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What does it mean, "gravity is so strong that not even light can escape from a BH..."? This can physically interpreted IMHO that beyond the event horizon of the BH, light is phase transitioned (accelerated) to a FTL superluminal energy that breaks known physics. The inner of the BH therefore appears in our spacetime as... | 0 |
classical electromagnetism , from my understanding can be derived completely from coloumb's law , charge invariance, superposition principle and postulates of special relativity. Biot savart law and lorentz force law are all consequences of it . For derieving faraday's we need not do any experiments, flux rule : e=-d(p... | 0 |
The photoelectric effect is most probably seen when the incoming light has lower energy than the energy needed for both Compton scattering and pair production to happen. The probability of the photoelectric effect to occur also increases when the matter that light interacts with has a big atomic number and high atomic/... | 0 |
I was watching Susskind lectures in string theory. There he explains that open strings can both, split at any point, and also join at the ends when the ends touch at a single point. I have one question about each of these two processes. Is not the likelihood that the two ends of a string end up at the same spatial posi... | 0 |
If we shine a light that has less energy than the work function energy of the electron of an atom on a metal, the electron is not released but excited and electron gives off this excess energy as heat or collisions or light. We know that the intensity of incoming light does not affect whether the electron will be relea... | 0 |
Current in a wire is defined as the amount of charge that passes through a cross-section of that wire in a single second. By this definition alone, it is clear that a current relies on the motion of some charged particle. I believe it is possible that electrons could transfer energy to each other in every direction and... | 0 |
In cubical homology you have to consider the group of degenerate cubics and use the group of cubes module degenerate cubes (Massey). If you not do that you get the wrong homology for one point space. In singular homology (simplices) every textbook explain you do not need to do that because you are going to get the same... | 0 |
I have a set with unknown cardinality. The cardinality can be from a number a to a number b. How can I indicate the number a (or b)? The question in texts is: "What is the least value of n(A)" (A is a set with a few possible cardinalities) How can I properly express this in mathematical notation? I used the notation(If... | 0 |
Clearly particles individually pass through slits, be it a single or a double slit experiment. The fact that wave interference is evident in their trajectory may be due to their interaction upon entering the slits. If water particles, or sand particles can together form waves, one can assume they will act in such way t... | 0 |
[Note I am asking up to, but not including, consciousness as this bleeds into philosophy and is a much messier question] Assuming that the laws of physics have remained constant across space & time since the big bang, has the way the universe evolved been entirely predetermined? While impossible to know the physical pa... | 0 |
If the Hamiltonian manifold for the moving surface of the standing wave is smooth then it must be minimum surface of revolution. The frequency of the string is the capacity of the manifold that is a periodic solution to the Hamiltonian. The frequency cannot be a rational number frequency = velocity/wavelength because t... | 0 |
I. I thought he was swimming - I thought he swam for a while but found out it was a lie I thought he is swimming - I thought he was swimming now, but he's not. I thought he swam - I thought he once swam (don't know when). I thought he has swam - I thought he has already swum recently (a couple of minutes ago). I though... | 0 |
If electric fields are created by an accelerated charged particle, such as an electron, and magnetic fields are generated by electric fields in motion, what are the individual fields that make up Electromagnetic Radiation oscillating between? What is the Y axis in their sinusoid function measuring, and how does that co... | 0 |
Hi there Wise people of the internet, I am trying to do analysis some data gathered from a gamma scintillator setup, its stored in root. So i have to do some coincidence measurements, and i found that in Krane, you normally use a TAC (Time to Amplitude Converter) to check for it. However, this gaussian peak expected th... | 0 |
There is a group in which is embedded the set of rational functions considering the formal composition binary operation? More generally how (and when) can we extend a non-commutative monoid (or semi-group) into a group ? I was looking at the interesting case that the natural numbers can be extended to de integers by th... | 0 |
I am stuck writing this sentence: Relics such as Fortran, B, D and other programming languages continue to stay alive, stories of programming languages in the graveyard, and those that spurred magic have antecedent the continues quest of improving programming. EDIT: Clarity. Think of this as a battle cry among programm... | 0 |
