darklight03/StackMIAsub huggingface dataset:
In analyzing Compton scattering we consider the conservation of both energy and momentum. However, in analyzing the photoelectric effect only the conservation of energy is taken into account. In fact, if the momentum of the photon is taken into account there seems to be a violation of the law of conservation of momentum. The momentum of the photon is into the metal but the momentum of the ejected electron is out of the metal i.e. the momentum of the ejected electron has a component opposite to the momentum of the incident photon. A way out would be to argue that the process of ejection is complicated and that it is actually the atom which ejects the electron. There is, however, a problem here. Experiments with linearly polarized x-rays or ultraviolet light show that the electron is always ejected in a direction parallel to the electric vector i.e. perpendicular to the direction of the incident beam of photons. This suggests that it is indeed the photon which ejects the electron, not the atom. Is there an explanation in either quantum mechanics or classical electromagnetism or both?
I'm not sure if this is an appropriate question for here, so fair enough if it gets closed/down-voted. I'm self-studying mathematics, did basic real analysis (Riemann, not Lebesgue integraton) and complex analysis, and I'd like to move on to functional analysis shortly. I have two books: Muscat's Functional Analysis (Universitext), and Axler's Measure, Integration and Real Analysis (GTM). It seems to me that Muscat is rigorous but still lighter (not meant in a bad way) on the maths, but covers more topics than Axler. But Axler spends a lot of time on measure theory (of course) and does also cover the basics of Hilbert spaces and Banach spaces and linear operators on Hilbert spaces. At this moment I'm more inclined to start with Axler, since I suspect after covering Axler I'll have a stronger foundation and more easily pick up on the topics covered in Muscat but not in Axler. Would this be wise? Other recommended books welcome.
I was playing with numbers in my middle school math club and found a beautiful pattern. I presented my idea in front of my club mates. The teacher was impressed with my result and suggested I write a short manuscript and have it published somewhere. I thought it was a great idea so I agreed. It took over six months to get used to latex but I think it was worth it. I had some previous experience with the Lua language so I chose LuaLaTeX for my document. It was a short one page article but I was still satisfied with it. I searched for journals that had writings similar to my skill level. I sent them my LuaLaTeX file and the pdf file resulting from it via email. They reacted positively to the content but asked me to write in plain latex because they do not use LuaLaTeX. It took a few more weeks to convert to LaTeX. I have never maintained a journal so I didn't know that it wasn't as simple as just ordering the articles in the right order then gluing them together. Will I face similar barriers in high school, university and beyond if I keep using LuaLaTeX and not LaTeX? Are engines that extend beyond LaTeX such as LuaLaTeX and XeLaTeX to be avoided in environments where I do not produce all the content of the publication?
I encountered several times a certain type of sentences (in colloquial contexts) which were clearly grammatically incorrect but seems to be widely spread and, as a non-native English speaker, I would have liked to have more information about that. It is about the conjugation of the modal verb to be. I have heard many times the following type of sentences: "We was playing basketball" "They was eating dinner" "I were there" "He were with me" where the last two were used without any conditional of any sort. These sentences are obviously wrong in standard English (as far as I know) but still seem widely used in everyday life conversations. Therefore, my question was the following: How wrong are these grammatical structures ? How strange will I sound if I use them ? Also, is it some kind of dialect/slang and in what kind of context do we use them? Unfortunately I do not have any neat example to provide for I've heard it in an everyday life conversation with my office mates who are from Scotland. (I did not dare to ask them directly as I barely understand when they speak) If someone could provide me more information about them I would be grateful.
Imagine an air hockey table where there is a puck P and a rectangular slab S. Both are free to move as there is zero friction. The slab is at rest in the middle of the table. The puck is moving towards the slab and collides with it in an elastic collision. The puck is not spinning before the collision. As this is a closed system, the total momentum, angular momentum and energy of the system should be preserved. But after the collision the slab will be spinning about its centre of mass, but does that mean the puck will also be spinning in order to conserve the angular momentum of the system? I feel like the elastic (frictionless) collision won't be able to impart any spin on the puck. I understand that the puck and the slab will also change their linear momentum and energy but overall in the system these will be conserved. How would you calculate the velocity vectors and angular velocity for the slab and the puck post-collision? Assume the puck and slab are made of the same uniform-density material and the puck has radius r and the slab side lengths a and b.
