I work in tech but I at some point I almost took a career path into theoretical physics. I changed my studies very late to data science and machine learning (my last masters year) and before that I was in physics and solid state physics. I am looking for research papers that could be interesting to read for a non-expert like me. I am used to read physics research papers (mainly solid state physics and spintronics). I am looking for papers that do not have super heavy mathematical development. An example would be the famous "More is different" from Anderson. It describes very well what I'm looking for although I'd prefer some mathematical perspective if applicable (but again not too much of it). The idea is to read about topics like General relativity not from Wikipedia but from research papers that are either foundational or educative/pedagogic. No books however I don't have time for them, I am not looking to study these subjects but rather acquire some insight and basic knowledge. I am looking for now in papers in : general relativity


In papers studying or searching for topological order (intrinsic or symmetry-protected) in various condensed matter systems (e.g. Field-tuned and zero-field fractional Chern insulators in magic angle graphene), a common refrain of motivation goes as follows: Topological physics began with the experimental discovery of the integer and fractional quantum Hall effect, for very clean two-dimensional electron gases in a large magnetic field. The large magnetic field is unfortunate, and it would be nice to get rid of it. In fact the large field is not necessary, and equally interesting physics can arise in our system due to strong interactions, time-reversal breaking, etc. But I realized I never really understood the second point: why is a large B such a problem? What applications or lines of scientific inquiry does it challenge? To what degree are these challenges insurmountable?


My problem is as follows: I would like to use the upright fourier package symbols together with dsfont. This produces: I therefore want to scale the dsfont symbols down to the same size as the fourier letters (or vice versa scale fourier up). I tried using scalerel, however for some reason pdflatex and Anki (a flashcard software) doesn't seem to like this. It declares no error message, however it doesn't scale the symbol (or any others I've tried for that matter). Overleaf does scale with the same code, so it's not the code. Are there any alternatives to scalerel to scale dsfont (or scale the fourier package - although this isn't preferred). I've also tried using the mathalfa package to load dsfont (although this doesn't load), as it has a scale feature. Some notes: I'm using mathastext with nosmalldelims enabled, if this makes a difference. Any help is appreciated:)


I recently encountered a quirky situation. A student wrote a sentence, and it was much more technical than this example (actually for a literature review on microbiology), but this examples illustrates the basic issue: John said, "The cat is on the table" with great anger. A parallel example: John said, "The cat is on the table" when the cat climbed up. I was always taught to offset complete quotations with commas, but I tend to only see examples with the signal phrase beginning or ending the quotation (e.g. "The cat is on the table," said John...). While I understand the sentences could easily be rearranged to terminate with the quotations and to improve style (e.g. With great anger, John said...), how should these be punctuated as they stand? Should there be a comma where the quotation terminates? Obviously, if the quotation were a regular clause, no comma would be used (the first due to a prep phrase, the second a dependent clause following an independent). But how does the quotation impact this rule? Thank you!


Localized ferromagnetism refers to materials where the magnetic moments are primarily associated with localized atomic orbitals. Ferromagnets, such as those made of iron or nickel, are called itinerant because the electrons whose spins aligned to create the magnetic state are extended and are the same as the ones responsible for conduction. I don't understand how can localized ferromagnetism exist at all. From my understanding, a large gap insulator will always be non-ferromagnetic because the full valence band always have electrons paired up and leave no net electronic magnetic moment (while nuclear magnetic moment is negligible for ferromagnetism). The energy favorability of spin alignment is much smaller than overcoming the large band gap, so the spins are always paired. This means ferromagnetism can only occur in metal or small gap insulator where the energy favorability of having aligned spins is larger than the energy favorability of strictly filling the spin-unpolarized states below the fermi-level (before considering the interaction of electron spins). Is my understanding of the necessary condition for ferromagnetism correct? (ie, energy favorability of aligned electron being larger than band gap)? If this is correct, how can localized ferromagnetism exist at all?


I have been studying Condensed matter physics, and there are some basics that are confusing me. Basically, when we find the dispersion relations of electrons in a lattice using the nearly free electron model, I do not get what the dispersion relation signifies physically. If we have one electron per orbital, I know that the band will be half-filled, which means that we have electron waves with those particular k values, but then how is the concept of Bloch wave packets incorporated? In Yang and Grivin, when the Bloch envelope is introduced, they integrate Blochs function over all k values, from which I thought all these electron waves superpose to make a wave packet of some average momentum. But then going forward, when considering the dispersion relation I got confused that will we still be looking at electrons in particular states individually? When we consider Bragg reflection at the boundaries of the Brillouin zone, does it mean that electrons with that crystal momentum form standing waves, all the while the other electrons with the other k values continue to propagate, and they all superpose to form a resultant wave or is it like when the wave packets k value becomes the k value at the boundary, it forms a standing wave?


The potential at the surface of a conducting sphere is KQ/R where 'R' is the radius of the conducting sphere and even the electric field on the surface of the conducting sphere is maximum. And while taking a test charge from the surface to the inside the potential neither increases nor decreases because electric field inside the conducting sphere is zero, so whatever potential is on the surface, it is the same inside. But my question is why don't we need to do work on the test charge kept on the surface against the electric field on the surface to move the test charge to the inside of the sphere where electric field is zero? And if we would need to do work against electric field on the surface to move charge inside then potential inside will also be different from the surface. Thank You! NOTE- i am assuming that the conducting sphere is positively charged.


Recently I have been learning about optimisation-techniques and built a simple "gradient-descent brachistochrone solver thingy" to try out some methods. One thing currently still hurting the results is the distancing of the points. Obviously, if they are first generated an equal distance on the x-axis apart from each-other, when the resulting graph becomes non-linear the distances between them start to vary quite significantly. Especially between the first two points there is obviously a massive "hole". What I would like to as is, if there is any well-established method of spacing these points by their distance from each-other instead of evenly along one axis. This at first doesn't seem to be too hard a problem; just move them a bit to be at the right place for their heights. This doesn't really work however, because when changing the "x-position" of the points their height must be re-adapted as well, leading to another distance-discrepancy and so on and so forth. As all algorithms I came up with required an ungodly number of iterations to complete, and I was unable to find the correct keywords to google this problem, this is now asked in form of a post on this forum. Thanks, Robbe.


Now here is a potentially stupid question about something that has really been annoying me recently. In the pdf viewer panel in Texstudio, there are two buttons in the toolbar that change your type of cursor: a magnifying glass and a blue scroll 'cross-arrow'. If you never click on any of the two buttons, your cursor is the standard (Windows) white arrow cursor. I made the mistake of clicking on the magnifying glass, and there seems to be no way of getting my nice white arrow cursor back - instead I am stuck with either of the two Texstudio cursor types and I find them very ugly. There is no button to choose the standard cursor, nor does it revert if you open and close the panel, or Texstudio itself. Any help?


In order for the jars to be sterilized for pickling tomatoes, they need to be boiled. But when one of the inverted jars was standing on the rack in the pot, it started sucking in the water that was boiling there (and sucked out almost all the water). The jars were turned over so that their open side was completely submerged in the water (so that approximately the entire surface of the neck of the jar was in contact with the grid). An approximate drawing of what is happening from the side: I took the jar out, emptied the water out of it, and put it back, but after a while it filled up again. The plastic grate on which the jars stood was uneven, and had holes in it. Why does this happen? Thank you!


According to Kepler's First Law, the orbit of a planet is an ellipse round the sun with the sun at one focus. There's an inherent asymmetry in this. Instead of the sun being in the dead center, its shifted over a little bit. In the hydrogen atom, all the orbitals of the electron are symmetric about the proton at the dead center. Why is there no similar asymmetry? You can convert the function for the position of a classical simple harmonic oscillator with respect to time to a space dependent probability distribution where the probability is higher at the classic turning points where the velocity is at its lowest. The ground state of the quantum harmonic oscillator has a higher probability exactly between the classical turning points. The quantum solutions more closely match the classical probability at higher quantum numbers. I was thinking the classic "lopsidedness" of gravity could be recovered at higher quantum numbers for a Coulomb like potential Schrodinger Equation. But higher principle quantum numbers just enlarge the orbitals, they all remain symmetric about the proton. So that can't give you higher probabilities on an ellipse. The technique to recover classical behavior that works for the harmonic oscillator fails for Coulomb like potentials. Are there circumstances where any asymmetry appears in the probability of the electron, in particular, concentrations of probability along an ellipse?


The Question: Boolean algebra is to classical logic like what is to relevant logic? Context: I guess this is a terminology question, so there's not much I can add, except that I've been interested in paraconsistent logic for a long time. Is the answer de Morgan algebra or is that something else? Meta Question: Is the question well-formed? Perhaps the question ought to be something like, Boolean algebra is to classical logic like what is to paraconsistent logic? I don't know. But: If the answer to the Meta Question (MQ) is yes, then please answer the main question. If the answer to the MQ is no, then please feel free to tell me why and answer what I hope will be a clear intended question. If the answer to the MQ is nonclassical, please explain.


