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If you want to learn, you need to forget AI and work the problem out yourself. It sounds to me as if you have not done that. If you had, you would not be asking this question. Look up Raoult’s Law - then calculate for yourself the first part of the problem and show me how you did that. Then we can talk about the second part.

The intrinsic properties of a manifold depend entirely on the manifold itself without any reference to a higher-dimensional embedding manifold (a coordinate transformation is an embedding of a manifold into a manifold of the same dimension). The distance between two points of a manifold depends on the path between the two points. That is, for the expression: (ds)2 = guv dxu dxv ds is not an exact differential (there does not exist a function s(..., xu, ...) for which ds is the differential). This btw is why there is no absolute time in relativity. If ds were an exact differential, then: [math]ds = \dfrac{\partial s}{\partial x^u} dx^u[/math] and therefore: [math](ds)^2 = \dfrac{\partial s}{\partial x^u} \dfrac{\partial s}{\partial x^v} dx^u dx^v[/math] [math]g_{uv} = \dfrac{\partial s}{\partial x^u} \dfrac{\partial s}{\partial x^v}[/math] But the RHS of this expression, as a matrix, has zero determinant, contrary to the requirement that the metric tensor is invertible. [If the above LaTeX doesn't render, please refresh browser.]

What was the background of their upbringing? Is it ok for 40. 50, 60 year old millionaires and billionaires to fuck 12, 13, 14 year olds and then blackmail, even kill them, to keep their silence? Children that age in developed nations are not considered to have sexual autonomy, especially with people that could be their grand and great grandparent. I'm not interested in the facts of each case, the principle of protecting naive/disadvantaged children from elder abuse supersedes all that. There is no circumstance, with consent or not, where a child can be sexually available to a person several times their age.

There is so much wrong with what you've written. You are talking about pubescent and prepubescent GIRLS like they should be self aware and be sexually familiar.

Consent laws protect children in these instances, or are supposed to, unless the men are shielded by other men who see nothing wrong with having sex with children.

Well, no, not really. Switzerland is a different country. And this is basically the same argument as before, about the timing of something vs the underlying sentiment; I see you omitted the paragraph that follows your quote, that shows the result of a vote to prohibit new construction. These lame “arguments” are rather tedious. You’ve obviously convinced yourself of something, but it’s not based on any facts you’ve shared.

Yes, and it's a classic joke, which I first heard as a teenager, and not in a vaccine context. It went something like: Doctor: Your baby has jaundice. Mother: I want a second opinion. Doctor: Okay. Your baby is ugly!

https://chemistryhelpforum.com/t/useful-latex-code-for-chemistry-equations.147266/ For those who might be interested, here is an interesting thread about LaTex / Mathml and MatthJAX in chemistry. I wonder if there is anything there we could learn for this forum ?

No, AJB was a former, higly respected member of this forum. I havent found that to be the case. While D Lincoln explains when and how the Higgs was found, for example, 'Physics Explained' delves into how the Higgs field/mechanism works, how it gives rise to mass and the detected particle, all with a 'simplified' mathematical model. Check it out ... Just what I needed, another cat video. Like I already don't spend half of my internet time watching cat videos. As a former owner of two cats that I adored, I'm resisting the idea of getting another, as putting the previous ones 'to sleep' after 15 and17 years were two of the most difficult decisions I've had to make.

It's unfortunate that he seems like one of those children like Damian in "The Omen." He's liable to push you off a balcony while you're changing a lightbulb. I've noticed the Dutch are fairly immigrant friendly. If I could handle hot weather and bring my extended family, you would be calling me "neighbor."

Dear Mister President, pleased to see that you have recovered from the run of disabilities that I read in the press prevented you from serving in the military during the Vietnam war. As for Afghanistan, like many in the UK I have known someone who gave his life. He was a 19 year old friend of my grandson. It so happens his last moments were filmed and can be seen in the following video which was made by a BBC crew who were making a documentary in Afghanistan at the time. "

I was under the impression that a recent experiment has cast serious doubts on the viability of Bohmian mechanics: https://www.nature.com/articles/s41586-025-09099-4 This is essentially a direct conflict between what BM predicts in that situation, and the observed outcome. What it means is that, if I understand the implications correctly (and I’m not sure that I do), the concept of “particle” that BM is constructed on does not correspond to particles in the real universe.

“Bees use polarized sunlight scattered by the atmosphere in order to navigate; they always know where the sun is, even if it’s cloudy or behind a mountain. Then they waggle dance to inform their hive-mates about food source locations.” https://kottke.org/26/01/an-optical-compass-inspired-by-bee-vision 10.5 min video in link discusses polarization and why the sky gives us polarized light, and then gets into making an optical compass using this. It finishes up with how bees use this and the waggle dance they use to tell the hive where food is.

