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 |
|---|---|---|---|---|
81,209 | <p>I feel a bit ashamed to ask the following question here. </p>
<blockquote>
<p>What is (actually, is there) Galois
theory for polynomials in
$n$-variables for $n\geq2$?</p>
</blockquote>
<p>I am preparing a large audience talk on Lie theory, and decided to start talking about symmetries and take Galois theory... | KristianJS | 19,367 | <p>(This should really be a comment I think, but I'm not highly rated enough to leave one, so please bear with me)</p>
<p>A Galois Theoretic condition for a polynomial in two variables to be solvable by radicals is found in the following paper: <a href="http://arxiv.org/abs/math/0305226" rel="noreferrer">http://arxiv.... |
139,232 | <p>Let $O$ be an operad in $\mathtt{SETS}$. Assume that $O(0)$ is empty and $O(1)$ only consists of the identity. Assume for simplicity that $O$ is monochromatic, i.e. we have no labels on the in/outputs. Assume also for simplicity that the operad is plain, i.e. neither symmetric nor braided. So the operads in question... | James Griffin | 110 | <p>I've left my original answer as some people may find it of interest.</p>
<p>I have a candidate counterexample. The idea is to find a (non-symmetric) set operad in between the free operad $Free_2$ on a single arity 2 generator and the associative operad $As$. The example I've chosen is the operad $P$ which is isom... |
2,002,601 | <p>Given n+1 data pairs $(x_0,y_0)...(x_n,y_n)$ for j=0,1,2...,n we have
$p_j=\prod_{i\neq j}(x_j-x_i)$ and $\psi(x)=\prod_{i=0}^n(x-x_i)$.</p>
<p>I am having trouble determining what $\psi(x_j)$ is and what $\psi'(x_j)$ would be. </p>
<p>I feel like $\psi(x_j)= 0$ because it would contain the $x_j-x_j$ term... But ... | parsiad | 64,601 | <p><strong>Hint</strong>: $\tanh(nx)\rightarrow \operatorname{sign}(x)$ pointwise as $n\rightarrow\infty$. Now, use the fact that the uniform limit of continuous functions is continuous for a locally compact space (such as $\mathbb{R}$).</p>
|
2,002,601 | <p>Given n+1 data pairs $(x_0,y_0)...(x_n,y_n)$ for j=0,1,2...,n we have
$p_j=\prod_{i\neq j}(x_j-x_i)$ and $\psi(x)=\prod_{i=0}^n(x-x_i)$.</p>
<p>I am having trouble determining what $\psi(x_j)$ is and what $\psi'(x_j)$ would be. </p>
<p>I feel like $\psi(x_j)= 0$ because it would contain the $x_j-x_j$ term... But ... | hamam_Abdallah | 369,188 | <p>As $\tanh$ is an odd function, we will assume $x>0$</p>
<p>$$\tanh(nx)=\frac{1-e^{-2nx}}{1+e^{-2nx}}$$</p>
<p>thus</p>
<p>$\lim_{n\to\infty}\tanh(nx)=1$</p>
<p>if $x=0, f_n(0)=0$.</p>
<p>all the functions $f_n$ are continuous at $\mathbb R$.</p>
<p>the pointwise limit function is not continuous at $0$, th... |
145,286 | <p>Yesterday I got into an argument with @UnchartedWorks over <a href="https://mathematica.stackexchange.com/a/145207/26956">in the comment thread here</a>. At first glance, he posted a duplicate of <a href="https://mathematica.stackexchange.com/a/145202/26956">Marius' answer</a>, but with some unnecessary memoization:... | Szabolcs | 12 | <p>You are absolutely correct that this memoization is completely unnecessary.</p>
<p>What seems to happens is that from the second run onwards on the same data, the builtin functions become faster. I do not understand why (perhaps some internal caching), but it does show that it has absolutely nothing to do with the... |
145,286 | <p>Yesterday I got into an argument with @UnchartedWorks over <a href="https://mathematica.stackexchange.com/a/145207/26956">in the comment thread here</a>. At first glance, he posted a duplicate of <a href="https://mathematica.stackexchange.com/a/145202/26956">Marius' answer</a>, but with some unnecessary memoization:... | jkuczm | 14,303 | <p>I can't reproduce claimed speedup on <code>"11.0.1 for Linux x86 (64-bit) (September 21, 2016)"</code>.</p>
<p>In my tests, custom function wrappers without memoization (as <a href="https://mathematica.stackexchange.com/questions/145286/throwaway-memoization-makes-built-ins-faster#comment390746_145286">suggested by... |
3,831,387 | <p><span class="math-container">$X,Y\sim N(0,1)$</span> and are independent, consider <span class="math-container">$X+Y$</span> and <span class="math-container">$X-Y$</span>.</p>
<p>I can see why <span class="math-container">$X+Y$</span> and <span class="math-container">$X-Y$</span> are independent based on the fact th... | antkam | 546,005 | <p>(1) The short, short answer is that it is <strong>wrong</strong> to say</p>
<p><span class="math-container">$$\mathbb{P}(X+Y=u|X-Y=v)\neq \mathbb{P}(X+Y=u)\,\,\,\,\,\,\text{(this is wrong)}$$</span></p>
<p>because in fact, both sides <span class="math-container">$=0$</span>, as these are continuous variables.</p>
<p... |
1,842,826 | <blockquote>
<p>Explain why the columns of a $3 \times 4$ matrix are linearly dependent</p>
</blockquote>
<p>I also am curious what people are talking about when they say "rank"? We haven't touched anything with the word rank in our linear algebra class.</p>
<p>Here is what I've came up with as a solution, will th... | Ethan Bolker | 72,858 | <p>Without knowing how far you've gotten in your linear algebra class it's hard top produce a proof at the right level.</p>
<p>What's really going on here is that the four columns of a matrix with three rows are vectors in three dimensional space. Since the dimension of the space is three, any set with more than three... |
246,589 | <p>Solve the boundary value problem
$$\begin{cases} \displaystyle \frac{\partial u}{\partial t} = 2 \frac{\partial^2 u}{\partial x^2} \\ \ \\u(0,t) = 10 \\ u(3,t) = 40 \\ u(x, 0) = 25 \end{cases}$$</p>
| Frenzy Li | 32,803 | <p><em><strong>HINT:</em></strong> Let's make the boundary condition homogenous by $v(x,t)=u(x,t)-10-10x$.<br>
(<em>How to see that?</em>). The pde for $v(x,t)$ is thus: (Notice that $v_t=u_t$, $v_{xx}=u_{xx}$, <em>and why?</em>)</p>
<p>$$\begin{cases}
v_t - 2 v_{xx} = 0, & 0<x<3, t>0, \\
v(0,t)=v(3,... |
308,117 | <p>I have the matrix
$$A := \begin{bmatrix}6& 9& 15\\-5& -10& -21\\ 2& 5& 11\end{bmatrix}.$$ Can anyone please tell me how to both find the eigenspaces by hand and also by using the Nullspace command on maple? Thanks.</p>
| Amzoti | 38,839 | <p>Given the matrix </p>
<p>$$A = \left(\begin{matrix}6& 9& 15\\-5& -10& -21\\ 2& 5& 11\end{matrix}\right).$$</p>
<p>Find the Eigensystem by hand.</p>
<p>First, lets find the eigenvalues by solving $det(A - \lambda I) = 0$, so we have:</p>
<p>$$det(A - \lambda I) = \left|\begin{matrix}6 - \... |
4,136,248 | <p>Let <span class="math-container">$a,b\in\mathbb{R}^+$</span>.
