I'm looking for different criteria of identity for the notion of 'set'. I know that the standard criterion of identity is extensionality but I was wondering if there are others. I looked around but couldn't find any literature on other criteria. I thought you could help me find some material on that topic? Thanks for taking the time. All the best, David


In one study, for instance, bar codes were placed on bees, tracking their activities. Is this correct? Or do we say... In one study, for instance, bar codes were placed on bees to track their activities.


If i have an electron and an antimuon is it possible that they could annihilate into a electron neutrino and antimuon neutrino? Am i correct in saying both spin and charge are conserved in this reaction.


We've been learning in class that when approaching a perturbed Hamiltonian with degeneracies you should try to choose a basis made of eigenstates of both the unperturbed Hamiltonian and the perturbation. But at that point, why do you need perturbation theory? Don't you now have the exact solutions to the Schrodinger equation?


I understand that superconductors have zero resistivity, however, I wonder if there is a relation between resistivity and the work function of a specific material so that a superconductor has zero (or a low value) work function. If so, photons with very long wavelengths can release electrons from the surface of a superconductor. Am I right?


Phonetic spelling is when one spells a word according to how it is pronounced (e.g.; knight => nite). What word would be used to describe the pronunciation of a word based on how it is spelled, ignoring silent letters (e.g.; knife => cuh-nife)?


I understand that a graph is connected if there is a path between any two vertices. However, I am unsure about the number of components of a graph. Is a single vertex considered a component? If yes, is it connected? In particular, what is the number of components in the below graph? Is it one or three?


I wonder how to best phrase this. The idea was to ask someone about their time zone. "Where do you live that it's morning for you?" The 'that' feels off / like a literal translation from Czech. "Where do you live where it's morning for you?" The double 'where' feels weird too. Are these sentences correct / do they feel natural for native speakers?


This is probably a very naive question, but is there something about Godel encoding that is essentially arithmetical, or is it possible to construct analogous mappings between the objects studied in a theory and statements in the theory itself for non-arithmetical theories (i.e, theories whose objects of study are not numbers). Are there any notable examples of this being done?


I was wondering if the word normal is redundant in the definition of a simple group. I strongly feel that there must be a simple group with at least three subgroups. However, I'm unable to find any. Any help?


I used to use Microsoft Word to draw this type of graphs, but it is not so convenient and it is difficult to be precise and accurate in details. Is there any better tool?


I heard my teacher stating that the base form of a verb is not an infinitive itself, but it is used to construct one of the two forms of infinitives. Edit note This question has been linked to a previous one of mine. However, in that question I was asking about the difference between infinitives and bare infinitives. This is a different question altogether.


I've been looking to understand linear algebra(vector spaces and subspaces) from different sources, then I came across the statement " the union of x-axis and y-axis" to the question "Give a set that is closed under scalar multiplication but not addition". I don't really understand how


I was wondering what's the order between mode, median, arithmetic mean, geometric mean and harmonic mean in a distribution with positive skewness and in another with negative skewness. Which are the theorems used to prove their relationship?


I am looking for books and papers that treat the physics theory of classical mechanics in a rigorous, mathematical manner. I would love to read such a text. Are there even such texts out there? I have already read physics books on classical mechanics, I am now looking for a rigorous mathematical text on it.


In Baldwin's "Model Theory and the Philosophy of Mathematical Practice" I read this phrase. The interaction between the syntactically definable properties of theories and the Boolean algebra of definable sets in their models is crucial. I was wondering what is meant by this "interaction", to me it is not evident that there should be such relation.


I just started going through this http://www.damtp.cam.ac.uk/user/tong/qft.html of QFT and I got stuck at the very beginning. The Lagrangian for the field is derived by: I think that the second step in this derivation is integration by parts, however after trying multiple times I still don't get what I need to get. Could someone explain what I'm doing wrong and provide step-by-step derivation?


I was watching a Lecture by Douglas Hofstadter on "Albert Einstein on Light; Light on Albert Einstein". And there was a slide which said that Einstein had idea that All periodic phenomena should have quantized energy levels. Can some one explain such idea? Thanks.


Recently I saw a video saying that due to the Pauli's exclusion principle there can only be a fixed number of electrons in a place but for photons it doesn't hold true as it's spin is different. So then is the number of electrons in an unit volume is fixed ?


I want to select a finite number of points on compact Lie groups that maximally cover the manifold in some sense. I have heard that some related aspects are Voronoi diagrams on manifolds and lattice structures on Lie groups. Any relevant literature on these topics would be appreciated.


How does a blackbody actually heat up and change temperature if it emits all the radiation that it absorbs by definition? And by extension, if the absorptivity and emissivity of a body are equal, then how is the temperature actually changing?


Does anyone know any good resources on the the Poisson kernel? I'm really looking for a measure theoretic approach to harmonic boundary problems. If anyone knows any such books, that would be much appreciated.


I have an assignment that requires us to make a continuous rollercoaster. we have to find the derivative of the functions we us and show that the gradients match. I have part of my rollercoaster done but I was wondering whether or not you can somehow connect an exponential function to the rollercoaster. Heres what my graph looks like atm.


For some strange reason, all the figures in a LaTeX document that I'm preparing are being put right at the end. This is despite me specifying precisely the figure placement in the figure arguments (with a ! too). How do I stop this from happening?


I was told the framing of a knot is the linking number of the push-off. But I don't understand why the framing does not depend on the knot but only on the parallel copy. How about a Legendrian knot? (I heard this is more rigid than a topological knot)


I have once heard that the definition of energy is "the ability to do work". However, that is a counterfactual definition, because a physical system can have that ability without actually doing work. So, has someone proposed a non-counterfactual, non-modal definition of energy? I want a rigorous definition.


Russians often use the formulation "take an integral". Now I noticed it in an article, written by a Russian and I can't recall I have ever encountered it in English. Is it a possible alternative to the basic "calculate an integral"? What other options are there? "Evaluate", "find", ...?


Hello Physics StackExchange Team, I want to know whether it is possible to experimentally determine the velocity at which the free liquid surface is moving/rotating in a cylinder with a rotating disk. The velocity at the free surface will be lower (in comparison to the disc speed) but is there any way to experimentally capture/calculate it? Please help. Thanks.


I've read many definitions for a transcendental number and some of them say that a transcendental number is a number that is not the root of any integer polynomial, while other say is a number that is not the root of any rational polynomial. My question is why such restrictions? Can't we just give algebraic polynomial instead?


I've read that if a state is a product state, the reduced density matrices are pure and if the state is entangled, the reduced density matrices are both mixed. What would it mean if you had a system of two particles, and when you calculated the reduced density matrices, one was pure and one was mixed?


What might cause the pdflatex output from a memoir class book to have two columns per page rather than one? Multiple columns can be used in some articles but not in a book.


In school, I was taught that a line segment must be finite. But a fractal has an infinite perimeter. If you just took two points along that perimeter, and "stretched it out" wouldn't you have an infinite line segment?


It is my understanding that the way cosmologists formulate theories like big bang theory, steady state universe theory, heliocentrism, etc, is by observing the sky using telescopes and imagining explanations for what they see. Are there any good books that I can read to learn more about these theories and the scientific processes that lead to them, that go into details?


How do I use LaTeX on an Android device? Can I make animations using it? If yes, how? I am new to stuff like this. What are some good sources and app for using LaTeX on Android?


I have two circular coils facing each other in parallel. Through one coil current flows. I move the second coil in a parallel fashion. What is the direction of current induced on the moving coil? Acc. to me, there should be no induced current as in parallel movement, the flux doesn't change.


Could you please help me to know why transfinite induction only works on well-ordered set and not arbitrary set ? Why the fact that every susbset has a least element is necessary for transfinite induction principle ? Thank you very much!


In software engineering, I sometimes see "first cut" in a context where it could mean "first version". Is it some idiom? Where does it come from? Does it mean literally "first version", or is there any emotional subtext, or did I misunderstand its meaning completely?


In an article of wikipedia, it is stated that : Charge is the fundamental property of matter that exhibits electrostatic attraction or repulsion in the presence of other matter with charge. I can't understand why there is "electrostatic" mentioned in this statement. Isn't there attraction or repulsion between two moving charge too?


There is a well known algorithm (Bhaskara-Brouncker) to compute the continued fraction expansion of the square root of an integer, using integer arithmetic only, no floating point. Is such an algorithm known also for square roots of fractions of integers?


A word for when an artist builds up their ability to paint, but over time their art starts regressing. Or perhaps a child, once magnificent at playing the guitar, begins to pluck the wrong string and slur the notes, eventually becoming mediocre.


Why is the construction 'I would love to' so commonly used, and is it grammatically correct? If the word 'would' cannot be used with state verbs like love, think, etc, then isn't this technically wrong? And why is it used so often?


In logic, why do we talk of universal closure of a formula, and don't consider its "existential closure" (as far as I know)? I guess that one of the reasons may be that interesting systems like PA are written down much more succinctly and naturally without the universal quantifiers. I think there might be more to it than that, though.


I don't know if this is true or not, I tried to prove and I didn't succeed and the I tried to give a counter example and I also failed. so can someone please help


I have a latex file and a website. On the website I have an input field, in which you can write text. Then when you click a download-button on the website I want that it automatically creates a pdf (with latex) with the value of the input field.


I've been reviewing the double slit experiment lately and I believe it is missing a few crucial things. Thickness of the wall is my primary focus. How could I add thickness into the equation? I'm interested in seeing the effect on a model.


