docid stringlengths 8 16 | text stringlengths 3 29.7k | randval float64 0 1 | before int64 0 99 |
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science336 | I think that ignoring limits is problematic. If there was a limit of the function $f(x,y)=x/y$ for $x,y \to 0$ regardless of how the limit is performed, then one would define that value to be $f(0,0)$, even if everything else is strange. Since the limiting value depends on the way the limit is done, choosing a value fo... | 0.389874 | 99 |
science338 | I'll try it for 2D and then we can get 1D as a corollary [excercise!]... This is the only proof I know of, there may be a more intuitive (and less messy without tex!) proof out there, but I like this one- it uses generating functions in a really nifty way. Consider the probability of being at the origin after 2n steps ... | 0.369291 | 0 |
science339 | For an old example, Mersenne made the following conjecture in 1644: The Mersenne numbers, $M_n=2^n − 1$, are prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and no others. Pervushin observed that the Mersenne number at $M_{61}$ is prime, so refuting the conjecture. $M_{61}$ is quite large by the standards of... | 0.24222 | 0 |
science342 | If we use the idea of set exponentiation to define exponentiation of cardinals, we have the following natural idea: $$A^B:=\{f:B\to A\}$$ We define the exponential of cardinals as follows: $|A|^{|B|}:=|A^B|$. It's easy to check that this agrees with our intuition for exponentiation of natural numbers when $B$ is nonemp... | 0.348797 | 99 |
science344 | If I'm not mistaken, the integral domain of holomorphic functions on a connected open set $U \subset \mathbb{C}$ works. It is a theorem (in Chapter 15 of Rudin's Real and Complex Analysis, and essentially a corollary of the Weierstrass factorization theorem), that every finitely generated ideal in this domain is princi... | 0.273842 | 99 |
science345 | See Durrett, Probability: Theory and Examples (link goes to online copy of the fourth edition; original defunct link). On p. 164 Durrett gives a proof that simple random walk is recurrent in two dimensions. First find the probability that simple random walk in one dimension is at $0$ after $2n$ steps; this is clearly $... | 0.206115 | 0 |
science346 | Markov chains, especially hidden Markov models are hugely important in computation linguistics. A hidden Markov model is one where we can't directly view the state, but we do have some information about what the state might be. For example, consider breaking down a sentence into what is called "parts of speech" such as... | 0.33634 | 99 |
science347 | This is close to being the flipside of the geometric mean, which is the nth root of the product of the numbers, and can be expressed as the exponential of the sum of the logarithms. Another pair of dual mean measures is the regular mean and the harmonic mean (n divided by the sum of the reciprocals). I say the soft max... | 0.3271 | 99 |
science352 | Edited to clear some things up: Simplicial and singular (co)homology were invented to detect holes in spaces. To get an intuitive idea of how this works, consider subspaces of the plane. Here the 2-chains are formal sums of things homeomorphic to the closed disk, and 1-chains are formal sums of things homeomorphic to a... | 0.422543 | 99 |
science353 | Most of us know that, being deterministic, computers cannot generate true random numbers. However, let's say you have a box which generates truly random binary numbers, but is biased: it's more likely to generate either a 1 or a 0, but you don't know the exact probabilities, or even which is more likely (both probabili... | 0.245033 | 0 |
science354 | Assuming you have unlimited time and cash, is there a strategy that's guaranteed to win at roulette? | 0.117398 | 0 |
science355 | Fork in the road 2 You're once again at a fork in the road, and again, one path leads to safety, the other to doom. There are three natives at the fork. One is from a village of truth-tellers, one from a village of liars, one from a village of random answerers. Of course you don't know which is which. Moreover, the nat... | 0.301053 | 0 |
science356 | I've always thought this sort of puzzle is a wonderful example of the difference between behaviour as you approach a limit, versus behaviour AT the limit. You can make the puzzle more revealing like this: Say I have an infinite number of banknotes, each with a unique serial number. I give you bill number 1. Now, you ha... | 0.145264 | 0 |
science357 | Adding additional axioms would make more truths provable. But it wouldn't make all truths provable (unless the axiom was inconsistent with the already given ones, in which case all falsehoods would also be provable too). So adding additional axioms isn't going to help make all the truths provable. I guess you could jus... | 0.092186 | 99 |
