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 |
|---|---|---|---|---|
3,977,687 | <p>A coin of radius 1 cm is tossed onto a plane surface that has been tessellated by right triangles whose sides are 8 cm, 15 cm, and 17 cm long. Find the probability that the coin lands within a triangle.</p>
<p>I know that this has to do with similarity because the inner triangle that is formed by the area where the ... | heropup | 118,193 | <p>If <span class="math-container">$O$</span> is the center of the coin, the locus of <span class="math-container">$O$</span> inside a single <span class="math-container">$8$</span>-<span class="math-container">$15$</span>-<span class="math-container">$17$</span> right triangle <span class="math-container">$\triangle A... |
2,300,613 | <p>I tried to calculate few derivatives, but I cant get $f^{(n)}(z)$ from them. Any other way? </p>
<p>$$f(z)=\frac{e^z}{1-z}\text{ at }z_0=0$$</p>
| Luke | 450,423 | <p>$$
g(z) = a_0+a_1z+a_2z^2+...,\\
\frac{g(z)}{1-z}=a_0\frac{1}{1-z}+a_1\frac{z}{1-z}+a_2\frac{z^2}{1-z}+...
$$
Using the power series for $\frac{1}{1-z}$ gives
$$
g(z)=a_0(1+z+z^2+...)+a_1z\cdot(1+z+z^2+...)+a_2z^2\cdot(1+z+z^2+...),\\
g(z)=a_0(1+z+z^2+...)+a_1\cdot(z+z^2+z^3+...)+a_2\cdot(z^2+z^3+z^4...),\\
g(z)=a_0... |
3,395,098 | <p>I am trying to work out for what <span class="math-container">$\lambda_1, \lambda_2 > 0$</span> is it true that <span class="math-container">$f(y) = \lambda_1 e^{y-\lambda_1 e^y} + \lambda_2 e^{y-\lambda_2 e^y}$</span> is unimodal?</p>
<p>Experimentally it seems it is unimodal when <span class="math-container">$... | Szeto | 512,032 | <p>Define <span class="math-container">$f(x)=ae^{-ax}(1-ax)+be^{-bx}(1-bx)$</span> for <span class="math-container">$x>0$</span>, where <span class="math-container">$b>a>0$</span>. </p>
<p>The OP is interested in the roots of <span class="math-container">$f$</span>.</p>
<p>Let <span class="math-container">$r... |
945,104 | <p>7 people are attending a concert.</p>
<p>(a) In how many different ways can they be seated in a row?</p>
<p>(b) Two attendees are Alice and Bob. What is the probability that Alice sits next to Bob?</p>
<p>(c) Bob decides to make Alice a rainbow necklace with 7 beads, each painted a different
colour on one side (r... | Marc van Leeuwen | 18,880 | <p>Since André Nicolas only left (c) to be answered I'll just mention that one. Since no starting point for the beads is fixed, one can choose any one bead as reference point and slide it to the first position; I'll arbitrarily choose the orange bead for this (O looks a bit like $0$). Then each necklace is uniquele det... |
4,350,695 | <p>My book was introducing the concept of integrals and wrote this:</p>
<p><span class="math-container">$$\text{Area under the curve of $f(x)$}=\lim_{\Delta x\to0}\sum_{n=1}^{N}f(x)\Delta x\tag{1}$$</span></p>
<p>My problem with <span class="math-container">$(1)$</span> is that there is no <span class="math-container">... | peek-a-boo | 568,204 | <p>That is indeed sloppy notation. The proper way of writing it is to say the following. Suppose <span class="math-container">$f:[a,b]\to\Bbb{R}$</span> is a given bounded function. Let <span class="math-container">$P=\{x_0,\dots, x_N\}$</span> be a partition of the interval <span class="math-container">$[a,b]$</span>,... |
1,265,074 | <p>I simply don't know how to go about answering this question. I've done a good few other questions about point estimation, but I really don't know where I'm going with this one:</p>
<p><img src="https://i.stack.imgur.com/coWTD.png" alt="Unbiased estimator of 1 over lambda"></p>
<p>Thanks for the help!</p>
<p>EDIT:... | Michael Hardy | 11,667 | <p>What would go wrong if $1$ had not been added in the denominator is that the probability that the denominator is $0$ would be positive so the expected value would be $\infty\ne 1/\lambda$.</p>
<p>The problem is simply to find the expected value:
\begin{align}
& \operatorname{E}\left( \frac n {1+\sum_{i=1}^n X_i... |
1,383,781 | <p>Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ </p>
<p>We can show that </p>
<p>$\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ </p>
<p>Hence $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms</p>
<p>Is there some deeper implication regarding this pa... | Stephan Kulla | 32,951 | <p>As pointed out by <a href="https://math.stackexchange.com/users/75808/clement-c">Clement C</a> in the comments: Equivalent norms induce the same topology. Also the other direction of implication is true: When two norms induce the same topology then they are equivalent.</p>
<p>Take two norms $\|\cdot\|_1$ and $\|\cd... |
3,063,651 | <p>i am currently looking out for some possible topics i could study for my research project in high school. Algebra, trigonometry, Pythagoras’ theorem, geometry, circles and their properties, etc. and perhaps combined with a little knowledge from Physics i.e. Kinematics, Gravity, etc. could interest me. </p>
<p>Anoth... | J.G. | 56,861 | <p>What are the requirements of your research project? I'm guessing they can't expect a high schooler to discover something new; they probably just want you to work through a non-syllabus proof, possibly after you conjecture the result from special cases.</p>
<p>If you want to do something with a Rubik cube you could ... |
646,109 | <p>For function $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $f\left(x+y\right)=f\left(x\right)f\left(y\right)$
and is not the zero-function I can prove that $f\left(1\right)>0$
and $f\left(x\right)=f\left(1\right)^{x}$ for each $x\in\mathbb{Q}$.
Is there a way to prove that for $x\in\mathbb{R}$?</p>
<p>This ... | Hagen von Eitzen | 39,174 | <p>If $f(x_0)=0$ for some $x_0$ then $f(x)=f((x-x_0)+x_0)=f(x-x_0)f(x_0)=0$ for all $x$. As the zero function is excluded, $f(x)\ne 0$ for all $x$ and in fact $f(x)=f(\frac x2)^2>0$. Therefore we can define $g(x)=\ln f(x)$ and find the functional equation $$ g(x+y)=g(x)+g(y)$$ for $g$. As has been pointed out, this... |
3,861,324 | <p>Given a nonzero column vector <span class="math-container">$A$</span>=<span class="math-container">$[a_1 a_2.......a_n]^T$</span>. Find the non zero eigen values and eigen vectors for <span class="math-container">$A$$A^T$</span>.</p>
<p>I have no idea.what theorem should I apply or what I have to do to solve this. ... | Zhanxiong | 192,408 | <p><span class="math-container">$\newcommand{\rank}{\mathrm{rank}}$</span></p>
<p>This is a well known exercise in eigenvalue-eigenvector, for which you actually do not need to start with solving the equation <span class="math-container">$\det(\lambda I - B) = 0$</span> to determine eigenvalues (as you usually do). The... |
3,151,662 | <p>Consider <span class="math-container">$a_1,\dots,a_n\in\mathbb{R}^n$</span> and identify <span class="math-container">$a_j\in\mathcal{L}(\mathbb{R},\mathbb{R}^n)$</span> via <span class="math-container">$\varphi\mapsto \varphi1$</span>.</p>
<p>Also, consider <span class="math-container">$A\in\mathcal{L}(\mathbb{R}^... | J.G. | 56,861 | <p>Here's an application in calculus. The multivariate generalisation of integration by substitution viz. <span class="math-container">$x=f(y)\implies dx=f^\prime(y)dy$</span> uses the determinant of a matrix called a Jacobian in place of the <span class="math-container">$f^\prime$</span> factor. In particular, the cha... |
3,318,993 | <p>Consider the functions <span class="math-container">$x$</span> and <span class="math-container">$x^2$</span> on <span class="math-container">$\mathbb{R}$</span>. Clearly, they are linearly independent.<br>
But consider the following argument.</p>
<p>Consider the matrix
