darklight03/StackMIAsub huggingface dataset:
I am trying to solve six first order coupled ODE's, two of these are associated with a heat balance of a catalyst pellet, and four are mass balances. I have been trying to solve these equations using Orthogonal collocation (rather than using those algorithms used by NDSolve in Mathematica - my hope is to speed up the computation). I have found a lot of examples and lectures on using Orthogonal collocation to solve a single ordinary differential equation, however, I have not come across anything in the literature, as how to solve these equations when they are coupled. I therefore wanted to ask if anyone has experience with this, and could possibly point me toward some literature, and/or worked examples, or explain the twist on how to migrate from a single ODE to coupled ODE's. Best Regards
So here's what I'm trying to do: Given a collection of lat/long coordinates that form a polygon like below, I want to be able to select a point inside the polygon and determine which side of the polygon it is closest to, and determine its distance to this closest edge. My problem is that I need to somehow get rid of the extra coordinates so I can do computation on them -- if, for instance, I only had four points that formed a four sided polygon (i.e. the west-most and east-most points on O'Farrell, and the west-most and east-most points on Ellis), I could do this computation. For instance, putting in the point for 'JINS' as input would return the edge that runs along O'Farrell St. We can assume that the polygon will not be smooth, and that each polygon represents a city square block, so they're relatively simple polygons. Can anyone think of a solution to this problem (whether that be getting rid of extra points or finding some other way to do this computation)? Thanks.
I'm a student of Physics, however I usually study mathematics on texts aimed at mathematicians to gain a deeper understanding. Currently I'm studying differential geometry on Spivak's book and one of the main results I need is the relationship between vector fields and infinitesimal transformations, i.e.: the idea of infinitesimal generators. The only problem is that Spivak's way to get into this is a little more complex than what I need. Indeed he spends time with differential equations and topological properties of manifolds that are related to differential equations. These are interesting topics, but for now what I was really needing was this relationship of vectors and infinitesimal transformations and the understanding of where Lie Groups come into play. Is there a shorter path into these results without needing to go through all of that stuff on differential equations? Is there a more direct way to get into these topics? I ask that because perhaps Spivak just presented that way because he wanted to show how vector fields relates to differential equations in a more concrete way. Thanks very much in advance.
I am having a discussion/conversation with a very close friend, and this conversation concerns me personally (you might even go as far as saying this is my best friend). At one point in our conversation he starts to "spam" (not literally, the messages have meaning and are well-formulated) me with many messages, but I really have nothing to say back (or anything useful to say back). I should clarify by saying that I completely understand what he is saying, and I somewhat agree with the things he said. He then proceeds to send me another text saying: "Don't you have anything to comment?.." How could I respond nicely with a proverb/saying of any kind that means "silence is sometimes better than speaking when one doesn't have anything useful to say"... I know some sayings like "silence is gold" or "better to remain silent and be thought a fool than to speak and to remove all doubt", but they don't really work for me in this scenario. I hope I was clear enough. Thanks
I have been trying to find out what the definition of a noncommutative regular local ring is, but to no avail. In fact, how does one even begin to define Krull dimension for a noncommutative ring? Hence, I would appreciate it if someone could kindly provide definitions for the following, in the case when the ring under study is noncommutative: Regular. In the commutative case, the definition of regular involves localizing at prime ideals. However, in the noncommutative case, how do we do localization? Is Ore's Condition invoked somewhere? Regular local. In the commutative case, the definition of regular local involves Krull dimension. However, in the noncommutative case, do we have an analogue of Krull dimension? On a different note, in the commutative case, is it true that a local ring that is regular the same as a regular local ring? (This might seem to be a stupid question.)
The use of the term sir as a form of address for men, especially those of higher rank or status, is discussed in several prior questions including this one. They all indicate that the term is reserved to males, and that there are a number of related terms for females, such as ma'am. A review of the first dozen online dictionaries at onelook.com confirms that sir is strictly reserved for male addressees. For example ODO defines it as used as a polite or respectful way of addressing a man, especially one in a position of authority: excuse me, sir On several recent television shows in the US, the term sir has been used by a police officer to address his or her supervisor who was female. In the context, the use was sincere and was not objected to by the superior. Has the use of sir when addressing a superior female in a military or quasi-military setting become acceptable, or is this merely literary license?
Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an informal introduction to the subject for a theoretical physicist (having in mind the applications in physics, e.g. in the string theory)? What I want for a moment is to get some informal picture of the subject rather than being dug up into the gory details of the proofs and lost in higher and higher layers of abstraction of commutative algebra and category theory. The texts I have found so far are all rather dry and almost completely lack this informal streak, and all of them are geared towards pure mathematicians, so if there exists something like "Algebraic geometry for physicists" and "Kahler manifolds for physicists" (of course, they would probably have different titles :)), I would greatly appreciate the relevant references.
How would I say that a phrase or word has "lost it's meaning due to constant repetition." Take the word "awesome": "Awesome" used to denote a situation in which the speaker (or writer) was overwhelmed with "awe." "Awesome" is now used as a measure of how "interesting" something is and isn't terribly high on that rating scale. How would I describe the loss of meaning due to the constant repetition of the word "Awesome"? Take George Santaya's words "Those who cannot remember the past are condemned to repeat it": People often say this phrase to try and instigate action but because of it's overuse, it no longer has any meaning. It's just something people say when they talk about The Holocaust, or the bad grades they got last semester in college. How would I describe the loss of meaning due to the constant repetition of the phrase "Those who cannot remember the past are condemned to repeat it"?
Fat lot of good is a phrase that I grew up with and continue to use occasionally as in the following: He is working hard to fix the problem, but a fat lot of good it will do him without the proper tools. Which means that despite his best efforts, he is not likely to fix the problem until he starts using the proper tools. I've only heard the phrase used with a sarcastic tone as if the phrase is supposed to mean that it will do a lot of good. (Similar in tone to when someone says I could care less when they really mean I could not care less.) What is the origin of fat lot of good and was it ever in common usage with a positive meaning?
I would not be able to put this into symbols, but I ask here because I think it's the correct place to ask. Would the chance of my parked car getting damaged (bumped or scraped) by other cars parking nearby increase over time? Gambler's fallacy says: if something happens less frequently than normal during some period, it will happen more frequently in the future. (wikipedia) For every time I leave my car parked, there is a chance it will get damaged. If I don't want to commit the gambler's fallacy, I should consider the chance of damage the same every time I park. But if I park at the same spot every day for many years, the chance that my car would have been damaged after all those years, would surely be greater than if I just parked there one day, right? How does this not contradict the gambler's fallacy? My insurance company asks a higher premium if I park on the street all year round, than in a garage, so somehow they must figure that the chance is higher than if I just park on the street one day. How does this not contradict the gambler's fallacy?
The three laws are: First law: The velocity of a body remains constant unless the body is acted upon by an external force. Second law: The acceleration a of a body is parallel[disambiguation needed ] and directly proportional to the net force F and inversely proportional to the mass m, i.e., F = ma. Third law: The mutual forces of action and reaction between two bodies are equal, opposite and collinear. The first law had already been formulated by some philosophers prior to Newton, Hobbes said in the Leviathan '...[the proposition] that when a thing is in motion it will eternally be in motion unless somewhat else stay it, though the reason be the same (namely that nothing can change itself)...', given his reasoning, I think it safe to mean constant speed and direction, otherwise change is occuring and he explicitly rules that out. I think the same proposition is mentioned in Lucretious's De Rerum Natura. Are there any antecedents for the second and third law?
I was listening to The Infinite Monkey Cage on the BBC and they were talking about general relativity and gravity. They were saying that gravity is not a force as Newtonian laws describe and is at odds with your real world experience. They said that if you are sitting on a chair, you don't feel a force pulling you down onto the chair. You feel the chair pushing up onto your bottom - at least that's what it feels like when you sit down and try it. They seem to be saying that gravity is not a force pulling from the center of the Earth, it's me that's still and the chair pushing up against me due to acceleration. My question is if this description is accurate or have I misunderstood this? If you are sitting on a chair, is what we call gravity actually the chair accelerating up onto me due to the geometry of Space Time? Is there a better analogy for this?
I'm having trouble completing a proof that for positive integers a and b, that the least common multiple of a and b is ab/gcd(a,b).This is how I've approached it so far: For s = lcm(a,b) we have the following definition: i) a|s and b|s ii) for any integer k where a|k and b|k, s|k Thus the proof comes down to proving that s satisfies these two properties. For i) let gcd(a,b) = d. Thus s = ab/d. Since gcd(a,b) = d, it follows that b/d is an integer so a|ab/d since ab/d = ta for t = b/d. The same conclusion can be made for b so i) is satisfied. Now comes the confusion. I'm not not sure how to prove that if a|k and b|k then ab/d|k. I tried using am = k and bn = k for integers m and n but it lead to no avail. Should I approach the problem som other way?
