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 |
|---|---|---|---|---|
3,165,494 | <blockquote>
<p>Consider the function <span class="math-container">$\phi(t,x)$</span> where <span class="math-container">$x=(x_1,x_2) \in \mathbb{R}^2$</span>. How can we check if <span class="math-container">$\phi(t,x)$</span> is a flow for a vector field? <span class="math-container">$$\phi(t,x)=(x_1\cos(r^2t)+x_2\si... | Lutz Lehmann | 115,115 | <p>If you write <span class="math-container">$\zeta(t,z)=ϕ_1(t,x)+iϕ_2(t,x)$</span> where <span class="math-container">$z=x_1+ix_2$</span>, you will find that
<span class="math-container">$$
\zeta(t,z)=e^{-i|z|^2t}z
$$</span>
and as a consequence <span class="math-container">$|\zeta(t,z)|=|z|$</span> constant along the... |
2,702,329 | <blockquote>
<p>Let $A$ be an $m\times n$ matrix, let $v \in \mathbb{R}^m$, and let $S$ be the set of least squares solutions to the system $Ax=b$. Show that there exists a unique <em>minimal least square solution</em> to $Ax=b$; that is, there exists some $x\in S$ such that $||x|| \le||y||$ for all $y\in S$.</p>
</b... | copper.hat | 27,978 | <p>Since $x \mapsto \|Ax-b\|^2$ is convex, the set of minimisers $S$ is a closed, convex set.</p>
<p>(In particular, if $x_0$ is a solution, then $S = \{x_0\} + \ker A$.)</p>
<p>A standard Hilbert space theorem states that a closed convex set has a unique point of minimum norm.</p>
<p>The $x$ is the question is the ... |
1,501,736 | <p>Let $x_1 = 1$ and for $n \ge2, x_n = \sqrt{3 + x_{n-1}}$, I just want to know how to determine the limit. </p>
| Jack D'Aurizio | 44,121 | <p><strong>Hint:</strong> prove that your sequence is increasing and bounded, hence converging to its supremum, then notice that there is only one real number $r$ such that $r=\sqrt{r+3}$.</p>
|
477,250 | <p>Let $u=\{u_\alpha\}$ be an open cover of $[a,b], a <b $. Let $S=\{r \in [a,b]$ such that $[a,r]$ is covered by some finite collection of open sets belonging to $u\}$. Is $S$ non-empty ? How ?</p>
| Gerry Myerson | 8,269 | <p>Let $v=(1,1,1,1,1)$. Note $Av=v$. Deduce $A^{-1}v=v$. What does this say about row sums of $A^{-1}$?</p>
|
1,607,108 | <p>Using the ratio test:
$$\frac{1}{3}\lim\limits_{n\to\infty}\left|\frac{(n+1)^3(\sqrt{2}+(-1)^{n+1})^{n+1}}{n^3(\sqrt{2}+(-1)^n)^n}\right|$$</p>
<p>Without evaluating the limit, the numerator is greater then denominator and the series is divergent. Is there an easier method for checking the convergence of this seque... | Community | -1 | <p>To answer your question without resorting to any explanation beyond what you have provided:</p>
<p>No. This part is seriously off "Assume 2|x−1 and 4|x+1" without any explanation. When you say "assume this", "assume that", you have to cover all options.</p>
<p>2 is prime so it must divide either (x+1) or (x-1), th... |
11,435 | <p><strong>Question</strong>: What is the correct notion of a <em>product</em> of integral (or rational) polytopes which induces a factorization of its Ehrhart (quasi-)polynomial into two primitive Ehrhart (quasi-)polynomials corresponding to its constituent polytopes, viz., $L_{P \times Q}(t) = L_{P}(t) L_{Q}(t)$? </p... | Dan Moore | 1,782 | <p>The (Motivation) description appears to be similar to a direct sum (a.k.a. tegum product) as defined in Section 1.2 of this paper:</p>
<p><a href="http://people.reed.edu/~davidp/homepage/seniors/mcmahan.pdf" rel="nofollow">http://people.reed.edu/~davidp/homepage/seniors/mcmahan.pdf</a></p>
<p>The difference is tha... |
2,118,230 | <blockquote>
<p>Let $G$ be a group and let $H,K \le G$ be subgroups of finite index, say $|G:H| = m$ and $|G:K|=n$. Prove that $\mathrm{lcm}(m,n) \le |G:H \cap K| \le mn$. </p>
</blockquote>
<p>I was able to establish the upper bound, but I am having difficulty establishing the lower bound, so I consulted <a href="h... | Andrew Chiriac | 172,569 | <p>The theorem you are looking for is the following: If $G_1\geq G_2\geq G_3$, then $[G_1:G_3]=[G_1:G_2][G_2:G_3]$. The proof of this theorem can be found in most introductory texts on abstract algebra. </p>
<p>Recall that the least common multiple of two integers $a$ and $b$ is the smallest positive integer that is d... |
1,129,175 | <p>In my assignment I have to find a Jordan normal form for this matrix:</p>
<p><img src="https://i.stack.imgur.com/8yVcy.jpg" alt="Original Matrix"></p>
<p><img src="https://i.stack.imgur.com/eor4m.jpg" alt="Jordan Normal Form">
<img src="https://i.stack.imgur.com/wEPtW.jpg" alt="Nilpotent">
<img src="https://i.stac... | Peter | 82,961 | <p>Hint : The Caley-hamilton-theorem states that a matrix with characteristic equation
$$a_n\lambda^n+...+a_1\lambda+a_0=0$$
satisfies the equation
$$a_nA^n+...+a_1A+a_0I=0$$</p>
<p>Also note, that A can be written as AI.</p>
<p>Now, do you see how $A^{-1}$ can be expressed by powers of $A$ ,... |
1,129,175 | <p>In my assignment I have to find a Jordan normal form for this matrix:</p>
<p><img src="https://i.stack.imgur.com/8yVcy.jpg" alt="Original Matrix"></p>
<p><img src="https://i.stack.imgur.com/eor4m.jpg" alt="Jordan Normal Form">
<img src="https://i.stack.imgur.com/wEPtW.jpg" alt="Nilpotent">
<img src="https://i.stac... | Karl | 203,893 | <p>I had a lot of trouble getting my head around this myself and apologise if anything in my answer is incorrect.</p>
<p>Consider the matrix expression
$$\begin{bmatrix}
3&6\\
1&2
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}$$
I can take a factor of 3 out of the 1st row
$$
3\begin{bmatrix}
1&2\\
1&am... |
2,617,555 | <p>So the cost per hour of running a cruiser is $\$ \left(\frac {V^2}{40} + 10\right)$, where $V $is the speed in knots. So I’ve answered the first question showing the cost would be $\$\frac DV \left(\frac {V^2}{40} + 10\right)$. And then they asked me to find the most economical speed for running the cruiser, and I h... | Redsbefall | 97,835 | <p>Let your cost function be $C(V)$, which is dependent on $V$, the velocity of the cruise.
$$ C(V) = \frac{D}{V}\left( \frac{V^{2}}{40} + 10 \right) $$
with $D$ as a constant. </p>
<p><em>The most economical velocity</em> is the velocity that makes the <em>smallest cost</em>.</p>
<p>Since we already have the cost ... |
3,612,316 | <p>I am trying to interpret an equation, but can't understand how the less-than signs work:</p>
<p><span class="math-container">$\sum _{n=1}^{\infty } \frac{1}{n^2}<1+\sum _{n=1}^{\infty } \frac{1}{n (n+1)}=1+\sum _{n=1}^{\infty } \left(\frac{1}{n}-\frac{1}{n+1)}\right)=1+1=2$</span></p>
<p>The first part returns ... | CHAMSI | 758,100 | <p>Since <span class="math-container">$ \left(\forall n\geq 2\right),\ n^{2}\geq n\left(n-1\right) $</span>, we get that <span class="math-container">$ \left(\forall n\geq 2\right),\ \frac{1}{n^{2}}\leq\frac{1}{n\left(n-1\right)} $</span>, thus : <span class="math-container">$$ \sum_{n=1}^{+\infty}{\frac{1}{n^{2}}}=1+\... |
1,679,383 | <p>Find numbers $\alpha,\beta, \gamma \in \mathbb{C}$ so that the integral $$\int_{-1}^{1}|x^3-\alpha-\beta x-\gamma x^2|^2dx$$ is minimal.
