darklight03/StackMIAsub huggingface dataset:
Recently I became very much intrigued by algebraic topology and am spending quite some time learning it. My reasons are three-fold: it's a beautiful theory; it gives geometric justification to (or perhaps rather an application of) many purely algebraic structures; and it has fascinating applications in quantum field theory and condensed matter theory. Nevertheless, what I am familiar with currently are just basics: various homology and cohomology theories, homotopy theory and some standard applications (Brouwer, Borsuk-Ulam, etc., etc.). While these are of course interesting of and by themselves (and I expect spending a great amount of time on understanding all of this properly), I guess it is more or less understood for some fifty years now, so supposedly people work on topics far more advanced than this (or at the very least they use far more advanced tools to understand standard but hard problems). So, I'd also like to know what the field is about from the modern perspective (some interesting problems and research topics, advanced tools, etc.) so that I can see a little where will the study of the subject lead me in the long run. Sorry if the question is too broad but I am not sure where else to look (I've more or less browsed through all general articles on AT at wikipedia and tried to search MO too). I've heard few magic words like K-theory, sheaf cohomology, various spectral sequences, etc. but I don't understand these at all yet; more importantly my motivation to learn these things is lacking since I have no idea how or when these magic words are used (although I am pretty sure they are used a lot).
I have been inspired by some sci-fi cannons that seem to operate by initially spinning up a projectile inside the cannon, and then suddenly firing the projectile out at high speed. Now, I am wondering whether it is possible for such a cannon to perform practically. So, from an energy point of view, it appears that this hypothetical cannon is able to do work on this projectile by spinning it up using a torque, and this work increases the rotational KE. Then, there is some unknown process that cause the transfer of rotational KE into translative KE, causing the projectile to fly off with a particular velocity. It would be appear that the spinning-up function of the cannon would be an interesting way of having the KE of the cannonball stored up and ready for firing. However, even though translative KE and rotational KE are both KE, the same type of energy, it looks like extra work needs to be done to transfer between both types of KE. To do this, by applying the principle of superposition, it looks like you would need to combine the effects of undoing the spin with a reversed torque, and then applying a force, in the direction of the cannon firing, in a short space of time. Does this apparent backtracking of doing work not seem inefficient, making the spinning-up of the projectile a waste of resources? Are there any real, practical mechanisms that work especially well in converting rotational KE into translative KE? In short, can there be any practicality in a cannon that initially spins up it's projectiles?
So, to make it abundantly clear what I am asking for, here is an example picture edited (poorly) in a drawing software: At left, you can see my current TexStudio look: the are where real text is indeed entered had its colors properly customized with a .txsprofile object. However, the menu toolbar, the tabs area and so on are still colored in light-gray, while I would like them to have a darker look (as in the image to the right). Is it possible to achieve that somehow? I am aware that plenty of questions have been asked here and elsewhere on the topic of TexStudio's colors. For example: How do I change the colors of the application interface in TexStudio? Dark theme for Texstudio How can I set a dark theme in TeXstudio? How do I change color settings in TeXStudio? However, none of them really addressed the main point at hand: how to customize the colors of TexStudio's menus, toolbars, etc (not the colors of the text-editor part of TexStudio). The colors of the text-editor of TexStudio can be modified directly at the Syntax Highlighting options in the Tools>Configurations menu. However, rarely something is said about the other parts of TexStudio. When a question pops up about this, it gets either no answers or very low-detailed reactions. For instance: How do I change the colors of the application interface in TexStudio? Change background colour of toolbars in TeXstudio I realize that what TexStudio is doing is probably just retrieving the operational system color for it's menus, tool-bars and such. My question is whether there is anyway to bypass that and customize such colors.
