Assume we have a non empty, open convex set $ K \subseteq \mathbb{R}^d $. Then $\overline{K}$ is also convex. How can we prove that statement? I thought that breaking it into cases would help. If $x,y \in int(K)=K^o$ then from hypothesis the statement holds. If $x \in K^o $ and $y \in \partial K$ then from a theorem of my notes :P means $[x,y) \in K $. I have stacked how to show it with details that for each $x,y \in \overline{K}$ then $[x,y] \in \overline{K}$
Today I was going through my text book exercise and got hold of the following question.     Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ then prove that $\operatorname{Closure}(A)$ is also a convex subset of $X$.   I gave it a try but could not succeed.
My understanding is that in machine learning it can be a problem if your dataset has highly correlated features, as they effectively encode the same information.  Recently someone pointed out that when you do one-hot encoding on a categorical variable you end up with correlated features, so you should drop one of them as a "reference".  For example, encoding gender as two variables, is_male and is_female, produces two features which are perfectly negatively correlated, so they suggested just using one of them, effectively setting the baseline to say male, and then seeing if the is_female column is important in the predictive algorithm.  That made sense to me but I haven't found anything online to suggest this may be the case, so is this wrong or am I missing something?  Possible (unanswered) duplicate: Feynman diagram for attractive forces How to draw Brownian motions in tikz/pgf Intuition behind the definition of a measurable set This week I saw the definition of a measurable set for an outer measure.     Let $\mu^*$ be an outer measure on a set $X$. We call $A \subseteq X$ measurable if $$\mu^*(E) = \mu^*(A\cap E) + \mu^*(A^c\cap E)$$ for every $E \subseteq X$.   This is not the first time I've seen this definition. Unlike most other things in mathematics, over time I have gained absolutely no intuition as to why this is the definition.   The only explanation I've ever seen is that a set is measurable if it 'breaks up' other sets in the way you'd want. I don't really see why this is the motivation though. One reason I am not comfortable with it is that you require a measurable set to break up sets which, according to this definition, are non-measurable; why would you require that? Of course, you can't say what a non-measurable set is without first defining what it means to be measurable so I suppose no matter what your condition is, it will have to apply to all subsets of $X$.     Is there an intuitive way to think about the definition of measurable sets? Is there a good reason why we should use this definition, aside from "it works"? Do my animals need fences? I have a barn and a coop. Each has several animals in. Because fences keep deteriorating and get in the way, I didn't build any. The animals can roam freely around my farm.  Is there a downside to this? Will the animals eat my crops or escape if I don't enclose them? Noun for "person with intermediate skill" I'm looking for the noun form of "person with intermediate skill". For example, in the context of a particular activity, "person with no skill" might be designated a novice, and "person with much skill" might be designated an expert. However I know of no such word in between these two extremes.   My only thought is amateur, which has a distinct meaning from describing level of skill, although skill level can often be inferred. I'm looking for a better solution.  EDIT: due to the attention this question has received, I'll try to offer some clarification in order to reduce repeat questions.   "Journeyman" would certainly not work. I am a software developer by profession. However the word I'm looking for should hopefully be profession-agnostic. I am trying to designate the skill level of the person that some learning material might be appropriate for. The word that I'm looking for follows the preposition "for".   For Beginners  For [word for intermediate-skilled person] For Experts  Sorry, but even after all the great answers, I'm still learning towards "amateur" even though it is technically incorrect. Also, the word should not be so esoteric that only people with English degrees know what it means. Keep the ideas coming! What should the order of DNS servers be for an AD Domain Controller and Why? This is a  about Active Directory DNS Settings. Related:     Assuming an environment with multiple domain controllers (assume that they all run DNS as well):  in what order should the DNS servers be listed in the network adapters for each domain controller? Should 127.0.0.1 be used as the primary DNS server for each domain controller? Does it make any difference, if so what versions are affected and how? Can we have a red/status tag for "obsolete"? Given the changes that have been rolled out on Meta for the new close hold reasons, we've lost "too localized" here on meta.  However, "too localized" has a genuine use on meta (which is part change management system) to indicate those things which are truly obsolete (as evidenced by the text of the old close reason "refers to a specific point in time").  