qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,552,706 | <p>Let <span class="math-container">$A$</span> be a unital Banach algebra and <span class="math-container">$\omega: A \to \mathbb{C}$</span> a continuous functional. Let <span class="math-container">$\lambda \in \mathbb{C}, x \in A$</span>. I'm reading a proof that claims that (note that we identify <span class="math-c... | MaoWao | 112,915 | <p>If <span class="math-container">$|\lambda|>\|x\|$</span>, then
<span class="math-container">$$
(\lambda-x)^{-1}=\lambda^{-1}(1-\lambda^{-1}x)^{-1}=\lambda^{-1}\sum_{n=0}^\infty \lambda^{-n}x^n.
$$</span>
Thus
<span class="math-container">$$
\|(\lambda-x)^{-1}\|\leq |\lambda|^{-1}\sum_{n=0}^\infty |\lambda|^{-n}\|... |
4,339,263 | <p>Consider a manifold <span class="math-container">$N$</span> and a submanifold <span class="math-container">$M$</span>. We're assuming that <span class="math-container">$N$</span> and <span class="math-container">$M$</span> can be non-compact. Is it possible to find a metric <span class="math-container">$g$</span> of... | Jason DeVito | 331 | <p>Let <span class="math-container">$E\xrightarrow{\pi} M$</span> be a vector bundle. We will construct a metric on <span class="math-container">$E$</span> for which <span class="math-container">$M$</span> (viewed as the zero-section) is totally geodesic and for which the injectivity radius is positive.</p>
<p>We'll u... |
1,550,541 | <p>After trying for hours I decided to ask. Please can anyone help me with this problem.</p>
<p>"Two cards are drawn at random and are thrown away from a pack of 52 cards. Find the probability of getting an Ace from the remaining 50 cards."</p>
<p>Please explain the correct method to do this.</p>
<p>I'm getting answ... | Tanmay Chandak | 394,883 | <p>Basic definition - Probability is a function of 'knowledge'
so unless you know which 2 cards are thrown away at random, the probability remains the same. i.e., $\frac{4}{52}=\frac{1}{13}$.
had there been any information about those two cards, which are thrown away, the answer would have been different.</p>
|
275,325 | <p>I am suppose to use calculus to find the max vertical distance between the line $y = x + 2$ and the parabola $y = x^2$ on the interval $x$ greater then or equal to $-1$ and less then or equal to $2$.</p>
<p>I really have no idea what to do. I found the critical numbers and that didn't help at all. I guess all the i... | lab bhattacharjee | 33,337 | <p>Clearly, $P(t,t^2)$ is any point of the parabola.</p>
<p>The distance of the line: $y=x+2$ from $P(t,t^2)$ is $$\left|\frac{t-t^2+2}{\sqrt{1^2+1^2} }\right|=\frac{|t-t^2+2|}{\sqrt2}$$</p>
<p>Let $f(t)=t-t^2+2$</p>
<p>so, $f'(t)=1-2t$</p>
<p>For the extreme values of $f(t), f'(t)=0\implies t=\frac12$</p>
<p>Now,... |
275,325 | <p>I am suppose to use calculus to find the max vertical distance between the line $y = x + 2$ and the parabola $y = x^2$ on the interval $x$ greater then or equal to $-1$ and less then or equal to $2$.</p>
<p>I really have no idea what to do. I found the critical numbers and that didn't help at all. I guess all the i... | Elias Costa | 19,266 | <p>Hint. See the Math Java Applet in </p>
<p><a href="http://www.ies.co.jp/math/java/conics/draw_parabola/draw_parabola.html" rel="nofollow">http://www.ies.co.jp/math/java/conics/draw_parabola/draw_parabola.html</a></p>
<p>for a geometry ideia.</p>
|
3,968,475 | <p>The question is about classifying groups of order <span class="math-container">$256$</span> with at least one element of order <span class="math-container">$64$</span>, and <em>justify why the elements of the list are non-isomorphic</em>.
I'm done except for showing that
<span class="math-container">$$
\mathbb Z_{... | Pedro | 23,350 | <p>Draw the lattice of subgroups, maybe. The lattice of the Klein four group is not a total order, but that of the cyclic group of order four is. Then note how doing <span class="math-container">$G\times -$</span> changes the lattice.</p>
|
38,629 | <p>I've asked this question on <a href="https://math.stackexchange.com/questions/4533/sobolev-space-norm-and-beppo-levi-space-norm">math.stackexchange.com</a> but I'm not satisfied by the answers I got, so I've decided to ask here instead. As always I apologize if my notation is not precise enough. I am a computer scie... | Piero D'Ancona | 7,294 | <p>You should maybe clarify your question; what are the general definitions you are trying to compare, and what kind of result are you interested in?</p>
<p>The first quantity in your question, which you call Sobolev, when $1 < p < \infty $, is equivalent to the following:
$$ \|f\| _{L^p} ^p + \|\Delta f\| _{L^p... |
128,255 | <p>I'm really confused. In a book <a href="http://www.wiley-vch.de/publish/en/books/bySubjectEE00/bySubSubjectEE70/0-470-27680-0/?sID=h1tjs4msm9be2umsaod861gql5" rel="nofollow noreferrer">ISBN: 978-0-470-27680-8</a> is written:</p>
<blockquote>
<p>The Euclidean distance can be generalized as a special case of a fami... | Vokram | 28,024 | <p>If you are asking about the difference between a metric and a norm:</p>
<p>A norm and a metric are two related but different things. Generally speaking, a norm is a more "vector space" concept than metrics.</p>
<p>A norm assigns value to a <strong>SINGLE</strong> vectors (its length) in a vector space, while metri... |
128,255 | <p>I'm really confused. In a book <a href="http://www.wiley-vch.de/publish/en/books/bySubjectEE00/bySubSubjectEE70/0-470-27680-0/?sID=h1tjs4msm9be2umsaod861gql5" rel="nofollow noreferrer">ISBN: 978-0-470-27680-8</a> is written:</p>
<blockquote>
<p>The Euclidean distance can be generalized as a special case of a fami... | GEdgar | 442 | <p>Both are (almost) right. And they say the same thing. Just notation adapted to the particular problem.</p>
<p>In</p>
<blockquote>
<p>$$
D(\mathbf{x}_i,{\mathbf x}_j)=\left(\sum_{l=1}^d |x_{il}-x_{jl}|^{1/p}\right)^p \tag{1}
$$</p>
</blockquote>
<p>we have points ${\mathbf x}_i$ and ${\mathbf x}_j$ in $d$-spac... |
2,436,775 | <p>Question : </p>
<p>How much work is done in pulling an object constrained to move along the portion of the curve y = $x^2$; z = $x^3$ from (0; 0; 0) to (1; 1; 1) (positions in meters), if the rope pulling it is always in the direction < 1;-3;-4 > and the tension in the rope is constant at 100 Newtons?</p>
<p>At... | spaceisdarkgreen | 397,125 | <p>Your form for $\vec r(t)$ is correct. </p>
<p>To get the vector for $\vec F$, you can write it $$ \vec F = F\langle 1,-3,-4\rangle$$ where $F$ is a scalar (that way it goes in the right direction). And then you know its magnitude is $100$ Newtons, so to find $F$ you set $|\vec F| = 100$ and solve. </p>
<p>Then tak... |
2,677,401 | <p>Consider all $1000$–element subsets of $\{1,2,3,4,\dots 2015\}$. From each such subset select least element. Find Arithmetic Mean of all these elements.</p>
<p>I easily managed the trick but my approach got me a lengthy answer. I selected leaste numbers and multiplied them by their number of subsets possible such a... | mickep | 97,236 | <p>If you instead let $y(x)=x u(x)$ your differential equation takes the form
$$
\bigl(x(xu)^4+xu)\bigr)\,dx-x(x\,du+u\,dx)=0
$$
that is
$$
x^5u^4\,dx=x^2\,du.
