Going Back to Grad School

Question: I am seriously considering going back to graduate school for the PhD and I have two schools in mind. The problems I am facing are: age and I have no letters of recommendation. I do have a MS in Math and have been adjuncting since graduating in 1998. My question, what would be a good strategy for re-inserting myself into graduate school and be able to get funded. TIA.
Comment: I was 52 when I successfully applied to grad school to do my PhD. It was over 25 years since I completed my master's degree. I used two non-academic managers and a CTO for my letters. If you are planning on Math there is a GRE subject test you could take. Just go for it.
Comment: @Mathin3D You have a lot of questions here. I'm sorry your question got shut down. Perhaps if you asked a less personal situation type of question like, "Is age a factor in graduate school admissions?" and another question like, "What are some options for someone applying to graduate school with no letters of recommendation?" And then you can expand on your situation within your post while having a post that's better fit for a more general audience (ie. others with a similar question.) I wish you the best of luck in your graduate admissions!
Comment: Hello and welcome to Academia S.E.! As it stands, I'm of the opinion that this question is [a poor fit for this site](http://academia.stackexchange.com/help/on-topic) as it's highly dependent on your own personal situation. I'm voting to close.
Answer: One idea that could work is to try and do another masters in math somewhere that has a thesis requirement.  You can use this thesis to study a research problem that is currently being worked on by faculty at the schools that you want to get into.
If you can't do a second masters, how about an independent study somewhere with a prof. that could agree to supervise you for a semester or two?

Going Back to Grad School


Morphisms of $\infty$-groupoids

Question: As far as I understand, there are several ways of defining $\infty$-categories. One of them is to think of $\infty$-cateogries as $top$-enriched categories. Hence we can think of $\infty$-groupoids as generalizing topological groups. Functors between groupoids are the generalization of group homomorphisms. Hence my question is if $\infty$-functors of $\infty$-groupoids generalize continous group-homomorphisms? For instance, if $G,H$ are topological groups, and $BG,BH$ denote the associated topological groupoid/$\infty$-groupoids, do we have
$$\text{Hom}_{\infty\text{-gpd}}(BG,BH)\cong \text{Hom}_{cont}(G,H)?$$
Comment: Presumably you want functors that intertwine the basepoints in $BG$ and $BH$. Otherwise this fails even for finite groups and 1-groupoids. But even with that correction, one issue is that $\infty$-functors are looser than strict group maps: you would want instead to talk about maps of "homotopy groups" (with the word "coherent" put in for historical reasons).
Comment: It can be quite misleading to think of ∞-categories as topologically enriched categories. Rather, ∞-categories are enriched in ∞-groupoids. ∞-groupoids are the objects of the classical homotopy category, which can be defined in many different ways and doesn't have that much to do with topological spaces and continuous maps.
Comment: Specifically, when people say "$\infty$-categories can be models as $top$-enriched categories", what they have in mind is to start with the 1-category of $top$-enriched categories, and then invert (in some $\infty$- or model-categorical way) the "weak equivalences" of $top$-enriched categories (morally: the continuous functors which are essentially surjective and are "homotopy fully faithful" in that they induce homotopy equivalences on all hom spaces).
Answer: Only for special kinds of topological groups (like Lie groups or groups with an underlying topological manifold).  This fails horribly for e.g. profinite groups, which are detected as discrete groups by the topological nerve.
Comment: That's to bad, I was mostly thinking of profinite groups. Could you maybe elaborate a bit? My confusion mostly stems for my assumption that $\infty$-functors should correspond to continuous functors
Comment: @curiousmathguy There is a solution to this problem, Pyknotic/Condensed sets, spaces, ∞-categories.  (See here: https://arxiv.org/abs/1904.09966 ).  The pyknotic nerve of the associated topological groupoid  any compactly generated topological group will contain all of the information you wanted.
Comment: The point is that any map $|\Delta^n|\to G$ ($G$ profinite) must be constant, so the continuous map $G^{dis}\to G$ induces an equivalence of $\infty$-groupoids $BG^{dis}\to BG$, that's what (I think) Harry meant

Morphisms of $\infty$-groupoids


How comfortable would my 57cm circumference head fit in a 59cm - 62cm helmet?

Question: There's a nice, cheap used helmet on sale that is described as being 'large'. My head is 57cm, which is 'medium'. How well would this helmet fit (I guess it does vary between manufacturers), and how can I improve the fit if it doesn't?
Comment: We can't answer this for you.  You will have to try it on.
Comment: I don't know where you are but here in the UK you can buy helmets that comply with the relevant safety standards for £10 (€12 or $14) in a big supermarket, and you can try them on.  Of course they're not the best, but I'd trust them more than a helmet of unknown provenance.  The situation is similar in the bits of western Europe I know, I'd assume it's the same in the US/Canada etc.
Answer: I'd strongly advise against buying used helmets because you just don't know how well they've been looked after. If it's been dropped, it might be structurally compromised without your being able to see any damage. And, if you think about it, why would anybody sell a cycling helmet? I guess maybe if they've upgraded to something lighter and/or more aero but, in most cases, people will wear a helmet until they think it's worn enough that it needs to be replaced. If it's worn enough that the original owner doesn't want to wear it, you don't want to wear it either.
In any case, this one is too big for you. A circle with circumference 62cm has radius 9.9 cm and a circle with circumference 57cm has radius 9.1 cm. So, with the helmet centred on your head, you'd be able to move it roughly a centimetre in any direction.
Comment: Even assuming it was the right size, it's advisable to try a helmet on before buying in.  It's not something soft and flexible like a baseball hat.  Last time I bought a helmet I had to try on at least 4 or 5 different models of the correct size before I found something comformtable.  They all fit around my head properly, but had other issues that cause them to fit incorrectly.


Short circuit protection stage in the 741 op. amp

Question: The 741 op amp has a rugged design with a short circuit protection feature if any one could shed the light on this mechanism where I have found it to be on of the most power efficient protection circuits that would not drain the power of a battery.
I would really appreciate it and thank you in advance for you efforts!
Comment: [This pdf goes into some detail about how the short circuit protection works.](https://coefs.uncc.edu/dlsharer/files/2012/04/G5.pdf). I don't feel like summarising, so just the link.
Answer: If you look at the circuit diagram of a 741

you will see this at the top of the output stage...

The output path is through Q14 and R6. When the current though R6 is high enough it turns on Q15 which starts to turn off Q14. This limits the amount of current it can ever source to about ~0.6V/27R = ~20mA.
It is your basic current limiter circuit.
A similar circuit, PNP version, exits below that to limit the sink current.
Comment: I do something similar for resolver excitation circuits to ensure they are short-circuit protected


What is the difference between executable and relocatable in elf format?

Question: What is the difference between executable file in elf format and relocatable file in elf format?
Answer: as you know every compiled executable file is a binary file with address relative and absolute, so  relocatable format is a format in which function and other symbols still have there names definition in other word functions and variables are not bound to any specific address. Instead, the addresses are still symbols
look :
unix > gcc -c main.c
unix > readelf --symbols main.o
Num:     Value   Size     Type      Bind       Vis     Ndx  Name
  0:  00000000      0   NOTYPE     LOCAL   DEFAULT     UND
  1:  00000000      0   FILE       LOCAL   DEFAULT     ABS  main.c
  2:  00000000      0   OBJECT    GLOBAL   DEFAULT     3    buf
  3:  00000000      0   OBJECT    GLOBAL   DEFAULT     1    main

unix > gcc main.c -o main
unix > readelf --symbols main
Num:     Value   Size     Type      Bind       Vis     Ndx  Name
 53:  08048460      2     FUNC    GLOBAL   DEFAULT     13   __libc_csu_fini
 54:  08048462      0     FUNC    GLOBAL   HIDDEN      13   __i686.get_pc_thunk.bx
 55:  0804a018      4     OBJECT  GLOBAL   DEFAULT     13   bufp0

do you see what i'm talking about
most of the time we use them as static library
look to the example here :
#ifndef MATH_H
#define MATH_H
int add(int a, int b);
#endif

/* math_test.c */
#include <stdio.h>
#include "math.h"

int main(void)
{
  int result = add(1, 2);
  printf("result: %d\n", result);
  return 0;
}

try to compile it
unix > gcc math_test.c -o math_test
/tmp/cclRibQq.o: In function `main':
math_test.c:(.text+0x19): undefined reference to `add'
collect2: ld returned 1 exit status

that due to function add has no body in math_test.c so we can do the flowing:
int add(int a, int b)
{
  return a + b;
}

then compile it as a relocatable using gcc
unix > gcc -c math.c                      # Create relocatable obj file (math.o)
unix > readelf -h math.o | grep Type      # Read math.o with readelf
Type:            REL (Relocatable file)   # and verify its type

then  you can link it with linker like this :
unix > gcc math_test.c math.o -o math_test
unix > ./math_test
result: 3

a great article about the difference between executable file in elf format and relocatable file in elf format you can find here
Comment: Please summarize it here, so that the information is preserved if the link dies.
Comment: @sorak can you take a look and till me if any other thing is wrong to fix it thank you
Comment: Good work, thank you. I added syntax highlighting. Check out how that works.
Answer: As it can be seen in the image below, relocatable ELF goes as the input to the linker, whereas the executable ELF is the product of the linker.
Answer: ELF executable, as we can understand from its name, is a file that can be executed. This file can be generated from C code for example.
The process of relocation is fixing the addresses of labels and symbols which were created in the code. For example, if you write a program in assembly language and you look at the listing file of your source code, you'll find some places where [00000000] is written instead of a label mentioned at this line. This zeroes mean that he linker uses relocation in order to fix the address the its future value.
Answer: Relocatable doesn't have any Load address its only binary code sequence with offset (for example offset related to main() func). But,executable file having load address not just offset related to any of the function.
one more fundamental difference between them is Executable have bootstrap application but relocatable doesn't have it.


How can I make this possible?