Suppose that <span class="math-container">$\{x_n\}_{n=0}^\infty$</span> is a sequence satisfying
<span class="math-container">$$|x_n|\leq a|x_{n-1}|+b|x_{n-1}|^2, $$</span>
for all <span class="math-container">$n\in\mathbb{N}$</span>. How can we bound <sp... | Alexandre Eremenko | 110,120 | <p>Let us define the sequence <span class="math-container">$y_n$</span> by <span class="math-container">$y_0=|x_0|$</span>, <span class="math-container">$y_{n+1}=ay_n+by_n^2$</span>. Then we have
<span class="math-container">$|x_n|\leq y_n$</span> for all <span class="math-container">$n$</span>, since <span class="math... |
89,810 | <p>I have defined a table </p>
<pre><code>Table[Table[
Graphics3D[
Cuboid[radijDensity[[j, i]] {-Sin[kotiDensity[[j, i]]],
1 - Cos[kotiDensity[[j, i]]], 0}, {radijDensity[[j, i]]*
Sin[kotiDensity[[j, i]]],
radijDensity[[j, i]] (1 - Cos[kotiDensity[[j, i]]]) +
visina[[j, i + 1]], 0}]], {i... | BenP1192 | 30,524 | <p>Its difficult to work with your question since you don't define your functions and variables but hopefully this example will be enough.</p>
<p>Let's first make a table of cuboids with different z values (but using the same z-value within each cuboid so they are still rectangles). This examples uses the same x and y... |
1,988,563 | <blockquote>
<p>Use the formal defintion to prove the given limit:
$$\lim_{x\to\frac13^+}\sqrt{\frac{3x-1}2}=0$$</p>
</blockquote>
<p>Not sure how to deal with $\sqrt\cdot$. Appreciate a hint.</p>
| ec92 | 34,552 | <p>You want to show that for any $\epsilon > 0$, there is $\delta >0$ such that if
$$ 0< x - \frac13 < \delta, $$
then
$$ \sqrt{\frac{3x-1}{2}} < \epsilon. $$</p>
<p>This is equivalent to
$$0 < 3x - 1 < 2 \epsilon^2,$$
or
$$ 0 <x - \frac13 < \frac23 \epsilon^2,$$
so you can choose $\del... |
3,422,830 | <blockquote>
<p>In the polynomial
<span class="math-container">$$
(x-1)(x^2-2)(x^3-3) \ldots (x^{11}-11)
$$</span>
what is the coefficient of <span class="math-container">$x^{60}$</span>? </p>
</blockquote>
<p>I've been trying to solve this question since a long time but I couldn't. I don't know whether opening ... | trancelocation | 467,003 | <p>The highest exponent possible is <span class="math-container">$1+2+ \cdots + 11 = 66$</span>.</p>
<p>Now, to create the exponent <span class="math-container">$60$</span>, you can only leave out the factors containing <span class="math-container">$(1,2,3),(2,4),(1,5)$</span> and <span class="math-container">$6$</span... |
286,930 | <p>I have been assigned this problem for homework:</p>
<blockquote>
<p>Show that, if $a < b + \epsilon$ for every $\epsilon \gt 0$, then $a\le b$.</p>
</blockquote>
<p>I have tried to go about this using Induction, but I don't know what the base case would be. It is obvious to me in my mind, but I don't know ho... | Clayton | 43,239 | <p>Hint: Use contradiction. If $a>b$, then show $a$ is not less than $b+\frac{a-b}{2}$. This contradiction implies $a\leq b$.</p>
|
286,930 | <p>I have been assigned this problem for homework:</p>
<blockquote>
<p>Show that, if $a < b + \epsilon$ for every $\epsilon \gt 0$, then $a\le b$.</p>
</blockquote>
<p>I have tried to go about this using Induction, but I don't know what the base case would be. It is obvious to me in my mind, but I don't know ho... | Steven Gamer | 47,540 | <p>assume a > b. Then a-b > 0. Then let ϵ = a - b. Then a < b + a-b = a , a contradiction so a <= b.</p>
|
424,445 | <p>I'm studying Pattern recognition and statistics and almost every book I open on the subject I bump into the concept of <strong>Mahanalobis distance</strong>. The books give sort of intuitive explanations, but still not good enough ones for me to actually really understand what is going on. If someone would ask me "W... | Avitus | 80,800 | <p>As a starting point, I would see the Mahalonobis distance as a suitable deformation of the usual Euclidean distance $d(x,y)=\sqrt{\langle x,y \rangle}$ between vectors $x$ and $y$ in $\mathbb R^{n}$. The extra piece of information here is that $x$ and $y$ are actually <em>random</em> vectors, i.e. 2 different realiz... |
1,527,137 | <p>Usually one has the matrix and wishes to estimate the eigenvalues, but here it's the other way around: I have the positive eigenvalues of an unknown real positive definite matrix and I would like to say something about it's diagonal elements.</p>
<p>The only result I was able to find is that the sum of the eigenval... | B. S. Thomson | 281,004 | <p>Well you are not getting much help here and perhaps the homework deadline is looming--so here is a push.</p>
<p>The function $f:[0,T]\to \mathbb{R}$ has bounded variation there. You already have seen, for any $0\leq a<b\leq T$, that $$V(f,[a,b]) + f(b)-f(a)\geq 0$$ is trivial.</p>
<p>What you need to show now... |
104,297 | <p>How would I go about solving</p>
<p>$(1+i)^n = (1+\sqrt{3}i)^m$ for integer $m$ and $n$?</p>
<p>I have tried </p>
<pre><code>Solve[(1+I)^n == (1+Sqrt[3] I)^m && n ∈ Integers && m ∈ Integers, {n, m}]
</code></pre>
<p>but this does not give the answer in the 'correct' form.</p>
| rhermans | 10,397 | <pre><code>Last@Reap@Do[
If[
ReIm[(1 + I)^n] == ReIm[(1 + Sqrt[3] I)^m]
, Sow[{n, m}]
]
, {n, 100}
, {m, 100}
]
</code></pre>
<blockquote>
<pre><code>{{{24, 12}, {48, 24}, {72, 36}, {96, 48}}}
</code></pre>
</blockquote>
|
310,930 | <p>Let $U$ be the subspace of $\mathbb{R}^3$ spanned by $\{(1,1,0), (0,1,1)\}$. Find a subspace $W$ of $\Bbb R^3$ such that $\mathbb{R}^3 = U \oplus W$.</p>
<p>As I am having an examination tomorrow, it would be really helpful if one could explain the methodology for doing this problem. I am mostly interested in the m... | Gerry Myerson | 8,269 | <p>Can you see that $W$ must be one-dimensional? So you are just looking for a single (non-zero) vector that's not in $U$ --- then let $W$ be the span of that vector. </p>
|
786,655 | <p>Say we have two r.v X and Y which are independent and differently distributed ( for e.g X follows a bell curve and Y follows an exponential distribution with parameter $\lambda > 0$</p>
<p>What are the different methods to numerically compute the distribution X+Y, X*Y, X/Y, min(X,Y) etc...?</p>
<p>I read abou... | fgp | 42,986 | <p>If $X,Y$ are independent and have distribution function $F_X,F_Y$ and densities $f_X$,$f_Y$, you have $$\begin{eqnarray}
&P(X+Y \leq z) &=& \int_{x+y \leq z} f_X(x) f_Y(y) \,d(x,y) = \int_{-\infty}^\infty \int_{-\infty}^{z-x} f_X(x) f_Y(y) \,dy \,dx \\
&&=& \int_{-\infty}^\infty f_X(x) F_Y... |
32,809 | <p>Is it possible to define new graphics directives?</p>
<p>For example, suppose I want to be able to use the following code:</p>
<pre><code>Graphics[{ BigPointSize[0.07], SmallPointSize[0.04],
Red, BigPoint[{1,1}], BigPoint[{1,3}], SmallPoint[{3,1}],
Blue, SmallPoint[{2,2}], SmallPoint[{3,2}], ... | Carl Woll | 45,431 | <p>It is possible to use <a href="http://reference.wolfram.com/language/ref/Style" rel="noreferrer"><code>Style</code></a> options as "graphics directives", and <a href="http://reference.wolfram.com/language/ref/CurrentValue" rel="noreferrer"><code>CurrentValue</code></a> can be used to query the values of these option... |
3,464,885 | <p>I need help integrating this one.
<span class="math-container">$\int \frac{\sin(50x)}{1+\cos^2(50x)}\,dx$</span></p>
<p>I started with <span class="math-container">$u = 50x$</span> as my <span class="math-container">$u$</span>-sub</p>
<p><span class="math-container">$$\int \frac{\sin(u)}{1+\cos^2(u)}\,dx$$</span><... | user284331 | 284,331 | <p><span class="math-container">\begin{align*}
\int\dfrac{\sin u}{1+\cos^{2}u}du&=-\int\dfrac{1}{1+\cos^{2}u}d(\cos u)=-\tan^{-1}(\cos u)+C.
\end{align*}</span></p>
|
3,464,885 | <p>I need help integrating this one.
