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 |
|---|---|---|---|---|
1,728,662 | <p>Each set has bijective ordinal and cardinality is defined as the least one of such ordinals. I know that a set of ordinals is well-ordered by $\subseteq$ (inclusion) and thus has $\subseteq$-least element. However, I wonder which axiom guaranteed that the bijective ordinals above really construct to be a set. I worr... | BrianO | 277,043 | <p>Given a set $X$, by the Axiom of Choice (AC) there's some ordinal $\alpha$ that's bijectable with $X$. </p>
<p>By AC, $\mathcal{P}(\alpha)$ (the powerset of $\alpha$) is bijectible with some ordinal $\beta$. Now, $\alpha$ can be injected into $\beta$, but by Cantor's theorem, $\beta$ can't be injected into $\alpha... |
1,786,421 | <p>I have the following equality to prove. </p>
<p>Given $X \sim Bin(n, p)$ and $Y \sim Bin(n, 1 - p)$ prove that $P(X \leq k) = P(Y \geq n - k)$. I have been trying to come up with a solution but cannot find one. I am looking for suggestions and not a complete answer as this is a homework question.</p>
<p>What I did... | André Nicolas | 6,312 | <p>Hint: Alicia flips a biased coin $n$ times. The coin has probability $p$ of landing heads. Let $X$ be the number of heads she gets, and $Y$ the number of tails.</p>
|
1,650,277 | <p>Does the category of partial orders have a subobject classifier? (Edit: No, see Eric's answer.)</p>
<p>If not, what is a category which is "close" to the category of partial orders (e.g. it should consists of special order-theoretic constructs) and has a subobject classifier? Bonus question: Is there also such an e... | David | 297,532 | <p>For all $\alpha$, we have $X \cap X_{\alpha} = X_{\alpha} \in \tau_{\alpha}$.</p>
|
551,252 | <p>I'm sure I've made a trivial error but I cannot spot it.</p>
<p>Fix R>0
Consider the cube $C_R$ as the cube from (0,0,0) to (R,R,R) (save me from listing the 8 vertices) </p>
<p>Consider $S_R$ as the surface of $C_R$</p>
<p>Consider the vector field $v:\mathbb{R}^3\rightarrow\mathbb{R}^3$
given by $v(x,y,z) = (3x... | Robert Israel | 8,508 | <p>For example, the integral over the side $z=R$ is $0$, while the integral over
$z=0$ is $-R^3$ (integrating $-R$ over a square of side $R$).</p>
|
158,451 | <p>Suppose that the contents of an urn are $w$ red balls, $x$ yellow balls, $y$ green balls, and $z$ blue balls collectively, where $w \geq 3$, $x\geq 1$, $y\geq 1$, and $z\geq 1$. We draw balls randomly from this urn without replacement.</p>
<p>What is the probability of our having drawn at least 1 yellow ball by (an... | Community | -1 | <p>This follows from Cauchy–Schwarz inequality. The Cauchy–Schwarz inequality states that for any two vectors $a$ and $b$ in an inner product space, we have that
$$\lvert \langle a, b \rangle \rvert^2 \leq \lvert \langle a, a \rangle \rvert \lvert \langle b, b \rangle \rvert$$
In your case, the vector $a$ is taken as $... |
27,951 | <p>Something I notice is when there's an advanced/specialized question, it often receives very few upvotes. Even if it is seemingly well written. I try to upvote advanced questions <strong>that I might not even understand</strong>, if they appear well written. </p>
<p>Is this good behaviour? Should we encourage upvoti... | samerivertwice | 334,732 | <p>Not a perfect remedy for precisely this issue, but a nice feature that would help with this and bring other benefits for the community, would be to give a few users exceeding some reputation threshold or some maths skill threshold (e.g. holding certain badges) limited numbers of say $+5$ votes.</p>
<p>Not sure how ... |
2,861,443 | <p>Let $C_c(\mathbb{R})$ be the following:</p>
<p>$$C_c = \{ f \in C(\mathbb{R}) \mid \exists \text{ } T > 0 \text{ s.t. } f(t) = 0 \text{ for } |t| \geq T\}$$</p>
<p>Let $T_n \in L(C_c(\mathbb{R}))$ be a linear operator such that:</p>
<p>$$T_n u = \delta_n *u, \forall u \in C_c(\mathbb{R}),$$</p>
<p>where</p>
... | Kelvin Lois | 322,139 | <p>I guess the theorem that you mention is Proposition 3.18 on Lee's book.</p>
<p>For any smooth chart $(U_{\alpha},\varphi_{\alpha})$ of $M$, we have a map $\widetilde{\varphi}_{\alpha} : \pi^{-1}(U_{\alpha}) \to \Bbb{R}^{2n}$ defined as
$$
\widetilde{\varphi}_{\alpha} (v_p) = (x^1(p),\dots,x^n(p),v^1,\dots,v^n) .
$$... |
1,494,167 | <p>Using only addition, subtraction, multiplication, division, and "remainder" (modulo), can the absolute value of any integer be calculated?</p>
<p>To be explicit, I am hoping to find a method that does not involve a piecewise function (i.e. branching, <code>if</code>, if you will.)</p>
| ASKASK | 136,368 | <p>I'm going to go with no.</p>
<p>My (not super rigorous) proof would be that any function of the variable $x$ using only addition, subtraction, multiplication, and division would have to be a differentiable function on its domain, but $|x|$ is not a differential function on all of its domain, and its domain is all o... |
2,891,444 | <p>For the intersection of two line segments, how was it know to use the determinants shown <a href="http://mathworld.wolfram.com/Line-LineIntersection.html" rel="nofollow noreferrer">here</a>? </p>
<p>I'm trying to determine how it was shown that they could be used to compute the intersection point.</p>
| Deepesh Meena | 470,829 | <p>Suppose both proof readers missed the error the probability of this event is$$ = \frac{2}{100}\cdot \frac{2}{100}=\frac{4}{10000}=0.04 \% $$</p>
<p>thus the probability of both working independently and detecting the error is $100-0.04=99.96\%$</p>
|
524,073 | <p>Hey can some help me with this textbook question</p>
<p>Let $R^{2×2}$ denote the vector space of 2×2 matrices, and let</p>
<p>$S =\left\{
\left[\begin{matrix}
a \space b \\
b \space c \\
\end{matrix}\right]\mid a,b,c \in \mathbb{R}\right\}$</p>
<p>Find (with justication) a basis for $S$ and determine the dimensi... | bradhd | 5,116 | <p>Observe that</p>
<p>$$S = \{\left[\begin{array}{cc}a&b\\b&c\end{array}\right]: a,b,c\in\mathbb{R}\}$$
$$ = \{\left[\begin{array}{cc}a&0\\0&0\end{array}\right]+\left[\begin{array}
{cc}0&b\\b&0\end{array}\right]+\left[\begin{array}{cc}0&0\\0&c\end{array}\right]: a,b,c\in\mathbb{R}\}$$
... |
1,748,542 | <p>So a friend of mine has a little project going, and needs some help.</p>
<p>Basically, we want to create a function that takes two variables; One $X$, and one that we call $DC$ ("Difficulty Class, as this is for a pen-and-paper game).</p>
<p>The output should be $0$ if $X\leq (1/2)DC$, and it should be $100$ if $X... | Doug M | 317,162 | <p>How about:</p>
<p>$y = \begin{cases} 0,&x\leq \frac{DC}{2}\\\frac{100}{1.5^3}\times(\frac{X}{DC}-\frac{1}{2})^3,&\frac{DC}{2}<X\leq 2\times DC\\ 100,&X>2\times DC \end{cases}$</p>
|
392,020 | <p>It is well known that the minimum number of monochromatic triangles in a red/blue coloring of the edges of the complete graph <span class="math-container">$K_n$</span> is given by Goodman's formula
<span class="math-container">$$M(n)=\binom n3-\left\lfloor\frac n2\left\lfloor\left(\frac{n-1}2\right)^2\right\rfloor\r... | RobPratt | 141,766 | <p>Here are results, obtained via integer linear programming, for the first question for <span class="math-container">$n \le 10$</span> and <span class="math-container">$b$</span> blue edges, where <span class="math-container">$b \le \binom{n}{2}/2$</span>:
<span class="math-container">\begin{matrix}
n\backslash b &... |
3,857,698 | <p>Let <span class="math-container">$D_1, ..., D_n$</span> be arbitrary <span class="math-container">$n$</span> sets where <span class="math-container">$D_i \cap D_j \neq \emptyset$</span>. In the simplified case where <span class="math-container">$n = 2$</span>, we have that
<span class="math-container">$$
