darklight03/StackMIAsub huggingface dataset:
I tried to prove that the centers of two tangent circles are colinear with the point of tangency. If I just assume they have a common tangent at their point of tangency then it's pretty easy. However, how do we prove that two circles do indeed have a common tangent at their point of tangency? (without using the collinearity of the centers and the point of tangency to avoid circular logic)
In the Spanish language there is a word that is used to describe a person that has been tagged as a deadbeat; it is Largo. Largo in Spanish means Long as in the measurement of the distance between two points or places, etc. If you loan any money to this person, it will be a long time before you get paid back, if you get paid back at all, or if you receive any kind of payment. They can also be known as a Largo if they are sly, tricky, or conniving with bad intentions.
I'm looking for a formal language that has the following properties: Contains finitely many words (and you can prove it). Decidable/recursive (there's a Turing machine that always halts, that can determine whether any given word belongs to the language). It's impossible to compute the exact number of words it contains. Meaning, even if you've already found all the words in the language, you can never know if there's another word that you just didn't try yet. Such a language feels "illegal", but I didn't manage to disprove its existence.
A Chinese teacher of English asked me about the following, taken from an English test for Chinese people. It's quite tricky I think. I would like to know three things: Which answer or answers do you consider to be correct? If you consider an answer to be wrong, why do you think it is wrong? How would you reword these answers to make them acceptable? Nanjing is a popular city with a long history. Could you tell me ______ . A. where can visitors enjoy delicious Nanjing snacks? B. when is the best time to visit the Confucious Temple? C. how will people get to the Zhongshan Hill Scenic Area? D. why was the city also called Jianye in ancient times?
I am pretty proficient till Calculus II, but I want to learn about more advanced methods of computing integrals like Gamma function, Elliptic integrals and Leibniz integral rule. I also want to improve my mathematical problem solving ability with this. Should I read Advanced Calculus by Frederick S. Woods or Inside Interesting Integrals by Paul J. Nahi? Or would it be better to first learn multivariable calculus? (any other book suggestions are welcome)
A group action on a set is said to be transitive if it has only one orbit. Now, a group action on the same set is said to be regular if all the stabilizers of the elements of the set are trivial. My doubt is, if there is only one orbit, then, by orbit-stabilizer, all stabilizers must be trivial right? What am I missing here? Any hints? Thanks beforehand.
From my understanding, the frequency and wavelength of a mechanical wave can be observed by the motion of the particles (i.e., as the particles oscillate from their state of equilibrium, the wave propagates through the medium). But, with electromagnetic waves, how do we measure its frequency and wavelength? If an electromagnetic wave travels across a vacuum, isn't it pretty much invisible to the naked eye?
I am tasked to simplify this circuit to one source and one resistor, which is equivalent to the original but obviously easier to solve. My problem is the fact, that I don't know how the nodes A and B affect the process. I'm more used to the circuits where the A and B nodes are on one end of the circuit, but these are in the middle. Not sure how this is supposed to be interpreted.
I am wondering if this statement about parallel circuits and conservative fields is true?: The voltage drop over resistors in parallel is the same because the electric field is conservative, so the work done on a charge only depends on the start and end point. The field is conservative because the force it generates consists of the electrostatic force (Coloumb) and a magnetic force. We can show that the electrostatic force is conservative, and the magnetic force on moving charges is perpendicular to the velocity vector, so it does not do any work on the charge. So the only force that does work on the charge is the electrostatic force. Is this statement correct?
I'm playing around with modeling an archway of a building using a solid modeler, and have come to a problem that exceeds my mathematical ability. Given an ellipse that has a known major radius, and is tangential to a line of a known length, angle, and offset on the Y-axis, can I find the ellipse's minor axis? It's been decades since I took a math class. I can handle basic trig, but I can't figure out how to get started on this one. Edit: Thanks everyone! You can see your solution in the flying buttresses of this (very preliminary) render.
The company "Fluke" makes a DMM that they claim can measure AC voltage by simply clamping the connector around a live wire, and using the E-field to induce a voltage/current into the capacitor. I can understand the concept if we assume the person holding the DMM is grounded and the measuring loop is sufficiently sensitive. But what if the person holding the DMM is not grounded, or is in a vacuum, etc. Some people on the internet claim it works even if the DMM is hanging from the wire in open air. I have my doubts... So, I am wondering if someone can explain to me if current can flow in a capacitor that has no path to ground. It seems like an obvious "no."
I am trying to understand the Von Neumann decomposition, according to which every Von Neumann Algebra can be uniquely decomposed as integral (or direct sum) of factors. More specifically, I am trying to connect (if possible) the Von Neumann decomposition to the GNS theorem. Can someone please help me answer the following questions? Given a normal state, is it true that its GNS representation is always a factor? The answer would be different if the dimension of the algebra is either finite or infinite? Is it possible that the algebra obtained by applying GNS theorem on a irreducible state is not a factor? From the quantum mechanics point of view, why is the Von Neumann decomposition important?
