qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,113,934 | <p>How to prove that $\mathrm{Hom}(\mathbb{R})=\mathrm{Aut}(\mathbb{R})$ ?</p>
<p>(We treat it as field homomorphisms. )</p>
<p>I know that $\mathrm{Aut}(\mathbb{R})=\{\mathrm{id}\}$ and $\mathrm{Mon}(\mathbb{R})=\mathrm{Aut}(\mathbb{R})$.</p>
| Georges Elencwajg | 3,217 | <p>Let us show that every morphism of fields $f:\mathbb R\to \mathbb R$ is the identity, namely $f(r)=r$ for all real $r$. </p>
<p>0) Trivially $f(q)=q$ for all $q\in \mathbb Q$ </p>
<p>1) Notice that $f$ preserves the order relation in $\mathbb R$ : $x\leq y \implies f(x)\leq f(y)$.<br>
Indeed if $r\geq 0$ we... |
267,963 | <p>Why <code>HoldForm</code> do this?</p>
<p><a href="https://i.stack.imgur.com/DDFDT.png" rel="noreferrer"><img src="https://i.stack.imgur.com/DDFDT.png" alt="enter image description here" /></a></p>
<pre><code>HoldForm[a/b \[Integral]f[x] \[DifferentialD]x]
</code></pre>
<p>Output is this ugly looking formula:
<span ... | Alexei Boulbitch | 788 | <p>It is just enough to wrap a/b. Try this:</p>
<p><a href="https://i.stack.imgur.com/cCmGW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cCmGW.png" alt="enter image description here" /></a></p>
<p>Have fun!</p>
|
267,963 | <p>Why <code>HoldForm</code> do this?</p>
<p><a href="https://i.stack.imgur.com/DDFDT.png" rel="noreferrer"><img src="https://i.stack.imgur.com/DDFDT.png" alt="enter image description here" /></a></p>
<pre><code>HoldForm[a/b \[Integral]f[x] \[DifferentialD]x]
</code></pre>
<p>Output is this ugly looking formula:
<span ... | Lukas Lang | 36,508 | <h3>Why it happens</h3>
<p>When you enter an expression and evaluate it in Mathematica, <code>MakeExpression</code> is used to convert the box expression (i.e. the description of the exact input you entererd) into an expression. For example, when you enter <code>f[a]</code> into a Notebook, what Mathematica sees is <co... |
2,785,449 | <p>So, it is obvious that the empty set is a subset of every set, such that <strong>∀A:∅⊆A</strong>.</p>
<p>Therefore, if the null set is an element of some set A, such that <strong>A = {∅}</strong>, is the empty set considered a singleton element of A?</p>
<p>Thank you.</p>
| Anuska | 542,827 | <p>Yes $$\emptyset$$ is the one and only element of the set $A$. </p>
|
2,785,449 | <p>So, it is obvious that the empty set is a subset of every set, such that <strong>∀A:∅⊆A</strong>.</p>
<p>Therefore, if the null set is an element of some set A, such that <strong>A = {∅}</strong>, is the empty set considered a singleton element of A?</p>
<p>Thank you.</p>
| fleablood | 280,126 | <p>Divorce your mind from any thought that of an element being a set in its own right, from the role of the element as an element in another set. If $x$ is the only element in $A$ then $x$ is a singleton element of $A$. It doesn't matter <em>what</em> $x$ is and if $x$ just happens to be a set, its elements have noth... |
4,516,006 | <p>Let <span class="math-container">$a,~b\in\Bbb Q$</span> and suppose <span class="math-container">$\sqrt{a},~\sqrt{b}$</span> is irrational and <span class="math-container">$\sqrt{a}-\sqrt{b}\in\Bbb Q$</span>. I want to prove that <span class="math-container">$\sqrt{a}-\sqrt{b}=0$</span>; that is, <span class="math-c... | David Lui | 445,002 | <p><span class="math-container">$\sqrt{a} - \sqrt{b} = r$</span></p>
<p><span class="math-container">$\sqrt{a} = r + \sqrt{b}$</span></p>
<p><span class="math-container">$a = (r + \sqrt{b})^2 = r^2 + 2 r \sqrt{b} +b$</span></p>
<p>If <span class="math-container">$r \neq 0$</span>, then :</p>
<p><span class="math-conta... |
3,190,781 | <blockquote>
<p>Suppose the polynomial <span class="math-container">$P(x)$</span> with integer coefficients satisfies the following conditions:<br>
(A) If <span class="math-container">$P(x)$</span> is divided by <span class="math-container">$x^2 − 4x + 3$</span>, the remainder is <span class="math-container">$65x −... | ajotatxe | 132,456 | <p>Hint:</p>
<p>If you know that the remainder of the division of some polynomial <span class="math-container">$Q$</span> by, say, <span class="math-container">$x^2-5x+4$</span> is <span class="math-container">$7x-8$</span> you can find some values of <span class="math-container">$Q$</span> by substituting <span class... |
3,190,781 | <blockquote>
<p>Suppose the polynomial <span class="math-container">$P(x)$</span> with integer coefficients satisfies the following conditions:<br>
(A) If <span class="math-container">$P(x)$</span> is divided by <span class="math-container">$x^2 − 4x + 3$</span>, the remainder is <span class="math-container">$65x −... | Community | -1 | <p>You could use Polynomial Remainder Theorem here, it's just impractical. It states P(x) mod (x-b) is congruent to P(b). It never says: let b, be a number. <span class="math-container">$x^2-4x+3=x-(-x^2+5x-3)$</span>.</p>
<p>Easier to note the second divisor is <span class="math-container">$10x-10$</span> more than ... |
2,876,040 | <p>So i have the following problem: I need to cover the whole interval from $1$ to $10^{12}$ on a computer program. But i can´t do it on a for loop because it would be too slow. So can i iterate over a smaller amount ($\sqrt{n}$ for instance), and still cover the whole interval? </p>
<p>What i've done so far:</p>
<p>... | Ethan Bolker | 72,858 | <p>If you have to examine every one of those numbers you will have to execute some code $10^{12}$ times. It doesn't matter how cleverly you organize the loop.</p>
<p>Some languages (for example, Python) can optimize looping for you by clever indexing ("vectorizing") that saves a little of the looping overhead - but no... |
2,487,709 | <p>I have to proof that if for any $z$:
$$\cos(z+\omega) = \cos(z)$$</p>
<p>then $\omega = 2\pi k \quad (k = 0, \pm 1, \pm 2 \ldots)$</p>
<p>I have no idea how to prrof it strictly</p>
| Mark Viola | 218,419 | <p>If $f(t)=a+(b-a)H(t)$ and $x<0$, then $\int_{-\infty}^x \delta'(t)f(t)\,dt =\int_{-\infty}^x \delta'(t)a\,dt=0$ since $\delta'$ is of compact support on $\{0\}$.</p>
<p>Next note that there is no meaning to the square of the Dirac Delta as a distribution (<a href="https://math.stackexchange.com/questions/2221429... |
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>
| Raghav | 164,237 | <p>When You write 12 you actually mean 10+2. Or in general when you represent a number in decimal by writing $a_na_{n-1}a_{n-2} . . . a_2a_1a_0$ we actually mean $ \Sigma_{i=0}^n a_i10^i$.
Now, Perhaps It is easy for you to see that you can add term-wise.
For example
You can verify $a_i 10^i + b_i 10^i = d_i 10^{i+1} +... |
3,264,655 | <p>I've found myself where I've got a bidirectional graph of roughly 5500 nodes, 14000 edges.</p>
<p>I'd like to visit ~85 of these nodes, each <em>at least</em> once, starting and ending at the same node. I'd like to get a short route (don't need explicitly shortest, though that would be nice!). What graph processing... | Sudix | 470,072 | <p>Create a graph with only the nodes of interest, and calculate for each pair the shortest distance (there are polynomial procedures for calculating the transitive closure of a graph).</p>
<p>Now you're looking for a relaxed version of the Traveling Salesman Problem.</p>
<p>Therefore, in this modified version, you c... |
1,591,990 | <p>A while ago I saw this question <a href="https://math.stackexchange.com/questions/1561877/quartic-diophantine-equation-16r4112r3200r2-112r16-s2">Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$</a> which was very relevant to a undergraduate research paper I am currently working on. The answer given fo... | Tito Piezas III | 4,781 | <p>(<em>This is a long comment re MacLeod's answer</em>.)</p>
<p>We can also combine the two cases together. Assume a quartic polynomial to be made a square,</p>
<p><span class="math-container">$$pu^4+qu^3+ru^2+su+t=z_1^2\tag1$$</span></p>
<p>has a <strong><em>known</em></strong> rational point, call it <span class=... |
4,244,565 | <p>In resources like Wikipedia the following equation holds for <span class="math-container">$k>0$</span>.
