Loop cut and slide didn't complete the loop
Loop-cut won't go all the way around
Integral vanishes on all intervals implies the function is a.e. zero I am having trouble with the following problem:      $f:\mathbb{R}\to \mathbb{R}$ is a measurable function such that for all $a$:   $$\int_{[0,a]}f\,dm=0.$$   Prove that $f=0$ for $m$ almost every $x$ (here $m$ is the Lebesgue measure).    I have no problem proving this for $f$ non-negative, or under the assumption that $f$ is integrable. But the question only assumes that $f$ is measurable and no more.  My idea was the usual thing; we look at the set of points where $f$ is positive and negative and assume one of these has measure greater than zero. Then I wanted to estimate one of these by an open set, look at the integral on the open set and show that it had to be greater than zero, a contradiction. But a key part of this attack is the assumption of the absolute continuity of the integral, which only holds in the case where $f$ is integrable.  Alternatively, if it were integrable one could simply estimate $f$ by a continuous function, where the result is quite obvious.  Ultimately we are going to show that $f$ is integrable, but it is not clear to me how to show this before showing it is zero a.e. So there must be a simpler way. Does anyone have suggestions? Do hit points reset upon each use of wild shape even if the same form's assumed? Let's say I've used wild shape to assume the form of a mammoth that has 126 hit points. I take 26 points of damage, putting me at 100 hit points. If I use my next turn to take a bonus action to exit wild shape then an action to use wild shape to assume the form of a mammoth again, do I now have 126 hit points or 100 hit points? For example: $$7x \equiv 1 \pmod{31} $$ In this example, the modular inverse of $7$ with respect to $31$ is $9$. How can we find out that $9$? What are the steps that I need to do? Update If I have a general modulo equation: $$5x + 1 \equiv 2 \pmod{6}$$ What is the fastest way to solve it? My initial thought was: $$5x + 1 \equiv 2 \pmod{6}$$ $$\Leftrightarrow 5x + 1 - 1\equiv 2 - 1 \pmod{6}$$ $$\Leftrightarrow 5x \equiv 1 \pmod{6}$$ Then solve for the inverse of $5$ modulo 6. Is it a right approach? Thanks. I have a VF page built on Lightning Design system (LDS) framework. When I add the renderAs="PDF" attribute, pdf does not show the content with LDS CSS/Style.  Reference Link:   Note1: apex:page tag has all the usual attributes which is required for pdf *     (showHeader="false" sidebar="false" standardStylesheets="false"   docType="html-5.0" applyHtmlTag="false" applyBodyTag="false")   Note2: Salesforce supports external stylesheet. That is, CSS file stored in Static resource. Link of example:   I also copied the whole css content from the LDS CSS file (Path: 'assets/styles/salesforce-lightning-design-system-vf.min.css') and applied as inline CSS in VF page. But still, there is no luck.  My styleSheet is referred in VF page as below  apex:stylesheet value="{!URLFOR($Resource.SLDS221,      'assets/styles/salesforce-lightning-design-system-vf.min.css')}"   Page works absolutely fine if it is not rendered as PDF.  Any suggestions? The book Theory of Functions of a Real Variable by I. P. Natanson, proves that a denumerable or finite union of pairwise disjoint sets of cardinality $c$ has cardinality $c$.  The proofs given in the book are fairly easy, using the axiom of choice. But then it is left as an exercise a generalization of the above theorems for not necessarily pairwise disjoint sets.  I have searched all over the web, but all the proofs found so far are for pairwise disjoint sets.  The proof given in the book relies on the fact that each set $A_k$ can be matched with a set like $[a_{k-1},a_k)$, with $a_k$ point of a partition of $[0,1]$, such that $\bigcup_{k=1}^{\infty\lor n}A_k=[0,1)$. So it is easy to guess the bijective function, but I can't generalize this proof.  Any help is highly appreciated. Thanks and regards. I have seen  and altough I face the same problem, the answer to that question is not helpful to my case.   I have large spreadsheets which are changed frequently. So, according to the solution accepted to that question, it is hard to duplicate all populated columns and it is cumbersome to edit one more cell everytime I wish to make a change on cells background colors.  Isn't there a way to sum data based on a condition regarding cell background color in Google Spreadsheets? How to handle relative urls correctly with a nginx reverse proxy Is diamagnetism a static or dynamic effect? When we put a diamagnetic material in the presence of an external magnetic field $\vec B_0$, the magnetic field inside the material decreases to $$\vec B=(1+\chi_m)\vec B_0,$$ where the magnetic susceptibility $\chi_m$ is a small negative number. I am assuming the material is linear and isotropic.  