There's a clip going around on the internet of the driver Ross Chastain, 'wall riding' in a race that allowed him to pass many opponents on one turn. Here is the clip. What I do not understand is how did this work? The only explanation I can think of is that the normal force of the wall acted like a centripetal force, which meant he didn't have to brake on the turn, but this doesn't make too much sense to me since the frictional force of the wall on the car would slow him down quite a lot.
I'm working on a project involving non-standard dice where each face can have a different size, but the overall shape and weight distribution of the die are uniform. I'm interested in finding out how to calculate the probability of the die landing on a particular face based on the size of its faces. Assumptions: The die maintains a uniform shape and weight distribution. The die has 'n' number of faces, each potentially with a different area. We ignore the complexities of rolling dynamics and assume the die lands flat on its faces. I am looking for an equation or a model that can accurately calculate the probability of landing on a face, given the area/dimensions of each face.
Does anyone know where I could find a copy of Ron Graham's PhD dissertation, titled "On Finite Sums of Rational Numbers"? I found this paper, with a similar title, but I could not find Professor Graham's dissertation. I need to double-check some detail(s) in Professor Graham's dissertation which may not be in the published version. This is in consideration of my (current) approach to odd perfect numbers via Egyptian fractions.
I am not a pure mathematician but an engineering professor. Right now I am going through the first chapter of Topology without Tears by Morrison. One of the questions is as follows. A set T consists of a set of real numbers, the null set and the closed interval [-r,r] where r is irrational. The question is whether (T,R) form a topology where R is the set of real numbers. The answer seems to be an obvious Yes irrespective of whether r is rational or irrational since both intervals can be infinitely large and can contain arbitrarily large numbers. But I am suspicious as it seems to be too obvious and I think the actual answer is No.
I am new to linguistics and am currently learning how to figure out phrase markers for sentences. I am unsure about this sentence: She went off in search for her hedgehog. I know that "she" is a noun phrase, but I am unsure about the VP. Is the VP "went off in search for her hedgehog"? And does that split again into the three phrases: went off, in search and for her hedgehog? Does it look like this?
I am trying to find the directional vector to hit a moving target. For example, as seen in my drawing, Lets say a space ship at position Sxy needs to fire a projectile to hit the moving asteroid (currently at position Axy). The asteroid is moving with a linear velocity vector Av. The projectile has a constant speed (magnitude) of Ps. How can I solve for the projectile directional vector Pv? Both the asteroid and the projectile have a radius.
I am studying simplicial homology and I am a bit confused about the definition of an orientation of a (finite) geometric simplex. The definition says that two orderings of a simplex are equivalent if they differ by an even permutation, and so we get two equivalence classes - two orientations. My question is, what is an ordering of a simplex? Is it just a permutation of the vertices or does it have to satisfy some other rules? If it's defined to be a permutation of vertices, then we draw paths between vertices indicated by the permutation of vertices (I am thinking geometrically), then my lines might cross each other. Is it that this will not happen because vertices of a simplex are in general position? Thank you!
I've been studying the Lax equivalence theorem . I've learned that it allows you to establish the convergence of a scheme through proof of stability. The Wikipedia article says that the method it's important because "it is ordinarily difficult to establish [convergence] because the numerical method is defined by a recurrence relation while the differential equation involves a differentiable function". I used the equivalence theorem to establish the convergence of the Crank-Nicholson method in an assignment. I was wondering what a direct proof would look like (I'm assuming it depends on the scheme). Can you please provide any references for this? Thank you in advance!
With some formal systems, it's possible to enumerate all the theorems. This is the case for instance in propositional calculus or in some first-order theories with a recursively enumerable set of axioms (e.g. group theory, field theory, Peano arithmetic, etc.). I'm looking for a software that would implement such an enumeration. Or more basically, a software that would produce random theorems (which may or may not be of interest). A software doing that in any of the formal systems listed previously would be acceptable.
My teacher taught us that a square matrix is called a diagonal matrix only if all the entries besides the ones on the principal diagonal are zero. He further cautioned that there is no restriction on the entries of the principal diagonal and that they may be zero or non-zero, BUT there must be atleast one entry on the principal diagonal that is non-zero. We were specifically told that null matrix is not a diagonal matrix as there simply is no entry along the diagonal. However, most if not all sources on the internet say that null matrix is also a diagonal matrix as all the non-principal diagonal entries are zero. Which one of the above definitions is correct?