<span class="math-container">$\int \frac{\sin(50x)}{1+\cos^2(50x)}\,dx$</span></p>
<p>I started with <span class="math-container">$u = 50x$</span> as my <span class="math-container">$u$</span>-sub</p>
<p><span class="math-container">$$\int \frac{\sin(u)}{1+\cos^2(u)}\,dx$$</span><... | Kenta S | 404,616 | <p><span class="math-container">\begin{equation}
\begin{split}
\int \frac{\sin(50x)}{1+\cos^2(50x)}dx&=\frac1{50}\int \frac{\sin(u)}{1+\cos^2(u)}du\\
&=-\frac1{50}\int \frac{(\cos(u))'}{1+\cos^2(u)}du\\
&=-\frac1{50}\arctan(\cos(u))+C\\
&=-\frac1{50}\arctan(\cos(50x))+C\\
\end{split}
\end{equation}</spa... |
135,663 | <p>It is a problem for a Hatcher's book, and it is my homework problem.</p>
<p>It is a section 2.2 problem 3, stating:</p>
<p>Let $f:S^n\to S^n$ be a map of degree zero. Show that there exist points $x,y \in S^n$ with $f(x)=x$ and $f(y)=-y$. Use this to show that if $F$ is a continuous vector filed defined on the uni... | Community | -1 | <p>Assume $H \leq A_5$ with $|H| = 15$ and let $X:=\{gH \mid g \in G\}$. Then $\# X = 4$. $G$ acts op $X$ by left multiplication i.e. $g'(gH) = (g'g)H$. Let $\alpha \in A_5$ be a 5-cycle. Then $\langle \alpha\rangle$ does act on $X$,too. But the length of an orbit divides the group-order which is 5. But $\# X = 4 < ... |
604,070 | <p>While doing the proof of the existence of completion of a metric space, usually books give an idea that the missing limit points are added into the space for obtaining the completion. But I do not understand from the proof where we are using this idea as we just make equivalence classes of asymptotic Cauchy sequence... | Henno Brandsma | 4,280 | <p>One way to construct the completion of a metric space $(X,d)$ is by isometrically embedding it into a large metric space, that is known to be complete already. </p>
<p>A classical way is due to (IIRC) Banach: define $CB(X)$ to be the set of all bounded continuous real-valued functions on $X$ with metric from the su... |
2,820,779 | <p>So i have this integral </p>
<p>$$\int_{\sqrt[3]{4}}^{\sqrt[3]{3+e}}x^2 \ln(x^3-3)\,dx.$$</p>
<p>I was thinking of using u subsitution to make everything easier. </p>
<p>I made $u = x^3-3$ and $du = 3x^2dx$.</p>
<p>So I would then re-write my integral as </p>
<p>$$1/3\int_{\sqrt[3]{4}}^{\sqrt[3]{3+e}} \ln(x^3-3... | Bernard | 202,857 | <p>When you integrate by substitution, you have the express the differential form under the integral sign: $f(x)\,\mathrm dx$ as a differential form $\;g(u)\,\mathrm d u$, and replace the bounds for the integral in $x$ with corresponding bounds for the new variable $u$.</p>
<p><em>Some details in this case</em>:</p>
... |
2,647,123 | <p>I'm asked to to find a $3\times3$ matrix, in which no entry is $0$ but $A^2=0$. </p>
<p>The problem is if I I brute force it, I am left with a system of 6 equations (Not all of which are linear...) and 6 unknowns. Whilst I could in theory solve that, is there more intuitive way of solving this problem or am I going... | Henry | 6,460 | <p>Force the first element of $A^2$ to be $0$, for example by finding $b,c,d,e$ with $bd+ce \lt 0$ and let $a=\pm{\sqrt{-(bd+ce)}}$ </p>
<p>Then consider</p>
<p>\begin{bmatrix}a&b&c\\d&db/a&dc/a\\e&eb/a&ec/a\end{bmatrix}</p>
|
2,647,123 | <p>I'm asked to to find a $3\times3$ matrix, in which no entry is $0$ but $A^2=0$. </p>
<p>The problem is if I I brute force it, I am left with a system of 6 equations (Not all of which are linear...) and 6 unknowns. Whilst I could in theory solve that, is there more intuitive way of solving this problem or am I going... | Sarvesh Ravichandran Iyer | 316,409 | <p>To do this, first consider a trivial matrix with lots of zeros that does satisfy this condition. One easy one is the matrix with a single $1$ on the diagonal above the principal. That is:
$$
\begin{pmatrix}
0 & 1 & 0 \\
0 & 0& 0 \\
0 & 0 & 0
\end{pmatrix}
$$</p>
<p>Now, given any matrix $A$,... |
2,491,394 | <p>So, my problem is with Axiom 5 of the proof, where Gödel considers necessary existence as a property. However, by his own definition, a 'property' applies to objects including those whose necessary existence has not even been proven, as can be inferred from Theorem 1. This, to me, seems like the perfect example of q... | Nagase | 117,698 | <p>Note that Theorem 1 of your link actually states: $P(\phi) \implies \Diamond \exists x \phi(x)$, i.e. if $\phi$ is a positive property, then possibly there is something that instantiates it. Given this, Gödel needs an explicit axiom stating that being god-like is a positive property (Axiom 3 in your link: $P(G)$). S... |
65,480 | <p>The example question is </p>
<blockquote>
<p>Find the remainder when $8x^4+3x-1$ is divided by $2x^2+1$</p>
</blockquote>
<p>The answer did something like</p>
<p>$$8x^4+3x-1=(2x^2+1)(Ax^2+Bx+C)+(Dx+E)$$</p>
<p>Where $(Ax^2+Bx+C)$ is the Quotient and $(Dx+E)$ the remainder. I believe the degree of Quotient is d... | Pierre-Yves Gaillard | 660 | <p>There is a simple closed formula for the remainder $R$ and the quotient $Q$ of the euclidean division of a polynomial $P$ by a nonzero polynomial $D$. Here $P,D,Q,R$ are in $\mathbb C[X]$. </p>
<p>For any complex number $a$, any nonnegative integer $k$, and any rational fraction $f(X)\in\mathbb C(X)$ defined at $a$... |
3,460,426 | <p>I tried to take the <span class="math-container">$Log$</span> of <span class="math-container">$\prod _{m\ge 1} \frac{1+\exp(i2\pi \cdot3^{-m})}{2} = \prod _{m\ge 1} Z_m$</span>, which gives </p>
<p><span class="math-container">$$Log \prod_{m\ge 1} Z_m = \sum_{m \ge 1} Log (Z_m) = \sum_{m \ge 1} \ln |Z_m| + i \sum_{... | Community | -1 | <p>Let <span class="math-container">$p_k=P(X=k)$</span>. If all children have equal probability of being chosen, then the probability of being in a family with <span class="math-container">$k$</span> children is <span class="math-container">$$\frac{kp_k}{\sum ip_i}= \frac{kp_k}{1.8}$$</span> and so the expected number ... |
1,456,444 | <p>How can I go about solving this Pigeonhole Principle problem? </p>
<p>So I think the possible numbers would be: $[3+12], [4+11], [5+10], [6+9], [7+8]$</p>
<p>I am trying to put this in words...</p>
| Zach466920 | 219,489 | <p>Denote the set $[3,4,5,6,7,8,9,10,11,12]$ by $S$. This set has $10$ elements. </p>
<p>As you correctly noted, this set can be split into another set $T$ with $5$ elements, $[(3,12);(4,11);(5,10);(6,9);(7,8)]$, such that the binary sum of each of the components for each element is $15$. </p>
<p>If you pick $6$ inte... |
2,855,339 | <p>What would be the complement of...</p>
<p>$\{$x:x is a natural number divisible by 3 and 5$\}$</p>
<p>I checked it's solution and it kind of stumped me...</p>
<p>$\{$x:x is a positive integer which is not divisible by 3 <em>or</em> not divisible by 5$\}$</p>
<p>Why the word <em>or</em> has been used in the solut... | Christian Blatter | 1,303 | <p>Define $e(t):= t \>{\rm mod}\>1$. The orbit ${\bf x}$ of the three hands is then given by
$${\bf x}:\quad t\mapsto\bigl(e(t),e(12t),e(720t)\bigr)\ .$$
Instead we look at the orbit ${\bf x}'=(x_1,x_2)$ of the second and third hands relative to the first hand, given by
$${\bf x}': \quad t\mapsto\bigl(e(11t),e(7... |
1,303,772 | <blockquote>
<p>Show that $$-2 \le \cos \theta ~ (\sin \theta +\sqrt{\sin ^2 \theta +3})\le 2$$ for all value of $\theta$.</p>
</blockquote>
<p>Trial: I know that $0\le \sin^2 \theta \le1 $. So, I have $\sqrt3 \le \sqrt{\sin ^2 \theta +3} \le 2 $. After that I am unable to solve the problem. </p>
| lab bhattacharjee | 33,337 | <p>Let $\cos \theta ~ (\sin \theta +\sqrt{\sin ^2 \theta +3})=y$</p>
<p>$\iff \sin \theta +\sqrt{\sin ^2 \theta +3}=y\sec\theta$</p>
<p>$\iff \sqrt{\sin ^2 \theta +3}=y\sec\theta-\sin \theta$</p>
<p>Squaring we get $\sin ^2 \theta +3=y^2(1+\tan^2\theta)+\sin^2\theta-2y\tan\theta$</p>
<p>$\iff y^2(\tan^2\theta)-2y(\... |
1,960,911 | <p>I am trying to evaluate this limit for an assignment.
