darklight03/StackMIAsub huggingface dataset:
I am new in Linear Algebra and I have encountered the concept of Solutions of linear equations. According to the textbook I use, a solution of a linear equation is a vector whose components satisfy the equation. However, I am puzzled by the choice of "vectors" specifically. Why is it not defined as a point whose coordinates are the same as the values that satisfy the linear equation? After all, a vector is more than just its components; it is also a set of points and its "components" define its magnitude and direction. A point, on the other hand, is a single entity with no additional properties. No magic. I think It's just a bit more intuitive to think of it this way instead. I can also anticipate that Vectors have a deeper meaning that relates them to linear equation systems but I can't tell what exactly. I'm going through the chapters and I'm still waiting for the "Bingo! I figured it out."
I have been reading the Loop Antenna section of Antenna Theory by Constantine Balanis and trying to understand how exactly a ferrite core improves the performance of a small loop antenna. Balanis writes, The radiation resistance, and in turn the antenna efficiency, can be raised by increasing the circumference of the loop. Another way to increase the radiation resistance, without increasing the electrical dimensions of the antenna, would be to insert within its circumference a ferrite core that has a tendency to increase the magnetic flux, the magnetic field, the open-circuit voltage, and in turn the radiation resistance of the loop. How exactly does increasing the magnetic flux (adding the ferrite core) increase the antenna's radiation resistance? Balanis claims the voltage increases, and presumably deduces that the radiation resistance increases likewise from the equation V=IR. However, this would only follow if we knew that the current I did not increase as the flux increased, which is not obvious to me. It also is not obvious to me why the voltage increases in the first place!
I've been really stumped on this particular concept. In Case A, when a bar magnet is brought towards a copper coil around a soft iron core, in accordance with Faraday's Law of Electromagnetic Induction, the the pole facing the magnet acquires a North Polarity while the opposing pole acquires a South Polarity. Now by Lenz's Law, the direction of the induced EMF must oppose the cause that produces it, but the current is in the same direction as the moving magnet. The galvanometer deflects in the same direction too. Why doesn't this obey Lenz's Law? And additionally, in Case B, by merely reversing the winding of the copper coil, the current flows in accordance with Lenz's Law and the Principle of Conservation of Energy. My question is, why does changing the winding of a coil defy Lenz's Law in Case A and then follow Lenz's law in Case B? Something such as the Principle of Conservation of Energy and Lenz's Law should apply in any case. The method of winding shouldn't matter that much right?
Is it correct to say that, in English, when you use the Present Simple tense in the Interrogative Negative form you are either implying the negative or just confirming the affirmative (depending on the order of "Do", "subject" and "not"), but no other possibility? Like in the scene of The Lord Of the Ring - The Return of the King, when Eowyn is talking to Aragorn just before he leaves to see the dead army in the mountain. Eowyn: "You cannot abandon the men. We need you here." Aragorn: "Why have you come?" Eowyn: "Do you not know?" In this last line, given the context, she is clearly implying that she thinks he knows she loves him, and thus, she is asking it just to confirm it (as I understood it). But then, if she had used the other form "Don't you know?", it would change the meaning, as she would appear to be asking it to really have a first idea on his feelings, not presuming anything. Then my question is: can the Interrogative Negative have only these two possibilities?
Have studied traditional point-set topology, but find there's a fairly large gap between the preparation typical point-set courses give you, and the level assumed in algebraic topology texts. Looking for a good introductory book on topology that uses more categorical / modern language - something that would segue smoothly into, for example, Tammo tom Dieck's text Algebraic Topology - at least covering most of the topics mentioned in the first chapter of his book (subspaces, quotients, products, sums, compactness, proper maps, paracompactness, topological groups, transformation groups). Have perused the MIT textbook 'Topology - A Categorical Approach', which looks decent, but isn't so comprehensive. Have found one text that looks good (Grundkurs Topologie by Laures and Szymik) - but unfortunately it's in German! (Anyone know of an existing English translation?) Open to recommendations. Thanks.
I'm studying nonlinear control systems, especially the Pontryagin's minimum principle and its applications. Throughout my studies, the authors have always defined the control systems with state variables and output variables. However, in this article Optimal Control of an SIR Model with Delay in State and Control Variables and so many others, it doesn't seem like control systems in epidemiology has output variables equations (or at least clear ones). Why is so? And is it possible for a nonlinear control system not to have output variables? I have found this answer: It is possible for a nonlinear control system not to have an explicit output variable, depending on how the system is defined and what its purpose is. In a control system, the input variable is typically the signal or information that is used to control the behavior of the system, and the output variable is the variable that is being controlled or affected by the system. However, in some control systems, the purpose may not be to directly control an output variable, but rather to achieve some other goal. Is it correct?