I do not feel comfortable with such constructs that contain "one's work ...ing" as the following: "University of British Columbia marine biologist Amanda Vincent has won the prestigious Indianapolis Prize for her work protecting seahorses." "Learn more about her work protecting the rights of women and girls." No matter how I feel about it, there is no denying that it is done and it is fairly commonplace, so I am not going to say it is wrong. However, it does not sit well with me. How would you explain this phrase grammatically? Is "protecting" a present participle? If so, construct wise, "her work" will have to be the subject of "protecting," but meaning wise, it is clearly wrong; the assumed subject is she. The only explanation I can come up with is that this is an idiomatic expression. Maybe "protecting ..." is a gerund and it is an appositive to the immediately preceding "her work"? Your input will be much appreciated!
I know that we can consider an object as point object, if its size is negligible as compared to distance traveled by it in reasonable amount of time. But in my book Ncert there is questions which asks to determine which of the following are point objects: (a) a railway carriage moving without jerks between two stations. (b) a monkey sitting on top of a man cycling smoothly on a circular track. (c) a spinning cricket ball that turns sharply on hitting the ground. (d) a tumbling beaker that has slipped off the edge of a table It states that (a) and (b) can be considered as point objects but (c) and (d) cannot. Why cannot we consider them as point object if we do not the distance they have travelled and their size?
If we place a conductor between the plates of a capacitor, the conductor reaches an electrostatic equilibrium with the surrounding electric field. At this equilibrium state, the charges within the conductor have redistributed such that the electric field inside the conductor is nullified. Now, what happens if we separate this conductor into two halves, each containing the redistributed charges corresponding to one side of the conductor (meaning we have one half that is positively charged and one half that is negatively charged). We did not take the conductors out of the field. Would there be an electric field between the separated halves of the conductor if we look at the complete system (Capacitor and its field and conductors and their interaction)? My book says there is no field because it would cancel out the external field of the capacitor but I always thought that electric field lines always end on charges, which would not make a superposition possible.
I am working on an English-language online resource. It seems an obvious good idea to allow users to choose a version in British English or American English spelling. However, I've noticed that spell-checkers also provide options such as Canadian English, Australian English, South-African English, Jamaican English, Hong Kong English, and so on. I have always assumed that these variants are all basically British or American, with maybe a few minor details that are different. So my question is: Are any of the other variants of English spelling significantly different from British or American spelling? With significantly different, I mean that they don't just add a number of new words, like "wee" or "bairn" in Scotland, or French loan-words in Canada, but that basic English words are spelled differently, in the way that e.g. "colour/color", "analyse/analyze" and "theatre/theater" are spelled differently in the UK/USA. (To be clear: I'm only interested in how great the difference is to an outside observer, not how important it is to e.g. a Canadian person to be able to select "Canadian English" instead of having to select "British English". I'm also not interested in differences in grammar.)
Physicist Grigory Volovik has put forward some ideas about the universe undergoing a topological phase transition (especially in the early stages of the universe). He published a book called "The Universe in a Helium Droplet" where he explained his ideas. You can find a brief discussion about it here. In one discussion I had with Mr. Volovik, he mentioned that depending on the type of topological phase transition that could have occurred in the universe, all the fundamental symmetries of the universe (spacetime symmetries, translation symmetries, CPT invariance, internal invariances...) could be all emergent from a more fundamental state without symmetries (like in Holger Nielsen's random dynamics proposal where all symmetries in the universe would be emergent) I asked him if this was all speculation or if there was some truth behind and he replied that although we don't know if the universe actually took this "path", we know that this topological phase transition would be possible. But is this true? Would that be possible according to what we currently know about physics (although we don't know if this actually occurred at some point of the universe's history)? Or, on the contrary, we don't even know if these transitions are even possible to begin with?
There are examples in physics in which a simple law results from an immeasurably more complicated set of underlying interactions. Consider Hooke's law, for instance: there is a very simple equation that relates the extension of a spring to the force required to extend it further, yet the underlying physics when considered at the level of the trillions of individual electrons and ions that form the spring is of an entirely different order of complexity. Is it possible that we need to find some new theory to replace quantum mechanics which is as different to quantum mechanics as quantum mechanics is to Hooke's law, or are there any considerations that limit the additional complexity we might encounter in a more fundamental theory. For instance, our current model of reality assumes a set of fundamental particles, and there have been attempts to model the particles as vibration modes of strings. Might it be that strings are themselves composite entities composed of countless smaller parts- in the way that springs are composed of atoms- or do we have firm physical grounds to suppose that there is a fundamental limit to the divisibility of matter which prevents strings, for example, from being composit entities at a much more granular level.
