2:36 AM
Related to comparing various notions of (semi)continuity for functions and multifunctions.
> But if we treat a function as a special case of a multi-valued functions, all semi-continuity notions for multi-valued-functions boil down to ordinary continuity (but not upper-or semi-continuity).
We can look at relation between functions and multifunctions also in different way.
If we consider a (closed-valued) multifunction as a single-valued function into the space CL(X) of all closed sets, we get that the upper/lower hemicontinuity is equivalent to continuity of the corresponding single-valued map if we endow CL(X) with the upper/lower Vietoris topology.
I am pretty sure this works for closed-valued multifunctions and closed sets. I am not sure whether the same works for arbitrary multifunctions.
Although Wikipedia mentions this without any restrictions in the article Hemicontinuity.
> The upper and lower hemicontinuity might be viewed as usual continuity:
> Γ : A → B is lower [resp. upper] hemicontinuous if and only if the mapping Γ : A → P(B) is continuous where the hyperspace P(B) has been endowed with the lower [resp. upper] Vietoris topology.
But again, even in this viewpoint. semicontinuity (hemicontinuity) turns out to be continuity. (With correctly chosen topology on the target space.)
@MichaelGreinecker The above is also somewhat related to your comparison about semicontiuity notions for functions and multifunctions. But probably less relevant - which is why I commented here in chat and not under your post.
This also supports a bit your argument that the two things should be considered different enough not to be in the same tag. I'll leave it up to you whether it is relevant enough to be mentioned in your answer.

3 hours later…
6:14 AM
Seeing that there are almost 250 questions tagged and more than 30 questions tagged , it seems that the clean-up of is continuing quite well.
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Almost six years ago, Michael Hardy raised the issue of the "partitions" tag being used for some very different concepts, and subsequently edited its tag wiki excerpt to at least be clear about what the various concepts covered were. Two months ago the tag wiki was changed to be solely about in...

7:09 AM
BTW wouldn't it be reasonable to rename to ?
> Moderators can create any tag, even one that differs from an existing tag only by pluralization or hyphenation.

4 hours later…
11:06 AM
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I have seen some posts which mentioned they want only a hint. They want to think about their problem and solve it themselves. Nevertheless some people give an answer, good or poor, even Op insist to it just want a hint, and we post an answer with starting "It might be useful . . . " or something ...

2 hours later…
12:55 PM
A "tagging war" is (still) going on here: math.stackexchange.com/posts/2332199/revisions. – I am not an expert on abstract algebra, any opinions from someone with more experience on that topic?

12

If $A$ and $B$ are matrices such that $AB^2=BA$ and $A^4=I$, then find $B^{16}$ My Method: Given $$AB^2=BA \tag{1}$$ Post multiplying with $B^2$ we get $$AB^4=BAB^2=B^2A$$ Hence $$AB^4=B^2A$$ Pre Multiplying with $A$ and using $(1)$ we get $$A^2B^4=(AB^2)A=BA^2$$ hence $$A^2B^4=BA^2 \t... I think I vaguely recall that if there are too many edits on some post, a flag to make moderators aware of this is automatically raised. Several comments from that question has been moved to chat, most of them seem to be related to tags. To be honest, probably in there are reasonable arguments for to be included and also reasonable argument for the tag to be removed. But unless I am missing something, s are completely unrelated to that question. 2 hours later… 2:37 PM A new tag has been created - apparently by Guy Fsone. It already has 8 questions. 1 I'm having trouble solving the following PDE problem. We're in the open unit ball in the plane, centered at the origin,$$B=\{(x,y)\in \mathbb{R}^{2},\ \ x^2+y^2<1\}$$The following boundary problem is given$$\Delta u(x,y)=0,\ \ \text{in}\ B,\ \ \ u(x,y)=\sin(x)\ \ \text{on}\ \partial B$$... 1 Let u:\Omega\rightarrow \mathbb{R} be an harmonic function (this is a smooth function) such that$$\Delta u =0 \quad \mathrm{ in }\quad \Omega,$$where \Omega\subseteq \mathbb{R}^{2} is an open set. Suppose that 0\in \Omega and \rho>0 such that \mathcal{B}_{\rho}(0)\subset \Omega, whe... 0 I'm having a bit of a problem proving the equality:$$u(x) = \frac{1}{\omega_n r^{n-1}}\int_{\partial B(x,r)} u\, d\sigma = \frac{n}{\omega_n r^n}\int_{B (x,r)} u\, dV$$Which is the mean value theorem for Harmonic functions, where \omega_n is the area of S^n and B(x,r) is the ball in \... 4 A well-known feature of harmonic functions on (domains of) \mathbb{R}^n is the mean-value property: that is, if \Delta u = 0, then$$ u(x_0) = \frac{1}{\text{Vol}(\partial B_r(x_0))}\int_{\partial B_r(x_0)}{u\,dS} = \frac{1}{\text{Vol}(B_r(x_0))}\int_{B_r(x_0)}{u\,dV}. $$Is the same true on ... 2 Theorem If u\in C(\overline{B_R(x_0)}) and is harmonic in B_R(x_0), then$$|D^mu(x_0)|\leq\frac{n^m\exp(m-1)m!}{R^m}\max_\limits{\overline{B_R~(x_0)}}|u|$$We can prove the theorem by induction, but I am stuck at the first step of the proof. When m=1, we have$$\triangledown u(x_0)=\frac{n...

