darklight03/StackMIAsub huggingface dataset:
For a differentiable function f, we know that the gradient is orthogonal to the level sets of f. What if f is Lipschitz continuous? we know that its gradient exists almost everywhere. Is the gradient (when it exists) orthogonal to the level sets?
I am wondering when the Legendre transformation is irreversible. I know if Y is a linear function of x, then the Legendre transformation of Y is irreversible. However ,what if Y is a multivariable function? Could anyone give necessary and sufficient conditions when Legendre transformation is invertible? Thanks!!
I have to work with the radial velocity dispersion. Could someone help me to understand what this actually is and how I work with velocity dispersion? What happens if you multiply the radial velocity dispersion with the radius? Is this something that is often done? In which systems does the radial velocity dispersion change with the radius? How is it related to the velocity?
I know that the purpose of differentiation is to find the rate of change of a function. But i don't really understand how this is concept can be used to find the tangent of a specific point. It would be greatly appreciated if you can explain this to me.
You can use the Euler Lagrange equation to show how the shape of a suspended cable with no load is simply a catenary. However, if you suspend a much heavier load (i.e. a bridge) with the cable, the cable forms a parabolic shape. Does anyone know how to show this fact using the Euler-Lagrange equation?
Where did the slang term juice/sauce for steroids come from? I suppose that it may be a metaphor because of being liquid or maybe the similarity in colour. I cannot find any information on its origin, can you help me out?
I'm typesetting a multi-page text with a fancy frame on each page. As you can see on the attached picture, the inner corners of the frame are taking extra space and I would like to increase the margin for the first/last two lines (or for a certain distance from the top/bottom) on every page. Is there a simple way to do that?
I'm trying to find a Scottish colloquial idiom that means that people talk too much (especially in a gossipping, hot air, or inconsequential way). Perhaps this might be along lines of the English 'more babbling than a brook in flood'. I'm ideally after an idiom, not a single word.
I'm currently working on proving an iff statement and was wondering is it allowed for me to prove the two statements required to prove an iff statement using two contrapositive statements. (for example, I'd prove if A then B using if ~B then ~A) and similarly for the second statement Thank you!
I was wondering if there is any connection between number theory and operator theory. Especially the applications of Hardy spaces, de branges-Rovnyak spaces, Dirichlet spaces in number theory. For example to study modular forms, and etc.Something except Riemann Hypothesis. Does anyone know any books or papers about that?
Is it possible to separate fluids or at least alter their relative concentrations in the presence of a magnetic field if one fluid is diamagnetic and the other paramagnetic? If so by how much? Are there any studies or data on this? How could this effect be calculated?
If someone uses the name of a book/object with punctuation in it, does one need to capitalize the word after? Do I write: In the book Who's Afraid of Virginia Woolf? the main character... Or In the book Who's Afraid of Virginia Woolf? The main character...
I'm writing a speech about someone and I would like to describe them as intelligent using a similar phrase to the way that "she has fire in her belly" would describe her as being passionate and driven. Something along the lines of She has fire in her belly and {something} in her head.
I am writing a scientific article and I need to describe the two hatched areas in the Figure below. How do you call these? Upward hatching and downward hatching? Or what? E.g. The upward hatching corresponds to newly captured individuals, the downward hatching to the recaptured ones.
I am searching for introductory books that explains in pedagogical way how scattering experiments are done . I want to understand how the experimental apparatus works and how experiments are done. For example in electron electron scattering I want to know what are the initial condition of the electrons like impact parameter, momentum etc,and how the detectors are placed.
I would like to know which of the following sentences is correct: Ideally, we should leave tomorrow early OR Ideally, we should leave early tomorrow? I know an adverb of time comes after the verb or object and usually at the end of the sentence by I'm not quite sure in this sentence. I would really appreciate if someone would help me with this.
I took an English test where I had to complete sentences. After I finished the test I was told that my answer "If I didn't have..." was wrong and I should have written "Hadn't I had...". That sentence: "Hadn't I had the chance to make that acquaintance, I would never have been able to enjoy such a huge success."
Find all positive integers, if any, such that all of its divisors can be arranged in a rectangular grid where the sum of all rows and the sum of all columns are equal.
I am reading the Wikipedia article on mathematical structure, and it says "a structure is a set endowed with some additional features". Features can encompass a metric, a topology, etc... I'd like to know if there exists a rigorous definition of this meaning of structure, especially of this term "features". Could "features" mean just a signature along with an interpretation ? Thanks in advance.
Recently, I needed some generalized version of Gronwall's Lemma, which I couldn't find in a quick search. However, I discovered that MSE is full of questions differing only in details on this very topic. Hence, I was wondering if there is a general version of Gronwall's inequality which covers most (if not all) of the different cases.