Hmm, Project Anchor. Missing W perhaps?😁

You need to remember though that just because you don’t understand it, doesn’t mean it’s not useful or doesn’t work. It evidently does, because we are using those findings in practical applications. I myself do not understand in detail how a mobile phone is constructed, since electrical engineering is not my area of expertise. But it still works. The average person in any math or physics department at a university isn’t a genius, with very few exceptions - they’ve just decided to put in the work necessary to learn the concepts. In-depth mastery of any subject requires time and effort, that’s just how it is.

Ahh, fond memories of Hollerith pinch card machines from the mid 70s. And a stack of cards about 6 inches high, each with one line of FORTRAN IV code, solving for the energy levels of the Helium atom by numerical methods. Tedious and frustrating, but I would gladly do it every day again, if I could be 21 years old again.

“At Dahlgren, West devoted herself to solving one of science’s most complex challenges: accurately modeling the shape of the Earth. Her painstaking calculations and programming helped transform raw satellite data into precise geodetic models, enabling reliable satellite-based navigation. That work ultimately became the backbone of the Global Positioning System (GPS) — now essential to aviation, shipping, emergency response, smartphones, and daily life worldwide.” https://thezebra.org/2026/01/18/dr-gladys-west-mathematician-whose-work-made-gps-possible-dies-at-95/

LOL. That was posted by Adam Kinzinger on Bluesky. Here's two more:

Ha! "That's one very small step for a man, one giant leap of illogic for all magakind."

It should be noted that expansion has units of inverse time, not speed, although it is usually expressed as speed per distance. For a flat (three-dimensional) universe described by the FLRW metric, the recession speed at a particular time is directly proportional to the distance, and therefore there will always be some distance beyond which is receding faster than c. However, it should be noted that when we look outward, we are looking at the past, so the observed recession speed is not necessarily proportional to the observed distance.

Me too. Apart from the daily Warfarin and a precautionary dab of antiseptic on abrasions, I tend to let nature take its course unless things get scary.

Reginald LaNiger and his friend Emma Damme like to watch documentaries about ancient Rome... Men oil lats - Saturnalia predictability: a day til I bat cider pail an' rut as stallion, Em.

[math]\text{Let:}\ \ \ \ u=\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}[/math] [math]\text{Let:}\ \ \ \ a=\sqrt[3]{7+\sqrt{50}}[/math] [math]\text{Let:}\ \ \ \ b=\sqrt[3]{7-\sqrt{50}}[/math] [math](a+b)^3=a^3+3a^2b+3ab^2+b^3=a^3+b^3+3ab(a+b)[/math] [math](a+b)^3-3ab(a+b)-(a^3+b^3)=0[/math] [math]a^3+b^3=14[/math] [math]ab=-1[/math] [math]u=a+b[/math] [math]\text{Therefore:}\ \ \ \ u^3+3u-14=0[/math] [math]u=\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}=2[/math] [If the above LaTeX doesn't render, please refresh browser]

UHT goat milk might also be good if there are any digestive issues not resolved by the fermentation. It has the A2 casein, which some find easier to handle. (A2 beta-casein is what is found in human milk)

48 + 68 + 98 = (28 + 38)2 – (64)2 = (28 + 38 + 64) (28 + 38 – 64) = 8113 x 5521

Tried a one litre UHT batch without heating and got 100% conversion to homogenous yoghurt. Negligible whey separation. Not that I can't find a use for the whey, but it's good to have the flexibility of a choice.

Dead right. The actual paper is here: https://www.cell.com/matter/fulltext/S2590-2385(22)00153-9?utm_source=EA from which it can be see that this is Cu²⁺, acting as a binding agent by complexing with 4 imidazole units. The key structure appears to be shown here: Looks like another case of lazy journalism, leading to a misleading impression that these things have Jaws-like metal teeth when they don't. I'm not sure what "totally metal" is supposed to mean, but "containing 10% copper ions" wouldn't be as clickbaity.

Allow me to give the hint: 216 = 63 The hint for me was that this problem is going to have a simple solution and is therefore constrained so as to have a simple solution.

Quite. In many ways, it's a very free society here - gives you the opportunity to be an adult, as my friend put it many years ago. Only, there's no safety net for the reckless. Suits some people: others not so much.

You are going to see whatever the leader is telling you to see, didn't you get the memo?

Was this a request for a simple guide to manifolds and spaces ? I have less than zero interest in Grok, but I can help with this. Again +1 for well put insight. This is a new variation on Sir Oliver Heaviside's famous remark to the Royal Society Gentleman, Should I refuse my dinner because I do not fully understand the process of digestion ?