Question: I'm making a map.  I want the player to create a stone button but they are on adventure mode so i need a command that makes it possible to be placed on something so when they make the button it can be replaced with a button that can be placed on dirt.
Comment: Take a look at [this](http://gaming.stackexchange.com/questions/199411/how-can-i-allow-placement-of-certain-blocks-in-minecraft-adventure-mode) question.
Answer: Is this what you wanted? (Gfycat Link)
This is the contraption I used:

1 is Repeat, Unconditional and Always Active
<code>/testfor @p {Inventory:[{id:minecraft:stone_button}],CanPlaceOn:["minecraft:dirt"]}
</code>
2 is Repeat, Unconditional and Needs Redstone
<code>/testfor @p {Inventory:[{id:minecraft:stone_button}]}
</code>
3 is Impulse, Unconditional and Needs Redstone
<code>/clear @p stone_button
</code>
4 is Chain, Conditional and Always Active
<code>/give @p minecraft:stone_button 1 0 {CanPlaceOn:["minecraft:dirt"]}
</code>

How can I make this possible?


How should I mention my Chinese professor in a document?

Question: I really don't know Chinese but I just started to study in China online. I am preparing a report and I need to mention my professor's name. In Chinese it's written as 刘雨欣(liuyuxin), this means 刘(liu) is her surname. Should I write Professor Liu Yuxin or Professor Yuxin Liu?
Answer: Both 'Professor Liu Yuxin' and 'Professor Yuxin  Liu' are acceptable (apply the first name first or the last name first format depend on the audience). To be safe, just highlight the family name with an underline or bold fond.
It is easier just use the last name 刘教授 (Professor Liu)
You can call your professor 刘雨欣教授 or 刘教授 in Chinese but you cannot call her 雨欣.刘 教授 (apply the English first name first format directly into an all Chinese name)
Edit: capitalize the last name also work
Comment: Unfortunately, her whole name should be underlined and in bold so I can't change it. But if it makes no difference as you said, I will go with Prof. Yuxin Liu.
Answer:
Just to supplement, you may ask if your professor has a specific preference / or is famously known for her name in the Western or Chinese order. Consider Lee Kwan Yew (李光耀), but Yo Yo Ma (馬友友).

There may also be a good justification to adopt the Western order, especially if your professor's work needs to be cited in her surname. E.g., Yitang Zhang (張益唐) is a mathematician who developed groundbreaking work in the field of number theory, but there were subsequent improvements on his work. The academic field may therefore need to use the Western order for its members to unambiguously and correctly cite his work as Zhang's work.

Wikipedia concludes several rules of thumb which you may find useful.
Besides Tang Ho's suggestion, you may also capitalise the surname, as in 'Yuxin LIU', but that may be unnecessary.
Comment: For an unknown person like me, it really depends on the audience. I was in Hong Kong,  applying for an ID, I wrote my name in English (first name first ) order as Tang Kai Ho. The officer thought my last name was Tang and called me 鄧繼豪先生 which was her educated guess for a typical Hong Kong Chinese who named 'Tang Kai Ho'-- Not a single character was correct
Comment: Thanks for your clear explanation. Actually, before I sent an email with Dear Prof. LIU Yuxin, but in this document, all other names are in initial capital format.
Comment: @TangHo that is rather unfortunate. That's why I prefer adding a comma after my surname when filling out forms that don't separate first names and surnames.


What is the Difference Between Tax Class : Texable Goods And Shipping

Question: In my store i set shipping rates for individual states and countries based on zipcodes. Pls guide me what i select in the stage of tax class.
Comment: You can get more information from here : https://custommagentodevelopmentindia.wordpress.com/2013/06/06/complete-guide-to-setup-and-manage-taxes-in-magento-magento-development/
Answer: Product Tax Class is a type of Product that is being purchased. When you create a Product, you will assign a Product Tax Class to that Product.
Magento comes with two Product Tax Classes:Taxable Goods and Shipping.
The class Shipping is applied to shipping charges because some places charge sales tax on shipping. If your customer's sales tax is different for different types of Products, then you will need to create a Product Tax Class for each type of Product.
You can refer below links for more information

link
link


texture stretched

Question: made 1 model after duplicate remove doubles then delete the edges in center. it would make the texture stretch, i delete edges to reduce poly count but texture stretched , how to do it without texture stretch out.
Answer: For your model you want to make sure that you have properly applied the UV's for your model. Add seams to areas where it's needed and then unwrap the model each time you make changes to see the final result. verify that the UV's aren't stretching where they shouldn't. You can use the stretch tool to assist with this.

The darker the blue the cleaner the final result will be.

texture stretched


Can "happy" be a verb?

Question: Can we say "happy" in sentences like "Happy reading!" is a verb and means "enjoy"? If not, what is that?
Answer: The sentence, "Happy reading!" is the same as "I wish that you have happy reading!"
The I wish that you have is understood. In English we say, "Happy Holidays" which means we want you to have a happy holiday season. We're just lazy like that!
Answer: In:

happy reading

"happy" is an adjective that qualifies the noun "reading". "Happy" cannot be a verb, there's no verb in that sentence but the verb "have" is implied:

I wish you have a happy reading,

meaning : "I wish you enjoy your reading."
Answer: No happy isn't a verb.  It is an adjective.  The fact is that the statement happy reading actually means I wish you will/did enjoy your reading...  P.S. Remember that happy isn't a verb.  It's an adjective.

Can "happy" be a verb?


Why do I need internet to listen to my music library?

Question: I was taking a walk yesterday, and I was listening to music in my personal library. No Apple Music subscription. After awhile, my music stopped. I checked, and it said something about needing internet to access music and turning on cellular music in settings or something like that. It happened again today at the mall.  I'm confused... I thought that my personal library was on my device. Is it? By the way, I am using an iPod touch 6th generation and running (gulp) iOS 9 public beta. My best guess is it's the beta. Maybe they don't want all my music on my device in case I have to reset it. But wouldn't it be backed up to iTunes?
Comment: Tell me what you see here: If you go to the Music app -> My Music and then click on the header above the song list (to switch between Songs, Albums, etc) - the value for "Show Music Available Offline" is it Green for on or white for off? If white, that means it will also show music available via iCloud, and that could be causing the connection problem.
Comment: @smoosher I can't find the header of which you speak - Is there a title on it?
Comment: I will attach it as an answer so you can see the photo but please do not accept it as an answer - if it solves your problem, I'll reword it later.
Comment: @smoosher Hmmm... It says "Show iTunes purchases" on mine... and it's on.
Comment: could you please attach a screenshot of what you see to your question?
Comment: @smoosher Here you go. Maybe we should move this to chat.
Comment: Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/26369/discussion-between-smoooosher-and-oboy).
Comment: Agree entirely the old iPod I had for years worked so much better and easier to use!!
Answer: Yipee! I figured it out! My problem is, as I forgot to mention in the question, I just got a new iPod. Aaaaaaand, I never realized it, but apparently when you get a new device, you have to download all your music into your library. Otherwise, you can still listen to it, but only over internet. Probably to save space. Downloading your songs can be done by tapping on the song or album, and then tapping "Make available offline".
Thank you everyone for your help!
Answer: I solved the problem which happened after moving to ITunes 12 4 3 and Windows 10 by restoring the iPod to factory settings and then reloading the library.
I can't say I like the new ITunes programme. It certainly is inferior to what went before.


WordPress "Billing Name and Address" fields block not displaying

Question: On our contribution page, I can select the amount and confirm, but then on the final screen the Billing Name and Address fields block does not display at all.
Comment: Please include the version of CiviCRM and the payment processor you are using.
Comment: Does it display on the first page and allow you to submit values?
Answer: This site was hacked prior to me working on it. I ended up restoring a backed up version of the plugin folder and the "files" folder. Now everything is working fine.


Word for the fear of getting a haircut?

Question: The other day I met someone who was scared of getting a haircut. I was trying to think of a word to describe, but I couldn't think of anything. I couldn't find anything online either, so I decided to ask here. What is a word for the fear of getting a haircut?
Comment: Are you talking about an *urgent haircut*?  A haircut on the stock market?  Assassination?
Answer: The usual protocol for naming irrational fears is to add the Greek-derived suffix -phobia to the (classical) Greek word for what is feared. Classical Greek for haircut is κουρά, so with the usual Latinization we would get curaphobia. Unfortunately that form would appear to have been applied to a fear of paperwork by Urban Dictionary and its ilk. Tonsurephobia has been suggested, and attributed to "experts"; it combines a Latin term with the Greek suffix, which some regard as barbarous. Neither word is in the OED.
Comment: Here's another site with an article entitled: [Help With Tonsurephobia](http://www.thriftyfun.com/Help-With-Tonsurephobia-Fear-of-Hair-Cuts.html)
Answer: Fear of Getting a haircut is Tonsurephobia.

Tonsurephobia is the fear of getting a haircut. The word Tonsurephobia
  is derived from Greek words 'Tonsure' meaning ' to cut' and 'Phobos'
  meaning 'deep dread or aversion'.

[Fearof.net]
Comment: What is the link for "haircut" for? Please don't link anything that is not related with your answer. [If you use direct quotations, always put them in quotation marks or in a quotation block.](http://meta.english.stackexchange.com/questions/9602/what-is-plagiarism-how-do-i-avoid-it-how-do-i-address-it-when-i-see-it)
Answer: Smart me went to google translate to get the Greek word for haircut, which turned out to be κούρεμα (kourema).
Thus kouremaphobia.
And kourema mania should be haircut mania. Whereas, kouremaphilia would be love for haircuts.
And I further found that kouremata is various ways to cut and style your hair.
Would kourematomania be a situation where hair-styling is an obsession?
And since koureas equals barber, then the words koureasphobia, koureasphobic, koureasmania, koureasmanic, koureasphilia, koureasphilic, carry meanings respective to their suffixes.
Comment: Indeed κούρεμα is *Modern* Greek for *haircut,* including the kind imposed on holders of Greek bonds in recent years. But for purposes of coinage in English, tradition favors classical Ionic dialect (and not classical Attic, where the two differ, as witness *thalassemia,* not *thalattemia,* for the Mediterranean version of sickle-cell anemia).
Comment: The only ionic bond I was schooled in is between atoms or molecules. Just kiddin.

Word for the fear of getting a haircut?


