qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
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29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Gjergji Zaimi | 2,384 | <p>How many times a day is it impossible to tell the time by a clock with identical hour and minute hands, provided you can always distinguish between a.m and p.m? P.S. Ask them for a fast answer.</p>
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29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Pietro Majer | 6,101 | <p>Given three equal sticks, and some thread, is it possible to make a rigid object in such a way that the three sticks do not touch each other? (all objects are 1 dimensional; sticks are straight and rigid, and the thread is inestensible).</p>
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29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Anthony Leverrier | 1,606 | <p>1000 prisoners are in jail.
There's a room with 1000 lockers, one for each prisoner. A jailer writes the name of each prisoner on a piece of paper and puts one in each locker (randomly, and not necessary in the locker corresponding to the name written on the paper!).</p>
<p>The game is the following. The prisoners ... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | rgrig | 840 | <p>Start with four beads placed at the corners of a square. You are allowed to move a bead from position x to position y if one of the other three beads is at position (x+y)/2. In other words, you may `reflect a bead with respect to another bead.' Find a sequence of such moves that places the beads at the corners of a ... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Community | -1 | <p>Via the great Martin Gardner: A cylindrical hole is drilled straight through the center of a solid sphere. The length of hole in the sphere (i.e. of the remaining empty cylinder) is 6 units. What is the volume of the remaining solid object (i.e. sphere less hole)? Yes, there is enough information to solve this pr... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | BlueRaja | 2,883 | <p>Alice secretly picks two different real numbers by an unknown process and puts them in two (abstract) envelopes. Bob chooses one of the two envelopes randomly (with a fair coin toss), and shows you the number in that envelope. You must now guess whether the number in the other, closed envelope is larger or smaller... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Gil Kalai | 1,532 | <p>This is a hat problem I heard only two days ago: you have a hundred people and each one has a (natural) number between 1 and 100 written on his hat. (Numbers may repeat.) As ususal, everybody can see only the numbers on other people's hats. Give these guys a strategy for guessing so that at least one will surely mak... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Gerhard Paseman | 3,402 | <p>Here is a classic:</p>
<p>Plant 10 trees in five rows, with 4 trees in each row.</p>
<p>I like this because there are two basic approaches to the problem: the one almost everyone thinks of and uses to grind slowly towards a solution, and the one they should think of instead, which leads quickly to many solutions.<... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Anweshi | 2,938 | <p>A puzzle(rather, a tale to lure the reader into the domain of complex numbers) lifted from George Gamow's "<a href="http://rads.stackoverflow.com/amzn/click/0486256642">One, Two, Three, Infinity</a>":</p>
<p>There was a young and adventurous man who found among his great-grandfather’s papers a piece of torn parchme... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | BlueRaja | 2,883 | <p>You have a large pile of ropes and some matches. All you know about the ropes:</p>
<ul>
<li>Each rope has a different length</li>
<li>Each rope burns completely (starting from one end) in exactly 64 minutes</li>
<li>Each rope has non-uniform density, meaning it is thicker at some points than others. Consequently, ... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Jérôme JEAN-CHARLES | 3,005 | <p>I understand your feeling , I myself know lots of them . Among original ones </p>
<p><a href="http://rads.stackoverflow.com/amzn/click/1568812019" rel="nofollow">http://www.amazon.com/Mathematical-Puzzles-Connoisseurs-Peter-Winkler/dp/1568812019</a></p>
<p>This Peter Winkler does something that is rarely done and ... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | ThudnBlunder | 10,821 | <p>Quite simply, a monkey's mother is twice as old as the monkey will be when the monkey's father is twice as old as the monkey will be when the monkey's mother is less by the difference in ages between the monkey's mother and the monkey's father than three times as old as the monkey will be when the monkey's father is... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Jakob Katz | 3,210 | <p>Saw this recently:</p>
<p>There is a board with a grid drawn on it. You hammer a few nails into the board at intersection points and then stretch a rubber band around the nails. Then you observe that</p>
<p>(1) you can't take away any of the nails without changing the shape,</p>
<p>(2) the rubber band does not en... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Florian | 467 | <p>A table with three legs does not wobble.</p>
<p>How about a quadratic table with four legs? Can it be rotated to fix the wobbling?</p>
<p>(Please assume reasonable conditions on life the universe and everything.)</p>
|
1,744,760 | <h2>Question</h2>
<p>What do we gain or lose, conceptually, if we consider <em>scalar multiplication</em> as a special form of <em>matrix multiplication</em>?</p>
<h2>Background</h2>
<p>The question bothers me since I have been reading about <em>dilations</em> and <em>scaling</em> of geometrical objects in Paul Lock... | M. Vinay | 152,030 | <p><strong><em>Note:</strong> Some discussion with <a href="https://math.stackexchange.com/users/261373/siddharth-bhat">Siddharth</a> has revealed a flaw in my earlier answer, which I've corrected now (with a complete reversal of conclusion).</em></p>
<p>The set of all endomorphisms of an Abelian group $(V, +)$ forms ... |
2,623,621 | <p>I'm struggling to find the derivative of this function :</p>
<p>$$y=\bigg\lfloor{\arccos\left(\frac{1}{\tan\left(\sqrt{\arcsin x}\right)}\right)}\bigg\rfloor$$</p>
<p>I've been told it should be 0 but how can I find that ?!</p>
| egreg | 62,967 | <p>Suppose <span class="math-container">$x$</span> belongs to the closure of <span class="math-container">$S=\bigcup_{i\in I}S_i$</span>. Fix a neighborhood <span class="math-container">$U_0$</span> of <span class="math-container">$X$</span>, so <span class="math-container">$U_0\cap S\ne\emptyset$</span>. On the other ... |
191,373 | <p>I have a usual mathematical background in vector and tensor calculus. I was trying to use the differential operators of Mathematica, namely <code>Grad</code>, <code>Div</code> and <code>Curl</code>. According to my knowledge, the definitions of Mathematica for <code>Grad</code> and <code>Div</code> coincides with th... | Somos | 61,616 | <p>I suggest to modify your simple piece of code to make the output more informative. I define a function <code>dim[]</code> which is similar to the <code>Dimensions[]</code> function but is restricted to matrices and arrays. For example, try <code>Dimensions[x+y]</code> which returns <code>{2}</code> but <code>dim[x+y... |
3,648,462 | <p>I need to find a solution to this equation, being <span class="math-container">$z\in\mathbb{C}$</span>,
<span class="math-container">$$2e^3\cos (6z)-e^6=1.$$</span>
I don't know what method can I follow to solve it.</p>
| GReyes | 633,848 | <p>The sum of all the coefficients is the value of the polynomial at <span class="math-container">$x=1$</span> that is <span class="math-container">$3^{200}$</span>.</p>
