darklight03/StackMIAsub huggingface dataset:
Can you recommend me some papers or lectures on finding the solution of elliptic PDE which is the saddle point of its energy functional. I glance over some methods including mountain pass theorem and min-max scheme, but it's hard to select one paper or lecture to do some intensive readings, so I wonder if you can recommend me some classical papers or lectures on this topic?
I am currently working through the book Mathematical Logic by Mileti (online version) and just finished the second chapter (induction and recursion). I have to admit that it is quite challenging and whereas I think I can follow through, it really takes a lot of effort and time to thoroughly understand the ideas since they are so abstract for me. Has anyone worked through it and could let me know if the density and difficulty will continue in that rate or if one quickly gets used to it? Thanks
I was always fascinated with pure math, but lately I've been increasingly more interested on applications of math (preferably algebra/topology but other fields would be interesting too) in the real world. I am wondering if there are any books that discuss such applications, as I wasn't able to find any. To be clear, I'm not looking for detailed books at the research level. I'm looking for books that I can read in my leisure time just out of curiosity to see how concepts from algebra/toplogy/etc are related to the real world.
I am working on a user manual, compiling it from .rst source files with the pdflatex engine. My user manual has two versions: in English and in Russian. The issue that I have is connected with tables that take more than one page. I have 'continues on next page' and 'continued from the previous page' captions added automatically. These captions are not a problem in English. However, they are partially translated to Russian (the 'continues on next page' caption is left unchanged, see the images). Is it possible to edit them or to remove them finally? I guess it might be defined in the preamble but I have no idea how to do this.
This is a question I came across on Instagram today, and here's the diagram: (Note: The image is NOT to scale) I attempted to solve it first by amending the quadrilateral in various ways, but each of those methods lead to a strange contradiction, likely because the diagram is not to scale. I'm going to post my successful approach as an answer below, please let me know if my answer is correct or if there's something wrong with the method (the correct answer wasn't revealed, and if there are any other ways to approach this that I missed!)
Blum proved that any mathematical theorem can be converted into a graph such that the proof of that theorem is equivalent to proving a Hamiltonian cycle in the graph from Applied Cryptography by Bruce Schneier I'm struggling to understand what the statement means, and I was unable to find anything on it after a few quick google searches. Can someone point me in the right direction, to what this statement is even talking about?
Antennas work by accelerating electrons to emit EM radiation. In fact, my understanding is that any accelerating electron will emit EM radiation. But in relativity, no frame of reference is preferred. Therefore, an electron in an antenna can claim that every other electron in the universe is moving around it. So does that mean that when an antenna creates a signal, every other electron in the universe sends the same signal back to it?
I am trying to recreate this diagram in a standalone PDF using TikZ or PSTricks. Two extra requirements: I have to be able to alter the viewing angle arbitrarily I want to use relative coordinates rather than hard-coding the absolute coordinates of every cube. I'd like to be able to define cubes relative to nearby cubes using nodes since it will be easy to generalise this code to other shapes. I'm fairly experienced with TikZ, but just looking for a more elegant solution.
I will use an example to explain my question... Transphobia is an irrational fear of, aversion to, or discrimination against transgender people. In the above sentence, is the true definition that only the fear is irrational, or is the aversion and discrimination also irrational? Is the true meaning: Transphobia is an irrational fear of transgender people. Transphobia is an irrational aversion to transgender people. Transphobia is an irrational discrimination against transgender people. Or: Transphobia is an irrational fear of transgender people. Transphobia is an aversion to transgender people. Transphobia is discrimination against transgender people.
I've been shown an Out of Office template, and one of the sentences in it asks the sender to contact someone else within the company "in case of need", as the email will not be forwarded. I've never come across that phrasing before. Is it British English? Usually, personally, I'd go with "if/as required", although that might be a tad too strong in this context...
I have one cylindrical coordinate system attached with a laboratory device and another cylindrical coordinate system that is attached with an object in it. The object is symmetric about the z-axis in its own coordinate system, which is both displaced and tilted with respect to the laboratory system. How do I express the unit vectors in the object coordinate system in terms of unit vectors in the laboratory coordinate system? I can write the coordinate transformation in Cartesian coordinates and then convert the Cartesian coordinates into cylindrical coordinates but do not see how to do that with unit vectors.
hello, hello. I am reading the book "Geometry" by Tatiana Alekseyevskaya (Gelfand) and found myself having the question below. An external angle is defined in Wikipedia as an angle formed by one side of a simple polygon and a line extended from an adjacent side. How does that definition applies to the following polygon (delimited by the green line)? If I extend one side of it, the angle will end inside the polygon. I also have found similar questions have been asked over this forum, but I suppose mine is more related to the definition itself being applied, unless this polygon doesn't qualify as "simple" polygon as the Wikipedia definition states. Thanks! -- Caleb
Is it possible to have one master .cls file in Overleaf that works across different projects? For example, if I make a change in the .cls file while working in one project, it will be updated in all of the other projects that use that .cls file? I know that I could import the .cls file from another project, but I cannot update it within the project that has imported it.
