Two people, A and B, starts from two different points and move in a perfectly straight line in an infinite plane. When they move they leave a visible trace after them. Question: What is the probability that their path (of traces) will intersect at some point regardless of where they start? What I've tried so far is to draw two circles and split it in quadrants. That helped a little bit but didn't really solve the problem, just got an overview. Here's some examples of interesecting paths (first row) and non intersecting paths (second row) which gives you an idea of the criterias for when they intersect and don't intersect: How would you approach and solve this problem?


I don't see why or how this could be argued against or if it really matters, but that's why I ask the forum. I am just trying to understand the nature of the fields in the universe. Am I correct in thinking that the entire universe is just a few fundamental fields laid over each other, getting influenced by the different particles that happen to interact with them?


In a nuclear reaction, protons and neutrons are exchanged and the resulting elements have a number of different protons and neutrons that uniquely define the element that we are studying. However, can we say that a hydrogen atom without any electrons (what we would usually call an ionised atom of hydrogen) is really hydrogen? Or do electrons fundamentally indicate the element that results from the reaction? Can we really say that an atom of hydrogen without electrons is actually hydrogen?


My textbook says: Similarly, non-polar (i.e., covalent or organic) compounds like naphthalene, anthracene etc. are soluble in non-polar (i.e., covalent or organic) solvents like benzene, ether, carbon tetrachloride, etc. and are very little soluble in water. My question is that if we have a non polar compound, then it seems as if it should not dissolve to any extent. However, the quote says that non polar molecules are a little soluble in water. Thus I want to know why non polar solutes dissolve in water at all, even if it is only to a very lesser extent?


What is a word to describe how a cat looks when it's groomed? My husband described it as nappy, which I don't think would be the correct word. To give an image of what I am trying to describe, think of a medium-haired cat that licked its fur and looks messy, with wet fur sticking up a bit. It's a weird request and may be too specific, but I hope there's some kind of answer to what I am asking.


I just read a quote in which someone, on behalf of herself and her spouse, wrote "Our heart goes out for ....". I Googled to know if when it comes to speaking about heart not as a body organ, is it possible to use it in the singular form, but I could not find any reliable source for that. So, what is the correct form for such situations and does it apply for other organs, specifically brain?


If I shred a book into numerous pieces, and scatter them, so that the original text is completely unreadable, then for me, as a reader, the book has lost all its information. Its entropy has greatly increased. As a reader, the book's information rested upon a state of minimal entropy. However, my (limited) understanding of Shannon's information theory is that the 'information' in the shredded book has increased along with its entropy. Is there a way of reconciling the conventional/layperson's understanding of information as an ordered presentation of knowledge (like a well-written book) with the Shannon equation of information with entropy/disorder? Apologies if I have failed to grasp something fundamental.


In the image above, we have the principle of induction cooking. An alternating current is run through the coil, which causes a change in flux. This change in flux induces eddy currents in the conductive pan, and by Joule heating/resistive heating (P=VI), this causes the pan to heat up. So the only thing that you need is a pan that can conduct electricity/the eddy currents right? As long as the pan does not have a too low or a too high resistance. However, why is the effect specifically optimal for ferromagnetic materials/conductors? How does the ability to magnetize somehow enhance the effect of induction heating? Does the magnetization of the pan itself somehow enhance the eddy currents or something?


Often charge is defined as a measure of interaction with the electromagnetic field. However, electromagnetic field is often defined as a field produced by moving electrical charges, or relates to charges in some way or the other. Due to this circular definition, I am not able to understand what really is charge and its influence on electromagnetism, or any interaction. So what exactly is charge and what does it have to do with electromagnetism, or any other interaction?


There is a word in German, ausgezeichnet which vaguely sounds like the English phrase "out of sight" but that is usually translated as "excellent". I could see some non-German speaker hearing it and, knowing that it meant "very good" from context, deciding to use it and approximating its sound with "out of sight". Note that I am interested in whether the German word and the phrase are related, not the general history of "Out of Sight" Is there any evidence to support this etymology?


Assume I have a hemispherical container of water and an eyedropper. Is it possible to add water to the container one drop at a time, such that each drop continues to add kinetic energy to the water? When a water drop is added, it creates waves. I am wondering if it is possible to keep adding water drops in time with the waves, so that it acts like constructive interference, and adds more energy into the waves. I think this could be a very interesting sculpture, to have a computer-controlled eyedropper which slowly builds up the amplitude of the waves.


(I'm South Korean, so non-native question here.) I saw the sentence 'that looks about it'. Does it mean that something is seemingly almost done? I searched this expression on google but I couldn't find the exactly same pattern of the expression but I found at the Cambridge dictionary: that's (about) it : there's no more to be said or done So I guess the sentence 'that looks about it' means something like 'I think it's almost done'. Is my interpretation right? How often is this expression used?


In quantum field theory we often hear that particles sometimes behave like waves and sometimes behave like particles. In quantum field theory we say particles are fundamentally fields. Is it correct to say that particles and waves are limits of quantum fields? For example, if we take a quantum field and apply the limits appropriate for a double-slit set-up, would the field come out a wave?


I am no expert in the calculus of variations but I had some lectures about it and attended some talks. I noticed that searching for minima of functionals seems way more popular than searching for saddles. I know about some MiniMax theory or mountain pass theorems that can be used to search for saddles aswell but especially in the context of actual variational problems it seems to me that people are more often looking for minima. From a PDE point of view that seems kind of odd, because saddles could solve corresponding EL-equations aswell. Question: Are people actually more interested in minima? If yes, why? If you got any literature recommendations on saddles or similar topics, I would be very thankful aswell. Thanks in advance.


I am once again asking more software-related questions, but I am happy to learn a general answer as well. I'm curious about how the is_infinite function works in SageMath (or in GAP). Does it simply check whether we input something that is already known to be infinite or not? Or does it print "infinite" after a specific number of elements? Or does it actually check if the group is infinite? I assume it's not a numerical answer to check if the group is infinite or not since the program can't count until infinity. So, I'm wondering what the method is for checking if a group is infinite or not (in SageMath and in general). Thank you for your answer.


Can you refer me to books that are at the same advanced level of real analysis via measure theory as Folland's and Royden's books? I want to have a solid foundation and I've been referred to books like Cohn's and Axler's, but I don't have deep knowledge of real analysis (I'm still going to start studying introduction to analysis) and I don't know if these books combine measure theory and real analysis like the ones I mentioned in the title or whether they are purely measure theory. If possible it was a Dover book because they have affordable prices here in Brazil. Thanks to everyone who can help me.


I've always been taught that vectors that form a closed polygon represent an object being at equilibirium, that is there is no resultant force on the object. However, this has never been intuitive to me. How does one prove, that if the vectors form a closed polygon, the resultant force has to be zero? Furthermore why must the arrows form a closed loop, why is it not enough for the lines themselves to form a closed loop?


