Which in this case are orthogonal. Therefore ...

area of a rectangle....

geometers act like they've got the whole thing figured out. which they do. but who's to say that i don't also have the whole thing figured out. i have a deactivated laptop with drafts of maybe 5 preprints. refute that, geniuses. i have secret cheat codes to worlds you have never imagined.

So what is the correct formula for our solution $x$?

which would give an area $||a|| ||c||$

for the rectangle of $a \times c$

Now go back to $a\times(a\times c)$ and work it out correctly.

3d geometry is really something else. i don't mean to distract from the endgame here. it's just, i had nothing to prepare me for it. does anywhere do something about vector geometry before plummeting people into this? at berkeley as an undergrad the experience was akin to falling down a well. but my high school was terrible.

$\{(0,x), x\in [0,1]\}\cup \bigcup_{n\in \Bbb N} \{(x,1/n), x\in[-1/n,1/n]\}$

$C = \{\bar B(0,1/n),n\in\Bbb N\}$

if you say so

work out what $x$ would be @TedShifrin?

We want $\|a\times x_0\| = \|c\|$, so $x_0$ has to be a scalar multiple of $a\times c$.

@AlessandroCodenotti The antenna, but better drawn

At any rate (and I think I've said this before), I was not looking for the formula, although the formula is fine. I wanted the argument that the solutions of $a\times x = c$ form a line parallel to $a$ and the solutions of $a\cdot x = b$ form a plane orthogonal to $a$. Therefore the solutions of the two equations are given by intersecting a line and a plane orthogonal to it; i.e., there exists a unique solution.

i realized this at approximately 11:58 pm PT last night.

it allowed me to go to sleep.

(replace n with 2^n in the above sets)

Bravo, leslie :)

I never had that visulaized what you explained......as you could see by my diagram I was operating on the idea that $a$ and $x$ form a plane, but not a plane orthogonal to $a$

lord almighty....I thought about that incorrectly....and I still have to figure out an expression for $x$ for my own personal satisfaction.

baptism by fire for cross products and geometry....

gotta stay in the plane.

This one is an example where the number of components goes to infinity but the whole thing only has one component

No, we're in the plane orthogonal to $c$ for your stuff. The other equation gives the plane orthogonal to $a$. You really have to read more carefully ...

my daughter's about to come home from day care, and "after care" (i think they invented this phrase to make me feel like less of a parent), and demand cookies. let's think of defenses. the affirmative argument is that if you need a cookie, eat a cookie.

Ask not if you need a cookie, ask if the cookie needs you

if you look too closely, you become the cookie. that's what i'm afraid of.

Next the cookie will do good for the country.

my daughter has arrived home and is demonstrating her muscular coordination by throwing things about the living room.

cookies don't do this to people.

so hold up...that scalar $t$ that I found earlier can't be correct then because I had taken $a \cdot ((\mathbf{a \times c}) + t\mathbf{a}) = b$, but if $\mathbf{x} = (\mathbf{a \times c}) + t\mathbf{a}$, then I should've been doing $((\mathbf{a \times c}) + t\mathbf{a)} \cdot (\mathbf{a \times c}) = b$. In which case I run into an issue because $t$ disappears...

have you convinced yourself that the problem provides solutions, prior to potentially unique solutions? i hate myself, but geometry provides the way to this.

Yes...geometrically I can "see" what Ted was describing above

i hate myself even more now. it's OK. most of this gets offloaded onto my wife's therapist.

The scalar part $t$. in your thing comes when you dot with $b$. I'm talking about $x_0$.

but isn't $x_{0} = a \times c$ with some scalar that I'm missing?

as the point being used to describe my lines parallel to $a$

Yes, that is where a scalar is needed.

you need that scalar.

I DO need that scalar....but it wouldn't help when doing $\mathbf{a} \cdot (\mathbf{a \times c})$ because I would still get 0

bloody hell........this vector is in the plane orthogonal to $a$, so of course I won't get the $0$ I'm talking about...

it's true. a dot anything crossed with a is going to give you zero.

I'm going to ponder this while I go work on some Insel....sigh....

you need to work on not just insel, but friedberg and spence. your homework is never over.

until you become aged, as i have become. then you can just issue pronouncements.

yeah they're there too...

Mr 40 acting like 80.

Are manifolds diffeomorphic to $\Bbb R^{3,1}$ vacuum solutions to EFE or some variant?

what

what

what what

I see your what and raise

the question does not make sense

so nothing out of the ordinary

Welcome to my world.

