I understand that the state space representation is mathematically equivalent to the transfer function representation for linear systems, and that it allows us to solve the corresponding DE by finding the eigenvalues of a matrix. However, for nonlinear systems, the transfer function can only represent a linear approximation, while the state space form can represent the full system. But what's the advantage of using state space form for nonlinear systems, if we can't generally solve them by matrix methods? How does state space representation help us analyze or design nonlinear control systems any better than we could by sticking with the original DE representation? Some background: My impression was that the state space form of linear systems is essentially just syntactic sugar for the final result of transforming a nth order DE into a system of n first order DE's, and writing that system as a single matrix equation. It "hides" the derivatives under the extra parameterization variables. But for nonlinear systems, we can't just get a system of linear equations and write it as a single matrix equation that doesn't explicitly involve derivatives. So I don't see how the state space form simplifies anything for nonlinear systems.


In my school, I learned that when two blocks are placed on the ground with one block above the other, if a force is applied to the lower block, two opposing forces of friction act on it: one from the ground and the other from the upper block's surface. Consequently, according to Newton's third law, the upper block experiences a friction force in the forward direction. However, I have a question regarding this scenario. If the external force applied to the lower block is significantly less than the limiting friction of the ground, the lower block won't be set into motion due to the opposition from the static friction of the ground. In addition, I believe that the static friction of the upper block also plays a role in opposing the motion(as it does when the block do move). Consequently, the upper block should experience an equal and opposite reaction that sets it into motion as well. However, this doesn't seem to happen in reality. What misconception do I have in this situation?


In analyzing Compton scattering we consider the conservation of both energy and momentum. However, in analyzing the photoelectric effect only the conservation of energy is taken into account. In fact, if the momentum of the photon is taken into account there seems to be a violation of the law of conservation of momentum. The momentum of the photon is into the metal but the momentum of the ejected electron is out of the metal i.e. the momentum of the ejected electron has a component opposite to the momentum of the incident photon. A way out would be to argue that the process of ejection is complicated and that it is actually the atom which ejects the electron. There is, however, a problem here. Experiments with linearly polarized x-rays or ultraviolet light show that the electron is always ejected in a direction parallel to the electric vector i.e. perpendicular to the direction of the incident beam of photons. This suggests that it is indeed the photon which ejects the electron, not the atom. Is there an explanation in either quantum mechanics or classical electromagnetism or both?


I'm not sure if this is an appropriate question for here, so fair enough if it gets closed/down-voted. I'm self-studying mathematics, did basic real analysis (Riemann, not Lebesgue integraton) and complex analysis, and I'd like to move on to functional analysis shortly. I have two books: Muscat's Functional Analysis (Universitext), and Axler's Measure, Integration and Real Analysis (GTM). It seems to me that Muscat is rigorous but still lighter (not meant in a bad way) on the maths, but covers more topics than Axler. But Axler spends a lot of time on measure theory (of course) and does also cover the basics of Hilbert spaces and Banach spaces and linear operators on Hilbert spaces. At this moment I'm more inclined to start with Axler, since I suspect after covering Axler I'll have a stronger foundation and more easily pick up on the topics covered in Muscat but not in Axler. Would this be wise? Other recommended books welcome.


I was playing with numbers in my middle school math club and found a beautiful pattern. I presented my idea in front of my club mates. The teacher was impressed with my result and suggested I write a short manuscript and have it published somewhere. I thought it was a great idea so I agreed. It took over six months to get used to latex but I think it was worth it. I had some previous experience with the Lua language so I chose LuaLaTeX for my document. It was a short one page article but I was still satisfied with it. I searched for journals that had writings similar to my skill level. I sent them my LuaLaTeX file and the pdf file resulting from it via email. They reacted positively to the content but asked me to write in plain latex because they do not use LuaLaTeX. It took a few more weeks to convert to LaTeX. I have never maintained a journal so I didn't know that it wasn't as simple as just ordering the articles in the right order then gluing them together. Will I face similar barriers in high school, university and beyond if I keep using LuaLaTeX and not LaTeX? Are engines that extend beyond LaTeX such as LuaLaTeX and XeLaTeX to be avoided in environments where I do not produce all the content of the publication?


I encountered several times a certain type of sentences (in colloquial contexts) which were clearly grammatically incorrect but seems to be widely spread and, as a non-native English speaker, I would have liked to have more information about that. It is about the conjugation of the modal verb to be. I have heard many times the following type of sentences: "We was playing basketball" "They was eating dinner" "I were there" "He were with me" where the last two were used without any conditional of any sort. These sentences are obviously wrong in standard English (as far as I know) but still seem widely used in everyday life conversations. Therefore, my question was the following: How wrong are these grammatical structures ? How strange will I sound if I use them ? Also, is it some kind of dialect/slang and in what kind of context do we use them? Unfortunately I do not have any neat example to provide for I've heard it in an everyday life conversation with my office mates who are from Scotland. (I did not dare to ask them directly as I barely understand when they speak) If someone could provide me more information about them I would be grateful.


Imagine an air hockey table where there is a puck P and a rectangular slab S. Both are free to move as there is zero friction. The slab is at rest in the middle of the table. The puck is moving towards the slab and collides with it in an elastic collision. The puck is not spinning before the collision. As this is a closed system, the total momentum, angular momentum and energy of the system should be preserved. But after the collision the slab will be spinning about its centre of mass, but does that mean the puck will also be spinning in order to conserve the angular momentum of the system? I feel like the elastic (frictionless) collision won't be able to impart any spin on the puck. I understand that the puck and the slab will also change their linear momentum and energy but overall in the system these will be conserved. How would you calculate the velocity vectors and angular velocity for the slab and the puck post-collision? Assume the puck and slab are made of the same uniform-density material and the puck has radius r and the slab side lengths a and b.


I do not feel comfortable with such constructs that contain "one's work ...ing" as the following: "University of British Columbia marine biologist Amanda Vincent has won the prestigious Indianapolis Prize for her work protecting seahorses." "Learn more about her work protecting the rights of women and girls." No matter how I feel about it, there is no denying that it is done and it is fairly commonplace, so I am not going to say it is wrong. However, it does not sit well with me. How would you explain this phrase grammatically? Is "protecting" a present participle? If so, construct wise, "her work" will have to be the subject of "protecting," but meaning wise, it is clearly wrong; the assumed subject is she. The only explanation I can come up with is that this is an idiomatic expression. Maybe "protecting ..." is a gerund and it is an appositive to the immediately preceding "her work"? Your input will be much appreciated!


I know that we can consider an object as point object, if its size is negligible as compared to distance traveled by it in reasonable amount of time. But in my book Ncert there is questions which asks to determine which of the following are point objects: (a) a railway carriage moving without jerks between two stations. (b) a monkey sitting on top of a man cycling smoothly on a circular track. (c) a spinning cricket ball that turns sharply on hitting the ground. (d) a tumbling beaker that has slipped off the edge of a table It states that (a) and (b) can be considered as point objects but (c) and (d) cannot. Why cannot we consider them as point object if we do not the distance they have travelled and their size?


If we place a conductor between the plates of a capacitor, the conductor reaches an electrostatic equilibrium with the surrounding electric field. At this equilibrium state, the charges within the conductor have redistributed such that the electric field inside the conductor is nullified. Now, what happens if we separate this conductor into two halves, each containing the redistributed charges corresponding to one side of the conductor (meaning we have one half that is positively charged and one half that is negatively charged). We did not take the conductors out of the field. Would there be an electric field between the separated halves of the conductor if we look at the complete system (Capacitor and its field and conductors and their interaction)? My book says there is no field because it would cancel out the external field of the capacitor but I always thought that electric field lines always end on charges, which would not make a superposition possible.


I am working on an English-language online resource. It seems an obvious good idea to allow users to choose a version in British English or American English spelling. However, I've noticed that spell-checkers also provide options such as Canadian English, Australian English, South-African English, Jamaican English, Hong Kong English, and so on. I have always assumed that these variants are all basically British or American, with maybe a few minor details that are different. So my question is: Are any of the other variants of English spelling significantly different from British or American spelling? With significantly different, I mean that they don't just add a number of new words, like "wee" or "bairn" in Scotland, or French loan-words in Canada, but that basic English words are spelled differently, in the way that e.g. "colour/color", "analyse/analyze" and "theatre/theater" are spelled differently in the UK/USA. (To be clear: I'm only interested in how great the difference is to an outside observer, not how important it is to e.g. a Canadian person to be able to select "Canadian English" instead of having to select "British English". I'm also not interested in differences in grammar.)


Physicist Grigory Volovik has put forward some ideas about the universe undergoing a topological phase transition (especially in the early stages of the universe). He published a book called "The Universe in a Helium Droplet" where he explained his ideas. You can find a brief discussion about it here. In one discussion I had with Mr. Volovik, he mentioned that depending on the type of topological phase transition that could have occurred in the universe, all the fundamental symmetries of the universe (spacetime symmetries, translation symmetries, CPT invariance, internal invariances...) could be all emergent from a more fundamental state without symmetries (like in Holger Nielsen's random dynamics proposal where all symmetries in the universe would be emergent) I asked him if this was all speculation or if there was some truth behind and he replied that although we don't know if the universe actually took this "path", we know that this topological phase transition would be possible. But is this true? Would that be possible according to what we currently know about physics (although we don't know if this actually occurred at some point of the universe's history)? Or, on the contrary, we don't even know if these transitions are even possible to begin with?