\begin{spli... | Thomas | 284,057 | <p>From the point of view of group theory there is a deep reason : the group of similitudes ( ratio-of-lengths preserving maps) of a (Euclidean) plane is isomorphic to the group of affine (or anti-affine) transformation of a complex line <span class="math-container">$(z\to az+b$</span> or <span class="math-container">... |
3,021,631 | <p>I've been strongly drawn recently to the matter of the fundamental definition of the exponential function, & how it connects with its properties such as the exponential of a sum being the product of the exponentials, and it's being the eigenfunction of simple differentiation, etc. I've seen various posts inwhich... | AmbretteOrrisey | 613,228 | <p>Maybe it's not <em>too</em> bad actually. What what needs to be shown reduces to is that <span class="math-container">$$\forall m\in ℕ_1\&n\in ℕ_1 $$</span><span class="math-container">$$\sum_{p\in ℕ_0,q\in ℕ_0,r\in ℕ_0,p+r=m,q+r=n}\frac{(-1)^{p+q+r}(p+q+r-1)!}{p!q!r!} =0 ,$$</span> whence that</p>
<p><span cla... |
197,393 | <p>Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?</p>
| JSCB | 25,841 | <p><strong>Proof without word</strong></p>
<p>$\tan^{-1} 1+\tan^{-1} 2+\tan^{-1} 3 =\pi$.</p>
<p><img src="https://i.stack.imgur.com/Cvd3u.png" alt=""></p>
|
197,393 | <p>Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?</p>
| Tryst with Freedom | 688,539 | <p>Consider, <span class="math-container">$z_1= \frac{1+2i}{\sqrt{5}}$</span>, <span class="math-container">$z_2= \frac{1+3i}{\sqrt{10} }$</span>, and <span class="math-container">$z_3= \frac{1+i}{\sqrt{2} }$</span>, then:</p>
<p><span class="math-container">$$ z_1 z_2 z_3 = \frac{1}{10} (1+2i)(1+3i)(1+i)=-1 $$</span... |
88,880 | <p>In a short talk, I had to explain, to an audience with little knowledge in geometry or algebra, the three different ways one can define the tangent space $T_x M$ of a smooth manifold $M$ at a point $x \in M$ and more generally the tangent bundle $T M$:</p>
<ul>
<li>Using equivalent classes of smooth curves through ... | Paul Siegel | 4,362 | <p>What all three definitions have in common is that they each try to capture the first order behavior of a smooth function on $M$.</p>
<ol>
<li><p>The derivative of a smooth function $f$ along a curve $\gamma$ with $\gamma(0) = p$ depends on $\gamma$ only insofar as it depends on $\gamma'(0)$, and indeed it recovers ... |
2,576,466 | <p>One says a bounded $f$ is Riemann integrable on [a,b] if the Upper and lower Riemann integrals are equal. Another sufficient condition for Riemann integrablity is that the set of discontinuity of $f$ must be countable set. The following function is continuous only at $x=1/2$ and so the set of discontinuity of $f$ is... | Przemysław Scherwentke | 72,361 | <p>HINT: The upper Riemann integral is the integral of $\max(x,1-x)$ on $[0,1]$ and the lower: $\min(x,1-x)$. (Why?)</p>
|
1,041,134 | <p>I need to show if $a$ is in $\mathbb{R}$ but not equal to $0$, and $a+\dfrac{1}{a}$ is integer, $a^t+\dfrac{1}{a^t}$ is also an integer for all $t\in\mathbb N$.
Can you provide me some hints please?</p>
| Steven Alexis Gregory | 75,410 | <p>Let $T_n = r^n + \dfrac{1}{r^n}$, $T_0 = 2$, and $T_1 = \alpha$</p>
<p>\begin{align}
r + \dfrac 1r &= \alpha \\
r &= \alpha - \dfrac 1r \\
\dfrac 1r &= \alpha - r\\
\hline
r^{n+2} &= \alpha r^{n+1} - r^n \\
\dfrac{1}{r^{n+2}} &= \dfrac{\alpha}{r^{n+1}} - \dfrac{1}{r^n} \\
r^{n+... |
139,135 | <p>Suppose $R = \mathbb{Q}[x_1, ..., x_n]/I$, and $J \subset R$ is a given height one ideal. Is there a quick algorithm one could write to determine if $J$ is a principal ideal or necessarily not principal? Is it not possible to do this with Groebner bases?</p>
| Karl Schwede | 3,521 | <h2>Locally principal</h2>
<p>At least in height-1 ideal case, in a normal domain (or at least G1+S2 domain), the following should let you know whether the ideal is <em>locally</em> principal. </p>
<p>Let $J$ be the ideal in question and let $J_2$ be another ideal isomorphic to $Hom_R(J, R)$ (which you can do in a n... |
1,682,961 | <p>I was making a few exercises on set proofs but I met an exercise on which I don't know how to start:</p>
<blockquote>
<p>If $A \cap C = B \cap C $ and $ A-C=B-C $ then $A = B$</p>
</blockquote>
<p>Where should I start? Should I start from $ A \subseteq B $ or should I start from this $ ((A\cap C = B\cap C) \land... | Akiva Weinberger | 166,353 | <p>\begin{align}
A\cap C&=B\cap C\\
A-C&=B-C
\end{align}
Take the union of the left- and right-hand sides:
\begin{align}
(A\cap C)\cup(A-C)&=(B\cap C)\cup(B-C)\\
(A\cap C)\cup(A\cap\overline C)&=(B\cap C)\cup(B\cap\overline C)\\
A\cup(C\cap\overline C)&=B\cup(C\cap\overline C)\\
A&=B
\end{align}... |
293,234 | <p>I recently asked a question about <a href="https://math.stackexchange.com/questions/287116/proof-that-mutual-statistical-independence-implies-pairwise-independence">pairwise versus mutual independence</a> (also related to <a href="https://math.stackexchange.com/questions/281800/example-relations-pairwise-versus-mutu... | Community | -1 | <p>Let $K = \mathbb{Q}(\alpha^2, \beta^2, \gamma^2, \delta^2) = \mathbb{Q}(\sqrt[4]{2}, i)$ be the splitting field of $t^4 - 4 t^2 + 8t + 2$. Then $[K:\mathbb{Q}] = 8$ and $L/K$ is a Kummer extension formed by adjoining four square roots.</p>
<p>As per Kummer theory, we are thus interested in the subgroup of $K^*$ mod... |
146,075 | <p>Within my limited experience, I have only known free groups to occur through two mechanisms: as fundamental groups of trees (graphs) and ping-pong. And sometimes only through one way: the fact that sufficiently high-powers of hyperbolic elements in a Gromov-hyperbolic group generate a free group arises via ping-pong... | Andy Putman | 317 | <p>A theorem of Leininger-Margalit says that any two elements of the pure braid group either commute or generate a free group; see <a href="http://www.math.uiuc.edu/~clein/pbrels.pdf">here</a>. But I don't think there is any hope for classifying free subgroups that are generated by more than $2$ elements.</p>
|
2,835,290 | <p>I am given two variables $Y_1$ and $Y_2$ obeying an exponential distribution with mean $\beta= 1$</p>
<p>We are asked what the distribution of their average is and the solution must be found using moment generating functions.</p>
<p>The solution to this exercise says:</p>
<p><a href="https://i.stack.imgur.com/eOi... | xpaul | 66,420 | <p>Under $x\to\sin x$, one has
$$I=\int_{0}^{1}\ln\left(\frac{a-x^2}{a+x^2}\right)\cdot\frac{\mathrm dx}{x^2\sqrt{1-x^2}}=\int_{0}^{\pi/2}\ln\left(\frac{a-\sin^2x}{a+\sin^2x}\right)\cdot\frac{\mathrm dx}{\sin^2x}.$$
Let
$$I(k)=\int_{0}^{\pi/2}\ln\left(\frac{a-k\sin^2x}{a+k\sin^2x}\right)\cdot\frac{\mathrm dx}{\sin^2x}$... |
1,270,584 | <p>I tried googling for simple proofs that some number is transcendental, sadly I couldn't find any I could understand.</p>
<p>Do any of you guys know a simple transcendentality (if that's a word) proof?</p>
<p>E: What I meant is that I wanted a rather simple proof that some particular number is transcendental ($e$ o... | marty cohen | 13,079 | <p>There is a BIG difference
between showing that
a particular number
that naturally occurs
(like $e$ or $\pi$)
is transcendental
and showing that
some number is.</p>
<p>The existence of transcendental
numbers was first shown in 1844
by Liouville. In 1851 he proved that
$\sum_{k=1}^{\infty} \frac1{10^{k!}}
$ is transc... |
1,282,420 | <p>In his book of Differential Topology, Hirsch starts a little detour in his theory in order to present a way to see things in a general perspective. The precise fragment I'm referring to is the following:</p>
<p><img src="https://i.stack.imgur.com/rfzFr.png" alt="enter image description here"></p>
<p>I have never e... | oxeimon | 36,152 | <p>Perhaps this will help.</p>
<p>Basically an inverse limit is like this.</p>