This might be a stupid question, but, why is it that gamma rays are able to penetrate almost any barrier without question? We know that gamma rays are simply high frequency waves with massive amounts of energy. However, what processes can enable it to go through layers of bonded atoms of metals and just about anything else? On another note, we know that beta particles can pass through paper but can not penetrate a thin layer of metal. What is different about the bonds of metal so that an electron can't squeeze through? I assume it is because of the tightly bonded metallic bonds and the high volume of electrons that flow throughout the metal that causes this. However, if this is the case, what would a beta particle do if it is "rejected"? Would it join the flow of electrons in the metal, would it just stay on the same side of the sheet or would it rebound in the opposite direction due to the equal, opposite force exerted on it by the sheet?
I am studying interacting QFT in the context of quantum fields in curved backgrounds, and I am getting some confussion about the concept of particles. To study some gravitational phenomena involving particles (e.g. Unruh effect, Hawking radiation, etc.), it is typically sufficient to deal with free fields, which are expanded in mode functions and particle/antiparticle operators (i.e. its energy eigenstates form a Fock space). This can be done, in general, because the Hamiltonian of free fields is quadratic in the field operators, and therefore one can calculate a single-particle Hamiltonian which can be diagonalized, giving rise to a band structure by means of which we describe this Fock space (eg. for the case of Dirac fermions, the vacuum state, prior to a particle-hole transformation, amounts to a filled lower band/Dirac sea). However, when one considers an interaction term, the Hamiltonian is no longer quadratic in the fields, and this band-structure cannot be obtained by direct diagonalization of the single-particle Hamiltonian (I am not even sure whether the notion of band structure remains). As a consequence, I do not understand if particles/anti-particles can only be defined when the Hamiltonian of the theory is quadratic, i.e., when the evolution of the theory preserves tha Gaussianity of the states. If this is the case, I imagine that a mean field approximation, which turns the Hamiltonian back to a quadratic one, would recover the notion of particles, is this the case?
I'm working on a set of beamer class slides in TeXstudio. Usually upon compile, the preview slide visible on the right will be at the position of the cursor - instead, for me, it consistently jumps two slides behind. This is reflected in the behaviour of the preview when I enable the "scrolling follows cursor" option - there, too, the preview is always two slides behind where the cursor actually is. If the cursor is on one of the first three slides, the preview stays on the first one. I get this behaviour on two different machines, so I assume it's not some super specific local bug but something more broad. It also persists independently of whether or not the document is compiled in handout mode. Any help on resolving this would be very much appreciated, as this behaviour is quite annoying.
I noticed that in several 'throwing' sports like the javelin throw and the shot put there have been a few cases where competitors tried to introduce a technique where they throw the projectile further by rotating their entire body (these techniques were then banned for safety reasons). As an example, there is the cartwheel shot put, where the person performs a cartwheel and rotates their body before throwing. I am not sure what advantage this has apart from that the ball is pushed over a slightly longer distance so that there is more kinetic energy. In the spinning javelin throw, instead of throwing the javelin with a regular technique, the athlete spins around and slings it out of their hand at high speed so that it flies forward. Again, I am not sure why this gives a distance benefit but I assume that it's different to the rationale behind the cartwheel shot put?
I am wondering if the esteemed members of this forum can help me with these questions, which have bothered me for a long time and are what have brought me to this forum. One thing I struggle to understand and work with is the use of the "editorial we." I am a copyeditor and proofreader, and I have a client who consistently likes to use the first-person plural voice for his works. I have asked my colleagues, and I got mixed, unclear answers. Here are a few examples: We will be able to feel magnificent light in our souls... Does "souls" have to agree with "our"? Should it not be "soul"? The chair or pillow may not be so comfortable, our kids may get rowdy, our spouse may need our help with this and that... Here, I have no issue with "our kids," because a person can certainly have more than one child. But should "spouse" be plural like "our" and "kids"? It is a constant issue that we must face our entire lives. "Entire" here does not seem to fit with "lives." But then "our entire life" does not seem to be correct as the singular "life" does not agree with the plural pronoun "our." I would be grateful to learn more about the proper use of the "editorial we" in this regard and to hear suggestions on how to deal with this issue. Thank you all in advance.
