darklight03/StackMIAsub huggingface dataset:
I understand that a taylor series is used to approximate a given function at a point to any required degree of accuracy, I have not however understood why when we take the taylor series of a function at point, expanding to infinite terms it perfectly represents the function over the entire domain of the function. Is there a mathematical proof for why this results from approximating a function and its infinite derivatives at a single point
If you have a point mass moving with some velocity 'v' and then a force is applied perpendicular to 'v', physics/mathematics claims no change in the magnitude of 'v' , just a change in direction. But that doesn't make intuitive sense to me because wouldn't a force cause an instantaneous change in the point mass velocity perpendicular to 'v'? Therefore, wouldn't there be instantaneous velocity vectors perpendicular to each other and a resultant vector with a greater magnitude than 'v'?
My professor was discussing this question. After finding out that CR equations are satisified at origin for the function he used the definition of derivative to check for differentiabilty. Also, he mentioned that we can't compute partial derivatives of u(x,y) and v(x,y), and for example one of them turns out to be discontinious at origin then say that the function is bot differentiable. I couldn't understand his reasoning. Isn't that the sufficient condition for differentiabilty?Thoughts? Click here for question
I'm trying to write a book (documentclass "memoir") and I would like each chapter to begin alternatively with a greek or arabic letter. I have no problem with greek thanks to the package "babel", by contrast whatever attempt to write a-single-arabic-letter creates big conflicts in my template, such as compiling errors, and other problems (such as, the letter is automatically written on the right hand side: and it makes sense, of course, but I just need it as a symbol). I've looked through many other questions, which usually concern writing sentences, and I wonder if there is a simpler method that fits my case.
The way I have it is: the Observable Universe looks as follows. In some ball, all the galaxy clusters exist, then in a bigger concentric ball the dark ages exist (no galaxies), then on the surface of this larger ball at the edge of the observable Universe the Cosmic Microwave Background exists. This leaves no room for an infinite or curled back on itself Universe, so it must be a wrong picture. How do you picture it? It would be advantageous to get a correct picture.
When submitting a memorandum request for approval of something and providing options of for the signer... what is the best way to word the denial option ? Disapproved? Denied? Unapproved? This is intended to allow the approver to circle their response. Approved / Denied Approved / Disapproved Approved / Unapproved I feel disapproved is not the correct word... since they may disapprove of the request in theory but that doesn't mean they're saying no to the action. Denied seems to be more definitive. Unapproved works but seems lackluster in some way and may indicate the action was taken without prior approval or something similar to that.
I am writing my Doctorate thesis in Latex. I am using book class, which by default is twoside and openright. But for the preliminary pages, specifically for page number ii, certificate, it needs to be printed on a new page, but the result is coming out as a larger right-side margin page for being an even-numbered page. The same problem arises for the list of figures and tables also, they are also starting on an even page and the output is a bigger right margin. Please help to overcome the situation. Thanks in advance.
I'm doing undergraduate research on the history of Abstract Algebra (specifically permutation groups) and the notion of symmetric groups in indigenous artwork has come up several times. Is anyone aware of evidence of this 'intrinsic' human desire to construct groups I could point to? Or maybe some natural phenomena that conform to the definition of group? I'd like to have specific examples I can explain (and possibly map some ideas of formalized group theory onto). Thanks!
So I had taken a course on BEC and Cold Atoms. I have read about the properties of non-interacting Bose gas and I was a little concerned about what we mean by two wave functions (of bosons) being the same. Does it mean that they are physically located at the same point in space? I understand that they have the same spatial probability distribution. But it is more than that. I basically want to know given two wave functions how do I say they are the same? A more physical meaning of two wave functions being the "same". P.S. This is my first time posting. So please be kind.
Regarding an acceleration in an empty universe, from the special relativity we feel an acceleration in an empty universe since there is still the presence of an space-time with respect to which an acceleration is measured and the equivalence principle says that locally the effect of gravity and acceleration is indistinguishable i.e we feel the gravity whenever we are acceleration and conversely we feel acceleration when there is gravity that means according to equivalence principle we can't feel an acceleration in an empty universe since there is no gravity but special relativity says that we can feel an acceleration in an empty universe ,for me there seems a contradiction between special relativity and equivalence principle. Does anybody make clear this doubt?
I imagine a swinging pendulum being held outside a black hole supported by the normal foce of a jet propelled rocket. The rocket's velocity is approaching light speed, therefore hovering the swinging pendulum right above the event horizon in the strong gravitational potential outside of the black hole. My first question is: what would the frequency be of this pendulum and would this be the natural limit to swinging frequencies? How could the frequency possibly increase more in potential in nature (besides by shortening length)? And following the consequences of this: I believe photons and pendulums behave identically under the influence of a gravitational potential; does this then imply photons also have a natural cut-off limit for their frequency?