science359 | A function can be represented as a power series if and only if it is complex differentiable in an open set. This follows from the general form of Taylor's theorem for complex functions. Being real differentiable--even infinitely many times--is not enough, as the function $e^{-1/x^2}$ on the real line (equal to 0 at 0) ... | 0.364187 | 99 |
science361 | Gauss-Jordan elimination can be used to determine when a matrix is invertible and can be done in polynomial (in fact, cubic) time. The same method (when you apply the opposite row operation to identity matrix) works to calculate the inverse in polynomial time as wel. | 0.191336 | 99 |
science363 | Let $p$ be a prime congruent to 1 mod 4. Then to write $p = x^2 + y^2$ for $x,y$ integers is the same as writing $p = (x+iy)(x-iy) = N(x+iy)$ for $N$ the norm. It is well-known that the ring of Gaussian integers $\mathbb{Z}[i]$ is a principal ideal domain, even a euclidean domain. Now I claim that $p$ is not prime in $... | 0.215505 | 99 |
science364 | A matrix is invertible iff its determinant is non-zero. There are algorithms which find the determinant in slightly worse than O(n2) | 0.278024 | 99 |
science367 | Grigory has already answered your particular question. However, I wanted to point out that your question "How do you prove that a group specified by a presentation is infinite?" has no good answer in general. Indeed, in general the question of whether a group presentation defines the trivial group is undecidable. | 0.334836 | 99 |
science368 | 100%, for the same reason as the 1-D walk In fact (again for the same reason), your chances are 100% of eventually reaching X-greater heads than tails (or tails than heads), where X is any non-negative integer. | 0.542989 | 99 |
science372 | The following slideshow gives an explanation of how algebraic geometry can be used in phylogenetics. See also this post of Charles Siegel on Rigorous Trivialties. This is not an area I've looked at in much detail at all, but it appears that the idea is to use a graph to model evolutionary processes, and such that the "... | 0.232686 | 99 |
science375 | The trains take half an hour to collide, which, at a rate of 75kph, leads to the fly travelling 37.5km. | 0.200401 | 99 |
science377 | The representation theory of finite groups can be used to prove results about finite groups themselves that are otherwise much harder to prove by "elementary" means. For instance, the proof of Burnside's theorem (that a group of order $p^a q^b$ is solvable). A lot of the classification proof of finite simple groups rel... | 0.464935 | 1 |
science379 | Yes; it's possible to write a system of equations that can be solved to find the correct coefficients. Here's an example for the given formula. We're trying to find coefficients A, B, and C such that $A (\mathrm{Al}) + B (\mathrm{O_2}) \rightarrow C (\mathrm{Al_2 O_3})$ In order to do this, we can write an equation for... | 0.237478 | 99 |
science380 | You can use Pappus's centroid theorem as in my answer here, but it does not provide much insight. If instead of a cylinder and a cone, you consider a cube and a square-based pyramid where the "top" vertex of the pyramid (the one opposite the square base) is shifted to be directly above one vertex of the base, you can f... | 0.33258 | 1 |
science385 | A visual demonstration for the case of a pyramid with a square base. As Grigory states, Cavalieri's principle can be used to get the formula for the volume of a cone. We just need the base of the square pyramid to have side length $ r\sqrt\pi$. Such a pyramid has volume $\frac13 \cdot h \cdot \pi \cdot r^2. $ Then the ... | 0.204304 | 1 |
science386 | The games aren't just about winning or losing, but also about utility. Here is a more accurate table for chicken: B swerves B straight A swerves No gain for either A loses, B wins A straight B loses, A wins Both have large loss Here is one for prisoners dilemma: B silent B testify A silent Both have small loss A large ... | 0.470689 | 99 |
science389 | A probability problem I love. Take a shuffled deck of cards. Deal off the cards one by one until you reach any Ace. Turn over the next card, and note what it is. The question: which card has a higher probability of being turned over, the Ace of Spades or the Two of Hearts? | 0.006028 | 0 |
science390 | I don't know if you consider General Relativity "outside acadamia"(and I don't care to argue the point!) but if you do, the group of symmetries with respect to the Lorentzian Metric can be written as Matrices containing hyperbolic trig functions as elements. Note Kenny's comment. | 0.087408 | 99 |