<p>Fix a category (for example the category of sets, groups, rings, topological spaces, vector spaces). It's easy to define the inverse limit if your objects have an underlying "set", upon which possibly additional structure is imposed. (i... |
156,769 | <p>Let $\{a_{n}\}$ be a sequence of real numbers, where $0<a_{n}<1$, such that $\lim_{n\to \infty} a_{n}=0$, (then every subsequence will converges to zero). Is there any way to find a subsequence of $a_{n}$ which is decreasing to 0?</p>
| Arturo Magidin | 742 | <p>Yes: since the sequence converges to $0$, for every $\epsilon\gt 0$ there exists $N\in\mathbb{N}$ such that for all $n\geq N$, we have $0\lt a_n\lt\epsilon$.</p>
<p>So define the sequence recursively: take $a_1$. Then let $\epsilon = \frac{a_1}{2}$; we know there is an $n_1\gt 1$ such that $a_{n_1}\lt \frac{a_1}{2}... |
2,885,453 | <p>Evaluate $$\lim _{x \to 0} \left[{\frac{x^2}{\sin x \tan x}} \right]$$ where $[\cdot]$ denotes the greatest integer function.</p>
<p>Can anyone give me a hint to proceed?</p>
<p>I know that $$\frac {\sin x}{x} < 1$$ for all $x \in (-\pi/2 ,\pi/2) \setminus \{0\}$ and $$\frac {\tan x}{x} > 1$$ for all $x \in ... | jim | 289,829 | <p>Can you just use the known expansions for $\sin x, \tan x$ for small $x$? Then have $$\frac{x^2}{\sin x \tan x} = \frac{x^2}{(x - \frac{x^3}{6} + \cdots)(x + \frac{x^3}{3} + \cdots)}.$$ The <em>rhs</em> only including terms up to $x^2$ can be written $$\frac{1}{(1 - x^2/6)(1 + x^2/3)}$$ and the denominator to order ... |
2,309,613 | <p>Find a general solution to $x^2-2y^2=1$</p>
<p>I found that (3,2) is a solution.
Now what should I do?
I can not catch what the question really want.</p>
<p>It is about pell's equation. Would you give me a form of general solution? </p>
| fonfonx | 247,205 | <p><strong>Hints</strong></p>
<p>I would use the fact that $f$ is equal to its Taylor series on $G$ to write that $$f(z)=\sum_{n=0}^{\infty} a_n z^n$$</p>
<p>Then using the fact that $f(z) \in \mathbb R$ for $z \in \mathbb R$ you should be able to prove that the $a_n$ are real.</p>
<p>Using the fact that $f(z) \in i... |
125,862 | <p>I have been given that I am working with the space of all 2x2 matrices. The basis $B$ for this space is given as a set of four 2x2 matrices, each with an entry of 1 in a unique position and zeroes everywhere else (sorry about the description in words - I don't know how to format matrices for this site).</p>
<p>I h... | geo909 | 27,882 | <p>You essentially have two ways of representing a Linear Tranformation (say, $T$ from now on):</p>
<ol>
<li>Using a "formula" or a kind of description (e.g. "the transpose")</li>
<li>Using a matrix (which depends on the basis that we choose; see below)</li>
</ol>
<p>In the second case, when you want to evaluate $T(u... |
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... | algori | 2,349 | <p>There are $n$ balls rolling along a line in one direction and $k$ balls rolling along the same line in the opposite direction. The speeds of the balls in the first group and in the second group are equal. Initially the two groups of balls are separated from one another and at some point the balls start colliding. Th... |
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... | Qiaochu Yuan | 290 | <p>Adam Hesterberg told me this one ages ago. It apparently used to circulate around MOP. </p>
<p>Three spiders and a fly are placed on the edges of a regular tetrahedron, and travel only on those edges. The fly travels at the rate of $1$ edge/s, whereas the spiders travel at the rate of $1 + \epsilon$ edge/s for so... |
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... | Roland Bacher | 4,556 | <p>Not a very difficult one but I like it since it is even suitable for non-mathematicians: </p>
<p>A small boat carrying a heavy stone is floating in a swimming pool. What happens to the level of water (up, down or remains equal) in the swimming pool if one removes the stone from the boat and throws it in the swimmin... |
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... | Gerald Edgar | 454 | <p>I had a good outcome with this one once. It probably helped that the other two mathematicians had a drink or two before dinner. (Otherwise they would have solved it in 5 seconds...) When salad was served, somebody had oil and vinegar in separate little pitchers...</p>
<p>Suppose you have two containers, one with... |
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... | Alex R. | 934 | <p>Bob and Alice want to marry each other, so Bob decides to send Alice a ring. The problem is that they both live in different countries, and any valuables they send through the mail are sure to be stolen, unless they are sent in a locked box. The box can be locked by a padlock which can only be opened by the right ke... |
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... | Douglas S. Stones | 2,264 | <p>What is four thousand and ninety-nine plus one?</p>
|
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... | Thierry Zell | 8,212 | <p>Simple puzzles; unfortunately, I do not know how to formulate them in a whimsical fashion suitable for a dinner, they very much sound like math puzzles.</p>
<ol>
<li><p>Take n labeled points $x_1, \dots, x_n$ in the plane. How do you construct a n-gon $a1, \dots, an$ such that for all i, $x_i$ is the midpoint of $... |
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... | Jesus Martinez Garcia | 1,887 | <p>I think this has not been published yet. Apologies otherwise. I learnt it from Antonio Sánchez Calle in my first year of undergraduate and I had 3 non-mathematicians thinking about it for about 4 hours, so there is a guaranteed success if you tell around :)</p>
<p>5 people are shipwrecked in a deserted island. They... |
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... | Seven | 135,088 | <p>I disagree that "scalar multiplication is only a nice-to-have shortcut", or that is "superfluous conceptually". In fact the very definition of a vector space $V$ requires there to be a scalar multiplication.</p>
<p>After that comes the concept of a linear transformation, which again requires the scalar multiplicati... |
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... | OA Fakinlede | 65,652 | <p>I am limiting myself to fields defined in the three dimensional Euclidean point space. The curl of a tensor can be found in these simple steps:</p>
<ol>
<li><p>Take the simple composition of the second-order tensor, <code>T</code>, with the <code>LeviCivitaTensor[3]</code>. This is effected by the command, </p>
<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$$
th... | Community | -1 | <p>(note: this answer was formulated in response to the original version of the question, which has since been edited by a third party)</p>
<p>You're parsing the expression wrong; it's not</p>
<p>$$ \color{red}{\mathbf{ f : x }} \mapsto 2x-3 $$</p>
<p>instead, it is </p>
<p>$$ f : \color{red}{\mathbf{ x \mapsto 2x... |
915,414 | <p>I recently did some work to try to find $\int{\frac{dx}{Ax^3 - B}}$, but I'm always paranoid that my solution has some minor trivial error in the middle of the process that screwed up the end result entirely, so could someone please help check my solution?</p>
<p>The first step to my solution is to eliminate $A$ an... | rogerl | 27,542 | <p>Everything looks fine until your partial fraction decomposition. Indeed $P=\frac{1}{3}$, but to find $Q$ and $R$, start with
$$1 = \frac{1}{3}(u^2+u+1) + (Qu+R)(u-1)$$
and choose $u=0$ to get
$$1 = \frac{1}{3} - R,$$
so that $R = -\frac{2}{3}$. You then get $Q = -\frac{1}{3}$. So the decomposition is
$$\frac{1}{u^3... |
150,295 | <p>I will appreciate any enlightenment on the following which must be an exercise in a certain textbook. (I don't recognize where it comes from.)