$$\lim_{x \to \infty} \sqrt{x^2-6x +1}-x$$</p>
<p>I have tried to rationalize the function:
$$=\lim_{x \to \infty} \frac{(\sqrt{x^2-6x +1}-x)(\sqrt{x^2-6x +1}+x)}{\sqrt{x^2-6x +1}+x}$$</p>
<p>$$=\lim_{x \to \infty} \frac{-6x+1}{\sqrt{x^2-6x +1}+x}$$</p>
<p>Th... | marwalix | 441 | <p>It leads to</p>
<p>$$=\lim_{x \to \infty} \frac{-6+(\frac{1}{x})}{\sqrt{1-(\frac{6}{x})+(\frac{1}{x^2})}+1}$$</p>
<p>And so the limit is $-3$</p>
|
2,361,336 | <p>Note: This is <strong>not a duplicate</strong> as I am asking for a proof, not a criteria, and this is a specific proof, not just any proof – <strong>please treat like any other question on a specific math problem.</strong> Please do not close. thanks!</p>
<p><a href="https://i.stack.imgur.com/5R2aE.png" rel="nofol... | Vassilis Markos | 460,287 | <p>An easy way to prove that $Q(x)=0$ is by induction on $n$:</p>
<ol>
<li>For $n=1$ we have that $Q(x)=a_0+a_1(x-x_0)$ and the wanted limit is:
$$\lim_{x\to x_0}\frac{a_0+a_1(x-x_0)}{x-x_0}=0$$
Let $$g(x)=\frac{Q(x)}{x-x_0}\Rightarrow Q(x)=g(x)(x-x_0)$$
Now, since $\lim\limits_{x\to x_0}g(x)=0$ and $Q$ is continuous ... |
2,500,961 | <p>I've been able to find formulas all over the place for the sum and product of roots, but I haven't found anything that explains the significance of what they mean or how to interpret them to further gain understanding of the polynomial under evaluation. Is there any physical meaning? Do the values have any significa... | Math Model | 440,850 | <p><a href="https://www.mathsisfun.com/algebra/polynomials-sums-products-roots.html" rel="nofollow noreferrer">https://www.mathsisfun.com/algebra/polynomials-sums-products-roots.html</a></p>
<p>I had to google it, but I would check out that link. It does give you information about coefficients in the polynomial, which... |
1,966,122 | <p>$$\sum_{k=1}^{2n}\frac{(-1)^{k+1}}{k} = \sum_{k=n+1}^{2n} \frac{1}{k}$$</p>
<p>I am trying to prove this inductively, so I thought that I would expand the right side out of sigma form to get</p>
<p>$$\sum_{k=1}^{2n}\frac{(-1)^{k+1}}{k} = \frac{2}{2n(2n+1)} - \frac{1}{n}$$</p>
<p>which simplified to</p>
<p>$$\sum... | Kat | 297,952 | <p>Note that: \begin{pmatrix}
1 & 2 & -2\\
2 & 1 & 1
\end{pmatrix} is the matrix $|f|_{BE}$ where B is the given basis and E is the standard basis for $\mathbb R^2$. Now recall that for two given bases, we have the respective change of basis matrices. Having this in mind, what you can use to get $|f|_{B... |
3,888,259 | <p>The special linear group of invertible matrices is defined as the kernel of the determinant of the map:</p>
<p><span class="math-container">$$\det:GL(n,\mathbb{R}) \mapsto \mathbb{R}^*$$</span></p>
<p>In my mind the kernel of a linear map is the set of vectors that are mapped to the zero vector. So the map above wou... | Mummy the turkey | 801,393 | <p><span class="math-container">$\mathbb{R}^∗$</span> is the multiplicative group, so the identity is <span class="math-container">$1$</span> - in particular the kernel is the elements of the group homomorphism sent to it.</p>
<p>NB: the determinant map is not linear and indeed the group operation on <span class="math-... |
253,966 | <p>Just took my final exam and I wanted to see if I answered this correctly:</p>
<p>If $A$ is a Abelian group generated by $\left\{x,y,z\right\}$ and $\left\{x,y,z\right\}$
have the following relations:</p>
<p>$7x +5y +2z=0; \;\;\;\; 3x +3y =0; \;\;\;\; 13x +11y +2z=0$</p>
<p>does it follow that $A \cong Z_{3} \tim... | Hagen von Eitzen | 39,174 | <p>The trivial counterexample is that the trivial group is generated by $x=0, y=0, z=0$ and of course the given relations hold with $x=y=z=0$.
(Note that noone said that the set $\{x,y,z\}$ has cardinality $3$).</p>
<p>If you should insist on $x,y,z$ being distinct, observe that <em>any</em> quotient of $Z_3\times Z_... |
402,802 | <p>I have read that $$y=\lvert\sin x\rvert+ \lvert\cos x\rvert $$ is periodic with fundamental period $\frac{\pi}{2}$.</p>
<p>But <a href="http://www.wolframalpha.com/input/?i=y%3D%7Csinx%7C%2B%7Ccosx%7C" rel="nofollow">Wolfram</a> says it is periodic with period $\pi$.</p>
<p>Please tell what is correct.</p>
| Mark McClure | 21,361 | <p>The results coming from <em>any</em> software should be checked and considered from multiple angles. Part of the reason the graph is provided is precisely to help you do that. In this case, the graph makes it crystal clear that the result is twice the smallest period.</p>
<p><img src="https://i.stack.imgur.com/Ki... |
481,834 | <p>Let $A=(A_{ij})$ be a square matrix of order $n$. Verify that the determinant of the matrix</p>
<p>$\left( \begin{array}{ccc}
a_{11}+x & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22}+x & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & \cdots & ... | Marc van Leeuwen | 18,880 | <p>Write each column$~j$ of $A+xI_n$ as a sum of the column$~j$ of$~A$ and $x$ times column $j$ of$~I_n$. Now apply multi-linearity of the determinant with respect to the columns for each of the columns, to obtain a sum of $2^n$ determinants (each column was a sum of $2$ terms, and doubled the number of terms obtained ... |
397,347 | <p>I'm trying to figure out how to evaluate the following:
$$
J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx
$$
I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies J=-I'(1)$, but I couldn't figure out what $I(s)$ was. My other idea was contour integration, but I'm not sure how to deal... | Mhenni Benghorbal | 35,472 | <p>Using the change of variables $ u=e^{-x} $, we have </p>
<blockquote>
<p>$$\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx = \int _{0}^{1}\!{\frac { \left( \ln \left( u \right) \right) ^{3}
\ln \left( 1-u \right) }{u-1
}}{du}- \int _{0}^{1}\!{\frac { \left( \ln \left( u \right) \right)^{4} }{u-1
}}{d... |
76,778 | <p>I'm reading Yao's unpredictability -> pseudorandomness construction
and Goldreich/levin's pseudorandom permutation -> pseudorandom generator construction.</p>
<p>My question is:</p>
<p>is there a direct way to show that:</p>
<p>given a pseudorandom function, we can construct a pseudorandom permutation out of it?<... | Igor Rivin | 11,142 | <p>To expand very slightly upon @Steve's words of wisdom, see
<a href="http://en.wikipedia.org/wiki/Feistel_cipher" rel="nofollow">http://en.wikipedia.org/wiki/Feistel_cipher</a></p>
|
1,984,843 | <p>if $\cup$ is finite, say $n$, I came up with formula</p>
<p>$f(x) = n x + i$, where $x \in [\frac{i}{n}, \frac{i+1}{n}]$, $n$ is non negative integer and $i$ differs between $0$ and $n-1$.<br><br></p>
<p>I'm not sure whether it's correct to assume the bijection holds if $n$ approaches infinity.</p>
| mfl | 148,513 | <p><strong>Hint</strong></p>
<p>First of all define $f(1/n)=n-1,\forall n\in\mathbb{N}.$ Thus we have covered the extremes of the union of intervals.</p>
<p>Now, define $f$ on $(1/(n+1),1/n)$ to be a bijection between $(1/(n+1),1/n)$ and $(2n-2,2n-1).$</p>
|
2,916,246 | <p>I've found this to be difficult to solve:</p>
<p>$$ \frac{d^2 x }{dt^2} + (a x + b) \frac{dx}{dt} = 0 $$</p>
<p>I've done some reading, and I guess I could write this as:</p>
<p>$$ \frac{d^2 x }{dt^2} + b \frac{dx}{dt} + ax \frac{dx}{dt} = 0 $$</p>
<p>If I then treat $v(x) = \frac{dx}{dt}$ as an independent var... | Chinny84 | 92,628 | <p>We know the solution you have is wrong by putting it back in
$$
v' = -bv
$$
so we have in your original equation
$$
-bv + bv + axv = axv = 0
$$
this is in general not true.</p>
<p>Your issue was not converting the $x$ in terms of $v$. </p>
<p>To give a hint.
$$
x'' = \frac{dx}{dt}\frac{dv}{dx}
$$
this leads to
$$
... |
2,916,246 | <p>I've found this to be difficult to solve:</p>
<p>$$ \frac{d^2 x }{dt^2} + (a x + b) \frac{dx}{dt} = 0 $$</p>
<p>I've done some reading, and I guess I could write this as:</p>
<p>$$ \frac{d^2 x }{dt^2} + b \frac{dx}{dt} + ax \frac{dx}{dt} = 0 $$</p>
<p>If I then treat $v(x) = \frac{dx}{dt}$ as an independent var... | Donald Splutterwit | 404,247 | <p>The substitution $p=\frac{dx}{dt}$, so $\frac{d^2x}{dt^2}=p \frac{dp}{dx}$ and this gives
\begin{eqnarray*}
dp = -(ax+b) dx \\
\frac{dx}{dt} = -ax^2/2-bx+c.