When reading about the notion of affine connection, the Levi-Civita connection appears naturally as the unique affine connection that preserve the metric and is torsion free. In this case, it is possible to express the connection coefficients analytically as a function of the metric and its derivatives. What I am curious of are alternative affine connections, possibly admitting explicit formulas for the Christoffel symbols. In other terms, what are other connections that agree with the definition, but are not necessarily compatible with the metric and do rather reflect some other underlying structure? Thanks


I know what a collocation method is and how it can be useful for the direct transcription of optional control problems, using specific collocation points. I also know what a quadrature rule is and how it can be used to solve integrals numerically, using specific quadrature points. But I'm struggling to see the mathematical connections between these two concepts, that are obviously related (for example: the trapezoidal collocation method has to be related to the trapezoidal quadrature rule somehow). So basically I'm wondering: is there a one-to-one correspondence between one collocation method and the related quadrature rule? If so, is it possibile to show how they are reciprocally related, in general, from a mathematical point of view?


I am writing to ask for references. I have been trying for hours to find some paper/ book containing information on cycles in cubic graphs. It is known that to determine whether a given cubic graph contains a Hamiltonian cycle is NP-hard, but since it is not a tree we know that it must contain at least one cycle. But do we know we can find such cycles in cubic graphs in polynomial time? Do we know an upper/ lower bound on the size of the cycles in cubic graphs? I would be grateful if you share any references or any information on this topic.


In some parts of my document math environments, such as "align", are not recognized, e.g. there is no arrow on the left to hide it, there is no automatic indent, and the theme colors are not applied properly. The same happens with "subsection", "section", etc. However, in other parts of the document the environments work. What could be the reason and how do I fix this? I am using pdfLaTeX compiler. Here is an example of how it looks in the editor:


For example, prove that if A AND B and A AND C are logically equivalent. How would I go doing this? I thought of using two truth tables: A | B | A AND B T T T T F F F T T F F T A | C | A AND C T T T T F F F T T F F T Then concluding they are logically equivalent. However, they are from two seperate truth tables. Would that be correct? Or do I have to combine the truth tables?


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?


I read the following text on This Page: Does every polynomial have at least one imaginary zero? No. Real numbers are a subset of complex numbers, but not the other way around. A complex number is not necessarily imaginary. Real numbers are also complex numbers. Now, FTA states that: every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. my understanding is that, unless "Complex Root" means something different than "Imaginary Zero", the former page statement is wrong. Please clarify the situation. Thanks. Edit: After reading the good comments below, I came across this diagram which helps: Diagram-Classifying Numbers


Recently I saw two interviews, one with Victoria Beckham, the other one with Elton John. They were talking about their past experiences and that's the phrasing they used: But it wouldn't have happened if we HADN'T HAVE GONE to the Ukraine. (Elton) And I probably would still be that way if I HADN'T HAVE MET the girls. (Victoria) Is there any explanation behind the grammar written in capital letters?


This is a silly question, but can it be done safely? My friend wanted to have a soup with electrolysis causing it to taste differently on each side of the bowl. I think this is dangerous, because there is table salt in the soup and I suspect that during electrolysis chlorine gas is made. Maybe someone can clarify the soup example or provide an alternative?


I am studying some results about existence and uniqueness of solutions to some PDEs and many times Banach fixed point Theorem is used. I saw that the ideia is to consider the integral formulation and use the Theorem to prove existence and uniqueness in a ball but I can't understant how the uniquess extends to all the space. Any one can explain how to get uniquess in all the space and how existence and uniquess of solution to the integral formulation garantees the same for the PDE? Any help is appreciated.


I have seen that there exists an intimate relationship between electric and magnetic properties. From what I know is that electric field creates electric potential around the source's space which can allow a charged particle to gain some PE and makes it possible for that particle to have translational and rotational motion. I have searched quite a bit and could see that magnetic potential energy can only cause rotational motion, why not translational motion?


I've recently found a really old "Philosophy of Math" book in my University library, and in the book it says that the it has been proven that : if non-euclidean geometry is inconsistent, then so is euclidean geometry, and the reverse is also valid. But the book wasn't as helpful as it doesn't prove, or give some reference or name to this result. Does anyone have a proof for this or for something similar? Even a definition of what exactly "inconsistency" means in this geometric context would be welcome.


I am competing in the American Computer Science League (ACSL), and I get problems similar to the following. Look at this directed graph. Now tell me the number of cycles in said graph. These questions need to be completed quickly and by hand. Unfortunately, the ACSL wiki (which provides help and study resources for the test) says to find all cycles by inspection. This obviously is slow and possible erroneous. I want to be able to complete these questions with the utmost accuracy. Is there a method to, given a graph, calculate the number of cycles by hand?


The energy of photons is reduced by the expansion of space. Since the photons have been decoupled from matter their wavelengths have decreased from the visible range to the radio range. The same should hold for the neutrinos that emerged from the big bang. They still travel at a speed that's close to the speed of light. Almost all their energy is kinetic. Will their speed be reduced to, say, half the speed of light in the far future? If so, how long will that take?


Every number theory book seems to start with divisibility, congruency, the division algorithm, etc., while completely ignoring the rigorous construction of the natural numbers. It seems like a wasted opportunity to explain to the reader what's actually happening. I feel the same about combinatorics. Permutations, combinations, and the rule of product are derived heuristically rather than through rigorous arguments from set theory. Is there a particular reason why number theory books take this approach rather than a rigorous from first principles approach?


I am writing a an essay comparing The Great Gatsby and Pride and Prejudice in English (Australian English style), and to save on word count I wondered if it would be appropriate to refer to them, after the first few uses of the names, as Pride and Gatsby respectively. Is this acceptable to do in a relatively formal context, and if so, is there any procedure for first introducing the contracted names?


I just built Lyx on Manjaro linux from pamac (there is an entry from the AUR, see https://aur.archlinux.org/packages/lyx). When I try to view a document in pdf using pdflatex an window pops up saying an error has occured in the external program "pdflatex" and that I should see the logs. Unfortunately I can't find the logs for pdflatex... If I type pdflatex in a terminal, it is up and running. Also all of the document classes and modules required by the Help - Introduction, Tutorial, ... are missing. I wonder if I missed a package to install. Thank you for your answers !


I need to quote and excerpt a lot of the same passages from a diverse library in my work. I'd like to find an app that's basically a purpose-built database for this kind of thing -- where I can store the excerpted text, the BibTeX source, the page number, a brief description, maybe some keywords/tags/categories. Basically something like Jabref, but where elements are passages not sources. I found an app called TextCite, but it looks like it's no longer being updated and I had some trouble installing it. Any help is most appreciated.


In a nuclear fission reaction the total energy and thus the mass of the products are lower than the total energy of the reactants. I understand the the difference in this energy can be used in a nuclear reactor/atomic bomb. But in the physical sense how does this energy manifest, is the kinetic energy of the reactants or some EM wave emission or combination of both. Also if it is the kinetic energy of reactants, wouldn't that energy be already included in the total energy of reactants?


When observed from the ground frame (assuming no friction is present), the only forces that act on the rotating bead are weight and the normal force exerted by the loop towards the center of the loop. I suppose there is no component of the normal force in the direction of the bead's movement, so it shouldn't move upwards. I am unable to comprehend why the bead does move upward.


