Take a deck of cards, cut it in three, and rearrange the three packs in any order (for instance, the middle goes on top, the top in the middle, and the bottom stays). Repeat as many times as you want with different (and possibly uneven) cuts. What permutations can you generate? Motivation: given a set of symmetric matrices, the trace of their product does not change if they are subject to a permutation generated in the way above.


I may be misunderstanding the general idea and the following specific example, but I was told that Dirac predicted the existence of the positron based on a formula for the motion of the electron(?) and then feeding in a negative value or imaginary value and somehow this suggested antimatter being a real thing. Sure enough the particle was soon discovered; perhaps the discoverers were guided by Dirac's work. But just why would anyone expect this? Could it not be that an equation works in many cases but not all? Was Dirac surprised by the discovery of the positron or would not finding such a particle have been more of a surprise?


If we consider the addition of orbital sngular momentum L and spin angular momentum S to produce the total angular momentum vector J, then J is a member of the vector space that is constructed by taking the tensor product of the vector spaces associated with L and S. Why don't we take direct sum of the vector spaces of L and S instead to construct the J-space? Why are we particularly using tensor product to combine the spaces?


From my textbook, A noun phrase is headed by a noun. Modifiers include articles, adjectives and demonstratives. Qualifiers include prepositional phrases and relative/adjectival clauses. Given the following: Your bag is in your locker What is the grammatical name? I know that its grammatical/syntactic function is the complement of the preposition . The problem is that I suspect it is a noun clause because "your" is neither an article, adjective nor a demonstrative. It is a determiner. Also, a noun clause can be perform every function that a noun phrase can, or so I think.


Recently I learned about Markov chains. Consider a simple two-state Markov chain: This looks like a categorical diagram, with states as objects, transition probabilities as morphisms/arrows, and of course, identities as the same state transition probabilities. Although I don't know much, I also came across connections between two separate Markov chains, implying a functor equivalence. I wonder if there is such treatment in literature. Is there any benefit in approaching it this way?


Today i studied about mirror formula, its derivation and sign convention used while deriving it, a question which i have is, why do i have to use sign convention while doing the derivation ? focal length, object and image distance all are distance so what is the point of using sign convention ? and while solving problems based on mirror formula why we use the sign convention again ? I have seen some answers on internet, they say that we use sign convention two times to cancel the effect of sign convention, but if want to cancel the effect of sign convention in the end then why do we use it, can't we solve without it ?


When thinking about a "random" atom in space (say a Hydrogen atom), should I assume that the electron is in a ground state (or any particular excited eigenstate) or is it in a superposition of all energy levels before I measure the energy of the atom? On the one hand, If the latter is the case that would mean that there is a nonzero probability to measure any arbitrarily large energy for the system, which seems weird. On the other hand, should not any quantum system be assumed to be in a superposition of all the eigenstates before making a measurement, then is there any mechanism by which the superposition of such atom is collapsed even before measurement?


I'm not sure where else to ask this since I'm not very familiar with this area. What is the backdrop of a theatre play act called? Like when the actors are behind the curtains and their silhouettes are instead flashed on the white screens or curtains. The play or act is all going on in the background just like a puppet play... What's it called? May I please know if this really is a thing or not?


Baum-Welch is an instantiation of the more general Expectation-Maximization (EM) algorithm. But recently when I learned about it I couldn't figure out where the Maximization step is used. Because the Baum-Welch algorithm give me a sense of just iterating between computing some intermediate variables according to old parameters and updating parameters according to computed intermediate variables. Out of the reason, I searched this problem online, but I only find this article useful to me. My problem is a step in the derivative process troubled me(it's shown in the figure below). Could you please explain it to me? I asked my professor and she couldn't help me. enter image description here


Consider a vessel completely filled with water and a vertically oriented capillary. If the water vessel was open at the top, you would expect capillary action to occur, lowering the water level in the vessel until a force equilibrium is reached. However, if we seal the vessel, the water would have to increase its volume/reduce its density for capillary action to take place, as no air can enter the vessel to fill up the volume. Does the (high) bulk modulus of water also apply here, as this expansion is nothing else than negative compression, i. e. would the hydrogen bridge bonds mostly suppress capillary action?


Order the following functions in terms of little oh notation: where: Im actually confused on how to do it. I think its the functions that confuse me not the method on how to do it. I know that: But either my mathematical skills are failing me or there's a different way to do it, that I don't know. Any help is appreciated. Thanks a lot in advance.


I just wanted to bring up some discussion about an apparently essential concept for some fields in mathematics as so as for some in physics, as already mentioned in the title, I'm referring to the word "symmetry". I'm currently studying about Lie symmetry method to search for first integrals of ODE's, and already saw that the main problem would be, exactly, to compute the symmetries, which involves a PDE. In adition, I constantly hear "symmetry breaking" from some working with QFT. It sounds like a joke but, are we trying to find those symmetries for you guys to "break'em"? Despite the kidding, could these two contexts be related in some degree?


In an infinite universe composed of single point masses which can be simplified as a uniformly distributed mass density, what is the equation for the gravitational field intensity in Newtonian gravity? I assume that: The gravitational field intensity is constant through space since the matter density is constant. The net force on any test particle must be zero, because all forces must cancel out due to symmetry. The gravitational field intensity is infinite since the its the sum the gravitational field intensity of each mass point, and there are endless mass points at infinite distance. But it must converge to a specific value since contributions of far away point masses contribute less and less with the square of distance.


Is there a word that means "lost thing"? Ir perhaps a word for "forgotten thing/ something that has been forgotten"? Similar to how requirement means "that which is required" or necessity is "something necessary"? I'm ideally looking for a noun or other words that are constructed to from an adjective describing something to the noun that stands for it. Here are two sample sentences: This book contains a list of major [lost things] throughout history. The [forgotten things] inspire many mythological stories. I am also okay if there's an archaic or uncommon word for this!


I don't know much about this topic, but I read something saying that gravity is not a force using an example of inertial observation. I started thinking about the topic again when I was researching photon spheres and thought why does gravity (if it is a force) curve light when photons have no mass. The only possible answer would be that gravity is not a force but rather the warp of space around mass. Would this be right?


I don't understand why you can't invert a hessian matrix using gauss jordan method. Can't you integrate or differentiate an entire row (because they are linear operators) and then subtract/add/swap? To clarify, I mean inverting the generic hessian formulation before it is applied to any function. In this way, I can get the hessian inverse so I can apply it to other functions. otherwise, I'd need to calculate hessian of a function and then invert, which takes a larger time complexity.


Can we calculate the total observed time dilation on a spaceship and a planet by combining the planet's relativistic spin velocity and the spaceship's approach or recession velocity from the planet it came from and where it's heading? Would the spaceship see both planets as time-dilated due to their combined relativistic spin and approach/recession velocities? Should we consider both the spin velocity and the apparent recession/approach velocity, or just one of them, in such calculations?


As I understand, it is perfectly possible to explain the standard double slit experiment with a laser beam using only classical physics. On the other hand, this experiment is one of those experiments which is mentioned the most when quantum theory is introduced. So at what point of the experiment -- or through which variation -- does quantum theory come in where classical physics fails ?


In Baby Rudin a subset E of a metric space X is defined to be connected if "E is not a union of two nonempty separated sets." This may be nit picking, but should this not be E is not a union of two or more seperated sets ? Wikipedia agrees. To be super nit picking, by Rudin's definition (and taking it super literally) a set that is the union of three nonempty separated sets would be connected since... two is not equal to three.


According to the argument in this post, a theory of Quantum Gravity should not be compatible with the notion of time evolution. This is also called "The Problem of time". However, the target spacetime formulation of string theory does have a notion of universal background time given by the target spacetime. So it seems like it does not address the problem of time. The CFT part of AdS/CFT is also a QFT with a well-defined universal time. So does String theory not address this problem? If no, then what does it say about it?


I heard this dialog in a Western film: Ruffian: Are you Josey Wales? Josey: That['d] be me. Methinks Josey responded That be me and not That'd be me. Is the former even correct? Is That be me. also a sentence in subjunctive mood, just as That'd be me is. If That be me is correct, what would be the difference in meaning between That'd be me and That be me? Please, take note that this question is not a duplicate.


The intended meaning is that 'it', described simply, is 'everything'. Would I write: It is in short...everything. Or: It is, in short...everthing. Instinctively, I believe that a comma is needed before "in short". But the confusion arises here: Is it a common enough phrase that the comma is not needed? Is it a publisher's preference situation? Author's preference? I've seen the phrase 'in short' both with and without commas. Normally two commas are used, as it's a parenthetical, but the ellipsis makes the second one unnecessary.


In one of our informal math research club sessions we are thinking of hosting a math meme making competition. Our idea is that math memes will trigger curiosity and interest among the participants. This would not only help in questioning and understanding the mathematical ideas behind the meme but the participants will also have fun doing it. Lets say that I go ahead with this idea. What could be a possible way achieve best results?


What is the meaning of the phrase, 'as thus construed'? I was writing an essay and felt that phrase was appropriate, but I looked it up with quotes on google and found only references in some legal documents and academic papers. Is there a similar phrase that I am thinking of, or does this make sense by itself? For example, suppose I just described x a certain way. I then say, "x, as thus construed, is Y." Does this make sense? Or is there a phrase that is more accurate here?


so I am looking for good text books that treat signed measures, integration thereof and total variation etc. In particular also for finitely additive signed measures. I have found Dunford and Schwartz a little bit too general in that they dont necessarily consider (signed) measures but more generally set functions. I would also like the book to be as standard in notation and the used concepts as possible. So can you give me any suggestions for text books?


