This question seems like it should be easy, but it's destroying me. Discriminant is -84, minkowski bound is 5.8ish so that we only need consider p=2,3,5. Let root(-21)=r, then Kummer factorization gives that
2R=(2, 1+ r)^2,
3R=(3,r)^2
5R=(5,2+r)(5,2-r)
it seems apparent we are supposed to adjoin two of the square roots of 2,3,5, or 7 (or perhaps root -2,-3,-5,-7). correct me if this is wrong. By taking Norms we see any element in our extension generating a principal ideal that divides 2R has norm 2, similarly for 3R has norm 3, and for 5R has norm 5. This is as far as I've seriously gotten: I've tooled around for a while with adjoining pairs and checking for elements of valid norms (which I list below) but nothing has worked. I am thinking I may need to focus more on adjoining negative roots.
Considering real roots, elements of norm 2 are sqrt(2), sqrt(3)-1, sqrt(7)-sqrt(5), sqrt(5)-sqrt(3), and their conjugates (I'm definitely missing some good candidates but these seem reasonable). Elements of norm 3 are sqrt(3), sqrt(5)-sqrt(2), sqrt(7)-2, etc. Elements of norm 5 are sqrt(5), sqrt(7)-sqrt(2), 2sqrt(2)-sqrt(3), etc. Essentially I think that it is not possible to have an element of norm 5 by adjoining both sqrt(3), sqrt(7) but I honestly have no clue.
If some of this makes no sense, I apologize because these computations are proving pretty difficult for me to wrap my head around. Can someone suggest a valid pair along with perhaps some generators? Thank you!
I also need to pick up new knee pads since my old ones got shattered towards the end of my playing time. I’m in chicago so i wouldn’t be surprised if the rinks are still under some limited capacity near me.
The fog was so thick you couldn't be seen, very clever and I look forward to seeing your art on my next walk
I live in rural America. How many companies can I expect to offer toll road services to my house without government getting in the way?
The alternate United States of America is something occasionally used in fiction but what would be your new states [let's assume you could add 10 more states to the U.S.]?
What do you think the state would be like culturally, geographically etc., and in terms of demographics, political voting etc.? What would the food be like?
I'm interested to know.
I want to build this bed.
How do I go about adjoining the 4x4's? Just wood glue?
So today, a few students and I were joking, laughing. Joking about a repair guy who came in to fix something in the classroom. I said, "yup being polite and friendly went a long way to get everything repaired and fixed."
I teach high school, so the kids and I were embellishing, calling each other "queen". You know, just casual fun.
The teacher next door (we have an adjoining door...) He walked right into my classroom and said, "that isn't what happened. They never replied this and that. They NEVER DID THAT." He inserted himself into a conversation that was none of his business. And was "correcting" facts he overheard, that again, were none of his business.
The students were stunned and shocked. And they all stared at me with second hand embarrassment. I was embarrassed and also shocked at what he did and how aggressive he was.
I didn't say anything except. "Okay."
Then he calmed down and asked a very arbitrary question and went back to his room.
The kids then said things like: "Is he really listening to our conversations?!" "Eeeewww. What was that?!" "How rude." "Are you serious?! Did that really just happen!"
I said, "everyone calm down. Let's finish our work." And we stopped laughing and joking. He killed the happy classroom vibe.
Idk, what is wrong with this man and why he keeps trying to bring me down. It makes no sense!
Not only is eavesdropping so rude but it sets such a bad example!
The first week, I worked there, he called me a nobody. And when I try to explain something, he actually brushes me off. Waves his hand in the air and shakes his head. Cuts me off and doesn't let me finish speaking. He also said I have to come to work no matter what, even if a relative dies I need to learn how to hold it together and show up for work. When I voice my concerns he labels me "hysterical".
Has anyone else experienced another teacher with such a big fragile ego or dare I say, a misogynist?
"The couple came in on June 8 into hotel quarantine and then the single person came in on June 10.
"But they immediately became positive and then the couple later became positive.
"So, we're contacting all of the 30 people on that floor 5 and anyone who left quarantine during the period that I'm concerned about.
"So they're all being contacted.
"And the people on floor 5, their stay has been extended until we've been able to confirm that there's been no additional transmission.
"I mean, it's reassuring that it is one room to the room next door, and we've seen, unfortunately, that happen across the country in our hotels.
"We now, of course, have all of our staff fully vaccinated and all of our staff working in these get tested every single day.