In Tao's Fourier Transform preprint, https://www.math.ucla.edu/~tao/preprints/fourier.pdf, they show explicitly that the Laplacian can be viewed as a Fourier multiplier. In the same paragraph, they write ...This identity shows that the Fourier transform diagonalizes the Laplacian Why is the word "diagonalizes" used in this context? I know it from a Linear Algebra context, in which we diagonalize some matrix, but in this context it seems out of place. Am I missing something?


Just a naive question about univalent foundations. As far as I understand, we want to define our mathematical types like sets, groups, categories, etc. such that structurally identical objects are equal in the sense of internal equality. But are we free to choose what kind of isomorphisms we are interested in? For example, can we define both, the type of categories with ismorphisms as equality, and the type of categories with equivalences as equality? Or the type of topological spaces with homotopy equivalences as equality, and the type of topological spaces with homeomorphisms as equality? How would these differences actually look like in the definition?


I was just thinking about this studying for a biology test as for cells a greater surface area to volume ratio is desirable, so I wondered if there was a theoretically optimal shape that would maximize this ratio. I thought it might be that there is no such shape as you could just add folds to the surface indefinitely but I don't know if that is truly the case.


This question is similar to this post, but in my case the actual numbers do not matter, I'm interested in the equations. I have an arbitrary triangle, of which only angle BAC and side AB are known. The sum of sides BC and AC (therefore the perimeter) is known and fixed. I am looking for an equation that gives me the value of angle ABC. Here's a screenshot, please remember that the exact dimensions are arbitrary, and are only there to show which parts are known. Triangle drawn in CAD This is really driving me crazy, I would appreciate any help! Thank you!


I've mostly used and seen "utilitarian" in a negative sense: something that only exists for its function, and specifically for those functions. In other words, something that is single-purpose. utilitarian designed to be useful rather than decorative: But what if I wanted to describe something positively, could utilitarian be an apt adjective? I often see phrases like "utility focused" or "utility oriented" as positive, but utilitarian as negative. Aren't they the same? For example: "I worked at Big Tech Company where I designed and developed utility focused modules" sounds positive or neutral. "I worked at Big Tech Company where I designed and developed utilitarian modules" sounds pretty bad.


I am looking into uses of Fourier series. I learned that it can be used to approximate functions (link). However, I think there are ways to approximate functions with polynomials (not limited to Taylor series). What advantage does the Fourier series have over polynomials in approximation? I think there is a disadvantage to the Fourier series. While the way to compute the values of polynomials is obvious, computing the values of Fourier series needs approximate values of the trigonometric functions.


I was thinking of the trading of kinetic energy during a gravitational slingshot maneuver and wondered if the kinetic energy lost during that process makes any noticeable impact on the orbit of the planet. Since the planet we are performing this maneuver on loses kinetic energy, it is realistically possible to do enough slingshots that we noticeably change the orbit of the planet? And what would be the consequences of changing the orbit?


Taka a standard spinning bicycle wheel with its spokes and place it in a magnetic field that is parallel with the axle. I know that if the spokes were not made of metal but some insulating rods there would be an emf and a corresponding current induced in the rim. My question is what happens when the spokes are made of some conducting metal, will there be an emf induced across the spokes and what will the current be in the rim and spokes?


Do Boltzmann brain thought experiments suggest literally anything can form randomly? What are the limitations to what random fluctuations can form? Literally any physical, material object? Lastly, I am curious as to how this compares to an object with a high degree of complexity always existing. For example, is a bike more likely to always exist for no reason than it being created by random fluctuations? Or is it less likely?


In a scientific abstract, I have a sentence like this: Yet, numerous capture-mark-recapture data sets, across wide taxonomic range, feature transient individuals. I want to emphasize that there are numerous capture-mark-recapture data sets, and that they all together cover a wide taxonomic range. Not each one of them; each of those data sets is covering only few very specific taxa; but all together they cover the wide taxonomic range. I don't want the expression to be too ambiguous in this direction. Is this sentence correct and unambiguous, or could it be written better? It shouldn't be much longer since I am very close to the word limit. EDIT: would this one be better? Yet, numerous capture-mark-recapture data sets, from a wide taxonomic range, feature transient individuals.


I am very new to semiconductors and I'm trying to understand how photodiodes work for imaging sensors. If I am not wrong, a PN junction is required to detect electron/hole pairs that are photo-generated. The electric field in the depletion zone will move the charge carriers and create a current that can be exploited. Why not only use a semiconductor without doping? And then apply an electric field to the semiconductor: when light is emitted, it slightly improves the material's conductivity, and a current is detected. Why is it not that simple? Thanks


The statement reads as follows "If is invertible, there are no special solutions in the null space." According to me, its actually true because if the matrix is invertible, it means that it has full rank, and its null space contains only the zero vector. However, the questions asks you to explain why the statement is false. Is there a way for this to be false? Or, is the question wrong and the statement is true?


Not that I work in this field, but I have heard the discourse from several people that string field theory is useless. So I would like to ask two simple questions? What makes SFT so difficult? Especially, it seems to me that one can construct an action that satisfies general coordinate transformation invariance and gauge symmetry (more precisely BRST symmetry) by using the BV formalism, as shown by Witten, but why is this useless? When moving from QM to QFT has several practical benefits, for example, one can easily construct relativistic scattering problems in Lorentz-covariant form, but in what way is SFT superior to string theory?


A devout Christian, she plays tennis every weekend. The most beautiful of villages, Ullapool has hosted a food festival for the last twenty years. I see this kind of thing a lot in non-fiction writing, particularly things like short bios/blurbs and tourist writing, where people are trying to convey information but also make it attractive/enjoyable to read. The part before the comma feels like it's introducing the part after the comma, but actually the two things have no relation. Like One of the most infamous mass murderers of his generation, his favourite colour is red. Is it a recognised 'thing' and does it have a name?


I'm aware that a semantically complete theory can be undecidable. (I believe it's because only logically valid sentences need be provable for a theory to be semantically complete.) But is it possible for a syntactically/negation complete theory to be undecidable? And do "syntactically complete" and "negation complete" mean the same thing? I've heard that: Any consistent, negation-complete, effectively axiomatized formal theory T is decidable. So for a negation complete theory to be undecidable, it'd have to be either inconsistent or not effectively axiomatized. But I have no idea how a theory could be negation-complete without those two other criteria in the first place.


I am having a hard time understanding what eigenvectors and values intuitively represent in an adjacency matrix in a graph problem. I understand in a linear transformation, eigenvectors are vectors that are only scaled after the transform is applied and that the eigenvalues are those scale factors. Is it strictly for computational efficiency that we are so interested in the eigenvalues in graph theory, so we can utilize the eigenbasis for computations?


I have these two graphs here: I wish to determine if they are Isomorphic. I know that I need to find a one to one and onto function, however I can't find a way to do it. My questions are: I know that these two are isomorphic. What is the function f ? In general, is there a way to find these functions ? Maybe using matrices ?


