darklight03/StackMIAsub huggingface dataset:
Euclid based much of his geometry on a theory of magnitudes that looks roughly like this: A general theory of whole and part and how they are related in size (eg: the whole is greater than the part). A general theory of the properties of magnitudes (eg: equals added to equals are equal). A basic rule that allows one to determine in specific cases that one magnitude is equal to another (eg: all radii of a circle are equal; all right angles are equal). Euclid applied this theory to lines, angles, and figures (meaning the area). From these simple foundations, he is able to prove the equivalence of all sorts of things that are not related by the basic rule alone. I've been looking for a modern development of this theory, and though there is some interest in it (Robering), modern work on mereology and mereotopology that I've been able to find online all does things that seem to violate the spirit of Euclid's work; in particular: They treat points as parts of lines and lines as parts of the plane (Robering comments on this). Euclid did not do that. In Euclid the parts of a line are lines and the parts of a plane figure are plane figures. They assume a single universal whole of which everything is a part. This is related to the first issue. Obviously there isn't a single universal line of which all lines are a part. Similarly for angles. Also related to the first issue: the works on mereotopology make use of the concept of an interior part, which basically means, for example, the part of a line segment not including the endpoints or the plane figure excluding the boundary. This doesn't makes sense when points aren't parts of lines and lines aren't parts of plane figures. The works on mereology assume that any two objects form a whole. Euclid only considered connected wholes. They assume completed infinities. Euclid avoided those. Robering treats an angle as the infinite plane section enclosed by the rays of the sides of the angle. I assume this is to turn angles into parts of the plane so they can be part of the universal whole. They assume that there is no maximum-sized whole, but this doesn't work for angles if you treat them as Euclid does as their own kind of magnitude. Can anyone suggest a body of work that I can find online that deals with some of these issues?
The direction of polarization of a transverse wave is defined to be the direction perpendicular to the direction of propagation of wave or the direction of oscillation of wave, right? But in the case of Electromagnetic waves, a class of transverse waves, there are two directions of oscillations that are perpendicular to the direction of propagation of the wave, the direction in which the electric field oscillates and the direction in which the magnetic field oscillates. But while referring to the direction of polarization of an Electromagnetic wave, only the direction of oscillation of Electric field is called the direction of polarization. But why isn't defined to be the direction of Magnetic field? For this I thought like this: "For transverse waves like waves on a string, to determine the direction of polarization, just put a plane with slit in different directions and the direction perpendicular to the direction of slit in which the wave completely gets absorbed is its polarization direction." Now, if I apply the same logic to the Electromagnetic waves by replacing the plane-with-slit with a material that can interact with the electric or magnetic components of the electromagnetic field, i.e., a polarizer, then I can define exactly what the polarization direction of EM wave is. First let us introduce a polarizer that can interact with electric field. Generally these polarizers contain some long chain linear molecules placed in the same direction. If an EM wave is allowed to fall on this polarizer, with the normal of plane of polarizer being in the direction of propagation of EM wave, in different directions each time, then the electric component of EM wave gets blocked if the direction of electric component and the axis of the long chain linear molecules are parallel. Hence the polarization of EM wave is along the direction perpendicular to the axis of such long linear molecules or simply the direction of Electric field. Now if I conduct the exact experiment with such a polarizer that can interact with magnetic field then the polarization of the EM wave comes out to be the direction of oscillation of magnetic field. But when direction of polarization of EM wave is referred, only the direction of oscillation of Electric component of EM wave is considered. Why is that so? Don't the polarizers that can interact with magnetic fields exist? Or between the both directions of polarizations only the direction of Electric component of EM wave is considered conventionally because of simplicity? Or am I missing something?
Firstly: having a lot of difficulty figuring out how to articulate this question due to lack of general math knowledge. There are multiple questions posed below, but I feel like if I knew more they could be condensed into a single question and am hoping someone can suggest and edit to the effect of the below. Thank you in advance for your consideration and assistance on this point! Background I'm taking a course which includes descriptive statistics. In that course they describe the method of calculating covariance and provide that equation. I find myself wondering - why did they choose to define it as they did (using multiplication instead of addition between the terms in the numerator - not even sure if that description is accurate). My question? Are equations for things like covariance derived from looking at phenomenon and 'cracking the code' of how those phenomenon could be described mathematically via a proof? Or does one, at some level of mathematical skill, say: I want to model this phenomenon and I want that model to have these features and qualities to its output, therefore, I choose this particular structure to achieve that goal and then I prove that functionality through a proof? If the latter case - what progression of mathematical learning develops that skillset? Is the same skillset used in both cases? Why I ask If I understand the motivation of the creator of the covariance equation, I could compare it to my own motivation and perhaps come up with a different approach to the same problem that better fits my own goals because maybe our goals are similar but not the same... Thank you again for any advice on how to simplify this..
