From school you are used to think of function in their explicit form y = f(x) but you can easily turn that into the implicit form f(x) - y = 0 or more generally f(x, y) = 0. With that you can plot the graph of f(x, y) either as a 3D surface with f(x, y) being the height at point (x, y) or encode the function value at (x, y) into some color at (x, y). Where that surface is equal to zero, i.e. where it intersects the z = 0 plane, that are the points of y = f(x). Points (x, y) at which the value of f(x, y) has small non-zero magnitude are what the article calls low error points or regions, points or regions that almost satisfy y = f(x).
f(x, y) = 0 is true only for some combinations of x and y. It’s an equation to be solved, not a universal statement like ∀ x, y : f(x, y) = 0, nor a definition like f(x, y) ≔ 0 (or “≝”). The solutions to the equation are the points (x, y) where the graph has height 0. Which points these are depends on how f is defined.
For example, f might be defined as f(x, y) ≔ x² + y² – 1. Then the points (x, y) for which f(x, y) = 0 are those on the unit circle (those for which x² + y² = 1). The graph will have height 0 only for those points.
They're really two different types of equal signs.
f(x,y) = x+y might be better written as f(x,y) := x+y where := means "is defined as". Then f(x,y) = 0 is an equation that expands to x+y = 0, or in familiar intro algebra form, y=-x.
I will say that in programming it's commonly used as assignment, which isn't quite the same thing as definition. golang uses it to declare variables so that's pretty close
When we say "f(x, y) = 0" in this context, we also usually have a separate definition for f(x, y) provided, where that f(x, y) is not necessarily 0 at for all x,y. And so this constraint "f(x, y) = 0" means "find pairs of x and y such that it makes f(x, y) become 0".
If "f(x, y) = 0" is actually the definition of f(x, y), then yes, it would be a pretty boring graph.
But only if you are looking at the revised goals, if you look back at the original goals, things look different. It was supposed to fly around the moon with people on board two years ago.
If you want to look long term, well, they're still stellar :) Considering everything they're achieving, and how they're so much better than everybody else in the field.
It's a failure only if you look at a rather small time range and criteria. Which I don't think was a surprise for anybody - Elon is famous for going for moon shots and failing, but still delivering better than anybody else.
.NET supports this because [Visual] Basic supports it. This can be used from C# - and other languages - but there is no nice syntax supporting it.
// This also supports multidimensional arrays, that is why the parameters are arrays.
var array = Array.CreateInstance(elementType: typeof(Int32), lengths: [ 5 ], lowerBounds: [ -2 ]);
// This does not compile, the type is Int32[*], not Int32[].
// Console.WriteLine(array[0]);
array.SetValue(value: 42, index: -2);
Console.WriteLine(array.GetValue(-2));
What would happen in a hypothetical scenario where Microsoft cut off everything [1] they can for all of Israel - no Azure, no Office, no Outlook, no Exchange, no SQL Server, no Windows, no Xbox, no ...? Depending on how many things they can make unusable, I would imagine that this would be pretty bad, probably even causing some deaths because of affected infrastructure.
[1] Not sure what they could actually make unusable by revoking licenses, blocking logins, and whatnot. It probably also matters how quickly the effects are felt, Azure would be gone immediately but I am not sure how often Office checks whether its license has been revoked, if at all. If license checks make things stop working over weeks and months, it would still not be pretty, but it would provide at least some time to prepare and avoid the worst.
IDK but Mossad is quite possibly the world's most effective spy agency and SV software corporations rarely have effective safeguards to protect against rogue employees so we must conclude that there are many sleeper agents planted throughout major corporations on behalf of just about every intelligence agency in the world including but not limited to mossad.
I have not seen any hard evidence of this nor have i ever suspected a fellow employee at any of my employers of being a double-agent loyal to a state intelligence agency but it's easy enough to do that there must be hundreds, maybe even thousands of sleeper agents all over santa clara and redmond.
I do not understand how this could possibly work. From a camera with RGB filters we get essentially three different integrals over the spectrum per pixel, how would we recover the spectrum from that? Even assuming you can account for the spectrum of the light sources and the color filters in the camera - which should be doable with a color chart with known spectra - I do not see how you could go from three data points to a full spectrum without making assumptions about the possible object spectra.
EDIT: Okay, after going to the actual paper I at least get transmission mode - you photograph the color chart through the sample and this will of course imprint the absorption spectrum onto the know spectrum of the color chart and you can then look at the difference to the color chart without the sample in between. But I do not get the logic behind their reflectance mode.
