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Lost_in_Oregon: 12349876: ph0rk: You already live in an HOA if you live in a town or city or county or state. Eminent domain.

They can condemn your house too, though the governments standards are generally much, much lower than an HOA.

HOAs do not have such power, they cannot take your property for their use. A city can take it, but never heard of one doing it without trying to negotiate first, plus the usual reason is streets, sidewalks, roundabouts, etc.

Maybe you mean easements.

I wasn't talking road building type condemnation, I was talking about condemnation when you let your house get into such a shiatty condition that it is unsafe and if you let that happen to your house the government will come after you and kick you out. Like this article. "'Unsafe to occupy': Apartments condemned due to mold" http://www.tahoedailytribune.com/article/20060629/NEWS/106290041

And the point about HOA's was they will start biatching at you about a shiatty house long before the government generally speaking.

Ed Willy: How come we never hear of the Tea Party opposing HOAs? I can't think of a more obnoxious group that makes such bullshiat rules.

Because they all live in HOAs to keep out the blacks and the other coloreds.

Louisiana_Sitar_Club: I lived in a condo with an HOA for almost two years and I would never ever to it again. I put up a basketball hoop for my kid and the next morning the president of the HOA knocked on my door. When I opened up he pushed past me, kicked over a lamp and sodomized the porcelain dalmatian that I keep next to my fire place. Then he ate all of my left over Pasta Roni. There is something seriously wrong with those people.

Aritocrats!

vudukungfu: Tellingthem: I'm sorry if you willingly buy a place with an HOA then you must follow the rules no matter how silly there are. If you don't like your HOA then move. You would think after all these stories people would learn...

\I will never buy a place with one

Nothing personal, but Stuff a Sock innit.

When you ban children from drawing on the sidewalk with chalk, you are only a few years from watching the police draw around them in chalk on the very same spot.

People have HOAs to keep neighbors from doing things they don't like. If you move into one, you're saying you support that concept. If you then oppose a rule when it's applied to you, you're a dick.

zenobia: People have HOAs to keep neighbors from doing things they don't like. If you move into one, you're saying you support that concept. If you then oppose a rule when it's applied to you, you're a dick.

More of a dumbass.

Really, I don't understand why more people don't just join their HOA boards. They usually have like 3 people who are already sick of it, at least in non-mcmansion neighborhoods. All you gotta do if you want to make a power play is deal with the crazy nutball people that complain about shiat like chalk on sidewalks.

My mom and dad brought me up in a home with an HOA.

When they objected to the pink flamingo I'd insisted on 'adopting' from the dollar store, named and treated as a pet (couldn't have a real one thanks to my stupid baby brother's allergies,) Mom simply put tennis balls on the ends of Agador's wire legs and he moved inside with us. (I'd seen 'The Birdcage' as a child, because Mom and Dad are cool.) She also let me take him, on his leash, to the monthly HOA meeting and made the Board explain to four-year-old me, in front of everyone, why they hated Agador and I could not have him as a pet. They caved at once and gave him a special exception, but by then he was quite happy in my room and only went outside again for walks.

When they objected to the marigolds I'd planted for a first-grade project, Mom pointed out that there were many other households who had them and that if they wanted to explain to Channel Four why there were preschoolers crying about the mean old men taking away their flowers on the evening news, one of her bridesmaids was an anchorwoman and it'd been a slow news week. They caved and agreed to add marigolds to the list of 'approved flowers.'

When Dad's car broke down and he and I were fixing it ourselves (age eight, I mainly handed him wrenches,) they send a nasty letter the third day it was up on jack stands, saying that junk cars would be subject to a $10-per-day fine for EACH day spent up on blocks. Dad and Mom were both working two jobs and alternators are a little tricky to replace when you only have an eight-year-old helping you and just one hour free a day. He used one of his rare days off and we got it fixed, then Dad drove me to the HOA building, still covered in engine grease and wearing my most adorable too-big coveralls and let me walk in all by myself with their letter, throw it on the desk and say "It's done, are you happy now?!" (It was the first time I'd ever been authorized to be rude to adults and the memory still makes me grin.) We weren't charged a dime and the Board member on duty actually apologized. When my baby sister arrived, the HOA objected to the 'It's a Girl' wreath on the front door that Mom's coworkers had sent. My uncles had also sent my brother and I a BB gun each, since we were older siblings now and would need to protect the family. Mom and Dad provided cast-iron alibis backed up by the pastor who lived next door after the HOA Board's tires and windshields were all inexplicably shot out. Amusingly enough, the local Safelite and Firestone guys were old friends of Dad's and the entire Board wound up being fined$140 each for having broken windshields for two weeks, their 'junk' cars were towed and were kicked off the Board for unpaid fines.

My parents freakin' rock.

I have to say I am somewhat sympathetic. I too hate the chalk messes on the sidewalk. Its little different from grafitti and makes the neighborhood look trashy. It would be easy to blame the kids but it is actually the parents who faiedl to teach them basic respect for their neighbors.

That said, I would far rather have my testicles cut off with a dull, rusty butter knife and spoon-fed to me with a sidehelping of excrement than live under an HOA. While I dont like folks who cant control their kids well to keep them from scrawling on the sidewalk, putting up with such things is just the price you pay for living near others. Which is of course why I plan to eventually move onto several acres in the hills.

Some things just don't really matter. This is one of them. I live in a neighborhood without an HOA, but I am single with no kids. All the neighbors have kids, and as long as the kids are not causing damage to property, let them be. Little girls drawing with sidewalk chalk in the summer? Spare me.

I say let them be.... there's some amazing sidewalk art out there.

What is all this Fuhrer about?

Skywolf Philosopher: I would rather live on an island all by my lonesome then put up with zoning laws and HOA. Without those illegitimate sons of adulterous female dogs-bastard sonsabiatches- trying to control my property, I would love to move to a quiet little town where I don't so much as need a car to get around.

/But, thanks to those onion-headed farkons my dream is now to find a peaceful place in the boondocks where I can damn well have a junk car for parts and paint my shed green if it behooves me.

Yes, I would too!

Thorak: The ONLY reason for laws about lawn length or paint colors or what you can park in your driveway is busybodies who want to get in touch with their inner fascist.

And socialists who insist that it is your responsibility to support the price of their house.

When did we give ultimate deciding power to the most offendable?

What could possibly be more innocuous than sidewalk chalk? The kid is gonna get bored of that in a couple days, then it will rain, and they'll have their precious unmarked pavement again. Whatever happened to being slightly tolerant of those around you for the sake of getting along? Everybody can't get their way 100% of the time.

zenobia: People have HOAs to keep neighbors from doing things they don't like. If you move into one, you're saying you support that concept. If you then oppose a rule when it's applied to you, you're a dick.

Enjoy hosing the life blood of your crotch droppings off your sidewalk, then.

A scene from Scrooged ran through my head:

Grace: Oh, it's a painting, one of my kids did. See, there's Santa Claus and there's Mrs. Claus.
Frank: Honey, how many fingers does Mrs. Santa Claus have here?
Grace: Eleven.
Frank: Eleven. Right. (rips it down) It's crap. Lose it. I don't want it on the wall. (throws it in the trashcan)

Vangor: ph0rk: Because the HOAs wouldn't work if that were the case. People with money like HOAs.

Oh, I am not saying this would ever happen until we are rebuilding civilization, just fantasizing.

/sorry, couldn't resist

SpiderQueenDemon: My mom and dad brought me up in a home with an HOA.

When they objected to the pink flamingo I'd insisted on 'adopting' from the dollar store, named and treated as a pet (couldn't have a real one thanks to my stupid baby brother's allergies,) Mom simply put tennis balls on the ends of Agador's wire legs and he moved inside with us. (I'd seen 'The Birdcage' as a child, because Mom and Dad are cool.) She also let me take him, on his leash, to the monthly HOA meeting and made the Board explain to four-year-old me, in front of everyone, why they hated Agador and I could not have him as a pet. They caved at once and gave him a special exception, but by then he was quite happy in my room and only went outside again for walks.

When they objected to the marigolds I'd planted for a first-grade project, Mom pointed out that there were many other households who had them and that if they wanted to explain to Channel Four why there were preschoolers crying about the mean old men taking away their flowers on the evening news, one of her bridesmaids was an anchorwoman and it'd been a slow news week. They caved and agreed to add marigolds to the list of 'approved flowers.'

When Dad's car broke down and he and I were fixing it ourselves (age eight, I mainly handed him wrenches,) they send a nasty letter the third day it was up on jack stands, saying that junk cars would be subject to a \$10-per-day fine for EACH day spent up on blocks. Dad and Mom were both working two jobs and alternators are a little tricky to replace when you only have an eight-year-old helping you and just one hour free a day. He used one of his rare days off and we got it fixed, then Dad drove me to the HOA building, still covered in engine grease and wearing my most adorable too-big coveralls and let me walk in all by myself with their letter, throw it on the desk and say "It's done, are you happy now?!" (It was the first time I'd ever been authorized to be rude to adults and the memory still makes me grin.) ...

I was on your side until the last paragraph. You, your family, the pastor, and your Dad's old friends became bigger dicks than the HOA.

LiberalEastCoastElitist: When did we give ultimate deciding power to the most offendable?

What could possibly be more innocuous than sidewalk chalk? The kid is gonna get bored of that in a couple days, then it will rain, and they'll have their precious unmarked pavement again. Whatever happened to being slightly tolerant of those around you for the sake of getting along? Everybody can't get their way 100% of the time.

I think it was the 1980s, when the FCC decided that 3 nutjobs who didn't know how to turn the radio off spoke for the entire USA.

If I wanted my freakin' sidewalks tagged I'd live in the ghetto.

BroncoFan_17: If I wanted my freakin' sidewalks tagged I'd live in the ghetto.

Yes, a little girl drawing flowers in chalk on your sidewalk is placing you in socie-economic peril.

Thorak: The ONLY reason for laws about lawn length or paint colors or what you can park in your driveway is busybodies who want to get in touch with their inner fascist.

They aren't laws, they are contractual rules you agree to - and the reason for them is because if you white trash up your house it farks up your neighbors property value.

