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Lebesgue’s approach to integration was summarized in a letter to Paul Montel. He writes:

I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.
Siegmund-Schultze, Reinhard (2008), “Henri Lebesgue”, in Timothy Gowers, June Barrow-Green, Imre Leader, Princeton Companion to Mathematics

(Source: Wikipedia)




Cityscape 1 (Landscape #1)  by Richard Diebenkorn, 1963

Cityscape 1 (Landscape #1)  by Richard Diebenkorn, 1963


hi-res




I loved being a rifle company commander. Having the responsibility for 211 men. Being totally in charge of their welfare and their training. That was the happiest period of my life, professionally, looking back on things.




vraieronique:

Craftsman Selling Cases by a Teak-Wood Building by Edwin Lord Weeks
Oil on canvas, c.1885, 99.7 x 73 cm. Private collection.

vraieronique:

Craftsman Selling Cases by a Teak-Wood Building by Edwin Lord Weeks

Oil on canvas, c.1885, 99.7 x 73 cm. Private collection.


hi-res




risk, however measured, is not positively related to (rational) expected returns. It goes up a bit as you go from Treasuries, or overnight loans, to the slightly less safe BBB bonds, or 3 year maturities. But that’s it, that’s all you get for merely taking the psychic pain of risk.

Just as septic tank cleaners do not make more than average, or teachers of unruly students do not make more than average, merely investing in something highly volatile does not generate automatic compensation. Getting rich has never been merely an ability to withstand some obvious discomfort.







razanlillie:

i wish yall knew my father. ive watched him get beaten up for standing up to creeps
ive watched him get arrested n spend months in prison
i watched him get convicted for a drug case
ive watched all the demons n violence from our classist patriarchial family follow us over
ive watched him get out n open a flower store to make it a little n then having it burn down
i watch him go to court every month
i watch him hating himself
i watch him stripped of everything
ive watched his weight spiral out of control
ive watched him unable to eat or sleep for days
n wanting so badly to at least fulfill the role of breadwinner




The rank-nullity theorem in linear algebra says that dimensions either get

  • thrown in the trash
  • or show up

after the mapping.

image

By “the trash” I mean the origin—that black hole of linear algebra, the /dev/null, the ultimate crisscross paper shredder, the ashpile, the wormhole to void and cancelled oblivion; that country from whose bourn no traveller ever returns.

The way I think about rank-nullity is this. I start out with all my dimensions lined up—separated, independent, not touching each other, not mixing with each other. ||||||||||||| like columns in an Excel table. I can think of the dimensions as separable, countable entities like this whenever it’s possible to rejigger the basis to make the dimensions linearly independent.

image

I prefer to always think about the linear stuff in its preferably jiggered state and treat how to do that as a separate issue.

So you’ve got your 172 row × 81 column matrix mapping 172→ separate dimensions into →81 dimensions. I’ll also forget about the fact that some of the resultant →81 dimensions might end up as linear combinations of the input dimensions. Just pretend that each input dimension is getting its own linear λ stretch. Now linear just means multiplication.

image
linear maps as multiplication
linear mappings -- notice they're ALL straight lines through the origin!

Linear stretches λ affect the entire dimension the same. They turn a list like [1 2 3 4 5] into [3 6 9 12 15] (λ=3). It couldn’t be into [10 20 30 − 42856712 50] (λ=10 except not everywhere the same stretch=multiplication).

image

Also remember – everything has to stay centred on 0. (That’s why you always know there will be a zero subspace.) This is linear, not affine. Things stay in place and basically just stretch (or rotate).

So if my entire 18th input dimension [… −2 −1 0 1 2 3 4 5 …] has to get transformed the same, to [… −2λ −λ 0 λ 2λ 3λ 4λ 5λ …], then linearity has simplified this large thing full of possibility and data, into something so simple I can basically treat it as a stick |.

If that’s the case—if I can’t put dimensions together but just have to λ stretch them or nothing, and if what happens to an element of the dimension happens to everybody in that dimension exactly equal—then of course I can’t stick all the 172→ input dimensions into the →81 dimension output space. 172−81 of them have to go in the trash. (effectively, λ=0 on those inputs)

So then the rank-nullity theorem, at least in the linear context, has turned the huge concept of dimension (try to picture 11-D space again would you mind?) into something as simple as counting to 11 |||||||||||.




Define the derivative to be the thing that makes the fundamental theorem of calculus work.