I am in charge of deciding the name of a skill in a role-playing game that serves as a catch-all for abilities/skills that are otherwise unlisted in our handbook (think skills that will help players flesh-out their character's personality, quirks, and life). Such a skill could be a profession type, a craft (e.g.,whittling), or something more exotic like hyperlexia. Profession and Craft do not capture the spirit well enough. Ability seems too generic, as are its multi-word variants, e.g. Unlisted Ability. Knack seems closer to the mark but open to suggestions.


I am currently studying a paper on Wasserstein Fair Classification. Several places they mention the Wasserstein barycenter, weighted barycenter distribution or the Wasserstein barycenter distribution. What is it? How is it defined? I can only find very deep measure-theoretic explanations, which are both a bit out of my league as a statistician, who never enjoyed analysis very much, but they simply also lack an intuition behind.


I was wondering is it possible to try and do optimisation of an ellipsoid given the formulae for its volume and surface area? I would be trying to keep the volume constant whilst minimising SA. This is for one of my maths investigations, basically trying to minimise packaging of a kinder egg, so if this is somewhat of a dead end, what are some alternative approaches I can take forwards? Any help would be really appreciated! Update: Ive derived the equations for the SA and Volumes of an egg shape Would the Lagrangian Multiplier Method work here


I happened upon this video purportedly showing a coin on a horizontal table receding away from a lens would disappear from bottom up. The explanation via diffraction resolution limit is wrong. However, I would like to find the correct explanation of the phenomenon. When the camera zooms in and focuses on the coin, the coin appears clearly. This can be explain by the fact that part of the light from the coin still enters the aperture of the lens. However, how would one explain the abrupt disappearance of the bottom of the coin below the table upper edge when the focus of the lens is on the table edge?


My question is this: Why can't electric potential energy be defined at one point. Surely a point charge has some inherent electric potential energy cause by force applied by the source field. The applied force would cause the charge to accelerate. The acceleration approaches zero as you get farther from the field, so the velocity could also approach a known limit (depending on the function). That velocity can be used to calculate the maximum kinetic energy, which would equal the original electric potential energy imparted on the charge by the field. I know this is a classical treatment of the situation, but what's wrong with this line of thinking?


I have questions about the inner product and coordinate transformation matrix. For F inner product space V, consider vector a and b in V and its inner product <a,b> then the inner product is invariant about changing the Basis of V? And let B and B' be Basiss of V. Then can I think that the coordinate transformation matrix from B to B', denoted by R is a unitary matrix because it preserves the inner product <a,b>?


Electrons have an e-field, and a moving e-field causes a b-field. These fields are defined in such a way that it makes calculating things like forces much simpler. Is this just a model and there aren't actual 'waves' traversing around us, but rather mere forces caused by a movement of an electron? If the force caused by a charge wasn't viewed as a wave, would electromagnetism only be a model for calculating the effects of the charge, utilizing waves as a tool? If so, is it fine to propose that visible light is simply our brain's reaction to certain forces caused by moving electrons in a certain frequency?


I have always been told that if you press the brake pedal fully and fast it can make the wheels lock themselves, and that you shouldn't do so because it would take longer to stop the car than if you push the brake pedal progressively, why so? I can imagine that when done progressively the car won't start doing jumps due to the wheels block as if you would brake immediately but I can't see why.


The existence of dark energy leads negative pressure. Is the pressure can be anisotropic? What situation can cause anisotropy of pressure? Further, Is the pressure can be negative in some directions and be positive in other directions? Such as in spherical coordinates, the radial pressure is negative and the tangential pressure is positive. Will this happen when matter and Dark energy exist at the same time?


I want to determine the largest area ellipse inscribed in a rectangle, in order to write a code that can determine the lengths c and d given a and b (see image here). I know the obvious answer would be c and d are halves of a and b, but I want the derivation (and also the maximum area) so I can stop thinking about this, saddly I can't seem to figure this one out. I remember doing the opposite in school (rectangle inside ellipse) but can't do it properly now. Hope you can help me.


I'm quite close to release my first game, and I can't decide which is grammatically correct on my first user facing screen. My game will surely have some grammatical mistakes but I would appreciate it if the first screen would be more or less accurate ;P So the title wants to welcome the user into my game but I dunno which way I should go. Welcome to {Game Title} or Welcome in {Game Title} Any feedback is appreciated!


i would like to find a way to study physics online and get academic credentials in it and be able to contribute to the research and publish. I know some topics are experimental and require physical attendance, however there must some subfields in physics that can be studied online. Now there are many online masters in STEM fields like computer science from reputable universities. Is it the same for physics?


The Wikipedia page for locally constant function says that a locally constant function is constant on each connected component, but that the converse only holds if the space is locally connected. What would be an example of a function that is constant on each connected component but not locally constant? I first thought of the topologist's sine curve, which is not locally connected, but since this is a connected space, such a function would be constant on the entire space and therefore automatically locally constant.


I need to prove in graph theory that if L(G)=G then G is a combination if simple cycles. I already did it assuming G is connected, so feel free to use this fact as given. My only issue is that it may be possible that in a non-connected graph not every one of the connected components will satisfy L(G)=G. For example, if the Connected components in G are A,B,C it may be possible that L(A)=B, L(B)=C, L(C)=A.


Suppose there is constant diameter pipe where a liquid flows through it, and if pressure of a liquid is due to external force or its weight when it is constrained or semi constrained (according to the accepted answer on this question), how can someone visualize how does pressure drop occur due to frictional losses? Does the force on the liquid decrease somehow or does it become less constrained so it has lower pressure?


I am studying the spontaneous symmetry breaking in the mean-field Ising model and it's clear to me the necessity of taking first the thermodynamic limit and then the zero-field limit to see the phase transition, otherwise we only see zero magnetisation even under Tc. I was wondering if there was a phase space approach underlying this result and if the spontaneous symmetry breaking implied and ergodicity breaking as well, as I imagine that the ensemble average is anyway zero, but this is not what we observe physically. Can someone give me a hint?


I went through a quite a lot of posts here but did not find the information I wanted. I'm looking for a beginner / introductory (yet mathematically rigorous) book recommendation for differential geometry / topology required for General Relativity. I'm perfectly fine (in-fact I like) the so-called dry mathematics books (theorem-proof-theorem). What I need is - Very clear definition of Mathematical objects used in GR Crisp / rigorous proof in modern notation I need the book at advanced undergraduate or beginner graduate level. Thanks in advance.


Is there any way one can modify a known functional so as to get a solution for another quantity related to it? Because this is too general I would like to make it more clear with an example. Given the classical EM Lagrangian, we acquire the equations of motion for a particle in an EM field. Starting from this, is there anyway one can find a functional that when varied will give the solution for, let us say, the electric potential, for a given charge distribution and known boundary conditions?


To understand the attractive and repulsive force between two currents, I am looking at two electrons moving parallel and antiparallel to each other. How can an attractive (or less repulsive) Lorentz force between two charges and its directional dependence be explained? I would like to understand this in terms of dynamical effects of the electric field and not in terms of a magnetic field which is the common approach for correctly treating and not further looking at such details.


Pramod Achar's book on perverse sheaves has a quick reference in the back that looks like this. How can I achieve something similar? The important things to me are: the general spacing (although maybe I would enjoy less margins on the outside), the two columns, the numbering and labeling, the horizontal line separators. For language, I only care about English. I don't have any font preference.


Consider this fact that we all know from school mathematics: There is a one to one correspondence between real numbers and points of a line. But the problem is I have never seen a rigorous proof of this fact. This is what Apostol writes. And similar is the case with other modern analysis books. So my question is, where can I find the proof of this one to one correspondence. I don't care if geometry is in Hilbert's axioms or Tarski's or anyone else all I care about is correspondence.


I'm bit confused about the current flow in capacitors. My question is, let's assume those two electric circuits: As shown in the picture, the only difference is only the capacitance. I'm wondering about the current flow magnitude. In which of the circuits the current flow will be bigger? or will it be the same? To start, I assume both plates of the capacitor are uncharged. My guess is that right at the moment of closing the switch before the capacitors are charged from the voltage source, the current flow will be even. Once they are fully charged, of course, the current will be bigger in the system with the bigger capacitor. But till this point, what will be the current flow? thanks


it is said that the time period of a simple pendulum in space is infinite. it is because the gravity in space is zero and anything divided by zero is equal to infinity.how can time period be infinite? also, in space, everything just floats around. how can a pendulum in space do oscillations if there is no restoring force to bring the mass back to mean position. how can a simple pendulum exhibit SHM in space? if it cannot exhibit SHM , how is the time period formula for the simple pendulum applicable to the pendulum in space and what is the meaning of infinite time period. why do the time period of pendulum in space even exists?


I was wondering if it is posssible to write my own environment for writing redox equations in form of: O: ... R: ... ___________________________ S: ... O stands for oxidation, R for reduction and S for the summary equation. Also, there is line that should separate O and R from S. I have never written any environment in LaTeX, so could anyone just help me start? I've seen that there are some packages that are about chemistry, but none help me about this problem.