I understand that the going down property does not hold since $R$ is not integrally closed (in fact, it is not a UFD), but I have no idea how to show that $q$ is such a count... | Georges Elencwajg | 3,217 | <p>I'll show that the existence of $Q\in Spec(A)$ satisfying $Q\subsetneq P$ and $q= Q\cap R$ leads to a contradiction. </p>
<p>We have $X\cdot (X-1)\in q $ , so $X\cdot (X-1)\in Q$.<br>
Hence we have $(X-1)\in Q$ (since $X\notin Q$ because $X\notin P$).<br>
But this forces $Q=(X-1)A$, since $(X-1)A\su... |
915,016 | <p>Let $S$ be a non - empty set and $F$ be a field. Let $C(S,F)$ denote the set of all functions $f\in \mathcal F(S,F)$ such that $f(s)=0$ for all but a finite number of elements of $S$. Prove that $C(S,F)$ is a subspace of $\mathcal F(S,F)$</p>
<p>My progress:
I have shown that if $f,g\in C(S,F)$ and $a\in F$ then $f... | mvw | 86,776 | <p>Just checking if your solution candidate fullfills the PDE $(*)$ and boundary condition $(**)$. </p>
<p>The condition $(**)$ holds.</p>
<p>For the PDE we need to calculate $u_x$ and $u_t$. For calculating $u_t$ we use the <a href="http://en.wikipedia.org/wiki/Leibniz_integral_rule#General_form_with_variable_limits... |
915,016 | <p>Let $S$ be a non - empty set and $F$ be a field. Let $C(S,F)$ denote the set of all functions $f\in \mathcal F(S,F)$ such that $f(s)=0$ for all but a finite number of elements of $S$. Prove that $C(S,F)$ is a subspace of $\mathcal F(S,F)$</p>
<p>My progress:
I have shown that if $f,g\in C(S,F)$ and $a\in F$ then $f... | doraemonpaul | 30,938 | <p>Follow the method in <a href="http://en.wikipedia.org/wiki/Method_of_characteristics#Example" rel="nofollow">http://en.wikipedia.org/wiki/Method_of_characteristics#Example</a>:</p>
<p>$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$</p>
<p>$\dfrac{dx}{ds}=1$ , letting $x(0)=x_0$ , we have $x=s+x_0=t+x_0$</p>
... |
1,534,246 | <p>I'm trying to simplify this boolean expression:</p>
<p>$$(AB)+(A'C)+(BC)$$</p>
<p>I'm told by every calculator online that this would be logically equivalent:</p>
<p>$(AB)+(A'C)$</p>
<p>But so far, following the rules of boolean algebra, the best that I could get to was this: </p>
<p>$(B+A')(B+C)(A+C)$</p>
<p>... | Lucian | 93,448 | <p><em>I would like to add the following explanation to the above answers:</em></p>
<blockquote>
<p>The first two terms translate as “If <em>A</em>, then <em>B</em>, else <em>C</em> ”. Notice, therefore, that <em>B</em> and <em>C</em> cannot <br> simultaneously coexist, meaning that the third term can be safely igno... |
1,049,933 | <p>If $M$ is the transition matrix of a discrete Markov chain, and $M$ is both irreducible, symmetric and positiv-definite, is the resulting Markov chain necessarily aperiodic? </p>
<p>In my intuition, periodicity would correspond to an $-1$-eigenvalue of $M$, but I don't know if that is true or how to formalize it.</... | frogfanitw | 297,978 | <p>I believe there is a recursive formula for this:</p>
<p>Let L(k) = LCM({$p_1 - 1$, $p_2 - 1$, $p_3 - 1$, ..., $p_k$}</p>
<p>Then L(k+1) = L(k) ($p_{k+1}-1$) / GCD(L(k),$p_{k+1}-1$)</p>
<p>where GCD = greatest common divisor</p>
|
4,242,093 | <p><em><strong>Question:</strong></em></p>
<blockquote>
<p>Let <span class="math-container">$G=(V_n,E_n)$</span> such that:</p>
<ul>
<li>G's vertices are words over <span class="math-container">$\sigma=\{a,b,c,d\}$</span> with length of <span class="math-container">$n$</span>, such that there aren't two adjacent equal ... | 1Rock | 208,645 | <p>Your solution B doesn't work. The graph <span class="math-container">$G_n$</span> has <span class="math-container">$4\cdot3^n$</span> vertices rather than <span class="math-container">$n$</span> vertices, so you'd need to show each vertex had <span class="math-container">$4\cdot 3^n/2$</span> vertices to apply Dirac... |
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... | Kuifje | 273,220 | <p>Color classes are necessarily pairwise connected by at least one edge (otherwise you could merge them), and since there are $k$ color classes ...</p>
|
1,930,933 | <blockquote>
<p>Does there exist an $n \in \mathbb{N}$ greater than $1$ such that $\sqrt[n]{n!}$ is an integer?</p>
</blockquote>
<p>The expression seems to be increasing, so I was wondering if it is ever an integer. How could we prove that or what is the smallest value where it is an integer?</p>
| Ege Erdil | 326,053 | <p>This is impossible due to Bertrand's postulate, since there will always be a prime $ p $ in $ n! $ occuring with multiplicity $ 1 $ as long as $ n \geq 2 $. This actually implies that $ n! $ is never a perfect power for $ n \geq 2 $.</p>
|
786,301 | <p>Is there a systematic way to express the sum of two complex numbers of different magnitude (given in the exponential form), i.e find its magnitude and its argument expressed in terms of those of the initial numbers?</p>
| Alex G. | 130,309 | <p>Not really. One has to express the complex numbers as the sums of their real and imaginary parts, and then add componentwise, like usual. The best we can really do is make use of the triangle inequality:</p>
<p>$| |z_1| - |z_2| | \leq |z_1 + z_2| \leq |z_1| + |z_2|$</p>
|
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 $... | Mariano Suárez-Álvarez | 274 | <p>No, you cannot conclude that $S^1$ is not a retract of $\mathbb R^2$ that way. To prove that something is not a retract usually requires more machinery, and algebraic topology is more or less designed to be helpful for this. I'll explain an argument using the fundamental group $\pi_1$, but one could use other functo... |
1,003,020 | <p>Without recourse to Dirichlet's theorem, of course.
