joigus

In the theorem's case, it's important to realise that, \[D=\frac{\partial^{2}f}{\partial x^{2}}\frac{\partial^{2}f}{\partial y^{2}}-\left(\frac{\partial^{2}f}{\partial x\partial y}\right)^{2}=\left|\begin{array}{cc} \frac{\partial^{2}f}{\partial x^{2}} & \frac{\partial^{2}f}{\partial x\partial y}\\ \frac{\partial^{2}f}{\partial y\partial x} & \frac{\partial^{2}f}{\partial y^{2}} \end{array}\right| \] In the case you propose, \[ \left|\begin{array}{ccc} \frac{\partial^{2}f}{\partial x^{2}} & \frac{\partial^{2}f}{\partial x\partial y} & \frac{\partial^{2}f}{\partial x\partial z}\\ \frac{\partial^{2}f}{\partial y\partial x} & \frac{\partial^{2}f}{\partial y^{2}} & \frac{\partial^{2}f}{\partial y\partial z}\\ \frac{\partial^{2}f}{\partial z\partial x} & \frac{\partial^{2}f}{\partial z\partial y} & \frac{\partial^{2}f}{\partial z^{2}} \end{array}\right| \] The condition that these 2nd-order determinants not being zero amount to being able to invert the relation that gives you the second-order derivatives and ascertaining the nature of your critical point. The determinant being zero amounts to second-order equations not being solvable, and therefore the method being inconclusive. There's an elegant method to solve this kind of problem which is the method of Lagrange multipliers. And there's the method that @Genady proposes too.

Sorry. I meant the next-to-zeroth-order correction. If you Taylor-series expand you immediately see it's a maximum, because there0s no 1st-order correction and the 2nd-order correction is negative. That's what I mean, and sorry for the confusion.

https://www.wolframalpha.com/input?i=plot+(x^2%2By^2)exp((-x^2-y^2)) The intuitive idea you can get from highlighting x=1 and y=1 curves: And another analytic tool is using polar coordinates, \[ \left(x^{2}+y^{2}\right)\exp\left(-x^{2}-y^{2}\right)=r^{2}\exp\left(-r^{2}\right) \] and then do a Taylor-series expansion around r=1 in a small positive parameter \( \varepsilon \), \[ r=1+\varepsilon \] \[ \varepsilon>0 \] \[ r^{2}\exp\left(-r^{2}\right)=\left(1+\varepsilon\right)^{2}\exp\left(-\left(1+\varepsilon\right)^{2}\right)=\left(1+2\varepsilon+\varepsilon^{2}\right)\exp\left(-1-2\varepsilon-\varepsilon^{2}\right)= \] \[ =e^{-1}\left(1+2\varepsilon+\varepsilon^{2}\right)\exp\left(-2\varepsilon-\varepsilon^{2}\right)= \] \[ =e^{-1}\left(1+2\varepsilon+\textrm{order}\left(\varepsilon^{2}\right)\right)\left(1-2\varepsilon+\textrm{order}\left(\varepsilon^{2}\right)\right)= \] \[ =e^{-1}\left(1-4\varepsilon^{2}\right)+\textrm{order}\left(\varepsilon^{2}\right) \] And, as you see, the first-order correction is negative no matter what orientation with respect to the origin (just where the bump is) you place yourself. Same goes for 1-epsilon, of course.

Great topic that deserves more attention, though I'm glad it got the one it did. I don't know if my comment will be useful, but I'm thinking of chess as an interesting testing ground. Computers have far exceeded the capabilities of human minds. Computers play only on the grounds of pure combinatorics. Grand Masters, on the contrary, although they have powerful combinatoric minds by human standards, at some point through the complexity of the game, they must base a significant part of their reasoning on strategic, conceptual principles rather than pure if-then sequences. Strategic principles can deal with wide classes of combinatoric landscapes with the result of improving your chances of winning only on the average. It is not entirely impossible that computers get so good at calculating outcomes that the make our pattern-based reasoning obsolete. It is entirely possible that if we insist on computers being conceptual, we'll force them to play on somebody else's turf. On the game vs task discussion; I see no difference at all. But maybe I haven't thought about it hard enough.

