Show that if $f(x)$ is integrable in $[a,b]$, then $f(x)$ is continuous at at least one point in $[a,b]$.
$f \in {\mathscr R[a,b]} \implies f $ has infinitely many points of continuity.
What is the value of $1^i$? What is the value of $1^i$? $\,$ CSS :after not adding content to certain elements Validity of maximising log-likelihood for maximum likelihood estimation For reasons owing to mathematical convenience, when finding MLEs (maximum likelihood estimates), it is often the log-likelihood function---as opposed to the standard likelihood function---which is maximised.  From what I've gathered, this approach is deemed valid as a result of the monotonically increasing nature of the (natural) logarithm function.  My understanding of a monotonically increasing function is: for all $x$ and $y$ (defined on a subset of the reals), if $x \leq y$ then $f(x) \leq f(y)$.  This, however, does not appear to be the case for all log-likelihood functions; for example: for the log-likelihood $\text{Gamma}(3, 5)$ function, if $x = 0.15$ and $y = 0.46$, then $f(x) = -0.59$ and $f(y) = -1.19$.  Clearly, I've misunderstood this concept. Fundamentally, I guess I'm asking can somebody (preferably mathematically) demonstrate why:  $$ \hat{\theta} = \text{argmax} \text{ } L(\theta) = \text{argmax} \text{ } \text{log} \text{ } L(\theta) $$  where $\hat{\theta}$ is the MLE for a given likelihood function. Android reading from .txt File created in src folder Generate a comma-separated list of cell contents, excluding blanks To concatenate a certain four cells, I'd use:  =CONCATENATE(A2,",",C2",",D2,",",F2)   This would make it so that...   A2 = "Matthew" C2 = "Mark" D2 = "Luke" F2 = "John"   would result in Matthew,Mark,Luke,John.  But we run into problems with something like...   A2 = "Jesus" C2 = "" D2 = "Mary" F2 = "Joseph"   which would result in Jesus,,Mary,Joseph.  Here, the extra comma is undesired. Is there a way to gracefully handle this so that all non-blank cells are included in the comma-separated list, while avoiding the addition of unnecessary commas when some cells are blank?  Certainly, this can be done with a fair number of nested IFs, but I really want to avoid that if possible. Can it be done with native Excel functions, or perhaps an array formula? Or would one have to resort to VB Script for something like this? Question about the order of the product of two elements in an abelian group My reference for learning abstract algebra says that in an abelian group $G$ the product of two elements (say $a$ with order $n$ and $b$ with order $m$) has order $mn$ if $gcd(n,m) = 1$.   What I don't understand is where the $gcd(m,n)=1$ comes into use in proving this fact? For example, $ a^{mn} = (a^n)^m = (e)^m = e$ (identity element). A similar argument for $b^{mn}$ together with the fact that $G$ is abelian is enough to show that $(ab)^{mn} = e$. From here, how do I conclude for an integer $r &lt; mn$ that $(ab)^r \neq e$  Also, are there any other conclusion that can be drawn if $gcd(n,m) \neq 1$ ?  Thanks for any feedback in advance ! Should I have separate threads for rendering and logic, or even more?  I'm aware of the immense performance drop caused by data synchronization (let alone any mutex locks).  I've been thinking of taking this to the extreme and doing threads for conceivable every conceivable subsystem. But I'm worried that may slow things too. (For example, is it sane to separate the input thread from rendering or game logic threads?) Would the data synchronization required make it pointless or even slower? Find the Unit Vector of a Three Dimensional Vector What do the bars between keyframes mean? Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!) First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an eigenvalue of $BA$?  If it's not true, then under what conditions is it true or not true?  If it is true, can anyone point me to a citation?  I couldn't find it in a quick perusal of Horn &amp; Johnson.  I have seen a couple proofs that the characteristic polynomial of $AB$ is equal to the characteristic polynomial of $BA$, but none with any citations.  A trivial proof would be OK, but a citation is better. Elastic net penalty I have a question about the elastic net penalty as implemented in glmnet in R compared to the original paper by Zou and Hastie (2005). In glmnet the penalty is listed as   $$ (1-\alpha)/2||\beta||_2^2+α||\beta||_1. $$  but in the paper it is  $$(1-\alpha)||\beta||_1 + \alpha||\beta||_2^2.  $$  Does anyone know where the factor $\frac12$ some from? (Never mind the fact that the $\alpha$'s were swapped between the two parameterisations.) In both cases the penalties are multiplied by $\lambda$, but what are the mathematical/technical arguments for not using a simple convex combination of the lasso and ridge penalties? Is it possible to build a .dmg file (for distributing apps) from a non-Mac platform? And if yes, how? regular expressions / pattern matching on ubuntu command line Get referenced node fields in twig In Drupal 8, I have a content type, that has an "entity_reference" field and I wish to access the referenced node's fields in twig template. the kint() output for the field is:   In this list I can't find the referenced node's field values. How can I (if at all possible) load the node or node fields/values in twig?  Best,  Alari Can't transfer purchases from iPhone to iTunes in iOS 9 After upgrading to iOS 9, I can't transfer purchases from my iPhone to iTunes. It happens to just start the process, then it finishes without transferring anything. However, I can still transfer purchases from my iPad with iOS 8.4 like before.   iPhone 5c, iOS 9 iTunes 12.3 MacBook Pro 13" with Retina Display Mid 2014, OS X El Capitan   Apparently, not many people happen to use this "transfer purchases" feature so it didn't get any real attention. I used it a lot, I used to update my apps on my iPhone, transfer them to iTunes and updated my iPad apps using iTunes. Still, I can update my apps using iTunes and update apps on all my devices. How do I backup a LVM partition to an image for recovery purposes?  I am trying to run dd on /dev/sda2, but it crashes after ca 8 hours, at around 380G   sudo lvmdiskscan   /dev/centos/swap [       3.89 GiB]    /dev/sda1        [     500.00 MiB]    /dev/centos/root [      50.00 GiB]    /dev/sda2        [     465.27 GiB] LVM physical volume   /dev/centos/home [     411.38 GiB]    /dev/sdb1        [     931.51 GiB]    3 disks   2 partitions   0 LVM physical volume whole disks   1 LVM physical volume   Anyway, could running from a stick make any difference? We are beginning to go down the path of mobile browser support for an enterprise e-commerce webapp (Java/Servlet based).  Of course there are many decisions to be made, but it seems to me the cornerstone is to be able to reliably detect mobile browsers, and make decisions on the content to be returned accordingly.  Is there a standard way to make this determination (quickly) based on the http request, and ideally glean more information about the given browser and device making the request (screen size, html capabilities, etc?).  I would also appreciate any supplemental information that would be of use from someone who has gone down this path of taking an existing large scale enterprise webapp and architect-ing out mobile browser support from the development side.  [edit] I certainly understand the request header and the information about a database of standard user agents is a great help.  For those talking about 'other' request header properties, if you could include similar standardized name / resource of values that would be a big help.  [edit] Several users have proposed solutions that involve a call over the wire to some web service that will do the detection.  While I'm sure this works, it is not a good solution for an enterprise e-commerce site for two reasons: 1) speed.  A call over the wire for every page request to a third party would have huge performance implications.  2) dependency/legal.  We'd tie our website response time and key functionality to their service, which is horrible for legal and risk reasons. why __getitem__ cannot be classmethod? Get current editor toolbar tab via Python? Toy Line (Production) Run I am not sure about the meaning of this:          Transformers: Generation 1 (also known as Generation One or G1) was a children's toy line that ran from 1984 to 1991 and was produced by Hasbro.     Would the following rewrite be better?        Transformers: Generation 1 (also known as Generation One or G1) was a children's toy line the production of which ran from 1984 to 1991 and was produced by Hasbro.