In surface integrals the process is to use the dot product of the vector field with the unit normal vector first and multiply that with the area element. In the line integral the process is to use the dot product of the vector field with the length element. Why isn't the unit tangent vector used in the line integral computations while in the surface integral computations the unit normal vector is used?


While deriving the Planck's radiation formula, why do we use MB statistics when we calculate the average energy of oscillators? Shouldn't we use BE? Is this because temperatures concerned are very high and quantum stat goes to classical at high temperatures? But again, in a similar calculation, while calculating the lattice specific heat of the solid using Einstein's model or the Debye model, I have seen books calculate the average energy of oscillators using MB distribution only, but we use this formula to predict specific heats at very low temperatures also. So I don't understand, what is the reason?


The sentence is as follows: "Isn't it evident who/whom they have in mind for the position?" Depending on how you try to reword it to figure it out, the answer seems different. --Is it not evident that he is the one they have in mind for the position? (This seems to be a case for "who" here.) --Is it not evident that they have him in mind for the new position? (This seems to be a case for "whom" here.) Should it be who or whom? Why?


I know multiple singular subjects have been addressed here and singular compound subjects here. However, I came across a sentence with multiple subjects, one single paired with one plural. I could not find this question addressed. For example Does a table and four chairs sound like sufficient furniture for a breakfast room? That doesn't quite sound right to me, but then Do a table and four chairs sound like sufficient furniture for a breakfast room? also sounds a bit off. Which would be correct? Or are both incorrect? Purdue uni subject-verb agreement is confusing.


An intuition that I thought of about a year ago, but my level does not allow me to prove it. I think it needs advanced tools to prove it, or just a counterexample to refute it. Who can help, please, thank you. The locus of the point of intersection of the perpendicular tangents of a convex curve is the convex curve Of course, the intuition is only in this direction and does not constitute an obligation, as the cardioid is a concave curve. Despite this, tracing the perpendicular tangents to it gives a circle


Using Hugyen's principle and wave nature of light, refraction, diffraction are relatively simple to explain but I have been visualing a mental model for a couple of days where the photons on reach the boundary of two media, experience a difference in the structure of the medium of propagation and hence some sort of collision or change in the overall velocity vector of each photon occurs which leads to refraction. Can someone help me understand these phenomenon using particle nature?


how one would know that the given frame of reference will be inertial or non inertial, without the use of 'accelerometers'. earlier physicst has defined this concept and I am sure that they define it without the help of accelerometers. I just want to know about the logic behind the whole concept. And I will be not getting any accelerometers in my exam and not able to conduct any experiment in the frame. i want to know how teachers just say that this frame will be inertial or not without any accelerometer.


How come there is no loss due to coupling between the superconducting channel and the single particle channel? I imaging the two channels being two parallel wires, so the magnetic field from the current running in the superconducting state, should fully inclose the single particle channel generating a electric field, that at least would create a displacement current and introduce a loss. Why is that not so? Also is there no current fluctuation due to the current being carried by discrete charge quanta? I would assume that, that would also create a electric field fluctuation in the single particle channel causing a loss?


I posted about monoid coproduct to try to understand the correct way for constructing them. In the process and from the answer I got, I am further confused whether the product and coproduct of plain monoids compare free monoids, what are the subtle differences in terms of what they look like. I understand in both coproduct of monoids and free monoids, the process of word reduction to strings are applied. I don't know anything beyond that. I am hoping someone can explain to me their subtle differences. Thank you in advance.


If we add a constant to both sides of an equation, we will get a new equation with identical solutions. This is mostly true for multiplication and square roots, but there are some minor complications/exceptions. Taking the derivative of both sides will result in entirely different solutions. Is there a formal way to show how performing some operation/action on both sides of an equation will affect the solutions of the resulting equation?


This is a question about grammar, in particular about a specific language syntax. In the german langage it's legit to write negative prefixes in braces like "Die JSON (De-)Serialisation wurde implementiert." Is this valid in english? "The JSON (de-)serialization has been implemented." As a software developer I need to document the work I've done and the software functionality itself. I often have a lot of language constructs similar to the one above. And if it's legit to use this syntax of negative prefixes it would save me a lot of time.


I wonder whether there is a space filling curve that looks like the following if the width of the domain has an odd number of cells and looks like the following if the width of the domain has an even number of cells. I would like to fill a three dimensional grid none of whose sides (lengths) are guaranteed to be a power of two or even.


How is it paradoxal that a set of all sets exists in set theory? Russel's paradox is about the set of all sets that do not contain themselves cannot exist, that I understand. But what about the set of all sets, in a unrestricted manner? Is it the fact that it contains itself (which may be what allows to define the paradoxal set of Russel's paradox) that is problematic?


Is the result of a contour integral around a closed loop path unique? I'd think it'd be similar to a definite integral, which is unique. This came up regarding the proof of uniqueness of a Laurent Series - if the contour integral is unique, then the definition of the Laurent Series can be used to prove its uniqueness. Edit - as pointed out below, the Laurent Series aspect may be a limiting case, as it's within a specified annulus, so for any of the Cauchy coefficients, the same analytic annulus is used, so the same number of poles will always be within the inner radius.


Is it true that if a curve is increasing, Euler's method will always underestimate an actual solution? So if a curve is either increasing and concave down, or increasing and concave up, we can simply ignore concavity and justify the answer based on the fact that it's increasing? Euler's method will therefore underestimate whenever the function is concave down and increasing, or concave up and increasing?


There are two machine learning models, A and B. Model A classifies samples into 'O' and 'G'. Model B classifies the samples determined to be 'O' into 'S' and 'C'. Each model shows the probabilties of classification. Input samples pass through model A first, and if the result is O, then they enter model B. (If not they don't) In this case, how could I get the probabilities of classifications, G, S or C?


What is a good, up-to-date, well-regarded text on writing physics solvers for (e.g.) game engines or realtime simulations? In particular, treatment of numerical stability, convergence, asymptotic complexity and performance, contact forces, and the handling of constraints are relevant to me. I have solid familiarity of university level physics, calculus, PDEs and ODEs, and linear algebra; and these can be assumed as prerequisites. (An ideal book would be the analogue of Pharr and Humphreys, but for simulation) What text(s) should I consult?


In a Euronews headline I saw the following: Frontex: What would happen if the EU border agency quit Greece?. Shouldn't it be "quits Greece"? As far as I understood EU border agency is singular. Reopen note: Although the original close vote as a duplicate was accepted, Stuart F has since pointed out that the real issue here is that quit is probably past tense. I agree. The 'duplicate'tag here might wrongly lead readers to assume that the issue here is simply about singular/plural subjects.


When I'm not writing a math paper, I write way faster by typing directly into a word processor. I form and revise sentences faster on the screen then I can on paper. But typesetting math is slower than handwriting, and I think better with a pen. I already know it's best to handwrite symbol-heavy text before typing, but what about text that's symbol-light, like an abstract or introduction? Should I write almost the entire paper by hand, or only the math dense portions?


For no reason other than my own curiosity, I have been trying to figure out the solution to the following question: Given a bag of x marbles, what is the probability of having seen every different marble at least once after y draws with replacement? I am interested in making different graphs for specific values of x just to know what it would look like but I can only figure it out when x=y. The moment y>x, I do not know how to approach it. I have scribbled on more pages than is reasonable, so I hope I can find an answer here. Thanks!


We could say it was painted beautifully to convey that the way it was painted was beautiful. "beautifully" being the adverbal form of beautiful (an adjective). What if the adjective is "lovely": How would you convey that the way it was painted was lovely? Is it: it was painted lovelyly This sounds/feels weird and I have never heard/seen it used. Furtherm "it was painted lovely" seems an error of grammar. Is there a rule/guideline that adjectives ending in "ly" have no adverbal form and can't be used as adverbs? Or should we just use the adjectival form for the adverb even if it is grammatically incorrect?


I'm studying instability in Stellar Dynamics (in particular globular clusters instabilities). I have not understood clearly the definition of these concept. Dynamical instability is the study of the instability considering an adiabatic process: we perturb the system and see what happens. Thermal instability happens at much larger timescale than the dynamical one and is due to the exchange of energy between single stars (and since the thermal capacity of the system is negative, this process brings phenomena like the gravothermal catastrophe). Am I missing something?


I want to make a robot that solves a hand-made maze, so that I want to provide it a LIDAR component to measure the distance to read the environment. Before buying a component I want to know if it's capable of measuring distance to a rotated surface. Under rotated surface I mean a wall that is not perfectly aligned with the LIDAR emitter, see the image Is it even possible that the light beam ever reach the LIDAR's receiver and how accurate the measurement can be?


I have read everywhere that the gold foil experiment performed by Rutherford was done hoping that the alpha particles face only a little deflection/no deflection however this doesn't sit right with me, the old model of Thomson clearly shows that atom is a positive sphere with electrons embedded in it, what made him think that positively charged alpha particles moving through this atom wont face a repulsive force from the positively charged cloud?


What exactly would happen if we sent a frequency that was not one of the harmonics into a closed and open pipe? What would happen if we gradually increased this frequency? From my understanding, I think if we were to send down this frequency, a standing wave could not form, so when we gradually increase this frequency, as soon as it matches the harmonic a standing wave is produced and max amplitude creates a loud sound. Would this be correct?