I'm trying to understand how electromagnetic radiation is created and can propagate through the void. I do understand the concept of an electromagnetic field. But I don't understand how we get from a "field" to a "wave". I'm not really interested in detailed mathematics of how this happens, rather I'm looking for a complete high-level answer that to the extent possible: Explains every step of how we get from having nothing to having an electromagnetic wave that propagates through space (e.g wifi signal) and Provides clear and intuitive justification for any point/law/fact it uses. Preferably all of this should be included within the answer but I do appreciate inclusion of helpful links and references for additional context. Below is my current understanding along with some more specific questions. To create electromagnetic radiation: You make some charge move (e.g through an antenna by creating an oscillating dipole). This moving charge creates fluctuating electric fields and magnetic fields around it All good so far as this is what you expect a charge to do, however these fields will become weaker proportional to inverse square of the distance and you would expect them to basically disappear at a distance. Electric and magnetic fields somewhat magically interact and now you have a self-perpetuating wave that doesn't fade out like the field. I don't understand this part, e.g how we get from a "field" to a "wave" and unfortunately most of the resources I tend to skip over why. I believe there should be a better explanation. For example, according to Wikipedia, Faraday/Len/Lorentz laws have to do with this. However all these laws/theories require a conductive "circuit" that we don't have in the air in the vacuum of space (I do understand how these laws explain how your antenna would receive an electromagnetic signal). Considering magnetic and electric fields/forces will act on charged particles, this raises a few questions: Does the magnetic field produced by the antenna somehow create charge in the air surrounding it? If the whole wave propagation is based on interactions between the fields and charged particles, then how can a wave propagate through the void of space when there is no charge? Thanks!
I'm an undergrad with an electronics repair background trying to find an explanation for one of the fundamental aspects of a transformer. Every explanation I've found of a transformer's basic operation insists that there is essentially no power loss; that is that VxI in the primary = VxI in the secondary. This suggests to me that if there was effectively no current in the secondary (i.e. it was an open circuit or there WAS no secondary) that current would more or less stop in the primary as well. My instructor adamantly insists that this is not the case, and that current in the primary is constant no matter what might be happening in the secondary which boggles my dang mind as it seems to fly in the face of every explanation I see, and I have not been able to get a clear explanation out of him. If this were the case, then a device with a transformer would consume equal power regardless of whether it was operating or not! Right? It seems to me from an intuitive and practical perspective that the EMF on the secondary should be relatively constant with respect to the voltage across the primary, and that the currents through each should be proportional at any given time, so that a changing load across the secondary should somehow influence the current in the primary, but I can't work out or find an explanation of that... mechanism. I feel like it must have to do with the secondary influencing the magnetic field in the core... Please help! I've been wondering about this for years!
In QFT, many mathematical issues arise. Setting aside renormalization, these deal with rigorous constructions of objects underlying QFT: i) In the canonical quantization approach, the main issue comes from trying to multiply (operator-valued) distributions. My understanding is that mathematicians have formalized some settings in which this makes sense, but you have to be very careful ii) In the path integral approach, the main issue comes from defining the path integral (both in terms of defining a sensible measure on paths, as well as making the integral well-defined despite the presence of oscillatory integrals). My main question is: Are the two issues (one for the canonical approach and the other for the path integral approach) related? If so, intuitively (from a purely mathematical perspective) how is the problem of defining products of distributions related to the problem of defining path integrals? I'm particularly curious as to whether there's some intuition to be gained from the usual proof of the equivalence (between the two approaches) in non-relativistic QM (which begin's with Schrodinger's equation, inserts of bunch of intermediate states, and removes operators one by one). In the case of non-relativistic QM, my understanding is that the canonical approach can be made fully rigorous, while the path integral approach isn't quite so (one can use Wick rotation to compute the integral using the Weiner measure, then rotate back using some analytic continuation argument, but I have read that this is only justified in some cases). Given this, do we expect that the issues of rigor aren't quite the same in the QFT land as well? Disclaimer: Apologies for any inaccuracies in my characterization of anything, as I'm still a beginner grasping with many aspects of QFT I am aware that decades of work have gone into formalizing QFT rigorously, and have addressed many of of i) and ii) for different variations of QFTs. What I'd like to understand here in particular is, if these approaches have given insight into how the different issues I outlined above are related. EDIT: I edited the original question to focus on just one question. Originally, I also asked about renormalization (which is what one of the comments addresses)
Would the positive and negative charges line up on either end of the wire? Or would it induce a current? Or would the wire be unaffected by the magnet? This was deleted for being a homework question. I'm not aware of any homework questions like this. I thought this up in my own head. (I drew the diagram in a program called Biorender, which I use regularly.) This IS a question about the underlying nature of physics. The question I have is this, "If simply moving a straight wire through a field causes motional emf, why then does a loop of wire require an area and a certain angle to create a current?" Why wouldn't moving a loop create a current if moving a straight wire creates a potential difference? All of the explanations I can find out there separate the two concepts from each other. "Because Lenz's law states..." How are they tied together? It's still a wire through a field. Physically, what happens to the charges when a loop moves through a field as opposed to when a straight wire moves through a field? But before I can even get to how the two situations are the same or different, I had to clarify some things about how a current can be induced in a straight wire. The following image was meant as the answer to my question, but Stack Exchange seemed to feel it went in the question. Now it may not make sense why I needed an answer when the answer is below. But that's why the answer is in the question itself in case anyone is confused about that.