If I launch a ball into the sky it would reach a distance after which it would return into the ground transforming the potential energy into kinetic energy as it hits the ground This is similar to what happens at galactic scales, where material (like gases) from an outflow get expelled from the galaxy, they reach a distance where they turn back, and fall again into the galaxy, increasing their kinetic energy as they are attracted by gravity towards it However, the presence of dark energy at galactic scales, causes outflows to be less bound to their galaxy (and could even reach a distance where the influence of gravity and dark energy are balanced, beyond which, it would be expelled from the galaxy to never return: Is there a distance from a gravitational source where the influence of gravity and dark energy are balanced out?). This then can make these outflows reach a greater distance before turning back into the galaxy. Then, as the material would travel more distance towards the galaxy, and there would be a point in this trip where the influence of dark energy would be negligible (only leaving gravitational attraction), would the infalling material have more kinetic energy in this scenario (than in one without dark energy)?


I finished calculus books like Thomas and currently reading a book on advanced calculus and another one on real analysis. I noticed recently that I don't solve enough "hard problems". I usually just solve the exercises on my books. so I figured that I need problem book(s) on calculus. I want a book(s) that has many exercise on calculus topics like Limits(without l'hopital) ,derivatives, integrals , series and sequence , multiple integrals etc... I also want them to be challenging and interesting problems and not "too hard" I consider myself on an intermediate level as the advanced exercises on Thomas book seems very easy to me, I also don't want "forward substitute questions" which is plugging numbers in formulas or straight forward use of a theorem or equation as I want to increase my problem solving skills. I will appreciate any suggestions ,thank you in advance .


There is much talk of using lasers to bring down drones. That talk is followed by talk of protecting the drones by surfacing them with mirrors. Would that work or does light falling on a mirror impart all its energy to the mirror first before it can get re-emitted? I am aware that lasers themselves use internal mirrors to reflect the light so it would seem that if lasers don't destroy themselves then the energy is not absorbed: or lasers are cooled, I don't know which is the case. There are two questions here really: the Prime question is about the possible efficacy of protecting a drone by mirror surfacing. The more general 'underneath' question regards reflection of light. Is it so that all light falling on a surface must be absorbed by that surface before it can be 'reflected' ? It is my belief that it must. That 'reflection' is in fact re-emission.


The textbook I'm using to study integral calculus usually assumes for it's proofs that the function takes on only positive values. The author says that if we divide the x-axis into intervals, and pick the point in each interval for which the value of the function at that point is a minimum at the interval, then we can approximate the area under the curve using inscribed rectangles. Specifically, the author says, this would be the lower sum. The upper sum is achieved by picking the point at each interval for which the value at that point is the maximum of the interval. I was thinking: wouldn't this be the opposite for negative functions. Because the upper sum would be achieved by taking the minimum at each interval, at the lower by taking the maximum at each interval. If I were to prove the same theorem for negative functions as well, is this the only difference that it makes? Or is this not the case? Because if the upper sum is always the greater numerically, and the lower sum is always the less numerically. And if "negative area" is a thing, then the lower sum would still be the one that is formed by taking the minimum at each point, and the upper by taking the maximum. So I am confused with the exact definitions right now and would enjoy guidance. Thank you in advance.


If we spectroscopically observe a cloud of hot gas, which is on the whole not very absorbent, and which is not illuminated by a source behind it, we observe emission lines. How does this type of spectrum form? I had thought that those lines are those in which there are transitions of atoms is true, but I don't think that's enough. Why should all the material as a whole emit like that. Why are there these electronic transitions? And then: if there is a light source behind the material, one observes a spectrum that is in a way the negative of the one I put, i.e. the lines become absorption lines. I had thought that something different must be happening in the two cases, although I don't know what. And then again: if we knew nothing about electronic transitions and only wanted to consider the macroscopic properties of the gas (which could be composed of complicated molecules, in which there are not only electronic transitions but also other phenomena), could we still justify the fact that the emission and absorption spectra are the negative of each other? Thank you for any input; complex, articulate and in-depth answers are also welcome.


Saw this on reddit: A: I'm a gun owner and I think any sort of gun sticker on a vehicle is cringe. -> B: Ditto any sort of camo, esp. grey/urban camo prints, sure go ahead and tell the world you're itching for an excuse to defend yourself with a gun while you wait for your latte -> -> C: But the hunting community uses camo! Do you think hunters are looking to kill people? Etc, where C continued to belabor the edge case. Now, to me, C misunderstood what B was saying and just continued to argue when it was clear that B didn't think that hunters were included in the group that they were talking about. It feels like C went out of their way to think that B was saying something about ALL people who wear camo -- but to me unless someone says "ALL ___" they just mean "in general the people who ____". Is there a term for "deliberately misunderstanding what someone said so that you can argue about it"? I see it all the time in forums/reddit/insta. Strawmanning doesn't seem adequate -- that's more about mischaracterizing someone's point so you can dismiss it. My best attempt at this is "reaching for outrage" but I've never heard any say that.


In a recent Introductory course on logic, we were introduced to first order and propositional logic. The purpose of these concepts is portrayed as being a way for us to formalise the reasoning we use when we compose proofs in other branches of mathematics, but something about this bothers me, namely that in defining these theories, we extensively used set theory, and the axioms that it grants us, however one needs first order logic to actually describe the axioms of set theory, and so neither of these theories can truly "precede" the other. I read around a bit, and to some other question on this site, someone commented that this is fine, since for example we use the English language to describe the grammar rules of the English language, and we are satisfied with that, so w should be satisfied with this. But there's a reason language and mathematics are different fields of study, and I find this answer unsatisfactory. Is there some way to avoid this circular dependency in a meaningful way? Edit: from my understanding of the answers given to the purposed duplicates, the consensus is "no we cannot". Can we do one better and in fact prove that a sufficient formalisation without this property is not possible?


I am studying the solubility of gases in liquids (flowing then into the study of oscillations of gas bubbles out of the liquid phase). The task at the moment is to familiarize myself with the laws of solubility of gases in liquids, gather material, etc. However, the only law I've discovered is Henry's Law. So far I'm only studying the simple case like water as a solvent and air as a gas. (no electrolytes in the liquid, no chemical reactions or anything). But for such a simple case, apart from Henry's Law, I have found nothing. Also I could not find any literature that could help in studying this question. I am wondering if there are other laws or equations of gas-liquid solubility for the simple air-water case, or is there nothing else besides Henry's law?


I have been reading recently about tension. I don't exactly understand how it works. Here are my major doubts: Here, Tension is said to be acting in the opposite direction of mg. I will assume tension is not the net force across the rope because net force is zero due to newton's third law. Thus, tension must simply be the force exerted on an object by another one through a rope/chain/rod, etc.In this case, since in all cases of an object exerting force on another object, since the other object exerts an equal and opposite force, the tension here is really the reaction force to the force of gravity. However, if pulling a box along the ground with a force F, tension along the rope is said to be in the same direction as F, but can't it be said that it exists in both directions as both the pulling body and the mass exert equal forces on each other? What decides which direction is assigned to tension?


Hi I was working on a problem concerning euclidean topology. I was doing the following exercise : The first statement seemed not to be true because for (i) to be the basis some topology the intersection of any most belong to the basis. As the intersection of any two open squares parallel to the axes is not necessarily a square, possible a rectangle. I assumed it wasn't the basis for any topology. This however didn't make sense when I looked at (ii) as far as I know the collection of all open discs is a basis for the euclidean topology. But the intersection of two disc is not always a disc. Which would mean it isn't the basis for any topology which left me confused. Does anyone know where I could be wrong? Thanks in advance.


Original question: If continuity is the only requirement, here is the solution: Dimension of space of continuous functions But the proofs use trigonometric functions and polynomial functions, which are not monotonic. A further question is, what if I would like to impose additional features on the functions, say, concavity? It seems that one has to search for new bases if using constructive proof because the bases in the original proof may not satisfy the additional feature. I was wondering whether there is a generic way of proof to circumvent this problem (My guess is that no matter what the feature is [as long as the set is a linear space], the set of functions with this feature on an interval is infinite-dimensional). Edit: Thanks for pointing out it is not a vector space. So I changed my question (I thought this new question is equivalent to the original question but it turned out they are not). Edit again [revoked]: I think maybe I should add another constraint that the functions are bounded, or even more stringent: the codomain is an interval...Otherwise, the answer seems to be obviously "No".


I'm currently starting to self study probability and statistic, a friend recommend me to use a book he has but his book does not go deep in the theorem and formula, instead it just state the equation and when to use it along with some properties but not the proof for the equation (for example: the chapter about the Poisson random variable just tells you how to use it and when, but lacks the proof of how mathematicians arrive at that complex equation ). I would like a book that is rigorous and proof-based for every problem in it ( like Tom M. Apostol's calculus books for example), im a colleague student and has good calculus and linear algebra background so an more advance than regular books is ok with me. Do you have any recommendation ?


I'm new to Kalman filters. I have a use case similar to the one-dimensional train example. But I have railroad track with switches and mergers. So it's a non-trivial topology. I would like to model the system as a graph with N nodes and E directed edges. Each edge has length (weight) L. So the system state (e, x, v) consists of the current edge e, the distance x along the edge and velocity v. The edge e is a discrete (categorical) variable, which does not fit the standard Kalman Gaussian methods. Can the Kalman filter be extended to this? Measurements: I do have measurements of (e, x) and related position error err_x. There is no error related to the measured edge e. Prediction model: We can assume a constant speed model with zero-mean acceleration as in the Wikipedia example. In case of a prediction through a node with multiple outgoing edges (rail switch) I'm not sure if I should return a multi-modal distribution. I don't have a requirement that this should run in real-time, so it is possible to look ahead in the data and see which edge is taken. Or to use some kind of backwards smoothing is also ok. The graph is in priciple cyclic, but in practice length of the edges are way larger than the measurement errors.