I'll see your two what's and raise an eh?

the question does not make sense

EFE means Einstein field equations

$\Bbb R^{3,1}$ is Minkowski space

I'd say manifolds diffeomorphic to Minkowski space with zero curvature are solutions to the Einstein field equations

vacuum solution that is

the reason is because you can construct a Lorentzian manifold with zero curvature that is diffeomorphic to Minkowski space. This would be a vacuum solution to the Einstein field equations

solving the field equations gives a lorentzian manifold

you have to understand, i went to a horrible middle school. a kid was shot in my school district, in physics class, the year i was there. so, shove, hey einstein, where are my field equations. also i need 50 cents for the vending machine because i could use a mountain dew.

shove here's your lorentzian manifold. that kind of thing. you can imagine the rest.

sachs and wu was my resource for this thing when i was interested in it. it is not a great resource for people who are not mathematicians.

physicists seem not to be troubled by whether or not things satisfying their equations exist. you may have to depart from equations at some point. it is a semi religious thing.

my question doesn't make sense, I was just trolling sorry

we had a good laugh at our house over your trolling. well played. we're still recovering from the laughter. :)

my wife just thinks the word "lorentzian" is funny. i tried to tell her it was a real thing but she wasn't having it.

Tell her it's laurencium misspelled.

she also went to berkeley and could understand this.

i met glenn seaborg once. he's probably the only person i'll ever meet who was also an element.

One of those guys lived in the house behind us in Kensington when I was a kid; I forget which one .

one of my wife's friends in college used to housesit for don glaser.

i wonder if berkeley can get those people anymore. when i was a grad student, it was really hard for their physics department to recruit anybody. the facilities were horrible.

Lawrence Labs and Livermore are still a big draw, I think..

i had a good friend who came to visit, was interested, and saw leconte and whatever else and was like "nope." lawrence labs could be different.

livermore is too hot in the summer. i said it.

Try TX.

Or GA. Or ...

where my grandmother was from, mississippi.

i'll take livermore.

Yup.

mississippi is no fun at all

i'd like to go back when it's all over. i was planning on some kind of trip with one of my parents before the pandemic. my dad is now retiring so maybe we can get back to memphis, where he grew up. after i'm vaccinated

In this problem why are they assuming 100x when it comes to investment why not x?

What is the intellectual jump that is being taken please tell me in detail.

What is x here?

Left side is the passage and right is the follow

Up question

Please tell me in details

what do you mean they are assuming 100x?

@RajorshiKoyal the answer is 53:50. you can reduce the problem to a linear system in $I_P,I_Q$ and $I_P=212000, I_Q=200000$.

rhetorical, i guess.

I want that guy's setup. those look like in-built bookcases, not furniture.

i think people would respect me more on zoom if i had stuff like that

when i do zoom it's just a blank wall. no respect.

surely you can get a green screen background? maybe the screen savers of the '80s, remember those penguins?

i'm surprised there isn't more crapware for that sort of stuff.

my home office puts me right in front of some blinds, they mess up the background stuff. the zoom filters do appropriately put unicorn horns on my head.

did i say horns? i mean horn, singular. that's the entire purpose of the unicorn. no need for anybody to point that out

the eda world started with video conferencing in the 90's and quickly tired of it except for maybe c-level folks. so almost all calls are voice only.

i have never had a work video call. i turned down a facebook interview because they were going to do sykpe. stupid of me.

i used to love remote when everybody else was stuck in offices. now i hate it.

i'm trying to arrange a deposition right now. there will be a court reporter to write down what we say, but the vendor will not record the video. someone asked me, 'why can't we record the video?' i don't know, who cares. none of this matters.

weird. would seem an obvious thing to do for a depo.

you need transcript no matter what, so i suppose...

for court proceedings it is necessary. this is a proceeding in front of the US patent office. they do not care about video. you actually have to ask them for permission to file the video. they hate it. so we don't do it.

every watch a netflix series called broadchurch set in north england?

we'll probably record the zoom or whatever anyway.

isn't california a one party state?

my wife is a big fan of that. i love olivia colman from peep show mostly.

i wiretap everybody, irrespective of whatever. we might be a two party state. the problem with depos is 'videographer' is a separate line item and unless it's going in front of a jury, which it isn't, the client doesn't want to pay for it

i like the character colman portrays, sort like a lot of folks i knew growing up.

