Certain ETFs double the daily rate of return of certain indices.  Can you use such ETFs to beat the market (not on a risk-adjusted basis, just in terms of pure returns)?   I have  that      When based on high volatility indexes, 2x leveraged ETFs can also be expected   to decay to zero   But I don't see why this can be true.  My rationale is the following: My assumption is that typically, markets have a positive return expectation (otherwise, prices should drop such that this is true).  If, on a daily basis, the expected return is positive then it should be profitable to double that return. And if this is true for one day, then surely this should be true for longer periods of time.  If you can win by following such a strategy, then who loses?
I was reading this article about S&amp;P 500 short ETFs:     and I was confused when it said: "investors should note that such products are suitable only for short-term traders as these are rebalanced on a daily basis". What is ETF rebalancing exactly? and why does it make these ETFs unsuitable for long-term short trading?
Find this integral $$\operatorname I=\int\limits_{0}^{1}\dfrac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}\operatorname d\!x$$  My try: let $$f(x)=x^4-2x^3+2x^2-x+1$$ I found  $$f(1-x)=(1-x)^4-2(1-x)^3+2(1-x)^2-x+1=x^4-2x^3+2x^2-x+1=f(x)$$ so $$I=\int_{0}^{1}\dfrac{\arcsin{\sqrt{(1-x)}}}{x^4-2x^3+2x^2-x+1}dx$$ so $$2I=\int_{0}^{1}\dfrac{\arcsin{\sqrt{x}}+\arcsin{\sqrt{(1-x)}}}{x^4-2x^3+2x^2-x+1}dx$$ then I can't,Thank you very much I'm trying to hide the app icon from the title bar in GNOME in Ubuntu 18.04. I know there is  asking the same thing but it's from 2011 and I can't find what it requires to find in the shell css. I'm also using the Arc-Dark custom theme.   Here are two examples of what I want to remove:      To be clear, I don't want to resize it to fit, I want to straight away remove it completely. When speaking with my mother a couple of days ago, I read to her a message I was sending to my cousin on her behalf ending with: "... the birthday of your youngest." [implying her child]  She immediately leapt on this and said that as my cousin only had two children, the use of the superlative was not permitted and the comparative must be used instead: "... the birthday of your younger child."  She was adamant that this was a solid grammatical rule that she was taught throughout her education.  As a younger Briton, I have relatively little formal grammatical education to older people who were better taught in this regard, so I usually defer to her on grammatical rule knowledge.  Is she right?  Are superlatives not permitted when the domain of the object is only two?  To me it seems bizarre.  The minimum or maximum of any set does not only exist when there are different minima and maxima, or indeed something which is neither. Windows 7 Taskbar Icon Highlight Sticks After running Win 7 for some unknown period of time, the taskbar icon highlight starts to stick (video demo here: ). If I restart or kill explorer.exe and relaunch, it goes away.  Any idea what might be causing this? I'm reading  and I'm trying to understand the proof under "Degeneracy in One Dimension".  Here's what it says:     Considering a one-dimensional quantum system in a potential $V(x)$ with degenerate states $|\psi_1\rangle$ and $|\psi_2\rangle$ corresponding to the same energy eigenvalue $E$, writing the time-independent Schrödinger equation for the system:       $$-\frac{\hbar^2}{2m}\frac{ \partial^2\psi_1}{ \partial x^2} + V\psi_1 =E\psi_1$$       $$-\frac{\hbar^2}{2m}\frac{ \partial^2\psi_2}{ \partial x^2} + V\psi_2 =E\psi_2$$   Multiplying the first equation by $\psi_2$ and the second by $\psi_1$ and subtracting one from the other, we get:       $$\psi_1\frac{d^2}{d x^2}\psi_2-\psi_2\frac{d^2}{d x^2}\psi_1=0$$   Integrating both sides       $$\psi_1\frac{\partial \psi_2}{\partial x}-\psi_2\frac{\partial \psi_1}{\partial x}=constant$$   In case of well-defined and normalizable wave functions, the above constant vanishes, provided both the wave functions vanish at at least one point, and we find: $\psi_1(x)=c\psi_2(x)$ where $c$ is, in general, a complex constant. So, the two degenerate wave functions are not linearly independent. For bound state eigenfunctions, the wave function approaches zero in the limit $x\to\infty$ or $x\to-\infty$, so that the above constant is zero and we have no degeneracy.   