science391 | I've read a fair amount of Sets for Mathematics and found it to be a gentle introduction. http://www.amazon.com/Sets-Mathematics-F-William-Lawvere/dp/0521010608/ref=pd_sim_b_5 | 0.346795 | 0 |
science393 | If you know your relationship is going to be a polynomial, then there are some pretty (conceptually) simple ways you can do this. If you know what degree your polynomial is (line, parabola, cubic, etc.) then your job will be much easier. But if not, then you simply need to look at the number of points you have. If you ... | 0.49119 | 99 |
science394 | It is actually the other way round. A generating function is generally defined to have an addition operation where the components are added and a multiplication operation like that you mentioned. Once we have made these definitions, we observe that polynomials obey the same laws and so that it is convenient to represen... | 0.270176 | 99 |
science395 | Casebash is correct that this is a definition and not a theorem. But the motivation from 3.48 (Defintion of product of series) of little Rudin may convince you that this is a good definition: $\sum_{n=0}^{\inf} a_n z^n \cdot \sum_{n=0}^{\inf} b_n z^n = (a_0+a_1z+a_2z^2+ \cdots)(b_0+b_1z+b_2z^2+ \cdots)$ $=a_0b_0+(a_0b_... | 0.360424 | 99 |
science396 | Given a list of terms of a sequence as you describe, one technique that may be of use (supplementary to Justin's answer) is finite differences. Calculate the differences between successive terms. If these first differences are constant, then a linear equation fits the terms you have. If not, compute the differences of ... | 0.210653 | 99 |
science397 | In the general case of trying to predict some infinite sequence of integers, there is no formula. This is because there is no reason to expect a pattern to continue, since all sequences are possible. However, you can say a certain number is more likely given a set of functions. For instance if you considered all Turing... | 0.4212 | 99 |
science398 | I believe the right generalization of dual spaces is that of a dual object in a tensor category, which I will assume symmetric for convenience. Recall what makes a dual space of a vector space work: We have a map $V \times V^* \to k$ (for $k$ the ground field). The problem is, this isn't a homomorphism in the category ... | 0.218035 | 99 |
science400 | Your trouble with determinants is pretty common. They’re a hard thing to teach well, too, for two main reasons that I can see: the formulas you learn for computing them are messy and complicated, and there’s no “natural” way to interpret the value of the determinant, the way it’s easy to interpret the derivatives you d... | 0.456271 | 0 |
science401 | These implications are reached by considering the three, different cases for the roots $\{ r_1, r_2, r_3 \}$ of the polynomial: repeated root, all distinct real roots, or two complex roots and one real root. When one of the roots is repeated, say $r_1$ and $r_2$, then it is clear that the discriminant is $0$ because th... | 0.279802 | 99 |
science403 | Suppose we shuffle a deck and get a permutation p. For each previous shuffling there is a 1-1/52! chance that p doesn't match it. Each previous shuffling is independent, in that regardless of what p and the other permutations are, the chance of p matching the shuffling is 1-1/52! When probabilities are independent we c... | 0.314351 | 99 |
science405 | The three triangle inequalities are \begin{align} x + y &> 1-x-y \\ x + (1-x-y) &> y \\ y + (1-x-y) &> x \\ \end{align} Your problem is that in picking the smaller number first from a uniform distribution, it's going to end up being bigger than it would if you had just picked two random numbers and taken the smaller on... | 0.043418 | 99 |
science407 | The answer is, it's just a fact “cone over a simplex is a simplex” rewritten in terms of the generating function: observe that because n-simplex is a cone over (n-1)-simplex $\frac{\partial}{\partial x}vol(\text{n-simplex w. edge x}) = vol(\text{(n-1)-simplex w. edge x})$; in other words $e(x):=\sum_n vol\text{(n-simpl... | 0.483889 | 99 |
science408 | Knuth says to look at it as generating all nested parentheses in lexicographic order. Look here for the details http://www-cs-faculty.stanford.edu/~uno/fasc4a.ps. | 0.444221 | 99 |
science409 | Basic Theory The way to solve this problem is to calculate how much each payment reduces your debt after you have been repaying your loan for $n$ years. Let $r=1+R/100$, ie. this converts the interest rate from a percentage to a value you can multiply your debt by to calculate how much you owe after adding one time per... | 0.036323 | 99 |
