What does the 1k resistor do in this schematic?
How does the pull down resistor work in this circuit?
Notation problem in integration: write $dx$ or ${\mathrm{d}}x$? Is the complement of a prime ideal closed under both addition and multiplication? How to write a unit-test / test class for trigger? How do I write a unit-test / test class for a trigger?  What are my options?   Inline with the trigger?  New Class? Was PreferenceFragment intentionally excluded from the compatibility package? I will soon buy a new iPhone. If I log in with my Apple ID in my new phone could I still access my Clash of Clans village without starting from scratch? Proof that the largest eigenvalue of a stochastic matrix is $1$ The largest eigenvalue of a  (i.e. a matrix whose entries are positive and whose rows add up to $1$) is $1$.  Wikipedia marks this as a special case of the , but I wonder if there is a simpler (more direct) way to demonstrate this result. Will the silhouette formula change depending on the distance metric? I am using Silhouette width to compute the best value for k in k-means. As I am performing document clustering, I am calculating the values of a and b as follows in Python:  a = distance(data[index], centroids[clusters[index]], metric=metric, p=p) b = min([distance(data[index], c) for i,c in enumerate(centroids) if i != currentindex]) score = float(b - a) / max(a, b) if max(a, b) &gt; 0 else 0.0   The original formula from the  is the following:    I am using a cosine similarity measure to compute the distances and was wondering if this formula needs to be changed or it can be left as is. In the above snippet of code, the function distance computes the cosine similarity metric. Proof of when is $A=X^TX$ invertible? Say we have an $n\times m$ matrix $X$. What are the specific properties that $X$ must have so that $A=X^TX$ invertible?  I know that when the rows and columns are independent, then matrix $A$ (which is square) would be invertible and would have a non-zero determinant. However, what confuses me is, what sort of conditions must we have on each row of $X$ such that $A$ would be invertible.   It would be very nice to have a solution of the form:   when $n &gt; m$ then $X$ must have... when $n &lt; m$ then $X$ must have... when $n = m$ then $X$ must have...   I think in the 3rd case we just need $X$ to be invertible but I was unsure of the other two cases. The foreach identifier and closures When defining a method on a class in Python, it looks something like this:  class MyClass(object):     def __init__(self, x, y):         self.x = x         self.y = y   But in some other languages, such as C#, you have a reference to the object that the method is bound to with the "this" keyword without declaring it as an argument in the method prototype.    Was this an intentional language design decision in Python or are there some implementation details that require the passing of "self" as an argument? A map is continuous if and only if for every set, the image of closure is contained in the closure of image How to Create a Grid Inside a Polygon that is Oriented with the Longest Side? I have been using a process called "Create Polygon Graticule" to create a grid within a polygon with assigned row and column attributes. It is a SAGA tool which I have been running in QGIS. It generates a grid based on the extent of the input polygon with the cell running N to S and E to W. It would be better for my analysis if the direction of the grid was parallel to the longest edge of the polygon. For reference I have attached images of the current automated output and an output that I manually altered to align the grid with the polygon. The grids originally covered the entire extent of the polygon but have been clipped to show only the cell parts that are inside.    Above image is what the tool out puts by default.   This is an image of a grid that was edited manually to fit the necessary parameters.  My question is, is there a way to produce grids that are aligned such as the one in the second image in an automated fashion? I will need to generate grids for a few thousand polygons. I'm reading a lot of documentation on sed, and am still stumped on my particular use case.  I want to replace this line in a conf file with my own line:  Replace this line: #maxmemory &lt;bytes&gt; with: maxmemory 26gb  This is what I tried:  sed s/maxmemory.*bytes.*/maxmemory 26gb/ /etc/redis/redis.conf  I get the error: sed: -e expression #1, char 30: unterminated `s' command  Which stumps me because I don't know what that means. So my question is:  How can I accomplish what I want? What does that error mean? (so I can learn from it) TOC entries and labels for included PDF pages I am embedding full pages from another PDF document (using the pdfpages package), but I would like to be able to declare my own section headings before each embedded document (for e.g. hyperlinking / TOC purposes). pdfpages has an experimental feature which lets me do this, but the problem is that I want to put all of these embedded documents inside a section, so I need to insert a section heading and a subsection heading on the first document I embed. Unfortunately pdfpages doesn't seem to be able to do this. Is there a way for me to do this?  Edit: Thank you for the suggestions. I am also in need of something similar for general labels — at the moment I am abusing the addtolist option of \includePDF but the objects I'm labelling aren't really figures or tables. What's in a name: hyperparameters So in a normal distribution, we have two parameters: mean $\mu$ and variance $\sigma^2$. In the book Pattern Recognition and Machine Learning, there suddenly appears a hyperparameter $\lambda$ in the  regularization terms of the error function.  What are hyperparameters? Why are they named as such? And how are they intuitively different from parameters in general? If the original question is "What time is it?" and I want to communicate it to another person, are these Embedded Questions OK?        I don't know what time is it.       or        I didn't know what time was it. (if communicated later) I don't understand philosophy very well, and so I am wondering whether you can "prove" anything in philosophy. It always seems you can go a layer down, and find another question, almost endlessly until you get to the question of "why/how/are we here?"  Can you prove, absolutely, and 100% that something is true, in the field of philosophy? How to do seperate render passes in blender? I a lot of VFX breakdown that uses something they call "render passes". From what I see, they are rendering the clip in seperate ways (like rendering the ambient occlusion, specular, shadows, depth of field, etc individually and then combine it). Is there any way I can do this? In terms of purely set theory, the axiom of choice says that for any set $A$, its power set (with empty set removed) has a choice function, i.e. there exists a function $f\colon \mathcal{P}^*(A)\rightarrow A$ such that for any subset $S$ of $A$, $f(S)\in S.$ Is this correct?  My question then is about proving this fact, so that we do not need to put it as an axiom. Now as per the research done on this single object- Axiom of Choice, I believe here that there should be some falsity in my argument. I do not find the mistake.  For any $S\in \mathcal{P}^*(A)$, since $S\neq \emptyset$, $\exists s\in S$. Define $f(S)=s$. Then $f$ is a choice function.  This was showing me that the axiom of choice is proved, but then why it had been put as an axiom? For example, in  book, the author asserts that      It is a metatheorem of mathematical logic that it is impossible to specify the function that assigns to each non-empty subset of $\mathbb{R}$, an element of itself.     There are several notes and books on axiom of choice, but here I am trying to understand through doing some argument for some problem, where problem actually arises. Is `sudo pip install` still a broken practice?