darklight03/StackMIAsub huggingface dataset:
We've learnt that friction is the opposition of motion and that friction appears the instant a force is applied on an object i.e when an object is at rest (with no force acting on it) then there is no frictional force. The moment a small amount of force is applied, friction becomes a factor. Therefore, friction is just the "equal and opposite" force between two bodies. Now, let an object be accelerated to a velocity 'v'. Then, let the acceleration cease. Ideally, the object will come to a stand still. However, if the acceleration is zero, doesn't that mean that there is no force => there will be no "equal and opposite" force i.e frictional force. And, only if there is an opposing force will there be retardation. Obviously, my reasoning is flawed, if not then an object that has been accelerated to a velocity will continue to move at a constant velocity. However, I don't get where my reasoning is flawed. Please do help...
I probably have seen this happen at various times in movies set in eras where people were very obsequious to royalty. The action I am trying to find the word for is a motion of the hand in a kind of circular motion or spiral (generally towards and away from oneself) while bowing towards someone, often while slowly moving backwards. My first thought was "genuflect" although to me that invokes the image of someone making the sign of the cross (although the dictionary doesn't seem to mention this, or in fact any hand movement - it seems to be closer to bowing then the hand movement) and I am looking for a word without any religious connotations. My second thought was "flourish" as that seems to be technically correct to some degree in the sense of "a bold or extravagant gesture or action, made esp. to attract the attention of others", but this word seems to have too many meanings that might confuse, and I am not sure anybody would recognize what I was talking about unless I put a lot of context into the sentence with it, or maybe even if I did!
I'm attempting a novel approach to some tough integration problems. I'm using the idea of series expansions to help integrate. In other words, I will attempt to approximate integration by integrating the series expansion of an integrand, rather than direct integration or standard numerical methods. I believe I can approximate integration of a series very easily, compared to the other methods. However, there's a catch. I will use at least two different series expansions. One for the lower limit of integration, and one for the upper limit. Now, when I attempt to integrate these expansions, the constant of integration comes into play, and it's not obvious what it is. Since I am using at least two different series expansions, the constant of integration may differ for each expansion. So I'm wondering if there is an easy way to get the constants of integration without much more work. Any help, ideas, or suggestions are welcome. EDIT A few additional notes... I know ahead of time that the series will converge. I consider that I could integrate in sections, like quadrature, while still using the series to aid in integration. However, I am considering the idea of only using only the endpoints, with two different series. So the constants of integration would be different for each series. If I could somehow find them or find how they differ relative to one another, that would save me the trouble of breaking the integral into sections and using something akin to conventional numerical methods.
In an earlier post - Phonetic understanding of tongue twisters - a comment was made that "hyphens ...(are) ...not needed in speech, so they must be extraneous". The phrase prompting this assertion was 'state of the art'. What does it mean to say that hyphens are not needed in speech? No one would say state hyphen of hyphen the hyphen art, of course. But when I say "This is a state of the art paper on tongue twisters" I make a point of saying the words 'state of the art' as a group and slightly apart from the run of the words on either side. If I didn't, and spoke the words in the same rhythm as the rest of the phrase, the meaning can easily be lost (and the sentence is certainly harder to read meaningfully at first sight). There is good reason to use hyphens, or some other notational device, in such cases, isn't there ? This is a state-of-the-art paper on tongue twisters This is a 'state of the art' paper on tongue twisters This is a state of the art paper on tongue twisters
In my Computer Science class, we were introduced to the Average Salary problem, where a group of people want to determine their average salary, but they don't want anyone to be able to determine the salary of anyone else. I proposed a solution which I later looked up and found to be a fairly common one, wherein everyone writes down several numbers on separate pieces of paper that add up to their salary. The papers are then collected in a hat, totaled, and divided by the number of people. My professor said that he was looking for a solution that only involved direct communication as to avoid the use of "trusted hardware". However, he also told me that my solution was flawed because some information is unnecessarily revealed, and, I must assume for the sole purpose of tormenting me, he said we would go over it later when he revealed the solution to the rest of the class. He also told me my solution was still inadequate when I said that the numbers could be both positive and negative, and everyone was to submit an arbitrary amount of numbers. My question is not what is the ultimate solution to this problem, but rather, what is wrong with mine? What information could be revealed from arbitrary numbers that when added up equal the total salary?
