darklight03/StackMIAsub huggingface dataset:
Papers often use one example throughout as their ongoing example. Similarly, it is possible to focus on one particular case and show a broader claim by using that case. I remember there was a really nice verb for using x at a test case, or perhaps x serves as a use case. I tried a thesaurus searching for exemplify (which is close, but not exactly right) and some other searches ... but none of these helped much. An example sentence might be: Testing food in general, we verbify eggs. or Eggs verbify food in our context. More elaborate examples: Using the RR dataset as a test case, we investigate how different aspects of metric design affect the computation cost. We perform an analysis study on efficiency, using the popular RR dataset as a test case. As a test case for investigating dataset efficiency, we analyze the results of RR.
Could somebody untangle following statement I found here: the integer cohomology groups correspond to the quantization of the electric charge. I know from pure mathematical side the meaning of cohomology groups, but not understand the translative part between physics and pure math at this point: Could somebody borrow some time to sketch how this identification/ correspondence is in detail established? ie how can the (presumably geometric) quantization procedure be "interpreted" / encoded purely in terms of certain cohomology group? And furthermore, what is here the precise meaning of the notion of "a charge" from viewpoint of pure mathematical terminology? Up to now I thought that a charge in mathematical physics can be recognized as a quantity comming from the existence of global symmetry of the given system, so in simple terms certain intergral/ number witnessing the existence of such global symmetry. Do they endow in the quoted sentence above certain "bulked" meaning to the term " charge" in order to making it to becomes the object subjected to quantization procedure?
I am trying to find some sort of motivation as to why we integrate manifold over differential form and why especially does it in some form corresponds to integrating the surface of the area. I have already passed my university courses that have covered these topics, including Stokes's theorem with proof (My uni has this approach of definition - theorem - proof structure in all courses with very little motivation). The issue is that sadly I've never intuitively seen that it could have anything to do with the real area if the manifold would be an earth-like object or any other intuitive geometric structure. Is there any free material/youtube class that illustrates that it truly corresponds well? I don't need to see proof as I quite understand the technical side of things. My issue also is that I cannot see differential form in any other way than only technical mathematical extremely abstract definition. Please keep in mind that my knowledge is limited to European standards of a bachelor's degree...
Let say we have some water in the sink and open the closure.. The water starts to move towards it in a whirpool like manner.. If we have a table tennis ball and leave it near the hole of the sink it will rotate like a planet instead move right towards the hole... If we suddenly open up more the hole the water will start to move faster towards the hole but even the water moves towards the hole the orbiting ball will still be able to detect the change of the water speed due to the hole of the sink being openned more... Can this be compared with gravitational waves where even the spacetime tissue moves towards the BH binary star the orbiting planet is still in position to detect this sudden perturbance moving in opposite direction regarding spacetime tissue simmilarely as the table tennis ball get the perturbance moving in oposite direction than the water moving in direcion of the sink hole?
Hi everyone, lets say we have a circle shaped space station and there was an accident where we got a hole (hole A on the picture) on a station and now most of the station is vacuum except for one enclosed part that is still filled with air. Now if we make holes B and C at the same time in the enclosed part, what will then happen with the air? Will it: because air moves in random directions, half of the air will go through hole B and then through hole A and other half will go through hole C, go through entire station and then through hole A because of difference in pressure all air will go through hole B and then through hole A (I am not sure if this is true or I am misremembering, but I read somewhere that air takes the shortest path to equalize pressure so that is the reason that all air will go through hole B? If this is not the true is there some way to influence all of the air to go through hole B, change the size of the holes, make a longer distance between holes C and A?) this and next scenarios are highly unlikely but I have left them just to cover all possibilities , all of air will go through hole C and then through hole A air will stay where it is and it will not leave the room
First, there are only three type of leaves: enter image description here If we assume lamination is compact ,all leaves is only can be a simple closed geodesic. I want to ask about how could union of uncountably many leaves be a minimal compact lamination,actually all leaves is complete simple geodesic,but all leaf in a compact minimal lamination in a punctured torus can only be a simple closed geodesic cause leaf of compact lamination can not goes up to cusp ,so it can only be closed simple geodesic ,so we take closure of any leaf ,it can only be itself, that contrict to minimal. cause i dont know that question above, i can not understand why we cut a punctured torus along a compact minimal lamination which is not a simple closed geodesic ,we can gent a ideal bigons which is patch two side of two ideal triangels.