|
2,004,895 | <p>In my textbook there is a question like below:</p>
<p>If $$f:x \mapsto 2x-3,$$ then $$f^{-1}(7) = $$</p>
<p>As a multiple choice question, it allows for the answers: </p>
<p>A. $11$<br>
B. $5$<br>
C. $\frac{1}{11}$<br>
D. $9$</p>
<p>If what I think is correct and I read the equation as:</p>
<p>$$f(x)=2x-3$$
th... | Barkas | 170,882 | <p>Answering your (main) question: Yes your interpretation of $f:x\rightarrow 2x-3$ being the same as $f(x)=2x-3$ is correct.</p>
<p>For your calculations: $\frac{7+3}{2}\neq10$.</p>
|
2,004,895 | <p>In my textbook there is a question like below:</p>
<p>If $$f:x \mapsto 2x-3,$$ then $$f^{-1}(7) = $$</p>
<p>As a multiple choice question, it allows for the answers: </p>
<p>A. $11$<br>
B. $5$<br>
C. $\frac{1}{11}$<br>
D. $9$</p>
<p>If what I think is correct and I read the equation as:</p>
<p>$$f(x)=2x-3$$
th... | DanielWainfleet | 254,665 | <p>As a matter of not merely style but of writing in complete sentences you should write $x=f^{-1}(7)\iff f(x)=7\iff 2x-3=7\iff (2x-3)+3=7+3\iff 2x=10\iff x=5.$ This reduces errors and shows the flow of the reasoning. In this case the sequence of "$\iff$" shows that the implications go in both directions. The importan... |
1,582,544 | <p>$X_1,\dots,X_n$ and $Y_1,\dots,Y_n$ are random variables that take values in $\{0,1\}$. Their distribution is unknown, each variable may have a different distribution and they might be dependent. The only known facts are:</p>
<ul>
<li>$\sum_{i=1}^n X_i = \sum_{i=1}^n Y_i$</li>
<li>For all $i$, $\Pr[X_i=0] \geq \alp... | Anders Muszta | 294,222 | <p>Since expectation is linear you can write $$\mathbb{E}\left(\sum_{i=1}^{n}X_{i}\right) = \sum_{i=1}^{n}\mathbb{P}(X_{i}=1) \leq n(1-\alpha)\ .$$ Similarly, $$\mathbb{E}\left(\sum_{i=1}^{n}Y_{i}\right) = \sum_{i=1}^{n}\mathbb{P}(Y_{i}=1) \geq n\alpha\ .$$
Since the two expectations are equal you have the following in... |
3,221,547 | <p>I'm trying to prove that there are no simple groups of order <span class="math-container">$240$</span>. So let <span class="math-container">$G$</span> be a simple group such that <span class="math-container">$|G|=240=2^4\cdot3\cdot5$</span>. Then
<span class="math-container">$$n_2\in\{1,3,5,15\}\quad n_3\in\{1,4,10,... | Hongyi Huang | 619,069 | <p>You may find <span class="math-container">$n_5\ne 6$</span> and <span class="math-container">$n_3\ne 4$</span> probably by the similar proof of <span class="math-container">$n_2 = 15$</span>: <span class="math-container">$G$</span> cannot be embedded to <span class="math-container">$S_4$</span> and <span class="math... |
20,364 | <p>The Stanford Encyclopedia of Philosophy's <a href="http://plato.stanford.edu/entries/category-theory/">article on category theory</a> claims that adjoint functors can be thought of as "conceptual inverses" of each other. </p>
<p>For example, the forgetful functor "ought to be" the "conceptual inverse" of the free-g... | Omar Antolín-Camarena | 1,070 | <p>In a nutshell, a natural bijection between $\mathrm{Hom}(f(x), y)$ and $\mathrm{Hom}(x, g(y))$ says that the "graph of $f$" is obtained from the "graph of $g$" by "reflecting in the diagonal", just like the relationship between the graphs of inverse functions in calculus.</p>
<p>In more detail: two functions $f \co... |
20,364 | <p>The Stanford Encyclopedia of Philosophy's <a href="http://plato.stanford.edu/entries/category-theory/">article on category theory</a> claims that adjoint functors can be thought of as "conceptual inverses" of each other. </p>
<p>For example, the forgetful functor "ought to be" the "conceptual inverse" of the free-g... | Community | -1 | <p>One of my friend was being confused with the same question and decided to ask the author of the article about it.</p>
<p>His reply was,</p>
<blockquote>
<p>If you take arbitrary abstract categories and stipulate that a pair of adjoint functors exists between them, the only thing you can hold on to is their abstr... |
2,390,036 | <blockquote>
<p><strong>Theorem</strong>. Let $G = (V,E)$ be a simple graph with $n$ vertices, $m$ edges and $\chi (G) = k$. Then, $$m \geqslant {k \choose 2}$$</p>
</blockquote>
<p>I tried proving myself but made little to no progress. I am aware of the inequality $$n/ \alpha (G) \leqslant \chi (G) \leqslant \Delta... | Mark Bennet | 2,906 | <p>Hint: If you have a hypothetical counter-example, can you show you have two colours which are not connected?</p>
|
2,742,497 | <p>When two events $A$ and $B$ are independent, $P(A)\cdot P(B)$ equals the probability of both events occurring together.</p>
<p>$$P(A|B) = \frac{P(A\cap B)}{P(B)}.$$</p>
<p>If $A$ and $B$ are not independent, $P(A\cap B)$ is either greater than or less than $P(A)\cdot P(B)$.</p>
<ol>
<li><p>I want to understand t... | quasi | 400,434 | <p>When $A,B$ are not independent, and if there is no more information, there's no natural interpretation of $P(A){\,\cdot\,}P(B)$.
<p>
The condition $P(A|B) > P(A)$ means $A$ is more likely to have occurred if it's given that $B$ has occurred, than if there is no information as to whether or not $B$ has occurred.
<... |
2,742,497 | <p>When two events $A$ and $B$ are independent, $P(A)\cdot P(B)$ equals the probability of both events occurring together.</p>
<p>$$P(A|B) = \frac{P(A\cap B)}{P(B)}.$$</p>
<p>If $A$ and $B$ are not independent, $P(A\cap B)$ is either greater than or less than $P(A)\cdot P(B)$.</p>
<ol>
<li><p>I want to understand t... | Graham Kemp | 135,106 | <p>For events $A,B$, the <strong>correlation</strong> is measured by $$\mathsf{Corr}(A,B)=\dfrac{ \mathsf P(A\cap B)-\mathsf P(A)\cdotp \mathsf P(B)}{\sqrt{\mathsf P(A)\cdotp\mathsf P(B)\cdotp(1-\mathsf P(A))\cdotp(1-\mathsf P(B))}}$$</p>
<p>Similarly the <strong>covariance</strong> is measured as: $$\mathsf{Cov}(A,B)... |
121,784 | <p>I hope this is not too trivial for this forum. I was wondering if someone has come across this polytope.</p>
<p>You start with the <a href="http://mathworld.wolfram.com/RhombicDodecahedron.html" rel="nofollow noreferrer">rhombic dodecahedron</a>, subdivide it into four parallellepipeds, <a href="https://i.stack.img... | Dr. Richard Klitzing | 118,679 | <p>The above mentioned <strong>Minkowski sum</strong> is just an example of <strong>Alicia Boole Stott's expansion</strong> operation(s), cf. <a href="https://en.wikipedia.org/wiki/Expansion_(geometry)" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Expansion_(geometry)</a></p>
<p>--- rk</p>
|
482,010 | <p>Sorry for my bad english And also my bad math literature in asking my first question.</p>
<p>Assume we have 15 glass full of water.</p>
<p>I need to count the number of possible ways which I can empty 5 glass in a way that no 2 empty glass remain in sequence.</p>
| Macavity | 58,320 | <p>Looking at it in reverse, we can first place $5$ empty glasses in a row. Now we must have $4$ full glasses in between them, to fulfil the constraint. We are now free to distribute remaining $6$ glasses in any of $6$ locations (in between or outside) the empty glasses. </p>
<p>So this is equivalent to putting $6$ id... |
84,312 | <p>In the topological sense, I understand that the unit circle $S^1$ is a retract of $\mathbb{R}^2 \backslash \{\mathbb{0}\}$ where $\mathbb{0}$ is the origin. This is because a continuous map defined by $r(x)= x/|x|$ is a retraction of the punctured plane $\mathbb{R}^2 \backslash \{\mathbb{0}\}$ onto the unit circle $... | Community | -1 | <p>Intuitively, a (continuous) retraction of $R^2$ onto $S^1$ would not be possible, because there would have to be a "tear" in the disc to accomplish this. The tear wouldn't be continuous, as nearby points need to go to nearby points. ..