I've seen diagrams for projectors where condenser lenses focus light directly onto the pupil of the projector lens, after going through the image source. Am I misunderstanding the diagram? If not, why does it focus directly onto the projector lens? I was under the impression that you'd want the condenser lens to form a real image somewhere before the projector lens - that way, the projector lens can image it, like this:


Being impressed by Barr and Diaconescu paper entitled "Atomic Toposes" (https://www.math.mcgill.ca/barr/papers/atom.top.pdf), I would like to ask whether it makes sense to investigate "Atomistic Categories", namely categories such that for any object the corresponding poset of subobject is atomistic (namely every element is arbitrary join of atoms). Have these categories been studied, are there non-trivial exaples (the category of sets is an example, but I'm searching for other examples!)


I am rather rusty and am reading Topology by Graham and Greene. One of the problems refers to a countable complete metric space. The rationals don't seem to be such a space, because the irrationals are adherent to them and one can form a Cauchy sequence of rationals that converges to an irrational. Thus, the rationals are not complete. Could someone please offer an example or two of countable complete metric spaces?


I want to say something along the lines of "We must protect everyone, especially children". But this doesn't exactly work for what I want to say. Because I do not mean that children are MORE deserving of protection in this instance. How can I exemplify a group without raising them up to an elevated status? I thought of "in particular" but this has the same problem.


In most material I can find on the cosmic distance ladder for a lay-level reader like myself, it never seems to explicitly say what distance its talking about. There is the proper distance when the light was emitted, and the comoving distance, where the light source is now. The light we are receiving however has traveled a different distance, the light travel time distance, which is less than the comoving distance and greater than the proper distance at emission time. What does the distance ladder actually measure? How could it be less than or greater than the distance light actually traveled?


Lenz's law was explained to my class using an example wherein there is a bar magnet passing through a current carrying loop in a gravity free space. Our professor told us that the induced magnetic force is repulsive in nature in order to perform negative work on the bar magnet. The kinetic energy lost by the bar magnet as a result of the repulsive force would be lost in the form of heat in the wires of some internal resistance. If the induced repulsion is because of the heating effect of current, won't Lenz's Law be invalid for resistance free wires, where resistance is zero ? If it is valid, why ?


In Landau's book of fluid mechanics, the most general formula for stress looks like this: let's call the eta in picture the first viscosity coefficient, call the zeta in the picture the second viscosity coefficient. I would like to know what are the typical values for these two coefficients for air. I am not sure if my memory is correct, but it seems that the second viscosity coefficient is often neglected. By the way, I am not sure if questions regarding fluid mechanics and CFD are appropriate to ask here, or if there is a more suitable place to ask these questions, if you know, please recommend some to me, thanks.


As an Electrical Engineer, I have been studying convex optimization for a while. During my study, I see that most textbook claim that both second order cone programming and geometric programming can be solved effectively with modern solver. However, I cannot find any document that compare them together. So my question is does second order cone programming harder to solve than geometric programming ? Futhermore, is there anyway to come up with a Big O complexity to measure the difficulty of solving them in terms of the number of variables and constraints ? Thank you for your enthusiam !


I was wondering if we could use background radiation to produce electricity (even a tiny amount). However, I've read somewhere that solar panels absorbing infrared (and lower frequencies photons) wouldn't be very efficient, as they are limited by the laws of thermodynamics. The arguments states that the panel needs to be cooler than the temperature of the body emitting the photon. How is this true? I cannot explain why it would be that way with the traditional band diagrams I've learned. As far as I understand it, the only limiting factor would be the band gap of the material that is used, and it needs to be low enough to capture low energy photons.


Recently, I find an interesting paper that mentioned the Definition of curvature of an affine optimal control system. It suddenly reminded me that many textbooks on Riemannian geometry only tell us about metrics, geodesics, parallel transport, and curvature tensors and son on, using some special surfaces with constant curvature as examples. So for a general control system, how do we calculate its curvature tensor? Is this a problem worth studying? Are there any more examples? Thank you all for your comments and help!


I have learnt that the intuitive idea behind inner product space is finding angles between vectors. But what does inner product actually mean physically or intuitively when it comes to matrices.Can I link inner product of matrices to concept of angles?.I can't imagine angles between matrices as I do for vectors. I really want to learn visually and not just theoretically. As I am amateur in this area my understanding might be wrong. So please do correct me.


I have a .pdf file, in which I have to embed all the fonts. I found out that I should use this command: gs -q -dNOPAUSE -dBATCH -dPDFSETTINGS=/prepress -dPrinted=false -sDEVICE=pdfwrite -sOutputFile=edited_file.pdf myfile.pdf but the pdf file loses its interactiveness. For example, I have hyperlinks that jump to references, or clickable table of contents that becomes not clickable anymore after executing this command. Can you please help me to find a solution.... I have only linux, so everything Windows-related is not an option.


If neutrino has mass, then it can change it's helicity. But all registered neutrinos have negative helicity while all anti-neutrinos have positive. Does it stands for all registered neutrinos including cosmic ones? I.e. does it mean that cosmic neutrinos had not enough time to switch their helicity? Or it was checked only for neutrinos in accelerator experiments, i.e. only for "young" neutrinos? What about solar neutrinos? Is their helicity known?


I keep coming across the term storying in articles on narrative research. The authors however do not explain how the term storying is different in meaning and connotation from the term storytelling or story. I am keen to understand the various theoretical angles that I need to explore to understand the term Storying? Should I connect the concept of reflexivity to the concept of Storying?


I'm working on a project that aims to discriminate semi-visible jets (SVJs) in a QCD background, through learning from simulated data in the context of the LHC. I know that these are related to hidden valley models, and probably also to dark matter. Can someone provide an intuitive explanation (even with some technicality) of why SVJs are important, and how these are originated? Also providing some insights about dark sectors, and potential implications for the discovery of dark matter. (sorry if I'm using an improper language, but I'm not a physicist.)


The problem I'm working on has a region that's largely accurately modeled by a GP model using a squared exponential kernel. However, there is a large region in the "truth" model that is uniformly zero, with all samples taken from this region being zero as well. However, the GP model in this region oscillates from positive to negative when it's necessary for the problem I'm working on that this region is accurately modeled as uniformly zero, or at least negative when the remainder of the data is positive. Adding a dot product kernel to the squared exponential kernel to create a custom kernel helps somewhat but can't eliminate the spurious oscillations.


this is my first post on Stack Exchange. I need a study guide for linear algebraic expressions. I am good at vector analysis and calculus, but not so good at linear algebra. Image : In the image above, is the part I have colored in blue a linear algebraic expression? I am having trouble searching and learning this. I can't type expressions by math symbols into the Google search bar. I can't find a proper interpretation of the formula and just figure out what it means from the context. I'm not even sure if that's a linear algebraic expression. Is my only option to take my time and study a linear algebra textbook sequentially? I apologize if this question doesn't fit into the Stack Exchange.


If we place a point charge in a grounded metal box and view a slice of the box, the lines of equipotential deform from circular (immediately around the point charge, as we'd expect) to square at around the perimeter of the box where it is grounded. Is there a mathematical relationship we can derive for the morphing of the field lines from circular to square? I am unsure if the change is a linear combination of the two functions.