According to general relativity, if an object keeps moving and warping space and then gets lost by entering a black hole, do all of the distortions in the space caused by that object get lost? Or do the distortions get back to where they began? There is an answer by "Navid" that says it is possible for the distortions that happened in space by the object to disappear. Well, first of all, my imagination says that space will keep warping if an object keeps moving. Meaning that the peak of the distortion keeps increasing; now if there is a case where the distortions will disappear, then wouldn't this come to mind that a new black hole would happen because there is a loss of space? By new black hole, I mean the disappeared or lost space between the destination black hole and the beginning position of the object. My other imagination is that when objects move in space, they carry the space with themselves and don't keep stretching or warping the space.


This question could be really out of the blue and might receive lots of downvotes, but bugging me quite a time and would appreciate your thoughts easily explained. We know that when we do work against nature force, we increase the potential energy of an object. Lifting a ball and putting it on the table increases potential energy of it as my putting work into it is transferred to the ball's potential energy. So as a sum up, for something to gain potential energy, some work must be done externally in opposite of nature. The same happens with charges. Taking away negative charge from positive increases potential energy of negative charge. It's all clear till this point, but if we imagine that worm by default appeared on a mountain, how would it have potential energy? As I understand, some external work must be done so the work is transfered/converted into its potential energy. What energy is transfered into worm's potential energy? I don't believe that gravitation itself gives potential energy to the worm.


Industrial printing is based on autotypical colour mixing, the simultaneous effect of subtractive and additive colour mixing. This makes it possible to render a large set of colours using only four standardized colours (CMYK): I do understand additive and subtractive colour mixing but I struggle to understand how two dots next to each other, i.e. with no overlapping, additively mix to another colour. This is illustrated in the lower part of the following diagram, e.g. a red dot and a green dot are perceived as a blue dot (provided they are small/far enough): I guess the answer revolves around the size of the dots, the amplitude of the resulting light wave and some properties of the human eye but I could not find any detailed physical/mathematical relation between them so far.


Assertion:Energy changes the kinetic only, and changes in potential require an intermediate change in kinetic. Reasoning: eg:when I throw a ball upward I do work/transfer energy,At the topmost point, there's a changed potential which is an effect of changing its position through its ke. eg:when I lift a weight at constant ke,Im transferring energy to increase the kinetic as I'm doing work against gravity , I can't change the potential without changing the kinetic. I did some research but the only answer I could find is an input of energy can increase both kinetic and potential but I don't see how it changes the potential without changing the kinetic. this seems to make sense for a lot of situations and i need to realize where im going wrong with this My question is: Is this valid and if it is does it hold for all fields in physics? Can energy only change the kinetic and require an intermediate change in kinetic to alter the potential?


This is my current understanding of convolution after having read through this blog post The convolution operator can be thought of as an operation of linear superposition. If we have the response of a linear system to a unit impulse, the overall response to an arbitrary input signal may be constructed by taking a linear superposition of the unit impulse responses accordingly translated and scaled. This can be done through the convolution integral. On the other hand, the convolution theorem allows us to perform the equivalent convolution operation by first taking the pointwise product of their Fourier transforms, then taking the inverse transform. In other words, the convolution operation is diagonalized in Fourier space, and acts on each Fourier component of the input signal by multiplying it by its eigenvalue, the corresponding Fourier coefficient from the unit impulse response. While I follow the logic leading up to either approach, my difficulty is in finding the connection between the two - how does it intuitivly make sense that superimposing unit impuse functions accordingly to the input signal has the equivalent effect to multiplying the Fourier spectra of the two functions?


There are two different senses in which we use the word "attribute"; for example, I can describe someone as "blond", which is a hair color. We say "blond" is a characteristic or attribute, but it is clearly the characteristic or attribute with regards to the characteristic or attribute "hair color" - "blond" is not how they smell or how tall they are, it's one of a set of attributes that fall under the attribute "hair color". I have called both "blond" and "hair color" with the same word, "attribute", here, and, as far as I know, all the popular synonyms of "attribute" really are synonyms - "characteristic", "trait", "property", etc. Is there any way or term or expression to verbalize the difference between an attribute in the sense of what it is - in this case, a person really being blond - and an attribute in the sense of what "category" that attribute belongs to - in this case, blond being a hair color? To put it another way, for the sentence "For the attribute "hair color", he has the attribute "blond".", I would like to find a different word for one or both of the uses of "attribute" here, so that the words used unambiguously convey that one is the actual attribute (blond), and the other is the "category of attributes" (hair color).


When conic sections are taught in high school, the concept of a focus is introduced from the geometric prespective. Well, at least if your teacher is any good. Later, once the algebraic equation of the conic is established, we find that this special geometric point can be in some way derived from the coefficients of the expressions in the algebraic equation for the conic. This is typically derived in school by moving back and forth between arguments involving that of geometric observation and that of algebraic manipulation. From an algebraic prespective, it seems quite unintuitive to me that there is a special point outside the curve which is so critical to understanding the curves' geometric properties. In a way, I see it as more of a meaningful quantity than a root. Now, my question is: Using advanced mathematics, what is the algebraic intuition behind the focus? Can we generalize this intuition to talk about foci of higher degree curves? And, are there analogoues to points like focus for higher degree algebraic curves?


The classical analogy for understanding gravity in Einsteinian physics is picturing a sort of fabric that sinks when an object rests on it. Thus, due to this curvature, objects will move towards each other because of the inherent structure of the fabric-object relationship. However, in this analogy, the objects move towards each other because gravity acts on them. In other words, the curvature itself is not enough to explain this phenomenon. Thus, howcome gravity be defined as the curvature of space-time if curvature itself is not enough to have an effect on two masses? I feel like this analogy tries to explain one concept (gravity) as the unison of two concepts (curvature and a force akin to gravity (redundant)). I mean, try doing this experiment in space. Set a fabric, add two masses and curvature but for some reason the masses don't move towards each other; because gravity doesn't act on them! (I know technically it does but for the sakes of the mental experiment it doesn't).


I have a doubt in this question, Diagram given below In the Question it is asked that "calculate the final velocity of the block in the figure" and in the solutions it is given that the work done by normal force exerted by the surface is not considered when applying the work-energy theorem but according to the proof of the work-energy theorem, the "'net' external force X displacement upto which the force was applied is equal to the change in kinetic energy of the object." (could use integration by taking the variable force to be constant for infinitely small time intervals) but by this definition normal force should also be added to calculate the net external force on the object as it also always acts in opposite directions to gravity (with different magnitudes obviously) So why is the normal force not considered here? Is the question/solution wrong? If it is correct, why? And what is the mathematical proof of not considering the normal force when applying the work-energy theorem?


Most introductory logic textbooks that I have skimmed through in a while, keep the terms 'sentence' and 'expression' undefined. I would intuitively see Earth is round. Why didn't Harry come to the party? Come here, Harry! x lives in Norway. all as sentences. However, 'x lives in Norway' is usually not taken to be a sentence in logic because it contains a variable. It rather gives sentences after variables have been replaced by constants. It seems to me that the term 'expression' has a broader meaning and sentences are special cases of expressions. But still what exactly is an expression? Would arh ahfb hghd udh be an expression? Would 'arh', 'udh', etc. be expressions? If 'arh ahfb hghd udh' is an expression , would it be a sentence? Is it necessary for an expression to have a meaning? All these troubles imo are stemming from a lack of definition of terms 'sentence' and 'expression'. How exactly can we define these terms?


So, as I have read and even been taught by my teachers, sign convention in trigonometric functions is based on the location of the respective x and y points denoting the coordinates of a particle going around a circle. Although I am kind of sure that I am right but I still want to confirm this one thing: When determining the slope of a graph, we encounter both obtuse and acute angles (the angle made by the tangent of the graph at a particular point on it with the x axis). So, is it just a coincidence that tan(a)(the angle made by the tangent) is positive in case of acute angles and negative in case of obtuse angles in both this graphical sense as well as tan(a) for a point on a circle? Because if a is obtuse(graph), that would mean that the quantity in the y axis is decreasing and if it is increasing, the angle is positive. So, I just want to ask that here tan(a) just happens to be equal to the slope of the graph right? I checked for other trigonometric functions and well sin(a) seems to be negative if the angle is obtuse in the graphical sense but positive in the circular sense. Edit: I have added a picture for clarity about my definition of graphical and circular sense.