I've read quite a few things about rarefaction waves in terms of giving weak and entropy solutions to certain PDE problems with fixed initial time data. I understand that the consideration of these solutions provide some sense of density that decreases with the passing of time, but I've not heard much else. My concerns... | 0 |
Papers often use one example throughout as their ongoing example. Similarly, it is possible to focus on one particular case and show a broader claim by using that case. I remember there was a really nice verb for using x at a test case, or perhaps x serves as a use case. I tried a thesaurus searching for exemplify (whi... | 0 |
Could somebody untangle following statement I found here: the integer cohomology groups correspond to the quantization of the electric charge. I know from pure mathematical side the meaning of cohomology groups, but not understand the translative part between physics and pure math at this point: Could somebody borrow s... | 0 |
I am trying to find some sort of motivation as to why we integrate manifold over differential form and why especially does it in some form corresponds to integrating the surface of the area. I have already passed my university courses that have covered these topics, including Stokes's theorem with proof (My uni has thi... | 0 |
Let say we have some water in the sink and open the closure.. The water starts to move towards it in a whirpool like manner.. If we have a table tennis ball and leave it near the hole of the sink it will rotate like a planet instead move right towards the hole... If we suddenly open up more the hole the water will star... | 0 |
Hi everyone, lets say we have a circle shaped space station and there was an accident where we got a hole (hole A on the picture) on a station and now most of the station is vacuum except for one enclosed part that is still filled with air. Now if we make holes B and C at the same time in the enclosed part, what will t... | 0 |
First, there are only three type of leaves: enter image description here If we assume lamination is compact ,all leaves is only can be a simple closed geodesic. I want to ask about how could union of uncountably many leaves be a minimal compact lamination,actually all leaves is complete simple geodesic,but all leaf in ... | 0 |
If I were to measure some quantitavie metric of a sample population and record its mean, and then I were to split by random selection all members of the population into two groups of equal size and record the means of each group on the same metric. Is their a statistical term/measure of the difference of those means fr... | 0 |
The court's ruling is claimed to be (...) because it has only counted specific actions to set up some (...) simple structure of justice that only encompasses the last hour neglecting the history of events. (...) = ? = "has only lasted for a short amount of time prior to being mentioned" Referring to current attitudes t... | 0 |
Simulate a magnet sticking to a metal door. The side that is against the door, means that side is the pulling side, and must be the inwards movement of a toroidal magnetic field, pulling the object inwards resulting in getting stuck to the object! Does this mean, the side that is against the object is the side where th... | 0 |
In school, I learned the mechanism of high and low pressure areas, which roughly goes like this: In the tropics, the sun warms up the air during the day. Water evaporates, so that the air gets warm and moist. Warm air is lighter than cold air, so it rises up, leading to a low-pressure area because the warm air is now "... | 0 |
My understanding is that in commercial nuclear reactor operations, fuel rods are not used up to the point where they're fully depleted and unable to support fission, but are replaced while they still contain an appreciable amount of fissionable isotopes, to ensure that the reactor stays in a stable operating regime at ... | 0 |
In biology, the scientific name of a species (known as the "binomial name" or just the "binomial" or sometimes even just the "binomen") is written as a pair of words in italics (or underlined, which is the equivalent of italics in handwriting). For example, modern humans belong to the genus Homo and, within this genus,... | 0 |
The following question was discussed in my Discrete Math class, but we couldn't reach a consensus. Think of a set as a collection of bins. Each bin contains exactly one object distinct from all others. While we are allowed to move the objects around between bins, we cannot remove any object, nor can we place two object... | 0 |
I have seen different explanations to understand why there are no local gauge invariant observables in gravity. Some of them explain that diffeomorphisms are a gauge symmetry of the theory and thus any observable evaluated at a spacetime point will be gauge dependent and therefore not an observable. This line of reason... | 0 |
I'm building the Proximal Policy Optimization algorithm from scratch (well, using PyTorch). I've been studying it by my own, but I'm a little bit confused in the optimization phase, here is the thing. From what I know... First, we initialize a Policy Network with random parameters. Second, we start with policy rollouts... | 0 |
I am currently very confused about the "topological" prerequisites of Lee's Riemannian Geometry (RG) book, An Introduction to Riemannian Manifolds. I have heard that this is a "truly introductory" text for beginners in RG. But as for prerequisites, in his preface Lee states that his other two books on Topological and S... | 0 |