Here I want to apply Hilbert space and projection lemma by considering the integral as the distance square between a point and a closed linear subspace. So how can I start my work?... | Disintegrating By Parts | 112,478 | <p>If $\mathcal{M}$ is the subspace of polynomials $p(x)=\alpha + \beta x + \gamma x^2$, then you are looking for $p\in \mathcal{M}$ closest to $x^3$. Equivalently, $(x^3-p)\perp\mathcal{M}$, which gives equations
$$
\int_{-1}^{1}(x^3-\alpha-\beta x-\gamma x^2)dx =0 \\
\int_{-1}^{1}(x^3-\alpha-\beta x-\ga... |
1,600,266 | <p>I'm trying to work out sum of this series $$1 + \frac{2}{2} + \frac{3}{2^2} + \frac{4}{2^3} + \ldots$$</p>
<p>I know one method is to do substitutions and getting the series into a form of a known series. So far I've converted the series into $$ 1 + \frac{2}{x} + \frac{3}{x^2} + \frac{4}{x^3} + \ldots $$ where $x=... | fleablood | 280,126 | <p>Okay. I found the text online and this is my interpretation.</p>
<p>However the formatting was hard to read, a page or two seems to be missing, and it's a subject I have never studied not even an iota....</p>
<p>but... I think</p>
<p>He's talking about writing a string parser. You give it a binary string and t... |
835,454 | <p>I was struggling with the following problem (from linear algebra):</p>
<blockquote>
<p>Let $V$ be the vector space of the $2 \times 2$ matrices with real coefficients. Consider the action of the group $SL_2(\mathbb{R})$ on $V$, namely for any matrix $T \in SL_2(\mathbb{R})$ and for every matrx $A \in V$ we define... | Olivier Bégassat | 11,258 | <p>Let $\Bbb k$ be a field, and let $V$ be a finite dimensional $\Bbbk$-vector space (of dimension $n$). The special linear group $SL(V)$ is generated <a href="http://en.wikipedia.org/wiki/Shear_mapping" rel="nofollow">shear mappings</a>. All shear mappings are conjugate in $GL(V)$, so that the linear maps "conjugation... |
2,598,848 | <p>I have a project I'm doing in desmos and want to know if this is possible.</p>
<p>I want to be able to set some parameter, a, equal to the solution of f(x) = 0. I plugged my specific equation into Maple and it could not arrive at an explicit solution for x.</p>
<p>However, when I type in the equation f(x) = 0 into... | Doug M | 317,162 | <p>The figure will be symmetric across the line $x = y$ (along with $x = 0$ and $y = 0$)</p>
<p>We can analyze $\frac 18$ the figure, and then use this symmetry to our advantage.</p>
<p>Suppose $a<b$</p>
<p>$\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1$ will be on the "inside" from $0$ to the point of intersection.</p... |
2,278,120 | <p>In one of my algorithm courses, there is this: </p>
<blockquote>
<p>A subset $S$ of vertices in a directed graph $G$ is
strongly connected if for each pair of distinct
vertices ($v_i$, $v_j$) in $S$, $v_i$ is connected to $v_j$ and
$v_j$ is connected to $v_i$.</p>
</blockquote>
<p>And then the following ex... | Matthew Leingang | 2,785 | <p>From your subset $S$ of vertices you can construct a set of pairs $P$, where $(v,w) \in P$ if there is a directed path from $v$ to $w$ (not necessarily confined to $S$). The set is strongly connected if
$$
\forall v,w\in S, ( v\neq w \Rightarrow (v,w) \in P \wedge (w,v) \in P)
$$
Let $S=G$ in the example graph.... |
195,432 | <p>Mathematica 12 has a new function <code>FunctionCompileExportLibrary</code>, which can export a pure function as a <code>.dll</code> file like this:</p>
<pre><code>FunctionCompileExportLibrary["function.dll", Function[
Typed[arg, "MachineInteger"], arg + 1]]
</code></pre>
<blockquote>
<p><code>"...\\function.... | sunt05 | 260 | <p>This is a follow-up of <a href="https://mathematica.stackexchange.com/a/195557/260">@ilian's answer</a>, which doesn't work under macOS but provides a promising basis. </p>
<p>The error reported by <a href="https://mathematica.stackexchange.com/questions/195432/how-to-use-the-library-exported-from-functioncompileex... |
244,048 | <p>A couple days ago I posted this on MSE (<a href="https://math.stackexchange.com/questions/1853774/determining-a-function-is-harmonic-from-mean-value-property-for-just-three-ra">here</a>) but in retrospect it might be more appropriate for this site.</p>
<p>This theorem is well-known (maybe it can be called Morera's ... | user111 | 89,429 | <p>I guess this is the theorem :</p>
<p>Let $f$ be an infinitely differentiable function defined in $\mathbb R^{n}$, and $u(x,r)$ the mean value of $f$ taken over the sphere with center at $x$ and radius $r$, and let $a$ and $b$ denote two fixed positive numbers. If $u(x,a)=u(x,b)=f(x)$ in $\mathbb R^{n}$, then $f$ is... |
116,494 | <p>I am trying to plot contour plots and some of them show a white patch because of a change in sign of the points plotted. How can I make the contour plot show this change in sign without having to take the absolute of the values?</p>
<p>The data is the following:</p>
<pre><code>axialP1plot= {{30, 0, 0.509185}, {60,... | m_goldberg | 3,066 | <p>As george2079 say in his comment to your question, add the option <code>PlotRange -> All</code></p>
<pre><code>ListContourPlot[axialP1plot,
PlotRange -> All,
ColorFunction -> "Rainbow",
Contours -> 10,
PlotLegends -> Automatic,
PlotLabel -> "Axial Force on Plate 1"]
</code></pre>
<p><a ... |
3,827,302 | <p>In part of Kenneth Falconer's proof of the countable stability of Hausdorff dimension on p. 49, sect 3.2 of <a href="https://www.wiley.com/en-us/Fractal+Geometry%3A+Mathematical+Foundations+and+Applications%2C+3rd+Edition-p-9781119942399" rel="nofollow noreferrer">Fractal Geometry</a>, I understand him to say that
<... | Kevin Arlin | 31,228 | <p>"Composability" and "invertibility" are not, as you've noted, really the relevant primitive notions in a quasicategory. But horn-filling accounts for all the possibilities you want. The way to make this all make sense is to consider your quasicategory as generalizing the nerve of a 2-category. Gi... |
3,411,982 | <p>I know the answer using
Total-none of it is <span class="math-container">$6 = 900−648=252$</span></p>
<p>My doubt is that if we you it like</p>
<p>when unit digit is <span class="math-container">$6 = 9\cdot10\cdot1 = 90$</span></p>
<p>when tens place is <span class="math-container">$6 = 9\cdot1\cdot10 = 90$</span... | Community | -1 | <p>If exactly one digit is a <span class="math-container">$6,$</span> there are <span class="math-container">$1\cdot 9\cdot 9+8\cdot1 \cdot 9+8\cdot 9 \cdot 1=225$</span> possibilities.</p>
<p>If exactly two digits are a <span class="math-container">$6,$</span> there are <span class="math-container">$9+9+8=26$</span> ... |
1,007,529 | <p>Suppose that the function $f:[-1,1]\rightarrow[-1,1]$ is continuous. Use the intermediate value theorem to prove that there exists a $c \in [-1,1]$ such that $f(c)=c^5$. You should carefully justify each of the hypothesis of the theorem. Question is to be done very formally.</p>
<p>First can I just say, shouldn't $... | Yiorgos S. Smyrlis | 57,021 | <p>Take $g(x)=f(x)-x^5$. </p>
<p>Then $g(-1)=f(-1)+1\ge 0$, while $g(1)=f(1)-1\le 0$. Using the Intermediate Value Theorem for $g$, we obtain that there exists a $c\in[-1,1]$, such that
$$
f(c)=c^5.
$$</p>
|
1,557,206 | <p>Given the series $\sum\limits_{n=1}^\infty a_n^2$ and $\sum\limits_{n=1}^\infty b_n^2$ converge. Show that the series $\sum\limits_{n=1}^\infty a_n b_n$ converges absolutely.</p>
<p>My idea so far:</p>
<ul>
<li>It's quite quite obvious that both given series converge absolutely</li>
<li>So the Cauchy-Produc tells ... | Michael Hardy | 11,667 | <p>\begin{align}
0 & \le (a+b)^2 = a^2 + b^2 + 2ab \\
0 & \le (a-b)^2 = a^2 + b^2 - 2ab \\[10pt]
\text{Therefore} \\
-2ab & \le a^2+b^2, \\
2ab & \le a^2 + b^2, \\[10pt]
\text{and consequently} \\
2|ab| & \le a^2 + b^2.
