I have wanted one general application of this idea in a spreadsheet. Specifically, I track some of my running, including speed (pace), distance, and time. Under different circumstances, I have exactly two of the three available and I want the third to be computed, but it varies which. I have found it fairly difficult to implement this kind of data entry in Google Spreadsheets and Excel, even know conceptually it's a very simple constraint "a*b=c" where I know some two variables.
As a more substantive comment: You may find the thesis "Propagation networks : a flexible and expressive substrate for computation" by Alexey Radul interesting. https://dspace.mit.edu/handle/1721.1/54635
You could create a table with 3 columns: distance, time, pace.
Set the display format for time and pace to "Duration".
Enter these formulas:
distance = time / pace
time = distance * pace
pace = time / distance
Drag fill everything down. At this point you get reference errors, but once you enter any two values (thereby overwriting the formulas in those cells), you get your result.
You just need two spreadsheet tabs. One for the "raw" input and one with a formula that either takes the input if it exists or falls back to the calculated version
In case anyone is curious, there is a discussion of AI in Section 5 (page 8 of the PDF). This direct link might work: https://arxiv.org/pdf/2511.14653#page=8
> Ask four mathematicians whether something is a Commutative Ring or not and they'll all agree.
Amusingly, there are at least 3 different definitions of commutative ring.
The principal issue is whether it must have a 1 (unity, ie a multiplicative inverse). Wikipedia https://en.wikipedia.org/wiki/Commutative_ring as well as most modern sources insist on this.
Finally, if you do have a 1, then sometimes people include the condition that 0 != 1, ie the trivial/zero ring is deemed not a [commutative] ring. This is somewhat hard to find, but is relatively common among people who specifically define the concept of "ring with identity" (eg Zariski+Samuel). I have also found it unqualified (ie, just in the definition of "commutative ring") in the wild, eg in "Handbook of Mathematical Logic" by Barwise or "The Math You Need" by Mack.
(I agree with people like Conrad and Poonen that rings should have a 1. And I guess that the zero ring is in fact a [commutative] ring.)
The moral of the story is: "be careful with simile and analogy". Otherwise you get your arse handed to you on a plate and your very reasonable argument gets lost in the weeds 8)
In these circles is is generally safer to stick with car analogies.
Suppose we designate a unit line segment, and from that we pick out a line segment of (what we now call) length 2 and separately a rectangle of area 3. To "an ancient" (like Aristotle), does it make sense to add the length and the area to get 5? If you then drew a line segment of length 5, would they say it has the same magnitude?
Today, as always, it is a serious mistake to add a length with an area.
Nowadays, everyone who attempts to do such things should understand that the value of a physical quantity is the product between a scalar (i.e. real number) with a unit of measurement.
The addition of distinct physical quantities is impossible, because the addition between distinct units of measurement is undefined (because there is no useful definition).
However, if you detach the scalars from the complete values of some physical quantities, those are elements of the field of the real numbers, so you can do any operations with them, not only addition, but you can compute arbitrary functions, like transcendental functions, e.g. logarithms (which you cannot compute using as argument a complete physical quantity, including a unit of measurement).
In the ancient Greek mathematics, the notions of physical quantity and measurement were not well formalized, even if they had some intuitive understanding, so they would not normally try to perform invalid operations with physical quantities. The concepts of physical quantity and measurement became well understood only in the 19th century, e.g. in the works of people like Weber and Maxwell.
Nevertheless, in the ancient Greek texts there are also examples of numbers that were detached from some geometric quantities and then used in unrelated operations where their origin was no longer taken into account. A simple example is the use in various arithmetic problems of several kinds of "geometric" numbers, e.g. square numbers, triangular numbers, cubic numbers or pyramidal numbers, i.e. numbers that were computed using formulae for areas or volumes, but which were used in applications were there were no areas or volumes involved.
Another example is the use of mechanical devices for the computation of some irrational or transcendental functions, which could be used for solving some famous problems like the trisection of an angle, the quadrature of a circle or the doubling of a cube. In such mechanical devices there were e.g. some lengths that were numerically equal with areas or volumes in the related problems. (While such mechanical devices, which were examples of analog computers, have been built and they have solved the problems, such solutions were considered cheating, because those problems had been formulated with the restriction of using only a straightedge and a compass for their solution, which is now known to be impossible.)
The essence of analog computing, whence the name "analog" comes, is that you have in the analog computer some physical quantities that are numerically equal (using some arbitrary units) with some physical quantities in the problem that must be solved by the computer, even if the nature of the physical quantities is very different. Analog computing is an important example of detaching the real number value from the complete physical quantity, and as I have said, there are examples of simple analog computers, dedicated to the computation of some irrational or transcendental functions, already since the Greek Antiquity.
That's not how I interpreted it: "...if Ukraine should become victim of an act of aggression or an object of a threat of aggression in which nuclear weapons are used." I interpreted that as either conventional aggression, or threat of nuclear aggression.
The language does seem really ambiguous though. I'm surprised it wasn't written more clearly.
Exchange trading happens in round lots that are usually 100 shares.
This is pretty much just a legacy thing, but so many technical systems have this assumption built in that while odd-lot trading (trades not in the round lot size) has become a little more common on the exchanges, it’s still treated weirdly by the various systems involved.
But also, it’s better for you as a retail investor, to get them from a middleman, because they will generally give you a better price than the exchange. They will give you a better price because retail traders tend on average to be worse at trading than the overall market. You should take advantage of that, regardless of your actual ability level.
Odd lots don't contribute to the NBBO, and placing an order for an odd lot doesn't have to execute within the NBBO. (People can trade "past" you, I am pretty sure ISO's don't need to clear you, etc). (Note these are rules for market participants, not retail customers). So for a firm trying to argue they provide excellent price improvement and execution efficiency for their customers, they can't "just" send the orders to the lits.