For some reason, TexShop started autocompleting any ampersand & by adding a period/full stop, which is not a feature that I consider desirable, given that I almost exclusively use the symbol for tabular and align environments. I feel that there is probably a simple way to turn this behavior off in the preferences, but I couldn't see how to do so. How do I change the default behavior?


When we think of functions, we refer to their oscillations or periodic components in terms of frequencies. The Fourier transform allows us to decompose a function into its constituent frequencies, providing a way to analyze the function in the frequency domain. For operators that act on infinite-dimensional function spaces, is there an analog of the frequency concept? Is there a transform or methodology that lets us analyze operators in a similar "frequency-centric" manner? I'm curious about both theoretical constructs and any potential applications such a perspective might offer.


From what I have read and seen online. The explanation of why there is a net reduction in the speed of EM waves through a medium is due to the interference with the oscillating EM fields produced by charged particles caused by the EM wave passing close by. I hope my understanding, if at least a base level, is correct so far. But would there be any reduction in the speed of light passing through matter made up entirely of neutrons? Either a neutron gas or possibly neutron degenerate matter in the cores of neutron stars. Thank you for your time.


I was wondering if there is an easy way to create diagrams like the following: where a, b, c are colors and strings connect like colors. In the paper I am reading, they don't allow crossings, but I would like to allow crossings between different colors. So something like this would be allowed (two colors red and blue) I tried to do this with the braid package, but the problem is that I don't know how to make those semi-circles Thanks in advance!


I'm writing an online game, and I want to tell users what their highscore is since the beginning of the hour. But I don't want to say something wordy like: Your highscore since the beginning of the hour is... At the moment I've got: Your highscore this hour is... I have 'today' and 'this week' for the other timeframes, but it still feels very unnatural. Is there any pithy phrase I'm overlooking? Many thanks!


Is there a ready-made LaTeX package that produces a calendar in a format known as Familienplaner in German speaking countries: Each month is set as a table on one sheet of paper. There is one row per day, and there are additional empty columns to be filled in (between one and five, typically). Sundays are specially marked (e.g., by red colour of the date, or by a grey row instead of a white one). Nice-to-have features: German language support for day-of-week and month names, week numbers, German holidays. P.S. Searching the net for "Familienplaner" results in a lot of free pdf offers for such calendars. P.P.S. The Familienplaner layout is different from the monthly calendar layout of this question: Calendar in LaTeX


I was wondering if the book Contemporary abstract algebra was suitable for self study. I've heard from people that the level is somewhat low and that things such as polynomials in several variables, particularly theorems on symmetric ones. Also,ive heard that applications from abstract algebra to linear algebra are lacking as well. Is this true? if so, which books would you reccomend for abstract algebra self study?


I am currently working on a BSM model for unification where I am adding scalars for unification. I add many scalars, so finding the most general gauge invariant lagrangian is becoming difficult. Is there a tool kit where I can enter the particle content, and under what representation of the gauge group they transform and get out the most general gauge invariant lagrangian? I have heard SARAH can do it, but I am unsuccessful with it. If anyone knows how to implement this in SARAH, that would be helpful, too.


Recently, I watched small dominoes toppling over, until they push over a massive one. However, I am slightly confused why this doesn't violate conservation of energy. Consider supplying some amount of energy to a small domino, eventually when it topples over the large domino, that large domino can do significantly more work, and should have more energy. This clearly cannot violate the laws of physics. So what is wrong with my logic?


Googling and Wikipedia, gives only the following construction A golden rectangle can be constructed with only a straightedge and compass in four simple steps: Draw a square. Draw a line from the midpoint of one side of the square to an opposite corner. Use that line as the radius to draw an arc that defines the height of the rectangle. Complete the golden rectangle. Question: What are the other known methods of constructing a golden rectangle with a ruler and compass?


I saw this solution to a problem on twitter, and the solution seems to of used the direct comparison test for the improper integral. I asked my professor about it, but they replied with "yes, just make sure to follow the given conditions." However, they didn't mention what conditions we would need to follow. If we can use Series Tests for Improper Integrals, how would we go about using them?enter image description here


I want to make a commutative diagram in the tikz (or tikz-cd) environment to demonstrate the universal property of the pullback. To be precise, I want to make the following diagram in LaTeX (the arrows on top need NOT be curved): I am not sure how to draw this. I know how to make general commutative diagrams, and there are answers to making pushouts with universal property diagrams (How to do the pushout with universal property?) but I can't figure out how to make the above pullback diagram. Any help will be appreciated.


I came across this question while studying for the SAT. After World War I, Percy Spencer was hired to work in the research _____ later known as Raytheon Manufacturing. A) lab of American Applicance Company: (a colon) B) lab of American Appliance Company-- (em dash) Though the answer choice states that B) is the correct answer, it does not specify the reason why A) is not preferable over B). I wonder if the usage of the colon in this sentence is correct. (As it sounds, to me, a little awkward to place a phrase like 'later known as --' after a colon.) I'd appreciate it if you could enlighten me. Thank you.


As in the picture, we have a room with vents at the bottom to let cold air in and at the top to let hot air out. Normally the air flow is limited by the temperature gradient (correct me if I'm wrong). If we place a hot object in the vent on top, won't that create a faster draft? That way we could get very high "wind" speed indoors to replace fans. Is this possible, and if yes, what's the limit for air flow per second?


Consider the following sentence Your time is limited, so don't waste it living someone else's. When I put it into a few grammar checkers like Grammarly, I get no errors. But I feel that there should be an error - like a word following "else's" (e.g., "Your time is limited, so don't waste it living someone else's life.", a quote due to Steve Jobs) Is there some sort of grammar error here, and if so what is it called?


Is Aristotelian categorical syllogism (as in Copi's Introduction to Logic) a special case of the first-order logic? (This is my thought after making efforts.) Is it exactly the specialization of the first-order logic without any function symbol (but only predicate symbols)? (This is my thought after making efforts.) If it is still a specialization of the above specialization, what kind of specialization is it exactly? How would you define it as a formal logic system? Thanks.


When a nucleus decays it is finding a more stable configuration, as all nature is. But is there a way of finding whether a specific configuration will decay? I know that there is a specific proton-to-neutron ratio range, of which outside nucleus decay. This can also be seen on the nuclides chart. I have also looked into concepts like nuclear binding energy and gluons. However, I never found a definitive answer to my question. My question: Is there a way, given the number of protons and neutrons (and electrons if need be), to find if the nucleus in this configuration will decay? (Please don't hesitate to use math to explain the concepts. In fact, I'd prefer it if I ended up with a formula.)


Today I was explaining to my friend about opposite categories, and how the definition of monomorphism leads to defintion of epimorphism in opposite category. He asked me quite an interesting question. If we have a morphism in a category, how does that relate to the morphism in the opposite category? For example, let's consider that category Top, and a homomorphism between two topological spaces. What would be exact expression for arrow in the opposite category? I think it'd be something like the inverse function but I am not sure.


