What is the interpretation of temperature in the Ising model? Can one describe a theoretical thermometer for the Ising model? As far as I know, in the theory of gases, temperature is interpreted as a quantity proportional to the mean kinetic energy. A thermometer is a device capable of converting the average collisions of particles into the movement of a dial. The higher the mean kinetic energy, the more movement the device registers, and the dial goes up. I can't grasp a similar idea for the Ising model.


I already know the term anaphora exists for repeating the same word for emphasis. I'm specifically interested in a term for repeating a word twice without additional clarification to express that something is especially strong or genuine. Examples "Do you like her, or do you like like her?" "Are you making quick bread, or are you making bread bread? "I'm so excited, I got a promotion promotion, with a pay raise and everything!" I don't know that I've ever seen this in formal prose; it seems unique to spoken English, but if anyone knows of any written examples please add them. If no term already exists, I'd like to coin the term "duplication duplication," to distinguish it from ordinary duplication.


Imagine that Alice is the President of Arstotzka. Alice has a lot of enemies but she's generally an upstanding President so her critics have a hard time building a campaign against her. Instead, the critics are patiently waiting in the shadows until Alice makes a minor mistake - perhaps she misspeaks at a conference or her car driver skips a stop sign. The critics then blow this incident out of proportion and use it to smear Alice before the next election. What's a word or expression specifically describing the act/strategy of waiting for a small transgression to happen?


Can you compute geodesics by treating it as a problem where you want to minimize the length of a curve through two points on a specified surface while having the constraint that the curve must reside on the specified surface? If so, can you explain how one could do so for a cylinder? Or is the calculus of variations the only method by which to do so? I tried setting up the constrained optimization problem for a cylinder but was unable to make any progress, leading me to think that maybe it's impossible to solve the problem without the calculus of variations.


I am working on my thesis and need to convert my paper to a different format. I'm new to using latex, only having started using it for this project, and am struggling to get the base version of this template working. No error occurs, but compilation appears to halt, judging by the fact that main.log log ceases to update after a certain point. The repository hasn't been updated in a while, so I am guessing that the setup may be outdated. I've tried removing some of the references to other sections, but that doesn't appear to make a difference. If someone could help me get this working, or point me in the right direction, I would greatly appreciate it. https://github.com/davidanastasiu/thesis_template-SJSU_CMPE


Show that, given four coplanar points, we can always draw two intersecting circles coplanar with the points, such that two of the given points are diameter endpoints of one circle, and the other two given points are diameter endpoints of the other circle. In this question, "intersecting" means that the circles share at least one common point. Example: I will post my answer. Alternative solutions are welcome. This question and answer serve to provide ideas that might help answer a harder question about five points and two non-intersecting circles.


Ive recently been introducing myself to Gaussian Processes. In Bayesian linear regression, one would expect that when adding new features, the likelihood on the training set would weakly increase due to the larger degree of freedom, of course leading to overfitting most likely. I was wondering if this is true in general for marginal likelihood for GPs, regardless of the choice of kernel/specific hyperparameters? If so, what is formal explanation for this? Can we prove it mathematically? Thanks :)


I heard that natives use Present Cont. to describe things in the future. As I understood, we use Present Cont. when we have arranged an action or there is at least one person with whom we agreed to have plans. But I saw this: I am teaching English tomorrow. I will be teaching English tomorrow. What is the difference and when I should use Present Cont. to describe action in the future?


I work in an office where the airconditioning is not working, so I brought a small mobile airconditioning device: it's a small machine, emitting cooled air. It looks like this: However, according to the laws of thermodynamics, the total amount of heat should always increase in a closed system. Therefore, regular airconditioning systems vent their waste heat to the outside, but this little mobile machine does not have any connection to the outside, so it has no means to discard its waste heat. So my question is: how can such a machine even exist?


Two congruent circles that touch at point H are given. Let the line p be their common tangent that doesn't pass through the point H. Construct a circle that touches both given circles and the line p. The most intuitive thing was to construct an equilateral triangle NOL (such that L is on p, and N and O are on the given circles) and find its circumscribed circle. However, the obtained circle intersects given circles at N and O, instead of just touching them. What am I doing wrong?


If a set is equipped with total order (reflexive, transitive, antisymmetric, strongly connected), does this necessarily mean there exists an injective mapping from that set to the reals? Alternatively, if a set has larger cardinality than the reals can it not be well-ordered? I'm struggling to think of a counter example. Most of the canonical examples of sets with larger cardinality than the reals (all functions, power set of reals) cannot be equipped with total order.


This question is bugging me from a very long time and I don't know whether it can be answered or not but still I am posting to get other's opinions. Is there any inherent property among all the uncountable sets which make them uncountable ? Or more precisely:- If I ever wanted to "make" an uncountable set, what ingredient should I put in so that I am guaranteed of the set being uncountable ? Edit :- My apologies for the wrong terminology used. By "ingredient" , I was just trying to refer to some property which is common to all the well known uncountable sets (e.g. the irrational set, set if real numbers) that I should keep in mind while forming a new set of numbers.


Imagine a uniform thin layer of a viscous gel sticking on a vertical wall. A small sphere is submerged in this gel. The gel has a yield stress (Bingham model). How can I calculate the yield stress that will cause the sphere to start sliding with the fluid on the wall? This yield stress will obviously depend on the weight and dimensions of the sphere.


I'm having a problem with the second part of the question, which shows that the sum of the vectors equals zero. I came up with a geometric proof, but I wondered how to prove it by using complex numbers and maybe links to the first part of the question. I was thinking about the equally distributed roots of unity, but don't know how to move on.


I understand how we can have three coin tosses (or three events if you prefer) such that all three tosses are pairwise independent, but not mutually independent. One trivial example of this would be to just have the third coin be the xor of the first two coins. My question is whether it's possible to have four or more coin tosses such that any two coin tosses are pairwise independent, but any three coin tosses are not independent. If this is not possible, what would a proof be like showing that it isn't?


I am a PhD student working on observational cosmology. My research is based on the experimental side of it (receiver design, etc.) but I would like to get deep into the CMB and CMB statistics. I have found several resources and have read some of the papers but I feel like they are a bit "unconnected". Does anyone know where I could start? I thought of just studying Ruth Durrer's (The Cosmic Microwave Background) book, but I am not sure if it will be complete enough.


I am studying blackbody radiation and modelling a cavity as a blackbody. However, I am encountering a number of confusions in this description: Many textbooks mention that the cavity consists of metallic walls that act as perfect reflectors. Light entering through the hole, after multiple reflections, get absorbed. However, if the cavity walls are perfectly reflecting, how is the radiation getting absorbed by the walls? For studying the radiation emitted by the blackbody, the textbooks model the radiation inside (emitted by walls due to heating) as standing waves. But how are standing waves being emitted by the cavity? Because standing waves are supposed to be confined inside. Can anyone please explain in simple terms?


For example, a carpenter works in carpentry and a plumber works in plumbing. So what trade does an electrician work in? Electrical? I searched the definition for "electrical" and found that it gives the part of speech as an adjective and not a noun like the dictionary does with carpentry or plumbing. This is confusing to me. I would like to understand and know how to talk about the trade for an electrician. I would appreciate it if someone could help clarify and explain this to me.


Imagine a mass hanging from a string attached to the ceiling of a lift and the lift descends with a certain acceleration. From the reference frame of the ground the tension in the string would be smaller than when the lift remained stationery. However, let's say your friend was inside the lift and when asked about the tension would say that the tension was same, regardless of the list ascending, descending or stationery. Isn't that really counterintuitive? Isn't the force not relative and the same, regardless of the reference frame?


For example: It is a thing that works producing stuff. This feels wrong to me, but I can't quite put my finger on what exactly is wrong about it. It seems like it's trying to be a participle phrase, but it's not necessarily modifying the current state of "it", and is, instead, describing what "it" is -- i.e. by working, "it" produces stuff. If it is, indeed, a participle phrase, then it should be able to be written as Producing stuff, it is a thing that works. But, to me, this doesn't seem correct either, so it leads me to believe that the very structure of the sentence is incorrect.


I have been taught that path difference and phase difference are essentially the same thing, except phase difference is measured in an angle unit while path difference is measured in multiples of wavelength. This seems incorrect. I think they are rather different concepts, with path difference representing the difference in distance a wave needs to travel in order to reach a point. Are they really the same thing like I have been taught?


For example, when referencing a webpage: Visit the "The Performers" page to learn about our musical lineup. In this case, I want to tell someone to visit a webpage entitled "The Performers" to find more information, but the repetition is throwing me off. If I were to remove the first "the" before "The Performers," though, then it would essentially read as "Visit page" without an article. So, is it okay to use the article of the webpage title as the article for the sentence, or must I include the first "the" before the webpage title? The two iterations: Visit the "The Performers" page to learn about our musical lineup. Visit "The Performers" page to learn about our musical lineup.


I think something is missing in the definition of homeomorphism I saw. It just said it maps the collection of open sets to the collection of open sets in a bijective way. What exactly makes this preserve topology? I can think of weird situations where each individual open set is mapped to totally disjoint open sets in a bijective way, throwing the topological structure out the window.


The definition of optical path length is the distance that light could have travelled in the same time, in vacuum. So, can we define something analogous for sound waves, like an "acoustical path length", with respect to a specific medium, say air? And if we were to define something similar, how far would it's applications reach? For example, can we use it in doppler effect in cases when the observer and source are in a different media? Or interference, when there is a change of medium in between? If so, kindly guide me on how to do it.