I recently asked this question about whether there was a "distance" between two galaxies where both the gravitational force and the influence of dark energy would be balanced. The answers and comments seem to indicate that there is indeed such a "radius" around a galaxy. I was very interested in this, so I contacted th... | 0 |
I'm an amateur and this is my first question here, I'm trying to formulate question about a general representation I have in mind after trying to grasp the idea of relativity and the concept of space-time. We always talk about the "speed of light", but it's a bit of a misuse of terminology, since we should really talk ... | 0 |
I understand that "laws of physics" is a bit of a misleading term since all they really are is just us applying a logical statement about observed physical phenomena in a way that allows us to predict or understand said physical phenomena. That said, what my question is getting at is whether there are any laws of physi... | 0 |
I am looking for a term to use as the name of a software project that I am working on. The project is a software tool, and this tool aims to be useful in virtually all software, so I am looking for a term that alludes to it being an indispensable, or perhaps even the most indispensable, tool in a profession, as in: the... | 0 |
In textbooks on many-body quantum physics (e.g. Fetter and Walecka), Feynman diagrams are typically introduced after formulating the Dyson perturbative expansion of the Green's function using Wicks theorem. Then the Feynman diagrams follow as a way to conveniently represent the resulting integral equations. In most lit... | 0 |
I think this is easier to understand the mapping of injective, surjective and bijective in terms of marraige proposal where men are from set A and women from set B but Is this analogy correct? Injective (One-to-One) Functions: If the marriage process between men from set A and women in set B is injective, this means: E... | 0 |
Trying to solve a problem with a colleague of mine, we prove a theorem that I'm sure someone else must have had to come across but couldn't find anything about it. We needed a way to tell how far away from equilateral was any triangle. A measure for any triangle of its non-equilateral-ness. One idea (that later turned ... | 0 |
I came across this sequence called Digital River, where the next number in the sequence is defined as the sum of the digits of the previous number, plus, the previous number itself. It caught my attention for some reason, and I wanted to analyse it. And I found some curious fractal-like patters. But let me begin by say... | 0 |
I'm sorry for mistakes if any; English is not my native language, but I'll try to explain myself as thorougly as possible. There is a geomerty topic about construction of different geometric shapes using just a straightedge and a compass (a pair of compasses, more accurately). E.g. a regular pentagon can be constructed... | 0 |
We have hydrogen inside a tube, and we induce a voltage on it; a current passes through it and light is emitted. The frequencies of light correspond to the differences of the eigenvalues of the energy operator, which is the observable in question, so it is customary to give a heuristic explanation that the electric ene... | 0 |
I have a bit of confusion because when doing QFT and QFT in curved spaces this particular issue seems to be avoided. I have this feeling that when we quantize a theory, we somehow choose a chart and we stick to it. This feeling comes from, for example, the way we deal with Lorentz transformations in QFT, namely via uni... | 0 |
What's the word(s) for a feeling of disappointment when you've lost something of financial value? For example, let's say I'd just got an expensive LCD monitor from a raffle, but I accidentally dropped it and it broke, and I now lament its loss. I guess "lament" works okay, but it's not really a colloquial word, and it ... | 0 |
There are a few river crossing problems that I have seen that share some common aspects. The cannibal and missionary problem is typical. All these problems involve moving everyone from one side of the river to the other side by using a boat to cross the river. Denote by complement, an arrangement that is equivalent to ... | 0 |
I saw other posts such as this one but I don't think it's quite the same question, or even if it is, the answer employs the operator formalism and I'm not sure I follow it. I'm wondering, if you have two multiparticle states - a multiparticle state being, in my mind, a complex probability amplitude for each possible co... | 0 |
Context: I ask this as a school teacher reaching past the boundaries of my expertise. A colleague was talking about the standard model with an advanced student, explaining how particles interact by exchanging gauge bosons, asking the student to imagine gauge bosons as little spheres. Of course they got to the issue tha... | 0 |
I am currently thinking through evaporation over lakes, specifically the Laurentian Great Lakes (a complex subject, I know). Particularly, I am trying to wrap my head around why evaporation peaks in the fall and winter. Based on what I have read, this fact is due to the vapor pressure gradient that exists between relat... | 0 |
As is well-known, classical conserved systems have conserved quantities by virtue of continuous symmetries, which can be derived from Lagrangian mechanics. For example, two masses on a spring can swap momentum between the two, but translational invariance ensures that the total momentum is conserved. But what if we int... | 0 |