For point one the reason is of course that π(p) has a [global] minimum at 2. Actually showing that is not that easy because the involved integrals have no closed form solution but in principle it is not too hard. The circumference of the circles is 2 π(p) and equals four times the length of the quarter circle in the first quadrant which has all x and y positive and allows dropping the absolute values. The quarter circle is y(x) = (1 - x^p)^(1/p) with x in [0, 1]. Its length is the integral of ds over dx from 0 to 1 where ds is the arc length in the Lp norm ds = (dx^p + dy^p)^(1/p) which yields ds = (1 + (dy/dx)^p)^(1/p) dx. For dy/dx you insert the derivative of the quarter circle dy/dx = -x^(p - 1)(1 - x^p)^(1/p - 1) and finally you have to compute the derivative of the integral with respect to p and find the zeros to figure out where the extrema are. Well, technically you have to also look at the second and third derivative to confirm that it is a minimum and check the limiting behavior. The referenced paper works around the integral by modifying the function in a way such that it still agrees with the original function in some relevant points but yields a solvable integral and shows that using the modified function does not alter the result.
Because the goal of the Zionists has always been to conquer all of Palestine and the State of Israel has been following those foot steps since day one. From the river to the sea. This has been declared illegal under international law more than half a century ago but Israel does not care about the law. Therefore Israel should be forced to comply, which means boycotts and sanctions or military force. And we should probably try boycotts and sanctions first before we send tanks. Which is unlikely to happen any time soon anyway as that would mean opposing the USA and we have seen in recent history what happens to people and countries supporting the Palestinians.
To the dead comment Israel agreed to the UN partition in 1947, and then Arabs started a war to kill all the Jews there.
You do not even understand what the UN general assembly does, it expresses majority opinions, it does not make legally binding decisions. That means the UN partition plan is only what a majority considered the best solution, not a legal decisions to divide Palestine. And the Palestinians vehemently opposed that solution and later violently its implementation.
And should one really be surprised that the Arab neighbors attacked Israel? The Jews had just occupied half of the Palestinian land, violently displaced hundred thousands of Palestinians, and established their own state on Palestinian land.
The German policy of more or less unconditional support for Israel is plain stupid. This policy exists because of the horrors that Germans have inflicted upon Jews but it now supports similar [1] horrors inflicted by Israelis upon Palestinians. I can not wrap my head around that. If anything, Germany should try to stop Israel with all available means to protect them from themselves. Germany should do the same as Ireland and so should everyone else.
[1] Feel free to mentally replace similar with any other word that you think more accurately compares the two scenarios.
It's very easy to wrap your head around - Germany is Israel's main arms supplier after the US.
Germany accounted for 30% of Israel's arms imports between 2019 and 2023. In 2023 it accounted for 47% of Israel's total imports of conventional arms.
Between October 2023 and May 2025, Germany greenlighted the delivery of weapons and military equipment worth €485m to Israel
They since stopped the export of weaponry 'used in gaza' but it's naive in the extreme to accept assurances from the incumbent Israeli government to the contrary.
N.B. Israel continues to illegally carry weapons through Irish airspace, and kindly refers our government agency to El Al's Legal department on inquiry.
That does not seem like nearly enough money to make a bad policy decision because of the money and that policy is probably much older than the arms deals.
Because that's not the reason (Israel primarily imports weaponry and munitions from the US and India [0]).
The issue is the other way around. A significant portion of Germany's ground AD and defensive systems are sourced from Israel- most notably the Arrow 3 missile shield [1] deal that recently went through. Germany is heavily dependent on Israeli cybersecurity companies as well [2]. Germany is also subsidizing Arrow 3 sales to Ukraine [3].
Protecting your nations citizens always trumps morality, and in Germany's case, it's become even more critical after what happened in Poland this week.
Politicians will always politick, but they do not tend to be the ones who make policy in a parliamentary system like Germany of Ireland.
Ireland basically has no army, and is entirely dependent on the UK for it's defense. As such, their politicans are free to say whatever (as long as it is not against the UK) because it's not going to come back and bite them in the behind. That said, that's now changing as the UK is trying to renegotiate the deal [0][1]
The Irish strategy is to make the pubs too attractive for any attacker to bother with armed conflict. ;)
The Irish position should not be underestimated. It tends to be a bellweather for what others will align with in the future. Ireland tends to use it's soft power very effectively at the global table.
> The Irish position should not be underestimated. It tends to be a bellweather for what others will align with in the future
This really overstates Ireland's position in foreign policy studies. No one at Bruegel, ECFR, Institute Montaigne, GMFUS, and the 2-3 other major EU think tanks that are the de facto voice of European policy are taking Irish policy into account. Ireland lost any chance it had of being at the table when the Eurozone crisis happened. Even Spain and Italy have barely rebuilt their credibility.
> Ireland tends to use it's soft power very effectively at the global table
How? Ireland barely comes up in most conversations aside from using IDA Ireland as a model for attracting services FDI.