Using the notion of topology as rubbersheet geometry, "tearing" (of the rubbe... |
1,946,637 | <p>I have come across this integral and have been unable to solve it so far.</p>
<p>$$I=\int_{-\infty}^{0.29881}e^{-0.5x^2}\,\mathrm dx$$</p>
| MrYouMath | 262,304 | <p>First consider the subsituted eqatuion, with $z^2=x$. I also replaced $j$ by $n$:</p>
<p>$$\sum_{n = 0}^\infty\frac{2^n}{3^n+4^n}(x)^n$$</p>
<p>The ratio test:</p>
<p>$$R_x=\lim_{n\to \infty}|a_n/a_{n+1}|=0.5\lim_{n\to \infty}\frac{3^{n+1}+4^{n+1}}{3^{n}+4^n}$$.</p>
<p>Now divide the expand the fraction with $4^... |
866,885 | <p>I try to write math notes as clearly as possible. In practice, this means using letters and notation similar to what the reader is already familiar with.</p>
<p>A real function is often $f(x)$, an angle is often $\theta$, a matrix has size $m\times n$, and $i$ is often an index. The full theoretical list is long an... | beep-boop | 127,192 | <p>If in doubt, use Wikipedia.</p>
<p>This gives a list of mathematical symbols and all the (widely-used) contexts in which they arise:</p>
<p><a href="http://en.wikipedia.org/wiki/List_of_mathematical_symbols" rel="nofollow">http://en.wikipedia.org/wiki/List_of_mathematical_symbols</a>.</p>
<p>Another good list for... |
2,584,405 | <p>During an excercise session in a basic course of probability it was shown that the secretary problem can be reduced to solving the following task: For a given natural $n$ optimize $2\leq k\leq n-1$ so that
$\frac{k-1}{n}\sum^n_{i=k}\frac 1{i-1} = \frac{k-1}{n}(\operatorname{H}_{n-1}-\operatorname{H}_{k-2})$ is the ... | Antonio Vargas | 5,531 | <p>A heuristic "justification" is that applying the approximation leads us to the guess $k \approx n/e + 1$, which is large when $n$ is large. So the large $k$ approximation is at least consistent with the result obtained.</p>
<p>Certainly more work needs to be done to justify that the maximum is not actually elsewher... |
2,779,841 | <p>I'm wondering how the following recursion for <span class="math-container">$D(n)$</span>, the derangement of <span class="math-container">$[n]$</span> distinct elements, can be proved</p>
<p><span class="math-container">$$D(n) = nD(n-1) + (-1)^n$$</span></p>
<hr>
<p>I see the argument for </p>
<p><span class="ma... | Peter Taylor | 5,676 | <p>It's a simple proof by induction from the recurrence you give:
$$\begin{eqnarray}D(n) &=& (n-1)(D(n-1) + D(n-2)) \\ &=& nD(n-1) - (D(n-1) - (n-1)D(n-2))\end{eqnarray}$$
So $$\begin{eqnarray}D(n) - nD(n-1) &=& - (D(n-1) - (n-1)D(n-2)) \\
&=& \cdots \\
&=& (-1)^{n-2} (D(2) - 2D(... |
3,361,489 | <p><strong>Question:</strong></p>
<p>Do there exist functions <span class="math-container">$f$</span> and <span class="math-container">$g$</span> such that
<span class="math-container">$$\lim_{x \to c} f(x) = 1 \text{ and } \lim_{x \to c} f(x) g(x) - g(x) \neq 0 \, ?$$</span>
(Allowing, of course, for <span class="ma... | Ingix | 393,096 | <p>Note that <span class="math-container">$f(x)g(x)-g(x) = (f(x)-1)g(x)$</span>. By the property you cited (limit of product equals product of limits, if they exist), that means, if <span class="math-container">$\lim_{x\to c}g(x)$</span> exists and equals <span class="math-container">$L$</span>, then by necessecity</p>... |
1,051,464 | <p>I was asked to find the truth value of the statement: </p>
<blockquote>
<p>$$
\pi + e \; \text{ is rational or } \pi - e\; \text{ is rational }
$$</p>
</blockquote>
<p>I am only allowed to use the fact that $\pi, e $ are irrational numbers and cannot use the theory of transcendental numbers.</p>
<p>Cannot proce... | drhab | 75,923 | <p>$2\sqrt2$ and $\sqrt2$ are two distinct irrational numbers s.t. the statement: '$2\sqrt2+\sqrt2$ is rational or $2\sqrt2-\sqrt2$ is rational' <strong>is not true</strong>.</p>
<p>$\sqrt2$ and $2-\sqrt2$ are two distinct irrational numbers s.t. the statement: '$\sqrt2+(2-\sqrt2)$ is rational or $\sqrt2-(2-\sqrt2)$ i... |
1,895,721 | <p>How to find $3\times3$ matrices that satisfy the matrix equation $A^2=I_3$? Can anyone please show me steps to do this question?</p>
| Caleb Stanford | 68,107 | <p><strong>Geometrically,</strong> the four (classes of) solutions will be:</p>
<ul>
<li><p>the identity ($I$);</p></li>
<li><p>reflection through the origin ($-I$);</p></li>
<li><p>reflection through any plane through the origin;</p></li>
<li><p>rotation of $180^\circ$ about any axis through the origin.</p></li>
</ul... |
2,000,556 | <p>Suppose $N>m$, denote the number of ways to be $W(N,m)$</p>
<h3>First method</h3>
<p>Take $m$ balls out of $N$, put one ball at each bucket. Then every ball of the left the $N-m$ balls can be freely put into $m$ bucket. Thus we have: $W(N,m)=m^{N-m}$.</p>
<h3>Second method</h3>
<p>When we are going to put $N$... | Astyx | 377,528 | <p>The mistake comes form the fact that the two possibilities of the second method are not disjoint. </p>
<p>For instance when putting the 6-th ball in a set of three buckets, if you have $(1,2,2)$ you can get $(2,2,2)$, or $(1,3,2)$ or $(1,2,3)$.</p>
<p>If you had $(0, 3, 2)$ you need to get $(1,3,2)$ which is a way... |
532,525 | <p>Is $P( A \cup B \,|\, C)$ the same as $P(A | C) + P(B | C)$ ?
Here $A$ and $B$ are mutually exclusive.</p>
| Stefan Hamcke | 41,672 | <p>You could write $\text{Cl}A+\text{Cl}B$ as the image of $\text{Cl}A\times \text{Cl}B=\text{Cl}(A\times B)$ under the addition map $+$. But $+$ is continuous, so $+(\text{Cl}(A\times B))\subseteq \text{Cl}(+(A\times B))=\text{Cl}(A+B)$.</p>
<p>You would still need to prove that $\text{Cl}A\times\text{Cl}B=\text{Cl}(... |
3,065,331 | <p>In <a href="https://rads.stackoverflow.com/amzn/click/0199208255" rel="nofollow noreferrer">Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers</a> one of the very first stated equations are, as in the title of the question,</p>
<p><span class="math-container">$$
\ddot{x} = \fr... | Lutz Lehmann | 115,115 | <p>On segments where the function <span class="math-container">$x(t)$</span> is monotonous, you can invert the direction of dependence and find <span class="math-container">$t(x)$</span>. Then also the derivative can be parametrized by <span class="math-container">$x$</span>, <span class="math-container">$\dot x=u(x)$<... |
697,458 | <p>Using a trivial example to illustrate the question -</p>
<p>$$\int_0^\infty 2 dx$$</p>
<p>$=2x \mid_0^\infty = 2(\infty) - 2(0)$</p>
<p>Can we actually say $2(\infty)$? It doesn't seem valid me.</p>
| Ruslan | 64,206 | <p>An integral, one (or both) of limits of which is infinite, is an <a href="http://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegrals.aspx" rel="nofollow">improper integral</a>. For such type of integral you can't take Newton-Leibniz formula <em>as is</em>, you'll have to use the definition of improper integral:... |