Imagine we have a hollow metallic toroid, with copper wire winded around it, which carries electric current. That implies we have magnetic field inside the hollow toroid. The toroid has vacuum inside. We have a set up of a high voltage supply and an electron gun that takes the free electrons from the metallic toroid an... | 0 |
I'm having trouble finding a good phrasing to describe a component of a system that is too important -- in a sense that it distracts a person from all the other components, that should have been, or were formerly, important aspects of the experience. The case where I'm trying use it is something like this: We tried add... | 0 |
Scenario Consider an empty universe with just the Earth, Moon and the Sun. The Earth and the Sun will rotate about their center of mass, which is inside the Sun. The Earth and the Moon will orbit their center of mass, which is inside the Earth. The Moon and the Sun will orbit about their center of mass, which is inside... | 0 |
Initially, I was looking for how centripetal force is produced on the surface of the rotating earth for a mass kept at any latitude. I went through the following threads - Which force provides the centripetal acceleration that makes objects on earth's surface rotate about Earth's axis of rotation? Is the normal force e... | 0 |
This Q&A did not answer my question. The voltage of a circuit is the difference in each Coulombs potential energy at the negative pole, compared to the positive pole. At the negative pole, there's a whole wire for the electrons to pass through under the influence of the Coulombic forces; a whole wire with atoms between... | 0 |
I'm now in the process of writing a report on a lab work i did with an Abbe refractometer. In all sources i found the working principle of this refractometer is described as such: "Light shines into a illuminating prism who's side that contacts the sample is roughed so light scatters uniformly in all direction into the... | 0 |
Whenever I Google to try to find an actual formal statement of the first incompleteness theorem (as opposed to all the oversimplified explanations that talk about "true but unprovable theorems" rather than theorems independent of the axioms), the definitions that don't just mention something like a system with "strong ... | 0 |
I have been using '[...]' to indicate skips in the middle of sentences, '[...].' to indicate that a single sentence has been skipped (or that a middle of sentence skip '[...]' is at the end of a sentence and so is punctuated), and '[....].' to indicate that two or more sentences have been skipped (or to indicate that a... | 0 |
In the past two months, I had found a wonderful pdf that went through a derivation of the determinant with calculating the area of a parallelepiped as its starting point. The document did not get into the weeds of calculating the determinant given a matrix or even focus explicitly on matrices at all; it was probably th... | 0 |
Motivated by question Can IC engines be modeled as Carnot engines?. I am wondering whether/how Carnot theorem could be generalized to other kinds of devices performing "useful work", such as, e.g.: Motor (or generator) fed by a battery Nuclear power generators Solar cells Water wheels I think that the theorem must be g... | 0 |
The Earth, effectively a non-inertial frame of reference, is where Newton concluded his laws of motion. However, Newton's first law only holds in an inertial frame of reference. In the process of inventing the Newton's laws of motion, since almost all (I suppose) the experiments were done in a non-inertial frame of ref... | 0 |
Let me try to illustrate what I mean. Consider e.g. a Solar radiation storm (Solar particle event) where high-energy protons are hurled at Earth from Solar flares. I've tried to illustrate my conception of this (I know the protons will typically not follow straight paths out from the flare due to the Parker spiral, but... | 0 |
My question, right up front, is: what is the term for a modifier that behaves this way? But "this way" takes some explanation, and that is the rest of the question. I am a mathematician, and my question makes the most sense in a context where words are formally defined anyway, but you can freely substitute "foo" and "b... | 0 |
So, I was reading some books by Stephen King, S.D. Perry, and a couple authors I really love. I notice they'll use pronouns or certains words twice in the same sentence. When I read it, it's pleasant and doesn't sound weird in my head at all. I assumed that if something is published under an author like Stephen King or... | 0 |
In my limited knowledge of statistics, I am confused between the meaning of Latent Variables and Nuisance Parameters. Here is my current understanding: Sometimes the variance parameters in a regression model (or the overdispersion parameters) are considered as Nuisance Parameters because we are not directly interested ... | 0 |
My partner frequently asks me questions that, when read literally, are questions about the past, but in intent and intended response are actually conditional questions: Did you have any thoughts about dinner? Did you want to have coffee? ...where the intent is not merely to inquire about the preferences I have establis... | 0 |