$$
I'm sure you can solve this differential equation.</p>
|
375,729 | <p>If $(X,\mathfrak D)$ and $(Y,\mathfrak C)$ are uniform spaces, $X$ compact and $f:X\rightarrow Y$ continuous, why $f$ is uniformly continuous? </p>
| vadim123 | 73,324 | <p>You have a two-dimensional eigenspace, so there should be two linearly independent eigenvectors, that each satisfy $x_1+x_2+x_3=0$ for you to choose.</p>
<p>More details: Every eigenvalue has an associated eigenspace. The dimension of this eigenspace is called the geometric multiplicity of the eigenvalue, while th... |
2,279,937 | <p>How to prove that $$\sum_{n=1}^{15} \frac{1}{n^3}\lt\frac 65$$ </p>
<p>I tried to compare this sum to the infinite sum, but <em>Apery</em>'s constant is just above $1.2$ so this approach doesn't work.</p>
<p>Then I typed this into wolfy and the sum seems as if it is just under $6/5$. </p>
<p>However, I was wonder... | zwim | 399,263 | <p>You can estimate the remainder of the series by comparing it to the integral.</p>
<p>$\displaystyle \sum\limits_{n=N}^{\infty}\frac 1{n^3}\ge\int_{N}^{\infty}\frac{dt}{t^3}=\bigg[\frac{-1}{2t^2}\bigg]_{N}^{+\infty}=\frac{1}{2N^2}$</p>
<p>$\displaystyle \sum\limits_{n=1}^{15}\frac 1{n^3}\le\zeta(3)-\frac 1{512}\sim... |
2,279,937 | <p>How to prove that $$\sum_{n=1}^{15} \frac{1}{n^3}\lt\frac 65$$ </p>
<p>I tried to compare this sum to the infinite sum, but <em>Apery</em>'s constant is just above $1.2$ so this approach doesn't work.</p>
<p>Then I typed this into wolfy and the sum seems as if it is just under $6/5$. </p>
<p>However, I was wonder... | StuartMN | 439,545 | <p>$\sum_{N+1}^{15} (1/x^3) $ < $\int_N^{15} (1/x^3) dx $ Try N=1 ,if it doesn't work,try N=2 etc this comes from the integral test . In fact you should increase the upper limit of the integral from 15 to +infinity to make it easier .I believe N=3 might work</p>
|
3,584,718 | <p>My uni is closed because of the pandemic and I'm home learning calculus. There is one problem I am really not sure how is supposed to be solved:</p>
<p><span class="math-container">$$\log(x)+\log(\sqrt[3]{x})+\log(\sqrt[9]{x})+\log(\sqrt[27]{x})+\ldots=6$$</span></p>
<p>I know that I am supposed to show my attempt... | RobPratt | 683,666 | <p>Use properties of log to rewrite the equation as
<span class="math-container">$$\log x + \frac{1}{3}\log x + \frac{1}{9}\log x + \frac{1}{27}\log x + \dots = 6.$$</span>
Equivalently,
<span class="math-container">$$\log x \sum_{k=0}^\infty \left(\frac{1}{3}\right)^k = 6.$$</span>
Next sum the geometric series to ob... |
145,191 | <p>I have a problem in vectors $\mathbf{x} = (x_1,\ldots,x_n)$ and $\mathbf{y}=(y_1,\ldots,y_n)$, where every $x_i,y_i\in\mathbb{R}_{\geq 0}$,</p>
<p>$$\begin{array}{ll} \text{maximize} & \displaystyle\sum_{i=1}^n {y_i}^u {x_i}^{1-u}\\ \text{subject to} & \displaystyle\sum_{i=1}^n x_i = 1\\ & \displaystyle... | OkkesDulgerci | 23,291 | <p>I tried something but not sure the result is right</p>
<pre><code>n = 2;
varx = Array[x, n];
vary = Array[y, n];
contsX = And @@ (a < # < b & /@ varx);
contsY = And @@ (c < # < d & /@ vary);
u = 0.4; a = 0; b = 1; c = 0; d = 1;
func = Sum[y[i]^u x[i]^(1 - u), {i, n}];
NMaximize[{func, Sum[x[i],... |
256,278 | <p>I was thinking about the following problem :</p>
<p>Define $ f:\mathbb C\rightarrow \mathbb C$ by </p>
<p>$$f(z)=\begin{cases}0 & \text{if } Re(z)=0\text{ or }Im(z)=0\\z & \text{otherwise}.\end{cases}$$</p>
<p>Then the set of points where $f$ is analytic is:</p>
<blockquote>
<p>(a) $\{z:Re(z)\neq 0$ an... | Sugata Adhya | 36,242 | <p>(a) is correct. </p>
<p>For $(a,0)\in\mathbb C$ with $a\neq0, u_x=0\neq 1=v_y$ at $(a,0)$ and for
$(0,a)\in\mathbb C$ with $a\neq0, u_y=1\neq 0=v_y$ at $(a,0)$. Further $f$ is not differentiable at $(0,0)$.</p>
<p>Since $C-R$ equation doesn't get satisfied anywhere in the real and imaginary axis (minus origin) it'... |
1,012,123 | <p><em><strong>The differentiable fuction $z=z(x,y)$ is given implicitly by equation $f(\frac{x}{y},z)=0$, where $f(u,v)$ is supposed to be differentiable and $\frac{\partial f}{\partial v}(u,v)\neq0$. Verify that
$$x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=0.$$</strong></em>
This is the exercise 6... | Christian Blatter | 1,303 | <p>I won't go into the calculations, but give an intuitive argument why
$$x{\partial z\over\partial x}+ y{\partial z\over \partial y}\equiv 0\tag{1}$$ should hold:</p>
<p>Along rays through the origin the quotient ${x\over y}$ is constant. Since $z=\psi(x,y)$ is determined from an equation of the form $f\bigl({x\over ... |
2,785,013 | <p>Following this <a href="https://www.derivative-calculator.net/#expr=5%5E%28xcosx%29&showsteps=1" rel="nofollow noreferrer">enter link description here</a>, where we're performing the derivative
$$\frac{d}{dx}5^{x\cos(x)},$$</p>
<p>The first step of the differentiation is: </p>
<p>$[a^u(x)]'$ = $ \ln(a) \,a^u(x... | user | 505,767 | <p>We have that</p>
<p>$$a^{u(x)}=e^{u(x)\cdot \log a}$$</p>
<p>and then</p>
<p>$$(a^{u(x)})'=(e^{u(x)\cdot \log a})'=e^{u(x)\cdot \log a}\cdot (u(x) \cdot \log a)'=a^{u(x)}\cdot u'(x) \cdot \log a$$</p>
|
1,606,727 | <p>If you have a set of $N$ items, how many subsets can you make?
For example, for the set existing of $3$ items (Item1, Item2 and Item3) the subsets are Item1, Item2, Item3, Item2+3, Item1+2, Item1+3 and Item1+2+3. Is there a general formula for this? </p>
<p>I think it might be $2^N-1$, but I'm not sure. </p>
<p>T... | Varun Iyer | 118,690 | <p>Yes there is a formula for the maximum number of subsets given a set of size $N$.</p>
<p>The formula is:</p>
<p>$$2^N$$</p>
<p>If you do not want to include the null set, then the answer would be $2^N - 1$.</p>
<p>If you wish to know the derivation of this formula, please see <a href="https://math.stackexchange.... |
1,606,727 | <p>If you have a set of $N$ items, how many subsets can you make?
For example, for the set existing of $3$ items (Item1, Item2 and Item3) the subsets are Item1, Item2, Item3, Item2+3, Item1+2, Item1+3 and Item1+2+3. Is there a general formula for this? </p>
<p>I think it might be $2^N-1$, but I'm not sure. </p>
<p>T... | Nick | 272,448 | <p>You know that there are $\binom n k $ subsets of $k \le n$ elements in a set of $n$. There are then subsets for $k=1,2,...,n$ and you have a total of $\sum _{k=1}^n \binom nk = (\sum _{k=0}^n \binom nk) -1$ subsets. The second sum (except for -1) is the Newton's formula for $(1+1)^n$. You have then a total of $2... |
753,997 | <p>Someone told me that math has a lot of contradictions. </p>
<p>He said that a lot of things are not well defined.</p>
<p>He told me two things that I do not know.</p>
<ul>
<li>$1+2+3+4+...=-1/12$</li>
<li>what is infinity $\infty$?</li>
</ul>
<p>Since I am not a math specialist and little. How to disprove the pr... | Ittay Weiss | 30,953 | <p>There are no known contradictions in mathematics. That does not mean there aren't, it just says we didn't find any. Considering the fact that thousands of mathematicians are creating new mathematics daily, and not a single one ever encountered any contradiction is quite overwhelming circumstantial evidence that ther... |
4,057,267 | <p>Given <span class="math-container">$u_t+cu_x=b(x,t)$</span> with initial value <span class="math-container">$u(x,0)=\phi(x)$</span>, I am asked to derive the following general solution
<span class="math-container">$$u=\phi(x-ct)+\int_{0}^t b(x-c(t-\tau),\tau)d\tau$$</span></p>
<p>I am not sure how to proceed with th... | JJacquelin | 108,514 | <p><span class="math-container">$$u_t+cu_x=b(x,t)$$</span>
You have correctly written the Charpit-Lagrange system of ODEs :
<span class="math-container">$$\frac{dt}{1}=\frac{dx}{c}=\frac{du}{b(x,t)}$$</span>
A first characteristic equation comes from solving <span class="math-container">$\frac{dt}{1}=\frac{dx}{c}$</spa... |
2,959,859 | <p>Let <span class="math-container">$\theta$</span> be a root of <span class="math-container">$p(x)=x^3+9x+6$</span>, find the inverse of <span class="math-container">$1+\theta$</span> in <span class="math-container">$\mathbb{Q(\theta)}$</span>.</p>
<p>So problems like this really annoy me but I did crappy on the last... | Will Jagy | 10,400 | <p><span class="math-container">$$ \left( \frac{ x^{2} - x + 10 }{ 4 } \right) $$</span></p>