<span class="math-container">$$A = \begin{bma... | Mark | 470,733 | <p>They are linearly independent as functions of variable <span class="math-container">$x$</span>. Yes, you are right that given a specific value of <span class="math-container">$x$</span> you can find non-zero <span class="math-container">$a,b\in\mathbb{R}$</span> such that <span class="math-container">$ax+bx^2=0$</sp... |
3,318,993 | <p>Consider the functions <span class="math-container">$x$</span> and <span class="math-container">$x^2$</span> on <span class="math-container">$\mathbb{R}$</span>. Clearly, they are linearly independent.<br>
But consider the following argument.</p>
<p>Consider the matrix
<span class="math-container">$$A = \begin{bma... | Arthur | 15,500 | <p>When discussing linear independence of the columns of a matrix, you must allow the coefficients in the linear combinations to come from the same space as the entries in the matrix (otherwise linear algebra doesn't in any way work the way you're used to). And clearly, if we allow <span class="math-container">$a$</spa... |
821,875 | <p>A school director must randomly select 6 teachers to participate in a training session. There are 30 teachers at the school. In how many different ways can these teachers be selected, if the order of selection does not matter?</p>
| StumpyLeg | 143,237 | <p>This is just "30 choose 6." If you have a graphing calculator, under the Probability menu there should be something like nCr. But your textbook should contain the relevant formula--and explain the basis for it.</p>
|
684,892 | <p>My progress:</p>
<p>Let's take $a \in \mathbb{Z}\left[\frac{-1 + \sqrt{-3}}2\right]$ such that $a \mid 2$, and function $l(x) = x \bar x$.</p>
<p>$a \mid 2$ $\Rightarrow$ $2 = ab$ $\Rightarrow$ $l(ab) = l(a)l(b) = 4 = l(2)$</p>
<p>If $z \in \mathbb{Z}[\frac{-1 + \sqrt{-3}}2]$, then $z = x + y\frac{-1 + \sqrt{-3}}... | Daniel Fischer | 83,702 | <p>Multiply with $4$ and complete the square, you are looking for integer solutions to</p>
<p>$$(2x-y)^2 + 3y^2 = 8.$$</p>
<p>That constrains $\lvert y\rvert \leqslant 1$ already. $5$ isn't a square, $8$ isn't a square, so $y = \pm 1$ and $y = 0$ are also ruled out.</p>
<p>Another proof is: If $x^2 - xy + y^2 \equiv... |
35,463 | <p>I was asked the following vector calculus problem:</p>
<blockquote>
<p>Let <span class="math-container">$D$</span> be the unit ball and let <span class="math-container">$S$</span> be the unit sphere in <span class="math-container">$\mathbb{R}^3$</span>. Suppose that <span class="math-container">$F:\mathbb{R}^3\righ... | Shuhao Cao | 7,200 | <p>Eric, I am simply rephrasing your second proof to give you a more PDE style approach to this problem, let $\vec{F} = \nabla u$, the problem you gave was equivalent to solve the following boundary value problem:
Find $u\in C^2(D)$ such that
$$
\left\{\begin{eqnarray}
-\Delta u &=& 0 \text{ in } D \\
\nabla... |
2,088,346 | <p>I've got the domain of function and I've attempted to find the first derivative at zero but it results in a quartic equation that is too difficult for me to solve. </p>
<p>$f'(x) = \frac{4x-3}{\sqrt{2x^2-3x+4}} + \frac{2x-2}{\sqrt{x^2-2x}}$</p>
<p>For $f'(x) = 0$:</p>
<p>$(4x-3)^2(x^2-2x) = (2-2x)^2(2x^2-3x+4)$</... | Matt | 263,495 | <p>I would suggest sketching your first square root and your second square root separately on a graph. The solution then doesn't require any sort of complicated algebra.</p>
<p><a href="https://i.stack.imgur.com/pQDNg.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pQDNg.png" alt="enter image descri... |
1,841,958 | <p>This is a claim on Wikipedia <a href="https://en.wikipedia.org/wiki/Partially_ordered_set">https://en.wikipedia.org/wiki/Partially_ordered_set</a></p>
<p>I am not sure how to make sense of the claim</p>
<p>What does it mean by ordered by inclusion? Inclusion as in $\subseteq$? </p>
<p>Can someone provide a small ... | Eric Wofsey | 86,856 | <p>Most of your questions have been answered well by user247327, so let me just answer your last question. The poset of all subspaces of a vector space $V$ is not linearly ordered as long as $\dim V>1$. For instance, if $v$ and $w$ are two linearly independent vectors, then $\operatorname{span}(v)$ and $\operatorn... |
1,039,141 | <blockquote>
<p>Let <span class="math-container">$X = \mathbb{R}$</span> and <span class="math-container">$Y = \{x \in \mathbb{R} :x ≥ 1\}$</span>, and define <span class="math-container">$G : X → Y$</span> by <span class="math-container">$$G(x) = e^{x^2}.$$</span>
Prove that <span class="math-container">$G$</span> is ... | Swapnil Tripathi | 117,387 | <p><strong>Hint:</strong> $g(x)=e^{x^2}$ is symmetric about the y-axis. Also $g$ is continuous and strictly increasing in $\mathbb{R}^+$ with $g(0)=1$</p>
|
3,201,797 | <p>I have three points <span class="math-container">$(x_1, y_1),~ (x_2, y_2),~ (x_3, y_3)$</span> that are on the same line. How to efficiently find which is the point in between.</p>
<p><a href="https://i.stack.imgur.com/e2wHq.png" rel="nofollow noreferrer">Example</a></p>
<p>Also, is there any efficient way to chec... | Vasili | 469,083 | <p>Find three distances and use the fact that the distance between end points is the greatest. You have points <span class="math-container">$A(x_1,y_1), B(x_2, y_2), C(x_3, y_3)$</span>. The square of distance between <span class="math-container">$A$</span> and <span class="math-container">$B$</span> is <span class="ma... |
3,201,797 | <p>I have three points <span class="math-container">$(x_1, y_1),~ (x_2, y_2),~ (x_3, y_3)$</span> that are on the same line. How to efficiently find which is the point in between.</p>
<p><a href="https://i.stack.imgur.com/e2wHq.png" rel="nofollow noreferrer">Example</a></p>
<p>Also, is there any efficient way to chec... | Intelligenti pauca | 255,730 | <p>Once you know that the three points are aligned,
compute
<span class="math-container">$$
t={x_3-x_2\over x_1-x_2}\quad
\text{(or}\quad
t={y_3-y_2\over y_1-y_2}\quad
\text{if $x_1-x_2\approx0$).}
$$</span>
Then:</p>
<ul>
<li>if <span class="math-container">$t>1$</span> then <span class="math-container">$P_2$</spa... |
706,546 | <p>If we have a non-zero real $n$ by $m$ matrix $M$, then there may exist a non-zero unit vector $v$ of $m$ elements so that $Mv = 0$. I understand we can't call this an eigenvector with eigenvalue $0$. </p>
<blockquote>
<p>Why is this not a sensible definition of an eigenvector of a rectangular matrix?</p>
</bloc... | Marc van Leeuwen | 18,880 | <p>The following observations suffice to prove the statement:</p>
<ol>
<li>A power $A^k$ is in the span of lower powers $A^0,\ldots,A^{k-1}$ if and only if there exists a (monic) polynomial$~P$ of degree$~k$ with $P[A]=0$.</li>
<li>If this happens for some $k=m$, it also happens for all $k>m$, so that by an immedia... |
3,364,316 | <p>While I'm reading E. Landau's <em>Grundlagen der Analysis</em> (tr. <em>Foundations of Analysis</em>, 1966), I couldn't understand the proof of <em>Theorem 3</em> at the segment of <em>Natural Numbers</em> which I've quoted below.</p>
<blockquote>
<p><strong>Theorem 3:</strong> <em>If</em><br>
<span class="math... | Keith Backman | 29,783 | <p>Let <span class="math-container">$g_n=p_{n+1}-p_n$</span>. Then <span class="math-container">$\frac{p_n}{p_n+p_{n+1}}=\frac{p_n}{2p_n+g_n}=\frac{1}{2+\frac{g_n}{p_n}}$</span>. The lim inf for increasing <span class="math-container">$n$</span> will occur when <span class="math-container">$g_n$</span> is as large as p... |
692,998 | <p>The inner product in a $L^2$ space can be defined as:</p>
<p>$$\langle f,g\rangle =\int_a^b \bar{f}(x)g(x)w(x)dx$$</p>