If I were to measure some quantitavie metric of a sample population and record its mean, and then I were to split by random selection all members of the population into two groups of equal size and record the means of each group on the same metric. Is their a statistical term/measure of the difference of those means from each other and/or the mean for the whole population. This comes as I'm trying to analyze an experiment in which both the control and sample group where told information regarding the experiment during a pretest where I fell like the information could have been left out due to the large size of the population. My rational behind that is the difference in the split means would decrease as the overall population increases and one could assume that both sample groups are evenly distributed.
The court's ruling is claimed to be (...) because it has only counted specific actions to set up some (...) simple structure of justice that only encompasses the last hour neglecting the history of events. (...) = ? = "has only lasted for a short amount of time prior to being mentioned" Referring to current attitudes that have arisen in the general span of an hour or day or other fairly newly developed engagement. "Of the moment" "According to recent neoteric dispositions" The structure is simple because it is founded on only one or two just-happened pieces of evidence. Adjacent to "transient" "transitory" "ephemeral" but in reference to a newly grown/spoken structure of logic. "recent" and "latest" seem too general and encompassing an indeterminate amount of time, whereas the word I'm looking for refers only to the nature of the just-referenced events, which is (...) Transient = lasting for a short amount of time (...) = has only lasted for a short amount of time
Simulate a magnet sticking to a metal door. The side that is against the door, means that side is the pulling side, and must be the inwards movement of a toroidal magnetic field, pulling the object inwards resulting in getting stuck to the object! Does this mean, the side that is against the object is the side where the field goes in through the magnet and will always be the actual South pole of any magnet? When the magnet is flipped onto the object, the field is in the opposite direction pushing against the object, which will be the North pole of that magnet. Does this mean that when holding a magnet and you can diffirentiate between the actual physical pushing and pulling side of a magnet, them to know that the pushing side must be North and the pulling side must be South? This has Nothing to do with: Stringing the magnet and it will find magnetic North or South and zero to do with earths magnetic field direction, but everything to do with the toroidal magnetic field of the magnet itself and which direction the magnets toroidal field flows?
In school, I learned the mechanism of high and low pressure areas, which roughly goes like this: In the tropics, the sun warms up the air during the day. Water evaporates, so that the air gets warm and moist. Warm air is lighter than cold air, so it rises up, leading to a low-pressure area because the warm air is now "missing", so we need air to horizontally flow in. The rising moist air cools down, so it starts to rain, that's why there are rainforests in the tropics. Some thousand kilometers to the north or south, the air is sinking to compensate for the rising air in the tropics. Sinking air warms up and gets dried, so it is very dry (Azores High), leading to the big deserts in the subtropics, e.g. the Sahara. I hope this is more or less correct. There is one thing I do not understand: Why is the air pressure lower in the tropics, and higher in the subtropics (Azores High)? My intuitive physical reasoning tells me the opposite, for various reasons: The reason warm air rises up is not that because "it is light", but that it is lighter than cold air, so the cold air can replace it by flowing under it and lifting it up. So the phrase "warm air rises up, leading to a low-pressure area because the warm air is now missing" is very misleading: it is the inflowing cold air that lifts the warm air up. So there is never a "moment" when we have a lower pressure - the warm air rises because it gets replaced by cold air, and there isn't a reason why the pressure should drop. The air pressure is more or less the combined weight of the air above us. The heavier the atmosphere above us, the higher the ambient pressure. Now, when the air warms up and gets lighter, it gets replaced by the cooler air flowing in. So after the warm air has risen up, we should have more air above us: the warm air that has risen up, plus the cold air that has flowed in horizontally. So the atmosphere above us is heavier in total, which again means we should have a high-pressure area. Another view could be that since the air in the tropics is warming up, it expands. This large-scale expansion should lead to a high-pressure area. After all, if you expand a large amount of gas, its pressure increases until it has had time to expand properly. I am very much aware that all these reasons must be wrong in some way. I don't doubt the existence of the Azores High or the rainforest. I just miss an intuitive understanding. How can the tropics Low and the subtropics High be explained in intuitive physical terms?
My understanding is that in commercial nuclear reactor operations, fuel rods are not used up to the point where they're fully depleted and unable to support fission, but are replaced while they still contain an appreciable amount of fissionable isotopes, to ensure that the reactor stays in a stable operating regime at all times. However, would it be possible, for any reason, to run a nuclear reactor without replenishing the fuel as long as any useful energy at all could be produced, that is, actually "running the fuel rods dry" (or, more formally, until criticality is irrecoverably lost and decay heat is the only remaining output left, at which point the reactor is essentially just a fancy spent fuel pool)? Could such an end state be safely achieved, with power just gradually fading away while the control rods extend out more and more to compensate for the declining reactivity? Or would unstable and potentially dangerous operation ensue, as in Chernobyl where operator response to a poisoned core started a catastrophic chain of events? The question is motivated by the realization that one of the safety-enhancing measures proposed for the RBMK reactor design after the Chernobyl disaster was to raise the enrichment grade of the fuel, apparently to reduce the type's susceptibility for core poisoning and the resulting power fluctuations (at least that's how I understood it). I'm interested if the converse is also generally true, that is, whether it would be hazardous to let the fuel deplete too much in modern PWR/BWR reactors that don't share the design flaws of the RBMK.