<p>==================================================</p>
<p><span class="math-container">$$ \left( x^{3} + 9 x + 6 \right) $$</span> </p>
<p><span class="math-container">$$ \left( x + 1 \right) $$</s... |
107,546 | <p>My matrix is</p>
<p>$\qquad A= \begin{pmatrix}
{1} & {2} & {3}\\
{4} & {1} & {0}\\
{0} & {5} & {4}
\end{pmatrix} $</p>
<p>I need </p>
<p>$\qquad A^n$</p>
<p>I tried</p>
<pre><code>MatrixPower[A, n]
</code></pre>
<p>I do not understand the result. Should it not go according to <code>... | halirutan | 187 | <p>You need to understand that <em>Mathematica</em> prefers to write some numbers in their closed form because with numerical values, you would loose information and probably precision. It is kind of why it is better to keep <code>Sin[4]</code> and not use <code>-0.756802</code>, because <code>Sin[4]</code> can probabl... |
7,420 | <p>I have a stand alone <code>Manipulator</code> (it is not in a <code>Manipulate</code>) and would like to know how to alter the style of the label. There are no explicit styling options for <code>Manipulator</code> so I tried wrapping it in <code>Style</code>:</p>
<pre><code>Style[
Manipulator[0.5, AppearanceElemen... | Mark McClure | 36 | <p>I suppose you could roll your own, as in </p>
<pre><code>s = 0.5;
Row[{Manipulator[Dynamic[s]], " ",
Style[Dynamic[s], FontSize -> 16]}]
</code></pre>
|
7,420 | <p>I have a stand alone <code>Manipulator</code> (it is not in a <code>Manipulate</code>) and would like to know how to alter the style of the label. There are no explicit styling options for <code>Manipulator</code> so I tried wrapping it in <code>Style</code>:</p>
<pre><code>Style[
Manipulator[0.5, AppearanceElemen... | Matariki | 379 | <p>Use <code>Slider</code> in which case you can use numerical values for <code>ImageSize</code>. </p>
<pre><code>Row[{Slider[Dynamic[n], {0, 100, 1}, ImageSize -> {150, 50},
ImageMargins -> 10], Style[Dynamic[n], FontSize -> 16]}]
</code></pre>
|
2,433,394 | <blockquote>
<p>If a simple group $G$ has a subgroup $K$ that is a normal subgroup of two distinct maximal subgroups ,prove that $K=\{e\}$.</p>
</blockquote>
<p><strong>Attempt</strong>:</p>
<p>Let $H_1,H_2$ be two maximal subgroups of $G$ in which $K$ is normal.</p>
<p>Now considering $H_1H_2$ in which $K$ is nor... | Michael Rozenberg | 190,319 | <p>$$|z_1z_2|=|(x_1+y_1i)(x_2+y_2i)|=\sqrt{(x_1x_2-y_1y_2)^2+(x_1y_2+x_2y_1)^2}=$$
$$=\sqrt{x_1^2x_2^2+y_1^2y_2^2+x_1^2y_2^2+x_2^2y_1^2}=\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)}=|z_1||z_2|$$</p>
|
2,839,457 | <p>Let $0 < \epsilon$ and $\delta < 1$, and let $Y$ be a random variable ranging in the interval $[0,1]$ such that $E(Y)=\delta + \epsilon$. Give a lower bound on $Pr[Y ≥ \delta + \epsilon/2].$</p>
<p>The standard application of Markov's Inequality gives the upper bound instead of lower. I tried to start with th... | Federico Fallucca | 531,470 | <p>When you identify the equation you have that </p>
<p>$6\sqrt{x^2+y^2}=6x+2y$</p>
<p>$\sqrt{x^2+y^2}=x+\frac{1}{3}y$ </p>
<p>So for $(x,y)\in \mathbb{R}^2$ such that $6x+2y\geq0$ you have that they are solutions of your equation if and only if </p>
<p>$ x^2+y^2=x^2+\frac{1}{9}y^2+\frac{2}{3}xy$</p>
<p>Then </p>
... |
2,036,864 | <p>Here is the question. </p>
<p>From past records, a clothing store finds that 55% of the people who enter the store will make a purchase. During a one hour period, 20 people enter the store. The random variable x represents the number of people who make the purchase. </p>
<p>Find the probability that :</p>
<ol>
<... | Brandon | 378,998 | <p>Do you know what the binomial theorem is?? You have all the info you need..</p>
<p>20 choose 7</p>
<p>Probability of success (p) = .55</p>
<p>Probability of failure (q) = .45</p>
|
2,863,179 | <p>I'll state the Cantor's theorem proof as is it is in my study texts:</p>
<p>Theorem (Cantor): Let $X$ be any set. Then $|X|<|\mathcal{P}(X)|$</p>
<p>Proof: Define map $\varphi:X\rightarrow\mathcal{P}(X)$ by $\varphi:x\mapsto\{x\}$. $\varphi$ is injective, thus $|X|\leq|\mathcal{P}(X)|$. Now suppose there is a b... | Fred | 380,717 | <p>That the specific mapping $ \varphi$ is not surjective, hence not bijective, is clear. </p>
<p>To prove that $|X|<|\mathcal{P}(X)|$ , it is enough to show that each(!) mapping $\psi:X\rightarrow\mathcal{P}(X)$ cannot be surjective.</p>
<p>In the proof above it is assumed that there is a surjective mapping $\ps... |
3,129,029 | <blockquote>
<p>The tournament involves <span class="math-container">$2k$</span> tennis players they play the tournament, each played with each exactly once. What is the minimum number of rounds you need to play to find 3 such that everyone plays with everyone?</p>
</blockquote>
<p>In each round, <span class="math-c... | nonuser | 463,553 | <p>Make a bipartite graph <span class="math-container">$K_{k,k}$</span>, here we have <span class="math-container">$k^2$</span> edges and no triangles.</p>
<p>So you have two groups of size <span class="math-container">$k$</span> with no plays beteween players in the same group and each pair or players from different ... |
73,204 | <p>How does one show that any graph with $n$ vertices and at least $n$ edges must have at least one cycle?</p>
| Brian M. Scott | 12,042 | <p>Here’s a slightly different approach. A <em>tree</em> is a connected graph without cycles. A <em>forest</em> is a graph without cycles; its connected components are trees. </p>
<ol>
<li>Show by induction on $n$ that a tree with $n$ vertices has $n-1$ edges. (HINT: What happens when you ‘pluck’ a leaf and its stem f... |
891,203 | <p>I am wondering about this: Assume you have a class $[f] \in \pi_1(B,b_0)$ and a covering map $p:E \rightarrow B$.
Now, I know that if you take any two paths $g,h \in f$ that are homotopic and they have liftings $\hat{g} , \hat{h}$, then if $E$ is not simply connected it might happen that the liftings are not closed... | mathlove | 78,967 | <p>Let $M$ be the intersection point of $QT$ and $PR$. </p>
<p>Since $M$ is on the $QT$, there exists $k\in\mathbb R$ such that
$$\vec{QM}=k\vec{QT}=k\left(\vec{QR}+\frac 12\vec{QP}\right)=k\vec{QR}+\frac{k}{2}\vec{QP}\tag1$$
On the other hand, since $M$ is on the $PR$, there exists $l$ such that
$$\vec{QM}=(1-l)\vec... |
302,519 | <p>My teacher gave us an homework. I solved it, but I don't think I have the right answer.</p>
<p><strong>PROBLEM</strong></p>
<p>We have three coins identical in appearance.</p>
<ul>
<li>Coin A falls on tails and heads with equal probability</li>
<li>Coin B falls twice as much on tails as heads</li>
<li>Coin C alwa... | user unknown | 8,692 | <p>I guess it is 61/78. </p>
<p>From the probabilities you get a total of 13/18 for a tail answer, but the dependent source is </p>
<ul>
<li>A: (3/13)*(3/6))</li>
<li>B: (4/13)*(4/6) </li>
<li>C: (6/13)*(6/6) </li>
</ul>
<p>which is (9+16+36)/(13*6) = 61/78 which is about 78%. </p>
|
1,478,460 | <p>For a linear autonomous system in the plane
$$ \mathbf{\dot{x}} = \begin{pmatrix} a & b\\ c & d \end{pmatrix}\mathbf{x} \qquad (a,b,c,d \in \mathbb{R})$$
with determinant $D = ad - bc$ and trace $T = a + d$ we have the characteristic polynomial
$$ \chi(\lambda) = \lambda^2 - T\lambda + D$$
and the eigenvalue... | amd | 265,466 | <p>In short, you’ve more or less answered your own question: it depends on the sign of $s$. </p>
<p>If the matrix of the equation is of the form $\pmatrix{\alpha&-\beta\\ \beta&\alpha}$, with $\beta$ positive, the phase curves are counterclockwise spirals with the direction of motion dependent on $\alpha$: Wh... |
3,777,750 | <p>I am analyzing the following system, where <span class="math-container">$I_{in}$</span> is a scalar parameter:
<span class="math-container">$$
\begin{aligned}
&\dot{V} = 10 \left( V - \frac{V^3}{3} - R + I_{in} \right) \\
&\dot{R} = 0.8 \left( -R +1.25V + 1.5 \right)
\end{aligned}
$$</span></p>
<p>It is a si... | neuronet | 52,369 | <p>My answer will be more intuitive, and largely a supplement to that of Hans Engler's.</p>
