darklight03/StackMIAsub huggingface dataset:
I tried finding a decent book to study nonstandard analysis from and found Goldblatt's Lectures on the Hyperreals. However, I was very disappointed to find out that the text is not rigorous at all --- Goldblatt doesn't even prove the transfer principle. Are there any completely rigorous treatments of the hyperreals? Ideally, I would like them to cover the same content that Goldblatt's book does: construction, nonstandard analysis and miscellaneous applications. However, I'd be more than satisfied with just construction and nonstandard analysis. I would also like the book to be more-or-less self-contained in this regard. That is, if the book simply states the transfer principle is a consequence of some stronger model-theoretic theorem, I don't gain anything from that. Of course, some knowledge of analysis, algebra and logic must be assumed, and I'm fine with all of those on a basic level. To clarify, I am also not interested in axiomatic approaches to these structures. So, for example, a construction of the hyperreals would be much preferred over an axiomatic introduction.
A popular misconception in the layman public is that the Big Bang was some sort "explosion" at a single point of space, where originally all matter was concentrated and then it "exploded" outwards. This is of course different from the modern general-relativity understanding of reality, which is that it is space itself which expands - not the content of the space moving, and the Big Bang did not start at a single point, but everywhere. My question is - what experimental evidence do we have that can convince people that the explosion model can't be right. Note that I'm not asking why GR guarantees that the space-expanding is the correct model, not the "explosion". I know that. I also know that GR has a lot of experimental evidence for its correctness at least in smaller scales. Rather I'm asking which evidence we have from astronomy, CMB measurements, or whatever, showing directly that the "explosion" model simply cannot be a valid explanation of the universe's history.
You know how when you talk to someone about a bad time you're going through, and they feel the urge to one-up you to achieve some sort of imaginary victory point? E.g. School is challenging because I have to catch a bus early in the morning. Pfft, that's nothing, I used to have to WALK to school. I did the dishes and the laundry this morning so I'm a little tired. You're tired from that? That's nothing, I swept and mopped my entire house AND my parents' house, I'M tired. Are there words/phrases that describe this behaviour? Either as verbs to describe the act itself, or as nouns to describe the competition. I've seen "struggle Olympics" used before and that's pretty accurate but I'm pretty sure that's just internet slang.
A sufficiently strong electromagnetic pulse can/will destroy smartphones and computers. I know somebody who went into MRI machine and forgot a Visa credit card in his pocket. The card was toast and he had to get a new one. A mobile phone in an MRI probably wouldn't fare better. But a big part of the human body itself is based on electric signals. The brain and nervous system, including the heart, works on electric signals. And those signals have to go to very precise places. There is an area of brain processing vision, another is responsible for speech, etc. Also the heart function depends on precisely timed signals traveling very specific routes. So it would seem that a trip to an MRI scan should totally fry anyone's possessing brain and heart. Except it doesn't. An MRI scan is harmless (if you are not allergic to those injections they give). Why? And then there are those electromagnetic pulse devices they show in Hollywood movies. While totally trashing electronics of bad guys, fellow humans are always shown unharmed. Again, why should the brain be different?
Suppose there are two arbitrary side lengths of a right angled triangle that are known to us. There are two possible cases here that I can see: Either one of the side lengths given is the length of the hypotenuse. Both the sides lengths given are the lengths of the legs of the right angled triangle Now, additionally one arbitrary acute angle measure is also known. There are again two possible cases that I can see: The angle lies between the two sides that are known to us (in which case it will then be confirmed that one of the sides given is the hypotenuse). The angle does not lie between the two sides that are known to us (in which case it will then be confirmed that the sides lengths given are the lengths of the legs of the right angled triangle). Now, I want to know if there is any possibility of determining the third side of the right angled triangle using the conditions given above (two arbitrary sides and one arbitrary acute angle) and without any additional conditions for eliminating the cases given above. If possible, please tell me the formulae/procedure to follow. Thanks in advance.
While the gravitational path integral is not a well-understood concept mathematically, a number of works (particularly in recent research connected to AdS/CFT) emphasize the importance of integrating over metrics on surfaces with nontrivial topologies as well as surfaces diffeomorphic to a flat spacetime. These metrics are obviously not continuously deformable to a flat spacetime metric. A superficially similar concept occurs in Yang-Mills with the notion of a large gauge transformation, which relates gauge configurations that are equivalent but not homotopically equivalent. In Yang-Mills one can also introduce a "topological term" which has no effect on the equations of motion, but computes the homotopy class of the gauge configuration. There is a somewhat analogous object in gravity, the Gauss-Bonnet term, which has no effect on the EOMs and which computes the Euler characteristic of a spacetime. My question is: is there a sense in which integrating over nontrivial topologies in a "gravitational path integral" approach to quantum gravity can be understood as including large gauge transformations in the gauge group of the theory?