Here are the parts I don't understand:   How does integrating $\psi_1\frac{d^2}{d x^2}\psi_2-\psi_2\frac{d^2}{d x^2}\psi_1=0$ yield $\psi_1\frac{\partial \psi_2}{\partial x}-\psi_2\frac{\partial \psi_1}{\partial x}=constant$.  Without having explicit formulas for $\psi_1(x)$ and $\psi_2(x)$, how do we just apparently pull them both out of the integral? I don't understand how the constant in the last equation vanishes just because $\psi_1$ and $\psi_2$ vanish at infinity. Even if it does vanish, how does that imply $\psi_1=c\psi_2$? How can I see my progress for different badges? I am trying to obtain more badges on Stack Overflow. For example, to gain Archaeologist I need to edit 100 inactive posts but I am not keeping track of the number of posts I've edited. For Deputy I need to raise 80 helpful flags, etc. How can I view all my statistics about the badges, such as how many more edits I need to do, etc. I'm not doing it for badges, but I can spend a bit more effort if I'm close to getting a badge. In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact.   Now because the basic open sets $X_f = \{\mathfrak{p} \in \operatorname{Spec} (A) : \{f\} \not\subseteq \mathfrak{p} \}$ form a basis for the Zariski Topology it suffices to consider the case when   $$X = \bigcup_{i \in I} X_{f_i}$$   where $I$ is some index set. Then taking the complement on both sides we get that   $$\emptyset = \bigcap_{i \in I} X_{f_i}^c$$  so there is no prime ideal $\mathfrak{p}$ of $A$ such that all the $f_i$'s are in $\mathfrak{p}$. Now from here I am able to show that the ideal generated by the $f_i$'s is the whole ring as follows. Since there is no prime ideal $\mathfrak{p}$ such that all the $f_i \in \mathfrak{p}$, it is clear that there is no $\mathfrak{p}$ such that $(f_i) \subseteq \mathfrak{p}$ for all $i \in I.$ Taking a sum over all the $i$ then gives $$\sum_{i \in I} (f_i) = (1).$$  Now here's the problem:          How do I show from here that there is an equation of the form $1 = \sum_{i \in J} f_ig_i,$ where $g_i \in A$ and $J$ some finite subset of $I$?       This part has been giving me a headache. I am not sure if the finiteness bit has to do with algebra, topology or the fact that we are dealing with prime ideals.  This is not a homework problem but rather for self-study.  $\textbf{Edit:}$ I have posted my answer below after the discussion with Dylan and Pierre. Can't get my model out of wireframe mode I'm new to Blender and this site, so I apologize in advance if I'm not following any of the proper etiquette.  I'm no artist, but I'm trying to learn Blender so that I have some programmer art for me to work around in Unity.    I followed a tutorial for making a basic human model, I rigged it with rigify, and now I'm starting to export it following yet another tutorial.  The thing is, when I was trying to rig the model, I couldn't get the model to go transparent so that I could see the bones inside the model.  Eventually, I managed to get it into wireframe mode, but I can't get it back into solid mode.    It's probably just a display thing and not a property of the model itself, but I want to be sure.  I'm looking at the display options under the "Figure" pane in Blender, and none of the Display check boxes are checked, including the one that says "Wire".  I don't want to import a plain wireframe model into Unity, and even if that doesn't happen, this is functionality of Blender that I should really know anyway, so I'd appreciate any help that anyone has to offer.  If necessary, I can upload the .blend file, but I'm not sure what site people would prefer that I upload it to. Simple way to monitor a multiple URLDownloadSubmit calls? How can I prove that all rational numbers are either terminating decimal or repeating decimal numerals? I am trying to figure out how to prove that all rational numbers are either terminating decimal or repeating decimal numerals, but I am having a great difficulty in doing so. Any help will be greatly appreciated. Guides on how to install drivers for a Brother all-in-one network printer are either for older devices or are rather complicated as they often involve editing or copying system files in my root directory.  