Suppose that I were to jump on the surface of the Earth. The moment I hit the surface, shouldn't the force Earth is applying on me be greater than the force I am applying to it so that I stop completely? If the forces were equal, wouldn't I still be moving?


I hear about "The Graph Isomorphism Problem", and am a bit confused as to where the issue is. Isn't the adjacency matrix ( up to permutation ) unique for a graph up to isomorphism? Can't the problem be solved as easily as checking if two graphs have the same matrix up to permutations?


Is there a concept of induced subgraphs for directed graphs? I would intuitively think it is nearly the same as for undirected graphs. I was just thinking that it would be a subset of the vertices such that all arcs from the original graph from one vertex in the subset to another in the subset exists in the induced subgraph edge set.


From my understanding, within the context of the Standard Model, charged lepton flavor conservation is an accidental symmetry due to some renormalization condition. How does this work in the case of pair production? Would the result be different if instead the photon pair produced to an electeon neutrino and a muon anti-neutrino?


Suppose you have a closed differentiable plane curve. Are the tangent lines to the curve at the most distant points on the curve always parallel? What if we assume that the curve is convex? I don't see why this should be the case, but it also seems like it might be the kind of weird geometric property that unexpectedly turns out to be true.


In any path-connected metric space, can we connect any two points with an injective path? This seems possible, for instance by "removing" the points where the loop is not injective, but I cannot prove it. Is it true? If yes, how to prove it? If not, do you have a counterexample? Thank you very much, AF


If a quadratic form is represented as a matrix, it may not be symmetric, but there must exist a representation that is symmetric (or Hermitian if complex). But is the reverse true? That is, do all symmetric and Hermitian matrices represent a quadratic form (except maybe in the case where matrices are all zeros)?


As the question states, how would you calculate the t-statistics in GLS? Is it just the same as in OLS? And are these results still reliable? I have heteroskedastic data and so am using GLS, but I'm not sure if the calculations for t-statistics change or the values are less reliable?


What is the difference between "a memoir" and "a biography"? According to my study, "a biography of a person A" is normally written by someone else not A and "a memoir of a person A" is normally written by A himself. Also, "a biography" is quite detailed from birth to death while "a memoir" is not too detailed.


I'm curious as to the purpose of relativity and why the universe would function this way as opposed to a universe with an aether. So what would be different if we had an aether?


The product of two matrices A and B with dimensions (n,k) and (k,m) results in a matrix C (n,m). I remember that my professor used to call this rule after a specific person (I guess the inventor of that rule). Do you know how the name of the person is? Or how do you call that rule in general?


I know that Galois groups let us determine when polynomials are solvable by radicals, and that Lie groups can be used to solve some differential equations. Does there exist an analogous group theory method that tells something about the solutions of recurrence relations? I am interested in both existence/Galois type results and symmetry methods of solution.


I just learnt about Feynman Diagrams, and I've been going through many examples to practice the reading of Feynman Diagrams myself. I have already found diagrams for Positron Emission, Electron Capture, and Beta Emission decays but yet have not seen one on Alpha emission or Gamma decays. Can these decay mechanisms be expressed through Feynman diagrams, or am I just missing something here?


Assuming a charged particle moves at speeds near the speed of the light. Will the electric field generated by that particle get affected by length contraction and time dilation? In other words, will this cause a more dense field closer to the particle which declines faster with distance compared to a stationary charged particle?


I'm trying to optimize plasma cleaning process to be signal driven instead of fixed time. I have a vacuum chamber with hydrocarbon contaminants, I use air plasma for cleaning. Is there an option to terminate plasma, based on the amount of contaminants left in the chamber?


I would like to know if the thermal motion of molecules stays constant if the pressure is increased at constant temperature or not. The viscosity increases with pressure, thus the friction between the molecules also. If the thermal energy is constant, will the motion of the molecules decrease due to higher friction?


I have a question about whether or not to use the plural for a noun in the following sentence: Men, woman, and horses grew smaller as they moved on. Since there was only one woman in the group, I think it is neither necessary nor correct to match the plural nouns with "women". Am I right? I appreciate any assistance you can provide.


I've heard that very massive stars can sometimes collapse into black holes without creating supernovae. How does this happen? (I suspect it's something to do with the relative lack of Urca process neutrinos which are active in the cooling of neutron stars).


I am trying to develop a more intuitive understanding of the field produced by magnetization and polarization. I have the following image, and I can derive it from Maxwell's equations what the fields inside and outside are given uniform polarization, but is there a way to understand them more intuitively?


I would like to use the QT Deuce font in overleaf. It is in the LaTeX font catalogue, but the instructions for usage on that site use the fontspec package which requires XeTeX or LuaTeX rather than the pdflatex that Overleaf uses. Is it possible to use QT fonts in overleaf?


In the book entitled "Algorithmic and Combinatorial Algebra" written by Bukot and Kukin and they showed that there exists a finitely presented Lie algebra with unsolvable word problem. Can we conclude same result for the case of Lie superalgebras?


According to Birkhoff's Theorem, any spherically symmetric body will not emit gravitational waves. I can understand this for an object that is contracting and expanding because from far away the gravitational force is the same. But now imagine, we have a spherically symmetric body that is linearly accelerating through free space. In this case, would gravitational waves be emitted ?


Cayley graphs are necessarily vertex transitive, but the converse is not true in general. Is there any general/asymptotic result that discusses the percentage of Cayley graphs in vertex-transitive graphs? Or would this be a hard problem?


Fix a finite-dimensional real inner product space. Does a self-adjoint operator have a symmetric matrix with respect to all bases, or just with respect to orthonormal bases? Also, does this change at all when one admits a pseudo-Euclidean inner product, rather than the (standard) Euclidean inner product?


How the slang word "jacked" (having well-developed muscles) is formed? The word jack itself may be related to drugs, however I don't think that the meaning comes from there. There is a theory that it has formed from the word jagged (having low body fat), but I cannot find a proof to that.


When reverse bias is applied, why can't electrons from p region that are currently occupying the holes jump into the conduction band of the n region, which according to the image (taken from hyperphysics) would not require extra energy, thus completing the circuit? But this doesn't seem to happen since the current is negligible during reverse bias,why?


Prove that for any finite collection of points in the plane, not all collinear, there is a triangle having three of the points as its vertices, which contains none of the other points in its interior.


my professor asked me to read about Tate's Thesis, but I have looked everywhere and I couldn't find the thesis itself. I found lecture notes and introductions on the the thesis. I am confused. Is Tate's Thesis a thesis? If so, where would I be able to find it? I really appreciate your help!


They thought that the general would lead them to victory. vs. They thought that the general would have led them to victory. Is there a difference between the two sentences above? Is one grammatically incorrect? There is no particular context. I am more interested in learning the difference between the two (if any) and how the subtlety in wording affects the overall meaning.


I'm trying to understand the nature of polar molecules. Have there been experiments where we can say to have actually measured the dipole electric field generated by a polar molecule such as hydrogen fluoride HF? The idealized experiment I have in mind for example is tracking the deflection angle of a free test particle (electron) as it passes near a single HF molecule...


I would appreciate if someone could help me with this problem. I have tried to find a general expression for the multiplication of three matrices but wasn't successful in my pursuit. enter image description here


John Taylor's Classical Mechanics says this... I was wondering if the second condition already implies the first? I mean, are there situations where the first condition is violated even though the second condition is not? And if so, how are the forces in that situation non-conservative even if they satisfy the second condition?


In an infinite-dimension space, the relative interior of a non-empty convex set may be empty. I was wondering whether there is a concept (as a generalization of relative interior) with the following properties: for any non-empty convex set, Its "concept" is non-empty. Whenever the relative interior of this set is non-empty, its "concept" is the same as the relative interior.


What is the motivation for studying D-modules? All I know is that they are some algebraic way of discussing differential equations. They are showing up in seminars, and I haven't yet been able to convince myself that D-modules are useful. So what kind of questions were D-modules made to answer, and how is it useful to view differential equations in this way?


I coded a small example of RSA in Python. When filling p and q, I mistakenly put in two numbers that were not prime numbers. And it worked fine anyway. My question is, why is it so important for RSA to take prime number as inputs?


I'm not sure if the use of "easy to move" in this sentence is correct. They say that our uniform looks cool and easy to move. It sounds strange to me. I can use the word "comfortable", but are there other words or ways to say it? I'd really appreciate your help. Thank you.


Quick question - if the formal name for the solid counterpart of a sphere is a ball, and the formal name for a "half-sphere" is a hemisphere, what is the formal name for the solid counterpart of a hemisphere? Half ball? Hemi-ball? Just "solid hemisphere"? Is there even an agreed-upon formal name for this solid?


Does anyone know how such a matrix is referred to in the literature? Ultimately my goal is to look for an analytic expression for its determinant. But because I don't know what it's called, it's been very difficult to go through the literature... Thank you so much for your help.


At least for German-language documents (and especially when using microtype) LaTeX, to my taste, puts too much emphasis on preventing hyphenation. Is there any parameter that can be changed so that, in favor of more consistent white spaces, more hyphenation is done?


There is an unhappy colleague in our company and I want to talk about this issue in my next meeting. I want to mean "A bad apple ruins good apples" but this sounds so negative. I want to model it positively. How do I do it?


I was learning basic statistics and I read a proof that the standard deviation is always less than or equal to the range. I don't understand how range can ever be equal to the standard deviation. I could only think of one case: when all data points have the same value. Is there any other case possible in which the equality holds?