I had a friend over recently and he expressed happiness for our family and home life. He was envious but with no negative connotations. It is something he wants for himself but is happy we have it. What would you call that?
And in general, what else can we say about the set of normal numbers? We know that a normal number must be irrational, and hence it is possible that the set of normal numbers is countable or uncountable, since the irrational numbers are uncountable. I have been unable to find much literature or research about this topic.
I was curious about solving the reflection and transmission amplitudes for the dirac delta potential barrier using fourier transform. I'm able to take the fourier transform but I'm unable to interpret the pole prescription that comes up. I didn't completely understand the following answer: Poles for a particle scattered in a delta potential
I want to create my own latex package with a .sty file in it, and install it via tlmgr command. I've looked it up in Google, and found only this page https://tug.org/texlive/pkgcontrib.html, where I found no detailed procedure to create a new package. Is there any document about this kind of things?
Are there any simple obstructions for the existence of a simplicial map between two given simplicial complexes? A natural candidate would be some simple to compute function that is non-increasing under simplicial maps. In general I'm interested in finding introductory references that explicitly discuss simplicial maps and their properties.
If a website is blocked, and I wish for it to be unblocked, which of the following is the best way to put it? "The blockade will be lifted tomorrow" "The block will be removed tomorrow" "Lockdown will be lifted tomorrow" Or something else?
Is there a reasonably simple (pen and paper) yet good approximation for the refractive index of an ideal gas as a function of temperature, pressure, and density. How would one derive this? Thanks for any help.
I know the average potential over a spherical surface equals the potential at the center of sphere. But is it also true for average over spherical volume ? In both cases the sphere is empty of charge.
I am searching for literature explaining the Spin/Helicity-Density Matrix formalism in order to compute "spin-dependent" matrix elements and cross-sections in particle physics. I would prefer if the source would contain minimal examples for elementary processes within QED or even the Standard Model.
In the general treatment of geodesics in the Schwarzschild metric symmetry is used to constrain the geodesics to a plane which is usually chosen to be equatorial. Is there a way to generalize this, without restricting ourselves to the equatorial plane?
I understand that a proton is composed of three valence quarks and a very large number of sea quarks that briefly come into existence and then annihilate as quark-antiquark pairs. Is it possible for the antiquark to instead interact with and annihilate one of the valence quarks and for the remaining quark to replace the valence quark?
Consider a nonlinear dynamical system that evolves with a continuously differentiable vector field that is there exists a compact set within which the solution trajectories exist and are unique. Is it possible that all the stationary points of the dynamical system are stable? (Apart from the trivial scenario of having only one steady state.) Thanks in advance.
I have txt file with table format, I did it in python but now I have to export this table to Latex but I don't know how. I'm not good at all in latex so I'll appreciate if you help me. Thank you!
Which preposition would be grammatically correct? I am confident that I have the character and integrity to be an asset to your organization. I am confident that I have the character and integrity to be an asset of your organization. I am confident that I have the character and integrity to be an asset for your organization.
It is a well known fact that there is a linear algorithm to determine if two given trees are isomorphic. But my question is: is there a (simple) way to expand that result to forests (a disjoint collection of trees)? Thanks in advance!
The following line occurs in Keats's poem 'Ode to a Nightingale': And haply the Queen-Moon is on her throne It seems to me that 'haply' means either, as Merriam-Webster says, 'by chance', or, possibly, the same as 'happily'. So, which is it?
I tried to use Torricelli's law to find the velocity of the fluid but I have to know the pressure inside the bottle, I can not calculate the pressure difference because the surface pressure is unknown
I have seen articles online discussing the roundness of electrons. This goes a bit against my understanding of electrons as elementary point like particles. How can a point have a shape? What does roundness mean in this context?
I saved a .tex file as a pdf with TexStudio. It has made a weird conversion and I was hoping to recover my original .tex file by saving it back as a pdf. However I now have this when I open it in TexStudio. Is there a way to recover my original file ? Thanks
I am looking for an introductory linear algebra textbook for undergrads. The basics: matrices, eigenvalues, change of basis, exponents, pseudo-inversion, SVD and QR factorization. But what I really need is a good set of problems, preferably with solutions. And it would be great if it is online or an open book. Do you know such a book?
How are vector spaces fundamental to higher mathematics (functional analysis, number theory).I'm familiar with vector spaces, basis vectors, inner products, eigenvalues,groups, rings , fields, modular arithmetic etc. I've read Axler's Linear Algebra Done Right and a bit of group theory.
I think I know what the displacement current is. But I don't know why they use the word "displacement" exactly. What is the literal meaning of "displacement" of the displacement current? Why don't they call it "electric flux variation current"?
If hawking radiation is the result of a pair of virtual particles popping into existence just outside the event horizon, and then one falling into the black hole through the event horizon, and the other escaping, how can this process reduce the mass of the Black Hole?