And even if they could "just" do so, internal matching typically provides better price improvement on the NBBO than even the best execution you could get off the lits.
Edit: But yes TBC, you're correct that odd lot trades aren't unusual. But you're seeing trades there by actual market participants, not retail orders. They're not just trying to get those 2 shares, there's a broader strategy and they're aware of all the above nitty gritty.
In example 3, the NBBO for stock ABC is 495--500, but there is also an odd lot offer for 497 on exchange. If a Robinhood customer sends a market buy order, then Citadel is allowed to fill it for 499.999 even though it's better to send to the exchange. (And if they then pick up the odd lot themselves, it's easy arbitrage.)
By the way, while you're correct about some of your claims, odd lot executions definitely have to occur within the NBBO. (How could it be otherwise?) Otherwise, in the example above, Citadel would give an even worse price!
I mostly mean scenarios where your limit order might not be marketable, end up resting on the book, and then get traded "through". I'm speaking from the perspective of an actual direct market participant, where you're not using a market order but are trying to enter a position while adding liquidity/collecting a rebate. (Most exchanges reward participants who have some % of their trades as liquidity-added, with rebate tiers).
Round lots are excluded from the NBBO so that the NBBO can't be as easily influenced by quantities of shares that don't represent any material price signal. 1 share of practically anything but BRK class A represents ~nothing. Less than a round lot on a price level is basically no liquidity available at that level.
There are per ticker rules to allow odd lots on most US markets. AFAIK unless you're trading penny stocks, every stock out there is entitled for odd lots, and most trades are indeed odd lots, that has been the case for 10 years at least.
Even if there wasn't, I guess at least half the trading on stocks is through CFDs and not cash, so lots aren't even a thing for most investors.
In case anyone was curious, I computed some popular margins of victory defined as (%popular vote of winner) - (%popular vote of highest other candidate).
I got the following small margins of victory smaller than the 1.5% in 2024, five of which are positive and five of which are negative.
-7.8% in 1824, 1.4% in 1844, -3% in 1876, 0% in 1880, 0.6% in 1884, -0.8% in 1888, 0.1% in 1960, 0.7% in 1968, -0.5% in 2000, -2.1% in 2016.
In case anyone was curious, the interest being paid on the national debt is of the order of $1 trillion per year, while the amount collected in taxes (federally only) is of the order of $5.5 trillion per year.
federally only is about $1.6Trillion, the rest is state taxes and never goes near the federal government to cover the interest on their debt. ($1.8Trillion at a poultry 5% interest)
dont know where you are getting your numbers. sounds like are confusing revenue and spending. relying on some AI maybe?
US collects about $12Trillion in taxes total (30% ish of GDP), under $2trillion of that is given to the federal government for medicare, medicade and the military, they spend more than $5trillion, which is what they spend on medicare, medicade the military and the interest on the $37Trillion debt they have accumulated spending more than they were given by the states for medicare medicade and the military - mostly bank and insurance fund bailouts to prop up the failed US financial system, adding about $3trillion to the federal debt each year, which is why it has gone from $30trillion at the end of 2022, to $37trillion now.
Getting downvoted because I do my own research instead of believing the latest gormless chatbot, that's new.
Of that, individual income tax was about $2.4 trillion, payroll tax was $1.7 trillion, corporate income tax was $530 billion, and there's about $253 billion of "other."
How exactly are you saying they spent 2T more than they collected in revenue again? Is this a Joe Biden forget where he put it or smth?
Meanwhile
Debt now
https://www.usdebtclock.org/
$36.4T
=$33,167 end of FY 2023, spent $3.3T on interest, Collected and spent $2.2T on medicare,medicade and the military.
= $33.1 +3.3 -2.2 +2.2 = $36.4T
good luck have fun. Im out. enjoy your fantasy economics for the few months it has left. Last group of federated states with group finances in a similar position was the USSR circa early 1991, pop quiz, can you guess what I think happens to the US next?
You sure did edit the hell out of your post. Obviously you aren't operating in good faith. But I should have figured that when, in your first post, you claimed the US collects $12T in taxes without reference and then ignored my reference showing it false. Have a great week.
The states of the united states collect about $12T in taxes total.
That is GDP times taxation as a % of gdp
roughly $36T times 30%
Precisely what that is doesnt matter, could be $10T, could be $20T
The federal government collects its tax from the states which it does through programs approved in congress.
Those programs are medicare, medicade and military spending + a few hundred billion total in scraps like the FAA or NASA. in total that sums to around $2T, which is all the states are obliged to give the federal government from the taxes they collect, if they dont like it they can choose the nuclear option and simply exit the union - California has a reasonable campaign long time ongoing to do exactly that called calexit - although right at this moment it lacks momentum. According to wikipedia there are growing movements in Alaska, California, Texas, Louisiana, Florida, and New Hampshire to secede.
Those movements will grow very very quickly when the states are actually presented with the now inevitable choice of doubling what they give the federal government while the federal government stops spending on pretty much every federal program - which is the only way the federal government can sustain paying its bills now the interest on their debt is larger than they take in revenue.
> The federal government collects its tax from the states which it does through programs approved in congress.
You’re aware that income tax, corporate income tax and payroll tax are paid directly to the federal government from individuals and companies right? The states don’t collect on the federal government’s behalf.
>nuclear option
There is no nuclear option. The country decided that 160 years ago.
As a more substantive comment: You may find the thesis "Propagation networks : a flexible and expressive substrate for computation" by Alexey Radul interesting. https://dspace.mit.edu/handle/1721.1/54635