science410 | Let g(n,k) = # combinations of cakes. Notice that: g(n,1) = 1. (all the cakes are the same) g(n,2) = n+1. (e.g. for 5 cakes, the # of cakes of type 1 can be 0, 1, 2, 3, 4, 5) g(1,k) = k. g(2,k) = k*(k-1)/2 + k (the first term is two different cakes; the second term is when both cakes are the same), as long as k > 1. (o... | 0.040683 | 99 |
science411 | Using a method that's often called "stars and bars": We draw $n$ stars in a row to represent the cakes, and $k-1$ bars to divide them up. All of the stars to the left of the first bar are cakes of the first type; stars between the first two bars are of the second type; . . . . **|***||*| Here's an example with $n=6$ an... | 0.332754 | 99 |
science414 | Of course there are infinite equation (even if you require them to be infinitely differentiable...) that satisfy the given constraints. As Isaac and Justin already wrote, you may always find a polynomial of degree at most n-1 (where n is the number of points given) which satisfies the given data; but you cannot be sure... | 0.368875 | 99 |
science416 | Couldn't all transformation which send each point (x,y) to another point (x',y') which can be computed from the first one by performing only the four operations and extraction of square root? | 0.206132 | 99 |
science417 | Without the unit-length segment--that is, without something to compare the first segment to--its length is entirely arbitrary, so can't be valued, so there's no value of which to take the square root. Let the given segment (with length x) be AB and let point C be on ray AB such that BC = 1. Construct the midpoint M of ... | 0.165066 | 99 |
science418 | If you have a segment $AB$, place the unit length segment on the line where $AB$ lies, starting with $A$ and in the direction opposite to $B$; let $C$ be the other point of the segment. Now draw a semicircle with diameter $BC$ and the perpendicular to $A$; this line crosses the semicircle in a point $D$. Now $AD$ is th... | 0.361817 | 99 |
science420 | Parabolas of the form you described (y = ...) are symmetric over a vertical line through their vertex. Let's call that line x = k. This means that if the graph crosses the x-axis (meaning that $ax^2+bx+c=0$ has real solution(s)), they must be equidistant from x = k, so (k,0) must be the midpoint of the segment with end... | 0.509402 | 99 |
science421 | If you use the formula you describe for squares, your measurements are coming from one corner of the square. Imagine those measurements growing slowly. The square will grow, but only along the two sides opposite the corner from which you measured, so the derivative of the area formula is only the perimeter on those two... | 0.296902 | 99 |
science424 | Complementary to Mau's answer: Call a series $a_n$ absolutely convergent if $\sum|a_n|$ converges. If $a_n$ converges but is not absolutely convergent we call $a_n$ conditionally convergent The Riemann series theorem states that any conditionally convergent series can be reordered to converge to any real number. Morall... | 0.322974 | 0 |
science427 | Let's say you have a sequence of nonnegative numbers $a_1 \geq a_2 \geq \dots$ tending to zero. Then it is a theorem that the alternating sum $\sum (-1)^i a_i$ converges (not necessarily absolutely, of course). This in particular applies to your series. Incidentally, if you're curious why it converges to $\log(2)$ (whi... | 0.40866 | 0 |
science429 | I had an answer before, but I looked into it a bit more and my answer was incorrect so I removed it. This link may be of interest: Circle Packing in a Square (wikipedia) It was suggested by KennyTM that there may not be an optimal solution yet to this problem in general. Further digging into this has shown me that this... | 0.405653 | 99 |
science430 | A problem is in class NP if its solution may be verified in polynomial time, that is if the dimension of the problem is n you may be sure that for large enough n you need less than r·nk operations to verify the solution. A problem is in class P if its solution may be found in polynomial time, instead. A problem in P is... | 0.257348 | 99 |
science431 | To expand on Mau's answer, you should care about NP-complete problems because there is an entire family of them that spans a large number of seemingly basic algorithms across a wide range of disciplines. These aren't obscure problems, but extremely important and highly practical questions. For examples, consider the fo... | 0.082653 | 99 |
science432 | Any problem for which a solution (once found) can be quickly verified as a solution is said to be "in NP" (Here, "quickly" means in polynomial-time). Any problem for which a solution can be found quickly is said to be "in P." P is a subset of NP - that is, any problem for which a solution can be quickly found can also ... | 0.26361 | 99 |