Winston Smith '84: I was on your side until the last paragraph. You, your family, the pastor, and your Dad's old friends became bigger dicks than the HOA.

Don't worry, it was fiction. A dad whose old buddies work at repair shops is not a dad who needs a car up on jack stands for a week in order to replace an alternator. But the stories were entertaining, weren't they?

If it's a public sidewalk, how does the HOA have a say? Wouldn't it fall to the city to have a law against that?

Far as i know the kids are not a home owner so they are not subject to the HOA rules.

Thats awesome. Its like twirling a rock around your head by a line. In a cage swarming with birds. It kills so many birds with one stone.

RedPhoenix122: FirstNationalBastard: Self-righteous Parents vs. HOAs?

Can they annihilate each other?

I say we let them fight it out in court.

/And while they're doing that, pump lethal gas through the vents.

RedPhoenix122: FirstNationalBastard: Self-righteous Parents vs. HOAs?

Can they annihilate each other?

I say we let them fight it out in court.

/And while they're doing that, pump lethal gas through the vents.

majestic: Draw on your own shiat? I'm all about fark the hoa's, but, really, just draw on your own shiat. Not everyone likes finding pictures of pee-pees and wee-wees on their sidewalks.

It's amazing how often the first thing out of someone's mouth upon hearing about children doing something is insults towards the kids along with a word for penis or sex.

I was going to chime in here about how you have to pick your HOA wisely, they're not all power crazy. We used to live in one, but it was very mellow - dues are cheap and pay for trash collection, community pools and parks. And as far as I ever knew, no enforcement whatsoever, and probably not even any rules. Obviously if you're causing some destruction or disturbance in a park or pool, that's an issue. Anyway...

Holy shiat, that's my old neighborhood! The kids on my block used to do lots of sidewalk chalk too. I never imagined anyone might object to that. Seriously, wtf?

It sounds like the HOA may still officially decide this is a bs complaint, I certainly hope so.

slayer199: If it's a public sidewalk, how does the HOA have a say? Wouldn't it fall to the city to have a law against that?

In most such neighborhoods, the sidewalks, lawns, streets, curbs, alleys, signage and other areas are designated "common" and are thus joint property of the community. The cops can't give tickets for speeding, for example, tho we do get other city fire and police protection.

Approves

Straelbora: Seriously, if you're 'offended' by sidewalk chalk drawing by a 3 year old, you should move to Pakistan where everything can be treated as an outrage and you can just spend all your time rampaging through the streets.

Vangor: jayphat: Ed Willy: jayphat: Oh you bought a house in a HOA? This is how many farks I give.........

Why don't Republicans pushing "right-to-work" laws also push "right-to- live" laws that let you buy a house without joining the HOA?

Holy Fark. As someone who leans conservative, why the fark have I NEVER thought of that?

This is a rather awesome idea, actually.

Except for the fact that buying a house with an HOA is 100% VOLUNTARY. I've probably closed 5,000 home loans in 25 years and never, ever once did I see a buyer have a gun held to their head when they accepted the deed to their newly acquired property and the attached covenants and restrictions that come with being a member of the HOA.

Not once.

WHY IS THIS 3 YEAR OLD WASTING HER TIME DOING THIS
WHEN SHE COULD BE DOING HOMEWORK
OR LEARNING JAVA IN 22 DAYS

Some call it a popular summertime pastime for children and others are calling it distracting and offensive.

Really? Who? There are people in this world who think sidewalk chalk drawings are distracting and offensive?

I guess some people are neurotic to the point that they are annoyed by the existence of things, which infringe on their right to look outside and not see things.

Sometimes I wonder if an HOA actually makes people more neurotic, because it gives people an avenue to complain. I know that the opportunity to object to something will sometimes cause people to hate something they would otherwise like or not care about.

If you don't want to be subjected to insane rules and busy-body-ing, don't buy a house with a HOA. Yeah, you might get a car up on blocks on your block. Pick your poison.

/I prefer cars up on blocks.

jmr61: Vangor: jayphat: Ed Willy: jayphat: Oh you bought a house in a HOA? This is how many farks I give.........

Why don't Republicans pushing "right-to-work" laws also push "right-to- live" laws that let you buy a house without joining the HOA?

Holy Fark. As someone who leans conservative, why the fark have I NEVER thought of that?

This is a rather awesome idea, actually.

Except for the fact that buying a house with an HOA is 100% VOLUNTARY. I've probably closed 5,000 home loans in 25 years and never, ever once did I see a buyer have a gun held to their head when they accepted the deed to their newly acquired property and the attached covenants and restrictions that come with being a member of the HOA.

Not once.

Congrats, that was the douchey-est response I've seen so far in this thread.

\it's true
\\but still douchey
\\\disagree about the "gun to the head" tho - I've been to those signing parties
\\\\and they end up with the buyer just signing blank pcs of paper "in case we need it later"

ph0rk: Thorak: hey have zero business whatsoever making any "regulations" about private property use in any way.

If you have 10 acres out in the boonies, fine.

However, if I can see your magenta door and fusca shutters from my front steps, it bothers me.

It seems absurd, but some people in small 1/4 acre lot neighborhoods do some crazy, hideous things to their houses, and that affects the property value of their neighbors.

/This is why I like condos.

Yes, and my point is, so what?

What right do you have to control someone else's legal actions on their own property? We already have laws to cover things like public indecency and whatnot, or unsafe construction, or a host of other things, we don't need an HOA to provide additional rules.

lilplatinum: They aren't laws, they are contractual rules you agree to - and the reason for them is because if you white trash up your house it farks up your neighbors property value.

If it were a free and open choice, I'd have no problem with it. The issue is that I'm supposedly buying a house. If I could buy the house and THEN, if I chose to, enter into an HOA agreement, then I'd see no argument. But they're stuck right on the lease.

It's like if you bought a car but the dealership said "okay, you can't buy gas anywhere but Esso, use any tires but Michelin, and you'll bring it in to the dealership 4 times a year for a mandatory servicing. If you fail on any of these you will be fined monthly and it the fines accumulate enough we'll repossess the vehicle".

If you want to control what the person living in a house does to the property, the relationship you're looking for is landlord/tenant. Rent the houses. If you're selling them, you should be giving up all right to control what the tenant does to the property. If they choose to bulldoze the entire place and build a brand-new house on the property in a completely different style, there should be ABSOLUTELY NOTHING any neighbour can do so long as the house abides by the zoning and construction laws for the city. If I paint if with fuschia polka dots and cover the entire lawn with a rock garden, tough. Deal. if any of this hurts your property values, not my problem.

slayer199: If it's a public sidewalk, how does the HOA have a say? Wouldn't it fall to the city to have a law against that?

As I understand it an HOA is like a condo corporation. It's set in place by the developer to get around certain municipal requirements so they can build faster and cheaper. Anyone who buys in to the neighborhood pays fees like condo owners, and the roads, sidewalks, parks etc are communal property, owned by the HOA.

I warned ya, Skinner! Didn't I warn ya? That colored chalk was forged by Lucifer himself.

Would it make any difference if the kid draw pictures of the American flag?

The derp is strong in this thread.

Lived in a development with a HOA in Greeley: "Villa Serena", which I always referred to (to the dismay of the board) as "Villa Varicosa". Was 62 at the time, and the youngest resident, though many of the homeowners were under 50 yeas of age.

/Fark HOAs
//Lawn, off,stay.

Older than the internet:

Kids Drawings

NSFW language (but funny as Hell)

ph0rk: It seems absurd, but some people in small 1/4 acre lot neighborhoods do some crazy, hideous things to their houses, and that affects the property value of their neighbors.

If your neighbor paints his door magenta and this "affects" your property value, first I call BS, and second, so what? Your neighbor is not supposed to act in a way that maximizes your house value across the street. He's not supposed to mow your lawn either.

Xcott: ph0rk: It seems absurd, but some people in small 1/4 acre lot neighborhoods do some crazy, hideous things to their houses, and that affects the property value of their neighbors.

If your neighbor paints his door magenta and this "affects" your property value, first I call BS, and second, so what? Your neighbor is not supposed to act in a way that maximizes your house value across the street. He's not supposed to mow your lawn either.

Lots of cities don't agree with you. Irvine and Cerritos in California come to mind.

Chameleon: bojon: I remember when Stapleton was an airport surrounded by ghettos.

I have fond memories of stopping by that crazy trucker diner with the coffee pot-shaped sign just off the Stapleton exit for greasy breakfast before my flight. And the floor tiles that were exactly the same "mixed color vomit" shade as the tiles of my elementary school. Good times, good times...

Sapp Brothers truck stop! I used to hang out with Rich Sapp, a real party animal.

Not far from the diner was a strip club named "The Landing Strip."

ph0rk: Hrist: Don't think they won't do it if they had the chance.

"We shall have lots of rules!"

Of course, the solution most people first suggest to any malfunctioning bureaucracy is more rules.

Branbot: So people actually have to take HOA seriously? I was a bored kid in white suburbia. We vandalized and played every dumb game there is on our neighborhood. HOA hated us, they kind of had a creepy obsession with talking about us at every meeting after a while. Our parents never admitted that we were theirs. There were never ramifications for our actions. But I guess we were kids.

If the HOA chooses to, they can fine your ass. That shiat is actionable, no matter how retarded the reason for the fine, assuming it is in line with their charter.

And if you don't pay the fines they can foreclose on your house, often without a judicial hearing.

Let the snowflake chalk up the driveway and keep her off areas that don't belong to you. Common courtesy. It's your kid... you get to look at the crappy drawings.

/dislike HOAs
//dislike lazy parents more

slotz: /sorry, couldn't resist

I did offer to continue our conversation via email if you wanted.

jmr61: Except for the fact that buying a house with an HOA is 100% VOLUNTARY.

Not exactly 100%, but highly voluntary, yes. No one will force you to purchase a home, but tons of factors limit the availability of houses to people, and hard to know how tedious and obnoxious an association is until moving in. I would prefer homes were not attached with senseless rules disconnected to the usage of common areas or availability of services.

