darklight03/StackMIAsub huggingface dataset:
In the past two months, I had found a wonderful pdf that went through a derivation of the determinant with calculating the area of a parallelepiped as its starting point. The document did not get into the weeds of calculating the determinant given a matrix or even focus explicitly on matrices at all; it was probably the single best description of why the determinant tells us about area I had found and I just really found the document's approach to determinants extremely useful. I'm trying to relocate this pdf to no avail and was really helping someone could help me find it. What I remember is this: it begins with calculating the area of a parallelogram as its starting point. It describes how we want the area to change when a side is scaled and when the image is sheared (I very distinctly remembering these two things being bulleted in the first page of the paper) and goes onto describe how these properties would manifest in a general "area function," that is a function that takes in the vectors as inputs and returns some type of unsigned area. It proves several facts about these functions that begin to show how the determinant "appears" from answering this question naturally. After this point, they expand their scope to signed area functions and, building upon the results in the first part, show that any function which returns the signed area of a parallelogram is simply a multiple of the determinant. The paper rounds out with a discussion about expanding these ideas to higher dimensions; it describes the problem of measuring the area of a higher dimensional parallelepiped with regards to measuring the area of its shadow. One of the more distinct parts of this description is that it begins with a hypothetical scenario where you find a floating parallelepiped outside of your window one day and cannot interact with it but wish to find its area. My attempts at googling with these bits of information I remember have not been fruitful, so any help finding this pdf would be appreciated.
Motivated by question Can IC engines be modeled as Carnot engines?. I am wondering whether/how Carnot theorem could be generalized to other kinds of devices performing "useful work", such as, e.g.: Motor (or generator) fed by a battery Nuclear power generators Solar cells Water wheels I think that the theorem must be generalized in at least three ways: Operating media is neither a gas nor a liquid - that is the reasoning based on isothermic adiabatic expansions might not apply. Generalizing the concept of temperature (introducing "effective temperature"?) - e.g., in case of a batter or a water wheel, we do not have two reservoirs with different temperatures to properly speak of, but rather two reservoirs with different (chemical) potential. Generalizing the concept of useful work - solar cell and water wheel are not really transferring the energy between two reservoirs - the energy already flows, and the device simply diverts a part of this energy into work. But, since the energy flows anyway, it is not clear whether/how the part of it that is diverted is useful: e.g., how is the current generated by a solar cell is more useful than the heat generated in the surface illuminated (which may be also "useful" in everyday sense.) Perhaps, there is not much left of Carnot theorem with all these generalizations, and we simply need to consider it as limited to a particular class of phenomena? If so, are there other upper boundaries on converting energy to work (that would be applicable to the devices cited in the beginning?)
The Earth, effectively a non-inertial frame of reference, is where Newton concluded his laws of motion. However, Newton's first law only holds in an inertial frame of reference. In the process of inventing the Newton's laws of motion, since almost all (I suppose) the experiments were done in a non-inertial frame of reference i.e. the Earth, why were people confident enough to believe that Newton's first law is true (to some extent, I am not talking about relativity etc.) in an inertial frame of reference? I am not trying to say that Newton's laws of motion are lies. I just had a logical question: since the research was done in a non-inertial frame of reference, how can we invent the laws regarding motion in an inertial frame of reference? I suppose that's because the Earth can be approximated to an inertial frame of reference as the effect of the self-rotation (which causes the Earth to be a non-inertial frame of reference, in my opinion) of the Earth on the objects on the Earth is quite small. And, therefore when people did the experiments, the uncertainty caused by the self-rotation of the Earth is too small to be significant (or, maybe they didn't even find/realise such an uncertainty!). And therefore, by inference (or guessing?), we can invent the laws of motion regarding objects in an inertial frame of reference. In theory, if we do the same experiment in a true inertial frame of reference, we will effectively get the same/similar results. (This is just my explanation, which can be very debatable!) I want to hear how other people think about this question. Thank you!
Let me try to illustrate what I mean. Consider e.g. a Solar radiation storm (Solar particle event) where high-energy protons are hurled at Earth from Solar flares. I've tried to illustrate my conception of this (I know the protons will typically not follow straight paths out from the flare due to the Parker spiral, but it's a simplification): So the protons get captured by the field (given sufficiently low velocities perpendicular to the field, as far as I understand it) and are then led to the poles due to their drift velocity, as they will almost always have some velocity component that's not perpendicular to the field. Now, to me it seems (and the same applies to the plasma in coronal loops as far as I can tell) that there's a current along the field lines themselves due to the drift of the protons, in the direction they're traveling. This should itself induce a magnetic field surrounding the imaginary magnetic field lines at the centers of the helical proton motion as if the magnetic field lines themselves are current-carrying wires, should it not? Something like this: Am I correct in thinking about it roughly in this manner? If so, does that mean that these new magnetic fields could potentially themselves partially trap particles (although I assume the stronger original field would overwhelm it) and induce new magnetic fields around them in turn? Is there a limit to this "fractal" process of magnetic field lines acting as currents inducing magnetic field lines acting as currents, and so on?