When doing palatography, one needs something to paint the tongue with. It is usually a mixture of olive/sunflower oil and powdered charcoal. The question is, what can this substance be called? Is it a colouring, paint or/and dye? What is the most suitable word?


I'm seeing some results on Google but it seems quite an unusual word combination. Is it okay to use it or would it be considered "weird"? Context: Free for non-companies We believe Open Source should be free for individuals and non-profits. Only companies pay. The word "non-profit" isn't usable as it wouldn't convey the same message.


Pretty much the question. Is there a line (figurative) that separates when you'd use one term and when you'd use the other? Is there a distinctive characteristic of one that is not found in the other? Please help me understand. Thanks!


just a quick question about how to determine if a matrix has a JCF - I know that if the characteristic polynomial factorises as a product of linear factors it has one but what if you only know the minimal polynomial factorises as a product of linear factors? Is this enough to say the matrix has a JCF? Thanks


NO SHADOW of the fork! A distinct black shadow can be observed at the bottom of a glass before and after the fork enters the water, but the shadow disappears at the instant of entry, is this a phenomenon of refraction of light? (as shown in the figure) How does it work?


Could someone please tell me how we define CM modular form and non-CM modular forms in the most basic way?Also, could you provide the references to your definition? I have tried searching for the definition. But still the definition is not clear to me.


In the image below, the expression is depicted as a result of multiplying a polynomial by an infinite series, which is represented as an exponential function. Could someone kindly elucidate how the final equality is derived? Thanks in advance.


I'm reading an article where risk measures are defined on the space of random variable (not necessarily bounded), so I want to understand why risk measures are defined at first (for example in Follmer and Schied) for bounded r.v.


If the Earth were perfectly isolated from the rest of the universe - with no exchange of matter, energy, radiation etc - would we be able to recycle all stuff on Earth perfectly? For example, simple processes like electrolysis and burning of hydrogen. If no energy or matter is lost during these events, what's preventing us from reversing and recycling them indefinitely?


I know Pieri's formula for elementary symmetric polynomials and for complete homogeneous symmetric polynomials, but is there an analogue for power sum symmetric polynomial? It seems that it should be similar, but the summation should be carried out by other partitions. Maybe someone knows where to read about it?


How do we judge if a certain ruler-and-compass construction is the shortest? For example "the shortest construction of a regular pentagon." Is there a software that can come up with a construction of a geometrical item given a collection of input data?


Rotating a fluid filled cylinder and calculating the equation of it's free surface is an elementry problem in fluid mechanics. However I was wondering, if we rotate the same cylinder by an axis parallel to the main axis and passing through a point on it's circumference, what would be the equation of the free surface in this case, and what will it look like?


In a cloud chamber, atoms of air are ionized by passing radiation, providing nucleation sites that enable droplets to form. When those atoms are de-excited, one would expect them to release light; does this happen, and if so is this radiation detectable?


Which is correct? Is it, "Do you want your house painting" or "Do you want your house painted"? Examples of both can be found on the internet. Is there a difference between them? Can you use either?


I would like to know how to write half integrals and generally integrals of nth order in LaTeX, for example in Cauchy's formula for repeated integration, because I didn't find how to do that on the whole internet.


What does the MHD theory predict when two regions with magnetic field of opposed polarization interact? When we take in account the resistivity then we can observe a rearrangement of the topology due to magnetic reconnection but I am wondering what does the ideal MHD equations predict in that scenario.


I was wondering, where I made a mistake in this attempt to disprove the four color map theorem. I hope that you smart folks can help me out. I already apologize, if the tags are wrong. I am not really a math specialist.


I was wonder about following statement. "Harmonic function is radially symmetric." What is correct meaning about above statement? I think it means 'Harmonic function is invariant under rotation' i.e: Let u(x) is harmonic function and x is n-dimensional vector. If A is nxn orthogonal matrix, then u(Ax) is also harmonic function Is it correct?


I have a confusion. At one hand, magnetic moment is forced to align with an external static magnetic field; on the other hand, a magnetic moment can precess around an an external static magnetic field with a fixed angle. Clearly these two phenomenon cannot happen simultaneously. So which is correct?


I am reading again about Lie Algebra, and one main concept is that of the tangent space of a lie group. We consider it to be all the possible tangent vectors as a path crosses the identity element in the lie group... but why exactly the identity element? Why not consider tangent spaces at a generic point?


If an object is projected towards east or west from the equator, we get the coriolis force to be vertical. If this projectile, say, miss its target from below, can this be attributed to the coriolis force?


Can the definite article have two different meanings in front of identical words in one sentence? As in The bicycle is a vehicle but the bicycle in this room is just a decoration. Or perhaps this sentence should be better written as A bicycle is a vehicle but the bicycle in this room is just a decoration. ?


Does anyone know of a relatively simple and easy-to-use graphics app (like GeoGebra) that can graph hyperbolic geodesics on a hyperbolic paraboloid? I'm trying to graph the geodesics on the top image to the surface on the bottom image, specifically to best visualize the hyperbolic case of the parallel postulate. I haven't found a way to do it on GeoGebra.


I encountered this correction in IntelliJ, a tool for editing computer programs: Is this right? I feel like there is an implicit "that" in the sentence: I need to borrow some money so [that] I can buy some medicine for Tom. And therefore the second clause is really a dependent clause in disguise. What do the experts say?


I have to do an exploration in maths. I was thinking to do it on how the angle of incidence and reflection vary with the eccentricity of the ellipse. Do you think could be interesting? Do you have any suggestions for the method? I was thinking to extend it on ellipsoid billiards.


If electricity is the flow of electrons, how come it can flow at the speed of light? Shouldn't how fast it moves be limited to a speed lower than the speed of light because it has mass?


I have several times seen stated the fact that the complex projective space with the Fubini-Study metric is isotropic, but i can't seem to find the proof anywhere, can anyone suggest a book or answer with a proof?


I have been learning electrostatics, and I didn't quite understand why charges reside only on the surface. The charges are moved until they reach the surface after which they cannot move further and hence the field that had developed cancels the external electric field. How does this cancel the external electric field?


Lorentz scalars are invariant under Lorentz transformations, which are a subset of linear transformations. I wanted to know if it is possible, for a Lorentz scalar, to NOT be invariant with respect to another kind of linear transformation (e.g. scaling or whatever linear map other than Lorentz). Can you mathematically prove your answer?


x = f(a,b,c), y = g(a,b,c), and z = h(a,b,c) if a,b,c are sides of a triangle, then x,y,z are also sides of a triangle. How can you write sets of functions f,g and h?


I'm wondering if its ever possible to find two non-isotopic knots which have identical jones polynomials but different seifert genus? Attempting to google for this I found this example of non-isotopic knots with identical jones polynomials. But in the first page of the paper its clearly stated that all the knots have identical genus.


So, the thing is i was using this code on overleaf (it worked fine there) and i wanted to switch to texstudio because of the compiling limits on overleaf. So i copied the whole command from overleaf tex file and pasted in the texstudio. But it says "Error: File 'sn-jnl.cls' not found." when i run it. how do i fix it?


Is there a name for the particular part of a circle shown in red? I know the white sections are segments. If the vertex was at the center of the circle it would be a sector. I don't suppose that any mathematician would have been brazen enough to call it the "ice-cream cone."


Is it correct to say that if two parabolas touch, then tangents at the point of intersection have the same slope? If it is true, is there a neat geometric interpretation of the fact?


If I had a pair of entangled particles that were both in superposition, and I put one of them through a interference experiment, such as the double slit experiment, would I see an absence of interference the moment I measure the other particle? And how would this change if both were in an interference experiment and I measure one of them?


Why were multiple layers of absorbers and scintillators used in the CMS HCAL? Couldn't they have used just one absorber and one scintillator layer? Why was the choice made to have multiple layers?


I read book "A primer of infinitesimal analysis" John Bell. I was confused when I saw example with area under curve. In that example author mentioned about nondegenerate triangle of zero area. But triangle of zero area is degenerate triangle. How is it possible? How to imagine what author meant in that example? Thanks.


I have read that a net charge in a closed, finite universe would result in the electric field from the charge "winding" forever which would lead to infinite energy. However, wouldn't the electric field lines cancel out if they were winding around the surface of a sphere like in the diagram?


I was scrolling through a wiki article on terminal velocity when I spotted an upside down delta. What does this symbol mean? How is it applied in other contexts? EDIT: If possible could someone expand upon "a covariant vector made of space derivatives?" I sort of understand how a vector can be made up of partial derivatives, but what does covariant mean?


I am currently setting up a portable LaTeX environment (TeX Live) with VisualStudio Code and the LaTeX Workshop extension as the editor. How do I specify in the LaTeX Workshop settings the path to the LaTeX environment? This is for Windows. Changing the environment variable on every computer used is not desirable. I also couldn't find something in the settings of the LaTeX Workshop.


I am currently reading Falconer's book on Hausdorff dimension. My question is whether Hausdorff dimension is invariant under local isometry between smooth Riemannian manifolds? I think it should be yes, but I am not sure how to choose the covering( needed for calculation of dimension) using the local charts explicitly to get the invariance. Any help would be very much beneficial for my understanding.


I have an html table like <table>...</table> with some css formatting and I'd like to put that table into latex document. I don't want to put it as a figure because I will not ba able to copy its content. How can I do that (preservig formatting and styling)?


I am learning general relativity by myself. I wonder if the equivalence principle is simply equivalent to, in geometrical terms, the statement that a Lorentzian manifold is locally flat, which is a somewhat trivial result from the point of view from math.