\end{eqnarray*}
To integrate further depends upon the discriminant of the quadratic in $x$.</p>
|
3,637,283 | <p>How would I find the fourth roots of <span class="math-container">$-81i$</span> in the complex numbers? </p>
<p>Here is what I currently have: </p>
<p><span class="math-container">$w = -81i$</span> </p>
<p><span class="math-container">$r = 9$</span> </p>
<p><span class="math-container">$\theta = \arctan (-81)$</... | bjcolby15 | 122,251 | <p>Hints:</p>
<p>1) Rewrite <span class="math-container">$-81i$</span> as <span class="math-container">$0-81i$</span>.</p>
<p>2) Find the modulus <span class="math-container">$|z|$</span> and argument of <span class="math-container">$0-81i$</span>, using the formulas <span class="math-container">$$|z| = \sqrt {a^2 + ... |
2,638,028 | <p><strong>Question:</strong></p>
<blockquote>
<p>If $p,q$ are positive integers, $f$ is a function defined for positive numbers and attains only positive values such that $f(xf(y))=x^py^q$, then prove that $p^2=q$.</p>
</blockquote>
<p><strong>My solution:</strong></p>
<p>Put $x=1$. So, $f(f(y))=y^q$, then eviden... | Community | -1 | <p>Your solution is not conclusive, since $f(f(y))=y^q$ has infinitely many solutions. But if you want to make that professor unhappy with a short solution, you can do that: replacing $x$ by $f(x)$ in your equation, you get
$$f(f(x)f(y))=f(x)^py^q.$$ The LHS is symmetric in $x,y$, so you must have also $$f(x)^py^q=f(... |
1,507,710 | <p>I'm trying to get my head around group theory as I've never studied it before.
As far as the general linear group, I think I've ascertained that it's a group of matrices and so the 4 axioms hold?
The question I'm trying to figure out is why $(GL_n(\mathbb{Z}),\cdot)$ does not form a group.
I think I read somewhere... | Bernard | 202,857 | <p><span class="math-container">$\DeclareMathOperator{\GL}{GL}\GL_n(\mathbf Z)$</span> <em>is</em> a multiplicative group, by definition: it is the set of <em>invertible</em> matrices with coefficients in <span class="math-container">$\mathbf Z$</span>. </p>
<p>The problem is that it's not what you seem to think – the... |
1,982,102 | <p>If I wanted to figure out for example, how many tutorial exercises I completed today.</p>
<p>And the first question I do is <strong>question $45$</strong>, </p>
<p>And the last question I do is <strong>question $55$</strong></p>
<p>If I do $55-45$ I get $10$.</p>
<p>But I have actually done $11$ questions:<br>
$... | CiaPan | 152,299 | <p>If you started at the question 15 and finished at the question 15; how many question have you answered?</p>
<p>Imagine a list of exercises to be done in order. Some of them are marked as done already. You start from the first unmarked question.</p>
<blockquote>
<p>Every time you complete an excercise you mark it... |
1,439,004 | <p>I am trying to come up with a counting argument for: $\sum_{k=1}^{n}q^{k-1} = \frac{q^n-1}{q-1}$. I am trying to base it off of counting the left side as the sum of the (k-1) length words from an alphabet of size q for $k=1$ to $k=n-1$, but I can't seem to come up with a fitting argument to count the right side of ... | Brian M. Scott | 12,042 | <p>HINT: Fix a letter $a$ of your alphabet; there are $q^n-1$ words of length $n$ that contain at least one letter different from $a$.</p>
<p>Now count the same set of words according to the position of the last non-$a$ letter. If this is position $k$, $k$ can have any value from $1$ through $n$; how many words of len... |
73,991 | <p>I have the axiom from Peano's axioms:</p>
<p>If $A\subseteq \mathbb{N}$ and $1\in A$ and $m\in A \Rightarrow S(m)\in A$, then $A=\mathbb{N}$.</p>
<p>My book tells me that it secures that there are no more natural numbers than the numbers produced by the below 3 axioms (also from Peano's axioms):</p>
<p>$1\in \mat... | hmakholm left over Monica | 14,366 | <p>Yes, $S(n)$ is intended to represent $n+1$. Later, when addition is defined, "$n+1$" will turn out to <em>mean</em> $S(n)$.</p>
<p>As for the induction: Let
$$A=\{1,S(1),S(S(1)), S(S(S(1))),\ldots\}$$
This is clearly a subset of $\mathbb{N}$ as defined by the axioms; it satisfies $1\in A$ and for every $n\in A$ it ... |
4,251,233 | <p>Find</p>
<p><span class="math-container">$\int\frac{x+1}{x^2+x+1}dx$</span></p>
<p><span class="math-container">$\int \frac{x+1dx}{x^2+x+1}=\int \frac{x+1}{(x+\frac{1}{2})^2+\frac{3}{4}}dx$</span></p>
<p>From here I don't know what to do.Write <span class="math-container">$(x+1)$</span> = <span class="math-container... | Adam Rubinson | 29,156 | <p>Write <span class="math-container">$x+\frac{1}{2} = t.$</span> Then,</p>
<p><span class="math-container">$$\frac{x+1}{\left(x+\frac{1}{2}\right)^2+\frac{3}{4}} = \frac{x+\frac12 + \frac12}{\left(x+\frac{1}{2}\right)^2+\frac{3}{4}} = \frac{t}{t^2+\frac{3}{4}} + \frac{\frac12}{t^2+\frac{3}{4}}.$$</span></p>
<p>Can you... |
887,473 | <p>I have been struggling with the following claim:</p>
<p>Let $A_n$ be a sequence of compact sets and $A$ a compact set. $A=\lim\sup_n A_n=\lim\inf_n A_n$ iff $d_H(A_n,A)\to 0$ where $d_H(.,.)$ is the Hausdorff metric.</p>
<p>$\lim\inf$ and $\lim\sup$ are defined by $\lim\inf_nA_n=\left\{y\in Y:\forall \varepsilon&... | Daniel Fischer | 83,702 | <p>For the direction</p>
<p>$$A = \limsup_n A_n = \liminf_n A_n \implies d_H(A_n,A) \to 0,$$</p>
<p>you need some additional hypothesis on the ambient space, namely that it is compact.</p>
<p>Without that hypothesis, the implication does not hold. An explicit counterexample using compact subsets of $\mathbb{R}$ is $... |
458 | <p>If you go to the bottom of any page in the SE network (e.g. this one!), you'll see a list of SE sites. In particular there's a link to MathOverflow, that is potentially seen by a large number of people (many of whom are outside of our target audience).</p>
<p>When you put your cursor over that link, there's a hover... | Scott Morrison | 3 | <p>Research mathematics (at graduate level and above)</p>
|
713,098 | <p>The answer to my question might be obvious to you, but I have difficulty with it. </p>
<p>Which equations are correct:</p>
<p>$\sqrt{9} = 3$</p>
<p>$\sqrt{9} = \pm3$</p>
<p>$\sqrt{x^2} = |x|$</p>
<p>$\sqrt{x^2} = \pm x$</p>
<p>I'm confused. When it's right to take an absolute value? When do we have only one va... | hmakholm left over Monica | 14,366 | <p><em>By definition</em> the square root of a number is the <em>positive</em> number whose square is the original number. So we have $\sqrt9=3$ and $\sqrt{x^2}=|x|$ and no doubt about either.</p>
<p>There <em>is no number</em> whose square root is $-3$ (even if we move to complex numbers and consider principal square... |
2,715,374 | <p>We know that \begin{equation*}
a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots+\cfrac{1}{a_n}}}}}=[a_0,a_1, \cdots, a_n]
\end{equation*}</p>
<p>If $\frac{p_n}{q_n}=[a_0,a_1, \cdots, a_n]$.</p>
<blockquote>
<p>How to prove that $$
\begin{pmatrix}
p_n & p_{n-1} \\
q_n & q_{n-1} \... | Ethan Bolker | 72,858 | <p>The first line in your extended (non)proof is wrong when $x=n$. So it's not true for all $x$ and $n$.</p>
|
2,715,374 | <p>We know that \begin{equation*}
a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots+\cfrac{1}{a_n}}}}}=[a_0,a_1, \cdots, a_n]
\end{equation*}</p>
<p>If $\frac{p_n}{q_n}=[a_0,a_1, \cdots, a_n]$.</p>
<blockquote>
<p>How to prove that $$
\begin{pmatrix}
p_n & p_{n-1} \\
q_n & q_{n-1} \... | Eric Wofsey | 86,856 | <p>You can't multiply inequations like this: $a\neq b$ does not imply $ac\neq bc$. Indeed, if $c=0$, then $ac=0=bc$ is always true, even if $a\neq b$.</p>
<p>(If you know that $c\neq 0$, then this step would be valid, since if $ac$ were equal to $bc$ you could divide both by $c$ to get $a=b$ which contradicts the fac... |
2,953,371 | <p>How can I find the derivative of this function ?