In principle, the comparative form of polysyllabic adjectives has "more" before the word, rather than taking a suffix "-er". But why do adjectives ending in "-y" often take the suffix "-er"? For example, "luckier". (and the same for superlative) Regardless of which from "more lucky" and "luckier" is considered correct, many people use the latter anyway, and it also applies to other words that end with "-y". So why are polysyllabic words ending in "-y" an exception?


In first-order logic, one has a way of constructing all sentences out of some logical symbols, as well as function and relation symbols pertaining to a special kind of signature one is in. Now in the book of Mac Lane on category theory, there is a similar construction, where the atomic statements (instead of only being things like equality over a single structure) involve also things like the domain of an arrow, where there are both objects and arrows, for instance. I would like to ask whether there is some widely used general construction that generalises both.


What do you call it when someone is being back-handed in a way that looks and sounds nice outrightly but is intended to be mean? Passive-aggressive is too obvious. The equivalent to Bless her heart. The rudeness would be hard to tell to anyone who wasn't there to see or wasn't aware of the person's dislike for someone. Typically the person believes it goes unnoticed by the "victim."


A large regular hexagon is cut out of a triangular grid and tiled with diamonds (pairs of triangles glued together along an edge). Diamonds come in three varieties, depending on orientation; prove that precisely the same number of each variety must appear in the tiling. This puzzle is from Gurmeet.net's Tiling with Calissons puzzle page. The "Proof without Words" solution is accompanied by this image: My approach: I tried reading through the solution but it simply blows my mind. The proof without words, how did they even come up with that? Any clue how to start/proceed will help.


I have come across them in a thrift shop and completely fell in love with the method employed by her. Almost nothing is taken for granted and everything from the definition of subtraction to the addition propert of equality is first demonstrated (or proved) before being employed. I have picked up Serge Lang's Basic Mathematics with the hope of reviewing Dolciani's exposition of high school algebra through someone else's words but the book failed me as it doesn't go deeper than your usual half-baked modern algebra book. I don't want to move on to reading the first chapters of some abstract algebra books as I don't know the notation much. Do you guys know of any other book similar to Dolciani's in method of exposition?


On high slopes and rugged terrains, riding a bicycle while standing on the pedals is easier. Even though I cannot physically define what is "easy"; since it is a feeling that my body generates based on the ratio of work output to the energy expenditure and degree of tiredness of my muscles, I am pretty sure a humanoid robot moving exactly like us would also spend much less energy standing and cycling. Why is it so?


Is there a word for people you like but are a pain in the ass? Nothing too vulgar. I am not looking to call someone it, but affectionately think about them "that person is a complete [word]". It's for a friend (of the opposite sex, though I don't necessarily want to signal that here). oh God Debbie, can you stop being a [word] I was thinking of 'frienemy' but that may be the opposite. I was also thinking of 'difficult' but it's potentially offensive and patronizing.


I am studying trigonometry and have wondered if there are generalized power-reduction formulae for sin^n(x) and cos^n(x) yet the best that I found was formulae for odd and even powers on Wikipedia without any proof. Could you write a formula for sin^n(x) or cos^n(x)(if there are any) and a proof if it is possible? I will be thankful! (Also sorry for the wrong tags ) : . I just could not find five accurate tags for the question, my excuses.)


Does spacial coherence mean that if I took a cross section of the laser beam beam, then I can find a relation between the phase of the electric field between any two points across the cross section even after the elapse of a very long time? Whereas the highly temporal coherence means that I can predict the phase of the electric field at any point across the cross section even after let's say a big amount of time?


I have three scenarios: The following setup will experience an induced EMF since the area covered increases with time. (Shaded in orange) If instead of a wire/rod,we had a coil moving within the field, there would be no induced EMF for a constant magnetic flux density since area covered by coil is fixed and does not vary with time. The question is if I have something similar to the first scenario (in that a rod is used) but without borders, such that entire rod lies within the field (like the second scenario), would I get an induced EMF? If yes, what would be the change in area?


Essentially what the title asks-- are matrix models, such as BFSS, believed to capture in any way the large possible space of false string vacua, for instance as saddles in the action with nonminimal energy, in such a way that the dynamics of settling into a false vacuum/metastable state could in principle be probed with such models? This would be from the perspective of answering questions about the "cosmological prior" that people sometimes gesture to in the context of anthropic arguments for the values of certain couplings.


Let's take a hypothetical an experimental scenario where a stream of high-energy photons from a powerful source is directed via a system of mirrors and lenses in a two coherent rays to a vacuum camera. The photons collide in the vacuum camera, producing positron-electron pairs. The pairs are promptly separated before the annihilation by two coils, which lead into two vacuum tubes, that serve as entrances to electromagnetic toroidal containment units that "speed up" the particles, and prevent them from touching the walls of the units. What kind of problems would prevent such a setup from working as intended?


I'm studying method of undetermined coefficients and i can understant what is its idea. But i'm ambiguous when we can use it. Of course, it is obvious that the differential equation is nonhomogeneous. And wikipidia says as following: Undetermined coefficients is not as general a method as variation of parameters, since it only works for differential equations that follow certain forms. But my book or wikipidia don't say the equation need to be linear or non linear To apply the method of undetermined coefficients, the differential equation have to be linear? or non-linear or both?


So, I happened to be reading a grammar book in which I came across this weird looking sentence. This is the car of which parts are not available now. I think it should have been something like this: This is the car, parts of which are not available now. If the improvised version is correct, I do not know why. Can you guys help me with such constructions?


I need some advice on the use of these two words - update and refresh. I am currently working on the desktop app and I need to give a button a correct name. There is a big number of objects that an app stores in the database and multiple users can work with them simultaneously. If one user changes an object, it gets changed for all the other users. And this button can disable or enable this function. I consider two names. Data Auto Refresh and Data Auto Update. Which sounds more native? And is there any significant difference between these words?


I have a theorem in the form : if A is verified and B is verified and C is verified then D is verified. I'd like to show that we can't do without hypothesis C. How can I do this? Do I need to find an example where I have A, B and non(C) which are verified such that D is unverified? If I do this, what will I have shown?


I am not well-versed in the research on topological solitons but am interested to make a good sense of its implication. The highly interesting point in this new talk by Nick Manton was where he is noting that multiple vacua can be modeled using solitons. But are these multiple vacua simply degenerate in energy? Or we can consider non-degenerate multiple vacua in different regions of space?


I am reading Hiroshi Kunita's lecture Stochastic Differential Equations Based on Levy Processes and Stochastic Flows of Diffeomorphisms.It proves Ito's formula for cadlag semimartingale. Kunita's proof The proof is concise, but I doubt that it is not correct. Right in this picture, it constructs a function r(x,y)(in the first highlight), and claims that it suffices the identity(in the second highlight), but I just found the identidy is not valid since we can not get F by the integraton of the second order derivative F'' (this will only get F' instead of F). So I think this is proof is incorrect. Did I make some mistake?


In modern day PV cells you usually have a silver-based grid on the upper end of the cell where it's exposed to the sun creating the optimization game of how thin/thick you can make your busbars and fingers so that maximum current can be passed without covering too much of the silicone surface blocking the sun's exposure to the n-type surface. In the early silicon cells they used to design the cells in the following way: Source: https://www.pveducation.org/pvcdrom/manufacturing-si-cells/early-silicon-cells Why did they stop with this approach? Was it just fabrication costs? This seems more intuitive as you're maximizing the surface area of the layer exposed to the sun?


Following this post, I did this: [Options] --> [Adv. Editor] --> [Special options] panel --> [Go to error when displaying log] which however doesn't resolve my issue; after compiling, the cursor is always stuck at the same place. I'm suspecting some part of my code is colliding with this functionality, but cannot figure which. Edit: I tried unmarking the checkbox for "Go to error when displaying log" and it is now working! I thought it should be marked, but it seems not.


A recent paper "Gravitational Pair Production and Black Hole Evaporation" (discussed in short here) says that any spacetime curvature would produce Hawking radiation, no need for event horizon. If this is correct then a question arises - how negative energy would take energy from observable object? Fermion cannot be partially annihilated and full annihilation is way more energetic then Hawking photon. So the only source of energy would be object's heat. But would that mean that object with temperature near absolute zero would stop radiating? There is nothing about temperature in equation. Does it mean that the paper is wrong? Or is there any mechanism to provide Hawking energy from object's mass? P.S. "Black holes are white hot" Hawkins:


Here are two sentences that sort of mean the same thing, but need your help in identifying which one would be more appropriate. I do already have an idea as to which phrase would be more appropriate but I still chose to go with the other one. His mind pushed back, but he kept willing it to do all that it still could. His mind pushed back, but he kept wanting it to do all that it still could. Does using 'wanting' instead of 'willing' drastically change the meaning here? Since will is more of a statement of commitment, 'willing' might seem more appropriate here. But in some cases using want in its place doesn't change the meaning.


Inspired by Are the visible matter and dark matter separately segregated? I wonder if the constituents (I don't say particles, as we don't know) have actions and forces among themselves, that we cannot detect directly from our observable matter. Might it be that there is a mirror world with the same forces as in our observable world, but just within themselves? Maybe even more than one, given that the dark matter has more mass than the observable one. As of now, we can "observe" the dark matter only by interpreting gravitational effects. If there should be any internal forces in dark matter, there should be a difference in clustering. Unfortunately, this is hard to detect. Is there any theory about that, or experimentation going on, or even results?