We can only hope.

JamesL's trajectory on SFN is the closest to a perpetual motion I've seen here. Two cycles and counting... Plus perfect experiment on shooting yourself in the foot.

Did you say 'ta-ra'? Sorry, that was a cheap joke. Thank you for providing evidence that it doesn't work --as exchemist has explained. I would also demand of any claim that any of the thermodynamic principles fails, a complete thermodynamic explanation, as well as a thorough examination from mechanics, or field mechanics, or quantum mechanics if need be; of why that thermodynamic principle fails, or an exception is met in that particular case.

I agree --from every bit of evidence I know about. I think the reason underlying it is behaviour is very heavily influenced by environmental factors and ongoing processes in the mind --recent memories, interrelation/correlation of stimuli... In order for a recognizable pattern having a nearly 100 % molecular basis to reflect in behaviour --in the form of a fixed action pattern*, quite independently of those passing processes-- one seems compelled to think in terms of something really wrong in the genes or regulatory sequences, etc.--what I mean by 'molecular.' Otherwise, it's just too iffy. *I'm thinking OCD, PTSD, etc.

What I mean is that some particular behaviours can be traced back to a molecular basis, at least partially. This is especially true of some behavioural disorders. Some of them have a genetic basis, others can be partially explained by imprinting coming from environmental factors that get more or less permanently registered. But I'm talking here with a certain amount of diffidence, as I'm not an expert. I'm not aware of any molecular mechanism having been shown as being at the basis of sexual selection.

References? The OP is basing the premise on a social perception (my emphasis): Unfortunately there aren't, to my knowledge, any salient features results/mechanisms of evolutionary biology that allow one to talk with any degree of confidence on how this or that peculiarity "is regarded" this or that way. I don't think, for example --and correct me if I'm wrong-- that sexual selection has been understood to the point of being related to molecular bases, or even to evolutionary dynamics in the way of population dynamics, etc.

Now that Mistermark has left for a couple of weeks, There's still an echo of bells lingering in the air, sounding, Chamberlain, Chamberlain..., The politics of throwing the dog a bone. It's not like it hasn't been practiced before, It's not like we're clueless about the horrours it brought. Megalomania is easy to diagnose, and we know the profile very well. Putin is a shameless narcisist, and that's plain to see. He plays dominant male primate to a tee.

Masses in the standard model of particle physics come from so-called radiative corrections, which correspond to virtual processes happening all the time and contributing to the inertia of the particle. The nature of those virtual processes depends on the model of elementary particles. If you have a richer theory (more particles), the vacuum gets enriched by these new degrees of freedom, and in high-precission experiments you would be able to tell, because all the masses would be shifted by a predictible amount. It would produce small corrections, thereby the interest of the news. The W bosons are the electrically charged messenger particles of the weak interaction. Analogous to what photons are for electromagnetism.

Thanks. This kind of thinking resonates with what I think myself; that there probably aren't very many ways for life to emerge. It's not just about "oh, I can change this atom for another chemically similar one that plays the same role," etc. It's also about the abundances of those elements in galaxies. One is compelled to think that whatever happened during the Hadean Eon that jump-started abiogenesis, rates of molecular collisions must have played an important role. But it's so hard to be sure about any of these things.

That's pretty much what I meant, and I think you actually subtly implied it with your sequence "and then..., and then..." @Eise seems to have understood that as well: It's an ongoing process, the way I picture it. And I suppose it would be pretty difficult to pin down the very moment when that "cognitive event" ends, if there is such a thing.

Nice account. I suppose after the "Have I seen this before?" come a series of cross-talk events back and forth involving prefrontal cortex / hippocampus, and other parts of the brain? For example, the cortex asks back to the visual cortex "Does it have such and such feature?"?