\end{align}</p>
<p>So
$$
\sum_n |a_n b_n| \le \frac 1 2 \left( \sum_n a_n^2 + \... |
2,014,806 | <p>I was hoping to get some help on a review problem from my text book. The problem reads: </p>
<p>Let $A$ be $m\times{n}$ matrix. Prove that</p>
<p>a) $A^*A$ is self-adjoint.</p>
<p>b) All eigenvalues of $A^*A$ are non-negative.</p>
<p>c) $A^*A + I$ is invertible.</p>
<p>For part a), it seems to follow very clear... | Robert Lewis | 67,071 | <p>For part (b), if</p>
<p><span class="math-container">$A^\dagger A \vec v = \mu \vec v, \; \vec v \ne 0, \tag 1$</span></p>
<p>then</p>
<p><span class="math-container">$\mu \langle \vec v, \vec v \rangle = \langle \vec v, \mu \vec v \rangle = \langle \vec v, A^\dagger A \vec v \rangle = \langle A\vec v, A \vec... |
2,278,436 | <p>Does there exist integers $a, n > 1$ such that $1 + \frac{1}{1 + a} + \frac{1}{1 + 2a} + ... + \frac{1}{1 + na}$ is an integer? I have no clue how to begin. I've tried to simplify this somehow, but with no effect.</p>
| Robert Israel | 8,508 | <p>If there is a prime (or, for that matter a prime power) dividing only one of the denominators, then the sum is not an integer. Now by the prime number theorem in arithmetic progressions, for any fixed $a$ this holds for sufficiently large $n$, as there will be a prime $p = 1 + k a$ for some $k$ with $n/2 < k \le ... |
868,663 | <blockquote>
<p>Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer.</p>
</blockquote>
<p>I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.</p>
| Adriano | 76,987 | <p>Suppose we add $372$ and $594$. Using "the addition method that we know", we would first add $2$ and $4$ to get $6$. Then we would add $7$ and $9$ to get $16$, so we write down a $6$ then carry the $1$. Then we add the carried $1$ with the $3$ and $5$ to get $9$, for a final answer of $966$. Why does this work?</p>
... |
3,772 | <p>In <a href="https://math.stackexchange.com/questions/102764/non-polynomial-functions-on-fields-of-finite-characteristic">this</a> question, there are two answers, both of which I up-voted, but I didn't "accept" either since there didn't appear to be a reason why one stands out as clearly preferable to the other. Ap... | Willie Wong | 1,543 | <p>Firstly, I fully endorse Rahul's comment. <code>:-)</code></p>
<p>Secondly, let me explain, or at least give my point of view, on why some users are getting worked up about the acceptance rate. There are two main ways of user-interaction which can be used to express gratitude for providing an answer beyond a simple... |
44,310 | <p>I wanted to draw (lattice-aligned) unit cubes inside a sphere with a given radius centered on the origin, so I wrote this program:</p>
<pre><code>Cube[x_, y_, z_] := Cuboid[{x - 1, y - 1, z - 1}, {x, y, z}];
Coords[r_] := {#1, #2, Floor[Sqrt[r^2 - #1^2 - #2^2]]} & @@ # & /@
With[{t = Sqrt[r^2 - 1]},
... | Michael E2 | 4,999 | <p><a href="http://reference.wolfram.com/mathematica/ref/Translate.html" rel="nofollow noreferrer"><code>Translate</code></a> is probably the most efficient way to represent and display such a figure.</p>
<pre><code>With[{r = 30},
Graphics3D[{
Translate[Cuboid[],
Union @@ (Permute[#, SymmetricGroup[3]] & /... |
1,016,585 | <p>I want to solve $$\int \frac{1}{\sqrt{x^2 - c}} dx\quad\quad\text{c is a constant}$$</p>
<p>How do I do this?</p>
<p>It looks like it is close to being an $\operatorname{arcsin}$?</p>
<hr>
<p>I would have thought I could just do:
$$\int \left(\sqrt{x^2 - c}\right)^{-\frac12}\, dx=\frac{2\sqrt{c+x^2}}{2x}\text{??... | k170 | 161,538 | <p>$$ \int \frac{1}{\sqrt{x^2-c}}dx = \int \frac{1}{\sqrt{c\left(\frac{x^2}{c}-1\right)}}dx=\frac{1}{\sqrt{c}} \int \frac{1}{\sqrt{\left(\frac{x}{\sqrt{c}}\right)^2-1}}dx $$
Let $u=\frac{x}{\sqrt{c}}$, then $du=\frac{1}{\sqrt{c}} dx$. So now we have
$$ \int \frac{1}{\sqrt{u^2-1}}du =\mathrm{arcosh}(u)+C = \mathrm{arcos... |
4,365,859 | <p>My professor had given us a problem. It goes like -</p>
<blockquote>
<p>Find equation of circle passing through intersection of circle <span class="math-container">$S=x^2+y^2-12x-4y-10=0$</span> and <span class="math-container">$L:3x+y=10$</span> and having radius equal to that of circle <span class="math-container"... | Cye Waldman | 424,641 | <p>In spite of your obfuscating figure, you are asking for the surface area of a torus whose inner radius, <span class="math-container">$R$</span> (to the center of the cross-section) and outer radius, <span class="math-container">$r$</span> (that <em>of</em> the cross-section) are the same. This is well known to be <s... |
4,365,859 | <p>My professor had given us a problem. It goes like -</p>
<blockquote>
<p>Find equation of circle passing through intersection of circle <span class="math-container">$S=x^2+y^2-12x-4y-10=0$</span> and <span class="math-container">$L:3x+y=10$</span> and having radius equal to that of circle <span class="math-container"... | robjohn | 13,854 | <h3>Comment on the Question</h3>
<p>I believe that this is the standard setup for surface of revolution about the <span class="math-container">$x$</span>-axis
<span class="math-container">$$
\int_{-a}^a\overbrace{\quad2\pi y\quad\vphantom{\frac{a}{\sqrt{a^2}}}}^{\substack{\text{account for}\\\text{revolution}}}\overbra... |
3,949,872 | <p>I am to expand <span class="math-container">$\log_3\frac{7x^2+21x}{7x(x-1)(x-2)}$</span> using the quotient rule of logs. I arrived at:</p>
<p><span class="math-container">$$\log_3(7)+\log_3(4)+\log_3(x)-\log_3(x-1)-\log_3(x-2)$$</span></p>
<p>Whereas my textbook solution is:</p>
<p><span class="math-container">$$\l... | 19aksh | 668,124 | <p>First isolate <span class="math-container">$7x$</span> from the numerator. You then have <span class="math-container">$\log_3\left[\frac{7x(x+3)}{7x(x-1)(x-2)}\right] = \log_3\left[\frac{(x+3)}{(x-1)(x-2)}\right] _{(x\ne0)} = \log_3(x+3)-\log_3(x-1)-\log_3(x-2)$</span></p>
|
3,949,872 | <p>I am to expand <span class="math-container">$\log_3\frac{7x^2+21x}{7x(x-1)(x-2)}$</span> using the quotient rule of logs. I arrived at:</p>
<p><span class="math-container">$$\log_3(7)+\log_3(4)+\log_3(x)-\log_3(x-1)-\log_3(x-2)$$</span></p>
<p>Whereas my textbook solution is:</p>
<p><span class="math-container">$$\l... | CACM6 | 814,401 | <p>You can factor the numerator and denominator, both have a common factor <span class="math-container">$7x$</span>, so you have
<span class="math-container">$$\frac{7x^2+21x}{7x(x-1)(x-2)}=\frac{x+3}{(x-1)(x-2)},$$</span> now you can use the <span class="math-container">$\log$</span> rules and get the answer that the ... |
1,531,525 | <p>I would like to compute the following sum:</p>
<p>$$\sum_{n=0}^{\infty} \frac{\cos n\theta}{2^n}$$</p>
<p>I know that it involves using complex numbers, although I'm not sure how exactly I'm supposed to do so. I tried using the fact that $\cos \theta = {e^{i\theta} + e^{-i\theta}\over 2}$. I'm not sure how to proc... | 2'5 9'2 | 11,123 | <p>If you are only looking for a hint, write your sum as $$\sum_{n=0}^\infty\frac{e^{in\theta}+e^{-in\theta}}{2\cdot2^n}$$ Break it up as two sums, each of which are geometric, so you can use the geometric series formula.</p>
|
209,259 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/16374/universal-chord-theorem">Universal Chord Theorem</a> </p>
</blockquote>
<p>I am having a problem with this exercise. Could someone help?</p>