The only thing worse than an HOA is someone who buys into the HOA with no idea what they're in for.

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text:
# Trying to recreate a solution - ODE

I'm trying to copy the method from this video https://www.youtube.com/watch?v=FTHLz2vgj2I

to solve the following differential equation: $\frac{dr}{d\theta}+r\tan \theta=\frac{1}{\cos \theta}$

Basically what the video uploader did:

He had a differential equation, he brought it to the form $y'+P(x)y=Q(x)$ and the then said:

$$u=e^{\int P(x)dx}, (uy)'=uQ$$

And then he solved for $y$.

I tried to recreate that solution with my question.

$u=e^{\int P(x)dx}=e^{\int \tan \theta d\theta}=e^{-\ln( \cos \theta)}=\frac{1}{\cos \theta}$

$$\frac {d(ur)}{d\theta}=\frac{d(\frac{r}{\cos \theta})}{d\theta}=\frac{r'\cos \theta+r\sin \theta}{\cos^2 \theta}=\frac{1}{\cos ^2 \theta}$$

this implies that $r' \cos \theta+r\sin \theta=1$, which is the exact same ODE that I was asked to solve...So the method in the video did not work here, why?

## 1 Answer

Hint:

The method works, you just have to integrate it up. If $$\frac{d(r/\cos\theta)}{d\theta} = \frac{1}{\cos^2\theta}$$

then

$$\frac{r}{\cos\theta} = \int\frac{d\theta}{\cos^2\theta}$$

Solve the integral (remember to include the integration constant) and you get the solution. This integral is well known, but if you don't know it try to compute $\frac{d\tan\theta}{d\theta}$ and compare. In the end if you do everything correctly you should get an answer on the form $r = A\cos\theta + B\sin\theta$.

format: plain


text:
MATHESIS - Integrating mathematical knowledge in a unique transversal curriculum (logic, arithmetic, algebra, analysis, geometry...)

E-BOOKS - Discover the e-books of the MATHESIS curriculum and master higher mathematics by yourself

The real numbers are all the “quantities” that we can order, and we can “construct” them in various ways thanks to set theory.

Numbers govern the world.” Pythagoras

The real numbers idealise all the “points” of the geometrical line, or from the arithmetic point of view all the possible “measures” of magnitude. The Ancients knew thanks to Pythagoras’ theorem that one could not measure the diagonal of the square of side 1 by a rational number, i.e. fractional, of the form $$\dfrac a b$$ for integers $$a$$ and $$b$$ (one says that this diagonal is incommensurable). Today, we would say that the length of this diagonal is $$\sqrt 2$$.

A host of “quantities” have since been discovered that are not rational numbers, such as $$\pi$$, the ratio of the perimeter of a circle to its diameter, or $$e$$, the basis of the exponential function. Real numbers theorise all those quantities that can be “ordered”, like rational numbers, but which exceed them, as it were. They can be constructed in a precise mathematical sense, from rational numbers, in several ways: the most famous are undoubtedly the method of Cauchy sequences, and that of Dedekind cuts.

## 1.There are “holes” in the set of rational numbers

### 1.1.The problem with the axiomatisation of the continuum

As we saw with the natural numbers (see What is a natural number?), in order to theorise a type of mathematical object we are interested in the set of objects of this type, of which we try to identify the fundamental properties (“postulates” or “axioms”). It is the same for real numbers: even if the intuition of what they are is more or less accessible to us, the mathematical theory about them will consist in describing a set, noted $$\mathbb R$$, of real numbers.

Contrary to the case of the natural numbers, however, it is not obvious to give elementary intuitive properties which would describe this set by clearly differentiating it from the set $$\mathbb Q$$ of rational numbers. For example, the idea of “continuity” of a geometrical line can be expressed by saying that for two points A and B of this line, there always exists a point C strictly comprised between A and B: one can “fill” indefinitely the spaces determined by two points. Now, this property is already verified for rational numbers: if $$x,y$$ are such numbers, with $$x<y$$, we can easily show that there always exists a rational number between the two, that is to say a rational number $$z$$ such that $$x<z$$ and $$z<y$$ (for example, $$z=\dfrac{x+y}{2}$$).

This property of the set $$\mathbb Q$$ has caused this set to be called in the past the “first order continuum” (see for example Henri Poincaré, in “La science et l’hypothèse”). The idea is to differentiate this continuum from a “second order” continuum, which would correspond precisely to the set $$\mathbb R$$ of real numbers. This also means that this first idea does not work to identify the set $$\mathbb Q$$.

### 1.2.Some quantities of geometry are not measurable by rational numbers

Let us return to the example of the length of the diagonal of the square of side $$1$$: this length is $$\sqrt 2$$, a number which can be shown not to be rational, which Pythagoras already knew and which was at the time a tragedy for his philosophy, because certain geometrical relations did not correspond to known numbers! On the other hand, this number is a solution of the equation $$x^2-2=0$$, since by definition the square of $$\sqrt 2$$ is $$2$$.

There are therefore equations with integer or rational coefficients, which have no solution in $$\mathbb Q$$: we expect some of these solutions to be real numbers, as is the case for $$\sqrt 2$$: we can indeed place $$\sqrt 2$$ on the scale of magnitudes. Other numbers, such as $$\pi$$ and $$e$$, must also represent quantities, but are not even solutions of an equation. In any case, there are “holes” in the set $$\mathbb Q$$, in the sense that some numbers are “missing” to measure certain magnitudes, i.e. certain “numbers” that can be situated in relation to all rational numbers.

It is to the French mathematician Augustin Louis Cauchy that we attribute a construction, perhaps the most famous, of the real numbers from the rational numbers, that is to say a description using the set $$\mathbb Q$$ and the resources of naive set theory (see the What is a set?). This construction, like that of the German mathematician Richard Dedekind, exploits this idea of “holes” in the set $$\mathbb Q$$.

## 2.Filling the holes with Cauchy : building the real numbers

### 2.1.Sequences of rational numbers “as close as we want”

The idea of the Cauchy construction is to consider that the “holes” existing in the set $$\mathbb Q$$ can be “approximated” indefinitely by sequences of rational numbers, i.e. infinite lists of numbers $$r_0,r_1,\ldots,r_n,\ldots$$, indexed by the natural numbers. There are indeed sequences of rational numbers which have a limit, i.e. which “approach indefinitely” a certain rational number $$r$$, in the precise mathematical sense that one can “approach as much as one wants the number $$r$$” provided that one “places oneself far enough in the sequence”; this idea finds a rigorous mathematical formulation thanks to set theory.

For example, the sequence $$1, \frac 1 2, \frac 1 3, \ldots,\frac 1 n,\ldots$$ has for limit the number $$0$$. Now, it turns out that these sequences, which are called convergent, have another analogous property: their terms are “as close to each other as we want” provided we choose them “far enough in the sequence”. This property, known as “Cauchy’s property” can also be formulated in a rigorous way, and if one notes the obvious analogy with the property of being a convergent sequence, this one is an intrinsic property of the sequence, not calling upon a possible “external” number.

### 2.2.Filling the “holes” with Cauchy sequences

Now, if all convergent sequences, i.e. which have a limit in the set $$\mathbb Q$$, have Cauchy’s property, the opposite is not always true : there are Cauchy sequences of rational numbers, which do not have a limit in $$\mathbb Q$$ ! The ingenuity of the Cauchy construction consists in considering that the “holes” of the set $$\mathbb Q$$ correspond to virtual limits of Cauchy sequences which do not have any… In other words, we can “represent” what we mean by a “real number” as a limit of such a Cauchy sequence, whether it exists or not.

Strictly speaking, several Cauchy sequences can intuitively determine the same virtual limit, when their respective terms are indefinitely close to each other, in other words when the difference of their respective terms has limit $$0$$: we say that two such sequences are equivalent. Thus, we can construct the real numbers as sets of equivalent Cauchy sequences, with the sole resources of set theory (and that of rational numbers, which is based on the theory of natural numbers).

### 2.3.One extends the “structure” of the set $$\mathbb Q$$ to a set $$\mathbb R$$ of real numbers}

In this approach, one can represent each rational number $$r$$ as a real number: one represents it as the set of sequences equivalent to the constant sequence $$r,r,\ldots,r,\ldots$$! Thus, we can consider that the set noted $$\mathbb R$$ of real numbers is an “extension” of the set $$\mathbb Q$$ of rational numbers. The arithmetical operations $$+$$ and $$\times$$ of the rational numbers extend to the set $$\mathbb R$$, as well as the order relation $$<$$ between the rational numbers, as well as several other properties. The demonstration of these facts is beyond the scope of this article.

The set $$\mathbb R$$ has in addition a fundamental property that the set $$\mathbb Q$$ already has, called Archimedes’ property: if $$x$$ is a real number, there always exists a natural number $$n$$ greater than $$x$$, i.e. having the property that $$x<n$$; this seemingly simple property is essential to real analysis, i.e. the study of functions of real numbers (polynomial functions, roots, sine, cosine, exponential, logarithm. …), as well as the property known as the “upper bound property”, which is not true in the set $$\mathbb Q$$ : any subset $$E$$ of the set $$\mathbb R$$, non-empty and having an upper bound (i.e. such that any element of $$E$$ is lower than a certain real number), has an least upper bound, i.e. one which is smaller than any other. This property is analogous to an essential property of the set $$\mathbb N$$ of natural numbers: any non-empty subset of $$\mathbb N$$ has a smallest element.

## 3.The real line is complete: there are no more holes

### 3.1.All Cauchy sequences of real numbers have a limit

Let us remark about Cauchy’s construction that since we have constructed the set $$\mathbb R$$ (also called “real line” because of its geometrical representation) by ” filling the holes ” of the set $$\mathbb Q$$, we could ask ourselves if we have filled “all the holes”. In other words, what guarantees us that there are no holes left in $$\mathbb R$$, in the same sense that there were in $$\mathbb Q$$ ? This question is reformulated with the concepts evoked previously in a precise way: whereas any Cauchy sequence of rational numbers does not have a rational limit, does any Cauchy sequence of real numbers have a real limit? If the answer is negative, we have only moved the problem. Fortunately, the answer is yes, and the proof relies on what is generally called a “diagonal argument”. This property is called the completeness of the real line, the least upper bound property can be deduced from it, and the abstract completeness property gives rise to the theory of spaces with similar properties to this line, called “complete metric spaces”.

