darklight03/StackMIAsub huggingface dataset:
I am an undergraduate CS student but I love mathematics and spend most of my time doing and reading math books. I realize that it's important to get into the habit of reading papers and journals so it will be easier to think of ideas for projects and research. I'd like to know how I should start reading papers and what papers are good for beginners? The areas of mathematics I really like are Discrete Mathematics, Combinatorics, Number Theory, Mathematical Induction, Problem Solving, etc. I like things like Calculus too but I feel papers on Calculus would be too difficult to understand. Also, are there any particular efficient methodologies for reading papers? Is knowledge better gotten from books or from papers? Note: I want to say that there are already threads asking which papers every mathematician should read, and which every computer scientist should read. The purpose of this thread is slightly different. It isn't asking which standard papers everyone should know. It's asking which papers allow for an incisive entry into deeper knowledge of the subject. P.S. : For the benefit of anyone who sees this thread later, I did find a wonderful journal called Crux Mathematicorum'' dedicated solely to problem solving! They allow free access to their back issues on their website. Other good journals I found wereParabola'' and ``Pi in The Sky,'' both of which may be read online for free.
I am trying to make a computer game, which involve radioactive hazard. The game is still in its early stage, and I want to add mechanisms which are as realistic and as fun as possible. So I'm trying to understand how radioactivity is "absorbed" in the air. For example (please excuse any physic mistake), if you have some alpha radioactivity, it will be absorbed by a very thin material, like a sheet of paper. For Beta, it's more like an aluminium sheet, and for gamma you're dead already. But how does the radioactivity weaken in the air ? Is it absorbed by random elements and the energy is lost in them ? I guess the explanation is similar to why Wifi just doesn't work at great distance, but then I'm not sure how it happens. Feel free to give any text explanation, however I'm open (even if a bit afraid) of mathematical explanation. I'm pretty sure there is a theorem or something about this, this would be even better (although explanation won't hurt).
I have a basic enough question. Assume that one has one of those ideal see saws, i.e. the teeter-totter pivots on point source, the balance is of uniform mass etc. Now assume that one places an object of mass m on one end. This will force the balance to tilt on side. Now suppose that one adds another mass of m on the other side. What happens? It seems that everyone agrees that the the balance would move to the stable equilibrium, that is the middle. What I don't understand is, what is the Force that moves the teeter-totter to the middle position? Once the second mass has been placed on the other side, the two net forces acting on the pivot is the same. And since the distance is the same, the torque is the same. So what force is it that moves the balance to the middle? And if no such force exists then every position of the balance is valid if the two weights are equal, then how do those mass balances work?
In a previous question, I learned that in order to detect an object in space, what matters is how much electromagnetic radiation it is giving off, and what sources of EM radiation the sensor can pick up. Given that the sensitivity of the sensor over the spectrum is a parameter for detectability, I would like to learn if some wavelengths of EM radiation are more difficult to detect than others, over increasingly longer distances in space. From my experience in biology, I know that longer wavelengths of lights from a source can be "pierce" deeper into tissue than shorter wavelengths (a bit counter-intuitive!). In fact, that's the premise of "two-photon microscopy" -- using wavelengths of light with such low energy that fluorescent dyes need to absorb two low energy photons in order to be excited. So, the probability of that happening is lower, but there are many benefits, including the fact that longer wavelength light can penetrate deeper into tissue. Does the physics behind such concerns at the microscale also affect detectability at astronomical scales? Is some EM radiation easier to detect than EM radiation with other energies? Can EM radiation of certain energies "travel" longer distances through imperfect, real space?
Good evening, I'm writing my thesis in Latex, with which I already had the pleasure to work for some other projects, but I'm now facing a problem I don't know how to solve. I'll try to get to the point as straight as possible: let's say I have a chapter of simulation results, which includes a huge set of images. This chapter is subdivided itself in some sections, among which the images are distributed. Now, what I would like to have is a way to have all the images related to section A inseted before section B starts and so on. I don't mean I want to precisely insert them where I want (the most of them are set of subfigures, difficult to be fit anywhere), it would be enough to be able to tell the compiler to put them all before the next section starts. Do someone of you have any suggestion on the matter ? Thanks a lot in advance Michele
This is kind of a big squooshy question (or series of questions), which I will try to cast in a more precise form. Apologies if I don't succeed. Context: I'm an amateur set theory/category theory enthusiast with special interest in stratified set theories. My knowledge is substantial in odd spots, but not very well rounded. Because of the ubiquity of talk of models in the literature, it seemed like a thing I should learn, but I'm having a difficult time because I feel like I don't grasp certain informal aspects of "model theory in practice", even though I technically understand about the first half of David Marker's "Model Theory: An Introduction", for instance. Questions: What are the big questions one wants to answer with models that would either be unapproachable or extremely painful without it? I gather questions of consistency are important ones, but are there other main ones? When does one want to know about the second order properties of elementarily equivalent models, and why? Are there even general answers to these questions, or am I asking things I would have to ask, say, another set theorist about to get anything informative? I hope this isn't too poor a question; I'm flailing a bit to articulate beyond "it's cool, but WHY???"
