darklight03/StackMIAsub huggingface dataset:
I am trying to check if my understanding is correct. This is not really a question but a request for validation. I hope this is allowed. When painting with a heat reflective paint there are two options A- The room does not contain a significant heat source. This will always lead to a reduction in ambient temperature as heat from the outside is not allowed in. Heat by convection and conduction will still penetrate but overall the energy transfer will decrease. B- The room contains a heat source emitting IR rays. The rays will keep bouncing over the painted walls until it is absorbed by the air or the body of the heat source itself. Depending on the ratio between the external and internal IR sources, a IR reflective paint can lead to either increase or decrease of ambient temperature
In algebraic number theory, we would like to study rings of algebraic integers but sometimes they are not PIDs and thus they don't possess good properties. Because of this, we have introduced the notion of Dedekind domains, in which rings of algebraic integer are included and one can study the nice properties of Dedekind domains to understand algebraic integers. While doing algebraic geometry and commutative algebra, I have encountered polynomial rings in several variables, whose properties are quite different from polynomial rings in one variable, as the former ones are just UFDs while the latter one is a PID. So this "degeneracy" of polynomial rings from PID to UFDs reminds me of the similar case of rings of algebraic integers but I don't think they are related since the dimension of polynomial rings depends on the number of variables , which does not fit the definition of Dedekind domains.But I wonder if there is a way to classify polynomial rings of several variables(especially those over an algebraically closed field) into a specific type of rings, such that this type of rings has nice properties just like Dedekind domains and from which one can classify the prime ideals as well as determine the irreducibility of polynomials easily?
I came across a question a while back. It stated that a oil droplet was suspended vertically within an electric field. The man who suspended it had left to eat a very LONG lunch, and came back to see that it was splattered on the top plate (nothing with the setup had changed), and the question asked us to state a physical reason for such an event. I thought it was because of a transfer of momentum from electrons moving from the negative plate on the bottom to the positive plate on top, which just so happened to collide with the oil droplet. Someone else answered saying that the droplet evaporated slightly, which caused it's mass to decrease and so the force acting upon it managed to accelerate it more than that of gravity, causing it to move upwards slowly. That was the correct solution, but I'm just a bit confused, since wouldn't it lose some charge when it evaporates as well? So overall the q/m ratio would still be the same? They said it was unlikely that it evaporated some of its charge, so I'm just wondering how the charge within an object is distributed. Is it distributed evenly throughout, or is it more concentrated in some areas?
I have learned some basic measure theory as covered in the first chapter of Durrett's Probability: Theory and Examples, which includes the construction of Lebesgue integral (from simple functions to bounded to positive to integrable ones), as well as some basic properties such as Fatou's Lemma, monotone convergence theorem, dominated convergence theorem, the Fubini theorem, etc. I will be officially taking Real Analysis (measure theory), next year. However, as sometimes functional analysis turns out to be sometimes illuminating for my undergraduate research in mathematical optimization, I want to have a basic pass through the most fundamental concepts of functional analysis. I have read some posts here stating that it's possible to learn functional analysis before measure theory, but the other order is preferred due to the need for integration. Since I don't have much time nor want to stumble upon something that requires measure theory, does anyone has some recommendation on a list of topics or theorems that might be needed for a basic read on functional analysis? Thanks in advance! Edit: I have some basic background in general topology.
Google dictionary (based on Oxford dictionary) has an entry for the verb "reveal" as follows: make (something) known to humans by divine or supernatural means. And it includes one example: "the truth revealed at the Incarnation" Based on the definition, I find it hard to understand the role of "the truth" in this example whether it's an object (as denoted by "something" in the definition) or a subject (that is supposed to "make(something) in the def.)? My best guess is that it's an object because after checking other dictionaries such as Collins, Merriam Webster, Cambridge, etc., I found that they all define "reveal" to be a transitive verb. So I guess that the "to be" may be omitted in this case and the full sentence should be: "the truth IS revealed at the Incarnation". However, no resources I found have mentioned such type of ellipsis like this one. So my question is, is my guess correct. If it's indeed correct could you give me some resource covering such type of ellipsis?
If I have a stick that has a spinning disc on the end of it, and I try to rotate the stick, I will feel more inertia the faster the disc is spinning because I have to transform its rotational energy into a different axis. It seems intuitive to me that this would apply to electromagnetic fields as well, since you have to move all the energy in the field to another axis. So if you had an MRI machine on an airplane and the plane tried to rotate, would it rotate more slowly when the machine was powered vs unpowered? If so, does this depend on whether the axis of the field is aligned with the axis of rotation? If they were aligned so that the dipoles are parallel to the body of the plane, could it roll without extra inertia? I may be describing the Einstein-de Haas effect or gyromagnetism, but I am confused whether they are describing the same thing as me in different words or just a related concept.