There are examples in physics in which a simple law results from an immeasurably more complicated set of underlying interactions. Consider Hooke's law, for instance: there is a very simple equation that relates the extension of a spring to the force required to extend it further, yet the underlying physics when considered at the level of the trillions of individual electrons and ions that form the spring is of an entirely different order of complexity. Is it possible that we need to find some new theory to replace quantum mechanics which is as different to quantum mechanics as quantum mechanics is to Hooke's law, or are there any considerations that limit the additional complexity we might encounter in a more fundamental theory. For instance, our current model of reality assumes a set of fundamental particles, and there have been attempts to model the particles as vibration modes of strings. Might it be that strings are themselves composite entities composed of countless smaller parts- in the way that springs are composed of atoms- or do we have firm physical grounds to suppose that there is a fundamental limit to the divisibility of matter which prevents strings, for example, from being composit entities at a much more granular level.


Gravitational waves carry energy. The sticky bead argument shows that this energy can be extracted: https://en.wikipedia.org/wiki/Sticky_bead_argument But Lee Smolin points out that "In principle, nothing can screen out the force of gravity or stop the propagation of gravitational waves, so nothing can be perfectly isolated. I discovered this important point during my PhD studies. I wanted to model a box that contained gravitational waves bouncing back and forth inside, but my models kept failing, because the gravitational waves passed right through the walls". Gravitational shielding is considered to be a violation of the equivalence principle and therefore inconsistent with both Newtonian theory and general relativity, and there is no experimental evidence for it. If energy from gravitational waves can be extracted, does this mean that partial gravitational shielding exists?


I read this from Div, Grad, Curl and All That: The second reason for introducing the electrostatic field is more basic. It turns out that all classical electromagnetic theory can be codified in terms of four equations, called Maxwell's equations, which relate fields (electric and magnetic) to each other and to the charges and currents which produce them. Thus, electromagnetism is a field theory and the electric field ultimately plays a role and assumes an importance which far transcends its simple elementary definition as "force per unit charge". The first reason was basically about how finding the field first, then finding the net force on a charge due to the net field simplifies calculations, but I don't understand the second reason. What is this fundamental importance of an electric field that the author is referring to? (PS, I am unfamiliar with Maxwell's equations and classical field theory)


I am currently writing the conclusions of my bachelor's thesis on convergence spaces and there are a couple of points I would like to make, but lack the proper references to cite in order to do so. The first point I would like to make is that one of the starting points of General Topology was trying to axiomatize the notion of convergent sequences and hence the notion of a convergence space is much closer to the origins of topology. I recall reading somewhere that the first attempts of defining topological spaces were like that, but I can't find where. The second point is that the category of convergence spaces is much more adequate than Top, because it has exponential objects. I know there is a whole discussion of the importance of having cartesian closed categories of spaces, but I by myself cannot argument that, because my background on category theory is really small. Therefore, I need to cite someone who knows that for a fact and can give reasons why that is. Hopefully I am not breaking any rules by asking two different sources in the same post. I thank gratefully for any answer.


Crossposted on MathOverflow I am an undergraduate mathematics student with a keen interest in pursuing research in the formalization of natural languages (from a more mathematical-logical approach), yet there aren't many resources that provide an overview of this very technical field. I wish to be able to provide myself a general map to navigate this subject more clearly, so I will attempt explain my basic understanding of an overview of the fields that tackle with this subject, and I'd greatly appreciate any correction or addition to my limited insight. From my understanding, (loosely speaking) there are two prominent mathematical-logical approaches in formalizing natural languages: Categorial Grammar and Montague Semantics. While Categorial Grammar uses methods borrowed from category theory to (mainly) study the syntax of natural languages, Montague Semantics, as the name might suggest, (mainly) focuses on the semantics of natural languages by implementing methods from Lambda Calculus. In terms of subareas of each of these two fields, I have not seen much discussed in terms of the subareas of Montague Semantics; however, Categorial Grammar seems ripe with subareas (although I have heard some of the fields mentioned below are only closely related to Categorial Grammar rather than being a strict subfield of it): Combinatory Categorial Grammar Lambek Calculus Type-Logical Grammar Pre-Group Grammar Proof-Theoretic Semantics In addition to any corrections or additions, I would greatly appreciate any suggestions for resources, references or books that deal with these subjects or their prerequisites


It seems that there are many terms in linear algebra that have multiple names. For example, unitary and orthogonal both refer to the same general idea, a Hermitian is essentially a self-adjoint matrix, invertible and nonsingular, and there are definitely more that I can't think of off the top of my head. I've noticed that I've seen terms like orthogonal and self-adjoint in more classes/texts that feel like they consider linear algebra from a more abstract algebraic perspective, while I've seen terms like unitary and Hermitian in physics and more applied settings. I was wondering if there was some kind of history behind these terms? Why do we have multiple terms for the same thing? Is it just a coincidence that I've seen Hermitian and unitary in these kinds of settings, or was the subject simply considered with different motivations by different people studying different things? If this is the case, I would love it if anyone could suggest references for where these terms originated, and also if there are multiple motivations/perspectives for linear algebra, are there any references for the origins of these different motivations/perspectives? I recall one of my physics professors mentioning briefly that mathematicians and physicists had independently developed the same theory only to realize later that they had been working on the same thing all along. Is there anywhere I could learn more about that history? And are there fields other than linear algebra where this has happened?


I'm a current MA student doing research in formal semantics, which is an application of, among other things, logic and model theory to the study of the semantics of natural languages. I'd like to build up a stronger foundation in formal logic before tackling other topics / projects. I love the interplay between logic, algebra, and topology. Model theory is very interesting to me, and I'm keen to learn more on that. I'm keen on extensions of first-order logic, as well (higher order logic, modal logic, etc). I also have Formal Semantics and Logic, by van Fraassen, and I'd like that text to be more accessible to me. There's so many introductory texts to logic, and a lot of them seem to spend time on things like deductive systems and such, which I'm not very interested in. Given the somewhat eclectic list I gave above, are there any logic texts that would be a good match?


I have a conceptual question about graphs which I couldn't find the answer to. I am calculating some node centralities and using them as features for a machine learning problem. I am using Networkx python library. I noticed that for the degree centrality the library does the weighting of the values by the highest possible number of connections. In other words, the degree centrality of a node is defined by the number of connections the node has divided by the number of nodes in the graph minus one. Although, from graph theory literature, the degree centrality of a node is simply counting the number of edges a node has. As far as I understand. I wonder what is the implications of this weighting of the values by the library? Doesn't it completely changes the concept of the degree centrality?


I'm currently writing my master thesis on "Differentiable Stacks". I'm really fascinated by the idea of generalizing manifolds to include also orbifolds/leaf spaces of foliations/moduli spaces... I formally understand the construction of differentiable stacks as stacks on the category of smooth manifolds possessing a representable epimorphism. And I formally understand why they generalize smooth manifolds. What I'm having some trouble to get is: how does one come up with such a definition? I really read many sources on the topic and none of them justifies the choice of such a definition for differentiable stacks (I mean the existence of a representable epimorphism aka an atlas). This "intuition problem" is getting better as I familiarize more with this object but I feel like I'm just "getting used to it". I'd like to know how this object was born and what's the idea behind it.


According the classical physics, the electron should radiate energy and fall to the nucleus in a short period of time. However, this was not the case. Hence, Bohr proposed his theory, suggesting that electrons existed in specific orbits, where they did not radiate energy. These orbits had quantised or discrete energies. Moving between these orbitals meant the emission of specific amounts of energy, emission of photons. However what was so revolutionary behind this idea? It seems to me, he solved this radiation problem, by simply stating that it didn't happen, "electrons exist in specific obrits, where they don't radiate energy". Did Bohr know why this actually happened or did he just state it? Currently, I am struggling to understand what was so revolutionary about Bohr's contribution to the previous nuclear model of the atom.


There was an open dump yard a few miles away from where I used to live for an internship. It was not noticeable during the daytime, but once the sun sets, the dump yard reminded us of its presence through its stinking odor. The smell came every evening and left the following day as if on a schedule, and my colleagues shared the same experience. One of my friends said it might be because of the temperature, but that didn't make sense. Smells, or the gases that cause those smells, travel faster at higher temperatures, which is why food has a more intense smell when served hot, as opposed to when taken out of a fridge. That would mean that we would get the smell during the day, and not during the night, which is the exact opposite of what is happening in reality. What might be the reason then, why we can smell the dump yard at night, and not during the day? P.S: There is no activity going on in the dump yard, apart from people throwing garbage during random intervals throughout the day. It used to be a barren land, and at some point in history became the nucleation site of the garbage of an entire city.


We know that Carleman's condition is a sufficient condition for the determinacy of Hamburger moment problem and the Stieltje's moment problem. The first one look at measures on the real line, and the second one look at measures on the positive side of the real line. There is a third problem called Hausdorff moment problem that looks at measures on the bounded interval. The interesting thing about this problem is that if the measure that fits moments exists, it is also unique. My question: Is Carleman's condition also sufficient for the determinacy of the Hausdorff moment problem? I figure that it should be since Hausdorff moment problem looks at smaller set of distribution than the Hamburger moment problem. I looked around but I couldn't really find this anywhere. I just want to makesure that I am not missing something subtle.