Many textbooks derive the single-particle excitation energy gap from the BCS mean-field hamiltonian, and then stop there, implying that this gap is sufficient for superconductivity. To be frank, I cannot see why a gap implies superconductivity. Superconductivity is a transport phenomenon. But the BCS mean-field hamiltonian is just a static hamiltonian; it is not directly connected to transport, right?


Everytime I approach any projectile motion/kinematics problem, I get confused. I don't know how to translate the problem into an operational method, and every time I complete a problem, the next one is a new mystery to me. How should I tackle this issue and master this problem type?


My super basic question is, the (magnetic) force between two steady current loops obeys Newton's third but the (magnetic) force between two charges doesn't. This is surprising given that the former is built out of the latter, so is there any significance to this fact?


I'm currently learning about Radiation in my Physics course, and so far i was interested in betavoltaic batteries that convert eletrons to photons and then eletricity. I'm confused about how to calculate the energy per second (W) produced by eletrons converted to photons. Can someone help me on that?


I am very confused and probably understanding the concepts of renormalization wrongly: If the running coupling constant is a real experimentally observable quantity and the change with scale has a physical interpretation as "vacuum polarization", how can it get independent of the scale if we could sum up all orders of perturbation theory?


Can you explain to me the reason why Thomson Scattering can not explain what happens when light meets an electron at low intensity, and what does that have to do with light being a wave or particle or relativistic/QM effects?


I cannot seem to find or come up with an answer to the following question: In mathematics, is it possible to prove that there is only one (shortest) proof of a given theorem (say, in ZFC)? Are there any (preferably natural) examples?


When you write a spline curve as a linear combination of b-spline basis functions, it's called a "b-spline". The basis functions are generated recursively by the deBoor-Cox algorithm, starting with degree zero and working upwards to higher degrees. These b-spline basis functions are thus written in terms of (i.e., as a linear combination of) other b-spline basis functions, so they are also b-splines?


I need to run an experiment on the effect of temperature on refractive index of a liquid, and in order to make the change most visible and have the lowest percent uncertainty I need the change in refractive index to be as high as possible. What are some liquid with high changes in refractive index with change in temp, higher change then water ideally.


I have seen some mathematical texts that use words like Lemma, Theorem, Corollary, etc. What would be the appropriate description of such terms? Do they fall under some linguistic category? Is there any general term associated with these kinds of premises?


I'm writing a bullet point in a presentation. I want to say: Our partner and customer commitment remains unchanged. However, it feels like it reads better if I change it to: Our partner and customer commitment remain unchanged. I know if I read them separately, remains is correct. I've searched Google, and "remains" is more common. Should I use remain or remains?


My textbook states that when the diffraction gap is smaller, there is more spreading of the waves. However, it seems unintuitive to me, since why would waves spread less if there is more space for it to spread? Is there an intuitive way of understanding it?


I came across a thesis which has a feature in its references. Each reference includes the page number(s) of which the reference is cited. Is this possible to do using Latex? I found it really interesting to use. Apologies for the image being in French.


Say, you have a complex number in standard form and another in trigonometric form, can you do operations with them, for example adding/subtracting, multiplying/diving the two numbers, just like that, or do you have to convert one of them to the other form to be able to do operations?


Is there a name for white noise that has non-constant variance? I have some examples from experimental data where the variance of the white noise increases with time. However, I am not sure how to model this.


For propositional logic, for every formula, there is an equivalent formula in the CNF and DNF. These normal forms have the advantage of being representable in a "tabular" form rather than a "tree" form, which may be more computationally friendly. Is there something similar for linear logic? I am mainly interested in the additive-multiplicative fragment.


I understand that an algebraic group may not be a topological group because the continuity of multiplication with respect to the Zariski topology is weaker than that with respect to product topology. But is there a nice example that helps to explain this. Many thanks!


I'm looking for a selection of problem books on probability theory, at the level of student olympiads in mathematics. I've already looked at Albert N. Shiryaev Problems in Probability, but maybe you can suggest some more interesting options. Many thanks!


Was just curious, is it possible to obtain the eigenvectors if you only know the eigenvalues and the dimensions of a square matrix? Or what further information from the matrix can you gain from only knowing the eigenvalues?


I know that mechanical computers have been used for a number of different calculations, the most famous (at least on the internet) seems to be the firing control computer used by the US Navy. But were there any entirely mechanical devices (other than a slide rule) that could calculate logarithms?


I am trying to understand Schmidt decomposition. I am stuck in one sentence here. See the example picture. Here, I can understand everything except the line "For both HA and HB the Schmidt basis is diagonal (Hadamard) basis." Can anyone help to understand what the author tried to tell here by using this line?


For example, during refraction, when photons pass through a glass slab, they change direction. Why does this happen? Is the reason for this not applicable when photons strike perpendicular to the glass slab?


Is there a TeXShop directive (something like % !TEX TS-program = lualatex) that I can place at the top of my document and can tell TeXShop to compile using LuaLaTeX with latexmk? That is, a TeXShop directive that does the equivalent of latexmk -lualatex file.tex.


We derive the formula of escape velocity by using conservation of mechanical energy, where we consider that work done by external force is neglected. But to provide the velocity to send an object to infinite distance from gravitational pull of Earth, don't we have to apply an external force?


Is it possible, with LuaTeX, to retrieve the textual content of a PDF (using lua code I imagine)? For my purposes, it doesn't have to be "clean" (for example, the order doesn't really have to be respected), it's for "statistics" on the words used.


So the context is this: I taught my friend how to play a video game and he's gotten to a point where he's passed all of my stats and records. I'm looking for a word to use like ronin (samurai without a lord or master) and I'm curious as to if that exists. Thank you ahead of time


As the title says: Is there a distance at which the Kerr metric looks like Schwarzschild metric? Edit: if there isn't any such distance (smaller than infinity), can we measure from the outside, with which speed a black hole rotates?


In complex analysis the term branch is used to designate a holomorphic function defined on a region. The most prominent example is the principal logarithm, which is a branch of the logarithm. Does anybody know the history of why the word branch used? I'm guessing branch, tree, decisions, choice.


In the context of optical tweezers, if used in a vacuum and a particle starts off slightly to the right from the center of a beam, would the particle eventually become perfectly centered or would it continuously slingshot from right to left? Does this depend on the strength of the laser?


Is this just an extension of the idea that with a small enough perturbation, the Principle of Least Action will still hold, and the system will continue to take the most optimal path?


Since light travels at the speed of light towards a target, the spatial distance to the target seems to be zero, owing to Lorentz transformation. It seems that there is a logical flaw here. Can someone explain that? Question is similar to this: Wouldn't a photon disappear because of length contraction?


I am using a convex lens in a wavefront-measuring interferometer for testing microscope objectives, where the lens images the exit-pupil of the objective onto my camera. It is my understanding that the effects of diffraction from the convex lens will have some influence on my interferogram, especially around its edges. How can I actually take these effects into consideration?


I came across the concept of topology of time and causality in Reichenbach book, "Philosophy of Space and Time". It would be nice to have list of references of recent developments of the same. It could be textbooks, research papers or published articles.


I am searching for optimal control examples in fluid mechanics in which only ordinary differential equations are considered (i.e., in system dynamics or model of optimal control problem) with exact or approximate state and control variable plots. So far, I am getting examples containing PDEs. Does anyone know of any sources (books, journal articles) containing such examples?


On Wikipedia (article:https://en.wikipedia.org/wiki/List_of_trigonometric_identities) I came across this trigonometric identity in the infinite sums section In the article no proof was provided and so I would like to know if anyone here knows a proof of the identity.


Given a pair of inversely similar triangles ABC and AB'C', as shown in the graph, CB and C'B' intersect at F. X, Y, Z are the midpoints of BB', CC', and AF respectively. Prove that X, Y, Z are co-linear. This is a very elegant result, and somewhat alike Newton's line. All proofs welcome, and a geometrical solution would be very nice!


Why doesn't the first line have an auxiliary to make it interrogative? These lines are from Sorry Seems To Be The Hardest Word, featuring Elton John: What I got to do to make you care? What do I do when lightning strikes me?


I'm an undergraduate student and currently I'm approaching tensorial calculus. I was wondering: is there some geometric meaning to the operation of rising/lowering indices (and then if there was any geometric difference between vectors and covectors), or are they only mere formal operations? In case, why don't we simply use only vectors in the definition of tensors?


What are the ways to center headers only without using the titlesec and sectsty packages? If we take the titlesec package, then the question is how to prevent this package from automatically changing header styles?


What is the concept of hypersurface in general relativity? I know it could be characterized into three categories but how do we define hypersurface (in general) in physics? I didn't get what thing it is.


I have always believed --- from somewhere --- that in archaic English, cats were always referred to using feminine pronouns, regardless of what sex they actually were. But I thought to go and find a reference for it and I can't find anything. Was this ever a real thing, or was I just hallucinating?


If I lift a weight to a certain height, where does the gravitational potential energy at that height originate from? is kinetic energy converted into gravitational potential energy as the weight is decelarating as it approaches the maximum height? or is the chemical energy from my body (doing the work) shared between kinetic energy and gravitational potential energy? Hope this makes sense. thanks.


Why do electrons have discrete energy states when they are bound to a nucleas and not when they are free for example in an electron beam. Why doesn't an electron beam have certain specified energies like bound electrons do.


I have seen that a number of packages exist that define symbols. How can I find a list of just the core symbols provided by the Latex Team, but without the use of packages like amssymb, MnSymbol and so on.