<span class="math-container">\begin{equation}
\sum_{n=0}^{k} [x^n]=\frac{1-x^{k+1}}{1-x}\hspace{1cm}x\neq1
\end{equation}</span>
If you put the right hand side in a limit it holds for <span class="math-containe... | Robert Shore | 640,080 | <p>You're trying to say that there are <span class="math-container">$m!$</span> bijections <span class="math-container">$f:X \to Y$</span> if <span class="math-container">$\vert X \vert = \vert Y \vert = m$</span>. Prove this by induction on <span class="math-container">$\vert X \vert$</span>.</p>
<p>If <span class="m... |
3,249,926 | <p>I’ve tried applying Vandermonde’s identity, but got stuck. Any help would be appreciated!</p>
| PM. | 416,252 | <p><a href="https://i.stack.imgur.com/fuHBW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fuHBW.png" alt="enter image description here"></a></p>
<p>In this case you can consider the graph of <span class="math-container">$x^2$</span> to understand the relevant interval and the lower bound <span cla... |
403,977 | <blockquote>
<p>Let $u_n>u_{n+1}>0$ for all $n \in \Bbb N$, and suppose that $u_2+u_4+u_8+u_{16}+\dots$ diverges. Prove that $\sum_{n=1}^{\infty}\frac{u_n}{n}$diverges.</p>
</blockquote>
<p>Please provide me a hint or a full solution.</p>
| Tim | 74,128 | <p>Hint: think about the partial sums
$$ S_k = \sum_{n = 2^k+1}^{2^{k+1}} \frac{u_n}n$$</p>
<p>Notice that $$\sum_{n=1}^\infty \frac{u_n}n = u_1 + \frac{u_2}2 + \sum_{k=1}^\infty S_k$$
so it's enough to show that the sequence $\sum_{k=1}^\infty S_k$ diverges.</p>
<p>Notice that the terms in the partial sums $S_k$ are... |
2,228,913 | <p>For some geometric problem that I am trying to solve, I am interested
in characterizing the cross product between a Bezier curve and its second order derivative.</p>
<p>So, let's $\mathbf{c}(t)$ be a Bezier curve, and $\ddot{\mathbf{c}}(t)$ it's derivative.</p>
<p>Is $\mathbf{w}(t) = \mathbf{c}(t) \times \ddot{\... | Jean Marie | 305,862 | <p>@Henning Makholm One can say a little more.</p>
<p>Let us recall that a Bézier curve with degree <span class="math-container">$n$</span> (particular cases <span class="math-container">$n=2$</span>: quadratic, <span class="math-container">$n=3$</span> cubic) can be <strong>defined</strong> as any expression of the f... |
495,863 | <p>I am trying to show that this sequence $\{a_n\}$ = (2n+1)/(3n+5) does not converge to $42$ if it is not bounded above. I have already showed that it converges to $2/3$. For this I want to use a proof by contradiction,i.e, I assume that the sequence does converge to $42$, which will lead to a contradiction to the ini... | Pete L. Clark | 299 | <p>Your approach seems distinctly strange. For one thing, if the sequence converged to $42$, then it <em>would</em> be bounded above!</p>
<p>On the other hand, you have a specific sequence that you already know is converging to $\frac{2}{3}$, so assuming that it converges to something else is simply contradictory (I ... |
3,926,580 | <p>I'm trying to prove with squeeze theorem that the limit of the following series equals 1:</p>
<p><span class="math-container">$$\frac{1+\sqrt{2}+\sqrt[3]{3}+...+\sqrt[n]{n}}{n}$$</span></p>
<p>For the left side of the inequality I did:</p>
<p><span class="math-container">$$\frac{1+\sqrt{1}+\sqrt[3]{1}+...+\sqrt[n]{1... | Yiorgos S. Smyrlis | 57,021 | <p>One may combine the following two facts:</p>
<p><a href="https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation" rel="nofollow noreferrer">1.</a> If <span class="math-container">$a_n\to a,\,$</span> then <span class="math-container">$\,\frac{1}{n}(a_1+\cdots+a_n)\to a$</span>.</p>
<p><a href="https://math.stackexchange... |
15,875 | <p>It is, of course, one of the first results in basic complex analysis that a holomorphic function satisfies the Cauchy-Riemann equations when considered as a differentiable two-variable real function. I have always seen the converse as: if $f$ is <em>continuously</em> differentiable as a function from $U \subset \mat... | lhf | 589 | <p>There's also the <a href="http://en.wikipedia.org/wiki/Looman%E2%80%93Menchoff_theorem">Looman–Menchoff theorem</a>.</p>
|
15,875 | <p>It is, of course, one of the first results in basic complex analysis that a holomorphic function satisfies the Cauchy-Riemann equations when considered as a differentiable two-variable real function. I have always seen the converse as: if $f$ is <em>continuously</em> differentiable as a function from $U \subset \mat... | paul garrett | 12,291 | <p>Thinking of the Cauchy-Riemann operator as an elliptic partial differential operator, the basic elliptic regularity result implies that any <em>distribution</em> satisfying the C-R equation is a holomorphic function. For example, locally integrable suffices. This result was used in Gunning' "Riemann Surfaces", for e... |
1,826,094 | <p>$i\cdot\bar{z} = 2+2i$</p>
<p>I know that $\bar{z} = a-bi$ so then i get $i(a-bi)=2+2i$
Then $ai+b=2+2i$ (because $i^2=-1$)</p>
<p>When 2 complex numbers are equal you usually can equal their parts
Ex: $2+2i=a+bi$ so $a=2$ and $b=2$</p>
<p>But in this case I aint got $a+bi$ I have $ai+b$</p>
<p>The answer is $z=... | Thomas | 26,188 | <p>It looks like you are trying to solve
$$
i\bar{z} = 2 + 2i.
$$
You can start by multiplying both sides by $i$:
$$
i^2\bar{z} = 2i + 2i^2 = -2 + 2i
$$
This gives you
$$
-\bar{z} = -2 + 2i.