I am new to algebraic geometry. I can't understand why an etale space is considered locally homeomorphic to its base space. The analogy I have heard is that etale space can be visualized as "stacked over" the base space, or that it is like a "puff pastry". I understand that we can (locally) map this stacked/pastry space to the base space, but I don't understand how an inverse map could exist. It seems to me like the inverse map would be multi-valued as it takes a point in the base space to several points in the etale space. As an example, consider the base space to be a circle and an etale space to be a circle winding around itself twice. The map from the etale space takes two points to one point of the circle. Doesn't this mean that the inverse map is multi-valued? Michael
I know that white light, upon entering another medium from air/ vacuum, disperses into its constituent colours. Essentially when travelling in the air, all of the constituent colours have the same speed but when entering another medium, they travel with different speeds and hence they are refracted by different amounts - red the least and violet the most among the seven colours. So we are able to see the constituent colours (especially in the case of a prism since they are diverging.) But I came across the following situation where green light (mono-chromatic) changes into red, as perceived by the human eye as well as the camera. Attaching an image of what happens: Why does this happen? I am a high school student and I haven't read anything so far that could explain my observation. Also as far as I know the properties of light are dependent on its frequency and frequency does not change from medium to medium. I saw the other posts but it would be nice if someone explained it in a much simpler language.
We commonly anthropomorphise or personify non-living things by giving them human characteristics: "the angry storm". We zoomorphise things by giving them animal characteristics: "the storm roared". But what word would describe giving general (not necessarily animal) attributes of living things to non-living things: "the storm died"? A more specific example would be treating geomorphological landforms as if they have a life-cycle, but in the context of drawing parallels with the life-cycles of trees and forests. The additional human or animal connotations of "anthropomorphise" and "zoomorphise" do not apply, so it would be good if there were a more general word for this practice. This is very similar to the question "Equivalent for "personify" that's not human-specific". However the accepted answer in that case was "zoomorphise" since the OP was talking about a self-driving car being "hunted" [like an animal], whereas I am looking for the right word for a more general case.
Semantic consistency of a theory T is defined by there being a model M in which all theorems of T are true. Syntactic consistency of a theory T is defined by there being no formulas A such that both A and it's negation ~A are provable. So let's assume semantic consistency and then assume syntactic inconsistency. Since T is inconsistent, both A and its negation ~A are provable. But since it's also semantically consistent, there must be a model where all of its theorems are true. So there must be a model M where both A and ~A is true, which is not possible. So semantic consistency implies syntactic consistency. Is this conclusion right or am I severely misunderstanding something? I thought that semantic consistency only implied syntactic consistency under the additional assumption of soundness, but soundness was not assumed here. Is this conclusion a consequence of defining semantic consistency too broadly? Should it be limited to there being a model where the axioms are true?
In classical mechanics , the Laplace-Runge-Lenz (LRL) vector is a characteristic feature of the Kepler problem. This enables a very simple discussion of the properties of the orbit for the problem. It is an extra conserved quantity besides the total energy and angular momentum for a particle moving under the influence of an inverse-square force field. More importantly, this is associated with some symmetry of the problem. However, if we consider motion in the gravitational field of a black hole, the analysis of the orbits of test particles become more complicated and we require to use general relativity. So, is there any analog of the LRL vector in general relativity? The description of black hole geometry by pseudo-Newtonian potential (e.g., the Paczynski-Wiita potential for non-rotating black holes) is a well-known concept. Is the LRL vector applicable in such systems for motion around non-rotating black holes having spherical symmetry?
So I've recently started taking maths seriously in hopes of doing physics at university and have gotten pretty good at algebra and learned some very basic calculus but I'm having trouble finding a good introductory geometry textbook. I know this question has already been asked at least twice before and I looked at the at some of the recommendations, but I feel they're geared toward more undergraduate study as I'm assuming the guys who asked the questions already did high school geometry which isn't what I'm looking for. I'm looking for comprehensive high school equivalent introductory geometry textbook that's fairly challenging and would set me up for learning more in depth topics that crop up in physics later. (I say fairly challenging because, like I said, I feel like I'm pretty good at algebra). Consider me somebody who has never done geometry before because that's basically what I am. Thanks. (PS. I've never asked for book recommendations before so forgive me if I'm being too vague, I'll gladly amend my request if anyone needs clarification)
I am struggling to see the motivation behind Radon measures. In many places I see that they are described as measures that "interact nicely with the underlying topology", but without further elaboration. On Wikipedia it states A common problem is to find a good notion of a measure on a topological space that is compatible with the topology in some sense. One way to do this is to define a measure on the Borel sets of the topological space. In general there are several problems with this: for example, such a measure may not have a well defined support. Yet they go on do define a Radon measure as being defined on Borel sets so how does that solve the problem they mentioned? What is the usefulness of Radon measures and why are they defined as they are?