How to understand the concept of compact space

Question: the definition of compact space is: A subset K of a metric space X is said to be compact if every open cover of K contains finite subcovers. What is the meaning of defining a space is "compact". I found the explanation on wikipedia : "In mathematics, more specifically general topology and metric topology, a compact space  is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space. " I can not understand it, or at least I can not get this by its definition. Can anybody help me?
Comment: Well suppose you have compact metric space $X$, consider for any $\varepsilon>0$ the covering $\{B_\varepsilon(x)|x\in X\}$. It has a finite subcovering. Now take any infinite sequence $(x_i)_{i \in \mathbb{N}}$. Each $x_i$ is contained in one open set of this finite subcovering. Hence (for fixed $\varepsilon>0$) there must be a point $x$ such that infinitely many $x_i$ are contained in $B_\varepsilon(x)$.
Comment: The intuition of the idea comes from metric space theory: suppose you took an 'infinite number of steps' to get from point A to point B, then each step could be covered by a ball around the centre of the step. The worst you could have done to take an infinite number of steps would have been to 'traverse the whole space' (so your balls cover the space), but then compactness would guarantee that a finite number of them would have done the job anyway. Very roughly. Really what you're after is the equivalence of sequential and covering compactness. See: any metric space textbook ever.
Comment: How the standard definition of compactness came about is a bit mysterious to me as well, although perhaps less so than the  definition of a topology itself:

http://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets
Comment: Closed; mathoverflow is for questions of interest to research mathematicians.
Answer: Some heuristic remarks are helpful only to a subset of readers. (Maybe that's true of all heuristics, as a meta-heuristic - if everyone accepts a rough explanation, it's something rather more than that.)
Non-compactness is about being able to "move off to infinity" in some way in a space. On the real line you can do that to the left, or right: but bend the line round to fill all but one point on a circle (which is compact) and you see the difference having the "other point" near which you end up. This example of real line versus circle is too simple, really. Another way you can "go off to infinity" in a space is by having paths branching out infinitely (as in König's lemma, which supplies another kind of intuition).
Compactness is a major topological concept because the various ways you might try to "trap" movement within a space to prevent "escape" to infinity can be summed up in a single idea (for metric spaces, let's say). The definition by open sets is cleaner, but the definition by sequences having to accumulate on themselves (not necessarily to converge, but to have at least one convergent subsequence) is somewhat quicker to say. If you restrict attention to spaces that are manifolds, you can think of continuous paths and whether they have to wind back close to themselves or not.
Comment: Your answer gives me a very intuitive view to compactness! Thanks very much!
Answer: Terry Tao has a nice explanation in the Princeton companion to maths. The article's also on his website:
http://www.math.ucla.edu/~tao/preprints/compactness.pdf
Comment: +1. I have been searching, for a while, for a good motivation of compactness. I read the article by Terry Tao and it is really illuminating. Thanks for the link!
Answer: Etymology may help. Compact, from the Latin Compactus, past participle of the verb Compingere: "to pack together closely and firmly". You may have an idea of how strong is the hypothesis of compactness if you look at what happens when even total boundedness is lacking.
Just consider e.g. the possibly most familiar infinite dimensional object, the separable Hilbert space H=l2. Its unit closed ball is not compact. A continuous real valued function on it may be unbounded, or bounded without minimum and maximum value. A linear form on H may be everywhere discontinuous and locally unbounded. There is a continuous transformation of the unit closed ball with no fixed points. The ball itself retracts on its boundary. Infinitely many disjoint unit open balls may be packed within a ball of radius $1+ \sqrt{2}$. The linear group GL(H) is connected. There are injective linear continuous transformations of H that are not surjective....
Answer: In elementary analysis one learns that every continuous function
from a closed bounded interval to $\mathbb{R}$ is bounded, but this
is not the case for open or unbounded intervals.
A little later one learns that each continuous function
from a closed bounded subset of $\mathbb{R}^n$
to $\mathbb{R}$ is bounded, but this fails for other subsets of
$\mathbb{R}^n$. The key properties are that a subset of
$\mathbb{R}^n$ is compact iff it is closed and bounded
(Heine-Borel theorem) and that the image of a compact space
under a continuous map is compact. Compactness is a sufficient
condition on a space to ensure that all continuous functions
to $\mathbb{R}$; moreover compactness is a purely topological
property, definable in terms of open sets.
The wikipedia quotation is a bit vague, but it refers to a property
called sequential compactness, which all compact metric spaces have.
This means that for any sequence of points in the space $X$ there is
a subsequence converging to a point of $X$. A metric space is compact
if and only if it is sequentially compact, but this does not hold
for non-metrizable topological spaces.
I don't like the wikipedia quote, as it sort of suggests that sequences
in compact spaces must be convergent; this is not so, though they must have
convergent subsequences.
Answer: A very nice exposition was given by Hewitt in this article in the Monthly; the central thesis is that compactness facilitates proving results that are nigh on trivial for finite sets in wider settings.
Comment: Yep.  I've heard people informally describe a compact set as one that is "almost finite".
Answer: In some way, you can ALWAYS think about compactness as "sequential compactness" provided you allow for things a bit more general than sequences- nets. Compactness is equivalent to the statement that every net as a convergent subnet. If a space is metrizable, this is equivalent to the analogous statement, where you only need to consider sequences. I prefer, in fact, to think in terms of ultrafilters and say that a space is compact if and only if every ultrafilter has a limit point. I suggest reading up about nets and ultrafilters. I found http://www.math.ksu.edu/~nagy/real-an/1-02-convergence.pdf a nice quick introduction to give you the idea (but it does not classify compactness). A more comprehensive reference might be http://alpha.math.uga.edu/~pete/convergence.pdf (but note here compact means compact Hausdorff, while quasi-compact means compact and possibly not Hausdorff).


How do I enable access to Google API sensitive scopes with 'Unverified' mobile app under development?

Question: I am developing a mobile app (Flutter) that is designed to access a user's Google Calendar events. I have learned that this has a newer process that requires verification of the Oauth consent screen in some circumstances.
I have configured everything on the 'Oauth Consent Screen' as explained in these instructions:
https://support.google.com/cloud/answer/6158849?hl=en&ref_topic=3473162
I have read through the FAQ here:
https://support.google.com/cloud/answer/9110914?hl=en&ref_topic=3473162
I have read that without 'verification' you can use restricted scopes up to 100 users and the Google signin process will display 'Unverified App' as explained here:
https://support.google.com/cloud/answer/7454865
My experience is different and I am hoping to get some guidance. As an aside, I also use GSuite.

I marked my app under development as 'External' because I need to test the app with beta users (less than 10 users) that will be random email addresses -not- within Gsuite, including personal gmail addresses.
When I did this, it automatically set the status of my app to 'Being Verified' for Verification Status
I received an email that in part said: *Thank you for your patience while we reviewed your project.
Unfortunately, testing/development apps are inapplicable for the Verification process.
If your app is only for testing purposes, we recommend that you continue to use your app with the "unverified app" screen intact. *
Ok fine, but the thing is I do 'NOT' get an 'Unverified app' screen, what I do get is a consent screen that does -not- list the sensitive scopes I selected in my consent screen.

Here are the scopes I have selected in the consent screen
Here is how my consent screen appears
In my app, of course - attempting to access the calendar API results in "Insufficient Permission: Request had insufficient authentication scopes."
From what I gather, there is scenario where I create a project that remains 'unverfied' but can access sensitive scopes. Is this correct? Can someone point me to what might be wrong or point me to instructions for setup of Oauth for a mobile app under development?
Answer: My case was a bit different, but I ended up with that I sent wrong request. Scope in my original request was wrong, so Google did not grant needed rights.
On the other hand, while application (registered in Google) is under development we can configure any sensitive scopes without additional checks from Google side. It is working like that at least for me.


Wait for UnityMainThreadDispatcher to return Unity

Question: I'm trying to move my pathfinding over to a separate thread. But I still have 1 function that I need to call from the Main thread at each node in the search. I found the UnityMainThreadDispatcher here which aims to fix this problem by running the given action on the main thread. However, I can't continue in my function until I have the value returned from this action.
Here is the code that i'm currently trying to make work.. When I debug this code, a valid returnVal is returned in the callback, but when I try and wait for this value, i.e. by using a while loop, I get stuck in an infinite loop. Why is this?
var latlon = Vector2d.zero;

UnityMainThreadDispatcher.Instance().Enqueue(() =>
{
    Debug.Log("Running on Main thread");
    _monoBehaviour.StartCoroutine(
        GetLatLonFromUnityCoords(point, returnVal =>
            {
                    latlon = returnVal;
                    Debug.Log($"Calculated latlon: {latlon}");
            }
    ));
    Debug.Log("Finished on Main thread");
});

while (latlon = Vector2d.zero) { }

// latlon has been returned, continue

I know this isn't good to do, what can I do instead?
Thank you
Answer: I've tried a few different Unity multiple thread plugins and haven't had great experiences. I'm not saying "UnityMainThreadDispatcher" can't be a good solution, but it seems like running Unity specific functionality on multiple threads would be a smoother, better documented process if the Unity team tackled it. Also, it's good to avoid including extra plugins if you can help it. That said:
Depending on the contents of your one "function that I need to call from the main thread", you could slice up your code into more modular pieces and make that main thread call in between firing async tasks. For example, I have a process heavy task that requires a loading bar for the user. I create a global variable that an await Task.Run() builds on, then I make updates/retrieve data from the Unity API on the main thread (including updating the progress bar), and then I go into the next await Task.Run().


Uhh... Hello? Why are there so few questions here?

