The inverse question has been asked in different fashions in this site, see for example , , or . There have been questions that seem to address my own, such as , but when I tried using command \textstyle inside an align environment I don't get the desired result. How can I locally set math to display as inline mode, in particular with regards to limits in summations and integrals, within the align environment? Note that I want to avoid having to use commands such as \nolimits before each symbol. Rather, I would like to set locally the behaviour of an environment instance. The following seems to work: \begin{align} \textstyle \bar{x}=\sum_{i&gt;0}x_i \end{align}  However, the following does not: \begin{align} \textstyle \bar{x}&amp;=\sum_{i&gt;0}x_i \\ \bar{y}&amp;=\sum_{i&gt;0}y_i \end{align}  In fact, as soon as I include an ampersand for alignment purposes, it seems the \textstyle command is ignored.
I'm currently typing a mathematical document which counts many sum symbols. I frequently use the align environment to render equations, although the default behaviour of this environment is to render the equations in displaystyle, which in my opinion wastes vertical space. I therefore have added \textstyle at the beginning of each line of math in this environment, but I'm looking for a more convenient solution.   So is there an environment that is similar to align with the exception it displays textstyle math by default ? Thank you.
What factors determine enemy levels or: How do enemy levels scale to the player character's? How did Harry's blood being inside Voldemort save him? There are, of course, an incredible amount of theories on how Harry Potter survived. Problem is that the most important one, the one that is canon, is the one I don't get: how Harry's blood inside Voldemort saves him?  Because all that I have read so far, leaves me with questions. I understand why Voldemort wanted Harry's blood, but why does it save Harry? Why is that necessary to keep the sacrifice alive, while it's still inside Harry? And if Harry lost it at 17 (I have seen some different ideas on this as well), why does Voldemort still have it at 69 (give or take a few)?  To clafiry, the parts I'm talking about:     "For a fleeting instant, Harry thought he saw a gleam of something like triumph in Dumbledore's eyes"    (GoF, right after Harry tells Dumbledore that his blood now goes through Voldemort.)  And:     "He took your blood believing it would strengthen him. He took into his body a tiny part of the enchantment your mother laid upon you when she died for you. His body keeps her sacrifice alive, and while that enchantment survives, so do you"   (Albus Dumbledore in DH)  I just don't get why the blood transfusion thing suddenly gave Harry the chance to survive the whole thing, and it's never really been satisfyingly answered, not to my knowledge at least, as every answer I've read raises more questions. How can i use ArcGIS server 10 mapservices in OpenLayers2? If $x_1,...,x_n$ are positive real numbers and if $y_k=1/x_k$, prove that $$\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2.$$   I've been learning induction, and I've come across this problem that I really can't even break down and begin to think about. I've been told it has something to do with Cauchy-Schwarz, but I cannot figure out how to apply it. I would appreciate help figuring out how to go about and formulate this proof. Thanks! Can I remove a digital signature from a DLL? Does every Abelian group admit a ring structure? Given some Abelian group $(G, +)$, does there always exist a binary operation $*$ such that $(G, +, *)$ is a ring? That is, $*$ is associative and distributive:  \begin{align*} &amp;a * (b * c) = (a*b) * c \\ &amp;a * (b + c) = a * b + a * c \\ &amp;(a + b) * c = a * c + b * c \\ \end{align*}  We also might have multiplicative identity $1 \in G$, with $a * 1 = 1 * a = a$ for any $a \in G$. Multiplication may or may not be commutative.  Depending on the definition, the answer could be no in the case of the group with one element: then $1 = 0$. But the trivial ring is not a very interesting case. For cyclic groups the statement is certainly true, since $(\mathbb{Z}_n, +, \cdot)$ and $(\mathbb{Z}, +, \cdot)$ are both rings. What about in general? Is there some procedure to give arbitrary abelian groups ring structure? As a student, I make mistakes all the time, and stealing my idea will not face heavy retaliation. I don't mind if someone comes up with concurrent works, but I do mind if my key idea is leaked. Many graduate students in my field therefore only use pre-print services if their papers are either publicly presented or conditionally accepted. I am looking for a website that can do most of what ArXiv does, with an additional private option or limited visibility, for example an option to make the paper unindexed by search engines. This way, my paper cannot be seen for at least a few months. By doing so, I can at least claim that I am one of the first few researchers who got the idea right. And most importantly, I can privately correct my mistakes, if any. Publishing mistakes is embarrassing and sometimes harms one's career. Many mistakes will be revealed during the peer-review processes and I have a chance to correct them before publicizing. Please kindly think twice before giving ethically and &quot;politically correct&quot; comments like no one will scoop me, everyone will cite me, and the scoopers will be punished. At least in my field, scooping and idea-stealing is commonly observable, and big-names don't care about publicizing pre-prints because they have resources to punish the scoopers. I do understand that keeping my work private might harm the community as the information becomes less transparent. However, it could also be possible that my un-reviewed flawed works misguide the community, and a paper reviewed will benefit the society more. I personally wasted six months because of someone else's mistake in a published work.  