I am still trying to get a good grasp on the motivations behind various concepts in Differential Geometry. But I am struggling to come to terms with how certain concepts have this added attribute of being coordinate independent? How does one identify such objects, be it a tangent space or a covariant derivative. How does one go about trying to prove that a certain geometric object is coordinate independent? How is coordinate independence a part of the "geometry" of a given surface or is it? P.S.: Actually is the concept of a coordinate system part of the intrinsic or extrinsic geometry? I think its the former, but sometimes embedded spaces tend to make me think twice. Edit: I would appreciate if the covariant derivative could be used as an example.
I am going through the chapter on compactness and completeness from Sternberg's Advanced Calculus and trying to build an intuition for what many of this topological properties mean, and which imply which. The book defines these concepts in the setting of metric spaces, but most of what I found online is in the about topology, and from what I see (correct me if I am wrong) it doesn't change the general picture much. I've made this diagram to see whats the relationship of the different concepts and have examples of each. I don't know if it is correct. For example, Is it true that a bounded complete metric (sub)space is compact (and therefore totally bounded)? Then why bother defining total boundedness? If you don't think anything else I wrote is a valid question stick to answering that, although pointing out any misconceptions I might have is appreciated. For intuition about compactness I've found this posts really helpful. It helps me to think that there are (at least) two different kinds of infinity: one in the sense of largeness (of which boundedness is the opposite), and another in the sense of denseness (of which discreteness is the opposite).
My question concerns the theory proposed in this classic paper by Misner and Wheeler. In the paper, the authors propose the idea of "charge without charge"--namely, that positive and negative particles might really be the ends of a wormhole, with field lines going into a mouth interpreted as a "negative" particle and the outgoing field lines at the other end as the "positive" particle. However, I noticed that the paper didn't mention whether or not the wormholes were traversable. If the wormhole was not traversable by an external material source (i.e., they have unstable inner horizons, have curvature singularities, etc.), then could the field lines technically travel through the wormhole's neck, or would they, too, be blocked? I know that the "charge without charge" idea isn't probable due to the tiny wormhole tunnels collapsing to form black holes, but my question also concerns larger wormholes as well. Thus, could field lines travel through a macroscopic non-traversable wormhole?
I am wondering what some applications of POVMs are in mathematics (or mathematical physics)? I am going through Berberian's 'Notes on Spectral Theory', which shows how we can write a normal operator as an integral over a spectral measure. Because it is not that much extra work, he treats operator valued integrals in generality, allowing for integration over a POVM. As it is however, I can't find any examples or motivations for integrating over POVMs. In quantum mechanics I have come across using POVMs to represent the most general form of measurement, but in that case a POVM is defined as a series of positive operators which sum to the identity. I suppose if you allow for a continuous range of results then this sum would become an integral, but is there anything else to it?
I'm having a bit of trouble explaining to a friend whether or not there's a big difference between the three modifiers in the title. Same and very on their own are different enough, but when combined, I find it difficult to draw a proper line on their meanings. Consider the following: I lived in the same house you're talking about. I lived in the very house you're talking about. I lived in the very same house you're talking about. Here, I understand there is a nuance in sentences one and two, though I have trouble explaining just what it is. "The very same" sounds like "the exact one", but wouldn't that be what "same" means anyway? Plus, that last sentence truly boggles the mind. How do you explain the grade of intensity expressed in sentence three? How do you explain each modifier?
I'm trying to translate a video on TED into my native language (Latvian). At the very start there is an expression I'm unfamiliar with - "animal warmth". I think I kind of understand the idea intuitively, but I can't think of any similar expressions in my native tongue (short of direct translation). It would be nice if someone could explain the concept to me, or give some more examples of usage. Here's the context: ... Because in my family, reading was the primary group activity. And this might sound antisocial to you, but for us it was really just a different way of being social. You have the animal warmth of your family sitting right next to you, but you are also free to go roaming around the adventureland inside your own mind.
Is there a single word, or commonly-used term, to describe the act of baiting another person into calling bullshit, when in fact you're not bullshitting? Conceptually, this either a sub-type, or the direct opposite, of bluffing, which is the act of pretending you have better cards than you actually hold, in order to scare your opponent into folding. The word I'm looking for will describe (metaphorically) the act of pretending you have worse cards than you actually hold, in order to bait your opponent into calling you. I'm looking for a word which implies "actively misleading" (as in lying); examples might be a basketball player faking left (in order to make his guard shift left, so he can go around to the right) or generally faking out. This is not restricted to gambling (just as "bluffing" is not restricted to poker), I'm only using gambling terminology to make my meaning clear.