Double-slit experiment image source: Wikipedia The double-slit experiment can be regarded as a demonstration that light and matter can display characteristics of both classically defined waves and particles. It also displays the fundamentally probabilistic nature of quantum mechanical phenomena. In a double-slit experi... | 0 |
I am working on a problem where I need to extract a connected tree of nodes based on certain attributes while optimizing for the minimum number of nodes. Some attributes of the nodes are known in advance, such as resource capacity. However, I do not know in advance how many nodes will be selected in the final tree, esp... | 0 |
Sorry if this equation is not phrase in precise mathematical form. I am open to suggestions to improve the explanation, and I have tried to formulate the problem as precisely as I could. I was talking to a friend--postdoc in PDEs--today and he was asking me about obtaining some Yahoo data on stock prices for his studen... | 0 |
I have seen that it is possible to approximate the metric in the presence of a gravitational field by the Rindler metric: Does a uniform gravitational field exist? Is there any acceleration in a uniform gravity field? Applying the principle of equivalence to an accelerated frame Rindler Coordinates and homogeneous Grav... | 0 |
This question was inspired by the interesting discussion here: Why isn't the T in "relative" flapped? It seems like the adverb already and the two-word phrase all ready should be pronounced differently, but as far as I can tell, both sound exactly the same. For comparison, consider the other phrase/adverb pair all read... | 0 |
Euclid based much of his geometry on a theory of magnitudes that looks roughly like this: A general theory of whole and part and how they are related in size (eg: the whole is greater than the part). A general theory of the properties of magnitudes (eg: equals added to equals are equal). A basic rule that allows one to... | 0 |
The direction of polarization of a transverse wave is defined to be the direction perpendicular to the direction of propagation of wave or the direction of oscillation of wave, right? But in the case of Electromagnetic waves, a class of transverse waves, there are two directions of oscillations that are perpendicular t... | 0 |
Firstly: having a lot of difficulty figuring out how to articulate this question due to lack of general math knowledge. There are multiple questions posed below, but I feel like if I knew more they could be condensed into a single question and am hoping someone can suggest and edit to the effect of the below. Thank you... | 0 |
I'm trying to understand how electromagnetic radiation is created and can propagate through the void. I do understand the concept of an electromagnetic field. But I don't understand how we get from a "field" to a "wave". I'm not really interested in detailed mathematics of how this happens, rather I'm looking for a com... | 0 |
I'm an undergrad with an electronics repair background trying to find an explanation for one of the fundamental aspects of a transformer. Every explanation I've found of a transformer's basic operation insists that there is essentially no power loss; that is that VxI in the primary = VxI in the secondary. This suggests... | 0 |
In QFT, many mathematical issues arise. Setting aside renormalization, these deal with rigorous constructions of objects underlying QFT: i) In the canonical quantization approach, the main issue comes from trying to multiply (operator-valued) distributions. My understanding is that mathematicians have formalized some s... | 0 |
Would the positive and negative charges line up on either end of the wire? Or would it induce a current? Or would the wire be unaffected by the magnet? This was deleted for being a homework question. I'm not aware of any homework questions like this. I thought this up in my own head. (I drew the diagram in a program ca... | 0 |
I have been trying to learn about lattice path integrals. Unfortunately, majority of the literature on this topic is in regard to Lattice Quantum Field Theory and Lattice Quantum Chromodynamics. That is fine, however what I desire is literature on evaluating plain, standard quantum-mechanical path integrals on a discre... | 0 |
I've started to study in details quantum optics and I find difficulties in linking the concepts of coherence and correlation among fields, especially because I'm building right now a background on classical optics aven if I already have a strong background in QFT. As far as I have understood, roughly speaking, coherenc... | 0 |
After reading the first two answers to this question, I've become interested in understanding the concept of (co)tangent complex as a way to get some intuition about homotopical algebra, being somewhat more used to the algebro-geometric framework than to the algebro-topological one. More specifically, I'd like to under... | 0 |
There is a particular Twitch streamer from a video game I played, MermaidonTap. If you subscribed and follow her, not all but most of her public streams, she uses "fuck" and a lot of the word "cunt." I am a fellow American citizen, but was not born in US but in an Eastern Asian country, and am trying to learn English. ... | 0 |
As far as I know, the first statement of the correspondence is between two formal theories named simply typed lambda calculus and intuitionistic propositional logic, which maps types to formulas and terms to proofs. We also have other statements for higher order logics and type theories. But it is also famous to replac... | 0 |