Gravitational waves carry energy. The sticky bead argument shows that this energy can be extracted: https://en.wikipedia.org/wiki/Sticky_bead_argument But Lee Smolin points out that "In principle, nothing can screen out the force of gravity or stop the propagation of gravitational waves, so nothing can be perfectly isolated. I discovered this important point during my PhD studies. I wanted to model a box that contained gravitational waves bouncing back and forth inside, but my models kept failing, because the gravitational waves passed right through the walls". Gravitational shielding is considered to be a violation of the equivalence principle and therefore inconsistent with both Newtonian theory and general relativity, and there is no experimental evidence for it. If energy from gravitational waves can be extracted, does this mean that partial gravitational shielding exists?
I read this from Div, Grad, Curl and All That: The second reason for introducing the electrostatic field is more basic. It turns out that all classical electromagnetic theory can be codified in terms of four equations, called Maxwell's equations, which relate fields (electric and magnetic) to each other and to the charges and currents which produce them. Thus, electromagnetism is a field theory and the electric field ultimately plays a role and assumes an importance which far transcends its simple elementary definition as "force per unit charge". The first reason was basically about how finding the field first, then finding the net force on a charge due to the net field simplifies calculations, but I don't understand the second reason. What is this fundamental importance of an electric field that the author is referring to? (PS, I am unfamiliar with Maxwell's equations and classical field theory)
I am currently writing the conclusions of my bachelor's thesis on convergence spaces and there are a couple of points I would like to make, but lack the proper references to cite in order to do so. The first point I would like to make is that one of the starting points of General Topology was trying to axiomatize the notion of convergent sequences and hence the notion of a convergence space is much closer to the origins of topology. I recall reading somewhere that the first attempts of defining topological spaces were like that, but I can't find where. The second point is that the category of convergence spaces is much more adequate than Top, because it has exponential objects. I know there is a whole discussion of the importance of having cartesian closed categories of spaces, but I by myself cannot argument that, because my background on category theory is really small. Therefore, I need to cite someone who knows that for a fact and can give reasons why that is. Hopefully I am not breaking any rules by asking two different sources in the same post. I thank gratefully for any answer.
Crossposted on MathOverflow I am an undergraduate mathematics student with a keen interest in pursuing research in the formalization of natural languages (from a more mathematical-logical approach), yet there aren't many resources that provide an overview of this very technical field. I wish to be able to provide myself a general map to navigate this subject more clearly, so I will attempt explain my basic understanding of an overview of the fields that tackle with this subject, and I'd greatly appreciate any correction or addition to my limited insight. From my understanding, (loosely speaking) there are two prominent mathematical-logical approaches in formalizing natural languages: Categorial Grammar and Montague Semantics. While Categorial Grammar uses methods borrowed from category theory to (mainly) study the syntax of natural languages, Montague Semantics, as the name might suggest, (mainly) focuses on the semantics of natural languages by implementing methods from Lambda Calculus. In terms of subareas of each of these two fields, I have not seen much discussed in terms of the subareas of Montague Semantics; however, Categorial Grammar seems ripe with subareas (although I have heard some of the fields mentioned below are only closely related to Categorial Grammar rather than being a strict subfield of it): Combinatory Categorial Grammar Lambek Calculus Type-Logical Grammar Pre-Group Grammar Proof-Theoretic Semantics In addition to any corrections or additions, I would greatly appreciate any suggestions for resources, references or books that deal with these subjects or their prerequisites
It seems that there are many terms in linear algebra that have multiple names. For example, unitary and orthogonal both refer to the same general idea, a Hermitian is essentially a self-adjoint matrix, invertible and nonsingular, and there are definitely more that I can't think of off the top of my head. I've noticed that I've seen terms like orthogonal and self-adjoint in more classes/texts that feel like they consider linear algebra from a more abstract algebraic perspective, while I've seen terms like unitary and Hermitian in physics and more applied settings. I was wondering if there was some kind of history behind these terms? Why do we have multiple terms for the same thing? Is it just a coincidence that I've seen Hermitian and unitary in these kinds of settings, or was the subject simply considered with different motivations by different people studying different things? If this is the case, I would love it if anyone could suggest references for where these terms originated, and also if there are multiple motivations/perspectives for linear algebra, are there any references for the origins of these different motivations/perspectives? I recall one of my physics professors mentioning briefly that mathematicians and physicists had independently developed the same theory only to realize later that they had been working on the same thing all along. Is there anywhere I could learn more about that history? And are there fields other than linear algebra where this has happened?