### 3.2.The completeness of $$\mathbb R$$ allows the definition of the functions of analysis

This completeness property allows the definition and study of many real functions, such as trigonometric functions (sine, cosine, tangent…) and their reciprocals, polynomial functions, power and root functions, exponential and logarithmic functions…, the knowledge of which is important both for mathematical science itself and for applications of analysis and geometry to other sciences. For example, as we have already discussed the integer $$2$$ has no square root in the set $$\mathbb Q$$. In contrast, any real number $$x$$ has a root of any order $$n$$ (for a given natural number $$n$$) if $$x\geq 0$$, and of any odd order if $$x<0$$. Such a root is a real number $$y$$ such that $$y^n=x$$, and this notion can be defined thanks to the least upper bound property, which follows from the completeness of $$\mathbb R$$.

### 3.3.The quantities of geometry, classical and modern, are measurable by real numbers

On this subject, on the level of the relation between arithmetic and geometry let us note that the new properties of the real numbers make it possible to solve the antique problem of the measurement of the diagonal of the unit square thanks to the number $$\sqrt 2$$, but also to measure the diagonal of any rectangle, and the circumference of an unspecified circle thanks to the number $$\pi$$: the application of rational numbers to the measurement of certain ratios of quantities associated with plane figures (constructed for example with a ruler and a compass) finds in the theory of real numbers the basis of a vast extension which concerns the application of the measurement of quantities to any real “shape”, in the plane or in higher dimensions. This is linked to the possibility of integration and differentiation theories, and constitutes an essential progress of modern mathematics compared to ancient mathematics. Differential geometry, a geometric theory classically associated with mechanics and physics in general, is made possible by these theories, and thus essentially by the completeness of the real line.

## 4.Another way to represent real numbers : Cantor, Dedekind and axioms

### 4.1.Other ways to fill in the gaps: Cantor and Dedekind’s methods

There are other ways to “construct” a set of real numbers. One of them, that of Georg Cantor, the father of set theory, consists in considering “adjacent” sequences of rational numbers, i.e. sequences varying in opposite directions and approaching each other indefinitely. Another related method, that of Richard Dedekind, consists in considering “cuts” of the set $$\mathbb Q$$, that is to say decompositions of this set into a lower part and an upper part. In both cases, the same idea of “filling the holes” is exploited : in the first case, there can be holes between two adjacent sequences, in the second case one can cut the set $$\mathbb Q$$ at the level of a “hole” !

### 4.2.The same essence of a “complete archimedean field”

What is important to remember is that these constructions lead to the same mathematical “object”, in the sense that we mentioned in the article on natural numbers: even if the sets of real numbers “constructed” by these methods are strictly speaking different, they are mathematically indistinguishable. It is moreover remarkable that one can explicitly give an abstract list of axioms which characterise (any version of) the set $$\mathbb R$$ ! This is what is cleverly called a complete archimedean field.

### 4.3.Real numbers in other universes

Finally, let us note that Dedekind’s construction, for example, can be reproduced in certain mathematical universes called “topos” when these possess a “fairly rich structure”. This means that real “numbers” can be defined or constructed in universes other than a universe of sets, using a more demanding logic, called “intuitionistic”, than the usual logic; we then obtain objects of another nature, which are sometimes similar to ” true” real numbers! For example, continuous real functions are real numbers “in another universe”. But this is another story.

## In short

If the intuition of real quantities exists, and if their quest is born from an observation of the incompleteness of an arithmetical or geometrical situation, discovering that they could be thought of as numbers exceeding but extending the rational numbers was not so obvious. Determining the set of real numbers is not as straightforward as defining the set of natural numbers, but with the resources of integers and set theory, we end up with the unambiguous definition of a formidable and perfectly axiomatised mathematical object, which solves the problem of measuring geometric quantities by numbers that can be ordered in a continuum.


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## Return to Question

2 deleted 256 characters in body

A complex manifold $X$ is said to be weakly pseudoconvex if there exists on $X$ a smooth plurisubharmonic exhaustion function $\psi$.

For example, Stein manifolds are weakly pseudoconvex (in this case $\psi$ can be chosen to be even strictly plurisubharmonic), compact complex manifolds are weakly pseudoconvex (take $\psi\equiv 0$), etc...

Now, let $X$ be a non-compact weakly pseudoconvex manifold. The question is the following:

Does there exist a smooth plurisubharmonic exhaustion function $\varphi$ on $X$ such that $d\varphi$ is never zero outside some compact set $K\subset X$?

This is indeed true for $X$ a Stein manifold or, equivalently by the solution of the Levi problem, for strongly pseudoconvex manifolds: these are in fact embeddable as close submanifolds of some affine complex space and $\varphi=|z|^2$ will do the job.

Thanks in advance.

1

# Plurisubharmonic exhaustion functions without critical points at infinity

A complex manifold $X$ is said to be weakly pseudoconvex if there exists on $X$ a smooth plurisubharmonic exhaustion function $\psi$.

For example, Stein manifolds are weakly pseudoconvex (in this case $\psi$ can be chosen to be even strictly plurisubharmonic), compact complex manifolds are weakly pseudoconvex (take $\psi\equiv 0$), etc...

Now, let $X$ be a non-compact weakly pseudoconvex manifold. The question is the following:

Does there exist a smooth plurisubharmonic exhaustion function $\varphi$ on $X$ such that $d\varphi$ is never zero outside some compact set $K\subset X$?

This is indeed true for $X$ a Stein manifold or, equivalently by the solution of the Levi problem, for strongly pseudoconvex manifolds: these are in fact embeddable as close submanifolds of some affine complex space and $\varphi=|z|^2$ will do the job.

Thanks in advance.


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1. ## Maximum Principle

The maximum principle for harmonic functions asserts:
Let $u \in C^2(U) \cap C(\bar{U})$ be harmonic in $U \subset \mathbb{R}^n$.
Then
1) (Weak maximum principle)

$\max_{\bar{U}}u=\max_{\partial U}u$;

2) (Strong maximum principle)

If $U$ is connected and there exists a point $x_0 \in U$ such that

$u(x_0)=\max_{\bar{U}}u$,

then $u=const$ in $U$.

I wonder if I got this right. The weak principle means that the maximal value on $\bar{U}$ of a harmonic function is necessarily attained in a boundary point, i.e. there are no solely interior points which give maximal values for a harmonic function. Although it may be that there are points $x \in \partial U$ and $y \in U$ such that

$u(y)=\max_{\bar{U}}u=u(x)$,

in which case the shape of $U$ may cause the function to be constant (i.e. in case $U$ is connected and so 2) holds); but if $U$ is any open set nothing specific is known about $u$, i.e. it may rather be non-constant.

I wonder why the conclusion of the strong version holds just for $U$ and not for $\bar{U}$. I mean 1) holds anyway and $u \in C(\bar{U})$. Doesn't this, together with 2), imply that $u$ is constant on the boundary? The proof of the strong version, however, works only with $U$, so that it's no big deal to accept it as it is.


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## Algebra: A Combined Approach (4th Edition)

Published by Pearson

# Chapter 7 - Test: 2

a) 115 b) 103

#### Work Step by Step

a) When x = 200, C = $\frac{200(100) + 3000}{200}$ C = 115 b) When x = 1000, C =$\frac{1000(100) + 3000}{1000}$ C = 103

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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# Is Bekenstein entropy limit inconsistent with universal continuity?

It is unknown whether the universe is discrete or continuous in its intricate quantum level structure.

See for example: Can universal continuity be experimentally falsified?

Is the universe finite and discrete?

How could spacetime become discretised at the Planck scale?

Is time continuous or discrete?

Even so, all branches of modern physics rely heavily on fully continuous structures.

From the Bekenstein bound applied to a black hole, we know that the Information entropy that can be contained inside a black hole is finite and proportional to the surface area of the event horizon.

From the No hair theorem/conjecture, it is believed that the black hole is uniquely described by mass/energy, linear and angular momentum, position, and electric charge, which amounts to a total of 11 real numbers. Possibly, if magnetic monopoles exist, we can add an additional number for magnetic charge.

Most physicists will argue that these 11 numbers are continuous (i.e. not bounded rational approximations).

With an assumption of real continuity, as the black hole undergoes change, for example taking on additional mass over a period of time, the 11 numbers will change as time flows over a continuous infinitude of real numbers, with no smallest increment of time. Each of the 11 numbers then, must assume values that are rational, irrational, transcendental, non-computable and non-definable, as they continuously sweep through the real number field.

In fact, if any number is sampled at random, i.e. at a random time, it will almost surely (i.e with probability one) be non-computable and non-definable. A non-computable and non-definable number has infinite Kolmogorov complexity and carries infinite entropy, as its shortest description is its own random and infinite digit sequence. How is that consistent with the starting assumption of bounded entropy?

• Does this really have anything to do with black holes? There are plenty of other things in physics that are continuous (including any wavefunction), but which are considered to have finite entropy - this argument would apply equally well to all of them. Oct 6, 2015 at 7:15
• True. At the time of writing, it seemed the argument is easier/clearer for black holes, but it could be made for any sphere. Oct 6, 2015 at 15:48
• Edited title and tags to remove reference to black holes. Oct 7, 2015 at 1:47
• Kolmogorov complexity $\neq$ Bekenstein entropy. Jan 9, 2016 at 17:12
• Your assertion that "A non-computable and non-definable number has infinite Kolmogorov complexity and carries infinite entropy" is simply false. It might carry entropy in the sense of information theory and encoding, but it certainly doesn't carry entropy in the sense of thermodynamics - that e.g. a thermodynamics system with temperature $z\ \mathrm{K}$ for $z$ some non-definable number should have infinite entropy because of that temperature is patently absurd. Jan 11, 2016 at 14:25

Your argument completely fails to take quantum mechanics into account.