<span class="math-container">$$f(x)= (4x^2 + 2x +5)^{0.5}$$</span></p>
| Toffomat | 380,397 | <p>It helps to look at the definition of a convolution: Given two functions <span class="math-container">$f$</span> and <span class="math-container">$g$</span> (assumed from <span class="math-container">$\mathbb R$</span> to <span class="math-container">$\mathbb R$</span>), the convolution <span class="math-container">... |
676,573 | <p>Exercise: Write the polynomial $1 + 2x -x^2 + 5x^3 - x^4$ at powers of $(x-1)$.</p>
<p>I presume this exercise is solved using Taylor Series, since it belongs to that chapter, but have no idea how to solve it. Otherwise, it's very straightforward.</p>
<p>Note: The above exercise is <strong>not</strong> homework.</... | Ant | 66,711 | <p>Another way to do that is to do just what you would do with any other function:</p>
<p>Calculate $$p(1), p'(1), p''(1)...$$ (strait-forward since $p$ is a polynomial)</p>
<p>and use the derivatives to build the appropriate taylor series around $x_0 = 1$</p>
<p>Indeed, it is considerably faster (in this case) than... |
3,516,241 | <p>Consider the equation:</p>
<p><span class="math-container">$$ x ^ 4 - (2m - 1) x^ 2 + 4m -5 = 0 $$</span></p>
<p>with <span class="math-container">$m \in \mathbb{R}$</span>. I have to find the values of <span class="math-container">$m$</span> such that the given equation has all of its roots real.</p>
<p>This is ... | user289143 | 289,143 | <p>You need also to consider <span class="math-container">$\Delta$</span> to be positive in order for the solutions to be real.</p>
<p><span class="math-container">$\Delta = (2m-1)^2-4(4m-5)=4m^2-4m+1-16m+20=4m^2-20m+21$</span> </p>
<p><span class="math-container">$m_{1,2}=\frac{10 \pm \sqrt{100-84}}{4}=\frac{10 \pm ... |
2,892,342 | <p>Given two adjacent sides and all four angles of a quadrilateral, what is the most efficient way to calculate the angle that is made between a side and the diagonal of the quadrilateral that crosses (but does not necessarily bisect) the angle in between the two known sides?</p>
<p>Other known information:</p>
<ul>
... | Exodd | 161,426 | <p>Well, your quadrilateral can be inscribed inside a circle, since the sum of opposite angles is 180, so the angle $\theta_1$ is equal to the angle $ACD$, that can be computed through 1 cosine rule and 1 sine rule.</p>
<p>and don't call your sketch awful. It's quite pretty ;)</p>
|
1,282,843 | <p>I'm having trouble proving the following statement:</p>
<blockquote>
<p>$x(u, v) = (u − u^ 3/ 3
+ uv^2 , v − v^ 3/ 3
+ u^ 2 v, u^2 − v^ 2 )$ is a minimal surface and x is not injective</p>
</blockquote>
<p>Proving that $x(u,v)$, which is also known as the Enneper surface, is minimal is not a problem. However, ... | Badshah Khan | 698,917 | <p>Just use the counter example... i.e.
<span class="math-container">$$x( \sqrt3 , 0) = (0,0,3)= x(- \sqrt3 , 0)$$</span>
With
<span class="math-container">$$( \sqrt3 , 0) \ne (- \sqrt3 , 0)$$</span></p>
|
3,271,675 | <p>Let <span class="math-container">$p$</span> be a prime of the form <span class="math-container">$p = a^2 + b^2$</span> with <span class="math-container">$a,b \in \mathbb{Z}$</span> and <span class="math-container">$a$</span> an odd prime. Prove that <span class="math-container">$(a/p) =1$</span></p>
<p>Could anyon... | Mark Bennet | 2,906 | <p>To explain further, we have that <span class="math-container">$p\equiv b^2 \bmod a$</span> from the given equation.</p>
<p>Quadratic reciprocity tells us that if <span class="math-container">$a$</span> and <span class="math-container">$p$</span> are odd primes and either leaves remainder <span class="math-container... |
1,714,902 | <p>(Question edited to shorten and clarify it, see the history for the original)</p>
<p>Suppose we are given two $n\times n$ matrices $A$ and $B$. I am interested in finding the closest matrix to $B$ that can be achieved by multiplying $A$ with orthogonal matrices. To be precise, the problem is</p>
<p>$$\begin{align}... | Community | -1 | <p>We may assume that $A,B$ are non-negative diagonal. We calculate the extrema of the function $f:(U,V)\in O(n)^2\rightarrow tr((UAV^T-B)(VAU^T-B))$.</p>
<p>Note that $H_1$ is in the tangent space to $O(n)$ in $U$ iff $U^TH_1\in SK$ (it is skew), that is $H_1=UH$ where $H\in SK$.</p>
<p>Then $\dfrac{\partial f}{\par... |
3,426,441 | <p>I'm working through some notes and trying to understand a piece of the following statement:</p>
<p>Suppose that the bivariate random variable <span class="math-container">$(X,Y)$</span> is uniformly distributed on the square <span class="math-container">$[0,1]^2$</span>, that is the joint probability distribution f... | Charith | 321,851 | <p>Will it be sufficient to write the function as </p>
<p><span class="math-container">$f:A\rightarrow B$</span> such that<br>
<span class="math-container">$f(x)=ln(x)$</span>
? </p>
|
3,426,441 | <p>I'm working through some notes and trying to understand a piece of the following statement:</p>
<p>Suppose that the bivariate random variable <span class="math-container">$(X,Y)$</span> is uniformly distributed on the square <span class="math-container">$[0,1]^2$</span>, that is the joint probability distribution f... | William Elliot | 426,203 | <p>Yes, a function can be extended to subsets of the domain.<br>
That is called the set extension of the function.<br>
If A is a subset of the domain of a function f,<br>
f(A) or for careful notation, f[A] = { f(x) : x in A }.<br>
The inverse set extension is f<span class="math-container">$^{-1}$</span>(B) or<br>
f<spa... |
4,159,771 | <p>I understand the geometric intuition behind determinants but what is the real life use of it? I'm not looking for answers along the lines of "it helps to find solutions to linear systems" etc, unless this is one of those concepts that is useful because it allows us to do "more math". I'm more int... | Vercassivelaunos | 803,179 | <p>Area integrals and volume integrals over complicated areas or volumes (like spheres, balls, ellipses, hyperboloids, etc.) can be calculated using the transformation theorem, which is a generalization of integration by substitution. It uses the determinant of the differential/Jacobian of the coordinate transformation... |
3,121,361 | <p>Given <span class="math-container">$G$</span> has elements in the interval <span class="math-container">$(-c, c)$</span>. Group operation is defined as:
<span class="math-container">$$x\cdot y = \frac{x + y}{1 + \frac{xy}{c^2}}$$</span></p>
<p>How to prove closure property to prove that G is a group?</p>
| Arthur | 15,500 | <p>Hint: show that <span class="math-container">$a_n-1\leq 1+\frac12+\frac14+\cdots+\frac1{2^{n-1}}$</span></p>
|
3,121,361 | <p>Given <span class="math-container">$G$</span> has elements in the interval <span class="math-container">$(-c, c)$</span>. Group operation is defined as:
<span class="math-container">$$x\cdot y = \frac{x + y}{1 + \frac{xy}{c^2}}$$</span></p>
<p>How to prove closure property to prove that G is a group?</p>
| Michael Rozenberg | 190,319 | <p>By the binomial theorem <span class="math-container">$$1<a_n=1+1+\frac{1}{2!}\left(1-\frac{1}{n}\right)+\frac{1}{3!}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)+...+\frac{1}{n!}\left(1-\frac{1}{n}\right)...<$$</span>
<span class="math-container">$$<2+\frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{n!}<2+\... |
2,555,499 | <p>Let $v_1=(1,1)$ and $v_2=(-1,1)$ vectors in $\mathbb{R}^2$. They are <strong>clearly linearly independent</strong> since each is not an scalar multiple of the other. The following information about a linear transformation $f: \mathbb{R}^2 \to \mathbb{R}^2$ is given: $$f(v_1)=10 \cdot v_1 \text{ and } f(v_2)=4 \cdot ... | user21820 | 21,820 | <p>This is really a matter of convention, which varies over time and from place to place, but in modern mathematics precedence is generally as follows (from highest to lowest, and with those of the same precedence put on the same line) where "$∙$" denotes the positions of the subexpressions:</p>
<ul>
<li><p>Brackets: ... |
3,100,831 | <p>What is the domain of <span class="math-container">$g(x)=\frac{1}{1-\tan x}$</span> </p>
<p>I tried it and got this. But I'm not really sure if it is right. Is that gonna be like this ? <span class="math-container">$(\mathbb{R}, \frac{\pi}{4})$</span></p>
| Kavi Rama Murthy | 142,385 | <p>It is <span class="math-container">$\mathbb R \setminus \cup_{\{n \in \mathbb Z\} }(\{n\pi +\pi /2\} \cup \{n\pi +\pi /4\})$</span></p>
|
206,780 | <p>Let $f:X\to Y$ is a measurable function. Banach indicatrix
$$
N(y,f) = \#\{x\in X \mid f(x) = y\}
$$
is the number of the pre-images of $y$ under $f$. If there are infinitely many pre-images then $N(y,f) = \infty$. </p>
<p>Let $X\subset\mathbb R^n$, $Y\subset\mathbb R^m$ with Lebesgue measure.</p>
<p><em>I am inte... | Nikita Evseev | 15,946 | <p>This is an attempt to rid of demand of continuity.