We're going to go over the problems in class but I'd prefer to know the answer today.</p>
<p>Let $S = \{3n+2 \in \mathbb P: n \in \mathbb N_{\ge 1}\}$</p>
<p>edit:</p>
<p>The original question is "the set of all primes of the form $3n + 2$, but I was only consid... | user2345215 | 131,872 | <p>If $n$ in your question is even, the number can't be a prime. So it suffices to prove there are infinitely many primes of the form $6n-1$.</p>
<p>Assume there are only finitely many of them let $n_1\ldots,n_k$ be their representants. Then
$$6\bigl((6n_1-1)(6n_2-1)\ldots(6n_k-1)\bigr)-1$$
is a number of the form $6n... |
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>
| Learnmore | 294,365 | <p>Note that $x^2+1=0$ is a annihilating polynomial of $A$. Now minimal polynomial of a matrix divides a annihilating polynomial.</p>
<p>So the possible choices are $x=-1,x=1,x^2+1$.</p>
<p>Hence the matrices in first two cases will be $A=I,A=-I$.</p>
<p>Consider the third case.Since the minimal polynomial and chara... |
1,103,478 | <p>$ r = 2\cos(\theta)$ has the graph<img src="https://i.stack.imgur.com/yvLb1.png" alt="enter image description here"></p>
<p>I want to know why the following integral to find area does not work $$\int_0^{2 \pi } \frac{1}{2} (2 \cos (\theta ))^2 \, d\theta$$</p>
<p>whereas this one does:</p>
<p>$$\int_{-\frac{\pi}{... | rlartiga | 93,314 | <p>Start with $\theta=0$ you get $r=2$ then move to $\theta=\frac{\pi}{2}$ you will get $r=0$ (then you have from here the upper half of the circle). From $\frac{\pi}{2}$ to $\pi$ you have a negative $r$ which don't have really much sense. Then you have to consider it from $\frac{3\pi}{2}$ to $2\pi$ to complete the low... |
3,532,173 | <p>I have seen this problem somewhere on the internet but I could not prove it.</p>
<p>Let <span class="math-container">$$I_{0}=\int^{\infty}_{0}\frac{\sin x}{x}dx$$</span> and then define
<span class="math-container">$$I_{n+1}=\int^{I_{n}}_{0}\frac{\sin x}{x}dx.$$</span></p>
<p>Show that
<span class="math-container... | approxolotl | 747,548 | <p>Denote the integral by <span class="math-container">$S(I_n)$</span></p>
<p><span class="math-container">$$
I_{n+1}+I_n=S_{n+1}+S_n\\
I_{n+1}-I_n=S_{n+1}-S_n \\
\rightarrow
2 I_{n}= 2 S_{n}
$$</span>
writing (Taylor expansion)
<span class="math-container">$$
S_{n}=I_{n-1}-\frac1{18}I_{n-1}^3+O(I_{n-1}^5)
$$</span></... |
156,013 | <p>I would like to <code>FoldList</code> a simple function, with desired output:</p>
<pre><code>f[a,b,1]
f[b,c,1]
f[c,a,1]
f[f[a,b,1],f[b,c,1],2]
f[f[b,c,1],f[c,a,1],2]
f[f[c,a,1],f[a,b,1],2]
f[f[f[a,b,1],f[b,c,1],2],f[f[b,c,1],f[c,a,1],2],3]
f[f[f[b,c,1],f[c,a,1],2],f[f[c,a,1],f[a,b,1],2],3]
f[f[f[c,a,1],f[a,b,1],2... | Alucard | 18,859 | <p>here is another option, though it is not as good as the code provided by Carl </p>
<pre><code>t3[j_List, l_, m_] := Append[ Take[RotateLeft[j, l], 2], m]
h[p_List, i_ ] := f @@@ MapIndexed[ t3[p, #, i] & , Range[0, 2]]
FoldList[h, {a, b, c}, {1, 2, 3 }]
</code></pre>
|
3,443,226 | <blockquote>
<p>Prove that
<span class="math-container">$$
\int_{0}^{2\pi}\frac{d\theta}{(a+\cos\theta)^2}=\frac{2\pi a}{(a^2-1)^{\frac{3}{2}}}.
$$</span></p>
</blockquote>
<p>This is an exercise in Stein's <em>Complex Analysis</em>.</p>
<p>By letting <span class="math-container">$z=e^{i\theta}$</span>, we have
... | José Carlos Santos | 446,262 | <p>Let <span class="math-container">$R(x)=\frac1{(x+a)^2}$</span>. You want to compute <span class="math-container">$\int_0^{2\pi}R(x)\,\mathrm dx$</span>. Now, let<span class="math-container">$$g(z)=\frac1zR\left(\frac{z+z^{-1}}2\right)=\frac{4z}{\left(2az+z^2+1\right)^2}.$$</span>Then<span class="math-container">$$\i... |
166,666 | <p>For which values of the coefficient $c$ does the quantity
$$
\cos\alpha\cos\beta- c\sin\alpha\sin\beta
$$
depend on $\alpha$ and $\beta$ only through their sum?</p>
<p>(I'll post a quick answer below. This will be the first time I've posted a question with intent to immediately post an answer.)</p>
| celtschk | 34,930 | <p>Well, let's define the new quantities $s=\frac{\alpha+\beta}{2}$ and $d=\frac{\alpha-\beta}{2}$. With those quantities the expression reads $$E:=\cos(s+d)\cos(s-d)-c\sin(s+d)\sin(s-d).$$ The question now is: For which values of $c$ is this expression independent from $d$?</p>
<p>Let's apply the standard addition th... |
2,767,392 | <p>I have the following curve:</p>
<p>$x^4=a^2(x^2-y^2)$</p>
<p>Prove that the area of its loop is $\frac{2a^2}{3}$.</p>
<p><strong>My approach</strong></p>
<p>This curve has four loops. So the required area should be:</p>
<p>$4\int_{0}^{a}\frac{x}{a}\sqrt{a^2-x^2} dx$</p>
<p>But, After solving, the area turned o... | epi163sqrt | 132,007 | <p>Here is answer where we follow the advice in the solution and use $(1-x)^n$ and $\frac{1}{1-x}$, but <em>inside out</em>. It is convenient to use the <em>coefficient of</em> operator $[x^k]$ to denote the coefficient of $x^k$ in a series. This way we can write for instance
\begin{align*}
[x^k](1+x)^n=\binom{n}{k}\ta... |
811,535 | <p>I want to show that every prime power $p^k$ that divides $\binom{2m}{m}$ is smaller than or equal to $2m$.</p>
<p>As a first step, I looked at
$$\binom{2m}{m}
= \frac{(2m)!}{(m!)^2}
= \frac{2m(2m-1) \ldots (m+2)(m+1)}{m!} \, .$$
Here I'm essentially stuck. I can apply the prime factorization to numerator and denomi... | Shaun | 104,041 | <p><strong>Hint:</strong> Let $\alpha_{n, p}$ be the largest natural number such that $p^{\alpha_{n, p}}\mid n!$ for $n\in\mathbb{N}$ and $p$ prime. Then $$\alpha_{n, p}=\sum_{k=1}^{\infty}{\left[\frac{n}{p^k}\right]},$$ where $[x]$ is the integer part of $x$.</p>
|
866,144 | <p>Prove that
$$1-2^{-x}\geq \frac{\sqrt 2}{2}\sin\left(\frac{\pi}{4} x\right)$$
for $x\in[0,1]$.
Any suggestions please?</p>
| AsdrubalBeltran | 62,547 | <p>If $$F(x)=\frac{\sqrt{2}}{2}\sin\left(\frac{\pi}{4} x\right)+2^{-x}$$</p>
<p>$F(x)>0$.
