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 |
|---|---|---|---|---|
1,921,687 | <p>given transformation to solve the following :
$$3x^5-y(y^2-x^3)y'=0 \qquad u=x^3, v=y^2 $$ </p>
<p>i get
$$v'=2yy'$$
$$u'=3x^2x'$$
we have </p>
<p>$$uu'-(\frac{v-u}{2})v'=0$$
how can I solve this equation to get a general solution of $(y^2-2x^3)(y^2+x^3)^2=c$. Thank you.</p>
| operatorerror | 210,391 | <p>$$u=x^3\Rightarrow \mathrm du=3x^2\mathrm dx \Rightarrow dx=\frac{\mathrm du}{3x^2}$$ and similarly $$v=y^2\Rightarrow \mathrm dv=2y\mathrm dy\Rightarrow \mathrm dy=\frac{\mathrm dv}{2y}$$</p>
<p>But replacing $\mathrm dx$ and $\mathrm dy$ transforms your ode into something quite nice:
$$
\frac{3x^5}{3x^2}\mathrm d... |
1,921,687 | <p>given transformation to solve the following :
$$3x^5-y(y^2-x^3)y'=0 \qquad u=x^3, v=y^2 $$ </p>
<p>i get
$$v'=2yy'$$
$$u'=3x^2x'$$
we have </p>
<p>$$uu'-(\frac{v-u}{2})v'=0$$
how can I solve this equation to get a general solution of $(y^2-2x^3)(y^2+x^3)^2=c$. Thank you.</p>
| Argento | 258,242 | <p>$$\frac{du}{dv}=\frac{(1-\frac{u}{v})}{2(\frac{u}{v})}$$
let $$w=\frac{u}{v} ,u'=w+vw'$$
we obtain $$\frac{2w}{(1-2w)(1+w)}dw = \frac{1}{v}dv$$
then
$$\int \frac{2}{3(1-2w)}-\frac{2}{3(1+w)}dw=\int\frac{1}{v}dv$$
therefore
$$ (y^2-2x^3)(y^2+x^3)^2=c$$</p>
<p>Thank you everybody. I just found it.</p>
|
2,133,683 | <p>1)</p>
<ul>
<li>this answer is incorrect. correct answer is $^{1029}P_{10}$.. Not sure why permutation instead of combination. </li>
</ul>
<p><em>Count the number of different ways that a disk jockey can play 10 songs from her station's library of 1029 songs.</em></p>
<p>For this one I did, $\binom{1029}{10}$ and... | B. Goddard | 362,009 | <p>For 1, order <em>does</em> matter, so it's a permutation. (Unless it's a country station, in which case all songs are really the same, so the answer is $1$.) </p>
<p>For 2, you need to count 3-letter and 4-letter combinations separately. I think the answer should be:</p>
<p>3-letter: $2\times 26 \times 26$</p>
... |
1,726,228 | <p>Here, $A\in \mathbb{K}^{n,n}$ ($n$ by $n$ matrix) where $\frac{1}{2} \in \mathbb{K}$ where $\mathbb{K}$ is a field.</p>
<p>I think this is true. Since, using the fact that $\det(AB)=\det(A) \cdot \det(B)$ for $n$ by $n$ matrices, we can arrive at $[\det(A)]^3=1$ so $\det(A)=1$.</p>
<p>Am I right? </p>
| mathreadler | 213,607 | <p>$${\bf A}^3 = {\bf I} \implies \lambda_k({\bf A}^3) = 1 \Leftrightarrow \lambda_k({\bf A})^3 = 1, \forall k$$
As others mention this is a restriction on $\lambda$ which all are in the scalar field of matrix elements.</p>
<p>However the determinant can be written as the product of eigenvalues
$$\det({\bf A}) = \prod... |
26,083 | <p>For example, Wikipedia states that etale cohomology was "introduced by Grothendieck in order to prove the Weil conjectures". Why are cohomologies and other topological ideas so helpful in understanding arithmetic questions?</p>
| Thomas Riepe | 451 | <p>New <a href="http://arxiv.org/abs/1009.4827" rel="nofollow" title="link">lectures by Atiyah</a> on physics-inspired questions in geometry mention some fascinating connections with number theory, e.g. the question about a "quantum analogue of the Weil conjectures"and an "infinite dimensional version" of them. </p... |
1,217,654 | <p>I took the Laplace transform and solved for $Y$ which resulted in $Y=\frac{1+e^{-s}}{s^2-4}$. I began to break up the problem separating the result into two equations but the fact that there is a $1$ added to the $e^{-s}$ is messing me up. The answer I got was $y(t)= \frac{1}{8} u_1(t)e^{-2t}$ but I'm not sure if th... | PeterJL | 227,079 | <p>First, note that we always have $[x] \le x < [x] + 1$. Thus, $0 \le x - [x] < 1$ for all real $x$. To show your inequality, let $u = a - [a]$ and $v = b - [b]$. My claim is that, if $u + v < 1$, then $[a] + [b] = [a + b]$, and if $u + v \ge 1$, then $[a] + [b] < [a + b]$. Can you show that?</p>
|
1,217,654 | <p>I took the Laplace transform and solved for $Y$ which resulted in $Y=\frac{1+e^{-s}}{s^2-4}$. I began to break up the problem separating the result into two equations but the fact that there is a $1$ added to the $e^{-s}$ is messing me up. The answer I got was $y(t)= \frac{1}{8} u_1(t)e^{-2t}$ but I'm not sure if th... | String | 94,971 | <p>Working along similar lines to an answer I gave to <a href="https://math.stackexchange.com/questions/514407/prove-that-lfloor-x-rfloor-lfloor-y-rfloor-le-lfloor-xy-rfloor/514419#514419">another question</a>, note that $f:\mathbb R\rightarrow\mathbb Z$ given by $f(x)=\lfloor x\rfloor$ is an increasing function, that ... |
1,217,654 | <p>I took the Laplace transform and solved for $Y$ which resulted in $Y=\frac{1+e^{-s}}{s^2-4}$. I began to break up the problem separating the result into two equations but the fact that there is a $1$ added to the $e^{-s}$ is messing me up. The answer I got was $y(t)= \frac{1}{8} u_1(t)e^{-2t}$ but I'm not sure if th... | Barry Cipra | 86,747 | <p>By the definition of $[x]$ as the greatest integer <em>less than or equal to</em> $x$, we have</p>
<p>$$[a]\le a\quad\text{and}\quad [b]\le b$$</p>
<p>and thus</p>
<p>$$[a]+[b]\le a+b$$</p>
<p>Now $[a]+[b]$ is certainly an integer (because it's the sum of two integers). Therefore, by the definition of $[x]$ as ... |
16,342 | <p>Why do you need logarithms?