"We'll retest all of the staff again whether or not they're at work or on leave, so we'll go back to all of our staff and see if there's any positive cases there.
"But so far, with all of that daily testing, there hasn't been any"
From the ABC Live Feed
Haven't bought any since my kids took up the sport a few years back.
This is a question in a book I’m reading:
“An eccentric woman has booked three adjacent and adjoining hotel rooms. When she checks in, she tells the receptionist that if he needs her, she will always be in the room next door to the room she was in the night before. The receptionist thinks nothing of this until an hour later when he realizes that her credit card has been declined, and he must now go find her. The problem is that he is very busy and only has time to knock on one door per day. How many days does he need to guarantee that he finds her?
What if it were four rooms?
Five rooms?
What if it were 17 rooms, and she is checked in for 30 days—can he find her before she leaves?”
Three rooms is simple enough. I’d check the middle room, if she wasn’t there then I know she was in one of the sides, which means she’ll go next door which would put her back in the middle the next day to find her.
I’ve been thinking about the four rooms and am stuck.
I would check one of the two middle rooms, say the second from the left. If she wasn’t in it, then that means she’s in one of the other rooms:
- far left: next day she’s in the second from the left
- second from the right: she could be in the second from the left, or the far right
- far right: she’s now in the second from the right
She couldn’t possibly be far left, so that eliminates that choice. I feel like I should go second from the left again, since two possibilities would lead to her being there. If I go there again and she’s not there, it means she could only have been in the second from the right or the far right…
- second from the right: either second from the left, or far right.
- far right: second from the right.
Still sort of the same situation… She could be in three places, but now equally likely. If I go second from the left again, and she’s not there, it doesn’t change anything. If I go second from the right, and she’s not there then she was either in the second from the left or the far right, which then opens up the far left again which doesn’t seem like what I want.
Any help with how I should be thinking about this?
I though I would share the algorithm I'm using to generate levels for my current roguelike work-in-progress. I needed fairly small levels with rooms that didn't need narrow hallways to connect them. My solution was to start with one room and, until I reach the desired size for the level, pick an existing room at random and build a new room of off one of its four walls. This way, every room will share a wall with at least one other room, which allow a doorway connect them without needing long tunnels.
I wrote the algorithm in gdscript for the Godot engine, but I've translated it into pseudocode here.
# First we define some necessary classes.
class Rect2:
var position
var size
var end
func Rect2(_position, _size):
position = _position
size = _size
end = position+size
func position(): # getter for position
return position
func position(val): # setter for position
position = val
end = val+size
func size(): # getter for size
return size
func size(val): # setter for size
size = val
end = position+val
func end(): # getter for end
return end
func end(val): # setter for end
end = val
size = val-position
class Vector2:
var x
var y
# you will need to impliment vector math operators +, -, *, /, +=, and -=
func Vector2(_x,_y):
x = _x
y = _y
const ROOM_MAX_SIZE = 10
const ROOM_MIN_SIZE = 5
const DESIRED_FILL_RATIO = .55
var NUM_ENEMIES = 0
var width as int
var height as int
var placed_rooms = []
var filled_area = 0
var total_area = -1
var doors = [] # array of Vector2
var start as Vector2 # where the player begins the level
var end as Vector2 # the stairs going to the next level
# extent variables => used to make sure that the dimensions of the level do not excede its height or width
var left_extent as int
var right_extent as int
var top_extent as int
var bottom_extent as int
const directions = ["North", "East", "South", "West"]
func generate(w, h):
width =I'm rewatching the series and I am on season 1 episode 5. During the conversation between Mary and Cora, we see Edith's face reflected in a picture frame in Mary's room. On my previous rewatches I thought their rooms were adjacent and somehow had a hidden window between them covered by a picture. On this rewatch we see Edith's face reflected in the frame as if she were in the room! The wall paper at the moment of reflection is the same wallpaper as in Mary's room. The scene then changes to Edith looking into Mary's room from the hallway while eavesdropping and then turning away. The walls in the hallway are not wallpapered. I find it hard to believe Mary's room would have a window that would enable people walking down the hall to look into her room. Was this a glitch or an artistic attempt at "walls have ears," "fly on the wall," or "little picters (pitchers originally) have big ears." The scene is at approximately 30:34 to 30:37. Can somebody please help; not understanding the perspective has been bothering me for a while!