My question, right up front, is: what is the term for a modifier that behaves this way? But "this way" takes some explanation, and that is the rest of the question. I am a mathematician, and my question makes the most sense in a context where words are formally defined anyway, but you can freely substitute "foo" and "bar" for any technical jargon. A ring is defined to be a set with two operations that satisfy certain properties. Among these properties, there is not universal agreement: should, or should not, the ring be required to admit a multiplicative identity ('unit')? For a mathematician who does not require this, it is easy to indicate when they wish temporarily to impose the hypothesis: they can just refer to a "unital ring". A mathematician who will almost always consider only unital rings might decide to make that part of the definition of a ring, so that they can say "ring" where a more permissive mathematician would say "unital ring". (This lightweight controversy is discussed in the Wikipedia article.) However, this latter kind of mathematician, on making the rare encounter of a ring that does not have a unit, must either make up an entirely new term for it, or call it a "non-unital ring". This usage is almost universally understood, but somewhat puzzling: while it makes sense to understand, e.g., a "commutative ring" as being a structure that it is a ring, and also satisfies the requirement that its multiplication be commutative, there is no way that one can so interpret "non-unital ring" if a ring, by definition, has a unit. So, what's happening here is that "non-unital" is not refining "ring" by adding conditions, but changing the meaning of "ring" by dropping existing conditions. That is, you know something about a commutative ring even without cracking open the definition of "ring" (namely, that it has a commutative operation); but, to know something about a non-unital ring, you must not only crack open the definition of "ring", but recognize which among the properties one is expected to remove. I'm looking for a word describing the behavior or function of modifiers that behave like "non-unital", in the sense outlined above.
So, I was reading some books by Stephen King, S.D. Perry, and a couple authors I really love. I notice they'll use pronouns or certains words twice in the same sentence. When I read it, it's pleasant and doesn't sound weird in my head at all. I assumed that if something is published under an author like Stephen King or Neil Gaiman, it must be grammatically correct to use something like "he" or "her" twice in the same sentence. I'm not sure if this is why it doesn't sound strange in my head when I read it, but I've provided an example below. Here it is (bolded words are the repeated pronouns): My example: Martin stared at the ceiling above and grimaced, counting the number of bumps and flecks embedded in its expensive paint. He gave up after some number he couldn't remember, tossed to his side, and yawned. The smell of curing lacquer and stain drifted off a nearby nightstand, burning his nose. And here is another example from It, by Stephen King: When he grinned, there was a ghost of the handsome man he would become in the lines of his face. There examples in multiple Stephen King novels of names repeated in the same sentence as well. I'm using Stephen King as an example, but note that almost every published author I've read does this in wildly different genres and styles. Anyway, is using a pronoun more than once in the same sentence grammatically correct? I know sometimes things are grammatically correct, but discouraged or frowned upon, but that's not my question, here. tl:dr; Is it grammatically correct to repeat a pronoun in the same sentence in any circumstance, and if not, why is there published material that violates this rule?
In my limited knowledge of statistics, I am confused between the meaning of Latent Variables and Nuisance Parameters. Here is my current understanding: Sometimes the variance parameters in a regression model (or the overdispersion parameters) are considered as Nuisance Parameters because we are not directly interested in estimating them In a Gaussian Mixture Model, the weights of each Gaussian Mixture are called Latent Variables because we do not observe them directly (although this point somewhat confuses me because we dont observe any parameter directly, latent or non-latent ... all we observe is data) From an estimation perspective: We remove the nuisance parameters (via marginalization, factorization) because they can bias the estimates. I don't fully understand this, but I think it has to do with degrees of freedom, MLE produces biased variance estimates (in complex regression models, we can not a priori know the correction factor needed to remove the bias). Later, if required, we estimate the variance parameters using a more complicated form of MLE called RMLE We remove the latent variables (again via marginalization) ... because they make the optimization problem easier? (i.e. reduce the dimensionality of the function being optimized) Is this the only reason we remove latent variables? Or is there some other reason ? (ex: the estimation of latent variables is biased?) From here, it looks like these concepts are almost interchangeable: A nuisance parameter is latent and a latent variable is a nuisance. They both complicate the estimation process for similar reasons and we try to remove them through clever math tricks (e.g. marginalization). I feel I am missing something here - any one have hints?
