darklight03/StackMIAsub huggingface dataset:
In Slavery in Massachusetts, Thoreau writes: But it chanced the other day that I scented a white water-lily, and a season I had waited for had arrived. It is the emblem of purity. It bursts up so pure and fair to the eye, and so sweet to the scent, as if to show us what purity and sweetness reside in, and can be extracted from, the slime and muck of earth. I think I have plucked the first one that has opened for a mile. What does for a mile mean in this sentence? I looked up for in a dictionary, and methinks the author might mean the first lily that he came across, and that was opened, after walking a mile. Or does Thoreau mean something else here?
Shouldn't acoustic oscillations created by primordial matter anisotropies create multiple peaks and troughs when those oscillations are frozen by decoupling ? It's not unthinkable to imagine that the exact calculations of the expected amplitudes of each will lead to a result where the first peak is much more significant than the rest, but do those other peaks and troughs theoretically exist, just at very small currently non measurable amplitudes ? In other words, the BAO peak corresponds to the first peak in the CMB power spectrum, but am I wrong in assuming the other peaks and troughs are theoretically also present in present matter distribution, just at negligible amplitude ? As a bonus, if the answer is indeed yes, why are the other peaks so negligible ? I understand that the first one has the greatest amplitude, but why are the remaining ones so vanishingly small as to remain undetected in current matter distribution, despite being nowhere near insignificant in the CMB power spectrum ?
A current-carrying loop of wire is placed in a uniform external magnetic field as shown. If the current in the wire is traveling counterclockwise in the picture, what do you predict the loop will do when released? The answer provided for this question: From the RHR for loops, place the heel of the right hand in the plane of the loop so that the fingers are curled in the direction of the current. The extended thumb points in the direction of the B-field. Since it is anti-parallel to the external B-field, the loop will flip over and then remain at rest. I don't understand what causes that. When you apply the Right-Hand Rule or Flaming Left-hand Rule I prefer, the force induced is inward around the loop. How come it makes it flip over? I must be missing something deep here. What is the underlying concept that causes the wire to flip over?
If I take a pair of classical particles with some energy interacting via Lennard-Jones potential, then the motion of the two particles will be in such a way that the distance oscillates around the equilibrium point (if the energy is not too high compared to barrier depth). If I consider a large number of Lennard Jones particles packed into a crystalline lattice, then intuitively I expect that each pair of particles should have a distance close to the equilibrium distance of the pair potential. In other words the "lattice constant" should be close to the equilibrium of the pair potential. But is it actually the case ? Can we consider a pair of neighbors in a Lennard-Jones lattice to be roughly an equilibrium of the pair potential with small corrections of further particles ?
The situation: I'm trying to generate my resume with Latex, and get a pdf with all the text - no problems here. When I copy all of this text into a plain .txt file though, the formatting transforms all whitespaces (newlines etc.) into a single space, it seems. For example, lists are not vertically placed anymore, but placed sequentially. Does this maybe have something to do with how Latex files are built up (compiled) differently than ordinary text documents, causing it not to have any classical newlines characters, for example? In that case, would there be some way to insert those, allowing for a proper copy-and-paste functionality? I tried to make use of the pachage cmap, which assists with Unicode mapping, but could not get it to work. Anyone who knows a fix to this, possibly with cmap, or possibly another package?
I'm working on performing a linear stability analysis on a system with fluid undergoing applied vibrations. Now, usually these systems are analyzed with no fluid velocity. Under these conditions, it can be shown that the system reduces to the form of a Mathieu equation (or a modified Mathieu equation in the case of viscous fluid). For Mathieu equations, it's known that there are discrete regions where the system response is either Harmonic or Subharmonic resonance, which is in line with the Faraday instability that's observed on the fluid interface. However, what happens when the system equations cannot be reduced to the form of Mathieu equations (for example, for complex fluids or fluids with flow)? Can the system allow resonance at frequencies outside of Harmonic and Subharmonic, or is there something more fundamental that says that these are the only allowable options? Should I also consider the possibility that the response might not even be periodic? Analysis procedure for inviscid Newtonian fluids Analysis procedure for viscous Newtonian fluids
All explanations of jet propulsion that I've seen are formulated as "due to conservation of momentum, air with momentum coming out of one end means the rocket must gain momentum in the opposing direction". However it can't be the case that particles simply exiting some area with a momentum means all of a sudden something else must gain opposing momentum. If we had a straight pipe open on both ends, particles suddenly exiting one side (maybe a chemical exploding inside or the like) does not mean the pipe will all of a sudden start moving in the opposing direction. You need one side of the pipe to be closed in order for the pipe to move opposingly. Similarly it can't be the case that particles suddenly exiting an inflated balloon or rocket means the object must propel in the opposing direction due to some 'rule'. I'm assuming what is actually happening is a case of 'particle collision averages'. In the case of a balloon the air inside is constantly bouncing off the walls in all direction. So when I let go of an inflated balloon I have disturbed this average. There are now particles hitting a side of the balloon with no opposing force, as the particles opposing that wall are now simply exiting the balloon. Assuming my above formulation is correct, does this mean all statements regarding conservation of momentum are simply a rule of thumb for this above 'particle collisions average' explanation?