my cat's name is olivia, so there's that, too. my dad has been trying to get me to watch broadchurch.

i used to ask permission to put my ex-ceo on edge when she gave me grief.

do you mind if i record this...

i did hire her, so no one to blame.

i say this not as a member of the bar of any court ,but if you remotely feel challenged, record everything. the damages in a lawsuit over that will probably be negligble, even if you shouldn't be doing it.

erase that from the timeline, i didn't say it.

i didn't hear it.

my poor son is getting rejects or waitlists from the colleges he wants.

college in america is a mess.

i googled it, california is a two-party state. i've had some very funny phone calls with people, where the other side phoned someone in a one-party state, and we have records, telling us not to tell the court something. "Why? why not?" no answer.

then we put the whole recording in front of the court.

my daughter was turned down by ucla, ucb but accepted by oxford. go figure.

god, college. my daughter is 2.5 years old. i'm assuming the system won't exist anymore when she's grown.

all is can say, is start thinking about her education now.

or send her abroad.

i'm assuming that there's no such thing as retirement, and that colleges won't exist. we're socking away the money but it will be a distaster in any case. i can't save at this level of inflation.

i retired, lost most of my money on 'safe investments', so am unretired, probably for as long as someone will pay me.

we're raising her bilingual, english/spanish. so less costlier abroad options might exist. but it is a disaster.

send her abroad for college. so much simpler. and better for everyone.

it was a disaster when i went through the UC system, and it's now like 1.5x-3x more expensive? how? the professors aren't being paid more.

i don't understand how it is so bad.

diluted by breath requirements.

i graduated in 2001. this is 20 years. how did it all change so badly. and it wasn't great then.

student loans did it.

we tried to get some math classes to qualify for breadth requirements when i was a grad student. the panel didn't approve any of the requests.

created a market that should not have existed.

breath requirements are a joke.

they basically told us, the breadth requirements exist to create hiring in other departments, not your department, pipe down please.

if it created an obvious balanced education i might be persuaded.

we had a pretty good syllabus put together, about math in various civilizations that were somewhat erased by colonization. based on primary sources and not a lot of speculation, just literal pictures of the primary documents and math.

it was like "yeah, breadth doesn't mean this."

i remember mentioning sin & cos in an introductory cs class and getting, "Joe, we didn't study calculus". i wondered how one could get into a cs class without knowing trig.

that was in '84 if i remember correctly.

we had a funny thing at berkeley in the late 90s. the CS department was very competitive. the math department was not. so you'd have people failing out of the qualifying requirements for computer science and then majoring in math because they'd done the other stuff. "oh, shit, i got a D in math. now i guess i have to major in math."

it drove the postdocs and graduate students crazy. if you go to princeton or wherever you imagine teaching a room full of mini-you's. this was a roomful of people who could not write code.

awwmemes.com/t/john-venn

i remember as a grad student trying to explain to my other companions, you don't understand. they majored in math because they got bad grades in math.

if that isn't real, it should be real.

:-).

fisher of statistics fame has a blue plaque just outside of hampstead heath. i showed my wife that before she got her statistics degree. maybe when this shit is all over we'll go back.

i wish i was a real mathematician. my levels of abstraction only go so far.

i still have to visit hamilton's plaque on broom bridge outside dublin.

ahh, statistics.

darrell huff.

he was at iowa! where i was a visiting professor for a time.

i'd love to take my mother to ireland. her relatives come from outside killarney. nobody in living memory, though. one generation back.

its not fun to visit now, covid stuff.

kerry, west cork, the west of ireland in general is beautiful

then again, so are oregon & washington

there's dublin and there's the rest of the country.

my preference is the rest.

i was born in dublin city in what was known as the old coombe.

perversely i am a true dubliner.

my mom's relatives come from killarney and cork. my dad's irish relatives come from carrickfergus.

there is some connection with filoli gardens and a place called muckross house in killarney.

i wish i lived in oregon or washington. the east bay area is foggy enough to pass for those regions.

i have a relative who was the groundskeeper for muckross something in killarney. i have photos of it. and the shed that the guy lived in.

wow, that's quite a spread for your folks.

oh, i see, their relatives.

muckross abbey. is that a thing? something like that. it looks damp.

i prefer long beach, palm trees and it never freezes.