I am new, not have expert level knowledge. But I have studied about black hole information paradox. The black hole doesn't kept any information which passes event horizon. Which can break physics. Is there any theory about Information Paradox, to resolve it.


What is a good term for plants whose parts, either as tubers, roots, or fruit/seeds are harvested and consumed? I exclude plants such as hemp and flax whose fibres are used, or, for example, poppies for opium.


i.e. is the comma to signify the boundary between the instruction and the recipient of it (that is to say, "Hey, stupid! Keep it simple"), or is it serving the function of an "and" (as is sometimes used in US English but rare in British).


I have tried to search a lot about the laws that predicted this behaviour of black body. But haven't really found out. And the ones I found like equipartition theorem, i find them contradicting with classical physics itself. How exactly it was predicted that with increase in temperature, there would be increase in intensity of radiation at each frequency.


I've realized that there are many results in set theory rely on the theorem that states "the union of countably many countable sets is countable". However, all of the proofs of this theorem I've found so far, including ones on SE, rely on the axiom of choice. So, I'm wonder without AC is it still true or false or neither provable nor refutable?


What's the difference between these? I will go to the bathroom and brush my teeth I will go to the bathroom and will brush my teeth Perhaps one of these sentences is grammatically incorrect. Which one is desirable to use in technical writing?


I've noticed that both are used though "point in" is seemingly far more prevalent. Is there any difference or it's down to one's preferences? E.g. There's no point in talking to you. vs. There's no point talking to you.


Is there a phrase with a similar meaning to the phrase Don't put all your eggs in one basket which doesn't make reference to eggs? I'm writing a story in which one character is a bird and I feel it doesn't make sense for them to use this specific phrase about eggs, given that eggs are essentially bird pregnancies


If the activity involved only me and one other friend, e.g. playing tennis, I know "I played tennis with Jake" is correct. However, is it possible to also say something like "We played tennis with Jake"? If not, is there a slight adaptation in the structure which allows for the use of "We"?


I've heard of the term escortee, but Googling it doesn't turn up any definition. If not that, then what could I use in this sentence: I escorted the girl through the school; she is a ________.


I do not know exactly what shape is a spoon. I feel like it is an elliptical paraboloid but really I don't know. I really need to do this for my math project. Any help would be very appreciated! Image of a sample shape


Is it possible to give an example of countable set, with uncountable derived set, such that intersection with this set and its derived set is empty? I know that if given set is uncountable, then it is impossible. But what if set is countable?


Do internal forces of an atom lead it to move, or can an atom only move when an external force is applied to it? For example, if we had a perfect vacuum with the exception of one arbitrary atom, would it move due to internal forces?


I am proofreading an article and have come across: "to a focus on a fulfilled (upper)middle-class life". At first I was simply going to put a space between the closing parenthesis and the "middle" but am now second-guessing myself. My question is, in keeping the parentheses, would it be: "(upper) middle-class" or "(upper-)middle-class" or some other form?


When a railway is laid in a cold environment, should tensile or compressive force be applied? How does this work? I think the answer should be tensile. The railways expand while heated so applying tensile force would mean there is less extension and therefore less stress. Is this a correct explanation?


I have a problem where I have to study "the geometric properties of the Brachistochrone curve in non-Euclidean spaces". But I am confused about the definition of the Brachistochrone Problem/curve in any non-Euclidean space. Please offer me some help.


I considered the word "circumstantial" or phrase "circumstantially true" but I feel like there's another word that better describes this. In an example; if you step in a puddle and your feet get wet then that is the truth in your case, but it is not the case that anyone who steps in a puddle will get their feet wet.


How would I go about drawing something that looks like this (please excuse the extremely lousy Paint image) I am sorry I cannot provide any exact code, but I am basically clueless as to how I'd do this. I hope you have understanding. Thanks in advance.


I've been learning about the density wave theory of spiral arms, and also how the gravitational potential of galaxies is non-axisymmetric, resulting in a sinusoidal spiral potential. I've then learnt that the spiral arms/density waves occur at this spiral potential minima. Why is this?


Is there any word for someone who doesn't accept others' favors? They have no reason for it, they just say no. Perhaps it's because they would feel insecure. Example: -Why does she always say no when I offer her snacks or drinks? -You know she is a [word]


I've been learning spectral methods mainly through reading the book "spectral mehtods in matlab" by Trefethen. I have a simple question: Could spectral methods solve first-order differential equations? It seems using spectral methods requires two initial conditions.


Recently, I was reading about a article which tells about something known as "Susskind Complexity". The article states that the interior volume of a black hole grows forever. How/why does the internal volume of a black hole grow forever? What is the mechanism/reason behind this?


How to prove that a Hilbert space is the directed colimit of its finite-dimensional subspaces? Does this imply that the category of Hilbert spaces (and bounded linear maps) is the Ind-completion (see here) of that of finite-dimensional Hilbert spaces?


I am looking for a distribution whose mean and median are different, preferably a distribution that share some properties with Gaussian distribution. I kinda know that there is something called skewed distribution or alike that probably would work but I have no idea how it is defined and/or how the expectation/median is estimated. Thanks a lot.


I know "No, it can't be" can mean "It's impossible that it is", but I don't think "No, it may not be" can also mean "It's impossible that it is". I know that "No, it may not be" can mean "It's possible that it is not", but can't the sentence mean "It's impossible that it is" like the case of 'can'?


So, I'm preparing for a PTE test and came across this problem in the Fill in the Blanks section. Gators can move through the water oh so silently, barely ___ (making/creating/adding/showing) a ripple. Why is the answer creating the ripple (not making the ripple) here? I just wanted to know the reasoning behind selecting the word "creating" instead of "making".


Can I use parse as a verb in this example? The clouds began to parse and the day opened up. Meaning the clouds began to separate from one another and dissolve. Parse is commonly used to describe analyzing text or sentences but can also be defined as to resolve into its elements.


If we have a beam of photons (high energy X-ray) and a beam of protons both with the same energy, which one will get further inside the human body and why? Can this be explaind due to LET (Linear energy transfer)?


In a problem, I have the expression of acceleration and velocity in Cartesian coordinates , and it ask me to calculate the tangential and normal acceleration, so we don't know how I can do that, can any one help me?


I have a problem, where I am asked to give a precise definition of the Sierpinski-triangle. The problem describes the usual algorithm to construct the triangle. However, I am clueless on how I should give a different definition. Is there some definition using the union of the points of the Sierpinski-triangle?


From what I 'understand', Noether's second theorem applies to infinite-dimensional symmetry groups. A classic, even historical, example is the invariance group of Riemannian spacetimes, i.e. the set of spacetime diffeomorphisms of GR. Does Noether's second theorem also apply to quantum field theory (= the 'gauge groups')? Or is it confined to general relativity?


I'm wondering what native English speakers call a person (man or woman) who is attracted to a no-make-up look and dislikes it when people wear a high amount of makeup. I can think of no-makeup lover or not into makeup. But is there a specific word being used?


It is well known that the quaternions, octonions, and sedenions are well studied, but I don't find any articles or books in which other hypercomplex numbers are studied. Does anyone know a book or an article in which other hypercomplex numbers are studied? And if so, what is the usage of numbers like pathions, voudons, etc...?


Looking for a reference for a textbook that discusses classical conditions for contraction mapping, such as diagonal dominance. It should contain theorems similar to the one in the picture (which I found on this post without a source).


Basically it means that the vowels must come in the order - E, U, A, I, O and can be place anywhere in the word. I tried to do it by assuming that each of the vowels can come in five different spots. But I didn't get anywhere...


We don't pronounce "th" in "pothead" as a single logical sound, or "ph" (as "f") in haphazard. They are consecutive letters pronounced individually. Is there a term to describe such a language phenomenon? I'm guessing no but it can lead to some humorous situations (I think on the Simpsons, policeman Lou said "poth-ead" which I found entertaining).


What is the adverb that is commonly used when describing something rapidly/drastically increasing? It's on the tip of my tongue and I can't get it out. It's not tenfold, rapidly, drastically, dramatically, stratospherically, etc. but similar. I think it has a connotation of quickly multiplying. (It's a single word.) It's driving me insane--any idea what it is?


My understanding is for a rigid body the angular velocity between two points fixed on the body is the same for all points. In other words the angular velocity vector would be the same. I can't find a proof of this. To be clear I'm talking translating the origin on the body, not relocating the secondary point you're computing the angular velocity of.


I know that a charge moving parallel to a straight current carrying conductor experiences a force towards that wire due to magnetic lorentz force. I am more curious about the idea where the charge is stationary and the wire is moved parallel to it in the opposite direction at the same speed. Will there be a force on the charge this time?


In Spanish there is a suffix, -udo, that's used for "in adjectives derived from nouns, it indicates abundance, big size, or intensity of the meaning of the root." Is there something like this in English?


I'm thinking of doing a project with some undergraduates soon involving edge ideals, particularly binomial edge ideals. Do you have a gentle introductory reference to this topic suitable for undergraduates? We can assume that the students are comfortable with basic algebraic objects such as rings and ideals, as well as what graphs are and some of their basic theory.


I'm not sure if there's a better way to say this: It would have been better if my brother had died rather than struck a bargain in the red desert. Am I using that right? Ugh! Help me please. This is the first line of my story and my critique group is arguing about which sentence is correct.


I learnt that an optic fibre uses the concept of total internal reflection (TIR) to transmit data at high speed, but why do they not use just simple mirrors instead of using refractive medium and making a light incident at an angle more than the critical angle, so that total internal reflection can occur?