<p>Suppose $a \in (0,1)$ is a real number which is not of... | fgp | 42,986 | <p>Look at $f(x) = \sin(2\pi x)$. For which values of $a$ can you find an $x \in [0,1]$ with $x+a \in [0,1]$ and $f(x) = f(x+a)$? In particular, if you additionally require $a > \frac{1}{2}$, can such an $a$ exist at all?</p>
<p>Once you've answered that you've solved your problem for some values of $a$. Which are ... |
224,474 | <p>I want to generate the following matrix for any n:</p>
<pre><code> Table[A[i, j], {i, 0, n}, {j, 0, n}];
</code></pre>
<p>Where,</p>
<pre><code> A ={{0,1,0},{0,0,2},{0,0,0}} when n=2;
A={{0,1,0,0},{0,0,2,0},{0,0,0,3},{0,0,0,0}} when n=3;
A={{0,1,0,0,0},{0,0,2,0,0},{0,0,0,3,0},{0,0,0,0,4},{0,... | Pillsy | 531 | <p>I sometimes like using <code>Module</code> to define local functions (more precisely, local symbols with <code>DownValues</code>) to use the ridiculous generality Mathematica provides for such definitions:</p>
<pre><code>ClearAll[specialMatrix];
specialMatrix[n_Integer?Positive] := specialMatrix[n] =
Module[{a, be... |
199,946 | <p>I came across this step in an inductive proof, but my algebra skills seem a bit rusty..</p>
<p>$7(7^{k+2}) + 64(8^{2k+1}) = 7(7^{k+2}+8^{2k+1})+57(8^{2k+1})$</p>
<p>How did they do this?</p>
<p>Note: The point was to show that the expressions are divisible by 57.</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ With $\rm\: A = 7^{\,k+2}\:$ and $\rm\: B = 8^{\,2k+1}\:$ it is more easy to comprehend</p>
<p>$$\rm 7\, A + 64\ B\ =\ 7\,(A + B) + 57\ B $$</p>
|
1,221,206 | <p>In this computer, numbers are stored in <span class="math-container">$12$</span>-bits. We will also assume
that for a floating point (real) number, <span class="math-container">$6$</span> bits of these bits are reserved for
the mantissa (or significand) with <span class="math-container">$2^{k-1}-1$</span> as the exp... | Rellek | 228,621 | <p>In a summation, any constant can be pulled out (your constant is 3). If you do this, you will see that this is a geometric series. This means that each term only differs by a constant from the preceding term. In this case it is easy to see that this constant will be 1/4. Note that $$\frac{1}{4^n} = \frac{1^n}{4^n} =... |
388,875 | <p>For $V$ a vector space.</p>
<p>Let $T\colon V\rightarrow V$ be a linear transformation of a finite dimensional inner product space.</p>
<p>How can I prove that $\ker(T^* T + TT^*) = $\ker(T)$\cap$$\ker(T^*)$?</p>
<p>And what is the meaning of $\ker(T^*T + TT^*)$ ?</p>
<p>Is $\ker(T^*T + TT^*)=\ker(T^*T)+\ker(TT... | Hagen von Eitzen | 39,174 | <p>$\ker(T^*T+TT^*)$ is the set of all $x\in V$ such that $$(T^*T+TT^*)x=0, $$
that is
$$ T^*Tx+TT^*x=0.$$
This is not the same as the set of all vectors $x$ that can writtten as $x=y+z$ with $T^*Ty=0$ and $TT^*z=0$. I.e., in general $\ker(T^*T+TT^*)\ne\ker(T^*T)+\ker(TT^*)$.</p>
|
497,609 | <p>There are 30 balls in a bag. There are 6 balls each of 5 different colors. What is the minimum number of times I have to pick a ball to pick 2 balls of each color?</p>
<ol>
<li>24</li>
<li>25</li>
<li>26</li>
<li>27</li>
<li>28</li>
</ol>
<p>Is the question correct? Someone asks me this and I'm confused.</p>
| obataku | 54,050 | <p>What we're counting is the minimum number of balls we could blindly pick yet be entirely sure we have two of each color. For example, liken it to picking socks blindly from a drawer -- we want to determine how many socks we have to pull out blindly to ensure we have a pair. </p>
<p>Think of the worst case scenario,... |
1,256,688 | <p>Given a sequence of real numbers, a move consists of choosing two terms and replacing each with their arithmetic mean. Show that there exists a sequence of 2015 distinct real numbers such that after one initial move is applied to the sequence — no matter what move — there is always a way to continue with a finite se... | futurebird | 2,563 | <p><strong>(1) Suppose that I want to prove that if the statements A, B,C hold true, then Z holds.</strong> </p>
<p>So you want to prove this:</p>
<p>(A and B and C) ==> z</p>
<p>Here is the truth table: </p>
<pre><code>(A and B and C) | Z | (A and B and C) ==> Z
-----------------------------------------... |
27,896 | <p>How would I solve these differential equations? Thanks so much for the help!</p>
<p>$$P'_0(t) = \alpha P_1(t) - \beta P_0(t)$$
$$P'_1(t) = \beta P_0(t) - \alpha P_1(t)$$</p>
<p>We also know $P_0(t)+P_1(t)=1$</p>
| Ross Millikan | 1,827 | <p>Use $P_0(t)+P_1(t)=1$ to turn it into $P_0'(t)=\alpha-(\alpha+\beta)P_0(t)$, which you should be able to solve.</p>
|
27,896 | <p>How would I solve these differential equations? Thanks so much for the help!</p>
<p>$$P'_0(t) = \alpha P_1(t) - \beta P_0(t)$$
$$P'_1(t) = \beta P_0(t) - \alpha P_1(t)$$</p>
<p>We also know $P_0(t)+P_1(t)=1$</p>
| Aryabhata | 1,102 | <p>There is a general method to solve such equations, if we view them as a linear system of equation</p>
<p>$$y'(x) = A y(x)$$</p>
<p>When $A$ is a matrix with constants, the solution can be written in terms of the <a href="http://en.wikipedia.org/wiki/Matrix_exponential" rel="nofollow">exponent matrix</a> $e^{Ax... |
1,060,933 | <p>It is necessary for me to find unit outward normal vector for the curve:</p>
<p>$$\gamma=(x(t),y(t))$$</p>
<p>where
$$x(t)=(0.6)\cos(t)-(0.3)\cos(3t)$$
and
$$y(t)=(0.7)\sin(t)+(0.07)\sin(7t)+(0.1)\sin(3t)$$
I know how to find unit outward normal vector for this: using</p>
<p>$$T=\frac{\gamma'(t)}{||\gamma(t)||... | John Hughes | 114,036 | <p>$$
\gamma'(t) = (-\sin(t) + 1.5 \sin(3t), \cos(t)+7\cos(7t)+3\cos(3t))\\
\| \gamma'(t) \| = \sqrt{(-\sin(t) + 1.5 \sin(3t))^2, (\cos(t)+7\cos(7t)+3\cos(3t))^2}
$$
In general, for a vector $(a, b)$ in the plane, $(-b, a)$ is perpendicular to it. </p>
<p>So your normal vector is
$$
N(t) = \frac{\pm (-\cos(t)-7\cos(7... |
2,193,306 | <p>I have to evaluate the integral of the $\sec x$ function and I do it as follows $$\int\sec xdx=\int\frac{dx}{\cos x}=\int\frac{\cos^2 x+\sin^2 x}{\cos x}dx=\sin x+\int\frac{\sin^2x}{\cos x}dx$$ Now we make a change of variables $\cos x=t$ so our integral becomes $$\sin x-\int\frac{\sqrt{1-t^2}}{t}dt$$ Now we make an... | Jacky Chong | 369,395 | <p>Hint: Use the trick
\begin{align}
\sec x= \frac{\sec x+\tan x}{\sec x+\tan x} \sec x.
\end{align}
then $u$-sub.</p>
|
3,787,576 | <p><span class="math-container">$$
\mbox{Prove}\quad
\int_{0}^{1}{\mathrm{d}x \over
\left(\,{x - 2}\,\right)\,
\sqrt[\Large 5]{\,x^{2}\,\left(\,{1 - x}\,\right)^{3}\,}\,}
=
-\,{2^{11/10}\,\pi \over \,\sqrt{\,{5 + \,\sqrt{\,{5}\,}}\,}\,}
$$</span></p>
<ul>
<li>Being honest I havent got a clue where to start. I dont thi... | Lai | 732,917 | <p><span class="math-container">$$
\begin{aligned}
\text { Let } x&=\sin ^2 \theta, \quad \textrm{ then } d x=2 \sin \theta \cos \theta d \theta, \textrm{ and }\\
I&=\int_0^{\frac{\pi}{2}} \frac{2 \sin \theta \cos \theta d \theta}{\left(\sin ^2 \theta-2\right) \sqrt[5]{\sin ^4 \theta \cos ^6 \theta}}\\
&=2 ... |
4,205,179 | <p>Solve the equation</p>
<blockquote>
<blockquote>
<p><span class="math-container">$\sin x-\cos x-4\cos^2 x \sin x=4\sin^2 x$</span></p>
</blockquote>
</blockquote>
<p>My attempt:</p>
<p>I have rewritten the equation as:
<span class="math-container">$$\sin x=\frac{4\cos ^2 x-\cos x-4}{4\cos^2 x-1}$$</span></p>
<p>I tr... | Theo Bendit | 248,286 | <p>Try using the half-angle substitution: <span class="math-container">$t = \tan(x/2)$</span>. Then the following relations hold:
<span class="math-container">\begin{align*}
\cos(x) &= \frac{1 - t^2}{1 + t^2} \\
\sin(x) &= \frac{2t}{1 + t^2}.