If I have an action that is not invariant under a change of coordinates. Does this effect the renormalizability of the theory? The procedure of renormalizing a field theory essentially boils down to carefully keeping track of observables while changing the length scale of the theory. If I have an action that is not diffeomorphism invariant changing the scaling of the action has physical meaning and also has consequences on the parameterization of the theory. How does the renormalization process differ between actions with and without coordinate invariance?
My electromagnetic waves professor said any time an electromagnetic signal is received by an antenna, a transmission also occurs. I have not independently verified this, but I believe it is likely true. Does someone know if this is true, and can the basic physics be explained, please? Note: This question was inspired by my own comment on Antennas and relativity. I'm thinking of this more from a Maxwell's equation POV then relativity, however. ANOTHER NOTE: Per comments below, transmission is usually called "scattering" when unintentional.
As an example, why does glass tend to break by means of intergranular fracture? My analysis so far led me to believe that the tendency can be related to the intermolecular bond strength in a given crystal grain and the intergranular bond strength. A stronger bond means that better load bearing. If the intermolecular bond is stronger than the intergranular bond, then a material is more likely to experience intergranular fracture. But is this correct?
So, I know the proof for Halting Problem is not recursive using diagonalization. We prove it using proof by contradiction. First we assume HP is recursive which implies there is a Total Turing Machine. And prove it is not possible by contradiction. But the proof just says Halting Problem is not recursive. It doesn't say Halting Problem is recursive enumerable. So how to prove Halting Problem is recursive enumerable
So in math, analysis in particular, divergent series are not particularly interesting since most sequences are divergent. However convergent series are interesting and they are useful. However, mathematicians have realized they had to be careful and define two types of convergence. Most of the theorems rely on absolute convergence. Therefore my question is, if your series turns out to be conditionally convergent, are you stuck? That is, can you not conclude anything further or can you still work with a conditionally convergent series?
Consider how the wizarding world is hidden from the muggle world in the Harry Potter/ Wizarding World universe. Or in Hell Boy II. The troll market is hidden under a bridge. Red can see it but it's veiled from the human eye. Shadow Hunters. Clary can see the runes outside of the nightclub but Simon can't see them. I have been referring to it as a troll market concept but I feel like there is a better word(s) for it.
So , I will be completing S.L Loney Coordinate geometry book very soon and I feel like I need more practice in Conics especially in Ellipse and Hyperbolae section , Can anyone suggest me a problem solving book in Conics (Not theory, Because S.L loney has a very vast theory and sufficient for my preparation) and the level of the book must up to a competitive level exam or Olympiad. Thank you:)
From Boston Legal (or your everyday conversation): "He and I will be going on a little vacation." "Where?" "Probably best you don't know." Being short for "It would probably be best if you didn't know," the spoken version changes the past tense to the present in the second half of the sentence. Is this something a grammarian would find fault with? Or dismiss it as colloquial?
From how I've learned it in school the magnetic vector potential is used as a mathematical tool to simplify problems with current-carrying wires in classical electromagnetism, but is never treated as bearing any physical meaning. After doing some research on it myself, I found Feynman's lecture on the subject, which outlines how this vector potential actually takes center-stage in QED, not only featuring in its primary equations for concepts like phase change but also as capable of producing physical changes such the Aharonov-Bohm effect that can't be explained by the magnetic field alone. Are there any comparable physical effects the vector potential has in classical electromagnetism?
I understand that thunder clouds are a congregation of millions of water droplets. As water evaporates from the ground, these water molecules rise up and collide with water molecules of the cloud, sometimes ripping off electrons. This creates a seperation between positively charged water droplets moving up inside the cloud, leaving electrons behind. Why do the electrons stay at the bottom of the cloud and not move up due to attraction to the positively charged water droplets?
I'm new to fractals and especially the Mandelbrot Set. I've noticed these never ending and self-similar spirals all around the Mandelbrot set, just like the one below: At lower max-iterations, there is a black point in the middle of the spiral, which suggests me that there could be a minibrot there. I've read that minibrots are centered on c values with super-attractive orbits. However, since the spiral never ends, I cannot precisely get the middle point and calculate its orbit. Is there really a minibrot infinitely small at the end of the spiral?
I've been learning about set theory and have heard of something called a "group". I didn't understand what it was, even after researching it, but I did find out one piece of information: All groups are sets but not all sets are groups My question is: what is an example of a set that is not a group? That would help me better understand it.