I am currently studying intro to analytical number theory by Tom Apostol. It is a very def-thm-proof kind of textbook, and even after taking notes of definitions and theorems, i still struggle with exercises. Can someone please suggest a method for me to get more out of the book? Also, does anyone have lecture notes that can be paired with the book?


I would like to learn what prerequisites are needed in order to be able to work on a postgraduate thesis on the topic: Invariant Einstein metrics on compact simple Lie groups and Stiefel manifolds. I have noted Lie groups, differentiable manifolds, general theory of relativity, and tensor analysis.


Assuming that a sinusoidal wave with an amplitude a million times bigger than the diameter of a black hole enters a black hole, what happens to those parts of the amplitude that are not inside the black hole? The other question I have is, "Does the black hole change the amplitude of a frequency that enters it?"


Sorry everyone. I do not really know if this belongs here, but I can't find a solution by myself, since I have no idea of what could be happening here! I have the fonts installed, but TexStudio says they are missing! How could I start solving this problem?


According to wikipedia: https://en.wikipedia.org/wiki/Affinity_laws, when shaft speed is constant, the pressure is proportional to square of impeller diameter. I am wondering why this is true. Surely, for larger diameter, more air is pumped, so we need a stronger force. But why pressure also need to increase?


Do we know absolutely nothing about what might be the state of matter-energy inside the black hole? Are there existing theories or research that explore the possibility of a metastable vacuum inside black holes?


Say I have a bunch of colors in math mode. How do I remove all of these colors (reset to default or black) efficiently? (If the colors are in text i.e. not math mode, I can just do this: Select text > Text Style > Color > Default > Apply. But when the colors are in math mode, this doesn't work.) Related


In programming it is rather the norm than the exception to have functions whose input are different data types. For example a function who produces a substring of a given string would have the following signature in Java: public static String substring(String s, int n) {..} Are there examples of such functions in Mathematics and what is their application?


According to my understanding we get a lot of energy during nuclear reaction because mass transforms to energy. According to picture below total mass remains the same after reaction. Is the picture wrong?


Minimal sufficient statistics are supposed to have the lowest dimension among all sufficient statistics. Then, is any sufficient statistic having the same dimension (as the mss) a minimal sufficient statistic? If the answer is no, is there any simple counterexample?


Consider these four situations: a tennis ball falling on a wooden floor a tomato falling on a wooden floor a book falling on a wooden floor a tennis ball falling on a soft mattress Why only in the first situation the object bounces?


It seems that paths of the Brownian motion are always continuous (please correct me if i am wrong). Levy process is the generalization of Brownian motion by allowing for jumps at random times. So why always people talk (e.g. using Donsker's theorem) about the relation between random walk and Brownian motion (and not between Levy process and random walk)?


Is there a word (or maybe a metaphor) to cover the idea of religion, mysticism, occult, and philosophy all together? In other words, how would you name a bookshelf with books and other texts by or about the following authors? Thomas Aquinas George Gurdjieff Aleister Crowley Aristotle


Let's say a box is moving to the right and friction is slowing it down. The friction force vector pointing to the left and the object moving to the right, should the vector be drawn on the left side of the object like this: : Or on the right side of the object like this:


Is there a specific name for the following graph? "Acyclic tree" is not unique and "one-dimensional lattice" doesn't sound quite right. Saying "line graph" is, in my opinion, ambiguous. The reason I ask this is because I need to reference this graph without the use of this picture, or any pictures for that matter.


In quantum field theory, consider acting on the vacuum with a local unitary operator that belongs to the local operator algebra associated with a region. In such a way, can we obtain a state that is orthogonal to the vacuum? i.e., Can the one-point function of a local unitary be zero? If so, any examples?


The Hauptvermutung (ie. the question in the title) is known to be false for PL manifolds and topological manifolds, but I can't find a result for smooth manifolds (with boundary), though I recall reading it is true. If not, is something close to this true?


I am trying to self-study probabilistic measure theory after completing my undergrad degree, and I am curious if there are more interesting applications of measure theory aside from Lebesgue integration ? It seems like (correct me if I am wrong here) measure on its own is a rich field before delving in integration of measurable functions.


If someone says something such as, "I don't know what else to do," it kind of implies a request for some suggestions. In the case the person doesn't actually want ideas for what to do, would this be considered a rhetorical statment?


As far as I know, the magnetic properties of a bar magnet is due to the alignment of electronic spins of all the atoms. According to the uncertainty principle the electronic spin is randomly up or down. So, why a bar magnet has certain poles while it may be expected its poles interchange? I appreciate any help about my misunderstanding.


In Latin, it simply means "an amount", which can be of anything and of any size. In modern English, especially in physics, it means the smallest amount physically possible, i.e. a physically indivisible amount of something, especially pertaining to energy.


I would like to justify text in lstlisting. If the line gets too long, there doesn't seem to be any way to automatically add a new line. Is there a way to justify text?


In the sentence below, 'do' refers to 'sense the environment' ,which is evident. But can 'do' refer to only a verb 'sense'? Share what you think about this, and please explain why it is or isn't okay. It is lazy and dangerous for the ecologist to assume that all other organisms sense the environment in the way we do.


I have a question about the use of "as well as" and "as did". Which would fit better in the following sentence? When the earthquake began, at least fifteen men and women fell to the ground, as did many children playing in the park that day. I appreciate any assistance you can provide.


How can Dirac neutrinos exist if neutrinos have no charge? As far as I'm aware, the antiparticle of a particle is its charge conjugate while all of its other characteristics remain same. How then can a neutrino be distinct from an antineutrino?


I've stumpled upon a peculiar usage of the Past Perfect Tense. At the end of the WWII, the Cold War had begun. The world would lay divided between two superpowers. This context seems suitable rather for the Past Simple Tense. I would be thankful for any hints as to why this tense was used. Have a good day, Damian


What are some famous open problems and conjectures in Lie symmetry theory? Is there some kind of list of these problems available or possibly a historical survey of the development of theory of symmetry groups?


What's the phase difference between A and B on the following diagram. Where it is a standing wave. This question doesn't even make sense to me as from watching animations of standing waves, points A and B cannot even appear at the same time. We cannot have two minimums next to each other. So what is it asking?


If i were to make an observation of the orbital motion of the planet Mercury from an inertial frame of reference, would I observe the precession of Mercury's perihelion? or would I observe it moving in the way predicted by Newton's mechanics? And if I will observe the precession of Mercury's perihelion, would my frame of reference still qualifies as being inertial?


I learnt that a point source produces spherical wavefronts and i know the reasoning behind that too and while studying the concept again a question popped in my mind, that "Are those spherical wavefronts equidistant ?", most online sources don't have an answer. A clear explanation for the question would be greatly appreciated.


Is there a list giving an estimate of the abundance of every element of the periodic table in the Earth's Core - possibly differentiating by inner core and outer core? I am specifically interested in estimates for extremely rare trace elements.


A steady state is Nash if it is strict and pure. And there can be multiple steady states, but not all of them satisfy the conditions of a Nash Equilibrium. Is this right?


in textbooks I find only results about the weak convergence of univariate stochastic processes. I am looking for a reference text which treats also the convergence of multivariate (vector) stochastic processes. Any hints? Thanks. Ciao Karl


I like to use the cochineal font for my document, however for this font I understand there are no corresponding mathematical fonts. So I am asking if anyone knows of any font similar to cochineal that also supports math fonts or a math font similar to cochineal fonts. I am using the KOMAscript class for the document. Thanks for all.


How do I get LyX to show the reference in full (without the "..." ellipsis)? Example: Below, I want LyX to show me "Ref: VeryLongLabelNameABC" instead of "Ref: VeryLongLabelNameA..." (which in this case is actually even longer and takes up more space)?


Electric potential is the amount of Work required to move a unit positive charge from infinity to a region of an electric field. Why do we need a positive charge for that? Can't we use negative charge?


I couldn't find this sort of question anywhere, but this is similar to possession, like when you say "Amanda's book" instead of "the book of Amanda," but those two have slightly different meanings. What's it called when you make the noun go after the word "of" and it describes the first noun?


I've come across a question that asks what the subatomic particles are. The answer being the proton, neutron and electron. However, from my understanding subatomic particles also include quarks etc. So what do main subatomic particles even mean? Could someone please provide a cited definition for subatomic particles?


It is possible to create fire by friction. For example - by striking a stone with another stone or by striking a matchstick on a rough surface. My question is: Can we say that, where there is fire there must be some kind friction in the fuel , at atomic or molecular level?


In the above question , one way of approach is definitely multiplying the matrices and comparing the respective elements then solving the equations. But I am not able to figure out a more efficient solution?


We can't use computers for factorising the number, as it's from a math contest where no computers are allowed. So I need to find a way with some manipulations. I can't find a way to approach it.


My question is the following. It is known that any Von Neumann algebra can be uniquely decomposed as integral over algebra factors. It is also know that any mixed state can be uniquely expressed as sum of irreducible states. There is any connection between these two decompositions? Maybe by using the GNS theorem?


I don't know much about logic, but similar to how certain theorems "require" their proofs to use the axiom of choice, are there theorems which "require" induction? Is there a way to detect them? Thanks!


By using the kernel of both transformations, we can easily check that they're not injective functions (due to the fact that kernel does not only contain the zero vector by seeing both operators as linear transformations) So, what am I interpreting wrong? I'm a bit confused on this. Thanks in advance for the help!