I was trying to find some references for modelling the equations of motion of a simple dynamical system (say a pendulum on a moving mass) when I realized that the very vast majority of the material you find online or even in textbooks suffer from the following problems: Reference frame poorly defined, wildly assigned or even implicit. No idea which point is considered to be origin, which direction positive, etc. Free body diagram completely unprincipled and/or confusing. Confusion between magnitude of a vector (e.g., Force) and the vector itself (not mentioning switching back and forth) No general form for basic laws of motion. Sometimes when the authors apply Newton's second law, for instance, they are applying it implicitly along some dimension. Constants are stated and used without derivation or motivation, e.g., moment of inertia are mysterious constants that you would look up from a table And many more problems. Does there exist any textbook on modeling classical dynamical systems (pendulum, masses, multi-link robot) from scratch?
While reading the Wikipedia article on infinite sets I found the following quote: A set is infinite if and only if for every natural number, the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset. This raised an interesting question; without AC, is it possible to have a set where, given any natural number, you can find a subset with cardinality great than that number, but not necessary one equal to it? For example a set where all definable subsets have even cardinality? Edit to clarify the question: are there sets that satisfy the intuitive meaning of infinat but not the definition as quoted above? FWIW, id be interested in both the case where "subsets larger than n exist" requires those subsets be finite and where they don't. (Though I suspect the second case is uninteresting as most of the interesting properties would be trivially true.)
Let's say the origin is an equilibrium position for a particle. If we have a case such that a slight displacement of the particle in x direction makes it return to the mean position (stable equilibrium) and a slight displacement of the particle in y/z directions makes it go further away from the mean position (unstable equilibrium). The directions of displacement for which particle is in unstable equilibrium are more than directions of displacement for which particle is in stable equilibrium. I have seen this question but the answer didn't clearly state whether there will be an equilibrium type defined for that position or will that position have no type of equilibrium (and the comments on that answer about Unstable equilibrium "winning" over Stable equilibrium confused me even more). So what will we actually call the equilibrium type at the origin? Or will we just define its equilibrium associated with direction (like stable in x direction and such)?
My understanding from popular science articles is that the boundary is a field theory that has no gravity and has one less spatial dimension than the bulk. However, I am not sure I understood this picture correctly. I just read a recent article at quanta magazine that states: "A solar system in the central anti-de Sitter region, for instance, can be described as a collection of particles scattered around the boundary that obey only quantum theory and have no sense of gravity or space-time at all". So my question is, did the quote mean no curved spacetime, or there is not even a minkowski spacetime associated with the physics in the boundary? I dont even understand how it is possible to have a quantum theory of particles without either space or time.
I am trying to understand Turing's halting problem proof by applying the same paradox to a similar problem where, instead of determining if a given code will halt, you instead determine if it will return True or False (assuming it will always return eventually). Obviously such a machine can exist, as it is simply a standard compiler. However, when run through the paradox: We assume h is a function that determines if the return is True or False We construct a machine P around h in the following way: def P(i): return not h(i, i) P(P) As we can see, if P returns True then P returns False, and vice versa. Thus compilers cannot exist, as a compiler is an example of h. Can someone please help me find the error in reasoning?
According to this and this answer, and as far as I understand these answers, dark matter halos cannot collapse to a black hole because, due to uncoupling from the EM field, they are unable to radiate their kinetic energy, and hence, getting closer to some gravitational center point also means that they get faster and so they resist further "collapse". But what about the motion perpendicular to the galactic plane? I would naively expect the dark matter to gravitationally fall down on the galactic plane on both its sides, until it concentrates there. Depending on whatever the type of dark matter may be, this may cause other forces (e.g. weak interaction) to take over (possibly at nuclear distances) or it may oscillate until it becomes spherical again. One way I imagine that this might happen is a small-scale alternating velocity variation in the dark matter field, so that dark matter is alternately falling to/moving away from the mid plane from location to location, and these regions simply pass by one another infinitely. If this alternating pattern in the velocity field exactly balances, the dark matter halo is able to maintain a spherical shape. But even the slightest imbalance might result in a global oscillatory motion between spherical and disc-like. Have the available observations been investigated with respect to these possible oscillations of the dark matter halos? And isn't it likely that such oscillatory motion (if it existed) would eventually stop due to second order dissipation (dark-matter to ordinary matter to radiation).