Question: There are only like 273 questions excluding this one. Mean while, there are over 18,000,000 on SO. Why is it so lonely here? Is anybody home?
Comment: Earth science may not be as popular as programming in general. On the other hand, about 10 million of the 18 million questions on SO are really bad quality, many of the others are mediocre. There is a danger that smaller communities may never really fly and the way out could be increasing the scope, for example allow scientific software usage questions maybe or fusing with another similar community. I personally would be happy if there was only a single natural sciences exchange. But then others may like it differently.
Answer: You are on Earth Science Meta.  This is the site to ask question about the site itself.  Those are naturally fewer in number.
There are currently 5,008 question on the main Earth Science page.  This is much less than on Stack Overflow.  This is because the site is lesser known, newer, and covering a smaller topic.  Earth Science will always be smaller than Stack Overflow.  This is not a problem, as long as Earth Science gets a critical mass of users and a good signal to noise ratio.
Note that Stack Overflow has 41,486 question on meta vs. 18,912,994 on main, or about 0.2% meta vs. main questions.  We have 274 meta questions vs. 5,008 main questions, or about 5.5% meta vs. main, so relatively speaking, our meta is much more active than the one on Stack Overflow.  This is probably also normal because we are a smaller site.
Comment: If $N$ is the number of main-site questions, then SO has $9.5 \sqrt{N}$ meta questions and we have only $3.9 \sqrt{N}$.


Beta multiplier reference

Question:
In the beta multiplier reference circuit, what is the function of start-up circuit? Can we use the beta multiplier without the start-up circuit?
Comment: The loop around the bottom left corner from MSU3-d to MSU1-gd to MSU3-g with Iref Ref =Vref offset
Comment: For fast startup you need a fast negative feedback bias for n and p. This circuit has a current source with negative feedback to do this . Can you see it?
Comment: Can you please specify which negative feedback are you referring to?
Answer: Zero current in the left-hand branch and zero current in the right-hand branch is a possible steady-state operating point. If you powered up this reference without a startup circuit the reference current would be zero.
MSU2 is a small pull-up, which turns on MSU3, which kickstarts the feedback loop in the beta reference. One the reference hits steady-state, MSU1 overdrives the pull-up and turns of MSU3, leaving the startup circuit high-impedance.
Comment: Can you please clarify what will be the gate voltages of pmos (M3,M4) and nmos (M1,M2) of beta multiplier during the steady state?
Answer: I'm now studying this topic, so I'll try to add something more about it; maybe it could be useful for future users..
Beta-Multiplier is a self-biasing circuit and, as in any self-biased circuit, there are two possible operating points: the desired one and the unwanted one, where zero current flows in it.
In the circuit above, the unwanted state occurs when the gates of M1/M2 are at ground while the gates of M3/M4 are at VDD.
In this situation, the gate of MSU1 is at ground and so it is off.
The gate of MSU2 is somewhere between VDD and (VDD - |VTH,P|).
So MSU3, which behaves like an NMOS switch, turns on and leaks current into
the gates of M1/M2 from the gates of M3/M4.
This causes the current to snap to the desired state and MSU3 to turn off.
Now Beta-Multiplier works in the desired state and the start-up circuit is off.
PS: This circuit is suitable if we are using long-channel MOSFETs.
Indeed, if we use short-channel MOSFETs, the performance of this version are poor, because Iref becomes sensible to the variation of VDD.
To overcome this, a different version can be used (with an additional differential amplifier) to have a better power-supply sensitivity.

Beta multiplier reference


How ~てある and ~ておいた differs?

Question:
晩ご飯が作ってある
Dinner has been made. (intentional)
晩ご飯を作っておいた.
(I) made dinner (so I can eat it).

They seem quite the same.
Btw what are some cases in which -てあった are used? I can't recreate one.
Answer:
1)「晩{ばん}ご飯{はん}が作{つく}ってある。」
Dinner has been made (and it is still there untouched.)
2)「晩ご飯を作っておいた.」
Someone made dinner some time ago (so that I/someone could eat it later on).

The difference in meaning between the two sentences would depend heavily on the current situation of the meal that one made.
Sentence #1, even without any further context, would always mean that the meal (or at least the speaker's portion) is still there uneaten and available as the speaker utters the sentence.  This is because of the use of 「~~ってある」.  "Dinner was made and it is still there."  The temporal point of reference is NOW.
Sentence #2 without any further context, however, leaves one unsure of what has happened to the dinner that was made.  Someone may already have eaten it or it may still be there untouched.  The temporal point of reference is in the PAST.
Thus, the two sentences would describe practically the same situation only on the condition that the dinner in question is still there.
Moving on to 「~~てあった」...

「晩ご飯が作ってあった。」

means:

"Dinner had been made."  (You found that situation some time ago.)

The temporal point of reference is in the past.  For instance, you could say "When I got home from school yesterday, dinner had already been made."  The meal was there ready to eat.  Once again, the sentence says nothing about whether or not you actually ate it.
Answer:
~ておく is used for doing something in advance, so one doesn't have to do it later

~てある simply describes the state of something already having been done

晩ご飯が作ってある。
Dinner is ready.

晩ご飯を作っておいた。
I already prepared dinner (in advance) [so that we wouldn't have to make dinner now].

The difference in meaning is I think the emphasis of ~ておく on the fact that something was done as a preliminary measure / pre-emptively / in advance.
There is also an important difference in grammar: In the first sentence 晩ご飯 is the subject; in the second sentence 晩ご飯 is the object and the subject is missing (and only implied by the context).
Answer: According to "A Dictionary of Basic Japanese Grammar", page76:

ある: Something has been done to something, and the resultant state of that action remains.
Examples:
  それはもうジョンに話してある。
  It has been told to John already.
飲み物はもう買ってあります。
  Drinks have already been bought.
窓が開けてある。
  The window is open.
おく: do something in advance for future convenience.
Examples:
  話しておく: talk in advance.
  食べておく: eat something in advance.

Verbて+ある: represents a state.
Verbて+おく: represents an action.

How ~てある and ~ておいた differs?


What actually happens to a light ray on a Schwarzschild black hole horizon?

Question: I know the Schwarzschild event horizon is a null surface generated by null geodesics. But what does that actually mean in terms of the path of a light ray that reaches it? Does that mean the geodesic trajectory of light on the surface will be along the surface? Which way will the light ray go? And what equation shows this?
Comment: Related: http://physics.stackexchange.com/q/46258/2451
Answer: The incoming light ray can pass through the horizon.  The easiest way to see this is to imagine light cones in a 2d (1 space + 1 time) spacetime.  As you get near the horizon, the light cones are more and more tipped up.  A light cone in 1+1 dimensions consists of two lines.  At the horizon, imagine one of these null lines being the null geodesic horizon generator through that point.  The other one represents the path of an incoming light ray.  This is nicely illustrated on the Eddington Finkelstein diagram in this wikipedia article.
The Eddington Finkelstein coordinates are a system that's non singular on the horizon (Kruskal Szekeres is another).  In this coordinate system, the incoming light ray is just represented by the equation $v=const$.
Comment: Thank you, I know this - but how does the fact that the horizon is a null surface and null geodesics are tangent to it affect the movement of light? Will it move tangent to the surface?
What happens if I, falling into a black hole, turn on a flashlight tangent to the horizon just as I reach the surface?
Comment: @Judy: exactly.  You can think of the light ray just sitting at the surface at a constant $r$, neither escaping nor falling into the horizon, until it intersects with future timelike infinity.  At least to the extent where you can take these sorts of metaphors seriously.
Answer: Provided you're willing to consider only radial light rays, i.e. normal to the event horizon, I calculated the behaviour at the event horizon in my answer to Why is a black hole black?.
You have to give some thought as to what you mean by the velocity of the light ray at the event horizon. The calculation I did uses Gullstrand-Painlevé co-ordinates and these don't correspond to physical co-ordinates measured by any observer. Nevertheless the result is quite convincing i.e. that for a light ray directed outward the velocity at the event horizon is zero.
I may be misinterpreting your question (if I am ignore this paragraph) but you seem to be suggesting that the null geodesics run tangential to the surface, i.e. constant $r = r_s$, so a light ray tangential to the event horizon would circle the event horizon. That doesn't happen. There are no null geodesics tangential to the event horizon. The closest distance that a null geodesic at constant $r$ exists is as $r = 3M$ i.e. the last stable orbit.
Comment: If there are no null geodesics tangent to the event horizon than why is it a null surface?
Comment: From Wikipedia (http://en.wikipedia.org/wiki/Null_surface): *In the theory of relativity, a null surface is a 3-surface whose normal vector is everywhere null (zero length with respect to the local Lorentz metric), but the vector is not identically zero. For example, light cones are null surfaces.* The event horizon is a null surface because light rays normal to the surface have a velocity that is zero at the surface. See the link in my article for the proof that the velocity of a radial light ray is zero at the event horizon.
Answer: Before you reach the horizon region you can find an interesting region, called photon sphere that will be full of unstable photon orbits, that spire down the black hole or escape to infinity.
On the other hand, to an in-falling observerat black hole, nothing special happens at the horizon. At least classically! The same is true for light. This is a result of equivalence principle. In particular, an in-falling observer is unable to determine exactly when he crosses the event horizon, as it is impossible to determine the location of the boundary from local observations. The only reason it is special is that it is a causal boundary i.e. once you cross it there is no way to go back.
In the quantum case, this question is subject to a recent heated debate, to what exactly happens at the horizon. This debate started with the information loss paradox. One of the logical possibilites, could be that our current local description fails near the horizon and we are sensitive to non-local quantum gravity. For example look at Black Holes: Complementarity or Firewalls?


Do physical laws require 2nd derivatives

Question: Is it a happy coincidence that second degree differential equations approximate reality or a necessity?  They describe how a system will evolve from one state to the next, but surely the ultimate laws of nature will not depend on two prior initial conditions. Surely reality can compute its next state from the current state alone.
Comment: _But surely the ultimate laws of nature will not depend on two prior initial conditions._ Why? What do you mean?
Comment: Why 2nd derivatives? 0th and 1st derivatives are common. 4th derivatives occur in elasticity theory. ... Reality does not compute anything. We do all the computations.
Comment: Possible duplicates: http://physics.stackexchange.com/q/18588/2451 , http://physics.stackexchange.com/q/4102/2451 and links therein.
Comment: Of course,such as the Newton's second law.
Answer: Actually if you think of classical (and non-relativistic) physics, the canonical equations of motion of the Hamiltonian formalism
$$
\dot p=-\frac{\partial H}{\partial q}\, ,\qquad \dot q=\frac{\partial H}{\partial p}
$$
are first order.  In addition, the Hamilton-Jacobi equation
$$
H+\frac{\partial S}{\partial t}=0\, ,\qquad p=\frac{\partial S}{\partial q}
$$
is also of first order.
It is true that in Lagrangian mechanics the equations of motions are of second order, but my personal appreciation is that the Hamiltonian formalism - because the equations of motions are expressed in a more symmetrical way amenable to symplectic transformations and because it leads to deeper results such as Liouville's theorem and other associated invariants - is more fundamental.
Comment: Are you saying the Hamiltonian formalism retains more symmetry than the Lagrangian formalism? I thought it is the other way around. Certainly, if a theory contains a symmetry it would be a symmetry of the Lagrangian.
Comment: @flippiefanus Good point.  I clarified my answer following your observation.  The Hamiltonian formulation of classical mechanics of point particles at least is an example where the equations need not be 2nd order.  Granted from field theory perspective the Lagrangian formulation is much more flexible.