$$
or
$$
\bar{z} = 2 - 2i
$$
Now just take the complex conjugate on both sides:
$$
z = \overline{\bar{z}} = \overline{2 - 2i} = 2... |
47,177 | <p>Recently I've been reading "The Wild Book" which applies semigroup theory to, among other things, chemical reactions. If I google for mathematics and chemistry together, most of the results are to do with physical chemistry: cond-mat, fluids, QM of molecules, and analysis of spectra. I'm more interested in learning ... | Fujita | 58,711 | <p>(1) Because my monograph:</p>
<p>S. Fujita, <a href="http://www.springer.com/chemistry/organic+chemistry/book/978-3-540-54126-4" rel="noreferrer">Symmetry and Combinatorial Enumeration in Chemistry</a> (Springer 1991)</p>
<p>has been referred to as an endeavor of dealing with topics of organic chemistry on the bas... |
1,482,564 | <p>In school I'm learning about concavity and finding points of inflection using the second-derivative test. A typical question will look like</p>
<blockquote>
<p><strong>Determine the open intervals on which the graph is concave upward or concave downward.</strong></p>
</blockquote>
<p>Here is how I would solve a prob... | math_nix | 234,249 | <p>No.</p>
<p>Consider $f(x) = x^4$. </p>
<p>In order for there to be a change in concavity there needs to be a change of sign for the second derivative around our potential inflection point. However, $f''(x)=12x^2$ is positive to both sides of $x=0$ so it's not an inflection point.</p>
<p>You can visually tell this... |
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>$$T=\frac{\gamma'(t)}{||\gamma(t)||... | Mikhail Katz | 72,694 | <p>Since your curve is in the plane, you can find the normal vector without differentiating. If the components of the unit tangent vector are $(a,b)$ then the components of a unit normal are $(-b,a)$. "Outward" doesn't really make sense until you have a closed curve, and involves global considerations.</p>
|
1,060,933 | <p>It is necessary for me to find unit outward normal vector for the curve:</p>
<p>$$\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>This problem [restated as : Given a point, $(x_0, y_0)$ on the given curve, compute the unit normal to the curve at $(x_0, y_0)$.] cannot be solved. Why? Because there might be two distinct normal lines to the curve at $(x_0, y_0)$. The following picture shows this:
<img src="https://i.stack.imgur.com/IP0lP.png" alt... |
95,168 | <p>I try to prove the following</p>
<p>$$\binom{2\phi(r)}{\phi(r)+1} \geq 2^{\phi(r)}$$</p>
<p>with $r \geq 3$ and $r \in \mathbb{P}$. Do I have to make in induction over $r$ or any better ideas?</p>
<p>Any help is appreciated.</p>
| sdcvvc | 12,523 | <p>Combinatorial proof of ${2n \choose n+1} \geq 2^n$ where $n \geq 2$:</p>
<p>Let's take set $\{x_1,y_1,\dots,x_{n-2},y_{n-2},a,b,c,d\}$ which has $2n$ elements; select three elements out of $\{a,b,c,d\}$ and for all $i$, a single element of $\{x_i,y_i\}$, you'll select $n+1$ in total. So</p>
<p>${2n \choose n+1} \g... |
680,501 | <blockquote>
<p>Let $M$ be an $R$-module and let $F$ be a free $R$-module of finite rank. Let $\phi : M \to F$ be an epimorphism. Then show that $M$ has a submodule $F' \cong F $ such that $M=F' \oplus \ker\phi$.</p>
</blockquote>
<p>I am new to Module theory. If I apply fundamental theorem of homomorphism then $M/\... | user149792 | 149,792 | <p>In the interests of completeness, we provide a complete solution.</p>
<p><strong>Lemma.</strong> Let $X_1, X_2, \dots$ be a sequence of Bernoulli random variables $($not necessarily independent, not necessarily with same $p$-value$)$. Define$$Y = \sum_{i=1}^\infty {{2X_i}\over{3^i}}.$$Then $\sigma(Y) = \sigma(X_1, ... |
4,426,699 | <p>I want to solve this functional</p>
<p><span class="math-container">\begin{equation}
L(y)=\int_{-1}^7\sqrt{1+y'^2} \ dx
\end{equation}</span></p>
<p>with IC: <span class="math-container">$y(0)=1,\ y(1)=2$</span></p>
<p>I start using the Euler Lagrange equation</p>
<p><span class="math-container">\begin{equation}
\f... | Luthier415Hz | 924,287 | <p>I found a second method:</p>
<p>Using the formula</p>
<p><span class="math-container">\begin{equation}
F-y'F_{y'}=C
\end{equation}</span></p>
<p>I get:</p>
<p><span class="math-container">\begin{equation}
\begin{array}
a\sqrt{1+y'^2}-y'\frac{y'}{\sqrt{1+y'^2}}=C \\
-y'^2+1+y'^2=C\sqrt{1+y'^2}\\
1=C(1+y'^2)\\
Cy'^2+C... |
3,984,652 | <p>Let <span class="math-container">$(X_n,d_n) $</span> a family of metric spaces and <span class="math-container">$(X=\prod_{n \in \mathbb N} X_n,\mathcal d ) $</span> a metric space with <span class="math-container">$\mathcal T_p$</span> the product topology is equal to the topology defined by the distance <span cla... | Prime Mover | 466,895 | <p>Obviously an <span class="math-container">$n$</span> digit number is represented as <span class="math-container">$a_{n-1} a_{n-2} \ldots a_0$</span> not <span class="math-container">$a_n a_{n-1} \ldots a_1$</span>.</p>
|
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>
| B. Goddard | 362,009 | <p>The sphere is considered to be a 2-dimensional surface embedded in 3-dimensional space. But the embedding is sort of extraneous. It's not as handy, but the same map embeds in 2-dimensional space.</p>
|
812,143 | <p>City a is separated by a 2km wide river and are located as shown in Figure 1 (not drawn to scale). A road is to be built between city A to B that crosses a bridge straight across the river. </p>
<p>Use the Sign Test to determine the type of stationary point </p>
<p>Test to see if you have a minimum</p>
| David | 119,775 | <p>Move $B$ $2$km closer to the river, in a direction perpendicular to the river, and ignore the river. Then you won't even need to use calculus.</p>
<p>Specifically, the shortest distance between $A$ and$B$ is now the straight line distance
$$\sqrt{17^2+12^2}=\sqrt{433}$$
and the value of $x$ will be given by simila... |
2,866,768 | <p>Question: Use the trigonometric identity $\cos(2A)=1-2\sin^2(A)$ to show that $$\sin\left(\frac{\pi}{12}\right)=\sqrt{\frac{1}{2} - \frac{\sqrt{3}}{4}}$$</p>
<p>What are good strategies to figure out this question in Particular?</p>
| Mohammad Riazi-Kermani | 514,496 | <p>$$\cos(2A)=1-2\sin^2(A)$$</p>
<p>$$\cos(\pi /6)=1-2\sin^2(\pi /{12})$$</p>
<p>$$1-2\sin^2(\pi /{12})=\frac {\sqrt 3}{2}$$</p>
<p>$$2\sin^2(\pi /{12})=1-\frac {\sqrt 3}{2}$$
$$
\sin^2(\pi /{12})=\frac {1}{2}-\frac {\sqrt 3}{4}$$
$$\sin(\pi /{12})=\sqrt {\frac {1}{2}-\frac {\sqrt 3}{4}}$$</p>
|
3,669,266 | <p>The equation <span class="math-container">$(1-x)^n = x$</span> has a solution in <span class="math-container">$x' \in (0,1)$</span> and indeed the solution <span class="math-container">$x' \to 0$</span> as <span class="math-container">$n \to \infty$</span>. (Consider <span class="math-container">$f(x) = (1-x)^n$</s... | Aphelli | 556,825 | <p>Then <span class="math-container">$\ln{x_n}=n\ln(1-x_n)\sim -nx_n$</span>. </p>
<p>Write <span class="math-container">$x_n=e^{-y_n}$</span> with <span class="math-container">$y_n \rightarrow \infty$</span>. Then, taking logarithms again, <span class="math-container">$y_n+\ln{y_n}=\ln{n}+o(1)$</span>. Thus <span cla... |
4,143,759 | <p><strong>What I understand so far :</strong></p>
<p>Intuitively this question gives me a yuuuuuge hint, it tells me at t=0, t = -4. So that let's me know that I have a phase shift of -4 immediately.</p>
<p>I understand what a period is, I understand what a phase shift is, I understand what amplitude and vertical disp... | David G. Stork | 210,401 | <p><span class="math-container">$$V = \frac{4 \pi}{3} r^3$$</span></p>
<p>so</p>
<p><span class="math-container">$$dV = \frac{4 \pi}{3} 3 r^2 dr$$</span></p>