Consider an electron spin. If you measure the spin of an electron using a standard measurement corresponding to an observable, you can only get two answers, $|v\rangle$ and $|-v\rangle$, where $v$ is some direction in three-dimensional space. And yet, you can prepare the electron so its spin is in any of a continuum of directions (any direction in 3-dimensional space), and you can measure the spin of an electron along any axis in 3-dimensional space.

Further, the von Neumann entropy of the electron spin is one bit, despite the fact that it can be pointing any direction of space. And if you encode information in the spin of an electron, there is a theorem that you can only retrieve one bit.

Now, let's see how your arguments apply to the spin of an electron, which is a very well-understood phenomenon.

From the Bekenstein bound applied to a black hole, we know that the Information entropy that can be contained inside a black hole is finite and proportional to the surface area of the event horizon.

We know that the Information entropy that can be contained in the spin of an electron is finite and equal to one bit.

the black hole is uniquely described by mass/energy, linear and angular momentum, position, and electric charge, which amounts to a total of 11 real numbers.

The spin of an electron is uniquely described by a 3-dimensional unit vector, which can be parameterized by 3 real numbers the sum of whose squares is one – so two real parameters.

How is that consistent with the starting assumption of bounded entropy?

How can two real numbers be characterized by one bit?

In fact, if any number is sampled at random, i.e. at a random time, it will almost surely (i.e with probability one) be non-computable and non-definable.

It's not clear whether quantum gravity lets you measure quantities with arbitrary accuracy. You certainly cannot measure the spin of an electron with arbitrary accuracy.

What your argument fails to take into account is that quantum information theory is different from classical information theory, and is extremely non-inutitive.

We don't know how to quantize space-time, but we can say that your argument that there must be a contradiction somewhere is not consistent with what we know about quantum mechanics.

From the Bekenstein bound applied to a black hole, we know that the Information entropy that can be contained inside a black hole is finite and proportional to the surface area of the event horizon.

And if you have astrophysical sources of collapse, then an external observer can always choose a coordinate system where the black hole has never formed and the where no event horizon has formed yet. And hence where there is no inside. So it is your choice to have them or not. And hence no physical conclusions can be drawn.

From the No hair theorem/conjecture it is believed that the black hole is uniquely described by mass/energy, linear and angular momentum, position, and electric charge, which amounts to a total of 11 real numbers.

The no hair theorem is about solutions that real solutions come closer and closer to approximating as time progresses.

With an assumption of real continuity, as the black hole undergoes change, for example taking on additional mass over a period of time,

Which is a coordinate v system artefact. You can choose one where it never forms. And as long as one person can do that, no physical conclusion can be drawn from assuming it does form.

In fact, if any number is sampled at random, i.e. at a random time,

And that is non physical as well. All you can do is interact with something based on a the material available, you don't have the ability to invoke random times. Sure if something isn't generic you shouldn't expect to be able to see it in a controlled setting. But you can't just a sum E you can do things if you can't actually do them.

How is that consistent with the starting assumption of bounded entropy?

Entropy is about knowledge, and groupings of possible states consistent with coarse graining knowledge. How is that related in the slightest?

• If physics gets to use all the tools from continuous analysis, then you get to use AC also, and whether you like it or not you will have non-measurable sets and non-computable numbers, and every time you sample a value it will be non-computable. Of course it is unphysical and the actual universe doesn't behave like this. But that is what you get if you insist spacetime is actually a differentiable 4-manifold, rather than just conveniently modelled as such. Oct 8, 2015 at 5:26
• @HalfdanFaber I disagree that you can sample the way you claim you can sample. Even if the universe were continuous, you are just assuming you can sample in a some indeterminate way without actually saying how you would do it. And I called you on it. Physics can use continuous results to approximate discrete events in some situations and to some level of accuracy, but that's doesn't magically allow us to do things we can't do, such as make magic samples. Oct 8, 2015 at 5:31
• I complete agree. If physics is merely using continuous approximations to model discrete events or structures, then you cannot sample in any meaningful way, at least in the way I am suggesting. However, in current physics, spacetime as a differentiable 4-manifold is not considered to be an approximation. Oct 8, 2015 at 5:49
• @HalfdanFaber You argument is specious and fallacious. Even if the universe is continuous, you only have a discrete set of finite descriptions of experiments and that is what you can use to collect data. You don't have magic powers. And isomorphisms require morphisms. But experimentally we don't have continuum many anything, so there isn't even an invertible map, let alone a structure preserving such map. There aren't continuum many experimental designs because you don't have magic powers. There aren't continuum many descriptions. Your concept of time isn't what you measure in an experiment. Oct 8, 2015 at 17:29
• @Timeaus, OK, I see. You may find this surprising, but I agree with your line of thought. Thanks for clarifying. I think you are saying that you remain agnostic about the physical inner structure, but choose to employ a well-working continuous model and then restrict probing to what you can actually measure. I think that is a consistent approach. Thanks for your answer. Oct 8, 2015 at 18:20

Is Bekenstein entropy limit inconsistent with universal continuity?

Yes, but that doesn't mean the Bekenstein bound is correct and everything is fine. Entropy can be considered as "sameness", related to available energy. And the expression $S \leq \frac{2 \pi k R E}{\hbar c}$ has a c in it, but the coordinate speed of light at the event horizon is zero.

It is unknown whether the universe is discrete or continuous in its intricate quantum level structure.

A photon has its E=hf quantum nature, but that doesn't mean it approaches you in steps. It's quantum field theory, and the wave nature of matter. IMHO there's no evidence at all for a universe that has some "discrete intricate quantum level structure".

Even so, all branches of modern physics rely heavily on fully continuous structures.

I'd say the universe appears to be continuous.

From the Bekenstein bound applied to a black hole, we know that the information entropy that can be contained inside a black hole is finite and proportional to the surface area of the event horizon.

We don't actually know that, it's hypothetical.

From the No hair theorem/conjecture

Which is also hypothetical.

it is believed that the black hole is uniquely described by mass/energy, linear and angular momentum, position, and electric charge, which amounts to a total of 11 real numbers. Possibly, if magnetic monopoles exist, we can add an additional number for magnetic charge.

There's potential issues with the angular momentum and charge and magnetic monopoles. But I'll go with the flow.

Most physicists will argue that these 11 numbers are continuous (i.e. not bounded rational approximations).

Note that a particle such as an electron has unit charge. But again, I'll go with the flow.

With an assumption of real continuity, as the black hole undergoes change, for example taking on additional mass over a period of time, the 11 numbers will change as time flows over a continuous infinitude of real numbers, with no smallest increment of time.

Fair enough.

The 11 numbers must then each assume values that are rational, irrational, transcendental, non-computable and non-definable, as they continuously sweep through the real number field.

OK.

In fact, if any number is sampled at random, i.e. at a random time, it will almost surely (i.e with probability one) be non-computable and non-definable. A non-computable and non-definable number has infinite Kolmogorov complexity and carries infinite entropy, as its shortest description is its own random and infinite digit sequence. How is that consistent with the starting assumption of bounded entropy?

The number does not actually exist. A bl;ack hole exists, and a photon exists. This photon can fall into the black hole increasing its mass/energy by E=hf, and E can take any value. But we can't say that a black hole consists of n photons or is anything to do with statistical mechanics or information theory. Note the black hole information paradox: "The black hole information paradox[1] is an observational phenomenon that results from the combination of quantum mechanics and general relativity which suggests that physical information could permanently disappear in a black hole, allowing many physical states to devolve into the same state". For all you know the photon totally loses its identity, and the black hole is like one big boson, where everything is the same. For all you know the black hole could be something like a BEC, and subject to a Bosenova.

Sorry this doesn't give you anything definite, but so much of this stuff is hypothetical.


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# Complex Number Power Formula

A complex number is the form of $x$ + $y^{i}$, where x and y are the real numbers and i is an imaginary number. The i satisfies $i^{2}$ = $-1$.

The complex number power formula is given below.

$\LARGE z^{n}=(re^{i\theta})^{n}=r^{n}e^{in\theta}$

Solved Examples
Question 1:Compute: $(3+3i)^{5}$

Solution:

Here is the exponential form of 3+3i

$r=\sqrt{9+9}=3\sqrt{2}$

$\tan\theta=\frac{3}{3}\Rightarrow\arg z=\frac{\pi}{4}$

$3+3i=3\sqrt{2}e^{i\frac{\pi}{4}}$

Now, $(3+3i)^{5}=(3\sqrt{2})^{5}e^{i\frac{5\pi}{4}}$

$=972\sqrt{2}\left ( \cos \left ( \frac{5\pi}{4} \right )+i\sin \left ( \frac{5\pi}{4} \right )\right )$

$=972\sqrt{2}\left ( -\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i\right )$

$=-972-972i$


text:
#### Which of the following statements about hydrogen is incorrect? Option 1) Hydrogen has three isotopes of which tritium is the most common. Option 2) Hydrogen never acts as cation in ionic salts. Option 3) Hydronium ion, H3O+ exists freely in solution. Option 4) Dihydrogen does not act as a reducing agent.

As we learned in

Acidic Nature -

It is a weak acid turn blue litmus red only in pure state.

- wherein

Hydrogen has 3 isotopes Tritium, Deuterium and Protium of which Protium is the most common.

Option 1)

Hydrogen has three isotopes of which tritium is the most common.

This option is correct

Option 2)

Hydrogen never acts as cation in ionic salts.

This option is incorrect

Option 3)

Hydronium ion, H3O+ exists freely in solution.

This option is incorrect

Option 4)

Dihydrogen does not act as a reducing agent.

This option is incorrect


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# Affinity

## Characterization

The affinity model of our intrinsics aims to characterize errors or abberations in the image plane that appear as affine transformations of our image coordinates. These can be caused by various effects depending on the type of camera and underlying electronics that power the camera. Primarily, we categorize these into scale and shear effects.