Imagine the following conditional: If washing machines are humans, washing machines are quadrupeds. It seems to me that the truth value of the conditional as a whole is ambiguous. Since its antecedent is false, logic tells us that the conditional is (vacuously) true. But in fact, the conditional as a whole does seem false: if we grant that washing machines are humans, then washing machines are clearly bipeds. But how can the same conditional be both vacuously true and (at least intuitively) false?
I just read this answer to "What exactly is a Photon?" which has me a bit confused. It seems to be arguing that "photon" is just a catch-all term for any sort of interaction with the EM field and the implication is that it's not even a particularly useful concept, in contrast to the fundamental particles of other fields. But how is that any different from other elementary particles, when all particles are just excitations in quantized fields? I also found several threads talking about whether photons have wavefunctions or not, such as this one: Wave function of a photon? and I'm seeing conflicting information. If photons don't have wavefunctions, are they even particles at all, in the sense of being localized disturbances in a field?
I am doing some interactive plots about cosmological horizons and in my research I stumbled upon the sound horizon, the baryonic acoustic oscillations and how it had an impact on the formation of the first structures in the universe. Some people have calculated the radius of the sound horizon at the time of recombination using the same equation used in the particle horizon but with the speed of "sound", which itself is a function of the density of matter and radiation. But I've never seen anyone actually plot the radius into the future like some have done with other cosmological horizons. Is there a reason for that? Does it not play an important role in the universe anymore? Was it only important at the time of recombination?
The nlab claims that the maximal ideal theorem in Heyting Algebras (i.e. for every proper ideal in a Heyting Algebra, there is a maximal ideal that contains it) implies the axiom of choice. Sadly, it does not cite any proofs and I've been unable to prove this, I'd appreciate any proofs/references. I find this particularly interesting as it's known that the maximal ideal theorem for Boolean Algebras is strictly weaker than the axiom of choice. Yet, the same theorem applied to the apparently slightly bigger class of Heyting Algebras suffices to prove choice.
I have difficulty understanding when we use "an analogy", "a metaphor" and "a figure of speech". I feel like in a casual conversation, people just use any of these words to say something that is different from its normal use. Does a native English speaker distinguish the difference between them in a casual conversation?. Is it correct to say: "I am like a bee in a bottle is an analogy" or "I am like a bee in a bottle is a metaphor" or "I am like a bee in a bottle is a figure of speech"
In the study of the geomagnetic field, an expansion in spherical harmonics is used to represent the scalar magnetic potential: the first terms give the dipole approximation, then the quadrupole, etc. For instance, we can find here (International Geomagnetic Reference Field) reference data for the magnetic field, given as coefficients of a spherical harmonics expansion. My question is: how are those coefficients calculated? I was thinking of the Monte Carlo method, but I couldn't find any information on that. Note that I would also accept answers related to the gravitational field as the question also applies there.
I'm an English learner, and sometimes I cannot understand why some words are used to express a meaning. For example, the word "Down". What's the difference between "I walked on the street" and "I walked down the street" or "Alan drove down the street" and "Alan drove on the street" ?? As far as I know, these verbs are not phrasal verbs, so how can I know when to add this "adverb particle-like" with a verb ?
I saw this in a textbook and I want to confirm if it's true If a body of mass m rolls down an inclined plane of height h and length l (hypotenuse), the work done is mgh. It is the vertical height of the inclined plane that is used in the calculation, not the length (hypotenuse) of the inclined plane. Please how true is this because I believe we should be using a little bit of trigonometry since the object also moved a horizontal distance. Please if true, how?
I'm trying to understand how car wheels work. In a perfect world the bottom of the wheel has no velocity. The wheels are in a state of pure rolling, which means there is no static or kinetic friction because the bottom contact point is not in motion. This would mean that in a world without air resistance a driver wouldn't have to push on the gas pedal, as there aren't any forces opposing the cars forward velocity. Is this right?
I have a question about the EPR phenomenon of the quantum-mechanics. When you observe the EPR phenomenon, you separate two spins in the singlet state. Then, you and your fellow independently observe the direction of each spin, and you compare the directions. If one spin is in New York, and another is, for example, in Tokyo, how do you assign the directions? Because the earth is round, I do not know which direction is opposite to another. In reality, it is difficult to make such a large separation without losing correlation. However, in principle, it is irrelevant how large the distance is. Thank you for your attention.
I found this problem online somewhere (cannot remember the exact source, sorry) and initially I thought it looked fairly simple and fun to solve. I've now sat for several hours and even asked some friends to help, but none of us can actually solve it (without complex algebraic equations). I am by no means an expert on geometry or trigonometric (or mathematics for that case), so I would assume there are some identities or theorems that would make this problem easier to solve that I just don't know. Can anyone help please. The problem is to solve for x in the below diagram (excuse the bad drawing in Paint):
Generally speaking, the universe is electrically neutral and the universe abhors an unbalanced charge. Wherever there is a positively charged object, you can bet there is a negatively charged object not far away. However, no laws of physics prevent the accumulation of positive charge, say, so long as the negative charge is sent somewhere else. This must be true on a cosmic scale as well. Hence my question: What is the most electrically charged celestial body in the universe, and what is its charge, i.e. positive or negative? Dipoles don't count.