2,156,606 | <p>I am stuck in this exercise of calculus about solving this indefinite integral, so I would like some help from your part: </p>
<blockquote>
<p>$$\int \frac{dx}{(1+x^{2})^{\frac{3}{2}}}$$</p>
</blockquote>
| rubik | 2,582 | <p>Let $x = \sinh u$. Then the integral becomes
$$\int \frac{\mathrm dx}{(1+x^{2})^{\frac{3}{2}}} = \int \frac{\cosh u}{\cosh^3 u}\mathrm du = \int\operatorname{sech}^2 u\,\mathrm du$$</p>
<p>Now, $\operatorname{sech}^2 u$ is the derivative of a well-known hyperbolic function. Can you spot it?</p>
<hr>
<p><strong>Pr... |
1,531,050 | <p>a. Say $X_n$ and $Y_n$ are two sequences such that $|X_n-Y_n| < \frac{1}{n}$ and $\lim X_n$ = $z$. Prove $\lim Y_n$ = $z$.</p>
<p>b. Say $X_n$ takes values in closed interval $[a,b]$ and $\lim X_n$ = $z$. Prove $z$ takes values in closed interval $[a,b]$.</p>
<p>c. Does b. hold when the interval is open?</p>
| egreg | 62,967 | <p>The hypothesis is the same as saying $|Y_n-X_n|<\frac{1}{n}$, or
$$
X_n-\frac{1}{n}<Y_n<X_n+\frac{1}{n}
$$
Can you say what
$$
\lim_{n\to\infty}\left(X_n-\frac{1}{n}\right)
\qquad\text{and}\qquad
\lim_{n\to\infty}\left(X_n+\frac{1}{n}\right)
$$
are?</p>
<p>For (b), prove that $z=\lim_{n\to\infty}X_n<a$ ... |
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | Edan Maor | 122 | <p>A probability problem I love.</p>
<p>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.</p>
<p><strong>The question</strong>: which card has a higher probability of being turned over, the Ace of Spades or the Two of Hearts?</p>
|
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | Moor Xu | 30 | <p>Here are three of my favorite variations on the hats and prisoners puzzle that I've collected over time:</p>
<ol>
<li><p>Fifteen prisoners sit in a line, and hats are placed on their heads. Each hat can be one of two colors: white or black. They can see the colors of the people in front of them but not behind them,... |
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | iamtheneal | 1,101 | <p>The Blue-Eyed Islander problem is one of my favorites. You can read about it <a href="http://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/" rel="noreferrer">here</a> on Terry Tao's website, along with some discussion. I'll copy the problem here as well.</p>
<blockquote>
<p>There is an island... |
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | Michal | 103,004 | <p>Is it possible to divide a circle into a finite number of congruent parts some of which don't touch the center?</p>
|
566 | <h3>We all love a good puzzle</h3>
<p>To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that driv... | user1713538 | 571,954 | <p>For solving any mathematics series below algorithm can be used. Even for some cases it will not satisfy your expected answer, but it will be correct in some other way.</p>
<p>Steps are as below:<br/>
1) Get difference between the numbers as shown below:<br/>
2) Keep making difference until it seems same(difference ... |
2,674,938 | <blockquote>
<p>Let $L:C^2(I)\rightarrow C(I), L(y)=x^2y''-3xy'+3y.$ Find the kernel of the linear transformation $L$. Can the solution of $L(y)=6$ be expressed in the form $y_H$+$y_L$, where $y_H$ is an arbitrary linear combination of the elements of ker L.</p>
</blockquote>
<p><strong>What I have tried:</strong></... | Martin Argerami | 22,857 | <p>The second derivative of $x^r$ is $r(r-1)x^{r-2}$. So the characteristic equation you should get is $$ r^2-4r+3=0,$$ with roots $1$ and $3$. This gives you a fundamental set of solutions
$$
y_1(x)=x,\ \ y_2(x)=x^3.
$$
Because your equation is linear homogeneous of order two, any solution will be a linear combinatio... |
844,887 | <p>I am trying to prove by induction that every non-zero natural number has at least one predecessor. However, I don't know what to use as a base case, since 0 is not non-zero and I haven't yet established that 1 is the number following zero.</p>
<p>My axioms are: </p>
<ol>
<li>$0$ is a natural number. </li>
<li>if $... | Carl Mummert | 630 | <p>The fact that every number is either 0 or a successor follows almost embarrassingly quickly from induction on the predicate
$$P(x) \equiv (x = 0 ) \lor (\exists y)[x = S(y)].$$</p>
<p>Clearly $P(0)$ holds. Also $P(S(y))$ holds for all $y$, so $(\forall y)[P(y) \to P(S(y))]$ also holds. Now apply induction.</p>
<p... |
2,309,418 | <p>How can I integrate this function?</p>
<p>$$\int \left( {3x^3+3x^2+3x+1 \over (x^2+1)(x+1)^2}\right)dx$$</p>
<p>Using a previous example I have found:</p>
<p>$$\int \left({3x^3+3x^2+3x+1 \over (x^2+1)(x+1)^2}dx\right) ={A\over x^2+1}+{B \over x+1}+{Cx+D\over (x+1)^2}$$</p>
<p>And then:
$$\int 3x^3+3x^2+3x+1 ={A(... | fonfonx | 247,205 | <p><strong>Hints</strong></p>
<p>The first step is roughly correct, you have to find the values of A, B, C and D such that</p>
<p>$$\int \left({3x^3+3x^2+3x+1 \over (x^2+1)(x+1)^2}dx\right) =\int \left({Ax+B\over x^2+1}+{C \over x+1}+{D\over (x+1)^2})\right)dx$$</p>
<p>To find the values of the constant, several met... |
2,050,939 | <p>My daughter got the following optional question for 8th grade homework mixed into her simultaneous equations questions. All of the other questions were simple for me to solve using substitution or elimination which is what she is being taught at the moment but this one seems to me to be at a much higher grade.</p>
... | Christian Blatter | 1,303 | <p><em>Claim.</em> The last ball is red with probability ${1\over2}\,$, irrespective of the initial numbers of balls.</p>
<p><em>Proof.</em> The game consists of several rounds. A <em>round</em> begins in
state $1$ with $m\geq1$ red balls and $n\geq1$ blue balls. According to the rules we randomly pick balls unti... |
2,084,671 | <p>I've tried simplifying the term by <strong>substituting</strong> $y := x^2$ and also to use a certain <strong>algorithm</strong> that outputs the f(a), but it gets too big for the input r, that is a variable and not a number to begin with (added little example):</p>
<p><a href="https://i.stack.imgur.com/8C9sV.png" ... | Jimmy R. | 128,037 | <p>\begin{align}x^4-6qx^2+q^2&=x^4-6qx^2+9q^2-8q^2\\[0.2cm]&=(x^2-3q)^2-(\sqrt{8}q)^2\\[0.2cm]&=\left(x^2-3q-2\sqrt{2}q\right)\left(x^2-3q+2\sqrt{2}q\right)\end{align} Now, note that $$r^2=\left(\sqrt{q}+\sqrt{q}\sqrt{2}\right)^2=q\left(1+2\sqrt2+2\right)=(3+2\sqrt{2})q$$ to conclude.</p>
|
1,593,339 | <p>What is $y$ in $$J^\frac{1}{2}f(x)=y$$
$$f(x)=w\sin(x)$$
where $w$ is a constant?</p>
| JJacquelin | 108,514 | <p>Hint : </p>
<p>The fractional derivative of the sinusoidal functions :</p>