Taking our generalized projection model, we can characterize these effects in the form of:

$\begin{bmatrix} x \\ y \end{bmatrix} = f \begin{bmatrix} X_c/ Z_c \\ Y_c / Z_c \end{bmatrix} + \begin{bmatrix} c_x \\ c_y \end{bmatrix} + g_{\mathsf{affinity}}(\bar{\textbf a})$

It is worth mentioning that while $$g_{\mathsf{affinity}}$$ can be modeled according to effects in $$x$$, effects in $$y$$, or both, the Tangram Vision Platform uses the convention of only applying our affinity model to the $$x$$ dimension for the sake of simplicity.

## Scale

The scale model aims to describe differences in pixel scaling in the $$x$$ (column) and $$y$$ (row) dimensions of our image plane.

In many perception systems, this is modeled as $$f_x$$ and $$f_y$$. See our explanation of the projection model for why Tangram doesn't follow this paradigm.

Scale effects may be due to:

1. Measured physical differences in pixel size (i.e. rectangular pixels). This is actually quite rare.
2. Differences between the clock frequencies of the analog CCD / CMOS clock and the sampling frequency of the Digital-Analog Converter (DAC) in the frame grabber.
3. Projective compensation effects resulting from an insufficient range of depths in the set of object space points used to calibrate. This is often the case if flat checkerboards are used for calibration.

The way that we model this effect is with a single linear coefficient $$a_1$$, as follows:

$g_{\mathsf{affinity}}(a_1) = \begin{bmatrix} a_1 (x - c_x) \\ 0 \end{bmatrix}$

Read the Tangram Vision Blog post on focal length for a more complete explanation of these effects.

## Shear

Shear effects model a non-orthogonality between pixels. In effect, we would be modeling our image plane being shaped more like a rhombus than a square or rectangle.

For most full-frame cameras, this effect is not typically worth modeling, nor is it something that can be typically observed in practice. However, shear effects can be apparent in many line or push-broom cameras. This effect can be modeled as follows:

$g_{\mathsf{affinity}}(a_2) = \begin{bmatrix} a_2 (y - c_y) \\ 0 \end{bmatrix}$

## Scale and Shear

Combining the two possible models above, we get a unified affinity model.

$g_{\mathsf{affinity}}(a_1, a_2) = \begin{bmatrix} a_1 (x - c_x) + a_2 (y - c_y) \\ 0 \end{bmatrix}$


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# Unicoherent

Unicoherent

A topological space $X$ is said to be unicoherent if it is connected and the following property holds:

For any closed, connected $A, B subset X$ with $X=A cup B$, the intersection $A cap B$ is connected.

For example, any closed interval on the real line is unicoherent, but a circle is not.

References

*MathWorld|urlname=UnicoherentSpace|title=Unicoherent Space|author=Insall, Matt

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# Logarithmic derivative and the riemann zeta function

I'm trying to prove the following theorem.

Theorem (Zero free region):

There exists $C>0$ such that $\sigma > 1-\frac{C}{log(\vert t \vert +4)}\Rightarrow \zeta(\sigma + i t)\neq 0$.

In the proof I have a problem with one of the arguments, namely:

• If $0<\delta\leq 2$ then $\frac{-\zeta'}{\zeta}(1+\delta) = \frac{1}{1+\delta -1} + O(1)$.

My question is simply why? I think it has something to do with the fact that $\zeta(s)$ has a simple pole for $s = 1$ but is otherwise analytic in the half-plane $\sigma >0$ (when we denote $s = \sigma + it)$. Someone who can clarify this?


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# Completeness (statistics)

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 Title: Completeness (statistics) Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Completeness (statistics)

In statistics, completeness is a property of a statistic in relation to a model for a set of observed data. In essence, it is a condition which ensures that the parameters of the probability distribution representing the model can all be estimated on the basis of the statistic: it ensures that the distributions corresponding to different values of the parameters are distinct.

It is closely related to the idea of identifiability, but in statistical theory it is often found as a condition imposed on a sufficient statistic from which certain optimality results are derived.

## Contents

• Definition 1
• Example 1: Bernoulli model 1.1
• Example 2: Sum of normals 1.2
• Relation to sufficient statistics 2
• Importance of completeness 3
• Lehmann–Scheffé theorem 3.1
• Basu's theorem 3.2
• Notes 4
• References 5

## Definition

Consider a random variable X whose probability distribution belongs to a parametric family of probability distributions Pθ parametrized by θ.

Formally, a statistic s is a measurable function of X; thus, a statistic s is evaluated on a random variable X, taking the value s(X), which is itself a random variable. A given realization of the random variable X(ω) is a data-point (datum), on which the statistic s takes the value s(X(ω)).

The statistic s is said to be complete for the distribution of X if for every measurable function g (which must be independent of θ) the following implication holds:[1]

E(g(s(X))) = 0 for all θ implies that Pθ(g(s(X)) = 0) = 1 for all θ.

The statistic s is said to be boundedly complete if the implication holds for all bounded functions g.

### Example 1: Bernoulli model

The Bernoulli model admits a complete statistic.[2] Let X be a random sample of size n such that each Xi has the same Bernoulli distribution with parameter p. Let T be the number of 1s observed in the sample. T is a statistic of X which has a binomial distribution with parameters (n,p). If the parameter space for p is (0,1), then T is a complete statistic. To see this, note that

\operatorname{E}(g(T)) = \sum_{t=0}^n {g(t){n \choose t}p^{t}(1-p)^{n-t}} = (1-p)^n \sum_{t=0}^n {g(t){n \choose t}\left(\frac{p}{1-p}\right)^t} .

Observe also that neither p nor 1 − p can be 0. Hence E(g(T)) = 0 if and only if:

\sum_{t=0}^n g(t){n \choose t}\left(\frac{p}{1-p}\right)^t = 0.

On denoting p/(1 − p) by r, one gets:

\sum_{t=0}^n g(t){n \choose t}r^t = 0 .

First, observe that the range of r is all positive reals. Also, E(g(T)) is a polynomial in r and, therefore, can only be identical to 0 if all coefficients are 0, that is, g(t) = 0 for all t.

It is important to notice that the result that all coefficients must be 0 was obtained because of the range of r. Had the parameter space been finite and with a number of elements smaller than n, it might be possible to solve the linear equations in g(t) obtained by substituting the values of r and get solutions different from 0. For example, if n = 1 and the parametric space is {0.5}, a single observation, T is not complete. Observe that, with the definition:

g(t) = 2(t-0.5), \,

then, E(g(T)) = 0 although g(t) is not 0 for t = 0 nor for t = 1.

### Example 2: Sum of normals

This example will show that, in a sample of size 2 from a normal distribution with known variance, the statistic X1+X2 is complete and sufficient. Suppose (X1, X2) are independent, identically distributed random variables, normally distributed with expectation θ and variance 1. The sum

s((X_1, X_2)) = X_1 + X_2\,\!

is a complete statistic for θ.

To show this, it is sufficient to demonstrate that there is no non-zero function g such that the expectation of

g(s(X_1, X_2)) = g(X_1+X_2)\,\!

remains zero regardless of the value of θ.

That fact may be seen as follows. The probability distribution of X1 + X2 is normal with expectation 2θ and variance 2. Its probability density function in x is therefore proportional to

\exp\left(-(x-2\theta)^2/4\right).

The expectation of g above would therefore be a constant times

\int_{-\infty}^\infty g(x)\exp\left(-(x-2\theta)^2/4\right)\,dx.

A bit of algebra reduces this to

k(\theta) \int_{-\infty}^\infty h(x)e^{x\theta}\,dx\,\!

where k(θ) is nowhere zero and

h(x)=g(x)e^{-x^2/4}.\,\!

As a function of θ this is a two-sided Laplace transform of h(X), and cannot be identically zero unless h(x) is zero almost everywhere.[3] The exponential is not zero, so this can only happen if g(x) is zero almost everywhere.

## Relation to sufficient statistics

For some parametric families, a complete sufficient statistic does not exist. Also, a minimal sufficient statistic need not exist. (A case in which there is no minimal sufficient statistic was shown by Bahadur in 1957. ) Under mild conditions, a minimal sufficient statistic does always exist. In particular, these conditions always hold if the random variables (associated with Pθ ) are all discrete or are all continuous.

## Importance of completeness

The notion of completeness has many applications in statistics, particularly in the following two theorems of mathematical statistics.

### Lehmann–Scheffé theorem

Completeness occurs in the Lehmann–Scheffé theorem, which states that if a statistic that is unbiased, complete and sufficient for some parameter θ, then it is the best mean-unbiased estimator for θ. In other words, this statistic has a smaller expected loss for any convex loss function; in many practical applications with the squared loss-function, it has a smaller mean squared error among any estimators with the same expected value.

### Basu's theorem

Bounded completeness occurs in Basu's theorem,[4] which states that a statistic which is both boundedly complete and sufficient is independent of any ancillary statistic.

Bounded completeness also occurs in Bahadur's theorem. If a statistic is sufficient and boundedly complete, then it is minimal sufficient.

## Notes

1. ^ Young, G. A. and Smith, R. L. (2005). Essentials of Statistical Inference. (p. 94). Cambridge University Press.
2. ^ Casella, G. and Berger, R. L. (2001). Statistical Inference. (pp. 285-286). Duxbury Press.
3. ^ Orloff, Jeremy. "Uniqueness of Laplace Transform".
4. ^ Casella, G. and Berger, R. L. (2001). Statistical Inference. (pp. 287). Duxbury Press.

## References

• Lehmann, E.L.;
• Lehmann, E.L.;
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

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text:
# [time-nuts] Lady Heather on low power CPU/Linux?

John Miles jmiles at pop.net
Thu Feb 23 23:50:00 UTC 2012

> I still haven't seen the source of LH or serial.exe , any hints ?

server.cpp and the rest of the Heather sources are in the same directory as
the .exe, normally c:\program files\heather or c:\program files
(x86)\heather.  (You basically end up with a copy of my development
directory in that folder.)

> Please point me to a zip or tgz , not a windows installer

Sorry, what you see is what you get.  A full refund is always
unconditionally available if the software is not fit for your intended
purpose. :)

-- john, KE5FX


text:
Ultraviolet bands of mercury bromide

Krishnamurthy, VG (1958) Ultraviolet bands of mercury bromide. In: Zeitschrift für Physik A: Hadrons and Nuclei, 152 (2). pp. 242-248.

