all the substitutions come from and make this proof so seemingly easy to complete.   Is there a general idea I can follow to complete such proofs? Because I know that the last step is to show that the given sum is smaller than $\epsilon$. I also know that I should bring the condition $a_n &lt; M$ into place somewhere, but I don't know where.  If I choose an $n \geq N: |a_n-a|&lt; \epsilon'$ what does that tell me about $|a_i-a|$? I know that they are seemingly the same just with a different index. How can I check dependency list for a deb package I have a  beautiful  inequality, but I can only prove part of cases. Given any positive real numbers $a,b,c$, we have   $$(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$$  How can we prove this statement?