Finding a Parabola from its height and second y-intercept

Question: I would like to know how to find the quadratic equation for a parabola that opens down and intersects the origin along with the vertex being in the first quadrant given the maximum of the parabola and what the x value of the second intercept is? Thanks
Comment: You mean "second x- intercept".
Answer: A parabola that opens up or down has equation
$$y = a(x-c)^2 + d$$
with $a,c,d$ real numbers. If $a\gt 0$, then the parabola opens up. If $a\lt 0$, then the parabolar opens down. The vertex of the parabola is at $(c,d)$.
If you know the parabola opens down, then you know $a\lt 0$. If the vertex is on the first quadrant, then you know that $c\gt 0$. If you know the maximum of the parabola, then you know the value of $d$ (it's the maximum). If you know that it goes through the origin, then you know that $0= a(0-c)^2 + d = ac^2+d$, so you know that $ac^2 = -d$. You need to know the value of $c$ (the location of the vertex) or the value of $a$ in order to completely determine the parabola.
If you happen to know that the other intersection with the $x$-axis occurs at a point $(b,0)$, $b\gt 0$, then the vertex will lie halfway between $(0,0)$ and $(b,0)$ (since the axis of the parabola is vertical), so $c = \frac{b}{2}$ and this gives you the necessary information ot find $a$, namely we have $a = -\frac{4d}{b^2}$, so the equation becomes
$$ y = -\frac{4d}{b^2}\left(x - \frac{b}{2}\right)^2 + d,$$
where the parabola goes through $(0,0)$ and $(b,0)$, opens down, and has highest value $d$. You can verify that $y=0$ when $x=0$ and when $x=b$, and that the maximum value is attained at $x=\frac{b}{2}$, where we have $y=d$.
Comment: That was exactly what I needed. Thanks
Answer: Another way of doing this: A parabola, opening downward, with x-intercepts at 0 and $x_0$ can be written in the form $y= -Cx(x- x_0)= -C(x^2- x_0x)$.  Completing the square, $y= -C(x^2- x_0x+ x_0^2/4- x_0^2/4)= -C(x- x_0/2)^2+ Cx_0^2/4$.
The highest point is at $x= x_0/2$ and that height is $Cx_0/4$.  I we want that height to be h, we must have $C= 4h/x_0$.
The parabola that crosses the x-axis at 0 and $x_0$ with height h is $y= -(4h/x_0)x(x- x_0)= -\frac{4h}{x_0}x^2+ 4hx$


Is there a quiet area to do some work on a laptop on a table before passing the security at SFO airport?

Question: I wonder whether there is any quiet area to do some work on a laptop on a table before passing the security at SFO airport. I know that after the security, there are some tables that one can use to work on a laptop, but I don't know about before passing the security.
Answer: I don't think there are quiet areas in airports, except for maybe some lounges during low occupancy hours.
At the SFO airport there are some tables and seating area at the food court of the international departure hall that is before security. I don't think there's much in terms of seating space or quiet in the domestic terminals before you go through security.
Comment: "I don't think there are quiet areas in airports": sometimes there is, e.g. SLC airport top floor above the departure area. It has a few relatively quiet tables.


Are topics experienced later in the day more likely to be present in dreams?

Question: Is there any evidence suggesting that thoughts or experiences from late in the day are more likely to be dreamed about the following night compared with similar experiences earlier in the day?
Comment: Couldn't find anything on the subject.  There has been research done on the effects of thought suppression on dreams:  http://www.mendeley.com/research/the-tendency-to-suppress-inhibiting-thoughts-and-dream-rebound/ and http://www.ncbi.nlm.nih.gov/pubmed/15043639.  In that last link, the researcher did ask part of the control group to think about a person before bed.
Comment: Interesting question. I'm no expert on dream research, but I remember hearing about some dream norms that might be relevant to your question: http://www2.ucsc.edu/dreams/Norms/index.html and there's a discussion of dream research methods [here](http://www.all-birds.org/Dreams/dreams017.htm)
Comment: @Bradford `did ask part of the control group...` well they're not really the control group anymore then...
Answer: The probability of what you'll dream is really dependent on a ton of factors. Stress can harm your ability to remember, and process, your memories during sleep. If you are highly stressed due to hunger later in the day, you brain may be more likely to process the memories of events before hunger set in. However, the opposite may be true for others. schizophrenics for example, have problems with how they process memories. Our brains can accidentally be configured to process memories in an atypical/unhealthy way. Also, extreme time-dilation can affect us. Personally, while in basic training, we got a lot less sleep than I was used too. I started noticing that the days seemed to blend together, and time moved quicker. My dreams, during that period, never seemed to follow any pattern based on time that I can remember, and it's likely true for most people in similar situations.
This article talks about sleep deprivation and how it affects the dreaming process/length:
http://www.scientificamerican.com/article/strange-but-true-less-sleep-means-more-dreams/
This article details the effects of memory processing during sleep:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC534695/
This study looked into how alcohol affects memory storage:
http://pubs.niaaa.nih.gov/publications/arh27-2/186-196.htm
Obviously there are many things besides alcohol which can affect your ability to form and store memories, so taken together, all of these indicate that the order of the events of our day, processed in our dreams, is affected by how you formed those memories while actually experiencing them awake.
Comment: Thanks for answering and glad you made it to the site (I use jonsca on A51 and many other places, in case this seems random).  I think you bring up some great points, but it would be great if you had some references for these as well.
Comment: And thanks for inviting me...though I was already here... :P I will definitely make sure I start regularly providing outside references.
Comment: "If you are highly stressed due to hunger later in the day, you brain may be more likely to process the memories of events before hunger set in." @JRFerrell Is there evidence supporting this? "Our brains can accidentally be configured to process memories in an atypical/unhealthy way" And this?


Passing a boot menu option to GRUB on a Live USB drive?

Question: I have an ubuntu system, and I want to have a semi-automated way of running a clonezilla unattended image / re-image from a USB drive.
To that end, I've modified the GRUB menu items to be along these lines: (One for Restore, one for image):
menuentry "RESTORE IMAGE" --id live-default {   $linux_cmd /live/vmlinuz boot=live union=overlay username=user config components noswap edd=on nomodeset enforcing=0 noprompt ocs_prerun="mount -L Images /mnt" ocs_prerun1="mount --bind /mnt/images /home/partimag/" ocs_live_run="ocs-sr -g auto -e1 auto -e2 -r -j2 -c -k0 -scr -icds -p reboot restoredisk ask_user ask_user" keyboard-layouts=en-us ocs_live_batch="yes" locales=en_US.UTF-8 vga=788 ip= nosplash net.ifnames=0 i915.blacklist=yes radeonhd.blacklist=yes nouveau.blacklist=yes vmwgfx.enable_fbdev=1   $initrd_cmd /live/initrd.img }

What I want to be able to do from the running machine is something like the following:

Set next boot (using efibootmgr?) to USB drive
touch or set something to choose either the RESTORE or IMAGE option on the USB drive GRUB menu item.
Reboot into USB drive, run the selected boot path

The dumb way I can think of this is modifying the USB grub.conf each time to set the default menu item. There has to be a better way, right?
Answer: Step 1: Use efibootmgr you mentioned
Step 2: There is a set default="0" in /clonezilla-usb-drive/boot/grub.cfg
Replace it with your menuentry title or id or index
https://www.gnu.org/software/grub/manual/grub/grub.html#default


Authentication flow leading to wrong authenticator app - what happened, and how to prevent?

Question: Someone I know ended up installing a sketchy authenticator app while going through Microsoft's SSO authentication flow, and I'm trying to understand how this happened exactly.

The user's organization started requiring 2FA, so the login flow requested setting up a second factor. Somewhere, a QR code was displayed in relation to the MS Authenticator app. I cannot say more because we cannot reproduce this scenario anymore (a second factor has been set up in the meantime).
After scanning the code (through the camera app on a Pixel phone), the user's phone ended up on a Google search for the following query (ids redacted): phonefactor://activate_account?code=123456789&url=https%3A%2F%2Fmobileappcommunicator.auth.microsoft.com%2Factivate%2F987654321%2FWEU
One of the top paid results (ads) was a link to the Play store for something named "Authenticator App".
Upon opening, the app showed a full-screen ad with misleading information (requesting the user's phone number and trying to trick them into a subscription).

How did this happen?
My hypothesis is that the SSO authentication flow showed a QR code containing a deep link into Microsoft's Authenticator app - however, since that app was not installed, the camera app fell back to googling for the string found in the QR code. Attackers know of this pattern and buy highly targeted ads trying to lead misguided users to their fraudulent app.
How to prevent?
As far as I understand, the SSO flow could just have used HTTP(S) links pointing to a domain owned by them, and have the Authenticator app register deep links for that pattern. For users who don't have the app installed, a server could redirect requests to the Play store.
Does my reconstruction of events sound accurate? Am I missing something, e.g. why the developers went with a custom URL scheme instead of HTTP(S)?
Answer: Your analysis seems correct. I can see how a flawed link would yield a search result that would cause the user to download any app that seems to match the use case eg "Authenticator App"
In terms of prevention, I think your best defense is user awareness training to make sure they only download the authorized MS authenticator and install it before they start the registration process. The process would be more secure, and users better served if they are sent to a page and app that you authorized, and have them install that as they begin the registration flow.