I am working in an algorithm to order a bed of close-packed spheres. In the case where I have got four spheres, I understand that the fifth sphere position and radius is determined by the positions and radii of the four other spheres. It seems that there would be different solutions: one that produces a sphere that is similar in size to the other four, and a second solution in which the four spheres are encompassed in a bigger sphere. What interests me is the first solution. Any ideas what equation system solves this? The input data for the systems would be the positions of the original four sphere centres and their respective radii, and the output would be the position and radius of the fifth sphere. It is also important to notice that the four original spheres can but do not need to be tangent between them. Thanks a lot in advance!
From what I understand, each particle has an energy called kinetic energy. When we consider a system of particles, in addition to their individual kinetic energy, there is an energy associated with the systems which depends on their relative configuration which we call the potential energy. Mathematically a system of particles can be analysed at the individual particle level by assuming that system of particles induces a field and associate potential energy for each particle and then sum up over particles to get the potential energy for the system of particles. So I thought that potential energy is some sort of energy associated with the field (which is a mathematical construct). But later I studied that fields are real and they have their own energy. So is the potential energy the same as field energy, if not how are they related? Additionally, just as the energy of a particle is decomposed as potential and kinetic energy, can we do the same for the field, i.e., potential field energy and kinetic field energy? If so, then how is the potential energy of the particle related to the potential energy of the field? Also, potential energy in my mind is related to the shape a system of particles takes. Changing shape releases or absorbs energy. Entropy is a concept which is also related to the structure/order of the system. So how does the entropy relate to the potential energy?
Can we answer the question: "What are all the types?" i.e. can we recursively generate all the types in a given type theory according to a certain set of rules? Pardon for asking such a weird question. Question/personal background: I'm leaving this question intentionally vague with regards to what I mean by "type theory" because I guess I'm just interested in the status of this question for type theories in general. My background is pure math and I am comfortable with proofs in ZFC and FOL but I can't write a single line of code in any programming language. I hope to understandthe basics of using type theory for proof like Lean does, and started reading about type theory yesterday in preparation, although I've heard snippets about type theory in passing elsewhere.
Let's say you're looking at your keyboard on your phone.. let's take the letter g. Light from g is going in every direction.. we know this because we can see g from all angles above the phone. So light from g is also hitting your eye where you would see all the other keys,screen and background objects however we don't see g in those areas.. just wherever g is located. What happened to the images of g everywhere else but where we see it? We should be seeing g in all places but something happened and we don't.. if you're getting what I'm saying I you should be as perplexed as I am.. also knowing this now think about how a camera can take a picture with light from all objects being scattered in all directions.. it shouldn't be able to take a picture like it does.. how on earth was it even made???? It shouldn't be able to work but it does!!! Does this prove something else about life?
In the same vein/spirit as the UIPAC books for standardising systematic names and formulae of chemicals, the SI units for units of physical measurements, or the INN for generic drug names, have there been efforts to write every statement in mathematics (eg. axiom, postulate, result, theorem) in a standardised language? This standardised language might be a formal logic like those touched upon in mathematical logic textbooks or monographs, or it might simply be a set of standardised terms or statements in a natural language like English. (I'd like to emphasise that a standardisation effort need not necessarily be one in a formal language.) I'd like to know if there existed, or there exists currently, a consortium, an organisation, an initiative etc. like this.. I tried searching for this on google but couldn't find one so far... Ermm, do put aside the debate on whether such an effort is desirable or not.. I just like to know if there is/was such an effort, not whether it is a good/bad thing.. I've come across TPTP.org, but not sure if this is the largest/most well-known/most followed/most active effort so far in mathematics.. Do let me know if there's any better..
In google translate, the word "bizarre" means "very strange or unusual, especially so as to cause interest or amusement.". But I believe that this description is more suited for "eccentric". For example, the character of Jack Sparrow from Pirates of the Caribbean is eccentric. On the other hand, AI chat GPT says that "bizarre" usually means something odd and unsettling. And I have seen people make use of the word "bizarre" for criticizing something, rather than an expression of fascination. In Meriam-Webster site, "bizarre" means odd or eccentric in stye. This means "bizarre" and "eccentric" share the same meaning? But "Jack Sparrow is such a bizarre person" and "Jack Sparrow is such an eccentric person" don't seem to resonate. I got different vibes from them. Or is it just my feeling? Both means "unconventional"? What are their subtle differences, and can I have one example for each that would enlighten me about their differences?