I have been trying to learn about lattice path integrals. Unfortunately, majority of the literature on this topic is in regard to Lattice Quantum Field Theory and Lattice Quantum Chromodynamics. That is fine, however what I desire is literature on evaluating plain, standard quantum-mechanical path integrals on a discrete spatial-lattice, ideally in continuous-time. This could be a simple chain of atoms or a crystal lattice, but it would be a system where the lattice actually exists and is not just used as a regularisation tool for a continuum quantum field theory. For example, what is the "path integral" (I imagine this would instead be a summation now instead of an "integral") for a free particle on a lattice? Can it be evaluated in that it has a closed-form solution? Even if an ideal closed-form expression cannot be obtained, how far can one go? (E.g. reduce to discrete Gauss-sum or a Jacobi-theta function) In the continuum case, this is just a Gaussian functional integral, which results in a Gaussian. What would it be if space is discretised? Then what is the path integral for a harmonic oscillator on a lattice? What if you add a force term? What about a general quadratic Lagrangian? I have struggled to find mention of these problems in the literature. The standard approach is to introduce the typical time-sliced path integral derivation, and then jump to the Klein-Gordon equation. Are there any resources that do not jump to field theory and instead just investigate these lattice problems for simple quantum mechanical toy models? As a slight aside, I was wondering also if you could start from a typical continuum space path integral, for say a free particle, and add a delta-functional constraint to restrict the continuous path to lattice sites? This would enter the action then as a Lagrangian multiplier and may be evaluable. Thanks!
I've started to study in details quantum optics and I find difficulties in linking the concepts of coherence and correlation among fields, especially because I'm building right now a background on classical optics aven if I already have a strong background in QFT. As far as I have understood, roughly speaking, coherence is meant to be some kind of measure on how similar the properties of one or more waves are at two different spacetime points. To describe with a quantity this feature, we make use of correlation functions to build the coherence factor (I'm thinking about the second degree of coherence): the closest the factor to one, the greater coherence my wave(s) shows between the two points. Furthermore, correlation functions usually arise in photodetection experiment, stemming for the importance of coherence itself in any process regarding the measurement of EM field (I know there are alternative ways to takle detection, but I'm sticking the question to the very basics of quantum optics). Now, this means that correlation functions give me the "degree" of dependence between the photodetection probabilities at two different points. This means that they have more or less the same role that in any other QFT. But now, this seems to me conflicting with the nature of coherence itself: more specifically, in (free) QFT we have the so called "miracolous cancellations" where spacelike correlation functions vanish due to commutation relations, but in the optical theory spacelike correlation functions are at the grounds of spatial coherence measurements, and thus are usually non-zero. I don't get what am I missing, probably I'm misunderstanding some basic stuff regarding coherence itself, and I'm hoping to get enlightned about it. Thanks for the help.