<p><a href="https://i.stack.imgur.com/c6Id8.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/c6Id8.jpg" alt="enter image description here"></a></p>
<p>Copy from page 46, Eqs. (14-4) and (14-5) in the technical report "Use... |
76,378 | <p>I am trying to simplify the following expression I have encountered in a book</p>
<p>$\sum_{k=0}^{K-1}\left(\begin{array}{c}
K\\
k+1
\end{array}\right)x^{k+1}(1-x)^{K-1-k}$</p>
<p>and according to the book, it can be simplified to this:</p>
<p>$1-(1-x)^{K}$</p>
<p>I wonder how is it done? I've tried to use Ma... | robjohn | 13,854 | <p>Note that
$$
\begin{align}
\sum_{k=0}^{K-1}\binom{K}{k+1}x^{k+1}(1-x)^{K-1-k}
&=\sum_{k=1}^{K}\binom{K}{k}x^{k}(1-x)^{K-k}\\
&=\left(\sum_{k=0}^{K}\binom{K}{k}x^{k}(1-x)^{K-k}\right)-(1-x)^K\\
&=(x+(1-x))^K-(1-x)^K\\
&=1^K-(1-x)^K
\end{align}
$$</p>
|
76,378 | <p>I am trying to simplify the following expression I have encountered in a book</p>
<p>$\sum_{k=0}^{K-1}\left(\begin{array}{c}
K\\
k+1
\end{array}\right)x^{k+1}(1-x)^{K-1-k}$</p>
<p>and according to the book, it can be simplified to this:</p>
<p>$1-(1-x)^{K}$</p>
<p>I wonder how is it done? I've tried to use Ma... | Tigran Hakobyan | 18,140 | <p>It follows from the Binomial Formula, <a href="http://en.wikipedia.org/wiki/Binomial_theorem" rel="nofollow">http://en.wikipedia.org/wiki/Binomial_theorem</a>.</p>
|
3,372,305 | <blockquote>
<p>When it is allowed to pull a r.v. out of the expectation, i.e.</p>
</blockquote>
<p><span class="math-container">$E(XY|Y)\overset?=YE(X|Y)\tag1$</span>.</p>
<p>or even </p>
<p><span class="math-container">$E(Y^2|Y)\overset?=Y^2\tag2$</span></p>
<p>in the computation rules it is written that if, </... | Bill Moore | 515,499 | <p>This is the derivative you want:</p>
<p><span class="math-container">$$\frac{\partial}{\partial x}\bigg(x^TAx \bigg)=x^T(A+A^T) ~~~~~(1)$$</span></p>
<p>further, If matrix A is symmetric, then <span class="math-container">$A=A^T$</span>. </p>
<p>Thus, for a symmetric matrix (1) becomes:</p>
<p><span class="math... |
813,121 | <p>I have to calculate this : $$ \lim_{x\to 0}\frac{2-x}{x^3}e^{(x-1)/x^2} $$ Can somebody help me?</p>
| Christopher A. Wong | 22,059 | <p>Hint: It may be fruitful to substitute $\alpha = 1/x$, in which case you obtain the limit</p>
<p>$$ \lim_{ \alpha \rightarrow \infty} \left(2 - \frac{1}{\alpha} \right) \alpha^3 e^{\alpha - \alpha^2} $$</p>
<p>I should note that, here, I'm taking your limit to in fact be the limit as $x$ approaches $0$ from the po... |
173,745 | <p>I am trying to prove the following: 'If $(X_1,d_1)$ and $(X_2,d_2)$ are separable metric spaces (that is, they have a countable dense subset), then the product metric space $X_1 \times X_2$ is separable.' It seems pretty straightforward, but I would really appreciate it if someone could verify that my proof works.<... | Francis Adams | 29,633 | <p>Two other thoughts about the answers so far:</p>
<ol>
<li><p>Kevin in his answer gives you a metric that works for the countable product. But, you can actually use $\displaystyle{d(x,y)=\sum\frac{1}{2^i}d_i'(x,y)}$ where $d_i'$ is any metric that is compatible with $d_i$ and bounded by 1. So for example $d_i'(x,y... |
2,237,963 | <p>One-point compactification of $S_{\Omega}$ is homeomorphic with $\bar S_{\Omega}$.</p>
<p>Let $X$ be a topological space. Then the One-point compactification of $X$ is a certain compact space $X^*$ together with an open embedding $c : X \to X^*$ such that the complement of $X$ in $X^*$ consists of a single point, ... | Michael Rozenberg | 190,319 | <p>We'll prove that
<span class="math-container">$$\sin1>\frac{4}{5}$$</span> by hand.</p>
<p>Indeed, let <span class="math-container">$f(x)=\sin{x}-x+\frac{x^3}{6}-\frac{x^5}{120}+\frac{x^7}{5040},$</span> where <span class="math-container">$x\in\left[0,\frac{\pi}{2}\right].$</span></p>
<p>Thus, <span class="math... |
2,812,518 | <p>I am self learning calculus and have a problem that may be simple here but I cant find an answer on the web so here it is:</p>
<p>If we have the equation:</p>
<p>$$y = y^2x$$</p>
<p>and differentiate it (implicitly and using product rule) we get:</p>
<p>$$\frac{dy}{dx} = \frac{y^2}{1-2xy}.$$</p>
<p>However, $y... | egreg | 62,967 | <p>The equation is
$$
y(1-xy)=0
$$
so the curve is the union of the line $y=0$ and of the hyperbola $xy=1$.</p>
<p>The function is locally invertible at every point; the derivative at points with nonzero $y$-coordinate is $y'=-1/x^2$; the derivative at points with zero $y$-coordinate is $y'=0$.</p>
<p>If you do impli... |
164,103 | <p>I have some questions. </p>
<hr>
<p>The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, \mathbb{P}_{U_2}$ be the corresponding Prikry forcings. Let $G\times H$ be $\mathbb{P}_{U_1}\times \mathbb{P}_{U_2... | Yair Hayut | 41,953 | <p>This should be a comment - but it is too long:</p>
<p>Assume that $U = U_1 = U_2$. I want to show that $\mathbb{P}_U ^2 \cong \mathbb{P}_U\times \mathbb{C}$ where $\mathbb{C}$ is the Cohen forcing.</p>
<p>Let $\{ {\alpha^0}_i\}_{i <\omega}, \{ {\alpha^1}_i \}_{i<\omega}$ be the two Prikry sequences. </p>
<p... |
1,706,310 | <p>Suppose X and Y are compact and they are Hausdorff topological spaces. Let $f : X \rightarrow Y$ be a continuous surjective function. Prove that any $U \subset Y$ is open if and only if $f^{−1}(U)$ is open.</p>
<p>I got stuck here because I don't know how to use compactness. Can someone help please?
Thanks</p>
| egreg | 62,967 | <p>Since $f^{-1}(Y\setminus U)=X\setminus f^{-1}(U)$ and a set is open if and only if its complement is open, you can prove the equivalent assertion</p>
<blockquote>
<p>Any $C\subset Y$ is closed if and only if $f^{-1}(C)$ is closed.</p>
</blockquote>
<p>Suppose $C\subset Y$ is closed. Then $f^{-1}(C)$ is closed be... |
1,785,044 | <p>Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side (say $AB$) of the polygon is shorter than the circular arc with the same endpoints ($\stackrel{\frown}{AB}$). Summin... | Jack D'Aurizio | 44,121 | <p>You may use a general fact:</p>
<blockquote>
<p>If $A,B\subset\mathbb{R}^2$ are two convex bounded shapes and $A\subset B$, the perimeter of $A$ is less than the perimeter of $B$.</p>
</blockquote>
<p>Proof: if $A\neq B$, you may "cut out" a slice of $B$ without touching $A$. By convexity, the perimeter of the "... |
1,149,685 | <p>Suppose that 4 guests check their hats when they arrive at a restaurant, and that these hats are returned to them in a random order when they leave. Determine the probability that no guest will receive the proper hat.</p>
<p>My attempt:
So I know that there are 4! ways of "arranging" the hats.