I want to learn calculus by myself, I searched a lot on internet as well as math stack exchange for suggestions for best calculus books, I know a lot of famous books like Stewart Thomas but I do not like the way it is presented. I want to learn rigorous mathematics from books like Apostol Spivak Courhant Serge-Lang, but I am not sure if I can take them, the level of proofs and logic they need, I am afraid if I buy them and not able to solve because high advanced level they are, some motivations is needed and also guide me if I need some more books as prerequisite to understand Apostol Spivak Courhant I am reading books like how to solve it by George Polya, if you can suggest some books which refine logic and thinking please suggest me. I am also someone who enjoys mathematics but not want to learn just basics. its more like dream if I one day able to read and understand books like real analysis by Walter Rudin THANK YOU


I am trying to write down a complete/detailed definition for the parity symmetry. Symmetry as a concept is different in mathematics and in physics. There are also many other concepts which differ in their use in physics and mathematics i.e : symmetry group, discrete/continuous group, continuous and discrete symmetry etc. I am trying to consider the following concepts and phrases for the definition: Parity symmetry. Parity transformation. Invariant property (invariance and symmetry while similar they also differ in use) parity symmetry group. discrete symmetry group. Then: In physics: "Parity symmetry it describes the invariance of a system, it's properties, under the parirty transformation, a spatial transformation, represented via the symmetry group of parity, a discrete symmetry group." Is my definition, while including the above listed phrases accurate? Also, can someone show me the fact that there is a group for parity? I.e elements, neutral element, inverse etc? In other words, which are the elements and the operation in the symmetry group of parity?


As we know, two bodies undergo radiative heat exchange due to each emitting a spectrum of light according to its temperature (blackbody radiation). When one body is hotter than the other, it emits a higher magnitude of light at each frequency. Thus the heat flow is from the hotter body to the colder body, even though light is exchanged in both directions. My question: as the photoelectric effect showed even a large magnitude of light of the 'wrong' frequency can't eject an electron, has the experiment been done where a large magnitude (high amount of W, i.e. not according to blackbody radiation) of low-frequency light was directed to an object to observe if the large amount of low-frequency light heats the object? Or is it not possible for other reasons?


Suppose you trying to sell the idea of the Yoneda embedding to perhaps a rather mixed bunch of students (so you can't presuppose too much mathematical background). Still you can say Think of a group-as-a-category (one object, all the arrows isomorphisms). Then applying the Yoneda embedding theorem we get ... [a bit of chat] ... hey, Cayley's Theorem. Think of a poset-as-a-category. Then applying the Yoneda embedding we get ... [a bit more chat] ... hey, the familiar result that a poset is isomorphic to a certain bunch of subsets of its objects (upper sets) ordered by inclusion. Those are in fact the usual textbook offerings. But what third or fourth examples of such embeddings (not requiring the fully caffeinated Lemma) might work as equally accessible? Or maybe not quite as accessible but more interesting??


(Look at the picture) Let's assume there is an horizontal plane impacted by diagonal airflow with components from coming downwards and ahead. If we say that the airflow is fully deviated by the horizontal window (which isn't allowed to move through space), the outcoming airflow has only an horizontal component greater than the one it had before. So, if the wing generated a force in order to change airflow direction, the airflow generated an opposite force on the wing. According to this theory as long as the horizontal component of the airflow isn't reduced, the wing is pushed both upwards and forewards, even though the airflow comes from ahead (which is counterintuitive). Is that correct or am I missing something, since I would normally think the wing must be pushed backwards?


I'm finally closing some gaps in sound waves, so forgive me for lots of questions. In metal, it's said sound travels fastest. The reason is molecules are tightly packed(more dense) in metals than in air lets say, and collisions with each other will be fast as each molecule doesn't have to travel long before it collides with another one, hence it travels faster. Though, I wonder even though it will be faster, we should have disadvantage here which is the signal created by source won't be much weak than it would be in air. My logical thinking is since molecules are tightly packed, it will be harder to transfer energy from particle to next particle(more tightly packed, harder to vibrate the same frequency as created by source). So even though sound travels fast in metal, it should give us weaker signal in the end than in air. Would this logical thinking be correct somehow?


I'd like to know how the parallel transportation behaves in non-Levi-Civita connections and how does one realize it formally. I know that parallel transportation along some piece-smooth curve is defined through moving by geodesics: say, in case of conformal connection, one transports a vector along a short geodesic close to a piece of the given curve preserving the angle at each moment of moving, so, for example moving by a horizontal line in the flat realization of the Lobachevsky plane (where geodesics are semicircles and vertical lines) one gets that a vector rotates uniformly. But how does one understand the parallel transport in general (of a general connection given on a principal bundle)? So, are there any axioms of the parallel transport except being an isomorphism between model spaces in close points on a smooth manifold and "additivity: moving along one piece of a curve and then the second" and "inverse: going in both sides" axioms?


I heard that if a dish of mercury is heated by a moving flame placed under it, the mercury will spin around - Sanderson [Ivan T. Sanderson, biologist and paranormal researcher] then goes on to make the basic observation that a circular dish of mercury revolves in a contrary manner to a naked flame circulated below it, and that it gathers speed until it exceeds the speed of revolution of said flame. Now, as in this question Why is mercury magnetic? , if an electric current is passed through the mercury in the presence of a magnetic field, the mercury will spin around, but I can't find any reference to a version with a flame. Is this true/would this theoretically be true? If so, why? Or was the person who said this just misremembering the electric current version?


While calculating the electric potential at a point near charged bodies such as a uniform ring, hollow shell and solid sphere, iv'e seen that the potential at a point is equal to: V = KQ/D(avg) Here D average denotes the average distance between point of potential measurement and the elements of the object. It is somewhat like the distance between the "Centre of charge of the body" and the point of measurement. This seems to work out for my few cases which ive tried. It also sometimes works for electric field at a point by replacing D with D squared, but fails in some cases such as with a uniform ring. My doubt: Is this approach of calculating "Centre of charge" always applicable for potential at a point? Why so? And if yes, why does it fail for electric field strength at a point?


I read several times about global warming leading to more exteme weather events, i.e. flooding, droughts and even winter storms occuring at higher rates and with more intensity. So, higher temperature supposedly leads to an increase in the variance of the probability distribution of the weather. This is not obvious at all. Naively, one might think that climate change would just shift the temperatures by several degrees. So, is there any physical mechanism, explaining the increase in the variance of the weather? I thought about the equilibrium vapor pressure of water at higher temperatures but I did not come useful conclusions. Btw, a standard physical model is the damped harmonic oscillator. Is there any similar model to demonstrate the effects of globabl warming or meteorolgy and climate change? I'm thinking of something like a closed box with stone islands and water under some radiation.


Suppose a player wishes to move on a square lattice graph without diagonals. Vertices on this graph have a chance (fixed or time-dependent, e.g. cumulative density of X~Po(t) with range n to infinity) to become a "point of interest". Upon visiting such vertex, it will revert to a "normal" vertex, which can become a "point of interest" again. The player wishes to visit all "points of interest", but travelling through the least number of edges. The Elevator Algorithm seems to solve a simplified version of this problem, where the "player" can only travel in two directions. My guess is that, if the player starts from the center of the lattice graph, the optimal choice would be either try to walk over all the vertices exactly once for multiple rounds (Hamiltonian path maybe?) or to stand still. However, this guess only works for the fixed-chance scenario. Thus, I wish to know how an optimal strategy would be derived, given a certain number of lattices are already "points of interests", especially when vertices convert to "points of interests" with a time-dependent chance.


I was recently reading about the Hafele-Keating experiment and asking, how does time in the plane which has flow westwards could have passed faster than on the surface of the earth if the frame of reference was here. It was then I realised that the frame of reference in the experiment was chosen to be the center of the earth. But now I'm asking what would have happened, if one really chose the frame of reference to be on the surface of earth. Then, the plane should have had a velocity greater than that of the inertial observer on the surface of the earth in both direction (in one even greater due to earth's rotation) and time should have passed in both situations slower with reference to the observer on the surface of the earth. I'm not sure if these assumptions are correct, as I've just started to learn SRT on my own.


Isobars are atoms (nuclides) of different chemical elements that have the same number of nucleons. According to the https://en.wikipedia.org/wiki/Mattauch_isobar_rule if you have two adjacent elements on the periodic table have isotopes of the same mass number, one of these isotopes must be radioactive. The decay can happen by positron emission, Electron capture of Beta decay. When electron capture occurs, there will be a hole in the first electron shell that will quickly be filled by an electron from a higher shell, giving of what's called a https://en.wikipedia.org/wiki/Characteristic_X-ray Something I've realised though is that if the Isobar with more protons is the heavier Nuclide, but the mass difference is less then the Characteristic X-ray of the lighter Nuclide, then it would be impossible to decat via capture of an electron from the innermost shell. Are there any Nuclides where this is the case?


This may be a very basic question, please excuse my lack of knowledge but I don't seem to understand the concept of anti-matter gravity. Upon research, many sources align with the conclusion that anti-matter reacts to gravity similarly to matter. ie. that it space-time warps around its mass. If we consider a particle/anti-particle pair we would expect and conclude that there is no gravity (warp in space-time) by this. ie. Just the vaccum of space. However, if we consider them as separate particles, then their corresponding gravities -pulling spacetime in the same direction, would add together (since gravity is not reversed for anti-matter). We would expect to increase the total gravity as they approach each other (greater density of mass in space-time). However, as we know, a pair would have no gravitational effects. It's hypothetical but to clarify; consider a plane vacuum with one particle and one antiparticle beginning to collide. Disregarding their electrical attraction, they each warp space-time in the same direction due to their masses. However, after the collision (as a pair) there is no gravity. So, how would their gravitational potentials dissipate and react as they approach each other and collide.