<p>For Legendre polynomials, we define it as:</p>
<p>$$\langle P_m,P_n\rangle =\int_0^1 \bar{P}_m(x)P_n(x)dx$$
so $w(x)=1$.</p>
<p>But there are case in which $w(x)\neq 1$. For example, Laguerre... | typedrums | 596,646 | <p>I think you mean this question in one of two ways.</p>
<p>Either you mean why do e.g. Laguerre polynomials have that specific weight? </p>
<p>The answer to which is just that you start with the weight and the polynomials follow from it. You can often in principle retrieve the weight from the polynomials though. Th... |
4,274,314 | <blockquote>
<p>Find all <span class="math-container">$f: \mathbb{R} \to \mathbb{R}$</span> such that
<span class="math-container">$$f\bigl(xf(y)+y\bigr)+f\bigl(-f(x)\bigr)=f\bigl(yf(x)-y\bigr)+y $$</span>
for all <span class="math-container">$x,y \in \mathbb{R}$</span>.</p>
</blockquote>
<p>Help me solving this. My ex... | Sathvik | 516,604 | <p><span class="math-container">$$f(xf(y)+y)+f(−f(x))=f(yf(x)−y)+y$$</span>
Define <span class="math-container">$f(0)=c$</span> for some <span class="math-container">$c\in \mathbb{R}$</span>.</p>
<hr />
<ul>
<li><span class="math-container">$(x,y)\equiv (0,0)$</span>
<span class="math-container">$$f(0)+f(-f(0))=f(0)\im... |
207,515 | <p>Suppose I have the following list, </p>
<pre><code>l = {{"b", "c", "d"}, {"e", "b"}, {"a", "b", "d", "e"}}
</code></pre>
<p>and further suppose I have the following association, </p>
<pre><code>l1=<|1 -> "a", 2 -> "b", 3 -> "c", 4 -> "d", 5 -> "e"|>
</code></pre>
<p>I wonder how can I replac... | Lukas Lang | 36,508 | <p>The following gives you a "reversed" version of your association, with keys and values flipped:</p>
<pre><code>l1 = <|1 -> "a", 2 -> "b", 3 -> "c", 4 -> "d", 5 -> "e"|>;
lookup = First /@ PositionIndex@l1
(* <|"a" -> 1, "b" -> 2, "c" -> 3, "d" -> 4, "e" -> 5|> *)
</code></... |
207,515 | <p>Suppose I have the following list, </p>
<pre><code>l = {{"b", "c", "d"}, {"e", "b"}, {"a", "b", "d", "e"}}
</code></pre>
<p>and further suppose I have the following association, </p>
<pre><code>l1=<|1 -> "a", 2 -> "b", 3 -> "c", 4 -> "d", 5 -> "e"|>
</code></pre>
<p>I wonder how can I replac... | Suba Thomas | 5,998 | <pre><code>l /. Reverse /@ Normal[l1]
(* {{2, 3, 4}, {5, 2}, {1, 2, 4, 5}} *)
</code></pre>
<p>or</p>
<pre><code>l /. AssociationMap[Reverse, l1]
(* {{2, 3, 4}, {5, 2}, {1, 2, 4, 5}} *)
</code></pre>
|
167,326 | <p>Let $ S = \sum_{i=1}^n X_i$ where:</p>
<ul>
<li>Each $X_i$ is independently 3 or 9 (with equal probability), and</li>
<li>The sample size $n$ is itself an independent random variable where $N \sim \text{NegativeBinomial}(r,p)$ e.g. $r = 5$ and $p = \frac34$</li>
</ul>
<p>Let $W = \begin{cases}S-10 & S > 1... | JimB | 19,758 | <p>Here is a basic principles approach to obtaining the probability mass function for both $S$ and $W$ (which uses the clarification asked for by @wolfies):</p>
<pre><code>(* Set parameters *)
r = 5;
p = 3/4;
pBin = 1/2;
(* Function to generate the combinations of n (sample size) and
k (number of successes) that r... |
112,651 | <p>What is known about the set of well orderings of $\aleph_0$ in set theory without choice? I do not mean the set of countable well-order types, but the set of all subsets of $\aleph_0$ which (relative to a pairing function) code well orderings. And I would be interested in an answer in, say, ZF without choice. My ... | Stefan Geschke | 7,743 | <p>There are two answers already, but I think this argument is simpler than both of the previous answers.</p>
<p>Any two permutations of $\omega$ give two different wellorderings of order type $\omega$.
We show that there are $2^{\aleph_0}$ permutations of $\omega$.
Given a function $f:\omega\to 2$ let $\sigma_f$ be t... |
2,435,596 | <p>Suppose we have an unsigned $8$ bit number (min=$0$, max=$255$).</p>
<p>the result of "$200 + 200$" overflows to $144$</p>
<p>the result of "$100 - 200$" (under?)overflows to $156$</p>
<p>Is there are mathematical symbol to represent this?</p>
| Siong Thye Goh | 306,553 | <p>You can denote it using $x \pmod {y+1}$ as long as your minimum is $0$ and the maximum value is $y$.</p>
<p>Example $1$:
$$200 + 200 = 400 = 256 + 144$$</p>
<p>$$200 + 200 \equiv 144 \pmod {256}$$</p>
<p>Example $2$:
$$100 - 200 = - 100 = -256 + 156$$</p>
<p>$$100 - 200 \equiv -100 \pmod {256}$$</p>
|
3,608,441 | <p>It is well known that we can define <span class="math-container">$e^x$</span> by the following limit</p>
<p><span class="math-container">$$e^{x}=\lim_{n\to\infty}\left(1+{x\over n}\right)^n$$</span></p>
<p>I would like to show that the RHS sequence is always less than or equal to <span class="math-container">$e^x$... | Gabriel Romon | 66,096 | <p>Here's an elementary proof that only requires Bernoulli's inequality. </p>
<p>Let <span class="math-container">$\displaystyle u_n:=\left(1+\frac xn\right)^n$</span>
and let <span class="math-container">$N$</span> be the smallest integer strictly greater than <span class="math-container">$|x|$</span>. Consider <span... |
3,608,441 | <p>It is well known that we can define <span class="math-container">$e^x$</span> by the following limit</p>
<p><span class="math-container">$$e^{x}=\lim_{n\to\infty}\left(1+{x\over n}\right)^n$$</span></p>
<p>I would like to show that the RHS sequence is always less than or equal to <span class="math-container">$e^x$... | pre-kidney | 34,662 | <p>To show that <span class="math-container">$(1+x/n)^n\leq e^x$</span> for all <span class="math-container">$x\geq -n$</span> is equivalent to showing that
<span class="math-container">$$
1+\frac{x}{n}\leq e^{x/n},\qquad x\geq -n.
$$</span>
This is a special case of the following inequality
<span class="math-container... |
1,283,325 | <p>I got two questions about $p$-adic numbers:</p>
<blockquote>
<ol>
<li>I often read that the field $\mathbb Q_p$ is much different than the field $\mathbb R$.</li>
</ol>
</blockquote>
<p>An element of $\mathbb Q_p$ is of the form $\sum_{i=-k}^{\infty}a_ip^i$ where $a_i\in \{0,...,p-1\}$.</p>
<p>But isn't thi... | Crostul | 160,300 | <p>As for the first question, the answer is no. For example, what real number should represent this
$$\sum_{n=0}^{\infty} 5^{n!}$$
$5$-adic number? Note that in real numbers this series is obviously divergent.</p>
<p>As for the second question: you should know that $\Bbb{Z}_p$ is a local ring whose unique maximal idea... |
1,858,297 | <p>Suppose the diameter of a nonempty set $A$ is defined as </p>
<p>$$\sigma(A) := \sup_{x,y \in A} d(x,y)$$</p>
<p>where $d(x,y)$ is a metric.</p>
<p>Is $\sigma(.)$ a 'measurement'? I.e., how do I prove the countable additivity for this particular case?</p>
| Ethan Bolker | 72,858 | <p>It's not even finitely additive. If $X$ and $Y$ are two disjoint closed intervals on the real line then the diameter of their union is not the sum of their diameters.</p>
|
1,070,870 | <p>"Write down (say, as a power series) a holomorphic function $f(z)$ on $D(1, 1)$ which satisfies $f(z)^5 = z$ and $f(1) = 1$. What is the result of analytically continuing $f$ along a path which travels once counterclockwise around the origin, returning to the point $1$? What about if you go $N$ times counterclockwis... | JohnD | 52,893 | <p>Unsure of your notation/assumptions, but here's a hint: </p>
<ol>
<li>For real matrices, $$\text{rank}(A^*A)=\text{rank}(AA^*)=\text{rank}(A)=\text{rank}(A^*)$$</li>
<li>For complex matrices, $$\text{rank}(A^*A)=\text{rank}(A)=\text{rank}(A^*)$$</li>
</ol>
<p>Mouse over for more after you've pondered it a bit:</p>... |
3,450,598 | <blockquote>