This might be a very simple question, but the answer is not entirely obvious to me and I've been unable to find it stated anywhere. Consider two sets of generators (vectors), which are reduced in the sense that no two generators are parallel to each other (if one starts with two parallel generators, reduce that set by adding them together to make a single one). Then, if these two sets are not identical, can they possibly represent the same zonotope? I believe the answer is no. Am I wrong? A proof one way or the other would be appreciated.


Outlook is questioning "equipment register" and asking me to double check if I am using singular or plural (error underline on the word register). This has me really confused and I can't find any information about what could be wrong with this phrase. For context, the phrase is part of a list of other items and I am using it to mean a register of equipment, like you would have a register of users, or a register of stock...etc. I get no grammar errors if I replace "equipment" with other words (although obviously I haven't exhausted all possible words!) If I write "An equipment register" the error remains. If I write "One equipment register" it goes away. Thanks!


It is well-known the notion of weak limit in category theory. To obtain it, just take the notion of a limit and drop out the uniqueness property in the universal property. Now, as the universal property is crucial when proving the uniqueness (up to isomorphisms) of a limit, I expect the non-uniqueness of weak limits, in general. Well, it is possible to find an explicit example of category that admits weak pullbacks, where a cospan diagram admits two non-isomorphic weak pullbacks?


While starting out on Lang's Algebra I have encountered the term "exact" sequence of homomorphisms. And I understand its definition and all the demonstrated examples but I would like to know more on this, like what are the things we get to know out of an exact sequence and does it aid in any kind of constructions or whatsoever. Is it possible to generalize them and how does that look like ?. For my background, I have only been introduced to Groups, Rings and Fields in my introductory algebra course but I would be delighted in seeing a bigger picture on these structures.


I understand the fact that if there was no metallic ball in it, then when the ice melts, the level of water would neither rise nor fall. Now let us consider the case where there is a metallic ball in it. When the ice melts, the volume of water added to the vessel would be less than the volume displaced by the ice structure when it was floating. This should mean that the water level will fall if i am not mistaken, further which the metallic ball would sink. When that happens, it is displacing the same amount of volume as it did when it was within the ice. So shouldn't it mean that the water level would fall ?


My intuition: In a system without air resistance, I would expect an arrow shot at an angle with its head pointing upwards to follow a ballistic trajectory without rotating around the horizon - because gravity can't induce torque on the body. So the arrow will impact the ground on its tail - not its head. Is this correct? If yes, what causes the same arrow to impact the ground on its head when there is air resistance?


I don't think I understand correctly what it means for a Boolean formula to be reversible. By my current understanding, if a Boolean formula is satisfiable, then there exists a setting of variables such that the formula evaluates to true. If you were to reverse said formula, simply flipping each variable in the setting would also satisfy the reverse formula. You can similarly show that the reverse formula is satisfiable if the original formula. Overall, the reversible formula is satisfiable if and only if the original Boolean expression is. Since the author claims that determining reversibility is NP-complete, I think I'm missing something here. This is from the paper: On the Complexity of Polyhedral Separability


Working on a thesis, I was wondering how to correctly hyphenate (if at all) the term "high data rate" in the following sentence: High data rate ECUs are connected directly to the backbone. The term "data rate" as such is not hyphenated. But compounds with "high" are hyphenated, e.g. as in "high-income", "high-stress". However, writing "high-data rate" doesn't seem right to me. What would be the correct hyphenation in this case?


I've been reading a bit about how it is necessary to consider anharmonic effects in crystals if one wants to properly understand things like thermal expansion etc. So for example here: So the cubic term is said to represent 'asymmetry of mutual repulsion' and the fourth-order one 'large amplitude weakening' of vibrations. How could one come to understand these intuitively? How does one look at a cubic etc. term and come up with this explanation?


I heard a while ago that the negation of the axiom of choice leads to the existence of a partition of the real numbers, into more partitions then there are real numbers. I found this post which says You don't want non-measurable sets? This means that you can partition the real numbers into strictly more parts than real numbers, but no part is empty. Yes, let that sink for a moment: you can partition the reals into more parts than numbers! I looked for a proof of this online, but could only find the result mentioned. A couple referenced this link, but it's locked behind a paywall. So, why does the axiom of choice lead to this counterintuitive result?


So I was searching online for methods to approximate Pi and found this continued fraction that supposedly approximates to Pi when continued infinitely. I've tried searching all over the internet for how this was derived or why this works but could only find something similar but not close enough. I've got some basic knowledge of continued fractions but do not have experience with numbers that change throughout, how would one derive this equation? Thanks


To compile my document (even a little one), my computer needs to load a lot of packages. During compilation (with pdflatex) I can see in the terminal that most of the compiling time is used to load the packages. It would save me lot of time (and ressources) if it was possible to keep the loaded packages in memory and resume compiling operations after that step when I don't modify the preamble. Is this feasible?


Suppose one measures the position, then momentum, then position of a particle, and that all these measurements are done in quick succession of one another (ie. arbitrarily close to zero-time as possible). After the momentum measurement, the particles position is very uncertain. Therefore, the second measurement of the particles position can be quite different from the first. But does this not violate causality, since the particle can move a great distance in practically no time?


The energy that an electromagnetic field can deliver to a system is proportional to the square of the field. However, we use the terminology watts per meter square to represent the amount of energy going past to represent the intensity of EM waves. How do you give a physical meaning to the 'amount of energy that the field carries past us per second' if you are only given the amount of energy that an EM field can deliver to a system? Why is it necessary to adopt watts per meter square for intensity, whereas I believe the energy that can be delivered to a system is perfectly fine for defining intensity


In the case of a direct bandgap semiconductor, the recombination of an electron and a hole generates a photon that has energy equal to the bandgap of the semiconductor, accounting for energy conservation. But in this case, as the electron and hole have the same momentum, 'K' there is no difference. But the generated photon does have a momentum. So how is the momentum conserved in the radiative recombination in the case of direct band gap semiconductors?


There is a Spanish joke, Ella es mi amiga vieja, disculpe, mi vieja amiga. The joke basically says, "She is my friend that is old, excuse me, my old friend", making fun of the person's age. Would there be a way to translate this into English? Of course, saying something like "She is my friend old..." doesn't work in English, though the joke does also work in languages like French. The translation I wrote may be considered to work, but I was looking for something smoother. Edit: my attempt: She is my old af friend, I mean my extremely old friend, I mean my old pal.


I would like to be able to understand what the fission process is like (ideally, from total ignorance). What references could I read? Would it be more convenient to read chemistry books than physics books? From what I understand, the fission process involves "breaking" the nucleus of an atom. Regarding this: Is any atom fissile? Is there some non-fissile atom? Can this process be understood mathematically? Thank you so much.


I couldn't find any similar topics and was curious if anyone had thoughts on the following: Could a magnet (ceramic and/or neodymium etc.) have an adverse affect on a Police Taser, if held to the Taser or held in proximity to the Taser. If yes, is there a material that the magnet could be covered in, to insulate those effects? (i.e. by effect, meaning unintentional discharge of the Taser, or causing the electronics on board to malfunction.) Thanks for any color you may have on this, much appreciated.