I know foliations as a particular topic in differential geometry. I understand the definition and basic properties of a foliation from the DG point of view, including the Frobenius theorem. While trying to understand why this concept is important I came across this Wikipedia page on integrability conditions for differential systems, but couldn't understand much of it. I do not want to too diverge from my other studies now, yet want to understand this association quickly and completely. Can you suggest me a source that is solely focused on the relation between foliations and systems of PDEs?


I was going through this paper in which the authors state that : In Landau theory, a continuous phase transition is associated with a broken symmetry. The phase transition in a black hole system can also be characterized by the symmetry and order degrees as in a conventional thermodynamic system. The authors don't explicitly state which symmetry is broken for the black hole phase transition. Also, would such symmetry breaking lead to any chaotic effects? Any explanation in this regard would be truly beneficial.


I am attaching the solution of a school question asking to calculate the inverse of a matrix. However, the solution proposed does not fit in anything I ever seen about calculating inverse matrices. I did not found any property that fits the final part of the solution (the one I highlighted in the image below) although the solution is correct, I would like to know which property or reasoning supports the final solution of such a problem. Thanks.


I read an article that referred to the idea that a double slit experiment near the event horizon of a black hole observed by someone inside the black hole creates a paradox because the inside observer can break the interference pattern, thereby bringing information of their presence out of the black hole. This seems odd to me. The significant part of "observation" here, as far as I understand, is that it requires interaction. The inside observer can't interact with the particle to measure it's position. He can only detect it's position if something outside the black hole interacted with the particle, which means the superposition would have collapsed whether or not he was there. Am I missing something? Edit: Link: https://www.sciencenews.org/article/black-hole-paradoxes-quantum-states


I have often heard that two physically distinguishable mixed quantum states produce different density matrices. However, how would I prove it? I know that they have to differ on the main diagonal, because these elements correspond to measurement probabilities in the standard basis. Moreover, I have seen someone using density matrices to show that two mixed states are physically indistinguishable. Does this process always work? In other words, is it true that for any two physically indistinguishable quantum mixed states, the density matrix is the same? And how to prove it?


I'm learning some math by myself and I thought of this question that has to do with how useful complex numbers are: I'm looking for a specific example of an equation that has the following properties: The equation is true, i.e., its LHS and RHS are always equal, no matter the values of its variables, if any. The LHS and the RHS are algebraic expressions (no limits, derivatives, integrals, sin, cos, etc...) The equation only involves real numbers. Its shortest equality proof involves complex numbers. The proof ends with no complex numbers. It has a longer equality proof that doesn't involve complex numbers. It's a short equation (this property is subjective). Please, let me know if my question is wrong or if no such equation can exist.


I want to understand in some detail why superconducting qubits need periodic calibration. The usual, hand wavy explanation is environmental effects that tend to vary from time to time. However, I suspect that the actual picture is more complex than that. I want to understand in particular which effects are purely external, like the earth magnetic field (in lack of a better example) and which are internal to the superconducting device. I am especially interested in those factors that are internal to the qubit itself, like manufacturing defects and physical characteristics that may change in time. A good reference on this subject would be appreciated.


A spanning tree of a graph is a subgraph obtained by deleting only edges of the graph and which is also a tree. Why does one study "spanning tree" in graph theory? What are "spanning trees" real life application? We know the number of spanning trees of a graph is equal to any cofactor of the Laplacian matrix of the graph. Why is one interested in knowing number of spanning trees of a given graph? I would like to know enough in order to motivate a undergraduate or graduate student.


Hi I was trying to determine the basis of the topology on the following collection in R {[a,b), a <b}. Here is what I tried : Knowing that the basis for a subspace topology is equal to the set of all intersections of {[a,b), a <b} with the basis of the topology it is a subspace to. Now I also know that the basis for the euclidean topology on R is {(a,b), a <b}. Can I just take the intersection of {[a,b), a <b} and {(a,b), a <b} to get the basis? If so what does this equal? Or am I totally wrong? Thanks in advance.


I understand that Berry curvature sinks and sources correspond to Weyl points. However, I'm curious about the identity of points exhibiting a Berry curvature spiral, highlighted by red circles in the figure below. Could these be Dirac points? Is it possible to differentiate between Dirac and Weyl points based on the direction of the Berry curvature rotation? It's worth noting that these two spirals exhibit opposite rotation directions.


I read that a sound wave (a scalar wave) produces monopole radiation, an eletromagnetic wave (a vector wave) produces dipole radiation, and a gravitational wave (a second order tensor wave) produces quadrupole radiation. How do I make sense of multipole radiation physically? The Wikipedia article on multipole radiation is informative on the mathematics, but I can't seem to get a physical picture. Note: my question is different from this post, which asks about the pattern of lowest order multipole radiation produced.


first post here. I've been looking into acoustics a bit recently and I'm wondering if for a compressed solid, acoustic impedance should change along different axis in certain cases. Consider a cubic lattice of atoms with pressure being applied to the top and bottom, compressing the lattice. Atoms would be spaced further apart in planes parallel to the applied pressure. I'm hypothesizing that acoustic impedance would be higher for a sound wave travelling through the lattice perpendicular to the plane of applied pressure and lower for a sound wave travelling through the lattice parallel to the plane of applied pressure. Could we consider density as a vector to solve our impedance equation if the above holds true?


We know that Maxwell's electromagnetic theory was originally an ether theory. Later, the ether was denied. But Maxwell's electromagnetic theory can be transformed into a theory of "field". The theory opposite to Maxwell's theory is the theory of action and reaction. Or it could be the theory of action-at-a-distance (Wheeler-Feynman absorber theory). According to Maxwell's electromagnetic theory, electromagnetic waves can exist independently of their source or have their own degrees of freedom. According to the theory of action-at-a-distance, electromagnetic waves cannot exist independently of their source. Electromagnetic waves do not have their own degrees of freedom. These two theories have been debated for hundreds of years. Is there a formula that can distinguish which of these two theories is right or wrong. Just like Bell's formula?


Consider a few sentences: I read a lot. ?I read much. I don't read a lot. I don't read much. Do you read much? This seems to suggest "much" is a negative polarity item, but then we can say things like "Much of it has to do with you not being here." So how much of a negative polarity item is "much"? I am particularly interested in existing literature on this.


While reading the CMU FLAC overview, I stumbled upon the above image. While I have seen Greek numerals before, I had no idea that there was an entire algebraic symbology developed in Greek (though this isn't too surprising). To try and find more about this particular notation, I reverse Google Image searched it and found nothing (well, I was greeted to snippets written in the Tengwar, but that's besides the point). What is this notation? Can you direct me to an overview of the system?


As a thought experiment, say, for the sake of simplicity, we have a meson. This meson, which is traveling near light speed, is traveling towards a black hole. And skirts the event horizon in such a way where the anti-quark ends up inside it's event horizon, but the quark does not. What would happen? Would this create a free quark? That seems like the only logical thing to happen, but I know that would also break color confinement.


The operational principle of frequency combs is that you generate very short pulses (in time domain), and that in the frequency domain (due to Fourier's transform) the spectrum of such pulses is a comb. But this is just maths. And how it works from the physics perspective? Let's say I have a monochromatic continuous laser pointer that outputs only ONE wavelength. In front of the pointer I place a shutter that chops the beam into short pulses. How can those pulses have multiple optical frequencies (with many different wavelengths)? Or is my idea too simplistic and I'm forgetting something?


If so, then why? We know that a current carrying closed loop in a uniform magnetic field experiences no force but when it's suddenly pulled by an external force, according to Lenz law a magnetic force will start acting on it to oppose the external force and eventually reach steady state at infinity. How can this be? Edit: Iam sorry for not specifying the doubt clearly, here is the question in which I was confused as to how there can be a magnetic force acting on the ring


What does the Stark-Lo Surdo effect consist of in the interaction of the electromagnetic field and the active medium in a laser? I thank those who want to give a clear and concise answer. In my student notes from the seventies, I found the topic I was looking for, the calculation for which I attach. But what happens in practice (since I don't have an optics laboratory available)?


Am getting the following error when I try to do tlmgr install package: tlmgr.pl: package repository http://mirrors.rit.edu/CTAN/systems/texlive/tlnet (not verified: pubkey missing) It appears I need to get the keys somewhere, but I haven't been able to find them on CTAN. Where else can I find them? If it's any use, I just updated my system on Arch, so maybe some other fellow Arch users had a similar problem?


Any regular curve may be parametrized by the arc length (the natural parametrization or the unit speed parametrization). But I haven't seen an analogous development for regular parametric surfaces. I hope we can do this at least for orientable surfaces with no umbilical points. For such a surface, there will be two orthogonal lines of curvature through each point, and I suspect that there is a parametrization whose parametric curves coincide with lines of curvature. But I'm not sure how I should approach showing the existence of such a re-parametrization. Is there a such parametrization? If not, what are the conditions that we should impose to have such a parametrization?


When talking about black holes and singularities, most books say that combining relativity and quantum mechanics gives the answer of infinity in some equations. They also say that: Infinity is the answer you get when the universe is trying to tell you that you have made a mistake. Why can't equations have answers of infinity? For example, a singularity is infinitely dense, so why can't other things about it be infinite?