About two years ago, I absolutely fell in love with mathematics. Since then, I have studied math almost religiously, absorbing everything I can about every subject I can. I have now established what I would call an understanding of most undergraduate topics, up to intermediate complex analysis, some abstract algebra, multivariate calculus, etc. I really want to get into a good college for mathematics - specifically MIT. As a middle schooler, I have plenty of time before I have to submit an application, and I really want to make the most of that time. You can't exactly put down "was pretty good at math in middle school" on a college application, so I was wondering what things I can do now to get a leg up in the future, that I could put down in a college essay or in my application. So far I've started a mathematical blog, and I am working really hard to get published by the time I finish high school. My question is - what else can I learn/do/create to give myself a head start in college and in employment as a mathematician?
I recently came across the following problem from Paul Zeitz's book The Art and Craft of Problem Solving. Given the image below, can you find a way to connect corresponding blocks (i.e. A to A, B to B, C to C), without having any of the connecting lines intersect one another? The question was an interesting one for me, because for the longest time I was convinced that it was impossible, and when I finally became acquainted with the solution, it took me quite a while to "accept" it. Granted, I am not the sharpest tool in the shed, but upon introspection I also wonder if I am being hindered by the "intuition" I have come to develop, and implicitly "accept". I wonder if it would be a helpful exercise to perhaps go through experiences that help me dismantle this intuition. The most accessible way I can think of of undergoing such a process would be by reading helpful books, given my limited resources. While I think problem solving books such as the one I am reading right now is good for this purpose as a side-effect of its initial intention ("teaching how to problem solve"), I wonder if there are books that are geared specifically towards deconstructing and examining "intution"? Prospective answerers, please attempt to answer this refinement of the question instead.
I am writing an essay on a book that I read where many of the characters are not human and have artificial intelligence instead. When I try to describe these characters, though, I find myself using "android", "machine", and "robot," none of which seems correct. The characters are not humanoid, so I am not sure if android is the correct word for them (although I think it was used in the book). I also tried using "inhuman" and "nonhuman", but I feel like those may be too vague. I feel as if calling the characters "machines" sounds too insensitive since they express many traits that humans do and my essay is about how they are very similar to humans despite not being human. Does anyone have good synonyms for "android" or "robot" for me to describe these characters?
I'm trying to do Young's double slit experiment at home. Note that I don't have a laser, only a torch. I could get a bulb or use a candle though, if it helps I built the slits by cutting into a black chart paper with a knife. I tried to build a setup by placing a single slit, double slits and a screen one after the other, and shine a torch through the single slit. I tried varying distance between single-slit/double-slit and double-slit/screen which did not help. I did not observe any interference pattern. All I got was two parallel bright fringes instead, like the ones you would get by shining a torch through two very thick slits. I think that is precisely the problem, that the slits are pretty thick. What is the optimal slit width I should have to observe an interference pattern? How do I build that with a chart paper? I have seen videos of this experiment online, where people use pencil leads and hair(!!), but they both use a laser. I don't want to use a laser(just because I don't have one, and I'm probably too lazy to go and get one!). Note that you may suggest using any other materials that might be easily available at home.
I'm going to be a teaching assistant and I'm currently looking for books/reviewed articles/journals written by mathematicians or people who taught mathematics (at a university level) about pedagogy and/or their experience of teaching mathematics. I know that these readings can't replace the experience of teaching but I think sharing experiences can't be bad for my future students. To be more precise about what I'm looking for, I consider the following questions very interesting : What mathematical concepts are difficult for most students and why ? What are the basic errors of a new teacher ? What is a good course in mathematics ? What is a good exercise session ? I know that there are plenty of different answers of this question and I'm looking for different opinions to build mine. Thank you PS : References in French or English only.
As written in the title, there's a specific word that is not too common in English that's used to describe the feeling you can get when you finally resolve your long-term disdain for someone, or reaching some sort of civility between you and them. I can't remember it but I would know it if I saw it. Example: He attained [a feeling of] ____ after talking to his estranged father after many years. I'm not looking for synonyms of peace or closure, and nothing on Thesaurus.com has helped me. It is a very specific, unique word. It's like closure, peace of mind, or inner peace, but I've only seen it used in the context of sunsetting a toxic, detached relationship. Edit: The actual meaning of the word I was looking for was peace of mind for any period of negativity. It's not necessarily unique to interpersonal relationships.