\end{align*}</span>
Then the equation becomes:
<span class="math-cont... |
4,205,179 | <p>Solve the equation</p>
<blockquote>
<blockquote>
<p><span class="math-container">$\sin x-\cos x-4\cos^2 x \sin x=4\sin^2 x$</span></p>
</blockquote>
</blockquote>
<p>My attempt:</p>
<p>I have rewritten the equation as:
<span class="math-container">$$\sin x=\frac{4\cos ^2 x-\cos x-4}{4\cos^2 x-1}$$</span></p>
<p>I tr... | Aaron | 9,863 | <p>Let <span class="math-container">$\omega=\cos(x)+i\sin(x)$</span>, so that <span class="math-container">$\cos(x)=(\omega+\omega^{-1})/2$</span> and <span class="math-container">$\sin(x)=(\omega-\omega^{-1})/2i$</span>. Then the equation we are trying to solve, when expressed in terms of <span class="math-container"... |
114,831 | <p>$$A_n=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n}$$
Try to prove $$\lim_{n \to \infty}n(\ln 2-A_n) = \frac{1}{4}$$</p>
<p>I try to decompose $\ln 2$ as $$\ln(2n)-\ln(n)=\ln\left(1+\frac{1}{2n-1}\right)+\dots+\ln\left(1+\frac{1}{n}\right)\;,$$ but I can't continue, is that right?</p>
| Did | 6,179 | <p>Note that
$$
\log2-A_n=\int_n^{2n}\frac{\mathrm dx}x-\sum_{k=1}^n\frac1{n+k}=\sum_{k=1}^n\int_0^1\frac{1-x}{(n+k-1+x)(n+k)}\mathrm dx.
$$
To get an upper bound, use $n+k-1+x\geqslant n+k-1$, hence
$$
\log2-A_n\leqslant\sum_{k=1}^n\frac{1}{(n+k-1)(n+k)}\int_0^1(1-x)\mathrm dx.
$$
The integral is $\frac12$ and the... |
217,711 | <p>Set
$$
g(x)=\sum_{k=0}^{\infty}\frac{1}{x^{2k+1}+1} \quad \text{for} \quad x>1.
$$</p>
<p>Is it true that
$$
\frac{x^{2}+1}{x(x^{2}-1)}+\frac{g'(x)}{g(x)}>0 \quad \text{for}\quad x>1?
$$
The answer seems to be positive. I spent several hours in proving this statement but I did not come up with anyth... | Iosif Pinelis | 36,721 | <p>Writing $k=\sum_{j=1}^k 1$ and then summing by parts, one has
$$\sum_{k\geq 1} \frac{(-1)^{k-1}k}{(x^{k}+1)^2}
=\sum_{j\ge 1}(-1)^{j-1}s_j
=\sum_{k\ge0}[s_{2k+1}-s_{2k+2}],
$$
where
$$s_j:=\sum_{k=j}^\infty(-1)^{k-j} a(k)=\sum_{m\ge0}(-1)^m a(m+j)
=\sum_{k\ge0}[a(2k+j)-a(2k+1+j)]
$$
and
$a(q):=\dfrac1{(x^q+1)^2}$... |
1,933,410 | <p>I'm reading a book about differential geometry and there's a part where he talks about the standar spherical coordinates on $S^2$, which are given by:</p>
<p>$$x:(0,2\pi)\times (0,\pi)\to S^2, x(\theta, \phi) = (\cos\theta\sin\phi,\sin\theta\sin\phi, \cos\phi)$$</p>
<p>Shouldn't it be in $S^3$?</p>
| Emilio Novati | 187,568 | <p>No. It is a subset of $\mathbb{R}^3$, so we have three coordinates $ (x_1,x_2,x_3)$ but it is a surface because, as you can see: $x_1^2+x_2^2+x_3^2=1$ so it is the surface of a sphere in the $3D$ space. </p>
|
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | lhf | 532 | <p>Also, differential Galois theory for differential equations.</p>
|
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Mariano Suárez-Álvarez | 1,409 | <p>One set of problems to which homological algebra applies surprisingly well is those posed by topology, as evinced by algebraic topology :P</p>
<p>This is of course quite anhistorical...</p>
|
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Qiaochu Yuan | 290 | <p>This example is more closely related to <a href="https://mathoverflow.net/questions/5449/combinatorial-results-without-known-combinatorial-proofs">a question of mine</a>, but I'll give it here anyway.</p>
<p>A graded poset $P$ is <em>Sperner</em> if no antichain is larger than the largest level $P_i$ of $P$. This ... |
8,741 | <p>Here is a topic in the vein of <a href="https://mathoverflow.net/questions/1890/describe-a-topic-in-one-sentence" title="Describe a topic in one sentence"> Describe a topic in one sentence</a> and <a href="https://mathoverflow.net/questions/4994/fundamental-examples" > Fundamental examples </a> : imagine that you a... | Martin Brandenburg | 2,841 | <p>As for <b>category theory</b>, I don't think that there is a motivating example which has not already a category theoretic flavor. The leading theme is to unify and then generalize constructions resp. arguments, which come up in all areas of mathematics. Historically, natural transformations were introduced for the ... |
3,104,058 | <p>I need to make a part of a program in java that calculates a circle center. It has to be a circle through a given point that touches another circle, and the variable circle center has the possibility to move over a given line.</p>
<p><a href="https://i.stack.imgur.com/RQgBC.png" rel="nofollow noreferrer"><img src="... | John Hughes | 114,036 | <p><strong>A strong hint, but not a complete solution</strong></p>
<p>Let
<span class="math-container">$$
\newcommand{\bu} {{\mathbf u}}
\newcommand{\bv} {{\mathbf v}}
\begin{align}
s &= \|B - C\|\\
\bu &= \frac{B - C}{s}
\end{align}
$$</span>
The point <span class="math-container">$P$</span> is then <span cl... |
443,013 | <p>If a linear map $A:\mathbb{R}^m\rightarrow \mathbb{R}^n$ is injective, then there exists $c>0$ such that $|Ax|\geq c|x|$ for all $x\in\mathbb{R}^m$</p>
<p>Could someone give any solution or hint?</p>
<p>Thanks.</p>
| Nick Peterson | 81,839 | <p><strong>Hint:</strong> Consider
$$
\alpha:=\inf_{\|x\|=1}\|Ax\|.
$$
This must exist and be non-negative, since $\|Ax\|\geq 0$ for all $x\in\mathbb{R}^m$. If $\alpha>0$, then you can use linearity to prove the desired result. What happens if $\alpha=0$? </p>
|
443,013 | <p>If a linear map $A:\mathbb{R}^m\rightarrow \mathbb{R}^n$ is injective, then there exists $c>0$ such that $|Ax|\geq c|x|$ for all $x\in\mathbb{R}^m$</p>
<p>Could someone give any solution or hint?</p>
<p>Thanks.</p>
| user71352 | 71,352 | <p>First note that if such an inequality holds on the unit sphere in the Euclidean norm then it is true for all $x\in\mathbb{R}^{n}$. This is because if we have</p>
<p>$\vert\vert Ax\vert\vert\ge C\vert\vert x\vert\vert=C$ for some constant $C>0$ on the unit sphere then we simply note that for any $x\in\mathbb{R}^{... |
4,009,117 | <p>Suppose <span class="math-container">$X$</span>, <span class="math-container">$Y$</span>, and <span class="math-container">$Z$</span> are exhaustive events with <span class="math-container">$Y$</span> and <span class="math-container">$Z$</span> being mutually exclusive. Can <span class="math-container">$Y$</span> an... | Arturo Magidin | 742 | <p>It depends on exactly what your definition of limit is.</p>
<p>In calculus, limits of real-valued functions are often defined in such a way that in order for the limit as <span class="math-container">$x\to a$</span> to exist, the function must be defined on some interval of the form <span class="math-container">$(a-... |
3,069,369 | <p>I know how to perform polynomial regression. But is there any method to use for estimating the degree of the polynomial that is best suited? Some kind of meta-regression.</p>
<p>With best suited I mean the grade that has the highest probability of being the true degree of the source for the data.</p>
<p>For exampl... | David G. Stork | 210,401 | <p>Starting with <span class="math-container">$2 x^4+x^3-7 x^2-x+5$</span> and using the insight that <span class="math-container">$x=1$</span> must be a factor:</p>
<p>Divide out:</p>
<p><span class="math-container">$${2 x^4+x^3-7 x^2-x+5 \over x-1} = 2 x^3+3 x^2-4 x-5$$</span></p>
<p>Notice again, from the sum of ... |
853,735 | <p>Can anyone show, step-by-step, how the expression on the LHS can be turned into the expression on the RHS?</p>
<p>$x^ay^b=a^ab^b(a+b)^{-(a+b)}(x+y)^{a+b}$</p>
| Juanito | 153,015 | <p>Take x=0, y=1, a=1, b=1. The LHS is not equal to RHS.
So, the identity is False.</p>
|
4,581,481 | <p>How would one go about ALGEBRAICALLY finding the range of a semicircle?</p>
<p>eg. <span class="math-container">$y = \sqrt{4-x^2}$</span></p>
<p>Since there is no plus-minus sign, we know that <span class="math-container">$y$</span> must at least have a range of <span class="math-container">$[0, \infty)$</span>. Yet... | N. F. Taussig | 173,070 | <p>Since <span class="math-container">$x, y$</span> are real numbers, <span class="math-container">$y = \sqrt{4 - x^2}$</span> is only defined if <span class="math-container">$4 - x^2 \geq 0$</span>.