<p>You know <span class="math-container">$dV/dt$</span>. Solve for <span class="math-container">$dr/dt$</span>.</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... | David Steinberg | 1,231 | <p>One can use the machinery of the fundamental group and of covering spaces to easily prove that any subgroup of a free group must be free.</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... | Douglas Zare | 2,954 | <p><strong>Problem:</strong> Suppose you care about the real world and objects you can hold in your hands. Show that any <a href="http://mathworld.wolfram.com/FlexiblePolyhedron.html" rel="noreferrer">flexible polyhedron</a> maintains a constant volume while it is flexed. This was known as the Bellows Conjecture.</p>
... |
2,159,092 | <p>$\sum _{k=1}^{n}\mathrm{sin}kz=\frac{\mathrm{sin}\frac{n+1}{{2}}\cdot \mathrm{sin}\frac{nz}{2}}{\mathrm{sin}\frac{z}{2}}$
Proof for $z\neq 0$</p>
| Derek Elkins left SE | 305,738 | <p>Before focusing on the specific question, I'd like to provide some context. First, every (non-trivial) topos has a Boolean subtopos. This is essentially what you say, and it roughly corresponds to the double negation interpretation of classical logic into intuitionistic logic. However, one of the things that makes... |
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="... | user376343 | 376,343 | <p><strong>Geometric solution</strong></p>
<p>Locus of points equidistant to a circle and a point (outside the circle) is a hyperbola.<br />
These equidistant points are centers of circles through the given point, and touching the given circle.</p>
<p><a href="https://i.stack.imgur.com/wPGR8.png" rel="nofollow noreferr... |
3,718,255 | <p>So i solved the first part of this question, that there's 588 ways for choosing 5 digit integers, but i can't solve "Find the number of integers in (i) which satisfies "if there's 8, it's followed by 6"
i tried to solve it using different cases:</p>
<p>Case 1) there's two 8s and two 6s => (3 * 3!)/... | fleablood | 280,126 | <p>Add up the ways to do it with no <span class="math-container">$8$</span>, one <span class="math-container">$8$</span> or two <span class="math-container">$8$</span>s.</p>
<p>If there are no <span class="math-container">$8$</span>s you must choose <span class="math-container">$5$</span> components (with order) from <... |
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 <... | zhw. | 228,045 | <p>Another approach: Note the set <span class="math-container">$\{x_1,x_2,\dots\}\cup \{x\}$</span> is bounded, hence is contained in <span class="math-container">$[-M,M]$</span> for some positive <span class="math-container">$M.$</span> Since <span class="math-container">$[-M,M]\times [0,1]$</span> is compact, <span c... |
3,631,714 | <p>If <span class="math-container">$f: \mathbb{R}^n \to \mathbb{R} $</span> is a function on the Schwartz space <span class="math-container">$\mathcal{S} (\mathbb{R}^n) $</span> we define the Fourier transform of <span class="math-container">$f$</span> as the function
<span class="math-container">$$ \hat f(\xi)= \int_{... | Mathsmerizing | 757,478 | <p>Well, going by the video you posted, I will assume that you wish to explore more competition oriented resources. The one resource for IIT-JEE exam is <a href="https://www.iitmind.com/2018/12/tmh-iitjee-mathematics-pdf.html" rel="nofollow noreferrer">TMH Jee Advanced</a> and this is the book I recommend the most for ... |
3,631,714 | <p>If <span class="math-container">$f: \mathbb{R}^n \to \mathbb{R} $</span> is a function on the Schwartz space <span class="math-container">$\mathcal{S} (\mathbb{R}^n) $</span> we define the Fourier transform of <span class="math-container">$f$</span> as the function
<span class="math-container">$$ \hat f(\xi)= \int_{... | Confused Simpleton | 722,188 | <p>These books covers the essence of what you are looking for. Almost everybody in 11th and 12th study from these books. </p>
<p><a href="https://jeemain.guru/best-books-for-mathematics-part-1/" rel="nofollow noreferrer">https://jeemain.guru/best-books-for-mathematics-part-1/</a></p>
<p>PS- The books written by NCERT... |
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... | mmfrgmpds | 284,638 | <p>To find the distributional equivalent of <span class="math-container">$f''$</span>, one calculates, say <span class="math-container">$\varphi\,' \in \mathcal{C^\infty_c([a-1,1])}$</span>, and calculates the Schwartz bracket quantity,
<span class="math-container">$$
\begin{align}
\langle f'',\,\varphi'\rangle &= ... |
200,414 | <p>Some people may carelessly say that you need calculus to find such a thing as a local maximum of $f(x) = x^3 - 20x^2 + 96x$. Certainly calculus is <em>sufficient</em>, but whether it's <em>necessary</em> is another question.</p>
<p>There's a global maximum if you restrict the domain to $[0,8]$, and $f$ is $0$ at t... | Mathemagician1234 | 7,012 | <p>There's actually a famous book by Ivan Niven on precisely this subject i.e. finding extrema by purely algebraic or geometric methods. It's called, simply enough, <em>Maxima and Minima without Calculus</em>. It's currently published by the Mathematicial Association of America. I think you'll find it fascinating and f... |
17,398 | <p>I am teaching my daughter, who is currently about <span class="math-container">$46$</span> months old, additions. She is very curious and asks a lot of <em>good</em> questions. For example, when I told her that <span class="math-container">$2+6=8$</span> and <span class="math-container">$4+4=8$</span>, she asked me ... | yomplex | 12,996 | <p>A <em>good</em> answer to this question is one that (is correct, and) <em>she</em> finds convincing.</p>
<p>As kids are growing up and making sense of the world around them, <em>experimentation</em> is often one of their key sources of truth. Perhaps, experimentation can also serve as a reliable and effective gatew... |
3,131,791 | <p>I read this:</p>
<blockquote>
<p><strong>Definition 1.1</strong>. <em>The</em> complex projective line <span class="math-container">$\mathbb{CP}^1$</span> <em>(or just</em> <span class="math-container">$\mathbb{P}^1$</span><em>) is the set of all ordered pairs of complex numbers</em> <span class="math-container">... | Travis Willse | 155,629 | <p>Suppose <span class="math-container">$F$</span> is homogeneous of degree <span class="math-container">$d$</span>, so that
<span class="math-container">$$F(x_0, x_1) = \sum_{i = 0}^d a_i x_0^i x_1^{d - i} .$$</span>
Any other nonzero point <span class="math-container">$(x_0', x_1')$</span> on the line spanned by <spa... |
2,501,450 | <p>I was trying for a while to prove non-existence of theu following limit:</p>
<p>$$\lim_{n\to\infty}(-1)^n\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}$$</p>
<p>Unfortunately, with no results.</p>
<p>My hope was to show, that:</p>
<p>$$\lim_{n\to\infty}\frac{2^n+4n+6}{2^n(\sqrt[n]{5}-1)}\not=0$$</p>
<p>But showing that wa... | DRPR | 347,579 | <p>Hint:Divide the numerator and denominator by $2^n$. You dont need Lhopital rule .</p>
|
1,216,576 | <p><img src="https://i.stack.imgur.com/p4xbH.png" alt="enter image description here"></p>
<blockquote>
<p>$ABC$ and $BDE$ are two equilateral triangles such that $D$ is the midpoint of
$BC$. If $AE$ intersects $BC$ at $F$, show that $Area(\triangle BFE)=2Area(\triangle FED)$</p>
</blockquote>
<p><strong>Solution... | Salomo | 226,957 | <p>Imagine there are two group of people, with each group $n$ people. Now, we want to choose $n$ people from these $2n$ people. On one hand, the number of combinations we can have is $\sum^n_{i=0} \binom{n}{i} \binom{n}{n-i}=\sum^n_{i=0} \binom{n}{i}^2$, since we can choose $i$ from one group and $n-i$ from another gro... |
2,804,086 | <p>A distribution is an element of the continuous dual space of some function space. Let us take the Schwartz space $\mathcal{S} := \mathcal{S}(\mathbb{R}^n)$ just as an example. A distribution $\phi \in \mathcal{S}'$ is then a map
$$ \phi: \mathcal{S} \rightarrow \mathbb{C}.$$</p>
<p>My question is this: how do I int... | Abdelmalek Abdesselam | 244,562 | <p>If the motivation is Wightman functions, it's better to use the differences $x_1-x_n,\ldots,x_{n-1}-x_n$ with a single point of reference $x_n$ rather than the consecutive differences $x_1-x_2,
\ldots,x_{n-1}-x_n$.