<p>Prove that <span class="math-container">$\sum_{i = m}^n a_i + \sum_{i = n + 1}^p a_i = \sum_{i = m}^p a_i$</span>, where <span class="math-container">$m ≤ n<p$</span> are integers, and <span class="math-container">$a_i$</span> is a real number assigned to each integer <span class="math-container">$... | Community | -1 | <p><span class="math-container">$$\sum_{i=m}^n a_i+\sum_{i=n+1}^{p}a_i=\sum_{i=m}^n a_i+a_{n+1}-a_{n+1}+\sum_{i=n+1}^{p}a_i
=\sum_{i=m}^{n+1} a_i+\sum_{i=n+2}^{p}a_i$$</span></p>
<p>is a base case and by induction,<span class="math-container">$$\sum_{i=m}^n a_i+\sum_{i=n+1}^{p}a_i
=\sum_{i=m}^{n+k} a_i+\sum_{i=n+k+1}^... |
1,626,821 | <p>Is there a name for a vector with all equal elements? If so, what is it?</p>
<p>For example,</p>
<p>$$ (7, 7, 7, 7, 7) $$</p>
| Zubin Mukerjee | 111,946 | <p>This is <a href="https://mathoverflow.net/questions/9898/notation-for-the-all-ones-vector">relevant</a>. Given how much of a hullabaloo there is over just the all-ones vector, I'm guessing there is no standard name for the more general vectors with all equal elements. </p>
<p>If we use $\vec{1}$ for the all-ones ve... |
40,348 | <p>I'm trying to prove the following statement (an exercise in Bourbaki's <em>Set Theory</em>): </p>
<p><em>If $E$ is an infinite set, the set of subsets of $E$ which are equipotent to $E$ is equipotent to $\mathfrak{P}(E)$.</em> </p>
<p>As a hint, there is a reference to a proposition of the book, which reads: </p>
... | Mike F | 6,608 | <p>Another approach would make use of the fact that $\kappa_1 + \kappa_2 = \max(\kappa_1,\kappa_2)$ when $\kappa_1,\kappa_2$ are cardinals at least one of which is infinite. From this it follows that, for any subset of $S \subset X$, either $S$ or its complement has cardinality $|X|$. Since more straightforward cardina... |
1,613,645 | <p>Let's get started:</p>
<p>$$\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi} |x|e^{-inx} dx$$</p>
<p>since $|x|$ is an even function:</p>
<p>$$= \frac{1}{\pi}\int_0^{\pi} xe^{-inx} dx$$</p>
<p>Integration by parts yields:</p>
<p>$$e^{-inx}\Big|_0^{\pi} + \frac{1}{in} \int_0^\pi e^{-inx} dx = (-1)^n - 1 + \frac{1}{in} \l... | Igor | 165,838 | <p>First you can set up a proportion to find the central angle it has moved,</p>
<p>35/60=x/360→x=210</p>
<p>Then you can use the arc length formula
arc length = 2πr(x/360) = 2π(15)(210/360)=35π/2.</p>
|
3,673,014 | <p>If you take an <span class="math-container">$2r\times 2r\times 2r$</span> cube, and divide it to 27 equal cubes, and then remove all the "axis" cubes (all the cubes which are straight left, straight right, straight up, etc. from the middle cube) then divide each cube into 125 equal cubes and remove all the axis cube... | amd | 265,466 | <p>In the method that you propose to use, the expression that defines the set is a convex combination of the vertices of the polyhedron. It’s a bit unfortunate that it was presented to you in simplified form since that obscures the simple construction: It’s just a linear combination <span class="math-container">$\sum_i... |
2,238,734 | <p>Let G be a group of rationals under addition, if $G_1$ and $G_2$ are two non empty subgroups of G, then prove that $G_1 \cap G_2 \neq${0}</p>
| Community | -1 | <p>It is the first definition. Because you know that $\mathbb{R^n}$ is a vector space, the second thing you wrote is trivial.</p>
<p>To show that $W$ is a subspace of $V$, you have to show it is closed under scalar multiplication and vector addition, so you have to show that:</p>
<p>$\forall x,y \in W: x+y \in W$</p>... |
1,983,745 | <p>I need to choose weather this is a product notation or a summation. I can figure out which one it is.</p>
<p>I have this expression:</p>
<p>$$2 \times 4 \times 6 \times 8 \times 10 \ldots \times 40$$</p>
<p>The answer is either:</p>
<blockquote>
<p>$$\sum_{m=2}^{40} m$$</p>
</blockquote>
<p><strong>or</strong... | Siong Thye Goh | 306,553 | <p>Neither of your proposal is correct.</p>
<p>For your first guess, it means $$2+3+4+\ldots+ 40$$</p>
<p>For your second guess, it means $$2(3)(4) \ldots (40)$$</p>
<p>You are multiplying even numbers, it should be</p>
<p>$$\prod_{i=1}^{20} (2i)$$</p>
|
1,983,745 | <p>I need to choose weather this is a product notation or a summation. I can figure out which one it is.</p>
<p>I have this expression:</p>
<p>$$2 \times 4 \times 6 \times 8 \times 10 \ldots \times 40$$</p>
<p>The answer is either:</p>
<blockquote>
<p>$$\sum_{m=2}^{40} m$$</p>
</blockquote>
<p><strong>or</strong... | Nambiar M. | 374,565 | <p>Neither is correct. It is in fact the product notation, as summation notation would be $2+4+6+8+...+40$, however, the expression is incorrect. The correct expression is $\prod_{m=1}^{20}2m$. There is a way to do it with summation notation, but I don't think that's what you're looking for.</p>
|
3,197,540 | <p>Let a function be defined as:</p>
<p><span class="math-container">$ f(x)=x^2\sin{\left(\frac 1x\right)}$</span> for <span class="math-container">$x \neq 0$</span> and
<span class="math-container">$ f(x)=0$</span> for <span class="math-container">$x=0$</span></p>
<p>I'm trying to prove that f is differentiable at ... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Use that <span class="math-container">$$\sin(x)-\sin(y)=2 \sin \left(\frac{x}{2}-\frac{y}{2}\right) \cos
\left(\frac{x}{2}+\frac{y}{2}\right)$$</span></p>
|
2,317,391 | <p>This question popped up somewhere on the internet and I thought it was interesting. I attempted to solve it but I don't know if it is correct.</p>
<blockquote>
<p>Find the derivative of $$F(x)=\int_{\cos{x^3}}^{\int_{1}^{x} {1/(1+t^2)dt}} {\sin{w} dw}$$</p>
</blockquote>
<p>$$\begin{align}
\implies F(x) & =-... | Yes | 155,328 | <p>Yes, your concept is correct. But note that we have
$$
F(x) = \int_{g(x)}^{f(x)}\sin w dw = \int_{a}^{f(x)} \sin w dw - \int_{a}^{g(x)} \sin w dw
$$
where $f,g$ are the given functions appearing in the original upper and lower integral limits, respectively; note that $f,g,F$ are "nice" enough to allow us to write
... |
1,322,016 | <p>Find the vertical asymptotes (if any) of the graph of the function. (Use $n$ as an arbitrary integer if necessary.)</p>
<p>$$s(t)= \frac{8t}{\sin{t}}$$</p>
<p>$t= ?$, where n cannot $=?$</p>
<p>I need a general rule for the asymptotes with where the exception of $n$ is. </p>
| RowanS | 246,774 | <p>We are not allowed to divide by $0$ so $sin(t) = 0$ is an asymptote. So $t = n\pi \quad \forall n \in \mathbb Z - \{ 0 \} $ </p>
|
1,322,016 | <p>Find the vertical asymptotes (if any) of the graph of the function. (Use $n$ as an arbitrary integer if necessary.)</p>
<p>$$s(t)= \frac{8t}{\sin{t}}$$</p>
<p>$t= ?$, where n cannot $=?$</p>
<p>I need a general rule for the asymptotes with where the exception of $n$ is. </p>
| Irddo | 71,356 | <p>So, you have a vertical asymtote, very times that $f(x)\to+\infty$ or $f(x)\to-\infty$, when $x\to a\in \mathbb{R}$, so $x=a$ will be a vertical asymtote.</p>
<p><strong>Hint:</strong> For which $x\in \mathbb{R}$, you have the $f(x)\to+\infty$ or $f(x)\to-\infty$? </p>
<p>(Try to annul the denominator of the funct... |
2,439,340 | <p>How would one proceed to prove this statement?</p>
<blockquote>
<p>The set of the strictly increasing sequences of natural numbers is not enumerable.</p>
</blockquote>
<p>I've been trying to solve this for quite a while, however I don't even know where to start.</p>
| David K | 139,123 | <p>For any infinite subset of the natural numbers, you can list its members in increasing order, and then you have a sequence that is strictly increasing.</p>
<p>Moreover, if you take two distinct infinite subsets of the natural numbers,
you get two different sequences.