@copper.hat How did you reduce this to the values that you highlighted in the problem. A detailed explanation to this is most welcome.

when they open up the beaches i'm going to teach my daughter how to surf. she needs enormous amounts of sunscreen. she calls it "sunsheen" because she is two years old.

donegal has some great surfing.

a bit cooler than hre.

my daughter use to refer to her privates as her bagina.

when she was 2. just to be clear :-)

my daughter calls it her butt. she said, somewhat quizzically, "i don't have a tongue on my butt like you do." when she was two.

@RajorshiKoyal just write down the equations. ${R_P-I_P \over I_P} = 0.4$, ${R_Q-I_Q \over I_Q} = 0.45$.

that's funny. the term fanny means bagina in Ireland/UK.

ah yes. the "fanny pack." that some people have in the US. they have no idea.

$R_Q-I_Q = R_P-I_P+5200$, $I_P = I_Q+12000$.

Then again, when I TAed I did ask my class for a rubber.

oh no. you didn't.

A very cute girl helped me navigate out if that one.

I still keep in contact with her.

there is a mockingbird going into full song right now. it's been dark for hours. 11:15 p.m. and he's running through all of his calls. i've heard chirps, robin song, phoebe song, hawk screeches, traces of car alarms. it's annoying but also impressive. if i were a mockingbird, i would be very impressed.

@RajorshiKoyal Eliminate $R_P, R_Q$ from the equations and rearrange into a linear syste, and solve.

We used to have a corncrake in the field beside us in one of the places we lived in Ireland. My granny used to take me out and night to listen and tell me that it won't be long before they will be very rare.

mockingbirds & their ilk are incredible.

he's very impressive. in the daytime they also do a flight display, up and down from a perch. it's something else.

he's now doing robin calls. the last 5 minutes he's been doing a robin.

we're on a hill and our second story is level with the trees he is in. it's very loud right now.

the cat isn't pleased.

the indoor-only cat would like to go out and play. perhaps practice the hunting maneuvers that she has perfected when ambushing us in the middle of the night

cats view me as talon sharping material.

i have numerous dog bites, but am more comfortable with dogs & large animals.

as a new parent, you might appreciate this: youtube.com/watch?v=pzRhlwJ49Os

laughing out loud

i resent the actor in that advertisement because he looks younger than me and has more hair than i do, but i will non-ironically follow the guidance of that advertisement.

:-). i resent anyone who as a functioning left hip :-)

still wondering why education and health are such challenges in the moon shot country.

Gentlemen..........So I have two truth or false questions about direct sums.....they have solution attached, but I'm not understanding the reasoning for them.

my answer will be probabilistic.

my parents both have fairly good health care because they served in the military. the reasons why this system (though imperfect) might not be extended to all people are unknown to me. it has something to do with politics and abstract notions of freedom. money from private health insurers may be involved. shrug, not my problem.

I love probability..........

So that is the definition of direct sum and the only theorem we have established

that's quite a bit of stuff, though.

I do'nt see how anything in the theorem could be applied though. All I see that could work is using the definition.

@dc3rd the first follows directly from definition?

@copper.hat Thanks a lot. I arrived at the answer. However it is particularly confusing when I find it difficult to deal with the terms given there to arrive at the conclusion

Profit percentage and profit are defined in a very confusing sense. How do you particularly understand whether or not I am supposed to ignore it totally or take that as reference?

I would've thought so @copper.hat, but I'm not sure where they get this union from in the solution also $\sum W_{k} = \{0\}$

@RajorshiKoyal It is a little confusing.

@dc3rd what union are you referring to?

Yes but why should I not take that into account?

in the first image I posted there is a small explanation about why the solution is True and for the second one why it is false

@copper.hat

unless that is a typo...because there was nothing about unions defined anywhere

or there is some set theory trick I'm supposed to pull out my magic hat.

@dc3rd there is no union, i'm not sure what bothers you?

If $V$ is the direct sum of $W_1,...,W_k$ then the intersection of any two must be the trivial subspace otherwise $V$ is not a direct sum.

I posted the three images, the first image has a union. $W_{i} \cup \sum_{i \neq k}W_{k} = \{0\}$

@dc3rd That's a typo.

They mean $\cap$.

they definitely mean cap

So I'm not losing my mind then............

i mean, you might be, but not for this reason.

Nope. It is an awkward definition of direct sum imo.

well I am studying mathematics...................so there is that

sanity is just mind control

the definition using the sum instead of the individual subspaces no?

whatever you do, don't become a prisoner of reality.