science433 | Another option would be to use eigendecomposition . It allows you to raise the eigenvalues in the diagonal of the decomposition A = VDV^-1 to a power. It changes the problem from matrix multiplication to the multiplication of the eigenvalues. Once in eigendecomposition form you could perform the same addition-chain exp... | 0.27148 | 99 |
science434 | Given that the reals are uncountable (which can be shown via Cantor diagonalization) and the rationals are countable, the irrationals are the reals with the rationals removed, which is uncountable. (Or, since the reals are the union of the rationals and the irrationals, if the irrationals were countable, the reals woul... | 0.398639 | 0 |
science435 | If you were to try to measure instantaneous speed as you described, you would in fact have traveled 0 miles in 0 time and 0/0 is undefined If, however, you look at your average speed over smaller and smaller periods of time around the instant you care about--that is, (distance traveled from $t=t_0$ to $t=t_0+\epsilon$ ... | 0.184886 | 99 |
science437 | I think there is a very clear meaning in the physical world: If, at some moment, you were going 40 mph, if you were to stop de/accelerating and just hold that velocity, you would cover 40 miles in 1 hour. | 0.10288 | 99 |
science439 | A state-variable kind of approach: (represents the physical states of a system) Newton's first law of motion says that without external forces, masses will move at constant velocity. Every change in velocity (for a car, whether a change in speed due to braking, or a change in direction due to friction of the tires and ... | 0.441697 | 99 |
science440 | BlueRaja's answer is certainly more complete than this one (and gives good references), but here's a rough overview of linear programming. Suppose that you have a linear function (in high school courses, it's typically a function of two variables) that you want to optimize on a convex "feasible" region bounded by linea... | 0.423518 | 99 |
science441 | With two stamps, you can do it in linear pseudo-linear time - O(totalCost/costOfLargerStamp) - by simply enumerating every possibility (there is only one possible count of the smaller stamp for each count of the larger stamp). In general, however, solving this is equivalent to solving a general integer linear programmi... | 0.371992 | 99 |
science443 | The Dirac delta "function." It's not a "function," strictly speaking, but rather a very simple example of a distribution that isn't a function. | 0.280477 | 99 |
science444 | This is similar to other answers. Imagine another car beside yours. That car is covering a distance of 40 miles over the next hour at a constant velocity. At the point at which you keep pace with that car (relative velocity = 0), you are traveling at 40 mph. | 0.020576 | 99 |
science447 | Yes it can. In fact, as Jamie Banks noted, a determinant is an intuitive way of thinking about volumes. To summarise the argument, if we consider the vectors as a matrix, switching two rows, multiplying one by a constant or adding a linear combination will have the same effect on the volume as on the determinate. We ca... | 0.276902 | 99 |
science448 | It is important to understand that NP-completeness is defined in terms of input size, not in terms of either cost to be paid, C, or the number of types of stamps, S, when we are dealing with integers rather than real numbers. This algorithm can actually be solved in time CS. We take the first stamp value an mark all mu... | 0.523488 | 99 |
science449 | The Weierstrass function is continuous everywhere and differentiable nowhere. The Dirichlet function (the indicator function for the rationals) is continuous nowhere. A modification of the Dirichlet function is continuous at all irrational values and discontinuous at rational values. The Devil's Staircase is uniformly ... | 0.109088 | 99 |
science450 | I would recommend against getting a degree from any online-only university. Even if you happen to find one that's not shady, everyone else who hasn't heard of it will assume it is some kind of diploma mill without bothering to do much research. Instead, I think you'd be better off going to a nearby university you're in... | 0.093427 | 99 |
science452 | $$\begin{align*} \sum F &= ma\\ \frac{dv}{dt} &= a\\ &= \frac{\sum F}{m}\\ \sum F &= mg - kv\\ \frac{dv}{dt} &= g - \frac{k}{m} v\end{align*}$$ This is a differential equation with a solution of $$\begin{align*} v &= A + B \space exp\left(\frac{-k}{m} t\right)\\ \frac{dv}{dt} &= - B \cdot \frac{k}{m} \cdot \exp\left(\f... | 0.410266 | 99 |
science455 | $dm$ takes density fluctuations into account, it's just $dm = \rho(\vec r) d^3 \mathbf r$. E.g. for a homogeneous cylinder with the rotational axis align parallel to the z-axis it is $\rho = \frac{m}{V}\cdot\theta(R-r)\theta(a^2-z^2)$, where $\theta(x) = \begin{cases} 0 \;\text{ for } x<0 \text{ and}\\ 1 \;\text{ for }... | 0.245131 | 99 |