A textbook I'm using to refresh some basic grammar states that indirect objects can be identified by it's answering of questions such as 'to whom', 'to what' etc. (fair enough) and they always come before direct objects in a sentence (this raises questions for me). So the text would identify the pattern in: The teacher... | 0 |
About a year ago, I came across a really cool property of the envelope curve of a parabola that I couldn't prove. I'm posting it now for help: If we have a straight line and a circle that belong to one plane, then the enveloping curve of the parabola whose focus is a moving point on the circumference of the circle and ... | 0 |
My question probably sits more on the applied mathematics side. There is a spring mass system, for the case where the mass moves into the spring (vertically downwards) the spring experiences compression and the second order equation for a spring mass system will provide a solution. However, when the mass moves vertical... | 0 |
As a rule of thumb, vapor condensation usually happens at the interface between the system and the heat reservoir. Now, according to my analysis below, it is the only way for vapor to condense, which implies the near impossibility of condensation in the bulk of vapor. : As the temperature drops, within the vapor system... | 0 |
I am using ZFC as a tool to demonstrate my problematic logic. In zfc we construct a proof system for zfc in zfc (a simulation of a proof is what I mean); we will call it inner proof system. We establish that if there is a proof in this inner proof system that concludes A then A holds. Turing machines can be formalized ... | 0 |
"For classical (non-quantum) systems, the action is an extremum that can never be a maximum; that leaves us with a minimum or a saddle point, and both are possible." The above statement is an excerpt from the "Introduction" (preface) of the book "THE PRINCIPLE OF LEAST ACTION - History and Physics" by ALBERTO ROJO & AN... | 0 |
I am an A-level physics student, and I've been taught that temperature is the average kinetic energy of a particle. So when gas particles are heated, they move faster. This makes sense as an airplane traveling faster does make the nearby air warmer when measured from the plane. Say I release a box of room temperature a... | 0 |
Many electromagnetic interactions are modeled as exchanges of a real photons: e.g. an excited electron can relax and emit a photon. Somewhere else, a photon and an electron can interact, "consuming" the photon and leaving the electron in a more excited state. Electromagnetic radiation is modeled as a flow of real photo... | 0 |
When calculating area of a hole in an irregular surface for water flow calculations, what defines that area? I need to calculate the amount of water passing through holes in surfaces in a fixed time, knowing the pressure on either side. This is mostly a straight-forward process (discounting the discharge coefficient, b... | 0 |
What is electric current? An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. How do we generate electric currents? We know charge is the property of a particle due to ... | 0 |
The way I work on homework questions, especially for analysis and topology, might be a little different (or maybe not). I would remember the questions and think of them when I run, take a shower, and during other periods when I don't need to use my brain; later I will write out the solution without much thinking. This ... | 0 |
In machine learning sometimes we build models using hundreds of variables/features that we don't know (at least at first) if they might have a relation with the target. Usually we find that some of them do and others don't. Some of them even have a true relation that we couldn't think of at the begining. Once we build ... | 0 |
In Einstein's original thought experiment involving "a (very long) train running along a [straight] railway embankment", of essential importance appears the prescription that "[E]very event which takes place along the [railway track] line also takes place at a particular point of the train." The constituents of the tra... | 0 |
I'm currently taking this college calculus course, and this exercise has stumped me. It is in German, but hopefully what it's asking is fairly clear. To summarize, the problem first asks me to prove that the preimage of an intersection of a family of sets is equal to the intersection of a preimage of a family of sets, ... | 0 |
How is quantum entanglement different from a controlled experiment where a pineapple is smashed at a high speed to a perfectly symmetric object and measuring one of its piece in air for spin, speed, direction etc. and finding correlations with other pieces of pinapple in mid air? In quantum mechanics information cannot... | 0 |
It occurred to me that the limits of possibility to the nature of the universe is it is either deterministic ie we are all at the will of natural laws that determine the outcome of events from the moment of inception and we are philosophically dust in the wind. Or is the world random and our future is uncertain and ind... | 0 |
The proofs presented in lectures, textbooks ect. are usually cleaned up versions that show just the necessary steps for logically proving the theorem, not the thought process that went into the proof. To give a concrete example, I'm working through an (overall quite good) MIT Open Course Ware class on real analysis. I ... | 0 |
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