 PDF 8full_text.pdf - Published Version Restricted to Registered users only Download (346kB) | Request a copy

Abstract

The ultraviolet bands of mercury bromide have been excited in uncondensed discharge and photographed with a quartz Littrow spectrograph. The class II system, lying between\lambda 2900 å to 2700 å, suggested byWieland as due to the triatomic molecule, has been studied in detail and ascribed to the diatomic molecule. The bands in the regionlambda 2900 å to 2770å have been analysed into two systems which may form the two components of a2 II –2 \sigma electronic transition with a2 II interval equal to 969·4 cm–1.Another system most probably due to2 \sigma–2 \sigma has been observed in the region\lambda 2770 to 2720.

Item Type: Journal Article Copyright for this article belongs to Springer. Division of Physical & Mathematical Sciences > Physics Mr. Khaleel Ahmed Haveri 22 Jan 2010 07:55 19 Sep 2010 05:33 http://eprints.iisc.ac.in/id/eprint/20527


text:
# Identify the Endothermic and Exothermic Reaction. B. 2 K C L O 3 ( S ) → δ 2 K C L ( S ) + 3 O 2 ↑ - Science and Technology 1

MCQ

Identify the endothermic and exothermic reaction.

$\ce{2KClO_{3(s)}->[\Delta] 2KCl_{(s)} + 3O2}$

#### Options

• Exothermic reaction

• Endothermic reaction

#### Solution

$\ce{2KClO_{3(s)}->[\Delta] 2KCl_{(s)} + 3O2}$- Endothermic reaction

Concept: Energy Change in Chemical Reactions : Exothermic and Endothermic Chemical Changes
Is there an error in this question or solution?

#### APPEARS IN

Balbharati Science and Technology Part 1 10th Standard SSC Maharashtra State Board
Chapter 3 Chemical reactions and equations
Q 8.2 | Page 46
SCERT Maharashtra Question Bank 10th Standard SSC Science and Technology 1 Maharashtra State Board 2021
Chapter 3 Chemical reactions and equations
Solve the following Questions | Q 4.3


text:
<< back

## C2 DETEST

##### Description:

A radioactive decay chain; Bateman's equation with different decay constants. Please note that the supplied exact solution is correct only for certain initial conditions (e.g., those from the IVPs).

decoupled

<< back


text:
# Search

You can find the results of your search below.

## Fulltext results:

syntax @wiki
You can mark something as deleted as well. You can mark something as deleted as well. **Paragraphs** are created from blank lines. If... ply www.google.com - You can set the link text as well: [[http://www.google.com|This Link points to goog... imply www.google.com - You can set link text as well: [[http://www.google.com|This Link points to goog
mark_twain @air
that, but then you must stick to the book. Very well, I begin to cipher out the German for that answer... out on the hypothesis that it is masculine. Very well -- then the rain is der Regen, if i... hat verb is from the reader's base of operations; well, in a German newspaper they put their verb away o... >PARTED."

However, it is not well to dwell too much on the separable verbs. One is
veg @eat
Best if chilled for a few hours.\\ \\ Goes very well with salmon fillet, lightly seasoned with salt, p... . Cook it about 20 min. (the mushrooms have to be well done to be edible) till you´ve got a thick sauce,... the onions or leave whole. Grill on high heat as well, add the halved tomatoes later.\\ When everything... \ salt** \\ \\ Season the tomatoes with salt, mix well and leave for a while.\\ Then add a mix of:\\ \\
e oil and salt after baking. \\ This freezes very well, just slice it before. \\ \\ //Adapted from Jami... east\\ 200ml lukewarm water** \\ \\ Mix and knead well, form a ball and leave in a warm place for at lea... e (fennel, anise and cumin)** \\ \\ Mix and knead well, form a ball and leave in a warm place for at lea... e in a warm and undisturbed place.\\ Stir Hermann well every 12 hours and feed him every 24 hours - 100g
chicken @eat:meat
minutes, then add the tomato purée and stir until well combined. * 3 Add all the dry spice, mix well. Now add the chicken and make sure the masala is coatin... nd reduce the heat, and cook until the chicken is well cooked, about 10 minutes. If you like more gravy,... for ca. 15 min.\\ Add\\ \\ **1 can coconut milk, well shaken\\ 50 ml water** \\ \\ and simmer gently fo
sweet @eat
in 4 equal batches to the carrot mixture, mixing well after each addition. - Pour the cake mixture in... thick and creamy.\\ Melt the chocolate and add as well as the oil.\\ Mix flour, salt, baking powder and ... ter, soft\\ 50g sugar\\ a little salt** \\ \\ Mix well and gradually add\\ \\ **5 egg yolks\\ 200g flour
===== Some sources of well written articles… ===== Are there well written articles expressing something sensible from the right? Why is everything I find that is well written so much leaning to the left? Just me? <h
quotes @dob
in one grave, the epitaph on its headstone might well be: `It seemed a good idea at the time."' - Dame ... by the 23 percent who are apparently doing quite well for themselves." --Jerry Garcia <X.corp>is takin... If you were my wife, I'd drink it. you're drunk! well you're ugly - tomorrow, I'll be sober, but you'll
chutney @eat
min\\ 2-3 teaspoons brown sugar (white will do as well)** \\ \\ Put everthing into a blender and blend u... oth - easier is impossible :-D\\ This goes really well with a curry dish or as a dip with tortilla chips
technology_ai_distribution_of_capital @blog
ion 101, Pawel Sysiak - Medium, July 2016]])) as well as other articles that question if technological
rubberbandits @dob
he wants a lift back to the hotel, and if it goes well, finger and a shift\\ She says Fitzy drives a Mit
of articles that caught my interest over time as well as the key points in each article that grabbed my
raspberry.pi.and.ubuntu @dob:tech
cation" together with SSH/SFTP option. All goes well - BUT no sound on HDMI. Problems is caused by PI
currency_variation_stock_price @blog


text:
💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Find the limit, if it exists. If the limit does not exist, explain why. $\displaystyle \lim_{x \to 0^-}\left(\frac{1}{x} - \frac{1}{|x|} \right)$

$\lim _{x \rightarrow 0^{-}} \frac{2}{x}$ which doesn't exist since denominator approaches 0 and numerator doesn't. To clarify this limit would be equal to negative infinity, but sinceinfinity is not a limit it would therefore be considered as not existent i.e. D.N.E.

Limits

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Discussion

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Campbell University

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Baylor University

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University of Michigan - Ann Arbor

Lectures

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Video Transcript

to evaluate this limit, we know that the absolute value of X this is equal to negative X. If Access less than zero and positive X if X is greater than zero. Now, since excess approaching zero from the left, then we're talking about absolute value of X equals negative X. And so in here we can write this as the limit As X approaches zero from the left of one over X one over negative x. Which you can simplify further into the limit As X approaches zero from the left of one over X plus one over X. Or this is the same as The limited sex approaches zero From the left of two over X Known that as X approaches zero from the left, the value of X is a very small negative number. So The value of two over X would be approaching negative infinity. And so this limit As X approaches zero from the left of two over X must be negative infinity.

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Anna Marie V.

Campbell University

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Baylor University

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text:
# Lunch with my travel agent

JeremyDover: Pat, thanks for taking me out to lunch today.

Pat: Very glad to do it! I know you've been mad at me lately. Maybe the flag thing was a bit much?

JeremyDover: Not really mad, just a bit frustrated.

Pat: Alright...I understand. What are you having?

JeremyDover: Do you have a recommendation?

Pat: Recommendation? I love the Ham on Rye.

JeremyDover: Sounds good. What about you?

Pat: I'll have the Pastrami Special on Kaiser, with the 31 seasoning.

JeremyDover: 31 seasoning?

Pat: Contains 31 different herbs and spices...so they say. Who will count?

JeremyDover: I think I might have the gazpacho too.

Pat: Oh, I forgot to say, I'm taking you to the cupcake place later too.

JeremyDover: What, are you trying to fatten me up for the next trip?

Pat: Did you hear my radio ad? We're going island hopping south of the equator this time!

JeremyDover: Are you just going to tell me where? Or is this going to be another puzzle?

Pat: Easy there! I'll send you the itinerary tonight.

Once Pat refused to tell me the destination in person, I knew it was going to be another puzzle. And this one's a doozy. All I got were a text file and an image, as below. Can you help me figure out where we're visiting this time?

Text File:

10111011010010110101001110111001001111101001111101001111111100110101010011111010
01101101001111100111011101001101111001010101100111110011111101001111111010010111
10100101010110011111001111101001010110110011111111001010111100111110010101011010
01011010100110110111001111111100111110100111110010111101001111101001011011100101
11011100111010110011111001111111010010111011100110110111001011110100110101110011
11100101010110010111101100101011011001111111001111100110110101001011111010011011
01110011011101001110110100101010111001111100101010110010111010100111111110010111
11010011111001111110100101101010011101101001111101001111100111111101001110110100
11101101001101110100111110010111110100110101110011010110100101111010011011010100
11111001010110100101010111001110110100111011100111110010111110100110101010010111
01110011111111001111111001111100111110100111011010010111101001101111001101110111

Image:

Solver Notes: To the color blind, I again apologize. The ability to recognize flags is very important in the solution of this puzzle. To everyone I apologize for the bubblegum pink background on the image, but it clearly does not match any flag colors! As a general word of warning, there will be a phase of the puzzle where multiple interpretations are possible...you need to pick the most specific interpretation. But for the most part the puzzle will guide you to this. If you get to this phase, I'm willing to provide the reference I used...just ping in comments!

A final note: There are several references to a number of my previous puzzles in here, as well as some homages to others, particularly Stiv. Thanks to all of you for creating an awesome community!

HINT:

Based on Stiv's partial below, two recommendations: first, compare the decoded text with the image file. Are there any correspondences you can detect? Second, the text in the decoded message makes sense in the context of amateur radio. Recommend checking out Q-codes, and how amateur stations are identified.

HINT:

The image and the list of call signs in the decoded text need to be reordered with the same permutation.