Imagine there is no EM or Nuclear Force, and Gravity is the only force. Can gravity alone produce an atomic structure similar to a basic atom? In this scenario, the nucleus is just a single particle. The electron is another particle with only mass and angular momentum, like a planet orbiting a star. How feasible is it for Gravity alone to produce such a structure and hold it together, with energy and mass equivelent to those of simple hydrogen nuclei, electrons and matching speed/energy of orbits and angular momentum


Let's assume we have two charged free particles with the same energy coming toward each other. What will be the time-dependent perturbation in this case? More precisely, I am asking what will be the change in Hamiltonian due to which perturbation is happening? Is it the Columbian potential only which will change? If yes, then how will it change as a function of time? Am I missing something very basic here? To provide more context to this question, I am trying to figure out how the Hamiltonian changes in a muon production process through electron-positron scattering. That is The form of scattering amplitude M .


The matrix is not necessarily symmetric. The highest eigenvalue is positive and real (per Frobenius). Assume there is another eigenvalue with real part less than or equal to highest eigenvalue, real part positive, and higher than any other real part of any other eigenvalues. By "semidefinite" eigenvector I mean that the signs of the real part of all non-zero components are equal, since an eigenvector is invariant upon multiplication by a scalar.


"Export settings" disambiguation and meaning. The aforementioned quotation is ambiguous. As I see it, it could mean either of the following: Export (verb) settings ---> Action of exporting the settings Export (adjectival noun) settings ---> Settings of export Is there a way to disambiguate "export settings" or is the English language, in such cases, inherently ambiguous and the actual meaning has to be inferred from context?


The sentence: This essay will analyse this topic from both points of view and express my opinion. I think might be grammatically incorrect because it combines two separate actions while the coordination sounds ambiguous to me. Alternatively, are these the more correction versions? This essay will analyze this topic from both points of view and then express my opinion. This essay will analyze this topic from both points of view, offering an analysis, and conclude with my opinion.


In the case of a particle system the wave function or state vector is in a superposition of possibilities before measurement. But does this rule apply in the case of vacuum state? Is it in a superposition of (energy) possibilities before measurement? We might say that nothing is there to measure in the vacuum. In particular virtual particles in the vacuum are non observable by definition. But with gravity, the effect of vacuum on curvature, can be measured and we can get information about it. Does "before measurement" make sense here at all?


In the book: Logical Foundations of Proof Complexity (S.Cook), we have a proof for the Derivational Soundness and Completeness of LK assuming that the underlying vocabulary does not contain =. In this proof is said that the soundness direction holds also when he underlying vocabulary contains =. Theorem in the mentioned book My question is, why the soundness holds with = and completeness does not?


Is there a single word to describe all the elements in a list, except the last item? The head of a list refers to the first item in the list and the tail refers to the last item. In mathematics and computer science, the tail can also refer to all the elements after the first; I'm not sure if that carries over into other domains. (chatGPT suggested "penultimates", which made me chuckle)


I Would like to know a place (book, paper) that discuss thorougly the problem of degenerate elements in singular homology. I know that they are not a problem with simplices but you need to explicity deal with them in cubic homology (Massey for instance) I have read an terse old paper by Tucker (degenerate cycles bound) that address the problem in simplicial homology but I would like further details. In fact I am puzzled because all the standard texts tiptoe this matter (Hatcher, Munkres, Rotman, Vick). Thank you very much. To further clarify Is there any result that allows to forget degeneracy at once or should we check that is not a problem in any theorem (Hurewicz for instance


Considering, the phenomenon of surface tension, is the interface of the liquid more dense than the bulk phase? Considering, if it were more dense, the phenomenon of surface tension would be explained, more dense, would result in greater strength at the surface. However, i've found various answers that state the interface is of higher density, while answers that state the interface is of lower density. Which one is the definite correct one?


So imagine this situation. An object is experiencing two forces, one due to gravity in downward direction and other applied by us in upward direction such that our force is less than gravitational force. So the object will accelerate downward and its potential energy will decrease. Will the change in potential energy equals to negative of the work done by gravity? And If so then how, isn't it should be less than that because of our force, but then the change in potential energy will be negative of the work done by resultant force. Thanks if you answer it.


When using Set heading in the margin to the left of text this method to place paragraph headings in the margin, this works for single-column documents. But when the twocolumn option of geometry is used (in report class), the second column's heading overlaps into the first column's text. Is there a way to have the left column's headings in the left margin, and right column's headings in the right margin?


Is there a Hilbert curve for every space? as I understanding it in layman terms, it's a way to order a space sequentially which visits every place once. Does it work with continuous spaces, or just discrete? Are there certain limitations of dimensionality that you cannot use a Hilbert curve to explore? I guess I'm asking if everything can be serialized? What must I understand to understand the boundary of the Hilbert Curve concept?


Why after Big Bang it is not going to Heat Death in the easiest path, actually the energy can get distributed uniformly to all directions. Instead we see the energy in Big Bang is being converted to atoms, then the stars, galaxy etc. are formed. Isn't these formations making the path to Heat Death more distant? Any natural law explain these formation of concentrated energy structures like atoms, stars, galaxy etc.


As we know, a star that is too massive to become a white dwarf becomes either a neutron star or if it is even more massive, a black hole: neutron stars form with great gravity that comes with great mass, the gravity pushing down on its core, merging protons and electrons into neutrons making the core consist only of neutrons. Take this into consideration, is it possible that the same thing happens to stars that become black holes, since their mass (and thus gravity) is so large that even light can't escape?


The interaction (of an electron with an external electromagnetic field) is because of the electron's charge, but charge causes an electric field, so, by Syllogism one can say the interaction is because of the electron's electric field. Now we can say that, by linearity, the external field and the field of the electron sums together. So how does a force arise from this summing process?


We know for a fact that if an electron absorbs a photon with enough energy it gets excited to a higher energy level, which increases that electron's total energy and we know that energy can present itself either with rest mass energy, kinetic energy and potential energy. And we know from Bohr equations that the higher the r(radius) is lower the linear velocity is, so its kinetic energy should decrease and be converted into potential energy? If so , if it is just a matter of energy conversion between kinetic and potential energies how does the total energy of an excited electron increase? How does the photon energy that excites an electron to a higher energy level shared between the potential and kinetic energy of that electron?


"Inclusion" means for something to be part of a group "Exclusion" means for something to not be part of a group Whether something is in the group or out of the group is obviously modified be the "in" or "ex" prefix, so the root word should be "Clusion" to mean "part of a group?", but I can't find any references to the word "Clusion" and im trying to come up with something that isn't super esoteric.


Say you want to wear a shirt: it would be odd to say, "I'm going topful". Is the reason due to topless being more out of the norm, and thus requiring a term? Maybe in places where not wearing a shirt is more normal, topful may be a more likely word to exist. Or maybe topful sounds like it would be totally covered, so the term isn't used. Or maybe the suffix 'less' often isn't paired with a converse version with 'ful'. Or maybe it's a combination of these things. In any case, any ideas why topful is not said? Topful is indeed a word, but that doesn't seem to apply here: Full to the top, or brim; brimfull. (Webster)


I want to understand what are the techniques one can explore to debug linear programming related optimization problem which is turning out to be infeasible. It turns out infeasible saying so and so equations lead to infeasibility but the problem is that the equation is formed using multiple inputs and has a logic behind it. How do I know which exact input caused the resultant equation to be such that it leads to infeasibility


A Device independent test is a procedure used to characterise quantum resources with the minimal level of trust. If one wants to test correlations like entanglement in a device-independent way, we get non-local correlations. Then my question is: given that Quantum Discord is considered as a form of quantum correlation that has no classical analogous, is it possible to test in a device-independent way? Has anybody addressed this question in the past years? If the answer is no, then a second question comes: is quantum discord really a notion about "correlations", if it can not be tested in a device-independent experiment?


I'm writing some text, with a float at some point. I don't mind where my float is placed exactly, but - I do mind if it breaks up individual paragraphs into two pieces, before and after the float. I just read this: How to protect text from being split by a float? where the (single and accepted) answer suggests some measure of clearing floats, or postponing the presentation of any floats (using placeins or flafter). But - that's not what I want; I'm ok with floats either appearing or not-appearing after or earlier than this point - I just want paragraphs to remain intact. What should I use instead?


I need advice on the statistical method, and I've included some preliminary findings from my dataset, which includes mean, standard deviation, and variance. To distinguish between artifact-related and non-artifact-related data, I would like to receive the results from the mean, standard deviation, and variance formulation. Can someone provide me with a statistical formula that will enable me to distinguish between artifact-free findings and those that are not?


At the moment I am trying to simulate charged droplets and their berakup in an homogeneous electrical field in vacuum. I am using a VoF-Method, but as the droplets get smaller and smaller I would need a finer mesh and smaller timestep. So I was thinking of using an Eulerian-Lagrangian approach by identifying droplets below a certain size and turn them into Lagrangian particles. The question then becomes how will I handle breakups for the Lagrangian particles. From what I gatthered there exist droplet breakup models between fluids like KH-RT, TAB, or PilchErdmann. Is there something similar for just electrostatic breakup?