The probability of ea... | user903142 | 903,142 | <p>Pr</p>
<p>n
i=1
Ai</p>
<p>=
n
i=1
Pr(Ai) −</p>
<p>i<j
Pr(Ai
∩ Aj ) +</p>
<p>i<j<k
Pr(Ai
∩ Aj
∩ Ak)
−</p>
<p>i<j<k<l
Pr(Ai
∩ Aj
∩ Ak
∩ Al) + . . .</p>
<ul>
<li>(−1)n+1 Pr(A1 ∩ A2 ∩ . . . ∩ An). We'll use this probability union formula for the solution. Let's assume that a first man will recieve a pr... |
2,441,488 | <p>We are not unfamiliar to imaginary angles, so what can be $\tan(a+ib)$</p>
<blockquote>
<p>If$$\tan(a+ib)=x+iy$$then find $$x,y$$</p>
</blockquote>
<p>My attempt,</p>
<p>Let $x$ and $z$ be two complex bombers such that,
$$\tan(z)=x$$
$$z=\tan^{-1} (x)$$
$$z=\tan^{-1}(re^{i\theta}).$$
But this doesn't seem to ge... | Michael Rozenberg | 190,319 | <p>I think you mean $\{a,b\}\subset\mathbb R$.</p>
<p>If so we have:</p>
<p>$$\tan(a+bi)=\frac{\tan{a}+\tan{bi}}{1-\tan a\tan{bi}}=\frac{\tan{a}+i\frac{e^b-e^{-b}}{e^b+e^{-b}}}{1-i\tan{a}\cdot\frac{e^b-e^{-b}}{e^b+e^{-b}}}=...$$
I hope now it's clear.</p>
<p>We know that $$e^{iz}=\cos{z}+i\sin{z}.$$
For $z=xi$ we ob... |
2,441,488 | <p>We are not unfamiliar to imaginary angles, so what can be $\tan(a+ib)$</p>
<blockquote>
<p>If$$\tan(a+ib)=x+iy$$then find $$x,y$$</p>
</blockquote>
<p>My attempt,</p>
<p>Let $x$ and $z$ be two complex bombers such that,
$$\tan(z)=x$$
$$z=\tan^{-1} (x)$$
$$z=\tan^{-1}(re^{i\theta}).$$
But this doesn't seem to ge... | adfriedman | 153,126 | <p>We get a rather pretty formula in terms of double angles after a fair chunk of work, giving
$$\boxed{\tan(a+ib) = \frac{\sin(2a)+i\sinh(2b)}{\cos(2a) + \cosh(2b)}}$$</p>
<p>The brief gist is just repeated application of the hyperbolic-to-trig identities $\sin(ix) = i\sinh(x)$ and $\cos(ix)=\cosh(x)$, and numerous h... |
1,513,078 | <blockquote>
<p>Suppose $b\in\mathbb Z$. Find all possible remainders of $b^3$ divided by $7$. </p>
</blockquote>
<p>I know $b^6=1\bmod7$ which means $(b^2)^3=1\bmod7$ so all square numbers^3 leave $1$ as a remainder, but how to continue?</p>
| peter.petrov | 116,591 | <p>It doesn't matter if they ask you about $b^3$ or about $b^5$ or about say $b^5+b^2+11$. The method is always the same: if your modulo is N, you just take the numbers $0, 1, 2, ..., N$ and you just put them in the formula (for $b$), and you observe what the outputs (modulo N) are. </p>
<p>The reason for this is th... |
125,399 | <p>How can I solve the following integral?
$$\int_0^\pi{\frac{\cos{nx}}{5 + 4\cos{x}}}dx, n \in \mathbb{N}$$</p>
| draks ... | 19,341 | <p>Evaluating some $n$ (<a href="http://tinyurl.com/c4synlh" rel="nofollow">$n=5$</a>), points at something like $\displaystyle \frac{(-1)^n \pi}{6\cdot 2^{n-1}}$...(tbc)</p>
|
125,399 | <p>How can I solve the following integral?
$$\int_0^\pi{\frac{\cos{nx}}{5 + 4\cos{x}}}dx, n \in \mathbb{N}$$</p>
| robjohn | 13,854 | <p>To elaborate on Pantelis Damianou's answer
$$
\newcommand{\cis}{\operatorname{cis}}
\begin{align}
\int_0^\pi\frac{\cos(nx)}{5+4\cos(x)}\mathrm{d}x
&=\frac12\int_{-\pi}^\pi\frac{\cos(nx)}{5+4\cos(x)}\mathrm{d}x\\
&=\frac12\int_{-\pi}^\pi\frac{\cis(nx)}{5+2(\cis(x)+\cis(-x))}\mathrm{d}x\\
&=\frac12\i... |
125,399 | <p>How can I solve the following integral?
$$\int_0^\pi{\frac{\cos{nx}}{5 + 4\cos{x}}}dx, n \in \mathbb{N}$$</p>
| fjaclot | 635,050 | <p><span class="math-container">$2 I(n+1) + 5I(n) + 2I(n-1) = 0$</span></p>
<p>d’ou <span class="math-container">$I(n) = (-1/2)^n . I(0) = (-1/2)^n . \pi/3$</span></p>
<p>fjaclot</p>
|
13,432 | <p>I want to turn a sum like this</p>
<pre><code>sum =a-b+c+d
</code></pre>
<p>Into a List like this: </p>
<pre><code>sumToList[sum]={a,-b,c,d}
</code></pre>
<p>How can I achieve this?</p>
| J. M.'s persistent exhaustion | 50 | <p>Still another route:</p>
<pre><code>Last[CoefficientArrays[#]] Variables[#] &[a - b + c + d]
{a, -b, c, d}
</code></pre>
|
1,206,528 | <p>Find the matrix $A^{50}$ given</p>
<p>$$A = \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix}$$ as well as for $$A=\begin{bmatrix} 2 & 0 \\ 2 & 1\end{bmatrix}$$</p>
<p>I was practicing some questions for my exam and I found questions of this form in a previous year's paper.</p>
<p>I don't know how to ... | science | 212,657 | <p><strong>Hint:</strong> Diagonalize the matrix you have been given and then take powers.</p>
|
1,206,528 | <p>Find the matrix $A^{50}$ given</p>
<p>$$A = \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix}$$ as well as for $$A=\begin{bmatrix} 2 & 0 \\ 2 & 1\end{bmatrix}$$</p>
<p>I was practicing some questions for my exam and I found questions of this form in a previous year's paper.</p>
<p>I don't know how to ... | achille hui | 59,379 | <p>This is for the updated version of question where $A = \begin{bmatrix}2 & -1\\0 & 1\end{bmatrix}$. For the original version of $A$, the derivation is similar.</p>
<p>The characteristic polynomial for matrix $A$ is</p>
<p>$$\chi_A(\lambda) = \det\begin{bmatrix}\lambda-2 & 1\\0 & \lambda - 1\end{bmat... |
1,721,055 | <p>This afternoon I've been studying the pythagorean identities & compound angles. I've got a problem with a question working with 2 sets of compound angles:</p>
<blockquote>
<p>Solve, in the interval $0^\circ \leq \theta \leq 360^\circ$, $$\cos(\theta + 25^\circ) + \sin(\theta +65^\circ) = 1$$ </p>
</blockquote... | Simply Beautiful Art | 272,831 | <p>Since you've reduced it down to $X\sin(\theta)+Y\cos(\theta)=Z$, the solution may be found with some trigonometric identities:</p>
<p>$$\sin(\theta+\alpha)=\sin(\theta)\cos(\alpha)+\cos(\theta)\sin(\alpha)$$</p>
<p>$$\frac{\sin(\theta+\alpha)}{\cos(\alpha)}=\sin(\theta)+\tan(\alpha)\cos(\theta)$$</p>
<p>Returning... |
186,395 | <p>i want write a module to find the integer combination for a multi variable fomula. For example</p>
<p>$8x + 9y \le 124$</p>
<p>The module will return all possible positive integer for $x$ and $y$.Eg. $x=2$, $y=12$.