Lol.............I'm gonna steal that

"sanity is just mind control"

none of us are real, you know that? we're your computer or phone or device come to life, or simulating life, just to show you the way.

I would say $V$ is the direct sum of $W_i$ iff every $v \in V$ has a unique representation as $w_1+\cdots+ w_k$ with $w_i \in W_i$.

as in the definition of direct sum I'm used to seeing doesn't define it as $W_{i} \cap \sum_{i \neq k}W_{k} = \{0\}$, but $V = \sum W_{i}$ where $W_{i} \cap W_{j} = \{0\}$

I spent the day helping a nephew in law with his 3rd year mechatronics problems. i have power supply problems crawling out of my head. math is so much more refreshing.

@dc3rd that is not sufficient.

what's missing?

i spent from approximately 6pm-9pm mediating a battle between my daughter and my cat over a toy. the toy contained catnip, which is suggestive of whose toy it actually was. but there was a difference of opinion.

Take the plane. The every line through the origin is a subspace. Pick any subset of these lines, then $W_i \cap W_j $ is trivial for any two, but this is far from a direct sum (which needs some form of uniqueness).

i would deactivate the cat before it becomes expensive :-).

my bark is far worse than my bite.

@dc3rd do you see what i am saying?

Yes, I'm just trying to go over how to see the uniqueness portion of your explanation.

I do see how the intersection of any two of them is trivial.

@copper.hat Can you elaborate slightly on how to eliminate revenue from the given terms?

@RajorshiKoyal Do you agree with the equations?

I'll probably end up reposting to CS se, but I thought you guys in chat should get a load of this before I take it down:

Not really I do not understand what the questioner means ny revenue. A detailed reply to this will most certainly be welcome.

From the profit percentage equations you get ${R_P \over I_P} = 1.4$ and ${R_Q \over I_Q} = 1.45$.

@RajorshiKoyal You might be better off asking on the main site, it is hard to answer here.

Ok thanks a lot but even if I take ratios how do I eliminate RP and RQ last question

Ratios mon

@dc3rd It is not enough to have pairwise triviality.

$R_P = 1.4 I_P$ and similar.

that just clicked in my mind as you typed it

There was entanglement between your brain and their brain because you both got two entangled photons from the sun today.

@dc3rd There is usually a definition of an internal direct sum and an external direct sum/

I was going to say that is why "we need the sum of all the others intersecting the individual set"

probably will get to that eventually.

I prefer the unique representation definition, I think it captures the point more clearly.

I agree in my notes I had written that at the bottom of the "formal" definition

Or just go for the direct sum of 2 subspace and then show the order does not matter.

and as we speak, I see why the second question is false....for the pairwise intersection reasoning above

Wine is coming out of the fridge as we speak...

Well it is bed time for me on the East Coast, you West Coast folks can enjoy the rest of the evening. And thank you for the help @copper.hat, even though I should stick around because if the wine is coming out there might be some interesting war stories from back in the day coming out.....

@dc3rd :-) I am afraid my drinking stamina has evaporated, I am a one glass weenie now.

;-)

@StudySmarterNotHarder I am sure it is clear to you but I have no idea what is going on. I used to do formal verification in the hardware design space.

@copper.hat see notes added at end. I pointed out where it is noteworthy

@copper.hat it would be instructive for my next post on CS se, if I know exactly where things are not clear on my post :)

Oh, that's enter a context, such as with open('file.txt', r) as file: print(file.read()); I used it as a mechanism so that proofs cannot be faked.

@copper.hat would C++ be more your forte?

Because I'm thinking of writing a kernel in C++, then a frontend in Python so that the uesr can interact with it from a good iterpreter

I am familar with .py (and c++) and have used many proof systems. i am just not getting how do do a proof.

Or you could address it in C++ itself, but there's no loss in speed with Python merely as a frontend

Proof of the statement S and simply put p = Proof(S) is just variable asignemnt

It's power is not in the syntactic sugar, it's not competing on that front

it's competing at how easily we get to the nitty gritty of say rings and groups, in < 600 lines of code

i don;t get any of the power, so that is lost on me.

It's got a lower learning curve than Coq

Coq is wayyy complex in language specification

so you insist, but unless you can communicate that greatness it will remain with you.

LOL true!

:D

i have used temporal proof systems and i 'get them'. i am not 'getting' what it is you are doing.