The Rindler coordinates are calculated in an inertial rest frame, from which the worldlines of accelerating observers are determined. How does this coordinate space look like in the frame of an accelerating observer? I was able to find a YT video where the lecturer shows the image below, but I was not able to find more information regarding it.


I want to make a cv. I Found this this (Simple CV) template I want to use but i can't change the language to Greek. I tried using the command usepackage[ ]{babel} but didn't work can someone help ?


If I am standing still, and a car is driving away from me at a constant speed. What would the relativistic effects be? Would it appear to move slower as it got farther away? With each second would I be seeing farther back in time when compared to the cars current location?


I'm reading some Shakespeare and noticing past tense verbs are written as deceiv'd and search'd etc rather than the modern deceived and searched. When did the shift take place in English to the modern way of writing it? Do we know how it happened?


This is a screenshot of a page of a textbook I am writing. The image is a jpg file that was inserted using the environment wrapfigure of the package wrapfig. How can I force the paragraph starting with "Una rappresentazione grafica ..." to be typeset at the normal page width?


I.e. compute the Euclidean path integrals of QED/the statistical field theory of electrodynamics? I have never seen anyone discuss this anywhere and I am wondering why? What if there is just an electromagnetic field at some temperature where we don't need quantum effects? That can happen right?


We know that a real sequence which is increasing and bounded above converges to its supremum. Is this possible for a non - increasing sequence as well? State that sequence. I am not able to think of a suitable example for this. Please help me.


I would like to know; if it is possible to force magnet wedges (as seen in picture) together to form a "Uni-Pole Radial Ring Magnet?" Also, if it is possible to force smaller "Uni-Pole Radial Ring Magnets" (like the one shown below) together to form a larger one? The magnets are neodymium.


When I read Mathematical Typography, I noticed that the prototype font presented (the predecessor of the Computer Modern font family) had different numbers (bottom). My question is, are there any versions of the font available anywhere that still has this number style?


Many results about single-sorted FOL generalize to the multi-sorted setting (see here). Are there aspects that do not generalize or are importantly different when working in a multi-sorted setting? I am starting to work in the multi-sorted setting and am wondering if there is anything to look out for if one is used to working in the single-sorted setting.


If it can be shown that a given NP-complete problem such as Clique cannot be reduced to a given P-complete problem, such as Horn-SAT, then we can conclude that P does not equal NP?


How do you visualize an electro-magnetic wave? If there is a very narrow beam (like the one produced by may be a maser?) confined within a few centimeters (in its wave-front) with amplitude say x, what does that mean in terms of its spatial presence?


Just out of pure curiosity, are there any examples of the consequences of accepting (or accepting its negation) the continuum hypothesis outside set theory? I know for Zorn's lemma, which is equivalent to AC, there are many applications in algebra, analysis, and topology, but I did not really find many for the continuum hypothesis beyond pure set theory.


I'm wondering if there are number systems with bases other than integers? For example, with a fractional, imaginary, irrational, transcendental basis, or with the basis "infinity"? If there are, then how is the translation made in and between them?


The potential difference across the terminals of a battery with resistance is equal to the emf when there is no current.But when there is a current there the potential difference decreases or increases depending on the direction of the current.How does this change in voltage occur.Does the charge density at the plates of the battery change? Or, something else happens?


I am working on a thesis and I am using Euler's criterion. I am interested in it's origin, so the first publication of it by Euler. I found some references, though I did not find the original work of Euler. It's not linked in mathscinet, either. Does anybody know where I can see the first usage?


I've written a .sty file, named creator.sty. When I was using TexLive, there is a folder where all .sty files are stored. I'm now changing to Tectonic, and I cannot find such a folder, and I don't want to put creator.sty into every project I develop. Is their any way to install a custom package by Tectonic?


Recently I came across this question, where the top voted answer claimed that all homogeneous equation represent a set of straiight lines passing through origin. I was wondering if this was true generally, and if so, how can we prove this?


Judging by the wiki page I get the feeling that this type of heat pump would not function if it's radiator were hotter than the heat source, but I can't quite put into words why and I feel like there might be something that I'm misunderstanding. Edit: Here is the link to the wikipedia page on it: https://en.wikipedia.org/wiki/Absorption_refrigerator


I was studying vectors and came up with this question Show that the line joining the midpoints of two non parallel sides of a trapezium is parallel to the parallel sides and is equal to half of their sum. Is this anyhow mid point theorem for trapezium or is it applicable here?


Are "the roof of the building" and "a house of stone" grammatically different prepositional phrase constructions? Or are they both understood to be possessives with the latter not in the usual sense of ownership/possession but something that "belongs" to a particular class or category?


Whenever I slightly touch the touchpad on my Mac Texmaker chooses to "unfocus" the cursor so I cannot write. Then I need to click the text window again to keep writing. I have tried to find out how to stop this from happening, but I can not find the place to fix this in the options. Any help is greatly appreciated.


Does the fractional quantum Hall effect always occur when the integer quantum Hall effect occurs? In other words, is there an example of a material where the IQHE can be measured without the "noise" of the FQHE, allowing us to see the idealized IQHE as shown here?


Given N nodes I randomly pick a permutation of the nodes and use this permutation to define an assignment: I associate one and only one node to each node. This procedure naturally defines a graph made of loops (auto loops are allowed). I was wondering what's the average number of loops in such a graph


I am able to add the figure in a lyx document, by using float and insert graphics. But when I view in the pdf, I don't see the figure in the inserted location. Instead I see the location of the file inside a blank box. See below. I want to know why this is the case and any suggestions on how to fix it


I'm surprised that I can't find any research on this topic. Maybe it's too obvious? Kinoshita proved that contractible continuum do not have FPP, but his example is not locally connected. Maybe if we add this to the conditions it will have FPP? UPD: Continuum as a nonempty compact connected metric space


I am studying the Koopman operator and its spectrum. Koopman spectrum is a new topic for me and It is hard to understand what kind of dynamical system shows the point spectrum and what does not. what are the dynamical systems with point Koopman spectrum? does a dissipative system or more specifically the damped pendulum show the point Koopman spectrum?


How is there so much pressure at the bottom of the ocean and therefore a greater force transmitted by collision, even though the temperature remains constant ? Shouldn't greater pressure and therefore greater molecular acceleration create a higher temperature ?


For the hydrogen atom, a simple separation of variables give the energy eigenvalue of the Schrodinger operator for one electron in a spherical potential. It is well known that there are no such explicit solutions for the helium atom. However, is there a rigorous mathematical proof that the helium atom's Hamiltonian operator indeed has discretized eigenvalues?


I've been trying to learn how to graph some miscellaneous functions and I came across this. I put it in Desmos and this is what I ended up with. Can you let me know how did the graph get its shape? And how can one graph such type of functions on their own?


Consider a damped sinusoid. It has a obvious absolute maxima, but it also has an infinite number of critical points. Is there a general solution to finding an absolute extrema given a function that has an infinite number of critical points?


I was reading multivariable calculus , where there were a few topological definitions on things like open ball, limit points, boundaries etc.i came across the statement "a limit point of a set need not belong to the set". Why is this true? can someone give a visual and mathematical proof of this.


I was doing my car license's questions, and it came up this question, why would the stoppage distance (the distance which the car stops after breaking) increase if you add more weight on top of the motorcycle? I thought it would decrease, since the friction force depends on the normal force which increases after adding weight to it, isn't it?


Can photons form black holes the same way as other matter? If there happens to be enough of them concentrated in an area of space so that enough energy exists within a radius to form an event horizon, will an actual inescapable black hole then come into existence?


While working with the Huygens-Fresnel integral, I learned about the Rayleigh integral again, and I can't tell where they apply and whether they can be replaced equivalently. Also, in terms of structured light in a focusing system, which is better to use. Of course, I'd love to get some references!


I have made predictions for the amount of conversions of a particular website. After predicting the amount for every day. I looked for ways to get a prediction interval for every day. The residuals are plotted as follows: Residuals From my knowledge this is not normally distributed. I am curious if there is a way to get prediction intervals.


Which of the two books do you recommend? I have been studying mathematics for three semesters and would like to improve my skills in demonstrating and understanding proofs more easily. I am open to other book suggestions.


As is known, non-zero infinitesimals exist. It can be proved. In the book "A Primer of Infinitesimal Analysis" John Bell introduced infinitesimals that indistinguishable from zero. He did it implicitly. How did he know that such infinitesimals exist? He didn't prove it. How to prove it? Thanks.


Has every infinite simple group a faithful irreducible representation? This question solves the finite case. However, the proof requires a non-trivial linear representation of a finite group. I want to know if the conclusion is true for an infinite simple group. If it is not, can you give me a counterexample?


Is there a popular English expression equivalent to this Russian proverb? It translates to: Water wears away a stone. And has the meaning that, step by step, eventually we (it) will get there (reach the goal, etc.). It can also be used to mean that harm or something undesirable can be inflicted over time.


When deriving the expression of momentum expectation value one gets to what's shown in the picture. However, every text I've seen so far simply neglect the first term inside the integral. I would comprehend if there were linear factors of the wave function itself, since it goes to zero. However this is not the case and I don't get why this term's cancelled.


I use a random number generator to generate normally distributed numbers. The problem is negative numbers are unphysical in that particular simulation. So I use "if/else" to discard them. My question is: What is the resulting distribution? It's obviously not Gaussian any more since it is cut on the left.