(There is some number $k$ that is in one of the ... |
231,036 | <p>I wonder if there is an example of rational homology sphere that is not a Seifert manifold. If there is, how can one construct such a rational homology sphere from a surgery of a knot in $S^3$?</p>
| Daniel Valenzuela | 52,936 | <p>By Thurston, all but finitely many $(p,q)$-surgeries on a hyperbolic knot in $S^3$ result in hyperbolic rational homology spheres for $p\neq 0$. In particular there are infinitely many integral homology spheres among them.</p>
|
3,950,098 | <p>I can evaluate the limit with L'Hospital's rule:</p>
<p><span class="math-container">$\lim_{n\to\infty}n(\sqrt[n]{4}-1)=\lim_{n\to\infty}\cfrac{(4^{\frac1n}-1)}{\dfrac1n}=\lim_{n\to\infty}\cfrac{\dfrac{-1}{n^2}\times 4^{\frac1n}\times\ln4}{\dfrac{-1}{n^2}}=\ln4$</span></p>
<p>But is there any way to do it without us... | Sewer Keeper | 213,667 | <p>Another approach using known limit</p>
<blockquote>
<p><span class="math-container">$$ \lim_{n \to +\infty} \frac{\mathrm{e}^{a_n}-1}{a_n} = 1 $$</span>
where <span class="math-container">$a_n$</span> is a sequence such that <span class="math-container">$ \lim_{n \to +\infty} a_n = 0$</span>.</p>
</blockquote>
<p><... |
3,380,081 | <p>Question: Suppose <span class="math-container">$n(S)$</span> is the number of subset of <span class="math-container">$S$</span> and <span class="math-container">$|S|$</span> be the number of elements of <span class="math-container">$S$</span>. If <span class="math-container">$n(A)+n(B)+n(C)=n(A\cup B\cup C)$</span> ... | Jack D'Aurizio | 44,121 | <p><a href="https://i.stack.imgur.com/vpzAQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/vpzAQ.png" alt="enter image description here"></a></p>
<p>The area of the <span class="math-container">$A$</span>-region can be computed as the difference between the area of a <span class="math-container">$4... |
621,461 | <p>I'm having trouble understanding division when the divisor is greater than the dividend, for ex 1/4.</p>
<p>I think of division as "how many times can the divisor fit into the dividend evenly". </p>
<p>Intuitively, when I see 1/4 in the context of slices of pizza, I think of it as 1 "out of" 4, but I can't seem to... | user21820 | 21,820 | <p>I'm not sure of your level, but it might not be helpful to think of $\frac{a}{b}$ as "the number of times" $b$ fits into $a$ simply because that is not so intuitive when it comes to non-integer number of times. Instead, $\frac{a}{b}$ (where $b \ne 0$) is the amount such that $\frac{a}{b} \times b = a$. In other word... |
1,984,076 | <p>How to prove that
$$
\sum_{k=1}^\infty\frac{k^k}{k!}x^k=\frac{1}{2}, ~\text{where}~~ x=\frac{1}{3}e^{-1/3}~?
$$
I found this sum in my notes, but I don't remember where I got it. Any hints or references would be nice.</p>
| Andreas | 317,854 | <p>Let's consider $\sum_{k=1}^\infty\frac{k^k}{k!}\big(\frac{1}{ae^{1/a}}\big)^k=\frac{1}{a-1}$ for $a > 1$ (as in the question). </p>
<p>Then $\frac{1}{a-1} = - 1+ \frac{1}{1-1/a} = \sum_{k=1}^\infty (\frac1a)^k$ by the geometric series. So one has to show </p>
<p>$\sum_{k=1}^\infty\Big[-1 + \frac{k^k}{k!}{e^{-k... |
3,936,187 | <blockquote>
<p>Consider the differential equation <span class="math-container">$$(1+t)y''+2y=0$$</span>
with the variabel coefficient <span class="math-container">$(1+t)$</span>, with <span class="math-container">$t\in \mathbb{R}$</span>.</p>
<p>Set <span class="math-container">$y(t)=\sum_{n=0}^{\infty}a_nt^n$</span>.... | mathcounterexamples.net | 187,663 | <p>Take <span class="math-container">$t=0$</span>. You get from the ODE <span class="math-container">$y^{\prime \prime}(0)+y(0)=0$</span>.</p>
<p>And answer b) is the only valid option <em>providing that at least one is valid</em>.</p>
|
670,292 | <p>Could someone assist with the following three surface integrals? </p>
<p><strong>Q1</strong> The portion of the cone $z=\sqrt{x^2+y^2}$ that lies inside the cylinder $x^2+y^2 =2x$. </p>
<p><strong>Q2</strong> The portion of the paraboloid $z=1-x^2-y^2$ that lies above the $xy$-plane.</p>
<p><strong>Q3</strong... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> Define
$$
f(x) = \frac{x}{1-x}
$$
Noting that $f(x)$ is continuous except at $x = 1$. Suppose that $b_n = \frac{a_n}{1+a_n} \to l \neq 1$, what can we say about the sequence $f(b_n)$?</p>
|
1,762,001 | <p>I recently watched a <a href="https://www.youtube.com/watch?v=SrU9YDoXE88" rel="noreferrer">video about different infinities</a>. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \ldots, \omega_\omega$, etc..</p>
<p>I can't... | PMar | 335,420 | <p>One issue not yet addressed is why we have both $\aleph$s and $\omega$s. These both exist because Cantor introduced two separate concepts about infinite sets - their <em>sizes</em> ($\aleph$) and their <em>order-types</em> ($\omega$). Everyone else explained sizes, so I won't go over that.</p>
<p>For order-types,... |
1,762,001 | <p>I recently watched a <a href="https://www.youtube.com/watch?v=SrU9YDoXE88" rel="noreferrer">video about different infinities</a>. That there is $\aleph_0$, then $\omega, \omega+1, \ldots 2\omega, \ldots, \omega^2, \ldots, \omega^\omega, \varepsilon_0, \aleph_1, \omega_1, \ldots, \omega_\omega$, etc..</p>
<p>I can't... | Mikhail Katz | 72,694 | <p>The preoccupation expressed in the OP's question goes back (at least) to the 17th century, when Nieuwentijt objected to Leibnizian hierarchy of infinite numbers by claiming that there should be only one level of infinity, and reciprocally, only one level of infinitesimal, say $\epsilon$ (though Nieuwentijt didn't de... |
3,568,230 | <p>My question is: why, in general we cannot write down an formula for the <span class="math-container">$n-$</span>th term, <span class="math-container">$S_{n}$</span>, of the sequence of partial sums?</p>
<p>I will explain better in the following but the question is basically that one above.</p>
<p>Suppose then you ... | G Cab | 317,234 | <p>Actually you can express the partial sums of the Harmonic sequence
<span class="math-container">$$
\sum\limits_{k = 1}^n {{1 \over k}} = \psi (n + 1) - \psi (1) = \gamma + \psi (n + 1)
$$</span>
through the <a href="https://en.wikipedia.org/wiki/Digamma_function" rel="nofollow noreferrer">Digamma Function</a>.</p>... |
180,169 | <p>Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot. </p>
| Robert Bryant | 13,972 | <p>The questions you are asking have been carefully studied in the literature, but, usually, with carefully chosen global hypotheses so that reasonable results can be achieved. You should look at the works of William Goldman, Mischa Gromov, and Scott Adams, just to name a few, as well as more recent authors studying p... |
234,348 | <p>There is a nice <a href="https://www.wolfram.com/mathematica/new-in-10/basic-and-formula-regions/compute-region-distance.html.en" rel="nofollow noreferrer">example</a> of how to generate an isometric visualisation of surfaces which are at a constant distance from a given region.</p>
<p>All these examples are just wo... | cvgmt | 72,111 | <p><code>Min</code> work or not?</p>
<pre><code>reg1 = Cuboid[{-5, -5, -12}, {5, 5, 12}];
reg2 = Cuboid[{-10, -10, -10}, {10, 10, 10}];
ContourPlot3D[
Min[RegionDistance[reg1, {x, y, z}],
RegionDistance[reg2, {x, y, z}]] // Evaluate, {x, -15,
15}, {y, -15, 15}, {z, -15, 15}, Mesh -> None, Contours -> {0.2... |
234,348 | <p>There is a nice <a href="https://www.wolfram.com/mathematica/new-in-10/basic-and-formula-regions/compute-region-distance.html.en" rel="nofollow noreferrer">example</a> of how to generate an isometric visualisation of surfaces which are at a constant distance from a given region.</p>
<p>All these examples are just wo... | Carl Woll | 45,431 | <p>If you try evaluating your <a href="http://reference.wolfram.com/language/ref/RegionDistance" rel="nofollow noreferrer"><code>RegionDistance</code></a> object for a point you will see that it doesn't work:</p>
<pre><code>RegionDistance[
RegionUnion[Cuboid[{-5,-5,0},{5,5,1}],Cuboid[{-10,-10,-10},{10,10,0}]],
... |
236,927 | <p>I thought this would be a hard problem but I found a link that seems to ask the answer to this question as a homework problem? Can somone help me out here, are there an infinite number of prime powers that differ by 1? or are there a finite number of them? If so which are they?</p>
| xtimz | 164,004 | <p>Finally, I know why this problem is elementary.</p>
<p>$p^r-q^s=1$ where $p$, $q$ are primes, and $r$, $s$ are integers greater than $1$.</p>
<p>If $p$ and $q$ are both odd primes, then $p^r$ and $q^s$ are both odd, then $p^r-q^s$ can only be even. So, either $p$ or $q$ is a even prime, that is $2$.</p>
<p>It see... |
945,736 | <p>:)</p>
<p>I have this matrix:</p>
<p>B = \begin{bmatrix}
0.626 & 2.56 & 2.15 & \\
0.835 & 6.66 & 5.16 & \\
0 & 0 & -1.65 &
\end{bmatrix}</p>
<p>I was wondering how to find a givens matrix such that I could apply it from the right side of the matrix and eliminate... | John Alexiou | 3,301 | <p>See <a href="https://stackoverflow.com/a/4361442/380384">https://stackoverflow.com/a/4361442/380384</a></p>
<blockquote>
<pre><code> | a b tx |
A = | c d ty |
| 0 0 1 |
</code></pre>
<p>which transforms the coordinates [x,y,1] into:</p>
<pre><code>[x',y',1] = A * [x,y,1]
</code></pre>
<p>Th... |
1,278,848 | <p>Based on <a href="https://math.stackexchange.com/questions/1267021/let-m-subseteq-mathbbrk-manifold-topology-vs-trace-topology/1267760?noredirect=1#comment2573732_1267760">this</a> question I'd like to know: Are there compact (sub)manifolds without boundary in $\mathbb{R}^n$ ? Because, as that question shows, the to... | Autolatry | 25,097 | <p>Yes; there are some. If $M$ is a compact, oriented $n$-manifold without boundary, then there must be some $n$-forms which do not arise from taking the exterior derivatives of $n-1$-forms. </p>
<p>If $M$ is pseudo-Riemannian, so we have a Hodge star to work with, this tells us that we always have some functions on $... |
2,948,327 | <p>Being an undergraduate student I find difficult to understand the perfect differences between normal and partial differential equations. Elaborate the answer </p>
| Community | -1 | <p>An <em>ordinary</em> differential equation involves a derivative over a single variable, usually in an univariate context, whereas a <em>partial</em> differential equation involves several (partial) derivatives over several variables, in a multivariate context.</p>
<p>E.g.