The following proof is essentially an adaptation of Banach's original proof in case of continuous function defined on segment <span class="math-container">$[a,b]$</span>. See also <a href="https://math.stackexchange.com/a/144832/23566">https://math.stackexchange.com... |
2,418,954 | <p>Using Vieta's formulas, I can get $$\begin{align} \frac{1}{x_1^3} + \frac{1}{x_2^3} + \frac{1}{x_3^3} &= \frac{x_1^3x_2^3 + x_1^3x_3^3 + x_2^3x_3^3}{x_1^3x_2^3x_3^3} \\&= \frac{x_1^3x_2^3 + x_1^3x_3^3 + x_2^3x_3^3}{x_1^3x_2^3x_3^3} \\ &= \frac{x_1^3x_2^3 + x_1^3x_3^3 + x_2^3x_3^3}{\left (-\frac{d}{a} \ri... | shrimpabcdefg | 473,212 | <p>If $x \neq 0$ is a solution to $at^3+bt^2+ct+d=0$ then since $a+b(\frac{1}{x})+c(\frac{1}{x})^2+d(\frac{1}{x})^3=0$, $\frac{1}{x}$ is a solution to $dt^3+ct^2+bt+a=0$. Thus if we have $\frac{1}{x_i}=y_i$ for $i=1,2,3$, </p>
<p>\begin{align*}
\frac{1}{x_1^3}+\frac{1}{x_2^3}+\frac{1}{x_3^3}&=y_1^3+y_2^3+y_3^3\\
&... |
2,979,315 | <p>Let <span class="math-container">$X$</span> be a continuous random variable with uniform distribution between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. Compute the distribution of <span class="math-container">$Y = \sin(2\pi X)$</span>.</p>
<p><span class="math-container">$... | Servaes | 30,382 | <p><strong>Hint:</strong> The composition of two reflections is a rotation.</p>
|
799,183 | <p>I'm trying to work out what the transformation $T:z \rightarrow -\frac{1}{z}$ does (eg reflection in a line, rotation around a point etc). Any help on how to do this would be greatly appreciated! I've tried seeing what it does to $1$ and $i$ but is hasn't helped me. Thanks!</p>
| user21820 | 21,820 | <p>What does negation do? Then what does taking the reciprocal do to the length and the angle? Note that you will need to know what <a href="http://en.wikipedia.org/wiki/Circle_inversion" rel="nofollow">inversion</a> is, to describe what happens to the length.</p>
|
2,261,500 | <p>I try to prove that statement using only Bachet-Bézout theorem (I know that it's not the best technique). So I get $k$ useful equations with $n_1$ then $(k-1)$ useful equations with $n_2$ ... then $1$ useful equation with $n_{k-1}$. I multiply all these equations to obtain $1$ for one side. For the other side I'm lo... | Lazy Lee | 430,040 | <p>According to the Bachet-Bezout Lemma, the gcd of two integers $gcd(a,b)=d$ is the smallest positive integer that can be written as $ax+by=d$ for some integers $x,y$. We split the discussion into two parts:</p>
<p><strong>If $k$ is even</strong>: Then there exists integers $\{x_i\}$ such that $$n_1x_1+n_2x_2=1$$$$n_... |
3,368,402 | <p>I am utilizing set identities to prove (A-C)-(B-C).</p>
<p><span class="math-container">$\begin{array}{|l}(A−B)− C = \{ x | x \in ((x\in (A \cap \bar{B})) \cap \bar{C}\} \quad \text{Def. of Set Minus}
\\
\quad \quad \quad \quad \quad =\{ x | ((x\in A) \wedge (x\in\bar{B})) \wedge (x\in\bar{C})\} \quad \text{Def. o... | J.G. | 56,861 | <p>Abbreviating and as <span class="math-container">$\land$</span> and not as <span class="math-container">$\lnot$</span>,</p>
<p><span class="math-container">$$x\in(A-B)-C\iff x\in A-B\land x\notin C \iff x\in A\land x\notin B\land x\notin C\\\iff x\in A\land x\notin C\land\lnot(x\in B\land x\notin C)\iff x\in A-C\la... |
737,692 | <p>I'm working on this question:</p>
<blockquote>
<p>Rewrite the following summation using sigma notation and then compute it
using the technique of telescoping summation.
$$\frac{1}{2*5}+\frac{1}{3*6}+\frac{1}{4*7}+...+\frac{1}{(n-2)(n+1)}+\frac{1}{(n-1)(n+2)} $$</p>
</blockquote>
<p>My work:
I replaced the ... | izœc | 83,639 | <p><strong>HINT:</strong> Your sigma expression is not correct (you can see this, as by plugging in $i=0$ the second term in the expression is undefined). Consider that in the sum
$$
\frac{1}{2*5}+\frac{1}{3*6}+\frac{1}{4*7}+...+\frac{1}{(n-2)(n+1)}+\frac{1}{(n-1)(n+2)}
$$
every term is of the form $\frac{1}{(n-1)(n+2)... |
898,495 | <p>A standard pack of 52 cards with 4 suits (each having 13 denominations) is well shuffled and dealt out to 4 players (N, S, E and W).</p>
<p>They each receive 13 cards.</p>
<p>If N and S have exactly 10 cards of a specified suit between them. </p>
<p>What is the probability that the 3 remaining cards of the suit a... | user2566092 | 87,313 | <p>When you condition, you get that you have 26 cards left and 3 of them are of the particular suit. There are $26 \choose 13$ ways of assigning these remaining 26 cards among E and W (because once you assign 13 cards to E, the remaining 13 cards automatically go to W. You get that one player has all 3 cards if either ... |
372,548 | <p><span class="math-container">$f:\mathbb R\to\mathbb R$</span> is a convex continuous function. We have a finite or a countable set of triples: <span class="math-container">$\{(x_n,f(x_n),D_n)\}_{n\in N}$</span>, where <span class="math-container">$D_n$</span> is the slope of a tangent line <span class="math-containe... | Rajesh D | 14,414 | <p>At zero time, assuming the bandwidth is <span class="math-container">$B$</span>, at next time instance, due to the appearance of the term <span class="math-container">$u_xu$</span>, the band width of <span class="math-container">$u_x$</span> also being <span class="math-container">$B$</span> at time step <span class... |
1,218,238 | <p>Describe explicitly a subgroup $H$ of order 8 of the permutation group $S_5$.</p>
<p>How could I find such a subgroup? I don't know how to start with. Should I start with some transition $(i,j)$ and use them to generate a subgroup?</p>
| Ross Millikan | 1,827 | <p>Your answer is correct unless $n=0$. In that case itself and the zero subspace are the same.</p>
|
1,209,934 | <p>So I am given two points $A=(-.5,2.3,-7.3)$ and $B=(-2,17.1,-0.3)$ and then using $AB = OB - OA$ to give me $(1.5,-14.8,-7)$. The plane is $$x+23y+13z=500$$ From there I got $r.n$ where $r=(1.5,-14.8,-7)$ and $n=(1,28,13)$. From here I do not know how to check if the vector is perpendicular to the plane.</p>
| ThunderGod763 | 781,395 | <p>We have the two points <span class="math-container">$A(-0.5,2.3,-7.3)$</span> and <span class="math-container">$B(-2,17.1,-0.3)$</span>, as well as the general equation of the plane <span class="math-container">$x+23y+13z-500=0$</span>. The vector from <span class="math-container">$A$</span> to <span class="math-con... |
1,097,579 | <p>How to solve $$\frac{dx}{2p}=\frac{dy}{2q}=\frac{du}{2(p^2+q^2}=\frac{dp}{2up}=\frac{dq}{2uq}=dt$$</p>
<p>as functions $$x=x(t), y=y(t), u=u(t), p=p(t), q=q(t)$$</p>
<p>My method is use the last three equalities to deduce $$\frac{d^2u}{dt^2}+u\frac{du}{dt}=0$$
But this nonlinearity troubles me...</p>
| math110 | 58,742 | <p>Hint: you equation is this following
$$y''+yy'=0$$
let
$$y'=p\Longrightarrow y''=\dfrac{dp}{dx}=\dfrac{dp}{dy}\cdot\dfrac{dy}{dx}=p\dfrac{dp}{dy}$$
so
$$p\dfrac{dp}{dy}+yp=0$$
so
$$p(\dfrac{dp}{dy}+y)=0\Longrightarrow p=0,\text{or},\dfrac{dp}{dy}=-y$$
if
$p=0\Longrightarrow y=C$</p>
<p>if $$\dfrac{dp}{dy}=-y\Longr... |
1,097,579 | <p>How to solve $$\frac{dx}{2p}=\frac{dy}{2q}=\frac{du}{2(p^2+q^2}=\frac{dp}{2up}=\frac{dq}{2uq}=dt$$</p>
<p>as functions $$x=x(t), y=y(t), u=u(t), p=p(t), q=q(t)$$</p>
<p>My method is use the last three equalities to deduce $$\frac{d^2u}{dt^2}+u\frac{du}{dt}=0$$
But this nonlinearity troubles me...</p>
| Sidharth Ghoshal | 58,294 | <p>Based on your last line:</p>
<p>Consider $F = u^2$</p>
<p>$$ \frac{df}{dt} = 2 u \frac{du}{dt} $$</p>
<p>Thus suppose we re-arrange</p>
<p>$$ \frac{d^2u}{dt^2} + u \frac{du}{dt} = 0 $$</p>
<p>Into</p>
<p>$$ \frac{d^2u}{dt^2} = - u \frac{du}{dt} $$</p>