It does not necessary be exactly $124$, could be any number less or equal to $124$. Must be as close... | NoChance | 15,180 | <p>Considering the equal case, if you analyze some of the data points that produce integer solution to the original inequality:</p>
<p>$$...,(-34,44), (-25,36), (-16,28), (-7,20),(2,12),(11,4),...$$</p>
<p>you can see that:
$$x_{i}=x_{i-1}+9$$</p>
<p>to get $y$ values, re-write the inequality as follows (and use the... |
1,429,000 | <p>Prove that for any integer $n$, the integer $n^2 + 7n + 1$ is odd.</p>
<p>I have $n=2k+1$ for some $k\in Z$</p>
<p>I really do not how to do this problem. any help in understanding would be greatly appreciated.</p>
| Ittay Weiss | 30,953 | <p>Option 1: plug in $n=2k+1$ into the form of the number you have, expand and show you get something of the form $2\cdot (some integer)+1$. Repeat by plugging in $n=2k$. Option 2: Do you know that the sum of three odd numbers is odd? Do you know that the square of an odd number is odd? Do you know that the product of ... |
1,429,000 | <p>Prove that for any integer $n$, the integer $n^2 + 7n + 1$ is odd.</p>
<p>I have $n=2k+1$ for some $k\in Z$</p>
<p>I really do not how to do this problem. any help in understanding would be greatly appreciated.</p>
| Guest | 102,415 | <p>You can distinguish two cases: $n=2k$ or $n=2k+1$ ($n$ even or odd, respectively) and substituting in $n^2+7n+1$ you should be able to rewrite the result as $2l+1$ for a certain $l$ in both cases.</p>
|
1,429,000 | <p>Prove that for any integer $n$, the integer $n^2 + 7n + 1$ is odd.</p>
<p>I have $n=2k+1$ for some $k\in Z$</p>
<p>I really do not how to do this problem. any help in understanding would be greatly appreciated.</p>
| Eric S. | 263,514 | <p>$$n^2+7n+1=n(n+7)+1.$$</p>
<p>Note that:</p>
<ul>
<li>if $n$ is even, $n+7$ is odd.</li>
<li>if $n$ is odd, $n+7$ is even.</li>
</ul>
<p>Thus $n(n+7)$ is the product of an odd and an even term, which is even. Thus $n(n+7)+1$ is an even term plus $1$, which is <strong>odd</strong>. </p>
|
850,162 | <p>How can i prove that there exist some real $a >0$ such that $\tan{a} = a$ ? </p>
<p>I tried compute $$\lim_{x\to\frac{\pi}{2}^{+}}\tan x=\lim_{x\to\frac{\pi}{2}^{+}}\frac{\sin x}{\cos x}$$ </p>
<p>We have the situation " $\frac{1}{0}$ " which leads us " $\infty$ " </p>
<p>$$\lim_{x\to\frac{\pi}{2}^{-}}\tan x=\... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>You are looking for the intersection of two functions $y_1=\tan(x)$ and $y_2=x$. You also know that $\tan(x)$ has discontinuities at $x=(2k+1) \frac {\pi}{2}$ (to be more precise, as gniourf_gniourf commented, the $\tan$ is not defined at these points). So, you have an infinite number o... |
1,102,310 | <p>For the quadratic function $$-ax^2 + 1$$ an upside down parabola with $y(0) = 1,$ is there a way to compute <em>a</em> such that the definite integral of $y$ between the roots ($x_1, x_2: f(x_1) \land f(x_2)= 0$) equals $1?$</p>
| abel | 9,252 | <p>the parabola is symmetric about $x = 0$ has roots $\pm \dfrac{1}{\sqrt a}$ so the area bounded by the parabola and $y = 0$ is $$\int_{-1/\sqrt a}^{1/\sqrt a}(1-ax^2) dx=2 \int_0^{1/\sqrt a} 1 - ax^2 dx = 2\left[ x - \dfrac{ax^3}{3}\right]_0^{1/\sqrt a}= 2(\dfrac{1}{\sqrt a} - \dfrac{1}{3\sqrt a}) = \dfrac{4}{3 \sqrt... |
2,579,137 | <p>According to the definition of harmonic number $H_n = \sum\limits_{k=1}^n\frac{1}{k}$.</p>
<p>How we can define $H_{n+1}$ and $H_{n+\frac{1}{2}}$? </p>
| xpaul | 66,420 | <p>Note that
$$ H_n=\sum_{k=1}^n\frac{1}{k}=\int_0^1\frac{1-x^n}{1-x}dx. $$
Hence
$$ H_{n+1}=\int_0^1\frac{1-x^{n+1}}{1-x}dx $$
and
$$ H_{n+1/2}=\int_0^1\frac{1-x^{n+1/2}}{1-x}dx. $$</p>
|
2,118,266 | <p>In Order Theory, what is the exact definition of a dual statement? And what is the duality principle for posets/lattices? I haven't been able to find an exact statement or definition in this regard.</p>
| William DeMeo | 10,915 | <p>Here's an example that might help you see what Hagen meant by "translates" in his answer.</p>
<p>Let $X = \{a,b,c\}$. Let $\leq$ denote the binary relation $\{(a,a), (a,b), (b,b), (c,b), (c,c)\}$. Then $\langle X, \leq \rangle$ is a poset, and the nontrivial pairs in the $\leq$ relation are $a\leq b$ and $c\leq b... |
282,780 | <p>Let $R = k[x_1, \ldots, x_n]$ for $k$ a field of characteristic zero and let $S \subset R$ be a graded sub-$k$-algebra (for the standard grading: $\deg x_i = 1$) such that $R$ is a free $S$-module of finite rank. Does this imply $S \cong k[y_1,\ldots,y_n]$?</p>
| David E Speyer | 297 | <p>It turns out that this result appears in Bourbaki, <a href="https://link.springer.com/chapter/10.1007/978-3-540-34491-9_3" rel="nofollow noreferrer">Groupes et algebres de Lie IV-VI</a>, Ch. 5, $\S$ 5, Lemme 1, with an elementary proof. (However, this proof uses characteristic zero, which Neil's answer does not.) A... |
921,868 | <p>I'm not quite sure whether this question belongs here, because it has no definite answer. But I'll give it a shot. If any of the mods objects, then I will, of course, respectfully delete this contribution.</p>
<p>A friend of mine, who knows a little arithmetic and only the bare rudiments of algebra, has asked me to... | James | 751 | <p>Something easily grasped by pretty much anyone are problems related to covering chessboards with dominoes. (I'm especially fond of these because learning about them was what initially got me interested in mathematics.)</p>
<p>A domino exactly covers two squares of chessboard. Can you always cover an $8\times 8$ ch... |
921,868 | <p>I'm not quite sure whether this question belongs here, because it has no definite answer. But I'll give it a shot. If any of the mods objects, then I will, of course, respectfully delete this contribution.</p>
<p>A friend of mine, who knows a little arithmetic and only the bare rudiments of algebra, has asked me to... | Frunobulax | 93,252 | <p><a href="http://weitz.de/math/" rel="nofollow noreferrer">Circle division by chords</a> is also a nice one IMHO. It is easy to explain, it shows that you shouldn't make assumptions too early, and the proof of the correct formula can be understood with high school algebra.</p>
<p><img src="https://i.stack.imgur.com... |
2,385,188 | <p>$c_R,R_f$ are known (constant part of return and risk free rate). Let $R=R_f(e^r-1)=c_R+\epsilon_R$, $r\sim N(\mu_r,\sigma_r)$, how to specify the distribution of $\epsilon_R\sim Lognormal(?,?)$ by mean and variance of a normal distribution?</p>
| DanielWainfleet | 254,665 | <p>$(P\implies Q)$ merely says that we cannot have $P$ true and $Q$ false. In other words, $(P\implies Q)$ is equivalent to $\sim (P\land \sim Q).$ </p>
|
2,385,188 | <p>$c_R,R_f$ are known (constant part of return and risk free rate). Let $R=R_f(e^r-1)=c_R+\epsilon_R$, $r\sim N(\mu_r,\sigma_r)$, how to specify the distribution of $\epsilon_R\sim Lognormal(?,?)$ by mean and variance of a normal distribution?</p>
| Janitha357 | 393,345 | <p>Suppose I tell you "If it rains in the evening then I will meet you." Then there are four possibilities (Look at the truth table). The only instance my statement is false is when it actually rains and I don't meet you. But my statement will still be true when it doesn't rain in the evening but I come and meet you, a... |
43,640 | <p>Consider the following problem:</p>
<p>Let ${\mathbb Q} \subset A\subset {\mathbb R}$, which of the following must be true?</p>
<p>A. If $A$ is open, then $A={\mathbb R}$</p>
<p>B. If $A$ is closed, then $A={\mathbb R}$</p>
<p>Since $\overline{\mathbb Q}={\mathbb R}$, one can immediately get that B is the answer... | t.b. | 5,363 | <p>A slightly more interesting example than Luboš's can be obtained by enumerating the rationals as $\mathbb{Q} = \{q_n\}_{n=1}^\infty$ and taking $A = \bigcup_{n=1}^{\infty} (q_{n} - \frac{\varepsilon}{2^{n+1}}, q_{n} + \frac{\varepsilon}{2^{n+1}})$. Then the Lebesgue measure of $A$ can be estimated by $\mu(A) \l... |
519,560 | <p>I need prove that:</p>
<p>$$\int_{0}^{2\pi} \frac{R^{2}-r^{2}}{R^{2}-2Rr\cos \theta +r^{2}} d\theta= 2\pi$$</p>
<p>By deformation theorem, with $0<r<R$.</p>
<p>Professor gave us the hint to use the function $f(z)= \frac{R+z}{z(R-z)}$, and define an adequate $\gamma : [a,b]\rightarrow \mathbb{C}$ circular cu... | Mhenni Benghorbal | 35,472 | <p><strong>Hint:</strong> Let </p>
<p>$$z=e^{i\theta}\implies dz= i e^{i\theta} d\theta , $$</p>
<p>then the new integral is</p>
<p>$$ \int_{|z|=1} \frac{R^{2}-r^{2}}{R^{2}-Rr(z+1/z) +r^{2}} \frac{dz}{iz}.$$</p>
<p>Now, you need to use residue theorem.</p>
|
2,104,382 | <p>I was trying to understand the proof of a theorem from Munkres's book.