More precisely, "to" is banned in the active form: The headmaster makes us honor our teachers but required in the passive (with no agent stated): We are made to honor our teachers I can just say it's something -- for lack of a better term -- mechanical: "to" is required to mark the infinitive in the passive voice. But then, why does "mechanics" matter in the passive voice but not in the active voice? More puzzling is that this phenomenon seems to apply to causative make, but not to causative force: The storm forced me to seek shelter I was forced (by the storm) to seek shelter Unlike make, "to" is required in the active voice of force. Thank you.
Imagine a stationary transmitter which vibrates the surface of the balloon and a stationary receiver half way around the balloon that can pick up these waves. Let the balloon expand. Will the frequency of the received waves change due to a Doppler effect? As the balloon expands, the tension in the surface will increase which could alter the velocity of the waves. This is not a Doppler effect and could be allowed for. I ask the question because the motion of both the source and the receiver due to the expansion is perpendicular to the direction the waves are travelling in, though there is also a velocity of recession, which is different to the velocity of expansion.
In the same context as of "Trendy names", why is "Brandy names" not used? Brandy comes naturally/intuitively as an adjective to brand in the minds of a new English speaker, instead of the commonly used words like brandworthy, brandable, branded. I want to understand the reason behind such usage. If we assume the word brandy does not denote a specific alcoholic beverage, is it grammatically correct to use "brandy" as an adjective of brand?
What are the different methods using contour integration that can be used to count the non-trivial zeros of the zeta function? I know of Backlund's exact formula which is similar to Riemann-von Mangoldt formula. What are the other ways? If there aren't many other different methods, then where can I find the exact derivation, like from scratch to end of Riemann-von Mangoldt formula? I have managed half of it till the point I can derive Backlund's exact formula. Also an evaluation of the limitations would be nice. I know Titchmarsh's works are useful but what other resources can help me?
In the sentence: "There were people everywhere. They thronged the streets, choking the alleys. They had come in their thousands to see the light show" I would group the infinitive of purpose and past perfect together, so that the sentence refers to a plan formed prior to the events of the story. At the same time, the arrival itself takes place before the crowds assemble and begin 'thronging'. Is my analysis off? EDIT: I apologise. I had to cut this short before clarifying. The sentence seems to be referring both to the arrival and a preformed plan.
I'm trying to prove the above statement, and I had a look at this site. This ends the proof with the following statement the rank is the number of non-zero eigenvalues But, this is what makes me confusing, because it's not all the case that rank of a matrix equals the number of non zero eigenvalues as far as I know. This shows a counter example of this. Any comment about the rank of idempotent matrix would be grateful.
I understand the general notion of twist like that of a mobius band(like a cylinder cut open and the glue the edges together with a "twist") , and the torsion in the context of algebraic structures like groups, in which a modulo-n torsion indicate when we multiply some element with n, we get the identity(please correct me if there is an error in my understanding). So in the context of topological groups, can these ideas be related, like in the case of a lie group, where we can talk about both twist and torsion and whether these ideas turn out to be related?
I want to add a citation inside a pdf figure, and would like the citation to be a link to the bibliography. I know embedding a pdf with includegraphics normally mangles the link, but I have found a solution (in pdflatex), using the pax package. But am open to any other solution. pax preserves the hyperlink, and I tested it is working with a www address. But for the internal link to bibliography, I am not sure what link address should I use. At the moment, I insert this link in the pdf figure "cite.citation_key" But the result is that the hyperlink points to /home/user/cite.citation_key instead of the bibliography. Does anyone know how to achieve this.
We were happy before. There was a market next to the school before. I saw a whale before. These are from a grammar book for kids that I'm proofreading, and this part is about "past simple tense". I would like to ask if these sentences are grammatically correct regarding the use of "before." It seems that "before" should be used with perfect tense? Should I make the following changes? Thank you! We were happy then. There was a market next to the school (sometime ago). I saw a whale (sometime ago).
We have the adjective "facial" for the noun "face", and I am wondering what the counterpart for the noun "head" is. I could not find it on Google, and ChatGPT told me "cranial" which seems a bit different from what I expected. Thank @Stuart F for the comment! My original thought was to make a joke on myself who can have some kind of "head expressions" (corresponding to "facial expressions") because I am a bit bald. And I am also happy to see that the vocabulary can be different in different contexts!
Newton's globe experiment: two globes that share all their features are connected with a rope in an otherwise empty universe. Newton introduced this experiment to show that even though the cases where the globes don't move and where they rotate are not distinguishable relationally (i.e. considering their distance to each other), we can nevertheless distinguish them due to inertial forces that only occur in the rotation case (tension on the rope between the two globes). My question: could Newton also have chosen an example using linear accelerated motion instead of angular accelerated motion? Inertial forces should occur there as well. If not, why does he need circular motion to make his point?