<span class="math-container">\begin{align*}
4 - x^2 & \geq 0\\
4 & \geq x^2\\
2 & \geq |x| && \text{... |
3,737,831 | <p>The problem is</p>
Let <span class="math-container">$\phi$</span> be an endomorphism of a <span class="math-container">$n$</span>- dimensional vector space V. Suppose that <span class="math-container">$\phi$</span> has n different eigenvalues. Prove that exists a vector <span class="math-container">$v$</span> of <... | alphaomega | 775,794 | <p>I know it's not exactly related but it's interesting to observe that the compact condition cannot be dropped. Consider for example,</p>
<ul>
<li><span class="math-container">$f(x,y)=(x+y)^2$</span>, <span class="math-container">$(x,y)\in [0, +\infty)^2$</span></li>
<li><span class="math-container">$x_n=\frac{1}{n}$<... |
1,117,924 | <p>How can I show with linear approximation that $y \approx x$ for small x? I know the rule $$f(x) \approx f(a) + f^{\prime}(a) (x-a),$$ but I don't know how to put it to use in this case.</p>
| A. Donda | 49,995 | <p>The function you are trying to approximate is
$$
f(x) = \ln(x)
$$
and you need an approximation around $a = 1$.</p>
<p>For the rule we need the derivative of the function, and we <a href="http://en.wikipedia.org/wiki/Logarithm#Derivative_and_antiderivative" rel="noreferrer">know</a> that the derivative of the natur... |
22,662 | <p>Suppose $V_1$ and $V_2$ are two $(g,K)$ modules of some reductive group $G$ with maximal compact $K$. Let $P$ be the minimal parabolic of $G$, $U$ its unipotent part, and $u$ its Lie algebra. Suppose the quotients $V_1/uV_1$ and $V_2/uV_2$ are isomorphic as modules for the Levi component of $P$, then what else do we... | Victor Protsak | 5,740 | <p>This is a comment, mostly on terminology that I think caused some confusion, not an answer, but I don't have enough "influence" to post this as a comment.</p>
<p>For real reductive groups, the Jacquet functor $V\mapsto J(V)$ (Jacquet-Casselman functor, Jacquet module, etc) is defined <em>differently</em> from the n... |
215,077 | <p>This question is partly connected with the following <a href="https://mathoverflow.net/questions/214931/connection-between-stallings-end-theorem-and-seifert-van-kampen-theorem">Connection between Stalling's end theorem and Seifert-van Kampen Theorem</a>. </p>
<p>By Stalling's Theorem a group with more than one ... | NWMT | 38,698 | <p>There is nothing <em>intrinsic</em> about whether a group splits as a free product with amalgamation or an HNN extension. For example free groups split both ways $$
\langle a,b \rangle = \langle a \rangle * \langle b \rangle = \langle a,b ; t \mid t^{-1} a t = b \rangle $$ as do many other groups. If you're cutting ... |
1,923,226 | <p>I have been trying to prove this with some continuity theorems but haven't put together a good proof yet. </p>
| kobe | 190,421 | <p>Let $m\in \Bbb Z$ and $\epsilon > 0$. Set $\delta = 1$. Show that for all $n$, $\lvert n - m\rvert < \delta$ implies $\lvert f(n) - f(m) \rvert < \epsilon$.</p>
|
1,923,226 | <p>I have been trying to prove this with some continuity theorems but haven't put together a good proof yet. </p>
| TheGeekGreek | 359,887 | <p>Let us take the Weierstrass' $\varepsilon$-$\delta$ criterion. If $f:E\subseteq \mathbb{R} \to \mathbb{R}$ is continuous at $x_0 \in E$, then $$\forall \varepsilon > 0 \exists \delta > 0 \forall x \in E \left( \left| x - x_0\right| < \delta \to \left| f(x) - f(x_0)\right| < \varepsilon \right)$$ Now Let ... |
4,153,364 | <blockquote>
<p>If <span class="math-container">$f(x) = x^2$</span> and <span class="math-container">$g(x) = \sqrt{x-1}$</span>, find <span class="math-container">$f(g(x))$</span> and specify the domain.</p>
</blockquote>
<p><strong>My solution</strong></p>
<p><span class="math-container">$$
f(g(x)) = \left(\sqrt{x-1}\... | gt6989b | 16,192 | <p>I think you have to differentiate between
<span class="math-container">$$
f(g(x)) = \left(\sqrt{x-1}\right)^2
$$</span>
and <span class="math-container">$h(x) = x-1$</span>. For <span class="math-container">$h(x)$</span>, indeed the domain are all reals, but <span class="math-container">$f(g(x))$</span> is defined t... |
4,153,364 | <blockquote>
<p>If <span class="math-container">$f(x) = x^2$</span> and <span class="math-container">$g(x) = \sqrt{x-1}$</span>, find <span class="math-container">$f(g(x))$</span> and specify the domain.</p>
</blockquote>
<p><strong>My solution</strong></p>
<p><span class="math-container">$$
f(g(x)) = \left(\sqrt{x-1}\... | Kevin Arlin | 31,228 | <p>Another way to put the answer, which has already been given, is that it’s <em>not</em> always the case that <span class="math-container">$\left(\sqrt{x-1}\right)^2=x-1$</span>! This equation holds if and only if <span class="math-container">$x\ge 1$</span>. If <span class="math-container">$x=0$</span>, for instance,... |
3,839,153 | <p>Let <span class="math-container">$f:[a-1,1] \to \mathbb R_{+}$</span> such that <span class="math-container">$f(x)=\begin{cases} 1-(a-x)^2 &a-1 \lt x \leq a+1 \\ 0 & a+1 < x \leq 1 \end{cases}$</span></p>
<p>for some real <span class="math-container">$a<0$</span>.</p>
<p>If I compute the first order de... | md2perpe | 168,433 | <p>You have <span class="math-container">$f'((a+1)^-) = -2$</span> and <span class="math-container">$f'((a+1)^+) = 0$</span> so <span class="math-container">$f'$</span> increases at <span class="math-container">$x=a+1$</span>. Therefore the sign of <span class="math-container">$f''$</span> at <span class="math-containe... |
1,401,661 | <blockquote>
<p>Dice are cubes with pips (small dots) on their sides, representing
numbers 1 through 6. Two dice are considered the same if they can be
rotated and placed in such a way that they present matching numbers on
the top, bottom, left, right, front, and back sides.</p>
<p>Below is an example of t... | Christian Blatter | 1,303 | <p>Put your dice on a table such that the $1$ is on the bottom for all of them. You still can rotate them freely around the vertical axis passing through the single dot on the bottom. Now begins the counting: You have five choices for the top face. The remaining four numbers can be paired in three ways to form pairs of... |
100,801 | <p>When doing calculations in notebook, I modify the code a lot, and there are several versions of similar codes in the same notebook. But I found myself easily forgot which version is the newest, and I have to confirm which cell is the newest.</p>
<p>So if mathematica can provide a menu options like "cell latest modi... | matheorem | 4,742 | <p>Albert Retey just created a great button that can give a cell's latest modification time directly.</p>
<p>Here is a another workaround. It is small trick I just found, but works perfectly : )</p>
<p>The frozen of mma when open Notebook History windows is due to the default setting is checked for "All cell" instead... |
553,103 | <p>consider the series of function $\sum f_n$ with $f_n(x)=\frac{x}{x^2+n^2}$.
It is easy to see that there is pointwise convergence on $\mathbb{R}$ (to a function that we'll call $f$) but not normal convergence. I want to know whether there is uniform convergence towards $f$ or not. </p>
<p>I tried to disprove unifo... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Without using Complex Calculus, divide the numerator & the denominator by $\cos^2x$ and substitute $\tan x$ with $u$</p>
|
553,103 | <p>consider the series of function $\sum f_n$ with $f_n(x)=\frac{x}{x^2+n^2}$.