How $F(0)=F(1)=1$, then by value mean theorem there is a $x_0\in [0,1]$ such that $F'(x_0)=0$
$$F'(x)=\frac{\pi\sqrt{2}}{8}\cos\left(\frac{\pi}{4} x\right)-2^{-x}\ln(2)$$
Note that $F'(0)<0$ and $F'(1)>0$, also $F'''(x... |
131,435 | <p>Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. <a href="http://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Popular_pages" rel="noreferrer">Here is a list of the 500 most popular math articles</a>. The... | Alexander Chervov | 10,446 | <p>+1 for question. Hope it will stimulate us to contribute more, it would be great. I feel great respect to those who contribute to Wikipedia. My experience about Wikipedia and its quality is highly positive, probably one of the reasons, is that many many articles are written by one of the Fields Medalists. </p>
<p>... |
597,949 | <p>$\begin{cases}x\equiv 1 \pmod{3}\\
x\equiv 2 \pmod{5}\\
x\equiv 3 \pmod{7}\\
x\equiv 4 \pmod{9}\\
x\equiv 5 \pmod{11}\end{cases}$ </p>
<p>I am supposed to solve the system using the Chinese remainder theorem but $(3,5,7,9,11)\neq 1$
How can I transform the system so that I will be able to use the theorem?</p>
| lab bhattacharjee | 33,337 | <p>Observe that $\displaystyle x\equiv4\pmod 9\implies x\equiv4\pmod3\equiv1$</p>
<p>Now, we can safely apply <a href="http://www.cut-the-knot.org/blue/chinese.shtml" rel="nofollow">C.R.T</a> on $$\begin{cases}
x\equiv 2 \pmod5\\
x\equiv 3 \pmod7\\
x\equiv 4 \pmod9\\
x\equiv 5 \pmod{11}\end{cases}$$ as $5,7,9,11$ are ... |
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>
| R.W | 253,359 | <p><strong>Hint 1:</strong> $\cosh^2(x)-\sinh^2(x)=1$ and use change of variables.</p>
<p><strong>Hint 2:</strong> </p>
<blockquote>
<p>$$\frac{\mathrm{d}}{\mathrm{d}x}\left(\tanh(x)\right) =\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\sinh(x)}{\cosh(x)}\right) = \frac{1}{\cosh^2(x)}\left(\cosh^2(x)-\sinh^2(x)\right)... |
354,885 | <p>Let <span class="math-container">$X_1,...,X_n$</span> be iid normal random variables. </p>
<p>I am looking for a strategy to establish the following limit for fraction of expectation values</p>
<p><span class="math-container">$$\lim_{N \rightarrow \infty} \frac{E(\prod_{1\le i < j\le n} \vert X_i-X_j \vert^{1/n... | Iosif Pinelis | 36,721 | <p>The <a href="https://en.wikipedia.org/wiki/Selberg_integral#Mehta's_integral" rel="nofollow noreferrer">Mehta integral</a> is
<span class="math-container">$$M_n(\gamma):=E\prod_{1\le i<j\le n}|X_i-X_j|^{2\gamma}
=\prod_{j=1}^n\frac{\Gamma(1+j\gamma)}{\Gamma(1+\gamma)}.$$</span>
So, your fraction under the li... |
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... | BlueRaja - Danny Pflughoeft | 136 | <p><strong>Frk n th rd 1</strong></p>
<p>Y'r n pth n n slnd, cme t frk n th rd. Bth pths ld t vllgs f ntvs; th ntr vllg thr lwys tlls th trth r lwys ls <em>(bth villgs cld b trth-tllng r lyng vllgs, r n f ch)</em>. Thr r tw ntvs t th frk - thy cld bth b frm th sm vllg, r frm dffrnt vllgs <em>(s bth cld b trth-tllrs... |
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... | BlueRaja - Danny Pflughoeft | 136 | <p><strong>Fork in the road 2</strong></p>
<p>You're once again at a fork in the road, and again, one path leads to safety, the other to doom.</p>
<p>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 ... |
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... | Ross Millikan | 1,827 | <p>From New Scientist some years ago: 20 teams play a round robin tournament, each gets 1 point for a win, 0 for a loss, and there are no ties. Each team's score is a square number. How many upsets occurred? An upset is defined as team A defeating team B where B scored more total points than A.</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... | Henry | 6,460 | <p>You are on the surface of a cube, starting at the midpoint of one of the edges. Which point(s) on the cube is furthest away from you if you are constrained to travel on the surface of the cube? </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... | MJD | 25,554 | <p>There is a square table with a pocket at each corner; in each pocket is a drinking glass, which you cannot see. Each glass might be right-side up ("up") or upside-down ("down").</p>
<p>You and an adversary will play the following game. You select exactly two of the pockets, withdraw the two glasses, thus learning t... |
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... | Asaf Karagila | 622 | <p><em>Not a mathematical puzzle in the classical sense, but it is an interesting variation on the men with hats problem. It has a much "quicker" solution which to some degree is a bit surprising. If you are willing to indulge me, I will also add a rich background story.</em></p>
<p>After Hilbert passed away in 1943 h... |
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></... | user577215664 | 475,762 | <p>Another way</p>
<p>$$x^2y''-3xy'+3y=0$$
For $x \neq 0$
$$y''-3 \left (\frac {xy'-y}{x^2} \right )=0$$
$$y''-3\left (\frac {y}{x}\right )'=0$$
$$y'-3 \left (\frac {y}{x} \right )=K_1$$
$$x^3y'-3{y}{x^2}=K_1x^3$$
$$\frac {x^3y'-3{y}{x^2}}{x^6}=\frac {K_1}{x^3}$$
$$\left (\frac {y}{x^3} \right )'=\frac {K_1}{x^3}$$
$$... |
724,302 | <p>I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is.</p>
<p>Is category theory a content in ZFC-set theory? (Just like measure theory, group theory etc.) If not, is it just another formal logic system independent from the standard ZFC-set theory?</p>
<p>Fol... | mathematician | 98,943 | <p>I know only very basic things about category theory, but from what I understand category theory is an axiomatic approach to proving things that various mathematical objects have in common. Like instead of proving a theorem about group homomorphisms and ring homomorphisms and maps between sets, you prove all those d... |
594,811 | <p>This is my first post, sorry for my naiveness..</p>
<p>I know a basic equation that relates Gram-schmidt matrix and Euclidean distance matrix:</p>
<p>$XX'=-0.5*(I-J/n)*D*(I-J/n)'$</p>
<p>Where $X$ is centered data (is $d \times n$), $I$ is identity matrix, $J$ is a matrix filled with ones (1), $n$ is the number o... | Aydin | 68,715 | <p>There are simple linear relationships between an Euclidean distance matrix and a Gramian matrix in both directions.