I know that "soliton" waves can consist of a crest without a trough. One would expect the reverse to be true as well. However, this Wikipedia excerpt says, So for this nonlinear gravity wave it is the total water depth under the wave crest that determines the speed, with higher waves traveling faster than lower waves. Note that solitary wave solutions only exist for positive values of H, solitary gravity waves of depression do not exist. Is this true, and if so, why?
I'm still introducing myself in derivation, antiderivation and integrals, because I haven't studied it in class yet (I will study it next year and derivation in some months), but well, about my question I have seen that integrals and antiderivatives, at least, looks like the same, so, is that true? and also I didn't understand at all why the integration gives you the area, I mean, realize that derivation gives you the value of the slope is easy, but why the antiderivation gives you the area is a bit harder for me to understand, thanks!
Alright so I am confused somewhat about Neutron Heavy Nuclei not decaying, or more specifically why it is that Neutron Heavy Nuclei sometimes decay by emitting Neutrons. Someone has already answered why neutron-heavy nuclei decay in general, weak force beta decay basically happens until it can't, but I do not understand why neutron heavy nuclei can, sometimes, emit Neutrons. I am talking about the light elements that have neutron upon neutron upon neutron. Hydrogen Isotopes beyond Tritium decay by Neutron Emission, why? Why doesn't it decay by beta decay? Why do other neutron heavy light elements perform neutron emission?
We know that flowing water remains liquid even in the below-zero temperatures (a good example is rivers in arctic regions). Of course water doesn't remain liquid forever if temperature goes down constantly. Also, I know water freezes from surface. But that's for stationary waters like lakes. Not for flowing waters like a dam output which is so fast. I think it depends on a sort of things, like the speed of molecules, the viscosity of water, etc. Can we formulate it to find the answer? Perhaps with such a formula we can find the freezing temperature of other flowing liquids as well. And a minor question: How does the density of flowing water behave in below-zero temperatures?
Consider a one dimensional chain of N classical point masses interacting with harmonic neighbor forces (with periodic boundaries for specificity). If the positions and velocities are prepared in a configuration corresponding to a specific oscillation mode, can the energy be transferred to other modes ? Or does the chain oscillate in that prepared mode forever ? Intuitively I do not see a reason why the energy should be transferred to other modes, but I am not sure. However supposing that is the case, does the chain return to that mode if small random disturbance is added at some time ? How is the stability of this mode to perturbations ?
I was planning on using whipping boy but it seems that means more like a scapegoat (ie someone who is innocent but made to take the blame for others). Moreover, I want something that means that everyone seems to point to one particular person as the bad example and turn a blind eye to the same behavior in others. The issue is more with reputation or perception from an outside group than guilt per se, but I don't want to suggest that the example person is innocent. The context is history.
I would like to calculate the repulsive force between two permanent magnets in a different variations of distance between the two polarity faces. Can the different distances associated to different repulsive forces be related to the magnetic field? Can the magnetic field have a variation when the distance between the magnets change? Can a tesla meter measure the magnetic field in the space between the two magnets? Can the repulsive magnetic flux between two permanent magnets be measured with a tesla meter? Thank you!
I was writing a Statement of Purpose letter and I am stuck with this dilemma. Here is what I wrote: I pray that this letter attains your approval. I first used "gains" but my mind instantly switched it to "attains", as the word seems more formal...and then I thought of "obtains", which only got me more confused. All I know is that these words have the same meaning of getting something but I'm still struggling with the context/condition it should be used on. Which one should I use?
I found that the definition of coherent phonon: A femtosecond laser pulse can initiate collective, in-phase atomic motions in solids called coherent phonons. My question is: what is the difference between coherent phonons and acoustic phonons? Acoustic phonons also mean the displacement of atoms has the same amplitude, direction and phase. Does it mean atoms move in phase and collectively for acoustic phonons? Thanks in advance.
Is there a word that symbolize a status of either unstarted, started or completed ? For example a football match can be either unstarted, started or completed. I would like another word than "status" to indicate those lifecycle adjective, as we have a lot of other "status" in this context. Something like "completionStatus" : but this is not really it, as this is just about completion. I thought about lifecycle but that's too biology oriented.
Throwing a small stone on the surface of a calm pond of water creates concentric ripples on the water surface starting from the point of impact and travelling outwards like transverse waves. At the same time the momentum of the sinking stone creates longitudinal underwater pressure waves that propagate outwards with the speed of sound in water. I am interested for an equation that describes the propagation speed relation of these two different types of waves created. Will the surface transverse waves have the same propagation speed with the underwater longitudinal pressure waves and if not how can the propagation speed of each type of wave calculated for water?