I have a particular model with two couplings, let's call it model A, for which I have the set of beta functions and fixed points. Now I am interested in a model where I have two copies of model A, each with a different fundamental field, let's call that model B. So, model B has two couplings for each field plus a fifth interaction that couples the two fields. I know this might highly depend on the specific scenario, but is there a general strategy as to how one would obtain the beta functions of model B from the beta functions of model A?
I am wondering that whether Kolmogorov spectrum, which describes the energy distribution in turbulence be used to describe the plasma fluctuation in the atmosphere of neutron star. As I know, Kolmogorov spectrum has been used in solar physics to model the small scale fluctuations, and there is also research on the turbulence in the core of neutron star and used Kolmogorov spectrum. However, I haven't found references concerning describing the fluctuation in the atomosphere of neutron star, can it use Kolmogorov spectrum or other methods?
I know the evolution of a main sequence star to a giant is a process that takes millions and millions of years, but how fast is the VISIBLE change? Basically, will our sun, for example, slowly grow to envelope the orbits of the inner planets, or will there be a sudden inflation of that star to a new much greater volume on a timescale of seconds or minutes or hours etc.?
All the sources I have read from say that radial component of acceleration of a body in circular motion only accounts for the change in direction . I do not understand how this is possible. Another thing I do not understand is what provides the torque for such a body to rotate. Could you please explain to me the forces and accelerations involved using the classic ball-and-rope example?
Calculus invention happened only few hundred years ago. So I assume that differential equations are being discovered and solved since that time. When I was in school I learnt that most differential equations cannot be solved analytically and are thus solved using numerical methods. The question in my mind is, what difference does it make if we can't solve a differential equation analytically? Is that bad? Are people doing research to find analytical solutions to old differential equations? Also, before computers existed, were people still able to use numerical methods to solve differential equations numerically? That would be quite a lot of work.
We have a compactness result, where the "flat limit" of an integral current is itself an integral current. (with some conditions) Now I am curious about submanifolds. I am expecting that an equivalent result does not hold for submanifolds in general. However I am having trouble finding a resource to verify this. Does anyone know of an example, book or article I could check out?
so I recently came across the term 'virtual mass' and when I looked up more about it, it just gave me some stuff about fluid mechanics that I dont understand properly. My understanding of virtual mass is that when a body is pushed through a fluid (let's say water) it moves slower than usual and behaves like hoe a big object with more mass would behave on land but then I realized that what I was thinking of is just upthrust/buoyancy so can somebody please explain virtual mass in simple (layman terms) words. (I just cant wrap my head around it)
The harmonics of a theoretically infinitely small diameter string are pure integer multiples of the fundamental frequency. However, a real string has a thickness, and when vibrating in a harmonic, the additional node or nodes should be taken into account when considering the string's length. This should mean that higher harmonics would have frequencies higher than the corresponding integer multiple of the fundamental frequency. What is the formula that allows for this calculation, and what effect might it have on the quality of the sound an instrument might produce?
In documentation for electrical vehicles there is a common acronym for "state of charge" that is written as "SoC" or "SOC". Are both versions equally ok, or are there rules guiding when to mix case, and when not to mix case? I hope this question doesn't fall under the category "Questions that are primarily opinion-based", which I learned are not be asked here (I just signed up).
I am a self learner and I just finished calculus books (Thomas calculus , Larson book ) and I want to learn further topics in integration so I searched for good books on integrals and I heard that the book A Treatise on the Integral Calculus by Joseph Edwards is very good but this book seems very old , so my question is this book still worth reading today or there are modern better books this ?
I have been trying to understand how to calculate the kinetic energy of a rotating rod that also has translational movement. I have written my question down on paper because I think it is more clear this way. In case B, I think I have miscalculated the kinetic energy when dividing it as the sum of the rotational and translational energy. I have been days trying to understand this high school level problem. I would be very thankful if someone can point out my error.
Let's say the original source says: "They were at the loud, busy, fair. They played many games there." My adaptation is: "They played many games [at the loud, busy, fair]. Since "at the loud, busy, fair" is taken verbatim from the original source, do I need to write my adaptation as: "They played many games ['at the loud busy fair']." to indicate that I have taken it verbatim?