In what situations do you use them?</p>
| MathematicalOrchid | 29,949 | <p>You can "undo" addition by performing subtraction. You can "undo" multiplication by performing division.</p>
<p>When it comes to exponents, $x^y \not= y^x$, so you need <em>two</em> different "undo" functions.</p>
<p>Suppose that you know the value of $v$, and you know that this value was calculated by $v = x^n$.<... |
119,027 | <p>I am trying to plot the equations that describe the Cartesian locations of Bi-Cylinder coordinates. The equations are as follows:</p>
<pre><code>x==(1.5 Sinh[n])/(Cosh[n]-Cos[p])
y==(1.5 Sin[p])/(Cosh[n]-Cos[p])
</code></pre>
<p>I keep getting the following error </p>
<blockquote>
<p>Options expected(instead... | Edmund | 19,542 | <p>You may use <code>Reap</code> and <code>Sow</code> with the functional height syntax of <code>DateHistogram</code>. </p>
<pre><code>{dh, {{bins, counts}}} =
Reap[DateHistogram[data, "Day", Function[{b, c}, Sow[b]; Sow[c]]]];
Show[dh, DateListPlot[Select[Last@# != 0 &]@Transpose@{Mean /@ bins, counts}]]
</... |
2,299,070 | <p>Find the sum of the following infinite series $$e^{-x}\sum_{i=0}^{\infty}\dfrac{i.x^i}{i!}$$</p>
<p>The summation looks like an exponential series but how to tackle that?$$ 0+\frac{x}{1!}+\frac{2x^2}{2!}+...$$</p>
| fractal1729 | 431,725 | <p>$$\sum_{n=0}^{\infty}\dfrac{nx^n}{n!} = x\cdot\sum_{n=1}^{\infty}\dfrac{x^{n-1}}{(n-1)!} = xe^x.$$</p>
<p>So $e^{-x}$ times that is just $x$. Is this what you're asking?</p>
|
1,835,414 | <p>What is the fastest method to find which number is bigger manually?</p>
<p>$\frac {3\sqrt {3}-4}{7-2\sqrt {3}} $ or $\frac {3\sqrt {3}-8}{1-2\sqrt {3}} $</p>
| Mythomorphic | 152,277 | <p><strong>Claim:</strong> </p>
<p>$$\frac {3\sqrt {3}-4}{7-2\sqrt {3}}<\frac {3\sqrt {3}-8}{1-2\sqrt {3}}$$</p>
<p><strong>Proof:</strong></p>
<p>Let $u=\sqrt 3-1$. Notice $1>u>\frac12$.</p>
<p>If</p>
<p>$$\frac{3u-1}{5-2u}<\frac{3u-5}{-2u-1}$$
$$\Leftarrow-6u^2-u+1>-6u^2+25u-25$$
$$\Leftarrow26>... |
2,060,944 | <p>I am trying to calculate the softmax gradient:
$$p_j=[f(\vec{x})]_j = \frac{e^{W_jx+b_j}}{\sum_k e^{W_kx+b_k}}$$
With the cross-entropy error:
$$L = -\sum_j y_j \log p_j$$
Using <a href="https://math.stackexchange.com/questions/945871/derivative-of-softmax-loss-function">this question</a> I get that
$$\frac{\partial... | martini | 15,379 | <p>The dimension mismatch appears when you are using the chain rule. In case of taking the derivative with respect to $W_i$ (which denotes the $i$-th row of $W$, right?), we have maps
$$ W_i \in \mathbf R^{1 \times k} \mapsto o_i = W_ix+b_i \in \mathbf R \mapsto L \in \mathbf R $$
hence a function $\mathbf R^{1 \times... |
3,790,099 | <p><span class="math-container">$\mathcal{P}(A \cap B) = \mathcal{P}(A) \cup \mathcal{P}(B) \implies A = B$</span></p>
<p>Let <span class="math-container">$S \in \mathcal{P}(A \cap B)$</span>.</p>
<p>Thus, <span class="math-container">$S \subseteq A \cap B$</span>.</p>
<p>Thus <span class="math-container">$S\subseteq A... | Brian Moehring | 694,754 | <p>Note that <span class="math-container">$$\mathcal{P}(X) \to {}^X\mathbb{F}_2 \\ A \mapsto \mathbf{1}_A,$$</span> which maps sets to indicator functions, is a bijection from the set <span class="math-container">$\mathcal{P}(X)$</span> to the ring <span class="math-container">$${}^X\mathbb{F}_2 = \{f : X \to \mathbb{F... |
49,633 | <p>I know the construction of the Hodge star operator in the context of (pseudo-)euclidean real vector spaces. Apart from the scalar product it involves a orientation of the vector space, which one has to choose (at least if one is not willing to deal with forms of odd parity). </p>
<p>While I was wondering how to ext... | Mariano Suárez-Álvarez | 1,409 | <p>Let $n=\dim V<\infty$. If you fix a non-degenerate bilinear form $V\times V\to k$, you get an isomorphism $\beta:V\to V^*$. Now fix a non-zero element $\omega\in\bigwedge^nV$. Multiplication in the exterior algebra gives you a non-degenerate pairing $ \bigwedge^kV\times\bigwedge^{n-k}V\to\bigwedge^nV$ which, usin... |
66,738 | <p>I've used <code>TreeForm</code>, and I appreciate that the syntax is fairly short, especially in comparison with <code>TreeGraph</code>.</p>
<p>Is it possible specify a different colors in <code>TreePlot</code> for coloring a section of the nodes and the paths to those nodes?</p>
| Dr. belisarius | 193 | <p>Anyway, Treegraph offers a lot of flexibility:</p>
<pre><code>nodes = {RandomInteger[#] , # + 1} & /@ Range[0, 30];
rn = Range@Length@nodes;
crules = Rule @@@ Partition[Riffle[rn, ColorData[15, "ColorList"]], 2];
g = TreeGraph[UndirectedEdge @@@ nodes, VertexSize -> 0.4, VertexStyle -> crules];
Highlight... |
66,738 | <p>I've used <code>TreeForm</code>, and I appreciate that the syntax is fairly short, especially in comparison with <code>TreeGraph</code>.</p>
<p>Is it possible specify a different colors in <code>TreePlot</code> for coloring a section of the nodes and the paths to those nodes?</p>
| berniethejet | 365 | <p>The terminal nodes of TreeForm are apparently all wrapped in an insidious HoldForm such that one cannot easily give them separate colors.</p>
<p>That is:</p>
<pre><code>ii = 0;
TreeForm[Nest[{#, ++ii} &, ii, 3],
VertexRenderingFunction -> ({Black,
Text[#2, #1,
Background -> (Hue[
S... |
2,160,944 | <blockquote>
<p>Prove by induction:
$$\sin{x}+\sin{3x}+\dots+\sin{(2n-1)x}=\frac{\sin^2{nx}}{\sin{x}}$$</p>
</blockquote>
<p>I tried the problem using the normal rule of induction(the first principle), but I failed.I failed to make the form $\sin^2{(m+1)x}$. Somebody help me.</p>
| Michael Rozenberg | 190,319 | <p>For $n=1$ it's obvious. </p>
<p>Thus, it remains to prove that $$\frac{\sin^2nx}{\sin{x}}+\sin(2n+1)x=\frac{\sin^2(n+1)x}{\sin{x}}$$ or
$$1-\cos2nx+\cos2nx-\cos(2n+2)x=1-\cos(2n+2)x,$$
which is obvious.</p>
|
742,577 | <p>It's been a while since I've had to do math and I've been stuck around a problem for two good hours. I hate asking questions but I can't figure it out. </p>
<p>I have the following problem: </p>
<p>Find the zero/domain of </p>
<p>$$ f(x) = \frac{9x^3-4x}{(x-3)(x^2-2x+1)} $$</p>
<p>So far, I've been able to find ... | 2'5 9'2 | 11,123 | <p>Suppose the die is a 1-inch cube. Draw a bunch of parallel lines spaced 1-inch apart. When you roll a die onto this field of lines, it will always cross one line (although it's not "impossible" to land right between two lines, there's $0$ chance that it will.) Sometimes, the die will cross <em>two</em> lines.</p>
<... |
94,014 | <p>Is there any reasonable approach, essentially different from Wightman's axioms and Algebraic Quantum Field Theory, aimed at obtaining rigorous models for realistic Quantum Field Theories? (such as Quantum Electrodynamics).</p>
<p>EDIT: the reason for asking "essentially different" is the following. It is possible t... | Igor Khavkine | 2,622 | <p>If I read your updated question correctly, you are asking whether people have considered non-linear modifications of quantum mechanics in order to accommodate interacting QFTs. I'm sure someone, somewhere has, but that's certainly not mainstream thought in QFT research, either on the mathematics or theoretical physi... |
4,335,831 | <blockquote>
<p>Let <span class="math-container">$f:\mathbb R\rightarrow \mathbb R$</span> be a differentiable function, and suppose <span class="math-container">$f=f'$</span> and <span class="math-container">$f(0)=1$</span>. Then prove <span class="math-container">$f(x)\neq 0$</span> for all <span class="math-containe... | yearning4pi | 702 | <p>Let <span class="math-container">$g(x) = f(x)e^{-x}$</span>. Then, <span class="math-container">$g$</span> is a differentiable function on <span class="math-container">$\mathbb R$</span>. By the product rule, <span class="math-container">$g'(x) = 0$</span>. So, <span class="math-container">$g$</span> is constant. Si... |
1,068,068 | <p>Can someone please explain to me how I am supposed to approach this question:</p>
<p>If $f: [0,1] \to \mathbb{ R}$ is continuous, and has only rational values, then $f$ must be a constant.</p>
| user197402 | 197,402 | <p>Hint: Suppose $f$ takes on two values $a,b\in\mathbb{Q}$ for which $a\neq b$. What does the intermediate value theorem tell us?</p>
|
65,608 | <p><em>Mathematica 10</em> generates a warning that it is unable to generate initial points for numerical optimization problems. I picked a particularly simple example. The problem goes away when <code>Abs</code> is dropped.</p>
<pre><code>NMinimize[{x + y, x >= 0 && Abs[x + 10 y + 100] <= 1}, {x, y}]
</... | Michael E2 | 4,999 | <p>You can get a glimpse into the workings of <code>NMinimize</code> by turning on the debug-printing:</p>
<pre><code>Block[{Optimization`NMinimizeDump`dbPrint = Print},
NMinimize[{x + y, x >= 0 && Abs[x + 10 y + 100] <= 1}, {x, y}]
]
</code></pre>
<p>It seems at a cursory glance that it decided to se... |
2,335,319 | <p>I am reading through Humphrey's <em>Introduction to Lie Algebras and Representation Theory</em> on my own and I am currently stumped by one of the exercises, namely Exercise 2 from Section 15.</p>
<p>Let $L$ be a semisimple Lie algebra over an algebraically closed field of characteristic zero. We want to prove that... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>Let us continue your work.</p>
<p>$$12|3 (l^2+2l+2)\implies $$
$$4|l^2+2l+2 \implies $$</p>
<p>$$2|l^2+2l+2 \implies $$
$$2|l \implies l=2p \implies $$
$$4|4p^2+4p+2 \implies 4|2$$</p>
<p>You can finish.</p>
|
547,151 | <p>I am attempting to show that if $R$ is commutative, local notherian ring with $J$ nilpotent then $R$ is Artinian.</p>
<p>Clearly as $\exists k$ such that $J^k=0$ then if there is no other ideal other than $R$ and $J^i$ then we are done but how do I show this?</p>
<p>That is how do I show that if $I$ is an ideal of... | Khosrotash | 104,171 | <p><img src="https://i.stack.imgur.com/dA0sq.png" alt="enter image description here"></p>
<p>root of trigonometric function is not unique</p>
|
3,007,962 | <blockquote>
<p>Prove
<span class="math-container">$$
\lim_{n \to \infty}\frac{q^n}{n} = 0
$$</span> for <span class="math-container">$|q| < 1$</span> using <span class="math-container">$\epsilon$</span> definition.</p>
</blockquote>
<p>Using the definition of a limit:</p>
<p><span class="math-container">$$
\... | Peter Szilas | 408,605 | <p>You have::</p>
<p><span class="math-container">$\dfrac{|q|^n}{n} < \dfrac{1}{n^2t}< \dfrac{1}{nt},$</span> <span class="math-container">$t>0$</span>.</p>
<p>Let <span class="math-container">$\epsilon$</span> be given.</p>
<p>Archimedean principle:</p>
<p>There is a <span class="math-container">$N$</spa... |
786,239 | <p>Can you find two functions $f$ and $g$ defined on a closed interval $[a, b]$, with real values, such that $\exists (x_n) $ an infinite sequence of distinct points in $[a, b] $ such that $$\forall n, f(x_n) =g(x_n) $$ but $$f\neq g$$</p>
<p><strong>EDIT</strong>: the question is not interesting as it is stated. I ... | Patrick | 142,001 | <p>Yes, (C) is correct. Just check the properties:</p>
<ol>
<li>$f(0,0) = K + 0 = K$</li>
<li>$f(x+t,y) = K + (x+t)y = K + xy + ty = f(x,y) + ty$</li>
<li>$f(x,t+y) = K + x(t+y) = K + xy + xt = f(x,y) + tx$</li>
</ol>
|
766,272 | <p>I am having a problem with this assignment. So the task says:</p>
<pre><code>Show that every planar graph with at least 4 vertices has at least 4
vertices of degree less than or equal to 5.