Looking at Cohen's success at proving independence of the Axiom of Choice and the Continuum Hypothesis, I was wondering if it was possible to mechanize forcing techniques for the purpose of proving independence (and hence undecidability) of conjectures. It would seem to be useful to test any sentence you're trying to prove or disprove with an automated theorem prover for independence. I'm imagining some automated forcing technique that constructs a model of ZFC plus the conjecture and its negation to prove it isn't a theorem of ZFC (or whatever foundation being used) rather than spinning forever on an undecidable statement, since the incompleteness theorems indicate that any statement in a sufficiently strong formal system can be undecidable. Its not obvious to me how to automate such a method, as its probably quite a bit more complicated than just implementing resolution, but it would be nice to have automated reasoning go down three paths of searching for proof of a conjecture, proof of its negation, and proof of its independence.
If I write My car can go pretty far and it gets good mileage I have combined two independent clauses to create a compound sentence. I might just as easily write My car can go pretty far. And it gets good mileage. But if I write Surprisingly, my car can go pretty far and it gets good mileage, meaning that it is a surprise my car has both of these attributes, then the independence of each clause seems diminished, because one without the other is not surprising. In other words, I cannot write Surprisingly, my car can go pretty far. And it gets good mileage. Is there a way to describe this difference? Is there an overriding term for two or more independent clauses that actually do not mean the same thing when not joined?
I was watching a BBC sitcom. The scene is set in a wedding ceremony. In the opening of the speech of the father of the bride, he tends to be humorous, and thus he says: "Welcome to the wedding of Laura and Paul, whether you're friends, family or freeloaders, loved ones or loathed ones, people we like or people we had to invite, and whether you're here for a free meal or a free drink, people who wouldn't have missed this special day for the world or people who had nothing better to do. You are all welcome." I am baffled by his using "wouldn't have missed." According to the context, I reckon what he means is more or less "people who wouldn't like to miss this special day for the world." But if he means exactly like that, why does he use the perplexing "wouldn't have missed?" Alright, he might be trying to be witty, implying "people who thought if they came to the wedding, they wouldn't have missed this special day after the wedding." Is my interpretation right? Even the the implication is like that, I was still wondering how it sounds like in a native speaker's ears? For me, a non-native speaker, all I received was only full of confusion. Not humorous at all. If I were in that wedding, I would get lost and stuck in this "wouldn't have missed" and miss out a bunch of the rest of the speech.
If a conformal mapping, e.g., a Schwarz-Christoffel mapping, maps the real line (from left to right) to a polygon, which is traced out from left to right, why is the upper half plane mapped to the interior of the polygon? Is it a preservation of orientation argument? I.e., since the upper half plane was to the "left" of the moving point in the z-plane (moving along the real axis), then the image of the upper half plane will again be to the left of the curve, which is also going from left to right, until it closes at the final vertex to form the polygon. I know that one can use a test point and then make a connectedness argument, but if the integral (or any tricky conformal mapping) were hard to evaluate, such as an elliptic integral, I don't feel that this is a wise move to make, and that there could possibly be a better way to argue why the image of the UHP is inside the polygon. Thanks,
So, I know that this question may be tough to answer, but I am asking this question in all seriousness, and I don't consider myself a newbie... Lately, I am trying to find a way to "generalize" my approach classical mechanics. Specifically, I need a system that can always guarantee a solution to any problem. In a sense, just like mathematicians like to generalize stuff. For example, if you get a system of linear equations, you can always solve it by using the Gauss method of substitution... I often have the problem of seeing why the solution provided by the textbook is right, but I don't see why my solution is wrong... I'm guessing this is the motivation behind my quest. I know a field as big as mechanics is hard to cover with one or two principles, but I need something to always help me set the correct system up, the rest being "just math". So is there a method that yields a uniform system? I have heard that Lagrangian mechanics can provide something like this, but I'm not sure whether to take it up... Can anyone post their experiences with that? Lastly, I would add that I am no stranger to advanced math topics. I like linear algebra and multivariable calculus... from my experience learning new math has always enhanced and deepened my understanding of physics. Don't hesitate if you have to offer something... maybe a book or similar resource...