<span class="math-container">$$\frac{dz(x)... |
869,341 | <p>What's the closed formula of this recurrence relation?
$$a_n = a_{n-1}+2a_{n-2}+2^n \text{ with } a_0=1, a_1=2 $$</p>
| Joel | 85,072 | <p>A generating function approach would make this straightforward:</p>
<p>$$G(x) = \sum_{n=0}^\infty a_n x^n = 1+2x+\sum_{n=2}^\infty a_n x^n$$
$$=1+2x+\sum_{n=2}^\infty (a_{n-1} + 2a_{n-2} + 2^n)x^n$$
$$=1+2x+\sum_{n=1}^\infty a_n x^{n+1} + 2 \sum_{n=0}^\infty a_n x^{n+2} + \sum_{n=2}^\infty 2^n x^n$$
$$=1+2x+x(G(x)-... |
869,341 | <p>What's the closed formula of this recurrence relation?
$$a_n = a_{n-1}+2a_{n-2}+2^n \text{ with } a_0=1, a_1=2 $$</p>
| Deathkamp Drone | 56,720 | <p>The best course of action for simple inhomogeneous recurrences is to make use of a smart "change of variables" (read: substitute with another recurrence relation) to turn it into a homogeneous recurrence. A nice observation here is to notice that <strong>$2^n$ is itself a recurrence relation</strong>, namely: </p>
... |
869,341 | <p>What's the closed formula of this recurrence relation?
$$a_n = a_{n-1}+2a_{n-2}+2^n \text{ with } a_0=1, a_1=2 $$</p>
| Mathsource | 12,624 | <p>This is nonhomogeneneous difference equations. First solve the homogeneous equation
$$
a_n - a_{n-1} - 2a_{n-2} = 0 \quad (1)
$$
Let $a_n = r^n$, so that $a_{n-1} = r^{n-1}$ and $r_{n-2} = r^{n-2} = 0$. Replacing in (1), we have
$$
r^n - r^{n-1} - 2r^{n-2} = 0 \quad r^2 - r - 2 = 0 \quad \Rightarrow \ r_1 = -1, \ r_... |
1,846,592 | <p>I know that a discrete topological space is where all singletons are open.</p>
<p>For example, $\mathbb{N}$ with the subspace topology inherited from $(\mathbb{R}, \mathfrak{T}_{usual})$. This is the case because we can find $\{n\} = (a,b) \cap \mathbb{N}$ which is open. Hence all singletons are open.</p>
<p>But a... | drhab | 75,923 | <p>Let $F$ be an arbitrary subset of $X$ where $X$ is equipped with discrete topology. </p>
<p>As you said: in a discrete topological space all singletons are open.</p>
<p>As you said: arbitrary unions of singletons are open so $F^c=\bigcup_{x\in F^c}\{x\}$ is an open set. </p>
<p>(You don't even need this subroute:... |
1,322,076 | <p>Hey can anybody help me with the following proof? I am trying to solve the following limit using epsilon delta and I have found the limit to be 1/3 using the squeeze theorem and have got to this thus far but am a bit confused where I go now as I have both a 3x and a sinx when trying to find an epsilon??
Thanks in ad... | RowanS | 246,774 | <p>You can easily show that $$\frac{\sin(x)+1}{3x+1}<\frac{2}{3x}$$
and then it is easy to show that $$\frac{2}{3x} < \epsilon $$</p>
|
1,516,450 | <p>When finding the Pythagorean triple where $a+b+c=1000$,
Wolfram alpha shows me that $a< -500\left(\sqrt{2} - 2\right)$</p>
<p>When I input $a^2+b^2=c^2, a<b<c$ and $a+b+c=1000$</p>
<p>How does wolfram arrive at that inequality:</p>
<p>$a< -500\left(\sqrt{2} - 2\right)$</p>
<p>Here is the link: <a hre... | lulu | 252,071 | <p>Note: seeing the edited form of the question, the following isn't really on point. It addresses an efficient algorithm for finding the triple, it does not speak to whatever Wolfram Alpha might be doing.</p>
<p>Playing with the code, the following emerges: Suppose $$a^2+b^2=c^2\;\;\;\&\;\;\;a+b+c=S$$
We remar... |
3,831,198 | <p>Suppose that <span class="math-container">$100$</span>kg of a radioactive substance decays to <span class="math-container">$80$</span>kg in <span class="math-container">$20$</span> years.</p>
<p>a) Find the half-life of the substance (round to the nearest year).</p>
<p>b)Write down a function <span class="math-conta... | Math Lover | 801,574 | <p>If <span class="math-container">$A_t$</span> is the amount left at time <span class="math-container">$t$</span>, <span class="math-container">$A_0$</span> is the initial amount and <span class="math-container">$k$</span> is the rate constant for the first order reaction, we know for the radioactive decay (which is f... |
3,015,149 | <p>I’m using the formula that the number of conjugacy class is given to be <span class="math-container">$\frac{1}{|G|}\sum|C_{G}(g)|$</span>, where <span class="math-container">$C_{G}(g)=\{h \in G ; gh=hg\}$</span>, which is a special result by Burnside’s theorem.</p>
<p>I found that the number of conjugacy class in <... | Doug M | 317,162 | <p><span class="math-container">$G = \{r,a|r^8 = a^2 = e, ara^{-1} = r^{-1}\}$</span></p>
<p><span class="math-container">$e$</span> commutes with all elements and is in a conjugacy class all by itself.</p>
<p><span class="math-container">$\{e\}$</span></p>
<p>Rotations fall into conjugacy classes that include their... |
120,992 | <p>An algorithm book <a href="http://rads.stackoverflow.com/amzn/click/1849967202" rel="nofollow">Algorithm Design Manual</a> has given an description:</p>
<blockquote>
<p>Consider a graph that represents the street map of Manhattan in New York City. Every junction of two streets will be a vertex of the graph. Neigh... | Brian M. Scott | 12,042 | <p>Each street crossing is a vertex of the graph. An avenue crosses about $200$ streets, and each of these crossings is a vertex, so each avenue contains about $200$ vertices. There are $15$ avenues, each of which contains about $200$ vertices, for a total of $15\cdot 200=3000$ vertices.</p>
<p>To make the description... |
2,724,686 | <blockquote>
<p>Set $B = \{1,2,3,4,5\}$, $S$ - equivalence relation. It is given that for all $x,y \in B$ if $(x,y)\in S$ and if $x+y$ is an even number then $x = y$. In such case is it true that:</p>
<ol>
<li>the number of elements in each equivalence class of $S$ is at most $2$</li>
<li>any relation $S$ wo... | Mark Bennet | 2,906 | <p>The answer to the first part is that if an equivalence class has three elements in it then two are odd or two are even, and the pair of common parity gives a counterexample to the condition.</p>
<p>For the second part, any equivalence class of size $2$ must consist of an odd number and an even number. Can you creat... |
4,562,451 | <p>I had this maths question:</p>
<blockquote>
<p>Given that <span class="math-container">$$8\sqrt{p} = q\sqrt{80}$$</span> where <span class="math-container">$p$</span> is prime, find the value of <span class="math-container">$p$</span> and the value of <span class="math-container">$q$</span></p>
</blockquote>