</code></pre>
<p>I know about the theorem stating that every planar graph has to have at least one vertex of degree 5 or les... | user135508 | 135,508 | <p>Assume to the contrary that no more than three vertices have degree 5 or less, so all but three have degree at least six. That mean $n \geq 6(m-3)+3 = 6m-15$ or $m \leq \frac{n+15}{6}$. But you have shown that $m \geq 3n$. Combining, we get $\frac{n+15}{6} \geq 3n$ or $17n \leq 15$, which is absurd.</p>
|
568,870 | <p>I've been asked to prove that</p>
<p>$$ \int^\infty_0 \cos(tx)\left (\frac{\sin(t)}{t} \right )^n \, dt = 0, \space \forall x > n \geq 2.$$</p>
<p>My approach so far has been to use a theorem proved in class that, for a random variable $X$ with characteristic function $\phi(t)$ and $a,b\in\mathbb{R}$,</p>
<p>$... | saz | 36,150 | <p><strong>Hint</strong> Let $X_1,\ldots,X_n$ be independent random variables such that $X_j$ is uniformly distributed on $[-1,1]$ for $j=1,\ldots,n$. It is not difficult so show that the Fourier transform of $X_j$ equals</p>
<p>$$\Phi_{X_j}(t) := \mathbb{E}e^{\imath \, t \cdot X_j} = \frac{\sin t}{t}.$$</p>
<p>There... |
2,478,173 | <p>I have a sample mean given by:</p>
<p>$$S_n=\frac{1}{n}\sum_{i=1}^nX_i$$
Where $X_i$ are i.i.d. Gaussian random variable, i.e., each of them has pdf:</p>
<p>$$p(X_i=x_i)=\frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x_i-\mu)^2/2\sigma^2}$$</p>
<p>The parameters $\mu, \sigma^2$ are, of course, mean and variance.</p>
<p>The ... | Flatfoot | 142,723 | <p>You can use the following <strong>Theorem</strong>:</p>
<p>If $X_1, X_2, ... , X_n$ are mutually independent normal random variables with means $μ_1, μ_2, ... , μ_n$ and variances $\sigma_1^2,\sigma_2^2,⋯,\sigma_n^2$, then the linear combination:</p>
<p>$$Y=\sum\limits_{i=1}^n c_iX_i$$ </p>
<p>follows the normal ... |
1,815,344 | <p>We want to prove there exists a finite field of $p^n$ elements ($p$ is prime and $n>0$). Take $q=p^n$ and $g(x)=x^q-x\in\mathbb{Z}_p[x]$, and let $E$ be a field that contains $\mathbb{Z}_p$ and all roots of $g(x)$. Let $F=\{a\in E:a^q=a\}$.</p>
<p>I understand that $F$ is closed under addition (since $E$ has cha... | Eric Wofsey | 86,856 | <p>If a subset $F$ of a field $E$ of characteristic $p$ is closed under addition, then it is automatically also closed under taking negatives. For if $x\in F$ then $px=0$ so $(p-1)x=-x$, and so $-x$ can be found by just taking a sum of $p-1$ copies of $x$.</p>
|
282,644 | <blockquote>
<p>Is the function $f:\Bbb R \rightarrow \Bbb R$ defined as $f(x)=\sin(x^2)$, for all $x\in\Bbb R$, periodic?</p>
</blockquote>
<p>Here's my attempt to solve this:</p>
<blockquote>
<p>Let's assume that it is periodic. For a function to be periodic, it must satisfy $f(x)=f(T+x)$ for all $x\in\Bbb R$, ... | Michael Albanese | 39,599 | <p>Let $f : \mathbb{R} \to \mathbb{R}$ be periodic with period $T$.</p>
<ul>
<li>The range of $f$ is precisely $f([0, T])$; in particular, if $f$ is continuous, the range of $f$ is bounded. </li>
<li>If $f$ is differentiable, then $f'$ is periodic with period $T$. </li>
</ul>
<p>Note that $f(x) = \sin(x^2)$ is differ... |
57,120 | <p>Let $A,B \subset \mathbb{R}$ and $m^*(A)=m^*(B)=1$ and $m^*(A\cup B)=2$.
Prove that $m^*(A\cap B)=0$.</p>
<p>I tried every way I can think of but I do not know how to figure this out.</p>
<p>Only properties that I am aware of are monotonicity, countable subadditivity, the outer measure of empty set is zero, trans... | Mark | 687 | <p>Lebesgue outer measure satisfies the inequality $m^* (A \cup B) + m^* (A \cap B) \le m^* (A) +m^* (B)$. Substitute to get the desired result.</p>
<p>The above inequality should not be difficult to prove no matter how you define Lebesgue outer measure. Basically it's an approximation argument. For instance, if you ... |
3,281,820 | <p>Let <span class="math-container">$X,Y$</span> be random variables <span class="math-container">$\Omega \to \mathbb{R}$</span> (where <span class="math-container">$(\Omega, \mathcal{F})$</span> is a measurable space). Let further <span class="math-container">$f,g:\Omega \to \mathbb{R}$</span> be two functions such th... | Abhishek Divekar | 244,555 | <p>I will add another answer which allows you to get a polynomial from a boolean expression when using True=+1 and False=-1 (@user3257842 has already answered for the case where True=1 and False=0). This is helpful in certain Machine Learning analysis.</p>
<hr />
<p>When using boolean True=+1 and False=-1, to get a pol... |
1,456,511 | <blockquote>
<p>How many ways to choose a 10 letter combination from {A B C D E F G} such that there is at most 3'A's and at least 1'F'.</p>
</blockquote>
<p>Here are my thoughts:</p>
<ul>
<li>Let X be the property that it would have at most 3'A's</li>
<li>Let Y be the property that it would have at least 1 'F'</li... | Cameron Buie | 28,900 | <p><strong>Hint #1</strong>: We only want to include such combinations with at least one F. How many of these are there?</p>
<p><strong>Hint #2</strong>: <em>Of the included combinations</em>, we must exclude those that have $4$ or more As. How many of those are there?</p>
|
1,017,126 | <p>Does the cardinality of a Null set is same as the cardinality of a set containing single element?