<p>Briefly: you are not seeing a supercritical Hopf Bifurcation but a <em>subcritical</em> Hopf bifurcation (as @Hans Engler pointed out in his answer), and the two recalcitrant facts you were trying to prove are only true for s... |
73,693 | <p>I have a function $\vec{F}_i(t)$, which is unknown, but I do know it's mean
$\langle \vec{F}_i(t) \rangle = \vec{0}$ and it's variance
$\langle \vec{F}_i(t) \cdot \vec{F}_j(t') \rangle = 2 k_B T \gamma \delta_{ij} \delta(t-t')$. </p>
<p>I am having a long expression and now want to evaluate it's mean - which mean... | bbgodfrey | 1,063 | <p>Given an expression,</p>
<pre><code>s = g[x] + Integrate[f[k] h[k], k] + Integrate[f[k] p[k], k]
</code></pre>
<p>the contraction of it with <code>f[t]</code> can be obtained with</p>
<pre><code>s[[Flatten[Take[Position[s, f[x_]], All, 1]]]] /. f[x_] -> c DiracDelta[t - x]
(* c*Integrate[DiracDelta[-k + t]*h[k... |
2,112,161 | <p>This is a question out of "Precalculus: A Prelude to Calculus" second edition by Sheldon Axler. on page 19 problem number 54.</p>
<p>The problem is Explain why $(a−b)^2 = a^2 −b^2 $ if and only if $b = 0$ or $b = a$.</p>
<p>So I started by expanding $(a−b)^2$ to $(a−b)^2 = (a-b)(a-b) = a^2 -2ab +b^2$. To Prove tha... | Thomas Andrews | 7,933 | <p>It might just be easier to use that $a^2-b^2=(a-b)(a+b)$.</p>
<p>So if $a-b=0$ then $(a-b)^2=(a-b)(a+b)$, and if $a-b\neq 0$ then $(a-b)^2=(a-b)(a+b)$ if and only if $a-b=a+b$.</p>
|
2,112,161 | <p>This is a question out of "Precalculus: A Prelude to Calculus" second edition by Sheldon Axler. on page 19 problem number 54.</p>
<p>The problem is Explain why $(a−b)^2 = a^2 −b^2 $ if and only if $b = 0$ or $b = a$.</p>
<p>So I started by expanding $(a−b)^2$ to $(a−b)^2 = (a-b)(a-b) = a^2 -2ab +b^2$. To Prove tha... | fleablood | 280,126 | <p>You have to prove two things:</p>
<p>1) If $b = 0$ or $b = a$ then $(a-b)^2 = a^2 - b^2$.</p>
<p>And </p>
<p>2) If $(a-b)^2 = a^2 - b^2$ then either $b = 0 $ or $b = a$.</p>
<p>To prove 1: we do what you did correctly:</p>
<p>If $b = 0$ then $(a - b)^2 = (a-0)^2 = a^2$. An $a^2 - b^2 = a^2 - 0^2 = a^2 - 0 = a^... |
115,325 | <p>I need a simple proof that a line cannot intersect a circle at three distinct points.</p>
| Aryabhata | 1,102 | <p>Without loss of generality, assume the circle is $x^2 + y^2 = r^2$ and the line is $y = mx + c$.</p>
<p>The x coordinates of the point of intersection satisfy $x^2 + (mx+c)^2 = r^2$ which is a quadratic and hence has at most $2$ roots.</p>
<p>Since given an $x$, the $y$ on the line is uniquely determined, we are d... |
3,035,549 | <p>I'm trying to prove the following inequality: </p>
<p><span class="math-container">$$\sin^{-1}(1)\geq\int_0^b1/\sqrt{1-x^2}\,dx +(1-b)\pi/2$$</span></p>
<p>for every <span class="math-container">$b \in [0,1)$</span>. </p>
<p>I'm given <span class="math-container">$\sin^{-1}(1) = \pi/2$</span> and <span class="mat... | Sameer Baheti | 567,070 | <p>I think <span class="math-container">$$\sin^{-1}(b) \leq b\frac{\pi}2$$</span> for every <span class="math-container">$b \in \big[0,1)$</span> since the graph of <span class="math-container">$y=b$</span> when scaled upto <span class="math-container">$y=\frac{\pi}2b\ $</span> will get above that of <span class="math... |
3,035,549 | <p>I'm trying to prove the following inequality: </p>
<p><span class="math-container">$$\sin^{-1}(1)\geq\int_0^b1/\sqrt{1-x^2}\,dx +(1-b)\pi/2$$</span></p>
<p>for every <span class="math-container">$b \in [0,1)$</span>. </p>
<p>I'm given <span class="math-container">$\sin^{-1}(1) = \pi/2$</span> and <span class="mat... | zhw. | 228,045 | <p>If <span class="math-container">$f$</span> is strictly convex on <span class="math-container">$[0,1],$</span> then <span class="math-container">$(b,f(b))$</span> lies below the line through <span class="math-container">$(0,f(0))$</span> and <span class="math-container">$(1,f(1))$</span> for <span class="math-contain... |
57,489 | <p>If I enter <code>x/x</code>, I get <code>1</code>. Such behavior leads to this:</p>
<pre><code>Simplify[D[Sqrt[x^2], x, x]]
</code></pre>
<blockquote>
<p>0</p>
</blockquote>
<p>The same would be even if I use <code>Together</code> instead of <code>Simplify</code>.</p>
<p>One could then think that $\sqrt{x^2}$ ... | Michael E2 | 4,999 | <p>Here are three approaches to the function:</p>
<pre><code>f1[x_] := Piecewise[{{Sqrt[x^2], x != 0}}, 0]
f2[x_] := Piecewise[{{Sqrt[x^2], x < 0}, {Sqrt[x^2], x > 0}}, 0]
f3[x_] := Piecewise[{{-x, x < 0}, {x, x > 0}}, 0]
</code></pre>
<p>Their second derivatives.</p>
<pre><code>Simplify@D[#[x], x, x] ... |
3,145,550 | <blockquote>
<p>Prove if <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are real numbers with <span class="math-container">$x \lt y$</span>, then there are infinitely many rational numbers in the interval <span class="math-container">$[x,y]$</span>.</p>
</blockquote>
<p>What I go... | Ash | 645,756 | <p>Okay, so I'll give it another shot given the feedback.</p>
<p>Proof:</p>
<p>Let <span class="math-container">$x,y \in \Bbb R$</span> with <span class="math-container">$x \lt y$</span> and <span class="math-container">$S = [x,y]$</span></p>
<p>Suppose there are only <span class="math-container">$n$</span> rational... |
437,596 | <blockquote>
<p>Show that $\mathbb{R}^n\setminus \{0\}$ is simply connected for $n\geq 3$.</p>
</blockquote>
<p>To my knowledge I have to show two things:</p>
<ol>
<li><p>$\mathbb{R}^n\setminus \{0\}$ is path connected for $n\geq 3$.</p></li>
<li><p>Every closed curve in $\mathbb{R}^n\setminus \{0\}, n\geq 3$ is nu... | Community | -1 | <p>You can sledgehammer proof your problem by induction on the number of points that you remove and the Seifert-van Kampen theorem. Say $x_1,\ldots,x_m$ are the points that you remove. We will induct on $m$. When $m = 1$ this is easy because $\Bbb{R}^n$ minus a point is homotopy equivalent to $S^{n-1}$ that is simply c... |
390,053 | <p>I'm trying to use topology to prove that: </p>
<p>$z^{n} + a_{n-1}z^{n-1} + ... + a_{1}z + a_{0} = 0$</p>
<p>has a solution in $\mathbb{C}$ if and only if, for each positive real number $c$, the equation </p>
<p>$z^n + \frac{a_{n-1}}{c}z^{n-1} + ... + \frac{a_1}{c^{n-1}}z + \frac{a_0}{c^n} = 0 $</p>
<p>has a sol... | Adam Saltz | 8,923 | <p>Notice that
$$(cz)^{n} + a_{n-1}(cz)^{n-1} + ... + a_{1}(cz) + a_{0} = c^nz^{n} + a_{n-1}c^{n-1}z^{n-1} + ... + a_{1}cz + a_{0}$$</p>
<p>Now divide through by $c^n$ to obtain $z^n + \frac{a_{n-1}}{c}z^{n-1} + ... + \frac{a_1}{c^{n-1}}z + \frac{a_0}{c^n}$.</p>
<p>Writing $p_1, p_2$ for the two polynomials of inter... |
3,883,504 | <p>I am trying to express the following: I have a set <span class="math-container">$A$</span> and the powerset (set of all subsets of <span class="math-container">$A$</span>) <span class="math-container">$P(A)$</span>. I have another set <span class="math-container">$S \in P(A)$</span>, and I want to get the sets in <s... | vvg | 820,543 | <p>You could use union of disjoint subsets to achieve the same thing: <span class="math-container">$X, Y \in P(A), X \sqcup Y \equiv A$</span></p>
|
498,674 | <p>I'm trying to interpret the very last sentence. He says nothing about $a_{1}$, so in the case of $a_{1}=0$ all the other coefficients can not be zero, which would imply linearly independence. The case where $a_{1}\neq 0$ all the other coefficients can not be zero because, if they were, we would have $a_{1}v_{1}=0$ w... | Rebecca J. Stones | 91,818 | <p>It seems to combine two cases together:</p>
<ul>
<li><p>If $a_0=0$, then since $\{a_1,a_2,\ldots,a_m\}$ contains a non-zero element and $a_0=0$, there must be a non-zero member of $\{a_2,a_3,\ldots,a_m\}$.</p></li>
<li><p>If $a_0 \neq 0$, then $a_1 v_1 \neq 0$ since $v_1 \neq 0$. Hence, if $a_1 v_1 + x=0$ where $x... |
4,631,361 | <p>Let <span class="math-container">$a, b, x, y>0$</span>, <span class="math-container">$a+b \ge x+y$</span> and <span class="math-container">$ab \le xy$</span> then <span class="math-container">$a^n+b^n \ge x^n+y^n$</span> where <span class="math-container">$n=2$</span>.</p>
<blockquote>
<blockquote>
<p><em>Is it t... | Joseph Harrison | 1,131,061 | <p>Assuming that the inequality holds for <span class="math-container">$n - 1$</span> and <span class="math-container">$n - 2$</span>, we can use induction.