Often we define a quantum phase of matter as a set of ground states of a gapped family of Hamiltonians, where the gap does not close anywhere in the family. Equivalently, we can define it as a set of states which are related by constant depth quantum circuits with local gates (equivalence of these two definitions is shown by the adiabatic theorem). Now consider a family of states which are ground states of a gapless family of Hamiltonians, and assume they have similar correlations and can be considered a "phase of matter" in the sense that all their observables vary analytically with the parameters of the Hamiltonian. Can we use the circuit definition of a phase of matter in the same way as the gapped case? That is, is it reasonable to define a "gapless phase of matter" in terms of states which are related to each other by a constant depth-quantum circuit? Maybe there are counter-examples to this which demonstrate this is not true. In particular, it seems reasonable to define gapless phases of matter in this way as (a) local unitaries do not change long-ranged entanglement properties, which intuitively define phase (b) unlike the Hamiltonian case, we do not need to worry about excitations to higher energy levels. Is there any literature on this at all?
This question might be very silly, but I am really confused about it from several days. Look transverse waves on a string propagate along the string due to the electromagnetic (EM) forces between adjacent particles of the string. So when one of the particles is displaced, it will drag nearby particles and thus the disturbance travels. In case of electromagnetic waves, the electric field values oscillate with time. But is there anything physical that connects the electric field at a point to that at its nearby points (just like we had for strings)? If not, then why does this information travel? And if yes, then what is that "thing"? Also, I would like to add that for strings the direction of displacement at nearby points was completely determined by the source point, but for EM waves the direction of electric field at one point does not necessarily imply that the electric field at nearby points will be in the same direction (take the case of unpolarised light). Why is that?
I am working on automated templates, and I want to use latexmake with luaLaTeX, as suggested in a comment on this question, where I attempted a similar setup with XeLaTeX. However, I am facing challenges with the configuration. My goal is to automate the process of typesetting my documents with luaLaTeX using latexmake, and I believe this will save me a significant amount of time. I have tried variations of latexmake main.tex -lualatex -outdir=output -auxdir=auxil in the terminal but it tells me that lualatex is not recognized as an argument For the code examples, I have a template that works and needs lualatex but needs several compilations. Hence the need for a latex maker. I also have tried to read the documentation for latexmake but I didn't find any solution and there isn't much content about this library online. If someone could provide guidance on how to configure latexmake to work with luaLaTeX, it would be greatly appreciated. Thank you!
For context, I watched PBS Spacetime's video on virtual particles (link goes to relevant timestamp) where they say that virtual particles aren't mathematically necessary, because the lattice version of nonperturbative QFT doesn't use them, and yet still makes all the same predictions as perturbative QFT. I was satisfied with that, until I had a brief exchange with someone in the comments of this answer where he says that, in most cases, it's impossible to actually do computations in nonperturbative QFT, and, when I asked if this was just due to not having sufficiently efficient algorithms, he said Note that in particular that even establishing the existence of a non-perturbative Yang-Mills QFT (which is what the standard model / QCD / QFD are) is a millennium problem. which implies that we don't currently even have a nonperturbative version of the standard model and that it's unclear whether one exists. However, PBS Spacetime is, in my experience, typically a reliable source for high-level explanations, so I wouldn't have expected them to mention nonperturbative QFT as the reason not to think virtual particles are physical if that theory wasn't actually useful for nontrivial calculation. Is it that most physicists think there probably is a nonperturbative version of the standard model and it just hasn't been discovered/created yet?
Let S be a compact orientable surface and U an open connected subset of S with finitely many ideal boundary points (or ends). U has a prime ends compactification which is a surface with boundary (following Mather). Let b be one of these ideal boundary points and Z(b) its impression in S. If Z(b) has more than one point, then there is a circle C(b) of prime ends associated to b. Each prime end e of C(b) has its impression Y(e) (another impression, the intersection of the closures of the sets of a chain that define e). This seem to be known by experts, but I could not find it written anywhere. Does anybody knows how to prove it? Knows a reference? I think I can prove it using scaffolds, a powerful tool created by Mather to prove various properties of prime ends. But it is a lot of work. Any good hint? Thanks.