Scientists are always saying that they cannot fully explain how lightning works, especially when it comes to the fact that there doesn't seem to be anywhere near enough potential difference between the cloud(s) and the ground to generate it. How large is the difference between the voltage theoretically required for lightning to strike and that which is actually observed?
I find it hard to choose the most appropriate one among these three sentences: she has been busy recently, she has prepared for an exam. she has been busy recently, she has been preparing for an exam. she has been busy recently, she is preparing for an exam.
What is the path followed by free electrons in a conductor under applied electric field? Is it a zig-zag path (due to collisions with atoms of the metal) or a curved path? Why (in each case)?
I am trying to study, through the Riesz representation theorem applied to a space of compactly supported continuous functions, what hypotheses must be met to establish that the measure given by the theorem (which is unique, regular and Borel) is a probability measure. I have tried to find literature on this but I cannot find it. Any help? Thanks.
I have almost completed my mock test syllabus, but complex numbers are pretty complicated for me. Can anyone give me some tips to approach this kind of problem? There are tons of questions like this in the previous year's papers. Please don't downvote me, I just want a tip to improve in these kind of problems
I was reading about the proof of the Mordell-Weil Theorem in "Rational points on elliptic curves" from Silverman and Tate and in Chapter III is the hight function described as an indicator on how "complicated" a rational number is. Does this mean that if the hight function of a rational number has greater value that it requires more numbers to write it?
I have been trying to find if a real experiment has ever been done to prove and calculate the magnetic field of a rotating charged disk, whether it's a conducting or a non-conducting charged disk. All I found on Google and YouTube are just thought experiments, so please if anyone had come across an actual experiment let me know.
I thought a lot about it but couldn't come to an answer. It's obvious that there must be an equation telling the answer. If one of the electrons were fixed, by Integral calculus, I would have found an answer, but when both the electrons are free to move, I am struggling to find a solution.
A cyclist and a car, which is twice as fast as the cyclist, are approaching an intersection of two different streets perpendicular to each other. At noon they were at an equal distance from the intersection. Twenty minutes after noon the car was at the intersection. How many seconds from then on do they need to be equidistant from the intersection again?
When I select text in Texstudio it remains highlighted even if deselected. The last selected section of text always has a light-blue highlighting. This is visually very irritating and I have no idea how to get rid of it. I might have turned it on by accident, hitting some shortcut that I am unaware of. Does anybody know how to turn this behaviour off?
I have learned from a knot theorist that quandles were developed in order to more sensitively distinguish between distinct knots. However, they seem to be an interesting algebraic object in themselves. Are there any applications of quandles to quantum theory?
what is the definition of maximal connected subgraph? Is every maximal connected graph of the form K_n (that is n nodes and between any pair of nodes there is an undirected edge)? Thanks in advance
From what I read meteors compress the air in front of them, increasing the air's temperature massively; this in turn is what heats the meteor. My question, which I can't seem to find a clear answer to is: does the resulting extremely hot air primarily heat the meteor through radiation or convection, or some combination of the two?
I was wondering if this sentence is grammatically correct: I am not glad. Such is the sentiment of fools. Am I using such correctly here? I spent a good hour researching its use and couldn't find what I was looking for. I thought that perhaps someone on here could provide me with the answer I seek.
I was studying Quantum Hall Effect and there I came out with a question that why the plateaus in the plot of Hall Resistivity are robust ? I know by solving Schrodinger equation and using Landau Gauge Condition we can arrive at the conclusion of landau levels, but I am not satisfied that why the plateaus are robust ?
Injective objects in the category of abelian groups are precisely the divisible groups. However, despite using injective resolutions of sheaves everywhere, I realized that I did not know a single example of an injective sheaf. I have searched for such examples but unfortunately found none on the internet. What are some easy examples of injective abelian sheaves (if there exist any)?
I'm trying to use Georgian language with vimtex. I found out that I have to compile with XelaTex or LuaTex but i don't know how to make vimtex to compile with them. I would like to keep default compiler and add Xelatex as optional, is it possible? I have tried to read official documentation but I can't understand much.
I'm writing a research paper and for that I'm using TeX Live and TeXmaker. Sometimes I need help (correction, improvement) from my professor. So we want a software or something, in which, if somebody edits, we can see it in real time and all edits will be updated and stored. I don't want Overleaf as it requires a permanent internet connection to edit.
I am studying Game Theory and I have some confusions. Is actually Carrot and Stick strategy is the same as Grim Trigger? Is the Carrot and Stick only for Multistage game? Can you please give me example of case using carrot and stick strategy? I know that it's very basic questions but I am struggling to understand this topic. Thank you!