Metadata rejected- replied but no response 2 days after?

Question: My app status changed to Metadata Rejected because they needed to ask me something. I answered 2 days ago and haven't gotten a reply nor has it changed back to in review. Is this normal?
Answer: What is normal? I am a developer and although Apple is extremely helpful, they don't have standards when it comes to deadlines. You are at the mercy of their prioritizing. It can take anywhere form 1-7 business days for metadata rejected apps. Fix the metadata they are referencing and submit back to them like you have done. There is no set time they guarantee to respond to you, but they will definetly respond. They won't leave you hanging.
Comment: @user3042719 again, 1-7 is a general consensus for metadata apps, but that's not a concrete number. It can take longer (holidays etc etc) the good thing about a meta rejected app is you don't have to submit a new binary and it is usually re-reviewed more promptly as opposed to an internal coding rejection
Comment: Additionally @user3042719 if you don't correct the deficiencies they referenced they won't re-review it. But you stated they had to ask you a question, in that case you will just have to stand by for the reply


WebGL Gradient Shader

Question: I am trying to learn WebGL and would like to have shader that gives a mesh a gradient effect from top to bottom. For example, the bottom of a ball or wall having no blue color and the top having all blue. I know I need to modify the fragment color with the y component of gl_Position but my implementations have thus far given me a black screen. Any help would be appreciated.
Answer: Are you sure that the fragments are getting actually drawn on the screen (disable culling, depth test), no GL errors ? If yes, only issue can be the Alpha setting with blending enabled. Try disabling GL_BLEND, or changing the value of Alpha to 1.0, like below, setting RGB to your colors:

gl_FragColor = vec4(R,G,B, 1.0);

WebGL Gradient Shader


Limit of a sequence of real numbers

Question: If $(a_n), (b_n)$ are two sequences of real numbers so that $(a_n)\rightarrow a,\,\,(b_n)\rightarrow b$ with $a, b\in \mathbb{R}^+$.  How to prove that $a_n^{b_n}\rightarrow a^b$ ?
Answer: Since $a_n\to a$ and $a>0$ by assumption, we have $a_n>0$ for $n\geq N$ for some sufficiently large positive integer $N$. So we can just consider $\log a_n$ for $n\geq N$. Note that $\log$ is a continuous function, we have
$$\lim_{n\to\infty}(\log a_n)=\log(\lim_{n\to\infty} a_n)=\log a.$$
Therefore, we have
$$\log\Big(\lim_{n\to\infty} a_n^{b_n}\Big)=\lim_{n\to\infty}(\log a_n^{b_n})=\lim_{n\to\infty}(b_n\log a_n)=(\lim_{n\to\infty}b_n)(\lim_{n\to\infty}\log a_n)=b\log a=\log a^b,$$
which implies that (by taking exponential on both sides)
$$\lim_{n\to\infty} a_n^{b_n}=a^b$$
as required.
Comment: (+1). For future readers: note that provided $b\ne0,$ this also works when $a=0$ (with $\log a=-\infty$).
Answer: Note: The statement doesn't require $b > 0$.  We don't assume it here.
If we take the continuity of $\ln$ and $\exp$ for granted, the problem essentially boils down to showing $b_n x_n \to bx$ where $x_n = \ln a_n, x = \ln a$ since $a_n^{b_n} = e^{b_n x_n}$.  This is what the work in the first proof below goes toward (the "add-and-subtract $b_n \ln a_n$" trick comes up elsewhere too).  But if you can use $b_n \to b \text{ and } x_n \to x \implies b_n x_n \to bx$ then the proof simplifies to the second one below.

Proof:  Given $\varepsilon > 0$, there exist $K_1, K_2 \in \mathbb{N}$ such that for $n \in \mathbb{N}$ we have $n > K_1 \implies |\ln a_n - \ln a| < \frac{\varepsilon}{2(|b|+1)}$ (since $a_n \to a \in \mathbb{R}^+ \implies \ln a_n \to \ln a$ by continuity of $\ln$ on $\mathbb{R}^+$) and
  $n > K_2 \implies |b_n - b| < \min(\frac{\varepsilon}{2 (|\ln a| + 1)},1)$ (by hypothesis).  Let $K = \max(K_1, K_2)$.  Then
  $$\begin{eqnarray}
n > K \implies |b_n \ln a_n - b \ln a| &=& |b_n \ln a_n - b_n \ln a + b_n \ln a - b \ln a|\\
&\leq& |b_n \ln a_n - b_n \ln a| + |b_n \ln a - b \ln a|\\
&=& |b_n|\,|\ln a_n - \ln a| + |b_n - b|\,|\ln a|\\
&<& (|b| + 1)\,\frac{\varepsilon}{2(|b|+1)} + \frac{\varepsilon}{2 (|\ln a|+1)}\,|\ln a|\\
&<& \varepsilon
\end{eqnarray}$$
  so $b_n \ln a_n \to b \ln a \in \mathbb{R}$ by definition.  But $x \mapsto e^x$ is continuous on $\mathbb{R}$, hence $a_n^{b_n} = e^{b_n \ln a_n} \to e^{b \ln a} = a^b$.

As mentioned at the top, using the theorem on the limit of a product gives this:

Short proof:
  $$\begin{eqnarray}
a^b &=& \exp\left[b \ln a\right]\\
&=& \exp\left[\left(\lim_{n \to \infty} b_n\right)\left(\ln \lim_{n \to \infty} a_n\right)\right] &\text{ assumption}&\\
&=& \exp\left[\left(\lim_{n \to \infty} b_n\right)\left(\lim_{n \to \infty} \ln a_n\right)\right] &\text{ continuity of }\ln\text{ at }\lim_{n \to \infty}a_n = a \in \mathbb{R}^+&\\
&=& \exp\left[\lim_{n \to \infty} b_n \ln a_n\right] &\text{ product of limits }\longrightarrow\text{ limit of product}&\\
&=& \lim_{n \to \infty} e^{b_n \ln a_n} &\text{ continuity of }\exp\text{ at }\lim_{n \to \infty}b_n \ln a_n = b \ln a \in \mathbb{R}&\\
&=& \lim_{n \to \infty} a_n^{b_n}
\end{eqnarray}$$

All the work and $\varepsilon$'s are still there, but now they're hidden in the theorems we used.
Comment: This seems to yield only that $a_n^b\to a^b$.
Comment: I am all for leaving *some steps to fill in* but this is not so here. In fact such a hint is dangerously close to suggesting a wrong answer (first step being that $a_n^b\to a^b$, second step being that $a^{b_n}\to a^b$, then what?).
Comment: Yes, there are some steps for him to fill in.  (Exercise for the reader!)  I wanted to stick with functions of one variable.
Comment: And now as show that $a_n ^{b_n}\rightarrow a^b$
Comment: I fleshed out the answer (this is what I meant by using continuity).  In retrospect, though, I see your point.  Thanks for getting me to improve this.
Answer: The function $f(x,y) = x^y = e^{y \ln x}$ is continuous on $\mathbb{R}_+ \times \mathbb{R}$, hence if $(a_n,b_n) \to (a,b)$ (with $a_n, a >0$, of course), then $a_n^{b_n} = f(a_n,b_n) \to f(a,b) = a^b$.
Comment: Why the downvote after over a decade?

Limit of a sequence of real numbers


home-server tag makes no sense

Question: From looking at home-server there's nothing there that can't just go in server
Answer: Merge it with server.
The only "difference" is that a home server will do a lot more stuff than say, a companies server. (as in, it'll run many different services as opposed to say, one or two.)
Answer: No, keep it!
We need our home server tag!
edit away with reasons to keep it folks

home-server tag makes no sense


How to apply border radius after drawing a rectangle?

Question: Photoshop doesn't support to apply border radius after drawing of rectangle. I found this link and tried myself but seems to be not working. Is there any step is missed or is there any other idea to apply border radius after drawing a rectangle?
Comment: I'm doing blur method here...
Comment: This is a duplicate, but it should be said that in Photoshop Creative Cloud you know have the option to change the properties of a rectangle after you draw the rectangle
Answer: Using script plugin we can add border radius in photoshop.
Here you can find the plugins.
http://photoshopscripts.wordpress.com/2012/11/29/rounded-rectangle-radius-resizer/
Installation
Copy and paste plugins here "C:\Program Files\Adobe\Adobe Photoshop CS4\Presets\Scripts".
Open Photoshop file , you can run the script from File->Scripts-> Rounded_Rectangle_Radius_Resizer.
Comment: I wouldn't be recommending a script when the OP doesn't know how to do it from the tools provided in the software...  just my 2
Answer: In the new photoshop cc, you can now change the border radius via the shape properties panel.


Essential fatty acids function

Question: Why are essential fatty acids so essential? I know that if taken less then it would cause diseases, but what metabolic role do they play? Are their importance is just because of their structural role?
Comment: ''Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.''
Answer: Essential fatty acids consist of two main EFAs - Omega 3 (Linoleic acid) and Omega 6 (Linolenic acid).
These two play vital roles in the following:

Growth and development
Brain functioning
Skin health
Hair growth
Metabolism
Reproductive system health
Cell membrane integrity.

The fact that are bodies can't make them makes them very important in our diet. They are most important in our cell membranes.
Omega 3 is an amazing anti-flammatory. It does this by blocking the inflammation pathways in the cell.
We need more Omega 3 than 6 - about a ratio of 2:1.
This is a pretty good page if you want more information.
Comment: thanks, but how do they achieve those functions?
Comment: @JM97 editing...

Essential fatty acids function


Why can't I get on my horse?