So I've recently seen a few people use the word "sufferer" to describe themselves having a certain mental disorder. I know that a person thinking that they are suffering a certain disorder may be quite subjective, but their usage is still questionable. The best exhibit I have for this question is somebody calling themself a "sufferer" of the mental disorder "misophonia". To save you a Google search, basically it means that you become pretty annoyed or even enraged at noises like people chewing ice, people chewing food loudly in general, and et cetera. I too have this disorder (if you can call it that) and I did not relate to the word "sufferer" at all, so I disagreed with the usage (not verbally; I didn't want to start an argument). To elaborate even more, I will include another example. I have seen people call themselves a "sufferer" of a disorder called "visual snow". Basically, this disorder causes static to cover one's vision in a way almost relatable to TV static (not really). It's very hard to explain and it is rare, but I suggest you Google it because it is interesting. Once again, I too have this disorder and have had it for my entire life. Seeing people describe their experience as "suffering" almost baffle me. I only notice my static if I am in a situation that has solid color (like pitch black or looking at a whiteboard) or if I purposefully pay attention to the static. It never obtrudes life in any way and many who tolerate this disorder will agree that it is not obtrusive. With all of this I do not understand why one would say they are suffering from a disorder that does not inflict physical, mental, and/or social harm. Each side of health is arguable, as neither of these technically cause harm on either end (referring to a commenter, you becoming enraged and punching a wall is not direct pain from the disorder "misophonia). So my main question is, after seeing that Google defines the "dated" usage of the word as a "tolerator", would the word "sufferer" be acceptable in cases in which one does not necessarily "suffer" anything? I'll apologize that this question is really subjective and may not have any answer, but really I am just trying to get a consensus. Thank you in advance!
This question isn't answered here: Differentiate between past and present just by pronunciation when word is followed by d- or similiar sound That question asks about what happens when the following word begins with a consonant in general. This one is asking about the various possibilities for an intervocalic tap if a following word is an unstressed function word beginning with /t/. It's also asking whether there's any possibility for regressive assimilation here. I read this in American accent book. I quote the text exactly how it is written in the book: The suffix -ed is not pronounced precisely when it is linked to another consonant. For example, mailed the sounds very much like mail the in the following sentences: I already mailed the letter. I will mail the letter. The suffix -ed is not heard at all when it is linked to /t/ or /d/. For example talked to sounds identical to talk to in the following sentences: I talked to her yesterday. I talk to her every day. Okay but then my question is, how do Americans distinguish between these two sentences: I try to call you. I tried to call you. In the first sentence the /t/ will be realised by a voiced tap because it occurs between two vowels. In the second will we get a regular [t] and then a voiced tap? Will we get a double tap or a double length tap? Or will we just get a single tap? Or maybe here we will just get two regular /t/s? Do native speakers differentiate between "I tried to call you" and "I try to call you" from the context?
For the sake of time, is it bad to just accept some fast paced class's theorem's (such as MIT's algebra class) as true even if you don't completely understand the proof or can't remember the proof off top of your head (after a while has passed?). I often find myself wasting too much time trying to memorize proofs when that's not the point of the class (and I can actually wait to memorize the proof later). Sometimes, I get caught up in one detail of the proof for hours and end up not having time to learn how to actually use the theorem and do the homework. Also, is it bad to gain a complete and working understanding of something after you take the class. I am not mentally capable of fully absorbing both what the class want's us to get and thinking about it enough to have a complete and sufficiently deep understanding of the subject all in one semester. However, I feel bad if I wait to think deeply about the class material until after the course is done, but I simply don't have enough time to fully understand some things during the semester. I always hear this advice on making sure you understand everything when you are studying to practice/do mathematics, but that seems not practical if you are taking four or more classes and struggling to make sure you understand what you need to for your other classes have other obligations to attend to.(and also if English is not your first language). I feel like a lot of the advice I hear is for native English speakers (I came to the U.S. when I was four, so I'm practically a native English speaker, but not when it comes to understanding things well the first time through in math, or at least making the understand be thorough by a native English speaker's standards.) The reason I ask this is because I can't say I fully understand something in English unless I can explain the proof verbally/descriptively to someone.
We make an important distinction between the topological insulators (which are essentially uncorrelated band insulators, "with a twist") and topological order (which covers a variety of exotic properties in certain quantum many-body ground states). The topological insulators are clearly "topological" in the sense of the connectedness of the single particle Hilbert space for one electron; however they are not "robust" in the same way as topologically ordered matter. My question is this: Topological order is certainly the more general and intriguing situation, but the notion of "topology" seems actually less explicit than in the topological insulators. Is there an easy way to reconcile this? Perhaps a starting point might be, can we imagine a "topological insulator in Fock space"? Would such a beast have "long range entanglement" and "topological order"? Edit: While this has received very nice answers, I should maybe clarify what I'm looking for a bit; I'm aware of the "standard definitions" of (symmetry protected) topological insulators and topological order and why they are very different phenomena. However, if I'm talking to nonexperts, I can describe topological insulators as, more or less, "Berry phases can give rise to a nontrivial 'band geometry,' and analogous to Gauss-Bonnet there is a nice quantity calculable from this that characterizes instead the 'band topology' and this quantity is also physically measurable" and they seem quite happy with this. On the other hand, while the connection to something like Gauss-Bonnet might be clear for topological order in "TQFTs" or in the ground state degeneracy, these seem a bit formal. I think my favorite answer is the adiabatic continuity (or lack thereof) that Everett pointed out, but now that I'm thinking about it perhaps what I should have asked for is -- What are the geometric properties of states with topological order from which we could deduce the topological order with some kind of Chern number (but without starting from a Chern-Simons field theory and putting in the right one by hand ;) ). Is there anything like this?