On the other hand the diamagnetism is explained in terms of Lenz law. When we change the magnetic flux of the external field over the material, atomic currents generate an induced magnetic field trying to restore the flux. But then the induced field could have any sign (in the appropriated direction), depending if we are increasing or decreasing the flux of the external field. It seems diamagnetism is a dynamic effect. How come the sign of $\chi_m$ is always negative? Moreover, the magnitude of $\chi_m$ should depend on how large is the variation of the flux but I do not see any suggestion of this when looking at tables of $\chi_m$. Let $A$ and $B$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$   I proved $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ by interpreting $AB$ as a composition of linear maps, observing that $\operatorname{ker}(B) \subseteq \operatorname{ker}(AB)$ and using the kernel-image dimension formula. This also provides, in my opinion, a nice interpretation: if non-stable, under subsequent compositions the kernel can only get bigger, and the image can only get smaller, in a sort of loss of information.  How do you manage $\operatorname{rank}(AB) \leq \operatorname{rank}(A)$? Is there a nice interpretation like the previous one? Is the usage of text embedded in an image a bad practice, for very controlled uses? What are your thoughts on writing your text in images?  I'm not saying throughout the whole website... But in certain key locations like image carousels. And I'm not talking about the special cases where you have font types that browsers can't render. I'm talking about text that uses standard fonts and is not too big (3/4 bullets).  Should I embed the text in the images that compose the carousel? Or should I apply the text to the images as "real text" HTML?  What are the advantages/ disadvantages of using text in an image? What is the correct formula for gravitational time dilation for a satellite in a circular orbit? How do I monitor network usage on a Windows server? Someone flagged my question as already answered, but it's not How to solve 5x5 grid with 16 diagonals? Best way to select a subset of items in a long list One of our clients wants us to build a web application to share information about the magazines he publishes. Each edition of a magazine contains a variable amount of advertisers. These advertisers come from a central pool of advertisers. This pool contains (at the moment) about 150 advertisers. Each edition of a magazine contains a subset (about 70-80) of these advertisers.  Our client wants us to come up with a way for him to add 70-80 advertisers from this list of 150 to an edition of a magazine. It needs to be as fast as possible, as easy as possible and as intuitive as possible.  What would be an ideal solution here? I was thinking about something in the lines of the image below.   List of available advertisers. List of selected advertisers. Advertisers have a delete button. Textbox with auto-suggest for advertisers in the list on the left. &lt;Enter> removes the advertiser from the left list and adds it to the right one. Textbox gets cleared and remains focus.   My colleagues suggested drag-and-drop or just one list with a checkbox for each advertiser, but those approaches force the user to do a lot of searching and clicking. With my approach the user can just keep on typing.  Any other suggestions for this situation? Determine when $\sum_1^\infty \frac{(2n)!x^n}{n(n!)^2}$ converges. By ratio test, when $|x|&lt;1/4$, the sum converges, $|x|&gt;1/4$ diverges. But I'm not sure about $|x|=1/4$. By Stirling's approximation $\frac{(2n)!(1/4)^n}{n(n!)^2}\sim\frac{4^n}{n{4^n}\sqrt{ \pi n}}\sim\frac{1}{\sqrt{\pi}n^{3/2}}$, so when $|x|=1/4$, the sum converges. But I haven't learned Stirling's approximation yet, so how to prove that when $|x|=1/4,$ the sum converges, without using it? Is it possible to change your spawn point? How to redirect output to a file and stdout In bash, calling foo would display any output from that command on the stdout.  Calling foo &gt; output would redirect any output from that command to the file specified (in this case 'output').  Is there a way to redirect output to a file and have it display on stdout? Formula for calculating $\sum_{n=0}^{m}nr^n$ According to the well ordering theorem "". Whenever we have a well ordering on a set, it is not difficult to construct a new well ordering with a largest element. My question is:     For a given infinite set, can we find a well ordering on it such that there is no largest element?