@s.harp Sorry, that I misinterpreted it. (I thought that $f$ is one fixed function) Now it is clear. Thanks a lot.
2 hours later…
user131753
09:10
Can anyone give me some intuition regarding the notion of a transportable concrete category? In Joy of Cats the definition is given as follows, "A concrete category $(\mathbf{A}, U)$ over $\mathbf{X}$ is said to be (uniquely) transportable provided that for every $\mathbf{A}$-object $A$ and for every $\mathbf{X}$-isomorphism $k:UA\to X$, there exists a (unique) $A$-object $B$ and a $\mathbf{A}$-morphism $j$ such that $U(B)=X$ and $U(j)=k$."
Probably this picture is intended to give a big-picture viewpoint of the concept but I couldn't understand it. Is there any special significance of the name? What does which category "transport" to which category?
Task
Let's observe in interval $[a,b]\subset \mathbb{R}$ defined real valued and continuous maps which form vector space $\mathcal{F}([a,b],\mathbb{R})$. Let $\langle ., . \rangle$ be some inner product.
Now let's observe equation
$$ Lf=u $$
where $L$ is a linear transformation in space $\m...
Nothing, i am reading a book. And i wanted to search more information about this "set of all functions from X to Y) but i didn't find anything using the writing method that was used in the book which was in German, so i thought that in English theres a different one that i can find in Wikipedia. thats why i asked.
Let $f:\Bbb R\to S^1$ be continuous injective function (where $S^1$ is unit circle in $\Bbb R^2$). How to show that $f(a,b)$ is open for any $(a,b)\in \Bbb R$ where $a=-\infty$ and $b=\infty$ are possibilities?
I went like this: Maybe, we can consider $f$ as function from $\Bbb R\to [c,d)$, then it is strict monotone (since continuous and injective), then I can prove that $f(a,b)$ open.
But, I can't justify why considering $f$ as function from $\Bbb R\to [c,d)$ is OK.
In chemistry; chirality is generally defined in 2 ways.
Lord Kelvin's definition: "I call any geometrical figure, or group of points, chiral, and say it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself" (source: 1. chirality.org , 2. Bi...
As well i dont know "mathematical chirality" its more difficult. But I've read mathematical chirality is related but different from chemical chirality (Lord Kelvin's one).
They renamed old months that had different names in honour of Caesar (July) and Augustus (August), they didn't introduce new ones, so they didn't shift september and the others
I mean they did at some point, but I'm not sure when that happened
> September (from Latin septem, "seven") was originally the seventh of ten months in the oldest known Roman calendar, the calendar of Romulus c. 750 BC, with March (Latin Martius) the first month of the year until perhaps as late as 451 BC.[2] After the calendar reform that added January and February to the beginning of the year, September became the ninth month, but retained its name. It had 29 days until the Julian reform, which added a day.
The "[2]" there is how you know it's legit
I should plan all future events with calendar reform in mind
"The competition, barring disaster or calendar reform, will be held on March 16 of next year…"
Though it's not a fair comparison: the "goal" for wheat can be thought of as "make lots of wheat plants" while the "goal" for humanity involves things like happiness and freedom
and isn't tied specifically to population
I need to finish that book
He describes the "population trap": any change that increases population at the cost of decreasing quality of life is irreversible
@AkivaWeinberger i cannot find any first and foremost papers on google scholar where a terminology first appeared or where a concept first described. it excessively focus on later papers.
Let $f:\Bbb R\to S^1$ be continuous injective function (where $S^1$ is unit circle in $\Bbb R^2$). How to show that $f(a,b)$ is open for any $(a,b)\in \Bbb R$ where $a=-\infty$ and $b=\infty$ are possibilities? I went like this: Maybe, we can consider $f$ as function from $\Bbb R\to [c,d)$, then it is strict monotone (since continuous and injective), then I can prove that $f(a,b)$ open. But, I can't justify why considering $f$ as function from $\Bbb R\to [c,d)$ is OK.
I was experimenting with discretizations of the gyroid minimal surface in Rhino (3d modelling software), and modelled this infinite polyhedron, and wondered what it is called.
I didn't find any documentation of it yet, but perhaps I am not searching for the right terms.
All its faces are the sam...
So I read all of this, and what I gleaned is as follows:
The methods used in producing large cardinals:
1. Make the first cardinal that is unreachable using a previous operation (formalised by the elementary embeddings and their critical points) 2. Use the reflection axiom to produce larger cardinals between CAI and the current position you are stuck in 3. Iterate whatever operation you are currently using, until it seemly get stuck (stationary sets, a-(insert large cardinal property) cardinal)
4. So huge it can partition things or make two things between the models cannot be distinguished (Ramsey cardinals mainly)
5. So huge that some class of cardinals are effectively measure zero in size (Measurable cardinals, supercompact cardinals) 6. Start iterate on the elementary embedding itself
7. And finally, let the universe to map into itself so it gets larger (Reinhart, Berkerley)
I think my favourite are the measurable cardinals and their generalisations. They do not just prove consistency of some theory beneath them, they also map out the distribution of many large cardinals
and most importantly, they deviate from the usual strategy of iterating an operation and find the next critical point
Sorry typo, CAI should be V
The Ramsay cardinals are one of the most well studied as they are tied to a famous open problem in combinitorics
> His New York Times obituary stated that "he added the middle initial himself, though it does not stand for a middle name",[1] an assertion that is supported by his obituary in The Guardian.[2]
Wait wow
OK so it really could be that
I read that clouds maintain the same fractal dimension (measure of "roughness") over 10 orders of magnitude, making them one of the most uniform fractal objects on the planet
Do you just measure the fractal dimension of various things and graphs?
How do you use that to make predictions?
> Early form of the name was spelled with an "s" (Benoist), but as with many words in the French language, the "s" was eventually replaced with a circumflex accent over the "i".
> ÉTYMOL. ET HIST. A.− 1. Ca 1130 beneeite (Couronnement de Louis, éd. E. Langlois, 27 : Quant la chapele fu beneeite); 1172-75 benëoit « sur qui a été répandue la bénédiction divine, bienheureux » (Chr. de Troyes, Chevalier Lion, 2380 dans T.-L. : benëoiz Soit mes sire Gauvains); ca 1190 « qui a été consacré par des cérémonies rituelles » ewe benëeite (M. de France, Purgatoire, 469, ibid.); 1150-1200 pain benëoit (Aliscans dans Bartsch, Chiertomathie, 19, 227), devenu iron. 1546 (Rabelais, Tiers Livre, éd. Marty-Laveaux, chap. 32, t. 2, p. 159 : et vous ayme tout mon benoist saoul), qualif…
Task
Let's observe in interval $[a,b]\subset \mathbb{R}$ defined real valued and continuous maps which form vector space $\mathcal{F}([a,b],\mathbb{R})$. Let $\langle ., . \rangle$ be some inner product.
Now let's observe equation
$$ Lf=u $$
where $L$ is a linear transformation in space $\m...
In chemistry; chirality is generally defined in 2 ways.
Lord Kelvin's definition: "I call any geometrical figure, or group of points, chiral, and say it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself" (source: 1. chirality.org , 2. Bi...
