It may be helpful to determine what question is being answered, literally or implicitly, by the statement. In my experience, "at" usually suggests a location while "in" suggests an activity.
By way of example, consider the following question/answer pairs:
Q: "Where is Ms. Smith?"
A: "She is at a shareholder meeting."
You would say "back in XXXX" to emphasize that it happened some time in the past. Otherwise there is no difference in meaning.
I graduated high school in 1984
I graduated high school back in 1984.
You can also say "way back in" to emphasize that it happened a (relatively) long time ago, as if it was ancient history.
I graduated high school way ...
Your proofreader is right.
The revised version is how this is normally written. I would understand your original text, but it would make me stumble. You should avoid wording that distracts your mathematician readers.
You could avoid the passive construction and replace some symbols by words by writing
For each j < n let h_j be a ...
The original quote has been so badly mangled by someone who simply failed to copy it out correctly in the first place.
It has then been handed round the internet by others who simply never bothered to check if it was correct.. or even if it made any sense at all.
Maybe they thought William Morris lived some time before Shakespeare & so "spoke funny", ...
Both prepositions are correct but have slightly different meanings here, depending on how the author considers the bus. The interpretation also depends on context1.
"On the bus" considers the bus functionally as a form of transport.
"In the bus" emphasises that the bus is a place.
So if I read that someone "fell asleep in the bus", my first impression is ...
The idiom is "The key to something", with "to" being a preposition, used idiomatically instead of "for". So we say "the key to the door" or "the key to my heart".
So after "the key to ..." we need the object of the preposition "to", this can be a noun or a gerund, but not a verb. So "...
Good answers all around. I'll add another permutation.
"Back in 2003" can give a very slight impression that you're trying to set the scene, so to speak, and help the listener remember what else was going on in 2003. "We founded this company back in 2003, before Twitter and Facebook existed!" or "I met my wife back in 1955, when the drive-in movie cost a ...
No, they're not the same.
Police shot the suspect.
This means that the person was actually shot. They were injured or killed by a bullet fired from a gun.
Police shot at the suspect.
This means that police shot in the direction of the suspect but did not necessarily actually hit them. This is appropriate to use if it's unknown whether the suspect was ...
In my own way is the idiomatically correct expression. I can't think of a situation where I would use on my own way.
Now, in my way and on my way are both valid expressions. You could say in my way instead of in my own way if you wanted to; the own just provides a bit of emphasis.
If I am on my way, I am proceeding to an expected destination. One more ...
First, the phrasal verb is indeed take off, which means:
take off (phrasal verb) To leave the ground and begin flight; to ascend into the air
Second, you can use a preposition after a phrasal verb:
The plane took off from the runway.
Third, we need to be careful about omitting the prepositions, because sometimes phrasal verbs can mean different ...
I have studied four Indo-European languages in addition to my native English. If there are any rules on which prepositions are proper in which situations, they are not easily discerned or explicated in any of those languages. In English, the predominant preposition relating to accessibility seems to be "to."
So "door to that room," "key to that lock," and "...
Here is how these two prepositional phrases are used in contemporary American idiom when the subject is Meetings.
Is Mr Jones available?
--No, he's in a meeting.
Were you at yesterday's meeting about the Pacific-Rim marketing strategy?
--Yes. That large conference room holds quite a few people, doesn't it!
"In a meeting" means to be in the room where a ...
Morning needs an article, so your example
I messaged you in the morning.
But since this is past tense, it would be better to write:
I messaged you this morning.
For the future it is OK though:
I will message you in the morning.
At it is idomatic, as I suspect you know. The OED defines at it under at as:
16b at it: hard at work, fighting, etc.; busy.
I point this out because I think the etymology of the idiom can be gleaned from the nearest definition of at:
16a: With actions in or with which one is engaged: as at dinner, at work, at play.
In other words, at means to be engaged ...