In the situation above, we have a power line which uses an alternating current. This alternating current causes a change in magnetic flux through the loop below the power line, which induces a current according to Lenz's law/Faraday's law, which eventually goes to the farmer's equipment. This is classed as power theft. However, I was wondering, does this power tapping by the farmer in some way reduce the power delivered by the power line or cause an energy loss from the power line? If so how? If it doesn't, then how is this situation exactly consistent with conservation of energy? Does the induced magnetic field in the farmer's loop somehow cause energy to get lost in the power line? (Where does the energy from changing magnetic flux come from in the first place?)
I have a hard time understanding time dilation and special relativity; each explanation seems to contradict the other, and don't explain the apparent paradoxes they cause. Say clock A orbits clock B at a very high speed. According to the explanations I've heard, B would perceive A as ticking slower than B, because of time dilation. But since A has equal right to claim to be stationary, it should observe B as orbiting A, thus ticking slower than A. If at any moment the clocks' time would be measured, would their respective elapsed time be different? In other words, is one clock actually slower than the other? This seems unreasonable, since the respective situations of the clocks are identical. If the answer is no, what exactly happens if the clocks suddenly stopped orbiting each other and become stationary relative to each other? If each clock has perceived the other as ticking slower for a while, would the clocks instantly jump to the same time? This also seems unreasonable, since orbiting each other for a longer time then would imply a different outcome when stopping, even if the events of stopping are identical. How can this be resolved?
Usually when we introduce postulates it is so they are of some use to us. I do not see the reason for these two. The Line Separation Postulate: Each point on a given line divides the line into three disjoint sets: the set containing the point, and the two half lines on either side of the point. The Plane Separation Postulate: Each line within any plane divides the plane into three disjoint sets: The line and the two half-planes on either side of the line. I understand the postulates and what they are saying, but I do not see how they can be used. And I have not yet found a proof that uses them in the book. I guess I am not convinced of their fruitfulness so if anyone can help convince me these are useful/necessary postulates that would be of great help.
Suppose that, as a manager, I am creating a campign to evaluate employees based on the NPS (Net Promoter Score) of their sales - the objective is to reward high perfoming employees. However, due to the different nature of each role, some employees have a lot more sales than others. What is the best way to create a ranking system? The data available is: NPS for each sale for each employee (the value can be null, if the survey has not been answered by the customer). I thought about using a bayesian ranking system, but I think that it would be unfair to low seeling employees which sell little beacuse of their specific role. In a more abstract sense, how can I compare samples of varying sizes without skewing the comparison towards bigger or smaller samples?
I've noticed while playing Minesweeper that when I have too few bombs, I get very easy to play games. In other words, I get games that can be solved with very simple algorithms. When I play games with too many bombs, I get unsolvable games or games that can't be solved without guessing. There seems to be a region in the middle(not too few, not too many), where some games are solvable but require increasingly complex algorithms. I've also read that Minesweeper is NP-complete, which I take to mean that there is no upper bound to how complex these algorithms can be as n, in an n-by-n game of Minesweeper, is allowed to grow. This seems relevant to information processing(halting problem?) but not exactly sure how. First, is my phrasing correct? And second, where can I search for clarification in my thinking?
In an office, there are typically various different kinds of environments in which each employee carries out their work. Some have their own rooms, some have cubicles, some only have a desk in an open landscape etc. I'm looking for a word for the place where an individual employee carries out their work, regardless of if it's in a room, at a desk etc. Things I considered: Workstation - probably the best I've got so far, but it sounds kind of techy to me (associating with a workstation computer perhaps) - does this work also for someone not using a computer such as writing by hand (!), manning a reception etc? Workplace - to me this sounds like a common word for the entire office/building? Am I wrong? Furniture - kind of encompasses all kinds of surfaces you can work at, but it sounds too broad (and plain silly!) Equipment - to me this sounds more like advanced/specialized lab gear (labs, fume hoods, ...)? Does anyone have a good suggestion? Imagine e.g. this sentence: "All employees in the office are guaranteed their own [ ... ] where they can carry out their work". And I don't want to go too broad with abstract terms such as "location", "spot", etc.
There are Two different defintions for the event horizon of a black hole. The Absolute horizon and the Apparent horizon. An apparent horizon is a surface that is the boundary between light rays that are directed outwards and moving outwards and those directed outward but moving inward. While and Absolute Horiozn is a boundary defined with respect to the external universe, inside which events cannot affect an external observer at infinity. Normally these horizons coincide but they can differ during event such ad black hole mergers. Technically, knowing exactly where the Absolute horizon is requires knowing the entire future history of spacetime. So it seems to me therefore that in a universe destined to even in a big crunch, black holes wouldn't technically have Absolute horizons, only Apparent horizons.
