Prove there exists some $a&gt;0$ such that the series $\sum_nf(an)$ diverges.  I think it can be useful to partition $[0, +\infty)$ on $[n, n+1)$ and choose some $a_n$ for every $n$. But I can't understand how to build $a$ from $a_n$. Also I think integral Koshie trapping is useful.
Hi can you help me solve this exercise? Thanks. Let $f: [0;+\infty) \to \mathbb{R}$ be nonnegative and continuous function. Suppose $\forall a &gt; 0$  $\sum_{n=1}^{\infty} f(na)$ is convergent. Prove that $\int_{0}^{\infty}f(x) dx$ is convergent. I tried to solve it by using the Riemann sum, but for fixed a it doesn't work. I have no other ideas.
How to fix 'android.os.NetworkOnMainThreadException'? Where to post questions asking for expert opinion? After my lie down, I posted  on SO. As I suspected would happen, it got moved to programmers. Unfortunately, the kind of people I'm interested in getting answers from (litb,skeet et al - the top SO programmers in their respective disciplines) don't frequent Programmers much. So my question is, where to ask somewhat non-specific questions of high-powered users? Theoretical proof of the constant of speed of light $c$ in vacuum in all frames of references Is there any theoretical proof for the constant speed of light in all frames of references? I know it is experimentally proven but just curious. Differences among Doctoral Degrees/titles Remount drive without physically disconnecting? So I have unmounted a usb external hard drive but I went to re-mount it.  Is there a command that I can use to remount the drive?  Only way I know to re-mount a drive is to disconnect the usb cable from my Mac and reconnect it. Question about "surely" and "certainly" Today I saw a sentence:  She surely can do it.  What does that mean? How does it differ from:  She certainly can do it. Surely she can do it.  This slight difference in meaning of those sentences I am not able to notice. Could you please help me? Thanks in advance! Show that if $fd'=f'd $ and the pairs $f, d $ and $f',d' $ are coprime, then $f=f' $ and $d=d' $. Show that if $fd'=f'd $ and the pair $f, d $ are coprime, that is gcd($d,f $)=1, as is the pair $f',d' $, then $f=f' $ and $d=d' $.  This should be simple, but I couldn't verify it. Thanks in advance! LaTeX template for resume/curriculum vitae If you have a TeX'ed resume, did you use a template or make your own? Are there any useful packages? What looks the most professional? How about special considerations for different areas of work (e.g. in academia)? Careers ad asking if I want to work where I already work? E: Problem executing scripts APT Update::Post-Invoke-Success error during apt-get update I cannot install any package. It seems the command apt-get update must be run but it throws some errors:  $ sudo apt-get update Get:1 http://security.ubuntu.com/ubuntu xenial-security InRelease [102 kB]      Hit:2 http://ve.archive.ubuntu.com/ubuntu xenial InRelease                      Hit:3 http://ve.archive.ubuntu.com/ubuntu xenial-updates InRelease              Hit:4 http://ve.archive.ubuntu.com/ubuntu xenial-backports InRelease            Fetched 102 kB in 23s (4337 B/s)                                                *** Error in `appstreamcli': double free or corruption (fasttop): 0x000000000210f4b0 *** ======= Backtrace: ========= /lib/x86_64-linux-gnu/libc.so.6(+0x777e5)[0x7fac8d8317e5] [...] Aborted (core dumped) Reading package lists... Done E: Problem executing scripts APT::Update::Post-Invoke-Success   'if /usr/bin/test -w /var/cache/app-info -a -e /usr/bin/appstreamcli;   then appstreamcli refresh &gt; /dev/null; fi' E: Sub-process returned an error code Show that $\lim\limits_{a \to 0^+} \int^1_0 ax^{a−1}f(x)dx=f(0)$. Suppose $f$ is Riemann integrable on $[0,1]$ and continuous at 0. Show that $$\lim\limits_{a \to 0^+} \int^1_0 ax^{a−1}f(x)dx=f(0).$$   Can anyone share some insight on this one? I have no idea where to begin. Maybe a substitution? optimality of the Carnot cycle It is well-known and is stated always and everywhere that the Carnot cycle is the most efficient thermodynamic cycle. Although stated as such it is rarely proven. One proof is in the article &quot;Engine Efficiencies and the Second Law of Thermodynamics&quot;, (AMERICAN JOURNAL OF PHYSICS VOLUME 37, NUMBER 11 NOVEMBER 1969), Tobin proves that among all engines performing a cycle, so that the engine returns to its starting state, the Carnot cycle is the most efficient where efficiency is defined as $\epsilon= 1-\frac{Q_{rejected}}{Q_{absorbed}}$, and the Carnot cycle is defined as one consisting of an isothermal-adiabatic-isothermal-adiabatic segment. The proof assumes the Clausius equality, that is in a reversible cycle $\oint \frac{\delta Q}{T} = 0$. Then using the geometry of the picture below a simple argument over the areas enclosed in the cycles IABCDJI vs. IAEFGDJI (heat absorbed) or IJDAI vs. GHEAIJDG (heat rejected) will show that any other cycle working between the same temperature limits such as EFGHE will have lower efficiency than the Carnot cycle ABCDA. But this proof assumes tacitly that pieces