I am simulating a simple Lennard-Jones fluid confined between two fixed walls and I am analyzing the autocorrelation function of the velocity along the direction of the confinement (normal to the walls). I observe an exponential like decay which is in line with the rough expectation, but in addition I observe periodic peaks. These peaks turn out to be the sound modes reflected back and forth between the two walls, as the time between the peaks increases when I increase the walls distance. And when I calculate the sound velocity for my system, it turns out to be equal to the walls distance divided by the time between the peaks. Now I know about the sound mode, I am not satisfied with just having the velocity autocorrelation for looking at the sound modes. I would like to visualize the mode with more geometric features such as the velocity field, as I expect rich structures. Of course the problem is that in such equilibrium simulations a snapshot of the particle velocities is just a random pattern, so some kind of averaging over the particle trajectories and velocities should involved to make the sound mode visible. Any suggestion ?


"Weed" (the annoying plant you don't want in your garden) and "weed" (the psychoactive drug) are treated differently grammatically. Just some example sentences "There are weeds in my garden" vs "There is weed in my garden" "There is a weed growing in this pot" vs "There is weed growing in this pot" "How many weeds are growing there" vs "How much weed is growing there" "I'm going to get rid of twice the weeds" vs "I'm going to get rid of twice the weed" "Here are two types of weeds" vs "Here are two types of weed" Basically, the two words are treated completely different grammatically. I was wondering how in particular the two words are categorized that links to their different treatment. (e.g. maybe one is a proper noun, and the other is not [I know that's not the case, but it's just an example of the kind of answer I'm looking for]). Or is this just some weird slang thing that only applies to the drug "weed"? P.S.: I swear I'm not high while asking this question >.<. The impetus was actually because an anime brought up "happa", which could either mean "leaf" or "weed" in Japanese, and I wasn't sure which they were referring to. And then I got down this line of thinking >.<.


I just thought of this question, and a quick wiki search did not turn up anything. So, here is the question: Rel is the category whose objects are sets, and whose morphisms are binary relations. Is there a way to differentiate between morphisms that are functions and morphisms that aren't using purely category theoretical means (that is, not using the set structure whatsoever). Followup question: If the answer turns out to be yes, let's call a generic morphism that follows the definition an f-morphism (for function). What are the f-morphisms in other, more commonly used categories, like Set, Top, Grp, Ring, etc.? What are the co-f-morphisms (i.e. morphisms defined dually to f-morphisms)? Note that I have not yet given the question serious thought, so it may be extremely difficult or extremely easy to answer.


I'm currently working on a variant of a non-convex low-rank matrix completion algorithm, whereby we take a uniform sample of entries in a (symmetric) matrix and look to complete said matrix. For various reasons, we're interested in trying to reduce the bandwidth of our matrix initialization for our algorithm, and our first idea was using the reverse Cuthill-McKee algorithm to do this (mostly just because it's pre-built in matlab). However, the output structure of the Cuthill-McKee algorithm provides a poor initialization for our algorithm. I'm mostly just curious if there are other bandwidth reducing algorithms that people use regularly. I did a brief literature search and found some papers using neural networks to try and learn permutations to decrease the bandwidth, which looks interesting but is more work than I think is worth for this particular problem.


In some linguistics papers that I have been reading, it has been argued that a false sentence has the same semantic status as a noun phrase that fails to refer. A classic example of an English noun phrase that fails to refer would be the present King of France, if uttered today. This is because, of course, France has no monarchy any more. However, one of the examples discussed by several papers is the highest prime number. Now, intuitively it seems to me, even as a lay non-mathematician, that there are obviously an infinitely large number of prime numbers. But that's just my intuition. My intuition as a schookid would have been that it would be easy to predict which numbers were going to be prime and which weren't. So much for intuition then. Do we know that there are an infinitely large number of prime numbers? Is it possible to explain how we know (or don't) to a non-mathematician like me?


It's commonly stated in the literature that the free distance of a convolutional code is the minimum Hamming distance between the all-zero path and any other (non-all-zero) path in the trellis originating in the zero state and ending in the zero state. (The free distance of a code is defined as the minimum Hamming distance between any two distinct codewords.) My question: Is this true in general? Or is it true only conditionally (e.g. when we assume that the code is terminated in the zero state)? If it's the latter, what's the condition for it to be true? It appears to me that a sufficient condition is that the convolutional code is a linear block code. This includes the finite length "zero-tail" codes and "direct truncation" codes. The former refers to codes which include all paths that end in the zero state; the latter means the codes allow the paths to terminate in all possible states. Am I mistaken?


I have a setup consisting of two lidar sensors with a known extrinsic calibration. Both lidars have messages with the same timestamps (no timedelay between them). My goal is to estimate odometry using the data from these lidars. To achieve this, I have performed a scan-matching algorithm (ICP) and obtained transformation matrices between the consecutive pointclouds. However, I'm facing difficulties in transferring the translation and rotation frames from one lidar to another to put into some filter. The lidars are mounted on a level arm and do not have the same translation due to the Euler rigid body equation. I would greatly appreciate your assistance in understanding how to appropriately transfer the translation and rotation frames between the lidars. Any insights, suggestions, or code examples would be highly valuable. Thank you in advance for your help!


I'd like track my LaTeX projects using git, especially when collaborating with other authors. I'd also like to track my texmf directory in a separate git repository. My desire is to add my texmf repository as a git submodule to each LaTeX project. However, I can't figure out how to tell latexmk (really, pdflatex) to see my project-local texmf. Ultimately, I want my LaTeX projects to be standalone, i.e. they depend on nothing other a TeX Live install and the files in the project directory so they can seamlessly be compiled on different machines or in a CI pipeline. How can this be done? Note: I have tried the solutions presented in similar questions, e.g. here (the TEXMFHOME environment variable seems to be ignored) and here (requires updating system texmf.cnf), to no success so far.


I have a relatively strong background in the theory of numerical analysis of partial differential equations (PDEs) and functional analysis, particularly applied to the numerical analysis of PDEs using the finite element method. I'm also very interested in graph theory. I was wondering if there are any theories or research areas that unify these fields and that I could explore given my background. Are there any connections between graph theory, functional analysis, and numerical analysis of PDEs that I should be aware of? I am aware that there is another question about this, but I am interested in it being in the context of PDEs and numerical methods. Also I am looking for open questions to initiate a research project, so I am seeking topics where the intersection is significant and there are open problems. Thank you in advance for your assistance.


In Borevich & Shafarevich's Number Theory, the authors define integral equivalence of quadratic forms as follows: Two forms of the same degree with rational coefficients are called integrally equivalent if each can be obtained from the other by a linear change of variables with rational integer coefficients. They further state a second definition In the case of forms which depend on the same number of variables, this is equivalent to saying that one of the forms can be transformed into the other by a linear change of variables with unimodular matrix. But I cannot see how to prove the equivalence of the two definitions. Specifically, it seems to me that to show the first definition implies the second, one has to prove that the two linear changes in the first definition are inverse of each other, or that both changes are nonsingular. Can anyone help? edit: Mr. Stucky's example in the comments has shown that the two changes occurred in the first definition need not be invertible. But still, I think to prove that the first definition implies the second, one has to show the existence of two nonsingular changes, as is pointed out by Stucky.


Here is the sentence in dispute: In humans, the femoral angle shows no correlation with femoral length. The question: why would 'femoral angle' receive a definite article, but not 'femoral length'? I feel like it does, but my co-author says no. I can't really justify it, but I feel like 'angle' somehow needs the article whereas 'length' does not. Thoughts? Thanks for all the answers. Yes, I am aware the anatomy may be esoteric, and apologies for that. In this case, the angle is measured in degrees, or fractions (in decimal) of degrees. The length is similarly measured in centimeters or fractions (in decimals) of centimeters. (And there really is only one of each on a person's leg.) So while I felt like angle requires a definite article because of the way I have always seen it used, in this case I can't logically defend the difference between angle and length.


In Reed-Solomon codes, the symbols of a code word contains multiple bits. Since the error correction and detection happens at the symbol level, it doesn't matter how many errors there are within the same symbol, it only counts as a single symbol error. Because of this, Reed Solomon codes are considered to be a great candidate for transmissions that are subject to burst errors. On the other hand, BCH codes are always considered to be a good candidate for random errors. But is that also true in an apples to apples comparison where you compare total number of message and parity bits (not symbols!) are the same for a Reed-Solomon code and a BCH code? Would there be any way in which the BCH would lose against the RS code in terms of error correcting capability?


I'm a graduate aerospace engineering who's really interested in the topic of numerical methods and in particular for structure preserving schemes. Hopefully at September I will start a phd in this subject, as a consequence of my work of thesis. However, I do feel my mathematical background is kinda poor, coming from an aerospace engineering master degree. Indeed, I'm already excited to follow some of the courses I will in my phd courses about advanced linear algebra and functional analysis. During the summer I have some spare time, and I would love to study some of the books I encountered during my work of thesis. In particular "Geometric Numerical Integration" by Hairer et al. As soon as I started reading this book I encountered some mathematical topics that I never encountered during my degree courses: Lie algebra, Lie groups (groups in general) and manifold. I would like to know what are the books you suggest to get into these topics. I saw that "Lie Groups, Lie Algebras, and Representations" by Brian C. Hall is a really good book as a starting point for this topic, but I know there's a more general introduction to lie groups which is related to manifolds. So which books do you suggest to introduce me to these topics ? Also, I would like to refresh some knowledge about the algebra courses I did in the past, but I don't really know which book I should refer to. Thank you so much for your help.