Citing the Eurozone crisis as if we were analogous on an economic or policy level to Spain/Italy/Greece is just farcical in the extreme. Given our population of ~5 Million we're probably punching above our capita to the largest extent of any EU member-state. Hell, even the Asylum laws governing Europe are named after us:
We are also the only EU country where the Constitution ordains a referendum to validate ratification of any amendments that result in a transfer of sovereignty to the European Union; such as the Nice Treaty which we can prevent from passing on an EU level.
Putting aside the multiple times we have held EU Council Presidencies, how about you take our two-year term on the UN Security Council from 2021 to 2022, where we got UN Security Council Resolution 2594 passed – the first ever Resolution on UN Peacekeeping transitions.
Since 1958, Ireland has maintained a constant presence on UN and UN-mandated peace support operations to the point where many English speakers in the South Lebanon do so with an Irish accent. 86 Irish soldiers have died in service of the UN since 1960.
We also have a particular legacy regarding the IDF and war crimes - Like in 1996 UN position 6-52, near Maroun al-Ras, a platoon of 33 Irish troops was surrounded and isolated from UN headquarters by a mechanised IDF unit. Or in May of this year when Irish peacekeepers in Lebanon came under fire from Israeli forces near a bombed out village at Yaroun
We have lost almost 50 troops in Lebanon alone. Approximately 50% of our casualties have been inflicted by Islamist resistance groups such as Hezbollah – the other 50% by the IDF and their paramilitary proxies in the area.
The Ultra-Orthodox can't even join the IDF. The issue is that the US HR Bill passed which legally equated Anti-Zionism with Anti-Semitism, and the US control the narrative in the english-speaking world around the Genocide. It's Orwellian in the extreme.
This statement betrays a deep lack of imagination about other things one could do against Jews. Killing them, for instance, or putting them in concentration camps are both far worse than any association you might choose to make with Israel.
> but it now supports similar [1] horrors inflicted by Israelis upon Palestinians
I would say that Israel is historically worried about the fact that his enemies want it to make it disappear. There are many things to criticize from them but this is the basic premise.
But the solution to that is not making some other people disappear in order to make some room for Jews. And they have already stolen half of Palestine and turned it into their own state and that state is - despite its illegal origin - now internationally recognized and therefore unlikely to be undone. They just have to be satisfied with what they took and stop attempting to take the rest of Palestine, too.
> Israel as a society is much more tolerant and does not make human shields with their own population.
It is an objective fact that Israel has killed almost 20,000 Palestinian children since the beginning of the latest conflict (post Oct 7, 2023).
For comparison, a total of less than 1,200 Israeli's were killed. Which is also unacceptable and Hamas should be condemned. But Israel's continued slaughter of innocent children has gone beyond "defending themselves" and is just as bad, if not worse, than what they claim the other side to be.
You blame Hamas for using people as shields, but do you think Israel is justified in killing all these innocent children in the first place?
> You blame Hamas for using people as shields, but do you think Israel is justified in killing all these innocent children in the first place?
No. I do not justify it. What I say is that Hamas is literally a group that does not care even about their own people as facts show. They usually falsify the data also. So I am not sure how that data is but I acknowledge this is wild.
When I think of these things I think that governments and rulers are what they are. But I wonder if you were a palestinian being a human shield of a piece of rubbish or were an Israelian that wants to take kids to school and you see how they launch missiles from the Gaza side as a routine from schools and hospitals, what would you do? It is a very desperate thing for a civil person.
I prefer to not make criminals out of whom are not criminals. So Hamas and probably Netanyahu are both criminals. But the poor people from Palestine are basically kidnapped by their own governors to an extent that is hardly bearable IMHO. Israel people are also people like you and I and they have a reasonable fear of being smashed. It is not for fun that Israel has spent a huge amount of its GDP in military stuff. They do not do it for fun, whatever people try to convince me of.
It is a really complex situation, that's all. No innocent deserves to die in either side. I just try to make reasonable descriptions of what I see. I am mostly neutral, even with my bias.
I appreciate you sharing your viewpoint. And I should clarify that when I say "Israel" I'm referring to the state and its military, not individual Israeli citizens who probably have nothing to do with the killings.
> It is a really complex situation, that's all. No innocent deserves to die in either side.
Absolutely no innocent deserves to die on either side, 100% agreed. I still don't think the killing children part is complicated at all - there's zero justification for it. But the overall situation is definitely very complicated due to a long history of conflict on all fronts.
> They usually falsify the data also. So I am not sure how that data is but I acknowledge this is wild.
The death toll numbers I used are from a peer-reviewed UK publication that more or less corroborates the numbers reported by Gaza itself.