I'm trying to create a cheap concave lens effect for a class I'm doing. It seems like a convex lens starts to create a similar effect anywhere passed twice it's focal length. It also makes everything upside down, but that's okay. Are the effects of a convex lens a demonstrational equivalent to a concave lens once you pass double the focal length? Ignoring the upside down aspect. This is for a hand made kaleidoscopes library program to give you some idea of the stakes. EDIT: I set up a magnifying glass and walked some distance away from it. What I see through it is: upside down, in focus, and inclusive of a wider view than what it blocks. I can see an entire shed within the lens, while it does not block me from seeing the shed around it.


Applied Force is our label for a contact force that a person exerts. When an applied force acts at an angle, it is actually a combination of two forces: normal and friction. The component of the applied force that is perpendicular to the surface is a normal force, and the component parallel to the surface is a friction force. I have two questions; When the applied force is at an angle then forces of friction and normal are between the surface of the object and the hand, not between the object and the ground so can we say that the parallel component of applied force is responsible for the acceleration of the object? (Is it okay to say that it is the friction force between our hand and the object that accelerates the object)? What will happen if an applied force acts on a thread also can we say that the applied force is the combination of normal and friction force between my hands and thread if so then tension force will be also called a combination of normal force and friction force or if I am wrong then please correct me.


I believe this question would have been asked before, but not like this. The popular answer to this question is that the slide-release action of a bow sets up vibrations in the strings, of which ultimately only the resonant frequencies would survive. Plucking a guitar string sets up a transverse oscillation on the strings at its resonant frequencies. But when you play a violin, you slide a bow over the strings. We know that the strings first attach to the bow, then release once the static friction is overcome, and this process repeats again and again. But how does this process ensure periodicity? We don't hear a discontinuous noise from a violin; rather we hear smooth continuous notes. How do the strings know when to catch the bow, and when to release them ?


Bell's therorem seems to disprove localism because measuring, let's say spin of an entangled electron, seem to communicate the measurement to it's another pair instantaneously. But isn't another thing possible? Maybe the electrons are not communicating anything and instead the two instruments which are measuring the electrons "know" at what angle the other instrument is measuring the other electron and that is a part of the measuring process of instrument and thus the Bell's equality is violated but local realism is still valid. Here the two instruments and the people who are performing the experiments are in some kind of weird sync where they cannot measure arbitrarily in any direction but instead the measurement angles are predetermined before hand and the instruments already know the angles with of each other which are used to measure, hence the spin directions can be correlated. This would mean that Universe is deterministic but also locally real. Is this at least theoretically possible? If it is possible wouldn't it be more saner theory to adopt rather than throwing local realism away? Why? Why not?


In "Gravitation" by Misner, Thorne and Wheeler the authors pose the following puzzle: The metric perturbation of the wave changes the scale of the distances slightly, but also correspondingly changes the scale of time. Therefore does not any possibility of any really meaningful and measurable effect cancel out? And they also give an answer: The widened separation between the geodesics is not a local effect but a cumulative one. [...] When one investigates the separation of geodesics [...] over a large number of periods he finds a cumulative, systematic, net slow bending of the rapidly wiggling geodesics toward each other. This small, attractive acceleration is evidence in gravitation physics [...]. Is LIGO measuring only a net acceleration (or maybe a net displacement as a cause of the net acceleration) of the mirrors instead of a wiggling? The LIGO collaboration states the opposite. On the FAQ page (https://www.ligo.caltech.edu/page/faq) you can find Back and forth the waves pass through (interfere with) each other as the arms themselves change length, causing light interference that ranges fully between totally destructive to totally constructive. In other words, instead of nothing coming out of the interferometer, a flicker of light appears. And how could we obtain the frequency of the gravitational wave when averaging over a large number of periods? But if I misunderstood the statements in the book "Gravitation", what is the correct solution of the puzzle stated in the beginning?


Consider a simple gas (or fluid) within a box at thermal equilibrium. I manage to give a kick to one particle within the gas, such that it acquires some momentum. After some time, it should be expected that this localized energy (or momentum) input would be just distributed such that Boltzmann distribution is recovered. But how does this recovery take place ? The simplest mode of transport of this momentum disturbance could be for example a sound mode, that is, a sequence of kicks in the same direction of the initial kick is generated starting from the kicked particle. But I can also imagine that after the first kick, momentum is transferred in random walk slow manner. Here the momentum disturbance would be tracing a some kind of random trajectory instead of a linear trajectory as it would be for a sound mode. How to know which kinds of modes are taking place (roughly depending on kick strength, density, temperature and so on..) ?


In this post Reconstructing isomorphisms via the bijection between the corresponding posets of subobjects I asked for the possibility of constructing an isomorphism via the order-preserving bijection between the corresponding posets of subobjects. In the case of the category of sets just work with the singletons to define the bijection - I'm working with classical NGB set theory - Now, an interesting counterexample has been exhibited in the aforementioned post. However, I've another question. Well, which are the categories sharing the same property as set (i.e. to give rise to an isomorphism between the corresponding posets of subobjects). I was thinking to toposes. Might it be a good way or are there also counterexamples coming from toposes? Actualy, there exists a paper by Barr and Diaconescu entitled "Atomic Toposes" (https://www.math.mcgill.ca/barr/papers/atom.top.pdf). Well, I think that what I'm searching for is related to the fact that the subobject lattice is atomistic, i.e. every element may be expressed as the join of its atoms. It is property related to my problem?


If something is said and people understand it, sounds like a word to me. If people didn't understand it, I would think they would rather say, "what do you mean?", than making it a word after a definition is explained. Or are there exceptions I'm not thinking of? Dictionary or not, if people understand something, even if just locally, it's a word. Of course, there is some irony in people seemingly claiming to have a mastery of language who do not know what the word 'word' means. One example may be this David Cross bit I've heard, where he said 'irregardless' isn't a word, even though it's been used for a long time and is understood. However, it is considered nonstandard in at least some dictionaries. There may be cases where something incorrectly thinks that a valid word isn't part of standard English even though it actually is. I would still include this in the scope of the question, because they they think it's wrong, whether they are right or not. If someone is making a mistake, if it's still understood, it'd still be a word, for that moment, at least. A more common example may be the use of "ain't". (Kids exclaiming 'that's not a word' out of disbelief would be an exception, but I think it's a rather different usage.)


I read this fascinating paper (RG). The total collision is a well-known singularity in Newton mechanics: the distances become zero and, therefore, the potential becomes infinite. In a paper before that one above, Paula Reichert describes a try to calculate through the singularity using shape dynamics. In shape dynamics, the absolute background space is abandoned and only relative positions and especially angles are considered. Unfortunately, shape dynamics alone turns out not to be sufficient to calculate through the singularity. Then, in the paper above, they make a reference to this paper where they calculate through the big bang singularity by adding a scalar "stiff matter" field in the Bianchi IX universe. How can one understand this scalar field? Since shape dynamics doesn't rely on an absolute space as a background, it comes to my mind that this scalar field may be interpreted as space itself, space that surrounds every single mass point as a field. Is that interpretation right or wrong? If wrong, how can one get an understanding for this scalar field?


I'm attempting to develop an understanding of how equations are developed and wondered whether or not all equations started their development in the quest to document observable phenomenon or are there any 'purely synthetic' equations, which are developed from the basis of a thought in order to produce numeric results which are useful in some way. Put another way: Do observations of phenomenon drive the development of equations or does a conceptual need for a tool drive their development instead? The reason I ask, is because as someone who is a novice in mathematics (arithmetic skill, elementary algebra and descriptive stats), the choice of the construction of equations / their form always leads me to wonder why this form was chosen. In particular as I delve further into stats I have seen descriptions where certain equations may be adjusted based 'on ones need' which further deepens the difficulty of just using a tool blindly and wondering at what point or in what way I can develop the skill to discern how to choose how to alter an equation to better fit a desired outcome... I attempted to articulate this originally HERE, then asked about whether it made sense to start a new question HERE, but for lack of feedback in either, I decided to open this query and post any positive results as link backs to those two questions for the community's later use. If there is a better way to approach asking this please let me know and I'll be happy to adjust accordingly.


In a laboratory a vessel was built which can sustain high pressure, thermostats and pressure gauge were connected, assuming closed system, dry ice was introduced in closed vessel, the temperature of vessel was increased the sublimation of dry ice resulted in formation of gaseous carbon dioxide increasing pressure in vessel, the increased pressure and temperature resulted in liquid carbon dioxide formation, after more heating critical temperature was reached and superfluid carbon dioxide was obtained, at these state gas and liquid phases became indistinguishable, ice was put upon vessel surface which resulted in decrease of temperature and the liquid carbon dioxide appeared again. My question is if we see typical pressure vs volume diagram it is observed that above critical conditions it is not possible to condense fluid by reducing temperature since the point will lie above the dome, then how come we see condensation of superfluid carbon dioxide in our experiment ? Note: the boundary liquid gas interface was seen disappearing in as it's typically seen in superfluids hence it's confirm that we had superfluid carbon dioxide.