In a textbook for thermodynamics, it considers a situation where work is done to a system by an irreversible work source through a thermally insulating piston, and it states "any irreversible work source can be simulated by a reversible work source". It briefly explains the reason; what the work source does is simply to apply force to the piston, and therefore it does not matter how the force is applied, whether or not it is applied by an irreversible work source. I have a difficulty to fully convince myself with the statement, and can anyone kindly help me with this? The textbook is a non-English one, and it is not available online as an electric file. The way how it defines the work source is simply that it is any system, which is connected with the system of our interest only through a thermally insulating piston. That is, there is no heat exchange between the two system. Let me rephrase my question: Suppose that system A is interacted with system B (irreversible work source) only with a thermally insulating piston. They exchange energy only through work, but not heat. My question is if it is possible to replace system B with a reversible work source in an indistinguishable manner, i.e., any mechanical reaction that system A receives from system B remains exactly the same. Thank you so much for your time.
As per my understanding: Multiple fermions cannot have the same quantum state (as per Pauli exclusion principle) Multiple fermions can occupy the same physical space as long as they have different quantum states (or numbers or properties such as spin) If both these statements are true then, part of the second statement "as long as they have different quantum states (or numbers or properties such as spin)" doesn't become necessary. Because first statement implies that "multiple fermions always have different quantum states". Hence, the second statement simply becomes "Multiple fermions can always occupy the same physical space" (For a moment let's consider only fermions, their quantum state and physical space they occupy. And not other factors like electromagnetic repulsion etc) However, at multiple places on the Internet it has been stated (and seems like widely accepted) that: Multiple fermions cannot occupy the same physical space as per Pauli exclusion principle, and that is why matter structures exists in the universe. Can someone please help me trying to figure out where am I making mistake?
According to wikipedia, the aurora borealis is primarily caused by charged particles from the solar wind being redirected to the poles by earth's magnetic field and slamming into the nitrogen and oxygen atoms in the upper atmosphere. This slamming gives the electrons in those oxygen and nitrogen atoms enough energy to escape their electron clouds. The light comes from when electrons emit photons in order to rejoin an atom. Hence the colors, green corresponding to oxygen atom readmittance and purple to nitrogen What I am wondering is what role dissipative absorption plays in the aurora borealis. I mean we have a bunch of accelerating charged particles right? Even if speed is constant their direction is changing due to the magnetic force and since they're charged they must be emitting some kind of em radiation right? Not to mention when the actual collisons take place between electrons. Not all of this em radiation will correspond to the energy levels of nitrogen and oxygen right? So shouldn't we see more colors as all sorts of em radiation ought to be emitted? If nothing else you would expect the electron clouds of oxygen and nitrogen to oscillate in response to the emitted em radiation of the incoming solar wind right? Why do we tend to see only green and purple?
Suppose I have a hose pipe connected to a tap. I turn on the tap and water flows out of the other end of the pipe. Now imagine I install two valves at different points along the pipe. Water will flow normally if both valves are open, but not if either one of them is shut. No doubt you can now see where this is going. If the spacing of the valves along the pipe is sufficiently long, and the duration for which the valves are open is sufficiently short, we can always imagine a second reference frame in which at any given time at least one of the valves remains shut, so how can we account for the fact that water flows in that frame even when the valves are not both open? I imagine the resolution is somehow to do with the fact that the flow requires a pressure wave to propagate along the pipe when a valve is opened, but I haven't been able to find a convincing resolution. Has anyone come across an analogous problem before?
I'm implementing physics for a computer game, and came accross something that looks unintuitive to me. Consider two bodies at rest: Let's say we momentarily apply the same amount of force to them, but in different locations: at the center of mass for one body, and off-center for the other. The second body will gain angular velocity, while the first one won't. Both will gain rightward velocity, in the same direction. But will the second body gain less velocity than the first? Or the same velocity? Naively I would expect less velocity, since some of the force was "expended" on giving it angular velocity, but I googled around and experimented in Algodoo, and it seems the velocity ends up the same. Is that correct? Is there an intuitive explanation for that?
I've got a few months to go before I enter in graduate school, but I have to say that my math background is not as good as I would like. Thus, I would like to take the opportunity to strengthen it and work on undergraduate analysis. I had in mind to work on the Rudin book I've heard so much about, the book is very concise and covers a wide spectrum of mathematical analysis (and a bit of topology) seen in undergraduate so I thought it was suitable for what I want to do with it. But I'd like to hear your opinions and recommendations if you've ever been in this situation and/or know of any other cool books (I'am also interested in functional analysis and especially Hilbert spaces so...) for the self-taught to get through the summer. Thank you a lot !
Let's assume we have a standard double-slit experiment, just for illustration's sake. When an electron collides with the detector, is its momentum parallel to the emitter's output, or is it something else? I know that the uncertainty principle forbids us from knowing too much about the momentum when we know the position, so let's assume we have a magic detector that will only give us the momenta and not the positions. This magic detector can detect the momentum of a single electron. If we shot one electron through this apparatus at a time, what would we see (on average)? I say on average, because there's a non-zero chance that the electron lines up with the emitter by chance. We should still be able to see the standard deviation of the momenta; if it's tight, then the momenta line up with the emitter, otherwise, it's relative to something else. I'll admit this question may be nonsense, but to my knowledge, the only limitations to this are engineering ones (due to the magic detector), not physical limitations. If this experiment is nonsense due to a physical limitation that I've misunderstood, I'd like to know why.