In the original version of Sequent Calculus by Gentzen for classical and intuitionistic propositional logic there is a structural difference: the classical version admits succedents with multiple formulas but the intuitionistic version admits only succedents with one formula. I'm searching for another formulation of Sequent Calculus in with succedents are restricted to one formula both for classical and intuitionistic logic. In particular I'm searching for a formulation in which Sequent Calculus for classical logic is obtained by Sequent Calculus for intuitionistic logic by adding some rule (i.e. excluded middle). Can someone suggest me some reference?


It is often said that the achievability proof for Shannon's coding theorem relies on the channel being discrete and memoryless. At the same time, following the classical proof (using random coding and AEP), I can't find any part of the proof that explicitly uses the fact that the channel is memoryless (except arguably perhaps the guaranteed existence of a supremum of the mutual information?), as AEP has been shown to work for any stationary and ergodic channel. Further, Shannon's original paper does not seem to make any mention of memory in the proof as far as I can tell. Is it true that the classical achievability proof relies explicitly on the channel being memoryless? If so, where in the proof is the memoryless assumption used?


Quick question with probably a simple answer. For context, I am currently in undergraduate classical mechanics studying potential energy. My question is, if a conservative force is one in which is only a function of position, can radioactivity be a conservative force? The example I thought of was if we take two identical particles which are undergoing radioactive decay, and we put one in the upper bound regions of the atmosphere, and the other at sea level, can we say that radiation is a function of position? I guess this also poses the question, can atmospheric pressure/temperature affect radioactivity? Feel free to slaughter my logic/understanding. Thanks :)


I know work done is equal to product of force, displacement and cosine of angle between them. But that formula works only when we assume that the force is constant during displacement and it acts so long the body moves and also when the direction of force and displacement is constant. But how should I calculate work done if the direction of force or displacement is changing. So in this case there would be no constant angle between them and so dot product won't work as I don't know the cosine value and at this point I have got stuck.


I have an object that takes a shape similar to a catenoid, which is known for its zero mean curvature. According to Laplace's law, the pressure difference is proportional to the mean curvature. This implies that if the mean curvature is zero, then the pressure difference is also zero. However, I am aware that when there is a non-zero pressure difference between the inside and outside of the object, which has left me puzzled. Is it possible to have a shape resembling a catenoid but with a constant pressure difference? Furthermore, how can I relate the curvature to the distance, r(z), from the z-axis to any point z on the boundary of this shape?


Reading through Einstein's Brownian motion paper "On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat", it seems the final argument is that he can calculate the Avogadro's constant by using data on the diffusion rates, particle size and fluid viscosity. But its hard to see the connection that ultimately atoms must exist. Can someone lay out a flowchart of step-by-step reasoning leading to that conclusion? I see all the math steps, but need a more philosophical type of breakdown. Also can't the random movement and diffusion law movement of the particles be just as well explained by fluctuations of pressure and density of continuous matter?


Is it correct to say: for any family of bounded operators (proper subset of B(H)) acting on a non-separable Hilbert space there is a non-trivial subspace of the Hilbert space that reduces the family? I know that this is true for any operator on the space so, I assume that it holds also for the von Neumann algebra generated by this operator. Can we generalize to non-commutative families? If the answers is positive (or positive with certain restriction imposed on the family) how can we show it? Is there any reference?


If you were to measure a regular hexagon, is there a way to rotate the hexagon so that if there were two lines bisecting the center of the hexagon, both lines would be the same length? Can we then use the lengths of those lines to determine the area of the hexagon? Edit: To clarify, I mean that I intend for the lines to be perpendicular.


There are four concepts which are studied in Calculus and Analysis: Convergence, Continuity, Differentiability and Integrability. In Calculus, you can define the latter three in terms of the first, but also you can define integrability without convergence using Darboux approach. In topology, you can define convergence and continuity in terms of neighborhoods, and in measure theory you can define integrability in terms of measurable functions. I wonder if there is a definition of differentiability without making any reference to the concept of limit or convergence, thank you so much by your help.


Consider the following diagram Here, the diode is in forward bias, and allows current to flow. However, I am slightly confused why this is the case. A diode is defined to only allow current to flow from the anode ( Positive ) to the cathode ( Negative ). When drawing a diode, the cathode is the line and the anode the base of the triangle. Conventional current flows from the positive terminal to negative terminal, but in reality the electrons flow vice versa. Hence, considering the above diagram, shouldn't the electrons not be able to flow through the diode? The only way I can see around this, is that diodes are also drawn, considering conventional current. Is this correct?


In the derivation of Torriceli's law it is assumed that the pressure at the tank hole is equal to atmospheric pressure. I believe this is true when the water starts flowing through the hole but I do not understand why this should hold during the whole purge procedure. Do you have ideas about the explanation? CFD calculations showing this behavior would be great. Many thanks.


The second law of thermodynamics states, that entropy (of the universe) always increases. Entropy can, however, be (locally) reduced when energy is provided. At the phase transition from a relatively unordered to a more ordered state (e.g. condensation of vapor to a liquid), the entropy is lowered. At the same time, energy is released from the system (thanks to the enthalpy of condensation). Naively, I would expect that I need to provide energy (instead of gaining some) to the system to lower the entropy. Why does this not contradict the second law of thermodynamics?


I know that the Heisenberg Uncertainty principle states that the position of an electron is uncertain, however, if an electron is created due to beta decay, then at what location is it more likely to begin its movement? Is it right inside the proton? Is it the outer edge of the proton? Is it adjacent to the proton? Is it near the proton? I mean is there a possibility of a beta particle from a nuclear decay to come into existence at the far end of the galaxy for example? What is the limiting condition for its position?


According to Fermat's principle, light should take the least time between two points. Therefore, is it correct to say that the angle of refraction is solely dependent on the difference between speed of light in two mediums? Surely the the position of the observer is a key factor too, given the findings of the double-slit experiment that show a photon is only actualised on measurement/observation. Does this not imply the necessity of an observer for the existence of an angle of refraction?


I've been reading more category theory as a prerequisite to understanding some more complicated theorems, and for the first time I'm running into arrows that are distinctly colored. Examples include these diagrams from Wikipedia for natural transformations and universal morphisms: The Wikipedia articles do not readily show me a link to where the coloring for these diagrams is explained, and I have not seen these colored diagrams in the textbook I have been reading for Category theory. Why are these arrows (and objects) colored blue and red? Better yet, (if possible) can someone give me a link/source/textbook that explains this coloring procedure? Thanks in advance.


I understand that the word spook is a racial slur that rose in usage during WWII; I also know Germans called black gunners Spookwaffe. What I don't understand is why. Spook seems to also mean 'ghost' in German. Did the Americans call them spooks because the Germans did? If so, why did the Germans call them that? Or, if the Germans called them that because Americans called them spooks, then why did the Americans call them that? And how did Americans know the Germans had a nickname for black gunners?


When researching the reflexive, symmetric, transitive, addition, subtraction, multiplication, division, and substitution properties of equality, the sources I found said they were true "for all real numbers". Are any or all of them not true for complex numbers? i.e., is the "for all real numbers" a necessary caveat, or should it say "for all numbers"? Also, I know the reflexive property applies to mathematical objects as well (any mathematical object is congruent to itself). Do the other properties of equality apply to all mathematical objects as well?


I am trying to calculate the determinant of the following matrix. I literally have no idea if there's a general approach for solving such strange looking determinants, but I decided to subtract the first row from each row after the second. I don't know what to do next... May you show me what we're supposed to do and how do we find the determinant? Thanks!


Assume a Gaussian surface in a non-uniform electric field that is directed along X-axis. Say the field is getting weaker as we go along positive X direction and it's constant along Y and Z directions. Then, if the Gaussian surface is a cube and charge enclosed is zero, the electric flux coming into the surface is more than that flowing out. So, shouldn't the net electric flux be non-zero contradicting Gauss's law? Does it mean such an electric field is impossible without a charge being distributed along the path?


The usual semantics of first-order theories involves models who have a structure-set. Hence it is required to use some sort of set theory as the metatheory. However it is also widely known that one can do logic with a minimal metatheory, say a fragment of Peano. If we are using such a metatheory, what will the semantics become ? What will replace the notion of "set-model" ?


Here is the problem, I thought it the I've been thinking for a long time, but I still don't have any ideas. Problem: Let A(S) be the group of all bijective functions on S with composition as its binary operation. then A(S) is finitely generated if and only if S is finite set. I know that proving from the right to the left is relatively easier, but when it comes to proving from the left to the right, I really don't have any ideas. I hope someone can help.


I am typesetting my thesis in latex overleaf. I am struggling with one issue. I want to highlight my section with blue rectangular background with white color for heading as can be seen in picture. For subsection I don't want a background rectangle I simply want subsection of a blue color as can be seen in attached picture. I will really appreciate it if someone help me regarding this. Thanks.


I was given an honors project to solve for the equations of motions of a pendulum with an oscillating fulcrum. I (somewhat) understand the procedure on how to solve it with lagrangian mechanics and the Euler-Lagrange equation, but I am stuck at a question. How do we know this cannot be solved with Newtonian mechanics? I tried to look up the answer on Google and even tried to solve for it myself but I obtained an equation with theta, acceleration in the x and y direction so Im not sure what to make of it. Does anyone know? Or are there any resources to understand the precise limitations of Newtonian mechanics? I am a physics undergrad who just finished electromagnetism and linear algebra, thank you!


