darklight03/StackMIAsub huggingface dataset:
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.)