<p>I di... | lone student | 460,967 | <p>The answer is "no". Because, we don't have an important restriction <span class="math-container">$q\in\mathbb Z^{+}$</span>. Otherwise, we have infinitely many solutions:</p>
<p><span class="math-container">$$q=2\sqrt {\frac p5}$$</span>
where, <span class="math-container">$p$</span> is an any prime number... |
1,218,354 | <p>I've read in some textbooks that $\vdash$ and $\vDash$ are symbols present only in metalanguage. From this, I infer that their use in object language is unacceptable.</p>
<p>I would like to know why. Can't we define them as relation symbols in a structure? Or introduce them in statements for the sake of formal proo... | goblin GONE | 42,339 | <p>I can explain it like this.</p>
<p>Given a set $X$, write $X^*$ for the collection of all finite sequences in $X$, including the empty sequence. Now let $L$ denote an arbitrary set, which we think of as a "language"; so you should be thinking of the elements of $L$ as "formulae". Then:</p>
<blockquote>
<p><stron... |
4,035,300 | <p>If someone could just do a very basic walkthrough on how you would go about answering this question it would be greatly appreciated as I'm practising for an exam!</p>
<p>'''</p>
<p>When a company bids for contracts it estimates the probability of winning each contract is <span class="math-container">$0.18$</span>, i... | Darth Geek | 163,930 | <p>We will prove the last statement.</p>
<p>Let <span class="math-container">$\alpha$</span> be a root of <span class="math-container">$p_1:= X^3 + X^2-1$</span>, i.e., <span class="math-container">$\alpha^3 +\alpha^2-1 = 0$</span>. We will prove that <span class="math-container">$\beta := \alpha^2$</span> is a root of... |
1,658,846 | <ul>
<li><p>How can one prove the existence of an order preserving bijection from $\mathbb{Q}$ to $\mathbb{Q}\backslash\lbrace{0}\rbrace$?</p></li>
<li><p>Can you give an example of such a bijection? </p></li>
</ul>
| hmakholm left over Monica | 14,366 | <p>Choose an irrational number $\alpha$.</p>
<p>Let $x_1, x_2, \ldots$ be a strictly <em>increasing</em> sequence of rational numbers that converge towards $\alpha$.</p>
<p>Let $y_1, y_2, \ldots$ be a strictly <em>decreasing</em> sequence of rational numbers that converge towards $\alpha$.</p>
<p>Then define $f:\mat... |
1,658,846 | <ul>
<li><p>How can one prove the existence of an order preserving bijection from $\mathbb{Q}$ to $\mathbb{Q}\backslash\lbrace{0}\rbrace$?</p></li>
<li><p>Can you give an example of such a bijection? </p></li>
</ul>
| Igor Rivin | 109,865 | <p>See the proof of theorem 3.7 here: <a href="http://www.math.wustl.edu/~freiwald/ch8.pdf" rel="nofollow">http://www.math.wustl.edu/~freiwald/ch8.pdf</a></p>
|
1,557,353 | <p>Context: I'm taking calc based physics, and we are supposed to be able to integrate moment of inertia for a cylinder. I referenced a mit vid, and though I have no education on multiple integrals, I got all but one thing. <a href="https://www.youtube.com/watch?v=iYFogDTPlRo" rel="nofollow">https://www.youtube.com/wat... | Hamed | 191,425 | <p>Other than David's geometrical method which is nicer, here is an algebraic method. (Remember $r$ is the radial length in the <strong>x-y</strong> plane)
$$
(x,y,z)=(r\cos\theta, r\sin\theta, z)
$$
Now $dx dy dz = Jdrd\theta dz$, where $J$ is the Jacobian and is defined as (this works for any change of coordinates)
$... |
1,557,353 | <p>Context: I'm taking calc based physics, and we are supposed to be able to integrate moment of inertia for a cylinder. I referenced a mit vid, and though I have no education on multiple integrals, I got all but one thing. <a href="https://www.youtube.com/watch?v=iYFogDTPlRo" rel="nofollow">https://www.youtube.com/wat... | root | 264,845 | <p>For more-dimensional change of coordinates (in this case: cartesian $\rightarrow$ cylindric) one uses, that (under certain circumstances you can find in every textbook) substitution is given by</p>
<p>$$ \int_{g(D)} f(\mathbf{y}) d\mathbf{y} = \int_D (f \circ g)(x) \left| \det \left( \frac{dg}{d\mathbf{x}}(\mathbf{... |
3,772,534 | <p>Tangents to a circumference of center O, drawn by an outer point C, touch the circle at points A and B. Let S be any point on the circle. The lines SA, SB and SC cut the diameter perpendicular to OS at points A ', B' and C ', respectively. Prove that C 'is the midpoint of A'B'.</p>
<p><a href="https://i.stack.imgur.... | Riemann'sPointyNose | 794,524 | <p>To expand a little on @Angina Seng's point, essentially the thing to notice is that if we can find the fourth root of <span class="math-container">${-1}$</span>, and replace <span class="math-container">$x$</span> with <span class="math-container">$x$</span> multiplied by this fourth root you get</p>
<p><span class=... |
1,690,346 | <p>How can we solve this integral?
$\int_0^1\:\frac{\ln(x)\:\Big[1+x^{-\frac{1}{3}}\Big]}{(1-x)\sqrt[3]{x}}\:dx$</p>
| Yuriy S | 269,624 | <p>We have</p>
<p>$$\int_0^1 x^a dx = \frac{1}{a+1}$$</p>
<p>$$\int_0^1 x^a \log x ~dx = -\frac{1}{(a+1)^2}$$</p>
<p>$$\int_0^1 x^b x^q \log x ~dx = -\frac{1}{(b+q+1)^2}$$</p>
<p>$$\int_0^1 \sum_{b=0}^{\infty} x^b x^q \log x ~dx = -\sum_{b=0}^{\infty} \frac{1}{(b+q+1)^2}$$</p>
<p>$$\int_0^1 \frac{x^q \log x}{1-x} ... |
1,859,178 | <p>Let parabola $\Gamma_{1}:$$y^2=4x$,and hyperbolic curve $\Gamma_{2}$: $x^2-y^2=1$.it is well known this two is symmetric.so the two point $A$ and $B$ about $x$ axial symmetry,or mean $x_{A}=x_{B}$.see figure,</p>
<p><a href="https://i.stack.imgur.com/wCPAN.png" rel="nofollow noreferrer"><img src="https://i.stack... | Giacomo Ascione | 354,026 | <p>I know it's rude to give complex solutions to real problems, but I think we can easily see where the second solution $x=2-\sqrt{5}$ came from.</p>
<p>We want to find solutions for: $$\begin{cases} y^2=4x \\ x^2-y^2=1 \end{cases}$$ Let's try find them in $\mathbb{C}$.</p>
<p>We got, as you said before, the equation... |
2,522,342 | <p>So far I have only got 9 from just guess and check. I am thinking of using Vieta's Formula, but I am struggling over the algebra. Can someone give me the first few steps?</p>
| user502959 | 502,959 | <p>Assuming that $x$ is a natural number, it's quite trivial, since $2^x-1,\ 2^x,\ 2^x+1$ are three consecutive numbers, so one of them is divisible by 3, and it obviously isn't $2^x$.</p>
<p>3 is the only prime divisible by 3, so:</p>
<p>$2^x-1=3$ leads to $x=2$ - good ($2^x+1=5$ - prime).</p>
<p>$2^x+1=3$ leads to... |
1,182,432 | <p>Is it possible, that everyone is a pseudo-winner in a tournament with 25 people?