I am having trouble understanding why is temperature coefficient for avalanche breakdown is positive? The explanation I found online says as the temperature increase the mean free path decreases making collision and generation of carrier pair easier. But the electric field require to accelerate charge carriers to point of collision remains constant . So why does avalanche breakdown voltage increases with temperature if the electric field remains same? Doesn't increase in voltage indicate it is slightly harder to obtain avalanche breakdown? If the mean free path is decreasing and charge carrier generation is getting easier shouldn't occur at same voltage and not increase further


The Coriolis force predicts that winds in the northern hemisphere should be deflected in a clockwise pattern and winds in the southern hemisphere should be deflected in an anti-clockwise pattern. Why is it that in the case of cyclones however, the cyclones spin anti-clockwise in the northern hemisphere and clockwise in the southern hemisphere? If it's indeed true then what is clockwise and anti-clockwise direction?!


What is it called when someone says no, but actually doesn't mean that? Imagine this situation: when two lovers have had an argument then one of them is trying to apologise but the other one (usually the girl) says "no, I don't forgive you." but she isn't really angry. She just want him to insist more. She says no in a some kind of seductive or cute way. What is this behavior called?


I understand why all non-degenerate energy states can be chosen to be real up to an overall phase (as is highlighted here Is a non-degenerate wavefunction real or complex?). However I've been told the argument holds in the opposite direction -- (that all energy eigenstates that can be represented as purely real are non-degenerate). Is this true? I'm having trouble finding a proof of this.


I know that as the kernel parameter for the RBF kernel increases, the Gaussian function becomes less peaked and broader. The reach of the points become larger meaning that farther datapoints have more weight. Intuitively, it makes sense that a larger reach means a smoother decision boundary. However, since I don't have an extensive background in mathematics, I find it difficult to come up with a more mathematical explanation. I was wondering whether anyone could help me with that?


I came across this question while studying for the SAT. What the humanities teach are valuable skills. These include the ability to think critically; to construct, analyze, and respond to arguments; to see both sides of an issue; and to understand the processes that have brought us to where we are today. (A) NO CHANGE (B) skills: these The answer sheet says that (A) is correct, but it does not specify why. I thought the answer is (B) as what follows correctly elaborates on the previous information.


This stems out of my personal curiosity and it's not related to any homework of sort. Suppose I have a table made of some uniform substance (like plastic), and then I strike some point of the table with a hammer. Will the disturbance/wave I produce travel always the speed of sound in that medium, or will there be cases in which the wave travels at a different speeds?


I have a weighted directed graph representing the intensity of movements of individuals between locations, with weights representing the mobility flow. I have another dataset consisting of particular paths that are taken by specific individuals. For example, for a person named Alice, I know how she moved from node A to node B, potentially via some intermediate stops through other nodes. What I want to evaluate is how likely this is given the mobility network (the weighted directed graph). Essentially, can I compute some kind of distance/statistical measure that tells me how "deviant" or how "consistent" these observed paths are, given the mobility network (taking into account the edge weights)?


I just discovered Lean and using computers for stating and proving theorems. The first question that came to my mind, can we write any proof with Lean? Or are there limitations (something that you can write a proof for on paper but not in Lean)? I'm not talking about practicality or feasibility. But does the Lean system have the capability? Edit: I posted the same question on proofassistants.stackexchange.


So I'm confused with The Second Law of Thermodynamics. It says that heat flows from the high temperature to low temperature. For example, if we put a glass of cold water on the dining room, the cold water will eventually have the same temperature as the dining room, because heat will flow from the air in the dining room into the cold water. The cold water "gains" heat and the air in the dining room "looses" heat. But, why does the temperature in the dining room remain the same? Shouldn't it become colder? The same goes for a thermometer when someone that has a fever uses it. Will their body temperature loose heat because the heat will flow from their body to the thermometer?


Geometrically, what is the difference between a diagonizable matrix and an orthogonally diagonizable matrix? I understand the difference algebraically, as explained here and many other places. But I'm struggling to see the difference geometrically: both will not rotate (assuming the field is the reals), but will stretch by different amounts along different axes (Wikipedia calls this a inhomogenous dilation). Graphically, I can't see the difference between, say, a circle or ellipse dilated inhomogenously along orthogonal axes versus non-orthogonal axes: in either case, it will remain an ellipse.


I'm a physics graduate that is joining the master degree program of my University. One of the areas of research that i'm very interested is Quantum Field Theory over curved surfaces, but the point is that i don't know how to begin to start studying these subjects. I imagine that i should star from classical field theory and go all through until reach in the quantum field theory over curved spaces. So, do you have suggestions of literature to start those subjects?


There is a word for animals like horses and cows that defecate wherever they happen to be when the need strikes them, versus animals like dogs and cats that seek out one place or another to do their business. It may be that all such animals are herbivores but "herbivore" is not the word I'm looking for as it relates to eating habits rather than pooping habits.


I am getting confused as I study the AdS/CFT correspondence, so I ask this question. CFT is given on the conformal boundary of AdS, which can be derived from Poincare coordinate patch to AdS. Would this mean that CFT has its AdS dual only on the subspace covered by this Poincare coordinate patch? Or does CFT has its AdS dual on the entire AdS space?


This Wikipedia's page says that: David Hume's problem of induction demonstrates that one must appeal to the principle of the uniformity of nature if they seek to justify their implicit assumption that laws which held true in the past will also hold true in the future. For which I'm confused about how to interpret the meaning of appeal in this sentence. Most of the dictionaries I consulted suggest that the phrase appeal to can only be followed by one or more people or organizations but here it is followed by to some law. So how should I interpret that sentence?


I just came across this sentence while studying for the SAT and wonder if it makes sense. The women soon disperse, SOME entering the shallow waters at the beach, OTHERS venturing out onto the rocks to access deeper waters. My first question is, it sounds a little awkward to add some additional fragments beginning with SOME and OTHERS here. Are they participial phrases? Can they be added like this beginning with SOME and OTHERS? My second question is, can they be added with just a comma? SOME ... , OTHERS ... I feel there should be some conjunctions to make it more natural. This makes me so confused. I'd really appreciate it if you could enlighten me. Thank you.


I am taking a graduate-level mechanics course right now, and we are working with the continuous limit of coupled harmonic oscillators. My professor mentioned that he prefers the "bloch wave method" to the determinant method for finding the wave equation, but I do not know what this is. There is no mention of it in Goldstein, to my knowledge. Does anyone know how this works?


For measuring the electric potential difference between two oppositely charged plates A and B, a test charge q is moved from plate A to plate B. My textbook says that the charge is kept in electrostatic equilibrium such that an equal and opposite force to the electrostatic force is applied on the charge to move it opposite to the electric field. However, I don't understand how it is possible for a charge to move if it is acted upon by two equal and opposite forces. My teacher said that the charge moves due to inertia, but I did not understand how, since the charge was initally moving towards the negative plate A.


One of the postulates of Copenhagen interpretation of quantum mechanics is that every observable has a Hermitian operator A. With these operators we can then find eigenvalues of observables, make a time-evolution of a wave function and, in general, calculate the properties of the system. However we also introduce anti-Hermitian operators. My question is: What is the use for anti-Hermitian operators, if observables are described with Hermitian operators? NOTE: I am just diving into the subject of Quantum mechanics, so I hope this question makes sense and is not "too basic".


I've used both Polish and Reverse Polish (RPN) notation. I get that using them one doesn't have to deal with varying evaluation order. While the benefits to automatic computation (as in software) is clear, it's unclear how this aided human reasoning. Can anyone explain how Polish Notation (or even RPN) simplified reasoning by people over infix notation? I'll be happy with any examples or even references to additional reading.


I have a Hotwheels track set. It has a looped structure such that at some point in the car's trajectory, it becomes upside down. But in such a situation, it doesn't fall to the ground. I understand this has something to do with its fast speed. Can someone please explain as to why exactly this happens? Why it doesn't fall down? Something like this. (arrow indicates the position I was talking about) I was wondering what kind of forces were acting on the car at that point. A theory based (and not mathematical) explanation would be helpful since I am not accustomed with mathematical explanations


A decimal representation of a rational number is necessarily either terminating or non-terminating and repetitive. This comes from the fact that when you are dividing there are only a finite set of possible remainders (remainder is always less than the divisor). As you go on dividing for decimal representation, the remainder has to repeat. Similarly, in finite fields, as you go on raising the power of an element, the product has to repeat because there are finite number of possible answers. This leads to Fermat's little theorem. In both cases, the finiteness of the possible set of answers is leading to these important and beautiful results. Can you suggest some more examples of this kind - understandable by school and junior college students?


(Disclaimer: I'm not all that familiar with any of the two topics) Consider the Ising model on some graph with, lets say, two heavily inter-connected components that are sparsely connected between each other. This "bottleneck" would imply (or would be implied by?) that the Cheeger constant of the graph (and thus the first eigenvalue of the Laplacian matrix) is small and also that it would be very difficult for the spins in one component to interact with the spins in the other component. My question then is: is there any result linking the Cheeger constant with the "magnetization" properties of the graph? Something like a bound on the temperature at which the phase transition occurs that depends on the Cheeger constant or vice versa? Thanks in advance!