I am teaching my students about the fairness criteria for voting system, working up towards arrow's impossibility theorem. One of the voting methods is called the pairwise comparison method: voters rank each of the candidates from most to least favourite. To tally the votes, talliers compare each pair of candidates. If candidate X is more often preferred than candidate Y, then X receives a point. (If they tie, they each get half a point.) At the end of the comparisons, the candidate with the most points is selected. We discuss criteria for a voting system to be fair. One criterion in particular is the "irrelevant alternative criteria" which states: If an election is held and a winner is declared, this winning candidate should remain the winner in any recalculation of votes as a result of one or more of the losing candidates dropping out. Can anyone think of an example of when the pairwise comparison method violates the irrelevant alternative criterion?
I am trying to grasp some aspects of the quantum entanglement, but the existing resources (including some of the links here) seem a bit confusing. I am trying to find an answer to the following questions. If two particles are entangled and then separated, will affecting one of them affect the other (for example, the particle is placed in a field that would set some property of it in a specific direction), or will it disentangle the system? If affecting one particle affects the other, then how is it not possible to use this effect to transfer information (once; for example by affecting the spin of the first particle to be up the other particle would have its spin down)? Is affecting the particle equivalent to measuring the property affected (for example the spin)? When a property of the particle is measured, does the particle get entangled with the measuring apparatus? More precisely, does interaction imply entanglement of a sort? I apologize if the questions are trivial or nonsensical, but I am asking as a layman in the field.
Here's what happens. I get stuck on some proof or some mathematical construction and I end up staring at the problem for hours, sometimes not making any progress. I do this because sometimes I think that I'm being lazy, I'm not thinking things through, or I'm just not thinking clearly. This approach is not practical because I only end up falling behind on other work. I don't like to look up solutions because I feel like, given enough time, I would be able to come up with the answer (or some good reasoning). But maybe I should start looking for answers after a shorter period of time. I don't know what the right thing to do is. Do you guys have similar problems? Should I feel bad because I have to look at solutions? Or is this just part of learning?
I'm searching for some good reading material on multifractal analysis. Preferably something accessible that doesn't put the stress too much on mathematical proofs but rather on applications. As long as it gives a good review of the status of the field, the interesting results and applications, I would be happy. Also, if anything related to multifractal analysis and statistics or time series comes up, I'll take it as well. Books, papers, internetpages, videos, etc... accepted! EDIT: Since the question has been bumped, I decided to put a bounty on it. But I also want to make a bit more precise what I'm looking for. I have always had the impression when encountering the multifractal techniques that people are able to compute a whole bunch of numbers with some nice and fancy formulas. But I have always missed an "understanding" of what the numbers mean. Why is it useful to do a mutlifractal analysis of fluid flow? Of species abundance distributions? Etc... I feel like the technique is purely descriptive with little theory backing up the connection with some deeper underlying structures. But that may just be due to my limited understanding of the field and that is precisely why I ask for pointers to where I can look for this.
Currently, I am attempting to learn noncommutative geometry. My interests mostly lie on the boundaries of pure mathematics and theoretical physics, so I am not only interested in the mathematical formulation of the theory, but also in the physical applications. I am familiar with differential/algebraic topology/particle physics and some basic notions of homological algebra, but am fairly weak in functional analysis. What books/references/review articles would one recommend as the best or easiest starting point to learning this subject? A book that is the most self-contained/pedagogical? (I am currently starting to read Basic Noncommutative Geometry by Khalkal, but was wondering if there were any books even more suitable for a beginner). As I want to get started with learning NCG as quickly as possible, are there any short review papers, notes, or specific chapters of texts where I can gain a bare minimum of prerequisites such as functional analysis to start reading a book on NCG instead of having to read an entire book on operator algebras before starting my study? Or some books that would be helpful that I could read concurrently? Thanks!
Everyone knows how Schrodinger's Cat is set up, so the question becomes whether there's a quantum/classical boundary and what that boundary is. Some people say everything is quantum while some may think there's a line that separates quantum/classical. So the cat in the box would have to be a mixture of live/dead cat until one of two things happened. Decay occurred and poison gas is released that kills the cat or it doesn't occur and the experimenter opens the box and sees a live cat. Wouldn't the wave function be a quantum ensemble of these two states described in Hilbert space until one of these measurements occurred? What state is the cat in before one of these measurements occur? Doesn't it have to be in a quantum ensemble of both states until one of these measurements occur?