I've been thinking about infrared radiation and noticing more and more how the human skin seems actually pretty sensitive to it. You can easily feel a bonfire from several meters away, far away from where any convection would heat your skin. When you open the hood of your car you can feel the heat from the engine even standing back a step or two (away from the updraft of hot air). Now try this: hold the palms of your hands against eachother a couple of inches apart and keep them like that for a couple of seconds. Then slowly (to avoid wind cooling) lift the other palm so they no longer face eachother. Do you feel it? For me there's a noticeable difference in warmth. Is that the skin detecting black body radiation from other skin? This could be easily blind-tested with a friend; you hold your palm out and look the other way, then see if you can correctly tell when your friend's palm is near you palm and when its not. Maybe the human skin is even able to detect black body radiation from another human standing behind her? Kind of like a sixth sense. Could explain the sensation of "i knew someone was there". I've noticed also that when you stand close to a concrete wall that was heated by the sun, but the sun has just set, you can tell which direction the wall is just from the heat on your body. Is this all placebo or does it actually work that way?
Has anyone actually developed a Program Synthesis system for creating computer programs automatically from a non-procedural specification that is taken from a fairly robust system of specifications? For example, I might say "Check if X is a factor of Y." or "List all prime numbers between X and Y." and out pops a program or two written in PHP that does just that. I would actually pay money for an example that shows that the answer is yes. Let's say you have relations for less than and multiplication, and you give a wff of logic. So it can be (exists A)MUL(A,X,Y) where MUL(A,B,C) iff A x B = C, for the first specification above. For the second specification with output of a number (the prime numbers) instead of TRUE or FALSE, we could say variable N means output all values of N such that the wff is TRUE. I have heard the claim that the answer is yes, but articles give no real examples of its being used. Either there is no example, or what they call an example merely displays a program and claims that it was or can be generated by some system that is imagined or even described in detail, but the example lacks any details at all. Can someone give a complete example of a system, a specification, the resulting program(s) and step-by-step exactly how that program is automatically constructed? And if an example is given, could it be a simple program with only a few steps to create, rather than dozens of pages of formulas that would take months to go through and is suspect of being obfuscation hiding the fact that it is not genuine?
I was recently in an argument with a friend who - equipped with an apparent understanding of the etymology of the words lend and borrow - insisted that to lend an object required not just the temporary exchange of its possession, but also a geographical displacement. He compared the words lend and borrow to take and bring which involve a transition to/from one's locus, which are apparently linked (though I can't independently verify) to our subject words grammatically. I argued that to so strongly stipulate (as he did) that an exchange was not a lending because the relocation was not significant (ie; it did not leave his personally defined location), is foolish given the complete generality / ambiguity of the word location. He insisted he could not lend me the salt-shaker as it traveled from his hand to mine because we were both common to the location 'this house', though I remarked that we sat in different locations; our respective chairs. After much debate, we concluded that to require lending to constitute a relocation is a poor definition, since there are (according to him) obvious stipulations as to the definition of the location. So... is any of this actually correct? Does lending actually have any requirement for a displacement of the lent object? If so, what exactly are these conditions (or rather, if they are as contextually defined as we both probably expect, what is a more appropriate word than 'relocation'?)? (I finally remarked that language is defined by the understanding of its speakers, and that I'd personally never heard a relocation was required in the lending of something. I also think the phrase "lend me your ears" doesn't beckon for their displacement...)