If a set A contains Null set as its subset, then the null set is taken into account to calculate the cardinality of set A or not?</p>
| John Hughes | 114,036 | <p>You could treat all of $CP^1$ except $z_2 = 0$ as being "parametrized" by $C$ via $z \mapsto (z, 1)$. Then the map you've defined just takes $(z_1, z_2) \mapsto z_1/z_z \in \mathbb C$. Now it's not so hard to compute the derivative. </p>
<p>You can then make a second coordinate chart for where $z_1 \ne 0$, via $z \... |
98,970 | <p><strong>Problem Description</strong></p>
<p>I have my data arranged in a tab delimited file and I'm trying to parse/access certain parts of it.</p>
<p>As an example, I'd like to extract the number of points and the sampling frequency as well as the time series which is saved like below (<code>>></code> = tab... | Jack LaVigne | 10,917 | <p>Once you have your data in the form of a string, Mathematica has very powerful string operations that can parse the data.</p>
<p>I am not claiming that this is the most efficient method but it will work with your data (or possibly some minor tweaks).</p>
<p>Starting with your first example:</p>
<pre><code>string1... |
1,627,907 | <p>Find the probability in the general case that in between any two red balls
are at least two blue balls.</p>
<p>So, at first, I tried to approach the problem by thinking about the probability
that no two red balls are drawn consecutively and I found that this probability
is</p>
<p>$P(\text{no two red balls drawn... | forallepsilon | 170,777 | <p>The set $J$ generates a $\sigma$-algebra, but $J$ itself is not one, since for example, the complement of an open interval is not an interval.</p>
|
644,714 | <p>given is
$f(x,y) = ( \frac{y}{x^2+1}, \frac{x^2}{y^2-1} ) $. I have to study the continuity of the function for$ (x,y) \to (0,1)$.</p>
<p>First function $f_1$ is continuous, since $lim f_1 = 1/1 = 1$ so the limit exists. And the function is defined on whole $\mathbb{R}^2$.</p>
<p>Secon function $f_2$.. Ok, here I ... | drhab | 75,923 | <p>Hint:</p>
<p>$h(x)=F(x^{-1})-F(5)$ where $F'(x)=10\times arctan(x)$.</p>
<p>The chainrule will do the rest.</p>
|
3,961,869 | <blockquote>
<p>Let <span class="math-container">$|f(x)-f(y)| \leqslant (x-y)^2$</span> for all <span class="math-container">$x,y\in \mathbb{R}.$</span> Show that <span class="math-container">$f$</span> is a constant.</p>
</blockquote>
<p>This seemed quite straightforward just using the definition of the derivative, bu... | Michelle | 718,613 | <p>We have
<span class="math-container">$$
|f(x)-f(y)| \leqslant (x-y)^2
$$</span>
so
<span class="math-container">$$
\frac{|f(x)-f(y)|}{|x-y|} \leqslant |x-y|
$$</span>
so
<span class="math-container">$$
\frac{|f(x)-f(y)|}{|x-y|} \to_{y \to x} 0.
$$</span>
Thus <span class="math-container">$f'=0$</span> and <span clas... |
3,961,869 | <blockquote>
<p>Let <span class="math-container">$|f(x)-f(y)| \leqslant (x-y)^2$</span> for all <span class="math-container">$x,y\in \mathbb{R}.$</span> Show that <span class="math-container">$f$</span> is a constant.</p>
</blockquote>
<p>This seemed quite straightforward just using the definition of the derivative, bu... | J.G. | 56,861 | <p>We needn't assume <span class="math-container">$f$</span> is differentiable, or even continuous. Just use the triangle inequality:<span class="math-container">$$|f(y+h)-f(y)|\le\sum_{k=1}^n|f(y+hk/n)-f(y+h(k-1)/n)|\le n(h/n)^2=h^2/n$$</span>for all <span class="math-container">$n\in\Bbb N$</span>, so <span class="ma... |
32,769 | <p>I have an ordinal list that I am trying to represent mathematically. The list is as follows:</p>
<p>10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000, 20000, 30000, 40000, 50000, 60000, 70000, 80000, 90000, 100000.</p>
<p>S... | Shai Covo | 2,810 | <p>See <a href="http://oeis.org/search?q=a037124&sort=&language=english&go=Search" rel="nofollow">OEIS sequence A037124: Numbers that contain only one nonzero digit</a>.</p>
|
4,169,114 | <p>So I am working on a problem, in that I need to integrate <span class="math-container">$\int_0^1 e^{-x^2}dx$</span>.</p>
<p>So I did it in this way:</p>
<p><span class="math-container">$$\int_0^1 e^{-x^2} dx=\int_0^1 \sum_{n=0}^\infty \frac{(-x^2)^{n}}{n!} = \sum_{n=0}^\infty \int_0^1 \frac{(-x^{2})^n}{n!} = \sum_{n... | Laxmi Narayan Bhandari | 931,957 | <p>Your solution is confusing and incorrect due to no use of brackets. Here's my solution.</p>
<p>We start by applying the power series expansion of <span class="math-container">$e^{-x^2}$</span>.</p>
<p><span class="math-container">$$\begin{align} \int_0^1 e^{-x^2}\, \mathrm dx &= \int_0^1 \left(\sum_{k=0}^{\infty... |
204,096 | <h3>Motivation</h3>
<p>The structuralist point of view on mathematical objects has two aspects:</p>
<p>On the one side, a mathematical object is seen as a concrete structure <strong>of</strong> abstract dots, e.g. a graph.</p>
<p>On the other side, a mathematical object is seen as an abstract dot <strong>in</strong> a ... | André Nicolas | 6,312 | <p>A polynomial over a ring $A$ is a sequence $a_0,a_1, a_2,\dots$ of ring elements such that all but finitely many elements in the sequence ae $0$. So set-theoretically, it is a certain kind of function from the ordinal $\omega$ to $A$. Then one defines addition and multiplication as usual, as if the sequence $(a_n)$ ... |
204,096 | <h3>Motivation</h3>
<p>The structuralist point of view on mathematical objects has two aspects:</p>
<p>On the one side, a mathematical object is seen as a concrete structure <strong>of</strong> abstract dots, e.g. a graph.</p>
<p>On the other side, a mathematical object is seen as an abstract dot <strong>in</strong> a ... | David Wheeler | 23,285 | <p>If $R$ is a commutative ring, and $X$ is any singleton set, then $R[X]$ is the free commutative $R$-algebra of rank $1$. This generalizes easily to the case where $|X| = n$, where we get that the polynomial ring in $n$ variables is the free commutative $R$-algebra of rank $n$. If we drop the commutativity requiremen... |
4,403,525 | <p>Geniuses!</p>
<p>We're having 2 dice with different sizes. For our example, dice A has 10 sides while dice B has 7 sides.
How to calculate the probability of Dice A being rolled higher than Dice B?</p>
<p>Thanks upfront!</p>
| Jack Gallagher | 1,020,750 | <p>It may be useful to draw a table for questions like these and see the values where the ten sided die is larger than the seven. The first die has 7 outcomes and the second has 10 so multiplying together gives 70 possible outcomes. If the ten sided dice is a 1 then there is 0 outcomes, if the ten sided dice is a 2 the... |
2,067,811 | <p>Let $A \in M_{n,n}(\mathbb{R})$. </p>
<p>What is the probability that, when I fill in the entries with random real numbers, $det(A) =0$? This looks like an interesting problem, as having determinant zero is equivalent to a lot of other things. For example, the answer to this question gives us information about the ... | gt6989b | 16,192 | <p>Even in the simplest case of $n=2$ over $\mathbb{R}$ you are asking what is the probability that $ad = bc$ where $a,b,c,d$ are random. This is clearly 0.</p>
<p>All others can be argued similarly. To make this interesting, ensure you are sampling entries from a finite distribution.</p>
|
1,767,773 | <p>I want to find whether the expression $D = \sqrt{5t^2 - 40t+125}$ is increasing or decreasing when $t=5$.</p>
<p>My logic is I want to find whether is $f'(5)>0$ or $f'(5) < 0$.</p>
<p>I need to use the chain rule $h'(x) = g'(f(x))f'(x)$</p>
<p>$g'(f(x)) = \frac{1}{2}(5t^2 - 40t+125)^\frac{-1}{2}$</p>
<p>$f... | Community | -1 | <p>Because $\sqrt{x}$ is strictly increasing, this question is equivalent to whether $f(t)=D^2=5t^2-40t+125$ is increasing or decreasing at $t=5$, $f'(5)=10>0$ so it's increasing at $t=5$.</p>
|
1,767,773 | <p>I want to find whether the expression $D = \sqrt{5t^2 - 40t+125}$ is increasing or decreasing when $t=5$.</p>
<p>My logic is I want to find whether is $f'(5)>0$ or $f'(5) < 0$.</p>
<p>I need to use the chain rule $h'(x) = g'(f(x))f'(x)$</p>
<p>$g'(f(x)) = \frac{1}{2}(5t^2 - 40t+125)^\frac{-1}{2}$</p>
<p>$f... | Brenton | 226,184 | <p>Your expression has a slight error. It should be:</p>
<p>$h'(t) = \frac{1}{2}(5t^2 - 40t+125)^\frac{-1}{2}(10t-40)$</p>
<p>Since all you need to know is the sign of $h'(t)$, you only need to look at the sign of $5t^2 - 40t+125$ for $t=5$ and $10t-40$ when $t=5$.</p>
<p>Since the last term is clearly $>0$, the ... |
224,559 | <p>Given two probability distributions $p,q$ on a finite set $X$, the quantity variously known as <strong>relative information</strong>, <strong>relative entropy</strong>, <strong>information gain</strong> or <strong>Kullback–Leibler divergence</strong> is defined to be </p>
<p>$$ D_{KL}(p\|q) = \sum_{i... | Tom Leinster | 586 | <p>Let $A_n$ be the set of pairs $(p, q)$ of probability distributions on $\{1, \ldots, n\}$ such that $q_i = 0 \implies p_i = 0$. (This is exactly the condition needed to guarantee that $D(p\|q) < \infty$.)</p>
<p><strong>Theorem</strong> <em>Let $(I: A_n \to [0, \infty))_{n \geq 1}$ be a sequence of functions. ... |
1,868,495 | <p>Consider $\lim_{ x \to -\infty} \sqrt{x^2-x+1}+x$ </p>
<p>Rationalising, one will get, $\lim_{x \to -\infty} \frac{1-x}{\sqrt{x^2-x+1}-x}$, which after taking x common and cancelling out gives $-\infty$.</p>
<p>Now, replace $x$ by $-x$, so the limit becomes, $\lim_{x \to \infty} \frac{1+x}{\sqrt{x^2+x+1}+x}$, whic... | zyx | 14,120 | <blockquote>
<p>Rationalising, one will get, $\lim_{x \to -\infty} \frac{1-x}{\sqrt{x^2-x+1}-x}$, which after taking x [highest common power] and cancelling out gives $-\infty$.</p>
</blockquote>
<p>If one "takes highest powers", replacing $\sqrt{x^2 - x + 1}$ by $|x|$ then the denominator becomes $(|x| - x)$ or $-2... |
5,249 | <p>Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ <a href="http://en.wikipedia.org/wiki/Derangement" rel="nofollow">derangements</a> of $\{1,2,\dots,n\}$ and that there are $D_n(q)=(n)_q! \left( 1-\frac{1}{(1)_q!}+\frac1{(2)_q!}-\frac1{(3)_q!}+\cdots+(-1)^{n}\fr... | Konrad Voelkel | 956 | <p>The "categorical" approach: Understand an object not via intrinsic properties (rather ignore them, they are not part of the categorical data) but via the morphisms.</p>
<p>By choosing a category to work in, we choose the class of properties that are interesting.