(pseudo-winner means that either he won against everyone, or if he lost against someone, then he beated someone else, who beated the one who he lost to).</p>
<p>In the "language" of graph theory: Is it possible in the directed K(25) (a... | Misha Lavrov | 383,078 | <p>The answer is a <a href="https://www.umop.com/rps25.htm" rel="nofollow noreferrer">well-known graph</a> generalizing a classical 3-vertex construction:</p>
<p><a href="https://i.stack.imgur.com/sZt9F.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sZt9F.jpg" alt="enter image description here"></a... |
2,935,693 | <p>I am trying to prove that </p>
<p><span class="math-container">$(a\to(b\to c))\to((a\to b)\to(a\to c))$</span></p>
<p>holds in natural deduction, in particular when I work backwards from a Fitch style proof I can only get so far:</p>
<p><a href="https://i.stack.imgur.com/w2BYf.png" rel="nofollow noreferrer"><img ... | Graham Kemp | 135,106 | <p><span class="math-container">$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}$</span>
How exactly does one go about proving a conditional statement? A Conditional Proof is usually used. Step one: assume the antecedant. Step two: derive the consequent. Step three: introduce t... |
1,025,321 | <p>$\ln(1+xy) = xy$</p>
<p>When I try to implicitly differentiate this I get</p>
<p>$\frac{1}{1+xy}(y + xy')$ = (y + xy')</p>
<p>At which point the $(y + xy')$ terms cancel out, leaving no $y'$ to solve for.</p>
<p>However, the answer to this is $-\frac{y}{x}$... How do you get this?</p>
| PtF | 49,572 | <p>As to the book recommendations I suggest the following:</p>
<p><strong>(i)</strong> A course in Differential Geometry (W. Klingenberg)</p>
<p><strong>(ii)</strong> Elementary Differential Geometry (A. Pressley)</p>
<p><strong>(iii)</strong> Differential-Geometrie und Minimal-Flächen (J. Jost) (in german but it's ... |
355,489 | <p>What are suggestions for reducing the transmission rate of the current epidemics?</p>
<p>In summary, my best one so far is (once we are down to the stay home rule) to discretize time, i.e., to introduce the following rule for the general populace not directly involved in necessary services:</p>
<p><em>If members o... | fedja | 1,131 | <p>Response to Steven Landsburg:</p>
<p>The main point Steven made was that M is at least as important as <span class="math-container">$A$</span>.
How to reduce M? (the interaction between public servants and ordinary people)? That, as Steven correctly pointed out, is in the hands of people. The servants just serve wh... |
669,207 | <p>For any function $f$ and any $x∈Dom(f)$, if for any neighbourhood $S$ of $x$,</p>
<p>$\qquad f(t)=0$ for some $t∈S$ </p>
<p>$\qquad f(u)=1$ for some $u∈S$ </p>
<p>then $ f$ is discontinuous at $x$.</p>
<p>Why is this true? I find it very hard to understand it :(.</p>
| Patrick Da Silva | 10,704 | <p>Inverse image applied to elements only makes sense if the function is injective, in which case we can restrict the range to the image and get a bijective map for which $f^{-1}$ is a well-defined map. Otherwise we reserve the notation $f^{-1}$ for the map given by $f^{-1} : \mathcal P(Y) \to \mathcal P(X)$ which send... |
2,875 | <p>I've heard that irreducible unitary representations of noncompact forms of simple Lie groups, the first example of such a group <code>G</code> being <code>SL(2, R)</code>, can be completely described and that there is a discrete and continuous part of the spectrum of <code>L^2(G)</code>.</p>
<ol>
<li>How are those ... | Hadi | 9,962 | <p>Regarding classification of irreducible unitary representations of <em>arbitrary</em> Lie groups, Michel Duflo showed (I guess in the early 80's) that at least for algebraic Lie groups the classification can be reduced to the case of reductive Lie groups. See Vogan's <em>Unitary representations of reductive Lie grou... |
692,085 | <p>It appears that taking the cube root of a negative number will yield a negative number, which when squared, will yield a positive number. But all the calculators and books I have seen show this particular problem yields a negative number. Any help here?</p>
| Thomas | 26,188 | <p>Indeed
$$
(-64)^{2/3} = 16.
$$
This can be seen two ways:
$$\begin{align}
(-64)^{2/3} &= ((-64)^{1/3})^2 = (-4)^2 = 16 \\
(-64)^{2/3} &= ((-64)^2)^{1/3} =(4096)^{1/3} = 16.
\end{align}
$$
Here remember that
$$
a^{1/3} = \sqrt[3]{a}.
$$</p>
|
692,085 | <p>It appears that taking the cube root of a negative number will yield a negative number, which when squared, will yield a positive number. But all the calculators and books I have seen show this particular problem yields a negative number. Any help here?</p>
| kitkat | 131,788 | <p>Well let's take this step by step though you do seem to have an understanding of it.</p>
<p>The cube root of -64 is -4. So yes It does yield a negative number. Then the square of that is 16, so like the previous person said perhaps you have not entered it into the calculator properly. </p>
<p>Ah technology, a bles... |
2,155,740 | <p>This is a homework problem for which I think I've missed the point or have incorrectly done the proof (or both). There are two parts to the problem: Let $(a_n)$ be a sequence with $a_n \ge 0,$ for all $n$.</p>
<p>Part 1:<br>
Suppose that $a_n \rightarrow 0$. Show that $\sqrt{a_n}\rightarrow 0$. I recognized that... | RRL | 148,510 | <p>Hint: </p>
<p>For an all-encompassing proof with $L \geqslant 0$ note that</p>
<p>$$|\sqrt{a_n} - \sqrt{L}|^2 \leqslant |\sqrt{a_n} - \sqrt{L}||\sqrt{a_n} + \sqrt{L}| = |a_n - L|$$</p>
|
1,607,190 | <p>Prove by induction that $8^{n} − 1$ for any positive integer $n$ is divisible by $7$. </p>
<p>Hint: It is easy to represent divisibility by $7$ in the following way: $8^{n} − 1 = 7 \cdot k$ where k is a positive integer.</p>
<p>This question confused me because I think the hint isn't true. If $n = 1$ and $k = 2$ f... | Dr. Sonnhard Graubner | 175,066 | <p>let $$T_k=8^k-1$$ then $$T_{k+1}=8(8^k-1)+7$$</p>
|
2,784,784 | <p>Let $f(x)=$$x-1 |x \in \mathbb{Q} \brace 5-x| x \in \mathbb{Q}^c$</p>
<p>Show that $\lim_{x \to a}f(x)$ does not exists for any $a \not= 3$</p>
<p>I first showed that $lim_{x \to 3}f(x)=2$. </p>
<p>I don't know how to approach this part. Can anyone please guide? I was thinking of using density theorem at first t... | Sarvesh Ravichandran Iyer | 316,409 | <p>Of course, you must use the fact that for any point, there is a sequence of rationals and another sequence of irrationals which converge to that point.</p>
<p>Here is the conventional limit definition :</p>
<blockquote>
<p>Given $f : D \to \mathbb R$ and $x \in D$, suppose there is a number $L$ such that for eve... |
4,150,776 | <p>Let <span class="math-container">$(a_n)_{n=1}^\infty$</span> Let be a positive, increasing, and unbounded sequence. Prove that the series:</p>
<p><span class="math-container">$$\sum_{n=1}^\infty\left(\frac{1}{a_{2n-1}}-\frac{1}{a_{2n}}\right)$$</span></p>
<p>convergent.</p>
<hr />
<p>We know that since <span class=... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>Since <span class="math-container">$a_n$</span> is positive and increasing , then <span class="math-container">$\frac1{a_n}$</span> is positive and decreasing. Moreover, we have</p>
<p><span class="math-container">$$\sum_{n=1}^{2N}\frac{(-1)^{n-1}}{a_n}=\sum_{n=1}^N \left(\frac1{a_{2n-... |
172,124 | <p>$$\int \frac{1}{x^{10} + x}dx$$</p>
<p>My solution :</p>
<p>$$\begin{align*}
\int\frac{1}{x^{10}+x}\,dx&=\int\left(\frac{x^9+1}{x^{10}+x}-\frac{x^9}{x^{10}+x}\right)\,dx\\
&=\int\left(\frac{1}{x}-\frac{x^8}{x^9+1}\right)\,dx\\
&=\ln|x|-\frac{1}{9}\ln|x^9+1|+C
\end{align*}$$</p>
<p>Is there completely ... | Venus | 146,687 | <p>Let we generalise the problem with a slightly different way. Consider
$$\int\frac{\mathrm dx}{x^n+x}$$
Making substitution $x=\frac{1}{y}$ and $z=y^{n-1}$ then reverse it back we get
\begin{align}
\int\frac{\mathrm dx}{x^n+x}&=-\int\frac{y^{n-2}}{1+y^{n-1}}\mathrm dy\\[9pt]
&=-\frac{1}{n-1}\int\frac{\mathrm ... |
185,549 | <blockquote>
<p>Suppose $F$ is a nondecreasing and right continuous function, and the sequence $\{x_n\}_{n\geq1}$ converges to $x$. Then $\liminf\limits_{n\to\infty}F(x_n)\leq F(x)$. </p>
</blockquote>
<p>How can I prove this?</p>
| Did | 6,179 | <p><strong>Hints:</strong></p>
<ul>
<li>For every $z\gt F(x)$, there exists $y\gt x$ such that $F(u)\leqslant z$ for every $u\leqslant y$. </li>
<li>Since $x_n\to x$, $x_n\leqslant y$ for every $n$ large enough.</li>
<li>Hence...</li>
</ul>
<p><strong>Edit:</strong></p>
<p>... $F(x_n)\leqslant F(y)\leqslant z$ for e... |
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