Almost every text on Category theory uses categories such as Ab, Grp, and so on as examples to work with but can category theoretic methods actually help us understand the structures better? In particular, does Category Theory aid us in proving some significant abstract algebraic results that are otherwise tedious to prove? Furthermore, is there any structural insight about abstract algebraic objects that is not readily apparent from algebra itself but becomes crystal clear with a category theoretic approach? I do not have a specific format for the answer in mind. As category theory is quite general, the answer may also include ways to say use results from other branches such as analysis or topology, so as to prove an abstract algebraic result which was tedious to prove using an algebraic approach. In this light, any insight about the relationship between category theory and abstract algebra is welcome.
I find it amusing why the infimum norm or supremum norm defined on function spaces work actually? It's difficult for me to reach to an intuition where I would see the infimum or supremum as a natural way to represent distance between two functions. Is there any intuition which can motivate the norms defined on function spaces ? How do we make sure that two functions having the same infimum or supremum don't deviate too much from each other and that the function which is a limit under such a norm satisfies the properties the functions in the sequence have ? What is the norm measuring in function spaces I went thorugh the above question but could not find an intuitive way to address the last question in the first paragraph of my post . Disclaimer : I am not a pure maths student , please pardon my immaturities . I was just trying to find an intuition here.
Im working on a nonlinear control based on Lyapunov theory and its working really well. I am able to implement it on a dynamical model of the system in simulink. However I think it has a really big limitation and that is: I cant guarantee that the real system will have the exact parameters that I am modelling. I want to make my lyapunov controller into a more robust controller by adding parameter estimation too it. I see some texts based around system identification, but I do not need to go that deep, I know what the system looks like. I just need some kind of method to zero in on the drift between my modelled and real components. Is this possible? I am controlling an DC to AC power converter. Physical System model: Model: Controller:
Often metaphors are likenesses where there's a direct connection. For example on the news somebody describes a crash/ earthquake/ explosion as It was like a bomb going off. What about where the two things are not similar. Her painted toe-nails were an explosion of colour. (Exploding feet anyone!) This poem Cherry Ripe by Thomas Campion is full of insinuations rather than direct likenesses. There is a garden in her face Where roses and white lilies blow; A heavenly paradise is that place, Wherein all pleasant fruits do flow: There cherries grow which none may buy Till 'Cherry-ripe' themselves do cry. Those cherries fairly do enclose Of orient pearl a double row, Which when her lovely laughter shows, They look like rose-buds fill'd with snow; Yet them nor peer nor prince can buy Till 'Cherry-ripe' themselves do cry. Her eyes like angels watch them still; Her brows like bended bows do stand, Threat'ning with piercing frowns to kill All that attempt with eye or hand Those sacred cherries to come nigh, Till 'Cherry-ripe' themselves do cry. My question is: Are there technical terms that distinguish between: (a) Direct likeness. eg sky-blue eyes (b) Not really like at all. eg eyes like angels (c) Over-arching 'big metaphor'. eg garden in her face
I am using TeXstudio with LaTex on Windows I also know about Git. In fact, I use Git to collaborate on RStudio but on TeXstudio I cannot. While typing on TeXstudio on my own without collaboration, I had my first problem saving different versions of .txt files on the same project, which sometimes confuses me. When I was writing my PhD thesis, I would send the PDF to my professor, which he could not edit. How do I solve these two problems? WHAT I WANT I have read different posts on reproducibility on TeXstudio, but I cannot figure out my foot on any. I want a step-by-step guide on achieving reproducibility and collaboration on TeXstudio using Git. I want to be able to revert to my previous changes without saving multiple documents.
Goldblatt gives a brief overview of adjunctions in his "Topoi", and one of the exercises asks to characterise the partial arrow classifier in terms of some universal arrow. Well, I gave it some thought, and I'm not even sure where to go from there. The main problem I have is that in the "there exists" part I have only one arrow (the PAC itself), and in the "such that for every" part I'm given two arrows (the "top-left" components of the PAC pullback), and I'm just not sure how to construct a category and a functor into/from it that'd allow for both. I tried something along the lines of the diagonalization functor used for (co)products derivation via adjoints, but it didn't get me far (or anywhere, frankly). So, how would I do that? Again, my exposition to adjoints is limited to several pages of "Topoi", so I likely miss some otherwise well-known results.