If we choose to work in the category of groups and gr... |
5,249 | <p>Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ <a href="http://en.wikipedia.org/wiki/Derangement" rel="nofollow">derangements</a> of $\{1,2,\dots,n\}$ and that there are $D_n(q)=(n)_q! \left( 1-\frac{1}{(1)_q!}+\frac1{(2)_q!}-\frac1{(3)_q!}+\cdots+(-1)^{n}\fr... | David Lehavi | 404 | <p>This relates to an earlier question you asked:</p>
<p><a href="https://mathoverflow.net/questions/1827/what-representative-examples-of-modules-should-i-keep-in-mind">What representative examples of modules should I keep in mind?</a></p>
<p>My answer is the same: look at the frontispieces of Miles Ried's book under... |
5,249 | <p>Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ <a href="http://en.wikipedia.org/wiki/Derangement" rel="nofollow">derangements</a> of $\{1,2,\dots,n\}$ and that there are $D_n(q)=(n)_q! \left( 1-\frac{1}{(1)_q!}+\frac1{(2)_q!}-\frac1{(3)_q!}+\cdots+(-1)^{n}\fr... | Autumn Kent | 1,335 | <p>Cuz otherwise it'd be like getting to know a bicycle without riding it.</p>
|
5,249 | <p>Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ <a href="http://en.wikipedia.org/wiki/Derangement" rel="nofollow">derangements</a> of $\{1,2,\dots,n\}$ and that there are $D_n(q)=(n)_q! \left( 1-\frac{1}{(1)_q!}+\frac1{(2)_q!}-\frac1{(3)_q!}+\cdots+(-1)^{n}\fr... | Pete L. Clark | 1,149 | <p>I want to answer your question twice: first with a "top-down" approach and second with a "bottom-up" approach. Let me limit myself to the first answer here and see how I do.</p>
<p>I claim the following analogy:</p>
<p>abstract groups : group actions on sets :: abstract rings : linear actions of rings on
abelian... |
5,249 | <p>Everybody knows that there are $D_n=n! \left( 1-\frac1{2!}+\frac1{3!}-\cdots+(-1)^{n}\frac1{n!} \right)$ <a href="http://en.wikipedia.org/wiki/Derangement" rel="nofollow">derangements</a> of $\{1,2,\dots,n\}$ and that there are $D_n(q)=(n)_q! \left( 1-\frac{1}{(1)_q!}+\frac1{(2)_q!}-\frac1{(3)_q!}+\cdots+(-1)^{n}\fr... | Joël | 9,317 | <p>If your friend knows well rings and ideals, he has certainly encountered quotient like $I/J$, hasn't he? What are they for him? sets? abelian groups? This point of view is wanting.
You can formulate basic and very useful results, such as (for $A$ local, noetherian): <em>if $I/mI$ has a set of $n$ generators, so has... |
3,829,161 | <p>Using the Axiom of choice, one can show that (see <a href="https://math.stackexchange.com/questions/288075/on-every-infinite-dimensional-banach-space-there-exists-a-discontinuous-linear-f?rq=1">here</a>) every infinite dimensional normed vector space has discontinuous functionals. My question is: Is this also true f... | Kavi Rama Murthy | 142,385 | <p>This proof fails because <span class="math-container">$V^{x}$</span> and <span class="math-container">$W^{y}$</span> actually depend on both <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.</p>
<p>If you call these <span class="math-container">$V ^{x,y}$</span> and <span class="ma... |
1,377,192 | <p>I have stumbled upon the following reasoning, but I'm not sure if it's correct. Here it goes:
Domain X</p>
<ol>
<li>$\forall x :\phi(x)⟹\gamma(x)$</li>
<li>Let $E\subseteq X⟹[\forall x\in E :\phi(x)⟹\gamma(x)]$</li>
<li>Suppose I know, by some property of $E$, that $\forall x\in E :\phi(x)$, that is for every $x\in... | M. Strochyk | 40,362 | <p>Taking logarithm and applying L'Hopital's rule, we have
$$\lim\limits_{x\to{\frac π{2}}^-}\ln\left[\left(\frac {2x}{\pi}\right)^ {\tan x}\right]=\lim\limits_{x\to{\frac π{2}}^-}\tan{x}\cdot\ln{\left(\frac {2x}{\pi}\right)}=\lim\limits_{x\to{\frac π{2}}^-}\frac{\ln{\left(\frac {2x}{\pi}\right)}}{\cot{x}}=\left[\frac... |
585,114 | <p>I have read a bit about Gauss, who was well known for being careful in only publishing work he had perfected (or in his own words "few, but ripe"). What is interesting to me about Gauss though is that all accounts of his students and contemporaries essentially make him appear flawless. Dedekind's recollections of ho... | Will Jagy | 10,400 | <p>You are seeing the down side of the "few but ripe" idea. He comes off badly in the episode with Bolyai. He did not publish his findings n a new, non-Euclidean geometry, over <a href="http://books.google.com/books?id=uCYPZ4nRqy0C&pg=PA182&lpg=PA182&dq=gauss%20howls%20of%20the%20boeotians&source=bl&... |
1,646,688 | <p>In my opinion, let $v\in$ Span($S_1\cap S_2$) and therefore $v\in$ Span$(S_1)$ and $v\in$ Span$(S_2)$. Write $v=c_1z_1+...+ c_nz_n$ where $z_k\in S_1\cap S_2$ and $c_k\in R$. Here I am feeling I have the wrong direction. Could someone suggest how to approach the question?</p>
<p>And how do you show that $W_1\cap W_... | Marko | 180,130 | <p>$W_1=W_2$ and there is no strict inequality between sets in your question. In general anything can happen. Even when $W_1=W_2$ you can have $S_1\cap S_2=\emptyset$. If $S_1\subseteq S_2$, then again you have an equality, since in this case $W_1\cap W_2=W_1$ and $\mathrm{Span}(S_1\cap S_2)=\mathrm{Span(S_1)}=W_1.$ In... |
1,070,699 | <p>Consider the matrix</p>
<p>$$
A= \begin{bmatrix}1/8 & \frac{-5}{8\sqrt{3}} \\
\frac{-5}{8\sqrt{3}} & 11/8
\end{bmatrix}
$$
which of the following transformations of the coordinate axis will make the matrix $A$ diagonal?</p>
<p>Rotation in $-60$ counter clock wise, $-30$ ccw, $30$ ccw, $60... | Sina | 143,830 | <p>You can use eigen value decomposition. We know that the eigen values and eigen vectors are computed by solving this equation:
$$Av_i=\lambda_i v_i \, , \quad i=1,\ldots,n $$
So if we re-write the equation for all $i$ in a matrix form, we will have:
$$AP=P\Lambda \Rightarrow P^{-1}AP=\Lambda$$
where $P=(v_1,\ldots,v_... |
1,264,613 | <p>While reading in a Discrete maths text book, there was this question:</p>
<blockquote>
<p>How many bit strings of length n are palindromes?</p>
</blockquote>
<p>The answer is:</p>
<blockquote>
<p><span class="math-container">$2^\frac{n+1}{2}$</span> for odd and <span class="math-container">$2^\frac{n}{2}$</span> for... | user3138972 | 329,828 | <p>Let $P_{n}$ be the number of palindromes of length n. Then the recursive formula for $P_{n}$ can be given as,
$P_{n} = 2.P_{n-2}$, with base case $P_{1} =1$ and $P_{2} = 2$,
that can be conversed easily into $P_{n} = 2^{n/2}$, if n is even or $P_{n} = 2^{(n+1)/2}$, if n is odd.</p>
|
2,070,706 | <p>I want to show that $$f: \mathbb{R}_{\geq0} \to \mathbb{R}_{\geq0}$$ $$x \mapsto x^s$$ is Holder continuous with Holder exponent $s \in \mathbb{R}$, where $0<s \leq 1$. So what I want to show is that $\exists \hspace{2 mm} C \in \mathbb{R}_{\geq0}$ sucht that for all $ x,y \in \mathbb{R}_{\geq0}, $<br>
$$|x^s -y... | fes | 400,212 | <p>Assume <span class="math-container">$x>y$</span>.</p>
<p><span class="math-container">$$x^s-y^s \leq C(x-y)^s$$</span></p>
<p><span class="math-container">$$\Leftrightarrow x^s \leq C(x-y)^s+y^s$$</span></p>
<p>Claim: this holds for all <span class="math-container">$(x,y)$</span> when <span class="math-container"... |
2,070,706 | <p>I want to show that $$f: \mathbb{R}_{\geq0} \to \mathbb{R}_{\geq0}$$ $$x \mapsto x^s$$ is Holder continuous with Holder exponent $s \in \mathbb{R}$, where $0<s \leq 1$. So what I want to show is that $\exists \hspace{2 mm} C \in \mathbb{R}_{\geq0}$ sucht that for all $ x,y \in \mathbb{R}_{\geq0}, $<br>
$$|x^s -y... | Divide1918 | 706,588 | <p><span class="math-container">$$|x^s - y^s||x^{1-s} + x^{1-2s}y^s + ... + y^{1-2s}| = |x - y|$$</span></p>
<p><span class="math-container">$\forall{x,y},\, \exists{m\gt0}\;\text{such that} \;m\le|x^{1-s} + x^{1-2s}y^s + ... + y^{1-2s}|$</span></p>
<p>If <span class="math-container">$|x - y|\lt 1$</span>, <span cla... |
155,365 | <p>It is well-known that for any presheaf $F \colon \mathcal{C}^{\mathrm{op}} \to \mathrm{Set}$, the category of elements (obtained by the so-called Grothendieck construction) of $F$ is a comma category $y/\ulcorner F \urcorner$ in the category $\mathrm{CAT}$ of </p>
<p>$$\mathcal{C} \xrightarrow{y} \widehat{\mathcal{... | Gerrit Begher | 1,261 | <p>The answer seems to be "no": Consider a strong-/pseudofuntor $F:C^\mathrm{op}\to \mathrm{Cat}$.</p>