What I was wondering is this. If I had a tube with water in it, with one end open and the other with a piston-like device attached to it, I can imagine that if I was to give the water a pulse, this would transmit down the tube as a mechanical wave as particles collide with each other - something like sound. But if we picture a scenario where instead of a pulse, I begin moving the piston down the tube at a constant velocity, what would the motion of the particles look like? Also what would this look like with air?


I think I understand Ampere's law in the way it was taught to me by my instructor. But as the idea was given that how Gauss's Law in Electrostatics relates to electric field lines produced by charges inside the gaussian surface under consideration. We weren't given any idea that how magnetic field lines even relate to Ampere's law. I have already searched for it on the internet including the posts present here but found nothing. All I ask for is idea that how Ampere's law relates to Magnetic field lines?. Thank you


I imagine that people will instinctively say, "There is no masculine or feminine speech in English," but I am not so sure. For instance, the stereotype is that men speak roughly and women speak softly. Then there is the usage of tag questions, fillers, and raising one's voice pitch so that the sentence sounds like a question. Then again, it may be a question of assertiveness. I hope someone knows what I am talking about!


Suppose we have a causal curve and we can cover the causal curve by convex normal neighborhoods. We also know that, in convex normal neighborhood there will exist a unique geodesic inside the neighborhood, connecting any two points in the neighborhood. So, my question is: Are we making the curve piecewise geodesic? If a curve is piecewise geodesic, can we say that the whole curve is a geodesic?


I have been trying to prove the stability of a queuing system (stochastic), and the state-space I have obtained is uncountable. I am aware of the Foster-Laypunov criterion, but as far as I know, it applies to systems with a countable state-space. I want to enquire of stability conditions for the case with an uncountable state-space, and also if Foster's criterion can be extended to this case. Please let me know! Thank you.


I recently stumbled upon the word biggety, meaning conceited or vain, and when I went to look it up, it seems the word is regional, which might explain why I'd never heard of it, except for one region it's supposedly used (Southern US) in is the region I've lived for over four decades. So, how common is it in the Southern US? How common is it in Midland US?


According to https://en.wikipedia.org/wiki/Closed_manifold "In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only non-compact components." Closed manifold is a compact manifold without boundary. Open manifold is a manifold without boundary that has only non-compact components. Questions: Is open manifold opposite or antonym of closed manifold? (It seems not, but why not?) How do we call "compact manifold with boundary"? What are the differences between open manifold and non-compact manifold? How is the concept "open" in open manifold compared with the "open" in open set in set theory with topology, and open cover in topology? These open seem to mean different things.


Godel's incompleteness theorem says that in mathematics there will be always true statements that cannot be proved and that adding that statement as a axiom generates other unprovable true statements. So, mathematics clearly has logical inconsistencies. I was wondering if it it is possible to find a language that doesn't satisfy Godel's incompleteness theorem or, to put it more simply, such that all its true statements are provable?


A thin charged wire conductor, with a semidiameter of a is placed into a vacuum at height h (h >> a) above a big grounded conducting plate. The wire is placed parallel to the x axis, and the area of the plate lies in Oxy-plane. The potential of the conductor compared to the plate is V. Find the equation for the areal charge density in the coordinate system (O). I tried solving this problem, I could find the equation for V using method of images but I don't know how to find the areal density. Can you help?


Is the Heisenberg's microscope gedanken experiment valid when considering spacetime kinematics? That is, if we consider a small region of space and try to measure its curvature, then we may use photons and see how the trajectory changes. On the other hand, the photon's energy changes the very curvature that we are about to measure. Here I assume that a high energy photon "maps" curvature more accurately. The ultimate goal would be to find an uncertainty principle that connects curvature and the rate at which the curvature changes.


I've gone to Wikipedia and watched a few videos regarding Linear Codes, and from what I see, the Parity Check Matrix is always given, and because of that the Generator Matrix (consisting of Identity Matrix, Parity Matrix) can be made. But what if the Parity Matrix isn't given, and instead, only k and n? I understand the general layout, but I do not see how to figure out the actual contents (or identity, I believe its called) of the Parity Matrix. I should add that the specific scenario I am asking about is in binary, so maybe its only doable there?


I've recently started learning about the physics of fluids and I've found that euler's equations exist and that I can use them to compute the flow of fluids with zero viscosity. But I have a problem that I don't see any easy way of adding an obstacle to a compessible flow as shown in this gif How should I do it? Is something similar possible also with navier-stokes equations? (I am aware that this questions is probably a duplicate of some older question, because this sounds as a really basic problem, but I've search for the anwer for some time and I even asked on the physics discord server and I found nothing)


Is there a way in TeX (preferably plain XeTeX, but other engines are fine) to automatically choose between different options (e.g. between "and" and "&") so that the badness of a paragraph is minimal? For example, for a small hsize, the following is fine | I read a book by Smith & John. | But if we increase the value of hsize, we have to add space: | I read a book by Smith & John. | So we prefer to use the other variant: | I read a book by Smith and John. | If I remember correctly, the first printers used this kind of trick to avoid line breaks.


I've heard about solitons in dense mediums (water), sparse mediums (acoustic) and optical fiber. But I can't find a good overview work on solitons in radio spectrum. Something like generating EM solitons with a peculiar antenna setup. It seems like an obvious thing for people to be working on, but looks like I just lack proper terminology to google it. Is there something that can be recommended, preferably with an experimental setup described?


Davisson and Germer's experiment confirmed the wave-particle duality of matter. In this experiment a slowed down electron beam is reflected by the surface of a nickel crystal, which is able to split the electron beam into several parts, each of which is characterized by a reflection angle, rising a diffraction pattern quite similar to that of the X-rays. Is what I'm about to write right? This phenomenon is possible because the surface of the nickel crystal is not regular, but is formed by microscopic grooves capable of rising to the phenomenon of diffraction.


I understand it is possible to find the shortest distance from a point to a straight line, but how do we extend this to any function. For a straight line it is simply the equation of the perpendicular which intersects the point of interest. If we have a cubic or a quartic function how would I model the shortest (straight line) distance to that function from somewhere on the plane? I anxiously wait a response.


Is there a way to make meaning of it? I was thinking that maybe because the covariance is divided by the percentile rank of std dev of A, the magnitude of A is to a degree bounded. But how does it change the covariance, as covariance is about the co-directions of A and B rather than the magnitude in the first place. Essentially, what does "covariance(A,B) divided by percentile rank of std dev(A)" reveal? Any thoughts would be greatly appreciated!


I have learnt that a planar graph is a graph which can be drawn on the plane without any of its edges crossing. What exactly is "the plane"? In plane geometry, we define a plane to be a flat surface of two dimensions; however here the use of "the" suggests we have a particular instance of a plane in mind, not just any plane. Is there a certain specific type of plane which is meant here, other than the general plane from plane geometry?


I am reading the paper "REPRESENTATION AND INDEX THEORY FOR TOEPLITZ OPERATORS" by Gerald Murphy and came across this passage: I was wondering why we care about the strict topology or why is it important for multiplier algebra. So I know how these are defined but I've never understood why we should use it. Am I supposed to assume we are talking about the strict topology every time I see the multiplier algebra? Now for this specific example given below for the Toeplitz matrix, why is it better to identify the operators on Hardy space with the multiplier algebra when talking about convergence?