I was going through the derivation of a mathematical equation for the upthrust exerted on a body which is given in my book. It says that the downward pressure exerted on the upper surface is less than the upward pressure on the lower surface. Thus, there is a net pressure acting in the upward direction and therefore, a net upward force. The lateral pressure gets counterbalanced. I am able to understand that the pressure in the downward direction is due to gravity and the lateral pressure is due to the fluid's tendency to flow. However, I am not able to understand how the fluid exerts pressure in upward direction. I have worked on the problem and arrived at two different explanation:- The pressure inside a fluid is due to the collision of the particles. Since, collision is random, pressure can be considered to be equal in all directions at the same horizontal level. The fluid exerts a pressure and thus, a force on the bottom of the container and the reaction of this force exerts upward pressure. Question:- How does a static fluid exert pressure in the upward direction? Which explanation of mine is correct?


While I was reading the book "In Search of Schrodinger's Cat" I found an interesting excerpt on how Max Planck used Boltzmann's statistical equations to solve the Blackbody radiation problem. The book mentions that there will be very few electric oscillators at very high energy end and at the lower end, electric oscillators would not have that much energy to add up to any significance. So most will be on the middle range. But my question is why and how this distribution of electric oscillators comes into picture. I mean why can't be the this distribution skewed, so that we may have electric oscillators concentrated at high ends. More specifically how this probability distribution was derived by Boltzmann? ( I am a layman interested in Physics so please be simple in your explanation)


I was reading a Mathematics books and it gave the axiomatic definition of a function as being a mapping from a set called "The domaine of the function" to another set called "The codomaine of the function", and at first I thought the codomaine is the image of the domaine by the function (i.e. the set that contains and only contains the images of every element of the domaine). Turns out that's the range or image of the domaine which is only a subset of the codomaine and is equal to the codomaine only if the function is surjective. My question is : Why not define a function as a mapping from a domaine (the set A) to the set B (defined as the set containing and only containing the image of every element of the domaine)? Why the need for the Codomaine set with extra elements? Isn't the complement of the range in the Codomaine irrelevant to the function? In other words, aren't all functions surjective in the end? Sometimes they say that the codomaine is the set of possible outcomes of a function, but I don't understand what they mean by "possible" in this context.


I have a hard time understanding GR. I understand a lot (from a math point) about (pseudo)Riemannian manifolds, and I also learned about Einstein's elevator thought experiment. So let me elaborate: From a physics point of view, you can take the elevator and derive that light has to bend, also that there has to be gravitational time delay. So far so good. Then almost all the literature I saw, turns to the next chapters, and assumes from the previous discussion that it is clear the (pseudo)Riemannian metric is all what matters now. For me, there is a bit gap, a 'how'/'why'/'what' in between. I do not see how the metric tensor relates to accelerated reference frames. I feel like I am missing something very obvious, but what is it? Can somebody elaborate?


I'm trying to create a transition effect in a situation when a character suddenly finds himself falling, and his last word, which ends in '-y', is transitioning into an unintelligible scream. However, English is not my native language, and when I've tried to write it as 'Maryeeee!!' it was criticized as awkward and weird. At the same time, the counter-suggestion of simply extending the '-y' at the end and writing 'Maryyyyy!' doesn't feel right to me. It may be grammatically correct, but my goal was to create an impression of a word seamlessly transitioning into a scream, preferably making it clear that it is an unintelligible scream and not just the character's speech trailing off, and if the resulting word would be a horrible mess, it was fine for as long as I achieved the desired effect. So the question is, how should I go about it?


I'm working on a problem where I need to convert an undirected and unweighted graph with cycles into a tree while preserving the edge information (all the edges from the graph are preserved in the resulting tree). For the resulting tree, the height and nodes duplication should be minimized. Nodes duplication can happen because in order to preserve the edge information from the graph, we will need to break the cycles by removing the edge(s) and then make a copy of the node(s) then connect it in the tree, or in other scenarios that I haven't thought of. I don't think minimum spanning tree algorithms will work because the algorithms will lose the edge information. So far my heuristic is: For choosing the root node of the tree, pick the vertex with the highest betweenness centrality in the graph. Because that vertex likely serves as a bridge which many shortest paths flow. When breaking the cycles, remove the edge whose nodes have the least total sum of degrees. This works for some examples I crafted.example My question is, how can I verify the heuristic is indeed optimal and provide a formal proof? Or is there a better strategy? Any insights, references, or alternative approaches would be greatly appreciated!


I am beginning to learn chemistry/physics, and I have recently read about JJ Thompson's experiment which led to the discovery of the electron. In every source that I've read, the writers note that Thompson used an anode and a cathode to conduct electricity. In addition, the magnet supposedly had N/S ends. I did not know what a charge was, so I looked it up. But, frustratingly, I always get answers that refer to electrons and protons. That is, things that are negative have more electrons than protons, and vice versa. This gives me no idea as to how Thompson inferred that electrons are negatively charged from his experiment. My question is, can you define charge without talking about electrons and protons? Thompson didn't know electrons existed, so it seems to me that he must've had some other working definition of charge in order to determine that electrons were negatively charged. What makes an object a cathode or an anode, without referring to protons and electrons? If that question doesn't make sense, is it possible to adapt it just for the sake of understanding the experiment?


My understanding of a simple k-vector is that it is the wedge product of k vectors. Also, two simple k-vectors are the same, when their magnitude, attitude and orientation match. Now my question is, could I just define a simple k-vector in this way? Meaning "a simple k-vector is an equivalence class of ordered k-tuple of vectors. Two tuples are equivalent if their attitudes and orientation match, and if the parallelograms they span attain the same magnitude." Especially a reference where this is stated would be greatly appreciated! I feel like I have read something like this somewhere, but I cannot find it anymore. I am only using k-vectors for something I am writing for university and having to explain the wedge product would deviate from my topic a little. That's why I am trying to avoid this definition.


So I am writing a program that works with regular polygons, and in part of that I need to represent circles that are inscribed and/or circumscribed upon the polygon. As this is programming, I need to refer to the relationship between the polygon and these circles. I have searched all around, but all of the places I have found that educate on calculating these circles never refer to this relational quality. As I said, I need a way to refer to this relationship and I cannot seem to find a word that represents whether a circle has the quality of being circumscribed or inscribed upon a polygon. Does any such word even exist? This pseudocode example might help clarify what I am trying to find: var myPolygon = new Hexagon(); var myCircle = CreateCircleOnPolygon(myPolygon, CircleType.Inscribed); ^^^^^^^^^^ ............ if (myCircle.CircleType == CircleType.Inscribed) { ... } ^^^^^^^^^^ ^^^^^^^^^^ The name emphasized in the above example is what I am currently using; however, considering that there are lots of different possible parts of my code that could need a different "type" of circle, the usage of the name CircleType is a little too broad in this context. I suppose the word "quality" is what I am going for, but I thought I would reach out to the language experts to see if there is a term I am unaware of that is more appropriate.


I can imagine a relatively simple experimental setup whose resulting data could easily be compared with theoretical predictions: Send two identical atomic clocks into orbit and settle them at rest relative to each other. Then synchronize them. Then use thrusters attached to one clock to oscillate it, causing it to experience a series of counterbalanced accelerations (not changing it's average distance from the non-accelerated clock). Then collect data from both clocks to measure their desynchronization over time. My preliminary calculations suggest that the energies involved to reach relative velocities between the clocks that should produce measurable desynchronization in a matter of days or weeks are feasible, since atomic clocks are so accurate. I know the twin paradox isn't really a 'paradox' and that the predictions in general relativity are completely coherent. I've read a bunch of responses to related questions, explaining exactly what is predicted. But I can't find any experiments that confirm those predictions directly.


As I understand, for elevation mapping using InSAR, one typically requires an out-of-plane baseline to create the required phase difference between images to detect objects at height. This usually requires either multiple satellites flying in formation or waiting for a repeat pass of a single satellite. This paper, A New Single-Pass SAR Interferometry Technique with a Single-Antenna for Terrain Height Measurements, suggests that one could achieve this in a single pass (along-track interferometry), if we are able to image at a high squint angle. The idea is that a grazing-angle difference is still present in the along-track case when squint angle is high. This is not the case for broadside imaging. However apart from this paper, I was not able to find any other sources to cross-reference the viability of this approach, nor does it seem that anyone else has reproduced the results. The mathematical principle behind seem sound to me. If it works, why is this concept not being used more often, and if it doesn't actually work, where is the error in the logic?


I would like to ask your opinion on the point that looks simple. Consider the group of orthogonal matrices of order n over the field R of reals, equipped with the topology induced by the Euclidean norm of matrices. Let g be one of such matrix and denote by X the topological closure of the cyclic group generated by g. My question: May I find a matrix g with the property that: if the identity matrix I is the accumulation point in X of some sequence of powers of g, then the sequence is definitely trivial, that is, all the elements (except finitely many) equal I? In other words, can I exclude that, for a suitable g, the identity I is obtained as the accumulation point of a non trivial sequence in X? I don't think such g exists but I'm not be able to exclude it with a direct argument. Thank you very much for help.