> The death toll numbers I used are from a peer-reviewed UK publication that more or less corroborates the numbers reported by Gaza itself.
I am quite skeptical of that data to be totally honest. What I found over the years is falsification after falsification. It is a fact that this has happened. We all know. But it is also the common thing that those numbers are inflated.
> I still don't think the killing children part is complicated at all - there's zero justification for it.
So if hamas has a child hostage in every arm warehouse (which is more or less what happens) and with every terrorist squad the only justified action for Israelis is... to die?
How does anyone not see that condemnation is not nearly enough? Hamas need eliminating, not merely "condemned".
Anyone arguing against the need to defeat Hamas in the place they launch attacks from and hold hostages in, is peddling a woefully broken, illogical argument. A war is happening. In war, you have enemies. "We should condemn our enemies" sounds like some kind of captain obvious baby talk.
I'd say the people that are historically and actively actually being disappeared are the Palestinians, the premise is that Israel since inception wants (edit: is) to eradicate Palestinians
Germany also felt guilty about the invasion of Russia. I understand this to be one of the reasons why Germany was keen to buy Russian guess, to make amends.
On the flip side Germany did a huge amount to bring solar power into large scale usage.
Remember there are are radical nationalist / racist / xenophobic groups in germany / anywhere.
Such groups are so strong in Germany that Hitler used them to strenghten the power of his discourse.
Recognizing Israel / the Jews are the ones now doing Genocide would *in the sight of the extremist groups* prove their belief that Hitler was right / the Jews are the devil. (I repeat, in their regard, not mine)
Therefore, my opinion is that the official stance in Germany must be "historical responsability" mainly not for the sake of the Jews, but for the stability of the society.
Yes, they are alreading sharpening their forks and oiling their torches. I think officially recognizing it ("Jews are bad") will drive normal people to don't consider them (the extremists / nazi) conspiracy nuts anymore, because of a "they / we were right all the time" discourse.
I apologize for basically repeating myself. I just wanted to refer to the conspiracy theories you mentioned because calling them that is, ironically, constructive.
You'd think those suffering genocide would do the one thing to stop it, namely releasing the hostages and ceasing their own officially chartered genocidal ambitions.
1. Strictly speaking, we’re not naming a table, we’re naming a relation.
And a relation is a set, hence plural.
2. It reads well everywhere else in the SQL query:
SELECT Employee.Name
, Manager.Name
FROM Users AS Employee
, Users AS Manager
WHERE Employee.ManagerID = Manager.ID
AND Employee.DateOfBirth IS NULL;
3. The name of the class you’ll store the data into is singular (User). You therefore have a mismatch, and in ORMs (e.g., Rails) they often automatically pluralize, with the predictable result of seeing tables with names like addresss.
The class User represents a single row, not the entire table, hence singular. If the O/R mapper or some other tooling has issues with singular and plural, then I agree, it might not be worth fighting the tools.
4. Some relations are already plural. Say you have a class called UserFacts that store miscellaneous information about a user, like age and favorite color. What will you call the database table?
I think having the table and the class name both in plural would be fine. That also seems rare enough in practice that I would not let this dictate the decision. In the given example I would also tend to record the user facts as a list of them. A user fact is a key value pair associated with an user, the keys living in their own table. Having the keys implicit as column names will also make some queries unnecessarily complicated and as the number of facts grows, the table will become increasingly wide.
Also sometimes we have singular names for collections of things, then it is fine to have a singular table name, you can name your Trees table Forrest if that makes sense in the domain.
I think that is the correct way to do it, you iterate over all the cats picking one cat at a time and it becomes quite obvious when you join a table to itself forcing you to do this. I am not writing that many SQL queries and I am certainly too lazy to always do that, especially if I am writing not too complicated ad hoc queries, but if I want the code to be as good as I can make it, then I always do this. Sure, it makes things a bit more verbose but you can also make the query more readable by picking a descriptive alias, FROM Users AS Manager, FROM Users AS NewEmployee, and so on.
Whether you alias your plural tables to singular nouns would probably be one of the very last things I would check out if I would need to assess your code.
And you should not but I think it is still useful. You will probably not even become consciously aware of the difference but Cat.Name will be ever so slightly easier to read than Cats.Name and maybe that difference in cognitive load is what makes you spot some issue that you would have missed if your brain got repeatedly slightly tripped up by incorrect grammatical numbers.
But if they think the bubble is the idea that "Modern AI will significantly transform business and the global economy, and that it'll lead to massive unemployment for knowledge workers", then—in my opinion—they're just wrong.
AI will one day transform the world but I do not believe modern AI will make that happen, that will have to wait for futuristic AI. I guess that makes AI not a bubble but I can still label modern, i.e. current, AI a bubble, right?