Background: My understanding is that model-theoretic semantics (MTS) and proof-theoretic semantics (PTS) differ in the following ways. In MTS, you first define the notion of truth in models and then having this notion you develop a "good" (sound, complete, decidable) proof system. In PTS, you just define a proof system and take this system itself to be the the definition of "truth", without even talking about models. So both MTS and PTS have proof systems, but these proof systems have different roles. I understand why proof systems are needed in PTS: because that's the only thing you have in PTS -- there are no models. But what is the motivation behind studying proof systems in MTS, given that in MTS there is already a notion of truth (with respect to a model) defined? Why do we need an extra apparatus of proof system specifically in MTS?
I'm confused by how thermometer works based on The Zeroth Law of Thermodynamics. The Zeroth Law of Thermodynamics said that "If a body, A, be in thermal equilibrium with two other bodies, B and C, then B and C are in thermal equilibrium with one another." Then, about thermal equilibrium, it said that "If when two bodies are placed in thermal communication, one of the two bodies loses heat, and the other gains heat, that body which gives out heat is said to have a higher temperature than that which receives heat from it." Then if we use a thermometer to measure our body temperature, if the temperature of our body is higher than the temperature of the liquid inside the thermometer, will the heat flow from our body to the liquid inside the thermometer to reach the thermal equilibrium? If yes, then will our body lose heat and temperature of our body decrease? How does the Zeroth Law of Thermodynamics on Thermometer actually works?
Greetings fellow physicists. I have some questions about the ability of different electromagnetic waves to pass through materials that I hope you can clarify. It seems that microwaves can go through concrete, wood, etc. since we can listen to the radio inside houses. However, it doesn't make sense to put a TV antenna indoors because the pictures might flake; so it is apparent that radio frequency doesn't have trouble passing through walls and so on, but the TV frequencies have. I'm guessing as the frequency increases, it gets harder for waves to get through non-conductive stuff. That would explain why we can't see inside boxes - visible light reflects instead of going through. Things get more confusing with X-rays and gamma rays. They seem to move through matter easily, like how X-rays image our skeletons and are used for security screening. Also, radiotherapy uses gamma rays to reach tumours inside patients. So I'm not sure if microwaves really penetrate concrete and other materials or not. But visible light, x-rays and gamma rays acting so differently is puzzling. Can anyone explain the physics behind why different waves can or can't pass through various materials? Some insight would be greatly appreciated!
In Australia, we use the expression Tall poppy syndrome for a "social phenomenon that occurs when someone's success causes them to be envied, resented, criticized or discredited." https://en.wikipedia.org/wiki/Tall_poppy_syndrome Conversely, is there a term or expression that's used for a social phenomenon where successful and popular figures are respected and blindly defended only because they are, well, popular and successful. I notice this behaviour in the YouTube world where smaller channels (or lesser known Youtubers) get extreme flak for criticizing popular channels, no matter how right they can be. Example: Critic: I cannot stand this person. His opinions have no basis. I can't believe he's so popular. Fan: Oh my god, you're such a pathetic jealous hater. Get a life! Critic: Wow, you sure suffer from [insert expression or word]. People like you are hilarious. I am fine with an idiom, a single word or an expression. Just whatever fits.
I would like to know of any techniques that can be used to measure the electric field strength precisely and accurately in both time and space. I know that there will be physical/ practical limitations to this and that it is not possible to measure at an infinitesimal position, but I would still be interested to know what techniques are out there. Assume the electric field I am interested in varies significantly in all spatial directions, and in time (I believe this means the use of a typical antenna is not possible as it sort of depends on a plane wave i.e. non-varying field in the z and x directions at a point in time, if the wave propagation is in the y direction). Assume that any electric fields are oscillating at frequencies at or below microwave frequencies. Thank you in advance for any insight.