My partner frequently asks me questions that, when read literally, are questions about the past, but in intent and intended response are actually conditional questions: Did you have any thoughts about dinner? Did you want to have coffee? ...where the intent is not merely to inquire about the preferences I have established prior to the time of questioning, but also about what my preference might be at the time of questioning. "Would you have any thoughts..." or "Did, or would, you have any thoughts..." might also capture intent, but awkwardly. I'm quite literal so I've had to train myself out of interpreting "did" as a past tense question of fact, but it occurred to me that it might be a regionalism or other speech pattern where there's a kind of merger going on between "did" and "would", or other shift where "did" is used conditionally in casual settings, maybe in cases where there's the possibility that the answer might have been determined in the past, but also might not have. Even in my examples above, "would" does feel a bit stuffy and formal by comparison, but I've never used "did" this way myself. Does this interpretation make sense idiomatically outside of the individual context of my partner? Does anyone know if there is any regionalism in play? My partner comes from a family that has lived in Connecticut for generations. My family is mostly from the midwest and south. Please feel free to correct me if I am misapplying "conditional" or can otherwise better describe the usage here, I will update as I can! Also, unfortunately I don't have the gift of search terms on this one, so I haven't been able to find any discussion of it elsewhere, but let me know if I can add any research or context.
Double-slit experiment image source: Wikipedia The double-slit experiment can be regarded as a demonstration that light and matter can display characteristics of both classically defined waves and particles. It also displays the fundamentally probabilistic nature of quantum mechanical phenomena. In a double-slit experiment using an electron beam an interference pattern is formed after experimenters record a large amount of electron detections. I have seen this answer by "anna v" which states an electron never travels through both slits only one slit per electron and the pattern formed is only a statistical probability distribution for the entire accumulation. But assuming in an experiment in which electrons travel one after the other and each electron travels only through one slit then how could the pattern on the screen be different from the one when we close one slit interchangeably and send electrons only through one slit at a time. (Actual experiments have shown patterns are different indeed) I think an electron through double-slit as a superposition of probabilities of spatial distribution. Like this picture below: But according to anna v picture that come to my mind is below (several electrons illustrated): So I have two related questions: Is the so-called wave nature of particles only a mathematical model or is there some physical nature (properties) to the probabilistic wave that passes through the double slit? Is stating whether the electron passes through both slits or it only passes through one slit just a personal opinion/interpretation that cannot be proven or disproven by observations? Edit: Evidence supporting simultaneous two (path) position: Using a Mach-Zehnder Interferometer to Illustrate Feynman's Sum Over Histories Approach to Quantum Mechanics One particle on two paths: Quantum physics is right by Vienna University of Technology Double-slits with single atoms: Selective laser excitation of beams of individual rubidium atoms by Andrew Murray, professor of atomic physics, University of Manchester, UK
I am working on a problem where I need to extract a connected tree of nodes based on certain attributes while optimizing for the minimum number of nodes. Some attributes of the nodes are known in advance, such as resource capacity. However, I do not know in advance how many nodes will be selected in the final tree, especially the intermediate nodes that ensure connectivity and may act as task-forwarding nodes. The goal is to formulate a mathematical optimization model with a set of constraints to select an ordered tree with the minimum number of nodes, where each node hosts specific tasks. Selecting individual nodes is straightforward, but I am struggling to write the constraints to ensure that: All selected nodes are connected. The hosted tasks follow a desired ordered sequence. For example, if nodes host tasks A, B, C, and D, task A should be in a node that comes before a node hosting task B, and so on. Multiple tasks can be placed in the same node if they do not exceed the resource capacity of the node and do not violate the ordered sequence of tasks. A node with task D may act as a root node in the tree. Most of the existing literature I found online assumes either the total number of nodes in the final tree or the total number of edges in the tree is known beforehand. In my case, it is possible that only a single node may satisfy the required attributes and host all the required tasks, or a set of constraints may be needed to satisfy the required attributes. Any hints or suggestions regarding how to formulate these constraints are highly appreciated.