The linear transformation $G\rightarrow D$ is given by
$$ D = \delta(G)\mathbf{1}^\mathrm{T} + \mathbf{1}\delta(G)^\mathrm{T} - 2G,$$
where $G=XX^\mathrm{T}$, and $\delta(G)$ denotes the diagonal eleme... |
4,561,863 | <p>This limit is proposed to be solved without using the L'Hopital's rule or Taylor series:
<span class="math-container">$$
\lim\limits_{x\to 0} \left( \frac{a^{\sin x}+b^{\sin x} }{2} \right)^{\frac1{x}},
$$</span>
where <span class="math-container">$a>0$</span>, <span class="math-container">$b>0$</span> are som... | insipidintegrator | 1,062,486 | <p><a href="https://math.stackexchange.com/questions/1987215/1-to-the-power-of-infinity-formula">This post</a> has multiple answers that explain why, if <span class="math-container">$\displaystyle\lim_{x\to a} f(x)=1$</span> and <span class="math-container">$\displaystyle\lim_{x\to a}g(x)=\infty$</span> then <span clas... |
4,561,863 | <p>This limit is proposed to be solved without using the L'Hopital's rule or Taylor series:
<span class="math-container">$$
\lim\limits_{x\to 0} \left( \frac{a^{\sin x}+b^{\sin x} }{2} \right)^{\frac1{x}},
$$</span>
where <span class="math-container">$a>0$</span>, <span class="math-container">$b>0$</span> are som... | ZGperx | 708,955 | <p>Let's prove this expression:</p>
<p>Given positive <span class="math-container">$a$</span> and <span class="math-container">$b$</span>:
<span class="math-container">$$\lim_{n\rightarrow\infty}\bigg(\frac{\sqrt[n]{a}+\sqrt[n]{b}}{2}\bigg)^n=\sqrt{ab}$$</span>.</p>
<p>In order to show this we are going to use the foll... |
1,790,222 | <p>I know that $[0,1]$ and a unit circle $\mathbb{S}^1$ are one-point compactifications of $\mathbb{R}$ under some suitable homeomorphism. But how does one construct the Stone–Čech compactification? </p>
| Henno Brandsma | 4,280 | <p>You cannot really construct it, as such. You can define it, and prove its existence (using the Axiom of Choice) but you cannot give a concrete, definable example of a point in the remainder $\beta\mathbb{R}\setminus\mathbb{R}$. It is common to define the half-line $\mathbb{H} = [0,\infty)$ and consider $\beta\mathbb... |
42,258 | <p>I have a Table of values e.g. </p>
<pre><code>{{x,y,z},{x,y,z},{x,y,z}…}
</code></pre>
<p>How do I replace the the "z" column with a List of values?</p>
| halirutan | 187 | <p>With the notation package something like this is easy. I would never use this by myself, because IMO such <em>sugar</em> can easily introduce bugs and undesired behavior if one is not cautious. I will paste a screenshot so that you see how I used the <code>Notation`</code> package, but first of all you have to load ... |
4,353,891 | <p>How to prove that <span class="math-container">$A=\{(x,y)\in \mathbb{R}^2: |x|+|y|^{1/2}<1\}$</span> is convex? I tried using the definition but couldn’t go far, since the second component involves square root(tried squaring, that made it complicated).</p>
<p>I plotted it in mathematica and the graph comes out to... | Essaidi | 708,306 | <p>It's not convex :<br>
<a href="https://i.stack.imgur.com/lqRPa.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lqRPa.png" alt="Surface <span class="math-container">$\{|x| + \sqrt{|y|} < 1\}$</span>" /></a></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... | J. M. ain't a mathematician | 498 | <p><code>Simplify[PowerExpand[Simplify[Sum[Binomial[K, k + 1]*x^(k + 1)*(1 - x)^(K - k - 1), {k, 0, K - 1}], K > 0]]]</code> works nicely. The key is in the use of the second argument of <code>Simplify[]</code> to add assumptions about a variable. and using <code>PowerExpand[]</code> to distribute powers.</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... | André Nicolas | 6,312 | <p>The following answer makes sense only with some background in probability.
Suppose first that $0 \le x\le 1$.</p>
<p>A possibly biased coin has probability $x$ of landing "heads." Toss the coin $K$ times. We compute the probability $P$ of <strong>one or more</strong> heads in two different ways. </p>
<p>The pro... |
2,976,613 | <p>I want to show, that <span class="math-container">$a:=\sum \limits_{n=0}^{\infty} \left(\dfrac{2n+n^3}{3-4n}\right)^n$</span> is not converging, because <span class="math-container">$\lim \limits_{n \to \infty}(a)\neq 0 \; (*)$</span>. Therefore, the series can't be absolute converge too.</p>
<p>Firstly, I try to s... | OgvRubin | 468,471 | <p>I'm a bit confused by your choice of notation for it seems like you write that a series diverges if its limit is not <span class="math-container">$0$</span> what is not true. So i think you mean that we want to show that </p>
<p><span class="math-container">$$\frac{(2n+n^3)^n}{(3-4n)^n}\not\rightarrow 0$$</span></p... |
2,237,963 | <p>One-point compactification of $S_{\Omega}$ is homeomorphic with $\bar S_{\Omega}$.</p>
<p>Let $X$ be a topological space. Then the One-point compactification of $X$ is a certain compact space $X^*$ together with an open embedding $c : X \to X^*$ such that the complement of $X$ in $X^*$ consists of a single point, ... | Travis Willse | 155,629 | <p>Drafting behind Michael Rozenberg's clever answer, appealing to the concavity of <span class="math-container">$\sin$</span> on <span class="math-container">$[0, \pi]$</span> quickly reduces the problem to showing the inequality
<span class="math-container">$$2 \sin 1 > \frac{8}{5} .$$</span>
From <a href="https:/... |
3,176,482 | <p>In my Econometrics class yesterday, our teacher discussed a sample dataset that measured the amount of money spent per patient on doctor's visits in a year. This excluded hospital visits and the cost of drugs. The context was a discussion of generalized linear models, and for the purposes of Stata, he found that a g... | BruceET | 221,800 | <p>I agree with @Michael that you might be thinking about a <strong>random sum of random variables</strong>. The number of visits <span class="math-container">$N \sim \mathsf{Pois}(\lambda)$</span> and the amount spent on the <span class="math-container">$i$</span>th visit is <span class="math-container">$X_i \sim \mat... |
4,320,849 | <p>I had this problem in an exam I recently appeared for:</p>
<blockquote>
<p>Find the range of
<span class="math-container">$$y =\frac{x^2+2x+4}{2x^2+4x+9}$$</span></p>
</blockquote>
<p>By randomly assuming the value of <span class="math-container">$x$</span>, I got the lower range of this expression as <span class="m... | soupless | 888,233 | <p>In general, if <span class="math-container">$\deg f = 0$</span> where <span class="math-container">$$f(x) = \frac{a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0}{b_nx^n + b_{n - 1}x^{n - 1} + \cdots + b_1x + b_0},$$</span> the limit of <span class="math-container">$f$</span> as <span class="math-container">$x$</s... |
794,301 | <p>I am trying to find out the sum (I just derived these from 2 + 0.5 + 0.125 + 0.03125 + ...):</p>
<p>$$\sum_{n=0}^{\infty} \frac{5^{2n-1}}{10^{2n -1}}$$</p>
<p>It's confusing me because it doesn't match $${ar}^{n-1}$$ the power by which $r$ is raised. </p>
| Henno Brandsma | 4,280 | <p>$\frac{a^k}{b^k} = (\frac{a}{b})^k$. Also $a^{2k} = (a^2)^k$, and $a^{2k-1} = \frac{1}{a} \cdot a^{2k}$. This should allow you to simplify it.</p>
|
125,399 | <p>How can I solve the following integral?