It is easy to see that there is pointwise convergence on $\mathbb{R}$ (to a function that we'll call $f$) but not normal convergence. I want to know whether there is uniform convergence towards $f$ or not. </p>
<p>I tried to disprove unifo... | Ron Gordon | 53,268 | <p>Hint: Let $z=e^{i x}$; $dx=-i dz/z$; $\cos{x}=(z+z^{-1})/2$. The integral is then equal to</p>
<p>$$-i \oint_{|z|=1} \frac{dz}{z} \frac{1}{1+\frac{3}{4} (z+z^{-1})^2} $$</p>
<p>Multiply out, determine the poles, figure out which poles, if any, lie within the unit circle, find the residues of those poles, multiply... |
17,398 | <p>I am teaching my daughter, who is currently about <span class="math-container">$46$</span> months old, additions. She is very curious and asks a lot of <em>good</em> questions. For example, when I told her that <span class="math-container">$2+6=8$</span> and <span class="math-container">$4+4=8$</span>, she asked me ... | Nick C | 470 | <p>I'm nearly <em>sure</em> I did this with my child when she was young.</p>
<p>First, establish that she understands that a number, like three, is equal to <span class="math-container">$1+1+1$</span>. Hold three fingers up and ask her "how many is this"?</p>
<p><a href="https://i.stack.imgur.com/HthuC.jpg" rel="nore... |
17,398 | <p>I am teaching my daughter, who is currently about <span class="math-container">$46$</span> months old, additions. She is very curious and asks a lot of <em>good</em> questions. For example, when I told her that <span class="math-container">$2+6=8$</span> and <span class="math-container">$4+4=8$</span>, she asked me ... | user21820 | 1,550 | <p>Here is a written explanation to accompany <a href="https://matheducators.stackexchange.com/a/17401">Nick C's excellent 'hand-waving' answer</a>. =)</p>
<p><span class="math-container">$2+6 \quad = \quad 1+1 \quad + \quad 1+1+1+1+1+1$</span></p>
<p>        <span class="math-container">$ = \quad... |
933,515 | <p>Let $T$ be a bounded operator acting on a Banach space $X$. The point spectrum $\sigma_p(T)$ is of $T$ is defined to be $$\sigma_p(T):=\{\lambda\in\mathbb C~|~T-\lambda\text{ has nonempty kernel}\}$$ and the residual spectrum $\sigma_r(T)$ is defined to be $$\sigma_r(T):=\{\lambda\in\mathbb C~|~T-\lambda\text{ has e... | Disintegrating By Parts | 112,478 | <p>The common classification of spectrum is the first one you gave. I'm not sure where $\sigma_{c}'(T)$ would have come from, or why it was defined that way, but it doesn't strike me as a standard or common definition, even though I can see a reason for the alternative definition.</p>
<p>If $T$ is closed, then this al... |
13,553 | <p>(This issue has been discussed before, specifically <a href="https://math.meta.stackexchange.com/questions/8485/how-civil-should-we-be/8493">here</a>. A different kind of censorship has been discussed <a href="https://math.meta.stackexchange.com/questions/8306/defacement-censorship-of-peoples-answers">here</a>.)</p... | Thomas | 26,188 | <p>In my little narrow minded world I can't comprehend why someone would suggest that we should allow swear words (or "bad" words in general). But then I am reminded that the world actually is bigger than the one I live in. I realize that my view of right and wrong doesn't necessarily overlap with other peoples views. ... |
3,459,178 | <p>Is there a general way to factor <span class="math-container">$ax^2 + bx$</span> into <span class="math-container">$($</span>some expression<span class="math-container">$)^2$</span>, where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are constants?</p>
<p>The reason I am askin... | Donald Splutterwit | 404,247 | <p><span class="math-container">\begin{eqnarray*}
ax^2+bx= a \left( x+ \frac{b}{2a} \right)^2 - \frac{b^2}{4a}.
\end{eqnarray*}</span></p>
|
2,327,181 | <p>I'm a bit confused about the definition of finite sets/intervals. I know that a set S is called finite when it has a finite number of elements, or formally, when there exists a bijection $f:S\to\{1,...,n\}$ for some natural number n. </p>
<p>However, the interval $(1,2)$ is called finite. I don't understand why; $(... | Asaf Karagila | 622 | <p>There are different ways to think about the size of a set. In the case of the real numbers, and specifically intervals, we can talk about their length (and generally, their Lebesgue measure in the case of measurable sets).</p>
<p>If you think about the real numbers as a model of time or space, then the distance bet... |
118,985 | <p>Here's a question I came across:</p>
<blockquote>
<p>count the number of permutations of 10 men, 10 women and one child,
with the limitation that a man will not sit next to another man, and a
woman will not sit next to another woman</p>
</blockquote>
<p>What I tried was using the inclusion-exclusion principl... | Robert Israel | 8,508 | <p>Try comparing to $e^{-tx}$ where $0 < t < 1$.</p>
|
118,985 | <p>Here's a question I came across:</p>
<blockquote>
<p>count the number of permutations of 10 men, 10 women and one child,
with the limitation that a man will not sit next to another man, and a
woman will not sit next to another woman</p>
</blockquote>
<p>What I tried was using the inclusion-exclusion principl... | Did | 6,179 | <p>For every $x\geqslant2$, $x^2\mathrm e^{-x}\leqslant16\mathrm e^{-2}\mathrm e^{-x/2}$ hence
$$\int\limits_2^{+\infty}x^2\mathrm e^{-x}\mathrm dx\leqslant16\mathrm e^{-2}\int\limits_2^{+\infty}\mathrm e^{-x/2}\mathrm dx=16\mathrm e^{-2}\cdot2\mathrm e^{-1}\lt2.
$$</p>
|
690,729 | <p>Vancouver is 300 km away from Seattle. Two friends, one leaving from each city on a bike want to find a campsite between the two cities. One cyclist starts from vancouver at 25 km/h. His friend will start 2.0h later from seattle at 32 km/h. How far from Vancouver do the friends meet?</p>
<p>Attempt:</p>
<p>dA + dB... | MCT | 92,774 | <p><strong>Hint</strong>: Suppose that the friends begin cycling at the same time, but instead of starting from $300$ kilometers away from each other, they start $250$ kilometers away from each other. If $x$ is the answer to this equation, the answer to your original equation will be $x + 50$.</p>
|
2,365,120 | <p>I'm doing some revision on indices and surds. How do you simplify </p>
<blockquote>
<p>$(xy^2)^p\sqrt{x^q} $</p>
</blockquote>
<p>Bit confused because my textbook says the answer is </p>
<blockquote>
<p>$x^{p+q/2}y^{2p}$</p>
</blockquote>
<p>I can understand simplifying but only when it's the same base. I'm ... | user247327 | 247,327 | <p>You need parentheses to clarify your expression, and you seem to have dropped a "^"- you mean (xy^2)^p sqrt(x^q). Apply the "rules of exponents": (xy^2)^p= x^p (y^2)^p= x^p y^{2p}. Also sqrt(x^q)= (x^q)^(1/2)= x^(q/2). </p>
<p>So (xy^2)^p sqrt(x^q)= (x^p y^(2p))(x^(q/2))= x^(p+ q/2) y^{2p} </p>
<p>Also "q/2" is... |
2,501,235 | <p>Many books define almost sure convergence as follows:</p>
<blockquote>
<p>The sequence of random variables ${(X_n)}_{n \in \mathbb{N}}$ defined on the probability space $(\Omega, \mathcal{F}, P)$ converges almost surely to a random variable $X$ defined on the same probability space, if
$$
P(\{ \omega \in \Omeg... | Michael Greinecker | 21,674 | <p>$$A=\{\omega\in\Omega\mid\text{ for all }\epsilon>0\text{ there is } N\text{ such that } |X_n(\omega)-X(\omega)|<\epsilon\text{ for all }n\geq N\}$$
$$=\{\omega\in\Omega\mid\text{ for all }m\text{ there is } N\text{ such that } |X_n(\omega)-X(\omega)|<1/m\text{ for all }n\geq N\}$$
$$=\bigcap_{m=1}^\infty\b... |
3,936,694 | <p>I would like to construct a deterministic TM that decides <span class="math-container">$L=$</span>{<span class="math-container">$<G,s,t,k>$</span> | <span class="math-container">$G$</span> is a directed graph that has a path of length at most <span class="math-container">$k$</span> from vertex <span class="mat... | Mike | 544,150 | <p>That's not true in general. Take <span class="math-container">$$y(t)=e^{-1/t}$$</span> and <span class="math-container">$$x(t)=-5t$$</span> say.