From many papers I've read in condensed matter physics, it seems "breaking time reversal symmetry" is often equivalent to "applying an external B field", while "breaking inversion symmetry" can be done (but not limited to) by "applying an electric field". Is this true in general? A side question: if a perturbation breaks the inversion symmetry of the original Hamiltonian, does that imply it mixes states of opposite parity? Similarly, if a perturbation breaks time-reversal symmetry, does that mixes time-reversal pairs? Looking forward to any counter-examples. Thanks!
At first glance, it seems like the "many" in "There are as many A's as B's" is an adjective. However, we can't substitute an arbitrary adjective in this sentence. Compare: "There are as many A's as B's." (correct) "The A's are as many as the B's." (incorrect) "There are as numerous A's as B's." (incorrect) "The A's are as numerous as the B's." (correct) I'm curious about how to describe the reason why two of these feel 'correct' (to me, as a native speaker) and two of them do not. Is 'many' here a different part of speech than an adjective? Or is "as many as" just its own grammatical construction with its own rules?
David Cameron, the previous Prime Minister that has been away from politics for many years, has just come back as the Foreign Secretary. The BBC refers to him as the "old new face". Politics aside, just as an English grammar question, I wonder why it's "old new face" instead of "new old face": is there some convention/rule to determine the order of the two adjectives? Or is it just slang? I did some searching but was overwhelmed by face recognition AI links...
I have an anodizing rack that has hundreds of prongs. These prongs are made of titanium and need to be squeezed together in order to place the parts on the rack. I am wondering how I would calculate the force needed to make the prongs touch so that parts can be added? I have seen some people saying to use F=kx treating the prongs as a spring, would this be the best way? Any help is appreciated.
I was at a jewelry workshop today and saw somebody place a gold ring under an intense flame. Weirdly, the ring looked normal when underneath the flame, but glowed "red-hot" as soon as the flame was removed. The latter behavior is what I expect of a hot object emitting blackbody radiation. It makes no sense to me why the ring would not look like a blackbody object in the presence of the flame (where it could only be hotter). Maybe ions in the flame are affecting the physics at the surface and making it effectively opaque? I didn't capture the effect on camera, but this video sort of shows the same thing happening, though not nearly as clearly as what I saw in person.
I'm trying to construct a table that looks like the following: I know I need to use multirow package in the tabular environment, but I can't figure out how since I couldn't find examples that are similar to my need. It seems like I need to use a multicolumn environment inside multirow for the 'Experiment' cells. Can someone guide me in the right direction? Thank you!
As I read on Wikipedia, a phosphor coating of a CRT screen lights up because an electron in one of the phosphor atoms emits light when it goes down from its excited state. When a single electron hits a phosphor coating, is it the case that exactly one atom of the screen will glow? If not, what is the shape of the spot on the screen for a single electron?
If we create a pair of rotary wings like those on a helicopter, and spin them in a vacuum, would it be possible for a virtual particle and anti particle pair to spawn, be split by an approaching wing, and take opposing paths around the wing, before re-combining and annihilating? Would this create a pressure differential as it does on macro scales in fluid dynamics? Does the particle and anti particle pair even need to be split or is it sufficient for multiple such pairs to spawn and move around the wing before annihilation? If we rotate the wings at relativistic speeds, does this increase the rate at which this occurs and the pressure difference generated?
Given that our best models so far suggest that the universe has been expanding a lot ever since the time the CMB was emitted, I'm wondering: How large was the (visible) universe back at the time it became opaque and emitted the CMB? Specifically, I'm curious as to the ("original") circumference of the spherical shell we "see" out there, in comparison to how far the light from it has traveled to reach us.
I was trying to figure out some details about a specific decay channel for a generic particle accelerator experiment. For instance if I have a proton proton collider with fixed energy, I would like to figure out what is the average decay length of a B meson, what would be the energy of its decay products (for a chosen channel) and similar things. I know there is a lot of MC software used in high energy physics but I was just wondering if someone had a recommendation for one which is free and is not hard to use and configure when doing such simple things.
In his classic poem which pretty much everyone knows by heart, Edgar Allan Poe writes as follows: Ah, distinctly I remember It was in the bleak December, And each separate dying ember Wrought its ghost upon the floor. Why? "... a bleak December" would have sounded (or would it?) more logical: not all Decembers are bleak. Or: "This occurred in bleak December" would have scanned just as well. What gives?