<p>The object "sets" are isomorphic: they are given by pairs $(x,a_x)$ with $x\in C$ and $a_x\in F(x)$. But arrows in the Grothendieck construction / category of elements are given by pairs</p>
<p>$$(f:x\to y,\varph... |
431,783 | <p>I am taking an <a href="https://class.coursera.org/calcsing-002/class/index" rel="noreferrer">online course</a> and we are currently learning Integration and this is my first time experiencing intergration, though I have some knowledge of it. I am having some difficulty understanding what a differential equation act... | Hendrik Kosmol | 84,294 | <p>The notion of a differential equation can be somewhat strange at first, so don't feel discouraged by not getting it instantly.</p>
<p>You know how in algebra you have equations like $2/x = 4$ and you want to know what (if any) numbers make this a true statement?
A differential equation is like that, but instead of... |
437,462 | <blockquote>
<p>Find the rectangle with the maximum area inside the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, whose edges are parallel to the axises. <strong><em>Hint: Find the function we need to maximize</em></strong>.</p>
</blockquote>
<p>Well in the answers it says that the function we need to maximize is... | Erdos Yi | 79,947 | <p>Consider the first quadrant for simplicity. </p>
<p>Let $x = a\cos \theta$, $y =b\sin \theta$, then the area of the rectangle in the first quadrant is
$$
A = xy = \frac12ab\sin(2\theta)
$$
As $0\le\sin(2\theta)\le 1$, we can see the maximum area of $A_{\max}$ is $\frac12ab$.
Because of the axial symmetry of the el... |
2,716,692 | <p>In solving the equation $z^7-1=0$, the obvious route is to get the root $z=1$.
The next step is to solve : $(z-1)(z^6+z^5+z^4+z^3+z^2+z+1)=0$. Now, it is difficult for me to solve, and lack of experience is the key. However, have found out that an approach given is to have $a=z+1/z$ to get $a^3+a^2-2a-1=0$. I want ... | Jack D'Aurizio | 44,121 | <p>There's a simple trick:
$$\sum_{n\geq 1}\frac{\sin(nx)}{3^n} = \frac{3}{2}\cdot\frac{\sin x}{5-3\cos x}\tag{1} $$
and
$$ \int_{0}^{\pi} x \sin(nx)\,dx = \frac{\pi(-1)^{n+1}}{n}\tag{2}$$
hence
$$ \int_{0}^{\pi}\frac{x\sin(x)}{5-3\cos x}\,dx = \frac{2\pi}{3}\sum_{n\geq 1}\frac{(-1)^{n+1}}{n 3^n}=\frac{2\pi}{3}\log\lef... |
3,481,384 | <blockquote>
<p>Find all integers <span class="math-container">$m,\ n$</span> such that both <span class="math-container">$m^2+4n$</span> and <span class="math-container">$n^2+4m$</span> are perfect squares.</p>
</blockquote>
<p>I cannot solve this, except the cases when <span class="math-container">$m=n$</span>.</p... | David G. | 733,021 | <p>I've found several infinite families of solutions. For any integer <span class="math-container">$k$</span> :</p>
<ul>
<li><span class="math-container">$m=k$</span> and <span class="math-container">$n=1-k$</span></li>
<li><span class="math-container">$m=k^2$</span> and <span class="math-container">$n=0$</span></li>... |
1,780,856 | <p>The least positive integer that is divisible by $2, 3 ,4,$ and $5,$ and is also a perfect square, perfect cube, $4^{th}$ power, and $5^{th}$ power, can be written in the form $a^b$ for positive integers $a$ and $b$. What is the least possible value of $a+b$?</p>
<p>The answer is $90$.</p>
<p>Even completely cheat... | sTertooy | 336,630 | <p>Let's call $a^b = x$. What are the prime factors of $x$? You know it must be divisible by $2$, $3$, $4$ and $5$, so it is of the form $x = 2^p 3^q 5^r$. All 3 of the exponents $p,q,r,$ must be divisible by $2, 3, 4$ and $5$, and the smallest integer that satisfies this is $60$. Thus:
$$x = 2^{60}3^{60}5^{60} = (2\cd... |
66,525 | <p>Let $f(x)=a+bx^2$. Define $f_n(x)$ to be the $n$-fold composition of $f$. That is
$$f_1(x)=f(x)$$
$$f_2(x)=f \circ f(x)$$
$$f_n(x)=f \circ f_{n-1}(x), n \ge 2$$</p>
<p>Is there a way to find a formula for $f_n$?</p>
<p>I tried to write down $f_2$, $f_3,\ldots$, but I don't see any pattern.</p>
| GEdgar | 442 | <p>After a simple change of variables, you will be iterating $z^2+c$ for some $c$. There <em>is</em> a simple formula for the $n$th iterate when $c=0$ or $c=-2$. But not otherwise. </p>
<p><strong>added</strong> </p>
<p>Double-angle formula: $\cos 2\theta = 2\cos^2-1$, so if we write
$z=2\cos\theta$, then we get ... |
191,738 | <p>I have the following limit:</p>
<p>$$\lim_{n\rightarrow\infty}e^{-\alpha\sqrt{n}}\sum_{k=0}^{n-1}2^{-n-k} {{n-1+k}\choose k}\sum_{m=0}^{n-1-k}\frac{(\alpha\sqrt{n})^m}{m!}$$</p>
<p>where $\alpha>0$.</p>
<p>Evaluating this in Mathematica suggests that this converges, but I don't know how to evaluate it. Any he... | Sangchul Lee | 9,340 | <p>Let $A_n$ denote the formula inside the limit. By noting that the double summation is taken for those non-negative integers $k, m$ with $l := k+m \leq n-1$, by changing the order of summation,</p>
<p>$$\begin{align*}
A_n &= e^{-\alpha\sqrt{n}} \sum_{m=0}^{n-1}\sum_{k=0}^{n-1-m} \binom{n-1+k}{k} 2^{-n-k}\frac{(\... |
2,711,319 | <p>I calculated $$(p_k \cdot p_{k+1})\mod p_{k+2}$$ for $k=1,2,...25$ and obtained a following numbers $$1,1,2,12,7,12,1,2,16,11,40,12,24,7,13,16,48,40,72,48,40,60,15,48,12$$</p>
<p>We can see that there are only $8$ odd numbers and that some numbers repeat.</p>
<blockquote>
<p>Is this expected behavior? Does ratio... | Robert Israel | 8,508 | <p>See <a href="https://oeis.org/A182126" rel="noreferrer">OEIS sequence A182126</a>. With $a_n = p_n p_{n+1} \mod p_{n+2}$, $a_n = (p_{n+2}-p_n)(p_{n+2}-p_{n+1})$ when $(p_{n+2}-p_n)(p_{n+2}-p_{n+1}) < p_{n+2}$. Cramér's conjecture implies this is true for sufficiently large $n$ (and it seems to be the case for $... |
868,582 | <p>Suppose that $u$ and $w$ are defined as follows: </p>
<p>$u(x) = x^2 + 9$</p>
<p>$w(x) = \sqrt{x + 8}$</p>
<p>What is: </p>
<p>$(u \circ w)(8) = $</p>
<p>$(w \circ u)(8) = $</p>
<p>I missed this in math class. Any help?</p>
| Lubin | 17,760 | <p>When the function isn’t too complicated, it <em>may</em> help to express it in words. So, your $u$ is “square the input, and then add $9$, to get your final output”. And your function $w$ is “add $8$ to your input, and then take the square root to get your final output”. And I’m sure you know that $u\circ w$ means t... |
172,715 | <p>Let $P$ be a convex lattice polytope. Then it has a polynomial Ehrhart function.</p>
<p>I am interested in what can be said about the Ehrhart polynomial when
$P$ has any of the properties</p>
<ul>
<li>is integrally closed</li>
<li>has a unimodular triangulation</li>
<li>has a unimodular pulling triangulation</li... | matthias beck | 3,193 | <p>I don't think you can generally conclude the nonnegativity of Ehrhart polynomials when expressed in the standard monomial basis; however, the more natural basis for Ehrhart polynomials is given by the binomial coefficients $\binom{n+d} d$, $\binom{n+d-1} d$, ..., $\binom n d$ (here $n$ is the variable of the polynom... |
3,329,085 | <p>The only subspace of <span class="math-container">$\mathbb{R}$</span> that is homeomorphic to <span class="math-container">$\mathbb{R}$</span> and complete (with the restricted metric) is <span class="math-container">$\mathbb{R}$</span>.</p>
<p>My work-</p>
<p>Let <span class="math-container">$A$</span> be a subsp... | Scientifica | 164,983 | <p>I don't know how to help continue your proof, but here's a different method:</p>
<p>Let <span class="math-container">$A$</span> be such a subspace. Since <span class="math-container">$\mathbb R$</span> is connected, so is <span class="math-container">$A$</span>. Therefore <span class="math-container">$A$</span> is ... |
3,329,085 | <p>The only subspace of <span class="math-container">$\mathbb{R}$</span> that is homeomorphic to <span class="math-container">$\mathbb{R}$</span> and complete (with the restricted metric) is <span class="math-container">$\mathbb{R}$</span>.</p>
<p>My work-</p>
<p>Let <span class="math-container">$A$</span> be a subsp... | Danny Pak-Keung Chan | 374,270 | <p>Let <span class="math-container">$d$</span> be the usual metric on <span class="math-container">$\mathbb{R}$</span>. Let <span class="math-container">$f:\mathbb{R}\rightarrow X\subseteq\mathbb{R}$</span>
be a homeomorphism such that <span class="math-container">$(X,d)$</span> is a complete metric space.
