darklight03/StackMIAsub huggingface dataset:
Google defines paradox in three ways, the first two being: a statement or proposition that, despite sound (or apparently sound) reasoning from acceptable premises, leads to a conclusion that seems senseless, logically unacceptable, or self-contradictory. a seemingly absurd or self-contradictory statement or proposition that when investigated or explained may prove to be well founded or true. These are almost opposite definitions. My English book defined it as the second, but I've always understood it to mean the first. Which is the most common usage?
What happens to the energy when waves completely cancel each other out via destructive interference? It seems like the energy just disappears, but that would violate the law of energy conservation. My guess is that the kinetic energy is transformed into potential energy. Or maybe what happens to the energy depends on the specific interference scenario? Can someone elaborate on that or correct me if I'm wrong?
I'm studying Complex Analysis and everything up to this point has been pretty straightforward to visualise, but I can't get my head around residues, especially as they seem to have two very different definitions (as a Laurent series coefficient and as an expression involving an integral on a closed path) that I can't understand why they equate. Can anyone give a fairly simple explanation of residues? Thanks!
A paracompact space is a space in which every open cover has a locally finite refinement. A compact space is a space in which every open cover has a finite subcover. Why must the product of a compact and a paracompact space be paracompact? I really have very little intuition about how to go about this question, so any hints or a proof would be greatly appreciated.
Is there a word that is similar to "marketing" that doesn't imply the idea of commercialization or profit? For example, a non-profit that assists domestic abuse victims may run a campaign where they advertise their services. They may put up posters, buy billboard space, and so forth. While this is marketing, people in that industry may be very reluctant to call it a "marketing campaign". This is because marketing tends to carry the connotation of trying to sell something. Is there a more neutral word?
I'm currently working on an English project and I have chosen to rewrite a story (Heart of Darkness by Joseph Conrad) in a biblical, King James style. I have done a large amount of reading but I do sometimes get confused, especially on certain sentence structures. Do you have any tips on grammar, vocabulary, or sentence structure that will help me make my writing seem more authentic?
Can someone please explain why the van der Waals force is attractive instead of repulsive? My understanding is that is has to do with the electron cloud creating a dipole, but I can't seem to understand why the force isn't repulsive (since like charges repel.) Since the electron cloud is moving, I would think it would cause a force that is attractive sometimes and repulsive sometimes, with the average effect being more repulsive. Obviously I am missing something!
I'm really sorry if this question has been asked before, I looked but couldn't find anything. I'm going through an elementary number theory book and in the first chapter it introduces the least integer principle. There is no proof accompanied with it, it just states: "a nonempty set of integers that is bounded below contains a smallest element." Is the reason the book doesn't give a proof because we take this as an axiom? Or is the proof so trivial that it need not be written? Thanks!
I've been reading a Real Analysis textbook that my friend loaned to me. I have come across a proposition that says that a totally bounded set is bounded, but a bounded set is not always totally bounded. This makes sense, however I am having trouble thinking of an example of a set that is bounded but not totally bounded. Could anyone shed some light on this? Thanks!
I am working on some optimization problems, and I am aware of the method of proving that the "angle of incidence equals the angle of reflection" using Fermat's principle and calculus. However, my textbook suggests that there is a simple way to prove this without calculus, but I'm a little unsure about how to do this. I would appreciate any advice on how to proceed.
Here are the definitions of the two words: Hypocrisy: the practice of claiming to have moral standards or beliefs to which one's own behavior does not conform; pretense. Affectation: behavior, speech, or writing that is artificial and designed to impress. However, I'm not sure about the difference between the usage/connotation of these two words? Is there any example where you would use hypocrisy instead of affectation or the other way round?
Wikipedia says: In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners. In a theory with unbroken supersymmetry, for every type of boson there exists a corresponding type of fermion with the same mass and internal quantum numbers (other than spin), and vice-versa. There is only indirect evidence for the existence of supersymmetry [...] I want a mathematical explanation of SUSY.
I know that when I'm solving a linear equation of the form ax = b (mod n), the gcd(a,n) tells me how many solutions there are. I don't really understand why this is. More than a proof, I'm interested in an explanation that will help me understand the relation between the gcd and number of solutions. Also, why does the gcd need to divide b for there to exist solutions? Thanks
I'm trying to wrap my head around function spaces. I get that you can define the inner product as the integral the multiplication of two functions over the entire domain because it satisfies the properties of the inner product. What I don't understand is what makes this a complete space? I don't understand the concept how a complete space works with functions. Any insight would be appreciated.