Perhaps another way to put it is, what exactly does it mean to quantize the EM field and why is it necessary? What mathematical properties does the quantized version of the field have that the classical version doesn't and vice versa? For context, I was reading a thread about where specifically classical E&M fails and we hence need QED to get accurate predictions, and stuff like the ultraviolet catastrophe and the photoelectric effect were mentioned. I get why those things necessitated the introduction of quantum mechanics and especially modeling photons as wavefunctions. What I don't get is why we can't just use "regular" relativistic QM with Maxwell's equations and the same mathematical E and B fields (which together form the EM field) from classical E&M. What goes wrong if we try to do this and how exactly does QED fix it? Or, to put it another way, what properties does a field modeling photons as wavefunctions need to have that the classical EM field doesn't have?


I am new in Linear Algebra and I have encountered the concept of Solutions of linear equations. According to the textbook I use, a solution of a linear equation is a vector whose components satisfy the equation. However, I am puzzled by the choice of "vectors" specifically. Why is it not defined as a point whose coordinates are the same as the values that satisfy the linear equation? After all, a vector is more than just its components; it is also a set of points and its "components" define its magnitude and direction. A point, on the other hand, is a single entity with no additional properties. No magic. I think It's just a bit more intuitive to think of it this way instead. I can also anticipate that Vectors have a deeper meaning that relates them to linear equation systems but I can't tell what exactly. I'm going through the chapters and I'm still waiting for the "Bingo! I figured it out."


I have been reading the Loop Antenna section of Antenna Theory by Constantine Balanis and trying to understand how exactly a ferrite core improves the performance of a small loop antenna. Balanis writes, The radiation resistance, and in turn the antenna efficiency, can be raised by increasing the circumference of the loop. Another way to increase the radiation resistance, without increasing the electrical dimensions of the antenna, would be to insert within its circumference a ferrite core that has a tendency to increase the magnetic flux, the magnetic field, the open-circuit voltage, and in turn the radiation resistance of the loop. How exactly does increasing the magnetic flux (adding the ferrite core) increase the antenna's radiation resistance? Balanis claims the voltage increases, and presumably deduces that the radiation resistance increases likewise from the equation V=IR. However, this would only follow if we knew that the current I did not increase as the flux increased, which is not obvious to me. It also is not obvious to me why the voltage increases in the first place!


I've been really stumped on this particular concept. In Case A, when a bar magnet is brought towards a copper coil around a soft iron core, in accordance with Faraday's Law of Electromagnetic Induction, the the pole facing the magnet acquires a North Polarity while the opposing pole acquires a South Polarity. Now by Lenz's Law, the direction of the induced EMF must oppose the cause that produces it, but the current is in the same direction as the moving magnet. The galvanometer deflects in the same direction too. Why doesn't this obey Lenz's Law? And additionally, in Case B, by merely reversing the winding of the copper coil, the current flows in accordance with Lenz's Law and the Principle of Conservation of Energy. My question is, why does changing the winding of a coil defy Lenz's Law in Case A and then follow Lenz's law in Case B? Something such as the Principle of Conservation of Energy and Lenz's Law should apply in any case. The method of winding shouldn't matter that much right?


Is it correct to say that, in English, when you use the Present Simple tense in the Interrogative Negative form you are either implying the negative or just confirming the affirmative (depending on the order of "Do", "subject" and "not"), but no other possibility? Like in the scene of The Lord Of the Ring - The Return of the King, when Eowyn is talking to Aragorn just before he leaves to see the dead army in the mountain. Eowyn: "You cannot abandon the men. We need you here." Aragorn: "Why have you come?" Eowyn: "Do you not know?" In this last line, given the context, she is clearly implying that she thinks he knows she loves him, and thus, she is asking it just to confirm it (as I understood it). But then, if she had used the other form "Don't you know?", it would change the meaning, as she would appear to be asking it to really have a first idea on his feelings, not presuming anything. Then my question is: can the Interrogative Negative have only these two possibilities?


Have studied traditional point-set topology, but find there's a fairly large gap between the preparation typical point-set courses give you, and the level assumed in algebraic topology texts. Looking for a good introductory book on topology that uses more categorical / modern language - something that would segue smoothly into, for example, Tammo tom Dieck's text Algebraic Topology - at least covering most of the topics mentioned in the first chapter of his book (subspaces, quotients, products, sums, compactness, proper maps, paracompactness, topological groups, transformation groups). Have perused the MIT textbook 'Topology - A Categorical Approach', which looks decent, but isn't so comprehensive. Have found one text that looks good (Grundkurs Topologie by Laures and Szymik) - but unfortunately it's in German! (Anyone know of an existing English translation?) Open to recommendations. Thanks.


I'm studying nonlinear control systems, especially the Pontryagin's minimum principle and its applications. Throughout my studies, the authors have always defined the control systems with state variables and output variables. However, in this article Optimal Control of an SIR Model with Delay in State and Control Variables and so many others, it doesn't seem like control systems in epidemiology has output variables equations (or at least clear ones). Why is so? And is it possible for a nonlinear control system not to have output variables? I have found this answer: It is possible for a nonlinear control system not to have an explicit output variable, depending on how the system is defined and what its purpose is. In a control system, the input variable is typically the signal or information that is used to control the behavior of the system, and the output variable is the variable that is being controlled or affected by the system. However, in some control systems, the purpose may not be to directly control an output variable, but rather to achieve some other goal. Is it correct?


This might be a stupid question, but, why is it that gamma rays are able to penetrate almost any barrier without question? We know that gamma rays are simply high frequency waves with massive amounts of energy. However, what processes can enable it to go through layers of bonded atoms of metals and just about anything else? On another note, we know that beta particles can pass through paper but can not penetrate a thin layer of metal. What is different about the bonds of metal so that an electron can't squeeze through? I assume it is because of the tightly bonded metallic bonds and the high volume of electrons that flow throughout the metal that causes this. However, if this is the case, what would a beta particle do if it is "rejected"? Would it join the flow of electrons in the metal, would it just stay on the same side of the sheet or would it rebound in the opposite direction due to the equal, opposite force exerted on it by the sheet?


I am studying interacting QFT in the context of quantum fields in curved backgrounds, and I am getting some confussion about the concept of particles. To study some gravitational phenomena involving particles (e.g. Unruh effect, Hawking radiation, etc.), it is typically sufficient to deal with free fields, which are expanded in mode functions and particle/antiparticle operators (i.e. its energy eigenstates form a Fock space). This can be done, in general, because the Hamiltonian of free fields is quadratic in the field operators, and therefore one can calculate a single-particle Hamiltonian which can be diagonalized, giving rise to a band structure by means of which we describe this Fock space (eg. for the case of Dirac fermions, the vacuum state, prior to a particle-hole transformation, amounts to a filled lower band/Dirac sea). However, when one considers an interaction term, the Hamiltonian is no longer quadratic in the fields, and this band-structure cannot be obtained by direct diagonalization of the single-particle Hamiltonian (I am not even sure whether the notion of band structure remains). As a consequence, I do not understand if particles/anti-particles can only be defined when the Hamiltonian of the theory is quadratic, i.e., when the evolution of the theory preserves tha Gaussianity of the states. If this is the case, I imagine that a mean field approximation, which turns the Hamiltonian back to a quadratic one, would recover the notion of particles, is this the case?


I'm working on a set of beamer class slides in TeXstudio. Usually upon compile, the preview slide visible on the right will be at the position of the cursor - instead, for me, it consistently jumps two slides behind. This is reflected in the behaviour of the preview when I enable the "scrolling follows cursor" option - there, too, the preview is always two slides behind where the cursor actually is. If the cursor is on one of the first three slides, the preview stays on the first one. I get this behaviour on two different machines, so I assume it's not some super specific local bug but something more broad. It also persists independently of whether or not the document is compiled in handout mode. Any help on resolving this would be very much appreciated, as this behaviour is quite annoying.


I noticed that in several 'throwing' sports like the javelin throw and the shot put there have been a few cases where competitors tried to introduce a technique where they throw the projectile further by rotating their entire body (these techniques were then banned for safety reasons). As an example, there is the cartwheel shot put, where the person performs a cartwheel and rotates their body before throwing. I am not sure what advantage this has apart from that the ball is pushed over a slightly longer distance so that there is more kinetic energy. In the spinning javelin throw, instead of throwing the javelin with a regular technique, the athlete spins around and slings it out of their hand at high speed so that it flies forward. Again, I am not sure why this gives a distance benefit but I assume that it's different to the rationale behind the cartwheel shot put?


I am wondering if the esteemed members of this forum can help me with these questions, which have bothered me for a long time and are what have brought me to this forum. One thing I struggle to understand and work with is the use of the "editorial we." I am a copyeditor and proofreader, and I have a client who consistently likes to use the first-person plural voice for his works. I have asked my colleagues, and I got mixed, unclear answers. Here are a few examples: We will be able to feel magnificent light in our souls... Does "souls" have to agree with "our"? Should it not be "soul"? The chair or pillow may not be so comfortable, our kids may get rowdy, our spouse may need our help with this and that... Here, I have no issue with "our kids," because a person can certainly have more than one child. But should "spouse" be plural like "our" and "kids"? It is a constant issue that we must face our entire lives. "Entire" here does not seem to fit with "lives." But then "our entire life" does not seem to be correct as the singular "life" does not agree with the plural pronoun "our." I would be grateful to learn more about the proper use of the "editorial we" in this regard and to hear suggestions on how to deal with this issue. Thank you all in advance.


