@G.Bergeron (1) It already means something else in topology, i.e., a locally flat submanifold N of a manifold M is something which admits charts (U, U \cap N) homeomorphic as pairs to (R^n, R^k)
(2) If you want to say a curvature vanishes at a point p of a Riemannian manifold M, you can just say it's "flat at p". (3) "Locally something" is usually used as an adjective for something which has that property "something" for some neighborhood about each point: but if the curvature is locally zero everywhere it's globally zero so just "flat" suffices.
@BalarkaSen And that is essentially the meaning I had for that terminology in mind...
@BalarkaSen Oh, not exactly, I would have used the term to describe a smooth part of a manifold, that is, it admits a diffeomorphic chart from an open set around that point to an open set of R^n...
@G.Bergeron Again, that doesn't make sense per se. If you want to say your manifold M admits a smooth chart at a point p, just say it's "smooth at p". If you want to say your manifold admits a smooth chart at every point p, that's just the same as saying "M is a smooth manifold", so the "local" adjective becomes redundant.
Now, you can have the situation that there is an equidimensional open smooth submanifold N of M (i.e., "the smooth bit of M"). I'd hardly call that that's equivalent to "M is locally flat/smooth/whatever"
@BalarkaSen I'd a agree with saying ''smooth at p''. But local refers to anything you can construct on some neighbourhoods of the points around which that thing is to be constructed.
Well, relaxation can heat things up along the way as energy get transferred while fulfilling certain constraints
It makes me wonder what happens for potential energy surfaces with the reaction path moving downhill and passing through a saddle point of very high entropy, will it get stuck?
@BalarkaSen Fair enough, our disagreement might stem from the (maybe abusive) use of the term ''locally'' even when there are (for instance) a measure zero set of points where the property in question does not hold.
actually I might considering pressing that button... at the cost of being extremely annoyed at the message discontinuity in the future
On an unrelated note, I am not fully nonsensing because I did proved that integral in math chat to go to zero (assuming I can actually resolve the tetration in terms of exponentials)
where $\int^{\int^{\int}} = e^{e^{\int \ln \int}\ln \int}$
Since $\int$ is a linear operator, I should be able to exponentiate it, the main trouble is whether it make sense under ln
suppose we can do so, then expanding the whole thing into sums eventually result in a series of cancellations and then this integral operator then evaluates anything to zero
(which details are in the math chat link above if interested)
Personally, that notation is wrote as a joke, but then I went curious and wonder if I can define it so that it make sense, we also discussed about exponential maps of linear operators some days earlier
$e^{\int}$ meanwhile should be well defined (up to a constant) since any integrable function is going to converge to some integrable function after being integrated, and its exponentiation is just some power series of integral operators
Okay so I read somewhere that a thermally conducting rod with higher $K$ (Thermal conductivity) absorbs negligible heat and conducts / transmits nearly all of its heat whereas its the opposite for a body with low $K$ . Now , since heat absorbed will cause the body to increase its temperature , can I say that bad conductors of heat get relatively hotter as compared to good conductors of heat (When same temperature difference is applied ) ?
I think if you plot the heat map of a conductor, you might see localised hot spots for the bad conductor, and the cold end should stay pretty cold
Likewise, for a good conductor, the heat map should look like a smooth gradient since the heat is being conducted quickly, thus evening out any local hot spots
I am not terribly sure if my intuition is correct, however
The heat equation suggests when $\alpha$ is increased, heat will diffuse to a wider area at shorter time, so the average temperature of the object should be lower
$\dot{u} = \alpha \nabla^2 u$
so a good conductor should felt cold since the heat is quickly transported away from the hot object
@Secret Okay , so what I get from this , if I observe certain sections of the rod nearer to the heat source , insulator must feel hotter since it absorbs most of heat + heat conducts slowly through it. And opposite for the sections of rod nearer to the end away from heat source.
@G.Bergeron But I derived all the formulae in analogy with electric current
For instance, you can play hot potato with your bare hands with a piece of coal (way above 100C) but try that with a 100C copper ball and the experience will be less than pleasant
@Tanuj Please be specific. Temperature difference between what and what. Which one is higher, etc.
@Tanuj I was never taught heat was like a fluid, though.
@Tanuj Actually I think a teacher even mentionned this failed phlogiston theory back when I was in high school
@Tanuj But you seem to be mixing concepts. Any material, when conducting heat from an infinite object at temperature Ta to a finite reservoir at temperature Tb with Ta > Tb will have to reach Ta before heating reservoir B to Ta...
@Tanuj And the reason I said an electrical analogy is flawed is because electrical potential is a form of energy with very low entropy while heat is a very high entropy one. Thus, any real material conducting a current will have losses which will show up as heat. But if the material is conducting heat already, it cannot really have losses (ignoring ambient atmosphere and radiative losses) as there is no other form of higher entropy energy state for these losses to go to.
On another note, this is why heat is almost always what energy losses in system becomes (wether mechanical, electric, chemical) as it often is the highest entropy state for that energy.
Short of spreading around to the environment, that is.
Can anybody suggest some good online resources to learn about angular displacement/momenten/velocity ? I'm deeply confused by Holliday Krane Chapter 8, where they define vector omega = d(phi)/dt, without defining d(phi) properly (they only thing that they told about d(phi) was that infinitesimal angular displacement are commutative, that too in a super hand wavy way) (sorry from mobile no latex)
That means 12th grade. No I don't understand for example for rotation around arbitrary axis, how you break d(vector phi) into d(vector phi x),d(vector phi y) , d(vector phi z) ?
@G.Bergeron I am not convinced vzn is sincere. He has come into the chat room with news of amazing experiments that challenge conventional thinking more times than I have count.