I'm trying to compile a document which needs the package minted, but it throws the same error over and over: ! Package minted Error: You must have `pygmentize' installed to use this package I can't seem to make it work. Other forums say to install the package "Pygments," which is a Python package, but MikTeX doesn't seem to have that package in its repositories. Am I missing something? Do I have to install another LaTeX motor or something? Edit: Before anybody asks, yes I'm using the "-shell-escape" flag to compile the document which minted requieres.


So, when we solve the optimization problem using Lagrange Multiplier method, I know that lambda can be positive or negative. Lambda is simply the rate of change in the optimal value when the constraint changes. So, I understand that lambda can be positive or negative. Now, my question is when we have inequality constraints. In that case, why do we have the requirement that lambda should be non-negative? I could understand the math behind it. But I am not able to follow up the intuition. Simply put, if Lambda is still the rate of change with respect to the change in the constraint, then why THEORETICALLY SPEAKING can't it be negative?


I have the following list of points : I'm trying to find the best regression model to fit these points. The logistic regression is not close enough to the points : I guess I need something closer to a spline, but I don't know how to compute a regression model based on a spline, all I can find are interpolation models. Also, I would like to be able to compute the derivative of the regression. With a spline interpolation, I don't know how to compute the derivative over x so that it appears as a function of one-variable. For context, this is in order to build a tool for acid-base titration for chemistry. Thanks !


How to punctuate a quotation which is an exclamation, at the end of a sentence which is a question? The quotation is "Fire!" The sentence is: Did he really shout "Fire!"? OK, this is simply the shortest example I can think of to illustrate the question, the actual quotation of concern is longer, and so is the question sentence. But if it is improper to use both the exclamation point and the question mark, I'd like to know why. Searching has only resulted in lots of hits regarding whether to put a single mark inside or outside the quotation.


Suppose I have a principal bundle, and some group G acting on the principal bundle. Is it always possible to find a G-invariant connection on the principle bundle? If G is compact, then I can imagine one can find such a connection by picking any connection to start with and then 'averaging' this connection over the group G. But what is G is not compact? Does an invariant connection still exist, in general? If not, what conditions do I need on G?


I recently asked this question about whether or not profinite groups admit maximal subgroups: And indeed, profinite groups admit subgroups of finite index, so taking any minimum index subgroup containing the finite index subgroup gives a maximal subgroup of the profinite group. Now I wonder if it's true that all subgroups of a profinite group are contained in a maximal subgroup, i.e., can you have infinite chains of infinite index subgroups which do not split off to maximal subgroups periodically as you go up the chain?


(There are a lot of questions about entanglement shown in the Similar Questions box. If there's already something about what I'm about to ask, please refer me there.) Suppose the particle paths are arranged to be of different length (e.g., detector A is close to the emitter but detector B is many miles away). And suppose that both detectors provide an exact time stamp of the detection event. Upon reading the logs of the two detectors, would you find that both detections registered at the same time, indicating that one particle got "stuck" in detector A until the other particle was detected at B?


I learned that the possessive apostrophe -s is used for living beings and the "of" form for non living things: The dog's name is Sherlock The color of the flower is yellow However, I have noticed that these rules are often ignored in spoken language: The road's name is Privet Drive The name of the dog is Sherlock Are these mistakes made by non-native speakers or are these just theoretical grammatical rules that are not really strictly applied by native speakers in their spoken language as well? Does one stand out directly as a non-native speaker with these errors?


I'm wondering if 'calls have grown for X to be Y-ed' is an acceptable substitute for 'there have been growing calls for X to be Y-ed'? To clarify, these are examples of the latter from the first page of Google search results: 'There have been growing calls by some eco-activists and developing countries for powerhouse states to foot the bill for climate change.' or 'There have been growing calls for St David's Day to be made a bank holiday in Wales'


I know that loop-level three-photon vertex in QED is zero since the contribution from fermion and antifermion cancel each other. Also, from what I know this has something to do with gauge invariance of the QED. However, my question is "is this true even when the gauge invariance is broken?". If the gauge invariance is broken, I guess the photon will not only get a mass, but also there will be additional contribution from longitudinal mode of the photon which I am not sure whether this will change the story.


I need a verb that describes asking someone to stop helping you, even though you know they mean well, because it's not helpful. So you ask them in a way that is almost pleading. It's stronger than just asking, but it's not quite pleading. I'm doing an analysis of a play, and one aspect of this analysis is that I have to categorize, line by line, what each character is trying to do in one action verb. It is proving to be harder than I thought.


When a capacitor is charged, electrons flow from the battery terminal to the plate of the capacitor, as the electrons flow through the wire, they must be having some kinetic energy? So, when the electrons finally stop at the plates of the capacitor, what happens to their kinetic energy? We don't normally assume the kinetic energy to get stored in the capacitor, we only consider the energy stored in the capacitor to be due to the decrease in potential energy of the charge as it flows from one terminal of battery to the other, then where does the kinetic energy go?


In the case of elastic scattering or absorption between photons and electrons, the incident light is either fully reflected or fully absorbed and hence it either retains its wavelength or ceases to exist. But how does inelastic scattering between them work? Specifically, if light has a wavelength, how exactly does that wavelength change in such a way that the light transfers only some of its energy and hence its wavelength increases?


Let A and B be solenoids with the same pole facing each other. Then will their repulsion cause the set up to jump upwards? Please correct me if I am wrong: Solenoid A exerts a downward force on Solenoid B, which in turn exerts the same force on the ground. Then due to newton's third law, the ground exerts the same force and the whole thing is pushed upwards. If this force is greater than gravity, the set up will jump upwards. The connecting material makes sure the solenoid B is also carried upwards with Solenoid A.


I am trying to find the minimum energy point of a catenary curve. Can anyone please let me know whether there is a potential energy or energy formula for a catenary curve ? Especially I would like to know when the end points of a catenary changes it's position, I would like to know how the energy of a particular point on a catenary changes and where that particular point will reach in position.


I am confused in which category to put this sentence. It's a polite request, so it looks like it's an example of Imperative Sentence. But it does express the desire to have mango (even though not directly with the use of "I wish" or "I desire"), so feels like it may be Optative Sentence as well. And the listener of this sentence will most likely respond in "yes" or "no", that makes it a question.


We know that thermal radiation is one of the ways energy can be transferred from one object to another. And we know, objects at any temperature emit electromagnetic radiation. So does that mean molecules and atoms are constantly losing energy? And why isn't this considered in the first law of thermodynamics? How can we even calculate the amount of thermal energy converted into electromagnetic energy (radiation).


I searched a lot on the internet but I couldn't find any helpful answers, so here I am. I'm trying to create a chessboard figure, using xskak package that contains a label for each square. An example of what I'm trying to achieve is shown in the xskak tutorial on overleaf, linked here. I'll also add a screenshot of the image I'm referring to: Thanks!


I've heard three seemingly contradictory facts, so apparently I'm not understanding at least one of them correctly. "Proper time along a path is the time elapsed for a clock that travels along that path." "Proper time along a null path is zero." "There are no preferred inertial reference frames, so if I travel at the speed of light, my own clock that I take with me will tick like normal for me." So how can it tick like normal but then not tick at all? Which of these three statements is incorrect?


A jack of all trades is a master of none, but oftentimes better than a master of one is apparently accredited to William Shakespeare. Just to clarify - I mean the FULL quote, not just 'Jack of all trades' part. Is there a book, poem, or work by Shakespeare that proves he is the author? I have searched through the complete works of Shakespeare and couldn't find any phrase like this.


Monoids represent maps from some mathematical object to itself. Groups represent the automorphisms of some mathematical object. What do abelian groups represent? One unsatisfying answer would be that they just represent automorphisms of nice enough mathematical objects. However it seems rather uncommon for the automorphisms of some object to form an abelian group unless they have very few automorphisms. Furthermore, the theory of abelian groups is very different from the theory of groups in general, and abelian groups appear frequently in other applications where they are not the automorphisms of some object in an obvious way, like homology groups. This suggests that maybe its just a coincidence that abelian groups are groups, and that they represent something else. What do they represent?


Consider the following: "Saranghe" not only means "I love you," but it also means "Let's be together till we die." In Telugu, "Intiki veltunnava?" means "Are you going home?" What I'm wondering is whether I should capitalize the i in the transliteration of "intiki veltunnava? In other words, which one should it be? In Telugu, "intiki veltunnava?" means... In Telugu, "Intiki veltunnava?" means... If Telugu had a distinction between uppercase and lowercase letters, I could just use the equivalent version in the English transliteration, but Telugu is written in an abugida script, so that isn't really applicable. The transliteration introduces a new element that I can't figure out how to resolve.


The Novikov self-consistency principle prevents a paradox in which a billiard ball is sent to its past through a closed timelike curve (allowed under general relativity) such that it collides with its past self and prevents it from entering. Imagine a setup where the ball, as soon as it exits in the past, hits a switch that closes a gate across the entrance. What's to stop that from creating a paradox? There's no alternate trajectory for the ball; the switch can be a plate that covers the exit so that the ball can't avoid hitting it.


Momentum is the defined as the product of mass and velocity and can be thought as measuring how much motion something has. However, it is not clear to me why we need momentum and why force is not sufficient to explain nearly all the phenomena. Consider a larger mass moving at the same speed as an object of smaller mass. We say the object of larger mass is harder to stop, as it has more momentum. However, could we not just say the object of larger mass exerts more force, so it requires a greater force to stop it? The use of momentum is not entirely clear to me.