So let me consider the statement that as elements in $S'(\mathbb{R}^{4n})$
$$
\mathcal{W}(x_1,\ldots,... |
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... | colormegone | 71,645 | <p>To stay consistent with your labelling, write the position of the Vancouver cyclist as $ \ x_A = 25 \ (t+2) \ $ and the position of the Seattle cyclist as $ \ x_B = 300 - 32t \ . $ We are calling the location of Vancouver $ \ x = 0 \ $ by doing so. Remember that the Vancouver cyclist is the one who has been pedall... |
4,083,588 | <p>Can somebody help me how to solve</p>
<p><span class="math-container">$x* 23 \equiv_{60} 1 $</span> with <span class="math-container">$ x \in \mathbb{N} $</span> and <span class="math-container">$x > 100$</span></p>
<p>What would be a good approach?</p>
<p>I know that x = 107 would be a solution. However how can ... | Doug M | 317,176 | <p>Use the Euclidean algorithm.</p>
<p>Since <span class="math-container">$\gcd(23,60) = 1$</span> there exists integers <span class="math-container">$m,n$</span> such that <span class="math-container">$23m + 60n = 1.$</span></p>
<p><span class="math-container">$m$</span> will be our inverse.</p>
<p>The Euclidean algor... |
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
$$
P(\{ \omega \in \Omeg... | Gabriel Romon | 66,096 | <p>Assuming $X$ is real-valued, $$\{ \omega \in \Omega: \lim_{n \rightarrow \infty} X_n(\omega) = X(\omega)\} = ((X-\limsup_n X_n)=0)\cap ((X-\liminf_n X_n)=0) $$</p>
<p>Since $\limsup_n X_n$, $\liminf_n X_n$ and $X$ are measurable, $((X-\limsup_n X_n)=0)\cap ((X-\liminf_n X_n)=0)$ is measurable.</p>
|
3,936,694 | <p>I would like to construct a deterministic TM that decides <span class="math-container">$L=$</span>{<span class="math-container">$<G,s,t,k>$</span> | <span class="math-container">$G$</span> is a directed graph that has a path of length at most <span class="math-container">$k$</span> from vertex <span class="mat... | Théophile | 26,091 | <p>Your claim is wrong. You can choose any nonnegative number as the limit:
<span class="math-container">$$\lim_{x\to0^+}\left(e^{-1/x}\right)^{ax} = e^{-a}.$$</span></p>
<p>This example is from the <a href="https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero" rel="noreferrer">Wikipedia article</a> on <span class="... |
686,167 | <p>I'm working on an assignment where part of it is showing that $S_k=0$ for even $k$ and $S_k=1$ for odd $k$, where</p>
<blockquote>
<p>$$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... | Robert Lewis | 67,071 | <p>To see that the given expression is equal to $1$, set</p>
<p>$\omega = e^{\frac{ik\pi}{n + 1}}, \tag{1}$</p>
<p>then the expression becomes </p>
<p>$\dfrac{1}{1 - \omega} + \dfrac{1}{1 - \bar{\omega}} = \dfrac{1 - \bar{\omega} + 1 - \omega}{(1 - \omega)(1 - \omega)} = \dfrac{2 - (\omega + \bar{\omega})}{(1 - \o... |
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... | Andreas Blass | 48,510 | <p>Since you wrote "short1 $\circ$ left1", you must be using $\circ$ to denote composition in the usual order for functions, where $f\circ g$" means first apply $g$ and then apply $f$. But in your "tests for composition", you seem to be using $\circ$ to mean composition in the opposite order, following the order of the... |
2,904,432 | <blockquote>
<p>Let the subspace <span class="math-container">$S=\{(x, y, z) \in R^3; x = -2z \}$</span></p>
<p>a) Determine its dimension and a basis B of S;</p>
<p>b) Complete B such that it becomes a basis M of <span class="math-container">$R^3$</span>.</p>
</blockquote>
<p><span class="math-container">$\operatornam... | CyclotomicField | 464,974 | <p>So a typical vector would normally be written as $\begin{bmatrix}x\\y\\z\end{bmatrix}$ but we can make a substitution here using $x=-2z$ so that our subspace vectors are written as $\begin{bmatrix}-2z\\y\\z\end{bmatrix}$. From here we separate the variables using addition then factor out the variables using scalar m... |
2,330,496 | <p>Players Ruby and Bob are given an undirected graph and a number $N$. First Ruby colors $N$ vertices red, then Bob colors $N$ vertices blue (they must be distinct from Ruby's choices). Afterward, all other points on the graph are given the color of whichever color they are closest to (shortest path) with ties left bl... | Alex Ravsky | 71,850 | <p>I can prove that for the first player (Ruby) is possible not to lose in some simple cases for the given graph $G$ with $n$ vertices. </p>
<ol>
<li><p><a href="https://en.wikipedia.org/wiki/Dominating_set" rel="nofollow noreferrer">Domination number</a> $\gamma(G)\le N$. Indeed, in this case at her move the first p... |
4,537,390 | <p>Let <span class="math-container">$C$</span> be a circle centred at <span class="math-container">$O$</span> and <span class="math-container">$P$</span> be a point inside the circle.</p>
<p>The line <span class="math-container">$OP$</span> intersects the circle <span class="math-container">$C$</span> at <span class="m... | Glorious Nathalie | 948,761 | <p>Let <span class="math-container">$Q_1 = (r, 0), Q_2 = (-r, 0), Q = r (\cos \theta, \sin \theta), P = (a, 0)$</span> where <span class="math-container">$ 0 \le a \le r $</span></p>
<p><span class="math-container">$PQ_1 = r - a$</span></p>
<p><span class="math-container">$PQ_2 = a + r$</span></p>
<p><span class="math... |
2,604,660 | <p>In a box there are $16$ ice-creams: $6$ lemon flavor,$4$ mint flavor and $6$ strawberry flavor.When we extract two ice-creams,what's the probability of getting two different flavors,given that at least one is strawberry flavor.</p>
<p>That's my solution : </p>
<p>$P(A)$ = "Different flavors" = $\frac{\binom{16}{2}... | Masacroso | 173,262 | <p>I dont see exactly what you did (edit: I will provide a complete anwer to clarify things). </p>
<p>There is a way to do it using the law of total probability: assuming that each ice-cream have the same probability to be taken then it is the probability that the first one will be some flavor and that the second will... |
26,736 | <p>I am still a newbie to <em>Mathematica</em>, so I am still getting used to the syntax and logic of the program - but I've been trying to do the following 3D plot:</p>
<pre><code>g[x_, t_] := y /. FindRoot[y == Tanh[(2*y + x)/t], {y, x, t}];
p1 = ContourPlot3D[{z ==
Sech[(2 g[x, t] + x)/t]^2/(t - 2 Sech[(2 g[x... | Silvia | 17 | <p>Michael's fantastic answer is a general way to accomplish implicit plots. But for the very problem in OP, the method I used in <a href="https://mathematica.stackexchange.com/a/23565/17">this</a> and <a href="https://mathematica.stackexchange.com/a/27235/17">this</a> answers is applicable, too.</p>
<p>First we defin... |
1,369,428 | <p>I am trying to solve the following equation;
$$\int_{-1}^{1}e^{i(x+a\cos x)} \, \mathrm{d}(\cos x)$$ or $$\int_{0}^{\pi}e^{i(x+a\cos x)} \sin x \, \mathrm{d}x$$</p>
<p>I tried this in <em>Wolfram Alpha</em>, but it says that answer cannot be obtained.</p>
| Mark Viola | 218,419 | <p>Let $F(a)$ be defined as</p>
<p>$$F(a)=\int_0^{\pi}e^{i(x+a\cos x)}\sin x\,dx \tag 1$$</p>
<p>Now, it is easy to show that the real part of $F$ is identically $0$. To do this, we write the real part of $(1)$ as </p>
<p>$$\text{Re}\lbrace F(a)\rbrace=\int_0^{\pi/2}\cos(x+a\cos x)\sin x\,dx+\int_{\pi/2}^{\pi}\cos(... |
176,240 | <p>I <a href="https://math.stackexchange.com/questions/2832912/sum-infty-infty-frac-exp-n21-4n2-in-closed-form/2832965#2832965">came across</a> a possible bug in Mathematica's evaluation of integrals that I thought I'd share. Consider the code:</p>
<pre><code>zmax = 4;
a = Integrate[Exp[-x^2] Abs[Sin[x]], {x, -z , ... | b3m2a1 | 38,205 | <p>I know halirutan has already provided an answer, but here's a nice way to deal with GIF import in a frame-wise manner (or really in chunks of frames).</p>
<p>Mathematica does a very dumb thing when it reads GIFs. It calls the function <code>System`Convert`CommonGraphicsDump`IEImageRead</code> which, if intelligent,... |
1,085,620 | <p>Which number between the interval 7902 & 7918 can be divided without remainder only to itself and to the number 1?</p>
| egreg | 62,967 | <p>The requirement is that the number is prime, so this excludes the even numbers, leaving
$$
7903;7905;7907;7909;7911;7913;7915;7917.
$$
We can remove $7905$ and $7915$ because they're divisible by $5$; $7911$ and $7917$ are divisible by $3$. Moreover $7903$ is divisible by $7$ and $7909$ is divisible by $11$; note th... |
2,668,779 | <p>The exercise consists of showing that the function $f(x,y)=x^4 + y^4$ has a global minimum and maximum under the constraint $x^4 + y^4 - 2xy = 2$.</p>
<p>In the solution to the exercise, it it follows that the constraint is compact if we can show that $\lim_{x^2 + y^2 \rightarrow \infty} x^4 + y^4 - 3xy - 2 \righta... | Atmos | 516,446 | <p>Consider the set $S$ formed of $x=(a,c) \in \mathbb{R}^2$ such as $\left|a\right|\geq \left|c\right|$.