I encountered a proof that the empty set is a subset of every set via this comment(Is "The empty set is a subset of any set" a convention?) which shows that it cannot be false that the empty set is a subset of every set. Without necessarily going into a proof of how the empty set is a subset of every set, I was wondering why the fact that it cannot be false that the empty set is a subset of every set shows that this is true- could it not be the case that the concept of subsets is meaningless with regards to the empty set, and it is not enough to show that it could not be false; that this statement could neither true or false as it has no meaning in this context? Also, I would appreciate some explanation as to how this condition holds "vacuously" as far as terminology, as I have learned that for an implication to be vacuously true, it is true when it's hypothesis is false. Thanks


I have a very abstract at the same time awkward question. In many formulas across physics we need to take several approximations and often we derive formula from previous formula which had certain approximations in them. By error analysis we know that errors keep getting carried forward. Hypothetically can there be situation in future when almost every advanced formula we use, which is a result of derivation of several heavily derived formulas(I am referring several heavily derived formulas as those which are derived from highly approximated formula), be filled error quite significantly high and our calculations are thrown off by huge percentage of error and inaccuracy. Can the error propagation in formula and theorems be a major problem we face in the future of physics? (Please note that I am not talking approximations involved in process of integration.) I am just curious to know to what level the approximations we take, effect the calculations we perform.I know this is a very awkwardly framed question. If you have any suggestions on improving the language of the question or any other changes, please let me know.


Per wikipedia: natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. Let's take a wine glass as an example. The wine glass sits on a table, it is not visibly moving. But since natural frequency exists, therefore it is oscillate on some minute scale. Where does the oscillation come from? Is it from the electrons and protons that are whizzing inside of the glass and getting excited due to stochastic heat? Is it because protons are hitting the surface of the glass? Or is it oscillating due to the minor tremor in the earth? Or perhaps the minute attractive force exerted upon it by all other objects? In each case, it is clear that there is a driving force: heat, proton's impact force, tremor. So clearly natural frequency must be associated with a type of oscillation that's not any one of the above. So where does the oscillation come from instead? It seems that given any source of this oscillation, you can identify a driving force. Hence such driving force is never absent, hence there is no natural frequency. Are all objects naturally oscillatory and that natural frequency is like a fundamental aspect of an object, such as its 'mass'? If so, can we identify this natural frequency in some way? E.g., I want to know my body's natural frequency at this moment.


My admittedly limited understanding of QM is that it is a matter of probabilities whether or not a photon is (re)transmitted through a polarising filter and that these are a function of the relative orientation of the polarization of the photon and that of the filter. Unfortunately, a single detection event cannot distinguish a very likely from a very unlikely one. Since the same applies to a case in which the photon is absorbed, it seems that not much information, if any, can be gained, when what would be nice is that we could learn something about the polarisation of the photon for example. Even if statistics are collected on streams of successive photons, they're likely each to be oriented differently in relation to the filter, so not much information can be gained by examining the statistics either.


Currently i include png/jpg files with graphicx package compiling with lualatex, no problem, in particular no error message because of bounding box not found. This question is about historic ways to compile because I don't want to break old code. First question: Is it true that when compiling into pdf with any of the modern compilers like lualatex with graphicx package command includegraphics, there is no special treatment necessary to get a bounding box? Second, I seem to have no problem with bounding box either when compiling into dvi. What makes me a bit nervous is, that I remember that message. I am sure i used package bmpsize and also command ebb to get the bounding box and this was necessary to avoid the error message cannot determine the size of the graphic But I cannot remember the details. My question is whether the issue with the bounding box is resolved by new compilers, even if compiling into dvi or xdv. I made a remark by hand, that bmpsize does not work with xelatex. Is it obsolete? May it be the case that behavior depends on the source of the jpg/png? Are there aspects related with the dvi to pdf converters? Are there other formats needing bmpsize or ebb? Please feel free to mention other aspects also. I am afraid about dropping ebb or bmpsize when there is still a use case. In the meantime I found out that my problem is specific for texlive distribution. Seemingly miktex neither provides bmpsize nor ebb.


The following pop science video discusses the experimental difficulty of measuring "quanta" of gravity. It seems to me that a significant argument involves how it would be difficult to investigate fluctuations of gravity on the plank scale, requiring detection methods that have densities that would collapse as a black hole. In optics, one way of "seeing" a photon is by interfering it with using homodyne detection to interfere it with a strong signal. The strong signal essentially magnifies the signal, allowing it to be seen by detectors that are much less sensitive than single photon level. Could such a scheme be used to measure gravitons? For example, if strong gravitational waves can be detected by LIGO, shouldn't a sufficiently strong wave already be measuring the "amplitude" statistics of the graviton? Similar to how a local oscillator (in a balanced homodyne detector) measures the shot noise (amplitude statistics) of the vacuum. I suppose the issue is that this needs to be a particuarly strong wave signal so be above experimental noise?


In the diagram, OA is perpendicular to PQ. O is the centre of the circle. AC and EP meet at B on the circle and QC and AE meet at D on the circle. The goal is to prove that AP=AQ. I had tried to prove this by drawing a line through E parallel to PQ and I had proved some of the triangles were similar. However, this is the part where I got stuck. I drew a line parallel to PQ through E. I named the meeting point between PE and QC "G", between the line produced at E and CQ "F" and between CA and the line produced at E "H". I then proved that triangle CBG is similar to DEG, CAD is similar to BAE, HBE is similar to ABP, FDE is similar to ADQ and FGE is similar to PGQ. It is at this point when I got stuck.


Recently, I tried to understand the effect of Quantum levitation. I read that there is some well-known fact: Relative distance between superconductor and magnet must be preserved, because of flux pinning mostly occurred in type II superconductor. My rough understanding about this is, quantisation of (penetrated) magnetic flux inside the superconductor, and continuity of magnetic flux depending on the distance from the magnet, induces the fixation of relative distance (It couldn't move continuously after fixation because there is no no-jump-but-continuous motion). I could understand the second part of the video. I guess it's because of the circular symmetry of magnetic field. There is nothing hard to understand the story. However, what about the first part? If we rotate the magnet, then what makes the superconductor follow it? Obviously, it's due to the preservation of relative distance in layman's terms. However, could we more elaborate on it? For instance, in the context of classical electrodynamics, someone could try to explain this by the induced electric fields and currents from time-varying magnetic fields. However, it may not be an answer because of the quantum nature of a superconductor. The quantised flux varies by the time dependent magnetic field, but could we describe such a variation in terms of the motion of the magnet?, and so, could we derive and describe the equation of motion of a superconductor?


I've noticed this quite often from Disney sound tracks. Lyrics are in a certain rhyming scheme - one line happens and the next line approaches the end of the line, but the last word subverts the rhyme. I have looked at this question and feel it's related, but not the same effect. Example: "In Summer" from Olaf in Frozen Da da, da doo, a bubba bubba boo The hot and the cold are both so intense Put them together ? it just makes sense! Rata ta ta dada dada da doo Winter's a good time to stay in and cuddle But put me in summer and I'll be a happy snowman! Obviously, the expectation is that the word is "puddle" It feels like a lyrical version of deceptive cadence, but I cannot find a succinct term for such an effect. The linked question accepts the answer "Broken Lyric" or "Enjambment", which may still be the answer here, but I am certain that these are two different effects.


I am a first year student of applied mathematics. I currently find myself always trying to explain any phenomenon with mathematical models- the few ones (models) that I encountered till now. I always try to fit phenomenas to models like a data scientist does when he fits data to models. Recently I found this book: Robert B. Banks Growth and Diffusion Phenomena: Mathematical Frameworks and Applications This book is different from other applied mathematics books in the sense that it focuses on a particular model unlike others that try to model the whole world in a single book. It particularly focuses on growth and diffusion phenomena. Which is , to me , great because this book studies these two models in great detail rather than just stating many models on the surface level. I also found some book that is specific to some single phenomenon like this book on Hysteresis Mathematical Models of Hysteresis and their Applications Can someone please recommend to me some books like these that describes a certain phenomena or discusses a certain mathematical framework to study in detail and apply some phenomena? Any book that discusses and models in detail some phenomena: could be growth,could be decline, could be (what happens in the limiting case)- type of thing, could be something else, could be anything...


The boundary of a simple closed curve has each point on it, touching two adjacent points one on each side. Can the entire boundary be represented as a series of circles of the same diameter "d" where each circle is touching the adjacent circle at exactly one point only without overlap in areas between the two circles. This will allow us to approximate the simple closed curve with a polygon where each side of the polygon is represented by a line joining the centers of adjacent circles and at a distance equal to the diameter "d" of the circle. Such approximation would imply that a inscribed square exists on the polygon (Toeplitz conjecture special cases) and therefore an "almost square" whose vertices are within the error range of the diameter of the circle must exist. Shrinking the size of the circles used will reduce the error of the "almost squares" to smaller and smaller values.


I'm working on an algorithm to allow a vehicle to reverse and then line up and stop on a given spot facing a specific direction. The best approach I've been able to come up with involves two internally tangent circles with the larger extending from the point of tangency of the vehicle. Vehicle starts at point A and is parallel with tangent line T. Vehicle reverses to B, then moves forward to C, then to A so that it is parallel with the vector displayed instead. r represents the minimum turning radius of the vehicle Given that line AC is always arranged vertically and knowing point A, but that the tangent line T can vary, is it possible to be able to determine points B and C? If not what other info is needed? Medium tangent Steep tangent Shallow tangent For simplicity I've made it so that the radius of the larger circle is twice the radius of the smaller. If this restriction is not required to solve the problem how could this larger radius be determined as well?