We go to sho... |
2,871,106 | <p>Because when we draw a graph there have no break point and breakthrough where we can notice that function jumped from his path or removed?? <a href="https://i.stack.imgur.com/XGkQ8.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XGkQ8.jpg" alt="https://i.stack.imgur.com/XGkQ8.jpg"></a> </p>
| Tobias | 114,571 | <p>Let $f(x)$ be a continuously differentiable function at $x=0$ with $f(0)=0$ and let
\begin{align*}
g(x) :=\frac{f(x)}{x} \text{ for } x\neq 0.
\end{align*}
The following formula is a direct consequence of l'Hopital's rule:
\begin{align*}
\lim_{x\rightarrow 0} \frac{f(x)}{x}=f'(0)
\end{align*}</p>
<p>Therefore you c... |
243,160 | <p>$A$ is defined as an $m\times m$ matrix which is not invertible. How can i show that there is an $m\times m$ matrix $B$ where $AB = 0$ but $B$ is not equal to $0$? </p>
<p>For the solution of this question I think giving an example is not enough because it is too easy to solve this by giving an example, so how can ... | J. Swain | 97,454 | <p>Since $A$ is not invertible, by the Invertible Matrix Theorem there is a nontrivial solution
$x_0$ of the homogeneous equation $Ax_0 = 0$. Take $x_0$ as the first column of $B$
and take the remaining columns of $B$ to be zero. That gives $AB=0$ but $B$ is not the zero matrix.</p>
|
3,708,953 | <p>Suppose you have two random variables </p>
<p><span class="math-container">$X:\Omega \to \mathbb{R}$</span> and <span class="math-container">$Y:\Omega \to \mathbb{R}$</span> </p>
<p>which are not necessarily independent. </p>
<p>How is the product <span class="math-container">$XY$</span> defined and how do I calc... | David K | 139,123 | <p>First to address an apparent point of confusion in the comments:
"<span class="math-container">$X(\Omega)=\{X(\omega_1)=x_1,X(\omega_2)=x_2\}$</span>
and <span class="math-container">$Y(Ω)=\{Y(\omega_1)=y_1,Y(\omega_2)=y_2\}$</span>."</p>
<p>That is <em>not</em> how random variables work in general.
In general, the... |
2,754,870 | <p>I've been studying naive set theory and I have been told that Russell's paradox causes problems in Cantor's set theory when sets get "too big". </p>
<p>I don't understand why this causes a problem. I know how the paradox effected Frege's work in terms of logic but not Cantor. </p>
<p>Any help will be appreciated, ... | alepopoulo110 | 351,240 | <p>These problems were all solved when axioms were stated for Set Theory. These axioms made it very clear what is and what is not a set. Objects that are not sets are called classes and they have this characteristic property: They cannot be elements of any set.</p>
<p>So Russel's paradox says something like this: Let ... |
3,892,622 | <p>The following function
<span class="math-container">$$f(x)=\left\{\begin{array}{ll} 50, &0\leq x\leq10 \\ 10+6x-0.2x^2, & 10<x\leq30\end{array}\right.$$</span>
gives me the number of products that are sold in the <span class="math-container">$x$</span>-th day of a month.</p>
<p>How am I supposed to find t... | Henry | 6,460 | <p>An approximation might be <span class="math-container">$$\int_{-0.5}^{10.5} 50 \,dx + \int_{10.5}^{30.5} (10+6x-0.2x^2)\, dx$$</span> which would suggest <span class="math-container">$\frac{4187}{3} = 1396-\frac13$</span>, so not far away</p>
|
883,903 | <p>I came across this problem whilst studying for a comprehensive exam in real analysis; for reference, see Exercise 1.24(A) in Folland's <em>Real Analysis</em>; it's a modification of that.</p>
<p>Consider the unit interval $I:=\left[0,1\right]$, and let $\mathcal{M}$ be the $\sigma$-algebra of all Lebesgue measurabl... | Yizi | 737,057 | <p>Here is one solution written by Xiudi Tang in his course notes:</p>
<p>It suffices to show that <span class="math-container">$m^*(A \cap E) = m^*(A)$</span> for any <span class="math-container">$A \in \mathcal{M}$</span>.</p>
<p>By the measurablity of <span class="math-container">$A$</span>, we have that <span cla... |
73,928 | <p>Suppose I wanted to use Mathematica graphics primitives to create a gradient of colors between two circular arcs. It's easy enough to make an area between two circles a solid color, but what if I wanted to have the area between two colors blur from black to white as you go out in radius?</p>
<p>VertexColor provides... | DavidC | 173 | <h2>With a little help from <code>LaminaData</code></h2>
<p><code>LaminaData</code> is a good place to start.