Then for $b \in \mathbb{R}$
$$
bx=\left(ba,bc\right)
$$
and $\left|ba\right| \geq \left|bc\right|$.</p>
<p>However, with $(4,1)$ and $(-3,2)$ which are both in the set
$$
(4,1)+(-3,2)=(1,3)
$$
which is not in the ... |
2,668,779 | <p>The exercise consists of showing that the function $f(x,y)=x^4 + y^4$ has a global minimum and maximum under the constraint $x^4 + y^4 - 2xy = 2$.</p>
<p>In the solution to the exercise, it it follows that the constraint is compact if we can show that $\lim_{x^2 + y^2 \rightarrow \infty} x^4 + y^4 - 3xy - 2 \righta... | M. Winter | 415,941 | <p>Choose any two linearly independent vectors $v,w$. Then the set</p>
<p>$$\{bv\mid b\in\Bbb R\}\cup\{bw\mid b\in\Bbb R\}.$$</p>
<p>satisfies your condition, but is not a vector space since $v+w$ is not contained (vector spaces must contain all sums of their vectors).</p>
|
2,668,779 | <p>The exercise consists of showing that the function $f(x,y)=x^4 + y^4$ has a global minimum and maximum under the constraint $x^4 + y^4 - 2xy = 2$.</p>
<p>In the solution to the exercise, it it follows that the constraint is compact if we can show that $\lim_{x^2 + y^2 \rightarrow \infty} x^4 + y^4 - 3xy - 2 \righta... | mathmandan | 198,422 | <p>Let $S$ be the union of the first and third quadrants (along with the $x$- and $y$-axes, let's say). Scaling by a real number $r$ will either keep each vector in its same quadrant (if $r>0$), or send the first and third quadrants to each other (if $r<0$), or send all vectors to the origin (if $r=0$). So $S$ ... |
1,726,540 | <p>How should $x^{\frac{1}{x}}$ be differentiated? I know the answer is
$$\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$$</p>
<p>but I do not understand how to get there.</p>
<h2>Attempt at solution.</h2>
<p>I believe the following is true:</p>
<p>$$
\begin{aligned}\frac{d}{dx}x^u&=ux^{u-1}\cdot u^\prime\\
\frac{d}{dx}a^... | C Marius | 327,443 | <p>$x^{\frac{1}{x}} = e^{\frac{1}{x} ln(x)}$ therefore $\left(x^{\frac{1}{x}}\right)' = \left(e^{\frac{1}{x} ln(x)}\right)' = e^{\frac{1}{x} ln(x)}\left(\frac{1}{x} ln(x)\right)' = e^{\frac{1}{x} ln(x)} \left(\frac{-1}{x^2}ln(x) + \frac{1}{x}\frac{1}{x}\right) = e^{\frac{1}{x} ln(x)}\left(\frac{1-ln(x)}{x^2}\right) = ... |
1,726,540 | <p>How should $x^{\frac{1}{x}}$ be differentiated? I know the answer is
$$\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$$</p>
<p>but I do not understand how to get there.</p>
<h2>Attempt at solution.</h2>
<p>I believe the following is true:</p>
<p>$$
\begin{aligned}\frac{d}{dx}x^u&=ux^{u-1}\cdot u^\prime\\
\frac{d}{dx}a^... | rahul joshi | 351,574 | <p>its very simple to get answer </p>
<p>Let $u=x^{\frac{1}{x}}$</p>
<p>take logarithm of both sides</p>
<p>$log\ u=log(x^{\frac{1}{x}})$</p>
<p>$log\ u=\frac{1}{x}log(x)$</p>
<p>differentiation wrt 'x'
$$\frac{d}{dx}log\ u=\frac{d}{dx}\frac{1}{x}log(x)$$
$$to\ differentiate\ this,\ use\ chain\ rule\ in\ LHS\ and\... |
1,726,540 | <p>How should $x^{\frac{1}{x}}$ be differentiated? I know the answer is
$$\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$$</p>
<p>but I do not understand how to get there.</p>
<h2>Attempt at solution.</h2>
<p>I believe the following is true:</p>
<p>$$
\begin{aligned}\frac{d}{dx}x^u&=ux^{u-1}\cdot u^\prime\\
\frac{d}{dx}a^... | Community | -1 | <p>There is a nice trick that can rescue you: differentiate all the functions obtained by considering all instances of $x$ but one as constants, and take the sum.</p>
<p>This gives ($X$ is a constant)</p>
<p>$$\left(x^{1/X}\right)'=\frac1Xx^{1/X-1}$$
and
$$\left(X^{1/x}\right)'=\ln X\left(\frac1x\right)'X^{1/x}=-\ln ... |
3,998,484 | <p>For the purpose of clarity, I do not want anybody to prove this statement, I am just looking to get some help translating it into the contrapositive using <em>symbolic logic</em>. Right now I have the following translation:</p>
<hr />
<p>Original Statement:</p>
<p><span class="math-container">$$(\exists k \in \mathb... | open problem | 876,065 | <p>You have fixed the homomorphism proof.</p>
<p>Continuing from there:</p>
<p>α((a,b))=e</p>
<p>Whenever 3b = e</p>
<p>This is true for b=0,9,18.</p>
<p>So <span class="math-container">$ker(\alpha) = Z_{9}\times (0,9,18)$</span></p>
|
332,130 | <p>Here is my proof, I would appreciate it if someone could critique it for me:</p>
<p>To prove this statement true, we must proof that the two conditional statements ("If $\mathcal{P}(A)⊆ \mathcal{P}(B)$, then $A⊆B$," and, If $A⊆B$, then $\mathcal{P}(A)⊆ \mathcal{P}(B)$) are true.</p>
<hr>
<p>Contrapositive of the ... | Brian M. Scott | 12,042 | <p>Your argument is fine, if a little more roundabout than necessary: both directions can be done easily with direct proofs. </p>
<p>Suppose first that $\wp(A)\subseteq\wp(B)$. $A\subseteq A$, so $A\in\wp(A)\subseteq\wp(B)$, so $A\in\wp(B)$, and hence $A\subseteq B$.</p>
<p>Now suppose that $A\subseteq B$. Then for a... |
750,906 | <p>I would like to show my Calc I class that $f(x)=f(a)+f'(a)(x-a)+(f''(c)/2)(x-a)^2$ for some $c$ in $(a,x)$ (for $f$ smooth). Just from the form of the statement, it seems as though this should be possible using only the ordinary MVT, with no integrals. However, I can't seem to get the symbols to behave properly; all... | solstafir | 142,542 | <p>Let $$g(x)=f(b)-f(x)-(b-x)f'(x)-\left(\frac{b-x}{b-a}\right)^2[f(b)-f(a)-(b-a)f'(a)]$$</p>
<p>Then $$g'(x)=\frac{2(b-x)}{(b-a)^2}[f(b)-f(a)-(b-a)f'(a)-\frac{1}{2}(b-a)^2f''(x)]$$</p>
<p>(Differentiate, then factor out $\frac{2(b-x)}{(b-a)^2}$.)</p>
<p>Applying the mean value theorem, $g(b)-g(a)=g'(c)(b-a)$ for so... |
750,906 | <p>I would like to show my Calc I class that $f(x)=f(a)+f'(a)(x-a)+(f''(c)/2)(x-a)^2$ for some $c$ in $(a,x)$ (for $f$ smooth). Just from the form of the statement, it seems as though this should be possible using only the ordinary MVT, with no integrals. However, I can't seem to get the symbols to behave properly; all... | xpaul | 66,420 | <p>Another way to solve the problem: Let
$$F(x)=f(x)-f(a)-f'(a)(x-a)-k(x-a)^2$$
where $k$ is a constant determined later. Then $F(a)=0$. Let $F(x)=0$; then we have
$$ k=\frac{f(x)-f(a)-f'(a)(x-a)}{(x-a)^2}. $$
Since $F(a)=F(x)=0$ and $F$ is differentiable in $(a,x)$, by Rolle's Theorem, there is $c\in(a,x)$ such that $... |
4,365,005 | <p>The set of all homomorphisms between two spaces <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> is denoted as <span class="math-container">$\text{Hom}(X,Y)$</span>, the set of endomorphisms of a space <span class="math-container">$X$</span> is denoted as <span class="math-containe... | John Bentin | 875 | <p>As mentioned in the comments, the notation <span class="math-container">$Y^X$</span> is standard for the set of maps from <span class="math-container">$X$</span> to <span class="math-container">$Y$</span>. Moreover, there is no other common notation for this.</p>