In a textbook for thermodynamics, it considers a situation where work is done to a system by an irreversible work source through a thermally insulating piston, and it states "any irreversible work source can be simulated by a reversible work source". It briefly explains the reason; what the work source does is simply to apply force to the piston, and therefore it does not matter how the force is applied, whether or not it is applied by an irreversible work source. I have a difficulty to fully convince myself with the statement, and can anyone kindly help me with this? The textbook is a non-English one, and it is not available online as an electric file. The way how it defines the work source is simply that it is any system, which is connected with the system of our interest only through a thermally insulating piston. That is, there is no heat exchange between the two system. Let me rephrase my question: Suppose that system A is interacted with system B (irreversible work source) only with a thermally insulating piston. They exchange energy only through work, but not heat. My question is if it is possible to replace system B with a reversible work source in an indistinguishable manner, i.e., any mechanical reaction that system A receives from system B remains exactly the same. Thank you so much for your time.


As per my understanding: Multiple fermions cannot have the same quantum state (as per Pauli exclusion principle) Multiple fermions can occupy the same physical space as long as they have different quantum states (or numbers or properties such as spin) If both these statements are true then, part of the second statement "as long as they have different quantum states (or numbers or properties such as spin)" doesn't become necessary. Because first statement implies that "multiple fermions always have different quantum states". Hence, the second statement simply becomes "Multiple fermions can always occupy the same physical space" (For a moment let's consider only fermions, their quantum state and physical space they occupy. And not other factors like electromagnetic repulsion etc) However, at multiple places on the Internet it has been stated (and seems like widely accepted) that: Multiple fermions cannot occupy the same physical space as per Pauli exclusion principle, and that is why matter structures exists in the universe. Can someone please help me trying to figure out where am I making mistake?


According to wikipedia, the aurora borealis is primarily caused by charged particles from the solar wind being redirected to the poles by earth's magnetic field and slamming into the nitrogen and oxygen atoms in the upper atmosphere. This slamming gives the electrons in those oxygen and nitrogen atoms enough energy to escape their electron clouds. The light comes from when electrons emit photons in order to rejoin an atom. Hence the colors, green corresponding to oxygen atom readmittance and purple to nitrogen What I am wondering is what role dissipative absorption plays in the aurora borealis. I mean we have a bunch of accelerating charged particles right? Even if speed is constant their direction is changing due to the magnetic force and since they're charged they must be emitting some kind of em radiation right? Not to mention when the actual collisons take place between electrons. Not all of this em radiation will correspond to the energy levels of nitrogen and oxygen right? So shouldn't we see more colors as all sorts of em radiation ought to be emitted? If nothing else you would expect the electron clouds of oxygen and nitrogen to oscillate in response to the emitted em radiation of the incoming solar wind right? Why do we tend to see only green and purple?


Suppose I have a hose pipe connected to a tap. I turn on the tap and water flows out of the other end of the pipe. Now imagine I install two valves at different points along the pipe. Water will flow normally if both valves are open, but not if either one of them is shut. No doubt you can now see where this is going. If the spacing of the valves along the pipe is sufficiently long, and the duration for which the valves are open is sufficiently short, we can always imagine a second reference frame in which at any given time at least one of the valves remains shut, so how can we account for the fact that water flows in that frame even when the valves are not both open? I imagine the resolution is somehow to do with the fact that the flow requires a pressure wave to propagate along the pipe when a valve is opened, but I haven't been able to find a convincing resolution. Has anyone come across an analogous problem before?


I'm implementing physics for a computer game, and came accross something that looks unintuitive to me. Consider two bodies at rest: Let's say we momentarily apply the same amount of force to them, but in different locations: at the center of mass for one body, and off-center for the other. The second body will gain angular velocity, while the first one won't. Both will gain rightward velocity, in the same direction. But will the second body gain less velocity than the first? Or the same velocity? Naively I would expect less velocity, since some of the force was "expended" on giving it angular velocity, but I googled around and experimented in Algodoo, and it seems the velocity ends up the same. Is that correct? Is there an intuitive explanation for that?


I've got a few months to go before I enter in graduate school, but I have to say that my math background is not as good as I would like. Thus, I would like to take the opportunity to strengthen it and work on undergraduate analysis. I had in mind to work on the Rudin book I've heard so much about, the book is very concise and covers a wide spectrum of mathematical analysis (and a bit of topology) seen in undergraduate so I thought it was suitable for what I want to do with it. But I'd like to hear your opinions and recommendations if you've ever been in this situation and/or know of any other cool books (I'am also interested in functional analysis and especially Hilbert spaces so...) for the self-taught to get through the summer. Thank you a lot !


Let's assume we have a standard double-slit experiment, just for illustration's sake. When an electron collides with the detector, is its momentum parallel to the emitter's output, or is it something else? I know that the uncertainty principle forbids us from knowing too much about the momentum when we know the position, so let's assume we have a magic detector that will only give us the momenta and not the positions. This magic detector can detect the momentum of a single electron. If we shot one electron through this apparatus at a time, what would we see (on average)? I say on average, because there's a non-zero chance that the electron lines up with the emitter by chance. We should still be able to see the standard deviation of the momenta; if it's tight, then the momenta line up with the emitter, otherwise, it's relative to something else. I'll admit this question may be nonsense, but to my knowledge, the only limitations to this are engineering ones (due to the magic detector), not physical limitations. If this experiment is nonsense due to a physical limitation that I've misunderstood, I'd like to know why.


I've been reading some category theory texts for my undergraduate monography, and I've found that one can talk about small and large categories using small/large sets, or do the same using sets/classes. Most introductory texts just shrug formalisms off, mention one of these formalizations and goes on. I've noticed that most category-focused authors and texts (such as Categories for the working mathematician and Sheaves in Geometry and Logic by Mac Lane) tend to prefer the small/large set approach, while texts that don't focus that much on categories (Topology and Groupoids by R. Brown) tend to prefer the class/set one. My question: Is there a categorical/set-theoretical reason to prefer the small sets approach? I'd assume that there's a benefit in having a set of objects even in large categories, but fail to see an actual difference. I found the paper Set Theory for Category Theory by Michael Shulman, but it was way out of the scope of what I can understand.


I'm looking for a real analysis book that covers measure theory and I've been recommended Folland's or Royden's books, but their price is quite high and I was wondering if any of the books below would serve as a cheaper substitute. Theory of Functions of Real Variables by Lawrence M. Graves; Real Variables with Basic Metric Space Topology by Robert B Ash; Theory of Functions of a Real Variable by I.P. Natanson; Real Analysis by Gabriel Klambauer; Nonstandard Analysis by Alain M. Robert; Foundations of Analysis by David F. Belding and Kevin J. Mitchell; Foundations of Mathematical Analysis by Richard Johnsonbaugh and W. E. Pfaffenberger; Mathematical Analysis by Tom M. Apostol; Elementary Real and Complex Analysis by Georgi E. Shilov; Foundations of Modern Analysis by Avner Friedman; Real Analysis by Norman B. Haaser; Introductory Real Analysis by A. N. Kolmogorov; Introduction to Analysis by Maxwell Rosenlicht.


I'm having a little bit of a hard time understanding some concepts to do with self-inductance in class. I understand mutual inductance. That's when a loop has some time-dependent current going through it, and because the current is time-dependent, the magnetic field the loop produces is also time-dependent, and the magnetic flux in a nearby loop is time-dependent. With self-inductance, though, is the idea that a single loop carrying a time-dependent current gives rise to a time-dependent magnetic field, and that very same time-dependent magnetic field gives rise to a change in flux in the single loop itself, which then generates an emf? If I'm correct in this reasoning, then that should mean that the new emf gives way to a current, which opposes the previously existing current. Why doesn't this just result in an infinite loop creating an opposing current, and then one that opposes that, and then one that opposes THAT...etc.?


Given a mapping f: R->R, what properties must f have in order for its range to be countably infinite? I'm personally not sure, especially in terms of formalization, but ideas of non-monotonicity or some construction of countably infinite equivalence classes come to mind. For context, in quantum mechanics it's said that a system's energy can only take discrete values, but this confuses me since energy is defined as the sum of a function of position and a function of momentum, which can both take any real value. The indexing of the sets doesn't make much sense to me, especially since the kinetic energy, the function of momentum, is very clearly uncountable since it's strictly monotonic. So even if the potential energy has a countable range, it doesn't make sense that the total energy would be since the union of a countable set and an uncountable set is uncountable. Like I said, I'm more concerned with the question I asked above rather than this confusion that created it, as I'm sure an answer will either lead me in the right direction, or at least give me the means to reasonably formulate a new question- so don't worry about addressing this. Any answers or information are greatly appreciated. Thank you!