0

Let $A \subset \mathbb{R}^2$ be open and connected and let $u \in C^2(A)$ be harmonic. Then $u$ satisfies $$u(x)=\frac{1}{2\pi}\int_0^{2\pi}u(x+r\hat n(\theta)) \, d\theta$$ I'm given the following proof: Let $x \in A$ and take $r>0$ so that $B_r(x) \subset A$ . Since $u$ is harmonic it satisfi...

2

A professor I talked to showed me a proof of the mean value property. (He actually showed it for functions solving the heat equation instead of Laplace's equation, but it seems like the argument is the same.) The proof involves distributions, which I am not very familiar with, so there is a step ...

1

From PDE Evans, 2nd edition, pages 25-26. THEOREM 2 (Mean Value Formulas for Laplace's equation). If $u \in C^2(U)$ is harmonic, then u(x)=\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_{\partial B(x,r)}u \, dS=\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_{B(x,r)} u \, d...

The tag with the same name was discussed before on meta and rejected:
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A search result for Mean Value Theorem gives us 2715 results, and results on the page are like ones I think we can include in the tag. The theorem is an important result in calculus, and questions relating to its applications, proofs. I think it would be useful if could have the tag, as it can gr...

Looking at the above question, it seems that the above tag is not for the mean value theorem from calculus, but for the mean value property of harmonic functions.
Using this query, it seems that the tag of this name has been created and removed before.

0

A tag named mean-value-theorem has been created recently. A tag with the same name was discussed before on meta and rejected: Tag proposal: mean-value-theorem. However, looking at the questions where the tag-creator added this tag, it seems that the intention was to created a tag for the mean val...

2:53 PM
@GuyFsone I see that you have created a new tag named (mean-value-theorem). I wanted to let you know that I made a post on meta about the new tag: Should we have a tag for mean value property of harmonic functions?Martin Sleziak 12 secs ago
I have considered posting in the tag management thread instead, but since the tag started growing relatively quickly, I thought that a separate question might be better, to get a bit more attention to the issue. Sorry for posting two tag-related questions in relatively quick succession and thus hogging a bit more space on meta from other posts. (BTW I have also pinged the tag-creator to let them know about this question.) — Martin Sleziak 20 secs ago
Is there a meaning difference between ''mean-value-property'' and mean value theorem.? I think it is same contain. but the appellation differ from authors to another. we can use both appellation as well for me it is ok — Guy Fsone 34 secs ago
At least we know from the above comment that the tag-creator knows about the question on meta.
Now I noticed that a previous instance of this tag was mentioned also here in this chat room: chat.stackexchange.com/transcript/3740/2014/9/21

2 hours later…
4:57 PM
@MartinSleziak The tag (exceptions) has been removed from that question. (So it should be completely away during next 24 hours.)
This seems quite reasonable to do - the tag-name is rather vague, I doubt such tag could be really useful.