I'm a current MA student doing research in formal semantics, which is an application of, among other things, logic and model theory to the study of the semantics of natural languages. I'd like to build up a stronger foundation in formal logic before tackling other topics / projects. I love the interplay between logic, algebra, and topology. Model theory is very interesting to me, and I'm keen to learn more on that. I'm keen on extensions of first-order logic, as well (higher order logic, modal logic, etc). I also have Formal Semantics and Logic, by van Fraassen, and I'd like that text to be more accessible to me. There's so many introductory texts to logic, and a lot of them seem to spend time on things like deductive systems and such, which I'm not very interested in. Given the somewhat eclectic list I gave above, are there any logic texts that would be a good match?
I have a conceptual question about graphs which I couldn't find the answer to. I am calculating some node centralities and using them as features for a machine learning problem. I am using Networkx python library. I noticed that for the degree centrality the library does the weighting of the values by the highest possible number of connections. In other words, the degree centrality of a node is defined by the number of connections the node has divided by the number of nodes in the graph minus one. Although, from graph theory literature, the degree centrality of a node is simply counting the number of edges a node has. As far as I understand. I wonder what is the implications of this weighting of the values by the library? Doesn't it completely changes the concept of the degree centrality?
I'm currently writing my master thesis on "Differentiable Stacks". I'm really fascinated by the idea of generalizing manifolds to include also orbifolds/leaf spaces of foliations/moduli spaces... I formally understand the construction of differentiable stacks as stacks on the category of smooth manifolds possessing a representable epimorphism. And I formally understand why they generalize smooth manifolds. What I'm having some trouble to get is: how does one come up with such a definition? I really read many sources on the topic and none of them justifies the choice of such a definition for differentiable stacks (I mean the existence of a representable epimorphism aka an atlas). This "intuition problem" is getting better as I familiarize more with this object but I feel like I'm just "getting used to it". I'd like to know how this object was born and what's the idea behind it.
According the classical physics, the electron should radiate energy and fall to the nucleus in a short period of time. However, this was not the case. Hence, Bohr proposed his theory, suggesting that electrons existed in specific orbits, where they did not radiate energy. These orbits had quantised or discrete energies. Moving between these orbitals meant the emission of specific amounts of energy, emission of photons. However what was so revolutionary behind this idea? It seems to me, he solved this radiation problem, by simply stating that it didn't happen, "electrons exist in specific obrits, where they don't radiate energy". Did Bohr know why this actually happened or did he just state it? Currently, I am struggling to understand what was so revolutionary about Bohr's contribution to the previous nuclear model of the atom.
There was an open dump yard a few miles away from where I used to live for an internship. It was not noticeable during the daytime, but once the sun sets, the dump yard reminded us of its presence through its stinking odor. The smell came every evening and left the following day as if on a schedule, and my colleagues shared the same experience. One of my friends said it might be because of the temperature, but that didn't make sense. Smells, or the gases that cause those smells, travel faster at higher temperatures, which is why food has a more intense smell when served hot, as opposed to when taken out of a fridge. That would mean that we would get the smell during the day, and not during the night, which is the exact opposite of what is happening in reality. What might be the reason then, why we can smell the dump yard at night, and not during the day? P.S: There is no activity going on in the dump yard, apart from people throwing garbage during random intervals throughout the day. It used to be a barren land, and at some point in history became the nucleation site of the garbage of an entire city.
We know that Carleman's condition is a sufficient condition for the determinacy of Hamburger moment problem and the Stieltje's moment problem. The first one look at measures on the real line, and the second one look at measures on the positive side of the real line. There is a third problem called Hausdorff moment problem that looks at measures on the bounded interval. The interesting thing about this problem is that if the measure that fits moments exists, it is also unique. My question: Is Carleman's condition also sufficient for the determinacy of the Hausdorff moment problem? I figure that it should be since Hausdorff moment problem looks at smaller set of distribution than the Hamburger moment problem. I looked around but I couldn't really find this anywhere. I just want to makesure that I am not missing something subtle.
I am simulating a simple Lennard-Jones fluid confined between two fixed walls and I am analyzing the autocorrelation function of the velocity along the direction of the confinement (normal to the walls). I observe an exponential like decay which is in line with the rough expectation, but in addition I observe periodic peaks. These peaks turn out to be the sound modes reflected back and forth between the two walls, as the time between the peaks increases when I increase the walls distance. And when I calculate the sound velocity for my system, it turns out to be equal to the walls distance divided by the time between the peaks. Now I know about the sound mode, I am not satisfied with just having the velocity autocorrelation for looking at the sound modes. I would like to visualize the mode with more geometric features such as the velocity field, as I expect rich structures. Of course the problem is that in such equilibrium simulations a snapshot of the particle velocities is just a random pattern, so some kind of averaging over the particle trajectories and velocities should involved to make the sound mode visible. Any suggestion ?