Say we want to inscribe a cicle with a regular polygon: The circumradius is is the distance to the vertices of the regular n-ngon. We can see on the image, this circumradius of a regular n-gon is greater at lower n. And as n goes to infinity, the circumradius should approach the radius of the inscribed circle. But what is the equation of the radius? I've tried calculating it and google it, and the condition must be, the apothem of the regular n-ngon should equal the radius of the inscribed circle. I would love your help on this.


If the eigenvalues of two matrices are known is there any (generalized) way to tell the eigenvalues of the sum of the two matrices? I have come to hear that no such theorem exists that can handle the general case. I want to ask if it is possible to find and prove such a theorem. If yes, what background in linear algebra is required for it?


Specifically, this is for the context of hydrodynamic levitation. Below is what I believe the forces acting on the ball are: What equation would be used to calculate R, the force from the Coanda effect? My logic above, by the way, is that since the ball exerts a force on the water that causes it to go to the right and downward, the water must be exerting an equal and opposite reaction force that causes the ball to be pushed upward and toward the left.


From my understanding, but please correct me if I am wrong, this equation is implicit (See the attached screenshot). However, the goal is to create an identical (or similar) shape that is defined using "parametric equations". Is there any good way to do that please? Thank you for the help. In addition, if it is possible to configure this shape to be in the direct "center" of the graph (instead of being on the side), that would be even better. In summary: Define this shape (does not have to be identical, just needs to be similar) using parametric equations Plot the equation/shape to be in the center of the graph (symmetrical by the y-axis) Thank you for any advice or help. A bean-shaped equation modelled in Desmos


The Coulomb's law is an expertimental law which calcuates the electrostatic force between distinct two electrically charged point particles (point charge) at rest. And Point particle doesn't take up space. However, there is following sentence in Content 'Limitation' in an the article in wikipedia: There are three conditions to be fulfilled for the validity of Coulomb's inverse square law: The charges must have a spherically symmetric distribution (e.g. be point charges, or a charged metal sphere). In here, i can't understand this sentence. Why charges must have a spherically symmetric distribution?


What happens to the temperature of a body (not in the body frame) as it approaches the speed of light? My thinking is that the molecules will necessarily 'prefer' (as a natural consequence of the basic effects of SR - yes, I am handwaving at the moment and not giving detail just to avoid extra text) a direction of movement, rendering the conventional notion of temperature less adequate.


I found this sentence in Oxford Dictionary in an entry for "draw up": Sarah drew herself up, full of indignation that he should presume to judge her. Also this one in the entry for "sully": They were outraged that anyone should sully their good name. My guess here is that "should" in the first example really means "should not", i.e, he should not judge her but I am not sure maybe some extreme meaning or pun is being used here beyond my pay grade in English grammar?


I have found a similar question here. However, my reputation is too low to append comment for requiring more information. As a result, I have to post a question. I read Pandas documentation and I encountered the following sentence: The axis labeling information in Pandas objects serves many purposes. I could catch the general idea of this sentence, except the "axis labeling information" part. If I was the writer, I would write as "the information of axis label ...". So, I have mainly two points to ask about: Is it OK to write "the information of axis label"? Why use "labeling" not "label"?


I am a computer science PhD student, and I need statistical manifolds theory for my work. I am currently reading Differential Geometry of Curves and Surface by Kristopher Tapp and Carmo. I plan to study Lee's smooth Manifold next. (My supervisor's recommendations.) Unfortunately I do not have the topology background to the depth I would like to have to; I studied Topology without Tears in my second year (not completely) but I do not remember all of what I read either. But I am keeping it alongside. Can anyone suggest a self-readable set of books to get to Stat. Manifold? I might end up needing quite a bit of Discrete Differential Geometry as well. Other suggestions about alternate study plans are also welcome.


I am currently conducting complex simulation involving deformation of a solid object ( which is a surface, or a shell using the vocabulary of structural mechanics) and some fluid discretized using a particle method ( namely SPH). I am used to look at theoritical results to understand a bit a behavior of the solutions ( existence, unicity, rate of convergence, potential numerical issues, ...) but for this case I could not find anything. Is anyone aware of some literature on the topic ? Thanks, Quentin


I have found answers to similar, much simpler questions on this site before (such as here How do you calculate probability of rolling all faces of a die after n number of rolls?), but can't find anything that deals with this exact generalization. After thinking about it some, this seems headache-inducingly more complicated to solve than these simpler variants, because the combinatorics are more nested and interfere with the simpler approaches used to solve those simpler problems (e.g. the fact that you want to roll each number b times instead of just once as in the linked question means that you can't use such a simple expression for not rolling that number in c rolls).


I can't seem to prove this but I also cannot find a counter example. I am dealing with the idea of a function that is constant on the imaginary axis and is defined through an unbound domain. Is it a constant function?If so, why? And if it isn't, can you show me such function? I would like to note that this is not a function with ImF being constant, but F is constant when fed imaginary numbers. Thank aplenty.


I know that in metric spaces sequentially compact is equivalent to compact. I also know that compact and sequentially compact for general topologies there is no relation between them. But I wanted to know if there is the same implication for Hausdorff I-countable spaces. The only thing I have found is that for Hausdorff spaces with the B-W property implies countably compact. Any books where I can research more on the subject?


I have learned that a blackbody object only absorbs light and thus does not reflect any light. The blackbody object will, however, emit light, and none of this light is due to reflection. Where does this light/energy come from? I assume, given an atom, that the light emitted is due to excitation and decay of electron states. Is this correctly understood, and can there be other methods of emission?


When capacitors are connected in series in a DC circuit, the voltage drop across individual capacitors at and immediately after the initial charging period is inversely proportional to the individual capacitance of each capacitor. But afterwards, this began to change due to leakage current. How does this affect individual voltages across each capacitor? Will this process eventually result in the capacitor with the lowest leakage current gaining a voltage drop nearly equal to the voltage drop across all of the capacitors in series?


I am trying to translate a book from English and am confused about the meaning of the word "block" as it is used in this context. We attempted to block for local spatial variation in the landscape immediately adjacent to stands (landscape patches) that were sampled by comparing the abundances of small mammals in the stands to a single type of habitat that is immediately adjacent to the target stand. [Landscape Ecology of Small Mammals] What is the intended meaning?


I am looking at GKP and cat states. Wigner plots are newish to me, and I'm trying to get an intuitive feel for the imformation they impart. Just by looking at a Wigner plot, can one discern whether the state shown would make a good qubit? What does one look for? Are certain, say, symmetries, important? E.g., I know the grid-like pattern of a GKP state means some error-correction is inherent. (Asking here too as well as on the quantum computing stack)


Suppose the left lead, right lead and the scattering region are all the same metallic system; such as iron. Why is the charge transimission coefficient must be a positive integer number at any fermi energy level? How is the charge transimission coefficient related to the number of corssing points between the fermi energy level and the eigen state in the band structure? Would anyone please give me some suggestions? Thank you in advance.


I am told many time that nothing can escape black-hole because black-holes escape velocity is more than speed of light. But we know object don't necessarily have to exceed speed of light to escape a gravitational body. Rockets can escape earth with lower than escape velocity if constantly thrusted. In the same way would it be possible to make an object cross event horizon from inside by constantly pushing?


I have some questions about the motion of the simple pulley system illustrated in the book Machinae novae. Are the following conclusion about the system correct? Since the length of the rope is constant, the carriage will move to right by the same amount, passenger is pulling the rope to the left, irrespective of the mass of carriage? The carriage will never toppel, irrespective of the amount of pulling force or location of the carriage along the path. Basically, only the angle of the pulling rope will change but the angle of rope supporting the carriage will always be perpendicular. The mass of the carriage will be limited by the breaking strength of the rope?


While we understand the universe to have had a beginning in the big bang theory, if gravity is quantum in nature yet still has an infinite range then surely once the distance between galaxies becomes great enough, pair production will begin in between those galaxies as a result of the gravitational potential energy being converted to stable particles? This is assuming that the expansion of the universe eventually reaches a steady rate and that dark energy is infinite. Am I missing something here? If so, what? I propose such a fate to the universe be named the "steady-state fate" if it is worthy of serious consideration.


I want to know when a vehicle is drifting, assuming it's rear wheel drive and has enough torque to give it some good grip but also has a fairly high steering angle, so the rear tires only slip or do they also roll? Does this change by the radius of the turn? I'm doing an analysis on differential stresses that's why I need to know if I should consider tire and ground friction a force in calculating the differential output or not.


I am a native English speaker but have never found a comfortable approach to requesting help. Would you please is more commanding/ authoritative in tone than I am looking for typically. Can you [please] is not actually grammatically correct: it literally is asking "Do you have the capacity to do X - as opposed to the actual intent of requesting their assistance (and not doubting their actual skills to perform). What might be other way(s) to lightly/tactfully/carefully request assistance?


In the lyrics of song Sorry by Halsey there's this line: I've missed your calls for months it seems Don't realise how mean I can be 'Cause I can sometimes treat the people that I love like jewelry Jewelry is a good and prestigious thing. Why does someone say I am a mean person for treating loved people like jewelry? My thought is that since jewelry makes the wearer more beautiful the wearer loves it for srlfish reasons. So I think the same applies here. It means they love people for selfish reasons, not for the sake of those people. Am I right or I'm missing something here?