I got a mouse which has wireless charging capabilities a while back. I would like to take advantage of this. However, I currently have an extremely nice mousemat from grovemade I grabbed in a sale a while back https://grovemade.com/mouse-pads/ which I would like to keep using. My idea is simply place a wireless charging mousemat below it. However, the grovemade mousemat has an aluminium plate in it and so the wireless charing will induce eddy currents. If you know how strong these eddy currents will be and/or whether that will make wireless charging impossible. I would appreciate any help. Thanks, Pioneer


This post, currently closed, asks for the intuition behind Liouville's theorem, which states that every bounded entire function is constant; I find the answers unsatisfactory and hand-wavy. I am looking for a more focused, precise description of what is going on. Is there a geometrical understanding of what is going on? I try to visualize a non-constant analytic function, to understand why unboundedness is necessary along some direction, but it isn't working for me.


There has been some recent buzz around generalized/higher form/categorical(?) symmetries in the physics community. I understand Seiberg's papers are a popular resource, and I am aware of McGreevy's review on the subject in the context of condensed matter. Nonetheless, what are some other resources I might look at if I want to learn about the subject? Specifically, where might I find problems on the subject (e.g., Andrew Lucas's lecture notes include one fairly introductory problem on the topic)?


I'm reading Billingsley's Probability and measure. I'm reading about extremal distributions, and I'm at where he is proving that Gumbel is one of the extremal distributions. I'm having a hard time understanding the proof and I'll appreciate it if someone helps me figure out the proof. The text highlighted in yellow is what i don't understand (it is quite half of the proof). Thank you in advance.


I'm new to optimization and I'm reading a paper that they reformulate an ILP problem into QUBO using penalty method. Let's say we use a solver to solve above QUBO, and the global optima of QUBO obtained from the solver is infeasible to the original ILP. (infeasible from my understanding is the QUBO solution does not satisfy all constraints of the ILP). My dumb question is: Can the local optima of QUBO obtained from the solver still be feasible to the original ILP? I don't have much background in optimization, so it would be nice if you guys can suggest some materials mentioning my question. Thank you in advance!


If someone sets up a table at the edge of a conference room selling their wares, it could be described as a "stall" or maybe a "kiosk". But what if someone instead sets up something like a few chess boards that people can play on? The word "stall" doesn't really seem appropriate to describe this ("a chess stall"?) but is there any word that would be more appropriate? I thought of "sideshow" but that doesn't seem right either because it's not really a show, it's an interactive game activity.


When using "too much", it is put after the verb & before the uncountable noun e.g. I drink too much coffee. However, this sounds strange in the sentence: I use too much social media. Instead I use social media too much sounds more natural. Can anyone explain why this one doesn't fit the rule? So far it seems that it only happens with the verb use. (Another e.g. would be I use my phone too much).


The Gauss-Bonnet theorem for compact regular surfaces is often only enunciated for totally regular surfaces, with no singular points. But what if I wanted to state it on a surface with a finite number of singular points but otherwise regular, like a cube? How is the Gauss-Bonnet modified? I know somehow the curvature has to be concentrated on the vertices, but I'd like some help in making that notion precise. Thank you.


Assuming "phreak" is a portmanteau of "phone" and "freak," giving us the term "phreaking" to describe the illicit act of hacking telephone networks, and "Phishing" pays homage to it by misspelling "fishing" to describe the practice of tricking someone into revealing sensitive information on computer networks, now we have "Vishing" (voice call phishing), "Smishing" (SMS text message phishing), and "Qishing" (QR code phishing) being used in the security industry. Is there a better term (besides "annoying") than "jargon" to describe these words?


In one of his lectures, L Susskind stated that he cannot make sense of a metric with more than one timelike dimension. I also have trouble imagining it, but is there a good mathematical or physical reason why this is not possible? Let us assume all extra timelike dimensions are compactified, so we cannot directly observe them. What kind of trouble or unphysical problems would this create? I am not asking for non-standard physics here, but just a more formal reason than intuition as to why we assume the extra dimensions are necessarily spatial (I have a basic understanding of why string theory needs extra spatial dimensions, but I do not understand if it forbides extra time dimensions). The question is not limited to string theory though.


I have seen several questions regarding the size of the absolute smallest black hole, the smallest stable black hole and similar. These made me wonder; what is the smallest stable black hole if it is immersed in a standard atmosphere? I assume there is some maximum rate at which the black hole can consume matter, and that this is related to its size. Therefore, since its Hawking radiation is inversely related to its size, logically there should be an unstable equilibrium point when these two equal each other. I could, of course, be completely wrong - however I'm afraid my knowledge of how to calculate anything regarding black holes is limited to working out the Schwarzschild radius.


Imagine, there is a truck of mass 'm' accelerating with an acceleration 'a', this means that the truck definately carries some force 'F'(let). Now, imagine the truck collides with a man standing stationary on the middle of the road. So, the truck will hit the man with force 'F' and due to Newton's third law the man also exerts a force 'F' on the truck, then my question is that, why don't the truck's net force becomes zero, and why it continues accelerating? Kindly explain, where am I going wrong. Note- there is no external force other than 'F' acting on the system.


We know that nothing including light can escape the gravitational pull of black hole. Now special relativity says that nothing travels faster than speed of light. Then how can effects of gravity due to the black hole be felt outside the black hole as nothing can escape black hole? This seems contradictory as anything is pulled inside the black hole due to gravity whereas nothing can escape a black hole.


I've seen that the Leibniz formula of determinants can be reduced to the Laplace expansion through consolidating terms. What I haven't been able to find online is a connection between any of those and the geometric interpretation of determinants, which is the volume (or measure) contained between the parallelopiped formed from the column (or row) vectors. How do I prove that the Leibniz formula, which is based on terms composed of permutations has anything to do with geometric measures?


I have just learned that linear frame dragging exists in General Relativity. I have also seen simulations where a periodically accelerated and decelerated mass causes a sort of gravitational dipole radiation, dragging nearby masses in the direction of the acceleration. My problem with this is that in my whole life I was taught that the leading order in gravitational radiation is quadrupole; dipole radiation can't exist because of conservation of linear and angular momentum. Is dipole radiation forbidden only in linearized General Relativity? What is the full picture here / what am I missing?


I am trying to understand the jump operators in the Lindblad equation. Specifically, if there is any condition of boundedness we need to impose on them. I ask this because, as was pointed out in this post, there should be no assumption on the norm of the jump operators. But for example in this review, the jump operators live in the space of bounded operators. So is boundedness of the jump operators a necessary condition? In Preskill's notes, there is a derivation of the Lindbladian in terms of the operator-sum representation which gives some normalization condition that the jump-operator should fulfill in terms of the Kraus operators but I am not sure if this is relevant to my question.


I think my formulation in the title may be a bit confusing. I mean a behaviour similar to "gaslighting" yourself into seeing/perceiving your own opinion from someone/something, even though that person/thing doesn't actually share your opinion. I've witnessed this a few times on social media when people interpreted satirical videos of political topics to match their own opinions/believes, even though the videos themselves were neutral/not actually picking a side in the political topic. Would be cool if there were a word for that.


In textbooks on Category Theory, monoids pop-up all over the place, we several easy examples such as integers, lists and so on. I was then wondering about comonoids. What are some "simple" examples of comonoids appearing in mathematics? I've tried searching, but most texts either only mentions the concept of a comonoid, or provide some convoluted example. I'm trying to get a better feel for what a comonoid is, thus I'm searching for easy to grasp examples.


A few days ago was the first time I ran into the word "Nigerien" meant to mean "something related to the African country of Niger" as opposed to "Nigerian" meaning "something related to the African country of Nigeria". Since this is very easily confused, for example by my spelling checker which didn't like "Nigerien" at all, and especially in spoken English, is that use actually correct, or is there a better word? And how would you pronounce it so that it doesn't get confused? (With the countries themselves, I thought NIger has an emphasis on the first syllable, and NiGERia on the second, but for Nigerien this sounds just wrong to me).


If there is a circular metal loop and a changing magnetic field perpendicular to the loop, there is an induced electric field that causes the electrons to move and cause a current. But my question is why is there even a current? Why don't the electrons just move and rearrange themselves to cancel out the induced electric field, just as they do in case of electrostatic shielding?


I am looking to replace the idiom "kangaroo court" in the following sentence: Class followed its usual script. The professor took center stage, exposing the deep racism, sexism, colonialism and homophobia of a previous generation and like well-rehearsed actors, us students assumed our roles as moral arbiters in a kangaroo court. I want the replacement to communicate the idea that the verdict or judgement is a forgone conclusion and that the trial/classroom is just a pretence of debate/justice as everyone already knows what the answer/verdict must be before discussion ensues. If no good replacement can be found you could rewrite the sentence entirely to help communicate the above point. Thanks a bunch.


We know that a capacitor and an inductor both offer some opposition to the flow of current through an AC circuit and their reactance is a measure of the amount of Ohmic opposition due to these circuit components. I am aware of how these quantities arise mathematically. However, my question is, physically why do these circuit components provide a type of opposition to current flow in the case of AC circuits? What is the actual cause for this reactance to appear?


Suppose you have a faucet that expels water at a rate r Liters/second. Will the rate at which water flows through some ring beneath the faucet be greater than r or equal to r? On one hand, if the rate at which water flows through the ring were greater than the flow rate of the faucet, wouldn't that violate conservation of mass? On the other hand, shouldn't water be traveling faster through the lower hoop, so it should have a higher flow rate? I've asked some people about this, and I've been told that the water column will inevitably become thinner as it falls such that the flow rate through the hoop will be the same as r. Is this true?