|
1,234,320 | <p>It is given that $4 x^4 + 9 y^4 = 64$.</p>
<p>Then what will be the maximum value of $x^2 + y^2$?</p>
<p>I have done it using the sides of a right-angled triangle be $2x , 3y $ and hypotenuse as 8 .</p>
| Rellek | 228,621 | <p>Since you tagged Lagrange Multipliers, I will use that method.</p>
<p>Allow that $F(x,y) = 4x^4+9y^4-64$ and $G(x,y)=x^2+y^2-C$, where $C$ is treated as a constant and is the maximum you wish to find.</p>
<p>We allow that $\nabla F(x,y) = \lambda \nabla G(x,y)$ and find two new equations:</p>
<p>$16x^3 = \lambda ... |
1,234,320 | <p>It is given that $4 x^4 + 9 y^4 = 64$.</p>
<p>Then what will be the maximum value of $x^2 + y^2$?</p>
<p>I have done it using the sides of a right-angled triangle be $2x , 3y $ and hypotenuse as 8 .</p>
| HK Lee | 37,116 | <p>$x^2=s,\ y^2=t \geq 0$ then $$ (2s)^2+ (3t)^2=8^2 \Rightarrow
2s=4\cos\ \theta,\ 3t=4\sin\ \theta,\ 0\leq\theta \leq
\frac{\pi}{2} $$</p>
<p>Then $$ s+t=2\cos\ \theta + \frac{4}{3}\sin\ \theta=\sqrt{2^2+
\bigg(\frac{4}{3}\bigg)^2} \sin\ (\theta+\alpha),\ 0< \alpha <
\frac{\pi}{2} $$</p>
<p>Let $\theta =\frac... |
2,558,619 | <p>Compute
$$
\int_{-\infty}^{\infty} t^2 \delta(\sin(t)) e^{-|t|} \mathrm dt
$$
In closed form, where $\delta(t)$ is the <a href="http://mathworld.wolfram.com/DeltaFunction.html" rel="nofollow noreferrer">Dirac Delta function </a>.</p>
<p>My attempt: </p>
<p>$$
\int_{-\infty}^{\infty} t^2 \delta(\sin(t)) e^{-|t|}... | Tom | 103,715 | <p>I'll comment on calculating the sum. First, the sum $\sum\limits_{n=0}^{\infty}n^2 e^{-\pi n}$ certainly does converge, just as $\int_0^{\infty} x^2 e^{-\pi x} \, dx$ converges. Here's how I would go about finding a closed expression which is equal that sum:</p>
<p>Note that $n^2 e^{-\pi n} = \frac{d^2}{d\lambda^2}... |
457,654 | <p>This question is a continuation of: <a href="https://math.stackexchange.com/questions/456118/finding-a-set-of-values-with-the-binomial-theorem">Finding a set of values with the binomial theorem</a></p>
<p>For $n \in \mathbb N$ and the function $p(x) = \left(x + \frac 1x\right)^{n}$.</p>
<p>By the binomial theorem... | Jyrki Lahtonen | 11,619 | <p>Obviously the polynomial $a(x)$ has to be coprime to $m(x)$ for otherwise the common factor will always prevent this. If the factorization of $m(x)$ to powers of irreducible polynomials is
$$
m(x)=\prod_i p_i(x)^{a_i},
$$
then by the Chinese remainder theorem
$$
\mathbb{F_p}[x]/(m(x))\simeq\bigoplus_i\mathbb{F_p}[x]... |
2,572,428 | <p>$$z^2=3+4i ,(z=x+iy)$$
Seems easy? That's what I thought!</p>
<p>I get a system of equations that I can't solve:
$x^2-y^2=3$ and $2xiy=4i$ which I then get $x=\frac{2}{y}$</p>
<p>I can't solve the system of equations?</p>
| WaveX | 323,744 | <p>Hint</p>
<p>$$3+4i = 4+4i-1$$ $$ = 4+4i+i^2 $$</p>
|
4,192,879 | <p>This question is about the <em>provided solution</em> to Question 11d from Chapter 8 of Spivak's Calculus. The relevant background material (i.e. 11a-11c) is as follows:</p>
<blockquote>
<p><span class="math-container">$(a)$</span> If <span class="math-container">$\{a_n\}$</span> is a sequence of positive terms such... | Hagen von Eitzen | 39,174 | <p>The map <span class="math-container">$(0,\infty)\times (0,\infty)\to (0,\infty)$</span>, <span class="math-container">$(x,y)\mapsto \frac xy$</span> is continuos. Use the <span class="math-container">$\epsilon\delta$</span>-definition of continuity (together with a simple lower bound on distances in <span class="mat... |
2,738,023 | <p>We know we can check if two vectors are 'orthogonal' by doing an inner product.</p>
<p>$a*b=0$</p>
<p>tells us that these two vectors are orthogonal</p>
<p>here comes the question:</p>
<p>if there a way to compute if they are 'parallel'? i.e., they are pointing at the same direction. </p>
| user1228123 | 148,520 | <p>Two parallel vectors $a$ and $b$ will have a null cross product:
$$a\times b=0\iff a\parallel b.$$</p>
|
633,482 | <p>Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube
can be placed into the box corresponds to a certain group of permutations of the vertices of the cube. This is the group
of <strong>group of rigid motions (or rotations) of the cube</strong>. (It should ... | Yiorgos S. Smyrlis | 57,021 | <p>The function $f(t)=\mathrm{e}^{at}$ can be viewed as the solution of the initial value problem:
$$
x'=ax, \quad x(0)=1.
$$
More generally, if $A\in\mathbb R^{n\times n}$ and $u_0\in\mathbb R^n$, then the solution of
the
initial value problem:
$$
x'=Ax, \quad x(0)=u_0,
$$
is $u(t)=\mathrm{e}^{tA}u_0$, where $\mathrm... |
673,229 | <blockquote>
<p>The medians to the two legs of a right angled triangle are $10$ and $4\sqrt{10}$. Find the hypotenuse of the right angled triangle.</p>
</blockquote>
<p>I'm confused. Can somebody please explain to me how to do this step by step? Not just the answer I want to know what you did to get the answer and w... | Kim Fierens | 172,717 | <p>Some additional examples that may be of interest: </p>
<ol>
<li>Spherical trigonometry.</li>
<li>Quaternion analysis (in the 19th century a rather hot and active area of research).</li>
<li>Synthethic projective geometry (in the tradition of Steiner and Poncelet).</li>
<li>Analytical theory of generalized continued... |
2,064,403 | <p>Example. Let $V$ be a finite dim vector space with two different bases</p>
<p>$S = \{ u_1,u_2 \} = \{ (1,2),(3,5) \}$ and $S' = \{ v_1, v_2 \} = \{ (1,-1), (1,-2) \} $</p>
<p>You can check that $v_1 = -8u_1 + 3u_2$ and $v_2 = -11u_1+4u_2$ and $P = \begin{bmatrix}
-8 &-11 \\
3& 4
\end{bmatrix}$ is the ... | Christian Blatter | 1,303 | <p>When a basis, or coordinate, transform is in action then we are not so much interested in how the new basis vectors are defined (this is dealt with at the start), but we need to transform the coordinates of vectors a thousand times. Therefore things are set up in such a way that the the formulas for the transformati... |
3,579,739 | <p>Its also mentioned that this question is an example of <strong>Sampling without replacement</strong></p>
<p>My question is , by the general method of how I do these kind of problems , I would assume that there are two possibilities</p>
<ol>
<li><p>You chose a defective bottle and then a non defective which will h... | Jamāl | 506,409 | <p>You can think of this problem in two ways. One way is the one you did which is perfectly fine.