In biology, the scientific name of a species (known as the "binomial name" or just the "binomial" or sometimes even just the "binomen") is written as a pair of words in italics (or underlined, which is the equivalent of italics in handwriting). For example, modern humans belong to the genus Homo and, within this genus, to the species Homo sapiens. The first word specifies the genus (meaning a group of related species) and has an initial capital, while the second word specifies the species, which doesn't have an initial capital, even if it is derived from the name of a person, e.g. the scientist who discovered the species. https://en.wikipedia.org/wiki/Binomial_nomenclature For example, that Wikipedia article says, "[...] the binomial name of the annual phlox (named after botanist Thomas Drummond) is now written as Phlox drummondii." Note that in the phrase "annual phlox", which is not in italics, "phlox" has a lower case initial. So in this case it seems that Wikipedia thinks it is acceptable to write "phlox" in all lower case. And yet I've often read that a genus name must always start with a capital letter. For example https://en.wiktionary.org/wiki/genus_name says, "genus name (plural genus names) (taxonomy) The scientific name of a genus, which is always capitalized; the generic name or generic epithet. Usage notes The scientific name (binomen) of a species is a two-part name and is typeset in italics, the genus name (the first name) has an initial uppercase (capital) letter and the species epithet (or specific epithet) is written with a lowercase (small) letter; for example, the scientific name of the wild Rock Dove is Columba livia.". First, notice how the dictionary quoted above wrongly has "wild Rock dove". It should be "wild rock dove" of course. Ha ha. Now, if you will, imagine people somewhere had started to refer to one or more locally found species of the genus Columba as "Columbas"/"columbas". This could happen because calling them "rocks" would be confusing. After all, rocks do sometimes fly, land, and so on. In writing they write them without italics or underlining and sometimes without an initial capital. So my question is: Would the latter be acceptable in formal writing among nonscientists? To me it seems a bit anomalous for something that is a type of animal to get a capital letter. It's as if biological jargon has spilled into ordinary English, and where it conflicts with the rules of the latter, it just brushes them aside. The italics/underlining rule is rarely followed by nonbiologists, so do we really have to always follow the capitalize all genus names rule? Can't we follow the English rule that says types of animal are not capitalized?
The following question was discussed in my Discrete Math class, but we couldn't reach a consensus. Think of a set as a collection of bins. Each bin contains exactly one object distinct from all others. While we are allowed to move the objects around between bins, we cannot remove any object, nor can we place two objects into the same bin. Someone hands us a new object. Can we place that object into a bin, after moving around some of the objects already sitting in bins? A. Yes, for some sets this works just fine. B. Sets are not composed of bins and this question makes no sense whatsoever. C. No, there is no room for another object, all bins are already filled. D. It depends on the new object. I'll try to summarize the general arguments for why each answer is correct or incorrect: A: A few of us think this is right, because we can insert the new object into the set iff it's not already in it. However, this is similar to answer choice D, which was given to be wrong (see below). B: Many of us shy away from the answer because it "sounds" incorrect, but I personally believe this is the right answer. This is because I disagree with the representation of a set as a collection of bins (each of which contains exactly one object), for the simple reason that the model does not seem to account for adding new bins (and hence does not account for adding new elements). C: Some thought this was correct, if we assume that the number of bins cannot increase (i.e. if we cannot add elements to the set). Some thought this was incorrect for the same reason; because it would mean we can't add any elements to the set at all. This answer was given to be wrong. D: Most thought this was correct - if the new object is different from all the objects already in the set, then we can add it, and otherwise we can't. However, this answer was given to be wrong. And the fact that this is wrong seems to also imply A is wrong (i.e. the intended answer isn't talking about whether or not the element is already in the set). Could someone give a justified answer to this question using concepts from Discrete Math and Set Theory?