When is it appropriate to use the terms Dear Sir or Madam and To whom it may concern? The rules I was taught state that Dear Sir or Madam should be used when you're writing a letter to a person about something that person has direct involvement in (e.g. returning a defective product to a customer service department). By the same rules, To whom it may concern would be used for situations in which the recipient is a third party to the topic of the letter (e.g. regarding a letter of reference or recommendation). Was I taught correctly? Is the rule actually something else? Or is this one of those situations where many people have an opinion but there is no real rule?
I've been trying to determine an explanation for the falsity of a logical statement for some time now and I've had no luck in figuring out exactly how to go about it. The statement in question goes as follows: Explain why the statement for any integers a, b, c, if ab = ac, then b = c is false. Perhaps I'm misunderstanding the way to determine the falsity of a logical statement? Any help is appreciated.
What is the difference between positive definite kernels and positive definite functions? As I understand it, a positive definite kernel is a positive definite function if it is translation invariant. Is this correct? If not, what is the actual relationship between these concepts? I'm reading Learning with Kernels and using this text, Wikipedia, and Google, haven't been able to find a good answer to this question. References are appreciated!
Often, I'll be driving down the road on a summer day, and as I look ahead toward the horizon, I notice that the road looks like there's a puddle of water on it, or that it was somehow wet. Of course, as I get closer, the effect disappears. I know that it is some kind of atmospheric effect. What is it called, and how does it work?
Just as there is a difference in meaning between "likely" and "liable" in terms of a desirable or undesirable outcome, is there any subtle diference between "apt" and "likely" ? Does the use of "apt to" and "likely to" in the following sentences convey the same degree of expectation ? "John is apt to call on us on Wednesday afternoon." "John is likely to call on us on Wednesday afternoon." My dog is apt to run away if I don't close the gate. My dog is likely to run away if I don't close the gate.
I recently went through Griffiths' Quantum Mechanics text and there is a chapter called the Adiabatic Theorem that includes Berry phase and the Aharonov-Bohm effect. As I found them very interesting, I would appreciate if anyone could provide me with some good sources(books, internet links etc) to learn more about them. Please keep in mind that I am only an undergraduate student, so difficult graduate stuff might not be of very use to me.
I'm currently learning about local extrema in serveral variables and have come across the second derivative test for classifying critical points of multivariable functions. I have read and understood the test (see link below), however I don't understand the idea behind it. Why is the critical point of a function a minimum if the eigenvalues of the Hessian matrix are all positive? I understand the idea behind the single variable case, however I am confused about the role of eigenvalues in the case of several variables. http://en.wikipedia.org/wiki/Second_partial_derivative_test Any insight into this would be much appreciated. Thanks.
Wikipedia's page for Imperial Units does not list Fahrenheit. The corresponding page for SI Units lists Kelvin as an SI unit, and Celsius as a derived SI unit. This leads me to believe that Fahrenheit does not belong to the Imperial System. Fahrenheit is listed as a page belonging to the Imperial Units category, also on Wikipedia. I know that the US mostly uses the Imperial system and also uses Fahrenheit for temperature. Meanwhile, the UK mostly uses the metric system and also uses Celsius for temperature. Is this just a coincidence, or is Fahrenheit actually a part of the Imperial system?
I'd like to use "digitally challenged" to refer to someone who is missing part or all of a finger, but it sounds like I'm talking about someone who doesn't know how to use computers. Are there any phrases, ideally somewhat humorous in nature, that I can use to refer to people who are missing part or all of a finger? Example usage: Steer clear of that izakaya. I've heard some of the patrons are [alternative to "digitally challenged"].
I'm preparing some marketing materials for my boss, and one section contains the phrase "Here are a few examples:". The list that follows contains two items, which strikes me as being incorrect, though I can't find much evidence of this. Is it semantically correct to refer to two items as "a few"? Is there a solid rule for this, or is it just personal preference? Edit - As Armen correctly pointed out, this question is about semantics rather than grammar. I changed my phrasing to reflect this.
At the moment I am following a course on variational methods for mathematicians. Last week we derived the Euler-Lagrange equations for a functional under a constraint. In this derivation we found that the Lagrange multiplier may depend on time. However, I could not remember having ever seen an example of a Lagrange multiplier dependent on time during my physics courses. Does anyone have a nice (mechanical) example?
I know that it is technically impossible to reach the speed of light in vacuum since the mass of the object travelling would reach infinity. However in a medium, would there be some sort of theoretical equivalent to a sonic boom that would occur? Since sonic booms are formed when sound waves can't catch up with an object, could the same happen with light waves? Would it look like an immense burst of light in the sky?