I was solving this question in my coursebook and was not able to understand it conceptually. Though I did understand the mathematical approach to it. Now I understand that when the rod is given an initial velocity towards right, due to its motion an EMF is induced across it. Current starts flowing until it reaches a steady state. However, once the current becomes constant, no potential difference exists and the current slowly starts dying out at which point the rod starts moving back to its mean position. Then the velocity is in the opposite direction and the 'polarity' of the induced EMF reverses and current starts flowing in the opposite direction. My questions are- Doesn't it take an infinite time for the inductor to reach its steady state (where it behaves as a connecting wire). So, wouldn't the rod move in the direction of its initial velocity forever? What will happen once a steady current is achieved in the circuit? Will it be correct to say that since the potential difference across the ends of the rod is zero, the current wants to decrease to zero immediately, but the inductor only supports a slow decay of current? Can we also say that the reason the circuit doesn't heat up due to short circuiting is because there is no actual battery in the circuit? I tried asking my teacher but I couldn't get a satisfactory answer.


Consider a circular horizontal plane (like a round tabletop) rotating around its center. Consider a body A resting on this tabletop. Since the tabletop is rotating around its center, the body is moving along with it in a circular motion around the center of the circle. I understand that in circular motion, there's always a centripetal force, pushing the object towards the center of the circle. And in this case, this force is the static friction force, operating on the radial axis towards the center. I have two questions regarding this: I was under the impression that a static friction force, or any friction force, always operates in the opposite direction of some other force which operates on the same body. Is this correct? If so, what is the additional force (opposite to the friction force) that I'm missing in the above scenario? Is the static friction force operating only on the radial axis towards the center? Or is there also static friction along another axis?


I'm wearing glasses with a sunglass clip-on. This means I have my regular glasses and, on top of them, I have a second pair of lenses that work as sunglasses and attach to my regular glasses using tiny magnets. I'm also in a car, and I'm looking through a window layered with Insulfilm. Curiously, when I look through the Insulfilm with the sunglasses on, I see a rainbow pattern. If I keep my glasses on, but remove the sunglass clip-on, the rainbow is gone. If I look through a window without Insulfilm, the rainbow also seems to be gone. If I look through the Insulfilm with the sunglasses clip-on, but without my regular glasses, the rainbow is there. I believe my sunglasses are polarized, and the rainbow fades away when I tilt my head a little. Why exactly is this phenomenon happening? I guess it has something to do with polarization due to the rainbow fading when I tilt my head, but I can't figure out why it is a rainbow and why I can't see it without the sunglasses.


This popular question got me wondering if there is a simpler way to reduce radon in homes. Our house has a fairly standard radon mitigation system. It has a lot of parts. There is a hole cut in the basement floor. There is a thick pipe which goes up through all the floors with lots of bends and turns, like the old Microsoft "pipes" screensaver. There is a powerful, always-running fan in the attic pulling the air up and out through a roof vent. There are also other components, such as a manometer and other sensors. It all has to be serviced by technicians. Like any system with a lot of components and a lot of puncture points through surfaces, there are many points of potential failure. Why not have a standalone unit in the basement with a fan that blows the air through an electrostatically-charged baffle of gills or fins or whatever, collecting the radon until it decays? At least then it won't end up in the lungs, where it causes the damage. We could put lead shielding around the unit, if necessary.


I found a few questions that are similar to mine, but I do not think that either of them answers exactly what I want: Is there a Birkhoff-like theorem for stationary axisymmetric metrics? or Gravitational Collapse: Kerr solution is a vacuum solution but not for any rotating body? My question is this: When we look at simple models of gravitational collapse for spherically symmetric matter distribution (such as Oppenheimer-Snyder) we always exploit the fact that the solution to the Einstein field equations just outside any spherically symmetric collapsing matter is Schwarzschild and that this is the only possible solution. This is thanks to Birkhoff's theorem. An analogous theorem exists also for charged matter. Unfortunately, no such theorem exists for rotating matter, as the spherical symmetry is broken. This means that we can't approach the gravitational collapse of rotating matter the same way we would for spherically symmetric charged or uncharged matter collapse. So how do theorists actually study rotating black holes? Do we need some sort of computational or numerical methods?


My friend was sitting in a wheelchair, with a cup of coffee in one hand. That made me wonder, how would the wheelchair move if he turned only one rear wheel? He tried it, using his right hand to turn the right rear wheel. The wheelchair starting moving in a circle. No surprise. But I was interested in the radius of the circle. The circular track of the right rear wheel on the ground, had a radius that was slightly larger than the distance between the rear wheels (i.e. the distance between the two points of contact between the rear wheels and the ground). What determines the radius of the circular track of the left rear wheel (the unturned rear wheel)? Possibilities that come to my mind are: Location of all four wheels The size and shape of the contact patch between the wheels and the ground The coefficient of friction between the wheels and the ground, and the weight of the system The diameter of the wheels Angular speed of the right rear wheel How the person is sitting within the chair (centred, or leaning to the left, leaning to the right) (Assume the wheels are in parallel planes.)


I apologize if this question has been asked before, but I haven't been able to find a post which covers this exact question. Let's say that we have a driving car which is moving forward. At some point, one of the rotating tires hit a stone on the road. Which way does the stone move, in relationship with the angle in which the tire hits the stone? Can the stone ever be thrown backward onto another car? Does the general direction (forward, backward, sideways) depend furthermore on the shape of the stone? Whether it is for example perfectly spherical or asymmetric? I am kind of struggling how I would answer such a basic question, since there seems to be many factors at play here. Can you even answer this question without fully knowing all the nuances and details?


Suppose we have an isolated n-type semiconductor having varying concentrations of carriers over x. After some finite time, the concentration becomes uniform for a given constant temperature. We always say that this is because of diffusion which allows the movement of charge carriers to move from higher to lower concentrations. But shouldn't this diffusion occur because of the repulsive forces between electrons? why do we say it's because of the random motion? (the random motion allowing for the carriers to rearrange should be because of the electric field produced by them due to the gradient of charges). If we see some excess charges added to an isolated conductor, then the electric field produced by these excess charges allows the charges in the conductor to redistribute to settle on the outer surface where the net force experienced by the excess charges is zero. Shouldn't the same principle apply to semiconductors? What exactly is happening inside? I hope to get a clarified answer. Diagrams will be highly appreciated.


I'm trying to deepen my understanding of Von Neumann entropy (out of interest in the quantum Bekenstein bound) by learning more about the reduced density matrix (aka local density matrix), which is given by the modular Hamiltonians (aka entanglement Hamiltonian). As I understand it, modular Hamiltonians are only known to be local (i.e. an integral of a local operator) under three specific situations. When the state is the vacuum state in a half-plane (the only case where the modular Hamiltonian is given by the traditional Hamiltonian) When the state is the vacuum state and is limited to a ball and the field theory is a CFT When the state is the vacuum state and is limited to an arbitrary piece of a null plane I could not, however, find any explanation for how or why these are the three known cases where it is local (and additionally why only the first one is given by the traditional Hamiltonian). I'd appreciate any insight you could provide on this matter.


Adjectives and adverbs can be formed by adding "y" or "ly" to e.g. a noun, such as: heart -> hearty, heartily However, sometimes these words are not in use, or make no logical sense, such as: chair -> chairy, chairly Although you could come up with a situation where you'd use "chairy", e.g. "this block of wood looks quite chairy", you'd probably be more likely to use "chairish" or "chair-like". And if you read all the books in the world, you'd probably never come across the word "chairly" or "chairily". What is the status of words like these? Do they exist because they are technically correct and their meaning is obvious? Do they not exist until someone (notable) uses them in a (notable) text? Or, to make the question more specific: if I were to create a spell checker, should I allow or disallow these words?


I am teaching students at a high school about bending of a beam (the beam is clamped at one end and then an applied force acts downwards close to the end of the beam to bend it). I thought that it did not require further explanation that the deflection of the beam due to the force at the end is inversely proportional to the stiffness of the beam, proportional to the size of the applied force, and that it is dependent on the distance from the fixed point where the beam is clamped. However, I have been asked to provide justification for these statements and not to appeal to physical intuition which is not obvious to students who are not seasoned physicists. What type of justification would be enough to use these statements at an elementary level? How could I explain that these statements are true?


Compressive strength is how much inward force a given area of material can withstand before failure. The force tries to compress the atoms closer together. Mohs hardness is the difficulty of a material to be scratched. Abrasion resistance. However, when an object scratches another, there first is a point of contact where the harder material is forced into the surface of the other, and then lateral forces shear off a grove from the softer material. But at this point of contact, the forces are compressive, trying to force the atoms together, until one of the materials give way. As such, I would expect materials with a high Mohs hardness to have a high compressive strength and vice versa. This holds for ceramics such as silicon carbide, aluminum oxide, and others which have both a high compressive strength and a high Mohs hardness. Does this trend hold for most/all materials and to be expected for the above reasons? Are there any counterexamples? If so, what could explain a high Mohs hardness and low compressive strength or vice versa?


I really don't like the phrase "excited for" which seems to have become very common in recent years, as in "I'm excited for the weekend...". My sarcastic reply would be "I don't think the weekend cares or appreciates you being excited on its behalf". Anyway, I'm aware I have to accept that language changes, even though here I think it's just bad grammar. The question is ... are there any studies looking at how (the wrong use of) this phrase came about? I have a theory based on absolutely nothing, that perhaps it came from Spanish where 'por' can be used in this sense, e.g. Google translates "excited about this weekend" to "emocionad(o/a) por este fin de semana". Is it possible that this usage came via speakers of Spanish, perhaps in the US?