I want to prove something interesting, but obviously that will take more than a day of coding. However, the shortest path is not coding everything using Type Theory background. That is my thesis essentially

I'm basically using substitution of expressions, just like humans do math

The expression tree is objects of classes all derived from Statement.

To make a statement, you can And together some other statements. In order to prove something, you must construct the conclusion and the expression you construct must match the statement's conclusion, exactly.

Id = Equality in the code, but I should probably refactor to Eq and _ProvenEq

That will enable me to get rid of the Proof._overrideEnter() / Proof._overrideExit()'s in the code. Those simply enable me to create Id objects but the user's proof cannot. So you cannot fake a proof out of thin air.

i am afraid your explanations are not making it past my cone of not understanding. i am going to sleep shortly. i hope you find another audience.

@copper.hat sorry to disappoint. I will do better with the explanation. Thank you for show me the way :)

pronounce xkcd and you are the Supreme

@Astyx beautiful

There's more

However I can't come up with an example where $X\setminus U$ hase $n$ components and $X\setminus\{x\}$ has more than $n$

Which I think is linked to my intuition about it being "increasing"

This one is better since you can take all the balls as your basis

I'm going to stop spamming now

I have a question

Fun fact: X is path-connected

In this if B is subset of A then p belongs to...

@Astyx yeah that sounds like it should'n happen

@Astyx @AlessandroCodenotti any hint

@LeonhardEuler it has four syllables.

hello everyone

hi

I successfully managed to prove the frullani integral formula

nevermind, I got my mistake

hi, is it generally true that if $X \cup_{f} Y$ is an adjunction space with $f : A \subset X \rightarrow Y$ an attaching map, then $(X \cup_{f} Y) / Y $ is the same space as $X / A$ (up to homeomorphism)?

@robjohn can you help with the above question

@porridgemathematics yes

@Astyx giving off Secret vibes

What is meant by 100%?

If we say percent then 100 is implied isnt it.

Please help.

a percent means a hundredth, so 100% is everything

my daughter's discovered that she can get negative attention by closing the door to the room where we keep the cat's litter box.

Is a colimit in Ab anything obvious ?

Slash, I mean anything concrete like a sum or smth

What is meant by average profit?

@EdwardEvans yes

what is it? A direct sum?

coproduct is direct sum, coequalizer is quotient by image of the difference, all other colimits are constructed from these

I see

my guess would be the sum of the profits in each of those three years, divided by three. it's a strange metric

at least, my ears would perk up if a company i was thinking about investing in told me about its average profits over many years instead of giving me those numbers. i guess i could imagine a loan agreement using a measurement like that to give a borrower some breathing room.

@EdwardEvans My cover has been blown

@Rover Not in my area of knowledge.

@Thorgott I asked because I thought $\Bbb Q/\Bbb Z = \varinjlim \frac{1}{n}\Bbb Z/ \Bbb Z$ but the text I'm reading says $\Bbb Q/\Bbb Z = \bigcup_n \frac{1}{n}\Bbb Z/ \Bbb Z$

both are true

a completely filtered colimit of subspaces is just a union

@robjohn ok

howdy @Astyx @robohn @Thor

Hi

@Thorgott okay this is what I thought

tyty

@EdwardEvans when they aren't yet submodules of each other then you need $\varinjlim$

if all the maps are injections then you can find your original modules inside the direct limit (which is a colimit), and then it will be the "same" as union

@leslietownes Why are you guessing it that way?

i was cleaning out my car, which has driven maybe 100 miles in the last year, and found a book on universal algebra under the passenger seat.

@user586228, just an assumption that "average X" without further qualification means a sum of the things described in X divided by their number. i do not feel strongly about this.

"average" is a funny word. it often means something more than that, but when it actually means that, i always get a little nervous. surely there's more. no, it's just a sum divided by a number. really? really. sometimes, anyway.

Oh, and howdy, @Edward, @Leaky, @leslie.

hi @Ted :)

@Leaky thanks hehe

good morning, ted. it is sunny and a balmy 71F in long beach. my daughter is trying to get the cat to attack her without success.

Describe that...

arguably the integral that you'd compute for that, in a continuous setting, is also a sum divided by a number, but i limit my previous remarks to the discrete setting, where it is often simpler.

and in the cases they bothered to tell us about in high school, where the acceleration was constant, it really was an average.

I'm pretty sure that if she pulls on the cat's tail, she'll get her wish.