Today my laptop battery died while I was finishing off a sticky note and so naturally I just went 'Oh, bollocks!'. After all, it didn't warrant one of the "big six"; it was more a "damn" moment. So I opted for "bollocks". I didn't think it was rude; it was just fun to say. I was around a colleague at the time and they gave me a bit of a look (and, frankly, an 'Excuse me?!'). They told me that it's just another way of saying "bullshit", yet based on their response, obviously not a euphemism. (I'm sceptical though. I know the dictionary says "balls", but it didn't say anything about "bullshit".) I've always thought that "bollocks" is just one of those great British words that are relatively harmless but fun to say, like "bugger" or "sugarbush". So, is it actually rude? Or is it just rude to some people? (If it's relevant, I live in Australia.)
I understand the phrase when it is used literally in this example. He's ill and doesn't know what he's saying. I perceive it to mean that a person is struggling with an illness that prevents him from thinking clearly and understanding his words. However, there have been times where the usage seems more vague to me, particularly when it is directed at an able-minded or sound-minded person. He doesn't know what he's saying. I don't think you know what you're saying. Okay. I think there are some possibilities. A person uses the phrase as a personal attack on the other person's intellect. A person uses the phrase to discredit the other person. A person is implying that the target person is oblivious to a connotation or perception of the target person has said. A person uses the phrase to expose evidence that the target person is a dilettante. Perhaps, the phrase means each one in different contexts, or perhaps, it means a combination of them?
Nearly four years ago, upon hearing of the observation of time dilation in two optical atomic clocks at an elevation one metre apart, due to acceleration towards earths centre of gravity by Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. I wondered where the point would be, at which time dilation caused by acceleration toward the earth gives way to time dilation caused by acceleration around the earth, via orbital velocity. (I mean the point at which the two effects are at equilibrium with each other.) I asked a friend who is a physicist, he got back to me almost immediately, fascinated by his back of an envelope calculations that this point is the Clarke orbit, or the geosynchronous orbit. If for instance the rotation velocity of earth were a slightly different speed, the result would be different, and no relationship apparent. I think there may be a very interesting relationship waiting to be revealed. Is there a physical or mathematical relationship? And if so, what is it? Not being a mathematician (I'm an artist) I'm not capable of doing the math, but this has fascinated me ever since.
While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, when studying the generalized transfer operator (or Ruelle operator) for the Gauss Map, Dieter Mayer points out that this operator is in fact nuclear (On the thermodynamic formalism for the Gauss map). While I can understand the definition of a nuclear operator, I still cannot get the real importance of being nuclear of order zero. Usually I'm interested in spectral gap properties for transfer operators, but is there any implication of the nuclear property? Also, any reference for nuclear operators and Fredholm kernels would be appreciated, since trying to learn directly from Grothendieck's works has been really difficult for me. Thanks in advance.
I don't really understand the reason why wave theory of light fails to explain the blackbody radiation. My textbook says the Planck's quantum theory explains blackbody radiation. It says "If we assume, he said, that radiation is emitted in packets of energy instead of continuously as in a wave, then we can explain the black body spectrum." So, the problem in the wave theory is that energy is absorbed or emitted continuously, not in multiple quanta. But why is continuous emission or absorption a problem when we deal with black body radiation? I was taught in class (or maybe I interpreted it this way) that if energy would be continuously radiated, then the intensity of radiation must increase on heating the black body and wavelength of light would stay same. But from experiment, wavelength changes. Hence it fails to explain it. But assuming that what I wrote above is correct, why can't the wavelength change? And even in Planck's theory, why can't simply the number of quanta emitted increase and frequency be constant, so it gives radiation of same wavelength? Please note that I have tried my best to look this up on the Internet, but all I see is explanations for back body radiations in terms of Planck's theory. If you find this question not framed well/not acceptable in its current form, please leave a comment so that I can edit it, before downvoting it to close it.