I am trying to prove the snake lemma. I want to show that the sequence of kernels form an exact sequence. Must I show that the functions that map one kernel to the other are R-module homomorphisms? Thanks in advance.
In Wikipedia's article "Complete Boolean algebra, the following example is given: The Boolean algebra of all Baire sets modulo meager sets in a topological space with a countable base is complete... This suggests that the Boolean algebra of all Baire sets modulo meager sets in a topological space without a countable base may not be complete. Is this true?
I want to model a pendulum consisting of two magnetic balls with the same 'sign' of the charge. How should I write the Lagrangian of the system? Will the Lorentz force appear in it? Also, what should I change in the Lagrangian in case of hitting one of balls with another?
I want to combine multiple departments of the same university in the institute section. Please refer to the image provided. This will save a lot of space and be much more clear as well. Please let me know what's the best way to go about doing this, as I currently have no idea. I'm using the LNCS template.
I'm not finding the answer to this on the internet after searching. When writing dialogue, do you use capital letters to spell out words? Jamie said, "I said P-L-E-A-S-E please, and don't you forget it!" Or should it be lower-case? Jamie said, "I said p-l-e-a-s-e please, and don't you forget it!"
I am a beginner with set theory. What is meant by " analytic set " ? I see this term in the context of set theory, calculus and real analysis but I have no idea what it means. See here : https://en.wikipedia.org/wiki/Borel_set where it occurs in the non-Borel set part.
I know that the particles that constitute air move freely about. There must be a significant amount of empty space between the bouncing particles. So why is it so hard to compress air without any machines/devices?
Why do vortices in a superconductor (i.e. magnetic flux-tubes) form triangular lattices? In one of the articles I found, I read that a square lattice would cause repulsion but a triangular lattice will cause attraction. Can anyone explain it?
Is there some way to argue that the Higgs VEV is unnatural, without even bringing up the Higgs boson mass? For example, can one show that the Higgs vev receives large corrections from new physics? The literature on renormalizing the Higgs vev doesn't seem to talk about this (or maybe I missed that part).
Some years ago, a weather satellite manufacturer in the SFO bay area was buying small quantities of piezoelectric acoustic guitar pickups from one of the big music supply houses, which greatly puzzled their sales rep. What were they doing with pickups intended to be glued to the underside of an acoustic guitar bridge?
I was trying to use the answer from here to ensure my tables fit to the width of my page. I'd like to redefine the tabular environment so that this change is applied automatically if the table's width exceeds the page width. Is there any way to do this?
I'm writing a report with the sentence: The vaccine has been shown to reduce diarrhea in an in vivo model. Colleagues use a sentence starting with The vaccine has shown to reduce... In my opinion, the vaccine obviously didn't show anything itself, so it makes sense to use passive voice. What is your expert opinion?
Is it true that the cardinality of a hypothesis class with finite VC dimension is less than the cardinality of real numbers? My intuition is that the number of functions in a hypothesis class with finite VC dimension cannot grow arbitrarily and there is a limit on that.
Is there a more accurate way to determine speed? I am asking because this has to do with an accident reconstruction where my son was killed. The investigators are trying to conclude his speed prior to impact and I want to know if this is the only way to determine speed.
I want to create a table of content like box on the left side of every slide with the name of every chapter/section and with the current chapter/section name highlighted while the other are slightly faded. Like in the following picture.
I am reading about the Douglas-Rachford Splitting algorithm and in that algorithm, the two operators resolvent and reflected resolvent are used extensively. I wanted to know if these operators have any graphical interpretations. e.g. What is the reflected resolvent of an operator? What does it reflect and etc.?
I want to know, are there any texts out there that take conditional probability as more primitive and define probability in terms of conditional probability? Perhaps they give some axioms for conditional probability. Anyway, I would be very interested in such texts.
Everyone floats in Dead Sea because it is so salty. Is it true in Great Salt Lake? How about Red Sea and Mediterranean Sea? Is it impossible for anyone to sink in these seas without added weight?
I'm wondering about lithium batteries in electric cars. Could time and technology improvements continue to improve charging speed, to the point where you could "refill" an electric car as fast as you can refill a gas tank? Or is there something potentially impossible to surmount there, like a law of thermodynamics?
I can't carry a heavy bag and a heavy heart, either. Is this sentence grammatically correct? Can I say this instead? I can't carry a heavy bag and a heavy heart at the same time.
I'm new to using chemfig. I'm trying to create chemistry handouts and I want to draw the tropylium cation: However I am unable to get the charge to come out of the ring. Any help would be appreciated.
I'm working on a project for which I originally put the title as: Methods for controlling the quality of PET bottle manufacturing process. However, now that I re-read it, I feel like its way more natural to say "of the PET bottle manufacturing process". I would normally go with the second variant, but is the original version grammatically correct too?