I heard that in molecular biology there is a widely-used service to upload your research data, like protein structure and sequencing, to get a timestamp. The uploaded materials can be set to be private for a period, and fully disclosed later. A lot of people do use this function. This is not exactly what I am looking for but shares some similar ideas. If I remember correctly, there is an interesting paper studying the scooping patterns with the aid of this database.  Solutions found!  and some other university-run pre-print services do have an embargo function. Moreover,  is a widely used repository, that, unlike arxiv, only displays the newest version to the general public, although still keeps a tag on the first submitted date! Subscribers currently in a Journey What ways could you create a data extension that contains all the subscribers (SubscriberKeys) that are currently “in” a Journey (Journey Builder)?  The Journey I would like to do this for is a complex activity with multiple decision branches resulting in multiple possible journey durations. For example; if the customer didn’t open Email1 after 2 days, then send Email2; else Exit Journey. If the customer didn’t open Email2 after 2 days, then send Email3; else Exit Journey. Wait 2 more days, then End Journey. The minimum journey duration for the above example is 2 days, while the maximum is 6 days.  Note that I don’t want a list of subscribers that “entered” a Journey, but a list of the ones that are “currently in” a Journey and have not yet exited. Answer must account for non-standard Journey exits including Unsubscribes/Contact Deletes, etc. Hat duplicates when info is expanded It seems like there is a bug that is causing a duplicate hat to appear when I expand info, like pressing &quot;(more)&quot; on a suggested edit. This bug does not seem to be specific to the suggested edits review queue though, as others have reported seeing this bug in other places, like code snippet expansion. Steps to repro:  Put on a hat. Navigate to one of your own suggested edits. (You can just also open the Suggested Edits review queue and go to a random suggestion, but there is no guarantee any of those users will be wearing a hat.) Click the &quot;(more)&quot; button to expand the top box.  What should happen: Hat should shift down with the rest of the content. What happens: All the content shifts down, but a duplicate hat remains in the initial place. Before expansion:  After expansion with duplicate hat: "Not possible" and "Impossible" When we say,      It is not fair.   or     It is unfair.   I'm not sure enough to say whether both of the sentences have the same meaning or not though superficially, there is no difference between them but if we say,     Something is not possible.   or     Something is impossible.   then there is really an observable difference in my first language. Hence, the question - is there a difference between the preceding two sentences in English? Making an Emission shader emit a different colour of light than the colour assigned to the object? What's the hardest or most misunderstood aspect of LINQ? Hi I want to try Ubuntu 20.04 without installing it due to some driver issues in my laptop. But I already have 2 OS ( Windows 10 and Ubuntu 20.04 [Dualboot] ), so will trying Ubuntu mess with the already installed Operating systems? And I know how to make live usb and booting from it but don't know how to exit it and then bring my laptop back to normal. Tried finding procedure on internet but reliable results are not found. Can $\pi$ or $e$ be a root of a polynomial with algebraic coefficients? Since $\pi$ and $e$ are transcendental, neither can be the root of a polynomial with rational coefficients. However, it is easy to construct a polynomial transcendental coefficients (with $\pi$ or $e$ as one of it's roots), namely $(x-\pi)$ and $(x-e)$. However, is it possible to construct a finite polynomial (one of whose roots is $\pi$ or $e$) with only algebraic coefficients? (i.e. something such as $y=x^5-(\sqrt 2 -1)x^2+x$ .) A polynomial with only radical coefficients would be the best.  P.S. The polynomial doesn't necessarily need to have BOTH $\pi$ and $e$ as it's roots, just one is interesting enough. Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible $VAR vs ${VAR} and to quote or not to quote Let $V$ be a subspace of $\mathbb R^3$ generated by the vectors $\{(1,1,1),(0,1,1)\}.$ There are $3$ vectors $u_1=(0,0,1),u_2=(1,1,0),u_3=(1,0,1)$.Then which of the following sets are connected?  $1.\left(\mathbb R^3\setminus V\right)\cup\{(0,0,0)\}$  $2.\left(\mathbb R^3\setminus V\right)\cup \{tu_1+(1-t)u_3:t\in [0,1]\}$  $3.\left(\mathbb R^3\setminus V\right)\cup\{tu_1+(1-t)u_2:t\in [0,1]\}.$  $4.\left(\mathbb R^3\setminus V\right)\cup \{(t,2t,2t):t\in \mathbb R\}$  Now $V$ is generated by two linearly independent vectors so is like $\mathbb R^2.$ So $\left(\mathbb R^3\setminus V\right)$ will be disconnected but giving the $(0,0,0)$ with them will make it connected.  No. $4$, I think will be connected also because $\{(t,2t,2t),t\in \mathbb R\}$ is in $V$ and contains the point $(0,0,0)$ Now, any neighbourhood of this point will have points from both $V$ and $\left(\mathbb R^3\setminus V\right)$ and so the union is connected.I'm not sure about no. $2$ and $3.$ Are static class variables possible in Python? Is it possible to have static class variables or methods in Python? What syntax is required to do this? If I had a very shallow question, then I am sorry. $x,y,z\in\mathbb{N}^{+}$ and$$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=4$$find $x,y,z$.  I try with AM-GM, just get$$ \frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}\geq\frac{3}{2}$$  This means that the equation must have a real solution, but can not be sure there is an integer solution.  Let: $x=ay=abz$, then the equation becomes:$$\frac{ab}{a+b}+\frac{b}{ab+1}+\frac{1}{a^2+ab}=4$$  Which makes the problem become non-homogeneous, and seems to become more difficult. I have no more ideas. Could anyone help me? Thanks a lot. I have an ArrayList that I want to output completely as a String. Essentially I want to output it in order using the toString of each element separated by tabs. Is there any fast way to do this? You could loop through it (or remove each element) and concatenate it to a String but I think this will be very slow.