When we have a set, is it correct to refer to another set as 'itself' when this other set is merely equal to it? More formally, I am asking whether or not two sets with the same elements can be considered to be two separate mathematical objects. I am a programmer so I am used to the "same object" being a distinct notion to "the object to which this object is equal". Therefore, this has led me to consider whether or not this concept is true within mathematics.


If I were to take a particle, moving at a non relativistic velocity, it will have a wavelength (according to De Broglie). However, If I were to change the referance frame of the motion, I know that the wavelength will change because the momentum changes. But will the change be synonymous with the shift of wavelength, if I did the same problem, just considering it wave and applying doppler's shift. If so, why? And if not so, then also, why? What would be the changes if I were to take a relativistic frame instead of a non relativistic one? PS: Due be noted that I am a high school students, who just has a surface level understanding of lorentz transformations and relativity.


I know that we call a quantity a vector if it has magnitude and direction and follows vector laws of addition(the triangle law and parallelogram law). But why only it should only follow addition laws and not other laws like cross product or dot product and why they are not given in its definition. Is it like that if a quantity follows vector laws of addition it also surely follows laws of cross product or dot product or etc.


I have been told that Quantum fields are the real "fundamental" of reality and particles are excitations in these omnipresent fields. I have also been told by no less credible sources that Quantum fields are instead simply a good mathematical "abstraction" and a fine way to quantum mechanically "describe" reality, namely elementary particles - as excitations in the Fock states of their respective quantum fields. I am not sure if this is a controversy, or even a debate in the scientific community, but I would like someone to elaborate as to whether or not those two views contradict each other, and if the do, to understand why.


I am working on a package that could be use in different languages. I'd like to delegate the standard language settings to babel, mainly the spacing around punctuation and the writing of dates. Is it a bad idea to impose the loading of babel to manage local settings? The user could load babel before mine, such as to use babel fine-tuning before working with my package?


It is quite easy to demonstrate that an object at rest that has balanced forces doesn't move. All the Internet examples of things that have balanced forced moving are large scale objects - cars and airplanes. I thought of something like a deployable parachute on a light object but it sounds complicated to set up. What is a good demonstation of balanced forces producing a roughly constant velocity?


In a p-type semiconductor there are always free electrons due to thermal generation allowing for conduction. But If such an electron falls to a hole, it can no longer conduct. So shouldn't more holes slow down conduction? If intrinsic semiconductors have many more electrons in the conductive band at any one point in time, I don't understand why a p-type semiconductor should be more conductive. (My humble guess is that, when there are less holes, an electron jumping to the conductive band due to thermal generation is far more likely to jump back to the same place, because the other holes are too far away)


For full context: I'm looking to categorize adults succinctly into binary categories: those with children, and those without So we could say there are two buckets of Adults here: "Childless" and ... "Childfilled"? Antonyms of "Childless" appear to just be fruitful, fertile, or around the ability to have a child I know an option would be to say that there are two buckets of Adults: "Those with Children" and "Those without Children", but it's too wordy for the system that we are trying to build Similarly, it could be that there are two buckets of adults: "Parents", and "Childless", but we found that to be too confusing to read. "Parents" is a noun while "Childless" is an adjective. Ideally, we are looking for two adjectives.


When writing the equations for center of mass we assume as it to be the point where all the forces act and write the corresponding kinematic equations, and in doing so we dont assume any distribution of forces (like constant or linear wrt length or any dimension of the body) But when we consider the case of a rod lying on a surface with a force acting as F(x) = Kx where x is the distance from one end of the rod, we see that the force is more concentrated above the center of mass and the net force doesnt necessarily pass through the com. I am unable to wrap my head around this difference in result. Need some help understanding where im going wrong in my thinking


The original Computer Modern Math Italic font has a modest amount of kerning listed in its TFM metrics (largely for punctuation, but also for some exceptions like lower case d). But when I use Latin Modern Math as a replacement using unicode-math in either of xelatex or lualatex, those kerns seem to be AWOL. The GPOS table in all of the unicode math fonts I've looked at has no kern tables. Is it really the case that OpenType math fonts don't have pair-based kerns? There's italic correction information in the MATH tables, of course, but I didn't see anything in the MATH table that corresponds to the kerns that seem to be missing.


I produced a set of lecture handouts using .cls files (ndjflart and dupart) that I'd previously used in the past for a paper I published in a journal . I've now been asked to convert those notes into a book. Is there a straightforward way to generate a memoir class file from them, so that I can have chapters, left/right page margins etc? Thanks a lot.


Is it possible to extract equivalent current sources from the partial measurements? I am specifically thinking of a scenario where we have a radiating source inside a closed surface and we measured radiated fields from that source. In principle, according to equivalence theorem or inverse source reconstruction method, if I can somehow measure all tangential fields over that closed surface, underlying current sources can be accurately extracted. But what if, only fields are measured over a part of a surface, is it still possible to extract equivalent sources correctly? What conditions do I need to enforce? Thank you


I am using Cascadia Code font as my tt font, and I wish to use Light series as upright font, LightItalic as italic font, Regular as bold font, and Italic as bold italic font. I am not considering using setmonofont from fontspec package since it's annoying to carry ttf files. Instead, I am using casacdia-code package. How to change default series only for ttfamily using cascadia-code package?


For context: This term or phrase is what it's called when a person (very often used in political debates) will try to take the rarest occurrence of any particular topic and present it as if it's the common practice in order to drive or support their argument. I would liken it to something like "anomaly argument" (except, not completely made up lol). I'll know it when I hear it but for the life of me, I just can't think of it. I hope my poor context helps just a little bit.


Is there a term/words to describe that you lost intrest in doing something because you have repeated it so many times? Examples for these scenarios: You liked hamburgers a lot, but because you have eaten them throughout the years so many times, you dislike them now That song you once listened to the first time sounded really good, but after listening to it for many hours, it becomes a pain in the ears or just a boring song


I kept a cold standing water in a glass for a while to warm it up. After a while I returned to see that small minute bubbles have formed in the glass. On closer inspection, the bubbles seem to not float to surface like fizzy drink but they have sticked to surface (mostly at bottom) while some but lesser bubbles have sticked to vertical surface of glass. I haven't seen or paid attention to this phenomenon, and also water is drinking water so it shouldn't be contaminated. I can't think of any physical phenomenon for this? So can anyone try to figure the root cause for this?


If relativity is symmetrical (triplets moving away and returning to the center) then what happens to the Doppler effect as seen by the moving triplets? A B C A should see C moving (Doppler) away/to faster than B C should see A moving (Doppler) away/to faster than B B should see A and C moving (Doppler) away/to at the same rate. Note that the clocks (long term) in each case should match the Doppler (short term).


Accoring to this question and a linked duplicate, it's been verified empirically up to some number that all twin prime averages greater than six, are the sum of two smaller twin prime averages. I was curious whether or not these formed a tree (with twin prime pairs at the vertices and the sum relation the edges). In order for these to form a tree, for every twin prime average, we would only be able to find one pair of smaller twin prime averages which summed to it. Can you find a counterexample, namely any twin prime average which is a sum of two smaller twin prime averages, more than one way?


When I was learning photoelectric effect, It was told that Photons incident on metallic plate in vacuum causes electron to emit if the energy of photon is greater than work function of metal, In this case to me, The emitted electrons in vacuum seemed like a similar condition like that of cathode ray tube, can these emitted electrons be called cathode rays and will they have all properties similar to cathode rays?


The two most popular ways to generate a topological manifold from others are picking a subspace of a topological space and computing the quotient space of a topological space. In differential geometry, it seems that looking at a subspace becomes far more popular than computing a quotient; an exemplary method is the preimage of a regular value of a differentiable function between manifolds. Is there an analogue result/ theorem for under which conditions a quotient space of some (differentiable) manifold is itself a (differentiable) manifold, the same way the regular value theorem spells out some conditions for which a subset of a differentiable manifold is itself a differentiable manifold?


For the converse, while I agree that the point will be on the pependicular bisector, we can't assume the line drawn is actually the perpendicular bisector... yet the above textbook example assumes it is. I don't see why, based only on XA=XB, we're allowed to conclude that XY is perpendicular to AB and YA is congruent to YB? Surely to make the given conclusions we'd need to state that if two points are equidistance from the endpoints of a segment, those points determine the perpendicular bisector - what am I missing?


I have been told that galaxies will never stop rotating because conservation of angular momentum But, there are planets inside of it can travel through dense nebulae and bodies of gas that would cause friction (and probably brake their travel speed through the Galaxy). Wouldn't that brake the galaxy's rotation? Also, gravitational waves can be emitted by rotating massive objects that have uneven mass distributions by so called quadrupoles or higher poles in their angular momentum. Also, in this study (https://www.space.com/dark-matter-slows-milky-way-rotation) they showed evidence that the galaxy rotational speed is decreasing due to dark matter. Then, would galaxies (and all their contents: nebulae, stars, planets...) ever stop rotating? Or they would only decrease their velocity but it would stabilize at some point?


I want to write a formula and put on its right side the description of its variables. Here I just screenshoted the two parts and put them side to side with figure environment but I want to write some "code" that puts it like that but not with screenshots or pictures or anything like that. Is it possible ? And if yes, how can I do that ?


I'm reading a textbook that gives both a "weak" and "strong" formulation of optimal control (first is a control function, second is the whole space, filtration, Brownian motion, and function etc). The textbook has a theorem about existence of optimal controls for the weak formulation and makes explicit that the strong formulation is not equivalent to a classical control problem. However, no existence theorem is present for the strong formulation.