I hope my question does not break the rules. I can't understand how this statement can be true in the book which I've mentioned: I cannot undestand why it's true because if I have two point charges with same sign and same magnitude then the electric field should be zero at point P: But according to gauss's law it's not zero and it equal to electric field due to one of these point charges. So this electric field is only caused by the charge inside the closed surface, not the charges outside it!


I am doing a lab experiment (Franck-Hertz) at uni and am supposed to find the temperature, at which electrons are emitted from a barium oxide cathode (without the necessity of an external E-field that provides additional energy). I am attempting to calculate the temperature T = W/kB from the work function W (i.e. the ionization energy). Is there any source, that contains such data? I tried NIST Chemistry WebBook, but they only have data for BaO in the gas phase. I need some value for solid BaO.


I am interested to see a derivation of Kepler's laws from Newtonian mechanics, which would be a suitable exercise or demonstration for a first-semester course in calculus. Therefore I'd like to avoid any use of the cross-product since this is usually only introduced much later in a calculus curriculum. I've looked at Newton's original derivations. They don't use any objects or theories which a first-semester calculus course couldn't use. But they do involve enough geometry that it seems "a bit much" for a first-semester calculus student. So I recognize that any such derivation will probably be hairy in one way or another. But in the off chance that there is some known elementary and simpler derivation, I wanted to ask. Thanks!


Given a container filled with identical objects, how could I find the number of objects in the container accurately purely by analyzing the sound produced on shaking? More specifically, which acoustic parameter should I focus on? Some ideas I have had so far are: Looking at the number of amplitude "spikes" since the more objects there are, the more collisions there are Using Fourier Transform to break down the combined sound into its constituent frequencies, finding the number of distinct frequencies. Unfortunately, I am not sure if the frequencies will be measured for each object, since the objects are identical


I have just switched to writing in LaTeX from markdown. I like to proofread by hand, and so in the past I used a template that increased the line spacing and changed the typeface to something monospaced just for proofreading. Now I cannot figure out how to change an entire document to being monospaced. I found several solutions that seem to only work on LuaLaTeX, but I would prefer to use pdfLaTeX.


Circle k and a line p that intersect at points R and Q are given. Inside the circle point A is given. Construct all the circles that touch the line p and the circle k and pass through the point A. I tried to solve this problem using homothety, but it didn't work out so I guess that's not the proper way to solve it. I would appreciate any ideas.


In the double-slit experiment, the interference pattern shows the probability of finding the particle (photon or electron) at a given volume. In a given experimental setting, this probability distribution doesn't change with time, so this probability distribution belongs to a stationary quantum state. I think that this doesn't mean that the Hamilton operator belonging to an experimental setting has only one eigenvalue and a single state that belongs to this eigenvalue because the pattern depends on the wavelength of the photon. If we used the superposition of two photons of different wavelengths, we would get a non-stationary interference pattern. Is there such a thing as "the superposition of two photons of different wavelengths"? Or the double-slit experiment can't produce non-stationary states?


Phonons are typically used to describe quantised vibrations in solids. However, is it legitimate to talk about phonons for e.g. a sound wave propagating in air? Contrary to photons that are particles that can exist both in free space as well as inside an optical cavity, it seems like phonons are quasiparticles defined only in a solid (i.e. a cavity for sound). Is this the case?


I've heard that because light takes time to travel from one place to another, we see objects in distant galaxies as they were when they released the light. new and advanced telescopes are able to see further and further into space, and therefore closer back in time to when the big bang happened. I've also heard that the laws of physics are slowly changing over time and that they might have been different in the early seconds of the big bang. In the future, could an advanced telescope be used to observe the light released by distant galaxies and other cosmic structures at the time of the big bang and observe physical laws that were different from the ones that we have in the present day?


what would happen if I took a toroidal core, generated a static toroidal magnetic field in the core and then rotated the core around it's symmetry axis within a larger stationary toroidal winding around the same core? Just for simplicity imagine a toroidal transformer with a rotating core inside. Would I get DC current generated in the stationary coil? If yes can anyone explain by what law and formula?


I haven't found satisfying answer yet. Beside those typical examples like how many colourings of a cube is there up to rotations or how many graphs are there on n vertices up to isomorphism etc., is there any "real" usage of Burnside's lemma in advanced math? All examples I could find seem to me more like funny riddles you can impress your friends with but they are not quite useful for anything.


If I have n cards and n stacks of cards, how many ways can I split the cards between the stacks if the order of the cards in the stack is significant but the order of the stacks is insignificant and stacks are allowed to be empty. And is there an algorithm to find them. So for example, with two cards (a, b) and two stacks the possible arrangements are: [ab][],[ba][], [a][b]. ([b][a] is equivalent to [a][b]) this might seem like a silly problem but I am very much stuck and very much desperate


Suppose I filled a bucket of water with a hose, now when it's filled, I remove the hose from the tap's end, while the other end is still submerged in water. Now if I throw the other end on the ground, it sucks all the water from inside the bucket spilling it out the ground till it's empty. Why does this happen? I assume there's something really simple about pressure and stuff going on which someone like me, who hasn't studied fluids in detail, can understand...


I am trying to first construct a fractional linear transformation, to map the sector to a half or full circle. From there it seems straightforward using rotations, translations, and standard mappings between circles to half-planes to create a map that takes the sector to the right-half plane. My question is, how do I construct the fractional linear transformation? Lang's Complex Analysis shows how to construction fractional linear transformations to map a triplet of points to another triplet of points, but I don't see how that would be useful here.


Good day all, I am looking for a way to estimate the composition of a thermal plasma formed by flowing gas between two charged electrodes. I understand that when plasma is formed this way, it would be composed of a number of species. Please is anyone able to point me in the right direction of finding an equation/correlation/data to allow me estimate the composition of the plasma for any gas? Thank you


I remember my high school teacher explicitly saying that it took me a long time to do x was the correct form and I took a long time to do x was not. In recent years, I seem to remember hearing the latter quite often in common language, though. How applicable is the latter phrase compared to the former (a) in everyday speech and (b) in essay writing?


I was discussing with some friends about English grammar, and we ended up confused about the accuracy of the sentences below "I know the motive for your rescheduling the class" "I know the motive for your rescheduling of the class" "I know the motive for you rescheduling the class" "I know your motive for rescheduling the class" Are all of them correct? Which one would be the most idiomatic?


When I was young, a workmate convinced me that there was indeed a word which described the behaviour of someone who would congratulate themselves for behaving the way a person normally ought to, anyway. I don't want to write that word here, because some people might think part of it seems racist. So instead, I'll ask as plainly as I can: Is there a word that means "Congratulating oneself or others, for behaving the way one ought to behave?"


Imagine a bicycle wheel level with the ground able to rotate around its center. Centered on one of the wheel "spokes" is a heavy marble which can slide from the center of the wheel to rim of the wheel and vice-versa. If the wheel is spun once very hard by hand, the marble will move from the center to the rim and return to the center as the wheel stops spinning. Question: Does the wheel spin longer with the heavy marble at the rim then if there was no marble on the spoke? I suspect it does spin longer. Please no formulas or highly technical discussion; a yes or no answer and a simple explanation would be appreciated. Thank you for sharing your knowledge of physics.


This is a screenshot from a Khan Academy video on friction at the nanoscale. I understand how the force of friction on the purple block acts opposite to the direction of the force applied explained by the nanoscale diagram, but I don't understand how the Newton third law pair is explained by the diagram. Wouldn't the movement of the top layer of atoms move the bottom layer as well? In that case wouldn't the two layers move in the same direction?


The motivation I was given for algebraic topology is to assign some algebraic objects as invariants to topological spaces. This way we can show that two spaces are not homeomorphic if they are assigned a different invariant. However it seems to me like in algebraic topology the invariants are always up to homotopy equivalence or maybe even only weak homotopy equivalence. This seems strange to me. Is there a reason why other kinds of algebraic invariants that can distinguish homotopy equivalent but not homeomorphic spaces not more widely studied? What is it about weak homotopy equivalence that is so special that seems to be the main focus of all algebraic topology?


In the book General Relativity by M.P Hoson, the author tries to calculate the velocity of a particle radially infalling in Schwarzschild spacetime. The velocity is measured by the observer sitting at rest at infinity. For the same, he takes the ratio of proper distance and proper time of the particle, both measured by the observer. To calculate the proper distance, the path traveled by the particle should be spacelike, whereas, to calculate the proper time, the path traveled should be timelike. Since the particle can travel with only one of the paths, how does the observer simultaneously calculate the proper time and proper distance along the particle geodesic


Provided everything remains constant, does the fringe separation, that is the distance between adjacent fringes become further apart for higher order maxima? Consider the above diagram. From the diagram it appears that the distance between adjacent maxima is getting larger with distance from the central maxima. My intuition agrees with this. However, I have heard that the distance between adjacent fringes is constant. Assume this is true. How so? Double slit interference pattern: Fringes are equally spaced and of equal widths. The above statement seems highly unintuitive. Shouldn't the spacing between adjacent fringes get larger, as the wave spreads out. How do I satisfy myself with the above statement? Proof?