<span class="math-container">\begin{align}
a^n + b^n &= a^{n - 1}(a + b) - a^{n - 1}b + b^{n - 1}(a + b) - ab^{n - 1} \\
&=(a + b)(a^{n - 1} + b^{n - 1}) -... |
68,173 | <p>I am working on tool for merging smaller textures into one larger for use on Android app.</p>
<p>I have $n$ rectangles of given size $(w_k, h_k)$, where $k=1,\ldots,n$ and I need to position them within master rectangle of size $2^l \times 2^m$, where $l, m \leq 9$ so none overlapping occur and $2^l \times 2^m$ has... | Henry | 6,460 | <p>We seem to have an number (I will use $k$) of identical $w \times h$ rectangles to fit inside a larger $2^n \times 2^m$ rectangles, and with a restrictions on $n$ and $m$.</p>
<p>For given $2^n$, we can fit $\lfloor 2^n / w \rfloor$ rectangles horizontally provided $w \le 2^n$. So we need at least $ \lceil k / \lf... |
107,972 | <p>The <code>Sign</code> function works in a very straightforward manner:</p>
<pre><code>Sign[8]
Sign[-8]
Sign[0]
</code></pre>
<blockquote>
<pre><code>1
-1
0
</code></pre>
</blockquote>
<p>Unfortunately, in my code I need <code>Sign[0]</code> to equal 1 or negative 1. I tried this but get a funny error:</p>
<pre><... | thedude | 27,670 | <pre><code>sign[n_] := 2 Boole[10^10 n > RandomChoice[{1, -1}]] - 1
</code></pre>
<p>or using the same idea,</p>
<pre><code>sign[n_] := Sign[10^10 n + RandomChoice[{1, -1}]]
sign[0] (*1*)
sign[0] (*-1*)
sign[-2] (*-1*)
</code></pre>
<p>unless $\left|n\right|<10^{^-10}$.</p>
<hr>
<p><strong>EDIT</strong></p>... |
1,463,457 | <p>Given a function </p>
<p>$$F(x)= \begin{cases} x^2 & \text{when }x \in \mathbb Q \\3x & \text{when }x \in\mathbb Q^c \end{cases}$$</p>
<p>Show that $F$ is continuous or not on $x=3$ with $\epsilon-\delta$.</p>
<p>I tried to deal with problems just like doing on Dirichlet functions. Mistakenly or not, I co... | D. A. | 275,736 | <p>let $\epsilon>0$,let $\delta = min(\frac{\epsilon}{7},1)$.
let $x$ satisfy $|x-3|<\delta$, if $x \in Q $, then we have:</p>
<p>$|F(x) - F(3)| = |x^2-9| = |x-3||x+3| \leq 7|x-3| < 7\delta \leq \epsilon$</p>
<p>if $x$ not in $Q$ we have:</p>
<p>$|F(x) - F(3)| = |3x-9| = 3|x-3| < 3\delta \leq 3 \frac{\ep... |
147,932 | <p>I can't really put a proper title on this one, but I seem to be missing one crucial point. Why do roots of a function like $f(x) = ax^2 + bx + c$ provide the solutions when $f(x) = 0$. What does that $ y = 0$ mean for the solutions, the intercept at the $x$ axis? Why aren't the solutions at $f(x) = 403045$ or some o... | Robert Israel | 8,508 | <p>Actually there <em>is</em> something special about the value $0$: the connection between the roots of a polynomial and its linear factors. That is, if $f(z)$ is a polynomial, $f(a) = 0$ if and only if we can write $f(z) = (z-a) g(z)$ for some polynomial $g(z)$.</p>
|
97,094 | <p>I have a list of inequalities, which include in total four parameters, that I would like to test by feeding the four parameters with random numbers. The goal is to find the best values of the variables such that they satisfy all inequalities.</p>
<p>The first problem I encounter is that I do not know how to "feed" ... | Community | -1 | <p>And the version with <code>FindInstance</code>:</p>
<pre><code>c = 2;
FindInstance[0 < a <= 5 && 1 < b < 2 && a + b < 8, {a, b}, c]
{{a -> 9/41, b -> 133/102}, {a -> 5, b -> 7/6}}
</code></pre>
<p>You can increase the variable <code>c</code> to get more solutions. if <cod... |
421,269 | <p>Given an infinite connected graph <span class="math-container">$G$</span> of bounded degree with vertex set <span class="math-container">$X$</span>, let <span class="math-container">$P_x^n$</span> the time <span class="math-container">$n$</span> distribution of the simple random walk started at the vertex <span clas... | tmh | 41,827 | <p>Too long for a comment, but I wanted to mention that we encountered some related issues in my recent preprint with Noah Halberstam <a href="https://arxiv.org/abs/2203.01540" rel="nofollow noreferrer">https://arxiv.org/abs/2203.01540</a> and thought that the work-arounds we found might be useful to you too.</p>
<p>We... |
1,588,361 | <p>I know if a matrix has a left and right inverse then the inverses are the same and are (is) unique and the original matrix is a square matrix, thus if I have a matrix which has multiple left inverses for example then it has no right inverse and is a non-square matrix. But if a matrix has a unique left inverse then ... | Waldeck Schützer | 367,297 | <p>In a ring of matrices, yes it does. If a matrix $A$, or any element of a ring, has a unique left inverse, say $BA=I$, then $$(I-AB+B)A=A-(AB)A+BA=A-A(BA)+BA=0+I=I.$$ The uniqueness implies that $I-AB+B=B$, so $AB=I$, and $B$ is also the unique right inverse. Obviously $A$ has to be square, otherwise $AB$ is meaningl... |
415,573 | <p>Show that there holds $$\sqrt{1+x} < 1+(x/2)$$ for all $x > 0$ .</p>
<p>I need guidance in doing this question. Can anyone help please? I'll be thankful!</p>
| peter.h | 81,601 | <p>Hint: Use <a href="http://en.wikipedia.org/wiki/Bernoulli%27s_inequality" rel="nofollow">Bernoulli's Inequality</a></p>
<p>Or consider the function $f(x) = 1 + \frac{x}{2} - \sqrt{1+x}$. $$f'(x) = \frac{1}{2} - \frac{1}{2}\cdot \frac{1}{\sqrt{1+x}}$$ Now $f'$ is $0$ when $1-\frac{1}{\sqrt{1+x}}=0 \implies x =0$. A... |
766,779 | <p>Hi this is a lagrangian optimization problem. Essentially as the title says, the question is asking us to maximize (if possible) $x^2+y^2+z^2$ on $x^2+y^2+4z^2=1$.</p>
<p>I started by the standard lagrangian method:$$L(x,y,z,\lambda)=x^2+y^2+z^2+\lambda(1-x^2-y^2-4z^2)$$
Which subsequently gives: $${∂L\over∂x}=2x-... | Daniel Node.js | 9,499 | <ol>
<li>$\cfrac{\frac{L^3}{2^3}}{6}$ distribute exponents. $\left(\cfrac{L}{2}\right)^3 = \cfrac{L}{2} \cdot \cfrac{L}{2} \cdot \cfrac{L}{2}$</li>
<li>$\cfrac{\frac{L^3}{8}}{6}$ compute $2^3$</li>
<li>$\cfrac{L^3}{48}$ rewrite as simple fraction. $\cfrac{\frac{a}{b}}{c}=\cfrac{a}{bc}$</li>
</ol>
|
4,503,490 | <p>Someone told me that "<span class="math-container">$L^1$</span> functions have to decay to 0 at <span class="math-container">$\pm\infty$</span>."<br />
I know that the <span class="math-container">$L^1$</span> function is defined as a collection of functions such that <span class="math-container">$\int_{X}... | geetha290krm | 1,064,504 | <p>I will write <span class="math-container">$Y$</span> for <span class="math-container">$\tau_b$</span>. Let <span class="math-container">$\phi (Y)=P(W_t>b|Y)$</span>. Then <span class="math-container">$\int_0^{\infty} P(W_t>b|Y=s)dF(s)$</span> is just a notation for <span class="math-container">$\int_0^{\infty}... |
4,503,490 | <p>Someone told me that "<span class="math-container">$L^1$</span> functions have to decay to 0 at <span class="math-container">$\pm\infty$</span>."<br />
I know that the <span class="math-container">$L^1$</span> function is defined as a collection of functions such that <span class="math-container">$\int_{X}... | CA-Math | 485,588 | <p>@geetha290krm: Thanks for your reply. It pointed me in the right direction, so I think I can formalise the answer as follows:</p>
<p>Given that <span class="math-container">$\mathbb{P}(W_{t}>b \mid \tau_{b}) := \mathbb{P}(W_{t}>b \mid \sigma( \tau_{b} ) )$</span> is a version of <span class="math-container">$\... |
2,756,762 | <p>I am trying to solve the following equation:
$$
z^3 + z +1=0
$$</p>
<p>Attempt: I tried to factor out this equation to get a polynomial term, but none of the roots of the equation is trivial.</p>
| lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>Like <a href="https://math.stackexchange.com/questions/2739299/how-to-solve-the-cubic-x3-3x1-0/2739305#2739305">How to solve the cubic $x^3-3x+1=0$?</a>,</p>