Is there a newer, more convenient way to install the drivers needed? What equipment is necessary/preferred for real estate photography? What equipment is necessary for real estate photography, and what is preferred?  I'm starting into real estate photography, and I'm trying to figure out what I need and what I will want to plan on adding to my kit as I go.  I'm thinking especially along the lines of lighting equipment and lenses (focal lengths).  For example, is an ultra-wide actually necessary? Or is a general walk around sufficient. Can't execute .out files, getting permission denied I have written a C++ program and complied it to produce a.out file. However, whenever I try to run it, I get Permission Denied. I read that we can use sudo, but I can't quite get it to work. I use something like, sudo "./a.out" but that too doesn't work.  Edit:  Here is the message I get when I try "./a.out".  bash: ./a.out: Permission denied Calculate battery life How do I summon a baby mob? In Minecraft Java Edition, one can summon baby zombies and drowned using the IsBaby NBT tag: /summon zombie X Y Z {IsBaby:1b}  Likewise, one can summon baby breedable mobs (Pigs, Cows, etc.) by giving them a negative Age NBT tag (the number of ticks until the mob grows up): /summon cow X Y Z {Age:-100000}  Since NBT is inaccessible from commands in Bedrock Edition, the above commands will not work there. How do I summon baby mobs in Minecraft Bedrock Edition? Proving that $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ if $n$ is greater than 2 Prove that:  If $n$ is greater than 2, then $(3n)!$ is divisible by $n! \times  (n + 1)! \times  (n + 2)!$  From Barnard &amp; Child's "Higher Algebra".  I know that the highest power of a prime $p$ contained in $N$! is  $$ \left\lfloor{\frac{N}{p}}\right\rfloor + \left\lfloor{\frac{N}{p^2}}\right\rfloor + \left\lfloor{\frac{N}{p^3}}\right\rfloor ... $$  I'm unable to show that the formula above gives a higher value for $N = 3n$ than for the sum of its values when $N$ = $n$ , $n + 1$ and $n + 2$, considering the condition that $n &gt; 2$. I have an app to deploy in production that uses 'honor system' security. That is, all users connect to the DB using a SQL user/passwd credential and the app manages permissions itself. The latter part doesn't bother me as much as the fact that the connection object contains embedded credentials and can be copied around freely. I'm try to find some way to limit connections to a more limited set of clients. I can create firewall rules to limit by IP, of course. Is there any way to 'prequalify' SQL logins either by Machine account or domain membership? Vote for Tag Synonyms Directly on the Page Before:  After:   The pros:  Speeds up the tag synonym voting process. Easier to locate (as opposed to navigating to each individual tag wiki page). Encourages users to vote on more than one at a time.  The cons:  Serial up/downvoting. But this really isn't even an issue because, (1) only experienced users can vote on tag synonyms, and (2) more than one vote is required to approve a tag synonym, so the occasional serial up/downvoter won't significantly impact the overall process. I am given two charged particles of same charge at a distance of $r$. They initially apply force $F$.  Now an infinite dielectric (of dielectric constant $4$) of width $\frac{r}{2}$ is introduced between the particles. What will be the new force?   I find this problem confusing because I have only been told about forces when its either fully dielectric or fully vaccum given by Coulomb's law. How do we get forces when only partial space is dielectric? Bibliography per chapter with continuous numbering I would like to create a document with two chapters. After the first chapter there should be a bibliography. The next chapter should be able to reference items of the former chapter, but only show the new items in its bibliography, starting with the next number from the first bibliography.  Example:     Chapter 1 - The Fu   Fu has been done by Bar [1]. Early versions ...   Bibliography   [1] Bar et. al., Doning the Fu      Chapter 2 - Compining Fu with Herp   Doning Fu as described in [1]   neglects the advantages of Herp pointed out by Derp [2].   Consequently ....   Bibliography   [2] Derp et. al., The Advantages of Using Herp   Please not too much discussion about if that is a friendly-to-use idea, it is not my own demand :)