Question: I'm unable to get on my horse. I can steal horses, but my horse won't let me on. I really don't enjoy stealing horses all the time, because then I have bounty and then they leave me when I'm done with the quest. So I was wondering if perhaps anyone else had this problem or how to fix it? =) thanks
Comment: It's a lot easyer to just steal a horse or if your is glitched steal a new one .
Answer: Try targeting the horse in console and using

disable
enable

If that doesn't work, try

kill
resurrect
disable
enable
Answer: Are you certain it's your horse? stupid question i know, prehapse your horse is glitched? Then again maybe you don't want to buy a horse if this is the case you are awarded a horse from the Dark brotherhood, it is the same horse you are awarded in Oblivian shadowmere just work up the ranks and you get a free horse and it is rather fast, it will follow you as well All you have to do is check stables if it is missing.
Answer: If you downloaded a new mod related to horses, check that. Also loading a earlier save game and see if you can get on that horse. If all else fails, you could try to disable all mods and check if it lets you on, or just resort to getting another horse if it gives you that much problems.
Answer: You can't go on your horse if you're in a tight space. Try to move it out of the tight space and then ride it again. This happened to me at Lakeview Manor when I purchased a horse from my steward and rode it in the stable area I couldn't until I move it out. if that doesn't work, then:

 YOU'RE DOOMED!
Answer: I installed 2 separate modules of Horsemen Combat, which uses the vanilla pony-type "horses" for its overhaul, whilst also running Horses Revamped - I couldn't ride on my horse again and I had to re-install and then after clean-saving UNinstalling the Horsemen Combat and it worked for 1 or 2 hours, then when weird stuff happens (like being saved from hunters on Frostfall or reloading a quicksave) I cannot ride on my horse again, as if it's being affected, even though I've searched out all the mods that seem to cause the problem. I've even re-installed Skyrim a few times, worked once but I don't think the last time until I got rid of some mod or another. It turns out Relationship Dialogue Overhaul prevents you from getting on your horse or accessing its inventory, too.
Comment: I don't think the issue had to do with mods, though.

Why can't I get on my horse?


Separated objects following the original one

Question: When i try to separate objects it all works out fine, but when i wanna move the object and keyframe the motion of it moving from the original one the original one just follows it. I tried uparanting, but it doesn't work. How do i solve this? Is this a bug or a feature?
Answer: Because when you separated the 2 objects they kept the same animation, and if you change the animation for one it will change it for the other, several objects can share the exact same animation. What you can do is:

Go in the Dopesheet, switch it to Action Editor

Create an new animation from the previous one (Create New Action), or remove all animation (X button).
Comment: Thanks! That worked perfectly!


GTA IV - Out of Commission - Helicopter bug

Question: I am on the last mission of GTA 4 - Out of Commission (PC) and cannot finish due to the bug with getting on the helicopter.
I have read some terrible hacks to get it to work e.g. use Fraps, but that doesn't even seem to work in all cases.
Is there a definitive fix for this?
Answer: I don't know if they will ever fix this, but here's a list o things you can try, worked for me:

used a wired keyboard;
remapped the jump key;
fps between 26 and 30.
Answer: Still not fixed in 2020(Complete Edition). I've tried several fixes, but what helped me is to turn on VSync in graphics settings:


Grammar checking program for LibreOffice

Question: Is there a tool for checking grammar in LibreOffice?
Comment: We are not familiar with this "Ginger in windows" of which you speak. Which grammar to you want to check? Natural language (which)? A computer language grammar?
Comment: No! just english grammar. Ginger is a grammar checking software in Windows. now i know there is extension in LibreOffice though, through @Err Hunter.
Comment: Sorry. Just asking about Language Tool Extension in LibreOffice.
Answer: LanguageTool extension for LibreOffice:
You need to install LanguageTool extension to  achieve grammar check in LibreOffice.
Get it from here : https://extensions.libreoffice.org/extensions/languagetool
And HowTo on install extension can be found here : http://www.libreoffice.org/assets/Uploads/EN_Documents/Installingextensions.pdf
Other extensions:
For more alternatives, have a look at this source link:
SOURCE : http://www.techrepublic.com/blog/five-apps/five-libreoffice-extensions-to-help-you-catch-grammar-problems/1280/
Answer: Firstly allow Universe repository and update:

How do I enable the "Universe" repository?

In LibreOffice Writer, you may install the desired language from menu Tools -> Language.

myspell

Documented Description:
  MySpell is a simple spell checker that uses affix compression and is modelled after the spell checker ispell. MySpell was written to explore how affix compression can be implemented.
The Main features of MySpell are:
Written in C++ to make it easier to interface with Pspell, OpenOffice, AbiWord, etc.
      It is stateless, uses no static variables and should be completely reentrant with almost no ifdefs.
      It tries to be as compatible with ispell to the extent it can. It can read slightly modified versions of munched ispell dictionaries (and it comes with a munched english wordlist borrowed from Kevin Atkinson's excellent Aspell.
      It uses a heavily modified aff file format that can be derived from ispell aff files but uses the iso-8859-X character sets only.
      It is simple with lots of comments that describes how the affixes are stored and tested for (based on the approach used by ispell).
      It supports improved suggestions with replacement tables and ngram-scoring based mechanisms in addition to the main suggestion mechanisms.
      Like ispell it has a BSD license (and no advertising clause).

GNU Aspell

Documented Description:
   GNU Aspell is a spell-checker which can be used either as a standalone application or embedded in other programs.  Its main feature is that it does a
   much better job of suggesting possible spellings than just about any other spell-checker available for the English language, including Ispell and
   Microsoft Word.  It also has many other technical enhancements over Ispell such as using shared memory for dictionaries and intelligently handling
   personal dictionaries when more than one Aspell process is open at once.

Available Aspell Dictionaries

Hunspell

Documented Description:
  Hunspell is a spell checker and morphological analyzer library and program designed for languages with rich morphology and complex word compounding or
   character encoding. It is based on MySpell and features an Ispell-like terminal interface using Curses library, an Ispell pipe interface and an
   OpenOffice.org UNO module.
Hunspell is the spell checker of LibreOffice, OpenOffice.org, Mozilla Firefox 3 & Thunderbird, Google Chrome, and it is also used by proprietary software packages, like Mac OS X, InDesign, memoQ, Opera and SDL Trados.
Main features:

Unicode support (first 65535 Unicode character)
morphological analysis (in custom item and arrangement style)
Max. 65535 affix classes and twofold affix stripping (for agglutinative languages, like Azeri, Basque, Estonian, Finnish, Hungarian, Turkish, etc.)
Support complex compoundings (for example, Hungarian and German)
Support language specific algorithms (for example, handling Azeri and Turkish dotted i, or German sharp s)
Handling conditional affixes, circumfixes, fogemorphemes, forbidden words, pseudoroots and homonyms.
Interfaces and ports: AndroidHunspellService (for Android, based on the Chromium fork of Hunspell), Enchant (Generic spelling library from the Abiword project), XSpell (Mac OS X port, but Hunspell is part of the OS X from version 10.6 (Snow Leopard), and now it is enough to place the Hunspell dictionary files into <code>~/Library/Spelling or /Library/Spelling</code> for spell checking), Delphi, Java (JNA, JNI), Perl, .NET, Python, Ruby, UNO, RichEdit.

Ispell

Documented Description:
  International Ispell (an interactive spelling corrector)
   Ispell corrects spelling in plain text, LaTeX, sgml/html/xml, and nroff files. [x]Emacs and jed have nice interfaces to ispell, and ispell works from
   many other tools and from the command line as well.
No ispell dictionaries are included in this package; you must install at least one of them ("iamerican" is recommended by default for no good reason);
   install the "ispell-dictionary" package(s) for the language(s) you and your users will want to spell-check.
It's a good idea to install "word list" package(s) for the same language(s), because they'll be used by ispell's (L)ookup command.
Comment: Grammar != spelling...
Comment: I believe this is a potentially useful answer to some totally different question about checking spelling. If it or something similar to it has not already been posted as an answer to such a question, then it probably should.


How to assess sunscreen effective via blueprinting?

Question: How do I assess the effectiveness of sunscreen creams by using the blueprinting process? I know blueprinting is done when something is sensitive to light and this may cause reactions to produce a product that causes Prussian Blue to appear. However I am unsure how to connect sunscreen and blueprinting - does it have to do anything with the colour in relation to its wavelengths and how it is connected to UV light?
Answer: From Wikipedia page on the blueprint process:

The blueprint process is based on a photosensitive ferric compound. The best known is a process using ammonium ferric citrate and potassium ferricyanide. The paper is impregnated with a solution of ammonium ferric citrate and dried. When the paper is illuminated, a photoreaction turns the trivalent ferric iron into divalent ferrous iron. The image is then developed using a solution of potassium ferricyanide forming insoluble ferroferricyanide (Prussian blue or Turnbell's blue) with the divalent iron. Excess ammonium ferric citrate and potassium ferricyanide are then washed away. The process is also known as cyanotype.

Exposing blue print paper to UV light will cause it to develop blue. Sunscreen is meant to block UV light so applying sunscreen over blueprint paper then exposing to UV light and developing will give you a semi-quantitative way to evaluate the sunscreen based on the intensity of the blue.
See: Colorimetry


Question on Access to Voter Lists on a State by State Basis

Question: In the United States, who (individual, group of individuals, political party) has access to state voter lists that indicate whether or not the given registered voters had voted in (i) the last general election (ii) in any specific general election since each given individual was registered?
Comment: Are you asking about "voter files" https://www.pewresearch.org/fact-tank/2018/02/15/voter-files-study-qa/
Comment: @James K Thank you for providing this link. Although I neglected to specify this (sorry) in the original post, I would like to know if there is a means of knowing---on a state by state bases---whom among  the registered voters in that state are recorded as having voted.
Comment: I don't think that the state maintains voter registration lists, I think it is by county. You see you don't register with a state typically, you register with a county. (that's not to say that a state may choose to compile a database of registered voters - but there is no apparent need to do so)
Answer: The question has not only who has the right, but also what is the liability to have the data.
Each state in USA and DC have their own laws and restrictions how voter data access is obtain and retain. Please take a look the updated Terms and Conditions
link provided by Walklsits - a product of ScaleCampaign.
The actual right for access is determine by either public access to everyone(free and open to the world), Freedom of information Act (FOIA) request, or severe restrictions (see Terms and conditions link above).
For example for free access take a look state of Ohio:
The process of obtaining FOIA for some states goes into few steps:
From SOS of each states, find the application under FOIA and submit request to obtain the data in question. Each state may restrict who as an individual has the right to submit FOIA for that state. That is done on state by state based. Below is an example of state of Michigan FOIA request that shows the answer and the actual data that was given in jump drive. Some states may issue CD via mail.