• Why this extended story scenario for a puzzle? Aug 14 '20 at 18:02
• Everything is in here for a reason. It might not be a good reason. It might even be to distract you. But it is there for a reason. Aug 14 '20 at 18:18
• @aminabzz -- There's nothing wrong with having a background story for your puzzle. Aug 14 '20 at 18:47
• Thanks for the nod, very honoured... :) I've spotted a couple of things about this which mean my evening now has a plan (and it's solving this). Will post my findings later if nobody else solves it in the meantime...
– Stiv
Aug 16 '20 at 15:47

## Solved at last!

The destination is (aptly, given the setting):

THE SOUTH SANDWICH ISLANDS - one of the world's remotest island groups.

Solution path (brace yourselves - it's a biggie...):

First, let's do the obvious thing here and identify all of the flags in the image. Note here that each column contains 7 flags (of countries, territories, island groups, states, counties, cities, and even one nautical signal flag); each one has been halved and both halves appear in the same column.

NB The white cross on red background is not Denmark - as you'll see later, we need the Sovereign Military Order of Malta as part of our solution...

Here, the rows in white are the top halves, while those coloured blue are the bottom halves.
Side note for OP (because I know you'll want to know...): It took me about 2 hours to nail down this list! Most of the national flags I could do from memory, but I had to cross-reference Wikipedia (and other) lists of flags for US states, UK counties, flags with the Union Jack on them, flags by colour combination and triband flags. Afterwards, there were still some I couldn't pinpoint this way, so I had to zoom in on details in the image like words in mottos on crests or unusual pictures within the flag and Google for those (hit and miss). Finally, I only had to reverse image search 4 flags as nothing else was turning up the answer (Ceuta, Chatham Islands, Dortmund and Tierra Del Fuego).

The most crucial thing to notice here is that:

Each row contains either all-top-halves of flags or all-bottom-halves. In fact, each row can be paired with one other so that all flag halves match across the two to form a set of complete flags. This is going to be useful later...

Let's turn away from the flags for a moment and focus on the message in binary. There are (at least) four clues in the text as to how to decode this:

1. If you take the first letter of each line spoken by Pat, this spells out VARICODE.

2. When Pat says she'll have 'the Pastrami Special on Kaiser, with the 31 seasoning', this is a cloaked reference to PSK31.

3. Pat mentions that she loves the Ham on Rye.

What is this leading us towards?

The world of ham radio (a colloquial name for amateur radio). From the linked Wikipedia article, PSK31 (or 'Phase Shift Keying, 31 Baud') is a "computer-sound card-generated radioteletype mode, used primarily by amateur radio operators to conduct real-time keyboard-to-keyboard chat". Varicode is the code used within this to encrypt each character, with full support for ASCII and all the main symbols you might wish to encrypt.

Most importantly for our purposes:

The binary message is actually encoded in Varicode. In this system, the string "1011101101" which appears at the start of the message means "START OF TEXT", and the string "1101110111" which appears at the end of the message means "END OF TEXT". Every occurrence of '00' marks a space between different characters. We can then decode the binary appearing between each of these spaces using the Varicode system to find the message (though I had to do this manually, as I could not find an online decoder without having to download files through GitHub).

The message then translates to the following:

 [START OF TEXT]
DE [SPACE] AA3PAT [RETURN]
QSO [RETURN]
GJ1O [RETURN]
A53R [RETURN]
ZD93A [RETURN]
1A04B [RETURN]
J491L [RETURN]
OY5I [RETURN]
VK9N2U [RETURN]
OX3K [RETURN]
GD2A [RETURN]
J22N [RETURN]
KL71V [RETURN]
CU2E [RETURN]
KP43I [RETURN]
A21S
[END OF TEXT]

This bears the hallmarks of a ham radio transmission. There are a few specific codes used in ham radio which we can spot at the very top:

"DE" means 'from'.

"AA3PAT" is the call sign of the station transmitting the signal. "AA" implies that this station is in North America. The "3" immediately following it shows that it is specifically within an area in the West of the USA or Canada. 'PAT' is a unique code specific to the broadcaster. Thus 'AA3PAT' here signifies that this transmission is from agent Pat, who is based in the AA3 region of North America:

Source: mapability.com

"QSO" is a standard 3-letter Q-code used in amateur radio communications, specifically to signify one station making contact with another. It is normal to find it at the start of a message, as here.

What this then leaves us with is:

A set of 14 codes between "QSO" and [END OF TEXT], each of which looks like it might be in the format of a radio station's call sign.

In particular, it's notable that the first part of each of the 14 call signs listed relates to a country or territory whose flag appears in the image (using this list):

GJ1O: GJ = Jersey (col 6, rows 4&6) - remaining 1O;
A53R: A5 = Bhutan (col 5, rows 7&10) - remaining 3R;
ZD93A: ZD9 = Tristan Da Cunha (col 9, rows 3&8) - remaining 3A;
1A04B: 1A0 = Sovereign Military Order of Malta (col 3, rows 7&10) - remaining 4B;
J491L: J4 = Crete (col 10, rows 1&14) - remaining 1L;
OY5I: OY = Faroe Islands (col 9, rows 5&11) - remaining 5I;
VK9N2U: VK9N = Norfolk Islands (col 11, rows 3&8) - remaining 2U;
OX3K: OX = Greenland (col 12, rows 9&12) - remaining 3K;
GD2A: GD = Isle of Man (col 1, rows 9&12) - remaining 2A;
J22N: J2 = Djibouti (col 1, rows 2&13) - remaining 2N;
KL71V: KL7 = Alaska (col 3, rows 1&14) - remaining 1V;
CU2E: CU = Azores (col 7, rows 3&8) - remaining 2E;
KP43I: KP4 = Puerto Rico (col 5, rows 1&14) - remaining 3I;
A21S: A2 = Botswana (col 12, rows 4&6) - remaining 1S.

What next? Well note that...

...there are not only 14 call signs in this list but also 14 rows in the flag image. Let's do a few steps at once here:

1. Match up each call sign in order to the corresponding row in the flag diagram (i.e. the first call sign, GJ1O, to the first row of half-flags; the second call sign, A53R, to the second, etc.);

2. Pair up the rows of the flag image to form the complete flags again (keeping the call signs matched to the rows as they move);

3. Then consider just the so far unused portions of the call signs, i.e. the final number-character pair. It is possible to arrange the rows of flags in such a way that the number parts of these pairs are put in increasing order and the letters beside them then spell out a phrase!

Reading down the characters at the far right of the diagram, we can see the phrase SOLVE A NURIKABI - so now we need to treat our image as a Nurikabe grid!

How do we do this? Perhaps the natural thing to do is:

to use the squares containing countries that were indicated among the list of call signs (highlighted orange in the diagram above) as those where we need to place the starting numbers.

What numbers do we use? Well, why not try using the numbers from the middle part of their call signs (thus far only used for ordering the letters to give the message in the previous step) - the starting grid would then look like this:

Following the usual deductive steps for this kind of puzzle, we end up at the following state:

So now what?! Well, squint and you might be able to make out...

...that the black squares appear to take the form of the number 3166. What's the significance of this? Well, this is surely a nod towards the ISO 3166 specification of alphanumeric country codes and sub-codes! The OP therefore expects us to convert some or all of the countries in the grid to one of their ISO 3166 codes, and then perform some further translation of some kind to perceive the next part of the message.

A message of sorts emerges if we specifically use the ISO 3166-1 alpha-2 2-letter codes for the geographical entities in the unshaded spaces (and not already replaced by a number for the nurikabe) of the diagram above. Using these codes, reading along and down the rows yields the letters NO/SH/AS/IN/SI/LI/CA/KI/CK/SH/AW/BO/TT/OM/ME/AT/MU/NI/CH. While this can potentially be parsed into several different phrases, the one we want (after a little trial and error) is:

NOSH AS IN SILICA KICKSHAW BOTTOM ME AT MUNICH.

Coupled with the OP's hint in comments below that we might find it helpful to add '(8)' to the end of this phrase, this suggests the enumeration of a cryptic crossword clue (something which forms the basis of many of the OP's other puzzles...).

So let's solve this final step of the puzzle...

'NOSH', here, is the definition - the answer will be a foodstuff.

'AS IN' is the connecting phrase between the definition and the wordplay.

'SILICA' is SAND...

'KICKSHAW BOTTOM' is W - the last ('bottom') letter of 'kickshaw'...

'ME AT MUNICH' is ICH - the German word for 'I' ('me').

So this cryptic clue yields up the answer SANDWICH - which is particularly appropriate given all the talk of sandwich fillings between the two characters in the flavour text!

For complete closure, where is the geographical location in question? After all, there are a few places in the world with this in the name. Well, there were a couple of pointers in the puzzle:

In the dialogue, Pat says "We're going island hopping south of the equator this time!" So we're looking for somewhere south of the equator (which rules out Hawaii, which was once christened 'The Sandwich Islands' by James Cook) and we're also looking to go island-hopping, suggesting more than one island by this name (which rules out Manuae and Efate, both previously known as 'Sandwich Island' to James Cook - boy, did that guy love a sandwich...).

This leads us to the territory of South Georgia and the South Sandwich Islands, which ticks all the boxes! And - wait for it - was also named by James Cook. Someone buy that guy some lunch already!

• Definitely great progress Stiv! But there is still a lot to do...one thing you can definitely count on is that there is a connection between the decoded text and the image. Signing off, de KB3YSM Aug 16 '20 at 20:54
• Again, excellent progress! The part you have added is the one I was most worried about, since there are numerous, often inconsistent, references out there. This is the one I used. That said, you are mostly there. Regarding your final comment, suggest checking out the second hint. It can help get you started. Aug 20 '20 at 23:20
• wow that looks insanely hard Aug 21 '20 at 5:07
• Yes, they are both correct. If you slightly broaden your idea of the correspondence, you'll see that this apparent inconsistency is in order. Aug 21 '20 at 12:52
• I so grateful you stuck with it...thanks to you! I'm glad you seem to have enjoyed how it comes full circle :-) Aug 25 '20 at 14:10