<p>Then we can re-write it as</p>
<p>$$ \frac{d}{dt} \le... |
1,443,680 | <p>In Quantum Mechanics one often deals with wavefunctions of particles. In that case, it is natural to consider as the space of states the space $L^2(\mathbb{R}^3)$. On the other hand, on the book I'm reading, there's a construction which it's quite elegant and general, however it is not rigorous. For those interested... | user91126 | 91,126 | <p><strong>First remark</strong>. QM postulates state that the Hilbert space is <em>separable</em>. Recall that <strong>Riesz-Fischer</strong> theorem ensures that separable Hilber spaces are all isometrically isomorphic, so it does not really matter which one you choose until you are speaking about the general theory.... |
1,196,317 | <p><a href="https://math.stackexchange.com/questions/1196261/let-g-be-a-group-where-ab3-a3b3-and-ab5-a5b5-prove-that-g-is/1196295#1196295">Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. How to prove that $G$ is an abelian group?</a></p>
<p>P.S Why cannot not we just cancel ab out of the middle of these... | Tim Raczkowski | 192,581 | <p>When we have $xyz=x'yz'$ and cancel out the middle factor, we can only do that if we know that the commutes with $x$ and $x'$, or with $z$ and $z'$. From being so accustomed to working with commutativity, we lose sight of this fact. In reality, we can only cancel from the left or right, and the only way we can do m... |
499,171 | <p>Let $\{x_n\}$ be "any" sequence containing all rationals. I have to prove that every real number is the limit of some subsequence. I know that rationals are dense in real. But, are not the order of the rationals in the sequence creating problem here ? How to pick rationals from this sequence. </p>
| Old John | 32,441 | <p>Suppose that $A$ is any real number, and suppose that $x_{n_k}$ is the subsequence (that we will construct).</p>
<p>Suppose we have found the terms $x_{n_1}, x_{n_2}, \dots, x_{n_m}$ and we show that it is possible to choose a term closer to $A$ than previous terms of the subsequence:</p>
<p>Let $\epsilon$ be the ... |
499,171 | <p>Let $\{x_n\}$ be "any" sequence containing all rationals. I have to prove that every real number is the limit of some subsequence. I know that rationals are dense in real. But, are not the order of the rationals in the sequence creating problem here ? How to pick rationals from this sequence. </p>
| user2345678 | 314,957 | <p>I think that i've managed another way to show this result. </p>
<p>We will use two facts: $1)$ Rationals numbers are dense in the reals and $2)$ if $(x_n)$ is a sequence in $\mathbb{R}$ such that every open ball (centered at $a$) contains terms $x_n$ of the sequence for arbitrarely large $n$, then $a$ is the limit ... |
297,036 | <p>If $f'(x) = \sin{\dfrac{\pi e^x}{2}}$ and $f(0)= 1$, then what will be $f(2)$?</p>
<p>This is what I tried to find the antiderivative of $f'(x)$ with u-substitution, </p>
<p>$$
\begin{align}
u &=\frac{\pi e^x}{2} \\
\frac{2}{\pi}du &=e^x dx
\end{align}
$$</p>
<p>I don't know what to do next.</p>
| userX | 61,346 | <p>another option.. you could go with a taylor expansion of sin(x) take the integral, and get f(2) [in form of an infinite series]. </p>
|
1,256,460 | <p>I want to solve the following problem: </p>
<p>$$u_{xx}(x,y)+u_{yy}(x,y)=0, 0<x<\pi, y>0 \\ u(0,y)=u(\pi, y)=0, y>0 \\ u(x,0)=\sin x +\sin^3 x, 0<x<\pi$$ </p>
<p>$u$ bounded </p>
<p>I have done the following: </p>
<p>$$u(x,y)=X(x)Y(y)$$ </p>
<p>We get the following two problems: </p>
<p>$$X''... | Community | -1 | <p>We have</p>
<p>$$\frac1t\int_0^t e^{-t/\tau}dt=\frac\tau t(1-e^{-t/\tau}).$$</p>
<p>Then with the initial temperature difference $\Delta_0=T_0-T_\infty$ and the difference at time $t,$$$\Delta_t=T_t-T_\infty=(T_0-T_\infty)e^{-t/\tau}=\Delta_0e^{-t/\tau},$$ the average is given by</p>
<p>$$\overline\Delta=\Delta_0... |
908,196 | <blockquote>
<p>Solve $x^2-1=2$</p>
</blockquote>
<p>I have no idea how to do this can somebody please help me? I have tried working it out and I could never get the answer.</p>
| Mary Star | 80,708 | <p>$$x^2-1=2 \Rightarrow x^2=1+2 \Rightarrow x^2=3 \Rightarrow \sqrt{x^2}=\sqrt{3} \Rightarrow |x|=\sqrt{3} \Rightarrow x=\pm \sqrt{3}$$</p>
|
908,196 | <blockquote>
<p>Solve $x^2-1=2$</p>
</blockquote>
<p>I have no idea how to do this can somebody please help me? I have tried working it out and I could never get the answer.</p>
| Eff | 112,061 | <p>I feel from the comments that you lack some understanding of equations in general, not just quadratic equation since this problem should be quite simple. Let's try and fix that!</p>
<p>First off, what does it mean to <em>solve</em> the equation $x^2-1=2$? It means to find the value(s) that we can plug into the $x$ ... |
908,196 | <blockquote>
<p>Solve $x^2-1=2$</p>
</blockquote>
<p>I have no idea how to do this can somebody please help me? I have tried working it out and I could never get the answer.</p>
| Community | -1 | <p>$x^2-1=2$ Initial Problem<br>
$x^2=3$ Add 1 to both sides.<br>
$x= \sqrt{3} , -\sqrt{3}$ Square root both sides and you have your two solutions</p>
|
745,436 | <p>I'm reading this pdf <a href="http://rutherglen.science.mq.edu.au/wchen/lndpnfolder/dpn01.pdf" rel="nofollow">http://rutherglen.science.mq.edu.au/wchen/lndpnfolder/dpn01.pdf</a> I understand some of the expression used in this but I don't understand the part $(m,n) = 1$</p>
<p>Is this a cartesian coordinate or some... | Yiyuan Lee | 104,919 | <p>It means that the <a href="http://en.wikipedia.org/wiki/Greatest_common_divisor" rel="nofollow">greatest common divisor</a> of $m$ and $n$, which is the largest integer dividing both of $m$ and $n$ is equals to $1$.</p>
<p>In otherwords, they are <a href="http://en.wikipedia.org/wiki/Coprime_integers" rel="nofollow... |
745,436 | <p>I'm reading this pdf <a href="http://rutherglen.science.mq.edu.au/wchen/lndpnfolder/dpn01.pdf" rel="nofollow">http://rutherglen.science.mq.edu.au/wchen/lndpnfolder/dpn01.pdf</a> I understand some of the expression used in this but I don't understand the part $(m,n) = 1$</p>
<p>Is this a cartesian coordinate or some... | user140943 | 140,943 | <p>$(m,n)$ is the gcd of m and n. </p>
|
39,597 | <p>There was a recent question on intuitions about sheaf cohomology, and I answered in part by suggesting the "genetic" approach (how did cohomology in general arise?). For historical material specific to sheaf cohomology, what Houzel writes in the Kashiwara-Schapira book <em>Sheaves on Manifolds</em> for sheaf theory ... | Tim Perutz | 2,356 | <p>What strikes me about the first fifty years of homology theory (from Poincaré to Eilenberg-Steenrod's book) is that the development was as much about stripping away unnecessary complication as about increasing sophistication. A famous example is singular homology, which was found very late, by Eilenberg. The ... |
3,479,953 | <p>Let <span class="math-container">$v={\{v_1,v_2,...,v_k}\}$</span> Linearly independent</p>
<p><span class="math-container">$\mathbb{F} = \mathbb{R}$</span> or <span class="math-container">$\mathbb{F}=\mathbb{C}$</span></p>
<blockquote>
<p>Prove that <span class="math-container">${\{v_1 + v_2 , v_2+v_3, v_3+v_4,.... | Alex | 48,061 | <p>Let <span class="math-container">$A$</span> be not the <span class="math-container">$n\times d$</span>-matrix, but the <span class="math-container">$n\times n$</span>-matrix <span class="math-container">$$A=[v_1 \ldots v_d\ \vec{0}\ldots\ \vec 0]^t.$$</span> Reduce <span class="math-container">$A$</span> to <a hre... |
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