As a general rule, please read the state law and be careful how you store, distribute/share voter data.
Answer: For the items you mention, (i) and (ii) I can tell you that I, a member of the general public, have access to this information on my home computer.
It is my understanding that databases like this are sold to interested organizations.
Whether this is true in all the thousands of US voting jurisdictions, I can not answer
Comment: In reference to your first paragraph, can you provide a link? Thank you.
Comment: Unfortunately I can't - because I don't remember the web address that I used to download the entire database for my county of about 750,000 people.  What I do know is that I have an excel spreadsheet with all the registered voters, addresses, date they registered, and specifics as to if they voted in each of the preceding 5 general elections, and if they voted in primaries if they used a R or D primary ballot. Databases like this are regularly used by political campaigns to target  likely R or D voters.


How to anti-alias an image in Flex?

Question: I have a Flex component with a background image. The image is sharp in the beginning, but is jagged whenever I scale the component using scaleX and scaleY. How would I make the image anti-alias so that, it it's scaled to 0.75, the lines are smooth, not jaggedy?
Here is the image
Here is the scaled version
And the unscaled (good) one
Answer: If you load the image with an Image component, you can cast the content property of the component to a Bitmap and then set smoothing to true. Unfortunately, the image component doesn't provide this functionality out of the box. However, it's rather easy to hack in.
Here is a tutorial to show you how to create such a component:
http://www.adobe.com/cfusion/communityengine/index.cfm?event=showdetails&productId=2&postId=4001
However, if this is set using the backgroundImage style of a component, you just might be out of luck unless you override updateDisplayList and perform the drawing of the bitmap yourself by using Graphics.beginBitmapFill (which does provide smoothing support).
Why smoothing of images doesn't have better support (such as different interpolation methods) in Flex (and subsequently, Flash) boggles my mind. At least pixel bender filters will help a bit by letting us implement such filters ourselves.
Comment: Great! Now I'll just do some layering to get rid of the backgroundImage.
Answer: If the dimensions of your Bitmap are both either n^2 or n^8, then the Flash player will automatically use a technique called mip-mapping that will dramatically improve the look (and performance) of scaled bitmap images.

How to anti-alias an image in Flex?


Retiring Windows file server - moving files to new NAS - need to add domain account to files and folders

Question: Summary: How can I automatically (script) check all the permissions for files and folders on a drive, and if a permission is tied to a local computer account have a new permission added: with a Domain Account with the same permissions.
Details:
I inherited a Windows 2008 file server that has many files and folders with Builtin (local) permissions like:
BUILTIN\Users :(ID)(...)(special access:)
BUILTIN\Administrators :(ID)(...)
To allow domain users access - the server has \Domain Users included in the local\Users and \Domain Administrators in the local\Administrators group.
I want to move the files to another device (NetApp) and retire the File Server.  When the File Server goes offline, the local permissions will be orphaned, and Domain Users will not have access to the files. (please correct me if I am mistaken)  The directory structure and permissions are somewhat complicated - this combined with the large quantity of files and folders makes it too time consuming to add the Domain credentials one at a time.
Is there a way to:

Loop thru each folder\file
If BuiltIn\Users has permissions on the folder\file
Add domain\Domain Users with the same permissions

Or maybe there is another solution?
EDIT:
I found this script that does part of what I am looking for. http://windowsitpro.com/windows/jsi-tip-0596-how-do-i-print-permissions-directory-tree
It goes thru all the folders and obtains the existing permissions.
I need to include files and a way to:
<code>If account is BuiltIn\Users:
    Add "mydomain\Domain Users" (with same permissions as BuiltIn\Users)
</code>
Answer: I found out that robocopy and Netapp are smarter than I gave them credit for...  When we began copying the files to the new location the Builtin User permissions were automatically changed from:
FileServer\User
to
NewFileServer\User
Problem solved.


Where are my deleted answers?

Question: My deleted recent answers link in the profile is showing only the recent 10 deleted answers. Where can I see all my deleted answers? Are they all permanently removed?
Answer: The deleted answers list only includes posts deleted in the past 60 days.
The rest are not permanently deleted, but you need to know where you posted them, as there currently is no way to list them all (unless you are a diamond moderator or Stack Exchange developer).
If you do remember what question they were posted on, you as the author can still see them by visiting that question, as can anyone with 10k+ reputation.
Comment: This is SUPER lame...I don't know why it's lame exactly, but I KNOW it's lame.  And SE bureaucracy :(
Comment: @lilredindy: this answer is no longer accurate, you can now view all your deleted posts, see [Show all of my question/answers to me even if they are deleted](https://meta.stackexchange.com/a/374184)
Answer: Another option:
If there was some reputation changes due to that post (or voting was performed on that post), you can go to your reputation tab. Here you can see all your posts where voting was performed (i.e. even if you didn't earn or lose reputation due to any reason). And yes don't forget to tick the option 'show removed posts'. You don't need to 10k privilege here.

Where are my deleted answers?


Is there a thing so-called integral over the projective line of $\mathbb R$?

Question: In projective geometry, a line can be viewed by its projective line, which has only one point at infinity, and if a point keeps moving to $+\infty$ it will return to $-\infty$. On the other hand many integrals are taken over the real line. I wonder if there is a connection between two concepts. Would it be an integral over a circle or something? Would this be affect if the integral is replaced with a sum over $\mathbb Z$? I understand the function under integration is a member of $\mathbb R^\mathbb R$, and the series under summation is a member of $\mathbb R^\mathbb Z$
Comment: Yes from $[x:y ] \mapsto [y:x]$ you see the topology of $\mathbb{P}^1(\mathbb{R})$ is of a circle and you can define an  integral accordingly, for example with $\int_{\mathbb{P}^1(\mathbb{R})} f(u)du = \int_{-\pi/2}^{\pi/2} f([\tan x:1])dx$
Answer: The issue is choosing a metric (alternatively, a measure) on the real projective line with respect to which the integration is to be performed.  It is natural to identify the real projective line with a circle in which case the natural metric is simply the least length of an arc joining the two points, i.e., the subtending angle $\theta$ in radians.
The way to define a distance between two points $A,B$ in the projective plane is to take the representing lines $a,b$ through the origin in the plane and set the distance between them be the angle between vectors $\alpha,\beta$ spanning $a,b$ respectively.  Thus one would set the distance between $A$ and $B$ to be $\arccos \alpha \cdot \beta$ where "$\cdot$" is the scalar product, if $\alpha$ and $\beta$ are unit vectors.
Comment: In projective geometry there is no distance; only relations of collinearity, concurrence, etc.  If you want to *integrate* you have to add an additional piece of information to the picture, namely the metric and/or measure. In some cases there are fairly natural ones, as in the case of the circle.
Comment: To answer your question "what exactly the circle", the simplest way of putting it is to say that it is the circle parametrized by the angle $\theta$ appearing as $\arccos \alpha\cdot\beta$ as above, as $\theta$ runs from $0$ to $\pi$.
Comment: Oh, so the model of projective line depends on how the metric is defined on it?
Comment: In the case of the metric induced from the euclidean $l^2$ norm, what exactly the circle we integrate? Can you give me an example?
Comment: The Fourier series works for periodic functions, which can be imagined as functions in circle. Is there a connection between the two?


How can I counteract the to bright tomato taste of a beef stew?

Question: I used a large tin of chopped tomatoes as the majority of the liquid (plus water) to make a beef stew. It doesn't taste right! I used lots of fresh spices; thyme, rosemary and sage. It really needs help. Can anybody suggest something, please?
Comment: Did you add other vegetables?
Comment: What type of stock did you use?
Comment: Just tell people it's a pot roast, not a stew  then they won't have a preconceived notion that it should taste like 'stew'.  (And I prefer pot roast anyway)
Answer: The thing to avoid with stews is boiling them as this tends to make the meat tough and drives off the more subtle aromatic flavours, cooking too hard can also make tomatoes taste a bit metallic.
If it tastes a bit too sharp there are a few this you can do to help it.

seasoning : just basic salt and pepper flavours will help as well as plain salt anchovies and fish sauce can work well with beef, if used moderately  these don't give the dish any sort of fishy taste. These also tend to enhance the savoury flavour of the beef but you do need to be a bit careful not to over-salt stews as this is difficult to fix.
sweetening : a bit of sweetener can counteract harsh flavours and enhance aromats. Something like honey or caramelised onions is generally better than refined white sugar, black treacle also works well with beef as it has a bit of bitterness to balance the sweetness.
Spice : even if you don't like hot food, paprika or black pepper can add extra depth to a stew without making it too hot as can fresh peppers. Aniseed flavours like tarragon, star anise and fennel seed also work  well with beef.

Smokey flavours can also work well and can be a good emergency fix for a stew which has gone a bit wrong eg smoked paprika or smoked oysters.
The key thing is to add seasonings gradually and taste as you go along, taking careful note of how they change the flavour often it sis not so much a case of tasting the seasoning itself as how it changes the whole balance.
Also stews get better with slow gentle cooking (not boiling) it's normal for a meat stew to taste bit weird after 30 minutes or so but after 45 minutes to an hour of gentle simmering is should start to come together so it's important not to season too much too early.
Comment: How do you prepare a stew without boiling it? They are normally not only boiled, but left on a soft boil for at least 2-3 hours to be good. If your meat gets tough in that time, you used the wrong meat.
Comment: As counterintuitive as it sounds - check if your dish needs another source of acidity (eg wine or vinegar) in addition to the tomatoes. Tomatoes have their way of making a dish taste properly SOUR without giving it enough actual acidity. Consider balancing with sweetness. Also: Add  sources of umami and meaty flavor (like soy sauce, doenjang/miso, strong broth, mushrooms). Also: Is your texture alright, or might it simply be too soupy (consider tomato concentrate or roux)?
Answer: try some worchestershire and a bay leaf and continue to cook and break the tomatoes down.