I've asked this question in Quora and the answers I got were: First answer: Using "more" and "less" helps maintain clarity and consistency in comparative forms. It provides a straightforward and predictable way to form comparatives and superlatives without relying on irregular or unpredictable suffixes. Second answer: Longer adjectives often have complex or multisyllabic structures, making it more challenging to add suffixes like "-er" and "-est" without affecting the pronunciation or flow of the word. Using "more" and "most" allows for a smoother and more natural-sounding comparative form. Third answer: Using "more" and "most" for more-than-one-syllable adjectives maintains consistency with two-syllable adjectives that also use "more" and "most" for comparison. This avoids creating separate rules or patterns for different types of adjectives Is there an identifiable reason for tending to restrict the forms -er, -est to single-syllable adjectives? It may be one of these I suppose; or something else.
I am actually having an introductory course in Special Relativity in which I was looking at the Michelson Morley experiment. And I have this silly confusion. The setup for the Michelson Morley experiment looks something like this in every textbook (assuming the observer is in the comoving frame) When this same setup is being observed by the observer in the hypothetical frame of "ether" (which this experiment was designed to detect) , then the vertical beam of the light is no more vertical but along the hypotenuse(the dashed red line) and this should be true so that the two observers agree on the results of the experiment. But I really don't understand why the light should not go vertically as the shown path of the light beam seems to violate the laws of reflection as the angle of incidence is no more equal to the angle of reflection. So how can then the two inertial frames be equivalent ? PS :- Nobody in the class bothered about this, not even the instructor of the course.
I know that the tangents from a point to a conic section subtend equal angles on the focus. However, I have mostly studied conic sections from the perspective of coordinate geometry, so even when there are properties common to all conic, I have to prove them separately for parabola, ellipse, hyperbola, and circle. The only common denominator between these figures I know of is that they are the locus of points which have a constant ratio of distance from a fixed line and a fixed point. However, I haven't been able to use that much to my aid, having hardly any experience in dealing with these figures in such a way. Observing the apparent simplicity of the result, is there a simple proof for the theorem? P.S.: I would be thankful if someone could suggest resources that deal with such results about conic sections, especially if they use synthetic geometry.
The Leibniz formula for determinants starts with multi-linearity and the alternating property and builds from there. I asked a question about why we should start with multi-linearity: What's so special about multi-linearity?. And the response (which had an excellent alternate motivation for the determinant which I wasn't aware of) seemed to be that there is indeed no good reason. The thing that keeps bothering me is that we are talking about linear maps here. And the determinant is really a property of the linear map. And given the definitions of linearity in the context of linear maps and the multi-linearity of determinants look exactly the same, I would assume a statement like this exists - "because the determinant is a property of linear maps, it clearly must satisfy the property of multi-linearity because _____". Is there no way to fill in the blank?
I just stumbled across this older video of a girl trying to "talk" to Alexa (the voice assistant). She says "I am trying to talk to you, hen". Now, I am not a native English speaker, but I am familiar with the endearing term "honey", also sometimes written "hunny" or even "hun" or "hon" depending on the pronunciation. However, the pronunciation "hen" made me think of the word for a female chicken. I've heard the word "chicks" to refer to females (but I think this is generally not an appreciated term? - again, not a native speaker), so there is a connection to use terms for poultry to refer to people. So my question is, in Scottish, what is the root/etymology of "hen"? Is it related to "honey" or to "hen" (chicken), or maybe something completely unrelated?
Consider a metric space with a path between any two points, so a real line segment of some length between them, and the length of this line is the same as the distance between the two points in the metric. This is called a geodesic metric space. Any two points are connected with a shortest path. There could also be multiple such paths as with the antipodes of a sphere. What I mean by homogenous is that any point can be mapped to another while preserving the structure of the space. Another way of looking at it is that the space is the same no matter what point is taken as the origin. Geometries like Euclidean and non-Euclidean geometry would be examples. However, I am curious about examples which are not locally Euclidean. So, they would behave kind of like Euclidean space in that there is a measurable shortest path between two points and everywhere in space is the same. So you could "move around" in a given "direction" in this space, but it does not locally resemble Euclidean space. Is this even possible? If so what characteristics would these spaces have and could they be classified? What about the underlying topologies of these spaces?
Say that I have two long cylinders of material as shown that are in a region of oscillating magnetic field where the field direction is aligned with the long axis of the cylinders. I want to calculate the inductive heating of each cylinder (let's pretend for a moment that none of the parameters like resistivity are temperature independent). I know how to calculate the skin depth and therefore the heating when only one is present, but don't know how to deal with two cylinders. Does the inner cylinder "steal" some of the magnetic flux by reducing the flux from its own induced current? If so, how much? I would think there would be time lag between the flux and the induced current. Any thoughts or references explaining how this can be attacked? Thank you