Rational numbers, rational functions, and Gaussian rationals are examples of fields of fractions. In each of those cases, one knows what the quotients are long before one hears of the idea of constructing the field of fractions of an integral domain. One case where one (typically??) does not know of such a thing in advance is the field of "convolution quotients"---the field of fractions of a ring of functions of a real variable in which the "multiplication" is convolution. But convolution quotients will not be appreciated by students who just finished a first-semester calculus course last week. Is there some example one could mention to such students where they wouldn't think they already know what is meant by division of the objects in question? Later edit suggested by answers and comments posted so far: I had in mind two or three purposes. One was that I wanted to mention this topic a bit obliquely in something the students are to read, and that had to be really terse, so I can't do anything really involved. Less than an hour after I posted the question, this ended up being a parenthetical comment on the course web site that said: "(for example, why is it that one can `divide' one divergent series by another?)". Here I had in mind the ring of formal power series suggested by Chris Eagle, but of course I needed to ruthlessly avoid mentioning power series. A second purpose concerned possible future uses. Not only in courses: if we get some good examples here, I'd like to add them to Wikipedia's article titled "field of fractions". A possible third purpose was just the satisfaction of knowing more than one decent example (since the only one mentioned above that's "decent" in the relevant sense is convolution quotients).
Assuming that antimatter is matter with time arrow reversed, would it be right to say that matter beyond black hole event horizon then would become antimatter because of space and time axes exchanged? Would not black hole then appear like a nice universe consisting from antimatter that slowly expands as matter falls into it? I do not claim anything just want to find out how wrong the idea is. Although, it is not really related to the question but I would like shortly explain where from my crazy idea that matter can indeed move to the opposite time direction is coming. I think that there was no Big Bang but initially was space filled with matter fluctuating back and forth in time (field fluctuating between matter and antimatter). Since there was no real matter - matter and antimatter fluctuated from vacuum and annihilated chaotically and hence there were no state transitions (movement) which we perceive as time - there was no time, in fact there was no matter either - just vacuum. But at some random event indicated as 'shortly after Big Bang', CP-symmetry got broken, which caused antimatter to disappear and gave rise to the time which we since perceive as going forward (if by some other event antimatter had won our time would go into opposite direction - but this does not mean 'back in time'). CP-violation caused universe expansion, but there was no initial rapid expansion, since universe did not arise from a singularity, but from a homogeneous space, which is in sync with recent Cosmic background radiation observations.
One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- (see below) &c&c, to be convinced that modified concepts are replete across maths, proliferating, and their diversity is likely accelerating. Shafarevich: "it is the destiny of mathematics to expand in all directions." This trend, coupled with the lack of standardized terminology, can make it difficult to compare results or in same cases even definitions. It seems clear that in general a modifier term doesn't categorically reveal whether the modified concept is a specialization or generalization of the underlying concept (eg, subset versus superset, or subcategory versus supercategory). In some cases the modified concept might not bear a sub/super relation to the underlying, for exmaple, co- and op- in category theory and universal algebra (what's the relationship of universal co-algebra to algebra or co-induction to induction?). So it appears we must be content with enumerating cases to discern the relation and then compare to see if a big picture emerges. Basic examples: Semigroups are generalizations of groups but inverse semigroups are specializations of semigroups. (Quasicrystals are crystals - this got the Nobel - but their symmetries don't satisfy the crystal restriction theorem, eg, translation invariance, so are not groups, but might be modeled by inverse semigroups [ML]). Quasimetrics are generalizations of metrics, but ultrametrics are specializations of the latter[VS] . Noncommutative geometry, Connes stresses, includes commutative geometry so it is a generalization. In the absence of an online OEIS-like database, would it be possible to crowd-source many more examples of mathematical concepts or categories noting sub/super (or other) relation to the underlying?
I think physicists can deal with this question best. I answered a question about "immortality" when some guy claimed I got it wrong that neurons die (I argued that even if you live a billion years you die slowly many times over because all your cells incl. nervous cells will have to be replaced, meaning you will regularly - but never suddenly - become a new person; also: continuous learning, and information storage is physical, and our brain has very limited capacity so memories will change), immortal means they are immortal. That got me thinking: Is it actually possible to have repair systems in each cell, or isn't it better to have a repair system on a much higher macro level and cell-level immortality is actually impossible? Background: The smaller the scale, the more events like quantum effects and Boltzman (energy) distribution (e.g. random atom movements, crystal structure defects, breaking bonds) dominate. Meaning small systems WILL break in unpredictable ways, so only large systems can live long and have reliable repair systems because the larger the system the less important those effects become. Summary: It is much better (or even possible) to have big long-living systems than small ones, because the inevitable repair system cannot be too small because it will be subject to random events of the micro-world. An "immortal" human would have a body-level repair system that repairs by letting broken cells die (we have that already) and build a new one, instead of having a repair system on the cell-level that would make cells immortal (too unreliable, won't work). Or in other words: The smaller the system the less likely a long life can be achieved. Note that I'm asking about complex dynamic systems, not static objects or information. Like living organisms or (complicated) machines, and about their chance to "live long".