Let us consider a metal clamp which is placed rigidy and is placed parallel eto the ground. Now on the both ends the plates of a capacitor are kept such that they are parallel to the ground. A dielectric is kept in between and slowly released from the clamp. Also a battery is connected to the circuit so as to always keep the voltage constant. So the question is that we had to prove that the dielectric executes periodic motion. My approach We can find the change in capacitance and hence change in work done by the battery. But i am confused so as to how a restoring force could act? I s it due to the battery?


If you have a square two-dimensional grid, that are lots of ways it could be triangulated -- for an individual square one could add an edge from upper-right to lower-left, or from lower-right to upper-left. For an ensemble of squares, a choice of extra-edge direction for each square, so lots of choices. I can think of a couple of reasons to triangulate a grid, to perform a linear interpolation on simplicies, and I read that algorithms for finding sinks and sources generally assume a triangular mesh. For these (and possibly for other) applications, are there better choices of square-grid triangulations? Pointers to the literature on this would be much appreciated.


I was studying duality maps in my Advanced Stat. Mech. class and it was told that all self-dual points need not correspond to critical point. I understand that critical points are points where analyticity of the partition function breaks down. If this point is unique then it can be given by the duality relationship between the constants which appear in the theory. But what happens if this point is not unique?


The system I am trying to model is essentially moving water interacts with ferrofluid. The ferrofluid is in a box that contains air and ferrofluid, such that the setup is buoyant. The ferrofluid chamber has outlets so that it can interact with water. Here's an example image of the system: I need confirmation in these aspects: That the right choice here is to model it like two phase flow not single phase or mixing model. That it can be modelled by either level set method or moving mesh method without compromises in results. If I do not want the fluids to mix, is there a way to still maintain pressure balance between them? Would a semi-permeable membrane work here? If not, is there any other way?


I have been having this doubt when applying the Divergence of a Electrical Potential evaluated in a point, and then doing the Divergence to get the Electric Field in that point. But is this valid? or I have to do the Divergence first and then evaluate? Because when I do the first thing I get the correct result, but the process of which I get the solution is triggering me a little


Is it possible to use punctuation marks such as commas, periods, and question marks just before colons or semi-colons? The error appears when accessing some URLs, e.g.: List items ... Can I set additional parameters for the following tasks?: List items ... I occasionally come up with such sentences due to perhaps my mother tongue (Japanese). However, I realised that such usages can't find anywhere so it might not be natural usage for native speakers. Any natural alternative ways to write it?


It is well known that a convex function is minimised over a convex set, if and only if there is a subgradient which is inwards normal to the set at that point. i.e the negative subgradient (direction of steepest decrease) points directly out of the set. Is there a corresponding rule for maximising a convex function? i.e is beig maximised at a point equivalent to having a subgradient that is normal outwardsto the set at that point. i.e the direction of steepest increase points directly out of the set.


A neutron is udd, then an u-anti-u starts to exist close to the neutron. Then the one d and u change places forming an uud (proton) and anti-ud which decays to an electron and electron-antineutrino. Now it is my reasoning that since this process exists the process can be sidestepped so that a udd decays directly to a uud and electron and electron-antineutrino. Is this true?


I wonder how would we go about to calculate centripetal/centrifugal acceleration for relativistic objects moving at near the speed of light , do we use the original acceleration = force/mass, or input gamma factor for it to be , Acceleration = (gamma) force / mass , Acceleration = Force / (gamma) mass Which one is valid ? If both aren't is there a particular way and/or equation to calculate centripetal and centrifugal acceleration for relativistic objects ? How ?


Is there a package or option that starts alternating row colors from within a table, so that no matter how many manually-colored rows are inserted, the alternating can be started again with the same color each time? (In a single table, I want to manually color a few "heading" rows, each followed by an undetermined number of alternating gray/white rows -- always starting with gray.)


This is something I have noticed on a fairly regular basis, and this is a recent example, where the questioner starts by saying "I'm working on" and ends up asking "How do we...?" Can someone help me with this circuitry? I am not an electrical engineer. I'm working on a brain/organoid computer interface system . . . How do we connect the cyclone . . . Is it some cultural thing? Is there a mechanism to it, like that if others are included they feel more inclined to help?


I'm writing math notes, and I would like to use the symbol for the letter T that appears in this picture (seen in a TeX.StackExchange post): click The symbol I want is in the line "Calligraphic: UPRIGHT: euler (Euler script)". How can I load it? Note that I'm not interested in changing all symbols in my document. I just want to use that T perhaps once in the entire document.


Elliptic Integrals and Jacobean Elliptic functions I know it is easy to determine u =F(x,k) where F is elliptic integral of the first kind knowing both x and k. I know it is also possible to determine x = sn(u,k) where sn is the Jacobean Elliptic function knowing only both u and k. Is there a function to determine the modulus k only knowing u and x?


Fox's Trapezoidal Conjecture asserts that the coefficients of the Alexander polynomial of an alternating knot alternate and the sequence of their absolute values forms a trapezoidal shape. The same is true for the Jones polynomial for some alternating knots that I checked. However, since I didn't find an analogous conjecture for the Jones polynomial, I expect that it is not true for some known knot. Which one? Was there proposed perhaps some more narrow version above conjecture for the Jones polynomial of alternating knots? (Either with stricter assumptions or with relaxed conclusions.)


What word describes someone who is extremely carefree to the point of risking their lives. The person is not even bad or annoying; in fact, they are vibrant & outgoing. Except they don't extremely carefree to the point of risking their lives. I was thinking of "extremely carefree" or even "unhinged". For example, That girl who got stabbed at the club was______; She was outgoing but extremely carefree to a scary point.


Here is the situation. Let's say I have have a mass of a given liquid and I heat it to create a gas. The gas has a lower density than air, so it will move higher and higer in the atmosphere. Then, I recondensate this gas into liquid. Suppose both process are ideal, so the energy I put in the system to cause evaporation is the same energy released by the system from the condensation. We now have the same liquid as before, but since it moved up in the atmosphere, it now has a muche bigger gravitationnal potential energy than it previously had. Of course, the energy is conserved, so this additionnal potential energy must come from somewhere, but where?


I am writing an application in which I complain that every time I point out to the cooks or their supervisor that the meal is not up to the mark,and actually show them how it is, they say that no body else complains, only I complain. However, whenever I ask my colleagues about this, they tell me that what I am saying is valid, and they also point it out all the time. So this is a habit of this staff to gaslight women employees like this. We have put up with this for years. Therefore, I'll really appreciate if someone can help in this. I need a synonym of gas-lighting which conveys the intent properly.


There is no scenario where we can use Coulomb's law. There is no static charge. Even if we consider the local charge density to be constant for a system of charges, the individual charges are still moving, and as we know they are not quite the same. Moving charges show different effects. Even if we think about putting a frame with the particle so that we can consider the charged particle as static that also does not help because there is no single frame for a system of charge Also we want to measure the effect in our lab frame. So, how is Coulomb's law still relevant?


Often you have to adjust the table of contents manually. With very large documents, it is then very time-consuming to compile the entire document each time just to be able to look at small adjustments to the table of contents. Is there any way to tell LaTeX to compile only the toc and not the document? Or maybe you can copy the toc file of the actual document, rename it to the jobname of a small test file and then let it compile, sort of without the associated document?


Say we have a hot cup of coffee that is giving off steam. It is releasing heat into the environment through radiation and conduction. The steam is from certain particles reaching a high enough kinetic energy as to escape the attractive, intermolecular forces that bind them intrinsically. If we physically inhaled steam, thereby increasing the rate of heat loss through steam, would it effectively cool the coffee?


In steady state condition of a conductor under normal temperature, we find that the derivative of drift velocity is zero by setting electric field equal to zero, and one concludes that current density is proportional to the electric field. But in case of superconductor, why don't we have such a steady state condition like thing or in other words, why cant we have a unique solution by setting electric field equal to zero, as in two fluid model of superconductor, one finds that the derivative of current density of super electron is proportional to electric field and not the current density.


Why are there two interior regions in this diagram? There seems to be two inner horizons, and two wormhole regions. Where do the two such regions comes from? What determines which reason someone falls into? Could a composite body fall into "both regions" with part of it going into one region and part into the other. The diagram seems to indicate that if you're precise enough you can fall almost directly from the Black hole to the white hole while spending an arbitrarily small amount of time in either of the Wormhole regions.


I would like to consider the sound incident to a water filled pipe wall. I think the pipe wall is typically considered as a rigid wall boundary, it means all the incident wave is reflected. Is this assumption is correct in real world? I think if the sound inside the pipe is large, we can hear the sound from outside the pipe. In practical, what kind of boundary condition should be assumed for the pipe wall?


Using thesisdown to write my Masterthesis I was wondering about the default indentations of the package: If you check the pictures below, all the pages (of the default 'knitted' thesisdown rmd-files) have different indentations (holds also for the toc). On a first glance my guess was that pages with a heading have a certain indentation, whereas the following pages don't have any indentation. This theory also holds for the toc. But imo it doesn't hold for the first picture. So I ask myself: is this indentation behaviour intended or is it an 'unwanted' side-effect? And is there a way to unify the indentations for all pages?