That said, there are things that are obsolete on meta, but we have no way of indicating that other than an "off topic" closure, which really sends the wrong message.  To mitigate that, I'd like a status-obsolete tag for meta sites which meta diamonds can apply to posts that have been made obsolete by other changes that were made to the system afterward. Every time i publish a new post i want to display an icon like a badge (NEW) and after a specific time it will disappear.  Lets say i publish a new post and i want to display that it is a new post. So i want to insert an icon that for example a star, in each new post in the blog list view into the metadata of the posts. Also i want to be removed after a specific time will pass. Lets say in one day or 24 hours the icon should be removed automatically.  How can i do this? ConTeXt: Flush right formula numbers in cases environment I would like to have the formula numbers of the cases environment flush right like regular formulas (not at a fixed distant from the cases like it is by default with the default setting numberdistance=2.5\emwidth). How can I do that?  \starttext  \startplaceformula     \startformula         \startcases             \NC x = 1 \NR[+]             \NC y = 1 \NR[+]         \stopcases     \stopformula \stopplaceformula  \stoptext     What I want (but with the left bracket):  \starttext  \startplaceformula     \startformula         \startalignment[n=1]             \NC x = 1 \NR[+]             \NC y = 1 \NR[+]         \stopalignment     \stopformula \stopplaceformula  \stoptext What's the difference between a method and a function? I have been trying around different ranges of summations:  $$ \sum_{(m,n) \in \mathbb{Z}^2} \frac{m^2 - n^2}{(m^2 + n^2)^2} = 0$$  That's not any good.  What about if we restrict to $m, n \in \mathbb{Z}$ as positive integers.    $$ \sum_{m &gt; 0, n &gt; 0} \frac{m^2 - n^2}{(m^2 + n^2)^2} = 0$$  Now here's an anti-symmetry.  I do not like taking out the symmetry, but perhaps I can ask about:   $$ \sum_{m &gt;  n &gt; 0} \frac{m^2 - n^2}{(m^2 + n^2)^2}  = \; ? \tag{$*$} $$  There doesn't seem to be a change of variables that can work.  And we've used symmetry about as much as we can.  This looks related to:  $$  \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$  Perhaps this other series ($*$) also has a special value. Example of a group $G$ such that $G = N_1\cdots N_n$ and $N_i \cap N_j = \{e\}$ for all $i \neq j$ but $G$ is not the internal direct product of them. I was trying to find an example of a group $G$ and normal subgroups $N_1 , \ldots , N_n$ such that $G = N_1\cdots N_n$ and $N_i \cap N_j = \{e\}$ for all $i \neq j$ and yet $G$ is not the internal direct product of  $N_1, \ldots , N_n$ .  It is easy to show that G is the internal direct product of normal subgroups $N_1 , \ldots , N_n$ iff  (i) $G = N_1\cdots N_n$  (ii) $N_i\cap (N_1 \cdots N_{i-1}\cdots N_{i+1} \cdots N_n) = \{e\}$ for $i = 1, 2, \ldots, n$.  So I guess the question boils down to produce an example of a group $G$ and its normal subgroups  $N_1 , \ldots , N_n$ such that   (1) $G = N_1\cdots N_n$  (2) $N_i \cap N_j = \{e\}$ for all $i \neq j$   but     (3) $N_i \cap (N_1 \cdots N_{i-1} N_{i+1} \cdots N_n) \neq \{e\}$ for at least one $1 \le i \le n.$ Thanks in advance for help.... Recursively delete empty directories in Windows Derivation of the Partial Derangement (Rencontres numbers) formula I'm looking for the method by which the partial derangement formula $D_{n,k}$ was derived. I can determine the values for small values of N empirically, but how the general case formula arose still alludes me.  Any links/books or an explanation will be appreciated.  The formula is:  $D_{n,k} = {n \choose k}!(n-k)$  Links: Case expressions in Access I already searched the internet but could not find a solution to this. Maybe it is the same question as  but there was no answer.  I want to render a video with tracking, but the output "loses" color information and gets kind of greyish. I first thought that it is related to the renderer, but it happens in Cycles and Eevee. Therefore i made a sample image, added it to the image sequence editor and rendered it as 1 frame. Resulting in the same problem.  I placed the original image beside blenders output so that you can see the difference. The only idea I came up with is that something with the lighting is not correct but I can't find a setting for that.  I am using Blender 2.80 with the default settings. Only set the output resolution to match the input resolution. How to avoid noisy renders in Cycles? I have a requirement to create triggers dynamically after a managed package is installed. I have read that the tooling API can be used to create triggers dynamically however it cannot be used in production. Is it possible to create triggers from a managed package in an org? I want to be able to create the trigger based on some user inputs. What are alternatives to ExtJS? So what I'm looking for is a javascript framework I can use that has several UI controls. I have taken a look at  but those controls are very basic compared to ExtJS. Are there any other competitive alternatives?     This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here. More info: .