"Longingly" is the adverb form of "longing". Depending on where you look, "longing" is described as an adjective, a verb, or a noun: Oxford defines it as either a noun or an adjective. Cambridge, Collins, and Merriam-Webster seem to define it exclusively as a noun. Wiktionary says it's either a noun or the present participle verb form of "long". Despite the disparate definitions, I can see justification for all three forms. This raises a question: which distinct form of "longing" is "longingly" derived from? Or is the adverb formed indistinctly? I know that adverbs ending -ly are typically formed from adjectives, but English is full of exceptions. I checked a few etymology sources but they either only had information on "longing" itself, or only described longingly as being formed from longing as a general word.


My question is closely related to the answer of this question: Why is general relativity background independent and electromagnetism is background dependent? General Relativity is often stated to be "background independent", because it calculates how spacetime is curved. That's in contrast to other theories like classical electrodynamics which act on the manifold without interacting with it. I understand that GR interacts with spacetime (and that that's a great advancement in comparison to the theories before) - however, I do not understand, why it's called "background independent". Matter curves spacetime. That means to me that there has to be a spacetime first which can be curved by matter. If the universe is empty (in GR), Minkowski spacetime is still there. If spacetime were produced by matter like the electromagnetic field is produced by its sources I would understand the term "background independence". But to me it's only interacting with the background, not background independent. (Like an artist who is forming any possible object out of clay is not independent of clay... Just acting on it in every possible way) How shall I understand the term "background independence"? Is the term not precise? Did I get something wrong?


Quite naturally, the observable Universe is the only bit of the Universe we can extract information from, as light from farther away has not reached us yet, and there are zones from which we'll never even be able to extract information, as they are causally disconnected from us due to their high speed. Therefore, how do we know the cosmological principle holds at larger scales if we can't observe them? It just seems to me a quite bold assumption to make that the entire Universe is homogeneous and isotropic. Perhaps it presents irregularities which we'll never know of, and which would be necessary for us to not mess up with our cosmological models. Frankly, I don't quite see the point of building cosmological models if we only know what seems to be a small portion of the Universe, just like it makes no sense to try to see the picture of a jigsaw puzzle just by examining one random piece of it.


Recently I came across Normal matrices and their properties, one of which states that their eigenvectors are the same as their adjoint and are orthogonal. I've gone across some proofs and I understand it but when I tried to prove the same using the inner product, It somehow states that the above is true for any arbitrary matrix. Can someone possibly help me point out where I'm going wrong? Let's say A is a matrix and B is its adjoint. If x is an eigenvector of A with eigenvalue k, then, < x | A | x > = < x | kx > = k< x | x > also < x | A | x > = < Bx | x > hence < Bx | x > = k< x | x > = < kx | x > So, x is also an eigenvector of adjoint of A.


In the English language there are many words where the letter "a" is pronounced as a short (continental European) "e". Or at least very close to it. However dictionaries point out that in these cases the correct pronunciation is more an "ae" sound. Several websites on English pronunciation take the same view. So I decided to test this assumption. Over a period of weeks, whenever I watched an English or American television program, I payed attention how these "a"-words where actually pronounced by native English speakers, and made notes of my observations. I found out that in fact there are many words that have the pure "e" sound. A few examples: back, cash, cat, fact, relax, track, crash. So what is going on here? Are this words pronounced incorrectly, or are the dictionaries and websites promoting viewpoints about pronunciation that are outdated? Or has it to do with regional variations (dialects)?


In Photoshop (and many other computer graphics applications) there are are various ways that images/layers can be composited with their underlying background. These are sometimes called blending modes. One blending mode in particular is multiply, which has the convenient feature that it eliminates white backgrounds from black line illustrations while properly dealing with anti-aliased edges (see the example below). Is there a way to do this in LaTeX? I have a large number of images with a white background that need including in two documents with different coloured backgrounds. I'm looking for a quick way to remove the white background from the images without having to manually add an alpha mask to every image in Photoshop. Normal blending mode: Multiply blending mode: Close up of anti-aliased edge, properly feathered:


I found this MC question that asks about a soft iron bar entering a solenoid: At first, my answer was (B) because the iron bar should increase the magnetic field induced by coil M and, according to Lenz's law, this magnetic field should oppose the original current, making the lamp dimmer. Also, the magnetic field created by M should then induce a current in N in the opposite direction to the current in M (also Lenz's law), so since the current in M is right to left, then it should be left to right in N. I wasn't too confident in my answer though. Then, I noticed that they had drawn a DC power source in M. This would indicate no changing magnetic field, thus no current induced into either M or N. This would mean that the lamp doesn't change brightness and there is no direction of current in N. Right? Is this just a mistake in the question, or am I missing something? Thanks to anyone who can help.


Given a path, how do the magnitude of the velocity and acceleration vector along the path correlate? I am confused due to the fact that the acceleration is the change of velocity over time and in general if you plot the velocity and acceleration over time, you can see a correlation (e.g. constant velocity means zero acceleration). So I expected the same relationship for the magnitude of the velocity and acceleration vector. Moreover, I got confused because the magnitude of a vector is always positive and in the general plots of the velocity and acceleration over time it is not the case (e.g. linear decrease in velocity means constant negative acceleration). Therefore, can someone clarify if there is a clear relationship between the magnitudes of the velocity and acceleration vector and how this is defined? And if not, is there a relationship between the velocity and acceleration vector, so you can verify if your velocity and acceleration vector are correctly calculated?


Relativistically, the electric field of a moving charge is not purely radially directed, but is instead concentrated perpendicular to the line of motion. So, a current loop consisting of electrons as charge carriers should generate a "charge separation" effect, where the negative electric field is concentrated perpendicular to the loop, and is not perfectly cancelled by the stationary positive ions in the (electrically neutral) conductor. Conversely, in the plane of the loop, the positive electric field should dominate. This "charge separation" effect is separate from the magnetic field produced by the current loop, as it should have the effect of accelerating stationary charges in the surrounding space, which are unaffected by a magnetic field, because their velocity is zero. I have never heard of such an effect, which should be small but measurable, hence I am asking the question.


A term in a Lagrangian is gauge invariant if one makes sure to use quantities which transform in proper representations of the group of gauge transformations. This means that one cannot write terms in the Lagrangian using a 'bare' gauge field, but one can use the associated field strength or act on a gauge field with a covariant derivative. On the other hand, if one writes a term using a 'bare' gauge field (for example, a combination of the gauge field and the associated field strength tensor), although it will not be gauge invariant, will this term still be invariant under diffeomorphisms of the spacetime on which the field theory is defined? Since diffeomorphism invariance is sometimes taken informally to be ''background independence of the field theory'', I guess that this term should be invariant under diffeomorphisms? Or does diffeomorphism invariance fail because it is a specific example of a gauge symmetry?


This may be an amateur question, however here goes. I understand the double slit experiment creates a situation where detecting which slit the particle/wave goes through causes a collapse of the wave function and for the particle to act like a particle and not cause an interference pattern. What I want to know is this: If say for example, the detector emits a signal to a speaker if it detects the particle and the speaker emits an audible sound, my understanding is that the wave function would collapse - we measured the outcome. However, what if we left the detector doing its thing, but turned off the speaker so that we didn't know? Would this still be considered "measurement" or does measurement require recording/knowledge (future included) of the outcome?


I'm considering to choose between "Convex Optimization" by Stephen Boyd and Lieven Vandenberghe, and Yurii Nesterov's "Lectures on Convex Optimization" to supplement for my university course. I have had Calculus and Linear Algebra, my course covers constrained and unconstrained convex optimization, linear programming and constrain programming so i want to know which one of these two books will be good for me. The lecture i'm having mostly just focus on the implementation of the method but not the mathematical proof behind it so i want to learn more about the math behind optimization algorithms. I've read a few section from each book and personally i think Yurii Nesterov's book is more rigorous and easy to understand, but i still want to ask because i don't have enough time to study both and i want a strong foundation on this subject for my AI major. P/S: I'm also really enjoy rigorous books like Apostol's Calculus I,II so i want to know which of the two books above have similar style.


I often see "functional" used as an adjective in situations where I think that "function" would actually be the better choice. Specifically, I am referring to translations of German compound nouns. Here is an example: Funktionsarchitektur. This is the architecture of the function/functions. To be specific, we are talking about a technical function in a machine or vehicle. This is the architecture of the devices, control units, etc. which perform the specific function. ABS would be one example. In this case I would use "function architecture", translating it as a compound noun in English as well. For me, "functional architecture" is an architecture which functions and not an architecture of/for functions. I see a lot of cases in which the adjective "functional" is used instead of expanding the compound noun. Any thoughts on this?


while working on problems related to fluid mechanics i came across a problem in which we were asked to find the kinetic energy of a sphere which is under pure rolling and has non viscous fluid filled inside it. here you can see the figure : so my initial thoughts were to write the moment of inertia of the whole body about center of mass and then proceed as normal. But to my surprise this was not correct so i thought more about this and made a conclusion that if the fluid is non viscous then maybe we don't have to consider its rolling motion. This assumption made me do this question but i am not sure whether it's correct or not. I want to know what you guys think about this assumption.