I interpret the expression 'If P then Q' as asserting that if P is true Q is automatically true. So, we would say 'If P then Q' is true only when it indeed is the case that P being true implies Q is true. However, in logic, the truth of 'If P then Q' is determined solely on the basis of the truth values of P and Q individually and not by verifying whether Q follows from P, or is implied by P. So I just don't get how we can decide the truth of if-then statements by just looking at the truth values of P and Q. For it to be true don't we need to prove somehow that the truth of Q follows from the truth of P? In one of the logic books that I read, they explained conditional statements in this manner: 'If P then Q' asserts that it is not the case that P is true and Q is false. I liked this. It makes me understand the truth table of conditional statements well. However, by this explanation, I'm not able to see why would one use the words 'If then' then. How the idea that it is not the case that P is true and Q is false follows from the meaning of words 'if then' (or 'implies' for that matter). Should I completely forget about the meaning of if-then sentences as used in ordinary language and assign them a new meaning?
One possible formulation of the second law of thermodynamics is that the work extracted during the change of a thermodynamic system between two thermodynamic states is at most equal to the free energy difference between those two states. One possible derivation of this empirical result from statistical mechanics is the derivation of Jarzinski relation in the canonical ensemble. Also, the Jarzinski relation was also derived in a stochastic setting by Crooks. For the sake of further simplifying the statement of the Jarzinski relation, I consider a Brownian particle fluctuating within a double-well potential in one dimension, i.e subject a Langevin equation on the real axis with known potential and noise. What I can say for this system is that the particle will be undergoing transitions between the two wells with a rate that is exponential in the barrier height of those two wells. Also, the equilibrium distribution of the particle will be a bi-modal having a peak at each potential minimum, and the height of each peak is given by the barrier height. Now which additional useful information does the Jarzinsky relation tell me about this simple system ?
I'd like to prove: Given two points A and B and a straight line connecting them AB, any smooth curve passing through A and B will, at some point, have a derivative that exceeds the slope of AB. My proof uses the Mean Value Theorem. So, any curve passing through A and B has, at some point, a derivative equal to slope of AB, call this point C. Consider lines AC and BC. Together with AB, they form a triangle. Noting the angles a and b, we know that the slope of either AC or BC exceeds the slope of AB, and, by the Mean Value Theorem, there exists a derivative on any curve passing through either A and C or through B and C that equals the slopes of the lines AC and BC. Therefore, any smooth curve passing through A and B will, at some point, have a derivative that exceeds the slope of AB. Perhaps this is trivial, but for me it is important.
I'm trying to understand where heat energy goes in a substance. I've seen that mainly it's translation at lower temperatures, rotation at mid range, and vibration at high range, but I'm not sure of that. Someone made the claim that an atom can expand as it is heated, which I don't think is right, but in a way, this is a good description of black body radiation? Does anyone know some approximation of percentages for how much heat energy goes towards motion (translation, rotation, vibration) and how much is radiated as blackbody? And, does this change between the different phases: solid, liquid, gas, plasma? I'm just looking for a general concept here, so no constraints. If there's some special constraints to note, then please list them and the energy distribution within that particular system for comparison.
Recently I saw a video talking about the Yoneda lemma from category theory being used in neuroscience. It was my first introduction to category theory. In category theory we have objects and maps between the same objects called morphisms. For example we have one type of object called set and its morphism would be the function. A function is a map from one set to another. Another example would be the object vector field with its morphisms being linear transformations. This just looks like we are renaming the same stuff. Is a vector space not just a type of set we construct to obey our desired axioms? If so then introducing linear transforms would be truly just renaming stuff. And if they are not sets, why not allow non linear transforms?
My first name is "Jean-Baptiste". "Baptiste" is not a second or middle name, however I noticed that it's not unusual for native English speakers to address me just as "Jean". I don't mind it at all, I'm not offended and find it actually charming, but it would be quite unusual for a speaker of my native language (France's French) to call me like that (either native speakers address me with my full first name, or they use various nicknames - and the nickname is never "Jean"). So I'm curious as to why many native English speakers seem to spontaneously call me "Jean". The two hypotheses I have is that either my full first name might be a bit difficult to pronounce (so just saying "Jean" is easier), or because they mistakenly think that "Baptiste" is my middle name - as compound first names are unusual in English (correct me if I'm wrong). I'd tend to think it's the second option, as I noticed that it also happens in writing, but I'd be interested in having the insight of people who know English language and English-speaking cultures better than I do. Maybe I'm completely on the wrong track! When it happens, I usually don't have the opportunity to ask people why they are calling me like that (e.g. in a professional context we usually don't have the time for that, plus in this kind of situation I don't want to make people feel like they offended me), that's why I ask this question here. Thank you,