After reading the first two answers to this question, I've become interested in understanding the concept of (co)tangent complex as a way to get some intuition about homotopical algebra, being somewhat more used to the algebro-geometric framework than to the algebro-topological one. More specifically, I'd like to understand this concept in order 'to do basic geometry - this time calculus - on a singular variety', as stated in the first answer (whatever this means), but not 'mechanically', instead trying always to keep an organizing point of view like the one described in the second answer. Also, I'd like to do so following the shortest possible path from basic algebraic geometry and basic category theory directly to the subject matter, with the smallest possible amount of detours, but comprehensively including all the needed basics. (I've studied some scheme theory and homological algebra before, including derived categories, and also ventured a little bit more deeply in the categorical world, but never dealt professionally with these topics and will have to recall a lot before being sufficiently at ease with them.) In this context, what I'm looking for is a double list of topics, one from algebraic geometry and the other from category theory, both ordered by degrees of complexity, designed to be studied in a parallel manner, showing the highest possible level of correspondence since the very beginning, and if possible accompanied with the most up-to-date literature available for this purpose. I'd be very grateful if someone would spend some time thinking about this and writing a nice answer.
There is a particular Twitch streamer from a video game I played, MermaidonTap. If you subscribed and follow her, not all but most of her public streams, she uses "fuck" and a lot of the word "cunt." I am a fellow American citizen, but was not born in US but in an Eastern Asian country, and am trying to learn English. I'm just wondering, is it normal for women to use the word "cunt" normally? I took university English and finally passed it on the fifth time growing up in America. So I'm trying my best to understand how the words in English come about and how certain words passed down to certain generations. As I have researched, the word, "cunt" is a derogatory term towards women and as well as another meaning as vagina. It just seems like, it's derogatory when men call women cunts. But what about women calling other people "cunts." I still find it derogatory, vulgar and I find it hard on my ears to hear words like "cunts" or "fucks" from women like her. Also, some of my teachers and some friends and my own mother, they both say that people who curse a lot and can NOT control their word usage, means they don't know any other language but vulgar words. They didn't go through college and university. They're not educated was what I heard and been told. Do you think this is also true or not? Overall, I'm just trying to understand the American English literacy. I don't mean to offend anyone but rather understand the meaning of the word being used at a person or in a public audience, like her on her public Twitch stream, MermaidonTap.
As far as I know, the first statement of the correspondence is between two formal theories named simply typed lambda calculus and intuitionistic propositional logic, which maps types to formulas and terms to proofs. We also have other statements for higher order logics and type theories. But it is also famous to replace the word "term" with "program" when people try to express the correspondence informally (like Wikipedia). I think that if we assume that programs and terms are same things here, then we can conclude that writing a program, which means expressing a program in a possibly Turing-complete language, is actually proving a mathematical theorem intuitionistically (without the use of law of excluded middle and other proof techniques that are banned for an intuitionist!) Is this true? I think that replacing the word "term" with "program" is misleading here! Because in any statements of the correspondence that I am aware of, we don't have a Turing-complete type theory, and I don't think that type theories are computational models like Turing machines and untyped lambda calculus. Also I think the fact that untyped lambda calculus is equivalent to Turing machine is also misleading to think that a term in a type theory is equivalent to some Turing machine, while as far as I know the equivalence between untyped lambda calculus and Turing machines is not necessarily a bijection, and even if it is, what does it have to do with simply typed lambda calculus?! All in all, don't you think that using the word "program" in the statement is wrong and misleading?
A textbook I'm using to refresh some basic grammar states that indirect objects can be identified by it's answering of questions such as 'to whom', 'to what' etc. (fair enough) and they always come before direct objects in a sentence (this raises questions for me). So the text would identify the pattern in: The teacher gave the students homework. as: S - TV - IO - DO but the pattern in: Tim kicked the ball to Ken. as: S - ITV - Prep. It's been a while, but I was taught the direct object received the action of the verb and the indirect object received the direct object, and also that a verb's classification of transitive or intransitive arises from how it is used in the sentence (i.e. it's not intrinsic to the word itself). So that I would have identified the second example's pattern as: S - TV - DO - IO because the preposition is receiving the DO and therefore is the indirect object. Since 'kicked' can be used with or without an object (i.e. The baby kicked.) I let it pass thinking the text and I could both be correct. But a third example from the text has me questioning how transitivity is assigned: Problems led to desperation. the text again gives the pattern as: S - ITV - Prep. But, 'Led' is almost never intransitive - not unless it's the answer to a question or given some additional context. And so this classification seems more forced to me. The text seems to be implying that the role the verb plays in the sentence depends on how you classify the thing it is acting on and that prepositions can not be indirect objects (despite receiving the direct object as well as answering the question 'to what'). Could someone please clarify, illuminate, or otherwise help me make sense of this?