Specifically in relation to meteorology. I was wondering if the angular momentum an object, lets say a parcel of air has due to the roation about the earths axis. Is it conserved if moved to a different lattitude. According to my understanding the conservation of angular momentum applies as long as there is no torque on it. Is moving an object in latitudinal direction a torque in this case?
Hi been trying to solve this question : Show that the following classes are not closed under ultraproducts: (a) the class of torsion groups; (b) the class of simple groups; Thought about showing that they are closed under isomorphism and to show that they are not axiomatizable hence they are not closed under ultraproducts, couldn't really proof that tho, so I would like some help! Thanks!
I am looking for resources treating the problem of simultaneous DC Stark and Zeeman Effects for arbitrary directions of the magnetic and electric fields (not necessarily orthogonal or parallel). I am in particular interested in the effect of such simultaneous fields on atoms with hyperfine structure such as alkali. Of course feel free to write an answer if you yourself know how to treat this problem.
You pull cards out of a shuffled deck until you get a jack and a queen. Given see the jack first, what is the expected number of cards between the jack and the queen? I believe this is a conditional expectation problem. Our goal is to subtract the expected position of the jack from the expected position of the queen. However, I am lost on how to find this other than writing out all possible combinations which isn't feasible. I know how to find the expected position of either card without the condition. Do I use this and compute variance in some way? Kind of lost, any guidance is appreciated.
Abbot et al. "Experimentally probing the algorithmic randomness and incomputability of quantum randomness" remark that "incomputability is a weaker property than Kolmogorov randomness". I understand that a Kolmogorov random infinite sequence is incomputable. The statement implies that there are incomputable sequences that are not Kolmogorov random. Why? (It's difficult for me to imagine what such a sequence would be like. If the answer is complicated, pointers to textbooks or literature at a similar level would be welcome.)
On a fire extinguisher's cover, "In case of fire, break glass" may be written. As you see, in short, instructional contexts, one often finds articles being omitted. Is this grammatically correct? Is there a name for this kind of writing? Must it instead be written as "In case of a fire, break the glass"? This omission of articles is very common in instruction manuals and tutorials.
This is a less concrete question I was just curious about. We always talk about straight lines of symmetry and reflections but can non-straight lines of symmetry exist when reflecting something, and if so what would they look like? Logically, I'm not sure if it makes too much sense to ask something like this since I guess for some reflections, the reflection of an object could overlap with its own reflection but I'm asking just for the sake of maybe discovering something new. I guess it's also difficult to visualise what that would look like for example, what would the following circle look like when reflected across the red curve? Any insight into this would be greatly appreciated!
I've been playing a game and after completing a mission, one character ask to another the following: How you doing on the feddy? From the context, I assume that it's somehow related to money, but I'm not sure-it's an odd word! The only site I found it was on Urban Dictionary. My question is, what does feddy really mean here, and how did it come to take on that meaning? (Here's the video that shows some more context.)
Would a high-pitched unhearable frequency be heard whilst the doppler effect is in play? For example, when a car uses its horn whilst travelling by, the pitch shifts as it passes - which is the Doppler effect. What would happen if someone were to play a frequency so high that humans couldn't hear it and went past a bystander? Would the bystander be able to hear the frequency as the pitch shifts? or would it not affect it? Just a random shower thought that was playing on my mind...
Does there exist a continuous function defined on an open interval and being differentiable in a unique point? I have found examples of real-valued functions which are differentiable in a unique point of their sources, but all of them are not continuous at the points where they are not differentiable. I would like to know if there are examples of these kind of functions in the continuous case.
Why are pressure hulls machined to be nearly spherical? According to Wikipedia, https://en.wikipedia.org/wiki/DSV_Limiting_Factor has a hull that is machined to be nearly spherical "for enhanced buckling stability". Is this really superior to an unmachined hull in terms of pressure resistance? I can see why it might be useful to trim any excess weight from a diving vehicle, but I don't see how removing material can ever improve pressure resistance. Does anyone have any insights?
So I'm learning data science concepts for analysis of stock price data and I've gone into a rabbit hole. I first came across numpy.triu() in the Numpy library which calculates the Triangle Matrix for a dataset to find large movements in stock price points over a range of time. Then I learned more about the concept and how values in the set are calculated. But now I'm wondering what the practical uses for this concept? I'm especially interested in uses in analyzing stock price data.