Just a couple of qualifications: I didn't mean to say panspermia isn't to play a role some day in long-term (mean-planet-life-wise \( \sim 10^{9} \) years) sense. It probably is, and a lot, IMO. For starters, it is an empirical fact that organic matter does make its way to neighbouring planets and moons, at least within the confines of Solar-System parameters. If you lower your standards for how "primitive" an "organism"* must be in order to be considered life --example: chemolithotrophs--, then it may well be that our concept of Goldilocks zone has to be recalculated to include much wider ranges within a solar system, as well as the different varieties of solar system that can harbour life this kind of "less-honourable" life. Panspermic events, in this view, would serve as bridges between communicating vessels of proto-life (planets and moons) to find their way to a more promising platform for multicellular/organelle-centred life. Trying to paint a vivid picture of what I think might be going on: Imagine that already gazillions of these proto-living forms are out there saying: We don't think of our arrangement as non-life; in fact, we're doing quite alright by our standards. We've lived here for 2 billion years within our Goldilocks zone, with our mind-bogglingly sluggish metabolism and reproductive cycle. If a higher-order Goldilocks zone opens up in any of the neighbours, higher-order, more sophisticatedly organised life will be seeded in those moons, don't worry. Enough of our spores are flowing around to guarantee that in, say, a couple billion years more, this higher form of life takes seed. I'm sure part of what I'm saying here, or similar, is being considered by people working on these subjects. * "Primitive" meaning things like: Doesn't have internal membranes/compartimentalisation equivalent to eukaryots Doesn't have to exploit any particularly profitable redox reaction => doesn't need to have a particularly fast metabolism/reproduction cycle Doesn't have more than order 103-104 pair bases in any of its nucleic acids (or the equivalent of DNA)

Agreed.

Nice info bites. And a majestic beauty. Thanks.

I'm always confused with panspermia. I've always seen it as taking the problem somewhere else. The formulation of plausible molecular mechanisms for life in a variety of plausible scenarios is what's key for me. Whether it happened in a young Earth or in a young Mars or Venus, is kinda lateral to me, as all young Solar-System planets that are candidates are similarly extreme and alien to us from the limited confines of eukaryot, multicellular life. I just love your last thought, and brings me memories of Carl Sagan. I have that feeling very often.

Well...?

If there are such things as personality disorders, and personality disorders there are, there must perforce be something we can call a personality, that can ultimately be attached to patterns of behaviour. I would suppose we speak of personality disorders when the patterns of behaviour of a person make them dangerous to themselves and/or others, or unfit to lead a normal reproductive life, socially harmonious life, etc. And I would suppose we speak of somebody's personality as such when patterns of behaviour --either considered healthy or pathological-- have enough differences among them that attaching a personality to a particular individual makes some empirical sense. Different aspects of personality are being addressed with several degrees of success by neuroscience, with considerable rate of ongoing success, I would say. But personality is extremely complicated. Genetics-hardwired responses, environment-development, environment-induced behaviour imprinting ulterior constraints on molecular mechanisms, etc. It's hugely complicated. What you won't find is a clear-cut definition of personality, IMO. Personality has emergence written all over it.

Ah, OK. Got it! So to me what's going on, schematically is, 2SO2 + O2 <> 2SO3 ------------------------------------------------------------------------------ First equilibrium: 0.3 0.2 1.5 Added SO2 (out of equilibrium): 0.3+0.1x 0.2 1.5 Second equilibrium: 0.3+0.1x-0.1 0.2-0.05 1.5+0.1 So I think the calculation should be, \[ 125=\frac{1.6^{2}}{\left(0.2+0.1x\right)^{2}\left(0.2-0.05\right)} \] Which is, I think, what @Dhamnekar Win,odd meant when they wrote, That is, 0.3-0.1+0.1x instead of 0.3-0.1 +x Am I right?

Silly me. You're right. So the total mass is constant? I didn't see that in the initial statement. By, "How many moles of sulfur dioxide must be forced into the reaction vessel", I understood new moles of sulfur dioxide are added to the equilibrium. From what you say, the new SO3 must come from the pre-existing equilibrium, right? I had difficulties implying that from the statement. Sorry if I sound obtuse.

Oh, I see; all of them are necessarily gases. The bit I don't understand is, what assumption from the exercise's statement is at the basis of O2 going down from 0.2 mol/L to 0.2-0.05 = 0.15 mol/L when the extra SO2 is put there? I seem to be missing something here...