EFG and GHE are such that $dS&gt;0$ and $dS&lt;0$, respectively. Here I am thinking of a nasty cycle such as shown in the next figure in red where the cycle cannot be neatly separated into two and only two monotonic parts having $dS&gt;0$ or $dS&lt;0$. There is no obvious reason why that restriction should hold in general. Clearly this monotonicity should hold if the engine has only temperature as its independent &quot;force&quot; but if the engine's state needs more than one independent force variable than the situation is more complicated. Instead it is reasonable to assume that any physical cycle should have finite number of monotonic pieces, kind of a &quot;thermodynamic Dirichlet condition&quot; but it could also have self intersections, little loops, and other ugly features when projected on the TS plane. How would then one prove the optimality of a Carnot cycle? Or ugly cycles on the ST plane are non-physical even if there are more than one thermodynamic variables? [] [] Explanation of terminal objects using limits I understand terminal objects, but I falter when trying to achieve them using the approach of diagrams, the category of constant diagrams $\mathcal{C}^{\mathcal{J}}$ and limits.  I have the empty category 0, now I have a diagram $D: \mathcal{J} \to \mathcal{C}$. However this diagrams image should be empty, because there are no objects or morphisms to map. At this point, I'm confused, because I can't get to the universal property $\forall c \in \mathcal{C}. \exists! f: c \to T$ anymore, because I'm missing T. (EDIT: I get that this is how it is supposed to be)  But continuing with the category of constant diagrams, even if I have a constant diagram mapping everything to one object / morphism, there is nothing to map, so the functors image is still empty. So now I'm missing both $T$, and any kind of $c$.  EDIT: It seems to me, have I chosen 1, instead of 0, I would have come up with that is considered the definition of the terminal category. What am I missing? In the Deadpool movie when Wade his mutant cells get activated, his skin changes so he looks like Freddy Krueger (As Weasel referred too). But he obtains healing abilities and his cancer gets cured because of it. He's become an immortal person which can heal any wound super fast.  What I don't understand is, why doesn't his skin heal so he would look like before? I remember that Francis (or Ajax) said that it's a 'side-effect' of mutating, but why wouldn't it heal when he has a healing ability? Position of \caption gets counter incremented twice bash: ./program: cannot execute binary file: Exec format error Coupon collector problem with partial collection of a specific set of coupons I am very new in probability and combinatorics and have a naive question around a variation on the coupon collector problem with partial collection.   Lets assume we have a box with 45 coupons labeled 1-45. Now in this case I would like to adjust the CCP such that I can calculate the expected value (amount of draws necessary) to collect 10 specific items. For example item 1-10. How do I adjust my model such that I can calculate the amount of draws necessary to collect each item n times.   I assume that I have to adjust CCP2 in following post () to include the probability that I catch one item is 10/45.   All tips and tricks are welcome!  Thanks for your help I am working on the problems from the textbook "Topology without tears". I am stuck with problem number $4$ in Exercise $2.2$. Could anyone suggest some hints on how to proceed? The question goes as follows.  A topological space $(X,\tau)$ is said to satisfy the second axiom of countability or to be second countable if there exists a basis $\mathcal{B}$ for $\tau$, where $\mathcal{B}$ consists of only a countable number of sets.   Prove that the discrete topology on an uncountable set does not satisfy the second axiom of countability. Let $(X,\tau)$ be the set of all integers with the finite-closed topology. Does the space $(X,\tau)$ satisfy the second axiom of countability?   For the first problem, I tried to argue by contradiction assuming that the basis for the topology is countable. But I did-not know what I need to look at to prove the contradiction. For the second one, I do not know where to start. I have not thought deeply on the second one though.  Thanks, Adhvaitha Using MySQL, I can do something like:  SELECT hobbies FROM peoples_hobbies WHERE person_id = 5;   My Output:  shopping fishing coding   but instead I just want 1 row, 1 col:  Expected Output:  shopping, fishing, coding   The reason is that I'm selecting multiple values from multiple tables, and after all the joins I've got a lot more rows than I'd like.  I've looked for a function on  and it doesn't look like the CONCAT or CONCAT_WS functions accept result sets.  So does anyone here know how to do this? What do I call someone who marries or has a sexual relationship with someone much younger than themselves? Their partner is someone who is at least 18 years old.    The term paedophile is not the correct word, because it refers to minors.