science456 | First, some background. A "line" (the object containing the path of least distance between two points) on a sphere is a great circle; a great circle is the intersection of a sphere with a plane passing through the center of the sphere. A spherical triangle is the triangle formed by the spherical line segments connectin... | 0.01316 | 99 |
science457 | This is a very interesting question. I don't have a complete solution yet, but I did get some results. Consider an arbitrary distribution of points. Let $P(i,b)=i+\frac{b}{2}-1$ where $i$ is the number of internal points and $P$ is the number of boundary points. Let $P(A)=P(i_A,b_A)$, where $A$ is a simple polygon with... | 0.024148 | 99 |
science460 | If W is randomly chosen with the PDF P(x), then the expectation value should be $E[e^{-\gamma W}]=\int_{-\infty}^\infty P(x) e^{-\gamma x} dx$ http://mathcache.appspot.com/?tex=%5cpng%5c%5bE%5Be%5E%7B-%5Cgamma%20W%7D%5D%3D%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20P%28x%29%20e%5E%7B-%5Cgamma%20x%7D%20dx%5c%5d And I think that... | 0.46733 | 99 |
science461 | Nonstandard proof Consider the semi-circle with endpoints A and C and center O and the inscribed angle ∠ABC (B on the semi-circle) together with the rotation image of both about O by 180°. The image of A is C and vice versa; let B' be the image of B. The image of a line under a 180° rotation is parallel to the original... | 0.375109 | 99 |
science462 | Look at this answer on MathOverflow: Yes, there is a way to guess a number asking 14 questions in worst case. To do it you need a linear code with length 14, dimension 10 and distance at least 3. One such code can be built based on Hamming code (see http://en.wikipedia.org/wiki/Hamming_code). Here is the strategy. Let ... | 0.54286 | 99 |
science465 | For the case of Bendford's Law, of course scale invariance is a necessary condition; it the law must be true either if we measure things in meters or in feet or in furlongs, thus multiplying given data for a constant, the only distribution which allows this is the logarithmic one. But its being necessary does not mean ... | 0.23298 | 99 |
science467 | There is, in fact, a general formula for solving quartic (4th degree polynomial) equations. As the cubic formula is significantly more complex than the quadratic formula, the quartic formula is significantly more complex than the cubic formula. Wikipedia's article on quartic functions has a lengthy process by which to ... | 0.134613 | 1 |
science468 | Yes, there is a quartic formula. There is no such solution by radicals for higher degrees. This is a result of Galois theory, and follows from the fact that the symmetric group $S_5$ is not solvable. It is called Abel's theorem. In fact, there are specific fifth-degree polynomials whose roots cannot be obtained by usin... | 0.16556 | 1 |
science470 | When you try to solve a degree $n$ equation, there are $n$ roots you have to find (in principle) and none of them is favoured over any of the others, which (in some metaphorical sense) means that you have to break an $n$-fold symmetry in order to write down the roots. Now the symmetry group of the n roots becomes more ... | 0.238783 | 99 |
science472 | Differentiation and integration is precisely why it is considered natural, but not just because $$\displaystyle\int \frac{1}{x} dx=\ln x$$ $e^x$ has the two following nice properties $$ \frac{d}{dx} e^x=e^x $$ $$ \int e^x dx=e^x+c $$ If we looked at $a^x$ instead, we would get: $$\frac {d} {dx} a^x= \frac{d}{dx} e^{x\l... | 0.349519 | 0 |
science473 | The Wikipedia article about e lists many properties of the constant that make it naturally occurring. I think the biggest reason it is natural when it comes to exponentiation/logarithms is that it is the only number that satisfies $$ \frac{d}{dt} e^t =e^t $$ while every other number satisfies $$ \frac{d}{dt} a^t = c \c... | 0.277424 | 0 |
science475 | The square of the discriminant is symmetric in the roots of the polynomial; if you permute the roots, it stays the same. By the fundamental theorem of symmetric functions, that means it can be expressed as a polynomial, with integer coefficients, in the coefficients of the original polynomial. (I am using both "coeffic... | 0.040616 | 99 |
science477 | The proof that uses the fact that shearing a parallelogram parallel to one of its sides preserves area is my favorite. Here's an animation from this site: | 0.0387 | 2 |
science479 | There is a combinatorial proof. If $A(x) = \sum_{n \ge 0} a_n \frac{x^n}{n!}$ is an exponential generating function for the number of ways $a_n$ to put a certain structure on a set of size $n$, where $a_0 = 0$, then $\exp A(x) = \sum_{n \ge 0} b_n \frac{x^n}{n!}$ has coefficients $b_n$ which count the number of ways to... | 0.23009 | 99 |