He wants to prove that a compact subset $Y$ of an Hausdorff space $X$ is closed. He simply proves that, for all $x\in X-Y$, there is a neighborhood which is disjoint from $Y$. Why is this sufficient to conclude that $X-Y$ is open?</p>
| m.s | 259,136 | <p>Because you can then write $X-Y$ as a union of open sets: Choose for every $x \in X-Y$ an open $U_x$ containing $x$, which is disioint form $Y$. Then $X-Y = \bigcup_{x\in X-Y}{U_x}$ is a union of opens, hence open.</p>
|
497,687 | <p>How would I integrate the following.</p>
<p>$$
\int_{\ln2}^{\ln3}\frac{e^{-x}}{\sqrt{1-e^{-2x}}}\,dx
$$ </p>
<p>I think I have to use the $\arcsin(x)$ formula.</p>
<p>Which means would I use $y=u^2$ with $u=e^{-x}$ but that does not seem to work.</p>
| peterwhy | 89,922 | <p>Let $\sin t = e^{-x}$. Then $\cos t \cdot dt = -e^{-x}dx$.
$$\begin{align}
\int^{\ln3}_{\ln2}\frac{e^{-x}dx}{\sqrt{1-e^{-2x}}}
=& \int^{\arcsin\frac{1}{3}}_{\arcsin\frac{1}{2}}\frac{-\cos t \cdot dt}{\sqrt{1-\sin^{2}t}}\\
=& \int^{\arcsin\frac{1}{3}}_{\arcsin\frac{1}{2}}\frac{-\cos t \cdot dt}{\cos t}\\
=&am... |
991,569 | <p>\begin{align}
x-\hphantom{2}y+2z+2t&=0 \\
2x-2y+4z+3t&=1 \\
3x-3y+6z+9t&=-3 \\
4x-4y+8z+8t&=0
\end{align}</p>
<p>Solve for x,y,z and t</p>
<p><img src="https://i.stack.imgur.com/y3R7x.jpg" alt="reduced row echelon form"></p>
| Antony | 180,827 | <p>$2x-2y+4z+4t=0$<br>
$2x-2y+4z+3t=1$<br>
Then we get that t=-1 .
And after that all equations become :
<br>$x-y+2z=2$<br> So we have $t=-1$ and $x=2+y-2z$ , where $y$ and $z$ can be everything</p>
|
455,302 | <p>Given: $x + \sqrt x = \sqrt 3$<br>
Evaluate: $x^3 - 1 / x^3$</p>
| mnsh | 58,529 | <p>Hint : </p>
<p>put $x=y^2$ and solve it by quadratic formula</p>
|
87,544 | <p>For teaching purposes, I want to create a <em>Mathematica</em> notebook that will "notice" when the user defines or redefines a variable or function of a particular name, so that it can check the value and take some appropriate action. For example, the notebook might be monitoring the symbol "foo", so that if the us... | Karsten 7. | 18,476 | <p>Something, that more or less does what you asked for, can be achieved by creating a hidden <code>InitializationCell</code> using a <a href="http://reference.wolfram.com/language/ref/DynamicWrapper.html" rel="nofollow"><code>DynamicWrapper</code></a></p>
<pre><code>DynamicWrapper["xxx",
If[foo == 23, MessageDialog... |
1,654,354 | <p>Why is $A=\{(x_1,x_2,...,x_n)|\exists_{i\ne j}: x_i=x_j\}$ a null set?</p>
<p>This claim was shown in a solution I ran into, and I don't see how it holds. I try to follow the formal definition of nullity, which is having the possibility to be covered by a collection of open cubes, whose volume is as small as desire... | saz | 36,150 | <p>As suggested in a comment, the claim follows if we can show that</p>
<p>$$A_{ij} := \{(x_1,\ldots,x_n) \in \mathbb{R}^n; x_i = x_j\}$$</p>
<p>is a null set for all $i \neq j$. Without loss of generality, I'll assume that $i=1$, $j=2$.</p>
<p>Since the countable union of null sets is again a null set, it suffices ... |
82,865 | <p>I've looked through the other posts on rotating a function around the x or y axis and haven't had any luck.</p>
<p>my question is this how do i rotate the area between the functions $f(x)=x^2+1$, and $f(x)=2x^2-2$, whereby $x>0$ and $y>0$ and the region is bounded by the y and x axis.</p>
<p>This is a plot o... | m_goldberg | 3,066 | <p>Not an answer but an extended comment on belisarius' answer.</p>
<p>I think the plot will look more like what I believe the OP is expecting if a few options are added.</p>
<pre><code>Show[RevolutionPlot3D[#, {x, 0, Sqrt[3]},
PlotRange -> {Automatic, Automatic, {0., Automatic}},
MeshStyle -> None,
Axes ... |
1,635,682 | <p>I have the functions
$$ f_n(x) = x + x^n(1 - x)^n $$</p>
<p>that $\to x$ as $n \to \infty $ (pointwise convergence).</p>
<p>Now I have to look whether the sequence converges uniformly, so I used the theorem and arrived at:
$$ \sup\limits_{x\in[0,1]}|f_n(x)-x| = |x+x^n(1-x)^n - x| = |x^n(1-x)^n| = 0, n \to \infty $... | BrianO | 277,043 | <p>You can express "$x$ loves everyone" with the formula $\forall y\, Loves(x,y)$, assuming, as we will in this example, that variables range over people only. You can express "if $x$ loves everyone then $x$ is good" with the formula $\forall y\, Loves(x,y) \to Good(x)$. [<em>Note: the scope of $\forall y$ is <strong>o... |
8,756 | <p>I'd like to suggest that after a Question receives an Accepted Answer, some consideration be given to revising the title (if appropriate) to reflect what the real issue turned out to be.</p>
<p>It seems to me users often pick titles when first posting a question that are uninformative and are worth revisiting once ... | MJD | 25,554 | <p>I am going to use this post to catalog words that should generally not appear in question titles:</p>
<ul>
<li><a href="https://math.stackexchange.com/search?q=+title%3Aanyone">anyone</a> (200 results)</li>
<li><a href="https://math.stackexchange.com/search?q=+title%3Adifficult">difficult</a> (<s>59</s> 185 results... |
2,037,428 | <p>Let $D$ be a compact, connected $Jordan$ domain in $R^n$ with positive volume, and suppose that the fuction $f:D →R$ is continuous. Show that there is a point $x$ in $D$ in which
$f(x)=(1/volD)$$\int$$f$ over $D$.
(Mean Value Property for integral)</p>
<p>I can prove it when $D$ is pathwise connected</p>
<p>But ... | Daniel | 391,594 | <p>Explaining better the second question, notice that since $D$ is a compact connected subset of $\mathbb{R}^d$ its image $f(D)$ through continuous function $f$ is also compact and connected. Thus, we have $f(D) = [\min_D f, \max_D f]$
or in other words:
$$ \min_D f \le f(x) \le \max_D f, \text{for every $x \in D$.}$$... |
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