I'm a physics undergraduate student who always enjoyed math, and briefly studied it at a university but for various reasons (laziness, youth) gave up and changed 'majors'. But I always wanted to go through an undergraduate math course in my own time, unconstrained by class, etc. Now that I've passed all my exams I was thinking of doing something over the summer. I had a look at Terry Tao's free lecture notes from an analysis course he taught and I was absolutely shocked at how good they are. I love the verbosity and how he motivates every bit of information. From what I read, he wrote an Analysis textbook which I intend to get. My question is, are there any other similar (in the sense of their exposition) textbooks for subjects such as Topology, Algebra (Linear and Abstract - from my brief studies I've come to believe that I'm an absolute algebra antitalent, but I'm hoping it's because I didn't have anything else than fairly dry lecture notes to study from, and let's be honest, I didn't study very much) and of course more advanced Analysis, Probability and Statistics?
Some actions (such as generating the Table of Contents) require two passes of the TeX compiler: during the first pass, some data get written to an auxiliary file, only to be retrieved during the second pass. Here are a few TeX.SE questions that require two-pass solutions: Highlight referenced equation number Backreferences for equations (To be completed... feel free to edit if you come across such a question) Two-pass stuff has piqued my interest; I have a few questions: Can I write (append) custom data to an existing auxiliary file (e.g. .aux)? Is that even a good idea? If not, can I generate my own auxiliary file (with a custom extension) to store/retrieve some data? What are good sources for learning the basics of writing to & reading from auxiliary files?
In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the following fact to commutative rings: A square matrix over a field has trivial kernel if and only if its determinant is non-zero. As Bill demonstrated in the above question, a related fact about fields generalizes directly to commutative rings: A square matrix over a commutative ring is invertible if and only if its determinant is invertible. However, the kernel being trivial and the matrix being invertible are not equivalent for general rings, so the question arises what the proper generalization of the first fact is. Since it took me quite a lot of searching to find the answer to this rather basic question, and it's excplicitly encouraged to write a question and answer it to document something that might be useful to others, I thought I'd write this up here in an accessible form. So my questions are: What is the relationship between the determinant of a square matrix over a commutative ring and the triviality of its kernel? Can the simple relationship that holds for fields be generalized? And (generalizing with a view to the answer) what is a necessary and sufficient condition for a (not necessarily square) matrix over a commutative ring to have trivial kernel?
People use the phrase "x strikes a chord with me" to address enthusiasm or personal movement. I know there is another question that addresses what this idiomatic phrase means, but I'm very curious as to where this came from and when? I've searched a number of English dictionaries in hopes that a definition of the idiom or simply the word chord would be affixed with the origin; I started with the Cambridge English dictionary and proceeded from there. I also tried many fruitless Google queries. If someone could point me towards a reliable resource, I'd have no problem doing further research. I know music is a very emotional endeavor, so I could see the connection there--considering chords are a significant component of music--but this is purely a personal inference. Does anyone know this idiom's origin?
They had cooks and drivers, and guards who occupied a gatehouse, armed with machetes. Seeing as I had regularly petitioned my parents for an electric fence, the business with the guards strikes me as the last word in quiet sophistication. - David Sedaris, Me Talk Pretty One Day I have three questions about this sentence. First, I think "Seeing as" in the sentence is working as a conjunction, but I only get the meaning vaguely. Are there other alternative conjunctions for it? Second, the context is that the author is envying his boy friend's family ["They"] for having had servants such as cooks, drivers, and guards. My question is, what does "the business" mean here? I think it is more like "the story about the guards". Is that correct? Third, I do not understand what "in quiet sophistication" means. I looked up the dictionary for the word "sophistication", but I think the words put together make a new meaning or something.
I'm writing up my Teaching Statement for an Assistant Professor position in the sciences. Because all I do is read and write science, I have no elegance in my writing at all! I'm trying to make the last sentence sound better: I have been fortunate enough to work with some great mentors in my life journey thus far. Integrity, compassion, and selfless care for students were role-modeled for me on a day-to-day basis, teaching me things far beyond academics. It has been a life-long goal to attempt to replicate them. I'm trying to express that it has been a life-long goal of mine to pass-on what great mentors in my life have done for me to future students. They took me under their wings and provided important role-modeling, perspective, and guidance in my life when I was a drifting early-twenty-something male. Is there a nice and succinct way of writing this? Is there a phrase or an expression that has the same meaning as the highlighted text in the provided context? Is there a word that could replace "attempt to replicate them" to make it more concise?