science480 | A nice application is Wedderburn's theorem: every finite skewfield is necessarily commutative. Here a skewfield is something which satisfies the same axioms as field, except that multiplication is not required to be commutative; the typical example are quaternions. To see this, let $F$ be a finite skewfield, $Z$ be its... | 0.089832 | 99 |
science484 | This is not really a proof, in the sense that it depends on certain physical assumptions, but it forces you to think very hard about what those physical assumptions are. I learned it from Mark Levi's The Mathematical Mechanic, although unfortunately I don't have a diagram. Consider a fish tank in the shape of a triangu... | 0.051901 | 2 |
science485 | As Akhil mentions, the keyword is elliptic regularity. Since I don't know anything about this, let me just say some low-level things and maybe they'll make sense to you. A differentiable function $f : \mathbb{R} \to \mathbb{R}$ can be thought of as a function which behaves locally like a linear function $f(x) = ax + b$... | 0.294307 | 2 |
science486 | Another definition of discriminant is as the resultant of the polynomial with it's first derivative (up to a scalar), and the resultant of two polynomials vanishes if and only if they have a common root. So when does a polynomial have a common root with it's derivative? That's when a zero is also a local max or min, pr... | 0.451088 | 99 |
science487 | Regarding the inability to solve the quintic, this is sort-of true and sort-of false. No, there is no general solution in terms of $+$, $-$, $\times$ and $\div$, along with $\sqrt[n]{}$. However, if you allow special theta values (a new operation, not among the standard ones!) then yes, you can actually write down the ... | 0.287103 | 1 |
science489 | Maybe this is helpful, a formula for the volume of Lp balls in R^n. | 0.131115 | 99 |
science493 | Answer edited, in response to the comment and a second wind for explaining mathematics: Let $F(x_0,x_1,x_2)=0$ be the equation for your curve, and take $(y_0,y_1,y_2)$ to be coordinates on $(\mathbb{P}^2)^*$. Also, assume that $F$ is irreducible and has no linear factors. Then $y_0 x_0+y_1 x_1+y_2 x_2=0$ is the equatio... | 0.222157 | 99 |
science494 | Benoît Mandelbrot Although he provided many valuable contributions to the field, I am most in love with his work on Fractals. I find math to be quite beautiful, and the Mandelbrot Set (magnified portion shown below) is a perfect example: | 0.000082 | 99 |
science498 | Well, class field theory states that the class number is the degree of the largest everywhere-unramified abelian extension of a number field (namely, the Hilbert class field). But class field theory really says a lot more: it says that there's an isomorphism between the Galois group and the ideal class group. And in ge... | 0.415504 | 99 |
science500 | On an intuition level: Complex numbers are somewhat equivalent to $\mathbb R^2$ (homeomorphic to a $2$-dimensional real space), on the other hand they are also a $1$-dimensional complex space. This gives them an extra structure with consequences like this theorem. Another view on this is that differentiable complex fun... | 0.212831 | 2 |
science501 | Suppose you have a large collection of books, all of the same size. Balance one of them on the edge of a table so that one end of the book is as far from the table as possible. Balance another book on top of that one, and again try to get as far from the table as possible. Take $n$ of them and try to balance them on to... | 0.392304 | 0 |
science503 | This isn't a complete answer, partly because I've had discussions with other grad students and we weren't able to work it out satisfactorily, but I've been told that you can actually put a topology on logical statements such that the compactness theorem translates to "The set of true statements is a compact subset of t... | 0.127612 | 0 |
science505 | The analogy for the compactness theorem for propositional calculus is as follows. Let $p_i $ be propositional variables; together, they take values in the product space $2^{\mathbb{N}}$. Suppose we have a collection of statements $S_t$ in these boolean variables such that every finite subset is satisfiable. Then I clai... | 0.496508 | 0 |
science506 | A further sense in which Higher Order Logics with standard or saturated semantics (HOL, hereafter) are less well-behaved than First Order Logic (FOL, hereafter) is a direct consequences of the failure of Completeness (and thus, as explained in other answers, of Compactness). The set of logical truths and the set of cor... | 0.426096 | 99 |
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