Sorry if this equation is not phrase in precise mathematical form. I am open to suggestions to improve the explanation, and I have tried to formulate the problem as precisely as I could. I was talking to a friend--postdoc in PDEs--today and he was asking me about obtaining some Yahoo data on stock prices for his students. He explained that he had some students working on stock price data and he was looking for software packages/libraries for time series analysis--like ARIMA, GARCH, ARMA, ETS, etc. I told him that as far as I remember, most many models to stock prices or asset prices use stochastic differential equation like the Black-Scholes models. So I was asking if he actually needed like a stochastic differential equation solver, rather than a time series package. He said something that surprised me. He said that many people used to use Black-Scholes models, but implied that they are not used as much any more. He basically said that many people lost money using these models. I myself don't work in mathematical finance or in banking at all. However, I was unaware of any major critiques of stochastic differential equations and their applications. From a formal perspective, I imagine that these SDE models have a set of parameters (means, variances, etc.) that users can tune against some data using an optimization method. I am not sure what optimization would make sense, I can imagine approximate bayes or something, could work, but there are probably a million choices. So is the claim of a bad fit, mean that the error between the model predictions and the ground truth data too high, or perhaps changing over time, etc. Also, are there a different set of finance models being used as replacements for Black-Scholes?
I have seen that it is possible to approximate the metric in the presence of a gravitational field by the Rindler metric: Does a uniform gravitational field exist? Is there any acceleration in a uniform gravity field? Applying the principle of equivalence to an accelerated frame Rindler Coordinates and homogeneous Gravity Field Gravitational field strength and Horizon in Rindler coordinates Now as some answers in the links have pointed out, this doesn't quite describe a uniform gravitational, because the acceleration described by the coordinates depends on one of the spatial coordinates. My question is, how can we refer to this as a gravitational field at all? The Rindler metric is derived from a coordinate transformation on inertial coordinates on Minkowski spacetime, so we know a priori the Riemann curvature tensor will be exactly zero in the Rindler metric. In doing this "Rindler approximation" to the gravitational field, say, near the surface of the Earth, we started out with a nonzero Riemann curvature tensor (indicating spacetime curvature exists), and then we obtained a situation in which the curvature tensor vanishes everywhere in the region we're approximating. Doesn't this render the approximation invalid? Even if you argue that the region of approximation is small (which makes sense), there is no sense in which we can make the curvature tensor outright vanish (because the Ricci scalar, which is a contraction of the curvature tensor, is supposed to be invariant). Accompanying this strange change in the curvature, objects that "fall into the Earth" followed geodesic paths prior to the approximation, and under the new approximation, the same objects are now undergoing proper accelerations, meaning they are no longer following geodesic paths. Is there a physical interpretation or the mathematical reasoning behind this change? It seems like we are replacing spacetimes outright as opposed to approximating them. Is this understanding correct?
This question was inspired by the interesting discussion here: Why isn't the T in "relative" flapped? It seems like the adverb already and the two-word phrase all ready should be pronounced differently, but as far as I can tell, both sound exactly the same. For comparison, consider the other phrase/adverb pair all ready and already, where there is a difference: "The suitcases are all ready" doesn't sound the same as "The suitcases are already..." because the stress pattern is different ([all ready] as opposed to [al ready]). Additionally, the "l" sound at the end of "all" in the phrase goes on for longer than the "l" sound in the adverb, corresponding to the separation between the two words. If someone said the second sentence (fragment) out loud, it would leave the listener asking "already what?" In contrast, I wouldn't be able to hear the difference if someone incorrectly substituted the adverb altogether for the phrase all together. They both have the same sequence of stressed and unstressed syllables ([all to ge ther] vs. [al to ge ther]). More surprisingly, the adverb also keeps the aspirated "t" sound found in "together," even though that t is between a stressed and an unstressed syllable. This seems very exceptional, since that context usually requires the substitution of the "flapped t" for the "aspirated t." But, if this were the case, then "altogether" could be spelled "aldogether" without changing its pronunciation, which it can't. ("Aldogether" just sounds like I have a stuffy nose.) This also contrasts with "relative"/"reladive" like in this question, where the two pronunciations are interchangeable (the flapped t one being more common when it's said faster). "Relative" is also not an exception to the rule cited in the accepted answer, since there the t comes between two unstressed (or at least only tertiary stressed) syllables. The cited paper states that flapping the t is optional in that case. I'm interested in what the context (either historical/usage-related or some less well known phonological rules, or something else?) could be behind the pair not being differentiable by sound, and only by spelling, but none of the answers to the other question mention the word "altogether." No one seems to have asked this specific question yet, as this one is about meaning and not pronunciation. Here is a reference for the pronunciation of "altogether," confirming it always has a distinct "t" and not a flapped t/d sound: https://en.wiktionary.org/wiki/altogether. This proves that I haven't just been mishearing and mispronouncing the word the entire time!