I'm writing an essay for medical school applications. Both my parents were physicians, so it is what I was familiar with growing up. Instead of pursuing medicine in college, I decided to pursue something else and try a few different careers after college. Now I'm back on the medicine path and I'm trying to explain why I needed to try other stuff first. I'm hesitant to use the word "comfort zone" because I think it implies that medicine would be the comfortable (i.e., easy) path, when that is not the case. Any other phrases come to mind that explain this sentiment?
I have been learning more about Hilbert spaces in an effort to better understand quantum mechanics. Most of the properties of Hilbert spaces seem useful (e.g. vector space, inner product, complex numbers, etc), but I don't understand why we need the inner product space to form a complete metric space. My Questions: Why do we need the inner product space to form a metric space? Why does it have to be a complete metric space?
I've been studying random walks as part of a finance course and I'm starting to become really interested, and I'd like to start a study of them from a mathematics (i.e. more formal/rigorous) point of view. However, my knowledge of mathematics is quite limited (calculus, basic linear algebra and probability, a bit of (stochastic) differential equations). So I was wondering what fields I need to familiarise myself with before I can effectively study random walks. Maybe there's even a good self-contained book on random walks for beginners?
In Mendelson's book ("Introduction to mathematical logic") he defines truth values for sentences in the propositional calculus using truth tables. However, it seems to me he assumes implicitly that every well-formed sentence (what he calls "statement form") has a unique parsing; i.e. it is impossible for the same statement form to arise in two different ways. This is of course correct, but it requires a proof. The omission of such proof (or even mentioning it is needed) is somewhat surprising for me as Mendelson's book is otherwise very explicit about everything. Am I missing something?
I have been trying to work this problem out for a bit and am stuck. Does anybody have any ideas how to proceed or solve this? I think this has something to do with rings and fields but I can't seem to find the connection. EDIT: I know the other proof listed as a duplicate here, but it uses Cauchy's Theorem and we are not allowed to use that because we skipped that chapter.
Does it sound right to say "be successful" instead of "good luck" as a quick reply? For instance: A : I have a tough exam tomorrow. B : I hope you'll be successful in your exam. I suppose using "be successful" as a part of a sentence sounds right, but isn't it kind of awkward or unnatural to say "be successful" as a short reply instead of "good luck"?
Apologies if this question has been posted somewhere else, but I have not managed to find an answer that works for me. I am trying to create a bibliography using Harvard style, using BibDesk to create the .bib file, and Biber and BibLatex. However, the bibliography contains the DOI, the URL, the ISSN, a further e-print URL - all things I do not want. My question is: is there a way to delete these fields in BibDesk, so that they are absent in the .bib file?
I came across this problem in my textbook, and I don't understand why the answer is 'as' and not 'that'. "We are attracted to people who have a sense of humor, who can laugh at the same things ____ we laugh at." In the sentence above, the word that goes in the blank is 'as' according to the textbook. Could someone explain why? Thank you in advance.
I want to apply the Biot-Savart law to calculate the magnetic field at a point created by current flowing through a square/rectangular conductor. More specifically, a trace on a printed circuit board. To me, a trace on a circuit board is the summation of many infinitesimal rectangular conductors. I have seen lots of examples for a wire (circular conductor), but not rectangular. How should I go about getting started with this problem?
I just came across the following sentence in the book "The theory of everything": If one keeps traveling in a certain direction on the surface of the Earth, one never comes up against an impassable barrier or falls over the edge, but eventually comes back to where one started. Using multiple "one"s repeatedly in a sentence sounds odd to me. Since I found it in a reputable book, I am wondering whether it sounds good to the ears of a native English speaker. Wouldn't it be better to replace the second and third "one" with "he", "she" or "they"?
Definition: dilemma a situation in which a difficult choice has to be made between two or more alternatives, especially ones that are equally undesirable. a difficult situation or problem. Source: Google. Example sentence: The dilemma of modern life: trying to find an outlet outside of your house. I guess this sentence fits the second definition. So is it correct? (Or should I use another word instead?)
When someone has unreasonable expectations and feels that he is owed by others, this trait is described as a "sense of entitlement". But how do you refer to the person who has this trait other than calling him "a person with a sense of entitlement"? If there is no single word to describe this kind of character, can you suggest a few equivalent phrases for "a person with a sense of entitlement"?
Let's say I want to describe a habit in the past (something that happened many times over and over again) in the form of a conditional statement. Is the following sentence grammatically correct? If I forgot to do my homework, I would pretend the dog ate it. Or is it If I had forgotten to do my homework, I would pretend the dog ate it. Or is there another, more correct form?
I have been trying to understand the notion of parallel transport and covariant derivative. I am unable to see why the change in a vector when it is parallel transported from one point to another shouldn't be a vector. If it is, why isn't the Levi-Cevita connection not a tensor ? Hence my questions are : What is a connection geometrically ? What is parallel transport in a particular coordinate system geometrically ?