Often it is said that Bell's theorem (and the observed violations thereof) rules out local hidden variable theories as the explanation for the seeming non-determinism found in quantum mechanics. I'm wondering if this extends also specifically to radioactive decay processes. So I guess this is really a question of whether the radioactive decay processes are currently understood to a level so that one can say that the seeming randomness boils down to some well-known quantum phenomena that would be covered by Bell's Theorem. Or, alternatively, that the decay processes are not understood to the extent that they are reducible to quantum phenomena and hence could have a different explanation. For example, one could imagine radioactive nuclei being a deterministic but chaotic system of interactions going through states, where sooner or later a state is reached where the decay occurs etc. The question is if such a view, where the individual decay is actually a deterministic phenomena (but appears as random since we dont know the current state of a particular atom) is compatible with current knowledge of radioactivity and quantum mechanics. The section https://en.wikipedia.org/wiki/Radioactive_decay#Theoretical_basis_of_decay_phenomena seems to leave the impression that the underlying mechanism of decay is not understood.
Sometimes people make insincere arguments to justify an action (or inaction), based upon the impact to a particular group. For example, a person might say "Building that sports arena near my house is irresponsible! Think of how many homeless people we could house with that money!" In truth, the person does not want the arena near his or her house, and does not really care about whether homeless people are housed or not. What is the term used to describe the homeless people in this situation? Another example might be large industrial agriculture organizations defending government policies "to protect the family farm" when these organizations are not family farms and do not care about protecting family farms. What is the term used to describe family farms here? The best term I can think of is "smokescreen", as in "Homeless people are being used as a smokescreen to hide other concerns about the sports arena" but this seems informal, and there are probably better phrases (formal or informal). What are they?
I'm building a website for a client in which customers can customise the shape of their board (curvature, length, width, thickness, and so forth) and the client has asked if we can calculate the volume of the board and display it in response to the user's customisations. This sounds like scary advanced calculus and/or differential geometry territory, but I said I'd try to find out the feasibility of doing so nonetheless. I'll mainly be using a set of bezier curves as a baseline to define the board shape and there will of course be curves in all dimensions. The nose will sometimes come to a point and other times be rounded off. The tail will come in a variety of shapes, including being rounded, coming to a point, or even having an inverted "fish tail" shape. Where would I even start with calculating the volume of a surfboard like this? I'd be looking to represent the volume in litres. Edit: I'm guessing I'd need to break the board down into chunks that minimise the number of simultaneous curves and calculate them individually. I'm thinking maybe it's also easier to calculate the volumes of each chunk relative to a uniform enclosing box and subtract each value from the total box volume to get the final volume...
I'm trying to write a database schema for measurements in a variety of categories, and am having trouble coming up with names for some basic elements. Let's assume we are gathering heart rate and blood pressure measurements from a group of people. Each observation is composed of the individual being measured, the time the measurement was taken, and the amount of the measurement. What are each of these called? What is the name for the type of measurement being taken (heart rate or blood pressure in this example)? The collection of individuals is the sample, correct? If so, what is the term for the collection of observations? Please feel free to revise the question, as my lack of knowledge makes it hard to ask accurately. I can come up with names myself, but after failing to find anything in some statistical glossaries, I'm curious if there are canonical names for these basic elements.
So this is my second query about the photovoltaic effect. I've looked into it more and understood it for the most part, but there's still something I don't completely get. After the electrons are excited into the conduction band, some sources say that they can't cross into the opposite material due to an electric field blocking their path and would need an external circuit, others say that some electrons do cross into the other material but their flow gets halted gradually into nothing by the formation of a depletion region, and would then need an external circuit to maintain a consistent flow. So which one is it? Another question is, what exactly causes the electrons to move in the first place? Their attraction to the holes in the opposite material? Simple as that? Much obliged, and thank you for your time. Bonus: If you could explain in a general context, rather than in relation to P-N junctions, you get extra brownie points.
I've been reading up on nuclear reactors, and understand explanations of how it works, how water is heated to steam, which turns the turbines, etc.. I understand all of the safety features, and how control rods are used, and what they do with spent fuel. However, what I can't figure out (after lookin at many websites and videos) is where the reaction actually starts. The fuel rods contain Uranium pellets, and then they're put in the reactor, where the reaction starts and neutrons start hitting each other to create heat? So is this happening all over the reactor, or just in each fuel rod assembly? Also, you can hold a Uranium pellet in your hand (ideally wearing a glove) and it's not dangerous, so what starts a reaction? A lot of people say the reaction can start by itself, so why do uranium pellets not suddenly heat up and start spreading radiation by themselves? Is it because they haven't been enriched? What if you dropped a Uranium pellet on the floor?