IIRC epidemiological lit also shows a stronger correlation with humidity than temp. In medical lit (or at least those that I remember) tend to focus on mucosal damages, rather than viral stability, though. Edit: I think one of the papers I read on that matter was this one:https://www.pnas.org/doi/abs/10.1073/pnas.1607747113

So basically it might help but at least it doesn't harm? If there are no downsides, I don't think it is a bad thing as such. The main worry I have with these things is that especially in recent times, they tend to get a life on their own and suddenly there will be trends on folks binging on casein (or whatever the hype of the week is). In a way I think it has become more important to work out the nuances as the internet has become a horrible amplifier of really bad ideas. Not that I am accusing this thread of anything like that, it is just a matter of professional struggle I am having.

Today I learned of white rainbows and how they can be formed.

No, I didn't, using pasteurised, it was fine. I think the culture soon overwhelms any other strains in there. I wouldn't bother heating with uht.

My app would activate every few days. I'm at work 12 hours, for 7 days every 2 weeks. Actually closer to 13 hours counting travel time. Most phones have 'health' apps or motion detectors; if no activity is detected by the phone in 12 hours, it should message your contact. Then again, I've been known to sleep for 12 hours as well ...

... and Nigeria too: Tinubu stops otter, rages Uganda dad nag. Use garret to spot subunit.

My question is does the electric company turn a profit? In my state utilities companies govern themselves. We have no idea what their expenditures are and how they figure what to charge. It would be hard to compare to other countries. Maybe compare the environmental resources used to the total utilities used a year. I heard about the resources a person uses versus the number of acres of resources per person on the show Explorations. For example someone in the US could pay less for the same amount of resources used by someone in Africa. And at the same time pay less while consuming more energy.

Regarding this bit, extremely low cost keys (that work) are likely OEM or surplus volume licensing keys that are resold (perhaps there are other sources, too). Technically, they violate the terms of service, as those keys are not supposed to be sold directly to user and could get deactivated. But I don't think it happens very often, unless the reseller is selling the same key multiple times, or perhaps a whole bundle gets somehow flagged.

Lmao @StringJunky beat me to the punchline

Pretty useless dam if it only lasts 50 years with at 30% chance of flooding. Must have been designed by a university maths dept. 😀

This is not about virtual particles though. It is standard undergraduate level chemical QM, involving bog-standard quantum systems, like vibrating chemical bonds or electronic states in atoms and molecules. As I said previously, although the idea of zero point energy often gets popularly associated with the energy of the vacuum, virtual particles popping in and out of existence and that weird QFT stuff, it is actually quite general to all kinds of mundane quantum system. Here's a simplified diagram of the energy well of a chemical bond between 2 atoms. The vibrational energy this system can have is restricted to certain quantised values, E₀ , E₁, E₂ etc. Notice that E₀ is not at the bottom of the well. That means there is still some kinetic energy left in the ground state - it is still vibrating a bit, in effect. You can't get that bit of energy out because it is already in the lowest allowed state. That is zero point energy, which the molecule will still have even at absolute zero. (The animation about the photon is just to point out if it doesn't have the correct energy it can't excite the vibration to the next level up. This is not important for the present discussion, it was just on the first decent diagram I found to copy.)

The rate at which it passes does, but not the existence of time itself. Religious belief is not empirical fact. You might take it as a given, but it is not. Yes, in college physics we had a lab where we did it. The rotating mirror experiment Markus described. If there was something missing there would be experimental anomalies. Which is not remotely the same thing. Works of fiction aren’t much of a rebuttal, because they’re literally made up.

No. There are time-dependent processes that do not involve motion, such as the decay of elementary particles for instance. Locally at any given location, all clocks always tick at exactly “one second per second” - so there is no meaningful way to say that it is different in different locations. The only thing that changes is the relationship of clocks in spacetime, but that’s not the same thing. Again, clocks don’t have different “speeds” - it’s only that clocks at different events are related in non-trivial ways. This may at first glance appear to say the same thing, but it doesn’t. I have personally done it twice - once in high school physics class with an apparatus basically consisting of an assembly of rotating mirrors, and once for fun using the classic setup involving marshmallows in a microwave. And there are many other ways to do it at home, it’s not really that difficult. Note though that the level of precision within such DIY tabletop experiments is naturally limited, so don’t expect too much in terms of accuracy of the final numerical value.

We just want to know stuff and we want to believe the stuff we know is correct; no time equals no history to learn from, just empty space...

But, there being an infinite number of universes, with infinite possibilities, there is a possible person with ALL the knowledge of the universe. Hopefully today you have learned that you should never use infinities to make a point, because, just like Physics theories, when infinities arise, your point ceases to be valid 🙂 .

No! Cast inedible Mac at a camel, Biden. It's a con.