<p>let $z=a\cos t,a\ne0$ such that $\dfrac{a^3}4=\dfrac{-a}3\implies a^2=-\dfrac43\implies a=\pm\dfrac2{\sqrt3}i$</p>
<p>$$-1=a^3\cos^3t+a\cos ... |
283,245 | <p>Draks gave the identity, <a href="https://math.stackexchange.com/questions/102413/higher-order-trigonometric-function">Higher Order Trigonometric Function</a> $$\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}=\frac{1}{m}\sum_{k=0}^{m-1} \exp( e^{i\frac{2k+1}{m}\pi}x )$$ How can this be proven?</p>
| Thomas Andrews | 7,933 | <p>Let $x_k=e^{i\frac{2k+1}{2m}2\pi}$. These are the $2m$th roots of unity that are not $m$th roots of unit, so they are the $m$ distinct roots of roots of $x^m+1=0$.</p>
<p>Now substituting above, and expanding $\exp$ we get:</p>
<p>$$\frac{1}{m} \sum_{k=0}^{m-1} \exp(x_k x) = \sum_{k=0}^{m-1} \sum_{j=0}^\infty \fra... |
1,399,402 | <p>Let $f:[0,\infty [\to[0,\infty [$ continuous such that $$\lim_{t\to\infty }\frac{f(t)}{t}=\ell\in[0,1).$$</p>
<p>Prove that $f$ has a fixed point, i.e. there is an $x\geq 0$ such that $f(x)=x$. </p>
<p>I don't really know how to solve this problem. My first intension was to use Brouwer, but it's only useable on a ... | Community | -1 | <p>Let us suppose $f(x) \neq $$x $ $\forall x>=0$. We have $f(0) > 0$ and because $f$ continuous:</p>
<p>1) $f(x) > x $ $\forall x>0$ </p>
<p>or </p>
<p>2) $f(x) < x $ $\forall x>0$.</p>
<p>(Because the continuous function $g(x) = f(x) - x$ is $\neq0 $ $\forall x$ so cannot change the sign)</... |
24,569 | <p>I am teaching a combinatorics class in which I introduced the notion of a "mass formula". My terminology is inspired by the <a href="http://en.wikipedia.org/wiki/Smith%25E2%2580%2593Minkowski%25E2%2580%2593Siegel_mass_formula" rel="nofollow">Smith–Minkowski–Siegel mass formula</a> for the total mass of positive-def... | Victor Protsak | 5,740 | <p>This doesn't qualify as a free reference, but "Graphs on surfaces and their applications" by Lando and Zvonkin has some nice examples. On p.46, after stating a theorem enumerating trees with a given "passport", the authors remark:</p>
<blockquote> We will often encounter enumerative formulas where the objects are n... |
56,373 | <p>ellipse described about the circle in which a regular pentagon is constructed mapped on an ellipse<img src="https://i.stack.imgur.com/aF9gG.jpg" alt="enter image description here"></p>
<p><img src="https://i.stack.imgur.com/7pK7f.jpg" alt="enter image description here"></p>
<p>The surface can be calculated from my... | Jiri Kriz | 12,741 | <p>The area of an ellipse sector from $0$ to $\theta$ degrees is $a b \theta/2$. The angle $\theta$ is measured along the circle with radius $a$. If a regular n-gon in inscribed into this circle then the angle of one sector is $\theta = 2\pi/n$. Hence, the area of an ellipse sector corresponding to one n-gon sector is ... |
4,331,390 | <p>Let <span class="math-container">$f_a(x) = axa^{-1}$</span> be an inner automorphism of G, where <span class="math-container">$a,x$</span> <span class="math-container">$ \in$</span> G. What is <span class="math-container">$f_a^{-1}$</span>?</p>
<p>Method 1.<br />
<span class="math-container">$f_a^{-1}(x) = (axa^{-1}... | Paul | 1,004,036 | <p>Method 1 is wrong.
The correct equation is</p>
<p><span class="math-container">${f_{a}(x)}^{-1} = (axa^{-1})^{-1} = ax^{-1}a^{-1} $</span></p>
<p>But in general you have that
<span class="math-container">$f_{a}^{-1}(x) \neq {f_{a}(x)}^{-1}$</span></p>
<p>(On the left-hand-side there is the image of x by the inverse ... |
3,567,076 | <p>I know that to determine the radius of convergence of the series
<span class="math-container">$$ \sum_{n=0}^\infty a_nx^n $$</span>
I need to find
<span class="math-container">$$ \lim_{k\rightarrow \infty} \left| \frac{a_{k+1}}{a_k} \right| = c$$</span>
Then the radius of convergence <span class="math-container">$R$... | Robert Israel | 8,508 | <p>No, that's not what you need. The Ratio Test is only one way to test convergence of a series. It doesn't always work. But in the case of a
series where some of the terms are zero, what might work is using the Ratio Test on consecutive <strong>nonzero</strong> terms. Thus your series
<span class="math-container">... |
577,639 | <p><strong>If $x$ & $y$ are natural numbers, and $56 x = 65 y$, prove that $x + y$ is divisible by $11$.</strong></p>
<p>Solution)</p>
<p>$56$ and $65$ are relatively prime</p>
<p>So, $65∣x$ and $56∣y$</p>
<p>Let $x = 65m$ and $y = 56n$</p>
<p>Then,
$56x = 65y$</p>
<p>$56.65m = 65.56n$,</p>
<p>$m = n$ </p>
... | LASV | 89,736 | <p>Here is an alternative proof:</p>
<p>$56x=65y$ implies $x\equiv -y \mod 11$ and the result is now clear.</p>
|
1,265,384 | <p>"Prove that if <span class="math-container">$det(A)=1$</span> and all the entries in <span class="math-container">$A$</span> are integers, then all the entries in <span class="math-container">$A^{-1}$</span> are integers."</p>
<p>I began by setting up the adjoint method for finding the inverse.</p>
<p><span class=... | lhf | 589 | <p>Here is a rewording of the adjoint argument.</p>
<p>The inverse matrix is obtained by solving a matrix equation <span class="math-container">$AX=I$</span>, which translates to <span class="math-container">$n$</span> linear systems. By Cramer's rule, the solution of each linear system has integer entries because <sp... |
602,143 | <p>Find the area bounded by the curves $y=e^x$, $y=xe^x$, and $x=0$</p>
<p>I know how to solve the integral for this, but I'm getting hung up trying to find the points of intersection for the two equations.</p>
<p>So I did what I normally do for these problems:</p>
<p>$$e^x = xe^x$$</p>
<p>I then did this (and this... | abkds | 112,225 | <p>Do this $(x-1)e^x=0$ , Then either of them should be zero . Now $e^x \ne0$ for any value of $x$ hence $x=1$ . Now use the limits $x=0$ to $x=1$ for the integration</p>
|
600,411 | <blockquote>
<p>Evaluate $ \iiint_D (x^2+y^2) \, ,\mathrm{d}V $, where $D$ is the
region bounded by the graphs of $y=x^2$, $z=4-y$, and $z=0$.</p>
</blockquote>
<p>How would I do this problem? I can't even visualize the region D. I tried plotting it into Wolfram Alpha but it doesn't understand that I need to 3D pl... | andraiamatrix | 114,466 | <p>To draw it, first try plotting the corresponding 2-dimensionl planes.</p>
<p>$y=x^2$ is just a parabola in the xy-plane. But notice z can be whatever it wants. This is how it looks in Wolfram: http://www.wolframalpha.com/input/?i=z+%3D+y-x^2 [<strong>Please don't edit this again. It will not work as a hyperlink bec... |
1,868,044 | <blockquote>
<p>For given positive integers $r,v,n$ let $S(r,v,n)$ denote the number of $n$-tuples of nonnegative integers $(x_1,\ldots,x_n)$ satisfying the equation $x_1+\cdots+x_n = r$ and such that $x_i \leq v$ for $i = 1,\ldots,n$. Prove that $$S(r,v,n) = \sum_{k=0}^m (-1)^k \binom{n}{k} \binom{r-(v+1)k+n-1}{n-1}... | Marko Riedel | 44,883 | <p>We have from first principles that this value is</p>
<p>$$[z^r] (1+z+z^2+z^3+\cdots+z^v)^n
= [z^r] \frac{(1-z^{v+1})^n}{(1-z)^n}.$$</p>
<p>This is</p>
<p>$$[z^r] \frac{1}{(1-z)^n}
\sum_{k=0}^n {n\choose k} (-1)^k z^{(v+1)k}
\\ = \sum_{k=0}^n {n\choose k} (-1)^k
[z^{r-(v+1)k}] \frac{1}{(1-z)^n}
\\ = \sum_{k=0}^n ... |
1,385,026 | <p>I came across something related to the degree of a splitting field for a polynomial over a field $K$. Let's suppose $p \in K[x]$ with degree $n$, and $p$ has irreducible factors $f_{1}, \ldots, f_{c}$ with respective degrees $d_{1}, \ldots, d_{c}$.</p>
<p>Ok, I know we can construct the splitting field as a tower ... | Divide1918 | 706,588 | <p>Let <span class="math-container">$L_i$</span> be a splitting field of <span class="math-container">$f_i$</span> over <span class="math-container">$L_{i-1}$</span>, whence <span class="math-container">$L_0$</span> is defined to be <span class="math-container">$K$</span> and <span class="math-container">$L_c=L$</span>... |