A few days ago, I came to an amazing intuition while using GeoGebra, and I could not prove whether it was previously discovered or is it new, please mention a source if it already exists... The sum of the lengths of the tangents of two circles touching a conic section, each of which touches it at two points, and starting from a point on the circumference of the conic section is equal to the distance between the centers of the two circles multiplied by the reciprocal of the coefficient of central difference, where the lengths of the two tangents are two vectors and the shortest of them will take a negative value if the point taken from the segment is not located between the two circles. These are illustrations that include different relative states: direct result: The length of a tangent to a circle tangent to a conic at two points that radiates from the point of tangency to another circle tangent to it at two points is equal to the distance between the centers of the two circles.


I've read some description of certain house design as below: Living and dining rooms are towards the front with kitchens to the rear, with bedrooms either off the hallway or upstairs if it's a two-storey terrace. and here's the corresponding house plan: the definition of "off" in google: as I understand, and which can be seen from the floor design, "bedrooms are off the hallway" sort of indicates that bedrooms are separated by and adjacent to the hallway, and here "off" is preposition, but I'm not sure which one of the two preposition definitions fits it. The definition "situated or leading in a direction away from (a main route or intersection)." seems close but I think it's used to describe something not adjacent to the main route such as in "my house is off the main road", which is obviously not the case in above house plan example. Thanks in advance for your help!


Suppose we have a closed circuit composed of a time constant voltage battery, a resistor and superconducting wires connecting them in series. In the equilibrium state, both the free charge the electric field in the interior of the superconducting wire should vanish, lest the free charge accelerates and there be no equilibrium/steady current. The electric field generated by the charges in the battery should be counteracted by nonzero surface charges on the superconducting wire in order to generate the zero electric field inside the wire. At the locations of the wire that are far from the battery and with small curvature which should approximate a straight wire, the surface charge density should be close to uniform. However, I have often read people claiming a superconducting wire with a constant equilibrium (direct) current should be free of surface charge. I would like to know what the bases is for their argument. Are there some references on this topic?


why potential at any point in a uniform electric field is not same i.e. why potential difference between any two points in the uniform electric field is not zero? According to me if i bring a test charge from infinity to a point A in uniform electric field [assume the field is due to a positive charge] then i will have to apply external force on the test charge against the electrostatic force by the uniform field such that external force is equal to electrostatic force so that kinetic energy of test charge would remain constant thus some potential energy will be stored in the test charge at point A.....NOW if i will bring the test charge from infinity to some another point in the uniform electric field...let the point be B then according to me i will do the same amount of work done on the test charge to bring it to point B because i will apply the external force that will also be equal to electrostatic force by the uniform electric field.....and because the electric field is uniform the test charge will experience the same amount of electrostatic force at each point on the uniform electric field i.e. electrostatic force at both the points A and B due the field will be same. Hence according to me the potential energy at each point on the uniform electric field will be the same thus the potential also. Kindly explain.... why not? thanks!


In the Hertz experiment, when light falls on the anode, electrons are emitted and they move towards the cathode and thus flow through the circuit. However, when reverse voltage is applied and the polarity of the electrodes is reversed, electrons being emitted from the anode (formerly, a cathode) only come out in the air/vacuum of the glass tube and do not flow in the circuit even though voltage is applied. How is it so? The emitted electrons should be repelled by the cathode and should enter the circuit through the wire attached to the anode as the increase in free electrons results in increase of current. The applied voltage should also direct the electrons to pass through the anode. In this way, a sort of capacitor might be created. Instead, the current is said to reach zero as stopping potential is reached. How can this be justified?


In fluids, in the state of equilibrium, there can only appear internal normal forces and these forces always have a tendency to compress the bounded volume in the fluid. Cited - Physics Galaxy ( Ashish Arora ) Would it be apt to say that in context with fluids, if I drop a block, the pressure exerted by the fluid on the block will be the effect of a compressive internal normal force over an area ? In addition, why is pressure independent of the amount of fluid above it, since it is the weight of the fluid that is causing the compressive force to act? What i think: Maybe somehow Pascal's principal acts on this, equalizing the forces and leading to pressure equalizing at all heights irrespective of the shape of the tube, if I'm right, could you elaborate on how it actually happens?


I want to recreate a plot where the area that is not visible from a certain point on earth (latitude and longitude) trough a full year. An example of this plots for example can be: Where in white is marked the zone a telescope is not seeing over all year. I searched in internet but i can not find anything related on how to do it, also Astropy don't seem to have any tool to do this calculus. My first idea for the approach can be first of all maybe represent in equatorial coordinates where this areas should be straight lines (but i don't know how to calculate the coordinates that this straight lines will have). And maybe after we can transform to galactic coordinates somehow. If someone have an idea on how to do this will be helpful.


Some examples needing an explanation, thank you for the help. "I came up with (an) idea, we can do..." We should use "an", shouldn't we? But why, if we have a specific idea explained further? What about "He gave me a/the idea to admit to the university" then? "The conclusion is ..." Why should we use "the", when it's the first mention of the conclusion. Maybe because that conclusion is specific, only for that situation. Then should i use "the" in the next sentence "I came up with the conclusion. The conclusion is..."? "I have a/the feeling i don't love him". "The" because i tell what specific feeling i have further? "I am gonna watch a/the video". What if only i know that video, and i want to watch a specific video, but another person in a dialogue doesn't know that video.


Consider for example the open sentence, x lives in India where 'x' is a variable, which can be substituted by various constants to make declarative sentences. For instance, Harshit lives in India Carol lives in India Zaid lives in India where 'x' has been replaced by constants 'Harshit', 'Carol' and 'Zaid', respectively. I don't understand how do we exactly come to know which constants can substitute the variables in an open sentence. I mean, if someone says, significant substitutes are names of people, in this example, I would say why not of pets? Why not just of people of a particular City or ethnicity? Since, this question is from logic, which is a prerequisite (AFAIK) of set theory, I would humbly expect no notions from set theory to be used.


In the most known incarnations of string theory, we have to compactify the needed, additional dimensions that are then taken to be a.) periodic and b.) "small" in some sense (most of the time their size/radius is taken to be on the order of the Planck length). Some models do propose so-called large extra dimensions but even in these the extra dimensions are compact. Now my question is, are there any string theory models in which some or all of the extra dimensions are non-compact and if there are, can you recommend me any literature? I am thinking of brane world scenarios such as this, but with six or at least some non-compact dimensions (but I would be glad about any example, even if it is not a brane world scenario).


Find the perspective sequence that maps collinear points A,B,C,D to D,C,B,A. Attempt: If we need to find a sequence of three perspectives that (A,B,C)->(A,C,B), where A, B, C are collinear, then first we mark p is a line through A, B, C, let S be some point that does not lie on p, let q be a line through A and S, let p' be a line through A different from p and q, let B' be an element of the line BS and C' an element of the line CS, and let S' be the intersection of the lines BC' and B'C. Then the first pespectivity maps us from the line p to the line p' and has its center in S: A->A, B->B', C->C', and the second one from p' to p and has its center in S': A ->A, B'->C, C'->B.


I would (like many before me) like to take my lecture notes using LaTeX. As writing pure LaTeX-code is very time consuming and not nearly as fast as handwriting, I can only really consider switching to LaTeX if I can use snippets and anything else that makes notetaking faster. I have discovered this post on how to get faster using vim, but at this point I don't really want to learn vim anymore. I know it is very hyped, and probably worth learning, but I have so much else do learn already that I don't know if I will ever find the time. At the moment I am very happy with VS Code (or really VSCodium) for all developing, scripting and stuff that I do, and as I can perfectly do just about everything within this one editor, I would very much like to keep it. Saying this, has anyone an idea or a hint for me on how to set up something like the blog post describes using VS Code? That would be very appreciated.


I am confused whether I should use "a" or "the" when I refer to an object given/identified by some other object. For example: The package is now in [a / the] deposit box with the number you received by SMS. For me this is the deposit box I previously stored the package in, but it seems to me, that for others (including the listener), it is still just some deposit box in the given vault: if I make a typo in the SMS message, my listener may hear from the front desk of the vault: A box with this number does not exist in our vault. Regardless of any such mistake, my assumption at the moment of speaking is that the listener will be able to find or identify the box unambiguously based on the number. Which is the correct article in such situations?


Take a primordial black hole and measure the Hawking radiation over a large amount of time by gamma-ray detectors, as well as a Large neutrino detector. Using theoretical calculations about the composition of Hawking radiation combined with the detected neutrino flux you can determine the neutrino luminosity. Combine that with the luminosity in the form of gamma rays and whatever particles the Black hole is hot enough to produce, and then compare that with the loss in mass over time. If you detect a discrepancy, it's likely the missing flux is in the form of gravitons. Would this experiment work? I did also consider using a primordial black hole cold enough to not emit neutrinos but the Hawking radiation calculator I used gave a black hole lifetime greater than most estimates of the lifetime of a proton.


Many problems in classical mechanics, classical and quantum field theory among others require the study of systems of partial differential equations where questions like existence and uniqueness cannot be answered with the typical counting-equations/variables arguments - questions which have physical relevance and that need to be addressed before going to numerics. For instance, problems in surface mechanics always involve the Gauss-Codazzi equations which is a non-linear overdetermined system of PDE. The most powerful technology to properly study general systems of PDEs are Exterior Differential Systems, and there are some canonical references in the mathematics literature like Bryant et. al. I found this challenging for a standard theoretical physics graduate because it requires a solid background in non-conventional math topics for physics like commutative algebra and number theory, so I wonder if there is any reference treating EDS for physicist.