The choice is entirely up to you. Usually, a comma is placed after an introductory adverbial (here: in this talk) if that adverbial is long. By placing a comma you then improve the readability of your sentence. In your case, the comma is not necessary, but if you do place it you're telling the reader to pause briefly when reaching the comma. Without the ...
Yes, the prepositional phrase, "in front of" can replace the preposition "before" in the examples you've given. Other examples include the following:
"Her name was before mine on the list." = "Her name was in front of mine on the list."
"That happened before my time with them." = "That happened prior to my time with them."
"He was in front of ...
Regard is tricky. The ‘rules’ below are those observed in formal writing; you may safely ignore them in conversation.
When used as a noun to express your attention to a particular topic, it should be used in the singular:
In regard to the appointment, yes! ... not regards.
With regard to the appointment, yes! ... not regards.
This is ...
As far as I know, accuse is always used with the preposition of. To say accuse someone for something is just grammatically wrong. If you accuse someone of doing something wrong or illegal, you tell them that you believe or suspect that they did it.
He was accused of embezzling millions of dollars through offshore accounts, but the court ...
Although in general we precede a singular, uncountable noun with a determiner such as the indefinite article a(n) or the definite article the, there are many instances in which we use the zero article (which means no article at all) before such a noun.
We use the zero article when the noun expresses a reference to a general class of things. The most common ...
You are confused between 'verb' and 'phrasal verb.'
You are absolutely right that 'off' here means 'away'.
I was walking off the road - away from the road.
But the main verb there is 'walking.' Here, the main verb, in fact a phrasal verb is 'take off.' The 'off' is a part of the verb. Thus, you require the destination as well. Here, the place is '...
You get into a box. (Although you can also get in it, which means the same thing.) Once you have gone into it, you are then in it.
Similarly, you can get out of a box. Once you have gone out of it, you are then outside it.
In the U.S., off of and off correspond to into and in. You can jump off of the roof. (You can also jump off it; this means the same ...
By itself, pick (in this sense) means to select:
Look at your choices carefully and then pick one.
Pick out means to select from a large group (at least, it does in my American experience).
She looked at a lot of dresses and picked out the one that she liked the best.
It can sometimes mean detect something from a collection (i.e., detecting something ...
The only time you consult with someone is when you're seeking advice from a person (not a book or internet article), and you're looking for general information. Any time you say "consult with", you can use "consult" instead. Using "consult" sounds more direct, which is why it's used when you're looking for very specific information. Here're a few examples:
implicitly means to move away from something, moving "away" is implied.
fled the city
fled from the city
have the same meaning, from is not necessary and some might consider it redundant, but both are correct. However, in the case of
fled to the city
the preposition to must be there to mean movement "towards", otherwise it will have the the ...
If one side of the bed is bordered by a wall, then "inside" and "outside" are natural and appropriate expressions in English - I don't think there are better ones. There are other ways to refer to these positions, but they are not better.
Note that you need the article "the" in all three cases.
"Inside" and "outside" use the preposition "on", while "...
As a rule of thumb, I would use "in" for larger items of clothing that clearly contain significant parts of the body (shirt, pants; even hats, shoes, and gloves) and "with" for smaller accessories (including earrings and necklaces). "Wearing" can be used for both, but has a more formal/stuffy tone. "With wearing" is ...
In your examples
A cookie in the shape of a foot.
is grammatically correct
A cookie in the shape of foot.
is grammatically incorrect, however
A cookie in the shape of feet.
is grammatically correct and could mean a single cookie which looks like more than one foot
or multiple feet made from a single type of cookie (dough)
Fookies can be made ...
For me (British), I would never say "on Christmas", because 'on' is used with a day, not a period. I would say "on Christmas Day", or "on New Year's Day", or "On Easter Sunday" but "at Christmas" or "at Easter".
As well as emphasizing that it happened some time in the past, "back in 2003" has a more informal feeling to it than "in 2003". You wouldn't be so likely to see "back in 2003" in a formal business document or legal document. On the other hand, "in 2003" might sound a bit too 'calculated' for a casual conversation; adding 'back' adds information about how you ...