511,730 | <p>I'm having issues with my logic on this problem:</p>
<blockquote>
<p>How many ways are there to form a list of three letters from the letters in the word COMBINATORICS if the letters cannot be used more often than they appear in COMBINATORICS? </p>
</blockquote>
<p>I'm trying to think of this as a set with 13 el... | Community | -1 | <p>Yes. The electric field might be generated for example by two charges: $+5C$ and $-4C$ separated by $2$ meters. There is a point in between them where the potential is $0$ (the electric potential is also $0$ in the infinite) but in that point the electric field vector is not null.</p>
<p>The electric field is relat... |
1,728,124 | <p>show if $A = B \oplus C$ then $B \cap C = \{0\}$.</p>
<p>attempt:</p>
<p>suppose, $ a \in A = B \oplus C$, then by definition we can write $ a = b + c$ for unique $b$ and $c$. Now I am trying to take a general $z \in B \cap C$ and show that it must be $0$, but I am having troubles</p>
| Arthur | 15,500 | <p>Take that $a=b+c$ that you have, where $a$ determines $b$ and $c$ uniquely. Let $z\in B\cap C$ be non-zero. What can you say about the expression
$$
(b+z)+(c-z)
$$
and how is that a contradiction?</p>
|
377,799 | <p>Does a matrix of the form <span class="math-container">$A_{ij} = v_i + v_j$</span> for some arbitrary vector <span class="math-container">$v$</span> have a particular name?</p>
<p>I am attempting to find the closed form solution (if it exists, although it <em>looks</em> like it might) for the <span class="math-conta... | Dirk | 9,652 | <p>I only have an answer for the first question: A matrix <span class="math-container">$A$</span> with entries <span class="math-container">$A_{ij} = v_i+w_j$</span> is called on <em>outer sum</em> of <span class="math-container">$v$</span> and <span class="math-container">$w$</span>. I would write this as
<span class=... |
3,222,926 | <p>This is the equation:</p>
<p><span class="math-container">$$\sin\theta=0.8\theta$$</span> </p>
| David G. Stork | 210,401 | <p>Clearly there are three solutions:</p>
<p><a href="https://i.stack.imgur.com/FP1Sd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FP1Sd.png" alt="enter image description here"></a></p>
<p>Clearly <span class="math-container">$\theta = 0$</span> is a solution. By the antisymmetry of the compone... |
3,222,926 | <p>This is the equation:</p>
<p><span class="math-container">$$\sin\theta=0.8\theta$$</span> </p>
| Claude Leibovici | 82,404 | <p>This equation being transcendental, it will not show analytical solutions and numerical methods are required.</p>
<p>You could approximate the solution using
the approximation <span class="math-container">$$\sin(\theta) \simeq \frac{16 (\pi -\theta) \theta}{5 \pi ^2-4 (\pi -\theta) \theta}\qquad (0\leq \theta\leq\... |
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | John D. Cook | 136 | <p>Donald Knuth's series of books <em>The Art of Computer Programming</em>. He even credits the source of the exercises at the end of chapters if he found them elsewhere.</p>
|
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | drbobmeister | 8,472 | <p>Anything by John Milnor. His little book <em>Morse Theory</em> is a very clear, concise introduction to certain essential aspects of differential topology and Riemannian
geometry, starting at a fairly elementary level and winding up with Bott periodicity
for unitary groups. In this vein, his <em>Characteristic Cla... |
16,719 | <p>What are your favourite <em>scholarly</em> books? My favourite is definitely G.N. Watson's <a href="http://books.google.com/books?id=Mlk3FrNoEVoC&printsec=frontcover&dq=watson+bessel&source=bl&ots=SOToCOngC4&sig=3jym5VIo2ESQR4IlZREpEdy6oXE&hl=en&ei=WeWKS9yGNYWWtgfhtN21Dw&sa=X&oi=... | Tom Dickens | 8,955 | <p>I vote for Titchmarsh's "The Theory of the Riemann Zeta function." Very complete (for its time) and lucid.</p>
<p>I would also add "A Panoramic View of Riemannian Geometry" by Marcel Berger.</p>
|
3,165,019 | <p><a href="https://i.stack.imgur.com/T0WSV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/T0WSV.png" alt="enter image description here" /></a></p>
<p>I converted both functions into polar form and got <span class="math-container">$z = r$</span> and <span class="math-container">$z = r^2 - r\cos\thet... | MarianD | 393,259 | <p><span class="math-container">$$f: \mathbb N\rightarrow 2\mathbb N$$</span>
<span class="math-container">$$n \mapsto 2n$$</span></p>
<p>It's a bijection from the set of all natural numbers <span class="math-container">$\{1, 2, 3, \cdots\}$</span> into its proper subset of all <em>even</em> natural numbers <span clas... |
18,831 | <p>I provided an answer (whether it really is one or not is a matter of dispute)
to the following question:
<a href="https://math.stackexchange.com/questions/1074861/why-isnt-a-noncommutative-ring-with-only-trivial-ideals-a-division-ring/1074867#1074867">Where does the proof for commutative rings break down in the non-... | quid | 85,306 | <p>The meaning of "not an answer" can be somewhat counter-intuitive and thus sometimes leads to confusions. </p>
<p>Text written in the answer box should be flagged as not an answer when
"[it] was posted as an answer, but it does not attempt to answer the question. It should possibly be an edit, a comment, another qu... |
4,009,431 | <p>Let <span class="math-container">$V$</span> and <span class="math-container">$W$</span> be finite dimensional complex vector spaces and let <span class="math-container">$f: V\xrightarrow[]{}W$</span> and <span class="math-container">$g: W\xrightarrow[]{}V$</span> be linear maps. Suppose that <span class="math-contai... | leoli1 | 649,658 | <p>Just define <span class="math-container">$V_0,V_1$</span> as in 2 and 3, i.e. <span class="math-container">\begin{align*}
V_0&=\ker g\circ f = \{v\in V\mid g(f(v))=0\}\\
V_1&=\ker (g\circ f-id_V)=\{v\in V\mid g(f(v))=v\}
\end{align*}</span>
Now check that these subspaces satisfy <span class="math-container">... |
4,057,243 | <p>My textbook lists two theorems and I'm not sure how I'm supposed to interpret them. I don't need a proof; I'm only trying to figure out what information I'm being told by each theorem.</p>
<blockquote>
<p>Let <span class="math-container">$p$</span> be a prime and let <span class="math-container">$a$</span> be an int... | Bill Dubuque | 242 | <p>No, you have misunderstood both. Let's consider a simple special case: <span class="math-container">$\,a=2,\, p=5.\,$</span> i.e. we consider the doubling map <span class="math-container">$\,n\to 2n\bmod 5\,$</span> whose action is as follows.</p>
<p><span class="math-container">$\qquad\qquad\bmod 5\!:\,\ \begin{al... |
3,892,668 | <p>Find the extreme values of <span class="math-container">$f(x,y)=x^2y$</span> in <span class="math-container">$D=\{x^2+8y^2\leq24\}$</span></p>
<p>It was easy to find using Lagrange multipliers the local extreme values on <span class="math-container">$\partial{D}$</span> since we have the condition <span class="math-... | alans | 80,264 | <p><span class="math-container">$24\geq x^2+8y^2=\frac{x^2}{2}+\frac{x^2}{2}+8y^2\geq 3\sqrt[3]{2x^4y^2}=3\sqrt[3]{2}(x^2y)^{\frac{2}{3}}$</span> by AM-GM inequality. Maximal value for <span class="math-container">$x^2y$</span> is attained for <span class="math-container">$x=4y=4$</span> and it is <span class="math-con... |
2,599,982 | <p>Can anybody help me finding a good way to (approximately) figure out the first, lets say $200$, <strong>positive</strong> roots of $$\tan(x) + 2 \ell x - \ell ^2 x^2 \tan(x) = 0,$$
where $\ell$ is just a constant?</p>