I need some help with the following question: In the game of "odd man out" each player tosses a fair coin. If all the coins turn up the same except for one, the player tossing the different coin is declared the odd man out and is eliminated from the contest. Suppose that three people are playing. What is the probability that someone will be eliminated on the first round? Hint: use the complement.
I've just switched from mac to windows and have just started using TeXworks. I like it, but I find that the default pdf viewer is difficult to use. For example, there doesn't seem to be a key command for going to the bottom or top of the current page; I can't view pages continuously; spacebar does nothing for navigation; and trackpad functionality appears to be broken (scrolling with the trackpad causes the pages to jump around in a seemingly random fashion). Is there a list of key commands somewhere? Or is there a way of using a different default pdf viewer?
What are the advantages and disadvantages of piecewise linear interpolation, cubic interpolation, and cubic spline interpolation? I know that piecewise linear interpolation is not smooth and may not be accurate for nonlinear functions. The other two are smooth. But I think my understanding is a little superficial. Especially the difference between the cubic interpolation and cubic spline though I know how to compute them. Could someone help me understanding the advantages and disadvantages of these three? Thanks so much!
While reading my copy of "Resnick and Halliday", I noticed that the book has made a distinction b/w the terms " degree celsius" and "celsius degree". When I googled this, I was able to find only one matching hit(on the first page), which stated that while degree celsius is the unit for the measured temperature, celsius degree is the unit of the difference in temperature. Is this correct? Please share your knowledge and help me.
A single toss of a fair coin cannot be predicted. But if we observe a large number of tosses, we can prove mathematically the law that roughly half of them will show up heads. The movements of individual molecules in a gas cannot be predicted and can be assumed to be random. But if we observe some macroscopic phenomena such as temperature or pressure, we can prove mathematically that some laws are satisfied. Individual quantum events are random. But if we observe a large number of such events, we discover experimentally that they satisfy the laws of quantum mechanics. Could the laws of quantum mechanics be proved mathematically as in the examples above?
Here's what I know about dark matter: Astrophysicists realized the movements of galaxies and other astronomical bodies cannot be explained by the gravitational effects of matter known to be there, so either there is a lot of matter missing that doesn't react with light or general relativity needs to be fixed. For the purpose of this question, assume the former. I've read that this "dark matter" can't be composed of baryons (quarks and leptons). My question is how do we know that it can't be baryonic? After all, air is invisible. Maybe that's a bad analogy, but you get my point: Why can't there be matter that is invisible, yet composed of baryons?
So, I have been solving a number of exercises involving contraction mapping. However, I am struggling to understand what exactly contraction mapping is at its most basic. All I really understand from the concept is that if we have two sets, then one specific point on the set is being mapped closer to another - I'm not sure if I have the right definition. Can somebody please explain this to me in the most layman way possible?
I was trying to understand the undecidable nature of the continuum hypothesis and came up with the following question: The set of circles with a rational diameter is countably infinite (with cardinality equal to the cardinality of integers). The set of circles with a rational circumference is countably infinite (with cardinality equal to the cardinality of integers). The cardinality of the union of these sets is clearly smaller than the uncountable set of irrational numbers, but why isn't it larger than the set of countable integers?
I often hear about string theory and its complicated mathematical structure as a physical theory, but I can't say that I've ever actually seen any of the related math. In general, I'm curious as to what the mathematics of string theory look like, can anyone point me to some references? In specific, I want to know if there is a fundamental equation in string theory that is assumed as a starting point for most problems, something comparable to Newton's second law in mechanics or the Schrodinger equation in QM?
Recently, I am studying the online course Effective Field Theory provided by MIT OCW. Prof. Stewart gives a nice picture to summarize the effective theories: As a newbie in this field (I only have a knowledge of QFT on the level of Peskin & Schroeder's book), I am quite confused with so many terminologies. What do they stand for and Why do we need these different theories? Could someone explain this picture in depth (especially on their advantages and limitations)? Any help will be greatly appreciated.
When a solid object moves in the atmosphere (e.g. a plane), in which direction does air fill up the "empty space" left by the translation of the object? Does this depends on the direction of movement of the object? (e.g. if the plane is moving horizontally, or is landing, or taking off) PS: I am aware of this question and I think this is not a duplicate of that one.
Here in Pittsburgh, we have lots of "Let's go Steelers!" (and some diehards who also say "Let's go Bucs!", but they're dying out). What does that phrase even imply? I assume it's similar to "Go Steelers", which I'm also not sure of the implications. "Go Steelers... to victory!" is a very strange way to phrase the sentence. Does anyone know where this phrase comes from?