I'm looking at a few English sentences, and I've realized that many verbs can be used as nouns in their infinitive form. For example. I am going on a run and in the present tense I am running. There are many more verbs that this can be done to. Are you going to take the jump? Why did you do three pitches? You know he would have hit one of them. (baseball) Can you please do your exercise elsewhere? It is too loud right here. So 'pitches' in the example is not in the true infinitive, but is in the plural infinitive, so this may not be a true example. 'jump' and 'exercise' are however. Normally, I would say that these are gerunds, however, after looking online for a while,. it seems that gerunds always use the present tense, and never the infinitive. What is the name for these verbs that can be nouns when in the infinitive form?


I'm writing longer mathematical texts (lecture notes) regularly, and I don't know where else to ask this question. I regularly find myself enunciating (in the typographical sense) every statement I make. This means that I will put any even vaguely mathematical statement, not only theorems, corollaries, definitions, etc., into an enumerated environment in Latex. As of now, I always use Remark for them, and I feel like I'm overusing it. Question: When do you write mathematical comments in some numbered environment, like Discussion, Observation, Remark, Idea, etc., and when do you write mathematical comments standalone in the running text? What other environments do you use to mark up mathematical comments? Edit: The reason why I enunciate this often is simple: Later on in the text or in the lecture, I might want to refer to this comment. E.g. in a mathematical comment, we combine all results in an informal discussion on how to compute fibres of maps between prime spectra. This result is somewhat important as it will come up every now and then, but its statement in the lecture was more or less like an argument, making it difficult to encase this as a corollary.


I have written a long book in latex memoir. I now want to feed its content to a machine translator into another language (say, German). I will use the automated translation for a first pass for a better human translation. Translators do not understand latex. There are a couple of methods to go about feeding them. I could use some online latex parsers, write a quick parser myself, etc. It occurs to me that I could also ask latex to create a pdf that is better suited to pdftotext (poppler). It should make big spaces between paragraphs, not break paragraphs across pages, etc., probably take out tables and figures, page headings, footings, and sidemargins, footnotes. (Eventually, I will also need to write a parse-match to integrate the translation back into a latex document that uses the original structure, but just replaces the text. Thus, I am thinking of adding a number for each paragraph, so that I can take the translation input and output and reconstruct a latex file.) Has someone written a style file to improve the results of pdf text extraction? or found a better solution?


I am trying to learn about neural networks. I was reading the paper An Efficient Graph Convolutional Network Technique for the Travelling Salesman Problem which uses graph neural networks such that given a graph we get a hamiltonian cycle which hopefully is nearly optimal. However, they use as ground truth the concorde algorithm, which when I ask google and chat-GPT seems seem to be an exact algorithm but in section E (solution visualization), there are some cases where the author say : The model predicts a shorter tour than Concorde for this instance by choosing a different contour around the nodes at the top-left of the graph But how can it be better? Doesn't concord compute the optimal solution already? Did I get something wrong about the traveling salesman problem?


i have taken two course in complex analysis, one variable and the other in several variable and the begining of complex manifolds, my understanding in one variable go well but in several variable our prof use two books, one of krantz and the other of Hoermander, i think due to my lack in the theory of integration(multiple and differential forms integration) i don't understand some proofs,in particular some construction for Hartog's extension theorem and the other about separate Holomorphicity And this brings me to my questions: do you think a good understanding of Lebesgue integral and Mesure theory can replace Riemann integration theory. what to study in complex analysis to develop geometric intuition,in particular for abstract algebraic geometry and sheaf theory, and if possible is there an approach that don't depend on functional analysis and PDE. and thanks


I use LaTeX for a lot of my classwork (worksheets, graphics, etc). One thing I have not transitioned is my exam writing. An example page of an exam might look like this. This is just a portion, doesn't have the header and such, I'm comfortable working with those. I've not moved to LaTeX because of my need to be placing the answer boxes and blanks in specific spots on the page, and it's frankly pretty easy to drag little rounded text boxes around MS Word to achieve my layout. But there could be a fair bit of programmatic benefit in being able to make some of those blanks more consistently, layout chemical equations more simply, etc. I also have a page of multiple choice for each exam that I'd love to shuttle off a lot of that layout work to LaTeX. Is achieving looks like this reasonable with enough flexibility to be able to shift around those blanks and such on a question-by-question basis? I've found this page about using a zero-sized box. Seems like it be applicable, but just want to check before I start down this path. Any resources or general code ideas for doing what I'd like would be appreciated. I'm sure in the future I'll come back with some MWE, but for now just gathering info on even executing this. Also if the answer is "Probably more work than it's worth and keep using MS Word", I can accept that too :)


I am posting the specific problem I am dealing with for reference and then explain the part I am confused with. My Understanding and approach to the problem I identified that in the ground frame the disc will have some inertia and thus simply remain stationary as the plank below slides away.When viewing from the plank's reference frame it's acceleration will be added to the COM of the disc in the opposite direction.So initially the instantaneous center of rotation is at the exact center. The Confusion I am unable to visualise how the instantaneous center of rotation will change as the COM gains velocity due to the acceleration. The solution that I found said that the points that have tangential velocity along the acceleration will have some added velocity while those against will have some net velocity in the direction of the acceleration.According to me if that happened then the rotation would completely stop which does not make any intuitive sense. Here's the solution for reference Conceptual Confusion The main concept I am struggling with is the instantaneous centere of rotation of the disc.I tried reading up about it on wikipedia but that doesn't clarify it for me.


This question is born out of the riddle where there are two characters, one that always lies and one that always tells the truth, and you want them to tell you which of two doors is the "good" or "bad" door. You can ask either character, "what would the other character say is the good door?" And the lier will answer with the bad door, and the truther will also answer with the bad door. My question is a variation, given two characters where each character might always tell the truth, or might always lie, but both characters think the other always tells the truth, are there any questions you can ask that allow you to determine more information about the composition? For example it could be that character A and B both always tell the truth (and each thinks the other always tells the truth) or it could be that character A and B both always lie (but again each thinks the other always tells the truth), or it could be that one character always tells the truth and the other always lies (but again they both think the other always tells the truth). Are there any questions that can be asked to determine the composition?


Is there a possible slight difference between protons and electrons behaviour while they are accelerated at speeds that cause relativistic changes to their mass. Why am I asking that? Because for a proton we know that it is made of several components which of them are the three quarks. If they have some speed inside the proton it is logical that this is not the same as they have no any intrinsic speed and that this speed does not matter as a portion that is added to the overall speed of the proton that is accelerated to a relativistic speed. Should that small part of the overall speed make a difference between the theoretical calculation of the particle speed to relativistic mass ratio and the empyrical results as the theoretical calculation threats the proton as an elementary particle because the intrinsic speeds of proton components are neglected in the formula?


An electromagnetic wave is usually illustrated with two sine wave shaped fields, one for the electric field and one for the magnetic field. As in the picture below: But as how I understand magnetism, it consists of two poles, north and south (alternatively described with field lines, with a direction, where north and south connects). So to focus only on the magnetic field (the red, B) in an electromagnetic wave; where are the poles located in such a sine wave (the red part)? Is only one of the poles illustrated, such that another sine wave could be added for the other pole? Update: In this video, Maxwell's second equation is described as the field lines in the magnetic field is always connected (north and south pole) - how these field lines are organized in a wave is what I don't understand. Later in the same video, the fields emitted from an antenna, are illustrated as "bubbles of fields" - quite different from how the "arrows/vectors" are illustrated in my picture above. Field lines from the video:


Lill's method is a visual way to find the real roots of a polynomial equation. It involves drawing a straight line with lengths equal to the coefficients of the polynomial. The solution can be found by using a slope of a right triangle. Lill's method - Wikipedia The values of a polynomial function can be "defined" as a distance between the vertice of the last polynomial coefficient and the path that we draw using right angles. Polynomial function's value Cubic function with defined function's value What I want to do is to find the local maximum and minimum in Lill's path. So, I'm curious if I can "define" a line that represents the first derivative of a cubic function and let it equal to zero, or if it can't be done. What I have done is by using the derivative's definition, and I got that a point (P in the picture) is the derivative of a function which its value is zero. But I think that's not the correct way to do it. My way to do the derivative in cubic function I drew the function and its derivative with the same "first coefficient" Please correct me if I'm wrong, Thanks.


I need to retrieve Roll and Pitch (but not Yaw) rotation values from rotation matrix (or euler angles, or quaternion, input can be different, fortunately, it can be converted from one form to another). [Info table] X points Left. Y points Up. Z points Forward. Rotation around local X axis is Pitch. Rotation around local Y axis is Yaw. Rotation around local Z axis is Roll. If I just use Euler Angles then, in some situations, I get wrong values. As I think, this due to rotation can be represented with multiple variations of Euler Angles, so it breaks when rotation around X axis (Pitch) and Z axis (Roll) is represented also using Y axis (Yaw). gif (How it works | How it should be, but I need rotation without Yaw axis) Also algorithm is running in time, so, solutions, computing some delta rotation, are also accepted.