<p>I believe there will be no analytic expression, so is there a better idea than just running <e... | Michael Behrend | 391,459 | <p>Set $x = 2u$; then it comes down to solving $2\ell u = -\tan u$ or $\cot u$. The graphs for $-\tan u$ and $\cot u$ are parallel curves at intervals of $\pi/2$ in $u$. This may or may not help.</p>
|
75,777 | <p>I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is uniformly distributed on the unit interval. With "elementary" I mean that it does not make use of complex analysis in partic... | Denis Chaperon de Lauzières | 20,038 | <p>Hardy and Wright give a proof based on continued fractions in <a href="https://books.google.com/books?id=P6uTBqOa3T4C&pg=PA520" rel="nofollow noreferrer">Section 23.10</a> of "An introduction to the theory of numbers" (reference valid at least in the 4th edition). </p>
<p>Interestingly, their <a href="https://... |
75,777 | <p>I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is uniformly distributed on the unit interval. With "elementary" I mean that it does not make use of complex analysis in partic... | David E Speyer | 297 | <p>There is a really easy proof that just uses the Pigeonhole principle. Let <span class="math-container">$\alpha$</span> be irrational.</p>
<p><b>Lemma:</b> For any <span class="math-container">$\delta >0$</span>, there is an <span class="math-container">$n>0$</span> such that <span class="math-container">$(n \a... |
152,949 | <p>I noticed that a factorization over algebraic fields is useless in Mathematica. Here is the example over the field containing I*Sqrt[3]:</p>
<pre><code>Pol=4 (3 I Sqrt[3] (-12 + 6 x - 4 x^2 + x^3) y^3 z +
9 (12 - 12 x + 12 x^2 - 6 x^3 + x^4) y^4 z^2 +
I Sqrt[3] (8 + x^3) y z^3 + (4 - 2 x + x^2)^2 z^6 -
3 y^2 (4 ... | Adam Strzebonski | 6,258 | <p>It is an input processing issue. Factor uses the extension generated by the algebraic numbers specified through the Extension option and algebraic numbers it finds in the coefficients of the input polynomial. Here it finds Sqrt[3] and I separately, so it uses the degree 4 extension generated by Sqrt[-3], Sqrt[3], an... |
2,718,302 | <blockquote>
<p>Show that the function $f(x)=\begin{cases}\frac{e^{1/x}-1}{e^{1/x}+1}, \:x\neq0\\0, \quad\:\:\: x=0\end{cases}$ is discontinuous at $x=0$.</p>
</blockquote>
<p><strong>My Attempt</strong></p>
<p>From the graph of the function it is clearly discontinuous at $x=0$, <a href="https://i.stack.imgur.com/f... | marty cohen | 13,079 | <p>$\dfrac{e^{1/x}-1}{e^{1/x}+1}
=\dfrac{e^{1/x}+1-1-1}{e^{1/x}+1}
=1-\dfrac{2}{e^{1/x}+1}
$
and
$\lim_{x \to 0+} \dfrac{2}{e^{1/x}+1}
=0
$
and
$\lim_{x \to 0-} \dfrac{2}{e^{1/x}+1}
=2
$.</p>
|
270,600 | <p>I'm a software engineer with math classes through differential equations about 15 years in my past, and I've gotten stuck trying to invert an equation.</p>
<p>The equation:
<span class="math-container">$y = x + (0.022 - x)^{1.414}$</span>.</p>
<p>In Mathematica form:</p>
<pre><code>sapcClamp[y] := y + ((22 / 100) - ... | Daniel Huber | 46,318 | <p>Instead of worrying about an analytical solution, it can be done numerically by an interpolating function:</p>
<pre><code>y[x_] = x + (22/1000 - x)^(1414/1000);
dat = Table[Reverse@{x, y[x]}, {x, 0, 0.022, 0.022/100}];
ifun = Interpolation[dat];
Plot[ifun[x], {x, y[0], 0.022}]
</code></pre>
<p><a href="https://i.sta... |
917,827 | <p>How do i simplify $\arccos(x)−\arcsin(x)$ for $x$ in $(−1,1)$</p>
<p>i got somewhere that...</p>
<p>$\sin(x)= \cos(\frac{\pi}{2}-x)$
so $\arccos(\sin(x))+x=\frac{\pi}{2}$</p>
<p>substituting that $\sin(x)=t \rightarrow \arcsin(t)=x$</p>
<p>$\arccos(t)+\arcsin(t)=\frac{\pi}{2}$</p>
<p>so working backwards
$\arc... | Robert Israel | 8,508 | <p>$\arccos(x)$ is the angle $\theta$ with $\cos(\theta) = x$ and $0 \le \theta \le \pi$, while $\arcsin(x)$ is the angle $\phi$ with $\sin(\phi) = x$ and $-\pi/2 \le \phi \le \pi/2$. Now $\sin(\theta) = \cos(\pi/2 - \theta)$ and $-\pi/2 \le \pi/2 - \theta \le \pi/2$, so $\pi/2 - \theta = \phi$. Thus $$\arccos(x) - ... |
1,393,694 | <p>I am having a rather tough time wrapping my head around any possible logical fallacy in my solution to the following question as my answer is wrong:</p>
<p>1 percent of children have autism. A test for autism is developed such that 90% of autistic children are correctly identified as having autism but 3% of non-aut... | Alex R. | 22,064 | <p>This is one of those "beating-a-dead-horse" questions. </p>
<p>A <em>function</em> is something that takes an input and produces <em>one</em> output. The squareroot function $f(x)=\sqrt{x}$ with $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ is a function because for every $x$ it gives <em>one</em> output. $f(4)=2$. If $... |
41,774 | <p>Let $x$ and $y$ be nonzero integers and $\mathrm{ord}_p(w)$ be the multiplicative order of $w$ in $ \mathbb{Z} /p \mathbb{Z} $. If $\mathrm{ord}_p(x) = \mathrm{ord}_p(y)$ for all primes (<strong>Edit:</strong> not dividing $x$ or $y$), does this imply $x=y$? </p>
| joriki | 6,622 | <p>[This is an answer to the original form of the question. In the meantime the question has been clarified to refer to the multiplicative order; this seems like a much more interesting and potentially difficult question, though I'm pretty sure the answer must be yes.]</p>
<p>I may be missing something, but it seems t... |
41,774 | <p>Let $x$ and $y$ be nonzero integers and $\mathrm{ord}_p(w)$ be the multiplicative order of $w$ in $ \mathbb{Z} /p \mathbb{Z} $. If $\mathrm{ord}_p(x) = \mathrm{ord}_p(y)$ for all primes (<strong>Edit:</strong> not dividing $x$ or $y$), does this imply $x=y$? </p>
| Zarrax | 3,035 | <p>A related fact: integers are determined by their remainders modulo all the primes: </p>
<p>Let $p_1 < p_2 < ...$ be a list of distinct primes. Then (via the Chinese remainder theorem for example) $Z_{p_1 \times ... \times p_n}$ is isomorphic to $Z_{p_1} \times .... \times Z_{p_n}$ via the map
$$a\pmod {p_1..... |
41,774 | <p>Let $x$ and $y$ be nonzero integers and $\mathrm{ord}_p(w)$ be the multiplicative order of $w$ in $ \mathbb{Z} /p \mathbb{Z} $. If $\mathrm{ord}_p(x) = \mathrm{ord}_p(y)$ for all primes (<strong>Edit:</strong> not dividing $x$ or $y$), does this imply $x=y$? </p>
| André Nicolas | 6,312 | <p><strong>Note:</strong> The material below makes only a small amount of progress towards a solution of the problem.</p>
<p>Qiaochu Yuan remarked that if the order of $x$ is equal to the order of $y$ modulo every prime $p$ that divides neither $x$ nor $y$, then the square-free parts of $x$ and $y$ must be the same. ... |
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... | Bryan Bugyi | 413,609 | <p>1) The correct answer is </p>
<p>$$P(1029,10) = 1029 \times 1028\times \cdots \times 1020$$</p>
<p>because order <em>does</em> in fact matter. The order in which the the disk jockey plays the songs does make a difference.
<br><br></p>
<p>2) For this question, it is important to understand that ABC and ABCD are en... |
4,563,516 | <h1>Question</h1>
<p>Evaluate <span class="math-container">$$\int_0^1 \cos^{-1} x\ dx$$</span> by first finding the value of <span class="math-container">$$\frac{d}{dx}(x\cos^{-1} x).$$</span></p>
<h1>My Working</h1>
<p>As the question said to evaluate <span class="math-container">$$\frac{d}{dx}(x\cos^{-1} x),$$</span>... | Cheese Cake | 947,878 | <p>After seeing the helpful comments, I can start using <span class="math-container">$u$</span>-substitution for the solution. Let <span class="math-container">$u=1-x^2$</span>, then</p>
<p><span class="math-container">\begin{align}
\frac{du}{dx}&=-2x\\
du&=-2x\ dx\\
\int_0^1\frac{x}{\sqrt{1-x^2}}&=-\frac{1... |
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