The below question was asked in national English exam in Turkey: Choose the best word or expression to fill the spaces in the passage. As Henry Hill, the actor Ray Liotta gives a complex portrayal of an Irish-Italian kid who succeeds in his youthful ambition ....... popularity as a wealthy gangster, only to have his position slowly eroded until it almost destroys him. A) to gain B)gaining C) to have gained D)having gained E) to be gaining My choice was b (gaining). But the correct answer is A (to gain). I thought the phrase "succeed in " takes gerund not infinitive. Why am I wrong for the answer.


I'm studying X-Ray reflectivity at interfaces and the book I'm reading reads: ... The ideally flat, but graded interface, and the ideally sharp, but roughened interface, will be considered... I'm struggling to understand the difference between the two kinds of interfaces mentioned. Please consider that English is not my first language. Exactly speaking, what do they mean with graded and sharp? How do these two surfaces differ from each other?


Imagine a system with a parallel plate capacitor with holes in the middle where the plates are charged before being disconnected and insulated. Neutral atoms are introduced between the plates and are subsequently ionized in some way. The created ions will gain momentum and travel through the hole after which their kinetic energy can be collected. As long as the kinetic energy gained by the ions is higher than the ionization energy we just gained energy. I know that this should not be possible so where am I wrong?


Assuming we can't block neutrinos, and most of them pass through the earth, how do we know that the change in the neutrino detector is not just happened randomly? Is the detection of neutrinos more frequent when earth is closer to the sun, or are we just always seeing the same amount of neutrinos in the detectors because we have no way to block them?


I'm doing mathematical analysis and I'm stuck with the following question: Give an example of a nowhere dense closed set of irrational numbers with no isolated points (that is, a perfect set of irrational numbers) I read this: Is this a perfect set? so apparently we can find such a set through specific enumeration of rational numbers, taking their vicinities union and taking its complement, but I struggle to find such enumeration that would not create isolated points.


I came across this interview question online: Suppose you sit on the road side and observe cars driving by. Assume the distribution of cars driving by is according to an exponential distribution. Now you observe a first car after sitting for x mins, a second car after sitting for y min. Given those observations, can you estimate the parameter in the distribution? I don't see how I am able to solve it without additional information. Thanks


I begin a PhD in coding theory next fall, and it concerns coding over certain types of rings (e.g. finite chain rings) instead of fields, and in consequence submodule codes. I have a fair amount of knowledge in comm. algebra and have had a standard undergraduate coding theory over finite fields course, so I was wondering: Is there a well-respected graduate text, or a good set of notes, covering more advanced tools in algebraic coding theory? Thanks, and let me know if I need to clarify anything.


If I understand correctly, massive objects cause time dilation, and so time seems to pass more slowly for observers closer to a massive object than those who are farther away. Do cosmologists take the local gravitational effects of earth, the sun, and the Milky Way into account when calculating the expected speeds of stars orbiting distant galaxies? Do galaxies that are more distant rotate faster, by our observations, than those closer to us, even if comparing galaxies of similar mass?


I am an undergraduate, studying physics. I have studied maths courses like Groups, Linear Algebra, Real analysis, Differential geometry and probability. I wish to get into mathematical physics, similar to what Arnold's book has to teach. But i find it a really difficult read since it's a graduate text in mathematics. Can anyone suggest me what stuff do i need to study before getting a good intuitive picture of Arnold's book. Or any resources that provide a road map, intuitive picture of the maths used in physics.


Now I am studying elementary number theory, I am interested in arithmetic function, I have studied Burton's Number Theory but I can't find Dirichlet Convolution as a particular topic, I will be highly glad if anyone suggest some good book to study arithmetic function where I can find Dirichlet Convolution in details with some theorem like 'the set of all arithmetic function forms integral domain under pointwise addition and Dirichlet Convolution'. Thank you.


I am trying to understand the concept of regular conditional distribution. I found the definition quit difficult to grasp . This is what i understood . A regular conditional distribution also called a probability kernel is a two variable function (x,A) -> K(x,A) x a scalar and A a set that verifies the following. when we fix x, we have a probability measure, when we fix A we get a measurable function. Can someone confirm this definition ? and give me an example of a non regular conditional distribution .


The uncertainty principle holds for pairs of certain observables, such as position and momentum. All these observables have a relation to spacetime. Other particle properties, by contrast, such as mass or electric charge can be measured at arbitrary precision. Quantum theories do not even model them as observables, but as parameters to the field. So, what is the philosophical argument for why there is a uncertainty principle for some quantities, but not for others?


I am currently using a software called "SCOUT" to thin film interference simulation, and I would like to know if there is a better software to do this type of simulations. My goal is to vary the thickness of a thin film in a multilayer, in order to create interference between the various thin films, and then create a specific color. So I want to freely change the thickness and get a graph of reflectance or transmittance corresponding to that change. The software should allow me to put my own materials with the respective reflectance and transmittance properties. I know some softwares, like "zemax" and "lumerical" but there are no videos doing what I intend. Does anyone have suggestions? Thank you!


I've got a Literate Agda document, which contains a large number of unicode characters. It builds nicely with XeLaTeX and life is good. The problem is that arxiv only supports pdflatex, and I'd like to upload the file to arxiv. Are there any tools/scripts that can process a latex document written for xelatex and add the necessary commands to compile it with pdflatex? Things like declareUnicodeCharacter for each used character. Basically, I'm just looking for a better way than "start compiling with pdflatex, get an error, add the unicode character, repeat". Most of the unicode symbols are math symbols, so it shouldn't be too hard to map from the characters to TeX commands automatically.


Hi, I'm taking a course of Electrical and Computer Engineering in my university. I don't understand why the partial circuit on the right side of the switch is ignored after the switch is closed. As you can see, the book says the network becomes (c) from (a) via the short circuit. What happens to the short circuit and the circuit on the right side? I think the resistor and the voltage source can still affect the network but it seems not..


I have two non-orthonormal basis vectors, and I want to represent a third vector as a pair of coordinates using the aformentioned basis vectors. How would I do that? The dot product, which usually transforms a vector into coordinates in each basis vector, doesn't work for non-orthonormal bases. How would I get coordinates in terms of my non-orthonormal basis vectors? I should also mention that I am coding, and so all my vectors are currently represented as pairs of numbers on an (x, y) plane. Please ask if you need any clarification, I explained it really poorly here. Also, I do not know very much linear algebra, so it would be appreciated if you answer in simpler terms :)


If Alice is located just beyond the event horizon of a black hole, and Bob is at some other point in space with de minimis gravitiational effect, what, if any, effect would it have on the "spooky action at a distance" between these two entangled entities? I'm a wannabe sci-fi writer looking to exploit untested but possibly plausible concepts derived from my own madness and admittedly poor understanding of the universe.


I am struggling with prooving convergence for an optimizer which uses adaptive step-size with heavy ball algorithm for convex and non-convex functions. In some literature, I could find a regret bound analysis/proof for convex functions and proving that the estimated gradient at t -> inf goes to zero. Could anyone please guide me through the process of convergence proof for non-convex functions or give me literature recommendations for the same. Thank you very much in advance.


I need help understanding how particles do what they do and maintain the structures they maintain if so many of them exist for such a short time? In the case of the nucleus and pions, pions only exist for mere nanoseconds. So if they decay, and their properties change, what is holding that nucleus together? Are the nucleons continually exchanging a stream of pions that perpetuates the interaction that holds the nucleus together? Like I said... Explain like I'm five.


I really like the greek letters of the default LaTeX font but I want the rest of the characters to be in Times New Roman using the Mathptmx font. I find the greek letters of the mathptmx package rather ugly since they're disproportionately bolder than the usual text. So may I know how can I change the greek characters only and retain the times new roman text? I have tried many suggestions from countless websites but to no avail. Alternatively, can I ask if there's a way to change the greek letters somewhat more similar to the greek letters in the books like those in University Physics by Young and Freedman?


When observing interference fringes one often uses a lens after the diffraction grating to focus the fringes on a screen like this: At first sight, one might think that after the lens the path difference between the rays is altered. I think this is not the case, but don't see a clear argument why. Maybe one could apply Fermat's principle, but this - as far as I know - is restricted to the situation that we compare the path from a point A to a point B (in this case P). But here we start with parallel rays and not from a single point A. So, is it right, that it doesn't alter the optical path difference and how to explain exactly why?


I had a debate with a buddy about this. He said you could get a chord by drawing the triangle formed by the two points and the center of the sphere and that chord corresponds to a single great circle arc. I can get to the chord but then projecting that chord on to a path on the surface of the sphere seems non obvious. So is it true that two points not colinear with the spheres center on the surface of a sphere have a unique great circle connecting them and if so how does one prove it?


As the diagrams of categories can be represented as graphs of objects and morphisms, I was wondering if (the diagrams of) higher-dimensional categories could be represented as hypergraphs, and if not why? And in the more general sense, what would be the best west to represent higher-dimensional categories (thinking of diagrams, but also data structures to represent them in programming languages)? And if we reverse the question, are there classes of categories whose diagrams would be best encoded as hypergraphs?


We know a way of measuring energy of a electromagnetic wave is the Poynting vector, which is independent of the frequency. But let's say we want to make two different electromagnetic waves, with different wavelengths and so different frequency, but with the same amplitude (so same Poynting vector). In my opinion the guy in the image is spending more energy for the wave with higher frequency. But the Poynting vector, doesn't care. So is there another property of Electromagnetic Waves that accounts for the energy that the guy used to make the wave?