@AlexKChen It'd be quite helpful to actually give details about your question... Like: motion in a plane, what are the coordinates, etc. I don't have that book in front of my eyes
@G.Bergeron its a big dealâ„¢ but ofc others will say coulda figured that/ whatever... kind of a theme around here... many around here very jaded even bordering on nihilistic at times :| o_O
@vzn Why do you publicly ask for comments (if at least implicitly) only to wave off anything that does not fit with your seemingly pre-constructed conclusions.
@vzn You seem to presuppose I don't have any knowledge about ML in physics, that I haven't thought about the topics before hand, discussed it with experts in the field, etc.
@G.Bergeron did not presuppose any of that. only reacting to your (unsurprisingly/ fitting around here) lukewarm comments. ps maybe you didnt notice the name of the researcher involved... deja vu for me at least o_O
@vzn The technical part is nice, the ML engineering and application aspect is really nice, but the bold claims challenging without basis foundationnal aspects of chaotic systems theory is what lukewarmed the whole thing. The article goes from nice to click-baity
@G.Bergeron youre obviously very smart/ well read but you think just like all the rest of them. do you have some original thinking going on? think you should look in the mirror with some of your very disparaging comments. ps bolbteppa/ semiclassical/ others were involved in threads, and theyre all oppositional to some of my thoughts...
@vzn I do have original line of thought going on, that's the whole point of research, isn't it? But then I won't broadcast anything other then to close friends before I have anything rigorous, solid to say and before I have tried my best to find any errors and ask said friends to do the same.
@G.Bergeron ok. think thats a shame, think cyberspace has some potential for sharing new thoughts outside of papers. but its obviously your call/ perogative, and do attest its the "safer" route. its true, sharing my original thoughts here has brought me almost nothing but vehement objection and namecalling etc. (it seems at times...)
@vzn That's because offloading thoughts on cyberspace is the easy part. Making everything rigorous well-defined and connecting it to the current body of knowledge is the extremely hard part.
@G.Bergeron agreed and other hand a newborn baby does not arrive fully formed and fully ready to face the world does it? you can see me struggling to add rigor at every moment.
though my attitude is really more along the lines of: "If you have built castles in the air, your work need not be lost; that is where they should be. Now put the foundations under them." (Thoreau)
@dmckee another pov, spend a lot of time working on bold, sometimes verging on wild hypotheses/ conjectures that may be true. if they are true, then others should think about them also. science does not progress/ evolve/ advance by past textbooks ("alone").
@G.Bergeron think you are wrong there. think that is part of the misconception. memes need to be developed. you are just seeing them developed before your eyes. yeah it can look messy. but science can be messy.
As dmckee and Semiclassical said, when you spend time grounding your ideas in order to broadcast them, you realized a lot of stuff does not work out as expected.
am tired of everyone around here ****ing on very solid work by many other top researchers and not taking it seriously, dismissing it, tossing it aside if its not "fully complete". think this crowd would have tossed aside einsteins original papers in the same spirit. yeah whatever, its obviously not finished yet
Do you think there is any possibility at all that people are simply concerned that you are running a wee bit ahead in drawing conclusions from what you read, maybe?
@dmckee another pov, think if anyone comes up with a major new correct theory, apparently those in this room will be the last on the planet to accept it or discuss it positively
@vzn Also, you have to realize the amount of claims we get all the time, by email, on this chat, in person, etc. I see it as a duty to at least consider seriously some of them. However, not a single one I checked personally turned out consistent.
@dmckee have seen ML go thru several "winters" also. cite the gartner hype cycle myself all the time. its as common as r e v o l u t i o n s. en.wikipedia.org/wiki/Hype_cycle
@vzn It was a waste of my time because when I pointed out serious problems with taking this paper as actually being about reality you entirely dismissed me and remained unmoved in your certainty, while never providing any rigorous counter arguments to what I brought forward.
@G.Bergeron do respect your opinion (bolbteppa + many others would probably agree). would like to work thru your objections at length (even more than already have done) but dont think you or anyone else really has the patience. have emailed the authors.
@G.Bergeron anyway, am all ears if you want to share a paper on what you think is the best/ most viable/ promising GR + QM direction/ angle. at least we agree on "1" )( scientific priority.
@dmckee there are only a few revolutions in play at any moment but there are lots of events that tie into them. thinking that "none are in play" is a sign of static thinking. btw every revolution cited by me is cited by other top-credentialed scientists etc...
@G.Bergeron Yeah. If you can't find stuff which 'obviously' indicates some new directions, it behooves you to look carefully at what you've got and try to test it as systematically as possible.
Which is hard to do with systems at strong coupling
I can guarantee you guys that if you pay Netflix engineer type money to young scientists, within 10 years someone will solve it. Nobody can dedicate the amount of time and effort required to do this unless they decided to be single and poor.
@vzn I can definitely tell you that the only way I'd be publishing anything (and I am capable) is if I can convince a company or some rich guy to pay what I think is fair for the kind of effort, level-headedness, and commitment that it must take. Nobody should have to work for less than they are worth
@vzn Following your analogy, it's more like humanity is massively dumping swords and battlecruisers and atomic bomb in the fire and at an industrial scale and you just came up with a kitchen wooden spoon...
Trust me, if people were properly motivated, we will have an abundance of riches in terms of credible ideas with predictive power and computational ease
@Cows theres huge $$$ in AI/ ML, a very scientific field, did you see this? A.I. Researchers Are Making More Than $1 Million, Even at a Nonprofit o_O nytimes.com/2018/04/19/technology/…
@Cows yeah am aware of that. ok let us know as soon as you can or whatever you can say. think its cool how many top companies youve interviewed with... not easy to get those interviews... :)