$$\int_0^\pi{\frac{\cos{nx}}{5 + 4\cos{x}}}dx, n \in \mathbb{N}$$</p>
| PAD | 27,304 | <p>Of course it can be solved easily with complex residues. Since $\cos x$ is even you replace it with 1/2 the integral from $0$ to $ 2 \pi$. Then you make the substitution $z=e^{i\theta}$. You end-up with an integral over the unit circle. You end-up with a function which has singularities only at $0$ and $-\frac{1}{... |
988,566 | <p>For x ∈ ℝ, define by:
⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋).</p>
<p>Use this definition to prove or disprove the following with a structured proof technique:
∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋.</p>
<p>I understand I need to start with assuming the domain to be true aswell as the antecedent, then equati... | kolonel | 104,564 | <p>Using Cauchy-Shwartz's inequality we have:
$$\begin{align*}
\mathbf{x}^\intercal H \mathbf{x} &= (1-\rho)\sum_{i=1}^n x_i^2 + \rho\left(\sum_{i=1}^n x_i\right)^2\\
&\geq \frac{(1-\rho)}{n}\left(\sum_{i=1}^n x_i\right)^2 + \rho\left(\sum_{i=1}^n x_i\right)^2\\
&=\frac{1+(n-1)\rho}{n}\left( \sum^n_{i=1} x_... |
265,377 | <p>Let $k$ be a finite field, and let $G$ be the absolute Galois group of $k$, which is isomorphic to $\widehat{\mathbb{Z}}$. Let $\mathcal{C}$ be the category of $G$-modules. Then, we have the following:</p>
<p>For a finite $G$-module $N$, we have
$$
Ext^r_{\mathcal{C}}(N, \mathbb{Z}) \simeq H^{r-1}(G,~N^D),
$$
wher... | nfdc23 | 81,332 | <p>As noted in R. van Dobben de Bruyn's answer, since $N$ is killed by some positive integer we can reformulate the problem as that of constructing natural isomorphisms ${\rm{Ext}}^i_G(N, \mathbf{Q}/\mathbf{Z}) \simeq {\rm{H}}^i(G, N^D)$ for any $i \ge 0$ and any finite discrete $G$-module $N$ (with $G := {\rm{Gal}}(k_... |
1,728,910 | <p>Whenever I get this question, I have a hard time with it. </p>
<p>An example of a problem:</p>
<p>In the fall, the weather in the evening is <em>dry</em> on 40% of the days, <em>rainy</em>
on 58% of days and <em>snowy</em> 2% of the days. </p>
<p>At noon you notice clouds in the sky. </p>
<p>Clouds appear
at noo... | Feras | 215,405 | <p>$P\left(S \mid C \right) = \dfrac{P\left(C \mid S \right)P\left(S\right)}{P\left(C\right)} = \dfrac{P\left(C \mid S \right)P\left(S\right)}{P(C|S)P(S)+P(C|D)P(D)+P(C|R)P(R)}$.</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>
| kglr | 125 | <pre><code>List @@ sum
</code></pre>
<blockquote>
<p>{a, -b, c, d}</p>
</blockquote>
<p>From the docs on <a href="http://reference.wolfram.com/mathematica/ref/Apply.html" rel="noreferrer">Apply (@@)</a>:</p>
<blockquote>
<p>f@@<em>expr</em> replaces the head of <em>expr</em> by f.</p>
</blockquote>
<p>So <code>... |
1,348,046 | <p>I ran into this sum $$\sum_{n=3}^{\infty} \frac{3n-4}{n(n-1)(n-2)}$$
I tried to derive it from a standard sequence using integration and derivatives, but couldn't find a proper function to describe it.
Any ideas?</p>
| mathlove | 78,967 | <p>Setting </p>
<p>$$\frac{3n-4}{n(n-1)(n-2)}=\frac{A(n-1)-B}{(n-2)(n-1)}-\frac{An-B}{(n-1)n}$$
gives you $A=3,B=2$, i.e.
$$\frac{3n-4}{n(n-1)(n-2)}=\frac{3(n-1)-2}{(n-2)(n-1)}-\frac{3n-2}{(n-1)n}.$$
Hence, we have
$$\begin{align}\sum_{n=3}^{\infty}\frac{3n-4}{n(n-1)(n-2)}&=\lim_{m\to\infty}\sum_{n=3}^{m}\left(\fr... |
1,206,528 | <p>Find the matrix $A^{50}$ given</p>
<p>$$A = \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix}$$ as well as for $$A=\begin{bmatrix} 2 & 0 \\ 2 & 1\end{bmatrix}$$</p>
<p>I was practicing some questions for my exam and I found questions of this form in a previous year's paper.</p>
<p>I don't know how to ... | paw88789 | 147,810 | <p>Look at the first few powers:
$$\left(\begin{matrix}2&-1\\0&1 \end{matrix} \right)$$
$$A^2=\left(\begin{matrix}2&-1\\0&1 \end{matrix} \right)\left(\begin{matrix}2&-1\\0&1 \end{matrix} \right)=\left(\begin{matrix}4&-3\\0&1 \end{matrix} \right)$$
$$A^3=\left(\begin{matrix}8&-7\\0&a... |
1,687,500 | <p>Prove that, any group of order $15$ is abelian (without help of Sylow's theorem or its application).</p>
<p>What I have done so far is, </p>
<p>by class equation we know that $|G|=|Z|+\sum\frac{|G|}{C(a_i)}$. Now if I can show that $|G|=|Z|$ then the theorem is proved. Now order of $|Z|$ can not be $3$ or $5$, bec... | Kuifje | 273,220 | <p>Assume we are dealing with a minimization problem. Any variable $x_r$ with negative ($\le 0$) reduced cost can enter the basis. Once this is done, another variable has to leave the basis. You can choose any variable $x_i$ such that
$$
\frac{b_i'}{a_{ir}'} \ge 0,
$$
i.e. any variable such that if $x_r$ replaces $x_i... |
1,358,927 | <p>I have to solve this problem using integration by parts. I am new to integration by parts and was hoping someone can help me.</p>
<p>$$\int\frac{x^3}{(x^2+2)^2} dx$$</p>
<p>Here is what I have so far:</p>
<p>$$\int udv = uv-\int vdu $$</p>
<p>$$u=x^2+2$$ Therefore, $$xdx=\frac{du}{2}$$
$$dv=x^3$$
Therefor, $$v=3... | Nathan Janos | 350,573 | <p>I believe what you are looking for is this:</p>
<p><span class="math-container">$$ x \bmod M = [\frac{1}{2} + \frac{i}{2\pi}\ln(-e^{-i2\pi x/M})]\times M $$</span></p>
<p>You can check it out graphed <a href="https://www.wolframalpha.com/input/?i=plot%20(1%2F2%20%2B%20(i%2F(2*pi))*log(-1*exp(-i*2*pi*x%2FM)))*M%20w... |
1,102,310 | <p>For the quadratic function $$-ax^2 + 1$$ an upside down parabola with $y(0) = 1,$ is there a way to compute <em>a</em> such that the definite integral of $y$ between the roots ($x_1, x_2: f(x_1) \land f(x_2)= 0$) equals $1?$</p>
| mickep | 97,236 | <p>Hint: $y=-ax^2+1$ is zero when $x=\pm1/\sqrt{a}$, so it might be so that you want to calculate the integral
$$
\int_{-1/\sqrt{a}}^{1/\sqrt{a}}1-ax^2\,dx.
$$</p>
|
2,579,137 | <p>According to the definition of harmonic number $H_n = \sum\limits_{k=1}^n\frac{1}{k}$.</p>
<p>How we can define $H_{n+1}$ and $H_{n+\frac{1}{2}}$? </p>
| Jack D'Aurizio | 44,121 | <p>Well, the question is more or less the same as <em>how do we define $n!$ for $n\not\in\mathbb{N}?$</em><br>
Intro: there are many functions agreeing with $n!$ over $\mathbb{N}$, but it we impose that </p>
<ol>
<li>$f(1)=1$ and $f(x+1)=x\,f(x)$ for any $x\in\mathbb{R}^+$</li>
<li>$\log f$ is convex over $\mathbb{R}^... |
301,318 | <p>As the title stated , what is the meaning of infinitely many ? When we say a set contains infinitely many elements, does this mean we cannot finish counting all the elements in the set ? Does infinitely many same as $\forall$ ? </p>
| Emily | 31,475 | <p>$\forall$ is just a shorthand way of saying "for all". It can be used for infinite or finite sets.</p>
<p>For example:</p>
<p>$$\forall x \in \{ 1, 2, 3\},\ x > 0.$$</p>
<p>"Infinitely many" means that there are not finitely many. In other words, "infinitely many" means that there does not exist some real inte... |
162,324 | <p>Let $x_n$ be a sequence in a Hilbert space such that
$\left\Vert x_n \right\Vert=1$ and $ \langle x_n,\ x_m \rangle =0 $, for all $n \neq m$.</p>
<p>Let $ K= \{ x_n/ n : n \in \mathbb{N} \} \cup \{0\} $.</p>
<p>I need to show that $K$ is compact, $\operatorname{co}(K)$ is bounded, but not closed and finally find ... | Michael R. Chernick | 30,995 | <p>Setting the partial derivative of L with respect to lambda f to 0 forces g(x,y)=0. Requiring partial of L with respect x and y to 0 will lead to a local extreme point subject to g(x,y) = 0. Because of the form of L this could be a minimum.</p>
|
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