I have seen different explanations to understand why there are no local gauge invariant observables in gravity. Some of them explain that diffeomorphisms are a gauge symmetry of the theory and thus any observable evaluated at a spacetime point will be gauge dependent and therefore not an observable. This line of reasoning then argues for Wilson loops, or asymptotic charges as good (non-local) observables in gravity. This explanation, in my opinion, is purely classical, it doesn't rely on the uncertainty principle, or commutation relations, etc. However, other explanations give the argument that any device trying to measure a local observable will have a finite size, and therefore a finite accuracy. If the device wants to probe local physics, it should be smaller. However, the uncertainty principle forces the device to collapse into a black hole before allowing the experiment to give us local information. Alternatively, it is also explained that the commutator of two (quantum) operators has to vanish for spacelike separations but that a metric that is dynamical and fluctuates will mess with the causality of the theory, therefore making the operators not observable. These arguments seem absolutely quantum mechanical. I have a couple of very similar questions: Is the statement "no local gauge invariant observables in gravity" true in classical GR, in quantum gravity or both? If it is true in both, why do people treat the statement "no local gauge invariant observables in quantum gravity" as something special? Do statements about observables in classical and quantum gravity mean different things?The arguments given to explain each one are pretty different and seem to involve different physics. The first one relies heavily on diffeomorhism invariance while the second one relies on holographic-flavoured arguments about how much information you can concentrate in a given volume before you form a black hole.
I'm building the Proximal Policy Optimization algorithm from scratch (well, using PyTorch). I've been studying it by my own, but I'm a little bit confused in the optimization phase, here is the thing. From what I know... First, we initialize a Policy Network with random parameters. Second, we start with policy rollouts. At each time step (t) of the episode, we compute the Value Functions (functions approximators) in order to get the Advantage Function A(s,a), also, we compute the clipped surrogate objective (J) of each time step (t). At the end of the episode, we sum all the clipped surrogate objective values, this will give us the expectation of the entire episode of the expected cumulative rewards. In the paper of PPO the equation contains the expected value of all time steps (t). Once we have the value of the expected clipped surrogate objective (sum of all clipped surrogate objective values) at the end of the episode, we compute the Stochastic Gradient Descent (SGD). In order to compute SGD we need a loss function, we have our expected clipped surrogate objective function, so we just do -(expected clipped surrogate objective function) and that will represented the loss, that is the same as computing the Stochastic Gradient Ascent in order to maximize the expected cumulative reward or the objective function. Now my confusion comes in here... I thought that the clipped surrogate objective function was computed at each individual time steps (t) and then at the end of the episode, we sum all of them in order to optimize it (compute SGD). The thing is that some authors say that the optimization process (SGD) is done at each time step (t) of the episode instead of at the end of the episode, but why? doesn't the clipped objective function equation in the paper contains an expectation symbol? if the computation is done at each time step (t) then the expected value symbol in the equation is redundant, isn't it? Also, they say that in order to compute SGD we need a loss, I thought that by setting the clipped surrogate objective function to negative (-) we can get the gradient of it and then minimize it using SGD that it would be the same as maximizing it, but the paper shows another equation as being the loss function that the optimization phase uses, how is this? So my questions are... When and how is the clipped surrogated objective function computed (at each time step (t) or at the end of the episode), is my implementation of the computation correct?. When and how is the optimization phase computed (at each time step (t) or at the end of the episode)? My thoughts about the loss function correct? or what does the paper means with this loss function they show? Thank you in advance:)
I am currently very confused about the "topological" prerequisites of Lee's Riemannian Geometry (RG) book, An Introduction to Riemannian Manifolds. I have heard that this is a "truly introductory" text for beginners in RG. But as for prerequisites, in his preface Lee states that his other two books on Topological and Smooth Manifolds are a sufficient preparation for his RG text. But from several online forums I've come to the realizations that this would be an "overkill" just for studying RG (e.g., one need not study all of his Topological Manifolds text to study most of his Smooth Manifolds text!). So, in general, his preface is confusing to me when discussing prerequisites. I'm confident with my analysis background (inverse/implicit function theorem, etc.) As for the differential geometry stuff, I'm thinking about going over doCarmo's Curves & Surfaces text. I'm a beginner in DG and I prefer E. Kreyszig's Differential Geometry as it introduces tensors early on (something I'm highly interested in learning about). I'm also aware of another "standard" text on RG by doCarmo as well. So, here's what I need guidance on: Which of the books among Lee and doCarmo on RG is preferable if one wants to get a decent exposure to RG so that they can pursue research in related areas? How much topology does one need to know in order to tackle the texts by Lee and/or doCarmo on RG? For differential geometry, is Kreyszig's DG book sufficient to prepare for Lee and/or doCarmo's RG text(s)? Or is it preferable to study do Carmo's DG before tackling RG? My ultimate goal is (broadly) mathematical physics, in particular, Relativity and Quantum Gravity. Moreover, I consider myself more of a "math"-person and so would not prefer texts that ruin mathematical rigor.