I have just started studying about affine schemes and I am searching for books about Spec as a functor from the category CRings to the category AffineSchemes can you suggest me some? Also I am searching about the geometric results of this equivalence of categories, any ideas?


a newbie here. For my thesis I needed to replicate this graph: I think it needs to be done in TikZ. It would be extremely helpful if I get an ipnut from the experts here. Thanks a bunch in advance.


I'm studying probability theory, and I have a question. From the definition of a random experiment, we don't know the outcome of an experiment, but we can calculate and get outcomes. How? If an event is random, how do we know its random variables and its outcomes? Which definition or axioms allow us to know outcomes?


In dynamical systems, what is the distinction between a homoclinic orbit and a limit cycle? It seems to me like a homoclinic orbit is effectively just a limit cycle with a particular fixed point along the way, but in both cases can't we describe the object as an attractor (or source, depending on stability) curve in phase space, periodically traversed?


It is well known that If in a triangle a median has the measure half the length of the side it is drawn, then the triangle is a right triangle. Is there any trigonometric proof of the above statement?


I know that perhaps some will answer with "the double slit experiment" but what is it that makes the double slit experiment something probabilistic? I cannot understand this and please I ask you to explain this as if I were a child


I would like the first line of footnotes to be indented, like below. It seems the easier approach is to override the template using YAML at the top of my input document, but I don't know what to insert in the YAML.


Today, l learned a new word 'supine'. It has multiple meanings, but I am asking about the opposite of the following meaning: Lying on the back facing upwards. In other words, what is the suitable word to describe the following: Lying on the stomach facing downwards


Is proving mathematical theorems one kind of computing? I read somewhere in a book that proofs are one kind of computation. I am not talking about formal proofs, I am talking about the informal natural language arguments that mathematicians use in their daily work.


I have a GitHub link and I want to use it as a reference. How can I put it in a .bib file and then cite it? I am using bibtex. An example GitHub link I want to reference: https://github.com/LLNL/zfp


I am a first year student. I am trying to find a precise definition of cardinal numbers. I am unable to find it anywhere. I am fine with finite cardinalities. I also know that different sets have different sizes. But how to have a precise definition of a cardinal number ?


This might be a random question, but I've always wondered, can the principle of induction be used not only for proving sum formulas, but something else? Is there anything else we can use it for?


The script fancy-preview is obsolete and no longer works in the recent TeXLive versions (see, for example, this question). According to the author of fancy-preview, there are some better options, but he never mentions what the option is. Since his words are not very convincing, do we have any alternative for fancy-preview?


I am interested in writing a chess book in LaTeX. Are there any existing LaTeX templates for writing Chess books in overleaf? Also, is there any streamlined way to use Chessbase directly for writing a chess book, whether it is in LaTeX or not.


Can anyone recommend a nuclear physics textbook which is focused on theory rather than engineering and comes with full solutions to all the problems provided in a separate solutions manual. I MUST have a full solution manual. It may be an introductory text or advanced. Either way I can use it. My query is very specific and there is no duplicate question.


To my knowledge the "white-sky albedo" is the surface albedo corresponding to diffuse illumination while "black-sky albedo" corresponds to the direct illumination. I am confused with the term "Lambertian albedo" used in some radiative transfer models (e.g libRadtran). Is it the same with the "white-sky albedo"? If not, what is their difference?


I think the body should not move since maximum static friction is greater than the applied force but my book says that it will roll. Also I have read that static friction is independent of the area of contact. Please tell me where am I going wrong?


As I know Newton physics is used for most engineering, aerospace, mechanical, naval etc. Where we must use Theory of Special Relativity and Theory of General Relativity to get correct results, only for astronomy, big distances/objects etc?


I can understand the logic behind spin-stabilizing bullets, which travel only a few hundred meters on fairly flat trajectories. But I cannot understand the logic for artillery. Won't spinning shells remain the same orientation throughout the trajectory, causing them to belly flop on the target?


Can I use when as a replacement for how often? Is it common to use when to ask about the frequency of an action? ex: Question: When do you do the dishes? Answer: Sometimes I do the dishes.


In the case of a free particle, if you have perfect knowledge of its momentum, the uncertainty in its position becomes infinitely large. Does this imply that there is a finite probability of finding the free particle anywhere in the entire universe?


As I do exercises in my physics book, there are a lot of questions that ask me to list the brightness of rays (reflected and refracted) in ascending order. A very difficult one is shown below. Can anyone explain why the answer is D? And also how I can do similar questions? Thanks!


Heisenberg's uncertainty principle states that we can't mutually determine both the trajectory and momentum of a subatomic particle. According to Einstein, If we take two particles with different masses,join them and let them spring apart. Their velocities and positions will be related , so can we calculate both velocity and position? IS the Heisenberg's Uncertainty principle disproved?


I have the structure for transitive phrasals and for prepositional verbs, but I am having some trouble when I have to draw the tree for a phrasal prepositional verb. I know for sure it must contain a V-core and a V-shell, but I don't know where the "DO" should be placed. Is the object part of the PP?


Here is the question I am trying to solve: Characterize paving matroids in terms of their collections of independent sets and in terms of their collection of bases. What exactly does it mean to characterize paving matroids in terms of their bases? could someone explain this to me please? what exactly should I do to solve this problem?


So I've heard that the definition of a singularity is a place where a geodesic cannot be extended smoothly. Which brings up the questions, have there been attempts to do this non-smoothly? Is the result trivial? If you give up smoothness but replace it by some weaker notion of differentiabilty can you do it?


There's a recurring preference for pionless effective field theory in recent papers modeling nuclei for fermion scattering processes (e.g., here). Why is pionless effective field theory commonly chosen? How does the exclusion of pions impact the theoretical framework?


When referring to the "politically organized body of people usually occupying a definite territory" in lieu of the proper name, should state be capitalized? Example: He will liaise with State and local governmental regulatory departments.


We can say, "I had a delicious breakfast" because of the adjective, as opposed to "I had breakfast", where we don't use an indefinite article. As in this former case we have ONE specific breakfast. Why can't we treat the word "weather" the same way? "It was such a great weather". Isn't it also ONE instance out of many?


According to my current understanding of special relativity, light would not 'see' the universe's time tick. In other words, the universe would remain a snapshot in time according to a photon from the instant the photon was created. Is that true? Also, would a photon 'see' ordinary (moving slowly) objects move at the speed of light? If not, then at what speed?


I'm installing a program called nauty using cygwin. It starts asking to run using "configure", so far everything has been fine. However, it asks to perform a "make" to compile the installation. I don't know what to write to perform the compilation. I hope you can help me on what I should write to perform the compilation and finalize the installation. I'm using windows.


Is there a word that refers to the organs/appendages used for processing food that do not imply that the organ/appendage in question is around a mouth or other orifice for food? Example sentence: The hammerhead worm's _______ are placed upon its head


There are mixed answers online. Some say 't' is silent. Others say 'e' is silent. Dictionaries don't seem to give a consistent pronunciation for this word, so it's hard know what is correct.


Concerning the Banach-Steinhaus theorem, what counterexamples arise if certain conditions are not met? In the context of the Banach-Steinhaus theorem, what counterexamples arise in the absence of completeness conditions (or when X lacks any Baire second category set)? Alternatively, when considering a non-linear family of operators, what specific instances of counterexamples manifest?


Is there a name for or relevant work on graphs constructed by taking a tree graph as input and changing the edge targets from the parent to the next-oldest sibling (if it exists)? It has come up in some work I'm doing and I'm not even sure what to search for in a literature search.


I'm struggling to find an answer to my question, which is about the difference between Magic Tree and Magical Tree For native English speakers, is there a preferable option that "sounds right"? If that's not the case, and there is a rule, when should I use each of them?


In finite-dimensions, the complexification of a real vector space is easily shown to be isomorphic to the original real space by using a proof which involves explicit use of a finite basis. Does the result also hold for infinite-dimensional spaces?


Someone has just messaged me saying as a professional business owner I should know the difference and that my sentence is wrong! I am not sure! Especially as I am from Yorkshire and we all know we speak differently up here! So why wasn't you good enough? Or So why weren't you good enough?


I have an ionic liquid solution phase and an organic phase. They are immiscible. When I place a small piece of nylon fibre in the liquid it always goes to the liquid/liquid interface instead of staying in the aqueous/organic bulk phase. What is the phenomena behind this, and to understand this, what are the concepts I have to search for?


I found in this site that if we use a set of colored lights we'll have colored shadows. But why is that? As far as I know a shadow is where there is no light. So why isn't the shadow that is produced by colored lights just black like a normal shadow that is produced by white light?


If an electrically conductive material has lots of protons, then how will it conduct electrons even though they should be getting attracted? Do the electrons somehow get pulled in the direction of the conductor?


Is it possible to have two table of contents in beamer which contain the contents of two different sections, e.g. the main part and the appendix part. One table of content only includes sections from the main part, and the latter one which is shown after the main part only shows the appendix sections?


given a parallelogram, show that the area of the parallelogram follows the relationship of two adjacent sides times sin of the angle between them. I tried dividing the Parallelogram into two congruent triangles, then finding out if I could get a relationship using the law of sines since I am stuck on how to integrate sin of the diagonal into the relationship.


I am studying mathematics at the undergraduate level and I am interested in the world of mathematical logic. I would like to know the opinion about which books or texts are recommended for an introduction to the subject and then a deepening? I am grateful to all those people who give me their opinion or recommendation of texts or notes, on mathematical logic.


I know how to find the quadratic residues and I know how to determine the quadratic nonresidues by simply taking the complement of the positive integers lower than p against the known residues; is that the only way to find nonresidues? It feels like there should be a method that's more direct.