Then <span class="math-container">$$\lim_{t \rightarrow 0^+} y(t) = \lim_{t \rightarrow 0^+} x(t)=0,$$</span> but <span class="math-container">$$\lim_{t \rightarrow 0^+} (... |
2,661,718 | <p>My basic understanding is that each time
I have 30% chance of winning the prize
so between 3 and 4 tries I should win it</p>
<p>cause .3 +.3 +.3 = 90% as I need to win only once with 3 try
.3+.3+.3+.3 = 120% with 4 try</p>
<p>I do remember a formula saying 1-(0.7)^3 = 65.7% chance but I dont remember what that ... | alepopoulo110 | 351,240 | <p>the derivative is $f'(x)=2x-2\log(1+x)-\log(1+x)^2$. By setting $t=\log(x+1)$ one gets $h(t)=2e^t-2t-t^2-2$. We get $h'(t)=2e^t-2t-2\geq 0$, by the Taylor expansion of $e^t$. But $h(0)=0$, so for $t<0$, which corresponds to $x<0$, we have $f'(x)<0$ and for $t>0$ we have $f'(x)>0$. Also $f(0)=0$, so $f... |
1,825,392 | <p>Let $f(z)$ be an entire function so that,</p>
<p>$$ \int \frac{|f(z)|}{1 + |z|^3} dA(z) < \infty$$</p>
<p>where the integral is taken over the entire complex plane. Show that $f$ is a constant.</p>
<p>I believe that the idea is to use the mean value property; that is:</p>
<p>$$f(z) = \frac{1}{\pi\delta^2}\in... | Hmm. | 227,501 | <p>Fix any $z_0\in \Bbb C$. $\forall r > 0$, we have by Mean Value Theorem, $$~~~~~~~~~~f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + re^{i\theta})\, d\theta$$ </p>
<p>$$\implies |f(z_0)| \leq \frac{1}{2\pi} \int_0^{2\pi} |f(z_0 + re^{i\theta})|\, d\theta$$ </p>
<p>For every $R>0$, we have,
$$\int_0^{R} \lvert ... |
294,383 | <p>Evaluate :
$$\int_{0}^{+\infty }{\left( \frac{x}{{{\text{e}}^{x}}-{{\text{e}}^{-x}}}-\frac{1}{2} \right)\frac{1}{{{x}^{2}}}\text{d}x}$$</p>
| xpaul | 66,420 | <p>We also can use this way to calculate instead of using complex analysis or special functions. Noting that
$$ \int_0^\infty te^{-xt}dt=\frac{1}{x^2} $$
we have
\begin{eqnarray}
&&\int_{0}^{\infty }{\left( \frac{x}{{{\text{e}}^{x}}-{{\text{e}}^{-x}}}-\frac{1}{2} \right)\frac{1}{{{x}^{2}}}\text{d}x}\\
&=&am... |
686,167 | <p>I'm working on an assignment where part of it is showing that $S_k=0$ for even $k$ and $S_k=1$ for odd $k$, where</p>
<blockquote>
<p>$$S_k:=\sum_{j=0}^{n}\cos(k\pi x_j)= \frac{1}{2}\sum_{j=0}^{n}(e^{ik\pi x_{j}}+e^{-ik\pi x_{j}}) $$</p>
<p>Here $x_j=j/(n+1)$.</p>
</blockquote>
<p>So, working through the al... | Michael Hardy | 11,667 | <p>$$
\frac{1}{1-e^{-a}} = \frac{1}{1-e^{-a}}\cdot\frac{e^a}{e^a} = \frac{e^a}{e^a-1} = \frac{-e^a}{1-e^a}.
$$</p>
<p>Hence
$$
\frac{1}{1-e^a} + \frac{1}{1-e^{-a}} = 1.
$$</p>
|
658,807 | <p>i am having difficulty understanding why the diagram on page 8 of this <a href="http://www.cs.toronto.edu/~sme/presentations/cat101.pdf" rel="nofollow">presentation</a> is a category. the author claims that it is on page 43.</p>
<p>it looks like the two smaller arrows on the left must be idempotent or we would have... | Andrej Bauer | 30,711 | <p>There is a problem when you say "it would also seem that composing one of the smaller arrows on the left with one of the longer (left pointing) arrows must result in the small arrow."</p>
<p>The category you've given is ok, but you are confused about the compositions. Given morphisms $f : A \to B$ and $g : B \to C$... |
2,975,647 | <p>Is this following statement valid (where both <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are positive)?</p>
<blockquote>
<p>If <span class="math-container">$x>y$</span>, then <span class="math-container">$\dfrac{1}{x} < \dfrac{1}{y}$</span>. </p>
</blockquote>
| user10354138 | 592,552 | <p>In fact, on <span class="math-container">$\{\lvert z\rvert=1,z\neq 1\}$</span>, <span class="math-container">$\sideset{_2}{_1}F(a,b;c;z)$</span> converges only conditionally if <span class="math-container">$0\geq\operatorname{Re}(c-a-b)>-1$</span> and diverges if <span class="math-container">$\operatorname{Re}(c-... |
2,061,655 | <p>Could we compute the limits
$$\lim_{x\rightarrow 0}\frac{\sin (x)-x+x^3}{x^3} \\ \lim_{x\rightarrow 0}\frac{e^x-\sin (x)-1}{x^2}$$
without using the l'Hospital rule and the Taylor expansion?</p>
| Vincenzo Oliva | 170,489 | <p>The first one is equal to $1+\lim\limits_{x\to0}\dfrac{\sin x - x}{x^3}.$ We can apply to this limit the technique in <a href="https://math.stackexchange.com/a/2028545/170489">this answer of mine.</a> </p>
<p>Let $L=\lim\limits_{x\to0}\dfrac{\sin x - x}{x^3}$. This is not infinite if it exists because near $0$ one ... |
2,176,503 | <p>A topological space $X$ is called quasi-compact if whenever ${(U_i)}_{i∈S}$ are a family of
open subsets such that $∪_{i∈S}(U_i) = X$ then there are a finite number of $(U_i)$’s which actually
cover X.</p>
<p>I'm thinking first we show that if $(U_{fi})$, $i ∈ S$ (where ${f_i}$, $i ∈ S$ a collection of elements of ... | Alex Mathers | 227,652 | <p>Affine schemes are Noetherian as topological spaces, meaning every descending chain of closed subspaces $Z_1\supseteq Z_2\supseteq\cdots$ stabilizes. This is the same as saying that each increasing chain of open subspaces $U_1\subseteq U_2\subseteq\cdots$ stabilizes.</p>
<p>Now try to show that if $X$ is not quasic... |
2,176,503 | <p>A topological space $X$ is called quasi-compact if whenever ${(U_i)}_{i∈S}$ are a family of
open subsets such that $∪_{i∈S}(U_i) = X$ then there are a finite number of $(U_i)$’s which actually
cover X.</p>
<p>I'm thinking first we show that if $(U_{fi})$, $i ∈ S$ (where ${f_i}$, $i ∈ S$ a collection of elements of ... | Andres Mejia | 297,998 | <p>Every descending chain of Zariski closed subsets in $X$,
$$X_1 \supseteq X_2,...$$</p>
<p>corresponds by inclusion reversal to
$$I(X_1) \subseteq I(X_2)...$$</p>
<p>However, we know that Polynomial rings are Noetherian, so the ascending chain of ideals stabilizes.</p>
<p>Hence, every descending chain of closed s... |
1,098,028 | <p>Let $T: \mathbb{R}_{\leq 3}[X] \rightarrow \mathbb{R}_{\leq 2}[X]$ be a linear map defined as $T(f(x)) = f'(x)$, and let $\beta$ and $\gamma$ be the standard ordered bases for resp. $\mathbb{R}_{\leq 3}[X]$ and $\mathbb{R}_{\leq 2}[X]$. Then the matrixrepresentation of the linear map is given as \begin{align*} A = [... | hmakholm left over Monica | 14,366 | <p>Do it one step at a time.</p>
<p>$f(-2)=0$ gives you one equation.</p>
<p>If $f$ has $y=6-6x$ as a tangent <em>at one of its zero crossings</em>, then the zero crossings in question must be at the point where $y=6-6x$ crosses the $x$ axis. Setting $f(x)=0$ there gives you another equation. Finally, the slope of $f... |
1,576,007 | <p>I just completed an exam and was curious as to the following question:</p>
<blockquote>
<p>How many surjections are there from $\{1, 2, 3, ..., n\}$ to $\{1, 2\}$ if $n \geq 3$?</p>
</blockquote>
<p>My assumption was the following:</p>
<p>The first element of the domain has the possibility to map to either 1 or... | sbares | 198,998 | <p>You had some of the right ideas, but you are not quite right. Let's start with finding the number of functions from ${1,2,...,n}$ to ${1,2}$ without any restrictions on the function. Then each of the numbers $1,2,...,n$ can be mapped to either $1$ or $2$, and there are hence $2^n$ such functions. How many of these a... |
3,208,003 | <p>How do you calculate the sum of this series?</p>
<p><span class="math-container">$$ \sum_{n=1}^\infty 0.5n \left(\frac{2}{3}\right)^n $$</span></p>
| Kandinskij | 657,309 | <p><strong>A solution withouth calculus</strong></p>
<p>We want to calculate the sum:</p>
<p><span class="math-container">$$\sum_{n=1}^{m} n a^n $$</span></p>
<p>Notice that:</p>
<p><span class="math-container">$$\sum_{n=1}^{m} n a^n=1a^1+2a^2+3a^3+...+m a^m =$$</span> <span class="math-container">$$=(a^1+a^2+a^3+.... |
3,415,211 | <p>I was solving my Rubik's cube when I though that if i toss my cube what is the probability of getting red colour twice in a row. So I calculate that it is 1/6 for first time, 1/6 possibility for second time getting a red and 1/6 * 1/6 = 1/36 is the answer.</p>
<p>But then I thought if the colour is unspecified i.e ... | José Carlos Santos | 446,262 | <p>Both are wrong. The first one is wrong because from the fact that <span class="math-container">$\frac1{x\log x}<\frac1x$</span> and from the fact that <span class="math-container">$\int_1^\infty\frac{\mathrm dx}x$</span> diverges you deduce nothing. And the second one is wrong because you have to deal not only wi... |
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