Not all questions start with Wh- words, so why don't we start a question with a question mark to make it more obvious that it is actually a question? For instance, when I'm reading a book which has a long interrogative sentence at the bottom of the right-hand page, it often isn't until I turn to the next page that I realize it is a question. Not that this represents too much of a problem, but from a beginner's perspective it must make it harder to interpret the words of the sentence in the correct context. Similarly, why don't we start exclamatory sentences with an exclamation mark? I know this is a trivial question, but I'm wondering whether some kind of historical or colonial/imperial decision might explain the absence of these marks in the situations I describe. P.S. I am a student programmer, so pardon my ignorance / funniness
We are regularly taught in high-schools and universities that, according to General Relativity (GR), gravity is nothing but a manifestation of space-time curvature (which, in its turn, is caused by matter and energy). However, GR is still only a model, which hasn't been challenged by experimental evidence/precision thus far. E.g., in wiki one might find a lot of alternatives to GR, some of which agree with observations not worse than GR (e.g., Brans-Dicke theory). There are theories which describe gravity not in terms of curvature, but in terms of torsion - but in reality gravity cannot be both at the same time! Besides, as far as I understand, curved space might be described as a curved surface in non-curved space of a higher dimension. So my question is: do I miss something and there are strong model-independent reasons to believe that gravity is geometry, or is it just that authors in most textbooks and articles imply that this is a model-dependent interpretation, without saying it explicitly?
Suppose the big bang did create equal portions of matter and antimatter and they exist right over top of each other, such that there is a constant inversion going on. Since a matter and antimatter universe would be identical, we wouldn't know whether we're living in one or the other; therefore, isn't it possible that at any given time we could be in one or the other, or oscillating between the two at a very high frequency, perhaps even at the smallest possible unit of time? Could such frequent inversions be responsible for what we call "inertia", since disturbances could cause a slinky-like reaction where the matter and antimatter pair are annihilating and recreating each other or swinging around each other in a tandem dance across space? Could such a phenomenon be responsible for quantum phenomena such as particles having no definite location, since they would be constantly coming into and out of existence in slightly different locations, with fairly regular offsets giving rise to an apparent quantum field? Wouldn't that explain entanglement as well? And if such a thing is occurring throughout the universe everywhere, perhaps it explains inflation as well, since it wouldn't be an expansion of space, so much as a shrinking of the size of every single particle in existence as the constant oscillation between matter and antimatter loses energy somehow...
My daughter is in high school and is working on a science project. She has really enjoyed writing the paper in LaTeX. Now, she has to make a tri-fold presentation of the major points. In the past, that has meant PowerPoint printed onto letter-size paper then pasting those pages (one slide per page) onto the board. How can we resize beamer to have at least the same aspect ratio as letter-size paper? Ideally, we would have the same dimensions, also, but that is not so important as we can scale at print time. I have used beamerposter in the past, but that generates just one page, I think. Any suggestions are appreciated. The default is to simply use beamer as-is and scale to as close to letter size as possible when printing.
There are a few tools that will automatically retrieve BibTeX-formatted references from MathSciNet. BibDesk is offered with TeXShop, but only works on a Mac. Are there similar tools that work for Unix/Linux or for Windows? There are a few applications written as stand-alones that get the job done: mscget (requires python), bibupdate, bibget (a shell script). JabRef can retrieve references from a number of archives, but MathSciNet is not listed among these resources. Some potential tools are mentioned in this answer to What are good sites to find citations in BibTex format?, listed under "Reference managers that allow BibTeX export/import". MathSciNet is not included among the archives covered, although it is mentioned as a resource from which BibTeX-formatted references can be downloaded manually. Note: I work for Mathematical Reviews / MathSciNet.
I have previously done a course on group theory and now I am doing a reading course on category theory. So as an interesting exercise I have been asked to write an exposition of group theory for someone who already knows category theory but doesn't know any group theory. I have been given the liberty to decide how I build the theory. I already have a vague idea of what is to be done. However, I would like to hear ideas about what should be done. So I solicit advice on things I should emphasize, the ways I can exploit the given familiarity with category theory for a more economic presentation and/or a exposition through "the path of least resistance". And please feel free to also mention any tips or precautions. Please refer me to material along this line. Thanks!
I often have problems with article words referencing to its correct referent. In the following statement for example: The touching scenes in the film leaves the audience a heavy feeling, encouraging them to consider charitable acts and the role it plays in our society. My intention is to have the word "them" linked to the audience, and the word "it" linked to the charitable acts. I am quite confident that the word "them" does link to the audience. But I am worry that the word "it" may not link to the charitable acts. I find that there is possibility the reader may confuse "it" referring to the touching scenes in the film instead of the charitable acts. Are the article words in the sentence that I have constructed above correctly linked to its referents? In general, is there a rule to watch out when identifying references in long and complex sentences?