Based on my intuition, once light is monochromatic it will stay that way unless you actively try to change its frequency. My intuition stems from the intricate optical laser setups in my lab that sometimes (often?) work with monochromatic light. Sound, on the other hand, seems to change its frequency easily when it interacts with the environment. Again, this is based on intuition so I'm not sure if its true. So my questions are Does light stay monochromatic once it is created? Does sound stay monochromatic once it is created? Why? Under what conditions do waves stay monochromatic?


Considering no air resistance, the acceleration time graph for a free falling object would be a horizontal line as the acceleration remains unchanged. When we do take into account air resistance, the graph becomes a curve with negative gradient. However, to me, this is unclear. I understand why this must be a negative relationship, but why can it not be a negative sloping line. When we graph acceleration against air resistance, we see this is a negative sloping line. Why must this be the case? I am looking for both an intuitive and mathematical answer.


We have a sentence like this: Transients are individuals just passing through, not contributing to the local population and its demographic rates, and they are thus very unlikely to be recaptured. Is it possible to remove "they are" behind the comma? So it would sound like this: Transients are individuals just passing through, not contributing to the local population and its demographic rates, and thus very unlikely to be recaptured. It would be much more elegant. But the catch also is that the "thus" doesn't refer to the most proximate sentence "not contributing to the local ....", but to the one at the beginning - "just passing through". Is that a problem somehow?


In this quote (from this essay), Russell talks about a theorem showing that "three or more lines meet in a point". "In geometry, instead of the tedious apparatus of fallacious proofs for obvious truisms which constitutes the beginning of Euclid, the learner should be allowed at first to assume the truth of everything obvious, and should be instructed in the demonstrations of theorems which are at once startling and easily verifiable by actual drawing, such as those in which it is shown that three or more lines meet in a point. " What theorem is he talking about, and what is its proof?


I'm new to manifolds and in my Computer Graphics class, we briefly explored the topic superficially through visual examples. From my intuitive understanding, we should be able to place a Euclidean grid on a manifold's surface or it should appear flat locally. However, I'm puzzled by how a pyramid with sharp edges and corners can still be a manifold. When we zoom in on the edges or corners, we only see flat areas along the faces of the pyramid, in any other direction it doesn't look flat. What is the difference between this case and two pyramids sharing a vertex (as an hourglass shape), which we classified as non-manifold? I'd appreciate if someone can explain what I'm missing in my intuitive understanding.


I've been looking into profinite groups, their topological subgroup lattices, etc. I asked the question does every profinite group admit maximal subgroups? I can't find an example of a profinite group which does not admit maximal subgroups, though I also cannot prove that each must have one. I think that if this is true, then it is related to profinite groups being residually finite, something about being able to tell apart elements using a projection onto a finite group, which must admit maximal subgroups. Can someone answer this question with the affirmative, or give a counterexample of a profinite group which has no maximal subgroups?


I know how to perform the double slit experiment in high school: a dark room, a source of light like a consumer electronics laser, a black board, and a card board with double slits. Is there way to extend this experiment to the one where we observe from which slit the photons pass (and causes them to not create the interference pattern anymore) without sophisticated measurement devices? Something that we can perform at home or at a typical high school? I have searched a bit but haven't found the answer. Does observing which slit the photons pass through require sophisticated measurement devices?


I recently found that a rope suspended from two points forms the shape of a catenary curve, which brought me to another question: What is the shape made by a rod of uniform density when only one of its ends is held horizontally? Obviously the material and width would affect the strength of the rod and thus the shape it makes, however I assume they must all resemble the same curve. (similar to how all shapes and sizes of ropes will resemble a catenary curve when held at two points) I would love any help with this question!


Suppose light from a laser passes through a single vertical slit. The width of the slit is certainly a factor how the light scatters after it goes through the slit; the thinner the slit, the wider the scattering. But my question is about the material surrounding the slit. How does the depth of the material influence the scattering pattern? And how does the composition of the material itself influence the scattering pattern? Does the nature of material around the slit significantly affect the scattering pattern?


I know that slope is the tangent of angle of inclination. But are they just two ways of defining the same thing? Or do they have any significant differences regarding what purpose they serve in mathematics? The way angles are defined as arc divided by radius, while slope the height divided by the base, doesn't show much difference. I only have a basic understanding of calculus.


So on the german wikipedia page of topological vector spaces it is written that, if a topological vector space is a Hausdorff space, then translation by a vector and dilation by a scalar are homeomorphisms. However, I have looked at several lecture notes and it is always stated (and proofed) that these two mappings are homeomorphism for general topological vector spaces (even those that are not Hausdorff). So I wanted to confirm that it is true that for any - possibly non Hausdorff - topological vector space the translation and dilation are homeomorhpisms. Also, can anybody think of a reason why on the german wikipedia they have restricted that statement to topological vector spaces with the Hausdroff property?


I have heard that light is a transverse wave which is necessary for effects like polarisation, but I am unsure of what exactly this means. Is this saying that light's electric and magnetic components oscillate in perpendicular directions of three dimensional space? i.e could the electric (or magnetic) component literally oscillate say 'up' and 'down' making the electric field strength of points in space above or below the wave stronger or weaker according to the light's phase? If not, then what could it mean for light to be a transverse wave, and how does polarisation work in this context? Thanks in advance for any help with understanding this.


Where do the electrons in a lightning bolt come from? There are problems with the explanations given up to at least two years ago. The explanation that the electrons come from inside the thunder cloud or are derived from cosmic rays are inadequate. There are a lot more electrons in a lightning bolt than those that come from these sources. Is there research which shows were the electrons actually come from ? Thanks.


So I know that the main trig functions are sine, cosine and tangent, while cosecant, secant and cotangent are simply known as the "reciporicals", but wouldn't it make more sense for sine, SECANT and tangent be the main trig functions while COsine, COsecant and COtan be the secondary ones? After all, secant is just the reciprocal of cosine, so everything you can do with cosine you can do with secant.


I was reading geometric optics when I came across the concept of virtual objects. I found the concept counter-intuitive, as far as reflection goes. (I perfectly understand the formation of such an object when the light ray is refracted through a glass slab.) Attaching the definition provided in my book: My confusion: A mirror forms an image when the rays originate from a certain object. So, how can an image of a virtual object form if the rays do not diverge from a particular point in the first place? What are some real-life examples of such a situation, as depicted by the figure below: I have checked answers to similar questions but my doubt is not cleared.


I don't know exactly how boomerangs work, but I imagine that the fact that they are less symmetric than a sphere is part of the reason they can be thrown in such a way that they return to whomever threw them. Eggs also appear to lack some symmetry, because one side is usually pointier than the other. Of course eggs aren't ass aerodynamic as boomerangs and here it is claimed that aerodynamics plays a vital role for how boomerangs work, so probably the answer to my question is no: Will boomerang work in vacuum chamber on Earth? But I am still curious as to what would happen if you did throw an egg (shaped object) really fast like a boomerang. Does anybody know?