I've been reading some category theory texts for my undergraduate monography, and I've found that one can talk about small and large categories using small/large sets, or do the same using sets/classes. Most introductory texts just shrug formalisms off, mention one of these formalizations and goes on. I've noticed that most category-focused authors and texts (such as Categories for the working mathematician and Sheaves in Geometry and Logic by Mac Lane) tend to prefer the small/large set approach, while texts that don't focus that much on categories (Topology and Groupoids by R. Brown) tend to prefer the class/set one. My question: Is there a categorical/set-theoretical reason to prefer the small sets approach? I'd assume that there's a benefit in having a set of objects even in large categories, but fail to see an actual difference. I found the paper Set Theory for Category Theory by Michael Shulman, but it was way out of the scope of what I can understand.
I'm looking for a real analysis book that covers measure theory and I've been recommended Folland's or Royden's books, but their price is quite high and I was wondering if any of the books below would serve as a cheaper substitute. Theory of Functions of Real Variables by Lawrence M. Graves; Real Variables with Basic Metric Space Topology by Robert B Ash; Theory of Functions of a Real Variable by I.P. Natanson; Real Analysis by Gabriel Klambauer; Nonstandard Analysis by Alain M. Robert; Foundations of Analysis by David F. Belding and Kevin J. Mitchell; Foundations of Mathematical Analysis by Richard Johnsonbaugh and W. E. Pfaffenberger; Mathematical Analysis by Tom M. Apostol; Elementary Real and Complex Analysis by Georgi E. Shilov; Foundations of Modern Analysis by Avner Friedman; Real Analysis by Norman B. Haaser; Introductory Real Analysis by A. N. Kolmogorov; Introduction to Analysis by Maxwell Rosenlicht.
I'm having a little bit of a hard time understanding some concepts to do with self-inductance in class. I understand mutual inductance. That's when a loop has some time-dependent current going through it, and because the current is time-dependent, the magnetic field the loop produces is also time-dependent, and the magnetic flux in a nearby loop is time-dependent. With self-inductance, though, is the idea that a single loop carrying a time-dependent current gives rise to a time-dependent magnetic field, and that very same time-dependent magnetic field gives rise to a change in flux in the single loop itself, which then generates an emf? If I'm correct in this reasoning, then that should mean that the new emf gives way to a current, which opposes the previously existing current. Why doesn't this just result in an infinite loop creating an opposing current, and then one that opposes that, and then one that opposes THAT...etc.?
Given a mapping f: R->R, what properties must f have in order for its range to be countably infinite? I'm personally not sure, especially in terms of formalization, but ideas of non-monotonicity or some construction of countably infinite equivalence classes come to mind. For context, in quantum mechanics it's said that a system's energy can only take discrete values, but this confuses me since energy is defined as the sum of a function of position and a function of momentum, which can both take any real value. The indexing of the sets doesn't make much sense to me, especially since the kinetic energy, the function of momentum, is very clearly uncountable since it's strictly monotonic. So even if the potential energy has a countable range, it doesn't make sense that the total energy would be since the union of a countable set and an uncountable set is uncountable. Like I said, I'm more concerned with the question I asked above rather than this confusion that created it, as I'm sure an answer will either lead me in the right direction, or at least give me the means to reasonably formulate a new question- so don't worry about addressing this. Any answers or information are greatly appreciated. Thank you!
I was solving this question in my coursebook and was not able to understand it conceptually. Though I did understand the mathematical approach to it. Now I understand that when the rod is given an initial velocity towards right, due to its motion an EMF is induced across it. Current starts flowing until it reaches a steady state. However, once the current becomes constant, no potential difference exists and the current slowly starts dying out at which point the rod starts moving back to its mean position. Then the velocity is in the opposite direction and the 'polarity' of the induced EMF reverses and current starts flowing in the opposite direction. My questions are- Doesn't it take an infinite time for the inductor to reach its steady state (where it behaves as a connecting wire). So, wouldn't the rod move in the direction of its initial velocity forever? What will happen once a steady current is achieved in the circuit? Will it be correct to say that since the potential difference across the ends of the rod is zero, the current wants to decrease to zero immediately, but the inductor only supports a slow decay of current? Can we also say that the reason the circuit doesn't heat up due to short circuiting is because there is no actual battery in the circuit? I tried asking my teacher but I couldn't get a satisfactory answer.