The other is to think about the possible outcomes you can have at the end of the two selections. There are <span class="math-container">$5\choose 2$</span> total number of selections you can make. Now assume that you have ... |
159,948 | <p>If I have this function (just and example) </p>
<pre><code> Plot3D[-Abs[Cos[ y] Sqrt[x] Sqrt[ y] Cos[ x]], {x, 0, 20}, {y, 0, 20}]
</code></pre>
<p>I am interested in finding the locations of the smallest (minimum) 6 values of this function? How can I do that. Apparently, Findminimum and its likes give me one va... | J. M.'s persistent exhaustion | 50 | <p>Another idea is to use the <code>MeshFunctions</code> option of <code>ContourPlot[]</code> (as previously shown <a href="https://mathematica.stackexchange.com/a/46569">here</a>) to extract putative positions for the minima, before polishing them with <code>FindMinimum[]</code>:</p>
<pre><code>(* pretend that the fu... |
199,374 | <p>Consider the two table set below</p>
<pre><code>t1= {0.44, 0.62, 0.77, 0.87, 0.93, 0.96, 0.98, 1}
t2= {0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25}
</code></pre>
<p>how can I create a plot which wil plot a rectangle with a heat map normalized to the values of the table?</p>
<p>The result should look like</p>
... | Carl Woll | 45,431 | <p>Another idea is to use <a href="http://reference.wolfram.com/language/ref/VertexColors" rel="nofollow noreferrer"><code>VertexColors</code></a> option of <a href="http://reference.wolfram.com/language/ref/Line" rel="nofollow noreferrer"><code>Line</code></a> (or <a href="http://reference.wolfram.com/language/ref/Rec... |
4,390,127 | <p>I ma trying to prove that if <span class="math-container">$A$</span> is an <span class="math-container">$n\times n$</span> matrix, then <span class="math-container">$f:\mathbb{R}^n\to\mathbb{R},\ f(\vec{x})=A\vec{x}\cdot\vec{x}=\vec{x}^T A\vec{x}$</span> is differentiable and <span class="math-container">$Df(\vec{a}... | Henno Brandsma | 4,280 | <p>Let <span class="math-container">$U$</span> be an open neighbourhood of <span class="math-container">$a$</span>. By local convexity (and regularity (we don't need Hausdorff, all uniform spaces are completely regular and regular, whether separated (<span class="math-container">$T_2$</span>) or nor) there is an open c... |
2,751,300 | <blockquote>
<p>Figure out with distinct real numbers the system of equations.</p>
<p><span class="math-container">$$\frac{xy}{x-y} = \frac{1}{30}$$</span>
<span class="math-container">$$\frac{x^2y^2}{x^2+y^2} = \frac{1}{2018}$$</span></p>
</blockquote>
<p>I multiplied x-y both side on the first equation and square on ... | Martin R | 42,969 | <p>Inverting the equations gives
$$
30 = \frac{x-y}{xy} = \frac 1y - \frac 1x \\
2018 = \frac{x^2+x^2}{x^2y^2} = \frac 1{y^2} + \frac 1{x^2}
$$
This suggests to substitute $a= \frac 1x, b= \frac 1y$:
$$
30 = b - a \\
2018 = b^2 + a^2 \,
$$
This leads to a simple quadratic equation for $a$ (or $b$) which has two dif... |
4,014,768 | <blockquote>
<p>Find the sationary points of the curve and their nature for the equation <span class="math-container">$y=e^x\cos x$</span> for <span class="math-container">$0\le x\le\pi/2$</span>.</p>
</blockquote>
<p>I derived it and got <span class="math-container">$e^x(-\sin x+\cos x)=0$</span>.</p>
<p><span class="... | InsideOut | 235,392 | <p>Hint:</p>
<p><span class="math-container">$$-\sin x+\cos x=0$$</span>
If and only if</p>
<p><span class="math-container">$$\tan x=1$$</span></p>
<p>Can you get it from here?</p>
|
4,014,768 | <blockquote>
<p>Find the sationary points of the curve and their nature for the equation <span class="math-container">$y=e^x\cos x$</span> for <span class="math-container">$0\le x\le\pi/2$</span>.</p>
</blockquote>
<p>I derived it and got <span class="math-container">$e^x(-\sin x+\cos x)=0$</span>.</p>
<p><span class="... | Jean-Claude Arbaut | 43,608 | <p>For all <span class="math-container">$x$</span>, <span class="math-container">$\cos(x+\pi/2)=-\sin(x)$</span> hence the equation is equivalent to
<span class="math-container">$$\cos(x)+\cos(x+\pi/2)=0$$</span></p>
<p>But we also have the identity</p>
<p><span class="math-container">$$\cos p+\cos q=2\cos\frac{p+q}{2}... |
4,038,685 | <p>Suppose that <span class="math-container">$u(x)$</span> is continuous and satisfies the integral equation</p>
<p><span class="math-container">\begin{equation}\label{1.4.1}
u(x) = \int_0^x \sin(u(t))u(t)^p dt
\end{equation}</span></p>
<p>on the interval <span class="math-container">$0 \leq x \leq 1$</span... | Matthew H. | 801,306 | <p>Assume there exists <span class="math-container">$t\in(0,1)$</span> such that <span class="math-container">$u(t)>0$</span>. Because <span class="math-container">$u$</span> is continuous and <span class="math-container">$u(0)=0$</span>, we can identify <span class="math-container">$a,b\in [0,1]$</span> such that <... |
1,519,070 | <p>How can I show that the sequence is defined recursively?</p>
<blockquote>
<p>Show that the recursively defined sequence $(x_n)_{n\in\mathbb{N}}$ with
$$x_1=1, \qquad\qquad x_{n+1}=\sqrt{6+x_n}$$
converges and determine its limit </p>
</blockquote>
<p><a href="https://i.stack.imgur.com/cL7BR.png" rel="nofollo... | Abr001am | 223,829 | <p>$\sqrt{6+\sqrt{6+\sqrt{6+...\sqrt{7}}}}<x$</p>
<p>If we go by induction :</p>
<p>$\sqrt{6+\sqrt{6+\sqrt{6+...\sqrt{7}}}}=U_n<x$</p>
<p>$(U_{n+1})²-6=U_n<x$</p>
<p>$U_{n+1}<\sqrt{x+6}$</p>
<p>$\sqrt{x+6}=x$</p>
<hr>
<ul>
<li>Now it is left to you calculating $x$ and $U_0$ testing part.</li>
</ul>
|
1,328,082 | <p>I've been trying to prove the following inequality, but until now I've had problems coming up with a solution:</p>
<p>$$
2^{mn} \ge m^n
$$</p>
<p>$m$ and $n$ can assume any natural number.</p>
<p>I wasn't able to find any counterexample that would invalidate this inequality, so I am assuming that this statement i... | coffeemath | 30,316 | <p>Let $A=\{1,2,...,n\}$ and $B=\{1,2,...,m\}.$ Then $m^n$ is the number of functions from $A$ to $B,$ while $2^{m \cdot n}$ is the number of relations from $A$ to $B$ (functions or not). And there's an injection from the set of functions to the set of relations.</p>
|
2,466,913 | <p>I am having trouble with a question on my homework and I have done some work but I'm not sure if I have completed all that is asked or how to take it further if needed.</p>
<p>Problem: </p>
<p>Let $\theta = \arccos(5x^3)$. Find $\frac{d\theta}{dx}$ in two ways:</p>
<p>(a) Use implicit differentiation starting fro... | Temidire Adesiji | 490,025 | <p>Note that every time you cut one piece of paper into $7$, the total number of pieces of paper reduces by $1$ $piece$ $of$ $paper$ increases by $7$. Thus, the student will have $7, 13 ,19, ...$ Which will give rise to a sequence of odd integers, Hence, Impossible to get $2016$ pieces of paper.</p>
|
3,886,643 | <p>I have doubts about how to solve this issue about bases in numbering systems:</p>
<p>"Determine the base b in <span class="math-container">$(104)_b = 8285$</span>"</p>
<p>can anybody help me?
Thanks.</p>
| md2perpe | 168,433 | <p>Since <span class="math-container">$(104)_b = 1\cdot b^2 + 0 \cdot b + 4$</span> you get the equation
<span class="math-container">$$b^2+4 = 8285,$$</span>
which you should be able to solve.</p>
|
2,585,828 | <p>I saw in an article that for every real number, there exists a Cauchy sequence of rational numbers that converges to that real number. This was stated without proof, so I'm guessing it is a well-known theorem in analysis, but I have never seen this proof. So could someone give a proof (preferably with a source) for ... | Paramanand Singh | 72,031 | <p>Based on the context gathered from your question and your comments it is clear that you are concerned about the equivalence of construction of reals as Dedekind cuts of rationals and as equivalence classes of Cauchy sequences of rationals.</p>
<p>We can do this in the following manner. First we show that correspond... |
3,140,052 | <p>I found a proof of this theorem in <a href="http://web.mit.edu/18.705/www/13Ed.pdf" rel="nofollow noreferrer">modern algebra</a> 2.30</p>
<blockquote>
<p>Proof: Set <span class="math-container">$S := \{ \text{ ideals } \mathfrak b \mid \mathfrak b ⊃ \mathfrak a \text{ and } \mathfrak b ∌ 1 \}$</span>. Then <span ... | Bernard | 202,857 | <ul>
<li><span class="math-container">$S$</span> is a set by the <em>Axiom schema of specification</em>.</li>
<li>You cannot apply <em>Zorn's lemma</em> to the interval <span class="math-container">$(0,1)$</span> for the usual order because not every chain in <span class="math-container">$(0,1)$</span> has an upperbou... |
22,269 | <p>How to verify, which version of MathJax is used here? It seems that v. 2.5 is not implemented yet. It has very useful features, as <code>\delimiterfactor</code>, which is needed here: <a href="https://tex.stackexchange.com/questions/284809/getting-large-braces-in-mathjax">https://tex.stackexchange.com/questions/284... | Davide Cervone | 7,798 | <p>MathJax does not implement the <code>\delimiterfactor</code> control sequence, but does have an internal value for <code>delimiterfactor</code> (the default value of 901 from TeX) that it uses in its computations for stretchy delimiters. That value can be set from Javascript, but not from within a TeX expression. ... |
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