I was reading the paper on the Hall Effect and found that the initial experiment was performed on a metal strip. The experiment on the metal failed to provide any useful results, and then it was stated that Owing probably to the fact that the metal disk used had considerable thickness, the experiment at that time failed to give any positive result. Later when the same experiment was performed with a gold leaf, the experiment gave satisfactory results. So, I wanted to know how thickness impacted the results of the Hall Effect Experiment.
Spatial gradients in the refractive index of a medium have an impact on the wavelength, frequency, and direction of waves propagating through them. What about media whose refractive index changes (spatially uniformly) with time? Is it correct to say that there will be no changes to wavelength or direction of propagating waves? What equations should be used to model wave propagation in a medium with time-varying refractive index? As a bonus, how would we model propagation if the index is changing non-uniformly in both time and space?
I was invited to review an article by one of the IOP journals (Journal of Physics Communications) a few months ago in which the author claimed that the EM fields, similar to photons, bend in a gravitational field. The calculations were slightly hard and I had not enough time to follow the calculations, and thus I had to decline to review the article. However, the article inspired me to consider that every single electric (or magnetic) field may have a photon-like behavior and it bends inside a gravitational field. Is this a rational deduction?
I want to design new characters (mainly ornaments but also mathematical symbols) for the CM modern font family using METAFONT. What is the best way to achieve consistency with the rest of the family? The first idea that came to my mind was to copy paste the cmbase.mf file from https://ctan.org/tex-archive/fonts/cm/mf but that's a lot of duplication and I am not sure I am legally allowed to do so.
I'm aware that in sentences using neither-nor constructions, the verb must agree with the subject closest to it. E.g.: Neither the dog nor the cats have been fed. Neither the cats nor the dog has been fed. However, when it comes to question forms, what should the verb agree with? Does the rule still apply? Has/have neither the dog nor cats been fed yet? To make it grammatically sound, should has or have be used in that sentence? Edit: The question should be "Has/have neither the dog nor THE cats been fed yet?", with the definite article "the" before "cats", which preserves the structure of the example I provided.
For a phonon mode to be able to emit electromagnetic radiation, it must have a dipole moment. However, it is not intuitive to me why the LA phonon does not have a dipole moment: suppose all atoms in a unit cell are oscillating in phase back and forth relative to the electronic background, isn't this itself a dipole moment? The radiation should came out perpendicular to the oscillatrion direction. Mathematically, when we determine whether a mode "has a dipole moment", what quantity are we actually talking about? Thanks.
I am confused about how to understand the sheaf on empty set (as an open subset) of the Zariski Topology over the Spec(Z), which is generated by the additive identity zero in Z. The sheaf I have in mind is the structure sheaf sending open sets to local rings. I read related pages and I cannot understand how sheaf over empty set is a terminal object. Is this terminal object an additive/multiplicative identity of the ring? Is terminal object an element of the local ring?
I've seen these words both used but not interchangeably. My guess is that fascistic refers to 'fascist in nature', e.g if a building is fascistic it is reflective of fascist ideology but not necessarily a product of a fascist regime while fascist means relating to actual instances of fascisism e.g, saying a building is fascist means it was a Nazi building or otherwise. I'm not sure though...
I have read that the Einstein-Cartan theory introduces torsion into general relativity in a way that produces coupling between gravity and the spin of particles. Then, the gravitational field is able cause the precession of this spin axis. But what about spinless particles? Wouldn't there be a difference between the way gravity acts on particles with spin vs no spin? I guess that even for particles with nonequal nonzero spin there would be a difference If gravity acts differently on the varied kinds of particles, the equivalence principle goes out the window. Plus, I don't know how to make the spin coupling disappear with a simple coordinate change
Strictly speaking, a stochastic process can characterize a collection of random variables whose index is not time-related. Please, are there examples in the literature where this has been considered? Otherwise, should we refer to those processes as random fields as opposed to stochastic processes, reserving the term "stochastic process" for the case wherein time is involved? I'm particularly curious about the case wherein there is a process over space as opposed to process over time.
I watched a video lecture that talked a bit about when proof by contradiction is and isn't a useful approach. This was useful, but it was from a heuristic approach which caused me to wonder is it the case that "P is provable" implies "there is a proof by contradiction of P"? I suspect this may vary by logic system. If that is the case I would still be interested in know what major systems do/don't have this property.
I am helping a friend edit a manuscript for an informal reference book about music. There are several instances where I am perceiving a mixture of tense but I'm not sure how or whether to correct it. Here is an example sentence: If we begin on the first note of the C major scale and skip every other note, the following chords get created: I suggested changing "get created" to "are created." The verbs begin and skip are clearly present tense, and I believe get created and are created are present perfect tense, so my analysis of mixed tense seems incorrect. Why does "get" seem like incorrect usage in this example?