This leads to two approaches, each of which ultimately displays the figure in <code>RegionPlot</code>.</p>
<p>Approach 1 is based on graphics objects (the intersection region of two disks). The graphics obje... |
264,591 | <p>i have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus book, i love maths, i need the right well of precalculus books. </p>
| William Martens | 829,250 | <h5>Hi, this is my preference (not everyone will agree)</h5>
<p>As no one (so far as I can see?) have recommended; I can say:</p>
<ul>
<li><p><span class="math-container">$George\, F\,.\, Simmons\ Calculus\ With\ Analytic\ Geometry$</span> 2nd edition</p>
</li>
<li><p><a href="https://rads.stackoverflow.com/amzn/click/... |
1,164,229 | <p>How do I solve: $$\lim_{x\to 0} (\cos 3x)^{1/ x^2}$$</p>
<p>I don't think I can use L'hopital because of the power? not really sure how I would rewrite this so that it is $f(x)/g(x)$ where $\lim f'(x) = 0$ and $\lim g'(x) =0$.</p>
<p>So I tried the following:</p>
<p>$$\cos(3x)^{1/x^2} =\cos(3x)^{x^{-2}}$$</p>
<p... | asosnovsky | 145,038 | <p>Here is how I'd do it:
(no Taylor expansion or $o$ order, just pure old brute force with L'hopitals rule) </p>
<p>$
\lim\limits_{x \to 0} (\cos (3x))^{1/x^2} =
\lim\limits_{x \to 0} e^{x^{-2} \ln (\cos (3x))} =
e^{\lim\limits_{x \to 0} x^{-2} \ln (\cos (3x))}
$</p>
<p>So now let's observe:</p>
<p>$
\lim\limits_{x... |
3,897,460 | <p>If <span class="math-container">$F$</span> has characteristic <span class="math-container">$P>0$</span> prove that <span class="math-container">$$A=\begin{pmatrix}1&\alpha\\0&1\end{pmatrix}$$</span> satisfies <span class="math-container">$A^p=I$</span>.</p>
<p>I am trying to solve this problem but I could... | VIVID | 752,069 | <p>You can find the <span class="math-container">$n^\text{th}$</span> power of your matrix:
<span class="math-container">$$
\begin{pmatrix}
1 & \alpha \\
0 & 1
\end{pmatrix}^2 =
\begin{pmatrix}
1 & 2\alpha \\
0 & 1
\end{pmatrix}
$$</span>
<span class="math-container">$$
\begin{pmatrix}
1 & \alpha \... |
1,197,795 | <p>A point is selected at random from the triangle {(x,y):0 <=x <= y <=1}. Let E be the event that a selected point has the x coordinate is less than 0.5 and F be the event that y-coordinate is larger than 2 times the x-coordinate. </p>
<p>a) sketch event E and F in the triangle
b) Determine if E and F are in... | Spenser | 39,285 | <p>$x\in\mathbb{R}[x]/(x^2)$ and $x\neq0$, but $x$ has no inverse since $x(a+bx)=ax\neq 1$ for any $a,b$.</p>
|
1,553,754 | <p>There is a highly believable theorem:</p>
<p>Let $A, B$ be disjoint sets of generators and let $F(A), F(B)$ be the corresponding free groups. Let $R_1 \subset F(A)$, $R_2 \subset F(B)$ be sets of relations and consider the quotient groups $\langle A | R_1 \rangle$ and $\langle B | R_2 \rangle$.</p>
<p>Then $\lang... | 1123581321 | 482,390 | <p>I'll try to give another proof of the fact <span class="math-container">$G_i=\langle X_i|R_i\rangle \Rightarrow *G_i=\langle \sqcup X_i| \sqcup R_i\rangle$</span>. Please let me know if it is coorect.</p>
<p>We have the following diagramm (I'm sorry for this bad picture but I could not type it here)</p>
<p><a href... |
2,849,851 | <blockquote>
<p>Prove that $(|x-1|^{\frac{6}{2}})^n = |x-1|^{3n}$ for every $x \in \mathbb{R}$ and $n \in \mathbb{N}$. </p>
</blockquote>
<p>I have some trouble and hard days to finish this problem. It seems so easy to think the way by using the exponentiation properties there, but this is one of real analysis probl... | Community | -1 | <p>First cleanup. As the property must hold for every $x$ and $x$ is only used in $|x-1|$, it suffices to prove for any positive $y$. Then $6/2=3$ and the identity simplifies to</p>
<p>$$(y^3)^n=y^{3n}.$$</p>
<p>This is a basic property of exponentiation.</p>
|
1,439,184 | <p>I am trying to write recursive algorithms for the following string operations:</p>
<p>1) An algorithm to reverse a string. </p>
<p>2) An algorithm to test if two strings are equal to each other. </p>
<p>3) An algorithm to check if a string is a palindrome (for example, "eye" or "racecar") - reads the same forward... | JMP | 210,189 | <p>Imagine we have a function that takes two string parameters, say <strong>first</strong> and <strong>last</strong>, and it returns <strong>last+first</strong> as a string.</p>
<blockquote>
<p>function reverseString(first,last) {return last+first;}</p>
</blockquote>
<p>So:</p>
<blockquote>
<p>reverseString('world','he... |
100,124 | <p>I'm going through a proof of the statement:</p>
<hr>
<p>Let $A$ and $B$ be commutative rings.
If $A \subseteq B$ and $B$ is a finitely generated $A$-module, then all $b \in B$ are integral over $A$. </p>
<hr>
<p>Proof:</p>
<p>Let $\{c_1, ... , c_n\} \subseteq B$ be a set of generators for $B$ as an $A$-module, ... | Georges Elencwajg | 3,217 | <p>You prematurely write "Then we must have that $\mathrm{det}(bI_n - (a_{ij})) = 0$".<br>
At that stage you can only deduce (by multiplying by the adjoint of your matrix on the left) that all the $det\cdot c_i =0$.<br>
However writing $1 = \alpha_1 c_1 + ... + \alpha_n c_n$ and multiplying by $det$ you do get </p>
... |
4,056,380 | <p>I am trying to prove the following statement from Axler's MIRA book and I would appreciate an hint about how to finish my proof (NOTE: <span class="math-container">$||$</span> refers to outer measure):</p>
<p>"<span class="math-container">$A\subset\mathbb{R}, t>0\Rightarrow |A|=|A\cap (-t,t)|+|A\cap (\mathbb... | David C. Ullrich | 248,223 | <p>It really is immediate from the definition. First a technicality:</p>
<blockquote>
<blockquote>
<p><strong>Lemma</strong> The definition of <span class="math-container">$|A|$</span> is unchanged if closed or half-open intervals are allowed as well as open intervals.</p>
</blockquote>
</blockquote>
<p>Proof: Say <spa... |
4,056,380 | <p>I am trying to prove the following statement from Axler's MIRA book and I would appreciate an hint about how to finish my proof (NOTE: <span class="math-container">$||$</span> refers to outer measure):</p>
<p>"<span class="math-container">$A\subset\mathbb{R}, t>0\Rightarrow |A|=|A\cap (-t,t)|+|A\cap (\mathbb... | tchappy ha | 384,082 | <p>I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.<br />
This exercise is Exercise 8 on p.23 in Exercises 2A in this book.</p>
<p>David C. Ullrich, thank you very much for your answer.</p>
<p>I slightly modified your answer.</p>
<p>If <span class="math-container">$|A|=\infty$</span>,... |
1,987,685 | <p>How do I find the following sum?
$\sum_{k=0}^n(-1)^k{2n\choose2k}$</p>
<p>Tried to simplify it somehow but got nothing less complicated.</p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One may observe that, by the <a href="https://en.wikipedia.org/wiki/Binomial_theorem#Theorem_statement" rel="nofollow">binomial theorem</a>,
$$
\begin{align}
\sum_{k=0}^n(-1)^k{2n\choose2k}&=\sum_{k=0}^n {2n\choose2k}i^{2k}
\\\\&=\sum_{k=0}^{2n} {2n\choose2k}i^{2k}
\\\\&=\sum_{p=0}... |
2,571,197 | <p>This is a problem from Berkeley problems in mathematics.</p>
<p>If $F$ is a subfield of $K$, and $M$ has entries in $F$, how is the row rank of $M$ over $F$ related to the row rank of $M$ over $K$?</p>
<p>where $M$ is a n by n matrix</p>
<p>The solution says "If a set of rows of $M$ is linearly independent over $... | user1551 | 1,551 | <p>The rank of a matrix is the size of its largest square submatrix with nonzero determinant. Since the determinant is determined only by the matrix's entries, it remains the same during a field extension. Thus the rank remains invariant too.</p>
<p>Alternatively, the rank of a matrix is just the number of nonzero dia... |
3,324,817 | <p>Angles <span class="math-container">$(\psi -\theta) $</span> relation is <em>unique</em> for Conics in a 3D situation. </p>
<p>// It is to seek <span class="math-container">$r- \psi$</span> of planar sections aka Conics on Gauss curvature <span class="math-container">$K=0$</span> cones for uniqueness and generality... | Blue | 409 | <p>It's not really clear what kind of relation OP expects, but here's something ...</p>
<p>Let the generators of a cone (with vertex at the origin and <span class="math-container">$z$</span>-axis as axis) make angle <span class="math-container">$\alpha$</span> with the <span class="math-container">$xy$</span>-plane. L... |
574,263 | <p>Give a 3x3 matrix that maps all points in $\mathbb{R}^3$ onto the line $[x,y,z] = t[a,b,c]$ and does not move the points that are on that line. Prove your matrix has these properties.</p>
<p>I'm not sure how to start this problem. My instructor told me the matrix should have a rank of 1, but I don't know what I sho... | AnyAD | 107,693 | <p>Look at it as a linear transformation, say $T$. Consider the dimension of the range of $T$,clearly equal the given line, spanned by the directional, so dimension $1$. Remember that the dimension of image of $T$ equals rank of $T$. </p>
|
4,606,642 | <p>We have the given problem</p>
<p><span class="math-container">$$xu_x-yu_y=u,\ x>0, \ y>0 \ (1)$$</span>
<span class="math-container">$$u(x,x)=x^2, \ x>0, \ (2)$$</span></p>
<p>They ask to check if the problem is well-posed and solve it next.</p>
<p>I know that a problem is well-posed if:</p>
<ol>
<li>It ha... | Giuseppe Negro | 8,157 | <p>The question itself is ill-posed. This is not a general boundary value problem or initial value problem. It is a specific request which you can explicitly solve. (You already solved it, actually, as <a href="https://math.stackexchange.com/questions/4606642/is-this-pde-xu-x-yu-y-u-well-posed#comment9707604_4606642">n... |
47,636 | <h3>Background</h3>
<p>Knots are typically written in 2 dimensions as a loop in the plane with normal crossings. One then asks when two such diagrams describe the same knot. Two diagrams describe the same knot when one can be made into the other by a sequence of <strong><a href="http://en.wikipedia.org/wiki/Reidermei... | sleepless in beantown | 8,676 | <p>Take a look at page 180 of <em>Low dimensional topology</em> by Tomasz Mrowka, Peter Steven Ozsvát, (following Ben Webster's comment about <strong>Movie Moves</strong> elucidated by Baez and Langford and their 30 basic movie moves, and by Carter and Saito who describe a 31st basic movie move.) A movie move is a seq... |
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